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--- abstract: 'We propose that the “isolated” planetary mass objects observed by Zapatero Osorio et al in the $\sigma$ Orionis cluster might actually be in orbit around invisible stellar mass companions such as mirror stars. Mirror matter is expected to exist if parity is an unbroken symmetry of nature. Future observations can test this idea by looking for a periodic Doppler shift in the radiation emitted by the planets. The fact that the observations show an inverse dependence between the abundance of the these objects and their mass may argue in favour of the mirror matter hypothesis.' address: | School of Physics\ Research Centre for High Energy Physics\ The University of Melbourne\ Victoria 3010 Australia\ foot, sasha, rrv@physics.unimelb.edu.au author: - 'R. Foot, A. Yu. Ignatiev and R. R. Volkas' title: | \ Do “isolated” planetary mass objects orbit mirror stars? --- A variety of observations strongly suggests the presence of a significant amount of dark matter (DM) in the universe. Galactic rotation curves and cluster dynamics cannot be explained using standard Newtonian gravity unless non-luminous but gravitating matter exists. Arguments from Big Bang Nucleosynthesis and theories of large scale structure formation disfavour the simple possibility that all of the DM consists of ordinary baryons. Candidates for the required exotic component in the DM abound: WIMPS, axions and mirror matter are examples. The observation of microlensing events from the Small and Large Magellanic Clouds is consistent with the existence of Massive Compact Halo Objects (MACHOs) in the halo of the Milky Way [@macho]. The inferred average mass is about $0.5 M_{\odot}$, where $M_{\odot}$ is the mass of our sun. The most reasonable conventional identification sees MACHOs as white dwarfs, although there are several strong arguments against this [@freese]. For example, the heavy elements that would have been produced by their progenitors are not in evidence [@freese]. This argues against the conventional white dwarf scenario, and in favour of exotic compact objects. In summary, there is strong evidence for exotic DM which is capable of forming compact stellar mass objects. Mirror matter [@mirror] is an interesting candidate for some of the required exotic DM [@blinnikov]. It can be independently motivated by the desire to see the full Poincaré Group, including improper transformations (parity and time reversal), as an exact symmetry group of nature. The basic postulate is that every ordinary particle (lepton, quark, photon, etc.) is related by an improper Lorentz transformation with an opposite parity partner (mirror lepton, mirror quark, mirror photon, etc.) of the same mass. Both material particles (leptons and quarks) and force carriers (photons, gluons, $W$ and $Z$ bosons) are doubled. Mirror matter interacts with itself via mirror weak, electromagnetic and strong interactions which have the same form and strength as their ordinary counterparts (except that mirror weak interactions couple to the opposite chirality). Because ordinary matter is known to clump into compact objects such as stars and planets, mirror matter will also form compact mirror stars and mirror planets. Since mirror matter does not feel ordinary electromagnetism, it will be dark. Gravitation, by contrast, is common to both sectors. Mirror matter therefore has the correct qualitative features: it is dark, it clumps, and it gravitates.[^1] Later, we will explain why the observed inverse dependence between the abundance of the these objects and their mass may already argue in favour of the mirror matter hypothesis. It has been speculated that MACHOs might be mirror stars [@f1], and one of us (RF) has proposed that the observed extrasolar planets [@extrasolar] might be composed of mirror matter [@f2]. The former is well motivated by the aforementioned difficulties in identifying MACHOs as conventional stellar mass objects such as white dwarfs. The latter is motivated by the fact that the detected extrasolar planets are rather massive and orbit very closely to the star, which are surprising characteristics. It is unlikely that ordinary planets of sufficient size could have condensed so close to the stars. If they are composed of ordinary matter, then they probably formed much further from the stars and then migrated in. Another possibility, though, is that they are composed of exotic material such as mirror matter. Because of the weak coupling between ordinary and mirror matter, there is no barrier to mirror planet condensation very close to an ordinary star. This idea should be testable by observing the opacity and albedo properties of the planets[@foot01]. Zapatero Osorio et al. [@zo] have recently presented strong evidence for the existence of “isolated planetary mass objects” in the $\sigma$ Orionis star cluster. These objects are more massive than Jupiter $M_{J}$, but not as massive as brown dwarfs ($\sim 5 - 15 M_J$ although there is some model dependence in the mass determination[@zo]). They appear to be gas giant planets which do not seem to be associated with any visible star. So far, eighteen such objects have been identified. Given that the $\sigma$ Orionis cluster is estimated to be between 1 million and 5 million years old, the formation of these “isolated planets” must have occured within this time scale. Zapatero Osorio et al. argue that these findings pose a challenge to conventional theories of planet formation because standard theories of substellar body formation (as well as new theories inspired by previous claims of isolated planet discovery), are unable to explain the existence of numerous isolated planetary mass objects down to masses $\sim$ few $M_J$. See Ref.[@zo] and references therein for further discussion. It is possible therefore that non-standard particle physics may be required to understand their origin. We speculate that rather than being isolated, these ordinary matter planets actually orbit invisible mirror stars. These systems could be, in a sense, just the mirror images of those ordinary star systems which have been speculated to feature large Jovian mirror planets in close orbit. Indeed, if there really are mirror planets in orbit around ordinary stars, then it is very natural to also expect mirror solar systems to sometimes contain large ordinary planets. It should be possible to test this idea by searching for a periodic Doppler shift in spectral lines emanating from these planets. We have that $$\frac{\Delta \lambda}{\lambda} = 2 \frac{v_r}{c},$$ where $\lambda$ is wavelength, $\Delta \lambda$ is the difference between the peak and trough of the periodic Doppler modulation of $\lambda$, $v_r$ is the maximum value of the component of the planet’s orbital velocity in the direction of the Earth, and $c$ is the speed of light. Suppose that a given planet is in a circular orbit of radius $r$ around a mirror star of mass $M$. Let $I$ be the inclination of the plane of the orbit relative to the normal direction defined by the Earth - mirror star line. Then $$v_r = \sqrt{\frac{GM}{r}} \sin I,$$ where $G$ is Newton’s constant. Combining these equations we obtain $$\frac{\Delta \lambda}{\lambda} \simeq 10^{-3} \sqrt{\frac{M}{M_{\odot}}} \sqrt{\frac{0.04\ A.U.}{r}} \sin I$$ as the level of spectral resolution required. Note that this is a few orders of magnitude larger than the Doppler shifts observed in extrasolar planet detection. However it is certainly true that the isolated planets are much fainter sources of light than the stars whose Doppler shifts have been measured so such a measurement may not be completely straightforward. However, it is worth noting that for the case of close orbiting ordinary planets where $r \sim 0.04\ A. U.$ (analogous to the close-in extra solar planets), the Doppler shift is quite large ($\sim 10^{-3}$) with a period of only a few days which should make this interesting region of parameter space relatively easy to test. Indeed, Zapatero Osorio et al. [@zo] have taken optical and near infrared low resolution spectra of three young isolated planet candidates (S Ori 52, S Ori 56, and S Ori 47). They have obtained absorption lines (at wavelengths $\sim 900$ nm), however their resolution was 1.9 nm[@zo] which is just below that needed to test our hypothesis. The higher resolution required has been achieved in the case of brown dwarfs[@brown] so we anticipate that it should be possible to test our hypothesis in the near future. One would also expect some ordinary matter to have accumulated in the centre of the mirror star. It is possible, but not inevitable, that this ordinary matter also observably radiates. If so, one would expect this radiation to experience a much smaller Doppler modulation compared to that from the planet. Because the planet and mirror star would not be spatially resolved, one observational signature would be that some of the spectral lines are modulated (those from the planet), while a different set are not (those from the ordinary matter pollutants in the mirror star). If the mirror star is invisible but opaque, then one would expect to see periodic planetary eclipses for some of these systems (those with $\sin I \simeq 1$). The eclipses should of course occur once per Doppler cycle, around one of the points of zero Doppler shift within a cycle. Obviously, such eclipses (along with the information provided by Doppler shift measurements) will be useful in distinguishing a mirror star from alternatives such as faint white dwarfs or neutron stars. However, it should be mentioned that standard objects such as white dwarfs and neutron stars are extremely unlikely candidates, because the age of the $\sigma$ Orionis cluser is estimated to be only 1 million to 5 million years old, while white dwarfs and neutron stars are typically billions of years old. Before concluding, we would like to point out an intriguing systematic in both the extrasolar planet and the Zapatero Osorio et al. data that may argue in favour of the mirror matter hypothesis. One envisages a universe that contained some admixture of ordinary and mirror matter from the earliest moments after the Big Bang. Eventually, both the smooth ordinary fluid and the mirror fluid condensed into large scale structures, stars and planets. Because gravitational condensation must be aided by non-gravitational dissipative effects to carry off kinetic energy, one does not expect the ordinary and mirror matter to have condensed in congruent locations, despite their common gravitational interaction. One expects instead a nonzero “segregation scale” $\ell$ to quantify the degree of spatial separation of condensed ordinary and mirror matter clouds or clumps. While we have too little information to theoretically calculate $\ell$, the qualitative expectation is a universe of cells of scale $\ell$, with a given cell being dominated either by ordinary matter or mirror matter. Provided that $\ell$ is much greater than a typical solar system scale, which is in fact observationally required,[^2] then the majority of hybrid ordinary-mirror systems should have disparate components: large ordinary objects with small mirror objects, or the other way around. The ordinary star plus mirror planet systems, and our proposed mirror star plus ordinary planet systems, have exactly this characteristic. Indeed one might expect the number of hybrid systems to increase as a function of the disparity between the components. Intriguingly, the observed extrasolar planets increase in number as their mass decreases. Even more interestingly, the Zapatero Osorio et al. objects also increase in number with decreasing mass: from Fig.2 of Ref.[@zo] we see that there are about as many objects between $8 M_{J}$ and $10 M_{J}$ as there are between $10 M_{J}$ and $20 M_{J}$ (taking 5 million years as the relevant lifetime). We predict, therefore, that an extended search would find greater numbers of these objects at even smaller masses. Of course if the “isolated planets” do orbit mirror stars then this suggests that the star forming region near $\sigma$ Orionis could also be a region of mirror star formation. This is certainly possible and was already envisaged many years ago by Khlopov et al.[@kh] where they argued that large molecular clouds (made of ordinary matter) could merge with large mirror molecular clouds in which case the formation of mixed systems (i.e. containing both ordinary and mirror matter) is enhanced. In conclusion, we have proposed that the “isolated” planetary mass objects observed by Zapatero Osorio et al. might actually be planets orbiting invisible mirror stars. This idea can be tested by searching for a Doppler modulation at the level of $10^{-3}-10^{-4}$ in amplitude. [99]{} C. Alcock et al., MACHO Coll., Ap. J. [**486**]{}, 697 (1997); [**542**]{}, 000 (2000); T. Lasserre et al., EROS Coll., astro-ph/9909505. See, for example, B. D. Fields, K. Freese, D. S. Graff, Ap. J. [**534**]{}, 265 (2000). T. D. Lee, C. N. Yang, Phys. Rev. [**104**]{}, 254 (1956); R. Foot, H. Lew, R. R. Volkas, Phys. Lett. B [**272**]{}, 67 (1991). S. Blinnikov, M. Yu. Khlopov, Sov. Astron. [**27**]{}, 371 (1983); E. W. Kolb, D. Seckel, M. S. Turner, Nature [**514**]{}, 415 (1985); M. Yu. Khlopov et al., Soviet Astronomy. 35, 21 (1991); H. M. Hodges, Phys. Rev. D [**47**]{}, 456 (1993); Z. G. Berezhiani, D. Comelli, F. L. Villante, hep-ph/0008265. R. Foot, H. Lew, R. R. Volkas, Mod. Phys. Lett. A [**7**]{}, 2567 (1992); R. Foot, Mod. Phys. Lett. A [**9**]{}, 169 (1994); R. Foot, R. R. Volkas, Phys. Rev. D [**52**]{}, 6595 (1995); see also, Z. G. Berezhiani, R. N. Mohapatra, Phys. Rev. D [**52**]{}, 6607 (1995). R. Foot and R. R. Volkas, Astropart.  Phys. [**7**]{}, 283 (1997); Phys. Rev. D[**61**]{}, 043507 (2000). S. L. Glashow, Phys. Lett. B [**167**]{}, 35 (1986); see also B. Holdom, Phys. Lett. B [**166**]{}, 196 (1986). R. Foot, S. N. Gninenko, Phys. Lett. B [**480**]{},171 (2000). Z. Silagadze, Phys.At. Nucl. [**60**]{}, 272 (1997); S. Blinnikov, Talk at “Baryonic Matter in the Universe and its Spectroscopic Studies”, Atami, Japan, 22-24 Nov. 1996, astro-ph/9801015; R. Foot, Phys. Lett. B [**452**]{}, 83 (1999). For a review and references see the extrasolar planet encyclopaedia, http://cfa-www.harvard.edu/planets/encycl.html. R. Foot, Phys. Lett. B [**471**]{}, 191 (1999). R. Foot, astro-ph/0101055. M. R. Zapatero Osorio et al., Science [**290**]{}, 103 (2000). See also, M. Tamura et al., Science [**282**]{}, 1095 (1998); P. W. Lucas. P. F. Roche, Mon. Not. R. Astron. Soc. [**314**]{}, 858 (2000). A. Thakrah, H. Jones and M. Hawkins, MNRAS, [**284**]{}, 507 (1997). A. Yu. Ignatiev, R. R. Volkas, Phys. Rev. D [**62**]{}, 023508 (2000). M. Yu. Khlopov et al., in Ref. [@blinnikov] [^1]: There are several other interesting implications of the mirror matter model. In particular, oscillations between ordinary and mirror neutrinos have been proposed as a solution to the solar and atmospheric neutrino problems [@mirrornu]. Ordinary - mirror neutrino oscillations also lead to interesting implications for early Universe cosmology[@fv]. Also photon-mirror photon kinetic mixing leads to potentially observable effects for orthopositronium[@gl] and can resolve the orthopositronium lifetime anomaly[@fg]. [^2]: For instance, one can deduce an upper bound of about $10^{-3}$ for the mirror matter content of the Earth [@ig].

--- abstract: 'We study a posteriori error analysis of linear-quadratic boundary control problems under bilateral box constraints on the control which acts through a Neumann type boundary condition. We adopt the hybridizable discontinuous Galerkin method as discretization technique, and the flux variables, the scalar variables and the boundary trace variables are all approximated by polynomials of degree k. As for the control variable, it is discretized by the variational discretization concept. Then an efficient and reliable a posteriori error estimator is introduced, and we prove that the error estimator provides an upper bound and a lower bound for the error. Finally, numerical results are presented to illustrate the performance of the obtained a posteriori error estimator.' address: - 'School of Mathematical Sciences, South China Normal University, Guangzhou 510631, Guangdong, China.' - 'School of Mathematical Sciences, South China Normal University, Guangzhou 510631, Guangdong, China.' author: - Haitao Leng - Yanping Chen date: '...' title: 'Residual-type a posteriori error analysis of HDG methods for Neumann boundary control problems' --- [^1] Introduction {#intro} ============ Many optimization processes in science and engineering lead to optimal control problems where the sought state is a solution of a partial differential equation. The complexity of such problem needs special care in order to obtain efficient numerical approximations for the optimization problem. One particular method is adaptive finite element method, which can reduces the computational cost and boosts the accuracy of the numerical solutions by locally refining the meshes around the singularity. Although the adaptive finite element method has become a popular approach for numerical solutions of partial differential equations since the work of Babuška and Rheinboldt [@br1978], it has only quiet recently become popular for constrained optimal control problems. The pioneer work concerning a posteriori error analysis for distributed optimal control problems is published by Liu and Yan [@ly2001] for residual-type error estimators and Becker, Kapp, and Rannacher [@bkr2000] for goal-oriented error estimators. Here, we further refer readers to [@hhik2008; @llmt2002; @yb2015; @yk2014; @zyy2014] for residual-type estimators and [@bv2009; @hh2010] for goal-oriented approach. Recently, in order to guarantee the performance of the a posteriori error estimator theoretically, many scholars have tried to prove the convergence of an adaptive finite element algorithm for distributed optimal control problems in [@gikh2007; @gy2017; @ksr2014; @lc2018]. Compared to distributed optimal control problems, there exists limited work on a posteriori error analysis for boundary optimal control problems. In [@ly2001a], the convex Neumann boundary control problem was considered on polygonal or Lipschitz piecewise $\mathcal{C}^2$ domain. Then a residual-type a posteriori error estimator was introduced, and the authors proved that the estimator provided an upper bound for the errors in the state and the control. In [@hiis2006], by introducing a Lagrange multiplier, the authors derived an efficient and reliable residual-type a posteriori error estimator for Neumann boundary control problems on polygonal domain. In [@krs2014], Kohls, Rösch and Siebert derived a unifying framework for the a posteriori error analysis of control constrained linear-quadratic optimal control problems for the full and variational discretizations. In [@by2017], Benner and Yücel investigated symmetric interior penalty Galerkin methods for Neumann boundary control problems with an extra coefficient in cost functional. By invoking a Lagrange multiplier associated with the control constraints, an efficient and reliable residual-type a posteriori error estimator was obtained for the errors in the state, adjoint, control and co-control. As for Dirichlet boundary control problems, we just mention [@cgn2017; @glty2018] and references therein for more details on a posteriori error analysis. Recently, the hybridizable discontinuous Galerkin (HDG) methods [@cgl2009], which keep the advantages of discontinuous Galerkin (DG) methods and result in a system with significantly reduced degrees of freedom, have been proposed for convection diffusion problem [@fqz2015], interface problem [@cc2019], flow problem[@npc2010], optimal control problem [@chsszz2018; @ghmszz2018], and so on. In [@cz2012; @cz2013; @cz2013a], Cockburn and Zhang studied HDG methods for second order elliptic problems, and an a posteriori error estimator with postprocessing solutions was obtained. To the best of our knowledge, there exists no work on residual-type a posteriori error analysis of HDG methods for boundary control problems. In this paper, we investigate a posteriori error analysis of Neumann optimal control problems under bilateral box constraints on the control. The HDG method is used as discretization technique, and the flux variables, the scalar variables and the boundary trace variables are discretized by polynomials of degree $k$. As for the control variable, we adopt the variational discretization concept proposed by Hinze in [@h2005] for approximation. Then an efficient and reliable residual-type a posteriori error estimator without any postprocessing solutions is introduced, and we prove that the error estimator provides not only an upper bound but also a lower bound up to data oscillations for the errors. Finally, numerical experiments are presented to validate the performance of the obtained estimator. The remainder of the paper is arranged as follows: In Section \[sec2\] we introduce the model problem and the associated optimality system. In Section \[sec3\] the discrete optimality system is given, and we prove that the discrete scheme has a unique solution. Then we prove the reliability and efficiency of the error estimator in Section \[sec4\] and Section \[sec5\] respectively. Numerical experiments are presented in Section \[sec6\] to validate the performance of the obtained estimator. Finally, some conclusions are provided in Section \[sec7\]. Throughout this paper, let $C$ with or without subscript be a generic positive constant independent of the mesh size. For ease of exposition, we denote $A\leq CB$ by $A\lesssim B$. The Neumann boundary control problem {#sec2} ==================================== Let $\Omega\in\mathbb{R}^d~(d=2,3)$ be a polygonal $(d=2)$ or polyhedral $(d=3)$ domain with boundary $\partial\Omega$. Before we introduce the model problem, let us summarize some notation. For bounded and open set $D\in\mathbb{R}^d$ or $D\in\mathbb{R}^{d-1}$, we denote the usual Sobolev spaces by $W^{s,p}(D)$ with norm $\|\cdot\|_{s,p,D}$ and seminorm $|\cdot|_{s,p,D}$. The Hilbertian Sobolev spaces are abbreviated by $H^s(D)=W^{s,2}(D)$ with norm $\|\cdot\|_{s,D}$ and seminorm $|\cdot|_{s,D}$. For $s=0$, $H^0(D)$ coincides with $L^2(D)$, and the inner product is denoted by $(\cdot,\cdot)_{D}$ for $D\in \mathbb{R}^d$ and $\langle\cdot,\cdot\rangle_D$ for $D\in\mathbb{R}^{d-1}$. Furthermore, we define $H(div,\Omega):=\{\textbf{v}\in (L^2(\Omega))^d:\nabla\cdot\textbf{v}\in L^2(\Omega)\}$. Based on the domain $\Omega$, we consider the following Neumann boundary control problem $$\label{cost} \min_{y\in H^1(\Omega),u\in U_{ad}}\mathcal{J}(y,u)=\frac{1}{2}\|y-y_d\|_{0,\Omega}^2+\frac{\alpha}{2}\|u\|_{0,\partial\Omega}^2,$$ subject to the elliptic equations \[state\] $$\begin{aligned} -\Delta y+y &=f\quad \rm{in}~\Omega,\label{state:1}\\ \nabla y\cdot\textbf{n}&=u+g\quad \rm{on}~\partial\Omega,\label{state:2}\end{aligned}$$ where the regularization parameter $\alpha$ is a positive constant, $y_d\in L^2(\Omega)$, $f\in L^2(\Omega)$, $g\in L^2(\partial\Omega)$, $\textbf{n}$ is the unit vector normal to the boundary $\partial\Omega$. The set $U_{ad}$ of constraints is given by $$U_{ad}=\{v\in L^2(\partial\Omega): u_a\leq v\leq u_b~a.e.~x\in\partial\Omega\},$$ where $u_a$ and $u_b$ are assumed to be constant, and that $u_a<u_b$. From [@l1971], we know that the Neumann boundary control problem (\[cost\])-(\[state\]) admits a unique solution $(y,u)\in H^1(\Omega)\times L^2(\partial\Omega)$, and there exists an adjoint-state $z\in H^1(\Omega)$ such that \[optimality\] $$\begin{aligned} -\Delta y+ y&=f\quad \rm{in}~\Omega,\label{optimality:1}\\ \nabla y\cdot\textbf{n}&=u+g\quad \rm{on}~\partial\Omega,\label{optimality:2}\\ -\Delta z+z&=y-y_d\quad \rm{in}~\Omega,\label{optimality:3}\\ \nabla z\cdot\textbf{n}&=0\quad \rm{on}~\partial\Omega,\label{optimality:4}\\ \langle\alpha u+z,v-u\rangle_{\partial\Omega}&\geq 0\quad \forall v\in U_{ad}.\label{optimality:5}\end{aligned}$$ Moreover, the variational inequality (\[optimality:5\]) is equivalent to the projection formula $$u=\Pi_{U_{ad}}\Big(-\frac{1}{\alpha}z|_{\partial\Omega}\Big),\label{equ}$$ where $\Pi_{U_{ad}}$ is the $L^2$-projection onto $U_{ad}$. Then let $\textbf{p}=-\nabla y$ and $\textbf{q}=-\nabla z$, the optimality system (\[optimality\]) can be rewritten in a mixed form as follows: \[mixed\] $$\begin{aligned} \textbf{p}+\nabla y&=0\quad \rm{in}~\Omega,\label{mixed:1}\\ \nabla\cdot\textbf{p}+y&=f\quad \rm{in}~\Omega,\label{mixed:2}\\ -\textbf{p}\cdot\textbf{n}&=u+g\quad \rm{on}~\partial\Omega,\label{mixed:3}\\ \textbf{q}+\nabla z&=0\quad \rm{in}~\Omega,\label{mixed:4}\\ \nabla\cdot\textbf{q}+z&=y-y_d\quad \rm{in}~\Omega,\label{mixed:5}\\ -\textbf{q}\cdot\textbf{n}&=0\quad \rm{on}~\partial\Omega,\label{mixed:6}\\ \langle\alpha u+z,v-u\rangle_{\partial\Omega}&\geq 0\quad \forall v\in U_{ad}.\label{mixed:7}\end{aligned}$$ The HDG discretization {#sec3} ====================== Let $\mathcal{T}_h$ be a conforming and shape regular partition of the domain $\Omega$. For each $K\in\mathcal{T}_h$, we denote $\partial K$ the set of its faces. Then we define $\partial\mathcal{T}_h=\{\partial K: K\in\mathcal{T}_h\}$. Denote $\mathcal{E}_h^o$ the set of all interior faces of $\mathcal{T}_h$ and $\mathcal{E}_h^{\partial}$ the set of all boundary faces of $\mathcal{T}_h$. Then we define $\mathcal{E}_h=\mathcal{E}_h^o\cup\mathcal{E}_h^{\partial}$. For any $K\in\mathcal{T}_h$ and $F\in\mathcal{E}_h$, $h_K$ and $h_E$ denote the diameters of the element $K$ and the face $F$ respectively. Furthermore, we define the mesh-dependent inner product by $$(w,v)_{\mathcal{T}_h}=\sum_{K\in\mathcal{T}_h}(w,v)_K,\quad \langle w,v\rangle_{\partial\mathcal{T}_h}=\sum_{K\in\mathcal{T}_h}\langle w,v\rangle_{\partial K}.$$ For vector-valued functions, the notations are similarly defined by the dot product. Based on the partition $\mathcal{T}_h$, we define the discontinuous finite element spaces for the flux variables, the scalar variables and the boundary trace variables as following $$\begin{aligned} &\textbf{V}_h^k=\{\textbf{v}\in(L^2(\Omega))^d:\textbf{v}|_K\in(\mathcal{P}^k(K))^d,~\forall K\in\mathcal{T}_h\},\\ &W_h^k=\{w\in L^2(\Omega):w|_K\in\mathcal{P}^k(K),~\forall K\in\mathcal{T}_h\},\\ & M_h^k=\{\mu\in L^2(\mathcal{E}_h):\mu|_F\in\mathcal{P}^k(F),~\forall F\in\mathcal{E}_h\},\end{aligned}$$ where $\mathcal{P}^k(S)$ is the set of polynomials of degree no larger than $k$ on the domain $S$. In this paper, we adopt the variational concept proposed by Hinze [@h2005] for the control variable, which suggests to approximate the state equation but not the control variable. Therefore the control variable will be implicitly discretized by formula (\[equ\]). Then the HDG scheme of the system (\[mixed\]) reads as follows: Find $(\textbf{p}_h,y_h,\widehat{y}_h)\in \textbf{V}_h^k\times W_h^k\times M_h^k$, $(\textbf{q}_h,z_h,\widehat{z}_h)\in \textbf{V}_h^k\times W_h^k\times M_h^k$ and $u_h\in U_{ad}$ such that \[HDG\] $$\begin{aligned} (\textbf{p}_h,\textbf{r}_1)_{\mathcal{T}_h}-(y_h,\nabla\cdot\textbf{r}_1)_{\mathcal{T}_h}+\langle\widehat{y}_h,\textbf{r}_1\cdot\textbf{n}\rangle_ {\partial\mathcal{T}_h}&=0,\label{HDG:1}\\ -(\textbf{p}_h,\nabla w_1)_{\mathcal{T}_h}+(y_h,w_1)_{\mathcal{T}_h} +\langle \widehat{\textbf{p}}_h\cdot\textbf{n},w_1\rangle_{\partial\mathcal{T}_h}&=(f,w_1)_{\mathcal{T}_h},\label{HDG:2}\\ \langle\widehat{\textbf{p}}_h\cdot\textbf{n},\mu_1\rangle_{\partial\mathcal{T}_h\backslash\partial\Omega}&=0,\label{HDG:3}\\ -\langle\widehat{\textbf{p}}_h\cdot\textbf{n},\mu_1\rangle_{\partial\Omega}&=\langle u_h+g,\mu_1\rangle_{\partial\Omega},\label{HDG:4}\\ (\textbf{q}_h,\textbf{r}_2)_{\mathcal{T}_h}-(z_h,\nabla\cdot\textbf{r}_2)_{\mathcal{T}_h}+\langle\widehat{z}_h,\textbf{r}_2\cdot \textbf{n}\rangle_{\partial\mathcal{T}_h}&=0,\label{HDG:5}\\ -(\textbf{q}_h,\nabla w_2)_{\mathcal{T}_h}+(z_h,w_2)_{\mathcal{T}_h}+\langle\widehat{\textbf{q}}_h\cdot\textbf{n},w_2\rangle _{\partial\mathcal{T}_h}&=(y_h-y_d,w_2)_{\mathcal{T}_h},\label{HDG:6}\\ \langle\widehat{\textbf{q}}_h\cdot\textbf{n},\mu_2\rangle_{\partial\mathcal{T}_h\backslash\partial\Omega}&=0,\label{HDG:7}\\ -\langle\widehat{\textbf{q}}_h\cdot\textbf{n},\mu_2\rangle_{\partial\Omega}&=0,\label{HDG:8}\\ \langle\alpha u_h+\widehat{z}_h,v-u_h\rangle_{\partial\Omega}&\geq 0,\label{HDG:9}\end{aligned}$$ for any $(\textbf{r}_1,w_1,\mu_1)\in\textbf{V}_h^k\times W_h^k\times M_h^k$, $(\textbf{r}_2,w_2,\mu_2)\in\textbf{V}_h^k\times W_h^k\times M_h^k$ and $v\in U_{ad}$. Similarly, we know that the inequality (\[HDG:9\]) is equivalent to the following projection formula $$u_h=\Pi_{U_{ad}}\Big(-\frac{1}{\alpha}\widehat{z}_h|_{\partial\Omega}\Big).$$ Here the normal component of numerical fluxes $\widehat{\textbf{p}}_h\cdot\textbf{n}$ and $\widehat{\textbf{q}}_h\cdot\textbf{n}$ is defined as $$\begin{aligned} &\widehat{\textbf{p}}_h\cdot\textbf{n}=\textbf{p}\cdot\textbf{n}+\tau_1(y_h-\widehat{y}_h)\quad \rm{on}~\partial\mathcal{T}_h,\\ &\widehat{\textbf{q}}_h\cdot\textbf{n}=\textbf{q}\cdot\textbf{n}+\tau_2(z_h-\widehat{z}_h)\quad \rm{on}~\partial\mathcal{T}_h,\end{aligned}$$ for stabilization parameters $\tau_1$ and $\tau_2$. For ease of exposition, we define operators $\mathcal{B}$ by $$\begin{aligned} &\mathcal{B}(\textbf{r}_1,w_1,\mu_1;\textbf{r}_2,w_2,\mu_2;\tau)\\ =&(\textbf{r}_1,\textbf{r}_2)_{\mathcal{T}_h}-(w_1,\nabla\cdot\textbf{r}_2)_{\mathcal{T}_h}+\langle\mu_1,\textbf{r}_2\cdot\textbf{n}\rangle _{\partial\mathcal{T}_h}\\ &+(\nabla\cdot\textbf{r}_1,w_2)_{\mathcal{T}_h}+(w_1,w_2)_{\mathcal{T}_h}\\ &+\langle\tau(w_1-\mu_1),w_2\rangle_{\partial\mathcal{T}_h}-\langle\textbf{r}_1\cdot\textbf{n}+\tau(w_1-\mu_1),\mu_2\rangle_{\partial\mathcal{T}_h},\end{aligned}$$ Then the HDG scheme (\[HDG\]) can be rewritten according to the operator $\mathcal{B}$: Find $(\textbf{p}_h,y_h,\widehat{y}_h)\in\textbf{V}_h^k\times W_h^k\times M_h^k$, $(\textbf{q}_h,z_h,\widehat{z}_h)\in \textbf{V}_h^k\times W_h^k\times M_h^k$ and $u_h\in U_{ad}$ such that \[bbs\] $$\begin{aligned} \mathcal{B}(\textbf{p}_h,y_h,\widehat{y}_h;\textbf{r}_1,w_1,\mu_1;\tau_1)&=(f,w_1)_{\mathcal{T}_h}+\langle u_h+g,\mu_1\rangle_{\partial\Omega},\label{bbs:1}\\ \mathcal{B}(\textbf{q}_h,z_h,\widehat{z}_h;\textbf{r}_2,w_2,\mu_2;\tau_2)&=(y_h-y_d,w_2)_{\mathcal{T}_h},\label{bbs:2}\\ \langle\alpha u_h+\widehat{z}_h,v-u_h\rangle_{\partial\Omega}&\geq 0,\label{bbs:3}\end{aligned}$$ for any $(\textbf{r}_1,w_1,\mu_1)\in\textbf{V}_h^k\times W_h^k\times M_h^k$, $(\textbf{r}_2,w_2,\mu_2)\in\textbf{V}_h^k\times W_h^k\times M_h^k$ and $v\in U_{ad}$. We assume that $\tau_1=\tau_2>0$ on $\partial \mathcal{T}_h$ and $0\in U_{ad}$. Then the system (\[bbs\]) has a unique solution. Since the system (\[bbs\]) is finite dimensional, we only need to prove that the system (\[bbs\]) just has the zero solution for the case of $f=y_d=g=0$. Let $(\textbf{r}_1,w_1,\mu_1)=(\textbf{q}_h,-z_h,-\widehat{z}_h)$ in (\[bbs:1\]) and $(\textbf{r}_2,w_2,\mu_2)=(-\textbf{p}_h,y_h,\widehat{y}_h)$ in (\[bbs:2\]), we have $$\begin{aligned} 0=&\mathcal{B}(\textbf{p}_h,y_h,\widehat{y}_h;\textbf{q}_h,-z_h,-\widehat{z}_h;\tau_1)+ \mathcal{B}(\textbf{q}_h,z_h,\widehat{z}_h;-\textbf{p}_h,y_h,\widehat{y}_h;\tau_2)\\ =&(y_h,y_h)_{\mathcal{T}_h}-\langle u_h,\widehat{z}_h\rangle_{\partial\Omega}\geq (y_h,y_h)_{\mathcal{T}_h} +\alpha\langle u_h,u_h\rangle_{\partial\Omega},\end{aligned}$$ from (\[bbs:3\]) and the assumption $0\in U_{ad}$. Hence $y_h=0$ and $u_h=0$. Furthermore, let $(\textbf{r}_1,w_1,\mu_1)= (\textbf{p}_h,y_h,\widehat{y}_h)$ in (\[bbs:1\]) and $(\textbf{r}_2,w_2,\mu_2)=(\textbf{q}_h,z_h,\widehat{z}_h)$ in (\[bbs:2\]), we have $$\begin{aligned} 0=&(\textbf{p}_h,\textbf{p}_h)_{\mathcal{T}_h}+(y_h,y_h)_{\mathcal{T}_h}+\langle\tau_1(y_h-\widehat{y}_h),y_h-\widehat{y}_h\rangle_{\partial\mathcal{T}_h},\\ 0=&(\textbf{q}_h,\textbf{q}_h)_{\mathcal{T}_h}+(z_h,z_h)_{\mathcal{T}_h}+\langle\tau_2(z_h-\widehat{z}_h),z_h-\widehat{z}_h\rangle_{\partial\mathcal{T}_h}\end{aligned}$$ Therefore $\textbf{p}_h=0$, $\widehat{y}_h=0$, $\textbf{q}_h=0$, $z_h=0$ and $\widehat{z}_h=0$. Then we conclude the proof. The residual-type a posteriori error estimator {#sec4} ============================================== Auxiliary results ----------------- Before we start to prove a posteriori error estimator for the model problem, we first provide some auxiliary results that will play an important role in the proof. For each element $K\in\mathcal{T}_h$ and face $F\in\mathcal{E}_h$, we denote $\Pi_j^o$ and $\Pi_j^{\partial}$ the $L^2$-projections onto $\mathcal{P}^j(K)$ and $\mathcal{P}^j(F)$ for the nonnegative integer $j$. Then, from [@cc2019] we have the following error estimates \[lem1\] For any $K\in\mathcal{T}_h$ and $F\in\mathcal{E}_h$, we have $$\begin{aligned} \|\Pi_j^o v\|_{0,K}\leq& \|v\|_{0,K}\quad \forall v\in L^2(K),\\ \|\Pi_j^{\partial}v\|_{0,F}\leq&\|v\|_{0,F}\quad \forall v\in L^2(F),\\ \|v-\Pi_0^o v\|_{0,K}\lesssim& h_K\|\nabla v\|_{0,K}\quad \forall v\in H^1(K),\\ \|v-\Pi_0^o v\|_{0,\partial K}\lesssim& h_K^{1/2}\|\nabla v\|_{0,K}\quad \forall v\in H^1(K).\end{aligned}$$ We conclude this subsection by introducing a lemma that has been proved in [@chz2017]. \[lem2\] Let $F$ be a face of the element $K\in\mathcal{T}_h$, $\textbf{n}_F$ the unit vector normal to $F$, and $s>0$. Assume that $v$ is a given function in $H^{1+s}(K)$ and $\Delta v\in L^2(K)$. For any $w_h\in\mathcal{P}^k(F)$, we have $$\langle \nabla v\cdot\textbf{n}_F,w_h\rangle_F\lesssim h_F^{-1/2}\|w_h\|_{0,F}(\|\nabla v\|_{0,K}+h_K\|\Delta v\|_{0,K}).$$ Reliability of the error estimator {#sec4:2} ---------------------------------- We begin this section by defining error estimators for each $K\in\mathcal{T}_h$ in the following $$\begin{aligned} \eta_{s,K,1}=&\|\textbf{p}_h+\nabla y_h\|_{0,K},\quad\quad \eta_{as,K,1}=\|\textbf{q}_h+\nabla z_h\|_{0,K},\\ \eta_{s,K,2}=&h_K\|f-\nabla\cdot\textbf{p}_h-y_h\|_{0,K},\quad\eta_{as,K,2}=h_K\|y_h-y_d-\nabla\cdot\textbf{q}_h-z_h\|_{0,K},\\ \eta_{s,\partial K}=&h_K^{-1/2}\|y_h-\widehat{y}_h\|_{0,\partial K},\quad\quad \eta_{as,\partial K}= h_K^{-1/2}\|z_h-\widehat{z}_h\|_{0,\partial K}.\end{aligned}$$ Furthermore, we define $$\begin{aligned} &\eta_s^2=\sum_{K\in\mathcal{T}_h}\{\eta_{s,K,1}^2+\eta_{s,K,2}^2+\eta_{s,\partial K}^2\},\\ &\eta_{as}^2=\sum_{K\in\mathcal{T}_h}\{\eta_{as,K,1}^2+\eta_{as,K,2}^2+\eta_{as,\partial K}^2\}.\end{aligned}$$ Next, we consider the following auxiliary problem: Find $\textbf{p}(u_h),\textbf{q}(u_h)\in H(div,\Omega)$ and $y(u_h),z(u_h)\in H^1(\Omega)$ such that \[auxi\] $$\begin{aligned} \textbf{p}(u_h)+\nabla y(u_h)&=0\quad \rm{in}~\Omega,\label{auxi:1}\\ \nabla\cdot\textbf{p}(u_h)+y(u_h)&=f\quad \rm{in}~\Omega,\label{auxi:2}\\ -\textbf{p}(u_h)\cdot\textbf{n}&=u_h+g\quad \rm{on}~\partial\Omega,\label{auxi:3}\\ \textbf{q}(u_h)+\nabla z(u_h)&=0\quad \rm{in}~\Omega,\label{auxi:4}\\ \nabla\cdot\textbf{q}(u_h)+z(u_h)&=y(u_h)-y_d\quad \rm{in}~\Omega,\label{auxi:5}\\ -\textbf{q}(u_h)\cdot\textbf{n}&=0\quad \rm{on}~\Omega.\label{auxi:6}\end{aligned}$$ Now the error $\|u-u_h\|_{0,\partial\Omega}+\|\textbf{p}-\textbf{p}(u_h)\|_{0,\Omega}+\|y-y(u_h)\|_{0,\Omega}+\|\textbf{q}-\textbf{q}(u_h)\|_{0,\Omega} +\|z-z(u_h)\|_{0,\Omega}$ can be bounded by $\|z_h-z(u_h)\|_{1,\Omega}$ and $\Big\{\sum_{K\in\mathcal{T}_h}\eta_{as,\partial K}^2\Big\}^{1/2}$. \[lem3\] Let $(u,\textbf{p},y,\textbf{q},z)$ and $(u_h,\textbf{p}_h,y_h,\widehat{y}_h,\textbf{q}_h,z_h,\widehat{z}_h)$ be the solutions of problems (\[mixed\]) and (\[bbs\]) respectively. Moreover let $(\textbf{p}(u_h),y(u_h),\textbf{q}(u_h),z(u_h))$ as defined above. Then the following error estimate holds $$\begin{aligned} &\|u-u_h\|_{0,\partial\Omega}+\|\textbf{p}-\textbf{p}(u_h)\|_{0,\Omega}+\|y-y(u_h)\|_{0,\Omega}+ \|\textbf{q}-\textbf{q}(u_h)\|_{0,\Omega}\\ &+\|z-z(u_h)\|_{0,\Omega}\lesssim\Big\{\sum_{K\in\mathcal{T}_h}\eta_{as,\partial K}^2\Big\}^{1/2}+\|z_h-z(u_h)\|_{1,\Omega}\end{aligned}$$ From (\[mixed\]), (\[auxi\]) and integration by parts to yield $$\langle z-z(u_h),u-u_h\rangle_{\partial\Omega}=\|y-y(u_h)\|_{0,\Omega}^2.\label{lem3-proof:1}$$ Obviously, $(\textbf{p}-\textbf{p}(u_h),y-y(u_h))$ is the solution of system (\[mixed:1\])- (\[mixed:3\]) with $g=-u_h$ and $f=0$, and $(\textbf{q}-\textbf{q}(u_h),z-z(u_h))$ is the solution of system (\[mixed:4\])-(\[mixed:6\]) with $y_d=y(u_h)$. Therefore we have $$\begin{aligned} \|\textbf{p}-\textbf{p}(u_h)\|_{0,\Omega}+\|y-y(u_h)\|_{0,\Omega}&\lesssim\|u-u_h\|_{0,\partial\Omega},\label{lem3-proof:2}\\ \|\textbf{q}-\textbf{q}(u_h)\|_{0,\Omega}+\|z-z(u_h)\|_{0,\Omega}&\lesssim\|y-y(u_h)\|_{0,\Omega},\label{lem3-proof:3}\end{aligned}$$ by the trace theorem. From (\[mixed:7\]), (\[bbs:3\]) and (\[lem3-proof:1\]), we obtain $$\begin{aligned} \alpha\|u-u_h\|_{0,\partial\Omega}^2\leq&\langle \widehat{z}_h-z(u_h),u-u_h\rangle_{\partial\Omega}\nonumber\\ \leq&\|z_h-z(u_h)\|_{1,\Omega}\|u-u_h\|_{0,\partial\Omega}\nonumber\\ &+\langle z_h-\widehat{z}_h,(\textbf{p}-\textbf{p}(u_h))\cdot\textbf{n}\rangle_{\partial\Omega}\nonumber\\ \lesssim&\|z_h-z(u_h)\|_{1,\Omega}\|u-u_h\|_{0,\partial\Omega}\label{lem3-proof:4}\\ &+\Big\{\sum_{K\in\mathcal{T}_h}\eta_{as,\partial K}^2\Big\}^{1/2}\Big(\|\textbf{p}-\textbf{p}(u_h)\|_{0,\Omega}\nonumber\\ &+\|y-y(u_h)\|_{0,\Omega}\Big),\nonumber\end{aligned}$$ by the trace theorem and Lemma \[lem2\]. Then we can conclude the proof by combining (\[lem3-proof:2\])-(\[lem3-proof:4\]). \[lem4\] Let $(\textbf{p}(u_h),y(u_h),\textbf{q}(u_h),z(u_h))$ and $(u_h,\textbf{p}_h,y_h,\widehat{y}_h,\textbf{q}_h,z_h,\widehat{z}_h)$ be the solutions of problems (\[auxi\]) and (\[bbs\]), then the following error estimates hold $$\begin{aligned} \|\nabla(y_h-y(u_h))\|_{0,K}\leq \eta_{s,K,1}+\|\textbf{p}_h-\textbf{p}(u_h)\|_{0,K},\label{lem4:1}\\ \|\nabla(z_h-z(u_h))\|_{0,K}\leq \eta_{as,K,1}+\|\textbf{q}_h-\textbf{q}(u_h)\|_{0,K},\label{lem4:2}\end{aligned}$$ for each $K\in\mathcal{T}_h$. Since $\textbf{p}(u_h)=-\nabla y(u_h)$ and $\textbf{q}(u_h)=-\nabla z(u_h)$ in each $K\in\mathcal{T}_h$, we can obtain the error estimates (\[lem4:1\]) and (\[lem4:2\]) directly by the triangle inequality. Now we are ready to prove a posteriori error estimators for $\|\textbf{p}_h-\textbf{p}(u_h)\|_{0,\Omega}+\|y_h-y(u_h)\|_{0,\Omega}$ and $\|\textbf{q}_h-\textbf{q}(u_h)\|_{0,\Omega}+\|z_h-z(u_h)\|_{0,\Omega}$. \[lem5\] Let $(\textbf{p}(u_h),y(u_h),\textbf{q}(u_h),z(u_h))$ and $(u_h,\textbf{p}_h,y_h,\widehat{y}_h,\textbf{q}_h,z_h,\widehat{z}_h)$ be the solutions of problems (\[auxi\]) and (\[bbs\]), then we have $$\begin{aligned} &\|\textbf{p}_h-\textbf{p}(u_h)\|_{0,\Omega}+\|y_h-y(u_h)\|_{0,\Omega}+\|\tau_1^{1/2}(y_h-\widehat{y}_h)\|_{0,\partial\mathcal{T}_h}\lesssim\eta_s,\label{lem5:1}\\ &\|\textbf{q}_h-\textbf{q}(u_h)\|_{0,\Omega}+\|z_h-z(u_h)\|_{0,\Omega}+\|\tau_2^{1/2}(z_h-\widehat{z}_h)\|_{0,\partial\mathcal{T}_h}\lesssim\eta_s+\eta_{as}, \label{lem5:2}\end{aligned}$$ for $\tau_1=\tau_2=h_K^{-1}$ on each $\partial K$ for all $K\in\mathcal{T}_h$, where $$\|\cdot\|_{0,\partial\mathcal{T}_h}^2=\sum_{K\in \mathcal{T}_h}\|\cdot\|_{0,\partial K}^2.$$ According to the definition of the operator $\mathcal{B}$ to infer that $$\mathcal{B}(\textbf{p}(u_h)-\textbf{p}_h,y(u_h)-y_h,y(u_h)-\widehat{y}_h;\textbf{r},w,\mu;\tau_1)=0,$$ for any $(\textbf{r},w,\mu)\in\textbf{V}_h^k\times W_h^k\times M_h^k$. Then from the above equality and the definition of the operator $\mathcal{B}$, we have $$\begin{aligned} &\|\textbf{p}_h-\textbf{p}(u_h)\|_{0,\Omega}^2+\|y_h-y(u_h)\|_{0,\Omega}^2+\|\tau_1^{1/2}(y_h-\widehat{y}_h)\|_{0,\partial\mathcal{T}_h}^2\nonumber\\ =&\mathcal{B}(\textbf{p}(u_h)-\textbf{p}_h,y(u_h)-y_h,y(u_h)-\widehat{y}_h;\textbf{p}(u_h)-\textbf{p}_h,y(u_h)-y_h,y(u_h)-\widehat{y}_h;\tau_1)\nonumber\\ =&\mathcal{B}(\textbf{p}(u_h)-\textbf{p}_h,y(u_h)-y_h,y(u_h)-\widehat{y}_h;\delta_{\textbf{p}},\delta_{y},\delta_{\widehat{y}};\tau_1),\nonumber\end{aligned}$$ where $\delta_{\textbf{p}}=\textbf{p}(u_h)-\textbf{p}_h-\textbf{r}$, $\delta_{y}=y(u_h)-y_h-w$, and $\delta_{\widehat{y}} =y(u_h)-\widehat{y}_h-\mu$ for any $(\textbf{r},w,\mu)\in\textbf{V}_h^k\times W_h^k\times M_h^k$. By integration by parts we yield $$\begin{aligned} &\mathcal{B}(\textbf{p}(u_h)-\textbf{p}_h,y(u_h)-y_h,y(u_h)-\widehat{y}_h;\delta_{\textbf{p}},\delta_{y},\delta_{\widehat{y}};\tau_1)\nonumber\\ =&-(\textbf{p}_h+\nabla y_h,\delta_{\textbf{p}})_{\mathcal{T}_h}+(f-\nabla\cdot\textbf{p}_h-y_h,\delta_y)_{\mathcal{T}_h}\nonumber\\ &+\langle y_h-\widehat{y}_h,\delta_{\textbf{p}}\cdot\textbf{n}\rangle_{\partial\mathcal{T}_h}+\langle \tau_1(\widehat{y}_h-y_h),\delta_y\rangle_{\partial\mathcal{T}_h}\nonumber\\ &-\langle(\textbf{p}(u_h)-\textbf{p}_h)\cdot\textbf{n}+\tau_1(\widehat{y}_h-y_h),\delta_{\widehat{y}}\rangle_{\partial\mathcal{T}_h}.\nonumber\end{aligned}$$ From (\[HDG:3\]), (\[HDG:4\]) and (\[auxi:3\]), we arrive at $$\begin{aligned} -\langle(\textbf{p}(u_h)-\textbf{p}_h)\cdot\textbf{n}+\tau_1(\widehat{y}_h-y_h),\delta_{\widehat{y}}\rangle_{\partial\mathcal{T}_h}=0.\nonumber\end{aligned}$$ Now we set $\textbf{r}=\Pi_0^o(\textbf{p}(u_h)-\textbf{p}_h)$ in the definition of $\delta_{\textbf{p}}$ and $w=\Pi_0^o(y(u_h)-y_h)$ in the definition of $\delta_{y}$. Then from Lemma \[lem1\] we have $$\begin{aligned} &-(\textbf{p}_h+\nabla y_h,\delta_{\textbf{p}})_{\mathcal{T}_h}+(f-\nabla\cdot\textbf{p}_h-y_h,\delta_y)_{\mathcal{T}_h}\nonumber\\ \lesssim&\Big\{\sum_{K\in\mathcal{T}_h}\eta_{s,K,1}^2\Big\}^{1/2}\|\textbf{p}(u_h)-\textbf{p}_h\|_{0,\Omega}\nonumber\\ &+\Big\{\sum_{K\in\mathcal{T}_h}\eta_{s,K,2}^2\Big\}^{1/2}\|\nabla(y(u_h)-y_h)\|_{0,\Omega},\nonumber\end{aligned}$$ and $$\begin{aligned} &\langle \tau_1(\widehat{y}_h-y_h),\delta_y\rangle_{\partial\mathcal{T}_h}\nonumber\\ \lesssim&\Big\{\sum_{K\in\mathcal{T}_h}\eta_{s,\partial K}^2\Big\}^{1/2}\|\nabla(y(u_h)-y_h)\|_{0,\Omega}.\nonumber\end{aligned}$$ By using Lemma \[lem2\] to yield $$\begin{aligned} &\langle y_h-\widehat{y}_h,\delta_{\textbf{p}}\cdot\textbf{n}\rangle_{\partial\mathcal{T}_h}\nonumber\\ =&\sum_{K\in\mathcal{T}_h}\sum_{F\in\partial K}\langle y_h-\widehat{y}_h,\delta_{\textbf{p}}\cdot\textbf{n}\rangle_F\nonumber\\ \lesssim&\Big\{\sum_{K\in\mathcal{T}_h}\eta_{s,\partial K}^2\Big\}^{1/2}\|\textbf{p}(u_h)-\textbf{p}_h\|_{0,\Omega}\nonumber\\ &+\Big\{\sum_{K\in\mathcal{T}_h}\eta_{s,\partial K}^2\Big\}^{1/2}\Big(\Big\{\sum_{K\in\mathcal{T}_h}\eta_{s,K,2}^2\Big\}^{1/2}+ \|y_h-y(u_h)\|_{0,\Omega}\Big).\nonumber\end{aligned}$$ Now we can obtain the approximation result (\[lem5:1\]) by combining Lemma \[lem4\], Young’s inequality and the above equalities and inequalities. Moreover, the error estimate (\[lem5:2\]) can be proved similarly. Compared to the error estimators introduced in [@cz2012; @cz2013; @cz2013a], the Lemma \[lem5\] provides an a posteriori error estimator without any postprocessing solutions, hence it is easer to calculate. Combining Lemma \[lem3\], Lemma \[lem4\] and Lemma \[lem5\] results in the following reliability estimate Let $(u,\textbf{p},y,\textbf{q},z)$ and $(u_h,\textbf{p}_h,y_h,\widehat{y}_h,\textbf{q}_h,z_h,\widehat{z}_h)$ be the solutions of problems (\[mixed\]) and (\[bbs\]). Then we have the following error estimate $$\begin{aligned} &\|u-u_h\|_{0,\partial\Omega}+\|\textbf{p}-\textbf{p}_h\|_{0,\Omega}+\|y-y_h\|_{1,\Omega}\\ &+\|\textbf{q}-\textbf{q}_h\|_{0,\Omega}+\|z-z_h\|_{1,\Omega}\lesssim\eta_s+\eta_{as},\end{aligned}$$ for $\tau_1=\tau_2=h_K^{-1}$ on each $\partial K$ for all $K\in\mathcal{T}_h$. Efficiency of the error estimator {#sec5} ================================= In this section, we will prove that, up to data oscillations, the estimator $\eta_s+\eta_{as}$ also provides a lower bound for the error. Especially, we will show that the local contributions of the estimator can be bounded from above by the local constituents of the error and the associated data oscillations. First of all, we define the data oscillations by $$\begin{aligned} osc^2(f,\mathcal{T}_h)&=\sum_{K\in\mathcal{T}_h}osc^2(f,K),\\ osc^2(y_d,\mathcal{T}_h)&=\sum_{K\in\mathcal{T}_h}osc^2(y_d,K),\end{aligned}$$ where $$\begin{aligned} osc(f,K)&=h_K\|f-\Pi_k^of\|_{0,K},\\ osc(y_d,K)&=h_K\|y_d-\Pi_k^oy_d\|_{0,K}.\end{aligned}$$ Obviously, $osc(f,K)$ and $osc(y_d,K)$ are of same order with $\eta_{s,K,2}$ and $\eta_{as,K,2}$ for non smooth $f$ and $y_d$ and of higher order for smooth $f$ and $y_d$. Next we denote by $\lambda_i^K$, $1\leq i\leq 3$, the barycentric coordinates of $K\in\mathcal{T}_h$ and refer to $E_K=27\Pi_{i=1}^3\lambda_i^K$ as the associated element bubble function. From [@hiis2006], we have \[bubble\] $$\begin{aligned} &\|p_K\|_{0,K}^2\lesssim(p_K,p_KE_K)_{K}\quad K\in\mathcal{T}_h,\label{bubble:1}\\ &\|p_KE_K\|_{0,K}\lesssim\|p_K\|_{0,K}\quad K\in\mathcal{T}_h,\label{bubble:2}\\ &\|p_KE_K\|_{1,K}\lesssim h_K^{-1}\|p_K\|_{0,K}\quad K\in\mathcal{T}_h,\label{bubble:3}\end{aligned}$$ for $p_K\in\mathcal{P}^k(K)$. Then the following error estimates hold. \[lem6\] Let $(u,\textbf{p},y,\textbf{q},z)$, $(\textbf{p}(u_h),y(u_h),\textbf{q}(u_h),z(u_h))$ and $(u_h,\textbf{p}_h,y_h,\widehat{y}_h,\\ \textbf{q}_h,z_h,\widehat{z}_h)$ be the solutions of problems (\[mixed\]), (\[auxi\]) and (\[bbs\]) respectively. Then we have $$\begin{aligned} \eta_{s,K,1}\leq&\|\textbf{p}-\textbf{p}_h\|_{0,K}+\|\nabla(y-y_h)\|_{0,K}\quad K\in\mathcal{T}_h,\label{lem6:1}\\ \eta_{s,K,2}\lesssim& osc(f,K)+\|\textbf{p}-\textbf{p}_h\|_{0,K}+\|y-y_h\|_{0,K}\quad K\in\mathcal{T}_h,\label{lem6:2}\\ \sum_{K\in\mathcal{T}_h}\eta_{s,\partial K}^2\lesssim &\|u-u_h\|_{0,\partial\Omega}^2+\|\textbf{p}-\textbf{p}_h\|_{0,\Omega}^2+ \|y-y_h\|_{1,\Omega}^2+osc^2(f,\mathcal{T}_h),\label{lem6:3}\end{aligned}$$ and $$\begin{aligned} \eta_{as,K,1}\leq&\|\textbf{q}-\textbf{q}_h\|_{0,K}+\|\nabla(z-z_h)\|_{0,K}\quad K\in\mathcal{T}_h,\label{lem6:4}\\ \eta_{as,K,2}\lesssim&osc(y_d,K)+\|\textbf{q}-\textbf{q}_h\|_{0,K}+\|z-z_h\|_{0,K}\nonumber\\ &+\|y-y_h\|_{0,K}\quad K\in\mathcal{T}_h,\label{lem6:5}\\ \sum_{K\in\mathcal{T}_h}\eta_{as,\partial K}^2\lesssim&\|u-u_h\|_{0,\partial\Omega}^2+\|\textbf{q}-\textbf{q}_h\|_{0,\Omega}^2+\|z-z_h\|_{1,\Omega}^2\nonumber\\ &+\|y-y_h\|_{0,\Omega}^2+osc^2(y_d,\mathcal{T}_h).\label{lem6:6}\end{aligned}$$ Obviously, the inequalities (\[lem6:1\]) and (\[lem6:4\]) can be obtained directly by the triangle inequality. According to the definition of $\eta_{s,K,2}$ and the triangle inequality we know that $$\eta_{s,K,2}\leq osc(f,K)+h_K\|\Pi_k^of-\nabla\cdot\textbf{p}_h-y_h\|_{0,K}.$$ Setting $p_K=\Pi_k^of-\nabla\cdot\textbf{p}_h-y_h$ we obtain $$\begin{aligned} h_K^2\|p_K\|_{0,K}^2\lesssim& h_K^2(\Pi_k^of-f+f-\nabla\cdot\textbf{p}_h-y_h,E_Kp_K)_K\\ =&h_K^2(\Pi_k^of-f,E_Kp_K)_K+h_K^2(\nabla\cdot(\textbf{p}-\textbf{p}_h)+y-y_h,E_Kp_K)_K,\end{aligned}$$ from (\[bubble:1\]). Then we can obtain the error estimate (\[lem6:2\]) by using (\[bubble\]), Young’s inequality and the above two inequalities. And the approximation result (\[lem6:5\]) can be proved similarly. Now we turn to prove the error estimate (\[lem6:3\]). From the proof of Lemma \[lem5\], we have $$\begin{aligned} \sum_{K\in\mathcal{T}_h}\eta_{s,\partial K}^2\lesssim&\|\textbf{p}(u_h)-\textbf{p}_h\|_{0,\Omega}^2+\|y(u_h)-y_h\|_{1,\Omega}^2\\ &+\sum_{K\in\mathcal{T}_h}\{\eta_{s,K,1}^2+\eta_{s,K,2}^2\}.\end{aligned}$$ Then by using (\[lem3-proof:2\]), (\[lem6:1\]), (\[lem6:2\]), the triangle inequality and the above inequality to infer that $$\begin{aligned} \sum_{K\in\mathcal{T}_h}\eta_{s,\partial K}^2\lesssim\|u-u_h\|_{0,\partial\Omega}+\|\textbf{p}-\textbf{p}_h\|_{0,\Omega}^2+ \|y-y_h\|_{1,\Omega}^2+osc^2(f,\mathcal{T}_h).\end{aligned}$$ Therefore the approximation result (\[lem6:3\]) is derived. And the inequality (\[lem6:6\]) can be proved similarly. Numerical experiments {#sec6} ===================== Now we provide two examples in order to examine the quality of the derived estimator. As we know, an adaptive algorithm consists of the loops “**SOLVE**$\rightarrow$**ESTIMATE**$\rightarrow$**MARK**$\rightarrow$**REFINE**”. In this section, a fix-point iteration algorithm presented in [@zilz2015] is used for solving the model problem. In step **REFINE**, the newest vertex bisection algorithm [@s2007] is employed, and the following marking strategy is used in step **MARK** $$\sum_{K\in\mathcal{M}_h}\eta_K^2\geq\theta\eta^2,$$ where $$\begin{aligned} \eta^2=\eta_s^2&+\eta_{as}^2,\\ \eta_{K}^2=\eta_{s,K,1}^2+\eta_{s,K,2}^2+\eta_{s,\partial K}^2&+\eta_{as,K,1}^2+\eta_{as,K,2}^2+\eta_{as,\partial K}^2.\end{aligned}$$ Furthermore, we define $$\begin{aligned} E=&\|u-u_h\|_{0,\partial\Omega}+\|\textbf{p}-\textbf{p}_h\|_{0,\Omega}+\|y-y_h\|_{1,\Omega}\\ &+\|\textbf{q}-\textbf{q}_h\|_{0,\Omega}+\|z-z_h\|_{1,\Omega}.\end{aligned}$$ Here, we note that the figures of convergence history are plotted in log-log coordinates. Based on the domain $\Omega=(0,1)^2$, we consider an example with $u_a=-0.1$, $u_b=0.1$ and $\alpha=1$. Let the functions $f$, $y_d$ and $g$ be such that the Neumann boundary control problem has the following exact solutions $$\begin{aligned} y=\sin(2\pi x_1)\sin(2\pi x_2),\quad z=\cos(2\pi x_1),\quad u=\Pi_{U_{ad}}\Big\{-\frac{1}{\alpha}z|_{\partial\Omega}\Big\}.\end{aligned}$$ ![Left: Convergence history for $k=1$. Right: Convergence history for $k=2$.[]{data-label="ex1f1"}](ex1_convergencep1.png "fig:"){width="6.5cm" height="5.4cm"} ![Left: Convergence history for $k=1$. Right: Convergence history for $k=2$.[]{data-label="ex1f1"}](ex1_convergencep2.png "fig:"){width="6.5cm" height="5.4cm"} ![Left: The effectiveness index for $k=1$. Right: The effectiveness index for $k=2$.[]{data-label="ex1f2"}](ex1_efficiencyp1.png "fig:"){width="6.5cm" height="5.4cm"} ![Left: The effectiveness index for $k=1$. Right: The effectiveness index for $k=2$.[]{data-label="ex1f2"}](ex1_efficiencyp2.png "fig:"){width="6.5cm" height="5.4cm"} ![Left: The profile of the numerical control for $k=1$. Right: The profile of the numerical adjoint state for $k=1$.[]{data-label="ex1f3"}](ex1_control.png "fig:"){width="6.5cm" height="5.4cm"} ![Left: The profile of the numerical control for $k=1$. Right: The profile of the numerical adjoint state for $k=1$.[]{data-label="ex1f3"}](ex1_astate.png "fig:"){width="6.5cm" height="5.4cm"} We test the example for $k=1$ and $k=2$. From the convergence history in Figure \[ex1f1\] for $\theta=0.2$ and $\theta=0.6$, we find that the error $E$ is equivalent to the estimator $\eta$ and the error $E$ and the estimator $\eta$ can achieve the optimal convergence order by adaptive refinement. Furthermore, the effectiveness index is presented in Figure \[ex1f2\], which indicates the obtained a posteriori error estimator is very efficient. Finally, the profiles of the numerical control and adjoint state are shown in Figure \[ex1f3\]. We consider an example with a boundary term $\int_{\partial\Omega}yg_1dx$ in the objective functional. Then the adjoint problem possesses the Nuemann boundary condition $\nabla z\cdot\textbf{n}=g_1$. Here the designed domain is given by $\Omega=(-1,1)^2\backslash([0,1]\\ \times(-1,0])$. The control constraints and the regularization parameter are set as $u_a=-0.2$, $u_b=0.2$ and $\alpha=1$. Furthermore let the functions $f$, $y_d$ and $g$ be such that the Neumann boundary control problem has the following exact solutions $$\begin{aligned} y(r,\theta_1)&=0.\\ z(r,\theta_1)&=r^{2/3}\cos\Big(\frac{2}{3}\theta_1\Big),\\ u(r,\theta_1)&=\Pi_{U_{ad}}\Big(-r^{2/3}\cos\Big(\frac{2}{3}\theta_1\Big)\Big),\end{aligned}$$ where $r=\sqrt{x_1^2+x_2^2}$, $\theta_1=\arccos(r^{-1}x\cdot e_1)$ and $e_1=[1,0]^{T}$. ![The profiles of the initial mesh (Top-Left), the adaptive mesh (Top-Right), the numerical control (Bottom-Left) and the numerical adjoint state (Bottom-Right) for $k=2$ and $\theta=0.4$.[]{data-label="ex2f1"}](initial_mesh.png "fig:"){width="6.5cm" height="5.0cm"} ![The profiles of the initial mesh (Top-Left), the adaptive mesh (Top-Right), the numerical control (Bottom-Left) and the numerical adjoint state (Bottom-Right) for $k=2$ and $\theta=0.4$.[]{data-label="ex2f1"}](adaptive_mesh.png "fig:"){width="6.5cm" height="5.0cm"} ![The profiles of the initial mesh (Top-Left), the adaptive mesh (Top-Right), the numerical control (Bottom-Left) and the numerical adjoint state (Bottom-Right) for $k=2$ and $\theta=0.4$.[]{data-label="ex2f1"}](control.png "fig:"){width="6.5cm" height="5.0cm"} ![The profiles of the initial mesh (Top-Left), the adaptive mesh (Top-Right), the numerical control (Bottom-Left) and the numerical adjoint state (Bottom-Right) for $k=2$ and $\theta=0.4$.[]{data-label="ex2f1"}](astate.png "fig:"){width="6.5cm" height="5.0cm"} ![The convergence history for $k=1$ (Left) and $k=2$ (Right).[]{data-label="ex2f2"}](convergencep1.png "fig:"){width="6.5cm" height="5.4cm"} ![The convergence history for $k=1$ (Left) and $k=2$ (Right).[]{data-label="ex2f2"}](convergencep2.png "fig:"){width="6.5cm" height="5.4cm"} ![The effectiveness index for $k=1$ (Left) and $k=2$ (Right).[]{data-label="ex2f3"}](efficiencyp1.png "fig:"){width="6.5cm" height="5.4cm"} ![The effectiveness index for $k=1$ (Left) and $k=2$ (Right).[]{data-label="ex2f3"}](efficiencyp2.png "fig:"){width="6.5cm" height="5.4cm"} The adjoint exhibits a typical singularity at the reentrant corner of the domain $\Omega$. In Figure \[ex2f1\], we show the profiles of the initial mesh, the adaptive mesh, the numerical control and the numerical adjoint state for $k=2$ and $\theta=0.4$. We can find that the mesh nodes are concentrated around the reentrant corner where the singularity is induced. Hence the obtained a posteriori error estimator can grab efficiently the singularity of the problem. In Figure \[ex2f2\], the convergence history for $k=1$ and $k=2$ is presented, which indicates that the estimator $\eta$ is equivalent to the error $E$ and the estimator $\eta$ and the error $E$ can achieve the optimal convergence order while $\theta$ is less than a certain value. In Figure \[ex2f3\], the effectiveness index for $k=1$ and $k=2$ are provided. We can find that the effectiveness index for $k=1$ is between 0.96 and 1 and the effectiveness index for $k=2$ is between 0.6 and 1.2, which means the obtained a posteriori error estimator is very efficient. Conclusions {#sec7} =========== In this paper, a Neumann boundary optimal control problem is considered. We use the hybridizable discontinuous Galerkin method as the discretization technique, and the flux variables, the scalar variables and the boundary trace variables are approximated by polynomials of degree $k$. Then an efficient and reliable a posteriori error estimator without any postprocessing solutions is obtained for the errors. Finally, two numerical experiments are provided to verify the performance of the obtained a posteriori error estimator. This work is just the first step for a posteriori error analysis of HDG methods for boundary control problems. Next we extend the method and the result to the more complicated situations for instance the Dirichlet boundary control problem and the Stokes optimal control problem. I. Babuška and W.C. Rheinboldt, Error estimates for adaptive finite element computations. SIAM J. Numer. Anal. 15 (1978) 736-754. R. Becker, H. Kapp and R. 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Yücel and P. Benner, Adaptive discontinuous Galerkin methods for state constrained optimal control problems governed by convection diffusion equations. Comput. Optim. Appl. 62 (2015) 291-321. H. Yücel and B. Karasözen, Adaptive symmetric interior penalty Galerkin (SIPG) method for optimal control of convection diffusion equations with control constraints. Optimization 63 (2014) 145-166. Z. Zhou, X. Yu and N. Yan, The local discontinuous Galerkin approximation of convection dominated diffusion optimal control problems with control constraints. Numer. Methods Partial Differential Equations 30 (2014) 339-360. Q. Zhang, K. Ito, Z. Li and Z. Zhang, Immersed finite elements for optimal control problems of elliptic PDEs with interfaces. J. Comput. Phys. 298 (2015) 305-319. [^1]: This work is supported by State Key Program of National Science Foundation of China (11931003) and National Nature Science Foundation of China (41974133, 11671157).

--- abstract: 'We first discuss why there remains continuing, strong motivation to investigate Hubble’s Constant. Then we review new evidence from an investigation of the Galactic Open Clusters containing Cepheids by Hoyle et al. that the metallicity dependence of the Cepheid P-L relation is stronger than expected. This result is supported by a new analysis of mainly HST Distance Scale Key Project data which shows a correlation between host galaxy metallicity and the rms scatter around the Cepheid P-L relation. If Cepheids do have a significant metallicity dependence then an already existing scale error for Tully-Fisher distances becomes worse and the distances of the Virgo and Fornax clusters extend to more than 20Mpc, decreasing the value of H$_0$. Finally, if the Cepheids have a metallicity dependence then so do Type Ia Supernovae since the metallicity corrected Cepheid distances to eight galaxies with SNIa now suggests that the SNIa peak luminosity is fainter in metal poor galaxies. As well as having important implications for H$_0$, this would also imply that the evidence for a non-zero cosmological constant from the SNIa Hubble Diagram may be subject to corrections for metallicity which are as big as the effects of cosmology.' author: - 'T. Shanks, P.D. Allen, F. Hoyle' - 'N.R. Tanvir' title: 'Cepheid and SNIa Distance Scales.' --- Status of Extragalactic Distance Scale ====================================== One major motivation for studying Hubble’s Constant is the complicated nature of the current standard model in cosmology, $\Lambda$-CDM. In this model, to order of magnitude, $\Omega_{baryon}\approx \Omega_{CDM}\approx \Omega_{\Lambda}$ and this seems unnatural. The coincidence between the CDM and Baryon densities worried a few when CDM was first introduced (Peebles 1984, Shanks 1985). The coincidence between $\Omega_{\Lambda}$ and the others worried many more (eg Dolgov, 1983, Peebles and Ratra, 1988 and Wetterich, 1988). These fine-tuning problems of the standard model are compounded by the fact that the inflation model on which the standard model sits, was partly based on a fine-tuning argument, the flatness-problem; to begin by eliminating one fine tuning problem only to end up with several gives the appearance, at least, of circular reasoning! Shanks (1985, 1991, 1999) noted that a simpler model immediately became available if H$_0$ actually lay below 50 kms$^{-1}$ Mpc$^{-1}$. An inflationary model with $\Omega_{baryon}$=1 is then better placed to escape the baryon nucleosynthesis constraint. Simultaneously, the low value of H$_0$ means that the X-ray gas in the Coma cluster increases towards the Coma virial mass and the lifetime of an Einstein-de Sitter Universe extends to become compatible with the ages of the oldest stars. Given the historical uncertainty there has been in the value of H$_0$, this provides clear motivation for investigating the distance scale route to a better determination of Hubble’s Constant. A New Era for Determining H$_0$ =============================== Some 25 galaxies have had Cepheids detected by HST. Seventeen of these were observed by the HST Distance Scale Key Project (Freedman et al, 1994). Seven were observed in galaxies with SNIa by Sandage and collaborators (eg Sandage et al, 1996) and M96 in the Leo I Group was observed by Tanvir et al (1995). In Fig. 1 we use these data to update the comparison of I-band TF distances of Pierce & Tully (1992) with HST Cepheid distances. As can be seen, the result implies that TF distance moduli at Virgo underestimated by $\approx$25$\pm$5%. This reduces Tully-Fisher estimates of H$_0$ from $\approx$85 to $\approx$65kms$^{-1}$Mpc$^{-1}$ (Giovanelli et al, 1997, Shanks 1997, Shanks, 1999, Sakai et al, 1999). The correlation of Cepheid residuals with line-width suggests TF distances may be Malmquist biased - possibly implying a bigger TF scale error at larger distances. This clear problem for TF distances, which previously has been the ‘gold standard’ of secondary distance indicators, warns that errors in the extragalactic distance scale may still be seriously underestimated! NGC7790 Cepheid metallicity dependence? ======================================= New JKT 1.0m + CTIO 0.9m + UKIRT UBVK photometry of Cepheid Open Clusters by Hoyle et al. (2001) has uncovered an anomaly in the NGC7790 UBV 2-colour diagram, in that the F stars in the cluster show a strong UV excess with respect to zero-age main sequence stars (see Fig. 2). The result is confirmed by independent photometric data (Fry, 1997, Fry and Carney, 1997) as shown in Fig. 7b of Hoyle et al (2001). If the UV excess is caused by metallicity then NGC7790 would have \[Fe/H\]$\approx$ -1.5 ! To keep the Galactic Cepheid P-L relation as tight as previously observed implies that Cepheids may have a stronger metallicity dependence, $\Delta M \approx -0.66 \Delta[Fe/H]$, than previously expected, in the sense that low metallicity Cepheids are intrinsically fainter. Currently we are obtaining metallicities for the F stars in NGC7790 in order to confirm this result. HST Cepheid metallicity dependence ================================== Meanwhile, Allen & Shanks (2001) have found an $\approx3\sigma$ correlation between dispersion around the Cepheid P-L relation and galaxy metallicity for HST Cepheid galaxies (see Fig. 3). This again suggests a strong Cepheid P-L metallicity dependence and tends to support the results described above. The more extended star-formation history of high metallicity galaxies may leave a wide range of metallicities than in low metallicity galaxies like the LMC, resulting in a higher P-L dispersion if Cepheid luminosities at given period depend strongly on metallicity. Allen & Shanks (2001) also obtain Cepheid distances via truncated maximum likelihood P-L fits to account for magnitude incompleteness caused by the non-negligible scatter in the HST P-L relations. They found that Cepheid galaxy distances at the limit of HST reach are too low. The higher than expected P-L dispersion for distant, metal-rich galaxies accentuates this effect. The conclusion is that current HST Cepheid distance moduli may be underestimated by more than 0.5 mag at the redshift of Virgo and Fornax due to both metallicity and statistical incompleteness bias. The TF distances discussed above are then underestimates by approximately 1 magnitude. Eight HST Cepheid galaxies also have Type Ia distances. Correcting the Cepheid scale for metallicity and incompleteness bias as above and then using these distances to derive peak luminosities using the SNIa data from Gibson et al (2000), implies a strong correlation between Type Ia peak luminosity and metallicity. Such a scatter in SNIa luminosities could easily be disguised by magnitude selection effects at moderate redshifts. At higher redshift the correlation is in the right direction to explain away the need for a cosmological constant in the Supernova Hubble Diagram results, since galaxies at high redshift might be expected to have lower metallicity. Thus the conclusion is that if Cepheids have strong metallicity dependence then so have SNIa and therefore SNIa estimates of q$_0$ and H$_0$ may require significant corrections for metallicity. Conclusions - Implications for H$_0$ and SNIa ============================================= Our conclusions are as follows:- - Key Project HST Cepheid distances imply Tully-Fisher distances at Virgo/Fornax are underestimated by $\approx25\pm5$%, reducing H$_0$ from $\approx$85 to $\approx$65kms$^{-1}$Mpc$^{-1}$. - TF distances may be Malmquist biased, suggesting there may be a bigger TF scale error at larger distances. - If the UV excess of F stars in open cluster NGC7790 is caused by low metallicity then Cepheids have a strong metallicity dependence, $\Delta M \approx -0.66 \Delta[Fe/H]$. - Current HST Cepheid distances may be significantly underestimated at Virgo/Fornax redshifts due to metallicity and magnitude incompleteness bias, implying that values of H$_0<$50kms$^{-1}$Mpc$^{-1}$ may still not be ruled out. - If Cepheids have a strong metallicity dependence then so have SNIa . Thus significant metallicity corrections may need to be applied to the Type Ia Hubble Diagram before reliable estimates of q$_0$ or H$_0$ can be made. We thank the HST Distance Scale Key Project for making the Cepheid data freely available. Allen, P.D. & Shanks, T. 2001, MNRAS, submitted. Dolgov, A.D., 1983, In ‘The Very Early Universe’, Eds. Gibbons, G.W., Hawking, S., & Siklos, S.T.C. CUP, pp. 449-458 Freedman, W. L. et al., 1994, ApJ, 427, 628. Fry, A.M., 1997, PhD thesis, Univ. of North Carolina. Fry, A.M. & Carney, B.W., 1997, AJ, 113, 1073. Gibson, B. K. et al, 1999, ApJ, 512, 48. Giovanelli, R., Haynes, M.P., da Costa, L.N., Freudling, W., Salzer, J.J., Wegner, G., 1997, ApJ, 477, L1. Hoyle, F., Shanks, T. & Tanvir, N.R., 2001 MNRAS submitted, astro-ph/0002521. Peebles, P.J.E., 1984, ApJ, 284, 439. Peebles, P.J.E. & Ratra, B. 1988, ApJ, 325, L17. Pierce, M.J. and Tully, R.B., 1992 ApJ, 387, 47. Sakai, S. et al, 1999, ApJ, 523, 540. Sandage, A.R., Saha, A., Tammann, G. A., Labhardt, L., Panagia, N., Macchetto, F. D., ApJ, 1996, 460, L15. Shanks, T., 1985, Vistas in Astronomy, 28, 595. Shanks, T. et al., 1991, In ‘Observational Tests of Cosmological Inflation’, Eds. Shanks, T., Banday, A.J., Ellis, R.S., Frenk, C.S. & Wolfendale, A.W. pp. 205-210 Dordrecht:Kluwer. Shanks, T., 1997, MNRAS, 290, L77. Shanks, T., 1999, In Harmonising Cosmic Distance Scales, eds Egret D. & Heck, A., PASP, pp. 230-234. Tanvir, N. R., Shanks, T., Ferguson, H.C. & Robinson, D.R.T. 1995, Nature, 377, 27. Wetterich, C., 1988, Nucl. Phys. B302, 645.

--- abstract: 'Local Fourier analysis is a commonly used tool to assess the quality and aid in the construction of geometric multigrid methods for translationally invariant operators. In this paper we automate the process of local Fourier analysis and present a framework that can be applied to arbitrary, including non-orthogonal, repetitive structures. To this end we introduce the notion of crystal structures and a suitable definition of corresponding wave functions, which allow for a natural representation of almost all translationally invariant operators that are encountered in applications, e.g., discretizations of systems of PDEs, tight-binding Hamiltonians of crystalline structures, colored domain decomposition approaches and last but not least two- or multigrid hierarchies. Based on this definition we are able to automate the process of local Fourier analysis both with respect to spatial manipulations of operators as well as the Fourier analysis back-end. This automation most notably simplifies the user input by removing the necessity for compatible representations of the involved operators. Each individual operator and its corresponding structure can be provided in any representation chosen by the user.' author: - 'Karsten Kahl[^1]' - Nils Kintscher bibliography: - 'Manuscript.bib' title: 'Automated Local Fourier Analysis (aLFA)[^2]' --- Introduction ============ Local Fourier analysis (LFA) is a powerful tool used in the construction and analysis of multigrid methods introduced in [@Bran1977]. The fundamental idea of LFA is to leverage the connection between position space and frequency space via the Fourier transform. That is, in case the involved operators can be described by stencils in position space, meaning that they are translationally invariant, their Fourier transform yields so-called symbols, which can be handled much more easily. In the context of multigrid methods LFA can be used to obtain precise approximations of the asymptotic convergence rate by assessing the spectral radius of the corresponding error propagation operator [@Saad]. This approximation of the convergence rate is (asymptotically) still valid if the translational invariance is slightly violated, as for example when lexicographic Gauss-Seidel type smoothers are used, or in the case of certain non-periodic boundary conditions [@Bran1994; @MR3866073; @Stevenson1990]. In other cases a similar convergence rate can usually be obtained by additional processing [@KahlKint2018; @TrotOostSchu2001]. Due to these facts, LFA is one of the main tools in the quantitative analysis of two- and multigrid methods. An introduction to LFA including several examples can be found in [@TrotOostSchu2001; @WienJopp2004]. Multigrid methods have first been considered for the solution of the linear systems of equations originating in the discretization of (elliptic) partial differential equations (PDEs) [@TrotOostSchu2001]. Due to the fact that the simplest tiling of space is rectangular and discretizations are particularly simple to carry out on such tilings, the usual multigrid components and the LFA have originally been designed and tailored for such discretizations (cf. [@TrotOostSchu2001; @WienJopp2004]). Several other geometries, including systems of PDEs, have been considered in the past as well. LFA has been carried out for operators defined on triangular tilings in [@GaspGracLisb2009] and on hexagonal tilings in [@ZhouFult]. Further, it has been applied to edge-based quadrilateral discretizations [@BoonLentVand2008], regular Voronoi meshes associated with acute triangular grids [@RodrSaliGaspLisb2014], edge-based discretizations on triangular grids [@rodrigo_sanz_gaspar_lisbona_2015] and jumping coefficients on rectangular grids [@RitExtending2017; @RittichBolten2018]. These papers do a complete two-grid analysis, and in some cases even a three-grid analysis, which was first introduced in [@WienOost2001]. While the concept of LFA is well understood, its application quickly becomes complex and involved the more frequencies get intermixed, e.g., by block smoothers, in a three-grid analysis, or in higher dimensional problems ($n > 2$). Thus, there exists software that automates the application of the LFA [@lfalab_HR; @WienJopp2004]. In contrast to the software described in [@WienJopp2004], which basically consists of a collection of templates corresponding to certain smoother and restriction/prolongation strategies for specific problems, the software [@lfalab_HR], freely available on GitHub,[^3] can be used to analyze arbitrary translationally invariant operators on rectangular grids. This software has for example been used to analyze colored block Jacobi methods in combination with aggressive coarsening applied to PDEs with jumping coefficients in [@RitExtending2017; @RittichBolten2018]. As the number of frequencies which get intermixed increases with the block-size of the smoother and the growing coarsening rate, a manual analysis of this problem would be laborious. In contrast to the approach developed in this paper, the bulk of the analysis in the previously mentioned references is mainly carried out in frequency space. LFA with an emphasize on position space is rare in the literature. In [@Huckle2008] under the name *compact Fourier analysis*, a position space oriented LFA is introduced where block Toeplitz matrices are used in order to capture the different classes of unknowns. This work has some aspects in common with the approach to LFA we develop in this paper, but lacks generality as it is limited to simple systems on rectangular grids. The LFA presented in this paper unifies the position space oriented approaches and allows the treatment of operators on arbitrary repetitive structures. To do so we introduce a mathematical framework for the analysis of translationally invariant operators which alter value distributions on lattices and crystals [@Ashcroft; @Schrijver:1986:TLI:17634]. These structures can be based on arbitrary sets of primitive vectors, including non-orthogonal ones, i.e., non-rectangular structures. These crystal structures, which naturally occur in the tight-binding descriptions of solid-state physics [@Ashcroft] or discretizations of systems of PDEs, enable the convenient and concise description of the resulting operators and allow for the automatic generation of their representations when enlarging their translational invariance, e.g., coarsening in the context of multigrid methods. Furthermore, they are very helpful in the representation of overlapping and non-overlapping block smoothers. Our framework is developed to such an extent that the only task required of the user is to provide a description of the occurring operators with respect to (potentially non-matching) descriptions of the underlying repetitive structures, i.e., each operator can be supplied in the simplest or most convenient representation. The remainder of the analysis can then be carried out automatically. In contrast to previously developed LFA, this is achieved by explicitly including a connection of the operator to its underlying structure. This allows us on one hand to automate the transformation of operators in position space, e.g., by finding a least common lattice of translational invariance of two operators and to rewrite their representations accordingly. On the other hand, this focus on structure yields a natural representation and discretization of the dual space that enables the automation of the frequency space part of the analysis as well. All these tasks can be carried out using basic principles and normal forms of integer linear algebra [@Schrijver:1986:TLI:17634]. In combination these developments alleviate the use of LFA by removing any manual calculations. That is, neither a mixing analysis nor transformations of operators to common (and rectangular) translational invariances have to be carried out by hand. While our automated LFA does not necessarily enlarge the set of methods that are analyzable by conventional LFA, it enables the reliable and easy-to-use analysis of complex methods on complicated structures (e.g., overlapping block smoothers and discretizations of systems of PDEs). An open-source Python implementation of the automated LFA framework [@aLFA_NK] is freely available on GitLab.[^4] The automation presented in this paper does have some limitation in terms of the smoothers that can be analyzed. Any sequential, i.e., lexicographic, smoother with overlapping update regions changes values in the overlap multiple times in one application. This cannot be easily translated to the structures introduced in this paper. While such smoothers have been analyzed in frequency space before (cf. [@MacLOost2011; @Molenaar1991; @Sivaloganathan:1991:ULM]), this particular treatment of sequential overlap is momentarily not covered in our framework. Note, that the mere presence of overlap is not the problem here. By introducing a coloring, such that the complete sweep can be split into a sequence of updates where each one of them only changes values at most once, automated LFA can be applied as we are going to demonstrate in this paper. Due to the fact that a coloring in overlapping approaches also favors parallelism over their sequential counterparts, we feel that this limitation is relatively minor when targeting actual applications. The paper is organized as follows. In \[sec:lattices\_crystals\] we introduce basic notation to describe the underlying structures of the operators we want to analyze: lattices and crystals. In the context of LFA we are interested in translationally invariant operators which alter value distributions on crystals. These kind of operators and their properties are specified in \[sec:operators\]. After that, \[sec:example\] illustrates the introduced notation by means of the discretized Laplacian and the red-black Gauss-Seidel method, which is the simplest method where a difference to the conventional LFA becomes apparent. In here, we manually rewrite the discretized Laplacian with respect to the translational invariance of the red-black splitting, such that the method can be analyzed via the results of \[sec:operators\]. contains the general theoretical results which are used to automate the complete procedure, removing the need to manually modify operators. In here, several results of integer linear algebra are used, which we review in \[sec:sublat\_quotient\]. Using these results we obtain the algorithms given in \[sec:algorithms\] which, in combination with the arithmetic of multiplication operators given in \[sec:rules\_of\_comp\], realize the automated LFA. Finally, \[sec:Application\] contains selected examples to demonstrate the merits of the developed approach. First, we analyze a $4$-color overlapping block Gauss-Seidel smoother for the tight-binding Hamiltonian of the carbon allotrope graphene and give some two-grid convergence results. Second, we reproduce the two-level analysis for the curl-curl problem found in [@BoonLentVand2008] as a further illustration of how our approach is applied to complex error propagators and to double-check its results. Lattices and Crystals {#sec:lattices_crystals} ===================== In order to be able to automate the process of LFA within a unified framework we first have to review some basic definitions of integer linear algebra and crystallography [@Ashcroft; @Schrijver:1986:TLI:17634]. An (ideal) crystal is an infinite repetition, defined by a lattice, of a structure element. Let ${\mathcal{a}}_1,{\mathcal{a}}_2,\ldots,{\mathcal{a}}_n \in { \mathbb{R}}^n$ be linearly independent. An $n$-dimensional lattice ${ \mathbb{L}}$ is the set of points $$\begin{aligned} { \mathbb{L}}={\{x=\sum_{\ell=1}^n j_\ell {\mathcal{a}}_\ell \in { \mathbb{R}}^n \, : \, j_1, j_2,\ldots, j_n \in { \mathbb{Z}}\}}. \end{aligned}$$ The vectors ${\mathcal{a}}_1,{\mathcal{a}}_2,\ldots,{\mathcal{a}}_n$ are known as the *primitive vectors* or *lattice basis*. Using matrix notation, i.e., ${\mathcal{A}}:=\mat[c]{{\mathcal{a}}_1&{\mathcal{a}}_2&\ldots&{\mathcal{a}}_n}$, we can abbreviate the notation by $${ \mathbb{L}}({\mathcal{A}}) := {\mathcal{A}}{ \mathbb{Z}}^n = { \mathbb{L}}.$$ Without loss of generality we restrict the definition of the second component of a crystal, the structure element, to primitive cells of the lattice. \[def:primitive\_cell\] A *primitive cell* ${\Xi}={\Xi}({\mathcal{A}}) \subset { \mathbb{R}}^n$ of a lattice ${ \mathbb{L}}({\mathcal{A}})$ is a (connected) volume of space that, if translated by all vectors of ${ \mathbb{L}}$, fills up ${ \mathbb{R}}^n$ completely without any overlap, i.e., $$\dot\cup_{x\in{ \mathbb{L}}} {\{x + \xi \, : \, \xi \in {\Xi}\}} ={ \mathbb{R}}^n.$$ A common choice of primitive cells is given by $${{ \mathcal{P}}}({\mathcal{A}}):={\mathcal{A}}[0,1)^n={\{y\in{ \mathbb{R}}^n \, : \, y = \sum_{\ell=1}^n \alpha_\ell {\mathcal{a}}_\ell,\, 0 \leq \alpha_\ell < 1\}},$$ i.e., parallelotopes spanned by the primitive vectors of ${ \mathbb{L}}({\mathcal{A}})$. The structure element of a crystal is now defined to consist of points ${\mathfrak{s}}_{1},\ldots,{\mathfrak{s}}_{m}$ that are contained in a particular primitive cell ${\Xi}$. Let ${ \mathbb{L}}({\mathcal{A}})$ be a lattice and ${\mathfrak{s}}\in {\Xi}({\mathcal{A}})^m$, $m \in \mathbb{N}$ be the *structure element*. A *crystal* is defined as the set of tuples $$\begin{aligned} { \mathbb{L}}^{{\mathfrak{s}}}({\mathcal{A}}):= {\{(x+{\mathfrak{s}}_1,x+{\mathfrak{s}}_2,\ldots,x+{\mathfrak{s}}_m) \, : \, x\in{ \mathbb{L}}({\mathcal{A}}),\ {\mathfrak{s}}=({\mathfrak{s}}_1,\ldots,{\mathfrak{s}}_m)\}}. \end{aligned}$$ The elements of ${ \mathbb{L}}^{{\mathfrak{s}}}({\mathcal{A}})$ are collectively written as $x+{\mathfrak{s}}=(x+{\mathfrak{s}}_1,x+{\mathfrak{s}}_2,\ldots,x+{\mathfrak{s}}_m)$. We define a crystal and the associated structure element to be a tuple instead of a set as we want to study value distributions on crystals and particular operators which manipulate them. For this purpose, the order of a structure element is of importance. To give an idea of the various occurrences of crystal structures in numerical applications we illustrate typical examples in \[fig:crystal\_examples\] together with their crystal representation. There are three main sources of repetitive structures that are well suited for crystal representations. First, the discretization of systems of PDEs on lattices lead to crystal structures, where the different species of unknowns typically constitute the structure element. Second, tight-binding Hamiltonian formulations in solid state physics for crystalline materials naturally imply a crystal representation based on the atomic structure. Last, colored domain decomposition approaches (e.g., red-black Gauss-Seidel) can be easily represented using crystals, where the smallest structure element typically consists of the union of one domain of each color. Sublattices and quotient spaces {#sec:sublat_quotient} ------------------------------- There are infinitely many representations of a crystal ${ \mathbb{L}}^{\mathfrak{s}}({\mathcal{A}})$. On one hand, the representation of any lattice in $n>1$ dimensions is non-unique, i.e., there exist different sets of primitive vectors that yield the same lattice structure. On the other hand, different representations of the crystal can be obtained by shifting the structure element or manipulating the underlying lattice structure, e.g., by using integer linear combinations of the primitive vectors and adjusting the structure element accordingly. In order to cope with this lack of uniqueness of the representation of crystals, illustrated in \[fig:crystal\_example\], we introduce basic results from integer linear algebra [@Schrijver:1986:TLI:17634], which resolve the relationship of lattice structures.[^5] An important tool in this is the notion of a sublattice. Let ${ \mathbb{L}}({\mathcal{A}})$ and ${ \mathbb{L}}({\mathcal{C}})$ be two lattices, if ${ \mathbb{L}}({\mathcal{A}}) \supset { \mathbb{L}}({\mathcal{C}})$ then ${ \mathbb{L}}({\mathcal{C}})$ is called *sublattice* of ${ \mathbb{L}}({\mathcal{A}})$. \[lem:sublattice\] A lattice ${ \mathbb{L}}({\mathcal{C}})$ is a sublattice of ${ \mathbb{L}}({\mathcal{A}})$ if and only if ${\mathcal{A}}^{-1} {\mathcal{C}}\in { \mathbb{Z}}^{n \times n}$. A key-role in the computational comparison of lattices play the *Hermite and Smith normal forms*. The Hermite normal form defines a canonical choice of primitive vectors, whereas the Smith normal form allows us to find least common sublattices. Using these canonical forms is a crucial ingredient in both the theoretical analysis and the automation of the LFA. \[def:unimodular\] A matrix $U \in { \mathbb{Z}}^{n\times n}$ is called *unimodular* if $\det(U)\in \{\pm 1\}$. \[def:HNF\] A matrix $H\in { \mathbb{R}}^{n\times n}$ is in *Hermite normal form* (HNF) if it is upper triangular, elementwise non-negative and its row-wise maximum is located on the diagonal. \[def:SNF\] A matrix $S\in { \mathbb{R}}^{n\times n}$ is in *Smith normal form* (SNF) if it is a diagonal matrix and the diagonal entries satisfy $ \nicefrac{s_{i+1}}{s_{i}}\in { \mathbb{Z}}$ for all $i=1,\ldots,n-1$. These entries are called the *elementary divisors*. \[thm:integer\_normalforms\] Let ${\mathcal{A}}\in { \mathbb{Q}}^{n\times n}$, then 1. there exists a unimodular $U \in { \mathbb{Z}}^{n \times n}$ such that $H = {\mathcal{A}}U$ is in HNF, 2. there exist unimodular $U,V \in { \mathbb{Z}}^{n \times n}$ such that $S = V {\mathcal{A}}U$ is in SNF. In addition both normal forms $H$ and $S$ are unique with respect to ${\mathcal{A}}$. Polynomial algorithms to compute the HNF and SNF can for example be found in [@ComputerAlgebraHandbook]. Implementations for the computation of these normalforms are for example part of the PARI software package [@PARI2]. Using these definitions and results one obtains a precise statement about the equality of two lattices. \[thm:lattice\_equiv\_unimod\] Let ${ \mathbb{L}}({\mathcal{A}})$ and ${ \mathbb{L}}({\mathcal{C}})$ be two lattices then the following statements are equivalent. 1. ${ \mathbb{L}}({\mathcal{A}}) = { \mathbb{L}}({\mathcal{C}})$. 2. ${\mathcal{A}}^{-1} {\mathcal{C}}\in { \mathbb{Z}}^{n\times n}$ and ${\mathcal{C}}^{-1} {\mathcal{A}}\in { \mathbb{Z}}^{n\times n}$. 3. There exists a unimodular matrix $U$, such that ${\mathcal{C}}={\mathcal{A}}U$. 4. ${\mathcal{A}}$ and ${\mathcal{C}}$ have the same HNF. Instead of analyzing infinite lattice/crystal structures, we limit ourselves to analyze finite dimensional periodic structures due to the fact that we are ultimately aiming to analyze finite dimensional problems. To this end, another helpful tool is the definition of crystal tori which are defined as quotient groups. Let ${ \mathbb{L}}^{\mathfrak{s}}({\mathcal{A}})$ be a crystal and ${ \mathbb{L}}({\mathcal{C}}) \subset { \mathbb{L}}({\mathcal{A}})$ be a sublattice. We define the *crystal torus* $T^{\mathfrak{s}}_{{\mathcal{A}},{\mathcal{C}}}$ by $$\begin{aligned} T^{\mathfrak{s}}_{{\mathcal{A}},{\mathcal{C}}}:={{\raisebox{.2em}{${ \mathbb{L}}^{\mathfrak{s}}({\mathcal{A}})$}\left/\raisebox{-.2em}{${ \mathbb{L}}({\mathcal{C}})$}\right.}}. \end{aligned}$$ For every $x+{\mathfrak{s}}\in { \mathbb{L}}^{\mathfrak{s}}({\mathcal{A}})$, their equivalence class $[x+{\mathfrak{s}}]$ is in $T^{\mathfrak{s}}_{{\mathcal{A}},{\mathcal{C}}}$. Furthermore, the elements of $T^{\mathfrak{s}}_{{\mathcal{A}},{\mathcal{C}}}$ are defined by the equivalence $$\begin{aligned} = [y+{\mathfrak{s}}] \quad \Longleftrightarrow \quad \text{there exists } z \in { \mathbb{L}}({\mathcal{C}}), \text{ such that } x = y + z. \end{aligned}$$ For theoretical and practical reasons, e.g., in \[thm:wavefunction\_ONB\_vec,alg:elements\_in\_quotient\_space\], it is necessary to be able to list all elements of a torus $T^{\mathfrak{s}}_{{\mathcal{A}}, {\mathcal{A}}M } = {\{[x]\in T^{\mathfrak{s}}_{{\mathcal{A}}, {\mathcal{A}}M } \, : \, x\in {{ \mathcal{P}}}({\mathcal{A}}M) \cap { \mathbb{L}}^{\mathfrak{s}}({\mathcal{A}})\}}$, $M \in { \mathbb{{ \mathbb{Z}}}}^{n \times n}$, uniquely. To illustrate this point, consider for example an arbitrary lattice ${ \mathbb{L}}({\mathcal{A}})$ with ${\mathcal{A}}\in \mathbb{R}^{2\times 2}$, and the lattice torus $T_{{\mathcal{A}}, {\mathcal{A}}M }$ with $$M = \mat[c]{m_1 & m_2} = \mat[c]{ 2 & 3 \\ 2 & -2 },\ m_{1},m_{2} \in \mathbb{Z}^{2},$$ as depicted in \[fig:HNF\_LIST\]. Even though we know that the quotient space consists of $|T_{{\mathcal{A}}, {\mathcal{A}}M }|=|\operatorname{det}(M)|=10$ different elements, there is no apparent canonical list of these elements. Fortunately, a canonical ordering of the lattice points on a torus can be formulated using the Hermite or Smith normal form of $M$. \[cor:quotient\_latticepointlist\] Let $T_{{\mathcal{A}},{\mathcal{C}}}$ be arbitrary, i.e., ${\mathcal{C}}= {\mathcal{A}}M$ for some $M \in { \mathbb{{ \mathbb{Z}}}}^{n \times n}$. - Let $H \in{ \mathbb{{ \mathbb{Z}}}}^{n \times n}$ be the HNF of $M$ with entries $H_{ij}$ (cf. \[def:HNF\]). Defining the index set $I=I_1 \times I_2 \times \ldots \times I_n$ by $I_\ell:=\{0,1,\ldots,H_{\ell\ell}-1\}$, we then obtain $$\begin{aligned} T^{\mathfrak{s}}_{{\mathcal{A}},{\mathcal{C}}} & = {\{[x_j+{\mathfrak{s}}] \, : \, x_j = {\mathcal{A}}j,\ j \in I\}} \end{aligned}$$ $\text{with } [x_j+{\mathfrak{s}}] \neq [x_{j'}+{\mathfrak{s}}] \ \Leftrightarrow\ j \neq j' \in I.$ - Let $S=U^{-1}MV \in { \mathbb{Z}}^{n\times n}$ denote the Smith decomposition of $M$ with diagonal entries $s_{i}$ (cf. \[def:SNF\]) and unimodular matrices $U,V$. Defining the index set $\tilde{I}=\tilde{I}_1 \times \tilde{I}_2 \times \ldots \times \tilde{I}_n$ by $\tilde{I}_\ell:=\{0,1,\ldots,s_{\ell}-1\}$, we then obtain $$\begin{aligned} T^{\mathfrak{s}}_{{\mathcal{A}},{\mathcal{C}}} & = {\{[x_j+{\mathfrak{s}}] \, : \, x_j = \tilde{{\mathcal{A}}} j,\ j \in \tilde{I}\}}\ \end{aligned}$$ $\text{with }\ [x_j+{\mathfrak{s}}] \neq [x_{j'}+{\mathfrak{s}}] \ \Leftrightarrow\ j \neq j' \in \tilde{I},$ where $\tilde{{\mathcal{A}}} := {\mathcal{A}}U$ denotes the altered lattice basis. Both statements are a direct consequence of the triangular or diagonal shape of the normal forms and \[thm:lattice\_equiv\_unimod\], i.e., lattices are not changed by unimodular column transformations. On the one hand we have $T^{\mathfrak{s}}_{{\mathcal{A}},{\mathcal{C}}} = T^{\mathfrak{s}}_{{\mathcal{A}},{\mathcal{A}}M } = T^{\mathfrak{s}}_{{\mathcal{A}},{\mathcal{A}}H }.$ The second statement follows from $\tilde{{\mathcal{A}}} S = {\mathcal{C}}V$ and hence $T^{\mathfrak{s}}_{{\mathcal{A}},{\mathcal{C}}} = T^{\mathfrak{s}}_{{\mathcal{A}}U , {\mathcal{C}}V} = T^{\mathfrak{s}}_{\tilde{{\mathcal{A}}} ,\tilde{{\mathcal{A}}} S }.$ In terms of the example of \[fig:HNF\_LIST\] we obtain: - The Hermite normal form $H$ of $M$ is given by $$H= \mat[c]{h_{1} & h_{2}} = \mat[c]{ 5 & 2 \\ 0 & 2},\ h_{1},h_{2} \in \mathbb{Z}^{2} .$$ Thus, a unique list of all representatives of $T_{{\mathcal{A}},{\mathcal{A}}M}$ is given by $$\begin{aligned} T_{{\mathcal{A}},{\mathcal{A}}M}= T_{{\mathcal{A}},{\mathcal{A}}H} &= {\{[x]=[j_1 {\mathcal{a}}_1 + j_2 {\mathcal{a}}_2] \, : \, (j_1,j_2) \in \{0,1,2,3,4\}\times\{0,1\} \}}. \\ \end{aligned}$$ - The Smith decomposition of $M$ is given by $$S= \mat[c]{s_{1} & 0 \\ 0 & s_{2}} = \mat[c]{ 1 & 0 \\ 0 & 10} =\mat[r]{ 1 & 0 \\ -4 & -1} M \mat[r]{ -1 & -3 \\ 1 & 2} .$$ Thus, another unique list of all representatives of $T_{{\mathcal{A}},{\mathcal{A}}M}$ is given by $$T_{{\mathcal{A}},{\mathcal{A}}M}= T_{\tilde{{\mathcal{A}}} , \tilde{{\mathcal{A}}} S}={\{[x]=[j_1 \tilde{{\mathcal{a}}}_1 + j_2 \tilde{{\mathcal{a}}}_2] \, : \, (j_1,j_2) \in \{0\}\times\{0,1,\ldots,9\} \}}, $$ where $\tilde{{\mathcal{A}}} = \mat{\tilde{{\mathcal{a}}}_1 & \tilde{{\mathcal{a}}}_2} = {\mathcal{A}}\mat[r]{ 1 & 0 \\ -4 & -1}$. All tori representations $T_{{\mathcal{A}},{\mathcal{A}}M}$, $T_{{\mathcal{A}},{\mathcal{A}}H}$ and $T_{\tilde{{\mathcal{A}}} , \tilde{{\mathcal{A}}} S} $ are depicted in \[fig:HNF\_LIST\]. In the remainder we drop the bracket notation for reasons of readability. Operators on Crystals {#sec:operators} ===================== Now that the basic notation of the underlying structure is in place we introduce notation for value distributions and operators on these structures. For aforementioned reasons, we restrict ourselves to the finite dimensional setting by only considering quotient groups of lattices $$\begin{aligned} T_{{\mathcal{A}},{\mathcal{Z}}}^{\mathfrak{s}}: = {{\raisebox{.2em}{${ \mathbb{L}}^{\mathfrak{s}}({\mathcal{A}})$}\left/\raisebox{-.2em}{${ \mathbb{L}}({\mathcal{Z}})$}\right.}} \end{aligned}$$ with ${ \mathbb{L}}({\mathcal{Z}}) \subset { \mathbb{L}}({\mathcal{A}})$ being an arbitrary sublattice of ${ \mathbb{L}}({\mathcal{A}})$. While it is significantly easier to be mathematically precise in this general finite setting than in an infinite setting it is also much closer to the targeted applications, namely finite dimensional approximations of PDEs and Hamiltonians. Eventually we only work with operators as part of numerical simulations, i.e., we face only a finite number of unknowns/lattice points anyway. The quotient group $T_{{\mathcal{A}},{\mathcal{Z}}}^{\mathfrak{s}}$, ${ \mathbb{L}}({\mathcal{Z}}) \subset { \mathbb{L}}({\mathcal{A}})$, precisely describes such an arbitrarily large but finite torus. To shorten notation we use $T_{{\mathcal{A}}}^{\mathfrak{s}}$ instead of $T_{{\mathcal{A}},{\mathcal{Z}}}^{\mathfrak{s}}$ whenever we do not specify ${\mathcal{Z}}$ explicitly. A crystal-operator is a linear function $$\begin{aligned} L : { \mathbcal{L}}( T_{{\mathcal{A}}}^{{\mathfrak{d}}} ) \longrightarrow { \mathbcal{L}}( T_{{\mathcal{A}}}^{{\mathfrak{c}}} ), \end{aligned}$$ where $T_{{\mathcal{A}}}^{{\mathfrak{d}}}$ corresponds to the crystal of the domain and $T_{{\mathcal{A}}}^{{\mathfrak{c}}}$ corresponds to the crystal of the codomain. The function spaces are defined by $$\begin{aligned} { \mathbcal{L}}( T_{{\mathcal{A}}}^{{{{\mathfrak{s}}}}} ) = {\{f=(f_1,\ldots,f_{|{{\mathfrak{s}}}|}) \, : \, T_{{\mathcal{A}}}\longrightarrow { \mathbb{C}}^{|{{{{\mathfrak{s}}}}}|}\}}. \end{aligned}$$ A value $f_j(x)$, $x\in{ \mathbb{L}}({\mathcal{A}})$, corresponds to a value at the position $x + {\mathfrak{s}}_j$. The function space is equipped with the scalar product $${\langle f,g\rangle_{}} := \frac{1}{|T_{{\mathcal{A}},{\mathcal{Z}}}|} \sum_{x\in T_{{\mathcal{A}},{\mathcal{Z}}}} {\langle f(x),g(x)\rangle_{2}},$$ where ${\langle f(x),g(x)\rangle_{2}} := \sum_{\ell=1}^{|{{\mathfrak{s}}}|} f_\ell(x) \overline{g_\ell(x)}$ denotes the Euclidean scalar product on ${ \mathbb{C}}^{|{{\mathfrak{s}}}|}$. In the context of LFA we are interested in operators which can be represented in (block) stencil notation. That is, translationally invariant operators that can be written as multiplication operators. As these two properties are in fact equivalent (cf. [@nla.cat-vn1897929 Theorem 3.16]), we can connect those operators to the notation of crystal structures. Let $L : { \mathbcal{L}}( T_{{\mathcal{A}}}^{{\mathfrak{d}}} ) \longrightarrow { \mathbcal{L}}( T_{{\mathcal{A}}}^{{\mathfrak{c}}} )$ be a crystal operator. The following statements are equivalent. 1. $L$ is a multiplication operator, i.e., there exist matrices $m_L^{(y)} \in { \mathbb{C}}^{|{{\mathfrak{c}}}|\times |{{\mathfrak{d}}}|}$ such that for each $x\in T_{\mathcal{A}}$ and $f\in { \mathbcal{L}}( T_{{\mathcal{A}}}^{{\mathfrak{d}}} ) $ we have $$\begin{aligned} (Lf)(x) = \sum_{y \in T_{\mathcal{A}}} m_L^{(y)} f(x+y). \end{aligned}$$ 2. $L$ is $({\mathcal{A}})$-translationally invariant, i.e., $${L}{\mathbb{T}}_{\mathcal{a}}- {\mathbb{T}}_{\mathcal{a}}{L}= 0 \quad \text{for all (primitive) vectors } {\mathcal{a}}\in { \mathbb{L}}({\mathcal{A}}),$$ where the translation operator is defined by $({\mathbb{T}}_{\mathcal{a}}f)(x) = f(x+{\mathcal{a}})$. For the analysis of such operators the concept of the dual lattice comes in handy as already considered in similar form in [@GaspGracLisb2009; @ZhouFult]. \[def:dual\_lattice\] Let ${ \mathbb{L}}({\mathcal{A}})$ be a lattice. Its *dual lattice* ${ \mathbb{L}}({\mathcal{B}})={ \mathbb{L}}({\mathcal{A}})^*$ is the set $$\begin{aligned} { \mathbb{L}}({\mathcal{A}})^*:={\{k \in { \mathbb{R}}^n \, : \, {\langle k,x\rangle_{2}} \in { \mathbb{Z}}\text{ for all } x\in{ \mathbb{L}}\}}. \end{aligned}$$ A lattice basis of the dual lattice is given by ${\mathcal{B}}= {\mathcal{A}}^{-T}$. The elements of ${ \mathbb{L}}({\mathcal{A}})^*$ may also be referred to as *wave vectors*. In addition to the dual space we introduce an orthonormal basis of wave functions that are compatible with the crystal structure introduced in \[sec:lattices\_crystals\]. This basis is an extension of the basis used in [@Kuo1989] to analyze the red-black Gauss-Seidel relaxation. Furthermore, similar basis functions have been used in the context of LFA, e.g. in [@BoonLentVand2008; @1803.08864; @RittichBolten2018; @1902.10248; @Brown2018LOCALFA]. \[thm:wavefunction\_ONB\_vec\] An orthonormal basis for the function space ${ \mathbcal{L}}(T^{\mathfrak{s}}_{{\mathcal{A}},{\mathcal{Z}}})$ with a structure element ${\mathfrak{s}}=({\mathfrak{s}}_1,\ldots,{\mathfrak{s}}_m)$ is given by the wave functions $e_{1,k},e_{2,k},\ldots,e_{m,k}$ defined by $$\begin{aligned} \left(e_{\ell,k}(x)\right)_j := \begin{cases} {e^{2\pi i{\langle k,x\rangle_{2}}}} & \text{ if }j=\ell, \\ 0 & \text{ else,} \end{cases} \end{aligned}$$ with $k \in T_{{\mathcal{A}},{\mathcal{Z}}}^*:= {{\raisebox{.2em}{${ \mathbb{L}}({\mathcal{Z}})^*$}\left/\raisebox{-.2em}{${ \mathbb{L}}({\mathcal{A}})^*$}\right.}} $. The statement follows by a straightforward, but lengthy calculation, by assuming without loss of generality that ${\mathcal{A}}^{-1}{\mathcal{Z}}$ is given in Smith normal form, making use of \[cor:quotient\_latticepointlist\] and the geometric sum formula. An illustration of a lattice torus $T_{{\mathcal{A}},{\mathcal{Z}}}$ along with its dual $T_{{\mathcal{A}},{\mathcal{Z}}}^*$ is given in \[fig:TorusAndDual\]. The orthonormal basis of \[thm:wavefunction\_ONB\_vec\] can be split into subsets with respect to the wave vector $k$, i.e., ${ \mathbcal{L}}(T^{\mathfrak{s}}_{{\mathcal{A}},{\mathcal{Z}}}) = \cup_{k \in T_{{\mathcal{A}},{\mathcal{Z}}}^*} \operatorname{span}(H_k) $ with $$\label{eq:space-of-harmonics} H_k = \operatorname{span}{\{e_{\ell,k} \, : \, \ell=1,\ldots,m\}}.$$ \[thm:L-invariance\] Let ${L}: { \mathbcal{L}}(T_{{\mathcal{A}},{\mathcal{Z}}}^{\mathfrak{s}}) \rightarrow { \mathbcal{L}}(T_{{\mathcal{A}},{\mathcal{Z}}}^{\mathfrak{s}})$ be a multiplication operator. Then the subspaces $H_k$ of \[eq:space-of-harmonics\] are $L$-invariant, i.e., $L(H_k) \subseteq H_k$. Let $e_{k}(x)$ denote an arbitrary value distribution in $H_{k}$. That is, there exist $\alpha_{1},\ldots,\alpha_{m} \in { \mathbb{C}}$ such that $$\begin{aligned} e_k(x)=\sum_{\ell}\alpha_\ell e_{\ell,k} = (\alpha_1 {e^{2\pi i{\langle k,x\rangle_{2}}}},\ldots,\alpha_m{e^{2\pi i{\langle k,x\rangle_{2}}}})^T \in \operatorname{span}(H_k). \end{aligned}$$ Then we obtain by direct calculation $$\label{eq:symbol} (L e_k)(x) = \sum_{y} m_L^{(y)} e_k(x+y) = \left(\sum_{y} m_L^{(y)} {e^{2\pi i{\langle k,y\rangle_{2}}}}\right) e_k(x).$$ Due to their $L$-invariance the subspaces $H_k$ are oftentimes referred to as *spaces of harmonics*. Thus we can easily represent any ${\mathcal{A}}$-translationally operator via its *symbols*, which are formally defined as follows. \[def:crystal\_op\_symbol\] Let $L:{ \mathbcal{L}}(T_{{\mathcal{A}},{\mathcal{Z}}}^{\mathfrak{s}}) \rightarrow { \mathbcal{L}}(T_{{\mathcal{A}},{\mathcal{Z}}}^{\mathfrak{t}})$ be a multiplication operator with $$\begin{aligned} (Lf)(x) = \displaystyle\sum_{y \in T_{{\mathcal{A}},{\mathcal{Z}}}} m_L^{(y)} f(x+y),\ m_L^{(y)} \in { \mathbb{C}}^{|{\mathfrak{t}}|\times |{\mathfrak{s}}|}. \end{aligned}$$ We define the *symbol* of $L$ according to \[eq:symbol\] by $$\begin{aligned} L_k:=\sum_{y \in T_{{\mathcal{A}},{\mathcal{Z}}}} m_L^{(y)} m_k^{(y)} \text{ with } m_k^{(y)} := {e^{2\pi i{\langle k,y\rangle_{2}}}}. \end{aligned}$$ In case ${\mathfrak{s}}= {\mathfrak{t}}$ the spectrum of $L$ can then be extracted from its symbols $L_{k}$. \[cor:eigenvalues\_crystal\_operator\] Let $L:{ \mathbcal{L}}(T_{{\mathcal{A}},{\mathcal{Z}}}^{\mathfrak{s}}) \rightarrow { \mathbcal{L}}(T_{{\mathcal{A}},{\mathcal{Z}}}^{\mathfrak{s}})$ be a multiplication operator with $$\begin{aligned} (Lf)(x) = \sum_{y \in T_{{\mathcal{A}},{\mathcal{Z}}}} m_L^{(y)} f(x+y),\ m_L^{(y)} \in { \mathbb{C}}^{|{\mathfrak{s}}|\times |{\mathfrak{s}}|}. \end{aligned}$$ Then ${\operatorname{spec}}(L) = \cup_{k\in T_{{\mathcal{A}},{\mathcal{Z}}}^*} {\operatorname{spec}}(L_k)$. Follows immediately due to the orthonormality of the basis $e_{\ell,k}$ (cf. \[thm:wavefunction\_ONB\_vec\]) and the $L$-invariance of the subspaces $H_{k}$ (cf. \[thm:L-invariance\]). \[rem:freq\_sampling\] The main purpose of ${\mathcal{Z}}$, i.e., the set of primitive vectors that define an arbitrary sublattice ${ \mathbb{L}}({\mathcal{Z}})$ of ${ \mathbb{L}}({\mathcal{A}})$, is to simplify the theory developed in \[sec:operators\] by turning an infinite dimensional setting to an (arbitrarily large) finite one. Though there is another interpretation to it as well. In \[cor:eigenvalues\_crystal\_operator\] ${\mathcal{Z}}$ explicitly specifies the resolution of the frequency space as seen in \[fig:TorusAndDual\], i.e., $$T_{{\mathcal{A}},{\mathcal{Z}}}^* = { \mathbb{L}}({\mathcal{Z}}^{-T}) \cap {{ \mathcal{P}}}({\mathcal{A}}^{-T}),$$ where the spectrum of the multiplication operator is sampled. Due to the reciprocal nature of the dual space, the larger $|\operatorname{det}({\mathcal{Z}})|$ is, the finer the resolution becomes. With these tools at hand we are able to fully analyze a single multiplication operator, but typically we are interested in analyzing a composition of several operators using LFA. As long as the corresponding domains and codomains of these operators are compatible we can use the rules of computation given in \[sec:rules\_of\_comp\] on the level of multiplication operators and/or on the level of the corresponding symbols. While computing the sum, a product and taking the transpose can easily be done on both levels, taking the (pseudo-)inverse is simple only on the level of symbols. The (pseudo-)inverse of a multiplication operator may have an arbitrarily large number[^6] of multipliers $m_{L^{-1}}^{(y)} \neq 0$ and thus there is no simple rule to compute it. In case a pseudo-inverse has to be used in the following we opt to employ the Moore-Penrose pseudo-inverse, which we denote by $S^{\dagger}$ for a multiplication operator $S$. In our framework, which tries to automate as much of the LFA as possible, we would like to allow for user friendly descriptions of all occurring operators. That is, it should be possible to describe operators in terms of their individual translational invariance and ordering of the structure element without having to worry about compatibility issues with other operators on the input level of the analysis. The process how to automatically make crystal representations of operators compatible is explained in detail in \[sec:crystal\_reps\_and\_isos\]. Before diving into the gritty details of this automation process we would like to illustrate the developments made so far with an example in order to convey the introduced notation. An example {#sec:example} ========== Consider the red-black Gauss-Seidel method applied to the discretized Laplacian on the unit square with periodic boundary conditions. The most fundamental representation of the discretized unit square with periodic boundary conditions is given by the torus $T_{{\mathcal{A}},{\mathcal{Z}}}$ with $$\begin{aligned} {\mathcal{Z}}=\mat[c]{1 & 0 \\ 0 & 1}, {\mathcal{A}}=\mat[c]{{\mathcal{a}}_1 & {\mathcal{a}}_2}=\frac{1}{h} \mat[c]{1 & 0 \\ 0 & 1}\text{ for some }h=\frac{1}{N}, N\in{ \mathbb{N}}. \end{aligned}$$ Then, the discretized Laplacian $L_h : { \mathbcal{L}}( T_{{\mathcal{A}},{\mathcal{Z}}}^{(0)} ) \longrightarrow { \mathbcal{L}}( T_{{\mathcal{A}},{\mathcal{Z}}}^{(0)} )$ using finite differences is given by $(L_h f)(x) := \sum_{y\in{ \mathbb{L}}({\mathcal{A}})} m_{L_h}^{(y)} f(x+y)$ with non-zero multipliers: \(c) at (0,0) [$=$]{}; ; ; \(c) at (0,) [$=$]{}; ; ; \(c) at (0,-) [$=$]{}; ; ; \(c) at (2.2\*,0) [$=$]{}; ; ; \(c) at (-1.9\*,0) [$=$]{}; ; ; The error propagator $G$ of the red-black Gauss-Seidel method can be written with respect to the crystal $T_{{\mathcal{A}},{\mathcal{Z}}}^{(0)}$ via $$\begin{aligned} G=(I-S_b^\dagger L_h)(I-S_r^\dagger L_h) : { \mathbcal{L}}( T_{{\mathcal{A}},{\mathcal{Z}}}^{(0)} ) \longrightarrow { \mathbcal{L}}( T_{{\mathcal{A}},{\mathcal{Z}}}^{(0)} ) \end{aligned}$$ with $$\begin{aligned} (S_r f)(x) := \begin{cases} \frac{4}{h^2} f(x) & x\in X_\text{red} \\ 0 & x\in X_\text{black} \end{cases} \text{ and } (S_b f)(x) := \begin{cases} 0 & x\in X_\text{red} \\ \frac{4}{h^2} f(x) & x\in X_\text{black} \end{cases} , \end{aligned}$$ where $X_\text{red}$ and $X_\text{black}$ correspond to the red and black unknowns of the torus $T_{{\mathcal{A}},{\mathcal{Z}}}^{(0)}$ as illustrated in \[fig:redblack\_laplacian\], b). In order to analyze this composite operator in our framework, we write all occurring operators as multiplication operators. It is now important that the red-black splitting of $T_{{\mathcal{A}},{\mathcal{Z}}}^{(0)} = X_{\text{red}} \cup X_{\text{black}}$ implies the crystal $ T_{{\mathcal{C}},{\mathcal{Z}}}^{\mathfrak{s}}$ with $$\begin{aligned} {\mathcal{c}}_1 = {\mathcal{a}}_1 + {\mathcal{a}}_2, \text{ } {\mathcal{c}}_2 = {\mathcal{a}}_1 - {\mathcal{a}}_2, \ {\mathfrak{s}}:= (0, {\mathcal{a}}_1) = { \mathbb{L}}({\mathcal{A}}) \cap {{ \mathcal{P}}}({\mathcal{C}}) \end{aligned}$$ such that $T_{{\mathcal{C}},{\mathcal{Z}}}^{(0)} = X_{\text{red}}$, $T_{{\mathcal{C}},{\mathcal{Z}}}^{({\mathcal{a}}_1)} = X_{\text{black}}$ and $T_{{\mathcal{A}},{\mathcal{Z}}}^{(0)} = X_{\text{red}} \cup X_{\text{black}} \cong T_{{\mathcal{C}},{\mathcal{Z}}}^{\mathfrak{s}}$ (cf. \[fig:redblack\_laplacian\], b)). With respect to this crystal the operators $S_r$ and $S_b$ can be written as multiplication operators $\hat{S}_r, \hat{S}_b : { \mathbcal{L}}( T_{{\mathcal{C}},{\mathcal{Z}}}^{{\mathfrak{s}}} ) \longrightarrow { \mathbcal{L}}( T_{{\mathcal{C}},{\mathcal{Z}}}^{{\mathfrak{s}}} )$ with $$\begin{aligned} (\hat{S}_r f)(x) = \mat[c]{\frac{4}{h^2} & 0 \\ 0 & 0} f(x) \text{ and } (\hat{S}_b f)(x) = \mat[c]{0 & 0 \\ 0 & \frac{4}{h^2}} f(x). \end{aligned}$$ We now have described all individual operators of $G$, each one defined with respect to its own (minimal) translational invariance, but the domains and codomains are not identical, i.e., ${\mathcal{A}}\neq {\mathcal{C}}$, such that we cannot directly use the computation rules described in \[sec:rules\_of\_comp\]. In order to carry on with the analysis we have to rewrite the operators with respect to a common crystal structure. In this example we construct this structure by hand, but this process can be automated as explained in \[sec:crystal\_reps\_and\_isos\]. As the crystal $T_{{\mathcal{C}},{\mathcal{Z}}}^{\mathfrak{s}}$ is yet another representation of $T_{{\mathcal{A}},{\mathcal{Z}}}^{(0)}$, the operator $L_h$ can be rewritten with respect to this crystal (cf. \[fig:redblack\_laplacian\]) as $ \hat{L}_h: { \mathbcal{L}}( T_{{\mathcal{C}},{\mathcal{Z}}}^{{\mathfrak{s}}} ) \longrightarrow { \mathbcal{L}}( T_{{\mathcal{C}},{\mathcal{Z}}}^{{\mathfrak{s}}} )$ with $(\hat{L}_h f)(x) := \sum_{y\in{ \mathbb{L}}({\mathcal{C}})} m_{\hat{L}_h}^{(y)} f(x+y)$ and non-zero multipliers: Thus, the spectrum of the error propagator of the red-black Gauss-Seidel method applied to the Laplacian $$\label{eq:RBGEP} \hat{G} = (I-\hat{S}_b^\dagger \hat{L}_h)(I-\hat{S}_r^\dagger \hat{L}_h)$$ can now be obtained elementwise for each fixed $k \in T_{{\mathcal{C}},{\mathcal{Z}}}^* = {{ \mathcal{P}}}({\mathcal{C}}^{-T}) \cap { \mathbb{L}}({\mathcal{Z}}^{-T})$ by first computing the individual *symbols* $I_k, (\hat{S}_r)_k, (\hat{S}_b)_k$ and $(\hat{L}_h)_k$ and assembling the symbols $\hat{G}_k$ according to \[eq:RBGEP\] and the rules in \[sec:rules\_of\_comp\], followed by the computation of the eigenvalues of the matrices $\hat{G}_k$. The resulting spectral information of the discretized Laplace operator $\hat{L}_h$ and the error propagator $\hat{G}$ is illustrated in \[fig:redblack\_spectrum\], where it is sampled on the dual lattice $T_{{\mathcal{C}},{\mathcal{Z}}}^*$. Note, that one naturally obtains two eigenvalues per sampled wave vector $k$. In case of the spectrum of the red-black Gauss-Seidel error propagator, one of the two eigenvalues is equal to zero for all wave vectors $k$. Crystal representations and natural isomorphisms {#sec:crystal_reps_and_isos} ================================================ In general, we are given several multiplication operators which make up the error propagator of an iterative method, each defined with respect to its own (minimal) translational invariance. In order to analyze the method we thus need to find a common denominator, i.e., a lattice basis corresponding to the collective translational invariance, and rewrite the operators accordingly. The following theorem yields a set of primitive vectors of such a collective translational invariance for two arbitrary lattices, if it exists. \[thm:lcm\_mat\] Given two $n$-dimensional lattices ${ \mathbb{L}}({\mathcal{A}})$, ${ \mathbb{L}}({\mathcal{B}})$. If there exists an $r\in { \mathbb{Z}}$, such that $M = r {\mathcal{A}}^{-1} {\mathcal{B}}\in { \mathbb{Z}}^{n\times n}$, then there is a lattice ${ \mathbb{L}}({\mathcal{C}})$ with ${ \mathbb{L}}({\mathcal{C}}) \subset { \mathbb{L}}({\mathcal{A}})$ and ${ \mathbb{L}}({\mathcal{C}}) \subset { \mathbb{L}}({\mathcal{B}})$ with $|\det({\mathcal{C}})|$ as small as possible. A lattice basis of ${ \mathbb{L}}({\mathcal{C}})$ is given by $$\begin{aligned} {\mathcal{C}}= {\mathcal{B}}T^{-1} N_{\mathcal{B}}, \quad \end{aligned}$$ where $N_{\mathcal{B}}$ is a diagonal matrix with $(N_{\mathcal{B}})_{i,i} := r \cdot \operatorname{gcd}(r,s_{i})^{-1}$, where $S= V^{-1} M T^{-1} = \operatorname{diag}{(s_{1},\ldots,s_{n})}$ is the SNF of $M$ (cf. \[def:SNF\]) and $\operatorname{gcd}(r,s_{i})$ denotes the greatest common divisor of $r$ and $s_i$. Consequently, we write ${ \mathbb{L}}({\mathcal{C}}) = \operatorname{lcm}( { \mathbb{L}}({\mathcal{A}}), { \mathbb{L}}({\mathcal{B}}))$ and call it the *least common multiple* of ${ \mathbb{L}}({\mathcal{A}})$ and ${ \mathbb{L}}({\mathcal{B}})$. Due to \[lem:sublattice\] it is sufficient to find integral matrices $N_{\mathcal{A}}, N_{\mathcal{B}}$, such that $$\begin{aligned} { \mathbb{L}}({\mathcal{C}}) = { \mathbb{L}}({\mathcal{A}}N_{\mathcal{A}}) = { \mathbb{L}}({\mathcal{B}}N_{\mathcal{B}}) \end{aligned}$$ with $|\det(N_{\mathcal{A}})|$ and $|\det(N_{\mathcal{B}})|$ as small as possible. Using \[thm:lattice\_equiv\_unimod\], i.e., $ { \mathbb{L}}({\mathcal{B}}) = { \mathbb{L}}({\mathcal{B}}V_1), \quad { \mathbb{L}}({\mathcal{C}}) = { \mathbb{L}}({\mathcal{C}}U_1), $ for any unimodular matrices $U_1,V_1$, we can assume the equality $$\begin{aligned} {\mathcal{A}}N_{\mathcal{A}}= {\mathcal{B}}U N_{\mathcal{B}}\end{aligned}$$ for any unimodular $U$ and $N_{\mathcal{B}}$ in Hermite normal form (cf. \[thm:integer\_normalforms\]). Plugging in the Smith decomposition $VST$ of $M=r {\mathcal{A}}^{-1} {\mathcal{B}}$ and defining $U:=T^{-1}$, we find $$\begin{aligned} \begin{array}{rcccl} N_{\mathcal{A}}& = & {\mathcal{A}}^{-1} {\mathcal{B}}T^{-1} N_{\mathcal{B}}&= & \frac{1}{r} VS N_{\mathcal{B}}. \end{array} \end{aligned}$$ Both matrices $$\begin{aligned} N_{\mathcal{B}}= \mat{(N_{\mathcal{B}})_{1,1} & \ldots & (N_{\mathcal{B}})_{1,n} \\ & \ddots & \vdots \\ & & (N_{\mathcal{B}})_{n,n} } \text{ and } \frac{1}{r} S N_{\mathcal{B}}= \mat{\frac{s_{1}}{r}(N_{\mathcal{B}})_{1,1} & \ldots & (N_{\mathcal{B}})_{1,n}\frac{s_{1}}{r} \\ & \ddots & \vdots \\ & & (N_{\mathcal{B}})_{n,n}\frac{s_{n}}{r} } \end{aligned}$$ have to be integral with $$|\det(N_{\mathcal{B}})| = |\prod_{i=1}^n (N_{\mathcal{B}})_{i,i}| \text{ and }|\det(\frac{1}{r} S N_{\mathcal{B}})| = |\prod_{i=1}^n (N_{\mathcal{B}})_{i,i} \frac{s_{i}}{r} |$$ as small as possible. It can easily be verified that $$\begin{aligned} (N_{\mathcal{B}})_{i,i} := \frac{r}{\operatorname{gcd}(r,s_{i})} \end{aligned}$$ is the optimal choice for the diagonal entries. With this choice, the off-diagonal entries $(N_{\mathcal{B}})_{i,j}$ have to be integral multiples of $(N_{\mathcal{B}})_{i,i}$. Due to the fact that $N_{\mathcal{B}}$ is in Hermite normal form, the off-diagonal entries are zero. We now study different representations of the same crystal structure in order to derive a general way to rewrite a multiplication operator with respect to some coarser crystal structure corresponding to a sublattice, as has been done manually in \[sec:example\] for the discretized Laplacian. \[thm:crystal\_congruence\] Let ${ \mathbb{L}}^{\mathfrak{s}}({\mathcal{A}})$ be a crystal and ${ \mathbb{L}}({\mathcal{C}}) \subset { \mathbb{L}}({\mathcal{A}})$ a sublattice. Denoting $T_{{\mathcal{A}},{\mathcal{C}}}=\{{\mathfrak{t}}_1,\ldots,{\mathfrak{t}}_p\}$, [^7] the set $$\begin{aligned} {\Xi}({\mathcal{C}}):={\{x+\delta\in{ \mathbb{R}}^n \, : \, \delta\in{{ \mathcal{P}}}({\mathcal{A}}), x \in \{{\mathfrak{t}}_1,\ldots,{\mathfrak{t}}_p\}\}} \end{aligned}$$ defines a primitive cell of ${ \mathbb{L}}({\mathcal{C}})$, and the tuple $$\begin{aligned} {\mathfrak{u}}= ({\mathfrak{t}}_1+{\mathfrak{s}}_1,\ldots , {\mathfrak{t}}_1+{\mathfrak{s}}_m, {\mathfrak{t}}_2+{\mathfrak{s}}_1,\ldots,{\mathfrak{t}}_p+{\mathfrak{s}}_m)\in{\Xi}({\mathcal{C}})^{p\cdot m} \end{aligned}$$ defines a structure element of ${ \mathbb{L}}^{\mathfrak{u}}({\mathcal{C}})$ such that ${ \mathbb{L}}^{\mathfrak{u}}({\mathcal{C}}) \cong { \mathbb{L}}^{\mathfrak{s}}({\mathcal{A}}),$ meant as a one-to-$p$ correspondence. Without loss of generality we may assume ${\mathfrak{t}}_{j} \in {{ \mathcal{P}}}({\mathcal{C}})$, $j = 1,\ldots,p$. Then, each element in ${ \mathbb{L}}^{\mathfrak{s}}({\mathcal{A}})$ can be written as $$\begin{aligned} z=({\mathcal{A}}x+{\mathfrak{s}}_1,\ldots,{\mathcal{A}}x+{\mathfrak{s}}_m) \end{aligned}$$ and there is a unique $y$, such that ${\mathcal{A}}x={\mathcal{C}}y={\mathcal{C}}\lfloor y \rfloor + {\mathcal{C}}(y - \lfloor y \rfloor)$ with ${\mathcal{C}}\lfloor y \rfloor \in { \mathbb{L}}({\mathcal{C}})$ and ${\mathcal{C}}(y - \lfloor y \rfloor)={\mathfrak{t}}_j \in {{ \mathcal{P}}}({\mathcal{C}})\cap{ \mathbb{L}}({\mathcal{A}})$. Thus, we find $z$ as a unique part of the element $$\begin{aligned} {\mathcal{C}}\lfloor y \rfloor +{{\mathfrak{u}}}=(\ldots,{\mathcal{C}}\lfloor y \rfloor + {\mathfrak{t}}_j + {\mathfrak{s}},\ldots)=(\ldots,z,\ldots). \end{aligned}$$ This argument works in the other direction in the same way. \[rem:natural\_isomorph\] Note, that ${\mathfrak{u}}$, as defined in \[thm:crystal\_congruence\], is an explicit representation of $T^{\mathfrak{s}}_{{\mathcal{A}},{\mathcal{C}}}$, thus using this explicit representation \[thm:crystal\_congruence\] implies a congruence of the function spaces $$\begin{aligned} { \mathbcal{L}}(T_{{\mathcal{A}},{\mathcal{Z}}}^{\mathfrak{s}}) \cong { \mathbcal{L}}(T_{{\mathcal{C}},{\mathcal{Z}}}^{T^{\mathfrak{s}}_{{\mathcal{A}},{\mathcal{C}}}}) \end{aligned}$$ given by the natural isomorphism $\eta: { \mathbcal{L}}(T_{{\mathcal{A}},{\mathcal{Z}}}^{\mathfrak{s}}) \rightarrow { \mathbcal{L}}(T_{{\mathcal{C}},{\mathcal{Z}}}^{T^{\mathfrak{s}}_{{\mathcal{A}},{\mathcal{C}}}})$, $$\begin{aligned} { \mathbcal{L}}(T_{{\mathcal{A}},{\mathcal{Z}}}^{\mathfrak{s}}) \ni f(\cdot) \mapsto (\eta f )(\cdot)= (f(\cdot+{\mathfrak{t}}_1),\ldots, f(\cdot+{\mathfrak{t}}_p)) \in { \mathbcal{L}}(T_{{\mathcal{C}},{\mathcal{Z}}}^{T^{\mathfrak{s}}_{{\mathcal{A}},{\mathcal{C}}}}), \end{aligned}$$ as $(f)$ and $(\eta f)$ describe the same value distribution on the crystal. This congruence in turn implies that the coarsest possible crystal interpretation, i.e., ${\mathfrak{u}}\cong T^{\mathfrak{s}}_{{\mathcal{A}},{\mathcal{Z}}}$, is simply the complex coordinate space $${ \mathbb{C}}^{mn} = { \mathbcal{L}}(T_{{\mathcal{Z}},{\mathcal{Z}}}^{T^{\mathfrak{s}}_{{\mathcal{A}},{\mathcal{Z}}}}) \cong { \mathbcal{L}}(T^{\mathfrak{s}}_{{\mathcal{A}},{\mathcal{Z}}})$$ with $n:=|T_{{\mathcal{A}},{\mathcal{Z}}}|$. The scalar product on ${ \mathbcal{L}}(T^{\mathfrak{s}}_{{\mathcal{A}},{\mathcal{Z}}})$ then corresponds to the Euclidean scalar product on ${ \mathbb{C}}^{mn}$ up to a factor of $n$. Using the natural isomorphism of function spaces in \[rem:natural\_isomorph\], we can derive the transformations of multiplication operators when coarsening the underlying crystal representation corresponding to a sublattice. \[thm:operator\_sublattice\_coarsening\] Consider crystals ${ \mathbb{L}}^{\mathfrak{d}}({\mathcal{A}})$, ${ \mathbb{L}}^{\mathfrak{c}}({\mathcal{A}})$, a sublattice ${ \mathbb{L}}({\mathcal{C}}) \subset { \mathbb{L}}({\mathcal{A}})$ and a multiplication operator $$\begin{aligned} L: { \mathbcal{L}}(T_{\mathcal{A}}^{{\mathfrak{d}}}) \rightarrow { \mathbcal{L}}(T_{\mathcal{A}}^{{\mathfrak{c}}}), \quad ({L}f)(x):=\sum_{y\in T_{{\mathcal{A}}}} m_L^{(y)} f(x+y), \quad m_L^{(y)} \in { \mathbb{C}}^{|{\mathfrak{c}}|\times |{\mathfrak{d}}|}. \end{aligned}$$ Then, using $T_{{\mathcal{A}},{\mathcal{C}}}=\{{\mathfrak{t}}_1,\ldots,{\mathfrak{t}}_p\}$, the multiplication operator $$\begin{aligned} G:{ \mathbcal{L}}(T_{\mathcal{C}}^{\hat{{\mathfrak{d}}}}) \rightarrow { \mathbcal{L}}(T_{\mathcal{C}}^{\hat{{\mathfrak{c}}}}), \quad (G g)(x)=\sum_{y\in T_{\mathcal{C}}} m_G^{(y)} g(x+y), \quad m_G^{(y)} \in { \mathbb{C}}^{p |{\mathfrak{c}}|\times p|{\mathfrak{d}}|}, \end{aligned}$$ with block matrices $(m_G^{(y)})_{i,k} := m_L^{(y-{\mathfrak{t}}_i+{\mathfrak{t}}_k)} \in { \mathbb{C}}^{|{\mathfrak{c}}|\times |{\mathfrak{d}}|}$ fulfills the commutative diagram: $$\begin{tikzcd} { \mathbcal{L}}(T_{\mathcal{A}}^{{\mathfrak{d}}}) \arrow[r, "{L}"] \arrow[d, "\eta^{\mathfrak{d}}"] & { \mathbcal{L}}(T_{\mathcal{A}}^{{\mathfrak{c}}}) \arrow[d, "\eta^{\mathfrak{c}}" ] \\ { \mathbcal{L}}(T_{\mathcal{C}}^{\hat{{\mathfrak{d}}}}) \arrow[r, "G" ] & { \mathbcal{L}}(T_{\mathcal{C}}^{\hat{{\mathfrak{c}}}}). \end{tikzcd}$$ Here, the mappings for ${\mathfrak{s}}\in \{{\mathfrak{d}},{\mathfrak{c}}\}$, $$\begin{aligned} \eta^{\mathfrak{s}}: { \mathbcal{L}}(T_{\mathcal{A}}^{{\mathfrak{s}}}) \rightarrow { \mathbcal{L}}(T_{\mathcal{C}}^{\hat{{\mathfrak{s}}}}), \quad f(\cdot)\mapsto (f(\cdot +{\mathfrak{t}}_1),\ldots,f(\cdot+{\mathfrak{t}}_p)), \end{aligned}$$ denote the natural isomorphisms between the congruent crystal representations. A straightforward calculation for each block-row $i$ yields $$\begin{aligned} \begin{array}{rcl} [(\eta^{\mathfrak{c}}L f)(x)]_i & = & (Lf)(x+{\mathfrak{t}}_i) \ =\ \displaystyle\sum\limits_{k=1}^{p} \displaystyle\sum\limits_{y \in T_{\mathcal{C}}} m_L^{(y+{\mathfrak{t}}_k)} f(x+y+{\mathfrak{t}}_i+{\mathfrak{t}}_k) \\[\medskipamount] & = & \displaystyle\sum\limits_{k=1}^{p} \displaystyle\sum\limits_{y \in T_{\mathcal{C}}} m_L^{(y-{\mathfrak{t}}_i+{\mathfrak{t}}_k)} f(x+y+{\mathfrak{t}}_k) \ =\ \displaystyle\sum\limits_{y \in T_{\mathcal{C}}} \displaystyle\sum\limits_{k=1}^{p} (m_G^{(y)})_{i,k} f(x+y+{\mathfrak{t}}_k) \\[\bigskipamount] & = & [G (f(x+{\mathfrak{t}}_1),\ldots,f(x+{\mathfrak{t}}_p))]_i \ =\ [(G \eta^{\mathfrak{d}}f)(x)]_i. \end{array} \end{aligned}$$ Using \[thm:operator\_sublattice\_coarsening\] we now know how to rewrite multiple multiplication operators with respect to some common crystal structure with a coarser translational invariance. Due to the fact that we do not make any assumption on the initial representation of the crystal structures, the resulting structure elements of \[cor:quotient\_latticepointlist\] might differ in their orderings and might contain shifts with respect to the common shift invariance. To automatically remove these differences and determine the corresponding transformations of the associated multiplication operators we first define the notion of congruent structure elements. \[def:crystaltori\_congruent\] Two crystal tori $T_{\mathcal{A}}^{\mathfrak{s}}\cong T_{\mathcal{A}}^{\mathfrak{t}}$, ${\mathcal{A}}\in { \mathbb{R}}^n$ are *congruent* with respect to ${ \mathbb{L}}({\mathcal{A}})$ if the structure elements are of the same size, i.e., $|{\mathfrak{s}}| = |{\mathfrak{t}}| = m$, and there is a permutation $\pi:\{1,\ldots,m\} \rightarrow \{1,\ldots,m\}$ as well as shifts $y_j \in { \mathbb{L}}({\mathcal{A}})$, such that $$\begin{aligned} {\mathfrak{s}}_j = y_j + {\mathfrak{t}}_{\pi(j)} \end{aligned}$$ In order to introduce a unique representation for the sake of automation, we introduce the following *normal form* and the required transformations to transfer any operator to this form. \[def:crystal\_op\_and\_normalform\] Let $L: { \mathbcal{L}}(T_{\mathcal{A}}^{{\mathfrak{d}}}) \rightarrow { \mathbcal{L}}(T_{\mathcal{A}}^{{\mathfrak{c}}})$ be a multiplication operator. We say $L$ is in *normal form* if - the coordinates of the structure elements are found in the primitive cell spanned by the primitive vectors, i.e., ${\mathfrak{d}}_i,{\mathfrak{c}}_j \in {{ \mathcal{P}}}({\mathcal{A}}) = {\mathcal{A}}[0,1)^n \text{\ for each\ } i,j,$ - the structure elements ${\mathfrak{d}}$ and ${\mathfrak{c}}$ are sorted lexicographically.[^8] We now derive the implications of \[def:crystaltori\_congruent\] for multiplication operators when the structure element is element wise shifted or permuted. We do so in two steps, \[thm:shifted\_structure\_element\] and \[thm:permuted\_structure\_element\]. First, we show that a shift of an entry of the structure element in the codomain or domain results in a modification of the corresponding row or column of the non-zero multipliers, respectively. \[thm:shifted\_structure\_element\] Consider the two multiplication operators $ L: { \mathbcal{L}}(T_{\mathcal{A}}^{{\mathfrak{s}}}) \rightarrow { \mathbcal{L}}(T_{\mathcal{A}}^{{\mathfrak{t}}})$ and $G: { \mathbcal{L}}(T_{\mathcal{A}}^{{\mathfrak{t}}}) \rightarrow { \mathbcal{L}}(T_{\mathcal{A}}^{{\mathfrak{u}}})$ defined by $$\begin{aligned} (Lf)(x) = \sum_{y \in T_{{\mathcal{A}}}} m_L^{(y)} f(x+y), \quad m_L^{(y)} \in { \mathbb{{\mathcal{C}}}} ^{|{\mathfrak{t}}|\times |{\mathfrak{s}}|}, \\ (Gg)(x) = \sum_{y \in T_{{\mathcal{A}}}} m_G^{(y)} g(x+y), \quad m_G^{(y)} \in { \mathbb{{\mathcal{C}}}} ^{|{\mathfrak{u}}| \times |{\mathfrak{t}}|}. \end{aligned}$$ Further let $\hat{{\mathfrak{t}}}$ be a structure element which is obtained from ${\mathfrak{t}}$ when shifted element-wise along ${ \mathbb{L}}({\mathcal{A}})$, i.e., $$\begin{aligned} {\mathfrak{t}}& = ({\mathfrak{t}}_1,\ldots,{\mathfrak{t}}_m) = (\hat{{\mathfrak{t}}}_1 + y_1,\ldots,\hat{{\mathfrak{t}}}_m+y_m ) = \hat{{\mathfrak{t}}} +(y_1,\ldots,y_m), \end{aligned}$$ where $y_1,\ldots,y_m \in{ \mathbb{L}}({\mathcal{A}})$ and $m=|{\mathfrak{t}}|$. Then, the operators $\hat{L}$ and $\hat{G}$ given by $$\begin{aligned} \begin{array}{rcllcl} (\hat{L}f)(x) &=& \displaystyle\sum\limits_{y \in T_{{\mathcal{A}}}} m_{\hat{L}}^{(y)} f(x+y), & m_{\hat{L}}^{(y)} \in { \mathbb{{\mathcal{C}}}} ^{|{\mathfrak{t}}|\times |{\mathfrak{s}}|} &\text{ with }& (m_{\hat{L}}^{(y)})_{i,j} := (m_{L}^{(y + y_i)})_{i,j},\\ (\hat{G}f)(x) &=& \displaystyle\sum\limits_{y \in T_{{\mathcal{A}}}} m_{\hat{G}}^{(y)} f(x+y), & m_{\hat{G}}^{(y)} \in { \mathbb{{\mathcal{C}}}} ^{|{\mathfrak{u}}|\times |{\mathfrak{t}}|} &\text{ with }& (m_{\hat{G}}^{(y)})_{i,j} := (m_{G}^{(y - y_j)})_{i,j} \end{array} \end{aligned}$$ fulfill the commutative diagram: $$\begin{tikzcd} { \mathbcal{L}}(T_{\mathcal{A}}^{\mathfrak{s}}) \arrow[r, "{L}"] \arrow[rd, "\hat{L}"] & { \mathbcal{L}}(T_{\mathcal{A}}^{\mathfrak{t}}) \arrow[r, "G"] \arrow[d, "{\mathbb{T}}" ] & { \mathbcal{L}}(T_{\mathcal{A}}^{\mathfrak{u}}) \\ &{ \mathbcal{L}}(T_{\mathcal{A}}^{\hat{{\mathfrak{t}}}}) \arrow[ur,"\hat{G}"] \end{tikzcd}$$ The natural isomorphism between the two corresponding function spaces is given by $$\begin{aligned} {\mathbb{T}}: { \mathbcal{L}}(T_{\mathcal{A}}^{\mathfrak{t}}) \rightarrow { \mathbcal{L}}(T_{\mathcal{A}}^{\hat{{\mathfrak{t}}}}),\ f=(f_1,\ldots,f_m)\mapsto (f_1(\cdot - y_1),\ldots,f_m(\cdot-y_m)) = {\mathbb{T}}f \end{aligned}$$ as $f$ and $({\mathbb{T}}f)$ describe the same value distribution on the crystal.[^9] Again, a straightforward calculation yields $$\begin{aligned} \begin{array}{rcl} [({\mathbb{T}}Lf)(x)]_i & =& [(Lf)(x-y_i)]_i \ =\ \displaystyle\sum\limits_{y \in T_{{\mathcal{A}}}} \displaystyle\sum\limits_{j=1}^{|{\mathfrak{s}}|} (m_L^{(y)})_{i,j} f_j(x-y_i+y) \\ & =& \displaystyle\sum\limits_{y \in T_{{\mathcal{A}}}} \displaystyle\sum\limits_{j=1}^{|{\mathfrak{s}}|} (m_L^{(y+y_i)})_{i,j} f_j(x+y) \ =\ [\displaystyle\sum\limits_{y \in T_{{\mathcal{A}}}} m_{\hat{L}}^{(y)} f(x+y)]_i. \end{array} \end{aligned}$$ Analogously we find $[( G {\mathbb{T}}^{-1} g)(x)]_i = [\sum_{y \in T_{{\mathcal{A}}}} (m_{\hat{G}}^{(y)}) g(x+y)]_i$. Finally, we show that permutations of the entries of the structure element result in a transformation of the non-zero multipliers by corresponding permutation matrices. \[thm:permuted\_structure\_element\] Consider the two multiplication operators $ L: { \mathbcal{L}}(T_{\mathcal{A}}^{{\mathfrak{s}}}) \rightarrow { \mathbcal{L}}(T_{\mathcal{A}}^{{\mathfrak{t}}})$ and $G: { \mathbcal{L}}(T_{\mathcal{A}}^{{\mathfrak{t}}}) \rightarrow { \mathbcal{L}}(T_{\mathcal{A}}^{{\mathfrak{u}}})$ defined by $$\begin{aligned} (Lf)(x) = \sum_{y \in T_{{\mathcal{A}}}} m_L^{(y)} f(x+y), \quad m_L^{(y)} \in { \mathbb{{\mathcal{C}}}} ^{|{\mathfrak{t}}|\times |{\mathfrak{s}}|}, \\ (Gg)(x) = \sum_{y \in T_{{\mathcal{A}}}} m_G^{(y)} g(x+y), \quad m_G^{(y)} \in { \mathbb{{\mathcal{C}}}} ^{|{\mathfrak{u}}| \times |{\mathfrak{t}}|}. \end{aligned}$$ Let further $\hat{{\mathfrak{t}}}$ be a structure element which is a permuted version of ${\mathfrak{t}}$, i.e., $$\begin{aligned} \hat{{\mathfrak{t}}} & = (\hat{{\mathfrak{t}}}_1,\ldots,\hat{{\mathfrak{t}}}_m) = ({\mathfrak{t}}_{\pi(1)},\ldots, {\mathfrak{t}}_{\pi(m)}) = m_\pi {\mathfrak{t}}\end{aligned}$$ where $m=|{\mathfrak{t}}|$, $\pi:\{1,\ldots,m\} \rightarrow \{1,\ldots,m\}$ is a permutation and $m_\pi \in \{0,1\}^{m\times m}$ the corresponding permutation matrix. Then, the operators $\hat{L}$ and $\hat{G}$ given by $$\begin{aligned} \begin{array}{rcllcl} (\hat{L}f)(x) &=& \displaystyle\sum\limits_{y \in T_{{\mathcal{A}}}} m_{\hat{L}}^{(y)} f(x+y), & m_{\hat{L}}^{(y)} \in { \mathbb{{\mathcal{C}}}} ^{|{\mathfrak{t}}|\times |{\mathfrak{s}}|} &\text{ with }& m_{\hat{L}}^{(y)} := m_\pi m_{L}^{(y)},\\ (\hat{G}f)(x) &=& \displaystyle\sum\limits_{y \in T_{{\mathcal{A}}}} m_{\hat{G}}^{(y)} f(x+y), & m_{\hat{G}}^{(y)} \in { \mathbb{{\mathcal{C}}}} ^{|{\mathfrak{u}}|\times |{\mathfrak{t}}|} &\text{ with }& m_{\hat{G}}^{(y)} := m_{G}^{(y)} m_\pi^{-1} \end{array} \end{aligned}$$ fulfill the commutative diagram: $$\begin{tikzcd} { \mathbcal{L}}(T_{\mathcal{A}}^{\mathfrak{s}}) \arrow[r, "{L}"] \arrow[rd, "\hat{L}"] & { \mathbcal{L}}(T_{\mathcal{A}}^{\mathfrak{t}}) \arrow[r, "G"] \arrow[d, "p" ] & { \mathbcal{L}}(T_{\mathcal{A}}^{\mathfrak{u}}) \\ &{ \mathbcal{L}}(T_{\mathcal{A}}^{\hat{{\mathfrak{t}}}}) \arrow[ur,"\hat{G}"] \end{tikzcd}$$ Due to the fact that the natural isomorphism $p: { \mathbcal{L}}(T_{\mathcal{A}}^{\mathfrak{t}}) \rightarrow { \mathbcal{L}}(T_{\mathcal{A}}^{\hat{{\mathfrak{t}}}})$ is a multiplication operator defined by $(p f)(x) = m_\pi f(x)$ for all $x\in{ \mathbb{L}}({\mathcal{A}})$, the statement is true due to the rules of computation in \[lem:calc\_rules\_mult\_op\]. and \[thm:lcm\_mat,thm:crystal\_congruence,thm:operator\_sublattice\_coarsening,thm:shifted\_structure\_element,thm:permuted\_structure\_element\] allow for the automatic adjustment of crystal representations within the LFA. The corresponding detailed algorithms which make use of these results are given in \[sec:algorithms\]. Application {#sec:Application} =========== Before demonstrating the application of aLFA to some selected examples let us briefly recapitulate the individual parts of the framework. The introduction of crystal structures in \[sec:lattices\_crystals\] allow for a canonical description of translationally invariant operators introduced in \[sec:operators\]. Combined with the definition of the dual of a crystal structure in \[def:dual\_lattice\] and a corresponding orthonormal basis in \[thm:wavefunction\_ONB\_vec\], the symbol of any single multiplication operator that is translationally invariant with respect to an arbitrary lattice structure, can be expressed (cf. \[def:crystal\_op\_symbol\]). This combination of choice of basis functions and the explicit connection of operators to their underlying structure enables an automated mixing analysis. This part of the framework can thus be seen on one hand as a unification of positional approaches to LFA by introducing structures that allow for the native treatment of arbitrary translational invariances and on the other hand as a general strategy for the automation of the frequency part back-end of LFA. In order to deal with compositions of operators as encountered in the analysis of iterative methods, the tools provided in \[sec:crystal\_reps\_and\_isos\] allow for an automatic transformation of the underlying crystal structures into compatible representations and the corresponding transformations of the operators. Thus, the only task remaining for the user is to provide any, i.e., the simplest or most convenient, crystal representation of the operators. The following examples show how such a construction can be carried out and as such serve as a tutorial for the use of the algorithms in \[sec:algorithms\]. Annotated Jupyter Notebooks of the presented examples can be found in [@aLFA_NK]. Multicolored block smoother for the tight-binding Hamiltonian of graphene {#subsec:multicolor_graphene} ------------------------------------------------------------------------- In [@KahlKint2018] a multigrid method for the tight-binding Hamiltonian of the carbon allotrope graphene based on Kaczmarz smoothing is constructed and analyzed using conventional LFA. Due to the hexagonal structure of graphene, the lexicographic ordering of Kaczmarz and the mixing analysis of the coarse grid correction which involved a mixing of eight frequencies, the analysis turned out to be quite lengthy. In this subsection we now want to analyze a two-grid method for this problem where we replace Kaczmarz by an overlapping colored Gauss-Seidel method that allows for better parallelism in the application of the multigrid method. Thus, the goal of this example is two-fold, first we want to show that the tight-binding Hamiltonian can be easily expressed using the native crystal structure of graphene and second that even the analysis of an overlapping block smoother can be carried out with ease using aLFA due to the fact that only a representation of the involved operators is needed. The graphene structure can be described as a crystal ${ \mathbb{L}}^{\mathfrak{s}}({\mathcal{A}})$ where the underlying lattice is triangular, i.e., any three nearby lattice points form an equilateral triangle. We have $$\begin{aligned} { \mathbb{L}}^{\mathfrak{s}}({\mathcal{A}}),\quad {\mathcal{a}}_1 = (\frac{3}{2}, \frac{\sqrt{3}}{2}) a,\quad {\mathcal{a}}_2 = (\frac{3}{2},-\frac{\sqrt{3}}{2}) a, \end{aligned}$$ with the structure element $$\begin{aligned} {\mathfrak{s}}= ({\mathfrak{s}}_1,{\mathfrak{s}}_2), \quad {\mathfrak{s}}_1 = ({\mathcal{a}},0)=\frac{1}{3}({\mathcal{a}}_1+{\mathcal{a}}_2), \quad {\mathfrak{s}}_2=(2{\mathcal{a}},0)=\frac{2}{3}({\mathcal{a}}_1+{\mathcal{a}}_2). \end{aligned}$$ The parameter $a= 1.42$Å which denotes the distance of two neighboring atoms can be omitted as it does not occur in the tight-binding formulation. An illustration of the crystal has already been given in \[fig:crystal\_example\]. The (nearest neighbor) tight-binding Hamiltonian is a multiplication operator $$\begin{aligned} L: { \mathbcal{L}}(T_{{\mathcal{A}}} ^{\mathfrak{s}}) \rightarrow { \mathbcal{L}}(T_{{\mathcal{A}}} ^{\mathfrak{s}}), \quad (Lf)(x) := \sum_{y \in T_{{\mathcal{A}}} } m_{L}^{(y)} f(x+y) \end{aligned}$$ with non-zero multipliers \(c) at (0,0) [$=$]{}; ; ; \(c) at (-,) [$=$]{}; ; ; \(c) at (,) [$=$]{}; ; ; \(c) at (-,-) [$=$]{}; ; ; \(c) at (,-) [$=$]{}; ; ; as illustrated in \[fig:tbh\_stencil\_and\_4color\_hexagon\], a). #### Overlapping Hexagons We now present an overlapping block smoother for the tight-binding Hamiltonian $L$. Consider the non-disjoint splitting (or coloring) of the crystal into hexagons as depicted in \[fig:tbh\_stencil\_and\_4color\_hexagon\], b). This splitting has a translational invariance of ${\mathcal{C}}= 2{\mathcal{A}}$. Rewriting the graphene crystal $T_{\mathcal{A}}^{\mathfrak{s}}$ with respect to this coarser lattice ${ \mathbb{L}}({\mathcal{C}})$, we find $T_{\mathcal{A}}^{\mathfrak{s}}\cong T_{\mathcal{C}}^{\mathfrak{t}}$ with the structure element $$\begin{aligned} {\mathfrak{t}}=({\mathfrak{t}}_1,\ldots,{\mathfrak{t}}_8)= ({\mathfrak{s}}, {\mathfrak{s}}+{\mathcal{a}}_1, {\mathfrak{s}}+{\mathcal{a}}_2, {\mathfrak{s}}+{\mathcal{a}}_1+{\mathcal{a}}_2). \end{aligned}$$ The splitting is then given by the structure elements $$\begin{aligned} {\mathfrak{t}}^{(1)}&= ({\mathfrak{t}}_1^{(1)},\ldots,{\mathfrak{t}}_6^{(1)}) = ({\mathfrak{t}}_2,{\mathfrak{t}}_3,{\mathfrak{t}}_4,{\mathfrak{t}}_5,{\mathfrak{t}}_6,{\mathfrak{t}}_7), \\ {\mathfrak{t}}^{(2)}&= {\mathfrak{t}}^{(1)} + {\mathcal{a}}_1,\ {\mathfrak{t}}^{(3)}= {\mathfrak{t}}^{(1)} + {\mathcal{a}}_2,\text{ and } {\mathfrak{t}}^{(4)}= {\mathfrak{t}}^{(1)} + {\mathcal{a}}_1 + {\mathcal{a}}_2, \end{aligned}$$ such that $T_{\mathcal{C}}^{{\mathfrak{t}}^{(1)}} \widehat{=} \CGcaptionAab[]$, $T_{\mathcal{C}}^{{\mathfrak{t}}^{(2)}} \widehat{=} \CGcaptionAbb[]$, $T_{\mathcal{C}}^{{\mathfrak{t}}^{(3)}} \widehat{=} \CGcaptionAba[]$ and $T_{\mathcal{C}}^{{\mathfrak{t}}^{(4)}} \widehat{=} \CGcaptionAaa[]$. In the case of the nearest neighbor Hamiltonian any two hexagons of the same color do not interact. Thus, a relaxed sweep on the unknowns of one color is cheap and, in addition, yields a good degree of parallelism. There are two options to describe the multiplication operators of the multicolored block smoother. On the one hand, we can construct them from scratch by defining the multiplication operator structure as well as the non-zero multipliers. On the other hand, we can derive the structure and non-zero multipliers of the operators corresponding to the colored blocking from the tight-binding Hamiltonian $L$ by exploiting the fact that a colored block corresponds to the operator $L$ restricted to this block. In order to proceed with the latter approach, we first need to find the description $\hat{L}$ of the underlying operator, the tight-binding Hamiltonian $L$, with respect to the translational invariance of the splitting ${\mathcal{C}}$. After that the structure element needs to be adjusted such that all unknowns and their couplings among each other are found within the central multiplier $m_{\hat{L}}^{(0)}$. Consider the structure element ${\mathfrak{t}}$. As can be seen in \[fig:tbh\_stencil\_and\_4color\_hexagon\], b), the coupling among the unknowns ${\mathfrak{t}}^{(\ell)}$ which we want to update simultaneously are found in the multipliers: [c|c|c|c]{} ${\mathfrak{t}}^{(1)}$ & ${\mathfrak{t}}^{(2)}$ & ${\mathfrak{t}}^{(3)}$ & ${\mathfrak{t}}^{(4)}$\ ------------------------------------------------------------------------ $m_{\hat{L}}^{(0)}$ & $m_{\hat{L}}^{(0)},\ m_{\hat{L}}^{(\pm {\mathcal{c}}_1)}$ & $m_{\hat{L}}^{(0)},\ m_{\hat{L}}^{(\pm{\mathcal{c}}_2)}$ & $m_{\hat{L}}^{(0)},\ m_{\hat{L}}^{(\pm{\mathcal{c}}_1)},\ m_{\hat{L}}^{(\pm{\mathcal{c}}_2)},\ m_{\hat{L}}^{(\pm({\mathcal{c}}_1-{\mathcal{c}}_2))}$   Thus, in order to obtain suitable descriptions for ${\mathfrak{t}}^{(\ell)}$, $\ell\in\{2,3,4\}$, we need to consider shifted versions ${\mathfrak{t}}+ \tau^{(\ell)}$. For example $$\label{eq:shiftedhex} \begin{bmatrix} \tau^{(1)}, & \tau^{(2)}, & \tau^{(3)}, & \tau^{(4)}\end{bmatrix} := \begin{bmatrix} 0, & {\mathcal{a}}_1, & {\mathcal{a}}_2, & {\mathcal{a}}_1 + {\mathcal{a}}_2 \end{bmatrix} .$$ Finally, the error propagator corresponding to a single color relaxed block Gauss-Seidel update can be written as $$\begin{aligned} G^{(\ell)} = (I - \omega (S^{(\ell)})^\dagger \hat{L}) : { \mathbcal{L}}(T_{\mathcal{C}}^{{\mathfrak{t}}+ \tau^{(\ell)} }) \rightarrow { \mathbcal{L}}(T_{\mathcal{C}}^{{\mathfrak{t}}+ \tau^{(\ell)} }) \end{aligned}$$ with $(S^{(\ell)}f)(x) := (m_P m_{\hat{L}}^{(0)} m_P)f(x)$ where $m_P$ is the diagonal matrix $$\label{eq:mpii_hex} (m_P)_{ii} = \begin{cases} 1 & \text{ for }i\in\{2,3,4,5,6,7\} \\ 0 & \text{ else}. \end{cases}$$ We summarize the algorithmical steps to obtain the error propagators. The error propagator of a successive update of the four colors corresponds to the product of the error propagators $G^{(\ell)} = ( I - \omega (S^{(\ell)})^\dagger \hat{L}^{(\ell)})$. Now that all operators of the overlapping colored block Gauss-Seidel smoother are defined, we can compute its spectrum. The computation of the eigenvalues of the error propagator is carried out with \[alg:compute\_spectral\_radii\] via $$\textproc{ComputeSpectrum}(g,I,L,S^{(1)},\ldots,S^{(4)}).$$ The function $g$ denotes the composition of the error propagators $$\begin{aligned} (I,L,S^{(1)},\ldots,S^{(4)}) \xmapsto{\enskip g\enskip } \prod_{\ell=1}^4 ( I - \omega (S^{(\ell)})^\dagger L)=:G. \end{aligned}$$ Note, that in this algorithm all operators are checked for compatibility and any incompatibility with respect to domains and codomains is dealt with using the transformations introduced in \[sec:crystal\_reps\_and\_isos\]. For $\omega=\frac{1}{2}$ we obtain the plot in \[fig:hexagon\_smooth\_twogrid\], a). As the largest spectral radius is greater than one, this method cannot be used as a standalone solver. #### Coarse grid correction In [@KahlKint2018] a Galerkin coarse grid correction is used with a corresponding coarse grid correction operator that is defined by $$\begin{aligned} E = (I- P(L_c)^\dagger R L) \end{aligned}$$ with $L_c = RLP$ and $P=R^T$. Just as in the description of the smoother, the shift invariance of the coarse grid is ${\mathcal{C}}=2{\mathcal{A}}$. Defining the coarse structure element by $2{\mathfrak{s}}$, and with ${\mathfrak{t}}$ denoting the structure element of the fine crystal according to the coarse lattice basis, the restriction operator can be described as $$\begin{aligned} R : { \mathbcal{L}}(T^{\mathfrak{t}}_{2{\mathcal{A}}}) \rightarrow { \mathbcal{L}}(T^{2{\mathfrak{s}}}_{2{\mathcal{A}}}). \end{aligned}$$ The multipliers of the restriction operator according to [@KahlKint2018] then read This coarse grid correction is constructed in a way, such that it preserves the kernel of the tight-binding Hamiltonian $L$, where with ${\mathcal{B}}=\mat{{\mathcal{b}}_1 & {\mathcal{b}}_2} = {\mathcal{A}}^{-T}$, $$\operatorname{ker}(L)=\operatorname{span}{\{\mat{{e^{2\pi i{\langle K,x\rangle_{2}}}} \\ 0}, \mat{0 \\ {e^{2\pi i{\langle K,x\rangle_{2}}}}} \, : \, K \in\{\frac{1}{3} {\mathcal{b}}_1 + \frac{2}{3} {\mathcal{b}}_2,\frac{2}{3} {\mathcal{b}}_1 + \frac{1}{3} {\mathcal{b}}_2\} \}}.$$ The frequencies corresponding to the kernel modes are known as the *Dirac points*. The two-grid analysis can now be carried out using \[alg:compute\_spectral\_radii\] via $$\text{\textproc{ComputeSpectrum}}(f,I,L,S^{(1)},\ldots,S^{(4)}, R).$$ The function $f$ denotes the composition of the two-grid error propagator $$\begin{aligned} (I,L,S^{(1)},\ldots,S^{(4)},R) \xmapsto{\enskip f\enskip } GEG. \end{aligned}$$ A plot of the spectral radii of the two-grid error propagator is given in b) of \[fig:hexagon\_smooth\_twogrid\] which shows that the two-grid method converges with a convergence rate of $\rho_{\max} \approx 0.167$. Thus, this new method with overlapping colored block Gauss-Seidel smoothing not only yields opportunities for parallel computations, but also converges faster than the old approach which used Kaczmarz smoothing. In order to double-check the results of the developed theory, we show in \[tbl:graphene\_twogrid\_results\] that the asymptotic convergence rate of the two-grid method with random initial guess $x_0$ and right-hand-side $0$ coincides with the convergence rate obtained in the LFA with a relative accuracy of roughly $.002\%$. This comes as no surprise as the theory is exact for this problem with periodic boundary conditions and we have chosen the sampling of the frequency space in accordance with the problem size (cf. \[rem:freq\_sampling\]). iteration $i$ $||r_i||_2:=||b - A x_i||_2$ $\rho_i:=\frac{||r_i||_2}{||r_{i-1}||_2}$ $\frac{|\rho_i - \rho_{\text{analytic}}|}{\rho_{\text{analytic}}}$ --------------- ------------------------------ ------------------------------------------- -------------------------------------------------------------------- 398 1.161369e-312 0.16685534 0.00220% 399 1.937806e-313 0.16685539 0.00217% 400 3.233335e-314 0.16685545 0.00213% : Convergence history of the two-grid method applied to the tight-binding Hamiltonian of graphene with $(41\times 41)\cdot 2^3$ unknowns/atoms and periodic boundary conditions. The reported asymptotic convergence rate $\rho_i$ coincides with high precision to the convergence estimate $\rho_{\text{analytic}} = \sup_{k \in T^*_{2{\mathcal{A}},41\cdot 2{\mathcal{A}}}} {\{| \lambda | \, : \, \lambda \text{ eigenvalue of } G_kE_kG_k\}} = 0.16685901$ obtained in aLFA.[]{data-label="tbl:graphene_twogrid_results"} Two-level analysis for the curl-curl equation {#subsec:curlcurl} --------------------------------------------- In [@BoonLentVand2008] a complete two-level analysis is carried out for the $2$-dimensional curl-curl formulation of Maxwell’s equations using conventional LFA. In this subsection we want to reproduce the results of the two-grid method discussed in this paper making use of the native crystal structure of the staggered discretization of the curl-curl equations. The method consists of a so-called half-hybrid smoother, introduced in [@Hipt1998], and a Galerkin coarse grid correction. The degrees of freedom of the discrete curl-curl equation $$\begin{aligned} \label{eq:discrete_curl} (\frac{1}{h^2}K_{cc} + \sigma M)x= b \end{aligned}$$ are associated with the edges of a quadrilateral lattice ${ \mathbb{L}}(\frac{1}{h} {\mathcal{A}})$ with $${\mathcal{A}}= \mat{{\mathcal{a}}_1 & {\mathcal{a}}_2}= \mat{1 & 0 \\ 0 & 1}.$$ By multiplying \[eq:discrete\_curl\] with $h^2$, the grid size can be incorporated into the material parameter $\sigma_h := h^2\sigma$. Then the discrete operator $K=K_{cc} + \sigma_h M$ can be expressed as a multiplication operator by $$\begin{aligned} K: { \mathbcal{L}}(T_{\mathcal{A}}^{\mathfrak{e}}) \rightarrow { \mathbcal{L}}(T_{\mathcal{A}}^{\mathfrak{e}}) \quad \text{with} \quad {\mathfrak{e}}= ({\mathfrak{e}}_h,{\mathfrak{e}}_v) = (\frac{1}{2}{\mathcal{a}}_1, \frac{1}{2}{\mathcal{a}}_2), \end{aligned}$$ where the elements ${ \mathbb{L}}({\mathcal{A}}) + {\mathfrak{e}}_h$ and ${ \mathbb{L}}({\mathcal{A}}) + {\mathfrak{e}}_v$ correspond to the (midpoints of the) horizontal and the vertical edge, respectively (cf. \[fig:curlcurl\_crystal\]). According to [@BoonLentVand2008] the multipliers are given by: #### Hybrid smoother The (half) hybrid smoother uses an auxiliary space correction. That is, it can be seen as a two-grid method itself where smoothing replaces the exact inversion of the auxiliary space operator. The construction of this operator is as follows. As already stated, we want to reproduce the results in [@BoonLentVand2008]. Due to the fact that lexicographic Gauss-Seidel smoothers are used, we introduce the exact same ordering of the lattice points used in this paper, that is, the coordinates $y\in { \mathbb{L}}({\mathcal{A}})$ are ordered from bottom to top and left to right which leads us to the following definition for $x,y\in{ \mathbb{L}}({\mathcal{A}})$: $$x=(x_1,x_2) < y=(y_1,y_2) :\Leftrightarrow x_2 < y_2 \text{ or } \left(x_2 = y_2 \text{ and } x_1 < y_1\right) .$$ The error propagator of the smoother on the original crystal $T_{\mathcal{A}}^{\mathfrak{e}}$ is a node-based lexicographic Gauss-Seidel iteration. When updating a horizontal edge $x + {\mathfrak{e}}_h$, $x\in{ \mathbb{L}}({\mathcal{A}})$ it is assumed that all edges $y+{\mathfrak{e}}_h$ and $y+\hat{{\mathfrak{e}}}_v$, $\hat{{\mathfrak{e}}}_v:={\mathfrak{e}}_v+{\mathcal{a}}_1 - {\mathcal{a}}_2$, with $y<x$ are already updated. When updating the edge $y+\hat{{\mathfrak{e}}}_v$, it is additionally assumed that $x+{\mathfrak{e}}_h$ is already updated (cf. [@BoonLentVand2008 Figure 6.1]). In order to obtain the error propagator which exactly represents this ordering, it is convenient to rewrite the operator $K$ with respect to this representation of the structure element, that is $$\begin{aligned} K \cong \hat{K}: { \mathbcal{L}}(T_{\mathcal{A}}^{\hat{{\mathfrak{e}}}}) \rightarrow { \mathbcal{L}}(T_{\mathcal{A}}^{\hat{{\mathfrak{e}}}}) \quad \text{with} \quad \hat{{\mathfrak{e}}} = ({\mathfrak{e}}_h,\hat{{\mathfrak{e}}}_v) \cong {\mathfrak{e}}, \end{aligned}$$ which can be obtained with \[alg:struct\_elem\_change\] via $$\begin{aligned} \hat{K}=\text{\textproc{ChangeStructureElement}}(K,\hat{{\mathfrak{e}}},\hat{{\mathfrak{e}}}). \end{aligned}$$ The corresponding error propagator is then given by $$\begin{aligned} G_E = (I - S_E^\dagger K ): { \mathbcal{L}}(T_{\mathcal{A}}^{\mathfrak{e}}) \rightarrow { \mathbcal{L}}(T_{\mathcal{A}}^{\mathfrak{e}}) \end{aligned}$$ with the multipliers $$\begin{aligned} m_{S_E}^{(y)} = \begin{cases} m_{\hat{K}}^{(y)} & \text{if } y < 0, \\ \texttt{tril}(m_{\hat{K}}^{(y)}) & \text{if } y=0, \\ 0 & \text {else.} \end{cases} \end{aligned}$$ In here, only the lower triangular part $\texttt{tril}(m_{\hat{K}}^{(0)})$ of the central multiplier is used due to the fact that it represents a Gauss-Seidel sweep where a horizontal edge is updated before a vertical edge. The crystal, where the auxiliary space system is formulated, is given by $T_{\mathcal{A}}^{(0)}$, i.e., the nodal points between the edges. The transfer operator to this crystal is the discrete gradient operator defined by $R_N: { \mathbcal{L}}(T_{{\mathcal{A}}}^{\mathfrak{e}}) \rightarrow { \mathbcal{L}}(T_{{\mathcal{A}}}^{(0)})$ with non-zero multipliers $$\begin{aligned} \begin{array}{rccrcc} m_{R_N}^{(-{\mathcal{a}}_1)} &=& \mat{1 & 0} & m_{R_N}^{(0)} &=& \mat{-1 & -1} \\[1em] & & & m_{R_N}^{(-{\mathcal{a}}_2)} &= &\mat{0 & 1}. \end{array} \end{aligned}$$ Using a Galerkin construction, i.e., $P_N = R_N^T$, the coarse grid operator $K_N = R_N K P_N$ can be obtained using the computation rules in \[lem:calc\_rules\_mult\_op\]. Inversion of this auxiliary space operator is then approximated by Gauss-Seidel. Thus, the auxiliary space correction with a single smoothing step on the auxiliary space is given by $$\begin{aligned} G_N = (I - P_N S_N^\dagger R_N K) : { \mathbcal{L}}( T_{\mathcal{A}}^{\mathfrak{e}}) \rightarrow { \mathbcal{L}}(T_{\mathcal{A}}^{\mathfrak{e}}),\end{aligned}$$ where $S_N: { \mathbcal{L}}(T_{\mathcal{A}}^{(0)}) \rightarrow { \mathbcal{L}}(T_{\mathcal{A}}^{(0)})$ consists of the scalar multipliers $$\begin{aligned} m_{S_N}^{(y)} = \begin{cases} m_{K_N}^{(y)} & \text{if } y \leq 0, \\ 0 & \text {else.}\end{cases} \end{aligned}$$ With this definition of the auxiliary space correction, the half-hybrid smoother consists of the following steps. 1. Smooth on $Kx=b$: $x \leftarrow (I+S_E^\dagger K)x + S_E^\dagger b$, 2. Restrict the residual: $r_N \leftarrow R_N(b-Kx)$, 3. Smooth on $K_N x_N= r_N$ with zero initial guess: $x_N \leftarrow S_N^\dagger r_N$, 4. Prolongate to the primal crystal $x \leftarrow x + P_N x_N$. The smoother is analyzed analogously to \[subsec:multicolor\_graphene\] with \[alg:compute\_spectral\_radii\] by computing the spectral radii via $$\text{\textproc{ComputeSpectrum}}(g,I,K,S_E,R_N,S_N),$$ where $g$ denotes the composition of the error propagators $$(I,K,S_E,R_N,S_N) \xmapsto{\enskip g\enskip } G_N G_E =:G.$$ In \[fig:LFA\_curlcurl\] a) a contour plot of $\rho(G_k)$ with $\sigma_h = 0.01$, with respect to $k\in {\mathcal{A}}^{-T}[ -\frac{1}{4},\frac{3}{4})$ is given. It corresponds to the result given in [@BoonLentVand2008 Figure 6.2, right]. #### Coarse grid correction In [@BoonLentVand2008] a Galerkin coarse grid correction is used corresponding to the error propagator $$\begin{aligned} E = (I- P(K_c)^\dagger R K) \end{aligned}$$ with $K_c = RKP$ and $P=R^T$. In here, the coarse crystal is ${ \mathbb{L}}^{2{\mathfrak{e}}}(2{\mathcal{A}})$ and the original crystal ${ \mathbb{L}}^{{\mathfrak{e}}}({\mathcal{A}})$ with respect to the lattice $2{\mathcal{A}}$ is given by ${ \mathbb{L}}^{{\mathfrak{f}}}(2{\mathcal{A}})$ with $$\begin{aligned} {\mathfrak{f}}= ({\mathfrak{e}}, {\mathfrak{e}}+ {\mathcal{a}}_1, {\mathfrak{e}}+ {\mathcal{a}}_2, {\mathfrak{e}}+ {\mathcal{a}}_1 + {\mathcal{a}}_2). \end{aligned}$$ Then, according to [@BoonLentVand2008] the restriction operator is given as $R : { \mathbcal{L}}(T^{\mathfrak{f}}_{2{\mathcal{A}}}) \rightarrow { \mathbcal{L}}(T^{2{\mathfrak{s}}}_{2{\mathcal{A}}})$ with the multipliers: Here, each coarse horizontal and vertical edge is connected to its six nearest edges of the same type (cf. \[fig:curlcurl\_crystal\] b)). The prolongation and coarse grid operator $P$ and $K_c$ can be obtained via \[lem:calc\_rules\_mult\_op\]. The spectral radii of the two-grid method can then be obtained via $$\begin{aligned} \text{\textproc{ComputeSpectrum}}(f,I,K,S_E,R_N,S_N,R) \end{aligned}$$ where $f$ denotes the composition of the two-grid error propagator $$\begin{aligned} (I,K,S_E,R_N,S_N,R) &\xmapsto{\enskip f\enskip } G E G =: M. \end{aligned}$$ b) shows the spectral radii, $\sup_k \rho(M_k)$, of the two-grid error propagator as a function of $\sigma_h$. We used exactly the same orderings in pre- and post-smoothing, as described in [@BoonLentVand2008]. Furthermore, we double-checked the results with convergence tests similar to \[tbl:graphene\_twogrid\_results\]. However, the obtained results show major differences to the one in [@BoonLentVand2008 Figure 8.1, left], which leads us to believe that our assumptions on the lexicographic orderings employed in pre- or post-smoothing in [@BoonLentVand2008] are wrong, as these have a large impact on the convergence rates. Conclusion ========== In this paper we present an LFA framework that is applicable to arbitrary crystal structures, which are encountered in many applications, e.g., systems of partial differential equations, block smoothers or tight-binding formulations. This was achieved by introducing a rigorous notion of crystal structures and translationally invariant operators that manipulate value distributions on crystal structures. Based on these structures we introduced a complete framework to modify and transform these operators with respect to different crystal representations by using normal forms of integer linear algebra. This allowed us to automate the LFA in both position space, i.e., in terms of multiplication operators/stencils, and frequency space, i.e., in terms of canonical basis functions and samplings. In two examples we showed that the approach can be used for complicated operators, i.e., hexagonal grids, overlapping block smoothers and hybrid smoothers, without requiring insight into the frequency space back-end of LFA. An explicit mixing calculation is no longer needed, and the user only has to provide any representation of the individual operators. The transformation of the individual operators to a compatible representation and the subsequent frequency analysis is then carried out automatically. Even though we have limited ourselves in the examples in this paper to $2$-dimensional problems our automated LFA can be applied to operators in higher dimension, where the difference is a larger set of primitive vectors defining the underlying lattice. The automation presented in this paper does have some limitation. Each individual operator in the analysis is only allowed to change each value of the value distribution at most once. This limitation solely restricts the class of smoothers that can be analyzed with this approach. Any sequential, i.e., lexicographic, smoother with overlapping update regions changes values in the overlap multiple times in one application. This cannot be easily translated to a corresponding local multiplication operator, but it can be dealt with in frequency space (cf. [@MacLOost2011; @Molenaar1991; @Sivaloganathan:1991:ULM]). This particular treatment of sequential overlap is momentarily not covered in our framework. Note, that the mere presence of overlap is not the problem here. By introducing a coloring, such that the complete sweep can be split into a sequence of updates where each one of them only changes values at most once, automated LFA can be applied (cf. \[subsec:multicolor\_graphene\]). Due to the fact that a coloring in overlapping approaches also favors parallelism over their sequential counterparts, we feel that this limitation is relatively minor when targeting actual applications. An open-source implementation of the automated LFA framework [@aLFA_NK] is freely available on GitLab[^10]. Jupyter Notebooks for all examples from \[sec:example,subsec:multicolor\_graphene,subsec:curlcurl\] are included in this software package as well. Rules of computation {#sec:rules_of_comp} ==================== Calculus of multiplication operators plays a key-role in local Fourier analysis. In this section we list all elementary operations, such as addition and multiplication. Proofs for these rules can be obtained by straightforward calculation. \[lem:calc\_rules\_mult\_op\] Let two multiplication operators be given by $$\begin{aligned} \begin{array}{rcl} L:{ \mathbcal{L}}(T_{{\mathcal{A}},{\mathcal{Z}}}^{{\mathfrak{s}}}) \rightarrow { \mathbcal{L}}(T_{{\mathcal{A}},{\mathcal{Z}}}^{{\mathfrak{t}}}), & (Lf)(x) = \displaystyle\sum_{y\in T_{{\mathcal{A}},{\mathcal{Z}}}} m_L^{(y)} f(x+y),& m_L^{(y)} \in{ \mathbb{C}}^{|{{\mathfrak{t}}}|\times |{{\mathfrak{s}}}|},\\ G:{ \mathbcal{L}}(T_{{\mathcal{A}},{\mathcal{Z}}}^{{\mathfrak{u}}})\rightarrow { \mathbcal{L}}(T_{{\mathcal{A}},{\mathcal{Z}}}^{{\mathfrak{v}}}),& (Gf)(x) = \displaystyle\sum_{y\in T_{{\mathcal{A}},{\mathcal{Z}}}} m_G^{(y)} f(x+y), & m_G^{(y)} \in{ \mathbb{C}}^{|{{\mathfrak{v}}}|\times |{{\mathfrak{u}}}|}. \end{array} \end{aligned}$$ Then the following operators are multiplication operators as well: 1. If ${\mathfrak{s}}= {\mathfrak{u}}$ and ${\mathfrak{t}}={\mathfrak{v}}$, then $ L+G:{ \mathbcal{L}}(T_{{ \mathcal{A}},{ \mathcal{Z}}}^{\mathfrak{s}}) \rightarrow { \mathbcal{L}}(T_{{ \mathcal{A}},{ \mathcal{Z}}}^{\mathfrak{t}}) \text{ with } m_{L+G}^{(y)} = m_L^{(y)}+m_G^{(y)}.$ 2. If ${\mathfrak{v}}= {\mathfrak{s}}$, then $LG:{ \mathbcal{L}}(T_{{ \mathcal{A}},{ \mathcal{Z}}}^{{\mathfrak{u}}}) \rightarrow { \mathbcal{L}}(T_{{ \mathcal{A}},{ \mathcal{Z}}}^{{{\mathfrak{t}}}}) \text{ with } m_{LG}^{(z)} = \sum\limits_{y+w=z} m_L^{(y)} \cdot m_G^{(w)}.$ 3. The adjoint is given by $L^*:{ \mathbcal{L}}(T_{{ \mathcal{A}},{ \mathcal{Z}}}^{\mathfrak{t}}) \rightarrow { \mathbcal{L}}(T_{{ \mathcal{A}},{ \mathcal{Z}}}^{\mathfrak{s}}) \text{ with }m_{L^*}^{(y)} = (m_{L}^{(-y)})^*.$ \[thm:Eigenvalues\_Crystal\_Mult\_Op\] Let two multiplication operators be given by $$\begin{aligned} \begin{array}{rcl} L:{ \mathbcal{L}}(T_{{\mathcal{A}},{\mathcal{Z}}}^{{\mathfrak{s}}}) \rightarrow { \mathbcal{L}}(T_{{\mathcal{A}},{\mathcal{Z}}}^{{\mathfrak{t}}}), & (Lf)(x) = \displaystyle\sum_{y\in T_{{\mathcal{A}},{\mathcal{Z}}}} m_L^{(y)} f(x+y),& m_L^{(y)} \in{ \mathbb{C}}^{|{{\mathfrak{t}}}|\times |{{\mathfrak{s}}}|},\\ G:{ \mathbcal{L}}(T_{{\mathcal{A}},{\mathcal{Z}}}^{{\mathfrak{u}}})\rightarrow { \mathbcal{L}}(T_{{\mathcal{A}},{\mathcal{Z}}}^{{\mathfrak{v}}}),& (Gf)(x) = \displaystyle\sum_{y\in T_{{\mathcal{A}},{\mathcal{Z}}}} m_G^{(y)} f(x+y), & m_G^{(y)} \in{ \mathbb{C}}^{|{{\mathfrak{v}}}|\times |{{\mathfrak{u}}}|} \end{array} \end{aligned}$$ with corresponding symbols $L_k$ and $G_k$. Then we have the following statements. 1. Assuming that ${{\mathfrak{s}}} = {{\mathfrak{u}}}$ and ${{\mathfrak{t}}} = {{\mathfrak{v}}}$, the symbols of $L+G$ are given by $L_k+ G_k$. 2. Assuming that ${{\mathfrak{v}}} = {{\mathfrak{s}}}$, the symbols of $L\cdot G$ are given by $L_k\cdot G_k$. 3. The symbols of $L^*$ are given by $L_k^*$. 4. The symbols $(L^\dagger)_k$ are given by $(L_k)^\dagger$. Algorithms {#sec:algorithms} ========== [ ]{} [ ]{} ($\triangleright$ See \[def:crystal\_op\_and\_normalform\])[$G = \func{Normalize}{L}$]{}[ ${\mathfrak{u}}_j = {\mathfrak{s}}_j - {\mathcal{A}}\lfloor {\mathcal{A}}^{-1}{\mathfrak{s}}_j \rfloor $ for all $j=1,\ldots,|{\mathfrak{s}}|$ ${\mathfrak{v}}_j = {\mathfrak{t}}_j - {\mathcal{A}}\lfloor {\mathcal{A}}^{-1}{\mathfrak{t}}_j \rfloor $ for all $j=1,\ldots,|{\mathfrak{t}}|$ Sort ${\mathfrak{u}}$ and ${\mathfrak{v}}$ lexicographically $G = \func{ChangeStructureElement}{L,{\mathfrak{u}},{\mathfrak{v}}}$ ]{} [ ]{} ($\triangleright$ See \[cor:quotient\_latticepointlist\])[${\mathfrak{s}}= \func{ElementsInQuotientSpace}{{\mathcal{A}},{\mathcal{C}}}$]{}[ $H = $ Hermite normal form of ${\mathcal{A}}^{-1}{\mathcal{C}}$ $m = \prod_{i=1}^n h_{i,i}$ ${\mathfrak{s}}_i = 0$ for all $i=0,\ldots,m$ ]{} [ ]{} ($\triangleright$ See \[thm:operator\_sublattice\_coarsening\])[$G = \func{LatticeCoarsening}{L,{\mathcal{C}}}$]{}[ ${\mathfrak{e}}= \func{ElementsInQuotientSpace}{{\mathcal{A}},{\mathcal{C}}}$ ${\mathfrak{u}}= ({\mathfrak{e}}_1 + {\mathfrak{s}},\ldots, {\mathfrak{e}}_{|{\mathfrak{e}}|} + {\mathfrak{s}}), {\mathfrak{v}}= ({\mathfrak{e}}_1 + {\mathfrak{t}},\ldots, {\mathfrak{e}}_{|{\mathfrak{e}}|} + {\mathfrak{t}})$ $(m_{{G}}^{({y})})=0 \in { \mathbb{C}}^{|{\mathfrak{v}}|\times |{\mathfrak{u}}|}$ for all ${y}\in{ \mathbb{L}}({\mathcal{C}})$ ]{} [ ]{} [ ]{} ($\triangleright$ See \[thm:shifted\_structure\_element,thm:permuted\_structure\_element\])[$\hat{L} = \func{ChangeStructureElement}{L,{{\mathfrak{u}}},{{\mathfrak{v}}}}$]{}[ $m_\pi =0 \in \{0,1\}^{|{\mathfrak{s}}|\times|{\mathfrak{s}}| }$, $m_\sigma = 0 \in \{0,1\}^{|{\mathfrak{t}}|\times|{\mathfrak{t}}| }$ ($\triangleright$ Compute changes in ${\mathfrak{d}}$)[$(i,j) \in \{1,\ldots,|{\mathfrak{s}}|\}^2$]{}[ ]{} ($\triangleright$ Compute changes in ${\mathfrak{c}}$)[$(i,j) \in \{1,\ldots,|{\mathfrak{t}}|\}^2$]{}[ ]{} $(m_{\hat{L}}^{({y})})=0 \in { \mathbb{C}}^{|{\mathfrak{t}}|\times |{\mathfrak{s}}|}$ for all ${y}\in{ \mathbb{L}}({\mathcal{A}})$ $m_{\hat{L}}^{(y)} = m_\sigma \cdot m_{\hat{L}}^{(y)}\cdot m_\pi^{-1}$ for all ${y}\in{ \mathbb{L}}({\mathcal{A}})$ ]{} [ ]{} ($\triangleright$ See \[thm:lcm\_mat\])[${\mathcal{C}}= \func{LeastCommonMultiple}{{\mathcal{A}},{\mathcal{B}}}$]{}[ Find an integer $r$, s.t. $M=r{\mathcal{A}}^{-1}{\mathcal{B}}$ is integral Compute Smith normal form $S=V^{-1}MT^{-1}$ of $M$ $(N_{{\mathcal{B}}})_{i,i} = r \cdot \operatorname{gcd}(r,s_{i})^{-1}$, ${\mathcal{C}}= {\mathcal{B}}T^{-1} N_{{\mathcal{B}}}$ ]{} [^1]: School of Mathematics and Natural Sciences, University of Wuppertal, 42097 Germany, `{kkahl,kintscher}@math.uni-wuppertal.de` [^2]: This work was partially funded by Deutsche Forschungsgemeinschaft (DFG) Transregional Collaborative Research Centre 55 (SFB/TRR55) [^3]: [`github.com/hrittich/lfa-lab`](https://github.com/hrittich/lfa-lab) [^4]: [`gitlab.com/NilsKintscher/alfa`](https://gitlab.com/NilsKintscher/alfa) [^5]: All proofs of \[lem:sublattice,thm:integer\_normalforms,thm:lattice\_equiv\_unimod\] can be found in [@Schrijver:1986:TLI:17634]. [^6]: Bounded by the number of lattice points on the (arbitrarily large) torus. [^7]: Recall that a unique list of representatives $T_{{\mathcal{A}},{\mathcal{C}}} = \{{\mathfrak{t}}_1,\ldots,{\mathfrak{t}}_p\}$ can be found via \[cor:quotient\_latticepointlist\]. [^8]: In case ${\mathfrak{d}}_i={\mathfrak{d}}_j$ or ${\mathfrak{c}}_i = {\mathfrak{c}}_j$ for any $i\neq j$ a consistent ordering of $i,j$ has to be defined a priori. [^9]: The left-hand side of $[({\mathbb{T}}f)(x)]_i=f_i(x-y_i)$ corresponds to the value at position $x + {\hat{{\mathfrak{t}}}}_i = (x-y_i)+{\mathfrak{t}}_i$ which coincides with the position of the value of the right-hand side. [^10]: [`gitlab.com/NilsKintscher/alfa`](https://gitlab.com/NilsKintscher/alfa)

--- abstract: 'We measure the correlation between the arrival directions of the highest energy cosmic rays detected by the Pierre Auger Observatory with the position of the galaxies in the  Parkes All Sky Survey (HIPASS) catalogue, weighted for their  flux and Auger exposure. The use of this absorption–free catalogue, complete also along the galactic plane, allows us to use all the Auger events. The correlation is significant, being 86.2% for the entire sample of  galaxies, and becoming 99% when considering the richest galaxies in HI content, or 98% with those lying between 40–55 Mpc. We interpret this result as the evidence that spiral galaxies are the hosts of the producers of UHECR and we briefly discuss classical (i.e energetic and distant) long Gamma Ray Burst (GRBs), short GRBs, as well as newly born or late flaring magnetars as possible sources of the Auger events. With the caveat that these events are still very few, and that the theoretical uncertainties are conspicuous, we found that newly born magnetars are the best candidates. If so, they could also be associated with sub–energetic, spectrally soft, nearby, long GRBs. We finally discuss why there is a clustering of Auger events in the direction on the radio–galaxy Cen A and an absence of events in the direction of the radio–galaxy M87.' author: - | G. Ghisellini,$^1$[^1] G. Ghirlanda,$^1$ F. Tavecchio,$^{1}$, F. Fraternali$^2$ and G. Pareschi$^1$\ $^1$INAF – Osservatorio Astronomico di Brera, Via Bianchi 46 Merate, Italy\ $^2$Dept. of Astronomy, University of Bologna, via Ranzani 1, 40127 Bologna, Italy title: 'Ultra–High Energy Cosmic Rays, Spiral Galaxies and Magnetars' --- cosmic rays – galaxies: gamma–rays: bursts – galaxies: statistics — radio lines: galaxies Introduction ============ The origin of ultra–high energy cosmic rays (UHECR), exceeding 10 EeV (1 EeV=$10^{18}$ eV) has been a mystery for decades, but the recent findings of the large area detectors, such as AGASA (Ohoka et al. 1997), HIRes (Abu–Zayyad et al. 2000), and especially the Pierre Auger Southern Observatory (Abraham et al. 2004), began to disclose crucial clues about the association of the highest energy events with cosmic sources. The Auger collaboration (Abraham et al. 2007) found a positive correlation between the arrival directions of UHECR with energies grater than 57 EeV and nearby AGNs (in the optical catalogue of Veron–Cetty & Veron 2006). Although this result has not been confirmed by HIRes (Abbasi et al. 2008) and it has been criticised by Gorbunov et al. (2008), it received an important confirmation by George et al. (2008), who considered a complete sample of nearby hard X–ray emitting AGNs detected by the BAT instrument onboard [*Swift*]{}. This sample is much less affected by absorption than any optical sample although, to identify as such an AGN, one relies on optical identification. Moreover, George et al. (2008) found a correlation not simply with the AGN locations, but by weighting them for the X–ray flux and the Auger exposure. This association, if real, is surprising, since the large majority of the correlating AGNs are radio–quiet, a class of objects not showing, in their electromagnetic spectrum, any sign of non–thermal high energy emission: no radio–quiet AGN was detected by the EGRET instrument onboard the [*Compton Gamma Ray Observatory*]{} (Hartman et al. 1999). Therefore they must accelerate particles (protons, nuclei, and presumably their accompanying electrons) to ultra–high energies without any noticeable radiative emission from these very same particles. Radio–loud AGNs, instead, together with Gamma Ray Bursts (of both the long and short category) do show high energy non–thermal emission, and have been considered for a long time better candidates as UHECR sources (Vietri 1995; Waxman 1995; Milgrom & Usov 1995; Wang, Razzaque & Mészáros 2008; Murase et al. 2008; Torres & Anchordoqui 2004 and Dermer 2007 for reviews, and Nagar & Matulich 2008 and Moskalenko et al. 2008 for the possible association of the AUGER events with radio–loud AGNs). Note also that some short GRBs could be due to the giant flares of highly magnetised neutron stars (“magnetars”, as the 27 Dec 2004 event from 1806–20; Borkowski et al. 2004; Hurley et al. 2005; Terasawa et al. 2005), and that, at birth, a fastly spinning magnetar can be much more energetic than when, later, it produces giant flares (Arons 2003). The possibility that GRBs and magnetars are the sites of production of UHECR would directly imply the direct association of these events with (normal) galaxies. In this case the found association of UHECR with nearby AGNs might then be due to the fact that local AGNs just trace the distribution of galaxies. The aim of the present paper is to test this possibility directly correlating the locations of the ultra–high energy Auger events with a well defined, complete, and possibly absorption–free sample of galaxies. For this purpose we use the sample of  emitting galaxies, compiled using the Parkes 64–m radio telescope (Barnes et al. 2001; Staveley–Smith et al. 1996), which is conveniently located in the south hemisphere, as the Auger observatory. The entire sample covers the portion of the sky visible by Auger, making it possible to use, for the correlation analysis, all the 27 UHECR events with energies larger than 57 EeV detected by Auger, without excluding the galactic plane, as is instead necessary when dealing with AGNs or optically selected galaxies. Note that the presence of neutral hydrogen strongly favours spirals (or, more generally, gas–rich galaxies) with respect to elliptical galaxies. We use a cosmology with $\rm{h}_0=\Omega_\Lambda=0.7$ and $\Omega_{\rm M}=0.3$. Data ==== UHECR events ------------ The Auger Observatory (Abraham et al. 2004, 2008), operating in Argentina since 2004, is located at latitude $-35.2\degr$ and it has a maximum zenith angle acceptance of $60\degr$. The relative exposure is independent of the energy of the detected events and it is a nearly uniform in right ascension. The dependence on declination is given by Sommers (2001). The Observatory can detect Cosmic Rays from sources with declination $\delta<24.8^\circ$. The available Auger list of UHECR events (Abraham et al. 2008) comprises 27 events with energies in excess of $5.7\times{}10^{19}\ev$ from an integrated exposure of $9000\crexp$. The event arrival directions are determined with an angular resolution of better than $1\degr$. However, magnetic fields of unknown strength will deflect charged particles on their trajectories through space. The advantage of studying the highest energy events is that this deflection is minimised, but it can still be up to $\sim{}10\degr$ in the Galactic field. The 27 UHECR detected by Auger are distributed in the range $\delta\in[-61,9.6]$ (or at galactic latitudes $b\in[-78.6,54.1]$ – open circles in Fig. \[HIn\] and Fig. \[HIflux\]). HIPASS catalogue ---------------- We compare the arrival directions of Auger UHECRs with the locations of sources of the  Parkes All–Sky Survey (HIPASS – Meyer et al. 2004). This is a blind survey of sources in  covering the full southern sky at $\delta<25^\circ$ which is the same sky area accessed by the Auger Observatory. The full catalogue is composed by a list of 4315 sources at $\delta<2^\circ$ (HICAT – Meyer et al. 2004; Zwaan et al., 2004) and by its extension to the northern sky up to $\delta=25^\circ$ (NHICAT – Wong et al. 2006) which includes 1002 sources. All sources are shown in Fig. \[HIn\] with the 27 UHECR detected by Auger. The HICAT and NHICAT have different level of completeness. To have a catalogue complete in flux at the 95% level, we cut the HICAT at $S_{\rm int}>7.4$ Jy km s$^{-1}$ and the NHICAT at $S_{\rm int}>15$ Jy km s$^{-1}$ as discussed in Zwaan et al. (2004) and Wong et al. (2006). $S_{\rm int}$ represents the total  line flux. For the purposes of this paper we also considered the sources within 100 Mpc which is the maximum distance at which UHECRs of $E>57$ EeV can survive the GZK suppression effect (see e.g. Harari et al. 2006). We call this sample 95HIPASS: it contains 2414 sources from the HICAT and 290 sources from the NHICAT for a total of 2704 sources and covers the entire sky at $\delta<25^\circ$. We will also consider the southern sky sample alone which is more complete and can be cut at 99% completeness for $S_{\rm int}>9.4$ Jy km s$^{-1}$ (also by considering sources at $<$100 Mpc). This sample contains 1946 sources and is called 99HICAT. -0.5cm -0.5cm Analysis ======== To quantify the possible correlation between UHECR Auger events and the distribution of  local galaxies we use the method adopted by George et al. (2008). In order to quantify the probability that two sets of sources are drawn from the same parent population of objects we perform the two-dimensional generalisation of the Kolmogorov–Smirnov (K–S) test (Peacock 1983) proposed by Fasano & Franceschini (1987). In our case the test is used to compare two data samples, i.e. the UHECR and the  galaxies. This test can then measure either if UHECRs have a galaxy counterpart, and, viceversa, if a concentration of galaxies has an UHECR counterpart. The test relies on the statistic $D$, also used for the unidimensional K–S test, which represents the maximum difference between the cumulative distributions of the two data samples. For each UHECR data point $j$ we compute a set of four numbers $d_{j,i}$ ($i$=\[1,4\]) defined as the difference of the relative fraction of UHECR and  galaxies found in the four natural quadrants defined around point $j$. Hence, $D=\max(d_{j,i})$ for all the data points considered. Defining $Z_{\rm {n}}=D\sqrt{n}$, the strength of the correlation between two catalogues is the integral probability distribution $P(D\sqrt{n}>\rm{observed})$, where $n=N_1N_2/(N_1+N_2)$, and $N_1$ and $N_2$ are the number of data points in the two sets. This measurement can be used to determine the similarity of sets of positions on the sky. The probability can be computed analytically for large data sets ($n>$80 – Fasano & Franceschini 1987). In our case, having only 27 UHECR, we have to rely on Monte Carlo simulations. We generate a large set of random UHECR events according to the relative Auger exposure. For each synthetic UHECR sample we compute $Z_{\rm {n}}$ by correlating it with the catalogue of  galaxies. The probability of the observed $Z_{\rm {n}}$ is given by the number of times we find a value of $Z_{\rm {n}}$ larger than the observed one. This is the probability that the correlation between the (real) UHECR sample and the  galaxies is not by chance. Large (low) values of the probability indicate a good (poor) correlation between the Auger UHECRs and the given  galaxy sample. As noted by George et al. (2008), the two–dimensional K–S test can be performed with the number of data points or with the flux of the sources in the comparison sample. In our case $D$ represents the maximum difference between the number of UHECRs and that of the sum of the galaxies weighted for their flux and for the the relative Auger exposure. The advantage of using the weighted flux of the sources is that it accounts for their distance. George et al. (2008) found that the UHECRs are more correlated with the weighted flux of [*Swift*]{} AGN than with with their position. In Fig. \[HIflux\] we show the map of the flux of the HIPASS catalogue weighted for the Auger relative exposure. -0.5 true cm Results ======= We found that with the 95HIPASS catalogue (2704  sources complete in flux at 95%) the probability that UHECRs are correlated with  galaxies is 71.6% by using the weighted flux of the  sources. Considering the more complete 99HICAT (1946  sources complete in flux at 99%) distributed within 100 Mpc and the 25 UHECRs distributed in the same sky region we find a larger flux–weighted probability of 87.8%. This probability is slightly smaller than found with local AGN by George et al. (2008). However, having a large sample of  galaxies we can study if the correlation probability changes by considering different sub–samples of galaxies selected according to their distance or luminosity. We have considered 4 bins of distance with an equal number of sources ($\sim$500) per bin. The correlation probability shows a maximum of 95% (97.8% for the 99HICAT) for sources distributed between 37.8 and 55 Mpc. We show these results in Fig. \[bella\] (open circles and stars in the bottom panel). Similarly we defined four equally populated luminosity bins, or, equivalently, four bins of HI mass content, since we can use $M/M_\odot = 2.36\times 10^5 D_{\rm Mpc}^2 S_{\rm int}$ to estimate the  mass (here $S_{\rm int}$ is measured in \[Jy km/s\]). We find that the probability (left panel in Fig. \[bella\]) is maximised by the most  luminous or massive (in ) sources (98% and 99% for the 95HIPASS and 99HICAT sample, respectively, for $M> 1.1\times 10^{10}\, M_{\odot}$). Selecting those  galaxies located within two $20^\circ \times 20^\circ$ boxes centred on the radio–galaxies Cen A and M87 (green boxes in Fig. \[HIn\]), we can show where they lie in the luminosity–distance plane in Fig. \[bella\] (orange and green dots, respectively). While there is no clustering of points at the distances of Cen A and Virgo, we can see that  galaxies in the direction of Cen A do cluster at distances of 40–50 Mpc, where the Centaurus cluster is. This could explain why some UHECR events [*appear*]{} to be associated with the radio–galaxy Cen A, and none with M87: beyond Cen A there is the Centaurus cluster, richer of  emitting spirals than the Virgo cluster. The ratio of the integrated HI fluxes from the two $20^\circ\times20^\circ$ boxes (Virgo/Cen A) is 5.9. To this, we have to multiply by another factor 3 for the lower Auger exposure in the direction of Virgo. The sample has too few galaxies beyond 100 Mpc to test the GZK effect (that would be revealed by finding no correlation for these galaxies). Discussion ========== The 27 Auger events above 57 EeV, with a total exposure of $9000\crexp$ correspond to an integrated flux, in CGS units: $$F_{\rm A}(E>57\, {\rm EeV})\, \sim \, 1.1 \times 10^{-11} \,\,\, {\rm erg\,\, cm^{-2}\, s^{-1}} \label{fa}$$ This flux is smaller than the electromagnetic flux that we receive from nearby radio–quiet AGNs in hard X–rays (see e.g. Tueller et al. 2008). We now compare this flux with the expected flux of other candidate sources. We will consider flaring or bursting sources, that is impulsive events, but the spreading of the arrival times of UHECRs from a source located at a distance $D$, $\Delta t \sim D \theta^2/2 c$, due to even tiny magnetic deflections, ensure that we can treat all candidate sources as continuous. We will estimate the predicted flux in two different ways. First, assume that a class of sources is characterised by a pulse of emission of UHECRs, of average total energy $<E>$. Assume also that these events occur at a rate $R$ per galaxy, per year, and consider those events occurring within the GZK radius $D_c$. We have: $$F \, = \, <E> {R \over 3.15 \times 10^7} \, { N_{\rm g}(D<D_c) \over 4\pi (a D_c)^2}$$ where $3.15\times 10^7$ is the number of seconds in one year and $N_{\rm g}(D<D_c)$ is the number of galaxies within $D_c$ of $L_*$ luminosity. The average distance of the sources is $a D_c$ ($a=3/4$ for sources homogeneously distributed). Setting the mean local galaxy density $n_{\rm g}= N_{\rm g}/(4\pi D_c^3/3)= 10^{-2} n_{\rm g, -2}$ Mpc$^{-3}$ we have: $$F \sim 1.2 \times 10^{-57} <E> R\, n_{\rm g, -2} {D_{c, 100} \over a^2} \,\,\, {\rm erg\, cm^{-2}\, s^{-1}} \label{rate}$$ where $D_c =100 D_{c,100}$ Mpc. The second estimate on the predicted UHECR flux uses the electromagnetic flux as a proxy. Assume that we detect, for a typical member of a class of sources, an average fluence $<{\cal F}>$, and that there are $N$ events per year. If a fraction $\eta$ of these events comes from sources within $D_c$, we have $$F\, = \, {\eta <{\cal F}> N \over 3.15 \times 10^7} \label{m2}$$ This estimate is more appropriate when dealing with sources, such as long and short GRBs, whose fluences and occurrences are known, while Eq. \[rate\] is more appropriate when dealing with possible sources of unknown electromagnetic output, but predicted energetics and rates, such as newborn magnetars (Arons 2003) or giant flares from old magnetars. Let us consider the above classes of sources in turn, starting from short GRBs. In the BATSE catalog (cossc.gsfc.nasa.gov/docs/cgro/batse/BATSE\_Ctlg/flux.html) we have 490 short GRBs of total fluence $5.5\times 10^{-4}$ erg cm$^{-2}$ in 9 years of operation. Tanvir et al. (2005) correlated these short GRBs with local optically selected galaxies finding that a fraction between 5 and 25% of BATSE short GRBs might be nearby, i.e. at $z < 0.025$, corresponding to 109 Mpc. Considering that BATSE saw half of the sky and setting $\eta=0.1$ we have an average flux of $3.9\times 10^{-13}$ erg cm$^{-2}$ s$^{-1}$. Then, if the UHECR flux is similar to the electromagnetic one, short GRBs do not match the required flux. Classical long GRBs (namely, energetic GRBs at $z\gsim 1$) in the BATSE sample have a total fluence of 0.024 erg cm$^{-2}$ (for the listed 1490 long GRBs in the BATSE catalog), corresponding to an average (all sky) flux of $1.7\times 10^{-10}$ erg cm$^{-2}$ s$^{-1}$, larger than the one given by Eq. \[fa\]. However, for long BATSE GRBs, $\eta$ must be much smaller than 0.1, as directly suggested by the paucity of nearby events, and by the lack of correlation with nearby galaxies and clusters (Ghirlanda et al. 2006). While we cannot dismiss them as sources of UHECRs, it seems likely that classical BATSE bursts are too distant (but see below). Consider now giant flares from relatively “old” magnetars. The giant flare from SGR 1806–20 of Dec 27, 2004 emitted an energy $E\sim 10^{46}$ erg in less than a second. The radio afterglow convincingly demonstrated the formation of a (at least mildly) relativistic fireball. With the current hard X–ray instruments, such flares can be detected up to $\sim$30–40 Mpc (Hurley et al. 2005). Eq. \[rate\], with $D_{c,100}=0.3$, and $<E>=10^{46}$erg, would require $R\sim 1$ event per galaxy per year, while an approximate limit to the rate is $R< 1/30$ yr$^{-1}$ (see e.g. Lazzati et al. 2005). Finally, consider fastly spinning newly born magnetars, whose rotational energy can exceed $10^{52}$ erg, with a rate of $R=10^{-4}$ events per galaxy per year (Arons 2003). With the estimated galaxy density ($n_{\rm g, -2}\sim 0.7$ with $L\sim L_*$; Blanton et al. 2001) there should be 1 event per year within 100 Mpc. If each magnetar produces $10^{50}$ erg in UHECRs, then this class of sources can be the progenitor of the Auger events (Eq. \[rate\]). This is independent of collimation, since the reduced rate of events pointing at us is compensated by an increase of the apparent energetics. But if an equal amount of energy is released in electromagnetic form, at energies detectable by BATSE, then they should be a significant fraction of all BATSE GRBs. Since the birth of a magnetar should be accompanied by a supernova, [*these events should be associated with long, rather than short GRBs,*]{} for which no associated supernova has been seen. If the radiative output is isotropic, they will all be nearby, sub–energetic, GRBs. The required fluence of these sub–energetic nearby long GRBs, to match the UHECRs flux, should be $$<{\cal F}>\, \sim\, 3.15\times 10^{-4} \epsilon_{\rm CR} \, {F_{\rm A, -11} \over \eta N} \,\,\, {\rm erg\,\, cm^{-2}}$$ where $\epsilon_{\rm CR}$ is the ratio of the emitted energy in radiation and UHECR. If $\eta\sim \epsilon_{\rm CR}\sim 1$, these events constitute a sizeable fraction of the total fluence of all long BATSE GRBs in one year (which is ${\cal F} \sim 0.024/9\sim 2.7\times 10^{-3}$ erg cm$^{-2}$). Since we know that the large majority of long GRBs are not nearby, newly born magnetars should not constitute conspicuous events in hard X–rays. Their fluence must be mostly emitted in another energy range. GRB 060218 (Campana et al. 2006) with an energy of a few $\times 10^{49}$ erg, at a distance of 145 Mpc, could be one of these events, and Soderberg et al. (2006) and Toma et al. (2007) already suggested that this GRB was powered by a newly born magnetar. The spectrum of its prompt emission peaked at $\sim$5 keV, i.e. its fluence in relatively soft X–rays exceeded the 15–150 keV fluence. It was also very long, slowly rising, and would not have been detected by BATSE. Soderberg et al. (2006) pointed out that these sub–energetic long GRBs should not be strongly beamed (not to exceed the rate of SN Ib,c), and should occur at a rate of $230^{+490}_{190}$ Gpc$^{-3}$ yr$^{-1}$, corresponding to $R \approx 10^{-5}$ events per $L_*$ galaxy per year, about ten times larger than for classical long GRBs whose radiation is collimated into 1% of the sky. According to this rate, Eq. \[rate\] would then demand $<E>\sim 6\times 10^{50}$ erg in UHECRs to match the observed flux. Conclusion ========== We have correlated the cosmic rays with $E>57$ EeV detected by the Auger Observatory with a complete, absorption–free sample of  selected galaxies. We found a significant correlation when correlating the  [*flux*]{} of galaxies of our sample. When considering the largest 95HIPASS catalogue and the 27 UHECRs we find a weak correlation (probability of 72%), while a larger significance (87.8%) is reached if we consider the most complete 99HICAT sample of galaxies (though with 25 UHECRs). These probabilities are maximised by cutting the  sample in distance or luminosity bins: it becomes 99% when considering the 500 most luminous (or most  massive) galaxies (1/4 of the sample), and 98% when considering the 500 galaxies lying between 38 and 54 Mpc, where the Centaurus cluster of galaxies is. Thus there is the possibility that the UHECRs coming from the direction of Cen A are instead coming from the more distant Centaurus cluster. Galaxies of this cluster are richer in  than Virgo galaxies, explaining why there is no UHECR event from the direction of Virgo. This sample is formed by  emitting galaxies, therefore it is biased against ellipticals. The found correlation with these galaxies, per se, is not disproving the found correlation with AGNs (Abraham et al. 2007, 2008; George et al. 2008), since they also trace the local distribution of matter, as spiral galaxies do. On the other hand, it opens up the possibility, on equal foot, that UHECRs are produced by GRBs or newly born magnetars (see also Singh et al. 2004 who used AGASA events). With the caveat that it is premature, with so few events and big theoretical uncertainties, to draw strong conclusions, we have pointed out that although classical (i.e. energetic) long GRBs and short GRBs have difficulties in producing the required UHECR flux, newly born magnetars can. If so, they could also be a subclass of [*long*]{} GRBs, possibly sub–energetic and relatively nearby, powered by fastly spinning, newborn magnetars. The future increased statistics of UHECRs arrival directions will help to discriminate among the different proposed progenitors, especially if there will be (or not) an excess of events close to the radio core and/or lobes of Cen A. Acknowledgments {#acknowledgments .unnumbered} =============== We thank the referee for constructive comments. 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--- author: - Sebastian Peitz - 'Kai Sch[ä]{}fer' - 'Sina Ober-Bl[ö]{}baum' - Julian Eckstein - 'Ulrich K[ö]{}hler' - Michael Dellnitz bibliography: - 'arXiv\_bibliography.bib' title: A Multiobjective MPC Approach for Autonomously Driven Electric Vehicles --- Introduction {#sec:Introduction} ============ In many applications from industry and economy, the simultaneous optimisation of several criteria is of great interest. In transportation, for example, one wants to reach a destination as fast as possible while minimising the energy consumption. This example illustrates that in general, the different objectives contradict each other. Therefore, the task of computing the set of optimal compromises between the conflicting objectives, the so-called *Pareto set*, arises, leading to a multiobjective optimisation problem (MOP) or multiobjective optimal control problem (MOCP). Based on the knowledge of the Pareto set, a *decision maker* can design improved systems or even allow for changes in control parameters during operation as a reaction on external influences or changes in the system state itself. There exist various algorithms for the solution of MOCPs such as scalarisation techniques (cf. [@Ehr05] for an overview), evolutionary algorithms ([@CLV07]) or set oriented methods ([@SWO+13]). All approaches have in common that a large number of function evaluations is typically needed. Thus, the direct computation of the Pareto set is time consuming and a computation in real-time is not possible. However, in particular the design of optimal drive strategies requires online adaption of control strategies. This is even more the case now that autonomous driving and battery electric vehicles (EVs) with comparatively low ranges are both gaining increased attention, requiring advanced control algorithms. Control theory has been influenced significantly by the advances in computational power during the last decades. For a large variety of systems, it is nowadays possible to use model based optimal control algorithms to design sophisticated feedback laws. This concept is known as model predictive control (MPC) (see e.g. [@Mac02; @GP11]). The general goal of MPC is to stabilise a system by using a combination of open and closed-loop control: using a model of the system dynamics, an open-loop optimal control problem is solved in real-time over a so-called *prediction horizon*. The first part of this solution is then applied to the real system while the optimisation is repeated to find a new control function, with the prediction horizon moving forward (for this reason, MPC is also referred to as moving horizon control or receding horizon control). Due to the huge success of MPC, a large variety of algorithms has been established, where a first distinction can be made between linear and non-linear MPC. The first category refers to schemes in which linear models and quadratic objective functions are used to predict the system dynamics. The resulting optimisation problems are convex, i.e. global solutions can be computed very fast. Linear MPC approaches have been very successful in a large variety of industrial applications (see e.g. [@QB97] and [@LC97] for an overview in applications and theory). The advantage of non-linear MPC ([@GP11]), on the other hand, is that the typically non-linear system behaviour can be approximated in a more accurate way. Furthermore, special optimality criteria and non-linear constraints can be incorporated easily. However, the complexity and thus the time to solve the resulting optimisation problem increases such that it is often difficult to preserve real-time capability (see e.g. [@EPS+16]). Further extensions are, for example, *economic MPC* (see e.g. [@RA09; @DAR11]) or *explicit MPC* (see e.g. [@AB09]). In the first approach alternative, *economic* objectives are pursued instead of stabilising the system. In the second approach the problem of real-time applicability is addressed by introducing an offline phase during which the open-loop optimal control problem is solved for a large number of possible situations, using e.g. multi-parametric non-linear programming. The solutions are then stored in a library such that they are directly available in the online phase. Another way for optimal strategy planning is the concept *motion planning with motion primitives* going back to [@FDF05] (see also [@Kob08; @FOK12]). The challenge of online applicability is addressed with a two-phase approach similar to explicit MPC but here, valid control as well as state trajectories are obtained by combining several short pieces of simply controlled trajectories that are stored in a motion planning library. These motion primitives can be sequenced to longer trajectories in various combinations. In the online phase, the optimal sequence of motion primitives is determined from the motion planning library using e.g. graph search methods (see e.g [@Kob08]). To reduce the computational effort, the motion primitive approach extensively relies on exploiting symmetries in the dynamical control system such that a motion primitive can be used in multiple situations, e.g. by performing a translation or rotation under which the dynamics are invariant. In this article, we present a new algorithm for multiobjective MPC of non-linear systems. Problems with multiple criteria have been addressed by several authors using scalarisation techniques (see e.g. [@BP09] for a weighted sum or [@ZFT12] for a reference point approach). For non-convex problems, scalarisation approaches often face difficulties such that we here want to compute the entire Pareto set in advance. To this end, we combine elements from multiobjective optimal control, explicit MPC and motion planning with motion primitives. The resulting algorithm consists of an offline phase during which multiobjective optimal control problems are solved and stored in a library for a wide range of possible scenarios (i.e. constant velocity, braking, accelerating). Invariances in the optimal control problem are exploited in order to reduce the number of problems that need to be solved. In the online phase, the currently active scenario is identified and the corresponding Pareto set is selected from the library. According to a decision maker’s preference, an optimal compromise is then selected from the Pareto set and the first part of the solution is applied to the system. Similar to MPC, this is done repeatedly such that a feedback control behaviour is realised. The difference to other approaches is the possibility to interactively choose between different objectives such that the system behaviour can be modified easily. This can be very useful for autonomous driving, where one is interested in reaching a destination as fast as possible while minimising the energy consumption. The outline of the article is as follows. In Section \[sec:Problem\_formulation\], we introduce the multiobjective MPC problem and the concept of Pareto optimality before describing the algorithm in detail and comparing it to other MPC approaches. In Section \[sec:Application\_EV\], we describe the application of the algorithm to an electric vehicle. The aim is to realise autonomous driving where the passenger can decide between the objectives fast and energy efficient driving. We present the results in Section \[sec:Results\] before drawing a conclusion in Section \[sec:Conclusion\]. Problem Formulation and Methodology {#sec:Problem_formulation} =================================== Before describing the algorithm, we will briefly introduce the two main concepts we will be making use of, namely multiobjective optimal control and model predictive control. For more detailed introductions, we refer to [@Ehr05] and [@GP11], respectively. A *multiobjective optimal control* problem (MOCP) can be formulated mathematically using differential(-algebraic) equations describing the physical behaviour of the system together with optimisation criteria and optimisation constraints in the following way $$\min_{x,u,t_f} J(x,u,t_f) = \int_{t_0}^{t_f} C(x(t),u(t))\, dt + \Phi(x(t_f))\label{eq:J}$$ such that $$\begin{aligned} & \dot{x}(t) = f(x(t),u(t))\quad \forall t \in [t_0,t_f],\quad x(t_0)=x_0 \label{eq:diff}\\ & h(x(t),u(t)) \le 0 \quad \forall t \in [t_0,t_f],\label{eq:constraints} \end{aligned}$$ where $x(t)\in \mathcal{X}$ is the system state (e.g. the position and velocity of a car) and $u(t)\in \mathcal{U}$ the control (e.g. the engine torque or the steering wheel position). $\mathcal{X}$ and $\mathcal{U}$ are the spaces of feasible states and controls, respectively. The constraints may depend on the state as well as the control, e.g. limiting the velocity or energy consumption. $J$ describes criteria that have to be optimised. When there exists a unique solution $x(t) \in \mathcal{X}$ for every $u(t) \in \mathcal{U}$ and $x_0 \in \mathcal{X}$ and we fix the time frame, we can introduce a reduced objective $J: \mathcal{U} \times \mathcal{X} \rightarrow \mathbb{R}^k$, where $k$ is the number of objectives, and the corresponding reduced problem: $$%\min_{u} J(u) = \int_{t_0}^{t_f} C(x(t),u(t))\, dt + \Phi(x(t_f))\label{eq:Jred}. \min_{u} J(u, x_0) = \int_{t_0}^{t_f} C(\varphi_u(x_0,t))\, dt + \Phi(\varphi_u(x_0,t_f)). \label{eq:Jred}$$ Here $\varphi_u(x_0,t)$ is the flow of the dynamical control system . In many applications from industry and economy, one is interested in simultaneously optimising not only one but *several* criteria and hence, $k>1$ and $J$ is vector-valued. In this situation the solution does in general not consist of isolated optimal points but of the *set of optimal compromises*, the so-called *Pareto set* (cf. [@Ehr05] for a detailed introduction). The set consists of all functions $u(t)$ that are *nondominated*, i.e. for which there does not exist a solution $u^*(t)$ that is superior in all objectives (cf. Figure \[fig:MOP\]). For the solution of , we here use a scalarisation technique by which the Pareto set is approximated by a finite set of points that are computed consecutively by minimising the euclidean distance between a point $J(u, x_0)$ and a so-called *target point* $T$ which lies outside the reachable set in image space (see Figure \[fig:ReferencePoint\] for an illustration). Since a point computed this way lies on the boundary of the reachable set, there exists no point which is superior in all objectives and hence, the point is Pareto optimal. Starting with one point (e.g. the scalar minimum of one of the objectives), the next points can be computed recursively until the other end of the Pareto front (i.e. the other scalar minimum) is reached. In [@DEF+16], this method is used to compute the Pareto set for the conflicting objectives driven distance and energy consumption for EVs. The scalar optimal control problems are solved using an SQP method (cf. [@NW06]). ![Sketch of the MPC methodology. While the first part of the predicted control is applied to the system, the next control is predicted (via open-loop optimal control) on a shifted horizon.[]{data-label="fig:MPC"}](Figures/MPC.pdf){width="40.00000%"} The algorithm presented here builds on these results, but we need to extend them in order to construct a feedback controller. This is realised by an *MPC* approach, where the problem is solved repeatedly for varying time frames ($t_0 = t_s$, $t_f = t_{s+p}$, $s = 1,2,\ldots$) while the system is running at the same time. Then, the first interval of the predicted control, $u(t_s)$, is applied to the real system and the optimal control problem is solved again with a time frame shifted by one. The procedure is illustrated in Figure \[fig:MPC\]. The concept of MPC was initially developed to stabilise a system ([@GP11]), i.e. to drive the system state to a (potentially time dependent) reference state. However, stabilisation is not always the main concern. Considering the EV, for example, we only require a part of the state, namely the velocity, to remain within prescribed bounds, which then gives us the opportunity to pursue additional objectives such as minimising the energy consumption. This concept is known as *economic MPC* (see e.g. [@RA09; @DAR11]). The Offline-Online Multiobjective MPC Concept {#subsec:Algorithm} --------------------------------------------- Since MOCPs are considerably more expensive to solve than scalar problems, it is computationally infeasible to directly include them in an MPC framework. A simple way to circumvent this problem is to scalarise the objective function by introducing a weighting factor (i.e. $\widehat{J}=\sum_{i=1}^k \rho_i J_i, \rho_i\in[0,1]$). In this case however, an assumption has to be made in advance which can in practice lead to unfavourable results. A slight increase in one objective might allow for a strong reduction in another one, for example. Hence, we are interested in providing the entire Pareto set during the MPC routine. To avoid large computing times during execution, we therefore split the computation in an *offline* and an *online phase*, similar to explicit MPC approaches (cf. [@AB09]). The *offline phase* consists of several steps. First, various *scenarios* are identified for which MOCPs need to be solved. The scenarios are determined by the system states and the constraints. Secondly, in order to reduce the number of scenarios, the dynamical control system is analysed with respect to invariances, which are formally described by a finite-dimensional Lie group $G$ and its group action $\psi: \mathcal{X} \times G \rightarrow \mathcal{X}$. A dynamical control system, described by , is invariant under the group action $\psi$, or equivalently, $G$ is a symmetry group for the system , if for all $g\in G$, $x_0\in \mathcal{X}$, $t\in [t_0,t_f]$ and all piecewise-continuous control functions $u:[t_0,t_f]\rightarrow \mathcal{U}$ it holds $$\psi(g,\varphi_u(x_0,t)) = \varphi_u(\psi(g,x_0),t) \quad \forall g\in G. \label{eq:Invariance}$$ That means that the group action on the state commutes with the flow. Invariance leads to the concept of [*equivalent trajectories*]{}. Two trajectories are equivalent if they can be exactly superimposed through time translation and the action of the symmetry group. In the classical concept of motion primitives ([@FDF05]), all equivalent trajectories are summed up in an equivalence class, i.e. only a single representative is stored that can be used at many different points when transformed by the symmetry action. In other words, controlled trajectories that have been computed for a specific situation are suitable in many different (equivalent) situations as well. In our approach, we extend this concept by identifying symmetries in the solution of the MOCP with respect to the initial conditions $x_0$: $$\begin{aligned} \arg \min_{u} J(u, x_0) = \arg \min_{u} J(u, \psi(g,x_0)) \quad \forall g \in G. \label{eq:Invariance_MOCP} %&\arg \min_{u} \int_{t_0}^{t_f} C(\varphi_u(x_0,t))\, dt + \Phi(\varphi_u(x_0,t_f)) \notag \\ %= &\arg \min_{u} \int_{t_0}^{t_f} C(\varphi_u(\psi(g,x_0),t))\, dt + \Phi(\varphi_u(\psi(g,x_0),t_f)) \notag \\ %&\quad \hspace{6cm} \forall g \in G. \label{eq:Invariance_MOCP} \end{aligned}$$ This means that we require the *Pareto set* to be invariant under group actions on the initial conditions. If the objective function is also invariant under the same group action, then all trajectories contained in an equivalence class defined by will also be contained in an equivalence class defined by . However, this class may contain more solutions since we do not explicitly pose restrictions on the state but only require the solution of to be identical. Alternatively, if the objective function is linear in the states and the group action corresponds to translations in initial states, we do not require invariance of the objective function to satisfy . Identifying invariances according to , the number of MOCPs can be reduced. If the system is invariant under translation of the initial position $p(t_0)$, for example, we do not need to solve multiple MOCPs that only differ in the position. Once these equivalence classes have been identified, we can reduce the number of possible scenarios accordingly. We then solve the resulting MOCPs on the prediction horizon $T_p$, introduce a parametrisation $\rho$ (which can then be chosen by the decision maker in the online phase) and store the Pareto sets and fronts in a library such that they can be used in the online phase. Since in general there is an infinite number of feasible initial conditions, there consequently exists an infinite number of scenarios that we have to consider. In practice, this obviously cannot be realised and we have to introduce a finite set of scenarios. In the online phase, we then pick the scenario that is closest to the true initial condition. If a violation of the state constraints has to be avoided (the EV, e.g., is not allowed to go faster than the maximum speed), then a selection towards the ”safe” side can be made. In case of the EV, we would consequently pick a solution corresponding to a velocity slightly higher than the actual velocity. This way, the maximally allowed acceleration would be bounded such that exceeding the speed limit is not possible. The *online phase* is now basically a standard MPC approach, the difference being that we obtain the solution of our control problem from a library instead of solving it in real-time, similar to explicit MPC approaches: - measure the current system states that are necessary for the identification of the current scenario, - choose the corresponding Pareto set from the library, i.e. the one with initial conditions closest to the current system state. (Due to the approximation, we cannot formally guarantee that the constraints are not violated. However, as a start we consider applications where this is acceptable.) - choose one optimal compromise $u$ from the set, according to a decision maker’s preference $\rho$, - apply the first step (i.e. the sample time) of the solution $u$ to the real system and go back to 1. The resulting algorithm thus provides a feedback law. In the offline phase, we define the scenarios in such a manner that the system cannot be steered out of the set of feasible states. This means that only controls $u$ are valid that do not lead to a violation of the constraints. Additionally, we include scenarios which steer the system into the set of feasible states from any initial condition. In the literature, this is known as *viability*, cf. [@GP11]. In case of the EV, for example, we have to include controls such that the velocity can be steered to values satisfying the constrains from any initial velocity. The presented algorithm can be seen as an extension of (extended) MPC approaches to multiple objectives. We consider *economic* objectives (cf. [@RA09]) and do not focus on the stabilisation of the system. This allows us to pursue multiple objectives between which a decision maker can choose dynamically, e.g. in order to react on changes in the environment or the system state itself. In contrast to weighting methods, the entire Pareto set is known, providing increased system knowledge. Application to Electric Vehicle {#sec:Application_EV} =============================== In this section the algorithm is utilised to control the longitudinal dynamics of an EV, thereby extending prior work, see [@DEF+14] for a scalar optimal control problem, [@DEF+16] for a multiobjective optimal control problem and [@EPS+16] for a comparison of two scalar MPC approaches. Vehicle Model {#subsec:EV_model} ------------- The EV model is derived by coupling the equations for the electrical and the mechanical subsystem via efficiency maps. This yields a system of four coupled, non-linear ordinary differential equations for the system state $x(t) = \left( v(t), S(t), U_{d,L}(t), U_{d,S}(t) \right)$. Here, $v$ is the vehicle velocity, $S$ is the battery state of charge and $U_{d,L}$ and $U_{d,S}$ are the long and short term voltage drops, respectively. The system is controlled by setting the torque $u(t)$ of the front wheels. Additionally, the battery current $I(t)$ is computed from the state $x(t)$ via an algebraic equation and the position by integrating the velocity: $p(t) = \int_{t_0}^{t}v(\tau)d\tau$. For the derivation and the exact formulation of the dynamical system, we refer the reader to [@EPS+16]. Based on the system dynamics, we formulate the MOCP for the EV with variable final time: $$\begin{aligned} &min_{u} \left( \begin{array}{c} \label{eq:MOCP_EV_J} S(t_0)-S(t_f) \\ t_f - t_0 \end{array} \right), \\ \dot{x}(t)&=f(x(t),u(t)), \label{eq:MOCP_EV_diff}\\ v_{min}(t) &\le v(t) \le v_{max}(t), \quad &t \in [0,t_f] \label{eq:MOCP_EV_constraints1} \\ I_{min}(t) &\le I(t) \le I_{max}(t), \quad &t \in [0,t_f] \label{eq:MOCP_EV_constraints2} \\ x(0) &= x_0, \ p(t_f) = p_f. \label{eq:MOCP_EV_x0} \end{aligned}$$ We set the final position $p_f$ to $100$m, which means that we here define the prediction horizon based on the position. Correspondingly, the sample time is also specified with respect to the position, $\delta = 20$m. The conflicting objectives are to reach $p_f$ as fast as possible ($J_2$) while minimising the energy consumption ($J_1$). The battery current $I$ is limited in order to avoid damaging the battery which results in implicit constraints on the control $u$. The velocity constraints are part of the scenarios which are defined in the offline phase. Offline Phase: System Analysis and Solution of Multi- objective Optimal Control Problems {#subsec:EV_offline_phase} ---------------------------------------------------------------------------------------- In this section we describe how the different steps of the offline phase are applied to the EV. ### Symmetry Analysis {#subsec:EV_symmetries} \ The more invariances the MOCP possesses (in the sense of Equation ), the fewer problems need to be solved which significantly reduces the computational effort. Hence, we numerically analyse the system in this regard. Since the position $p$ does not occur in the dynamical system , the dynamics are obviously invariant under translations in $p$. Moreover, when exemplary looking at the velocity $v$ and the state of charge $S$ (cf. Figures \[fig:Invariances\]a and \[fig:Invariances\]b), we see that, on the one hand, the trajectories are almost invariant for a wide range of translated initial values of the state of charge $S(0)$. Note that this is not a strict invariance. However, as argued in Section \[subsec:Algorithm\], we do not require invariances according to Equation  but according to the weaker condition  which is satisfied much more accurately for the EV application. When looking at Figure \[fig:Invariances\]c on the other hand, we observe that the dynamics are clearly not invariant under translations in the initial velocity $v(0)$. After performing the same analysis with regards to the other state variables $U_{d,L}$ and $U_{d,S}$, we can conclude that we only need to define scenarios with respect to the initial velocity $v(0)$ and the active constraints $v_{min}(t)$ and $v_{max}(t)$. ### Constraints {#subsec:EV_constraints} A constraint on the velocity is given by the current speed limit $v_{max}(p)$ which depends on the current vehicle position. Since we need to avoid interfering with other vehicles by driving too slow, we define a minimal velocity $v_{min}(p) = 0.8\cdot v_{max}(p)$. (Here we have written the velocities as functions of the position because they are given by the problem formulation this way. In the MOCP, they have to be reformulated as functions of time.) Our *set of feasible states* is now determined by the velocity constraints, i.e. $v_{min}(t)\le v(t) \le v_{max}(t)$, which determine the different scenarios. We distinguish between four cases (see Figure \[fig:Constraints\]a). While the cases constant velocity (box constraints) and stopping ($v=0$ at the stop sign) are easily implemented, we introduce a linear constraint for the scenarios (b) and (c), respectively (see Figure \[fig:Constraints\]b) where, depending on the current velocity, a minimal increase $\overline{a}_{min} = (dv/dp)_{min}$ or decrease, respectively, must not be violated. An example is shown in Figure \[fig:PS\_accel\], where the Pareto set (\[fig:PS\_accel\]a) and the resulting velocity profiles (\[fig:PS\_accel\]b) are shown for the scenario $v(0) = 60\ km/h$ and $\overline{a}_{min} = 0.05\ \frac{km/h}{m}$. Note that here, we have chosen the control $u$ to be constant over the prediction horizon in order to reduce the numerical effort. As mentioned in Section \[subsec:Algorithm\], we cannot solve an MOCP for every initial condition. Solving an MOCP for every step of $0.1$ in the initial velocity leads to 1727 MOCPs in total. Online Phase: Multiobjective MPC with Paretooptimal Control Primitives {#subsec:EV_online_phase} ---------------------------------------------------------------------- The online phase is now exactly as described in Section \[subsec:Algorithm\]. In each sample time, the current velocity and the active constraints (for the current position) are evaluated in order to determine the valid scenario. The corresponding Pareto set is then selected from the library and according to the weighting parameter $\rho \in [0,1]$ determined by the decision maker, an optimal compromise is chosen which is then applied to the system. On a standard computer, this operation takes in the order of $10^{-3}$ seconds in Matlab. Results and Discussion {#sec:Results} ====================== ![Different trajectories computed by the MPC approach. The dashed lines use a constant weight $\rho$ whereas the green line possesses dynamic weighting ($\rho = 0 / 0.5 / 1.0$, respectively)[]{data-label="fig:Road_var_rho"}](Figures/road_allprefs_voverp_posdyn-eps-converted-to.pdf){width="48.00000%"} In Figure \[fig:Road\_var\_rho\], several solutions with different weights $\rho$ are shown for an example track including two stop signs. The *set of feasible states* is bounded by the red lines $v_{min}$ and $v_{max}$. The dashed lines correspond to constant weights, varying from $\rho=0$ (energy efficiency) to $\rho = 1$ (high velocity) and the solid green line is a solution where the weighting is changed from 0 over 0.5 to 1 during driving. We clearly see that the vehicle is driving according to the decision maker’s preference. This means that we have realised a closed-loop control for which the objectives can be adjusted dynamically. This can either be done manually or by an additional algorithm, which for example takes into account the track, the battery state of charge and the current traffic. The objective function values for the entire track and different values of $\rho$ are depicted in Figure \[fig:PF\]a. In order to evaluate the quality of our solution, we compare it to a control computed via dynamic programming (DP, see [@BD15] for an introduction and [@SG09] for the algorithm that is used): For computational reasons, the comparison is performed on a shorter track without stop signs and a relatively coarse discretisation leading to a 100-dimensional problem. In the DP problem, we use a simplified linear model (cf. [@EPS+16]) and the objective is a weighted sum of the MOCP , $J= t_f + \beta E(t_f)$, where $E$ is the consumed energy computed by integrating over the wheel torque and $\beta = 6\cdot10^{-5}$. In Figure \[fig:PF\]b, we see that the solution obtained via DP is superior to our MPC approach. This is not surprising since in MPC, we only consider a finite horizon such that the results are at best suboptimal ([@GP11]), whereas the entire track is considered at once in DP. Consequently, the DP algorithm is not real-time applicable and does not possess feedback behaviour. Additionally, we have until now only considered constant torques over the prediction horizon in our approach. We intend to refine the discretisation in future work and expect an improved performance. ![Validation of the approach versus a Dynamic Programming solution (blue). Green line: dynamic weighting according to the lower plot.[]{data-label="fig:Road_var_rho_dyn"}](Figures/Track_rhoDynamic_vs_DP-eps-converted-to.pdf){width="48.00000%"} When using a simple, manually tuned heuristic for the preference $\rho$ instead of fixed values (larger values for $\rho$ at low velocities, lower values at high velocities and linear changes in $\rho$ when approaching braking manoeuvres, see Figure \[fig:Road\_var\_rho\_dyn\], bottom), we see that we can improve the quality of our solution significantly which is now comparable to the global optimum obtained by DP. We see in Figures \[fig:Road\_var\_rho\_dyn\] (top) and \[fig:PF\]b, respectively, that the resulting trajectories as well as the function values $J_1$ and $J_2$ almost coincide. By this, we obtain two different ways to utilise the results. On the one hand, a decision maker can select the preference according to his wishes and on the other hand, $\rho$ can be determined by a heuristic, leading to solutions of a quality comparable to the global optimum. Conclusion {#sec:Conclusion} ========== We present an algorithm for MPC of non-linear dynamical systems with respect to multiple criteria. The algorithm utilises elements from economic and explicit MPC, multiobjective optimal control and motion planning. According to a decision maker’s preference, the system is controlled in real-time with respect to an optimal compromise between conflicting objectives. Using a simple heuristic for the weighting factor $\rho$, we obtain solutions of equivalent quality compared to a global optimum computed by open loop DP. In the future, we intend to analyse the proposed method from a more theoretical point of view, addressing questions concerning feasibility and stability for systems where these aspects are critical. Furthermore, we want to improve our control strategies by developing intelligent heuristics for the preference weighting function $\rho$.\   **Acknowledgement:** This research was funded by the German Federal Ministry of Education and Research (BMBF) within the Leading-Edge Cluster *Intelligent Technical Systems OstWestfalenLippe (it’s OWL)*.

[****]{}\ Vincenzo Branchina[^1]\[one\] Department of Physics, University of Catania and\ INFN, Sezione di Catania, Via Santa Sofia 64, I-95123, Catania, Italy Marco Di Liberto[^2]\[two\] Scuola Superiore di Catania, Via S. Nullo 5/i, Catania, Italy Ivano Lodato[^3]\[three\] Scuola Superiore di Catania, Via S. Nullo 5/i, Catania, Italy and\ INFN, Sezione di Catania, Via Santa Sofia 64, I-95123, Catania, Italy [Abstract]{}\ It has been recently claimed that dark energy can be (and has been) observed in laboratory experiments by measuring the power spectrum $S_I(\omega)$ of the noise current in a resistively shunted Josephson junction and that in new dedicated experiments, which will soon test a higher frequency range, $S_I(\omega)$ should show a deviation from the linear rising observed in the lower frequency region because higher frequencies should not contribute to dark energy. Based on previous work on theoretical aspects of the fluctuation-dissipation theorem, we carefully investigate these issues and show that these claims are based on a misunderstanding of the physical origin of the spectral function $S_I(\omega)$. According to our analysis, dark energy has never been (and will never be) observed in Josephson junctions experiments. We also predict that no deviation from the linear rising behavior of $S_I(\omega)$ will be observed in forthcoming experiments. Our findings provide new (we believe definite) arguments which strongly support previous criticisms. Introduction ============ The origin of dark energy is one of the greatest mysteries confronting theoretical and experimental physics. Different proposals for the solution of this so called cosmological constant problem are put forward and many review articles nowadays discuss and compare these alternative approaches (see for instance [@peeb],[@padam],[@cope],[@nobe]). Among many other issues, Copeland et al.[@cope] discuss the suggestion of Beck and Mackey[@bema1; @bema2] according to which dark energy can be (and has been) observed in laboratory experiments [@koch] by measuring the power spectrum of the noise current in a resistively shunted Josephson junction. If true, this would mark a dramatic progress in our understanding of the origin of dark energy. According to[@bema1; @bema2], this power spectrum is due to thermal and vacuum fluctuations of the electromagnetic field in the resistor and these experiments[@koch] provide a measurement of the electromagnetic zero point energies, which they consider as being at the origin of dark energy. These ideas have generated a certain debate and some authors[@jetz1; @doran; @maha] have argued against them. Beck and Mackey have rebutted these criticisms[@bema3] but they were again criticized in [@jetz2]. Copeland et al.close the section of their review devoted to this issue by saying that “time will tell who (if either) are correct”[@cope]. Needless to say, this issue is of the greatest importance and deserves further investigation. Scope of this work is to bring additional elements to the analysis of this problem in the hope that the question posed in[@cope] could finally find an answer. To this end, it is necessary to review in some detail the Beck and Mackey proposal[@bema1; @bema2]. These authors begin by considering the work of Koch et al.[@koch], where the spectral density $S_I(\omega)$ of the noise current in the resistor of a resistively shunted Josephson junction was measured and confronted against the theoretical prediction, \[spectr\] S\_I() = (+ ), and good agreement was found between experimental results and theory ($T$ is the temperature and $R$ the resistance of the resistive shunt). Eq.(\[spectr\]) comes from an application of the fluctuation-dissipation theorem (FDT)[@cawe] and we immediately recognize the term in parenthesis as the mean energy of a quantum harmonic oscillator of frequency $\omega$ in a thermal bath. Where does this Bose-Einstein (BE) distribution factor come from? Does it reflect an underlying “harmonic oscillator structure” of the system[@taylor]? If yes, which harmonic oscillators are involved in Eq.(\[spectr\])? A correct answer to these questions will turn out to be crucial in understanding the status of the Beck and Mackey proposal[@bema1; @bema2] and, we believe, in settling the controversy. Beck and Mackey interpret this factor as coming from the modes of the electromagnetic field in interaction with the charged particles [@bema2] and claim that this experiment provides a direct measurement of vacuum fluctuations of the electromagnetic field (the $\frac{\hbar\omega}{2}$ term). Moreover, they assume that dark energy originates from vacuum fluctuations of fundamental quantum fields and conjecture that only those fluctuations which can be measured in terms of a physical power spectrum are gravitationally active, i.e. contribute to dark energy. Then, by observing that for strong and electroweak interactions it is unlikely that a suitable macroscopic detector exists that can measure the corresponding vacuum spectra, they conclude that the only candidate where we know that a suitable macroscopic detector exists is the electromagnetic interaction[@bema2]. Accordingly, by noting that astrophysical measurements give $\varrho_{dark} \sim 10^{-47} GeV^4$ (in natural units), they argue that there should be a physical cut-off frequency $\nu_c =\frac{\omega_c}{2\pi}\sim 1.7\, {\rm THz}$ such that, for frequencies above this cut-off, the spectral function of the noise current in the Josephson junction should behave differently than in Eq.(\[spectr\]). According to their hypothesis, in fact, for $\omega \geq \omega_c $ the $\frac{\hbar\omega}{2}$ term should be absent. Coming back to the BE distribution factor in Eq.(\[spectr\]), Jetzer and Straumann[@jetz1] (see also [@kubo]) observed that this term simply comes from ratios of Boltzmann factors which appear in the derivation of the FDT and stressed that the $\frac{\hbar\omega}{2}$ term has nothing to do with zero point energies, while Beck and Mackey reply that this is contrary to the view commonly expressed in the literature[@bema3]. A simple look to the derivation of the FDT (see Section 2) shows that, as for the ratios of Boltzmann factors, Jetzer and Straumann[@jetz1] are definitely right. Nevertheless, in a sense that we are going to make clear in the following, there is an element of truth in the common lore according to which this factor can be regarded as due to a sort of underlying harmonic oscillator structure of the system (the resistive shunt in the case of the Koch et al. experiment[@koch]). In a recent paper[@noi] we have shown that whenever linear response theory applies, which is the main hypothesis under which the FDT is derived, any generic bosonic and/or fermionic system can be mapped onto a fictitious system of harmonic oscillators in such a manner that the quantities appearing in the FDT coincide with the corresponding quantities of the fictitious one (for completeness, in sections 2 and 3 we briefly review these results. For a comprehensive exposition, however, see[@noi]). This allows us to understand [*in which sense*]{} the harmonic oscillator interpretation can be put forward so that we shall be able to say whether the Beck and Mackey’s proposal is tenable or not. We shall see that it is not. The rest of the paper is organized as follows. In Section 2 we briefly review the derivation of the FDT and consider two convenient expressions for the power spectrum of the fluctuating observable and for the imaginary part of the corresponding generalized susceptibility respectively. In Section 3 we consider the special case of a system of harmonic oscillators in interaction with an external field and show how the above mentioned mapping is constructed. In Section 4 we apply the results of the two previous sections to our problem, namely the dark energy interpretation[@bema1; @bema2] of the measured power spectrum $S_I(\omega)$[@koch] and show that this interpretation is untenable. Section 5 is for our conclusions. The fluctuation-dissipation theorem =================================== In the present section we briefly review the derivation of the FDT (see[@kubo] for more details) and provide expressions for the spectral function and the imaginary part of the generalized susceptibility which will be useful for our following considerations. Consider a macroscopic system with unperturbed hamiltonian $\hat{H}_0$ under the influence of the perturbation \[inter\] = - f(t)(t), where $\hat{A}(t)$ is an observable (a bosonic operator) of the system and $f(t)$ an external generalized force[^4]. Let $|E_n\rangle$ be the $\hat{H}_0$ eigenstates (with eigenvalues $E_n$) and $\langle E_n|\hat{A}(t)|E_n \rangle =0$. Within the framework of linear response theory, the quantum-statistical average $\langle\hat{A}(t)\rangle_f$ of the observable $\hat{A}(t)$ in the presence of $\hat{V}$ is given by \[resp2\] (t)\_f = \_[-]{}\^t d t’ \_[\_[A]{}]{}(t-t’) f(t’) where $\chi_{_{A}}(t - t')$ is the generalized susceptibility, \[chi\] \_[\_[A]{}]{}(t - t’)=(t-t’) = -G\_R(t - t’) , with $\langle ... \rangle = \sum_{n} \varrho_n \langle E_n| ... |E_n \rangle$, $\varrho_n= e^{-\beta E_n}/Z$ , $Z=\sum_n e^{- \beta E_n}$, $G_R(t-t')$ being the retarded Green’s function and $\hat{A}(t)=e^{i\hat{H_0}t/\hbar}\hat{A}e^{-i\hat{H_0}t/\hbar}$. If we now consider the mean square of the observable $\hat A(t)$ and write the generalized susceptibility $\chi_{_{A}}$ as $\chi_{_{A}} = \chi^{'}_{_{A}} + i\,\chi^{''}_{_{A}}$ (with $\chi^{'}_{_{A}}$ the real part and $\chi^{''}_{_{A}}$ the imaginary part of $\chi_{_{A}}$), it is not difficult to show (see [@kubo] and [@noi]) that the Fourier transform $\langle \hat{A}^2(\omega)\rangle$ of $\langle \hat{A}^2(t)\rangle$ is related to the Fourier transform $\chi_{_{A}}^{\,''}(\omega)$ through the relation \[fddt\] \^2()= \_[\_[A]{}]{}\^[”]{}() = \_[\_[A]{}]{}\^[”]{}() [coth]{}( ) =2\_[\_[A]{}]{}\^[”]{}() (1 2 + ), which is the celebrated FDT. In Eq.(\[fddt\]) we recognize the ratio of Boltzmann factors alluded by Jetzer and Straumann[@jetz1]. Actually, it was already observed by Kubo et al.[@kubo] that the BE factor in Eq.(\[fddt\]) is simply due to a peculiar combination of Boltzmann weights and that there is no reference to physical harmonic oscillators of the system whatsoever. However, as we already said in the Introduction, there is an element of truth in the common lore which considers this term as due to a sort of harmonic oscillator structure of the system (the resistive shunt in the case of the Koch et al.experiment[@koch]). In order to show that, we now refer to our recent work[@noi], where we have derived the following useful expressions for $\langle \hat{A}^2(\omega)\rangle$ and $\chi_{_A}^{\,''}(\omega)$ : \^2()&=& \_[j &gt; i]{}(\_i - \_j) |A\_[ij]{}|\^2 [coth]{}( ) \[o5\]\ &=& [coth]{}()\_[j &gt; i]{} (\_i - \_j) |A\_[ij]{}|\^2 ,\[o6\]\ ”()&=&\_[j &gt; i]{}(\_i - \_j) |A\_[i j]{}|\^2 \[chii3\]. Clearly, from Eqs.(\[o6\]) and (\[chii3\]) the FDT (Eq.(\[fddt\])) is immediately recovered. However, what matters for our scopes are the explicit expressions in Eqs.(\[o5\]) and (\[chii3\]). Starting from these equations, in fact, we can easily show that it is possible to build up a mapping between the real system and a fictitious system of harmonic oscillators[@noi] in such a manner that $\chi_{_A}^{\,''}(\omega)$ and $\langle \hat{A}^2(\omega)\rangle$ are exactly reproduced by considering the corresponding quantities of the fictitious system. In the following section we outline the main steps for this construction (see [@noi] for details). The Mapping =========== In order to build up this mapping, we consider first a system ${\cal S}_{osc}$ of harmonic oscillators (each of which is labeled below by the double index $\{ji\}$ for reasons that will become clear in the following) whose free hamiltonian is: \[armonico\] H\_[osc]{} = \_[j &gt; i]{}( + q\_[ji]{}\^[2]{}), where $\omega_{ji}$ are the proper frequencies of the individual harmonic oscillators and $M_{ji}$ their masses. Let $| n_{j i}\rangle$ ($n_{ji}=0, 1,2,...$) be the occupation number states of the $\{ji\}$ oscillator out of which the Fock space of ${\cal S}_{osc}$ is built up. Let us consider also ${\cal S}_{osc}$ in interaction with an external system through the one-particle operator: \[armint\] V\_[osc]{} = - f(t) A\_[osc]{}, with \[onepart\] [A]{}\_[osc]{} = \_[j &gt; i]{} (\_[j i]{} [q]{}\_[ji]{} ). Obviously, the FDT applied to ${\cal S}_{osc}$ gives $\langle {\hat A}_{osc}^2(\omega)\rangle = \hbar \chi_{osc}^{\,''}(\omega) \,{\rm coth}\left( \frac{\beta\hbar\omega}{2}\right)$, but this is not what matters to us. What is important for our purposes is that, as shown in[@noi], for ${\cal S}_{osc}$ we can exactly compute $\langle {\hat A}_{osc}^2(\omega) \rangle$ and $\chi_{osc}^{\,''}(\omega)$. The reason is that for this system, differently from any other generic system, we can explicitly compute the matrix elements of ${\hat A}_{osc}$. The result is (compare with Eqs.(\[o5\]), (\[o6\]) and (\[chii3\])): \_[osc]{}\^2()&=&\_[j&gt;i]{} \_[ji]{}\^2 [coth]{}() \[(-\_[ji]{}) +(+\_[ji]{})\]\[osc5\]\ &=& [coth]{}()\_[j&gt;i]{} \_[ji]{}\^2 \[(-\_[ji]{}) - (+\_[ji]{})\]\[oscc5\];\ \_[osc]{}\^[”]{} ()&=& \_[j&gt;i]{} \_[ji]{}\^2 \[(-\_[ji]{}) - (+\_[ji]{})\]\[chi3\]. Naturally, comparing Eq.(\[oscc5\]) with Eq.(\[chi3\]) we see that for ${\cal S}_{osc}$ the FDT holds true, as it should. However, for our scopes it is important to note the following. For this system, the ${\rm coth}\left(\frac{\beta \hbar\omega}{2}\right)$ factor of the FDT originates from the [*individual contributions*]{} ${\rm coth}\left(\frac{\beta \hbar\omega_{ji}}{2}\right)$ of each of the harmonic oscillators of ${\cal S}_{osc}$. We can now build up our mapping. Let us consider the original system ${\cal S}$, described by the unperturbed hamiltonian $\hat H_0$, in interaction with an external field $f(t)$ through the interaction term $\hat V = - f (t)\,\hat A$ (see Eq.(\[inter\])), and construct a fictitious system of harmonic oscillators ${\cal S}_{osc}$, described by the free hamiltonian ${\hat H}_{osc}$ of Eq.(\[armonico\]), in interaction with the same external field $f(t)$ through the interaction term ${\hat V}_{osc}$ of Eq.(\[armint\]), with $\hat A_{osc}$ given by Eq.(\[onepart\]), where for $\alpha_{j i}$ we choose \[alfa\] \_[j i]{} = ()\^[12]{} (\_i - \_j)\^[12]{} |A\_[ij]{}| and for the proper frequencies $\omega_{ji}$ of the oscillators \[omega\] \_[ji]{}= (E\_j-E\_i)/&gt; 0, with $E_i$ the eigenvalues of the hamiltonian ${\hat H}_0$ of the real system. By comparing Eq.(\[oscc5\]) with Eq.(\[o6\]) and Eq.(\[chi3\]) with Eq.(\[chii3\]), it is immediate to see that with the above choices of $\alpha_{ji}$ and $\omega_{ji}$ we have: \^2()&=& \_[osc]{}\^2()\[cen1\]\ \_[\_A]{}\^[”]{} () &=& \_[osc]{}\^[”]{} ()\[cen2\]. Eqs.(\[cen1\]) and (\[cen2\]) define the mapping we are looking for. They show that it is possible to map the real system ${\cal S}$ onto a fictitious system of harmonic oscillators ${\cal S}_{osc}$, [S]{}\_[osc]{}, in such a manner that $\chi_{_A}^{''} (\omega)$ and $\langle \hat{A}^2(\omega)\rangle$ of the real system are equivalently obtained by computing the corresponding quantities of the fictitious one. The key ingredient to construct such a mapping is the hypothesis that linear response theory is applicable (which is the central hypothesis under which the FDT is established). Now, by considering the “equivalent” harmonic oscillators system ${\cal S}_{osc}$ rather than the real one, we can somehow regard the BE distribution factor ${\rm coth}\left( \frac{\beta\hbar\omega}{2}\right)$ of the FDT in Eq.(\[fddt\]) as originating from the individual contributions ${\rm coth}\left( \frac{\beta\hbar\omega_{ji}}{2}\right)$ of each of the oscillators of the equivalent fictitious system (see above, Eqs.(\[osc5\]), (\[oscc5\]) and (\[chi3\])). In this sense, this mapping allows for an oscillator interpretation of the BE term in the FDT. At the same time, however, the above findings clearly teach us that the BE distribution term in the FDT [*does not describe the physical nature of the system*]{}. It rather encodes a fundamental property of any bosonic and/or fermionic system: whenever linear response theory is applicable, any generic system is equivalent (in the sense defined above) to a system of quantum harmonic oscillators. Dark energy and laboratory experiments ====================================== We are now in the position to apply the results of the two previous sections to our problem. As we said in the Introduction, Beck and Mackey[@bema1; @bema2] interpret the Koch et al.experimental results[@koch] for the spectral density $S_I(\omega)$ of the noise current in a resistively shunted Josephson junction as a direct measurement of [*vacuum fluctuations of the electromagnetic field*]{} in the shunt resistor. Moreover, according to their ideas, these zero-point energies are nothing but the dark energy of the universe. In view of our results, however, this interpretation seems to be untenable. Eq.(\[spectr\]) for $S_I(\omega)$ comes from an application of the FDT to the case of the noise current in the shunt resistor. Therefore, according to our findings, the BE distribution factor which appears in $S_I(\omega)$ has nothing to do with thermal and vacuum fluctuations of the electromagnetic field in the resistor. Our analysis shows that this factor rather reflects a general property of any quantum system valid whenever linear response theory applies. The resistor (as well as any other generic system) can be mapped onto a system of fictitious harmonic oscillators in such a manner that the the power spectrum of the noise current and the related susceptibility can be reproduced by considering the equivalent quantities for the fictitious oscillators. It is in this sense, and [*only in this sense*]{}, that the BE factor can be interpreted in terms of harmonic oscillators, no other physical meaning can be superimposed on it. According to these considerations, we conclude that the claim that dark energy is observed in laboratory experiments[@bema1; @bema2] is based on an incorrect interpretation of the origin of the BE factor in the FDT. We believe that this should help in solving the controversy, which is left open in the Copeland et al.review[@cope], between the proponents[@bema1; @bema2] of the dark energy interpretation of the Koch et al.experiments[@koch] and the opponents[@jetz1; @doran; @maha; @jetz2]. In this respect, it is worth to stress that our analysis provides new arguments which strongly support the conclusions of these latter works[@jetz1; @doran; @maha; @jetz2]. A distinctive new element of our work, which in our opinion should greatly help in settling the question, concerns the interpretation of the FDT presented in section 5 of[@bema2]. These authors note that, although the FDT is valid for arbitrary hamiltonians $H$, where $H$ need not to describe harmonic oscillators, in the FDT appears a [*universal function*]{} $H_{uni}$, $H_{uni}= \frac12\hbar\omega +\hbar\omega/(exp(\hbar\omega/kT)-1)$, which can always be interpreted as the mean energy of a harmonic oscillator. Then, they identify the $\frac12\hbar\omega$ in $H_{uni}$ as the source of dark energy. The distinctive feature of our analysis is that, with the help of the formal mapping discussed in the previous section, which is valid for [*any generic system*]{}, the reason for the appearance of this [*universal*]{} function is immediately apparent. At the same time, however, this clearly shows that it cannot be claimed that the Koch et al.[@koch] experimental device is measuring zero point energies. As already noted in[@jetz1; @jetz2], these experiments simply measure a general quantum property of the system, the $\frac12\hbar\omega$ in $S_I(\omega)$ has nothing to do with zero point energies. Another very important point related to these issues concerns future measurements[@barb; @warb] of the power spectrum $S_I(\omega)$ for values of the frequency higher than those measured by Koch et al.[@koch]. In fact, according to Beck and Mackey[@bema1; @bema2], in forthcoming experiments[@barb; @warb], which are purposely designed to test a higher frequency range of $S_I(\omega)$, we should observe a dramatic change in the behavior of the spectral function $S_I(\omega)$ for these higher values of the frequency due to the presence of a cut-off which separates the gravitationally active modes from those which are not gravitationally active (see the Introduction). In view of our findings, however, we do not expect to observe in these experiments[@barb; @warb] any change in the behavior of $S_I(\omega)$. We simply state that such a cut-off does not exist. In this respect, we note that Beck and Mackey have recently proposed a new model for dark energy which should naturally incorporate such a cut-off[@bm3]. According to our analysis, this model seems to be deprived of any experimental and theoretical support. As a consequence of our results, a deviation of $S_I(\omega)$ from the behavior given in Eq.(\[spectr\]) could be observed only if the central hypothesis on which the derivation of the FDT is based, namely the applicability of linear response theory, no longer holds true in this higher frequency region. Summary and Conclusions ======================= With the help of a general theorem, which shows that (under the assumption that linear response theory is applicable) any bosonic and/or fermionic fermionic system can be mapped onto a fictitious system of harmonic oscillators, we have shown that the appearance of a Bose-Einstein distribution factor in the power spectrum of the noise current of a resistively shunted Josephson junction[@koch] has nothing to do with a real (physical) harmonic oscillator structure of the shunt resistor. We then conclude that, contrary to recent claims[@bema1; @bema2], experiments where this power spectrum was measured[@koch] do not provide any direct measurement of zero point energies and, as a consequence, no dark energy has ever been measured in these laboratory experiments. A direct consequence of our analysis is that, contrary to what is predicted in[@bema1; @bema2], we do not expect any deviation from the linear rising behavior of $S_I(\omega)$ with $\omega$. According to our analysis, in fact, the $\frac12\hbar\omega$ term in $S_I(\omega)$ has nothing to do with the dark energy in the universe, therefore we do not expect any cut-off which separates the gravitationally active zero point energies from the gravitationally non-active ones. Finally, our analysis suggests that the theory which should naturally incorporate such a cut-off[@bm3] is deprived of any experimental and theoretical foundation. We believe that our work provides a satisfactory answer to the intriguing and important question left open by Copeland et al. in the section of their review devoted to the possibility of measuring dark energy in laboratory experiments[@cope]: “time will tell who (if either) are correct”. According to our analysis, the opponents to the dark energy interpretation of the Koch et al.experiments[@koch] are correct. We would like to thank Luigi Amico, Marcello Baldo, Pino Falci and Dario Zappalà for many useful discussions. [99]{} P.J.E. Peebles, B. Ratra, Rev. Mod. Phys. 75, 559 (2003). T.Padmanahban, AIP Conf.Proc.[**[861]{}**]{}, 179 (2006). E.J.Copeland, M.Sami, S.Tsujikawa, Int.J.Mod.Phys.[**[D15]{}**]{}, 1753 (2006). S. Nobbenhuis, Found. Phys. [**36**]{}, 613 (2006). C.Beck, M.C.Mackey, Phys.Lett.[**[B605]{}**]{}, 295 (2005). C.Beck, M.C.Mackey, Physica [**[A379]{}**]{}, 101 (2007). R.H.Koch, D.J.Van Harlingen, J.Clarke, Phys.Rev.[**[B26]{}**]{}, 74 (1982). P.Jetzer, N.Straumann, Phys.Lett.[**[B606]{}**]{} 77 (2005). M. Doran, J.Jaeckel (DESY), [**[JCAP]{}**]{} 0608:010 (2006). G.Mahajan, S.Sarkar, T.Padmanahban, Phys.Lett.[**[B641]{}**]{}, 6 (2006). C.Beck, M.C.Mackey, Fluct.Noise Lett.7:C31 (2007). P.Jetzer, N.Straumann, Phys.Lett.[**[B639]{}**]{} 57 (2006). H.B.Callen, T.A.Welton, Phys.Rev.[**[83]{}**]{}, 34 (1951). J.C.Taylor, J.Phys.: Condens.Matter [**[19]{}**]{}, 106223 (2007). R.Kubo, M.Toda, N.Hashitsume [*Statistical Physics II*]{}, Springer-Verlag, Berlin (1985). V.Branchina, M.Di Liberto, I.Lodato, [*Fluctuation-dissipation theorem and harmonic oscillators*]{}, arXiv:0905.4254v1 \[cond-mat.stat-mech\]. Z.H.Barber, M.G.Blamire, [*Externally-Shunted High-Gap Josephson Junctions: Design, Fabrication and Noise Measurements*]{}, EPSRC grants EP/D029872/1. P.A.Warb, [*Externally-Shunted High-Gap Josephson Junctions: Design, Fabrication and Noise Measurements*]{}, EPSRC grants EP/D029783/1. C.Beck, M.C.Mackey, Int.J.Mod.Phys.[**[D17]{}**]{}, 71 (2008). [^1]: vincenzo.branchina@ct.infn.it [^2]: madiliberto@ssc.unict.it [^3]: ivlodato@ssc.unict.it [^4]: More generally, we could consider a local observable and a local generalized force, in which case we would have $\hat{V} = -\int d^3\,\vec r \hat{A}(\vec{r})f(\vec{r},t)$, and successively define a local susceptibility $\chi(\vec{r},t;\vec{r'},t')$ (see Eq.(\[chi\]) below). As this would add nothing to our argument, we shall restrict ourselves to $\vec r$-independent quantities. The extension to include local operators is immediate.

--- author: - | Dominic Joyce\ Lincoln College, Oxford, OX1 3DR, England date: May 1999 title: | Asymptotically Locally Euclidean\ metrics with holonomy ${\rm SU}(m)$ --- \[section\] \[thm\][Lemma]{} \[thm\][Proposition]{} \[thm\][Conjecture]{} \[thm\][Corollary]{} \#1[[(\[\#1\])]{}]{} \#1[\#1\^2]{} \#1[\#1 ]{} \#1[\#1 ]{} \#1[\#1 ]{} \#1[\#1 ]{} Introduction ============ Let $G$ be a finite subgroup of U$(m)$ acting freely on $\C^m\setminus\{0\}$. Then $\C^m/G$ has an [*isolated quotient singularity*]{} at 0. Suppose $(X,\pi)$ is a [*resolution*]{} of $\C^m/G$. Then $X$ is a noncompact complex manifold modelled at infinity on $\C^m/G$. In this paper we will study Kähler metrics $g$ on $X$ which are also [*Asymptotically Locally Euclidean*]{}, or [*ALE*]{} for short. This means that $g$ approximates the Euclidean metric $h$ on $\C^m/G$ by $g=h+O(r^{-2m})$, with appropriate decay in the derivatives of $g$. We are particularly interested in [*Ricci-flat*]{} ALE Kähler manifolds. The [*Calabi conjecture*]{} [@Cala1] describes the possible Ricci curvatures of Kähler metrics on a fixed compact complex manifold $M$, in terms of the first Chern class $c_1(M)$ of $M$. It was proved by Yau [@Yau] in 1976. The following theorem is a corollary of Yau’s proof. Let $M$ be a compact complex manifold admitting Kähler metrics, with $c_1(M)=0$. Then there is a unique Ricci-flat Kähler metric in each Kähler class on $M$. \[cyrfthm\] Our main results are Theorems \[alerfthm\] and \[alesumthm\]. Theorem \[alerfthm\] is an analogue of Theorem \[cyrfthm\] for ALE Kähler manifolds. It says that if $X$ is a resolution of $\C^m/G$ with $c_1(X)=0$, that is, a [*crepant resolution*]{}, then every Kähler class of ALE Kähler metrics on $X$ contains a unique Ricci-flat Kähler metric. Theorem \[alesumthm\] says that these metrics have holonomy ${\rm SU}(m)$. When $m=2$ the metrics were constructed explicitly by Kronheimer and others. Section 2 defines ALE metrics and ALE Kähler metrics, and §3 states the main results of the paper, postponing the proofs until §6, and gives some examples. Section 4 develops some analytical tools for ALE manifolds: Banach spaces of functions called weighted Hölder spaces, and elliptic regularity theory for the Laplacian $\Delta$ on them. In §5 we discuss $k$-forms and de Rham cohomology on ALE manifolds. Section 6 states a version of the Calabi conjecture for ALE manifolds. Only a sketch of the proof is given; a complete proof, following Yau [@Yau], will be given in the author’s book [@Joyc3 §8]. We apply this Calabi conjecture to prove Theorem \[alerfthm\], and then prove Theorem \[alesumthm\]. A number of other people have already written papers on noncompact versions of the Calabi conjecture, and I should at once admit that there is some overlap between their results and mine. In particular, Tian and Yau [@TiYa1; @TiYa2] and independently Bando and Kobayashi [@BaKo1; @BaKo2] prove the following result [@TiYa2 Cor. 1.1], [@BaKo2 Th. 1]: Let $X$ be a compact Kähler manifold with $c_1(X)>0$, and $D$ a smooth reduced divisor on $X$ such that $c_1(X)=\alpha[D]$ for some $\alpha>1$. Suppose $D$ admits a Kähler-Einstein metric with positive scalar curvature. Then $X\setminus D$ has a complete Ricci-flat Kähler metric. Also Tian and Yau give estimates on the decay of the curvature of their Ricci-flat metric. With a certain amount of work, the existence of the metrics of Theorem \[alerfthm\] follows from the theorem above. But our estimates on the asymptotic behaviour of the metrics are stronger than those proved by Tian and Yau. For example, we show that the curvature is $O(r^{-2m-2})$ for large $r$, but Tian and Yau only show that it is $O(r^{-3})$, which is not good enough for the applications we have in mind. In a sequel [@Joyc4] we will extend the material of this paper to construct a class of Ricci-flat Kähler metrics on crepant resolutions of [*non-isolated*]{} singularities $\C^m/G$, which we will call [*Quasi-ALE metrics*]{}. These metrics are not covered by the work of Tian and Yau or Bando and Kobayashi. The original motivation for this paper and [@Joyc4] is that ALE and Quasi-ALE metrics with holonomy SU(2), SU(3), SU(4) and Sp(2) are essential ingredients in a new construction by the author of compact manifolds with the exceptional holonomy groups $G_2$ and Spin(7), which generalizes that of [@Joyc1; @Joyc2]. This construction will be described at length in the author’s book [@Joyc3], which also discusses the results of this paper and [@Joyc4]. Asymptotically Locally Euclidean metrics ======================================== Suppose $G$ is a finite subgroup of ${\rm SO}(n)$ that acts freely on $\R^n\setminus\{0\}$. Then $\R^n/G$ has an [*isolated quotient singularity*]{} at 0. Let $h$ be the Euclidean metric on $\R^n$. Then $h$ is preserved by $G$, as $G\subset{\rm SO}(n)$, and so $h$ descends to $\R^n/G$. Let $r$ be the radius function on $\R^n/G$, that is, $r(x)$ is the distance from $0$ to $x$ calculated using $h$. We will define a natural class of noncompact Riemannian manifolds $(X,g)$ called [*ALE manifolds*]{}, that have one infinite end upon which the metric $g$ asymptotically resembles the metric $h$ on $\R^n/G$ for large $r$. Let $X$ be a noncompact manifold of dimension $n$, and $g$ a Riemannian metric on $X$. We say that $(X,g)$ is an [*Asymptotically Locally Euclidean manifold*]{} asymptotic to $\R^n/G$, or an [*ALE manifold*]{} for short, and we say that $g$ is an [*ALE metric*]{}, if the following conditions hold. There should exist a compact subset $S\subset X$ and a map $\pi:X\setminus S\rightarrow\R^n/G$ that is a diffeomorphism between $X\setminus S$ and the subset $\{z\in\R^n/G:r(z)>R\}$ for some fixed $R>0$. Under this diffeomorphism, the push-forward metric $\pi_*(g)$ should satisfy $$\nabla^k\bigl(\pi_*(g)-h\bigr)=O(r^{-n-k})\quad \text{on $\{z\in\R^n/G:r(z)>R\}$,} \label{aleeq}$$ for all $k\ge 0$. Here $\nabla$ is the Levi-Civita connection of $h$, and $T=O(r^{-j})$ if $\md{T}\le Kr^{-j}$ for some $K>0$. \[aledef\] If $G=\{1\}$, so that $(X,g)$ is asymptotic to $\R^n$, then we call $(X,g)$ an [*Asymptotically Euclidean manifold*]{}, or [*AE manifold*]{}. We shall call the map $\pi:X\setminus S \rightarrow\R^n/G$ an [*asymptotic coordinate system*]{} for $X$. Equation says that towards infinity the metric $g$ on $X$ (and its derivatives) must converge to the Euclidean metric on $\R^n/G$, with a given rate of decay. We will explain in §3 why we have chosen the powers $r^{-n-k}$ here. Let $(X,g)$ be an ALE manifold asymptotic to $\R^n/G$. We say that a smooth function $\rho:X\rightarrow[1,\infty)$ is a [*radius function*]{} on $X$ if, given any asymptotic coordinate system $\pi:X\setminus S\rightarrow\R^n/G$, we have $$\nabla^k\bigl(\pi_*(\rho)-r\bigr)=O(r^{1-n-k})\quad \text{on $\{z\in\R^n/G:r(z)>R\}$,}$$ for all $k\ge 0$. This condition is independent of the choice of asymptotic coordinate system, and radius functions exist for every ALE manifold. \[aleraddef\] A radius function is a function $\rho$ on $X$ that approximates the function $r$ on $\R^n/G$ near infinity. In doing analysis on ALE manifolds, we will find it useful to consider Hölder spaces of functions in which the norms are weighted by powers $\rho^\beta$ of a radius function. Note that by definition $\rho\ge 1$, so we do not have to worry about small values of $\rho$. Here is one way to think about ALE metrics. The manifold $X$ is noncompact, but it can be compactified in a natural way by adding the boundary ${\mathcal S}^{n-1}/G$ at infinity. So we can instead regard $X$ as a [*compact manifold with boundary*]{}. Then ALE metrics are metrics on $X$ satisfying a certain natural boundary condition. It is a general principle in differential geometry that most results about compact manifolds can also be extended to results about compact manifolds with boundary, provided the right boundary conditions are imposed in the problem. ALE manifolds are an example of this principle, because many results about compact Riemannian manifolds have natural analogues for ALE manifolds. Next we define [*ALE Kähler metrics*]{}. Suppose $G$ is a finite subgroup of U$(m)$ acting freely on $\C^m\setminus\{0\}$. Then $\C^m/G$ has an isolated quotient singularity at 0, and the standard Hermitian metric $h$ on $\C^m$ descends to $\C^m/G$. Let $r$ be the radius function on $\C^m/G$. Suppose $(X,\pi)$ is a [*resolution*]{} of $\C^m/G$, that is, $X$ is a normal nonsingular variety with a proper birational morphism $\pi:X\rightarrow\C^m/G$. Then we can consider metrics on $X$ which are both Kähler, and ALE. Let $(X,\pi)$ be a resolution of $\C^m/G$, with complex structure $J$, and let $g$ be a Kähler metric on $X$. We say that $(X,J,g)$ is an [*ALE Kähler manifold asymptotic to*]{} $\C^m/G$, and that $g$ is an [*ALE Kähler metric*]{}, if for some $R>0$ we have $$\nabla^k\bigl(\pi_*(g)-h\bigr)=O(r^{-2m-k})\quad \text{on $\{z\in\C^m/G:r(z)>R\}$,} \label{alekeq}$$ for all $k\ge 0$. We say that a smooth function $\rho:X\rightarrow[1,\infty)$ is a [*radius function*]{} on $X$ if $\rho=\pi^*(r)$ on the subset $\bigl\{x\in X:\pi^*(r)\ge 2\bigr\}$. A radius function exists for every ALE Kähler manifold. \[alekdef\] Because $X$ is a resolution of $\C^m/G$, it comes equipped with a resolving map $\pi:X\rightarrow\C^m/G$, which gives a natural asymptotic coordinate system for $X$. The consequence of using this preferred asymptotic coordinate system is that on an ALE Kähler manifold $(X,J,g)$, both the metric $g$ and the complex structure $J$ are simultaneously asymptotic to the metric and complex structure on $\C^m/G$. We also use $\pi$ to simplify the definition of radius function. In dimension 2 one can also desingularize $\C^2/G$ by [ *deformation*]{}. By adopting a slightly more general definition of ALE Kähler manifold we can include deformations and resolutions of deformations of $\C^2/G$, and most of our results also apply to them. This will be discussed in [@Joyc3 §8.9]. However, by [*Schlessinger’s Rigidity Theorem*]{} [@Schl], if $m\ge 3$ then an isolated quotient singularity $\C^m/G$ admits no nontrivial deformations. Ricci-flat ALE Kähler manifolds =============================== We now state some results on Ricci-flat ALE Kähler manifolds, and give some examples. The proofs will be deferred until §6. A resolution $(X,\pi)$ of $\C^m/G$ with $c_1(X)=0$ is called a [*crepant resolution*]{}, as in Reid [@Reid]. A great deal is known about the algebraic geometry of crepant resolutions, especially when $\dim X$ is 2 or 3. In particular, for $\C^m/G$ to admit a crepant resolution $G$ must be a subgroup of ${\rm SU}(m)$, and when $m$ is 2 or 3 a crepant resolution of $\C^m/G$ exists for every finite subgroup $G$ of ${\rm SU}(m)$. Our first proposition shows that Ricci-flat ALE Kähler metrics exist only on [*crepant resolutions*]{}. The proof is elementary, and we omit it. Let $G$ be a finite subgroup of ${\rm U}(m)$ acting freely on $\C^m\setminus\{0\}$, let $(X,\pi)$ be a resolution of $\C^m/G$, and suppose $g$ is a Ricci-flat ALE Kähler metric on $X$. Then $X$ is a crepant resolution of $\C^m/G$ and $G\subset{\rm SU}(m)$. \[alerfcrepprop\] Next we define [*Kähler classes*]{} and the [*Kähler cone*]{} for ALE manifolds. Let $(X,J,g)$ be an ALE Kähler manifold asymptotic to $\C^m/G$ for some $m>1$, with Kähler form $\omega$. Then $\omega$ defines a de Rham cohomology class $[\omega]\in H^2(X,\R)$ called the [*Kähler class*]{} of $g$. Define the [*Kähler cone*]{} $\mathcal K$ of $X$ to be the set of Kähler classes $[\omega]\in H^2(X,\R)$ of ALE Kähler metrics on $(X,J)$. It is not difficult to prove that $\mathcal K$ is an open convex cone in $H^2(X,\R)$, which does not contain zero. The following two theorems will be proved in §6. Let $G$ be a nontrivial finite subgroup of ${\rm SU}(m)$ acting freely on $\C^m\setminus\{0\}$, and $(X,\pi)$ a crepant resolution of $\C^m/G$. Then each Kähler class of ALE Kähler metrics on $X$ contains a unique Ricci-flat ALE Kähler metric $g$. The Kähler form $\omega$ of $g$ satisfies $$\pi_*(\omega)=\omega_0+A\,\d\d^c(r^{2-2m})+\d\d^c\chi \label{pisomeq}$$ on the set $\bigl\{z\in\C^m/G:r(z)>R\bigr\}$, where $A<0$ and $R>0$ are constants, $\omega_0$ is the Kähler form of the Euclidean metric on $\C^m/G$, $r$ the radius function on $\C^m/G$, and $\chi$ a smooth function on $\bigl\{z\in\C^m/G:r(z)>R\bigr\}$ such that $\nabla^k\chi=O(r^{\gamma-k})$ for each $k\ge 0$ and $\gamma\in (1-2m,2-2m)$. \[alerfthm\] Theorem \[alerfthm\] is the main result of this paper, and is an analogue of Theorem \[cyrfthm\] for ALE Kähler manifolds. We use the notation that $\d^cf=i(\overline\partial-\partial)f$, when $f$ is a differentiable function on a complex manifold. Then $\d^c$ is a real operator, and $\d\d^c=2i\partial\overline\partial$. Note that because $A<0$ in Theorem \[alerfthm\], the term $A\,\d\d^c(r^{2-2m})$ in is nonzero. Therefore $\pi_*(g)-h$ decays with order exactly $O(r^{-2m})$, and similarly $\nabla^k(\pi_*(g)-h)$ decays with order exactly $O(r^{-2m-k})$. Thus in Definition \[aledef\] the decay rates given in are [*sharp*]{} for all Ricci-flat ALE Kähler metrics, and cannot be improved upon. This is why we chose the powers $r^{-n-k}$ in our definition of ALE metrics. Let $G$ be a nontrivial finite subgroup of ${\rm SU}(m)$ acting freely on $\C^m\setminus\{0\}$, let $(X,\pi)$ be a crepant resolution of $\C^m/G$, and let $g$ be a Ricci-flat ALE Kähler metric on $X$. Then $g$ has holonomy ${\rm SU}(m)$. \[alesumthm\] For an introduction to holonomy groups of Riemannian manifolds, and the connection between Ricci-flat Kähler metrics and holonomy ${\rm SU}(m)$, see Salamon [@Sala]. Examples -------- ALE Kähler manifolds with holonomy SU(2) are already very well understood. Eguchi and Hanson [@EgHa] gave an explicit formula in coordinates for the metrics of ALE spaces with holonomy SU(2) asymptotic to $\C^2/\{\pm1\}$, and this was generalized by Gibbons and Hawking [@GiHa] to explicit expressions for ALE spaces asymptotic to $\C^2/{\bb Z}_k$ for $k\ge 2$. More generally, Kronheimer [@Kron1; @Kron2] gave an explicit, algebraic construction of every ALE manifold with holonomy SU(2), using the hyperkähler quotient. Thus, we can write down many explicit examples of ALE manifolds with holonomy SU(2). For $m\ge 3$, Calabi [@Cala2 p. 285] found an explicit ALE Kähler manifold with holonomy ${\rm SU}(m)$ asymptotic to $\C^m/{\bb Z}_m$, which we describe next. In the case $m=2$, Calabi’s example coincides with the Eguchi–Hanson metric. Let $\C^m$ have complex coordinates $(z_1,\ldots,z_m)$, let $\zeta={\rm e}^{2\pi i/m}$, and let $\alpha$ act on $\C^m$ by $\alpha:(z_1,\ldots,z_m)\mapsto(\zeta z_1,\ldots,\zeta z_m)$. Then $\alpha^m=1$, and the group $G=\langle\alpha\rangle$ generated by $\alpha$ is a subgroup of ${\rm SU}(m)$ isomorphic to ${\bb Z}_m$, which acts freely on $\C^m\setminus\{0\}$. Thus the quotient $\C^m/G$ has an isolated singular point at 0. Let $(X,\pi)$ be the [*blow-up*]{} of $\C^m/G$ at 0, so that $\pi^{-1}(0)\cong \bb{CP}^{m-1}$. It is easy to show that $X$ is in fact a [*crepant resolution*]{} of $\C^m/G$. Let $r$ be the radius function on $\C^m/G$, and define $f:\C^m/G\,\big\backslash\{0\}\rightarrow\R$ by $$f=\sqrt[m]{r^{2m}+1}+{1\over m}\sum_{j=0}^{m-1}\zeta^j \log\left(\sqrt[m]{r^{2m}+1}-\zeta^j\right). \label{calafdef}$$ To define the logarithm of the complex number $\sqrt[m]{r^{2m}+1}-\zeta^j$ we cut $\C$ along the negative real axis, and set $\log(R{\rm e}^{i\theta})=\log R+i\theta$ for $R>0$ and $\theta\in(-\pi,\pi)$. Then $f$ is well-defined, and it is a smooth [*real*]{} function on $\C^m/G\,\big\backslash\{0\}$, despite its complex definition. Define a (1,1)-form $\omega$ on $X\setminus\pi^{-1}(0)$ by $\omega=\d\d^c\pi^*(f)$. It can be shown that $\omega$ extends to a smooth, closed, positive (1,1)-form on all of $X$. Let $g$ be the Kähler metric on $X$ with Kähler form $\omega$. Then Calabi [@Cala2 §4] shows that $g$ is complete and Ricci-flat, with $\Hol(g)={\rm SU}(m)$. Equation is derived from [@Cala2 eqn (4.14), p. 285]. Note also that the action of ${\rm U}(m)$ on $\C^m$ pushes down to $\C^m/G$ and lifts through $\pi$ to $X$, and $g$ is invariant under this action of ${\rm U}(m)$ on $X$. This metric $g$ on $X$ is an [*ALE Kähler metric*]{}. To prove this, we show using that $$f=r^2-{1\over m(m\!-\!1)}\,r^{2-2m}+O(r^{-2m})\quad \text{on $\C^m/G\,\big\backslash\{0\}$, for large $r$.}$$ Now the Kähler form of the Euclidean metric on $\C^m/G$ is $\omega_0=\d\d^c(r^2)$. Hence $$\pi_*(\omega)=\omega_0-\frac{1}{m(m\!-\!1)}\,\d\d^c(r^{2-2m}) +\d\d^c\chi \quad\text{on $\C^m/G\,\big\backslash\{0\}$,} \label{pisomeqex}$$ where $\chi=f-r^2+{1\over m(m-1)}r^{2-2m}$. It is easy to show that $\nabla^k\chi=O(r^{-k-2m})$ on $\C^m/G\,\big\backslash\{0\}$ for large $r$, and it quickly follows that $g$ is an ALE Kähler metric on $X$, by Definition \[alekdef\]. Also, $g$ is one of the Ricci-flat ALE Kähler metrics of Theorem \[alerfthm\], and comparing with we see that $A=-\,{1\over m(m-1)}$, which verifies that $A<0$. \[alecalabiex\] For $m\ge 3$, the metrics of Example \[alecalabiex\] are the [*only*]{} explicit examples of ALE metrics with holonomy ${\rm SU}(m)$ that are known, at least to the author. It is possible to find these metrics explicitly because they have a large symmetry group ${\rm U}(m)$, whose orbits are of real codimension 1 in $X$. Because of this, the problem can be reduced to a nonlinear, second-order ODE in one real variable, which can then be explicitly solved. It is a natural question whether we can find an explicit, algebraic form for any or all of the other ALE metrics with holonomy ${\rm SU}(m)$ for $m\ge 3$, that exist on crepant resolutions of $\C^m/G$ by Theorem \[alerfthm\]. The author believes that general ALE metrics with holonomy ${\rm SU}(m)$ for $m\ge 3$ are essentially transcendental, nonalgebraic objects, and that one cannot write them down explicitly using simple functions. Furthermore, the author conjectures that for $m\ge 3$, the metrics of Example \[alecalabiex\] are the only ALE metrics with holonomy ${\rm SU}(m)$ that can be written down explicitly in coordinates. Analysis on ALE manifolds ========================= Let $(M,g)$ be a Riemannian manifold. Then the [*Hölder spaces*]{} $C^{k,\alpha}(M)$ are Banach spaces of functions on $M$, defined in Besse [@Bess p. 456-7]. When $M$ is compact, elliptic operators such as the Laplacian $\Delta$ have very good regularity properties on Hölder spaces. Here is a typical elliptic regularity result, following from [@Bess Th. 27 & Th. 31, p. 463-4]. Theorems of this kind are essential tools in analytic problems such as the proof of the Calabi conjecture. Let $(M,g)$ be a compact Riemannian manifold, let $k\ge 0$ be an integer, and $\alpha\in (0,1)$. Then for each $f\in C^{k,\alpha}(M)$ with $\int_Mf\,\d V_g=0$ there exists a unique $u\in C^{k+2,\alpha}(M)$ with $\int_Mu\,\d V_g=0$ and $\Delta u=f$. Moreover, $\nm{u}_{C^{k+2,\alpha}}\le C\nm{f}_{C^{k,\alpha}}$ for some $C>0$ independent of $u$ and $f$. \[mschauderthm\] However, if $(X,g)$ is an ALE manifold then the results of Theorem \[mschauderthm\] are false for $X$. This tells us that the $C^{k,\alpha}(X)$ are not good choices of Banach spaces of functions for studying elliptic operators on an ALE manifold. Instead, it turns out to be helpful to introduce [*weighted Hölder spaces*]{}, which we define next. Let $(X,g)$ be an ALE manifold asymptotic to $\R^n/G$, and $\rho$ a radius function on $X$. For $\beta\in\R$ and $k$ a nonnegative integer, define $C^k_\beta(X)$ to be the space of continuous functions $f$ on $X$ with $k$ continuous derivatives, such that $\rho^{j-\beta}\bmd{\nabla^jf}$ is bounded on $X$ for $j=0,\ldots,k$. Define the norm $\nm{\,.\,}_{\smash{C^k_\beta}}$ on $C^k_\beta(X)$ by $$\nm{f}_{C^k_\beta}=\sum_{j=0}^k\sup_X\bmd{\rho^{j-\beta}\nabla^jf}.$$ Let $\delta(g)$ be the injectivity radius of $g$, and write $d(x,y)$ for the distance between $x,y$ in $X$. For $T$ a tensor field on $X$ and $\alpha,\gamma\in\R$, define $$\bigl[T\bigr]_{\alpha,\gamma}= \sup\begin{Sb}x\ne y\in X\\d(x,y)<\delta(g)\end{Sb} \left[\min\bigl(\rho(x),\rho(y)\bigr)^{-\gamma}\cdot {\bmd{T(x)-T(y)}\over d(x,y)^\alpha}\right]. \label{aleholdereq}$$ Here we interpret $\md{T(x)-T(y)}$ using parallel translation along the unique geodesic of length $d(x,y)$ joining $x$ and $y$. For $\beta\in\R$, $k$ a nonnegative integer, and $\alpha\in(0,1)$, define the [*weighted Hölder space*]{} $C^{k,\alpha}_\beta(X)$ to be the set of $f\in C^k_\beta(X)$ for which the norm $$\bnm{f}_{C^{k,\alpha}_\beta}=\bnm{f}_{C^k_\beta} +\bigl[\nabla^k f\bigr]_{\alpha,\beta-k-\alpha} \label{aleholdnormeq}$$ is finite. Define $C^\infty_\beta(X)$ to be the intersection of the $C^k_\beta(X)$ for all $k\ge 0$. Both $C^k_\beta(X)$ and $C^{k,\alpha}_\beta(X)$ are Banach spaces, but $C^\infty_\beta(X)$ is not a Banach space. \[wholderdef\] This definition is taken from Lee and Parker [@LePa §9]. A function $f$ in $C^k_\beta(X)$ or $C^{k,\alpha}_\beta(X)$ grows at most like $\rho^\beta$ as $\rho\rightarrow\infty$, and so the index $\beta$ should be interpreted as an [*order of growth*]{}. Similarly, the derivatives $\nabla^jf$ grow at most like $\rho^{\beta-j}$ for $j=1,\ldots,k$. As vector spaces of functions $C^k_\beta(X)$ and $C^{k,\alpha}_\beta(X)$ are independent of the choice of radius function $\rho$. The norms on these spaces do depend on $\rho$, but not in a significant way, as all choices of $\rho$ give equivalent norms. There is also another useful class of Banach spaces on ALE manifolds, the [*weighted Sobolev spaces*]{} $L^q_{k,\beta}(X)$, which we will not define. They have similar analytic properties to the weighted Hölder spaces, and are described in [@LePa §9]. We have chosen to use weighted Hölder spaces instead, as they are often more convenient for nonlinear problems. Next we discuss the analysis of the Laplacian $\Delta$ on ALE manifolds. Much work has been done on the behaviour of $\Delta$ on weighted Sobolev spaces and Hölder spaces on $\R^n$, and more generally on AE manifolds. A useful guide, with references, can be found in Lee and Parker [@LePa §9]. Most of these results apply immediately to ALE manifolds, with only very minor cosmetic changes to their proofs. Let $(X,g)$ be an ALE manifold of dimension $n$ asymptotic to $\R^n/G$, let $\beta,\gamma\in\R$ satisfy $\beta+\gamma<2-n$, and suppose $u\in C^2_\beta(X)$ and $v\in C^2_\gamma(X)$. Then $$\int_X u\,\Delta v\,\d V_g=\int_X v\,\Delta u\,\d V_g. \label{intxuveq}$$ Let $\rho$ be a radius function on $X$. Then $\Delta(\rho^{2-n})\in C^\infty_{-2n}(X)$ and $$\int_X\Delta(\rho^{2-n})\,\d V_g= {(n\!-\!2)\,\Omega_{n-1}\over\md{G}}, \label{intderhoeq}$$ where $\Omega_{n-1}$ is the volume of the unit sphere ${\mathcal S}^{n-1}$ in $\R^n$. \[alede1prop\] Let $S_R$ be the subset $\{x\in X:\rho(x)\le R\}$ in $X$. Stokes’ Theorem gives that $$\int_{S_R}\bigl(u\Delta v-v\Delta u\bigr)\,\d V_g=\int_{\partial S_R} \bigl[(u\nabla v-v\nabla u)\cdot{\bf n}\bigr]\,\d V_g, \label{uvsreq}$$ where $\bf n$ is the inward-pointing unit normal to $\partial S_R$. But for large $R$ we have $\vol(\partial S_R)=O(R^{n-1})$ and $u\nabla v-v\nabla u=O(R^{\beta+\gamma-1})$ on $\partial S_R$, so that the r.h.s. of is $O(R^{\beta+\gamma+n-2})$. Since $\beta+\gamma<2-n$ we see that the r.h.s. of tends to zero as $R\rightarrow\infty$, and this proves . The point about the power $\rho^{2-n}$ is that $\Delta(r^{2-n})=0$ away from 0 in $\R^n/G$. Using the definitions of radius function and ALE manifold one can show that $\Delta(\rho^{2-n})\in C^\infty_{-2n}(X)$, as we want. Using Stokes’ Theorem again we find that $$\int_{S_R}\Delta(\rho^{2-n})\,\d V_g=\int_{\partial S_R} \bigl[\nabla(\rho^{2-n})\cdot{\bf n}\bigr]\,\d V_g.$$ But for large $R$ we have $\nabla(\rho^{2-n})\cdot{\bf n}\approx (n\!-\!2)R^{1-n}$ and $\vol(S_R)\approx R^{n-1}\Omega_{n-1}/\md{G}$. Thus, letting $R\rightarrow\infty$ gives . Let $n>2$ and $k\ge 0$ be integers and $\alpha\in(0,1)$, and let $\R^n$ have its Euclidean metric. Then - Suppose $\beta\in(-n,-2)$. Then for each $f\in C^{k,\alpha}_\beta(\R^n)$ there is a unique $u\in C^{k+2,\alpha}_{\beta+2}(\R^n)$ with $\Delta u=f$. - Suppose $\beta\in(-1-n,-n)$. Then for each $f\in C^{k,\alpha}_\beta(\R^n)$ there exists $u\in C^{k+2,\alpha}_{\beta+2}(\R^n)$ with $\Delta u=f$ if and only if $\int_{\R^n}f\,\d V=0$, and $u$ is then unique. In each case $\nm{u}_{\smash{C^{k+2,\alpha}_{\beta+2}}}\le C\nm{f}_{\smash{C^{k,\alpha}_\beta}}$ for some $C>0$ depending only on $n,k,\alpha$ and $\beta$. \[alede1thm\] This is an analogue for $\R^n$ of Theorem \[mschauderthm\]. If $u\in C^2_{\beta+2}(\R^n)$ for $\beta<-2$ and $\Delta u=f$, then by [@GiTr §2.4] we have $$u(y)={1\over(n-2)\Omega_{n-1}}\int_{x\in\R^n}\md{x-y}^{2-n}f(x)\d x, \label{intufeq}$$ where $\Omega_{n-1}$ is the volume of the unit sphere ${\mathcal S}^{n-1}$ in $\R^n$. This is [*Green’s representation*]{} for $u$. Let $\rho$ be a radius function on $\R^n$. Then $\md{f(x)}\le\nm{f}_{\smash{C^0_\beta}}\rho(x)^\beta$, so gives $$\md{u(y)}\le{1\over(n-2)\Omega_{n-1}}\,\nm{f}_{C^0_\beta} \cdot\int_{x\in\R^n}\md{x-y}^{2-n}\rho(x)^\beta\d x.$$ We split this into integrals over the three regions $\md{x}\le{1\over 2}\md{y}$, ${1\over 2}\md{y}<\md{x}\le 2\md{y}$ and $\md{x}>2\md{y}$ in $\R^n$. Estimating the integral on each region separately we prove $$\int_{x\in\R^n}\md{x-y}^{2-n}\rho(x)^\beta\d x\le \begin{cases} C'\rho(y)^{\beta+2} & \text{for $\beta\in(-n,-2)$,} \\ C'\rho(y)^{2-n} & \text{for $\beta<-n$.} \end{cases}$$ In case (a), if $\beta\in(-n,-2)$ then $\md{u(y)}\le C''\nm{f}_{\smash{C^0_\beta}}\rho(y)^{\beta+2}$ for some $C''>0$ depending only on $n$ and $\beta$, and so $u\in C^0_{\beta+2}(\R^n)$ and $\nm{u}_{\smash{C^0_{\beta+2}}} \le C''\nm{f}_{\smash{C^0_\beta}}$. One can extend this to show that $u\in C^{k+2,\alpha}_{\beta+2}(\R^n)$ and $\nm{u}_{\smash{C^{k+2,\alpha}_{\beta+2}}}\le C\nm{f}_{\smash{C^{k,\alpha}_\beta}}$ for some $C>0$ using the method of [*Schauder estimates*]{}, as in [@GiTr §6]. The difficulty in doing this is to correctly include the powers of $\rho$ involved in the weighted Hölder norm. To do this, for each $x\in\R^n$ we consider the ball $B_{\rho(x)/2}(x)$ of radius ${1\over 2}\rho(x)$ about $x$ in $\R^n$. On this ball we have $u=O\bigl(\rho(x)^{\beta+2}\bigr)$, $\nabla^jf=O\bigl(\rho(x)^{\beta-j}\bigr)$ for $j=0,\dots,k$, and $[\nabla^kf]_\alpha=O\bigl(\rho(x)^{\beta-k-\alpha}\bigr)$. Using the Schauder interior estimates on the unit ball in $\R^n$ and rescaling distances by a factor ${1\over 2}\rho(x)$, we show that $\nabla^ju=O\bigl(\rho(x)^{\beta+2-j}\bigr)$ for $j=0,\dots,k\!+\!2$ and $[\nabla^{k+2}u]_\alpha=O\bigl(\rho(x)^{\beta-k-\alpha}\bigr)$ on the interior of $B_{\rho(x)/2}(x)$. Thus $u\in C^{k+2,\alpha}_{\beta+2}(\R^n)$ and $\nm{u}_{\smash{C^{k+2,\alpha}_{\beta+2}}}\le C\nm{f}_{\smash{C^{k,\alpha}_\beta}}$, completing the proof of case (a). Next we prove (b). Suppose $\beta\in(-1-n,-n)$, $u\in C^{k+2,\alpha}_{\beta+2}(\R^n)$ and $\Delta u=f$. Then $$\int_{\R^n}f\,\d V=\int_{\R^n}1\Delta u\,\d V =\int_{\R^n}u\Delta(1)\,\d V=0$$ by Proposition \[alede1prop\], since $u\in C^2_{\beta+2}(\R^n)$ and $1\in C^2_0(\R^n)$ and $\beta+2+0<2-n$. Thus, given $f\in C^{k,\alpha}_\beta(\R^n)$, there can only exist $u\in C^{k+2,\alpha}_{\beta+2}(\R^n)$ with $\Delta u=f$ if $\int_{\R^n}f\,\d V=0$. So suppose that $\int_{\R^n}f\,\d V=0$, and define $u$ by $$u(y)={1\over(n-2)\Omega_{n-1}}\int_{x\in\R^n} \Bigl[\md{x-y}^{2-n}-\rho(y)^{2-n}\Bigr]f(x)\d x. \label{intufreq}$$ Since $\int_{\R^n}f\,\d V=0$ the term involving $\rho(y)^{2-n}$ in this integral vanishes, so the equation reduces to and thus $\Delta u=f$. From we see that $$\md{u(y)}\le{1\over(n-2)\Omega_{n-1}}\,\nm{f}_{\smash{C^0_\beta}} \cdot\int_{x\in\R^n}\Bigl\vert\md{x-y}^{2-n}-\rho(y)^{2-n}\Bigr\vert \rho(x)^\beta\d x,$$ and estimating as before shows that $\md{u(y)}\le C\nm{f}_{\smash{C^0_\beta}}\rho(y)^{\beta+2}$ when $\beta\in(-1-n,-n)$. Thus $u\in C^0_{\beta+2}(\R^n)$ and $\nm{u}_{\smash{C^0_{\beta+2}}}\le C\nm{f}_{\smash{C^0_\beta}}$. The rest of case (b) follows as above. Now we extend Theorem \[alede1thm\] to ALE manifolds. Suppose $(X,g)$ is an ALE manifold asymptotic to $\R^n/G$ for $n>2$, and $\rho$ a radius function on $X$. Let $k\ge 0$ be an integer and $\alpha\in(0,1)$. Then - Let $\beta\in(-n,-2)$. Then there exists $C>0$ such that for each $f\in C^{k,\alpha}_\beta(X)$ there is a unique $u\in C^{k+2,\alpha}_{\beta+2}(X)$ with $\Delta u=f$, which satisfies $\nm{u}_{\smash{C^{k+2,\alpha}_{\beta+2}}}\le C\nm{f}_{\smash{C^{k,\alpha}_\beta}}$. - Let $\beta\in(-1-n,-n)$. Then there exist $C_1,C_2>0$ such that for each $f\in C^{k,\alpha}_\beta(X)$ there is a unique $u\in C^{k+2,\alpha}_{2-n}(X)$ with $\Delta u=f$. Moreover $u=A\rho^{2-n}+v$, where $$A={\md{G}\over(n\!-\!2)\,\Omega_{n-1}}\cdot\int_Xf\,\d V_g \label{adefeq}$$ and $v\in C^{k+2,\alpha}_{\beta+2}(X)$ satisfy $\md{A}\le C_1\nm{f}_{\smash{C^0_\beta}}$ and $\nm{v}_{\smash{C^{k+2,\alpha}_{\beta+2}}}\le C_2\nm{f}_{\smash{C^{k,\alpha}_\beta}}$. Here $\Omega_{n-1}$ is the volume of the unit sphere ${\mathcal S}^{n-1}$ in $\R^n$. \[alede2thm\] The theory of weighted Hölder spaces on AE manifolds and the Laplacian is developed by Chaljub-Simon and Choquet-Bruhat [@ChCh], who restrict their attention to the case $n=3$. In particular, they prove part (a) of the Theorem for the case $n=3$, $k=0$ and $G=\{1\}$, [@ChCh p. 15-16]. Their proof uses a result equivalent to part (a) of Theorem \[alede1thm\] in the case $n=3$ and $k=0$. By using Theorem \[alede1thm\] together with the methods of [@ChCh] one can show that Theorem \[alede1thm\] applies not only to $\R^n$ with its Euclidean metric, but also to any ALE manifold $(X,g)$ asymptotic to $\R^n/G$. This proves case (a) of of the Theorem immediately. For case (b), let $f\in C^{k,\alpha}_\beta(X)$, and define $A$ by . Then by equation we have $\int_X\bigl[f-\Delta(A\rho^{2-n})\bigr]\d V_g=0$. Also $\Delta(\rho^{2-n})\in C^\infty_{-2n}(X)$ by Proposition \[alede1prop\], and so $f-\Delta(A\rho^{2-n})$ lies in $C^{k,\alpha}_\beta(X)$ and has integral zero on $X$. Since $\md{f}\le\nm{f}_{\smash{C^0_\beta}}\rho^\beta$ we have $\md{A}\le C_1\nm{f}_{\smash{C^0_\beta}}$ for $C_1=\int_X\rho^\beta\,\d V_g$, as we have to prove. Applying case (b) of Theorem \[alede1thm\] for $X$ to $f-\Delta(A\rho^{2-n})$, we see that there is a unique $v\in C^{k+2,\alpha}_{\beta+2}(X)$ with $\Delta v=f-\Delta(A\rho^{2-n})$, which satisfies $$\nm{v}_{\smash{C^{k+2,\alpha}_{\beta+2}}}\le C\bigl(\nm{f}_{\smash{C^{k,\alpha}_\beta}}+\md{A}\cdot \nm{\Delta(\rho^{2-n})}_{\smash{C^{k,\alpha}_\beta}}\bigr). \label{vckaleq}$$ Defining $u=A\rho^{2-n}+v$ gives $\Delta u=f$ as we want. Clearly $u\in C^{k+2,\alpha}_{2-n}(X)$, and the inequality $\nm{v}_{\smash{C^{k+2,\alpha}_{\beta+2}}}\le C_2\nm{f}_{\smash{C^{k,\alpha}_\beta}}$ then follows from and the estimate on $\md{A}$ above. Exterior forms and de Rham cohomology ===================================== Let $(X,g)$ be an ALE manifold asymptotic to $\R^n/G$. Let $H^*(X,\R)$ be the de Rham cohomology of $X$, and $H^*_c(X,\R)$ the de Rham cohomology of $X$ [*with compact support*]{}. That is, $$H^k_c(X,\R)={\bigl\{\eta:\text{$\eta$ is a smooth, closed, compactly-supported $k$-form on $X$}\bigr\} \over\bigl\{\d\zeta:\text{$\zeta$ is a smooth, compactly-supported $(k\!-\!1)$-form on $X$}\bigr\}}.$$ Both $H^k(X,\R)$ and $H^k_c(X,\R)$ are finite-dimensional vector spaces. Let us regard $X$ as a compact manifold with boundary ${\mathcal S}^{n-1}/G$. Using the long exact sequence $$\ldots\rightarrow H^k_c(X,\R)\rightarrow H^k(X,\R)\rightarrow H^k({\mathcal S}^{n-1}/G,\R) \rightarrow H^{k+1}_c(X,\R)\rightarrow\ldots,$$ the de Rham cohomology of ${\mathcal S}^{n-1}/G$, and the fact that $H^k_c(X,\R)\cong\bigl[H^{n-k}(X,\R)\bigr]^*$ by Poincaré duality for manifolds with boundary, one can show that $$\begin{split} & H^0(X,\R)=\R,\quad H^0_c(X,\R)=0,\quad H^n(X,\R)=0,\quad H^n_c(X,\R)=\R,\quad\text{and}\\ & H^k(X,\R)\cong H^k_c(X,\R)\cong \bigl[H^{n-k}(X,\R)\bigr]^*\cong\bigl[H^{n-k}_c(X,\R)\bigr]^* \quad\text{for $0<k<n$.} \end{split}$$ Now the material on weighted Hölder spaces of functions in §4 generalizes naturally to weighted Hölder spaces of $k$-forms on ALE manifolds $(X,g)$, so we may define the spaces $C^{l,\alpha}_\beta(\Lambda^kT^*X)$ and $C^\infty_\beta(\Lambda^kT^*X)$ in the obvious way. Similarly, the results of §4 on the Laplacian $\Delta$ on functions generalize to results on the Laplacian $\Delta=\d\d^*+\d^*\d$ on $k$-forms. These tools can be used to generalize the ideas of Hodge theory to ALE manifolds. In particular, one can prove the following result. Let $(X,g)$ be an ALE manifold asymptotic to $\R^n/G$ for $n>2$, and define $$\mathcal{H}^k=\bigl\{\eta\in C^\infty_{1-n}(\Lambda^kT^*X): \d\eta=\d^*\eta=0\bigr\}.$$ Then $\mathcal{H}^0=\mathcal{H}^n=0$, and the map $\mathcal{H}^k\rightarrow H^k(X,\R)$ given by $\eta\mapsto[\eta]$ induces natural isomorphisms $\mathcal{H}^k\cong H^k(X,\R)\cong H^k_c(X,\R)$ for $0<k<n$. The Hodge star gives an isomorphism $*:\mathcal{H}^k\rightarrow \mathcal{H}^{n-k}$. Suppose $1-n\le\beta<-n/2$. Then $$C^\infty_\beta(\Lambda^kT^*X)=\mathcal{H}^k \oplus\d\Bigl[C^\infty_{\beta+1}(\Lambda^{k-1}T^*X)\Bigr] \oplus\d^*\Bigl[C^\infty_{\beta+1}(\Lambda^{k+1}T^*X)\Bigr],$$ where the summands are $L^2$-orthogonal. \[alehodgethm\] This is an analogue of the Hodge Decomposition Theorem and Hodge’s Theorem. For the rest of the section we shall restrict our attention to ALE Kähler manifolds. If $(X,J,g)$ is an ALE Kähler manifold then we can define the weighted Hölder spaces of $(p,q)$-forms $C^{l,\alpha}_\beta(\Lambda^{p,q}X)$ on $X$ in the obvious way. The Laplacian $\Delta$ acts on these spaces by $$\Delta:C^{l+2,\alpha}_{\beta+2}(\Lambda^{p,q}X)\rightarrow C^{l,\alpha}_\beta(\Lambda^{p,q}X).$$ They have very similar analytic properties to the weighted Hölder spaces of functions on an ALE manifold discussed in §4. We can use facts about the Laplacian on weighted Hölder spaces of $(p,q)$-forms to develop an analogue for ALE Kähler manifolds of Hodge theory for compact Kähler manifolds. Let $(X,J,g)$ be an ALE Kähler manifold asymptotic to $\C^m/G$. Define $$\mathcal{H}^{p,q}=\bigl\{\eta\in C^\infty_{1-2m}(\Lambda^{p,q}X): \d\eta=\d^*\eta=0\bigr\}.$$ Then $\mathcal{H}^{p,q}$ is finite-dimensional, and the map $\mathcal{H}^{p,q}\rightarrow H^{p+q}(X,\C)$ defined by $\eta\mapsto[\eta]$ is injective. Define $H^{p,q}(X)$ to be the image of this map. Then $$H^k(X,\C)=\bigoplus_{j=0}^kH^{j,k-j}(X)\qquad\text{for $0<k<2m$.}$$ \[alekhodgethm\] In fact, if $X$ is a crepant resolution of $\C^m/G$ then $H^{p,q}(X)=0$ for $p\ne q$. Let $(X,J,g)$ be an ALE Kähler manifold, where $X$ is a resolution of $\C^m/G$. Then $H^{2,0}(X)=H^{0,2}(X)=0$, and each element of $H^{1,1}(X)$ is represented by a closed, compactly-supported $(1,1)$-form on $X$. \[alekh2thm\] Here is a sketch of the proof of this theorem. Since $X$ is a resolution of $\C^m/G$, it can be shown that the homology group $H_{2m-2}(X,\C)$ is generated by the homology classes of the exceptional divisors of the resolution. But $H_{2m-2}(X,\C)\cong H^2_c(X,\C)$. Thus $H^2_c(X,\C)$ is generated by cohomology classes dual to the homology classes $[D]$ of exceptional divisors $D$ in $X$. If $U$ is any open neighbourhood of $D$ in $X$, then we can find a closed $(1,1)$-form supported in $U$ representing the cohomology class dual to $[D]$. Therefore $H^2_c(X,\C)$ is generated by cohomology classes represented by closed, compactly-supported $(1,1)$-forms. It easily follows that $H^{2,0}(X)=H^{0,2}(X)=0$, and the proof is finished. Next we prove a version of the Global $\d\d^c$-Lemma for ALE Kähler manifolds. Let $(X,J,g)$ be an ALE Kähler manifold asymptotic to $\C^m/G$ for some $m>1$, and let $\beta<-m$. Suppose that $\eta\in C^\infty_\beta(\Lambda^{1,1}_{\R} X)$ is a closed real $(1,1)$-form and $[\eta]=0$ in $H^2(X,\R)$. Then there exists a unique real function $u\in C^\infty_{\beta+2}(X)$ with $\eta=\d\d^cu$. \[aleddcthm\] Let $\omega$ be the Kähler form of $g$. Then if $u$ is a smooth function on $X$ we have $$\d\d^cu\wedge\omega^{m-1}=-{\textstyle{1\over m}}\,\Delta u\,\omega^m. \label{ddcueq}$$ Also, if $\zeta$ is a real (1,1)-form on $X$ and $\zeta\wedge\omega^{m-1}=0$ it can be shown that $$\zeta\wedge\omega^{m-2}=-{\textstyle{1\over 2}}(m-2)!\,*\zeta \quad\text{and}\quad \zeta\wedge\zeta\wedge\omega^{m-2}= -{\textstyle{1\over 2}}(m-2)!\,\ms{\zeta}\d V_g, \label{zetomeq}$$ where $*$ is the Hodge star and $\d V_g$ the volume form of $g$. Equations and hold on any Kähler manifold of dimension $m$. Define a function $f$ on $X$ by $\eta\wedge\omega^{m-1}=-{1\over m}f\,\omega^m$. Since $\eta\in C^\infty_\beta(\Lambda^{1,1}_{\R} X)$, it follows that $f\in C^\infty_\beta(X)$. Now suppose for simplicity that $-2m<\beta<-m$. Then by part (a) of Theorem \[alede2thm\] there exists a unique function $u\in C^\infty_{\beta+2}(X)$ with $\Delta u=f$. Set $\zeta=\eta-\d\d^cu$, which is an exact 2-form in $C^\infty_\beta(\Lambda^{1,1}_{\R} X)$. As $\beta<-m$ we can use the last part of Theorem \[alehodgethm\] to prove that $\zeta=\d\theta$, for some $\theta\in C^\infty_{\beta+1}(T^*X)$. By we have $\zeta\wedge\omega^{m-1} =-{1\over m}(f-\Delta u)\,\omega^m=0$, so gives $$\d\bigl[\theta\wedge\zeta\wedge\omega^{m-2}\bigr]= \zeta\wedge\zeta\wedge\omega^{m-2}= -{\textstyle{1\over 2}}(m\!-\!2)!\,\ms{\zeta}\d V_g. \label{dgazetaeq}$$ Let $\rho$ be a radius function on $X$, and define $S_R=\bigl\{x\in X:\rho(x)\le R\bigr\}$ for $R>1$. Integrating over $S_R$ and using Stokes’ Theorem gives that $$-{\textstyle{1\over 2}}(m-2)!\cdot\int_{S_R}\ms{\zeta}\d V_g= \int_{\partial S_R}\theta\wedge\zeta\wedge\omega^{m-2}. \label{intsrzeta2eq}$$ But for large $R$ we have $\theta=O(R^{\beta+1})$, $\zeta=O(R^\beta)$ and $\omega=O(1)$ on $\partial S_R$, and $\vol(\partial S_R)=O(R^{2m-1})$. Thus the r.h.s. of is $O(R^{2\beta+2m})$. As $\beta<-m$, taking the limit as $R\rightarrow\infty$ shows that $\int_X\ms{\zeta}\d V_g=0$, and so $\zeta=0$ on $X$. Thus $\eta=\d\d^cu$, as we have to prove. We have proved the theorem assuming that $-2m<\beta<-m$, but we wish to prove it for all $\beta<-m$. If $\beta\le 2m$ and $\eta\in C^\infty_\beta(\Lambda^{1,1}_{\R} X)$ then $\eta\in C^\infty_\gamma(\Lambda^{1,1}_{\R} X)$ for any $\gamma$ with $-2m<\gamma<-m$, and so from above we have $\eta=\d\d^cu$ for some unique $u$ in $C^\infty_{\gamma+2}(X)$. However, if $u\in C^\infty_{\gamma+2}(X)$ and $\d\d^cu\in C^\infty_\beta(\Lambda^{1,1}_{\R} X)$, one can show that $u\in C^\infty_{\beta+2}(X)$ as we want. This is because $\d\d^cu$ is a stronger derivative of $u$ than $\Delta u$ is, and contains more information. Finally, we show we can modify any ALE Kähler metric to be flat outside a compact set. Let $\C^m/G$ have an isolated singularity at $0$ for some $m>1$, let $(X,\pi)$ be a resolution of $\C^m/G$ that admits ALE Kähler metrics, and let $\rho$ be a radius function on $X$. Then in each Kähler class there exists an ALE Kähler metric $\hat g$ on $X$ such that $\hat g=\pi^*(h)$ on the subset $\bigl\{x\in X:\rho(x)>R\bigr\}$, where $h$ is the Hermitian metric on $\C^m/G$ and $R>0$ is a constant. \[alekprop\] Let $g$ be an ALE Kähler metric on $X$, with Kähler form $\omega$. By Theorems \[alekhodgethm\] and \[alekh2thm\] there exists a closed, compactly-supported, real (1,1)-form $\theta$ on $X$ with $[\theta]=[\omega]$ in $H^2(X,\R)$. Define $\eta=\omega-\d\d^c(\rho^2)-\theta$. Then $\eta$ is an exact real (1,1)-form on $X$. Now the Kähler form of $h$ on $\C^m/G$ is $\omega_0=\d\d^c(r^2)$. So from the definition of ALE Kähler metric we see that $\omega-\d\d^c(\rho^2)\in C^\infty_{-2m}(\Lambda^{1,1}_{\R} X)$, and therefore $\eta\in C^\infty_{-2m}(\Lambda^{1,1}_{\R} X)$ as $\theta$ has compact support. Thus by Theorem \[aleddcthm\] there is a unique real function $u\in C^\infty_{2-2m}(X)$ with $\eta=\d\d^cu$, and we have $\omega=\theta+\d\d^c(\rho^2)+\d\d^cu$. Let $\mu:\R\rightarrow[0,1]$ be a smooth function with $\mu(t)=1$ for $t\le -1$ and $\mu(t)=0$ for $t\ge 0$. For each $R>0$ define a closed (1,1)-form $\omega_R$ by $$\omega_R=\theta+\d\d^c(\rho^2)+\d\d^c\bigl[\mu(\rho-R)\cdot u\bigr].$$ Then $\omega_R=\omega$ wherever $\rho<R-1$, and $\omega_R=\d\d^c(\rho^2)$ wherever $\rho>R$ and outside the support of $\theta$. It is easy to show that $\omega_R$ is a positive (1,1)-form for large $R$, which therefore defines a Kähler metric $g_R$ on $X$. Define $\hat g$ to be $g_R$ for some $R$ sufficiently large that $\omega_R$ is positive, $\rho\le R$ on the support of $\theta$ and $R\ge 2$. Then $\hat g$ is an ALE Kähler metric in the Kähler class of $g$, and where $\rho>R$ we have $\hat g=\pi^*(h)$, since the Kähler form of $\hat g$ is $\d\d^c(\rho^2)$, the Kähler form of $h$ is $\d\d^c(r^2)$, and $\rho=\pi^*(r)$ as $\rho>R\ge 2$. The Calabi conjecture for ALE manifolds ======================================= We can now state the following version of the Calabi conjecture for ALE Kähler manifolds. [**The Calabi conjecture for ALE manifolds**]{} *Suppose that $(X,J,g)$ is an ALE Kähler manifold of dimension $m$ asymptotic to $\C^m/G$ for some $m>1$, with Kähler form $\omega$, and that $\rho$ is a radius function on $X$. Then* - Let $\beta\in(-2m,-2)$. Then for each $f\in C^\infty_\beta(X)$ there is a unique $\phi\in C^\infty_{\beta+2}(X)$ such that $\omega+\d\d^c\phi$ is a positive $(1,1)$-form and $(\omega+\d\d^c\phi)^m={\rm e}^f\omega^m$ on $X$. - Let $\beta\in(-1-2m,-2m)$. Then for each $f\in C^\infty_\beta(X)$ there is a unique $\phi\in C^\infty_{2-2m}(X)$ such that $\omega+\d\d^c\phi$ is a positive $(1,1)$-form and $(\omega+\d\d^c\phi)^m={\rm e}^f\omega^m$ on $X$. Moreover we can write $\phi=A\rho^{2-2m}+\psi$, where $\psi\in C^\infty_{\beta+2}(X)$ and $$A={\md{G}\over(m-1)\Omega_{2m-1}}\cdot\int_X(1-{\rm e}^f)\d V_g. \label{aleccadefeq}$$ Here $\Omega_{2m-1}$ is the volume of the unit sphere ${\mathcal S}^{2m-1}$ in $\C^m$. It is easy to rewrite this in terms of the existence of ALE Kähler metrics with prescribed Ricci curvature, as in the original Calabi conjecture. The two cases (a) $\beta\in(-2m,-2)$ and (b) $\beta\in(-1-2m,-2m)$ come from Theorem \[alede2thm\]. By combining the method of Yau’s proof [@Yau] of the Calabi conjecture with the ideas of §4 on analysis on ALE manifolds, we can prove the Calabi conjecture for ALE manifolds. The conjecture will be proved in [@Joyc3 §8.5–§8.6], and we give only a sketch of the proof of part (a) here. We use the [*continuity method*]{}. Suppose $\beta\in(-2m,-2)$. Fix $f\in C^{3,\alpha}_\beta(X)$, and define $S$ to be the set of all $t\in[0,1]$ for which there exists $\phi\in C^{5,\alpha}_{\beta+2}(X)$ such that $\omega+\d\d^c\phi$ is a positive (1,1)-form and $(\omega+\d\d^c\phi)^m={\rm e}^{tf}\omega^m$ on $X$. Clearly $0\in S$, taking $\phi=0$. We prove that $S$ is both [*open*]{} and [*closed*]{} in $[0,1]$. Thus $S=[0,1]$ as $[0,1]$ is connected, so $1\in S$, and there exists $\phi\in C^{5,\alpha}_{\beta+2}(X)$ with $\omega+\d\d^c\phi$ positive and $(\omega+\d\d^c\phi)^m={\rm e}^f\omega^m$ on $X$. We then use Theorem \[alede2thm\] to show that if $f\in C^\infty_\beta(X)$ then $\phi\in C^\infty_{\beta+2}(X)$, and this completes the proof. To prove that $S$ is open, we fix $t\in S$ and show that $S$ contains a small neighbourhood of $t$ by considering the [ *linearization*]{} of the equation at $t$. This linearization turns out to involve the Laplacian of the metric with Kähler form $\omega+\d\d^c\phi$, and part (a) of Theorem \[alede2thm\] gives us what we need. To prove that $S$ is closed, we take a sequence $\{t_j\}_{j=0}^\infty$ in $S$ such that $t_j\rightarrow t\in [0,1]$ as $j\rightarrow\infty$. Let $\{\phi_j\}_{j=0}^\infty$ be the sequence of solutions to $(\omega+\d\d^c\phi_j)^m={\rm e}^{t_jf}\omega^m$. Then $\phi_j$ converges to some $\phi\in C^{5,\alpha}_{\beta+2}(X)$ as $j\rightarrow\infty$ with $(\omega+\d\d^c\phi)^m={\rm e}^{tf}\omega^m$, and thus $t\in S$. Therefore $S$ contains its limit points, and is closed. The difficult part in showing $S$ closed is finding an [*a priori estimate*]{} for $\phi_j$ in $C^{5,\alpha}_{\beta+2}(X)$. To do this we first follow Yau’s proof to get an a priori estimate in $C^{5,\alpha}(X)$. Then we use a ‘weighted’ version of Yau’s method to estimate $\phi_j$ in $C^0_\delta(X)$ for some small $\delta<0$. This can be improved to $C^{5,\alpha}_\delta(X)$, and then to $C^{5,\alpha}_{\beta+2}(X)$ by a kind of induction, decreasing $\delta$ step by step until $\delta=\beta+2$. This concludes our treatment of the Calabi conjecture for ALE manifolds, and we are now ready to prove Theorems \[alerfthm\] and \[alesumthm\]. The proof of Theorem \[alerfthm\] --------------------------------- Let $X$ be a crepant resolution of $\C^m/G$, where $G$ acts freely on $\C^m\setminus\{0\}$. By Proposition \[alekprop\], in each Kähler class of ALE Kähler metrics on $X$ we can choose a metric $\hat g$ with $\hat g=\pi^*(h)$ wherever $\rho>R\ge 2$, where $h$ is the Euclidean metric on $\C^m/G$. Let $\hat\omega$ be the Kähler form and $\eta$ the Ricci form of $\hat g$. Then $\eta$ is closed and $[\eta]=2\pi\,c_1(X)$ in $H^2(X,\R)$. But $c_1(X)=0$ as $X$ is a crepant resolution, so $[\eta]=0$ in $H^2(X,\R)$. Also, $\eta=0$ wherever $\rho>R$, since there $\hat g=\pi^*(h)$ and $h$ is flat. Thus $\eta$ is a closed, compactly-supported $(1,1)$-form on $X$ with $[\eta]=0$ in $H^2(X,\R)$, and by Theorem \[aleddcthm\] there exists a unique function $f\in C^\infty_\beta(X)$ for each $\beta<0$ with $\eta={1\over 2}\d\d^cf$. In fact $f=0$ wherever $\rho>R$, so $f$ is compactly supported. The Calabi conjecture for ALE manifolds holds by [@Joyc3 §8.5–§8.6]. Part (b) of the conjecture shows that there exists a unique function $\phi=A\rho^{2-2m}+\psi$ where $A$ is given by and $\psi\in C^\infty_{\beta+2}(X)$ for $\beta\in(-1-2m,-2m)$, such that $\omega=\hat\omega+\d\d^c\phi$ is a positive (1,1)-form and $\omega^m={\rm e}^f\hat\omega^m$. Let $g$ be the Kähler metric on $X$ with Kähler form $\omega$. Then since the Ricci form of $\hat g$ is ${1\over 2}\d\d^cf$ it follows by standard properties of the Ricci form that $g$ has Ricci form zero, and is Ricci-flat. On $\bigl\{z\in\C^m/G:r(z)>R\bigr\}$ we have $\pi_*(\hat g)=h$, so that $\pi_*(\hat\omega)=\omega_0$, and $\pi_*(\rho)=r$. Thus defining $\chi=\pi_*(\psi)$ gives . Since $\psi\in C^\infty_{\beta+2}(X)$ for $\beta\in(-1\!-2m,-2m)$, putting $\gamma=\beta+2$ we see that $\nabla^k\chi=O(r^{\gamma-k})$ for $k=0,1,2,\ldots$ and $\gamma\in(1-2m,2-2m)$, as we have to prove. From we see that $\nabla^k(\pi_*(g)-h)=O(r^{-2m-k})$ for $k\ge 0$, and thus $g$ is an ALE Kähler metric by Definition \[alekdef\]. Also, $g$ is unique in its Kähler class of ALE metrics because $\phi$ is unique. It only remains to prove that $A<0$. We can do this by giving an explicit expression for $A$. Let $\zeta$ be the unique element of $\mathcal{H}^{1,1}$ with $[\zeta]=[\omega]$. Then a calculation shows that $$A=-\,{\md{G}\over 2m(m\!-\!1)^2\Omega_{2m-1}}\int_X\ms{\zeta}\d V_g, \label{aconsteq}$$ where $\Omega_{2m-1}$ is the volume of the unit sphere ${\mathcal S}^{2m-1}$ in $\C^m$. Now $[\omega]\ne 0$ as this is outside the Kähler cone, so $\zeta\ne 0$ and $A$ is negative. This completes the proof of Theorem \[alerfthm\]. The proof of Theorem \[alesumthm\] ---------------------------------- First we show that $\C^m/G$ has no crepant resolutions when $m>2$ and $G\subset{\rm Sp}(m/2)$. Suppose that $m>2$ is even and that $G$ is a nontrivial finite subgroup of ${\rm Sp}(m/2)$ which acts freely on $\C^m\setminus\{0\}$. Then $\C^m/G$ is a terminal singularity, and admits no crepant resolutions. \[spm2prop\] Let $\gamma\ne 1$ in $G$. Then there are coordinates $(z^1,\ldots,z^m)$ on $\C^m$ in which $\gamma$ acts by $$\bigl(z^1,\ldots,z^m\bigr)\,{\buildrel\gamma\over\longmapsto}\, \bigl({\rm e}^{2\pi ia_1}z^1,\ldots,{\rm e}^{2\pi ia_m}z^m\bigr). \label{gajacteq}$$ As $\gamma$ acts freely on $\C^m\setminus\{0\}$ we can take $a_j\in (0,1)$ for $j=1,\ldots,m$. Since $G\subset{\rm Sp}(m/2)$ we know that $\gamma$ preserves a complex symplectic form on $\C^m$, and we can choose $(z^1,\ldots,z^m)$ so that this form is $\d z^1\wedge\d z^2+\cdots+\d z^{m-1}\wedge\d z^m$. Thus gives ${\rm e}^{2\pi ia_{2j-1}}{\rm e}^{2\pi ia_{2j}}=1$ for $j=1,\ldots,m/2$. But as $a_{2j-1},a_{2j}\in (0,1)$ this implies that $a_{2j-1}+a_{2j}=1$ for $j=1,\ldots,m/2$. Therefore $a_1+\cdots+a_m=m/2>1$ for all $\gamma\ne 1$ in $G$. So by Reid [@Reid §4] it follows that $\C^m/G$ is a [*terminal singularity*]{}, as defined in [@Reid p. 347]. Terminal singularities are essentially singularities which have no crepant partial resolutions. To be more precise, a crepant resolution of a terminal singularity has no exceptional divisors. Thus, if $X$ is a crepant resolution of $\C^m/G$ then $b_{2m-2}(X)=0$. By Poincaré duality for manifolds with boundary we see that $b_2(X)=0$, which is a contradiction, as $X$ must contain a complex curve. So $\C^m/G$ has no crepant resolutions. We now prove Theorem \[alesumthm\]. Let $X$ be a crepant resolution of $\C^m/G$, where $G$ is nontrivial and acts freely on $\C^m\setminus\{0\}$, and let $g$ be a Ricci-flat ALE Kähler metric on $X$. As $X$ is simply-connected, by general facts about holonomy groups we know that $\Hol(g)$ is a connected Lie subgroup of ${\rm SU}(m)$. Since $g$ is Ricci-flat it is nonsymmetric. Also $(X,g)$ is not a Riemannian product, because it is asymptotic to $\C^m/G$, which is not a product. Thus $g$ is irreducible. Therefore we may apply Berger’s classification of Riemannian holonomy groups [@Sala §10]. The only two possibilities are $\Hol(g)={\rm SU}(m)$ or $\Hol(g)={\rm Sp}(m/2)$. When $m=2$ the two groups coincide, so suppose $m>2$. The holonomy of the Euclidean metric $h$ on $\C^m/G$ is $G\subset{\rm SU}(m)$. Since $g$ is asymptotic to $h$ one can show that $G\subset\Hol(g)\subseteq{\rm SU}(m)$. Hence, if $\Hol(g)={\rm Sp}(m/2)$ then $G\subset{\rm Sp}(m/2)$. But Proposition \[spm2prop\] then shows that $\C^m/G$ admits no crepant resolutions, a contradiction. So $\Hol(g)\ne{\rm Sp}(m/2)$, and thus $\Hol(g)={\rm SU}(m)$, which completes the proof. The essential point in this proof is that there do not exist ALE manifolds with holonomy ${\rm Sp}(m/2)$ for $m>2$. 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--- abstract: | In this paper, we propose a new capsule network architecture called Attention Routing CapsuleNet (AR CapsNet). We replace the dynamic routing and squash activation function of the capsule network with dynamic routing (CapsuleNet) with the attention routing and capsule activation. The attention routing is a routing between capsules through an attention module. The attention routing is a fast forward-pass while keeping spatial information. On the other hand, the intuitive interpretation of the dynamic routing is finding a centroid of the prediction capsules. Thus, the squash activation function and its variant focus on preserving a vector orientation. However, the capsule activation focuses on performing a capsule-scale activation function. We evaluate our proposed model on the MNIST, affNIST, and CIFAR-10 classification tasks. The proposed model achieves higher accuracy with fewer parameters ($\times$0.65 in the MNIST, $\times$0.82 in the CIFAR-10) and less training time than CapsuleNet ($\times$0.19 in the MNIST, $\times$0.35 in the CIFAR-10). These results validate that designing a capsule-scale operation is a key factor to implement the capsule concept. Also, our experiment shows that our proposed model is transformation equivariant as CapsuleNet. As we perturb each element of the output capsule, the decoder attached to the output capsules shows global variations. Further experiments show that the difference in the capsule features caused by applying affine transformations on an input image is significantly aligned in one direction. author: - | Jaewoong Choi Hyun Seo Suii Im Myungjoo Kang\ Seoul National University\ [{chjw1475, hseo0618, a5828167, mkang}@snu.ac.kr]{} bibliography: - 'egbib.bib' title: Attention routing between capsules --- ![image]({"image/model_overview"}.pdf){width="0.95\linewidth"} Introduction ============ Convolutional neural networks(CNNs) have had much success in computer vision tasks [@AlexNet] [@VGGNet] [@ResNet] [@DenseNet]. The convolutional layer is an effective method to extract local features due to its local connectivity and parameter sharing with spatial location. However, the convolutional layer has a limited ability to encode a transformation. For example, if the convolutional layer is combined with a max-pooling layer, the extracted feature is local translation invariant. As CNN models become deeper [@ResNet] [@HighwayNetwork], the receptive field of each feature is getting larger. Then, the information loss from the translation invariance also increases. To overcome the transformation invariance of CNNs, the transforming autoencoder [@transformingautoencoder] uses the concept of “capsule”. A capsule is a vector representation of a feature. Each capsule not only represents a specific type of entity but also describes how the entity is instantiated, such as precise pose and deformation. In other words, the capsules are transformation equivariant. The CapsuleNet [@capsulenet] is a novel method that implements the idea of the capsules. By introducing the dynamic routing algorithm and squash activation function \[eq:squash\], CapsuleNet uses vector-output capsules as a basic unit instead of scalar-output features. $$\label{eq:squash} \textrm{squash}(\mathbf{s_j}) = \frac{||\mathbf{s}_{j}||^2}{1+ ||\mathbf{s}_{j}||^2} \frac{\mathbf{s}_{j}}{||\mathbf{s}_{j}||}$$ where $\mathbf{s_j}$ is a pre-activation capsule. However, CapsuleNet has a room for development. The number of parameters of CapsuleNet is much larger than that of comparable performance CNN-based models. Also, the dynamic routing is an iterative process. The reported accuracy of CapsuleNet on the benchmark datasets like CIFAR-10 is inferior to state-of-the-art CNN models. [@complex] In this paper, we propose a convolutional capsule network architecture comprised of building blocks of CNNs. We substitute the dynamic routing and squash capsule-activation function of CapsuleNet[@capsulenet] with attention routing and capsule activation. In the attention routing, the log probabilities of agreement coefficients between the $l$th layer and the $(l+1)$th layer are learned by a scalar-product between the capsules of the $l$th layer and the kernel of convolution. The kernel of convolution serves as an approximation of the reference vector to perform routing. By replacing an iterative process of the dynamic routing with forward-pass convolution, the attention routing is fast while maintaining spatial information. Two important properties of squash activation function \[eq:squash\] is that the squash activation function preserves a vector orientation and is a capsule-wise activation function, not an element-wise activation function such as $\textrm{ReLU}$ or $\tanh$. The dynamic routing is an unsupervised algorithm to find a centroid-like output capsule of the prediction capsules. Therefore, the squash activation function and its variant \[eq:squash\_variant\] [@complex] focus on preserving a capsules orientation. $$\label{eq:squash_variant} \textrm{squash variant}(\mathbf{s_j}) = \left( 1- \frac{1}{\exp{( ||\mathbf{s}_{j}||}) } \right) \frac{\mathbf{s}_{j}}{||\mathbf{s}_{j}||}$$ However, we focus on the capsule-wise operation rather than preserving orientation. The capsule activation performs an affine transform on the capsules and then applies an element-wise activation function. The capsules on the same capsule channel share parameters used in the affine transformation. Thus, the capsules on the same capsule channel are mapped to the same feature space, and the operation is parameter efficient. Therefore, the capsule activation is a capsule-wise function that does not preserve a vector orientation. Since the capsule activation applies a nonlinear transformation to a linear combination of the prediction capsules, parametrizing the routing process through the attention routing is compatible. We refer to our proposed model as *Attention Routing CapsuleNet (AR CapsNet).* We evaluate the AR CapsNet on three datasets (MNIST, affNIST, and CIFAR-10). The AR CapsNet significantly outperforms CapsuleNet in the affNIST and CIFAR-10 classification task and shows a comparable performance in the MNIST dataset while being faster and using less than half parameters than CapsuleNet. Moreover, the AR CapsNet preserves the transformation equivariant property of CapsuleNet. As we perturb each element of the output capsule, the decoder attached to the output capsules shows global variations as in [@capsulenet]. Further experiment showed that the affine transformations on an input image cause the feature capsules to change in the significantly aligned direction. From these experiments, we prove that the AR CapsNet encodes an affine transformation on the input image in some basis of capsule space. In addition, our proposed architecture is constructed in a convolutional manner so that it can be easily extended to a deeper network structure. Contribution ------------ - We propose a new architecture called AR CapsNet by introducing two modifications to the CapsuleNet [@capsulenet]. These modifications are the *attention routing* and *capsule activation.* - The capsule activation expands the concept of the existing capsule-wise activation functions such as the squash activation. The capsule activation performs an orientation-nonpreserving transform on the pre-activation capsules. The performance of the AR CapsNet demonstrates that the transformation equivariant features can be extracted even if the routing process is not restricted to the clustering approach and the capsule activation is not limited to the normalization. - The AR CapsNet shows better results on the affNIST, and CIFAR-10 classification tasks and comparable results on the MNIST classification task while using much smaller parameters than CapsuleNet. Also, the AR CapsNet preserves the transformation equivariant property of the CapsuleNet. As we perturb each element of the output capsule, the decoder attached to the output capsule shows global variation as in [@capsulenet]. - To investigate the transformation equivariance further, we suggest a new experiment. We observe the difference in the output capsule caused by applying transformations on an input image. In the AR CapsNet, these difference vectors are significantly aligned compared to a set of random vectors. These results demonstrate that transformation on an input image is encoded in some basis vector. Related Works ============= The CNN models that consist of convolutional layers and max-pooling layers have a local translation invariance. To overcome transformation invariance, CapsuleNet [@capsulenet] uses vector-output capsules and the dynamic routing in place of scalar-output features and max-pooling. By demonstrating that the dimension perturbation of digit capsules leads to a global transformation of the reconstruction image, CapsuleNet claims to have transformation equivariance. A number of methods to improve the performance of CapsuleNet have been proposed in [@EMrouting] [@wang2018optimization] [@SpectralCapsulenet] [@SegCaps] [@NeuralNetworkEncapsule]. In [@wang2018optimization], they interpreted the routing-by-agreement process as an optimization problem of minimizing clustering loss. They proposed another routing process from the point of view of clustering. Their approach achieved better results on an unsupervised perceptual grouping task compared to [@capsulenet]. The matrix capsules with EM routing [@EMrouting] proposed another routing method called EM routing. The EM routing measures compatibility between matrix capsules by clustering matrix capsules through Gaussian distributions. The matrix capsules with EM routing achieved the state-of-the-art performance on a shape recognition task using the smallNORM dataset. The spectral capsule networks [@SpectralCapsulenet] is a variation of [@EMrouting]. Spectral capsule networks use a singular value to compute the activation of each capsule instead of the logistic function in [@EMrouting]. Spectral capsule networks achieved better performance on a diagnosis dataset compared to [@EMrouting] and deep GRU networks while showing faster convergence compared to [@EMrouting]. The SegCaps [@SegCaps] applied a capsule network to the object segmentation task. The SegCaps introduced two modifications to the CapsuleNet and devised the concept of deconvolutional capsules from these modifications. The two modifications are the locally connected dynamic routing and the sharing of transformation matrices within the same capsules channel. The sharing of transformation matrices is equivalent to the convolutional transform of our conv caps layer except for the addition of biases. The EncapNet [@NeuralNetworkEncapsule] performs a one-time pass approximation of the routing process by introducing two branches. The master branch extracts a feature from the locally connected capsules as in [@SegCaps] and the aide branch combines information from all the remaining capsules. Also, they introduced a Sinkhorn divergence loss which works as a regularizer. The EncapNet achieved competitive results on CIFAR-10/100, SVHN, and a subset of ImageNet. Our proposed model uses attention architecture as a routing algorithm. The attention architecture learns a compatibility function between low-level features and high-level features. In [@BahdanauAttention], the output of attention architecture is a weighted sum of input features, and the weights are the compatibilities based on the input features and the RNN hidden state. The compatibility function is a feedforward neural network with a softmax activation function. In [@LuongAttention], they experimented on various kinds of attention architectures from global attention to local attention and three compatibility functions. One of the three compatibility functions was a softmax output of the scalar-products between a target hidden state vector and source hidden state vector. The transformer network [@Transformer] uses a similar attention architecture as in [@LuongAttention]. Transformer performs a scaled scalar-product between the keys and values and then applies a softmax activation function. Our proposed attention routing computes the scalar product between capsules and a kernel. ![Detailed operation process of conv caps layer. Conv Transform denotes the convolutional transform. The convolutional transform performs a locally connected affine transform on each capsule channel. The attention routing learns the agreement between the convolutional transformed capsules for each spatial location. The capsule activation applies a capsule-wise activation function on each capsule channel. []{data-label="fig:model"}]({"image/operation_diagram"}.pdf){width="0.95\linewidth"} Proposed Method =============== Our proposed architecture consists of *primary caps layer*, *conv caps layer*, and *fully conv caps layer*. We denote the $l$th capsule layer as $\mathbf{u}^{l}_{w,h,d,n}$, where $w, h, d$, and $n$ index the spatial width axis, spatial height axis, capsule dimension axis, and capsule channel axis, respectively. We refer to the capsules with the same capsule channel index as a *capsule channel* $\mathbf{u}^{l}_{(:, :, :, n)}$ [^1]. Primary Caps Layer ------------------ We denote the primary capsule layer as the $0$th capsule layer. Before entering the primary caps layer, we extract local features $\Tilde{x}$ from the input image $x$ by performing the convolution blocks composed of convolution layer and batch normalization.[@BatchNormalization] We consider the local features $\bf{\Tilde{x}}$ as a single capsule layer. In our primary caps layer with N channels of D dimensional output capsules, 3x3 convolution with kernels of filter size D and stride 2 is performed on the input capsules $\bf{\Tilde{x}}$ N times independently. Each output of a convolution layer is a capsule channel. $$\mathbf{s}_{(:,:,:,n_0)}^{0} = \textrm{ReLU}\left( \textrm{Conv}_{3\times3} \left( \Tilde{x} \right) \right)$$ Note that this is equivalent to performing a 3x3 convolution of $N\times D$ kernels and then reshaping the features to $(B, W, H, D, N)$ where $B$ denotes the batch size and $(W, H)$ denotes the spatial size of the capsule layer. Then, the capsule activation is applied to each capsule channel instead of the squash activation function in [@capsulenet]. Capsule Activation ------------------ The capsule activation takes an affine transformation on each capsule channel and then applies $\tanh$ activation function. The capsules on the same capsule channel share parameters of the affine transformation. Thus, the capsule activation is equivalent to taking 1x1 convolution with a kernel of filter size D and $\tanh$ activation function on each capsule channel. $$\mathbf{u}_{(:,:,:,n_0)}= \tanh\left( \textrm{Conv}_{1\times1} \left( \mathbf{s}_{(:,:,:,n_0)} \right) \right)$$ Each element of the output capsules of the capsule activation depends on the corresponding input capsule. Therefore, the capsule activation is a capsule-wise activation function. The $\tanh$ activation function normalizes each element of capsules, thus stabilizes the lengths of the capsules. Conv Caps Layer --------------- We denote the input to the $l$th conv caps layer as $\mathbf{u}_{w,h,d,n}^{l-1}$ which is the output of the $(l-1)$th conv caps layer. We first perform a *convolutional transform* on each capsule channel. The convolutional transform is a locally-connected affine transformation sharing parameters within the same capsule channel. In particular, the convolutional transform is a 3x3 convolution of $D^{l}$ kernels without activation function, where $D^{l}$ denotes the capsule dimension of the $l$th conv caps layer. $$\Tilde{\mathbf{s}}_{(:,:,:,m)}^{l, n} = \textrm{Conv}_{3\times3} \left( \mathbf{u}_{(:,:,:,m)}^{l-1} \right)$$ Each output of the convolutional transform is fed to the attention routing. The output of attention routing is a linear combination of the convolutional transformed capsules with the same spatial location. $$\mathbf{s}_{(w,h,:,n)}^{l} = \sum_{m =1, \cdots, N^{l-1} } c_{(w, h, m)}^{l, n} \cdot \Tilde{\mathbf{s}}^{l, n}_{(w,h,:,m)}$$ where the capsules $\mathbf{s}_{(w,h,:, n)}^{l} , \Tilde{\mathbf{s}}^{l}_{(w,h,:,m)} \in \mathbb{R}^{D^{l}}$. The weights $c_{(w, h, m)}^{l, n} \in \mathbb{R}$ are computed by the attention routing. The log probabilities $b_{(w, h, m)}^{l, n}$ are the scalar-product between a concatenation of capsules $[\Tilde{\mathbf{u}}_{w,h,:,1}^{l}, \Tilde{\mathbf{u}}_{w,h,:,2}^{l}, \cdots, \Tilde{\mathbf{u}}_{w,h,:,N^{l-1}}^{l} ]$ and a parameter vector $w_{n}^{l} \in \mathbb{R}^{D^l \times N^{l-1}}$. This operation can be implemented efficiently by 3D convolution on the convolutional transformed capsule layers with kernels $w_{n}^{l} \in \mathbb{R}^{1 \times 1 \times D^l \times N^{l-1}}$, stride=(1,1,1), and valid padding. $$b^{l, n}_{(:, :, :)} = \textrm{Conv3D}_{1\times1\times D^{l}} \left( \Tilde{\mathbf{s}}_{(:,:,:,:)}^{l} \right)$$ The weights $c_{(w, h, m)}^{l, n}$ are softmax outputs of the log probabilities $b_{(w, h, m)}^{l, n}$ along the capsule channel axis. $$c_{w, h, m}^{l, n} = \frac{ \exp( b_{(w, h, m)}^{l, n} ) } { \sum_{1 \leq m \leq N^{l-1}} \exp( b_{(w, h, m)}^{l, n} ) }$$ Note that the attention routing adjusts the weight $c_{w, h, m}^{l, n}$ for each spatial location $(w,h)$ corresponding to the convolutional transformed capsules $\{ \Tilde{\mathbf{u}}^{l}_{(w,h,:,m)} \}_{m}$ with the same spatial location. Finally, the capsule activation is performed on each capsule channel $ \mathbf{s}_{(:,:,:, n)}^{l} $. A set of convolutional transform, attention routing, and capsule activation is performed independently $N^{l}$ times. (*i.e*., each output of the convolutional transform, attention routing, and capsule activation is a capsule channel $ \mathbf{u}_{(:,:,:,n)}^{l}$ ) $$\mathbf{u}_{(:,:,:,n)}^{l} = \tanh\left( \textrm{Conv}_{1\times1} \left(\mathbf{s}_{(:,:,:,n)}^{l} \right) \right)$$ Intuitively, the dynamic routing uses a centroid of the transformed capsules as the reference vector to measure agreement by scalar-product. As the dynamic routing process iterates, the capsule with the higher agreement has a larger weight, and the reference vector evolves in that capsule direction. On the other hand, since the capsule activation in the conv caps layer do not preserve vector orientation, the output capsule $\mathbf{u}_{(:,:,:,n)}^{l}$ cannot approximate the centroid of transformed capsules $\{\Tilde{\mathbf{u}}^{l}_{(w,h,:,n)} \}$. Instead of measuring agreement between the transformed capsules and the output capsule $\mathbf{u}_{(:,:,:,n)}^{l}$, the attention routing parametrizes the routing process. The parameter vector $w_{n}^{l}$ which is the kernel of convolution serves as an approximation of the reference vector to perform routing. We propose replacing the dynamic routing of [@capsulenet] with the convolutional transform and attention routing. Compared to dynamic routing, our proposed operation is faster and more parameter efficient. Since dynamic routing is constructed in a fully connected manner, the transform weight matrices are assigned for each pair of the input capsule and output capsule. We share the weight matrices across the spatial location and keep the translation equivariance by performing 3x3 convolution on the $l$th layer in the convolutional transform.(Section \[Transformation Equivariance\]) Besides, the dynamic routing has an iterative routing process to compute the weight $c^{l}_{w, h, n}$. On the other hand, by introducing a trainable parameter vector, our proposed operation is a fast forward-pass. Fully Conv Caps Layer --------------------- The fully conv caps layer is almost the same as the conv caps layer and serves as the output layer of AR CapsNet. The convolutional transform combines capsule features from the all spatial location by applying a kernel of the same spatial size as the input with valid padding. Therefore, the output of the fully conv caps Layer has a shape of $(1, 1, D^{L}, N^{L})$. [**Input:**]{} ${\bf u}^{\ell =0} \in \mathbb{R}^{(w,h,D^{\ell=0},N^{\ell=0})} $ Margin Loss and Reconstruction Regularizer {#loss} ------------------------------------------ We adopt the margin loss and reconstruction regularizer in [@capsulenet]. Since the output capsules of capsule activation have a length of up to $\sqrt{D^L}$ where $D^L$ denotes the capsule dimension, we use the normalized length to predict the probability of the corresponding class of the dataset. $$||\mathbf{u}^{L}_{n}||_{\mathrm{nor}} = \frac{||\mathbf{u}^{L}_{n}||}{\sqrt{D^L}}$$ where $||\mathbf{u}^{L}_{n}||$ denotes the output capsules of the fully conv caps layer and $n$ indexes the capsule channel axis. We applied the Margin loss, $L_n$, for each class $n$ on the $||\mathbf{u}^{L}_{n}||_{\mathrm{nor}}$. $$\begin{aligned} L_n = T_n &\max(0, m^+ - ||\mathbf{u}^{L}_{n}||_{\mathrm{nor}})^2 \\ &+ \lambda (1-T_n) \max(0, ||\mathbf{u}^{L}_{n}||_{\mathrm{nor}}-m^-)^2 \end{aligned}$$ where $T_n = 1$ iff the corresponding class of output capsule is present and $m^+=0.9$ and $m^-=0.1$. The output capsules $\{\mathbf{u}^{L}_{n}\}_{n=1, \cdots, N}$ are fed to the reconstruction decoder. We used a decoder consisting of 3 fully connected layers as in [@capsulenet] except that our decoder has (512, 512, the number of input image pixel) nodes. We refer to the mean of L2 loss between an input image and the decoder output as a reconstruction loss. We add the reconstruction loss that is scaled down by 0.3 to the margin loss as a regularization method.[^2] Experiments =========== We evaluate our model on the MNIST, affNIST, and CIFAR-10 datasets. For each dataset, we split the training images into a training set (90%) and a validation set (10%). We choose the model with the lowest validation error and evaluate the model on the test set. Then, we compare the results with CapsuleNet [@capsulenet]. We use a Keras implementation[[^3]]{} for CapsuleNet. Before training the model on the image dataset, we divide each pixel value by 255 so that it is scaled in the range of 0 to 1. Then, we extract the local features $\Tilde{x}$ from an input image through two convolutional layers with batch normalization(BN) [@BatchNormalization] and ReLU activation function. These two convolutional layers use 3x3 kernels with a stride 1. Then, the features go through the AR CapsNet to obtain vector outputs. For each conv caps layer and fully conv caps layer, the dropout layer [@Dropout] of keep probability 0.5 is applied to the input capsules before the convolutional transform. We use the RMSprop optimizer with rho of 0.9 and decay of 1e-4 to minimize the loss defined in Section \[loss\]. We set the learning rate as 0.001 and batch size as 100 Method MNIST MNIST+ affNIST C10 C10+ ------------------------- ------------- --------------------- --------- ------------ ------------ CapsuleNet[@capsulenet] 99.45$^{*}$ 99.75 (99.52$^{*}$) 79.0 63.1$^{*}$ 69.6$^{*}$ CapsuleNet+ensemble(7) - - - - 89.4 Ours 99.46 99.46 91.6 87.19 88.61 Ours+ensemble(7) - - - 88.94 90.11 Classification Results on MNIST and affNIST {#Results_MNIST_affNIST} ------------------------------------------- [**Dataset**]{} The MNIST dataset is composed of $28 \times 28$ handwritten digit images. We adopted 0.1 translation as a data augmentation for the MNIST dataset. The affNIST dataset consists of $40\times40$ images, which are obtained by applying various affine transformations such as rotation and expansion to the images from MNIST. For the affNIST classification task, we trained our model with randomly translated MNIST images in horizontal or vertical directions up to shift fraction 0.2 as in [@capsulenet]. Any other affine transformations like rotations were not used in the training process. The affNIST dataset has a separate validation set, thus we chose the model with the lowest validation error based on the affNIST validation set. Then, we tested our model with the affNIST test set. [**Implementation**]{} For the MNIST and affNIST datasets, we used the AR CapsNet which consists of a primary caps layer, one conv caps layer and fully conv caps layer. Before entering the AR CapsNet, an input image goes through two convolutional layers of 64 channels (3x3 Conv - BN - ReLU). The primary caps layer has eight channels of 16-dimensional capsules, the conv caps has eight channels, and the fully conv caps layer has ten channels. Each capsule channels in the conv caps layer and fully conv caps layer has 32 dimensions in the MNIST and 16 dimensions in the affNIST. We decreased the spatial size of the capsule features by applying a 3x3 convolution of stride 2 in the convolutional transform of the conv caps layer. We trained our model for 20 epochs. [**Accuracy**]{} Our model shows a comparable accuracy with the substantial decrease in the number of parameters and training time. Our model with 5.31M parameters achieved 99.45% accuracy on the MNIST dataset without any data augmentation and 99.46% accuracy with data augmentation. (Table \[table:result\_table\]) The CapsuleNet with 8.21M parameters achieved 99.45% accuracy without any data augmentation and 99.52% with data augmentation. The reported accuracy of CapsuleNet on the MNIST dataset with translation augmentation is 99.75% [@capsulenet]. Also, the training took 37.2 seconds per epoch for our proposed model and 199.5 seconds per epoch for CapsuleNet when we experimented on GTX 1080 GPUs. In the affNIST experiments, there are two options to generate training images from the MNIST dataset. The first option is to create a larger dataset by generating a set of all the possible augmented data before training. The second option is to apply translation over the original dataset for each epoch. The reported accuracy of CapsuleNet is 79% and that of the baseline CNN model is 66% in [@capsulenet]. The experiment is performed on the former option.[[^4]]{} Our proposed model achieved 91.6% accuracy for the latter option. Under the comparable experiment, our model outperformed the CapsuleNet and the baseline CNN model. Since our model is transformation equivariant (Section \[Transformation Equivariance\]), our model is robust to affine transformations. Classification Results on CIFAR-10 {#Results_CIFAR} ---------------------------------- [**Dataset**]{} The CIFAR-10 dataset is a $32 \times 32$ colored natural images in 10 classes. We adopted 0.1 translation, rotation up to 20 degrees, and horizontal flip as a data augmentation for CIFAR-10. [**Implementation**]{} For the CIFAR-10 classification task, we added four conv caps layer between a primary caps layer and fully conv caps layer. We decreased the spatial size of the capsule features in the first conv caps layer as in Section \[Results\_MNIST\_affNIST\]. Each conv caps layer has eight channels of 32-dimensional capsules and is connected to the next conv caps layer with a residual connection [@ResNet]. Note that the residual connection in [@ResNet] connects the $l$th layer and $(l+2)$th layer, but our residual connection connects the $l$th conv caps layer and $(l+1)$ conv caps layer. We trained our model for 200 epochs. [**Accuracy**]{} The results in Table \[table:result\_table\] show that our model outperforms CapsuleNet with and without data augmentation. CapsuleNet with 11.74M parameters shows 63.1% accuracy in C10 and 69.6% accuracy in C10+. However, our proposed model with 9.6M parameters shows 87.19 % accuracy in C10 and 88.61% accuracy in C10+. In [@capsulenet], an ensemble of 7 models achieves 89.4% accuracy when the models are trained with $24 \times 24$ patches of images and the introduction of a *none-of-the-above* category. However, an ensemble of 7 AR CapsNet models trained with C10+ achieved 90.11% test accuracy. Note that C10+ only uses rotation, shift, and horizontal flip as data augmentation and not the cropping or the *none-of-the-above* category. Conv caps layer Caps dim Params C10 C10+ ----------------- ---------- -------- ------- ------- 16 7.3M 77.51 81.89 32 12.6M 77.44 81.97 16 3.5M 81.96 82.83 32 7.7M 82.39 83.92 16 3.7M 84.48 84.77 32 8.4M 85.46 87.01 16 3.8M 85.56 86.93 32 8.9M 86.56 87.91 16 4.0M 86.37 87.21 32 9.6M 87.19 88.61 : Test accuracy (%) on the MNIST and CIFAR-10 for various hyperparameters. In each experiment, we trained a model for 200 epochs and chose the model with the lowest validation error. For each hyperparameter setting, AR CapsNet shows stable performance without showing severe degradation. []{data-label="table:Hyperparam_variation"} Robustness to hyperparameters ----------------------------- [**Implementation**]{} The AR CapsNet requires a set of hyperparameters, such as the number of conv caps layer and the capsule dimension of each capsule layer. To test the robustness to hyperparameters, we evaluate the AR CapsNet in the CIFAR-10 classification tasks according to the various setting of hyperparameters. The evaluated AR CapsNet architecture is the same as the models mentioned in Section \[Results\_MNIST\_affNIST\] and \[Results\_CIFAR\]. The primary caps layer has eight channels of 16-dimensional capsules, and the conv caps layer has eight capsule channels. In the setting of hyperparameters, the *Conv caps layer* denotes the number of conv caps layer between the primary caps layer and fully conv caps layer. In every model with at least one conv caps layer, the first conv caps layer decreases the spatial size of the capsule layer by adopting a 3x3 convolution of stride 2 in the convolutional transform. The *Capsule dim* denotes the capsule dimension of the conv caps layer and fully conv caps layer. [**Robustness**]{} All the AR CapsNet models trained with CIFAR-10 dataset show decent performance in Table \[table:Hyperparam\_variation\]. Increasing capsule dimension and the number of conv caps layer lead to an improvement in the test accuracy. The AR CapsNet model with four conv caps layer shows 86.37% accuracy with 16-dimensional capsules and 87.19% accuracy with 32-dimensional capsules. The AR CapsNet with four conv caps layer and 32-dimensional capsules shows the best results of 87.19% in C10 and 88.61% in C10+. Also, the AR CapsNet model with no conv caps layer has more parameters than the model with four conv cap layer and shows the worst performance. The features in the primary caps layer has a large spatial size. Thus, the fully conv caps layer connected to the primary caps layer assigns excessive parameters, and this causes overfitting. ![Decoder outputs according to dimension perturbations. We observed the variations of decoder output as we perturbed one dimension of the output capsules by steps of $0.05\sqrt{D^L}$ from $-0.25\sqrt{D^L}$ to $+0.25\sqrt{D^L}$. The perturbation leads to the combination of variations in the decoder output images. (e.g., rotation, thickness, etc.).[]{data-label="fig:mnist_manipulate"}](image/manipulate-0.png "fig:"){width="0.95\linewidth"} ![Decoder outputs according to dimension perturbations. We observed the variations of decoder output as we perturbed one dimension of the output capsules by steps of $0.05\sqrt{D^L}$ from $-0.25\sqrt{D^L}$ to $+0.25\sqrt{D^L}$. The perturbation leads to the combination of variations in the decoder output images. (e.g., rotation, thickness, etc.).[]{data-label="fig:mnist_manipulate"}](image/manipulate-8_1.png "fig:"){width="0.95\linewidth"} ![Decoder outputs according to dimension perturbations. We observed the variations of decoder output as we perturbed one dimension of the output capsules by steps of $0.05\sqrt{D^L}$ from $-0.25\sqrt{D^L}$ to $+0.25\sqrt{D^L}$. The perturbation leads to the combination of variations in the decoder output images. (e.g., rotation, thickness, etc.).[]{data-label="fig:mnist_manipulate"}](image/manipulate-9_1.png "fig:"){width="0.95\linewidth"} Transformation Equivariance {#Transformation Equivariance} --------------------------- [**Dimension perturbation**]{} To prove that our proposed model is transformation equivariant, we executed experiments on the MNIST model as in [@capsulenet]. We observed the variations of decoder output as we perturbed one scalar element of the output capsules (Figure \[fig:mnist\_manipulate\]). The experiments in the [@capsulenet] perturbed one scalar element from -0.25 to 0.25. Since the output capsules of the AR CapsNet have lengths of up to $\sqrt{D^L}$ compared to 1 in [@capsulenet], we perturbed one scalar element from $-0.25\sqrt{D^L}$ to $0.25 \sqrt{D^L}$ where $D^L$ denotes the capsule dimension of output capsules. Figure \[fig:mnist\_manipulate\] shows that some dimensions of the output capsules represent variations in the way the digit of the corresponding class is instantiated. Some dimensions of the output capsules represent the localized skew in digit 0, the rotation and the size of the higher circle in digit 8, and the rotation, thickness, and skew in digit 9. Digit Rot+ x+ y+ Rot- x- y- ------- -------- -------- -------- -------- -------- -------- -- 8 0.89 0.89 0.91 0.86 0.87 0.86 5 0.89 0.78 0.73 0.84 0.88 0.86 0.88 0.86 0.84 0.83 0.83 0.84 (0.89) (0.88) (0.80) (0.81) (0.84) (0.84) : The average of relative ratio $\{r_i\}$ for each combination of digit and transformation. The avg represents the average of all 10,000 test samples for each affine transformation. We report the results of models trained on the MNIST+ dataset in the (). A high relative ratio implies the difference vectors are strongly aligned. For random vectors, the average of relative ratio $\{r_i\}$ is 0.311 and standard deviation is 0.262. []{data-label="table:SVD result table"} [**Alignment ratio**]{} Each scalar element of the output capsules represents a combination of variations such as rotation, thickness, and skew. (Digit 9 in Figure \[fig:mnist\_manipulate\]) Since the length of the output capsules is basis-invariant, the transformation on an input image could be represented in coordinates of any basis. To further test the transformation equivariance of the AR CapsNet, we tested whether the difference in the output capsules caused by applying a transformation on an input image is aligned in one direction. Let $\{T_i\}_{i=1, \cdots, N}$ be a set of affine transformations on an input image $x$. We denote the difference between the output capsules $\mathbf{u}_n^{L}(T_i (x))$ and $\mathbf{u}_n^{L}(x)$ as $\mathbf{v}_i(x)$ where n denotes the corresponding class of x. $$\mathbf{v}_i(x) = \mathbf{u}_n^{L}(T_i (x)) - \mathbf{u}_n^{L}(x)$$ We denote the concatenation of $\mathbf{v}_i(x)$ along the row axis as $\mathbf{V}$. In order to obtain a representative unit vector $\Tilde{v}$ of $\{\mathbf{v}_i(x)\}$, we apply a Singular-Value Decomposition(SVD) on matrix $\mathbf{V}$. $$\begin{aligned} \mathbf{c}, \Tilde{v} &= \arg\min_{c_i, \Tilde{v}} \sum_{i} || \mathbf{v}_i(x) - c_{i} \cdot \Tilde{v} ||_{2}^{2} \\ &= \arg\min_{c_i, \Tilde{v}} || \mathbf{V} - \mathbf{c} \cdot \Tilde{v}^{T} ||_{F}^{2}\end{aligned}$$ where F denotes the Frobenius norm, $\mathbf{c} = (c_1, \cdots, c_N)^{T}$, and $c_i \in \mathbb{R}$. The exact solution of this low rank approximation problem is the first right-singular vector $\Tilde{v}$ of $\mathbf{V}$. This experiment is similar to the Principal Component Analysis(PCA) except that we do not subtract the mean for each columns of $\mathbf{V}$. The align vector $\Tilde{v}$ corresponds to the principal vector of PCA. We observed the relative ratio $r_i$ of principal component of $\mathbf{v}_i(x)$ to the vector norms $||\mathbf{v}_i(x)||_2$. $$\label{relative ratio} r_i = \frac{|\mathbf{v}_i(x) \cdot \Tilde{v}|}{||\mathbf{v}_i(x)||_2}$$ We randomly chose 10,000 images from the test set. For each test image, we generated five images by applying an affine transformation and observed the relative ratio $r_i$. In Table \[table:SVD result table\], Rot$(\pm)$ represents $\pm\{5, 10, 15, 20, 25 \}$ degrees rotations and x$(\pm)$ represents a horizontal translation up to $\pm5$ pixels. y$(\pm)$ represents a vertical translation up to $\pm5$ pixels as well. We observed the average of relative ratio $r_i$ for each combination of digit and transformation. Table \[table:SVD result table\] shows the average of relative ratio $r_i$ for two digits (highest : digit 8, lowest : digit 5) and the average for 10,000 test samples for each transformation. As a reference, we generated five random vectors from the standard multivariate normal distribution. We conducted the same experiment for random vectors for 1,000 times as well. We obtained an average of 0.311 and a standard deviation of 0.262 for random vectors. Even for the worst-case digit 5, every transformation shows a significantly higher relative ratio $r_i$ of 0.73 in y$+$ than the random vectors. This result implies that the difference vectors are strongly aligned in one direction. Therefore, AR CapsNet encodes affine transformations on an input image by some vector components. Also, we report the results of models trained on the MNIST+ dataset in the (). The models trained on the MNIST+ show comparable relative ratio $r_i$ to those trained on the MNIST. This result shows that AR CapsNet encodes affine transformations even without observing transformations during training. ![Distribution of cosine similarity between unit align vectors $\Tilde{v}$ of positive and negative affine transformations. []{data-label="fig:cosine similarity"}]({"image/hist_ours"}.pdf){width="0.8\linewidth"} ![Output capsules $\mathbf{u}_n^L$ when the cosine similarity between align vectors of positive and negative transformations is -1 or 1. $T_{+}$ and $T_{-}$ represent the positive and negative transformation. **Left:** cosine similarity -1. **Right:** cosine similarity 1. []{data-label="fig:output capsule diagram"}]({"image/cosine_sim_diagram"}.pdf){width="0.85\linewidth"} An interesting observation is that AR CapsNet is transformation equivariant but do not distinguish the positive and negative transformations. Figure \[fig:cosine similarity\] shows the histogram of the cosine similarity between the align vectors of positive and negative transformation.[[^5]]{} We observed two peaks around -1 and 1. The cosine similarity of -1 and 1 imply that positive and negative transformations are encoded in one dimension. However, the cosine similarity of 1 suggests that the difference vectors of positive and negative transformations have the same direction.(Figure \[fig:output capsule diagram\]) We leave the explanation of this observation to future work. Conclusion ========== In this work, we suggested a new capsule network architecture called Attention Routing CapsuleNet (AR CapsNet). By introducing the attention routing and capsule activation, AR CapsNet obtained a higher accuracy compared to CapsuleNet while using fewer parameters and less training time. The attention routing is an effective way to route between capsules because it only compares capsules of the same spatial location. In addition, the attention routing does not require an iterative routing process as the dynamic routing does because it directly learns the weights between capsules. The capsule activation is based on the assumption that the capsule-scale activation can extract transformation equivariant features even if it is not orientation-preserving. This assumption distinguish the capsule activation from the squash activation function and its variant. While using the building blocks of CNNs, AR CapsNet is transformation equivariant. We showed that capsules have transformation information by manipulating the output capsules and then observing the decoder output images. Also, we observed the difference vectors between the output capsules of an original image and an affine transformed image. By showing that the difference vectors are strongly aligned in one direction, we proved that AR CapsNet encodes transformation information in some dimensions. There are natural variations of AR CapsNet such as introducing a feature compression by 1x1 convolution to the capsule activation and a transformer network [@Transformer] to the attention routing. We plan to study these variations in the future. [^1]: $\mathbf{u}^{l}_{(:, :, :, n_0)} := \{u^{l}_{w,h,d,n}|n=n_0\}$ [^2]: CapsuleNet [@capsulenet] scaled the reconstruction loss by 0.392. Since we use the mean of L2 loss and CapsuleNet use the sum of L2 loss, 0.392 = 0.0005 $\times$ 784. [^3]: https://github.com/XifengGuo/CapsNet-Keras [^4]: https://github.com/Sarasra/models/tree/master/research/capsules [^5]: The direction of the first right-singular vector $\Tilde{v}$ is given by $\mathbf{v}_i(x) \cdot \Tilde{v} > 0$ in for each positive and negative transformation.

[**Weighted $2$-Motzkin Paths**]{}\ William Y.C. Chen$^1$, Sherry H.F. Yan$^2$ and Laura L.M. Yang$^3$\ Center for Combinatorics, LPMC\ Nankai University, Tianjin 300071, P. R. China\ $^1$chen@nankai.edu.cn, $^2$huifangyan@eyou.com, $^3$yanglm@hotmail.com [**Abstract.** ]{} This paper is motivated by two problems recently proposed by Coker on combinatorial identities related to the Narayana polynomials and the Catalan numbers. We find that a bijection of Chen, Deutsch and Elizalde can be used to provide combinatorial interpretations of the identities of Coker when it is applied to weighted plane trees. For the sake of presentation of our combinatorial correspondences, we provide a description of the bijection of Chen, Deutsch and Elizalde in a slightly different manner in the form of a direct construction from plane trees to $2$-Motzkin paths without the intermediate step involving the Dyck paths. [**AMS Classification:**]{} 05A15, 05A19 [**Keywords:**]{} Plane tree, Narayana number, Catalan number, $2$-Motzkin path, weighted $2$-Motzkin path, multiple Dyck path, bijection. [**Suggested Running Title:**]{} Weighted $2$-Motzkin Paths [**Corresponding Author:**]{} William Y. C. Chen, Email: chen@nankai.edu.cn Introduction ============ The structure of $2$-Motzkin paths, introduced by Barcucci, Lungo, Pergola and Pinzani [@blpp], has proved to be highly efficient in the study of plane trees, Dyck paths, Motzkin paths, noncrossing partitions, RNA secondary structures, Devenport-Schinzel sequences, and combinatorial identities, see [@ds; @Klazar; @SW]. While it is most natural to establish a correspondence between Dyck paths of length $2n$ and $2$-Motzkin paths of length $n-1$, Deutsch and Shapiro came to the realization that direct correspondences between plane trees and $2$-Motzkin paths can have many applications. Recently, Chen, Deutsch and Elizalde [@cde] found bijections between plane trees and $2$-Motzkin paths for the enumeration of plane trees by the numbers of old and young leaves [@cde]. The main result of this paper is to show that the bijection of Chen, Deutsch and Elizalde, presented in a slightly different manner, can be applied to weighted plane trees in order to give combinatorial interpretations of two identities involving the Narayana numbers and Catalan numbers due to Coker [@coker]. This leads to the solutions of the two open problems left in the paper [@coker]. Recall that a [*$2$-Motzkin path*]{} is a lattice path starting at $(0,0)$ and ending at $(n,0)$ but never going below the $x$-axis, with possible steps $(1,1)$, $(1,0)$ and $(1,-1)$, where the level steps $(1,0)$ can be either of two kinds: straight and wavy. The [*length*]{} of the path is defined to the number of its steps. Deutsch and Shapiro [@ds] presented a bijection between plane trees with $n$ edges and $2$-Motzkin paths of length $n-1$. So the number of $2$-Motzkin paths of length $n-1$ equals the Catalan number $$C_n = {1 \over n+1} \, {2n \choose n}.$$ Recently, Coker [@coker] established very interesting combinatorial identities ((\[q1\]) and (\[q2\]) below) by using generating functions and the Lagrange inversion formula based the study of multiple Dyck paths. A multiple Dyck path is a lattice path starting at $(0,0)$ and ending at $(2n, 0)$ with big steps that can be regarded as segments of consecutive up steps or consecutive down steps in an ordinary Dyck path. Note that the notion of multiple Dyck path is formulated by Coker in different coordinates. The main ingredients in Coker’s identities are the Catalan number and the Narayana numbers: $$N(n,k)=\frac{1}{n}{n \choose k}{n \choose k-1},$$ which counts the number of all plane trees with $n$ edges and $k$ leaves [@SW; @stanley2]. It is sequence $A001263$ in [@sloane]. Coker [@coker] left the following two open problems: [**Problem 1.**]{} Find a bijective proof of the following identity $$\label{q1} \sum_{k=1}^n\frac{1}{n}{n \choose k}{n \choose k-1}4^{n-k}=\sum_{k=0}^{\lfloor{(n-1)}/{2}\rfloor}C_k{n-1 \choose 2k}4^k5^{n-2k-1}.$$ [**Problem 2.**]{} Find a combinatorial explanation for the following identity $$\label{q2} \sum_{k=1}^n\frac{1}{n}{n \choose k}{n \choose k-1}x^{2k}(1+x)^{2n-2k}=x^2\sum_{k=0}^{n-1}C_{k+1}{n-1 \choose k}x^k(1+x)^{k},$$ which is equation (6.2) in [@coker]. The above identity (\[q1\]) is a special case of the following identity: $$\label{q1x} \sum_{k=0}^n \, {1\over n} \, {n\choose k} {n \choose k-1} t^{n-k} = \sum_{k=0}^{\lfloor (n-1)/2\rfloor}\, C_k\, {n-1\choose 2k}\,t^k (1+t)^{n-2k-1},$$ where the left hand side of (\[q1x\]) is the Narayana polynomial, as denoted by ${\cal N}_n(t)$ in [@coker]. The identity (\[q1x\]) is the relation (4.4) in [@coker], which can be derived as an identity on the Narayana numbers and the Catalan numbers due to Simion and Ullman [@SU], see also [@CDD]. Remarkably, (\[q1x\]) has many consequences as pointed by Coker [@coker]. For example, it implies the classical identity of Touchard [@T], and the formula on the little Schröder numbers in terms of the Catalan numbers [@G]. The reason for the evaluation of ${\cal N}_n(t)$ at $t=4$ lies in the fact that ${\cal N}_n(4)$ equals the number $d_n$ of multiple Dyck paths of length $2n$. The first few values of $d_n$ for $n=0, 1,2,3,4, 5,6,7$ are $$1,\;1,\; 5,\; 29,\; 185,\; 1257,\; 8925,\; 65445,$$ which is the sequence $A059231$ in [@sloane]. From the interpretation of Narayana numbers in terms of Dyck paths of length $2n$ and of $k$ peaks, it is not difficult to show that $d_n={\cal N}_n(4)$. However, the right hand side of (\[q1\]) does not seems to be obvious, which is obtained by establishing a functional equation and by using the Lagrange inversion formula. The natural question as raised by Coker [@coker] is to find a combinatorial interpretation of (\[q1\]). Note that the enumeration of multiple Dyck paths has also been studied independently by Sulanke [@Sulanke] and Woan [@Woan]. The relation (\[q2\]) was established from the enumeration of multiple Dyck paths of length $2n$ with a given number of steps. Let $\lambda_{n,j}$ be the number of multiple Dyck paths of length $2n$ and $j$ steps, ${\cal P}_n(x)$ be the polynomial $${\cal P}_n(x)= \sum_{j=2}^{2n} \, \lambda_{n,j}\, x^j.$$ It was shown that $${\cal P}_n(x) = \sum_{k=1}^n \, {1 \over n} {n\choose k} {n \choose k-1} x^{2k} (1+x)^{2n-2k},$$ which can be restated as $${\cal P}_n(x) = x^{2n} {\cal N}_n((1+x^{-1})^2).$$ Coker [@coker] discovered the connection between ${\cal P}_n(x)$ and the polynomial ${\cal R}_n(x)$ introduced by Denise and Simion [@denise-s] in their study of the number of exterior pairs of Dyck paths of length $2n$. The polynomials ${\cal R}_n(x)$ have the following expansion: $${\cal R}_n(x) = \sum_{k=0}^{n-1} \, (-1)^k C_{k+1}{n-1 \choose k}\, x^k (1-x)^k.$$ It now becomes clear that the identity (\[q2\]) can be rewritten as $$\label{PR} {\cal P}_n(x) =x^2 {\cal R}_n(-x).$$ To give combinatorial interpretations of both (\[q1\]) and (\[q2\]), we apply the bijection of Chen, Deutsch and Elizalde [@cde] to weighted plane trees to get weighted $2$-Motzkin paths. Then we use weight-preserving operations on $2$-Motzkin paths to derive the desired combinatorial identities. More precisely, these weight-preserving operations are essentially the reductions from weighted $2$-Motzkin paths to Dyck paths and $2$-Motzkin paths. It would be interesting to find a direct correspondence on Dyck paths which leads to a combinatorial interpretation of (\[PR\]). Weighted $2$-Motzkin Paths ========================== Let us review a bijection between plane trees and $2$-Motzkin paths due to Chen, Deutsch and Elizalde [@cde], which is devised for the enumeration of plane trees with $n$ edges and a fixed number of old leaves and a fixed number of young leaves. Such a consideration of old and young leaves reflects to the four types of steps of $2$-Motzkin paths. For the purpose of this paper, we present a slightly modified version of the nonrecursive bijection in [@cde]. Our terminology is also somewhat different. For a plane tree $T$, a vertex of $v$ is called a leaf if it does not have any children. An internal vertex is a vertex that has at least one child. An edge is denoted as a pair $(u, v)$ of vertices such that $v$ is a child of $u$. Let $u$ be an internal vertex, and $v_1$, $v_2$, $\ldots$, $v_k$ be the children of $u$ listed from left to right. Then we call $v_k$ an [*exterior vertex*]{} and $(u, v_k)$ an [*exterior edge*]{}. If $k>1$, then the edges $(u, v_1)$, $(u, v_2)$ $\ldots$, and $(u, v_{k-1})$ are called [*interior edges*]{} and $v_1, v_2, \ldots, v_{k-1}$ are called [*interior vertices*]{}. An edge containing a leaf vertex is called a [*terminal edge*]{}. Let $u$ be the root of $T$, $(u, u_1)$ be the exterior edge of $u$, $(u_1, u_2)$ be the exterior edge of $u_1$, and so on, finally $(u_{k-1}, u_{k})$ be the exterior edge of $u_{k-1}$ such that $u_k$ is a leaf. The exterior edge $(u_{k-1}, u_k)$ is called the [*critical edge*]{} of $T$. To summarize, the edges of a plane tree $T$ are classified into five categories. - Non-terminal interior edges. - Non-terminal exterior edges. - Terminal interior edges. - Terminal exterior edges (which do not include the critical edge). - The critical edge. Note that the [*critical edge*]{} of $T$ is the last encountered edge when we traverse the edges of $T$ in preorder. From the above classification on the edges of a plane tree, it is easy to describe the Chen-Deutsch-Elizalde bijection between plane trees and $2$-Motzkin paths by the preorder traversal of the edges of $T$. To be precise, let $u$ be the root of $T$, $v_1, v_2, \ldots, v_k$ be the children of $u$, and $T_1, T_2, \ldots, T_k$ be the subtrees of $T$ rooted at $v_1, v_2, \ldots, v_k$, respectively. Then the preorder traversal of the edges of $T$, denoted by $P(T)$, is a linear order of the edges of $T$ recursively defined by $$(u, v_1) \, P(T_1) \, (u, v_2) \, P(T_2) \, \cdots \, (u, v_k) \, P(T_k).$$ [**The Bijection of Chen, Deutsch and Elizalde [@cde]:**]{} Let $T$ be any nonempty plane tree. At each step of the traversal of the edges of $T$ in preorder, \(i) draw an up step for a non-terminal interior edge; \(ii) draw a straight level step for a non-terminal exterior edge; \(iii) draw a wavy level step for a terminal interior edge; \(iv) draw a down step for a terminal exterior edge; \(v) do nothing for the critical edge. (400,60) (5,15)[(1,1)[10]{}]{} (15,25) (15,25)[(-1,-2)[5]{}]{} (5,15) (10,15) (5,10)[(0,1)[5]{}]{} (5,10)[(1,-2)[2.5]{}]{} (5,10)[(-1,-2)[2.5]{}]{} (5,10)[(0,-1)[5]{}]{} (5,10) (7.5,5) (2.5,0) (5,5) (5,5)[(-1,-2)[2.5]{}]{} (5,5)[(1,-2)[2.5]{}]{} (2.5,5) (7.5,0) (15,25)[(0,-1)[10]{}]{} (15,15)[(1,-2)[2.5]{}]{} (15,15)[(-1,-2)[2.5]{}]{} (15,15) (17.5,10) (12.5,10) (17.5,10)[(-1,-2)[2.5]{}]{} (17.5,10)[(1,-2)[2.5]{}]{} (20,5)[(0,-1)[5]{}]{} (15,5) (20,5) (20,0) (15,25) (15,25)[(1,-1)[10]{}]{} (25,15) (25,15)[(2,-1)[10]{}]{} (35,10) (25,15)[(1,-1)[5]{}]{} (30,10) (25,15)[(0,-1)[5]{}]{} (25,10) (35,10)[(0,-1)[5]{}]{} (35,5) (30,10)[(0,-1)[5]{}]{} (30,5) (45,10)[(1,0)[6]{}]{} (51,9)[(-1,0)[6]{}]{} (60,5) (60,5)[(1,1)[4]{}]{} (64,9) (64,9)[(1,0)[4]{}]{} (68,9) (68,9)(1,0)[4]{}[(1,1)[0.5]{}]{} (68.5,9.5)(1,0)[4]{}[(1,-1)[0.5]{}]{} (72,9) (72,9)[(1,1)[4]{}]{} (76,13) (76,13)(1,0)[4]{}[(1,1)[0.5]{}]{} (76.5,13.5)(1,0)[4]{}[(1,-1)[0.5]{}]{} (80,13) (80,13)[(1,-1)[4]{}]{} (84,9) (84,9)[(1,-1)[4]{}]{} (88,5) (88,5)(1,0)[4]{}[(1,1)[0.5]{}]{} (88.5,5.5)(1,0)[4]{}[(1,-1)[0.5]{}]{} (92,5) (92,5)[(1,1)[4]{}]{} (96,9) (96,9)(1,0)[4]{}[(1,1)[0.5]{}]{} (96.5,9.5)(1,0)[4]{}[(1,-1)[0.5]{}]{} (100,9) (100,9)[(1,0)[4]{}]{} (104,9) (104,9)(1,0)[4]{}[(1,1)[0.5]{}]{} (104.5,9.5)(1,0)[4]{}[(1,-1)[0.5]{}]{} (108,9) (108,9)[(1,0)[4]{}]{} (112,9) (112,9)[(1,-1)[4]{}]{} (116,5) (116,5)[(1,0)[4]{}]{} (120,5) (120,5)(1,0)[4]{}[(1,1)[0.5]{}]{} (120.5,5.5)(1,0)[4]{}[(1,-1)[0.5]{}]{} (124,5) (124,5)[(1,1)[4]{}]{} (128,9) (128,9)[(1,-1)[4]{}]{} (132,5) (132,5)[(1,0)[4]{}]{} (136,5) It is easy to see that we have obtained a $2$-Moztkin path. More precisely, a plane tree with $n$ edges corresponds to a $2$-Motzkin path of length $n-1$. The above bijection is denoted by $\Phi$. As a hint to why the above bijection works, one may check that for any plane tree $T$, $\#$ non-terminal interior edges $=$ $\#$ terminal exterior edges. We are now ready to assign weights to the edges of a plane tree $T$ in order to obtain combinatorial interpretations of the identities (\[q1\]) and (\[q2\]). The weights of the edges of a plane tree will translate into weights of steps of the corresponding $2$-Motzkin path. In fact, we will take slightly different formulations of (\[q1x\]) and (\[q2\]). For $n \geq 1$, we have $$\label{identity1} \sum_{k=1}^n\frac{1}{n}{n \choose k}{n \choose k-1}x^{k-1}=\sum_{k=0}^{\lfloor{(n-1)}/{2}\rfloor}C_k{n-1 \choose 2k}x^k(1+x)^{n-2k-1}.$$ Let $T$ be a plane tree with $n$ edges. We assign the weights to the edges of $T$ by the following rule: All the terminal edges except the critical edge are given the weight $x$ and all other edges are given the weight $1$. The weight of $T$ is the product of the weights of its edges. Then the left hand side of (\[identity1\]) is the sum of the weights of all plane trees with $n$ edges. By the above bijection $\Phi$, the set of weighted plane trees with $n$ edges is mapped to the set of $2$-Motzkin paths of length $n-1$ in which all the down steps and wavy level steps are given the weight $x$, and other steps are given the weight $1$. Consider the weighted $2$-Motzkin paths of length $n-1$ that have $k$ up steps and $k$ down steps. These $k$ up steps and $k$ down steps form a Dyck path of length $2k$. The binomial coefficient ${n-1 \choose 2k}$ comes from the choices of the $2k$ positions for these $k$ up steps and $k$ down steps. The remaining $n-2k-1$ steps are either wavy level steps or straight level steps. Since only a wavy level step carries the weight $x$, the total contributions of the weights of $n-2k-1$ level steps amount to $(1+x)^{n-2k-1}$. The $k$ up steps would contribute $x^k$. Therefore, the right hand side of (\[identity1\]) equals the total contributions of all the weighted $2$-Motzkin paths of length $n-1$, as desired. For $n \geq 1$, we have $$\label{identity2} \sum_{k=1}^n\frac{1}{n}{n \choose k}{n \choose k-1}x^{2(k-1)}(1+x)^{2(n-k)}=\sum_{k=0}^{n-1}C_{k+1}{n-1 \choose k}x^k(1+x)^{k}.$$ Given a plane tree $T$ with $n$ edges, we assign the weights to the edges of $T$ by the following rule: All the terminal edges except the critical edge are assigned the weight $x^2$, all the non-terminal edges are given the weight $(1+x)^2$, and the critical edge is assigned the weight $1$. Then the bijection $\Phi$ transform $T$ into a $2$-Motzkin path in which all the down steps and wavy level steps have the weight $x^2$ and all the up steps and straight level steps have the weight $(1+x)^2$. By the above weight assignment, the left hand side of (\[identity2\]) is then the sum of the weights of all plane trees with $n$ edges. We now proceed to show that the right hand side of (\[identity2\]) is the sum of weights of all $2$-Motzkin paths of length $n-1$. Consider a $2$-Motzkin paths that has $k$ up steps and $k$ down steps. Since the up steps have weight $x^2$ and the down steps have weight $(1+x)^2$, it makes no difference with respect to the sum of weights if one changes the weights of both up steps and down steps to $x(1+x)$. In other words, such an operation on the change of weights is a weight-preserving bijection on the set of $2$-Motzkin paths. Note that for any weight assignment, we may transform the sum of weights of all $2$-Motzkin paths of length $n-1$ to the sum of weights of all Motzkin paths of the same length by the following weight assignment: the up steps and down steps carry the same weight, and the horizontal steps in the Motzkin paths carry the weight as the sum of the weights of a straight level step and a wavy level step in the $2$-Motzkin path. Therefore, for our weight assignment the sum of weights of $2$-Motzkin paths of length $n-1$ equals the sum of Motzkin paths of length $n-1$ given the following weight assignment: up steps and down steps are given the weight $x(1+x)$, and the horizontal steps are given the weight $$x^2+(1+x)^2= 1 + x(1+x) + x(1+x).$$ So we have transformed the sum of weights of $2$-Motzkin paths of length $n-1$ to the sum of weights of all Motzkin paths of length $n-1$ in which all the up steps, down steps have weights $x(1+x)$, and the horizontal steps can be regarded as either a straight level step with weight $x(1+x)$, or a wavy level step with weight $x(1+x)$ or a special dotted step with weight $1$. We now get the desired sum as on the right hand side of (\[identity2\]) by considering the distribution of the special dotted steps, because the remaining steps (up, down, straight level, wavy level) all have the weight $x(1+x)$ and they form a $2$-Motzkin path. Setting $x={1/4}$ in (\[identity1\]) we obtain (\[q1\]). [**Acknowledgments.**]{} This work was done under the auspices of the National Science Foundation, the Ministry of Education, and the Ministry of Science and Technology of China. [99]{} E. Barcucci, A.D. Lungo, E. Pergola and R. Pinzani, A construction for enumerating $k$-coloured Motzkin paths, Lecture Notes in Computer Science, Vol. 959, Springer, Berlin, 1995, 254–263. W.Y.C. Chen, E. Deutsch and S. Elizalde, Old and young leaves on plane trees and $2$-Motzkin paths, preprint, 2004. W.Y.C. Chen, E.Y.P. Deng and R.R.X. Du, Reduction of $m$-regular noncrossing partitions, Europ. J. Combin., to appear. C. Coker, Enumerating a class of lattice paths, Discrete Math., 271 (2003) 13–28. A. Denise and R. Simion, Two combinatorial statistics on Dyck paths, Discrete Math., 137 (1995) 155–176. E. Deutsch and L.W. Shapiro, A bijection between ordered trees and $2$-Motzkin paths and its many consequences, Discrete Math., 256 (2002) 655–670. D. Gouyou-Beauchamps and B. Vauquelin, Deux propriétés combinatoires des nombres de Schröder, Theor. Inform. Appl., 22 (1988), 361-388. M. Klazar, On numbers of Davenport-Schinzel sequences, Discrete Math., 185 (1998) 77–87. R. Simion and D. Ullman, On the structure of the lattice of noncrossing partitions, Discrete Math., 98 (1991) 193–206. N.J.A. Sloane, The On-Line Encyclopedia of Integer Sequences, http://www.research.att.com/$\thicksim$njas/sequences. R. Sulanke, Counting lattice paths by Narayana polynomials, Electron. J. Combin., 7 (2000) \#R40. W.R. Schmitt and M.S. Waterman, Linear trees and RNA secondary structure, Discrete Appl. Math., 51 (1994) 317–323. R.P. Stanley, Enumerative Combinatorics, Vol. 2, Cambridge University Press, Cambridge, UK, 1999. J. Touchard, Sur certaines équations fonctionnelles, In: Proc. 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--- abstract: 'We investigate the in-medium masses of $D$ and $\bar{D}$ mesons in the isospin-asymmetric nuclear matter at finite temperatures arising due to the interactions with the nucleons, the scalar isoscalar meson $\sigma$, and the scalar iso-vector meson $\delta$ within a SU(4) model. However, since the chiral symmetry is explicitly broken for the SU(4) case due to the large charm quark mass, we use the SU(4) symmetry here only to obtain the interactions of the $D$ and $\bar D$ mesons with the light hadron sector, but use the observed values of the heavy hadron masses and empirical values of the decay constants. The in-medium masses of $J/\psi$ and the excited charmonium states ($\psi(3686)$ and $\psi(3770)$) are also calculated in the hot isospin asymmetric nuclear matter in the present investigation. These mass modifications arise due to the interaction of the charmonium states with the gluon condensates of QCD, simulated by a scalar dilaton field introduced to incorporate the broken scale invariance of QCD within the effective chiral model. The change in the mass of $J/\psi$ in the nuclear matter with the density is seen to be rather small, as has been shown in the literature by using various approaches, whereas, the masses of the excited states of charmonium ($\psi(3686)$ and $\psi(3770)$) are seen to have considerable drop at high densities. The present study of the in-medium masses of $D$ ($\bar{D}$) mesons as well as of the charmonium states will be of relevance for the observables from the compressed baryonic matter, like the production and collective flow of the $D$ ($\bar D$) mesons, resulting from the asymmetric heavy ion collision experiments planned at the future facility of the FAIR, GSI. The mass modifications of $D$ and $\bar{D}$ mesons as well as of the charmonium states in hot nuclear medium can modify the decay of the charmonium states ($\Psi^{''}, \chi_{c}, J/\Psi$) to $D\bar{D}$ pairs in the hot dense hadronic matter. The small attractive potentials observed for the $\bar{D}$ mesons may lead to formation of the $\bar{D}$ mesic nuclei.' author: - Arvind Kumar - Amruta Mishra title: D mesons and charmonium states in asymmetric nuclear matter at finite temperatures --- \#1 Introduction ============ The study of in-medium properties of hadrons is important for the understanding of strong interaction physics. The study of in-medium hadron properties has relevance in heavy-ion collision experiments as well as in nuclear astrophysics. There have been also extensive experimental efforts for the study of in-medium hadron properties by nuclear collision experiments. In these heavy-ion collision experiments, hot and dense matter is produced. By studying the experimental observables one can infer about how the hadron properties are modified in the medium. For example, the observed enhanced dilepton spectra [@ceres; @helios; @dls] could be a signature of medium modifications of the vector mesons [@Brat1; @CB99; @vecmass; @dilepton; @liko]. Similarly the properties of the kaons and antikaons have been studied experimentally by KaoS collaboration and the production of kaons and antikaons in the heavy-ion collisions and their collective flow are directly related to the medium modifications of their spectral functions [@CB99; @cmko; @lix; @Li2001; @K5; @K6; @K4; @kaosnew]. The study of $D$ and $\bar{D}$ mesons properties will be of direct relevance for the upcoming experiment at FAIR, GSI, where one expects to produce matter at high densities and moderate temperatures [@gsi]. At such high densities, the properties of the $D$ and $\bar{D}$ mesons produced in these experiments are expected to be modified which should reflect in experimental observables like their production and propagation in the hot and dense medium. The reason for an expected appreciable modifications of the $D$ and $\bar D$ mesons is that $D$ and $\bar{D}$ mesons contain a light quark (u,d) or light antiquark. This light quark or antiquark interacts with the nuclear medium and leads to the modifications of $D$ and $\bar{D}$ properties. The experimental signature for this can be their production ratio and also in-medium $J/\psi$ suppression [@NA501; @NA50e; @NA502]. In heavy-ion collision experiments of much higher collision energies, for example in RHIC or LHC, it is suggested that the $J/\psi$ suppression is because of the formation of quark-gluon plasma (QGP) [@blaiz; @satz]. However, in Ref. [@zhang; @brat5; @elena] it is observed that the effect of hadron absorption of $J/\psi$ is not negligible. In Ref.[@vog], it was reported that the charmonium suppression observed in Pb + Pb collisions of NA50 experiment cannot be simply explained by nucleon absorption, but needs some additional density dependent suppression mechanism. It was suggested in these studies that the comover scattering [@vog; @capella; @cassing] can explain the additional suppression of charmonium. An important difference between $J/\psi$ suppression pattern in comovers interaction model and in a deconfining scenario is that, in the former case, the anomalous suppression sets in smoothly from peripheral to central collisions rather than in a sudden way when the deconfining threshold is reached [@capella]. The $J/\psi$ suppression in nuclear collisions at SPS energies has been studied in covariant transport approach HSD in Ref.[@cassing]. The calculations show that the absorption of $J/\psi$’s by both nucleons and produced mesons can explain reasonably not only the total $J/\psi$ cross–section but also the transverse energy dependence of $J/\psi$ suppression measured in both proton-nucleus and nucleus collisions. In Ref.[@wang], the cross section of $J/\psi$ dissociation by gluons is used to calculate the $J/\psi$ suppression in an equilibrating parton gas produced in high energy nuclear collisions. The large average momentum in the hot gluon gas enables gluons to break up the $J/\psi$, while hadron matter at reasonable temperature does not provide sufficiently hard gluons. Due to the reduction in the masses of $D$ and $\bar{D}$ mesons in the medium it is a possibility that excited charmonium states can decay to $D\bar{D}$ pairs [@brat6] instead of decaying to lowest charmonium state $J/\psi$. Actually higher charmonium states are considered as major source of $J/\psi$ [@pAdata]. Even at certain higher densities, it can become a possibility that the $J/\psi$ itself will decay to $D\bar{D}$ pairs. So this can be an explanation of the observed $J/\psi$ suppression by NA50 collaboration at $158$ GeV/nucleon in the Pb-Pb collisions [@blaiz]. The excited states of charmonium also undergo mass drop in the nuclear medium [@leeko]. The modifications of the in-medium masses of $D$ mesons is large then the $J/\psi$ mass modification [@haya1; @friman]. This is because the charmonium states are made up of a heavy charm quark and a charm antiquark. Within QCD sum rules, it is suggested that these heavy charmonium states interact with the nuclear medium through the gluon condensates. This is contrary to the interaction of the light vector mesons ($\rho, \omega, \phi$), which interact with the nuclear medium through the quark condensates. This is because all the heavy quark condensates can be related to the gluon condenstaes via heavy-quark expansion [@kimlee]. Also in the nuclear medium there are no valence charm quark to leading order in density and any interaction with the medium is gluonic. The QCD sum rule approach [@klingl] and leading order perturbative calculations [@pes1] to study the medium modifications of charmonium, show that the mass of $J/\psi$ is reduced slightly in the nuclear medium. In [@leeko], the mass modification of charmonium has been studied using leading order QCD formula and the linear density approximation for the gluon condensate in the nuclear medium. This shows a small drop for the $J/\psi$ mass at the nuclear matter density, but there is seen to be significant shift in the masses of the excited states of charmonium ($\psi(3686)$ and $\psi(3770)$). The in-medium modifications of $D$ and $\bar{D}$ mesons have been studied using various approaches. For example, in the QCD sum rule approach, it is suggested that the light quark or antiquark of $D (\bar D)$ mesons interacts with the light quark condensate leading to the medium modification of the $D(\bar D)$ meson masses [@arata; @qcdsum08]. The quark meson coupling (QMC) model has also been used to study the D-meson properties [@qmc]. In the QMC model, the light quarks (u,d) and light antiquarks ($\bar u$,$\bar d$) confined in the nucleons and mesons interact via exchange of a scalar-isoscalar $\sigma$ meson as well a vector $\omega$ meson. The nucleon has a large reduction of mass in the dense medium arising due to the interaction of the light quarks (u,d) with the $\sigma$ field. The drop in the mass of $D$ mesons observed in the QMC model turns out to be similar to those calculated within the QCD sum rule approach. In the present investigation, we study the properties of the $D$ and $\bar{D}$ mesons in the isospin-asymmetric hot nuclear matter. These modifications arise due to their interactions with the nucleons, the non-strange scalar isoscalar meson $\sigma$ and the scalar isovector meson $\delta$. We also study the medium modification of the masses of $J/\psi$ and excited charmonium states $\psi(3686)$ and $\psi(3770)$ in the nuclear medium due to the interaction with the gluon condensates using the leading order QCD formula. The gluon condensate in the nuclear medium is calculated from the medium modification of a scalar dilaton field introduced within a chiral SU(3) model [@paper3] through a scale symmetry breaking term in the Lagrangian density leading to the QCD trace anomaly. In the chiral SU(3) model, the gluon condensate is related to the fourth power of the dilaton field $\chi$ and the changes in the dilaton field with the density are seen to be small. We study the isospin dependence of the in-medium masses of charmonium obtained from the dilaton field, $\chi$ calculated for the asymmetric nuclear matter at finite temperatures. The medium modifications of the light hadrons (nucleons and scalar mesons) are described by using a chiral $SU(3)$ model [@paper3]. The model has been used to study finite nuclei, the nuclear matter properties, the in-medium properties of the vector mesons [@hartree; @kristof1] as well as to investigate the optical potentials of kaons and antikaons in nuclear matter [@kmeson; @isoamss] and in hyperonic matter in [@isoamss2]. For the study of the properties $D$ mesons in isospin-asymmetric medium at finite temperatures, the chiral SU(3) model is generalized to $SU(4)$ flavor symmetry to obtain the interactions of $D$ and $\bar{D}$ mesons with the light hadrons. Since the chiral symmetry is explicitly broken for the SU(4) case due to the large charm quark mass, we use the SU(4) symmetry here only to obtain the interactions of the $D$ and $\bar D$ mesons with the light hadron sector, but use the observed values of the heavy hadron masses and empirical values of the decay constants. This has been in line with the philosophy followed in Ref. [@liukolin] where charmonium absorption in nuclear matter was studied using the SU(4) model to obtain the relevant interactions. However, the values of the heavy hadron masses and the coupling constants in Ref. [@liukolin], were taken as the empirical values or as calculated from other theoretical models. The coupling constants were derived by using the relations from SU(4) symmetry, if neither the empirical values nor values calculated from other theoretical models were available [@liukolin]. The $D$ meson properties in symmetric hot nuclear matter using SU(4) model have been studied in ref. [@amdmeson] and for the case of asymmetric nuclear matter at zero temperature in [@amarind]. In a coupled channel approach for the study of D mesons, using a separable potential, it was shown that the resonance $\Lambda_c (2593)$ is generated dynamically in the I=0 channel [@ltolos] analogous to $\Lambda (1405)$ in the coupled channel approach for the $\bar K N$ interaction [@kbarn]. The approach has been generalized to study the spectral density of the D-mesons at finite temperatures and densities [@ljhs], taking into account the modifications of the nucleons in the medium. The results of this investigation seem to indicate a dominant increase in the width of the D-meson whereas there is only a very small change in the D-meson mass in the medium [@ljhs]. However, these calculations [@ltolos; @ljhs], assume the interaction to be SU(3) symmetric in u,d,c quarks and ignore channels with charmed hadrons with strangeness. A coupled channel approach for the study of D-mesons has been developed based on SU(4) symmetry [@HL] to construct the effective interaction between pseudoscalar mesons in a 16-plet with baryons in 20-plet representation through exchange of vector mesons and with KSFR condition [@KSFR]. This model [@HL] has been modified in aspects like regularization method and has been used to study DN interactions in Ref. [@mizutani6]. This reproduces the resonance $\Lambda_c (2593)$ in the I=0 channel and in addition generates another resonance in the I=1 channel at around 2770 MeV. These calculations have been generalized to finite temperatures [@mizutani8] accounting for the in-medium modifications of the nucleons in a Walecka type $\sigma-\omega$ model, to study the $D$ and $\bar D$ properties [@MK] in the hot and dense hadronic matter. At the nuclear matter density and for zero temperature, these resonances ($\Lambda_c (2593)$ and $\Sigma_c (2770)$) are generated $45$ MeV and $40$ MeV below their free space positions. However at finite temperature, e.g., at $T = 100$ MeV resonance positions shift to $2579$ MeV and $2767$ MeV for $\Lambda_{c}$ ($I = 0$) and $\Sigma_{c}$ ($I = 1$) respectively. Thus at finite temperature resonances are seen to move closer to their free space values. This is because of the reduction of pauli blocking factor arising due to the fact that fermi surface is smeared out with temperature. For $\bar{D}$ mesons in coupled channel approach a small repulsive mass shift is obtained. This will rule out of any possibility of charmed mesic nuclei [@mizutani8] suggested in the QMC model [@qmc]. But as we shall see in our investigation, we obtain a small attractive mass shift for $\bar{D}$ mesons which can give rise to the possibility of the formation of charmed mesic nuclei. The study of $D$ meson self-energy in the nuclear matter is also helpful in understanding the properties of the charm and the hidden charm resonances in the nuclear matter [@tolosra]. In coupled channel approach the charmed resonance $D_{s0}(2317)$ mainly couples to $DK$ system, while the $D_{0}(2400)$ couples to $D\pi$ and $D_{s}\bar{K}$. The hidden charm resonance couples mostly to $D\bar{D}$. Therefore any modification of $D$ meson properties in the nuclear medium will affect the properties of these resonances. In Ref.[@haid1; @haid2], the $\bar{D}N$ interactions at low energies have been studied using a meson exchange model ($\omega$ and $\rho$) in close analogy with the meson exchange model for the KN interactions [@julich], supplemented with a short-distance contribution from one-gluon exchange. The scattering lengths for the I=0 and I=1 channels for the $\bar D N$ interactions arising from the one gluon exchange are seen to be very close to the values in Ref. [@MK]. Generalizing the SU(4) models with vector meson exchange potentials to SU(8) spin-flavor symmetry, that treats the heavy pseudoscalar and vector mesons on equal footing-as required by heavy-quark symmetry, in Ref. [@cgar], the charmed baryon resonances that are generated dynamically have been studied within a unitary meson-baryon coupled-channel model. Some of the resonances in this model are identified with the recently observed baryon resonances. In the present investigation, the $D(\bar D)$ energies are modified due to a vectorial Weinberg-Tomozawa, scalar exchange terms ($\sigma$, $\delta$) as well as range terms [@isoamss; @isoamss2]. The isospin asymmetric effects among $D^0$ and $D^+$ in the doublet, D$\equiv (D^0,D^+)$ as well as between $\bar {D^0}$ and $D^-$ in the doublet, $\bar D \equiv (\bar {D^0},D^-)$ arise due to the scalar-isovector $\delta$ meson, due to asymmetric contributions in the Weinberg-Tomozawa term, as well as in the range term [@isoamss]. We organize the paper as follows. In section II, we give a brief introduction to the effective chiral $SU(3)$ model used to study the isospin asymmetric nuclear matter at finite temperatures, and its extension to the $SU(4)$ model to derive the interactions of the charmed mesons with the light hadrons. In section III, we present the dispersion relations for the $D$ and $\bar{D}$ mesons to be solved to calculate their optical potentials in the hot and dense hadronic matter. In secton IV, we show how the in-medium masses of the charmonium states $J/\psi$, $\psi(3686)$ and $\psi(3770)$ in the present investigation, arise from the medium modification of the scalar dilaton field, introduced within the chiral model to incorporate broken scale invariance leading to QCD trace anomaly. Section V contains the results and discussions and finally, in section VI, we summarize the results of present investigation and discuss possible outlook. The hadronic chiral $SU(3) \times SU(3)$ model =============================================== We use a chiral $SU(3)$ model for the study of the light hadrons in the present investigation [@paper3]. The model is based on nonlinear realization of chiral symmetry [@weinberg; @coleman; @bardeen] and broken scale invariance [@paper3; @hartree; @kristof1]. The effective hadronic chiral Lagrangian contains the following terms $${\cal L} = {\cal L}_{kin}+\sum_{W=X,Y,V,A,u} {\cal L}_{BW} + {\cal L}_{vec} + {\cal L}_{0} + {\cal L}_{SB}$$ In Eq.(1), ${\cal L}_{kin}$ is the kinetic energy term, ${\cal L}_{BW}$ is the baryon-meson interaction term in which the baryons-spin-0 meson interaction term generates the baryon masses. ${\cal L}_{vec}$ describes the dynamical mass generation of the vector mesons via couplings to the scalar mesons and contain additionally quartic self-interactions of the vector fields. ${\cal L}_{0}$ contains the meson-meson interaction terms inducing the spontaneous breaking of chiral symmerty as well as a scale invariance breaking logarthimic potential. ${\cal L}_{SB}$ describes the explicit chiral symmetry breaking. To study the hadron properties at finite temperature and densities in the present investigation, we use the mean field approximation, where all the meson fields are treated as classical fields. In this approximation, only the scalar and the vector fields contribute to the baryon-meson interaction, ${\cal L}_{BW}$ since for all the other mesons, the expectation values are zero. The interactions of the scalar mesons and vector mesons with the baryons are given as $$\begin{aligned} {\cal L} _{Bscal} + {\cal L} _{Bvec} = - \sum_{i} \bar{\psi}_{i} \left[ m_{i}^{*} + g_{\omega i} \gamma_{0} \omega + g_{i\rho} \gamma_{0} \rho + g_{\phi i} \gamma_{0} \phi \right] \psi_{i}. \label{lagscvec}\end{aligned}$$ The interaction of the vector mesons, of the scalar fields and the interaction corresponding to the explicitly symmetry breaking in the mean field approximation are given as $$\begin{aligned} {\cal L} _{vec} & = & \frac{1}{2} \left( m_{\omega}^{2} \omega^{2} + m_{\rho}^{2} \rho^{2} + m_{\phi}^{2} \phi^{2} \right) \frac{\chi^{2}}{\chi_{0}^{2}} \nonumber \\ & + & g_4 (\omega ^4 +6\omega^2 \rho^2+\rho^4 + 2\phi^4),\end{aligned}$$ $$\begin{aligned} {\cal L} _{0} & = & -\frac{1}{2} k_{0}\chi^{2} \left( \sigma^{2} + \zeta^{2} + \delta^{2} \right) + k_{1} \left( \sigma^{2} + \zeta^{2} + \delta^{2} \right)^{2} \nonumber\\ &+& k_{2} \left( \frac{\sigma^{4}}{2} + \frac{\delta^{4}}{2} + 3 \sigma^{2} \delta^{2} + \zeta^{4} \right) + k_{3}\chi\left( \sigma^{2} - \delta^{2} \right)\zeta \nonumber\\ &-& k_{4} \chi^{4} - \frac{1}{4} \chi^{4} {\rm {ln}} \frac{\chi^{4}}{\chi_{0}^{4}} + \frac{d}{3} \chi^{4} {\rm {ln}} \Bigg (\bigg( \frac{\left( \sigma^{2} - \delta^{2}\right) \zeta }{\sigma_{0}^{2} \zeta_{0}} \bigg) \bigg (\frac{\chi}{\chi_0}\bigg)^3 \Bigg ), \label{lagscal}\end{aligned}$$ and $$\begin{aligned} {\cal L} _{SB} & = & - \left( \frac{\chi}{\chi_{0}}\right) ^{2} \left[ m_{\pi}^{2} f_{\pi} \sigma + \left( \sqrt{2} m_{k}^{2}f_{k} - \frac{1}{\sqrt{2}} m_{\pi}^{2} f_{\pi} \right) \zeta \right]. \end{aligned}$$ In (\[lagscvec\]), ${m_i}^*$ is the effective mass of the baryon of species $i$, given as $${m_i}^{*} = -(g_{\sigma i}\sigma + g_{\zeta i}\zeta + g_{\delta i}\delta) \label{mbeff}$$ The baryon-scalar meson interactions, as can be seen from equation (\[mbeff\]), generate the baryon masses through the coupling of baryons to the non-strange $\sigma$, strange $\zeta$ scalar mesons and also to scalar-isovector meson $\delta$. In analogy to the baryon-scalar meson coupling there exist two independent baryon-vector meson interaction terms corresponding to the F-type (antisymmetric) and D-type (symmetric) couplings. Here antisymmetric coupling is used because the universality principle [@saku69] and vector meson dominance model suggest small symmetric coupling. Additionally, we choose the parameters [@paper3; @isoamss] so as to decouple the strange vector field $\phi_{\mu}\sim\bar{s}\gamma_{\mu}s$ from the nucleon, corresponding to an ideal mixing between $\omega$ and $\phi$ mesons. A small deviation of the mixing angle from ideal mixing [@dumbrajs; @rijken; @hohler1] has not been taken into account in the present investigation. The concept of broken scale invariance leading to the trace anomaly in (massless) QCD, $\theta_{\mu}^{\mu} = \frac{\beta_{QCD}}{2g} G_{\mu\nu}^{a} G^{\mu\nu a}$, where $G_{\mu\nu}^{a} $ is the gluon field strength tensor of QCD, is simulated in the effective Lagrangian at tree level [@sche1] through the introduction of the scale breaking terms $${\cal L}_{scalebreaking} = -\frac{1}{4} \chi^{4} {\rm {ln}} \Bigg ( \frac{\chi^{4}} {\chi_{0}^{4}} \Bigg ) + \frac{d}{3}{\chi ^4} {\rm {ln}} \Bigg ( \bigg (\frac{I_{3}}{{\rm {det}}\langle X \rangle _0} \bigg ) \bigg ( \frac {\chi}{\chi_0}\bigg)^3 \Bigg ), \label{scalebreak}$$ where $I_3={\rm {det}}\langle X \rangle$, with $X$ as the multiplet for the scalar mesons. These scale breaking terms, in the mean field approximation, are given by the last two terms of the Lagrangian denstiy, ${\cal L}_0$ given by equation (\[lagscal\]). The effect of these logarithmic terms is to break the scale invariance, which leads to the trace of the energy momentum tensor as [@heide1] $$\theta_{\mu}^{\mu} = \chi \frac{\partial {\cal L}}{\partial \chi} - 4{\cal L} = -(1-d)\chi^{4}. \label{tensor1}$$ Hence the scalar gluon condensate of QCD ($\frac {\alpha_s}{\pi} \langle {G^a}_{\mu \nu} G^{\mu \nu a} \rangle$) is simulated by a scalar dilaton field in the present hadronic model. The coupled equations of motion for the non-strange scalar field $\sigma$, strange scalar field $ \zeta$, scalar-isovector field $ \delta$ and dilaton field $\chi$, are derived from the Lagrangian density and are given as $$\begin{aligned} && k_{0}\chi^{2}\sigma-4k_{1}\left( \sigma^{2}+\zeta^{2} +\delta^{2}\right)\sigma-2k_{2}\left( \sigma^{3}+3\sigma\delta^{2}\right) -2k_{3}\chi\sigma\zeta \nonumber\\ &-&\frac{d}{3} \chi^{4} \bigg (\frac{2\sigma}{\sigma^{2}-\delta^{2}}\bigg ) +\left( \frac{\chi}{\chi_{0}}\right) ^{2}m_{\pi}^{2}f_{\pi} -\sum g_{\sigma i}\rho_{i}^{s} = 0 \label{sigma}\end{aligned}$$ $$\begin{aligned} && k_{0}\chi^{2}\zeta-4k_{1}\left( \sigma^{2}+\zeta^{2}+\delta^{2}\right) \zeta-4k_{2}\zeta^{3}-k_{3}\chi\left( \sigma^{2}-\delta^{2}\right)\nonumber\\ &-&\frac{d}{3}\frac{\chi^{4}}{\zeta}+\left(\frac{\chi}{\chi_{0}} \right) ^{2}\left[ \sqrt{2}m_{k}^{2}f_{k}-\frac{1}{\sqrt{2}} m_{\pi}^{2}f_{\pi}\right] -\sum g_{\zeta i}\rho_{i}^{s} = 0 \label{zeta}\end{aligned}$$ $$\begin{aligned} & & k_{0}\chi^{2}\delta-4k_{1}\left( \sigma^{2}+\zeta^{2}+\delta^{2}\right) \delta-2k_{2}\left( \delta^{3}+3\sigma^{2}\delta\right) +k_{3}\chi\delta \zeta \nonumber\\ & + & \frac{2}{3} d \chi^{4} \left( \frac{\delta}{\sigma^{2}-\delta^{2}}\right) -\sum g_{\delta i}\rho_{i}^{s} = 0 \label{delta}\end{aligned}$$ $$\begin{aligned} & & k_{0}\chi \left( \sigma^{2}+\zeta^{2}+\delta^{2}\right)-k_{3} \left( \sigma^{2}-\delta^{2}\right)\zeta + \chi^{3}\left[1 +{\rm {ln}}\left( \frac{\chi^{4}}{\chi_{0}^{4}}\right) \right] +(4k_{4}-d)\chi^{3} \nonumber\\ & - & \frac{4}{3} d \chi^{3} {\rm {ln}} \Bigg ( \bigg (\frac{\left( \sigma^{2} -\delta^{2}\right) \zeta}{\sigma_{0}^{2}\zeta_{0}} \bigg ) \bigg (\frac{\chi}{\chi_0}\bigg)^3 \Bigg ) +\frac{2\chi}{\chi_{0}^{2}}\left[ m_{\pi}^{2} f_{\pi}\sigma +\left(\sqrt{2}m_{k}^{2}f_{k}-\frac{1}{\sqrt{2}} m_{\pi}^{2}f_{\pi} \right) \zeta\right] = 0 \label{chi}\end{aligned}$$ In the above, ${\rho_i}^s$ are the scalar densities for the baryons, given as $$\begin{aligned} \rho_{i}^{s} = \gamma_{i}\int\frac{d^{3}k}{(2\pi)^{3}} \frac{m_{i}^{*}}{E_{i}^{*}(k)} \Bigg ( \frac {1}{e^{({E_i}^* (k) -{\mu_i}^*)/T}+1} + \frac {1}{e^{({E_i}^* (k) +{\mu_i}^*)/T}+1} \Bigg ) \label{scaldens}\end{aligned}$$ where, ${E_i}^*(k)=(k^2+{{m_i}^*}^2)^{1/2}$, and, ${\mu _i}^* =\mu_i -g_{\omega i}\omega -g_{\rho i}\rho -g_{\phi i}\phi$, are the single particle energy and the effective chemical potential for the baryon of species $i$, and, $\gamma_i$=2 is the spin degeneracy factor [@isoamss]. The above coupled equations of motion are solved to obtain the density and temperature dependent values of the scalar fields ($\sigma$, $\zeta$ and $\delta$) and the dilaton field, $\chi$, in the isospin asymmetric hot nuclear medium. As has already been mentioned, the value of the $\chi$ is related to the scalar gluon condensate in the hot hadronic medium, and is used to compute the in-medium masses of charmonium states in the present investigation. The isospin asymmetry in the medium is introduced through the scalar-isovector field $\delta$ and therefore the dilaton field obtained after solving the above equations is also dependent on the isospin asymmetry parameter, $\eta$ defined as $\eta= ({\rho_n -\rho_p})/({2 \rho_B})$, where $\rho_n$ and $\rho_p$ are the number densities of the neutron and the proton and $\rho_B$ is the baryon density. In the present investigation, we study the effect of isospin asymmetry of the medium on the masses of the charmonium states $J/\psi, \psi(3686)$ and $\psi(3770)$. The comparison of the trace of the energy momentum tensor arising from the trace anomaly of QCD with that of the present chiral model gives the relation of the dilaton field to the scalar gluon condensate. We have, in the limit of massless quarks [@cohen], $$\theta_{\mu}^{\mu} = \langle \frac{\beta_{QCD}}{2g} G_{\mu\nu}^{a} G^{\mu\nu a} \rangle \equiv -(1 - d)\chi^{4} \label{tensor2}$$ The parameter $d$ originates from the second logarithmic term of equation (\[scalebreak\]). To get an insight into the value of the parameter $d$, we recall that the QCD $\beta$ function at one loop level, for $N_{c}$ colors and $N_{f}$ flavors is given by $$\beta_{\rm {QCD}} \left( g \right) = -\frac{11 N_{c} g^{3}}{48 \pi^{2}} \left( 1 - \frac{2 N_{f}}{11 N_{c}} \right) + O(g^{5}) \label{beta}$$ In the above equation, the first term in the parentheses arises from the (antiscreening) self-interaction of the gluons and the second term, proportional to $N_{f}$, arises from the (screening) contribution of quark pairs. Equations (\[tensor2\]) and (\[beta\]) suggest the value of $d$ to be 6/33 for three flavors and three colors, and for the case of three colors and two flavors, the value of $d$ turns out to be 4/33, to be consistent with the one loop estimate of QCD $\beta$ function. These values give the order of magnitude about which the parameter $d$ can be taken [@heide1], since one cannot rely on the one-loop estimate for $\beta_{\rm {QCD}}(g)$. In the present investigation of the in-medium properties of the charmonium states due to the medium modification of the dilaton field within chiral $SU(3)$ model, we use the value of $d$=0.064 [@amarind]. This parameter, along with the other parameters corresponding to the scalar Lagrangian density, ${\cal L}_0$ given by (\[lagscal\]), are fitted so as to ensure extrema in the vacuum for the $\sigma$, $\zeta$ and $\chi$ field equations, to reproduce the vacuum masses of the $\eta$ and $\eta '$ mesons, the mass of the $\sigma$ meson around 500 MeV, and pressure, p($\rho_0$)=0, with $\rho_0$ as the nuclear matter saturation density [@paper3; @amarind]. The trace of the energy-momentum tensor in QCD, using the one loop beta function given by equation (\[beta\]), for $N_c$=3 and $N_f$=3, is given as, $$\theta_{\mu}^{\mu} = - \frac{9}{8} \frac{\alpha_{s}}{\pi} G_{\mu\nu}^{a} G^{\mu\nu a} \label{tensor4}$$ Using equations (\[tensor2\]) and (\[tensor4\]), we can write $$\left\langle \frac{\alpha_{s}}{\pi} G_{\mu\nu}^{a} G^{\mu\nu a} \right\rangle = \frac{8}{9}(1 - d) \chi^{4} \label{chiglu}$$ We thus see from the equation (\[chiglu\]) that the scalar gluon condensate $\left\langle \frac{\alpha_{s}}{\pi} G_{\mu\nu}^{a} G^{\mu\nu a}\right\rangle$ is proportional to the fourth power of the dilaton field, $\chi$, in the chiral SU(3) model. As mentioned earlier, the in-medium masses of charmonium states are modified due to the gluon condensates. Therefore, we need to know the change in the gluon condensate with density and temperature of the asymmetric nuclear medium, which is calculated from the modification of the $\chi$ field, by using equation (\[chiglu\]). $D$ and $\bar D$ mesons in hot asymmetric nuclear matter ======================================================== In this section we study the $D$ and $\bar{D}$ mesons properties in isospin-asymmetric nuclear matter at finite temperatures. The medium modifications of the $D$ and $\bar D$ mesons arise due to their interactions with the nucleons and the scalar mesons and the interaction Lagrangian density is given as [@amarind] $$\begin{aligned} \cal L _{DN} & = & -\frac {i}{8 f_D^2} \Big [3\Big (\bar p \gamma^\mu p +\bar n \gamma ^\mu n \Big) \Big({D^0} (\partial_\mu \bar D^0) - (\partial_\mu {{D^0}}) {\bar D}^0 \Big ) +\Big(D^+ (\partial_\mu D^-) - (\partial_\mu {D^+}) D^- \Big ) \nonumber \\ & +& \Big (\bar p \gamma^\mu p -\bar n \gamma ^\mu n \Big) \Big({D^0} (\partial_\mu \bar D^0) - (\partial_\mu {{D^0}}) {\bar D}^0 \Big ) - \Big( D^+ (\partial_\mu D^-) - (\partial_\mu {D^+}) D^- \Big ) \Big ] \nonumber \\ &+ & \frac{m_D^2}{2f_D} \Big [ (\sigma +\sqrt 2 \zeta_c)\big (\bar D^0 { D^0}+(D^- D^+) \big ) +\delta \big (\bar D^0 { D^0})-(D^- D^+) \big ) \Big ] \nonumber \\ & - & \frac {1}{f_D}\Big [ (\sigma +\sqrt 2 \zeta_c ) \Big ((\partial _\mu {{\bar D}^0})(\partial ^\mu {D^0}) +(\partial _\mu {D^-})(\partial ^\mu {D^+}) \Big ) \nonumber \\ & + & \delta \Big ((\partial _\mu {{\bar D}^0})(\partial ^\mu {D^0}) -(\partial _\mu {D^-})(\partial ^\mu {D^+}) \Big ) \Big ] \nonumber \\ &+ & \frac {d_1}{2 f_D^2}(\bar p p +\bar n n )\big ( (\partial _\mu {D^-})(\partial ^\mu {D^+}) +(\partial _\mu {{\bar D}^0})(\partial ^\mu {D^0}) \big ) \nonumber \\ &+& \frac {d_2}{4 f_D^2} \Big [ (\bar p p+\bar n n))\big ( (\partial_\mu {\bar D}^0)(\partial^\mu {D^0}) + (\partial_\mu D^-)(\partial^\mu D^+) \big )\nonumber \\ &+& (\bar p p -\bar n n) \big ( (\partial_\mu {\bar D}^0)(\partial^\mu {D^0})\big ) - (\partial_\mu D^-)(\partial^\mu D^+) ) \Big ] \label{ldn}\end{aligned}$$ In Eq.(\[ldn\]), the first term is the vectorial Weinberg Tomozawa interaction term, obtained from the kinetic term of Eq.(1). The second term is obtained from the explicit symmetry breaking term and leads to the attractive interactions for both the $D$ and $\bar{D}$ mesons in the medium. The next three terms of above Lagrangian density ($\sim (\partial_\mu {\bar D})(\partial ^\mu D)$) are known as the range terms. The first range term (with coefficient $\big (-\frac{1}{f_D}\big)$) is obtained from the kinetic energy term of the pseudoscalar mesons. The second and third range terms $d_{1}$ and $d_{2}$ are written for the $DN$ interactions in analogy with those written for $KN$ interactions in [@isoamss2]. It might be noted here that the interaction of the pseudoscalar mesons with the vector mesons, in addition to the pseudoscalar meson-nucleon vectorial interaction, leads to a double counting in the linear realization of chiral effective theories. Further, in the non-linear realization, such an interaction does not arise in the leading or subleading order, but only as a higher order contribution [@borasoy]. Hence the vector meson-pseudoscalar interactions will not be taken into account in the present investigation. The dispersion relations for the $D$ and $\bar{D}$ mesons are obtained by the Fourier transformations of equations of motion. These are given as $$-\omega^{2}+\vec{k}^{2}+m_{D}^{2}-\Pi\left(\omega,\vert\vec{k}\vert\right) = 0$$ where, $m_D$ is the vacuum mass of the $D(\bar D)$ meson taken as 1869 MeV and 1864.5 MeV for the $D$ and $\bar D$ mesons respectively. $\Pi\left(\omega,\vert\vec{k}\vert\right)$ denotes the self-energy of the $D\left( \bar{D} \right) $ mesons in the medium. The self-energy $\Pi\left( \omega , \vert\vec{k}\vert\right) $ for the $D$ meson doublet $ \left( D^{0} , D^{+}\right) $ arising from the interaction of Eq.(\[ldn\]) is given as $$\begin{aligned} \Pi (\omega, |\vec k|) &= & \frac {1}{4 f_D^2}\Big [3 (\rho_p +\rho_n) \pm (\rho_p -\rho_n) \Big ] \omega \nonumber \\ &+&\frac {m_D^2}{2 f_D} (\sigma ' +\sqrt 2 {\zeta_c} ' \pm \delta ') \nonumber \\ & +& \Big [- \frac {1}{f_D} (\sigma ' +\sqrt 2 {\zeta_c} ' \pm \delta ') +\frac {d_1}{2 f_D ^2} (\rho_s ^p +\rho_s ^n)\nonumber \\ &+&\frac {d_2}{4 f_D ^2} \Big (({\rho^s} _p +{\rho^s} _n) \pm ({\rho^s} _p -{\rho^s} _n) \Big ) \Big ] (\omega ^2 - {\vec k}^2), \label{selfd}\end{aligned}$$ where the $\pm$ signs refer to the $D^{0}$ and $D^{+}$ mesons, respectively, and $\sigma^{\prime}\left( \sigma - \sigma_{0}\right) $, $\zeta_{c}^{\prime}\left(\zeta_{c} - \zeta_{c0}\right)$, and $\delta^{\prime}\left( = \delta -\delta_{0}\right) $ are the fluctuations of the scalar isoscalar fields $\sigma$ and $\zeta$ and the scalar-isoscalar field $\delta$ from their vacuum expectation values. The vacuum expectation value of $\delta$ is zero $\left(\delta_{0}=0 \right)$, since a nonzero value for it will break the isospin-symmetry of the vacuum. (We neglect here the small isospin breaking effect arising from the mass and charge difference of the up and down quarks.) We might note here that the interaction of the scalar quark condensate $\zeta_{c}$ (being made up of heavy charmed quarks and antiquarks) leads to very small modifications of the masses [@roeder]. So we will not consider the medium fluctuations of $\zeta_{c}$. In the present investigation, we take the value of the D meson decay constant, $f_D$ as 135 MeV [@weise]. Within the present model, the medium modification to the $D(\bar D)$ mesons due to the scalar interaction depends only on the fluctuations of the scalar $\sigma$ and $\delta$ fields in the asymmetric hot nuclear medium which are determined by solving the coupled equations (\[sigma\]), (\[zeta\]), (\[delta\]) and (\[chi\]). As the scalar term of (\[selfd\]) does not contain any free parameters, with the assumption that the fluctuation of the charm condensate has a negligible effect on the masses of the $D(\bar D)$ mesons, there are no uncertainties in the mass shift due to the scalar interaction in the present investigation, once the value for $f_D$ is chosen. In Eq. (\[selfd\]), $\rho_{p}$ and $\rho_{n}$ are the number densities of protons and neutrons given by $$\rho_{i} = \gamma_{i} \int \frac{d^{3}k}{(2\pi)^{3}} \left( \frac{1}{e^{\left( E_{i}^{*}(k) - \mu_{i}^{*}\right) /T} + 1} - \frac{1}{e^{\left( E_{i}^{*}(k) + \mu_{i}^{*}\right) /T} + 1}\right), \label{vecdens}$$ for $i$=p and n, and $\rho_{p}^{s}$ and $\rho_{n}^{s}$ are their scalar densities, as given by equation (\[scaldens\]). Similarly, for the $\bar{D}$ meson doublet $\left(\bar{D}^{0},D^{-}\right)$, the self-energy is calculated as $$\begin{aligned} \Pi (\omega, |\vec k|) &= & -\frac {1}{4 f_D^2}\Big [3 (\rho_p +\rho_n) \pm (\rho_p -\rho_n) \Big ] \omega\nonumber \\ &+&\frac {m_D^2}{2 f_D} (\sigma ' +\sqrt 2 {\zeta_c} ' \pm \delta ') \nonumber \\ & +& \Big [- \frac {1}{f_D} (\sigma ' +\sqrt 2 {\zeta_c} ' \pm \delta ') +\frac {d_1}{2 f_D ^2} (\rho_s ^p +\rho_s ^n )\nonumber \\ &+&\frac {d_2}{4 f_D ^2} \Big (({\rho^s} _p +{\rho^s} _n) \pm ({\rho^s} _p -{\rho^s} _n) \Big ] (\omega ^2 - {\vec k}^2), \label{selfdbar}\end{aligned}$$ where the $\pm$ signs refer to the $\bar{D}^{0}$ and $D^{-}$ mesons, respectively. The optical potentials of the $D$ and $\bar{D}$ mesons are obtained using the expression $$U(\omega, k) = \omega(k) - \sqrt{k^{2} + m_{D}^{2}}$$ where $m_{D}$ is the vacuum mass for the $D(\bar{D})$ meson and $\omega(k)$ is the momentum-dependent energy of the $D(\bar{D})$ meson. Charmonium masses in hot asymmetric nuclear matter ================================================== In this section, we investigate the masses of charmonium states $J/\psi$, $\psi(3686)$ and $\psi(3770)$, in isospin asymmetric hot nuclear matter. From the QCD sum rule calculations, the mass shift of the charmonium states in the medium is due to the gluon condensates [@leeko; @arata]. For heavy quark systems, there are two independent lowest dimension operators: the scalar gluon condensate ($\left\langle \frac{\alpha_{s}}{\pi} G_{\mu\nu}^{a} G^{\mu\nu a}\right\rangle$) and the condensate of the twist 2 gluon operator ($\left\langle \frac{\alpha_{s}}{\pi} G_{\mu\nu}^{a} G^{\mu\alpha a}\right\rangle$). These operators can be rewritten in terms of the color electric and color magnetic fields, $\langle \frac{\alpha_s}{\pi} {\vec E}^2\rangle$ and $\langle \frac{\alpha_s}{\pi} {\vec B}^2\rangle$. Additionally, since the Wilson coefficients for the operator $\langle \frac{\alpha_s}{\pi} {\vec B}^2\rangle$ vanish in the non-relativistic limit, the only contribution from the gluon condensates is proportional to $\langle \frac{\alpha_s}{\pi} {\vec E}^2\rangle$, similar to the second order Stark effect. Hence, the mass shift of the charmonium states arises due to the change in the operator $\langle \frac{\alpha_s}{\pi} {\vec E}^2\rangle$ in the medium from its vacuum value [@leeko]. In the leading order mass shift formula derived in the large charm mass limit [@pes1], the shift in the mass of the charmonium state is given as [@leeko] $$\Delta m_{\psi} (\epsilon) = -\frac{1}{9} \int dk^{2} \vert \frac{\partial \psi (k)}{\partial k} \vert^{2} \frac{k}{k^{2} / m_{c} + \epsilon} \bigg ( \left\langle \frac{\alpha_{s}}{\pi} E^{2} \right\rangle- \left\langle \frac{\alpha_{s}}{\pi} E^{2} \right\rangle_{0} \bigg ). \label{mass1}$$ In the above, $m_c$ is the mass of the charm quark, taken as 1.95 GeV [@leeko], $m_\psi$ is the vacuum mass of the charmonium state and $\epsilon = 2 m_{c} - m_{\psi}$. $\psi (k)$ is the wave function of the charmonium state in the momentum space, normalized as $\int\frac{d^{3}k}{2\pi^{3}} \vert \psi(k) \vert^{2} = 1 $ [@leetemp]. At finite densities, in the linear density approximation, the change in the value of $\langle \frac{\alpha_s}{\pi} {\vec E}^2\rangle$, from its vacuum value, is given as $$\left\langle \frac{\alpha_{s}}{\pi} E^{2} \right\rangle- \left\langle \frac{\alpha_{s}}{\pi} E^{2} \right\rangle_{0} = \left\langle \frac{\alpha_{s}}{\pi} E^{2} \right\rangle _{N} \frac {\rho_B}{2 M_N},$$ and the mass shift in the charmonium states reduces to [@leeko] $$\Delta m_{\psi} (\epsilon) = -\frac{1}{9} \int dk^{2} \vert \frac{\partial \psi (k)}{\partial k} \vert^{2} \frac{k}{k^{2} / m_{c} + \epsilon} \left\langle \frac{\alpha_{s}}{\pi} E^{2} \right\rangle _{N} \frac {\rho_B}{2 M_N}. \label{masslindens}$$ In the above, $\left\langle \frac{\alpha_{s}}{\pi} E^{2} \right\rangle _{N}$ is the expectation value of $\left\langle \frac{\alpha_{s}}{\pi} E^{2} \right\rangle$ with respect to the nucleon. The expectation value of the scalar gluon condensate can be expressed in terms of the color electric field and the color magnetic field as [@david] $$\left\langle \frac{\alpha_{s}}{\pi} G_{\mu\nu}^{a} G^{\mu\nu a}\right\rangle =-2 \left\langle \frac{\alpha_{s}}{\pi} (E^{2} - B^{2}) \right\rangle.$$ In the non-relativistic limit, as already mentioned, the contribution from the magnetic field vanishes and hence, we can write, $$\left\langle \frac{\alpha_{s}}{\pi} E^{2} \right\rangle =-\frac {1}{2} \left\langle \frac{\alpha_{s}}{\pi} G_{\mu\nu}^{a} G^{\mu\nu a}\right\rangle \label{e2glu}$$ Using equations (\[chiglu\]), (\[mass1\]) and (\[e2glu\]), we obtain the expression for the mass shift in the charmonium in the hot and dense nuclear medium, which arises from the change in the dilaton field in the present investigation, as $$\Delta m_{\psi} (\epsilon) = \frac{4}{81} (1 - d) \int dk^{2} \vert \frac{\partial \psi (k)}{\partial k} \vert^{2} \frac{k}{k^{2} / m_{c} + \epsilon} \left( \chi^{4} - {\chi_0}^{4}\right). \label{masspsi}$$ In the above, $\chi$ and $\chi_0$ are the values of the dilaton field in the nuclear medium and in the vacuum respectively. In the present investigation, the wave functions for the charmonium states are taken to be Gaussian and are given as [@friman] $$\psi_{N, l} = {\rm { Normalization}} \times Y_{l}^{m} (\theta, \phi) (\beta^{2} r^{2})^{\frac{1}2{} l} exp^{-\frac{1}{2} \beta^{2} r^{2}} L_{N - 1}^{l + \frac{1}{2}} \left( \beta^{2} r^{2}\right) \label{wavefn}$$ where $\beta^{2} = M \omega / h$ characterizes the strength of the harmonic potential, $M = m_{c}/2$ is the reduced mass of the charm quark and charm anti-quark system, and $L_{p}^{k} (z)$ is the associated Laguerre Polynomial. As in Ref. [@leeko], the oscillator constant $\beta$ is determined from the mean squared radii $\langle r^{2} \rangle$ as 0.46$^{2}$ fm$^2$, 0.96$^{2}$ fm$^2$ and 1 fm$^{2}$ for the charmonium states $J/\psi(3097) $, $\psi(3686)$ and $\psi(3770)$, respectively. This gives the value for the parameter $\beta$ as 0.51 GeV, 0.38 GeV and 0.37 GeV for $J/\psi(3097)$, $\psi(3686$ and $\psi(3770)$, assuming that these charmonium states are in the 1S, 2S and 1D states respectively. Knowing the wave functions of the charmonium states and calculating the medium modification of the dilaton field in the hot nuclear matter, we obtain the mass shift of the charmonium states, $J/\psi$, $\psi (3686)$ and $\psi (3770)$ respectively. In the next section we shall present the results of the present investigation of these in-medium charmonium masses in hot asymmetric nuclear matter. Results and Discussions {#results} ======================= In this section, we present the results of our investigation for the in-medium masses of $D$ and $\bar{D}$ mesons as well as of the charmonium states $J/\psi(3097)$, $\psi(3686)$ and $\psi(3770)$, in isospin asymmetric nuclear matter at finite temperatures. We have generalized the chiral $SU(3)$ model to $SU(4)$ to include the interactions of the charmed mesons. The values of the parameters used in the present investigation, are : $k_{0} = 2.54, k_{1} = 1.35, k_{2} = $-4.78$, k_{3} = -2.77$, $k_{4} = -0.22$ and $d = 0.064$, which are the parameters occurring in the scalar meson interactions defined in equation (\[lagscal\]). The vacuum values of the scalar isoscalar fields, $\sigma$ and $\zeta$ and the dilaton field $\chi$ are $-93.3$ MeV, $-106.6$ MeV and $409.8$ MeV respectively. The values, $g_{\sigma N} = 10.6$ and $g_{\zeta N} = -0.47$ are determined by fitting to vacuum baryon masses. The other parameters fitted to the asymmetric nuclear matter saturation properties in the mean-field approximation are: $g_{\omega N}$ = 13.3, $g_{\rho p}$ = 5.5, $g_{4}$ = 79.7, $g_{\delta p}$ = 2.5, $m_{\zeta}$ = 1024.5 MeV, $ m_{\sigma}$ = 466.5 MeV and $m_{\delta}$ = 899.5 MeV. The nuclear matter saturation density used in the present investigation is $0.15$ fm$^{-3}$. The coefficients $d_{1}$ and $d_{2}$, calculated from the empirical values of the $KN$ scattering lengths for $I = 0$ and $I = 1$ channels, are $2.56/m_{K}$ and $0.73/m_{K}$, respectively [@isoamss2]. In isospin asymmetric nuclear medium, the properties of the $D$ mesons ($D^{0}$, $D^{+}$) and $\bar{D}$ mesons ($\bar{D}^{0}$,$D^{-}$), due to their interactions with the hot hadronic medium, undergo medium modifications. These modifications arise due to the interactions with the nucleons (through the Weinberg-Tomozawa vectorial interaction as well as through the range terms) and the scalar exchange terms. The modifications of the scalar mean fields modify the masses of the nucleons in the hot and dense hadronic medium. Before going into the details of how the $D$ and $\bar{D}$ mesons properties are modified at finite temperatures in the dense nuclear medium, let us see how the scalar fields are modified at finite temperatures in the nuclear medium. In figures \[fig1\], \[fig2\], \[fig3\] and \[fig4\], we show the variation of scalar fields $\sigma$, $\zeta$ and scalar-isovector field $\delta$ and the dilaton field $\chi$, with temperature, for both zero and finite densities, and for selected values of the isospin asymmetry parameter, $\eta = 0, 0.1, 0.3$ and $0.5$. At zero baryon density, we observe that the magnitudes of the scalar fields $\sigma$ and $\zeta$ decrease with increase in temperature. However, the drop in their magnitudes with temperature is negligible upto a temperature of about 100 MeV (that is, they remain very close to their vacuum values). The changes in the magnitudes of the $\sigma$ and $\zeta$ fields are 2.5 MeV and 0.8 MeV respectively, when the temperature changes from 100 MeV to 150 MeV, above which the drop increases. These values change to about 10 MeV and 3 MeV for the $\sigma$ and $\zeta$ fields respectively, for a change in temperature of 100 MeV to 175 MeV. At zero baryon density, it is observed that the value of the dilaton field remains almost constant upto a temperature of about 130 MeV above which it is seen to drop with increase in temperature. However, the drop in the dilaton field is seen to be very small. The value of the dilaton field is seen to change from 409.8 MeV at T = 0 to about 409.7 MeV and 409.3 MeV at T = 150 MeV and T = 175 MeV, respectively. The thermal distribution functions have an effect of increasing the scalar densities at zero baryon density, i.e. $\mu_{i}^{*}=0$, as can be seen from the expression of the scalar densities, given by equation (\[scaldens\]). This effect seems to be negligible upto a temperature of about 130 MeV. This leads to a decrease in the magnitudes of the scalar fields, $\sigma$ and $\zeta$. This behaviour of the scalar fields is reflected in the value of $\chi$, which is solved from the coupled equations of motion of the scalar fields, given by equations (\[sigma\]), (\[zeta\]), (\[delta\]) and (\[chi\]), as a drop as we increase the temperature above a temperature of about 130 MeV. The scalar densities attaining nonzero values at high temperatures, even at zero baryon density, indicates the presence of baryon-antibaryon pairs in the thermal bath and has already been observed in literature [@kristof1; @frunstl1]. This leads to the baryon masses to be different from their vacuum masses above this temperature, arising from modifications of the scalar fields $\sigma$ and $\zeta$. For finite density situation, the behaviour of the scalar fields with temperature is seen to be very different from the zero density case, as can be seen from the subplots (b), (c) and (d) of figures \[fig1\], \[fig2\] and \[fig4\] for $\sigma$, $\zeta$ and $\chi$ fields respectively, where the fields are plotted as functions of temperature for densities $\rho_{0}$, $2\rho_{0}$ and $4\rho_{0}$ respectively. At finite densities, one observes first a rise and then a decrease of the scalar fields $\sigma$, $\zeta$ and $\chi$ with temperature. For example, at $\rho_{B} = \rho_{0}$, and for the value of the isospin asymmetry parameter, $\eta = 0$ the scalar fields $\sigma$, $\zeta$ and $\chi$ increase with temperature upto a temperature of about 145 MeV and above this temperature, they both start decreasing. At $\eta = 0.5$ the value of temperature upto which these scalar fields increase becomes about 120 MeV. For $\rho_{B} = 4\rho_{0}$ and $\eta = 0$ the scalar fields $\sigma$, $\zeta$ and $\chi$ increase upto a temperature of about 160 MeV. At $\eta = 0.5$ this value of temperature is lowered to about 120 MeV for $\sigma$, $\zeta$ and $\chi$ fields respectively. This observed rise in the magnitudes of $\sigma$ and $\zeta$ fields with temperature leads to an increase in the mass of nucleons with temperature at finite densities. The reason for the different behaviour of the scalar fields ($\sigma$ and $\zeta$) at zero and finite densities can be understood in the following manner [@kristof1]. As has already been mentioned, the thermal distribution functions in (\[scaldens\]) have an effect of increasing the scalar densities at zero baryon density, i.e., for $\mu_{i}^{*} = 0$. However, at finite densities, i.e. for nonzero values of the effective chemical potential, $\mu_{i}^{*}$, for increasing temperature, there are contributions also from higher momenta, thereby, increasing the denominator of the integrand on the right hand side of equation (\[scaldens\]). This leads to a decrease in the scalar density. The competing effects of the thermal distribution functions and the contributions from the higher moments states give rise to observed behaviour of the scalar density and hence of the $\sigma$ and $\zeta$ fields with temperature at finite baryon densities [@kristof1]. This kind of behaviour of the scalar $\sigma$ field on temperature at finite densities has also been observed in the Walecka model by Li and Ko [@liko]. The behaviour of the scalar fields $\sigma$ and $\zeta$ with the temperature is reflected in the behaviour of $\chi$ field, since it is solved from the coupled equations of the scalar fields. In figure \[fig3\], we observe that the value of the scalar isovector field, $\delta$ is zero at zero baryon density, since there is no isospin asymmetry at zero density. At finite baryon densities the magnitude of the scalar isovector field, $\delta$ decreases with increase in the temperature of the nuclear medium. However, for given temperature as we move to higher densities the magnitude of the $\delta$ field is seen to increase which means that the medium is more asymmetric at higher densities. In the isospin asymmetric nuclear medium, the scalar isovector $\delta$ meson attains a nonzero expectation value. However, the magnitudes of the $\delta$ field at a given baryon density and temperature of the medium, which are solved from the coupled equations, (\[sigma\]) to (\[chi\]), for the scalar fields, turn out to be small ($\le 4-5 MeV$),as can be seen from figure \[fig3\]. This causes changes in the $\sigma$ and $\zeta$ fields, as can be seen from figures \[fig1\] and \[fig2\], also to be small in the isospin asymmetric nuclear matter, since these solved from the equations (\[sigma\]) and (\[zeta\]), along with the equations for $\delta$ and $\chi$, given by (\[delta\]) and (\[chi\]). In figures \[fig5\] and \[fig6\], we show the variation of the energy of $D$ and $\bar{D}$ mesons due to Weinberg-Tomozawa term, the scalar exchange term arising from the explicit symmtery breaking term and the range terms, as functions of temperature and for the densities $\rho_{B} = 0, \rho_{0}$ and $4\rho_{0}$. These are plotted asymmetric nuclear matter with the value of the isospin asymmetry parameter as 0.5, and in the same figure, the results arising from these contributions for the isospin symmetric case ($\eta$=0) are shown as the dotted lines. At zero density, the contributions from the Weinberg-Tomozawa term is zero, since the densities of the proton and neutron,$\rho_p$ and $\rho_n$ are zero at zero density. The scalar fields attaining values different from their vacuum values at zero density at values of temperature higher than about 100 MeV, as already observed in figures \[fig1\] and \[fig2\], leads to a decrease in the $D^+$ and $D^0$ mesons due to the scalar meson exchange term and an increase in the contribution due to the range term, as can be seen in the subplots (a) and (b) of figure \[fig5\]. These two terms seem to almost cancel with each other, resulting in negligible modifications of $D^+$ and $D^0$ mesons. The drop in the masses of $D^+$ and $D^0$ are seen to be about 0.75 MeV and 0.75 MeV respectively at temperature T=150 MeV and about 5.4 MeV and 5.35 MeV at a temperature of 175 MeV. At finite densities, the contribution to the energies of the $D^+$ and $D^0$ mesons arising from the Weinberg-Tomozawa term as a function of temperature, is constant, since this term depends on the densities of proton and neutron only, which for a given baryon density and a given value of the isospin asymmetry parameter, are constant. At $\rho_B=\rho_0$ and T=0, the Weinberg-Tomozawa term gives a drop of about 23 MeV in the masses of $D^+$ and $D^0$ mesons from their vacuum values in isospin symmetric matter and about 31 MeV and 16 MeV for $D^+$ and $D^0$ mesons in the isospin asymmetric matter with $\eta$=0.5. The mass drop of $D^+$ meson of 23 MeV in symmetric nuclear matter at zero temperature may be compared with the value of about 43 MeV, which is the drop in $D^+$ mass in the Born approximation of the calculation of the coupled channel approach of Ref. [@mizutani6], when only the Weinberg Tomozawa interaction is included. In the presence of the scalar exchange term as well, the drop in the $D^+$ mass in the present investigation is seen to be about 144 MeV, which may be compared to the drop in the mass of the $D^+$ meson with Weinberg-Tomozawa as well as scalar interaction of about 60 MeV in Ref. [@mizutani6]. In the present investigation, we observe the scalar interaction to be much more dominant than in Ref. [@mizutani6]. The contributions to the energy of $D^+$ and $D^0$ mesons due to the explicit symmetry breaking term (scalar meson exchange term) are observed to increase with temperature upto a particular value of temperature, above which it is seen to decrease with temperature. This is a reflection of the behaviour of the scalar fields $\sigma$ and $\zeta$ shown in figures \[fig1\] and \[fig2\], whose magnitudes are seen to increase with temperature upto a particular value, above which there is seen to be a decrease in their magnitudes. The range term is seen to give an increase in the $D^+$ and $D^0$ masses and at $\rho_B=\rho_0$, there is seen to be a rise of about 71 MeV at zero temperature for $\eta$=0. This value is modified to about 67 MeV at T=150 MeV and about 70 MeV at T=175 MeV. For a density of $\rho_B=4\rho_0$ and at zero temperature, the drop in the D-meson masses in the symmetric nuclear matter is seen to be about 27 MeV due to the range term. For this density, the contribution of the $d_1$ and $d_2$ terms dominate over the first range term thus leading to an overall drop of the D-meson masses in the medium of about 347 MeV in the symmetric nuclear matter at zero temperature. It may be noted that the difference in the in-medium energy of $D^{+}$ and $D^{-}$ mesons arises from Weinberg-Tomozawa term only (see equations (\[selfd\]) and (\[selfdbar\])) whereas the the scalar meson exchange term and the range terms have identical contributions to the energy of $D^{+}$ and $D^{-}$ mesons. The repulsive contribution for the $D^-$ and $\bar {D^0}$ mesons for the Weinberg-Tomozawa term as compared to the attractive contribution for the $D^+$ and $D^0$ mesons, gives a rise in the masses of the $\bar D$ mesons (plotted in figure \[fig6\]) as compared to the masses of the $D$ mesons shown in figure \[fig5\]. At zero density, the contribution from the Weinberg-Tomozawa term is zero and hence the modifications for the $D^+$ and $D^-$ are identical, both having negligible changes in their masses due to the cancelling effects of the attractive scalar term and the repulsive range terms. At nuclear matter saturation density, when one takes into account only the contribution from the Weinberg-Tomozawa term, there is seen to be an increase in the mass of the $D^-$ meson of about 24 MeV. However, when we consider the contributions due to all the individual terms, the attractive scalar interaction is seen to dominate over the repulsive Weinberg-Tomozawa and the range terms, therby giving a drop of the $D^-$ mass of about 27 MeV in the symmetric nuclear matter for $\rho_B=\rho_0$. However, for higher densities, the $d_1$ and $d_2$ terms of the range term, which are both attractive, give a further drop of the mass of the $D^-$ meson and for density, $\rho_B=4\rho_0$, the overall drop is seen to be about 162 MeV for $\eta$=0. ![(Color online) The scalar-isoscalar field $\sigma$ plotted as a function of temperature at a given baryon density ($\rho_{B} = 0, \rho_{0}, 2\rho_{0}$ and $4\rho_{0}$), for different values of the isospin asymmetry parameter, $\eta$. []{data-label="fig1"}](f1.eps){width="16cm" height="16cm"} ![(Color online) The scalar-isoscalar field $\zeta$ plotted as a function of temperature at a given baryon density ($\rho_{B} = 0, \rho_{0}, 2\rho_{0}$ and $4\rho_{0}$), for different values of the isospin asymmetry parameter, $\eta$. []{data-label="fig2"}](f2.eps){width="16cm" height="16cm"} ![(Color online) The scalar-isovector field $\delta$ plotted as a function of temperature at a given baryon density ($\rho_{B} = 0, \rho_{0}, 2\rho_{0}$ and $4\rho_{0}$), for different values of the isospin asymmetry parameter, $\eta$. []{data-label="fig3"}](f3.eps){width="16cm" height="16cm"} ![(Color online) The dilaton field $\chi$ plotted as a function of temperature at a given baryon density ($\rho_{B} = 0, \rho_{0}, 2\rho_{0}$ and $4 \rho_{0}$), for different values of the isospin asymmetry parameter, $\eta$. []{data-label="fig4"}](f4.eps){width="16cm" height="16cm"} ![(Color online) The energies of $D^{+}$ meson ((a),(c) and (e)) and of $D^{0}$ meson ((b),(d) and (f)), at momentum $k = 0$, versus temperature, T, for different values of the isospin asymmetry parameter ($\eta = 0$ and $0.5$) and for given values of density ($\rho_{B} = 0, \rho_{0}$ and $4\rho_{0}$). The values of parameters $d_{1}$ and $d_{2}$ are calculated from $KN$ scattering lengths in I = 0 and I = 1 channels.[]{data-label="fig5"}](f5.eps){width="16cm" height="16cm"} ![(Color online) The energies of $D^{-}$ meson ((a),(c) and (e)) and of $\bar{D^{0}}$ meson ((b),(d) and (f)), at momentum $k = 0$, versus temperature, T, , for different values of the isospin asymmetry parameter ($\eta = 0$ and $0.5$) and for given values of density ($\rho_{B} = 0, \rho_{0}$ and $4\rho_{0}$). The values of parameters $d_{1}$ and $d_{2}$ are calculated from $KN$ scattering lengths in I = 0 and I = 1 channels.[]{data-label="fig6"}](f6.eps){width="16cm" height="16cm"} We study the density dependence of $D$ and $\bar{D}$ masses at finite temperatures at selected values of the isospin asymmetric paramater, $\eta$ and compare the results with the zero temperature case [@amarind]. The isospin symmetric part of the Weinberg-Tomazawa term gives a drop of the $D$ mass, as can be seen from the expressions of the self-energies of the $D$ and $\bar D$ mesons given by equations (\[selfd\]) and (\[selfdbar\]). However, the isospin asymmetric part of this term is seen to give a mass splitting for the $D^+$ and $D^0$, given by the second term of the Weinberg-Tomazawa term, giving a further drop of the $D^+$ mass, whereas the asymmetry reduces the drop of the mass of $D^0$. For the $\bar D$ mesons, the isospin symmetric part of the Weinberg Tomazawa term gives an increase in the mass and the isospin asymmetric contribution of this term gives a further rise in the mass of the $D^-$ mass, whereas it reduces the increase of the $\bar {D^0}$ mass in the asymmetric nuclear medium. For both $D$ and $\bar D$ mesons in the symmetric nuclear matter, the scalar meson interaction is attractive and identical (except for a small difference due to the difference in the vacuum masses of $D^\pm$ and $D^0 (\bar {D^0})$ mesons). One might observe from the expressions of the $D (\bar D)$ self energies, given by equations (\[selfd\]) and (\[selfdbar\]) that the non-zero value of $\delta$ meson arising due to the isospin asymmtery in the medium gives a drop in the masses of $D^+ (D^-)$ in the asymmteric nuclear matter, whereas this interaction is repulsive for $D^0 (\bar {D^0})$ mesons. The contributions to the $D$ and $\bar D$ masses due to the range terms are given by the last three terms of the self energies given by equations (\[selfd\]) and (\[selfdbar\]). The first of these range terms is repulsive, whereas the second and third terms are attractive, when isospin asymmtery is not taken into account. However, due to a nonzero value of the $\delta$ field arising from isospin asymmtery in the medium, the $\delta$ term of the first range term leads an increase in the masses of the $D^+(D^-)$ and a drop in the masses of $D^0 (\bar {D^0})$ mesons. The second of the range terms (the $d_1$ term) is attractive and gives identical mass drops for $D^+$ and $D^0$ in the $D$ doublet as well as for $D^-$ and $\bar D^0$ in the $\bar D$ doublet. This term is proportional to $(\rho_s^p+\rho_s^n)$, which turns out to be different for the isospin asymmetric case as compared to the isospin symmetric nuclear matter, due to the presence of the $\delta$ meson. This is because the equations of motion for the scalar fields for the two situations (with/without $\delta$ mesons) give different values for the mean field, $\sigma$ ($\sim ({\rho_s}^p +{\rho_s}^n)$). The last term of the range term (the $d_2$ term) has a negative contribution for the energies of $D^+$ and $D^0$ mesons as well as for $D^-$ and $\bar {D^0}$ mesons for the isospin symmetric matter. The isospin asymmetric part arising from the ($({\rho_s}^n -{\rho_s} ^p)$) term of the $d_2$ term has a further drop in the masses for $D^\pm$ mesons, whereas it increases the masses of the $D^0$ and $\bar{D^0}$ mesons from their isospin symmetric values. In Fig. \[fig7\], we show the variation of the energy of the $D$ mesons ($D^{+}, D^{0}$) at zero momentum with baryon density $\rho_{B}$ for different values of isospin asymmetry parameter $\eta$ and with the values of temperature as T = 0, 100, 150 MeV. The isospin-asymmetry in the medium is seen to give an increase in the $D^{0}$ mass and a drop in the $D^{+}$ mass as compared to the isospin symmetric ($\eta = 0$) case. This is observed both for zero temperature [@amarind] and finite temperature cases. At nuclear matter saturation density, $\rho_{B} = \rho_{0}$, the drop in the mass of $D^{+}$ meson from its vacuum value (1869 MeV) is 78 MeV for zero temperature case in isospin symmetric medium. At a density of $4\rho_{0}$, this drop in the mass of $D^{+}$ meson is seen to be about 347 MeV for $\eta$=0. At finite temperatures, the drop in the mass of $D^{+}$ meson for a given value of isospin asymmetry decreases as compared to the zero temperature case. For example, at nuclear saturation density, $\rho_{0}$, the drop in the mass of $D^{+}$ meson turns out to be 72, 65 and 62 MeV at a temperature of T = 50, 100 and 150 MeV respectively for the isospin symmetric matter. At a higher density of $\rho_B=4\rho_0$, zero temperature value for the $D^+$ mass drop of about 347 MeV is modified to 336, 313 and 294 MeV for T = 50, 100 and 150 MeV respectively. Thus the masses of $D$ mesons at finite temperatures and finite densities are observed to be larger than the values at zero temperature case. This is because of the increase in the magnitudes of the scalar fields $\sigma$ and $\zeta$ with temperature at finite densities as mentioned earlier. The same behaviour remains for the isospin asymmetric matter. This behaviour of the nucleons and hence of the D-mesons with temperature was also observed earlier for symmetric nuclear matter at finite temperatures within the chiral effective model [@amdmeson]. The drop in the mass of $D^{+}$ meson is seen to be larger as we increase the value of the isospin asymmetry parameter. We observe that as we change $\eta$ from 0 to 0.5, then the drop in the mass of $D^{+}$ meson is 95 MeV and 384 MeV at densities of $\rho_{0}$ and $4\rho_{0}$ respectively, for the zero temperature case. At $ T = 50 $ MeV these values change to 89 MeV at $\rho_{0}$ and 377 MeV at a density of $4\rho_{0}$. For a baryon density, $\rho_B=\rho_0 (4\rho_0)$, the drop in the $D^+$ mass is 83 (363) MeV at $T = 100$ MeV, and 84 (359) MeV at T=150 MeV. We thus observe that for a baryon density of $\rho_0$, for $\eta$=0.5, the drop in the $D^+$ mass from its value for $\eta$=0 is 17, 18 and 22 MeV for T=0, 100 and 150 MeV respectively, and for $\rho_B=4\rho_0$, these values are modifed to 37, 50 and 65 MeV respectively. Thus we observe that for a given value of density, as we move from $\eta = 0$ to $\eta = 0.5$ the drop in mass of $D^{+}$ mesons is larger at higher temperatures. ![(Color online) The energies of $D^{+}$ meson ((a),(c) and (e)) and of $D^{0}$ meson ((b),(d) and (f)), at momentum $k = 0$, versus the baryon density (in units of nuclear saturation density), $\rho_{B}/\rho_{0}$, for different values of the isospin asymmetry parameter ($\eta = 0, 0.1, 0.3, 0.5$) and for given values of temperature (T = 0, 100 MeV and 150 MeV). The values of parameters $d_{1}$ and $d_{2}$ are calculated from $KN$ scattering lengths in I = 0 and I = 1 channels.[]{data-label="fig7"}](f7.eps){width="16cm" height="16cm"} ![(Color online) The energies of $D^{-}$ meson ((a),(c) and (e)) and of $\bar {D^0}$ meson ((b),(d) and (f)), at momentum $k = 0$, versus the baryon density, expressed in units of nuclear matter saturation density, $\rho_{B}/\rho_{0}$, for different values of the isospin asymmetry parameter ($\eta = 0, 0.1, 0.3, 0.5$) and for a given temperature (T = 0, 100 MeV and 150 MeV). The parameters $d_{1}$ and $d_{2}$ are determined from $KN$ scattering lengths in I = 0 and I = 1 channels.[]{data-label="fig8"}](f8.eps){width="16cm" height="16cm"} The mass of the $D^0$ meson drops with density as can be seen from figure \[fig7\]. The drop in the mass of $D^{0}$ meson at density $\rho_{0}$ from its vacuum value (1864.5 MeV) is 78, 65, 62 MeV for temperature $T = 0, 100, 150$ MeV respectively at $\eta = 0$. At density $4\rho_{0}$ and isospin-asymmetry parameter $\eta = 0$, these values become 347, 313 and 294 MeV for temperatures T=0,100 and 150 MeV respectively. For the $D^{0}$ meson, there is seen to be an increase in the mass as we move from isospin symmetric to isospin asymmetric medium. For example, at zero temperature and baryon density equal to $\rho_{0}$ and $4\rho_{0}$, the rise in the masses of the $D^{0}$ meson are 21 and 91 MeV respectively as we move from isospin-symmetric medium ($\eta = 0$) to isospin-asymmetric medium ($\eta = 0.5$). At a temperature of 100 MeV, these values become 17 MeV at $\rho_{0}$ and 72 MeV at $4\rho_{0}$. For T=150 MeV, these values become 9 MeV at $\rho_{0}$ and 44 MeV at $4\rho_{0}$. Thus for $D^{0}$ mesons, the rise in the mass, is seen to be lowered at higher temperatures as we move from $\eta$ = 0 to $\eta$ = 0.5. The strong isospin dependence of the $D^+$ and $D^0$ meson masses should show up in observables such as their production as well as flow in asymmetric heavy-ion collisions planned at the future facility at FAIR, GSI. Fig.\[fig8\] shows the results for the density dependence of the energies of the $\bar{D}$ mesons at zero momentum at values of the temperature, $ T = 0, 100, 150$ MeV. There is seen to be a drop of the masses of both the $D^-$ and $\bar {D^0}$ with density. This is due to the dominance of the attractive scalar exchange contribution as well as the range terms (which becomes attractive above a density of about 2–2.5 times the nuclear matter saturation density) over the repulsive Weinberg-Tomozawa interaction [@amarind]. It is observed that the drop in the mass of $D^{-}$ and $\bar{D}^{0}$ in isospin symmetric nuclear matter is 27.2 MeV at $\rho_{0}$ and 162 MeV at $4\rho_{0}$, at zero temperature, from their vacuum values. As we go to higher temperatures, the drop in the masses of $D^{-}$ and $\bar{D}^{0}$ mesons decreases. For example, at $\eta$ = 0 and $\rho_{B} = \rho_{0}$ the drop in the mass of $D^{-}$ meson is 20.9, 14.3 and 10.8 MeV at a temperature of 50, 100 and 150 MeV respectively. For temperature, T=0,50,100 and 150 MeV, at $\rho_B=\rho_0$, the values of the drop in the $D^-$ mass are seen to be 27.2, 21.4, 14.6 and 15.5 MeV respectively, and the drop in the $\bar {D^0}$ mass are modifed from about 27, 21, 14.2 and 11 MeV to 23, 19, 14 and 19 MeV, when $\eta$ is changed from 0 to 0.5. The masses of $D^{-}$ and $\bar{D}^{0}$ mesons are observed to have negligible dependence on the isospin asymmetry upto a density of $\rho_{B}=\rho_{0}$. However, at high densities there is seen to be appreciable dependence of these masses on the parameter, $\eta$. As we change $\eta$ from 0 to 0.5, at $\rho_B=4\rho_0$, the drop in the mass of the $D^-$ meson is modified from 162 MeV to 139 MeV at zero temperature. At higher temperatures, T= 50, 100 and 150 MeV and at the density of $4\rho_0$, when we change $\eta$ from 0 to 0.5, the values of the $D^-$ mass drop are modifed from 149, 123 and 102 MeV to 128, 111 and 107 MeV respectively. It is seen that, at high densities there is an increase in the masses of both $D^{-}$ and $\bar{D}^{0}$ mesons in isospin asymmetric medium as compared to those in the isospin symmetric nuclear matter for temperatures T=0, 50 and 100 MeV. However, at $T = 150$ MeV, it is observed that for densities upto about 4.5$\rho_0$, the mass of $D^{-}$ meson is higher in the isospin symmetric matter as compared to in the isospin asymmetric matter with $\eta$ = 0.5. It is also seen that the modifications in the masses of $D^{-}$ mesons is negligible as we change $\eta$ from 0 to 0.3 upto a density of about 4$\rho_0$. For the $\bar{D}^{0}$ meson, one sees that the isospin dependence is negligible upto $\eta$=0.3. This is because the drop in the mass of $\bar{D}^{0}$ mesons due to isospin asymmetry given by Weinberg-Tomozawa term almost cancels with the increase due to the scalar and range terms as we go from $ \eta = 0$ to $ \eta = 0.3$. At zero temperature [@amarind] as well as for temperatures T=50 and 100 MeV, there is seen to be an increase in the mass of the $\bar{D}$ mesons ($D^-$,$\bar {D^0}$) as we go from isospin symmetric medium to the isospin asymmetric medium. This is because for T=0,50 and 100 MeV, the increase in mass of $\bar D$ given by the scalar exchange and the range terms dominate over the drop given by the Weinberg Tomozawa term as we go from symmetric nuclear medium ($\eta = 0$) to isospin asymmetric nuclear medium ($\eta$ =0.1,0.3 0.5). However, at $T = 150$ MeV, for $\eta$ =0.5, the drop given by Weinberg term dominates over the rise given by scalar and range terms for $\bar {D^0}$ and upto a density of about 4.5$\rho_0$ for $D^-$, and therefore mass of $\bar{D}$ meson decreases as we go from symmetric nuclear medium to isospin asymmetric nuclear medium in these density regimes. The medium modifications of the masses of $D$ and $\bar{D}$ mesons in the present investigation are due to the interactions with the nucleons and scalar mesons $\sigma$, $\zeta$ and $\delta$ in the hot nuclear medium. The values of the scalar fields in the medium are obtained by solving the the equations of motion for the scalar fields and the dilaton field $\chi$, given by the coupled equations (9) to (12). The temperature and density dependence of the dilaton field $\chi$ is seen to be negligible and thus the changes of the values of the scalar fields are observed to be marginal for the present investigation when the medium dependence of the $\chi$ field is taken into account as compared to when its medium dependence is not taken into account (the so-called frozen glueball approximation where the value of $\chi$ is taken to be its vacuum value). This leads to the modifications of the $D$ and $\bar D$ meson masses as marginal as compared to the case when the medium dependence of the dilaton field is not taken into account. For example, for temperature T = 0, the drop in the mass of $D^{+}$ mesons in isospin symmetric nuclear medium is about $81$ MeV and $364$ MeV at $\rho_{B} = \rho_{0}$ and $4\rho_{0}$ respectively in the frozen glueball approximation, which may be compared to the values of mass drop as $78$ MeV and $347$ MeV in the present investigation, when we take into account the effect of variation of the dilaton field with density. Hence one observes the difference between them to be marginal, of the order of about 5%. Similarly, the mass drop in the $D^{-}$ meson for the isospin asymmetric matter at T=0 in the frozen glueball approximation is observed to be about $30$ MeV and $184$ MeV at $\rho_{B} = \rho_{0}$ and $4\rho_{0}$ respectively. These are different from the values 27 MeV and 162 MeV of the present calculations with medium dependent dilaton field, by about 10%. In the present calculations, we have used the value of decay constant, $f_{D} = 135$ MeV. In isospin symmetry nuclear medium, for temperature T =0, we observe that the values of mass drops for $D^{+}$ and $D^{-}$ mesons are about 42 MeV and 4 MeV respectively at nuclear saturation density when we set $f_D$=157 MeV [@fd157]. These values may be compared to the values of 78 MeV and 27 MeV when the $f_D$ is taken to be 135 MeV. Hence by modifying the value of $f_D$ by about 15$\%$ leads to modifications of the mass shifts of $D^+$ and $D^-$ by about 46$\%$ and 85$\%$ at density, $\rho_0$ for symmetric nuclear matter at zero temperature. The mass drops for the value of the density as 4$\rho_0$ for symmetric nuclear matter at T=0, are modified from 347 MeV and 162 MeV to the values 238 MeV and 93 MeV for the $D^+$ and $D^-$ respectively, when we change the value of $f_D$ from 135 MeV to 157 MeV. Hence, for $\rho_B=4\rho_0$, the modifications for the $D^+(D^-)$ masses are about 30$\%$ and 40$\%$ when we change the value of the D-meson decay constant. Hence, there seems to be appreciable dependence of the mass shifts of the $D(\bar D)$ mesons with the D-meson decay constant. ![(Color online) The energies of $D^{+}$ meson ((a),(c) and (e)) and of $D^{0}$ meson ((b),(d) and (f)), at momentum $k = 0$, versus the baryon density (in units of nuclear saturation density), $\rho_{B}/\rho_{0}$, for different values of the isospin asymmetry parameter ($\eta = 0, 0.1, 0.3, 0.5$) and for given values of temperature (T = 0, 100 MeV and 150 MeV). The values of parameters $d_{1}$ and $d_{2}$ are calculated from $DN$ scattering lengths in I = 0 and I = 1 channels.[]{data-label="fig9"}](f9.eps){width="16cm" height="16cm"} ![(Color online) The energies of $D^{-}$ meson ((a),(c) and (e)) and of $\bar {D^0}$ meson ((b),(d) and (f)), at momentum $k = 0$, versus the baryon density, expressed in units of nuclear saturation density, $\rho_{B}/\rho_{0}$, for different values of the isospin asymmetry parameter ($\eta = 0, 0.1, 0.3, 0.5$) and for given values of temperature (T = 0, 100 MeV and 150 MeV). The values of parameters $d_{1}$ and $d_{2}$ are calculated from $DN$ scattering lengths in I = 0 and I = 1 channels.[]{data-label="fig10"}](f10.eps){width="16cm" height="16cm"} We next examine how the masses of the $D$ and $\bar{D}$ mesons change if we determine the parameters $d_{1}$ and $d_{2}$ from DN scattering lengths calculated to be $-0.43$ fm and $-0.41$ fm in the I = 0 and I = 1 channels respectively, in a coupled channel approach [@MK]. In figures \[fig9\] and \[fig10\], we show the variation of the energies of $D$ and $\bar D$ mesons respectively, at zero momentum, with baryon density $\rho_{B}$ for different values of the isospin asymmetry parameter $\eta$ and for the values of the temperature as T = 0, 100, 150 MeV. The values of $d_{1}$ and $d_{2}$ parameters determined from these values of the DN scattering lengths turn out to be $8.95/m_{D}$ and $0.52/m_{D}$ respectively, which can be expressed in terms of the mass of the kaon as $2.385/m_K$ and $0.14/m_K$. These values of parameters are smaller than the values of $d_{1}$ and $d_{2}$ as $2.56/m_{K}$ and $0.73/m_{K}$ respectively, when determined from the KN scattering lengths in $I =0$ and $I = 1$ channels. Since both of these terms are attractive, the masses of the $D$ ($\bar D$) mesons turn out to have a smaller drop when these parameters are fitted from DN scattering lengths as compared to when these are fitted from the KN scattering lengths. With the set of values of $d_{1}$ and $d_{2}$ parameters as fitted from the DN scattering lengths, the drop in the masses of $D^{+}$ meson in the isospin symmetric nuclear medium at nuclear matter saturation density $\rho_{0}$ turns out to be $51, 42$ and $40$ MeV, at the values of temperature as, $T = 0, 100$ and $150$ MeV respectively. These may be compared to the values of the mass drop of $D^+$ meson of 78, 65 and 62 MeV for T=0, 100 and 150 MeV, respectively, when the parameters are fitted from the KN scattering lengths. For $D^{0}$ mesons the values of mass drop are $50, 42$ and $40$ MeV at T=0, 100 and 150 MeV respectively, when $d_1$ and $d_2$ are fitted from the DN scattering lengths. These may be compared to the values of mass drop of $D^0$ meson of about 78, 65 and 62 MeV, when these are fitted to the KN scattering lengths. At higher densities, the $d_1$ and $d_2$ terms become more dominant and overcome the repulsive interaction of the first range term, leading to a drop of the D-meson masses due to the range term as well. At a density of $\rho_B=4\rho_0$, the drop in the $D^+$ mass is observed to be about 286 MeV, 256 MeV, 240 MeV for T=0, 100 MeV and 150 MeV, which may be compared to the values of 347 MeV, 313 MeV and 294 MeV for T=0,100, 150 MeV, when the parameters $d_1$ and $d_2$ are fitted from the KN scattering lengths. We plot the masses of the $\bar{D}$ mesons in figure \[fig10\], with the values of $d_{1}$ and $d_{2}$ are fitted from $DN$ scattering lengths [@MK]. At the nuclear saturation density $\rho_{0}$, the value of mass of $D^{-} (\bar {D^0})$ meson in isospin symmetric nuclear medium, is observed to increase by 2, 10 (10.5) and 12 (12.5) MeV at $T = 0, 100$ and $150$ MeV, respectively. However, at higher densities, the $d_1$ and $d_2$ terms become more dominant thus leading to a drop of the $D^- (\bar {D^0})$ masses in the nuclear matter. For $\rho_B=4\rho_0$, the mass of $D^- (\bar {D^0})$ meson is seen to decrease by 87 (86) MeV, 52 (51.5) MeV and 34 (33.6) MeV respectively. These may be compared to the results of the in-medium masses of $D^- (\bar {D^0})$ meson, when $d_1$ and $d_2$ are fitted from the KN scattering lengths. In the latter case, as already mentioned for T=0, 100 MeV and 150 MeV and for $\rho_B=4\rho_0$, there is seen to be a drop of $D^- (\bar {D^0})$ mass of 27.2 (27.2) 14.3 (14.2) and 10.8 (10.8) MeV at $\rho_B=\rho_0$ and 162 (162) 122.6 (122) and 101.7 (101.2) MeV at $\rho_B=4\rho_0$. As mentioned earlier, the in-medium mass of $D^{+}$ meson decreases with increase in the isospin asymmetry of the nuclear medium. However, with values of $d_{1}$ and $d_{2}$ parameters fitted from DN scattering lengths, the decrease in the mass with isospin asymmetry of the nuclear medium is observed to be smaller than that when these parameters are fitted from the KN scattering lengths. For example, at $\rho_{B} = 4\rho_{0}$ and $T = 0$, as we move from $\eta = 0$ to $\eta = 0.3$, the in-medium masses of $D^{+}$ mesons decrease by $27$ MeV with the values from the KN scattaring lengths and by $13$ MeV with the set of values of $d_{1}$ and $d_{2}$ parameters fitted from the DN scattering lengths. For the $D^{+}$ meson, the $d_{1}$ term gives a rise in the mass whereas the $d_{2}$ term gives drop in the mass when one has a nonzero value of the asymmetry parameter as compared to the symmetric nuclear matter. Also, the first range term gives an increase due to isospin asymmetry, whereas both the Weinberg Tomozawa term and scalar meson exchange term give drop in the mass of the $D^{+}$ meson. Due to the drop arising from the $d_{2}$ term, Weinberg Tomozawa term and the scalar term dominating over the the increase due to the $d_1$ term, the mass of $D^{+}$ meson decreases in the isospin asymmetric nuclear medium as compared to the mass in symmetric matter. When we use the values of the parameters $d_{1}$ and $d_{2}$ as calculated from the DN scattering lengths, then there is a smaller contribution to the drop of the mass due to $d_{2}$ term, as compared to the drop due to isospin asymmetry arising to this term when the parameters $d_1$ and $d_2$ are calculated from the KN scattering lengths. This is due to the smaller value for the $d_2$ in the former case. Therefore, with values of $d_{1}$ and $d_{2}$ parameters calculated from the DN scattering lengths, the drop in the mass of $D^{+}$ meson with isospin asymmetry of the medium is seen to be small. With the set of values of the parameters $d_{1}$ and $d_{2}$ as fitted from the KN scattering lengths, the mass of the $D^{-}$ meson is observed to increase with the isospin asymmetry of the medium. With the values of $d_{1}$ and $d_{2}$ parameters as fitted from the DN scattering lengths, there is seen to be larger increase in the mass of the $D^{-}$ meson due to the isospin asymmetry of the nuclear medium. For example, at $\rho_{B} = 4\rho_{0}$ and $T = 0$, as we move from $\eta =0$ to $\eta = 0.3$, the mass of the $D^{-}$ meson is seen to increase by about $7$ MeV when we use the values of $d_1$ and $d_2$ as fitted from the KN scattering lengths and by $32$ MeV with the values of $d_{1}$ and $d_{2}$ fitted from the DN scattering lengths. For the $D^{-}$ meson, there is an increase in the mass due to isospin asymmetry due to the Weinberg Tomozawa term, the first range term (term with coefficient ($-\frac{1}{f_{D}}$) and the $d_{1}$ term, whereas the scalar term and $d_{2}$ term give drop in the mass of $D^{-}$ meson as compared to the symmetric nuclear matter. The net effect is that the mass of the $D^{-}$ meson increases with the isospin asymmetry of the nuclear medium. With the values of $d_{1}$ and $d_{2}$ fitted from DN scattering lengths, because of the smaller value of $d_2$, the drop arising from the $d_{2}$ term due to isospin asymmetry in the medium is observed to be smaller than the case when the parameters $d_1$ and $d_2$ are fitted from the KN scattering lengths (same as for $D^{+}$ mesons). There is seen to be larger increase in the in-medium masses of $D^{-}$ mesons as a function of the isospin asymmetry of the nuclear medium when the parameters are determined from the DN scattering lengths as compared to the KN scattering lengths. With the values of $d_{1}$ and $d_{2}$ calculated from KN scattering lengths, the mass of the $\bar{D^{0}}$ meson increases with the isospin asymmetry $\eta$ of the medium as shown in figure \[fig8\]. However, if we use the values of $d_{1}$ and $d_{2}$, fitted from the $DN$ scattering lengths, then the mass of $\bar{D^{0}}$ mesons are seen to decrease with increase in the isospin asymmetry of the medium as shown in the figure \[fig10\]. For example, at $\rho_{B} = 4\rho_{0}$ and $T = 0$, as we move from $\eta = 0$ to $\eta = 0.3$, the $\bar{D^{0}}$ mass increases by about $10$ MeV when the parameters $d_1$ and $d_2$ are fitted from the KN scattering length, whereas the mass of $\bar {D^0}$ is seen to decrease by about $16$ MeV when $d_{1}$ and $d_{2}$ are calculated from the DN scattering lengths. The reason is, as a function of isospin asymmetry of the medium the $d_{1}$ and $d_{2}$ terms give rise to the masses of $\bar{D^{0}}$ mesons and the first range term gives a drop in the $\bar {D^0}$ mass. The values of $d_1$ and $d_2$ are larger when fitted from the KN scattering lengths as compared to when calculated from the DN scattering lengths. Hence in the former case, the the increase in $\bar {D^0}$ from isospin asymmetry arising from the $d_{1}$ and $d_{2}$ terms dominates over the drop given by first range term. Therefore, with isospin asymmetry of the medium the mass of $\bar{D^{0}}$ increases in the former situation. However, for the values of $d_{1}$ and $d_{2}$ parameters fitted from DN scattering lengths, the increase due to $d_1$ and $d_2$ terms is dominated by first range term. The Weinberg term gives a drop and the scalar term gives an increase in the mass of $\bar{D^{0}}$ meson with the isospin asymmetry of the nuclear medium. The net effect on the mass of the $\bar{D^{0}}$ meson is a drop with isospin asymmetry, $\eta$, of the nuclear medium. The parameters $d_{1}$ and $d_{2}$ have the same effect on the masses of the $D^{0}$ and $\bar{D^{0}}$ mesons of giving an increase in their mass in the isospin asymmetric medium as compared to the masses in the symmetric nuclear matter whereas the first range term gives a drop in their masses. The Weinberg Tomozawa term and the scalar meson exchange term lead to an increase in the mass of $D^{0}$ meson with isospin asymmetry. As can be seen from figures \[fig7\] and \[fig9\], the net effect on the $D^0$ meson mass is an increase with isospin asymmetry, but the rise is less for the case when $d_1$ and $d_2$ are calculated from the DN scattering lengths due to the smaller values of $d_1$ and $d_2$ as compared to when these parameters are determined from the KN scattering lengths. For example, at $\rho_{B} = 4\rho_{0}$ and $T = 0$, as we move from $\eta =0$ to $\eta = 0.3$, the in-medium mass of $D^{0}$ mesons increases by about $45$ MeV when the parameters are calculated from KN scattering lengths and by about $26$ MeV when fitted from the DN scattering lengths. ![(Color online) The optical potential of $D^{+}$ meson ((a),(c) and (e)) and of $D^{0}$ meson ((b),(d) and (f)), are plotted as functions of momentum for $\rho_{B}=\rho_0$, for different values of the isospin asymmetry parameter ($\eta = 0, 0.1, 0.3, 0.5$) and for given values of temperature (T = 0, 100 MeV and 150 MeV). The values of parameters $d_{1}$ and $d_{2}$ are calculated from $KN$ scattering lengths in I = 0 and I = 1 channels.[]{data-label="fig11"}](f11.eps){width="16cm" height="16cm"} ![(Color online) The optical potential of $D^{-}$ meson ((a),(c) and (e)) and of $\bar {D^{0}}$ meson ((b),(d) and (f)), are plotted as functions of momentum for $\rho_{B}=\rho_0$, for different values of the isospin asymmetry parameter ($\eta = 0, 0.1, 0.3, 0.5$) and for given values of temperature (T = 0, 100 MeV and 150 MeV). The values of parameters $d_{1}$ and $d_{2}$ are calculated from $KN$ scattering lengths in I = 0 and I = 1 channels.[]{data-label="fig12"}](f12.eps){width="16cm" height="16cm"} The mass modifications of $D$-mesons at finite density have been studied in the QCD sum rule and the mass shift at nuclear matter saturation density was found to be about $-50$ MeV [@arata]. In the QMC model, the mass shift was around $-60$ MeV [@qmc]. In the present investigation, at finite densities the magnitude of the scalar fields first increases with increase in the temperature upto a particular value of temperature after which it starts decreasing. This behaviour is then reflected in the variation of the nucleon mass with temperature at finite densities. In QMC model, the behaviour of the scalar field is also seen to be the same as in the present model [@amqmc]. However, in the QMC model, the nucleon mass is seen to monotonically rise with temperature and there is no change in this trend observed even upto a temperature of about 250 MeV [@amqmc] unlike in the present chiral model or in the Walecka model [@frunstl1]. This is because $\sigma$ field in the QMC model is not as strong as in the chiral model or the Walecka model. In QMC model, there are contributions to the masses of nucleons from the thermal excitations of the quarks inside the nucleon bag. This contribution of quarks dominates over the $\sigma$ field in QMC model [@amqmc]. The small attractive mass shift for the $\bar{D}$ mesons, obtained within our calculations are in favor of charmed mesic nuclei as suggested in the QMC model [@qmc]. This is, however, contrary to a repulsive potential obtained for the $\bar D$ mesons in the coupled channel approach [@mizutani8]. In our investigation, if we do not take into consideration the effect of the range terms on the in-medium properties of $D$ mesons then at nuclear saturation density $\rho_{0}$ and temperature $T = 0$, the mass of $D^{+}$ and $D^{-}$ mesons drop by $144$ MeV and $96$ MeV respectively in isospin symmetric nuclear medium ($\eta = 0$). The large mass-shift shows the absence of repulsion due to total range term at nuclear saturation density. However, in coupled channel approach of Ref.[@mizutani8], a repulsive mass-shift of $11$ MeV is given for $\bar{D}$ mesons. Figures \[fig11\] and \[fig13\] show the isospin dependence of the optical potentials for the $D$ mesons as functions of the momentum, for densities $\rho_{0}$ and $4\rho_{0}$ respectively and for values of the temperature as $T = 0, 100, 150$ MeV. These optical potentials are plotted for the values of $d_1$ and $d_2$ calculated from the KN scattering lengths. Figures \[fig12\] and \[fig14\] illustrate the optical potentials for the $\bar{D}$ doublet. The isospin dependence of optical potentials is seen to be quite significant for high densities for the D-meson doublet ($D^{+}, D^{0}$) as compared to those for the $\bar D$ doublet. This is a reflection of the strong isospin dependence of the masses of the D-mesons as compared to the $\bar {\rm D}$ as has been already illustrated in figures \[fig7\] and \[fig8\]. For the $\bar{D}$ mesons, it is seen, from figure \[fig8\], that the masses of the the $D^{-}$ meson and $\bar{D}^{0}$ meson for a fixed value of the isospin asymmetry parameter, $\eta$ are very similar, an observation which was seen earlier for the zero temperature case [@amarind]. These are reflected in their optical potentials, plotted in figures \[fig12\] and \[fig14\], where one sees a maximum difference of about 5 MeV or so between $D^-$ and $\bar {D^0}$ for $\rho_B=\rho_0$ and about 10 – 15 MeV for $\rho_B=4\rho_0$. The present investigations of the optical potentials for the $D$ and $\bar D$ mesons show a much stronger dependence of isospin asymmetry on the $D$ meson doublet, as compared to that in the $\bar D$ meson doublet, as was already observed for the zero temperature case. However, when the parameters $d_1$ and $d_2$ are fitted from the DN scattering lengths, then one sees a greater sensitivity to the isospin asymmetry in the masses of the $\bar D$ mesons as compared to the masses of the $D$ mesons, as illustrated in figures \[fig9\] and \[fig10\]. ![(Color online) The optical potential of $D^{+}$ meson ((a),(c) and (e)) and of $D^{0}$ meson ((b),(d) and (f)), are plotted as functions of momentum for $\rho_{B}=4\rho_0$, for different values of the isospin asymmetry parameter ($\eta = 0, 0.1, 0.3, 0.5$) and for given values of temperature (T = 0, 100 MeV and 150 MeV). The values of parameters $d_{1}$ and $d_{2}$ are calculated from $KN$ scattering lengths in I = 0 and I = 1 channels.[]{data-label="fig13"}](f13.eps){width="16cm" height="16cm"} ![(Color online) The optical potential of $D^{-}$ meson ((a),(c) and (e)) and of $\bar {D^{0}}$ meson ((b),(d) and (f)), are plotted as functions of momentum for $\rho_{B}=4\rho_0$, for different values of the isospin asymmetry parameter ($\eta = 0, 0.1, 0.3, 0.5$) and for given values of temperature (T = 0, 100 MeV and 150 MeV). The values of parameters $d_{1}$ and $d_{2}$ are calculated from $KN$ scattering lengths in I = 0 and I = 1 channels.[]{data-label="fig14"}](f14.eps){width="16cm" height="16cm"} We shall now investigate how the behaviour of the dilaton field $\chi$ in the hot asymmetric nuclear matter affects the in-medium masses of the charmonium states $J/\psi, \psi(3686)$ and $\psi(3770)$. In figures \[fig15\], \[fig16\] and \[fig17\], we show the shifts of the masses of charmonium states $J/\psi, \psi(3686)$ and $\psi(3770)$ from their vacuum values, as functions of the baryon density for given values of temperature T and for different values of the isospin asymmetry parameter, $\eta$. We have shown the results for the values of the temperature, T = 0, 50, 100 and 150 MeV. At the nuclear matter saturation density, $\rho_{B} = \rho_{0}$ at temperature T = 0, the mass-shift for $J/\psi$ meson is $-8.6$ MeV in the isospin symmetric nuclear medium ($\eta = 0$) and in the asymmetric nuclear medium, with isospin asymmetry parameter $\eta = 0.5$, it is seen to be about $-8.4$ MeV. For $\rho_{B} = 4\rho_{0}$ and at zero temperature, the mass-shift for $J/\psi$ meson is observed to be about $-32.2$ MeV in the isospin symmetric nuclear medium ($\eta = 0$) and in isospin asymmetric nuclear medium ($\eta = 0.5$), it is seen to be modified to $-29.2$ MeV. The increase in the magnitude of the mass-shift, with density $\rho_{B}$, is because of the large drop in the dilaton field $\chi$ at higher densities. However, with increase in the isospin asymmetry of the medium the magnitude of the mass-shift decreases because the drop in the dilaton field $\chi$ is less at a higher value of the isospin asymmetry parameter $\eta$. For the nuclear matter saturation density $\rho_{B} = \rho_{0}$ and at temperature T = 0, the mass-shift for $\psi(3686)$ is observed to be about $-117$ and $-114$ MeV for $\eta = 0$ and $0.5$ respectively, and for $\psi(3770)$, the values of the mass-shift are seen to be about $-155$ MeV and $-150$ MeV respectively. At $\rho_{B} = 4\rho_{0}$ and zero temperature, the values of the mass-shift for $\psi(3686)$ are modified to $-436$ MeV and $-396$ MeV for $\eta = 0$ and $0.5$ respectively, and, for $\psi(3770)$, the drop in the masses are about $-577$ MeV and $-523$ MeV respectively. As mentioned earlier, the drop in the dilaton field, $\chi$, at finite temperature is less than at zero temperature and this behaviour is reflected in the smaller mass-shift of the charmonium states at finite temperatures as compared to zero temperature case. At nuclear matter saturation density $\rho_{B} = \rho_{0}$, temperature T = 100 MeV, the values of the mass-shift for the $J/\psi$ meson are observed to be about $-6.77$ MeV and $-6.81$ MeV for isospin symmetric ($\eta = 0$) and isospin asymmetric ($\eta = 0.5$) nuclear medium respectively. At baryon density $\rho_{B} = 4\rho_{0}$, temperature T = 100 MeV, the mass-shift for $J/\psi$ is observed to be $-28.4$ MeV and $-27.2$ MeV for isospin symmetric ($\eta = 0$) and isospin asymmetric ($\eta = 0.5$) nuclear medium respectively. For the excited charmonium states $\psi(3686)$ and $\psi(3770)$, the mass-shifts at nuclear matter saturation density $\rho_{B} = \rho_{0}$ and temperature T = 100 MeV, are observed to be $-91.8$ MeV and $-121.4$ MeV respectively, for isospin symmetric nuclear medium ($\eta = 0$) and $-92.4$ MeV and $-122$ MeV for the isospin asymmetric nuclear medium with $\eta = 0.5$. For a baryon density of $\rho_{B} = 4\rho_{0}$, and temperature T = 100 MeV, the mass-shifts for charmonium states $\psi(3686)$ and $(\psi(3770))$ are seen to be $-386$ MeV and $-510$ MeV respectively, for isospin symmetric ($\eta = 0$) and $-369$ MeV and $-488$ MeV for isospin asymmetric nuclear medium with $\eta = 0.5$. For temperature $T = 150$ MeV and at the nuclear matter saturation density $\rho_{B} = \rho_{0}$, the mass-shifts for the charmonium states $J/\psi, \psi(3686)$ and $\psi(3770)$ are seen to be $-6.25$, $-85$ and $-112$ MeV respectively in the isospin symmetric nuclear medium ($\eta = 0$). These values are modified to $-7.2$, $-98$ and $-129$ MeV respectively, in the isospin asymmetric nuclear medium with $\eta = 0.5$. At a baryon density $\rho_{B} = 4\rho_{0}$, the values of the mass-shift for $J/\psi, \psi(3686)$ and $\psi(3770)$ are observed to be $-26.4$, $-358$ and $-473$ MeV in isospin symmetric nuclear medium ($\eta = 0$) and in isospin asymmetric nuclear medium with $\eta = 0.5$, these values are modified to $-27.6$, $-375$ and $-494$ MeV respectively. Note that at high temperatures, e.g., at $T = 150$ MeV the mass-shift in isospin asymmetric nuclear medium ($\eta = 0.5$) is more as compared to isospin symmetric nuclear medium ($\eta = 0$). This is opposite to what is observed for the zero temperature case. The reason is that at high temperatures the dilaton field $\chi$ has a larger drop in the isospin asymmetric nuclear medium ($\eta = 0.5$) as compared to the isospin symmetric nuclear medium ($\eta = 0$), due to the contributions from the $\delta$ field for the nonzero $\eta$, which is observed to decrease in its magnitude at high temperatures. The dependence of the wave function of the charmonium on the density and temperature of the medium can be introduced through the modification of the strength of the harmonic potential for the charmonium state [@friman] given as the parameter $\beta$ in equation (30). In the medium, one expects the strength of the confining potential to be smaller than in the vacuum as due to medium modifications of the hadrons, more decay channels can become accessible, which are not available in the vacuum. We observe that when the parameter is decreased by about 5$\%$, the mass drops of the charmonium states are increased by about 14$\%$ and a change of the parameter $\beta$ by 2$\%$ leads to a mass drop of the charmonium states larger by about 5$\%$. ![(Color online) The mass shift of $J/\psi$ plotted as a function of the baryon density in units of nuclear matter saturation density at given temperature, for different values of the isospin asymmetry parameter, $\eta$.[]{data-label="fig15"}](f15.eps){width="16cm" height="16cm"} ![(Color online) The mass shift of $\psi(3686)$ plotted as a function of the baryon density in units of nuclear matter saturation density at given temperature, for different values of the isospin asymmetry parameter, $\eta$.[]{data-label="fig16"}](f16.eps){width="16cm" height="16cm"} ![(Color online) The mass shift of $\psi(3770)$ plotted as a function of the baryon density in units of nuclear matter saturation density at given temperature, for different values of the isospin asymmetry parameter, $\eta$.[]{data-label="fig17"}](f17.eps){width="16cm" height="16cm"} The values of the mass-shift for the charmonium states obtained within the present investigation, at nuclear matter saturation density $\rho_{0}$ and temperature $T =0$, are in good agreement with the mass shift of $J/\psi, \psi(3686)$ and $\psi(3770)$ as $-8, -100$ and $-140$ MeV respectively, at the nuclear matter saturation density computed in Ref. [@leeko] from the second order stark effect, with the gluon condensate in the nuclear medium computed in the linear density approximation. On the other hand, in the present work, the temperature and density dependence of the gluon condensates are calculated from the medium modifications of the dilaton field within the chiral SU(3) model. In Ref. [@leeko], the masses of the charmonium states were calculated for the symmetric nuclear matter at zero temperature, whereas the present investigation studies the isospin asymmetry dependence of the masses of the charmonium states in the nuclear medium at finite temperatures, which will be relevant for the asymmetric heavy ion collision experiments planned at the future facility at GSI. The mass-shift for $J/\psi$ has also been studied with the QCD sum rules in [@klingl] and the value at nuclear saturation density was observed to be about $-7$ MeV. In [@kimlee] the operator product expansion was carried out upto dimension six and mass shift for $J/\psi$ was calculated to be $-4$ MeV at nuclear matter saturation density $\rho_{0}$ and at zero temperature. The effect of temperature on the $J/\psi$ in deconfinement phase was studied in [@leetemp; @cesa]. In these investigations, it was reported that $J/\psi$ mass remains essentially constant within a wide range of temperature and above a particular value of the temperature, T, there is seen to be a sharp change in the mass of $J/\psi$ in the deconfined phase. For example, in Ref. [@lee3] the mass shift for $J/\psi$ was reported to be about 200 MeV at T = 1.05 T$_{c}$. In the present work, we have studied the effects of temperature, density and isospin asymmetry, on the mass modifications of the charmonium states ($J/\psi, \psi(3686)$ and $\psi(3770)$) in the confined hadronic phase, arising due to modifications of a scalar dilaton field which simulates the gluon condensates of QCD, within a chiral SU(3) model. The effect of temperature is found to be small for the charmonium states $J/\psi(3097)$, $\psi(3686)$ and $\psi(3770)$, whereas the masses of charmonium states are observed to vary considerably with density, in the present investigation. The medium modifications of the masses of $D$ and $\bar {D}$ mesons as well as that of charmonium states could be an explanation for the observed $J/\psi$ suppression observed by NA50 collaboration at $158$ GeV/nucleon in the Pb-Pb collisions [@blaiz]. Due to the drop in the mass of the $D\bar D$ pair in the nuclear medium, it can become a possibility that the excited states of charmonium ($\psi^{'}, \chi_{c2}, \chi_{c1}, \chi_{c0}$) can decay to $D\bar{D}$ pairs [@amarind] and hence the production of $J/\Psi$ from the decay of these excited states can be suppressed. Even at high values of densities at given temperatures, it can become a possibility that $J/\psi$ itself decays to $D\bar{D}$ pairs. Thus the medium modifications of the $D$ mesons can modify the decay widths of the charmonium states [@friman]. In figures \[fig18\] and \[fig19\], we show the density dependence of the masses of the $D^{+}D^{-}$ as well as $D^{0}\bar{D^{0}}$ pairs calculated in the present investigation, for temperatures, T = 0, 100 and 150 MeV and for isospin asymmetry parameter, $\eta = 0$ and $\eta$=0.5 respectively. We also show the in-medium masses of the charmonium states $J/\psi, \psi(3686)$ and $\psi(3770)$ in these figures. We observe that in the isospin symmetric nuclear medium at zero temperature, the in-medium mass of charmonium $\psi(3770)$ is less than $D^{+}D^{-}$ and $D^{0}\bar{D^{0}}$ pairs above baryon densities $0.6\rho_{0}$ and $0.8\rho_{0}$ respectively and therefore its decay does not seem possible at densities higher than these densities in the nuclear medium. However, as we move to the isospin asymmetric medium ($\eta = 0.5$), the medium modifications for the masses of the $D\bar D$ pairs as well as of the charmonium states indicate that the decay of $\psi(3770)$ to $D^{+}D^{-}$ pairs can be possible above a density of about $2\rho_{0}$, but the decay to $D^{0}\bar{D^{0}}$ does not seem possible above a density of about nuclear matter saturation density. In the isospin symmetric nuclear medium, the decay of the charmonium state $\psi(3686)$ to $D^{+}D^{-}$ and to $D^{0}\bar{D^{0}}$ seem possible above densities of about 3.4 $\rho_0$ and 3.3 $\rho_0$ respectively. In isospin asymmetric nuclear medium ($\eta = 0.5$) the decay of the charmonium state $\psi (3686)$ to $D^{+}D^{-}$ pairs seem as possibilities above a density of about $2 \rho_{0}$. The effects of the temperatures on the decay of the charmonium states to $D\bar D$ pairs have also been illustrated in the figures \[fig18\] and \[fig19\]. The decay of $\psi (3770)$ to the $D\bar D$ pairs do not seem possible above a density of about $\rho_0$ even at T=100 and 150 MeV. However, for $\psi (3686)$, for T=100 MeV, the decay to the $D^+ D^-$ and $D^0 \bar {D^0}$ seem possible above densities of about 4$\rho_0$ and 3.8 $\rho_0$ for symmetric nuclear matter For T=150 MeV, these values are modified to 4.5$\rho_0$ and 4.3 $\rho_0$ respectively, for $\eta$=0. As we move to the asymmetric nuclear matter, the densities above which the decay of $\psi (3686)$ decaying to $D^+ D^-$ becomes possibile are 2.4 $\rho_0$ and 2.2 $\rho_0$ respectively. Similar to the zero temperature case, we do not see a possibility of $\psi (3686)$ to $D^0 \bar {D^0}$ for the asymmetric nuclear matter ($\eta$=0.5) at T=100 and 150 MeV. In the present investigation, the decay of $J/\psi$ to $D\bar D$ pairs does not seem as a possibilty even upto a density of about 6$\rho_0$. We observe from figures 18 and 19 that the temperature dependence is minimal for the decay of the charmonium states to the $D\bar D$ pairs even though the density dependence is quite appreciable. ![(Color online) The masses of the $D\bar{D}$ pairs \[$D^{+}D^{-}$ in (a), (c), (e) and $D^{0}\bar{D}^{0}$ in (b), (d), (f)\] in MeV plotted as functions of $\rho_{B}/\rho_{0}$ for isospin for the symmetric nuclear matter ($\eta$=0) and for temperatures, T = $0, 100, 150$ MeV.[]{data-label="fig18"}](f18.eps){width="16cm" height="16cm"} ![(Color online) The masses of the $D\bar{D}$ pairs \[$D^{+}D^{-}$ in (a), (c), (e) and $D^{0}\bar{D}^{0}$ in (b), (d), (f)\] in MeV plotted as functions of $\rho_{B}/\rho_{0}$ for isospin asymmetry parameter value $\eta = 0.5 $ and temperatures, T = $0, 100, 150$ MeV.[]{data-label="fig19"}](f19.eps){width="16cm" height="16cm"} The decay of the charmonium states have been studied in Ref. [@friman; @brat6]. It is seen to depend sensitively on the relative momentum in the final state. These excited states might become narrow [@friman] though the $D$ meson mass is decreased appreciably at high densities. It may even vanish at certain momentum corresponding to nodes in the wave function [@friman]. Though the decay widths for these excited states can be modified by their wave functions, the partial decay width of $\chi_{c2}$, owing to absence of any nodes, can increase monotonically with the drop of the $D^{+}D^{-}$ pair mass in the medium. This can give rise to depletion in the $J/\Psi$ yield in heavy-ion collisions. The dissociation of the quarkonium states ($\Psi^{'}$,$\chi_{c}$, $J/\Psi$) into $D\bar{D}$ pairs has also been studied [@wong; @digal] by comparing their binding energies with the lattice results on the temperature dependence of the heavy-quark effective potential [@lattice]. summary ======= We have investigated in a chiral model the in-medium masses of the $D$, $\bar{D}$ mesons and the charmonium states ($J/\psi$, $\psi (3686)$ and $\psi (3770)$) in hot isospin asymmetric nuclear matter. The properties of the light hadrons – as studied in $SU(3)$ chiral model – modify the $D(\bar{D})$ meson properties in the dense and hot hadronic matter. The $SU(3)$ model, with parameters fixed from the properties of the hadron masses in vacuum and low-energy KN scattering data, is extended to SU(4) to derive the interactions of $D(\bar{D})$ mesons with the light hadron sector. The mass modifications of $D^{+}$ and $D^{0}$ mesons is strongly dependent on isospin-asymmetry of medium when we determine the parameters $d_1$ and $d_2$ consistent with the KN scattering lengths. However, the sensitivity to the isospin asymmetry is seen to be more for the $\bar D$ doublet, when we fit the parameters to the DN scattering lengths as calculated in the coupled channel approach in Ref. [@MK]. At finite densities, the masses of $D (\bar D)$ mesons are observed to increase with temperature [@amdmeson] upto a temperature above which it is observed to decrease. The mass modification for the $D$ mesons are seen to be similar to earlier finite density calculations of QCD sum rules [@qcdsum08; @weise] as well as to the quark-meson coupling model [@qmc]. This is in contrast to the small mass modifications in the coupled channel approach [@ljhs; @mizutani8]. Also we obtain small attractive mass shifts for $\bar{D}$ mesons similar to the results obtained from the QMC model, which might lead to formation of charmed mesic nuclei. These results for the $\bar D$ mesons are contrary to the results from the coupled channel approach [@mizutani8], where the $\bar D$ mesons experience a repulsive interaction in the nuclear medium. In our calculations the presence of the repulsive first range term (with coefficient $-\frac{1}{f_{D}}$ in Eq. (\[ldn\])) is compensated by the attractive $d_{1}$ and $d_{2}$ terms in Eq.(\[ldn\]). Among the attractive range terms ($d_{1}$ and $d_{2}$ terms), $d_{1}$ term is found to be dominating over $d_{2}$ term. We have investigated in the present work, the effects of density, temperature and isospin asymmetry of the nuclear medium on the masses of the charmonium states $J/\psi$, $\psi(3686)$ and $(\psi(3770)$, arising due to modification of the scalar dilaton field, $\chi$, which simulates the gluon condensates of QCD, within the chiral SU(3) model. The change in the mass of $J/\psi$ with the density is observed to be small at nuclear matter saturation density and is in agreement with the QCD sum rule calculations. There is seen to be appreciable drop in the in-medium masses of excited charmonium states $\psi(3686)$ and $\psi(3770)$ with density. The mass drop of the excited charmonium states $\psi(3686)$ and $\psi(3770)$ are large enough to be seen in the dilepton spectra emitted from their decays in experiments involving $\bar{p}$-A annihilation in the future facility at GSI, provided these states decay inside the nucleus. The life time of the $J/\psi$ has been shown to be almost constant in the nuclear medium, whereas for these excited charmonium states, the life times are shown to reduce to less than 5 fm/c, due to appreciable increase in their decay widths [@friman]. Hence a significant fraction of the produced excited charmonium states in these experiments are expected to decay inside the nucleus [@golu]. The in-medium properties of the excited charmonium states $\psi(3686)$ and $\psi(3770)$ can be studied in the dilepton spectra in $\bar{p}$-A experiments in the future facility of the FAIR, GSI [@gsi]. The mass shift of the charmonium states in the hot nuclear medium seem to be appreciable at high densities as compared to the temperature effects on these masses, and these should show in observables like the production of these charmonium states in the compressed baryonic matter experiment at the future facility at GSI, where baryonic matter at high densities and moderate temperatures will be produced. The medium modifications of the $D$ meson masses can lead to a suppression in the $J/\Psi$ yield in heavy-ion collisions, since the excited states of the $J/\Psi$ can decay to $D\bar{D}$ pairs in the dense hadronic medium. The medium modifications of the masses of the charmonium states as well as the $D$ and $\bar{D}$ mesons have been considered in the present investigation. The isospin asymmetry lowers the density at which decay to $D^{+}D^{-}$ pairs occur. Due to increase in the mass of $D^{0}\bar{D}^{0}$ in the isospin-asymmetric medium, isospin-asymmetry is seen to disfavor the decay of the charmonium states to the $D^{0}\bar{D}^{0}$ pairs. At zero or finite temperatures, there does not seem to be a possibility of decay of $J/\Psi$ to $D^+ D^-$ or $D^{0}\bar{D}^{0}$ pairs. The isospin dependence of $D^{+}$ and $D^{0}$ masses is seen to be a dominant medium effect at high densities, which might show in their production ($D^{+}/D^{0}$), whereas, for the $D^{-}$ and $\bar{D}^{0}$, one sees that, even though these have a strong density dependence, their in-medium masses remain similar at a given value for the isospin-asymmetry parameter $\eta$. This is the case when we fit the parameters $d_1$ and $d_2$ from the KN scattering lengths. When we determine these parameters from the DN scattering lengths as calculated in Ref. [@MK], the masses of the $\bar D$ doublet are seen to be more sensitive to isospin asymmetry in the medium. The strong density dependence as well as the isospin dependence of the $D(\bar{D})$ meson optical potentials in asymmetric nuclear matter can be tested in the asymmetric heavy-ion collision experiments at future GSI facility [@gsi] in observables like the $D^+/D^0$ as well as $D^-/\bar {D^0}$ ratios. In the present work, we have investigated the in-medium masses of the charmonium states due to their interaction with the scalar dilaton field (simulating the gluon condensates of QCD) and $D (\bar D)$ mesons due to the interaction with nucleons as well as scalar mesons. The parameters of the asymmetric nuclear matter are fitted to the nuclear matter properties, vacuum baryon masses, the hyperon potentials as well as the KN scattering lengths in the chiral SU(3) model. 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  and -3.2mm A unified evolutionary scheme that includes post-AGB systems, barium stars, symbiotics, and related systems, explaining their similarites as well as differences. Can we construct it? We compare these various classes of objects in order to construct a consistent picture. Special attention is given to the comparison of the barium pollution and symbiotic phenomena. Finally, we outline a ‘transient torus’ evolutionary scenario that makes use of the various observational and theoretical hints and aims at explaining the observed characteristics of the relevant systems. stars: AGB and post-AGB, stars: binaries \[Sect:intro\] The term “after-AGB binaries” will be used in this paper to refer to binary systems in which at least [*one of the components*]{} has gone through the AGB phase, as distinguished from the term “post-AGB”, commonly used to denote the short transition phase between AGB and PN core (CSPN) stages of (single or binary) stellar evolution. In many of such after-AGB systems, the mass transfer from an AGB star has left its mark on the companion, enhancing its abundances with the products of the AGB nucleosynthesis, most remarkably C, F, and s-process elements (see Habing & Olofsson 2003 for a recent review). This pollution will manifest itself later in the companion’s evolution. An exemplary case of after-AGB systems are barium stars: G-K type giants remarkable for their overabundances of Ba (McClure et al. 1980). Related families include Abell-35 subclass of PNe (Bond et al. 1993), barium dwarfs (including the so called WIRRing stars, Jeffries & Stevens 1996), subgiant and giant CH stars, extrinsic S stars and d’-type yellow symbiotics. But not all of the after-AGBs need to be s-process rich. The post-AGB binaries are an interesting case, as they are all by definition after-AGBs: some of them do exhibit s-process enhancement while others do not (Van Winckel 2003, also this volume). Red s-type symbiotic stars (SyS) with massive white dwarf companions ($M_{\rm WD}>0.5 M_{\odot}$), another member of the after-AGB group, also do not exhibit s-process enhancement (Jorissen 2003a). Not much in this respect can be said about most binary CSPNe, as the unevolved companion is usually too faint to be seen. Finally, some of the cataclysmic variables (CV) with massive white dwarfs should also belong to the after-AGB family. \[Sect:links\] Peculiar red giants are a characteristic part of the after-AGB family and are in many respects closely related to symbiotic stars. The links between them have been reviewed previously (Jorissen 2003a, Jorissen et al. 2005); here we provide an updated discussion based on Fig. 1 which displays the various types of SyS and peculiar red giants in a metallicity – spectral-type plane. The vertical axis corresponds to metallicity which impacts (i) the taxonomy of the classes (CH giants for instance – box 1 – are the halo-equivalent of the disk barium stars – box 6); (ii) the efficiency of heavy-element synthesis (Clayton 1988), and (iii) the location of evolutionary tracks in the HR diagram (hence the correspondence between spectral type and evolutionary status, like the onset of TP-AGB, will depend on metallicity). Fig. 1 therefore considers three different metallicity ranges: (i) \[Fe/H\]$<-1$, corresponding to the halo population; (ii) $-1 \la [Fe/H] \la 0$, or disk metallicity; (iii) \[Fe/H\]$\ga 0$, solar and super-solar metallicities. The horizontal axis in Fig. 1 displays spectral type, which roughly corresponds to an evolutionary sequence at a given metallicity, passing from the red giant branch (RGB) to the thermally-pulsing AGB (TP-AGB) phases (between these two phases is the core He-burning phase which is hardly distinguishable from the lower RGB; CH giants probably belong to that phase). Symbiotic activity is expected in the middle of this sequence, because (i) at the left end, stars (like CH) are not luminous enough to experience a mass loss sufficient to power symbiotic activity; (ii) at the right end, the stars with the barium syndrome need not be binaries (above the TP-AGB luminosity threshold, heavy-elements are synthesized in the stellar interior and dredge-up to the surface, so that “intrinsic Ba” (or S) stars occupy the rightmost boxes – 4 and 8 – of Fig. 1), and hence need not exhibit any symbiotic activity. Those evolved giants which [*are*]{} members of binary systems (like Mira Ceti – box 9) will of course exhibit symbiotic activity. It is noteworthy that late M giants are inexistent in a halo population (hence the crossed box 5), because evolutionary tracks are bluer as compared to higher metallicities. Examples of such very evolved (relatively warm) stars in a halo population (box 4) include CS 30322-023 (Masseron et al. 2006) and V Ari (Van Eck et al. 2003). In the halo population, the most interesting issue is to understand the origin of the difference between yellow s-type SyS (box 2) and metal-deficient Ba stars (box 3): why do the latter not exhibit any symbiotic activity? The reason seems to reside in a difference in their orbital period distributions: yellow s-type SyS represent the short-period tail of the distribution (Fig. 2), where for a given mass-loss rate accretion will be more efficient, and hence will trigger symbiotic activity. [r]{}\[0pt\][68mm]{} In the intermediate-metallicity regime, the situation is quite clear: as the star evolves to the right along the spectral sequence, its luminosity (hence mass loss) increases, and for binary stars along that sequence, the symbiotic character will become stronger. The real issue here is to understand the origin of the difference between boxes 7 and 10/11: why are red s- (and d-)type SyS never exhibiting the barium syndrome, despite very similar locations in the HR diagram and similar orbital-period distributions (Fig. 2)? In former discussions of this issue (e.g. Jorissen 2003a), it had been suggested that binary stars with the barium syndrome (box 7) and SyS without it (box 10) differ in their metallicities. It is known that at high metallicities heavy-element synthesis is less efficient (Clayton 1988). However, there is so far [*no evidence for red symbiotic stars being on average more metal-rich than barium or extrinsic S stars.*]{} Schild et al. (1992) and Schmidt & Mikołajewska (2003) have compared the carbon abundances of SyS and normal giants, and found absolutely no difference, thus confirming at the same time the absence of any signature of internal nucleosynthesis and dredge-ups, or of pollution through mass transfer. Other solutions to this puzzle may be suggested, such as (i) s- and d-type SyS are not intrinsic barium stars, because they are not TP-AGB stars; (ii) neither are they extrinsic barium stars, because their companion never went through the TP-AGB phase, either because it is a He WD or because it is a main sequence star. Regarding item (i), it is very likely indeed that red s-type SyS are not TP-AGB stars, since they rather involve early M giants. The situation is less clear for d-type SyS, as they involve Miras which are often claimed to be TP-AGB stars. Nevertheless, many Miras do not exhibit signatures of heavy-element nucleosynthesis (they are not carbon stars and lack lines from the unstable element Tc; Little et al. 1987). Regarding item (ii), the possibility for hot companions to SyS to be He WDs is the most appealing since (a) the eccentricities observed for symbiotic systems are much smaller than those observed in pre-mass-transfer systems (M giants in the period range 200 – 1000 d have eccentricities up to 0.3; Jorissen et al. 2004), thus suggesting that mass transfer [ *has taken place*]{} in these systems; (b) the mass distribution of the hot components peaks between 0.4 and 0.5 $M_\odot$  (Mikołajewska 2003 and this volume), as expected for He WDs. Of course, an alternative explanation – like a main sequence companion – needs to be found for those SyS companions with masses exceeding 0.5 $M_\odot$ (T CrB, FG Ser, FN Sgr, AR Pav, V1329 Cyg; Mikołajewska 2003). Although a main sequence companion is quite unlikely in recurrent or symbiotic novae like T CrB and V1329 Cyg, the situation regarding the nature of symbiotic-star companions for non-nova systems is far from being settled, as mentioned by Mikołajewska (2003) while answering a question by one of the authors at the La Palma symbiotic-star conference: [ *the question of whether \[the companion to CI Cyg, Z And, FN Sgr\] is a disk-accreting main-sequence star or a quasi-steady hydrogen-burning white dwarf is open so long as we have no good theory to distinguish between these possibilities*]{}. Indeed, the nature of the companion to CI Cyg has changed over the years, from main-sequence accretor (Kenyon & Webbink 1984; Kenyon et al. 1991; Mikołajewska & Kenyon 1992) to hot and luminous stellar source powered by thermonuclear burning (Mikołajewska 2003)! The same move from main-sequence to white-dwarf accretor holds true for AR Pav (Kenyon & Webbink 1984,Quiroga et al. 2002). \[Sect:solar\_Z\] The evolutionary status of the rare set of yellow d’ SyS (box 12), which were all shown to be of solar metallicity, has recently been clarified (Jorissen et al. 2005) with the realisation that in these systems, the companion is [*intrinsically*]{} hot (because it recently evolved off the AGB), rather than being powered by accretion or nuclear burning. Several arguments support this claim: (i) d’ SyS host G-type giants whose mass loss is not strong enough to heat the companion through accretion and/or nuclear burning; (ii) the cool dust observed in d’ SyS (Schmid & Nussbaumer 1993) is a relic from the mass lost by the AGB star; (iii) the optical nebulae observed in d’ SyS are most likely genuine planetary nebulae (PN) rather than the nebulae associated with the ionized wind of the cool component (Corradi et al. 1999). d’ SyS often appear in PN catalogues. AS 201 for instance actually hosts [*two*]{} nebulae (Schwarz 1991): a large fossil planetary nebula detected by direct imaging, and a small nebula formed in the wind of the current cool component; (iv) rapid rotation is a common property of the cool components of d’ SyS (see Table 1 of Jorissen et al. 2005). It has likely been caused by spin accretion from the former AGB wind like in WIRRing systems (Jeffries & Stevens 1996; Jorissen 2003b). The fact that the cool star has not yet been slowed down by magnetic braking is another indication that the mass transfer occurred fairly recently (Theuns et al. 1996). Corradi & Schwarz (1997) obtained 4000 y for the age of the nebula around AS 201, and 40000 y for V417 Cen. The possible existence, in box 13, of binary systems of nearly solar metallicity with orbital properties typical of barium systems, but not exhibiting the barium syndrome, is still controversial, as discussed by Jorissen (2003b). \[Sect:orbital\] Intense AGB mass loss/transfer is not only important for chemical abundances; it does also influence the orbital properties of after-AGB systems. Four binary evolution processes are usually invoked when describing the after-AGB systems formation: (i) tidal interactions, (ii) wind accretion, including tidally enhanced winds (Companion-Reinforced Attrition Process or CRAP, Eggleton 1986), (iii) stable Roche-lobe overflow (RLOF), and (iv) common envelope (CE) evolution. Alas, current evolutionary computations fail to reproduce the correct ranges of orbital periods, eccentricities and s-process enhancement levels (e.g. Pols et al. 2003; Frankowski 2004). The basic reason for this problem is quite simple, as nicely put by Iben & Tutukov (1996): [*as a result of CE interaction, initially close systems become closer and, because of wind mass loss, initially wide systems become wider. \[...\] most known symbiotic systems belong to a rare population on the borderline between initially close and wide binaries.*]{} The models do not produce eccentric systems with periods below $\sim2000$ – 3000d and all systems below $\sim$1000d enter a CE and undergo a dramatic orbital shrinkage. The observed after-AGB systems with intermediate periods (100 – 2000d) have somehow avoided the catastrophic outcome of a CE, but the theoretical concepts proposed so far are not satisfactory in explaining this fact: (i) the inclusion of tidal forces affects only the detached evolution and does not improve the final results; (ii) CRAP does allow for slightly shorter final periods in detached evolution but still not below 2000d and any stronger effect would prevent TP-AGB, thus impeding s-process; (iii) stable RLOF occurs only for a narrow range of initial parameters; (iv) lowered binding energy of the AGB envelope due to the inclusion of ionisation energy as proposed e.g. by Han et al. (1994) is problematic (Harpaz 1998); (v) CE formalism based on angular-momentum instead of on energy (Nelemans et al. 2000) is promising, however, for the moment it lacks physical explanation. Also puzzling is high eccentricity (up to $e\!=\!0.4$) at periods down to 300d observed among post-AGB binaries, and to a lesser extent Ba stars and extrinsic S stars. The most promising explanation here is eccentricity pumping by a circumbinary disk (Waelkens et al. 1996; Artymowicz et al. 1991). Another suggestion is periastron mass loss eccentricity pumping (Soker 2000) but this mechanism can operate only for wide (detached on the AGB) systems. We suggest that these conundrums are part of a bigger puzzle together with the following observational and theoretical hints. First, some of the young after-AGB objects exhibit combined RS CVn and Ba star properties: X-rays, H$_\alpha$ emission and fast rotation combined with Ba enhancement and long orbital periods. The list consists of Ba stars 56 Peg (Frankowski & Jorissen 2006), HD 165141 (Jorissen et al. 1996), d’ symbiotics (Jorissen et al. 2005, see also the discussion in Sect. 2.3), WIRRing stars, (Jeffries & Stevens 1996), and Abell-35 CSPNe (Thévenin & Jasniewicz 1997). They form a strong evidence for fast rotation in young after-AGB systems, supposedly due to spin accretion from wind (Jeffries & Stevens 1996; Jorissen 2003b). Second, post-AGB systems, the youngest among the after-AGB family, are known to possess circumbinary disks (Van Winckel 2003). Dusty circumbinary disks, tori and bipolar outflows are common among bipolar and ring-like PNe, and have also been observed in some AGB stars, notably $\pi^1$ Gru (Sahai 1992) and V Hya (Knapp et al. 1999). The latter object is also remarkable for having fast rotation velocity (6-16 km s$^{-1}$) and a long secondary photometric period ($\sim 6200$d, in addition to the radial pulsation period of 530d), possibly due to a binary companion. Another notable factor is that dust formation and radiation-driven wind cause reshaping of Roche equipotentials and reduction of the effective gravity of the mass-losing star (Jorissen 2003b; Schuerman 1972; Frankowski & Tylenda 2001). \[Sect:scenario\] Gathering the observational and theoretical constraints described above, we propose a ‘transient torus’ scenario for explaining the observed orbital periods and eccentricities of “after AGB” binaries. This scenario can be divided into four phases, schematically represented in Fig. 3: 1\. Wind accretion. The system is well detached and the companion accretes mass and angular momentum from the giant’s wind. Spin accretion is especially efficient, proceeding through an accretion disk formed around the companion (Theuns et al. 1996; Mastrodemos & Morris 1998). Orbital evolution proceeds roughly as in spherically-symmetric wind case (Jeans mode), i.e., $a(M_1+M_2) =$ const and the eccentricity stays almost constant. 2\. (Near) RLOF with substantial $L_2/L_3$ outflow. Tidal forces and evolutionary expansion of the giant bring it closer to its Roche lobe. The outflow becomes concentrated in the direction to the companion, which happens even before the actual Roche-lobe filling (e.g. Frankowski & Tylenda 2001). The matter is ’funnelled’ through the vicinity of $L_1$. 3. Formation of a circumbinary torus. Matter escaping through the vicinity of $L_2$ (or $L_3$, after mass ratio reversal) forms a spiral around the system. But after one orbital period every portion of ejecta becomes shadowed from the giant by the newly ejected matter and ceases being accelerated outwards by the radiation pressure on dust. Part of the older ejecta gravitationally falls back onto the binary and collides with the new stream. A thick circumbinary torus is formed. 4\. Formation of a Keplerian circumbinary disk. The torus drags angular momentum from the binary and at the same time it is slowly pushed outwards by the radiation pressure on dust. The leftovers become a Keplerian disk. Only small part of the ejecta is pulled into Keplerian motion, so the angular momentum removal from the central binary is moderate and the orbital period can stay as long as a few hundred days. Point 2. in this sequence deserves particular consideration. At this stage the companion resides within the wind acceleration zone which is governed by the dust condensation radius, $R_{\rm cond}$. On TP-AGB $R_{\rm cond}$ is 2–5 $R_{*}$ (Gail & Sedlmayr 1988). 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--- abstract: 'Fermionic phase space representations are a promising method for studying correlated fermion systems. The fermionic Q-function and P-function have been defined using Gaussian operators of fermion annihilation and creation operators. The resulting phase-space of covariance matrices belongs to the symmetry class D, one of the non-standard symmetry classes. This was originally proposed to study mesoscopic normal-metal-superconducting hybrid structures, which is the type of structure that has led to recent experimental observations of Majorana fermions. Under a unitary transformation, it is possible to express these Gaussian operators using real anti-symmetric matrices and Majorana operators, which are much simpler mathematical objects. We derive differential identities involving Majorana fermion operators and an antisymmetric matrix which are relevant to the derivation of the corresponding FokkerPlanck equations on symmetric space. These enable stochastic simulations either in real or imaginary time. This formalism has direct relevance to the study of fermionic systems in which there are Majorana type excitations, and is an alternative to using expansions involving conventional Fermi operators. The approach is illustrated by showing how a linear coupled Hamiltonian as used to study topological excitations can be transformed to Fokker-Planck and stochastic equation form, including dissipation through particle losses.' address: - '$^{1}$ Centre for Quantum and Optical Science, Swinburne University of Technology, Melbourne 3122, Australia' - '$^{2}$ Centro de Investigaciones en Óptica A.C., León, Guanajuato 37150, México' author: - 'Ria Rushin Joseph$^{1}$, Laura E C Rosales-Zárate$^{1,2}$ and Peter D Drummond$^{1}$' title: Phase space methods for Majorana fermions --- Introduction ============ Due to the complexity arising from Pauli’s exclusion principle, Fermi statistics are a challenging topic in theoretical physics. As a result, new approaches to understanding fermion physics are important. While ideas from coherence theory play an important role in the physics of bosons, they have not yet been widely used to study fermions. Probabilistic methods are especially interesting, as they have the promise that they can be efficiently sampled. Here we show, by using fermionic Q-functions, that first order Fokker Plank equations can be obtained both for linear coupling and for dissipation. This makes simulations possible in regimes where topological excitations and Majorana fermions are expected to occur. The control acquired in experiments of ultra-cold fermionic atomic systems opens new areas that were not explored previously, including Majorana fermions [@Majorana:1937; @Alicea:2012; @Elliott:2015] and the unitary limit of strong interactions [@Bloch_RMP_2008_ManyBodyPhys_UG]. These complement studies of strongly correlated fermions in condensed matter physics, which are now investigating such exotic types of fermionic excitation as well. In particular, there has been a recent resurgence of interest in the work of Majorana [@Majorana:1937]. As well as the question of fundamental Majorana fermions, which is still not completely resolved, there is a growing interest in Majorana-like excitations and quasi-particles in condensed matter physics [@fu2008superconducting]. There have been several claims of experimental observations, typically in environments involving coupling of semiconductors to superconductors [@Mourik:2012; @Nadj-Perge:2014; @Xu:2015; @Rokhinson:2012]. Recent reviews regarding Majorana fermions focus on their possible observations in nuclear, particle and solid states physics [@Elliott:2015; @Beenaker:2013] or their possible relation to Random Matrix Theory [@Beenakker:2015]. Some of this work is also closely related to proposals for topological insulators and quantum computing [@DasSarma:2015]. While this development is still speculative [@Kitaev:2001; @Stern:2013], it does reveal an important issue. How can one evaluate topological or related quantum technology proposals? Many studies of this type omit crucial issues such as decoherence mechanisms. To understand decoherence and fidelity, which are essential to quantum technology applications, a method for treating such Hamiltonian terms theoretically is needed. Neither mean field theory nor perturbation theory is sufficient to treat these deeply quantum-mechanical systems. The use of exact methods on an orthogonal basis lacks scalability; thus, other methods will be needed. The general field of Majorana fermionic excitations in condensed matter systems is growing [@Alicea:2012]. The common problem that occurs theoretically is that if one wishes to go beyond mean field theory, then either one must work in a restricted Hilbert space of small dimension, or else there are sign issues from using Monte-Carlo methods. Recent methods in this direction, intended to overcome the sign problem, include using the symmetry of the Hamiltonian properties of Lie groups and Lie algebras [@Wang:2015] or treating fermionic systems interacting with spins [@Huffman:2016]. A different sign free approach is to consider the Majorana representation together with Quantum Monte Carlo simulations [@Li:2015; @Li:2016; @Wei:2016]. Finally we point out that extremely powerful methods for classical simulation of fermion chains with linear couplings are known to exist [@Terhal:2002; @Bravyi2012; @Pedrocchi:2015], originally based on work by Valiant [@Valiant:2001], although this technique has not been used to treat particle loss or gain. Just as the use of a coherent state basis and the corresponding phase-space methods proved essential to understanding laser physics [@WeidlichRiskenHaken1967], it is useful to develop a similar coherent formalism for Majorana physics. Phase space representations have been widely used in quantum optics [@Hillery_Review_1984_DistributionFunctions; @Gardiner_Book_QNoise] and ultracold atoms [@Blakie:2008; @Polkovnikov_2010_Phase_Rep_QDyn] in order to study dynamics. The advantage of using phase-space methods is that it is possible to obtain an exact Fokker Planck equation from the time evolution equation of the density operator. The Fokker Planck equation can be readily transformed to a stochastic differential equation [@Carmichael:1999_Book; @Gardiner_Book_SDE]. An advantage of this method is that it is possible to sample the stochastic equation probabilistically, allowing one to perform quantum dynamical calculations for large number of modes and particles [@drummond2016quantum]. These methods can be used to treat dissipation and losses. Such processes occur in many physical systems and represent, for example, the loss of atoms from a trap in the case of magnetically trapped atoms. For electrons they may originate in the transfer of electrons from one band to another. In general, such physical processes are part of the quantum dynamics, and are described by the use of master equations. Since these represent non-unitary processes on the subsystem of interest, they require appropriate techniques able to represent the entire density matrix. This is shown here through a simple example. The purpose of these approaches is to enable the use of probabilistic methods, which can overcome the exponential growth problem of large many-body Hilbert spaces. An example is the positive Q-function. For this representation, there have been applications using the SU(2) coherent states [@Arecchi_SUN; @Gilmore:1975], including simulations of Bell violations [@ReidqubitPhysRevA.90.012111]. Another application of the dynamics of the Q-function was in the study of quantum correlations of a degenerate optical parametric oscillator [@Zambrini2003]. These methods have been recently extended to the fermionic raising and lowering operators, resulting in fermionic analogs of the normally ordered P-function and the anti-normally ordered Q-function for bosons. Such phase space representations use as a basis the Gaussian operators [@Corney_PD_JPA_2006_GR_fermions; @Corney_PD_PRB_2006_GPSR_fermions; @Corney_PD_PRL2004_GQMC_ferm_bos], which are exponential of quadratic forms of creation and annihilation operators. A related concept is the fermionic Gaussian state, for which the density matrix is an exponential of quadratic forms of operators defined by a correlation matrix. Some applications of the fermionic Gaussian operators include the study of the Fermi Hubbard model [@Imada_2007_GBMC; @Corboz_Chapter_PhaseSpaceMethodsFermions; @Corney_PD_PRL2004_GQMC_ferm_bos], and fermionic molecular dissociation [@Ogren:2010_Qdynamics_fermions_MolDiss; @Ogren2011_fermiondynamics]. It was proved recently that the fermionic Q-function exists as a complete probabilistic representation [@FermiQ]. Hence, it is important to develop this method further. Earlier theoretical work used a Gaussian basis of Fermi operators [@Corney_PD_JPA_2006_GR_fermions; @Corney_PD_PRB_2006_GPSR_fermions; @Corney_PD_PRL2004_GQMC_ferm_bos]. By transforming the basis in terms of Majorana operators, the resulting differential identities can help to treat Majorana physics also. The fermionic Gaussian operators that we consider belong to the non-standard symmetry class D defined by Altland and Zirnbauer [@Altland_Zirnbauer:1997]. This non-standard symmetry class was originally proposed to study mesoscopic normal-metal-superconducting hybrid structures, which is exactly the type of structure that has led to recent experimental observations of Majorana quasi-particles. We show that, under a unitary transformation, we can write the class D operators in terms of an anti-symmetric matrix and Majorana variables [@Majorana:1937], and obtain a positive phase-space representation. Since the distribution is now defined in terms of the Majorana operators, we expect it will be useful in investigating the recently discovered Majorana fermionic quasi-particles in hybrid superconductor or ultra-cold atomic systems [@Mourik:2012; @Nadj-Perge:2014; @Liu:2012]. Other symmetries can also be used to identify classes of operator transformations, and to reduce the resulting dimensionality. These are not treated here, but allow this work to be extended. There has been much work exploring the connection of Class D symmetry with topological and Majorana-like excitations [@Beenaker:2013; @Lutchyn:2010; @Bagrets:2012; @DeGottardi:2013; @Gibertini:2013; @Morimoto:2013; @Budich:2013; @Chiu:2016; @Hegde:2016]. This relationship therefore has a strong physical underpinning. Moreover, the use of conventional methods such as mean field theory is not applicable when there are strong quantum correlations. Therefore, our approach is to investigate other mathematical relations involving differential identities, that can help these investigations in future. These identities allow the symmetry class to be utilized in the operator representation. The end result is that one can treat exponentially complex Hilbert spaces as dynamical equations in a continuous phase-space, which are generally simpler to solve. In this communication, we derive differential identities for the Gaussian phase space representation using Majorana operators. These identities are relevant to the general use of a Gaussian basis and to the fermionic Q and P-function. Differential identities allow simulation of the time evolution either in real or imaginary time. The latter approach has already been used to investigate the ground state of the Fermi Hubbard model [@Imada:2007_2DHM_Superconductivity; @Imada_2007_GBMC]. Thus, these differential identities are potentially relevant to the dynamics of Majorana systems or to systems that have Majorana operator excitations. We expect that using methods based on Majorana operators will further help to study other Fermi systems as well, because the class D symmetry has a fundamentally simpler form using these operators. This paper is organized as follows: In section \[sec:Gaussian-Operators\] we review the Gaussian phase space representation as well as we introduced the Majorana Gaussian operators. Section \[sec:Majorana-differential-properties\] presents our main result, which is the differential identities for the Majorana operators. These are given for the un-normalized and normalized form of the Majorana Gaussian operator. We obtain identities for both ordered and unordered Majorana operator products. In section \[sec:Phase-space-representations\] we present the corresponding phase space representations with Fermi and Majorana operators. We present the time evolution equation for the Majorana Q-function for one of these differential equations in section \[sec:Time-Evolution-MQf\]. In section \[sec:TimeEvOpenSyst\], we discuss the time evolution of the Q-function for an open quantum system. Finally, a summary of our results and conclusions are given in section \[sec:Summary\_Conclusion\]. Fermionic Gaussian operators\[sec:Gaussian-Operators\] ====================================================== Fermionic phase space representations that use a phase-space of Grassmann variables are known [@Cahill_Glauber_fermions_1999]. While this is useful for analytic calculations, it is an exponentially complex problem to directly represent the Grassmann variables computationally. Therefore, in order to avoid such exponential complexity, it is helpful to map the fermion problem into a phase-space of covariances, which are more computationally tractable. One method to achieve this is to use the fermionic Gaussian operators as a basis [@Corney_PD_JPA_2006_GR_fermions; @Corney_PD_PRB_2006_GPSR_fermions; @ResUnityFGO:2013; @FermiQ]. In this case, the phase space is a space of covariance matrices, which are complex variables. We emphasise that through using identities, any correlation order can be calculated, including the extremely high orders that can occur in some quantum technology and mesoscopic physics applications [@Opanchuk:2014; @ReidqubitPhysRevA.90.012111]. An alternative method is to derive the covariance matrix equations of motion from Grassmann variable equations [@dalton2016grassmann], which is not treated here. In this section, we review the properties of the fermionic Gaussian operators in the notation used in the paper. These were first introduced as variational BCS states [@bogolyubov1958zh; @valatin1958comments], and their algebraic properties as operator transformations have been studied extensively [@Balian_Brezin_Transformations; @Altland_Zirnbauer:1997]. We consider a fermionic system that can be decomposed into $M$ modes, where $\hat{\bm{a}}$ is defined as a vector of $M$ annihilation operators, while $\hat{\bm{a}}^{\dagger}$ is the corresponding vector of $M$ creation operators. The Fermi operators, $\hat{a}_{i}$ and $\hat{a}_{j}^{\dagger}$, obey the usual anti commutation relations: $$\left\{ \hat{a}_{i},\hat{a}_{j}^{\dagger}\right\} =\delta_{ij},\qquad\left\{ \hat{a}_{i},\hat{a}_{j}\right\} =0.\label{eq:FerComm}$$ The single mode number operator is $\hat{n}=\widehat{a}^{\dagger}\widehat{a}$, and the action of the operators on the single-mode number states $\mid n\rangle$ is that ${\displaystyle \widehat{a}^{\dagger}\mid n\rangle=}\left(1-n\right)|n+1\rangle$, and ${\displaystyle \widehat{a}\mid n\rangle=}n|n-1\rangle$. Using the Fermi operators, we define a $2M$ extended vector of creation and annihilation operators $\hat{\underline{a}}=\left(\hat{\bm{a}}^{T},\hat{\bm{a}}^{\dagger}\right)^{T},$ with the corresponding adjoint vector defined as $\hat{\underline{a}}^{\dagger}=\left(\hat{\bm{a}}^{\dagger},\hat{\bm{a}}^{T}\right)=\left(\hat{a}_{1}^{\dagger},\ldots,\hat{a}_{M}^{\dagger},\hat{a}_{1},\ldots,\hat{a}_{M}\right)$. We use the following notation throughout the paper: $M$ vectors are in bold type as $\hat{\bm{a}}$, $2M$ vectors are denoted with a single underline as $\hat{\underline{a}}$, $M\times M$ matrices are in bold type as $\mathbf{I}$, and $2M\times2M$ matrices are denoted with a double underline as $\underline{\underline{I}}$. The un-normalized fermionic Gaussian operators $\hat{\Lambda}_{f}^{u}$ are defined here [@Corney_PD_JPA_2006_GR_fermions; @Corney_PD_PRB_2006_GPSR_fermions] as the set of all possible normally-ordered Gaussian functions of these $2M$ dimensional operator vectors, $$\hat{\Lambda}_{f}^{u}\Biggl(\underline{\underline{\mu}}\Biggr)=:\exp\Biggl[-\hat{\underline{a}}^{\dagger}\underline{\underline{\mu}}\hat{\underline{a}}/2\Biggr]:.$$ Similar definitions were subsequently introduced elsewhere [@Kraus:2009; @Eisler:2015], except without normal ordering. Throughout the paper, normal ordering applies to the annihilation and creation operators, and is denoted by $:\ldots:$. This includes a sign change for every fermion operator swap, hence $:\hat{a}_{i}\hat{a}_{j}^{\dagger}:=-\hat{a}_{j}^{\dagger}\hat{a}_{i}$. Anti-normal ordering is denoted by $\left\{ \ldots\right\} $, hence $\left\{ \hat{a}_{j}^{\dagger}\hat{a}_{i}\right\} =-\hat{a}_{i}\hat{a}_{j}^{\dagger}$. We will also use nested orderings [@Corney_PD_JPA_2006_GR_fermions]. In this case the outer ordering does not reorder the inner one, but the sign changes still take place for each swap - for example $\left\{ :\hat{O}\hat{b}_{i}^{\dagger}:\hat{b}_{j}\right\} =-\hat{b}_{j}\hat{b}_{i}^{\dagger}\hat{O}$. A convenient notation is to introduce a basis of unit trace fermionic operators $\hat{\Lambda}_{f}$, where: $$\hat{\Lambda}_{f}=\sqrt{\det\left[\rmi\underline{\underline{\sigma}}\right]}:\exp\left[-\hat{\underline{a}}^{\dagger}\left(\underline{\underline{\sigma}}^{-1}-2\underline{\underline{J}}\right)\hat{\underline{a}}/2\right]:\,.\label{eq:NormalizedGO_sigma}$$ Here $\underline{\underline{J}}$ is a matrix square root of the identity: $$\underline{\underline{J}}=\left[\begin{array}{cc} -\mathbf{I} & \mathbf{0}\\ \mathbf{0} & \mathbf{I} \end{array}\right].\label{eq:I_GO}$$ The importance of expressing the Gaussian operators in this form is that these Gaussian operators using annihilation and creation operators are the basis for the fermionic Q and P-functions. The matrices $\mathbf{0}$ and $\mathbf{I}$ are the $M\times M$ zero and identity matrices, respectively, and $\underline{\underline{\sigma}}=\left(\underline{\underline{\mu}}+2\underline{\underline{J}}\right)^{-1}$ is the *covariance* matrix, which can be expressed in terms of an $M\times M$ matrix $\mathbf{n}$ and a complex antisymmetric $M\times M$ matrix $\mathbf{m}$: $$\underline{\underline{\sigma}}=\left(\underline{\underline{\mu}}+2\underline{\underline{J}}\right)^{-1}=\left[\begin{array}{cc} \mathbf{n}^{T}-\mathbf{I} & \mathbf{m}\\ \mathbf{m}^{+} & \mathbf{I}-\mathbf{n} \end{array}\right].\label{eq:Sigma-decomposition}$$ We use the notation $\mathbf{n}^{T}$ to indicate a transpose, $\mathbf{m}^{*}$ a complex conjugate, and $\mathbf{m}^{\dagger}$ a conjugate transpose. The most general fermionic Gaussian operator is non-hermitian. In this case $\mathbf{n}$ is not hermitian, and $\mathbf{m}^{+}$, $\mathbf{m}$ are independent antisymmetric complex matrices. To obtain hermitian Gaussian operators, one must impose the additional restrictions that $\mathbf{m}^{+}=\mathbf{m}^{\dagger}$ and $\mathbf{n}=\mathbf{n}^{\dagger}$ [@Corney_PD_PRB_2006_GPSR_fermions]. The $\bm{n}$ and $\bm{m}$ notation is used because these terms in the covariance matrix of a hermitian Gaussian density matrix are equal to the normal and anomalous Green’s functions, which are ubiquitous in fermionic many-body theory. We note that the covariance matrix correspond to the first-order correlations of the Gaussian operators in normally ordered form, since $\Tr\left[:\hat{\underline{a}}\hat{\underline{a}}^{\dagger}\hat{\Lambda}_{f}\right]=\underline{\underline{\sigma}}$ [@Corney_PD_JPA_2006_GR_fermions]. The Gaussian operators have a more widespread utility than just their obvious value in treating Gaussian states, which are the ground state solutions of quadratic Hamiltonians. The general fermionic Gaussian operators are a complete basis for all fermionic density matrices, including those that are not Gaussian, through taking linear combinations of Gaussian operators. On the other hand, the hermitian fermionic Gaussian operators can be regarded as physical density matrices by themselves. Several nonstandard symmetry classes were defined by Altland and Zirnbauer [@Altland_Zirnbauer:1997]. Here we focus on Class D, which corresponds to cases which do not have time-reversal or spin-rotation symmetry. The general hermitian Gaussian operators have the nonstandard Class D symmetry [@FermiQ]. Majorana Gaussian operators \[subsec:Majorana-Gaussian-operators\] ------------------------------------------------------------------ The work of Majorana [@Majorana:1937] showed that a more symmetric form of quantization was obtainable if the Fermi operators are transformed to an hermitian Majorana fermion operator basis. We use the normalization that is most common in modern papers, where the Majorana operators are obtained using a matrix transformation of the usual Fermi annihilation and creation operators [@Elliott:2015], $$\hat{\underline{\gamma}}=\left[\begin{array}{c} \hat{\bgamma}_{(1)}\\ \hat{\bgamma}_{(2)} \end{array}\right]=\underline{\underline{U_{0}}}\hat{\underline{a}}\,\,.$$ This uses a matrix $\underline{\underline{U_{0}}}$, given by [@Balian_Brezin_Transformations]: $$\underline{\underline{U_{0}}}=\left[\begin{array}{cc} \mathbf{I} & \mathbf{I}\\ -\rmi\mathbf{I} & \rmi\mathbf{I} \end{array}\right],\label{eq:U0}$$ which is not unitary, since $\underline{\underline{U_{0}}}^{\dagger}=2\underline{\underline{U_{0}}}^{-1}$, although $\underline{\underline{U_{0}}}/\sqrt{2}$ is unitary. This definition corresponds to introducing hermitian quadrature operators, $\hat{\bgamma}_{(i)}$, as: $$\begin{aligned} \hat{\bgamma}_{(1)} & = & \left(\hat{\bm{a}}+\hat{\bm{a}}^{\dagger}\right)\nonumber \\ \hat{\bgamma}_{(2)} & = & -\rmi\left(\hat{\bm{a}}-\hat{\bm{a}}^{\dagger}\right),\label{eq:p-var}\end{aligned}$$ which implies that the $\hat{\gamma}$ variables have the following anti-commutation relation [@Elliott:2015; @Beenaker:2013; @Beenakker:2015].: $$\left\{ \hat{\gamma}_{i},\hat{\gamma}_{j}\right\} =2\delta_{ij}.\label{eq:anti-com_MajoranaOp}$$ The relationship can be inverted, and the extended ladder operators can be expressed as: $$\widehat{\underline{a}}=\underline{\underline{U_{0}}}^{-1}\hat{\underline{\gamma}}.\label{eq:OpMajUn}$$ Using the transformation $\underline{\underline{U_{0}}}$ it is possible to write an un-normalized Gaussian operator as: $$\hat{\Lambda}^{u}\Biggl(\underline{\underline{Y}}\Biggr)=:\exp\Biggl[\frac{\rmi}{2}\hat{\underline{\gamma}}^{T}\underline{\underline{Y}}\hat{\underline{\gamma}}\Biggr]:.\label{eq:GOY}$$ We will call the above operators the un-normalized Majorana Gaussian operators. Here we have defined: $$\underline{\underline{Y}}=\frac{\rmi}{2}\underline{\underline{U_{0}}}\underline{\underline{\mu}}\underline{\underline{U_{0}}}^{-1}.\label{eq:YMat}$$ Provided that $\underline{\underline{\mu}}$ has the class D symmetry properties identified above, the matrix $\underline{\underline{Y}}$ is a real antisymmetric matrix. In the single mode case, one can define $Y\equiv Y_{12}$, and since all normally ordered powers beyond first order will vanish, $$\hat{\Lambda}^{u}\Biggl(Y\Biggr)=1+iY:\hat{\gamma}_{1}\hat{\gamma}_{2}:=1+2\hat{n}Y\,.$$ Unit trace Gaussian operators ----------------------------- We would like to express the unit trace Gaussian operators in terms of Majorana operators, together with a matrix $\underline{\underline{X}}$ that plays the role of a Majorana covariance. Covariance matrices and Majorana variables have been used in order to study topological edge states correlations in free and interacting fermion systems [@Meichanetzidis:2016]. They have also been used to study pairing in fermionic systems [@Kraus:2009]. For this purpose, we define the following real antisymmetric matrices: $$\underline{\underline{X}}=\rmi\underline{\underline{U_{0}}}\left[\underline{\underline{J}}-2\underline{\underline{\sigma}}\right]\underline{\underline{U_{0}}}^{-1},\label{eq:XMat}$$ and an antisymmetric real ’identity-like’ matrix $$\underline{\underline{\mathcal{I}}}=\rmi\underline{\underline{U_{0}}}\underline{\underline{J}}\underline{\underline{U_{0}}}^{-1},\label{eq:UnitFI}$$ which can be written explicitly as; $$\underline{\underline{{\cal I}}}=\left[\begin{array}{cc} \mathbf{0} & \mathbf{\mathbf{I}}\\ -\mathbf{I} & \mathbf{0} \end{array}\right].$$ Equation (\[eq:Sigma-decomposition\]) gives a relationship between the variance matrix $\underline{\underline{\sigma}}$ and the $\underline{\underline{\mu}}$ matrix. In terms of the corresponding antisymmetric matrices $\underline{\underline{X}}$ and $\underline{\underline{Y}}$, this relationship is given by: $$\underline{\underline{X}}=\underline{\underline{\mathcal{I}}}+\left(\underline{\underline{Y}}+\underline{\underline{\mathcal{I}}}\right)^{-1}.\label{eq:RelXY}$$ In the single-mode case, one has that: $$\underline{\underline{X}}=\left[\begin{array}{cc} 0 & 1-1/\left(1+Y\right)\\ 1/\left(1+Y\right)-1 & 0 \end{array}\right].$$ In order to define the normalized Gaussian operators using Majorana operators, we first consider the normalization factor. Since the determinant is invariant under unitary transformations we can write it as follows: $$\sqrt{\det\left[\rmi\underline{\underline{\sigma}}\right]}=\frac{1}{2^{M}}\sqrt{\det\left[\underline{\underline{\mathcal{I}}}-\underline{\underline{X}}\right]}$$ Therefore the unit-trace Majorana Gaussian operator in a variance form is: $$\hat{\Lambda}\left(\underline{\underline{X}}\right)=N\left(\underline{\underline{X}}\right):\exp\Biggl[\textcolor{blue}{-}\rmi\hat{\underline{\gamma}}^{T}\left[\underline{\underline{\mathcal{I}}}-\left(\underline{\underline{X}}-\underline{\underline{\mathcal{I}}}\right)^{-1}\right]\hat{\underline{\gamma}}/2\Biggr]:,\label{eq:MajOpX}$$ where $$N\left(\underline{\underline{X}}\right)=\frac{1}{2^{M}}\sqrt{\det\left[\underline{\underline{\mathcal{I}}}-\underline{\underline{X}}\right]}.\label{eq:NormGOX}$$ In the single-mode case, with $X=X_{12},$ the normalization factor is: $$N\left(X\right)=\frac{1}{2(1+Y)}=\frac{1-X}{2},$$ where the domain boundaries are such that $-1<X<1$, and: $$\hat{\Lambda}\left(X\right)=\frac{1-X}{2}+\hat{n}X\,.$$ We note that there is a particle-hole symmetry, since $\hat{\Lambda}\left(X\right)$ is invariant under the transformation $\hat{a}\rightarrow\hat{a}^{\dagger}$, $\hat{a}^{\dagger}\rightarrow\hat{a}$, $X\rightarrow-X$. Classical domains ------------------ After transformation to Majorana form, there are two possible fermionic representations with different phase-spaces. The fermionic P-representation corresponds to the complex antisymmetric matrices in Majorana space, while the Q-representation corresponds to the real antisymmetric matrices. As we show explicitly below, the natural phase-space of the Majorana Q-function is bounded. This is because the Gaussian basis that is used in the construction of the Q-function depends on an antisymmetric matrix $\underline{\underline{X}}$, which exists in one of the irreducible bounded symmetric domains of Cartan [@Cartan:1926; @Cartan:1927Bull; @Cartan:1935]. In the complex case, these symmetric spaces are called the classical domains by Hua [@Hua_Book_harmonic_analysis; @Xu:2015]. The Gaussian basis has deep links to Lie group theory, since we can define a $2M\times2M$ matrix $\underline{\underline{T}}=\exp\left[\underline{\underline{X}}\right]$ such that $\underline{\underline{T}}^{T}\underline{\underline{T}}=\underline{\underline{I}}$, where $\underline{\underline{I}}$ is the usual identity matrix. The $\underline{\underline{T}}$ matrices belong to the orthogonal group $O(2M)$, which is a Lie group. In the case that $\det\underline{\underline{T}}=1,$ the group is $SO(2M)$. The corresponding Lie algebra of this Lie group is isomorphic to the real $2M\times2M$ antisymmetric matrices. The classical domains  [@Cartan:1935] are given in terms of complex matrices $\underline{\underline{Z}}$. This leads to a consideration of the four classical domains, which in the classification of Hua [@Hua_Book_harmonic_analysis] are defined as [@Helgason_book_LieGroup; @Caselle_Magnea:2004_ReviewRMT_SymSpaces; @Cartan:1927Bull; @XuComplexDom]: 1. The domain $\mathcal{R}_{I}$ of $m\times n$ complex matrices with: $\underline{\underline{I}}^{(m)}-\underline{\underline{Z}}\underline{\underline{Z}}^{\dagger}>0.$ 2. The domain $\mathcal{R}_{II}$ of $n\times n$ symmetric complex matrices with: $\underline{\underline{I}}-\underline{\underline{Z}}\underline{\underline{Z}}^{*}>0.$ 3. The domain $\mathcal{R}_{III}$ of $n\times n$ skew-symmetric (anti-symmetric) complex matrices with: $\underline{\underline{I}}+\underline{\underline{Z}}\underline{\underline{Z}}^{*}>0.$ 4. The domain $\mathcal{R}_{IV}$ of $n$-dimensional vectors $z=\left(z_{1,}z_{2},\ldots,z_{n}\right),$ where $z_{k}$ are complex numbers, satisfying: $\left|zz^{T}\right|+1-2zz^{T}>0,\qquad\left|zz^{T}\right|<1.$ The most general Majorana Gaussian operators correspond to the complex domain $\mathcal{R}_{III}$, for $n=2M$, which we refer to as $\mathcal{D}_{C}.$ The subspace that corresponds to the hermitian Hamiltonians that we use here has a real phase space of $M(2M-1)$ dimensions, or $\mathcal{D}_{R}$, which is the real subspace of $\mathcal{R}_{III}$ . In our notation, the classical domain boundary of the space of the most general complex antisymmetric matrices $\underline{\underline{Z}}$ considered here is given by: $$\underline{\underline{R}}=\underline{\underline{I}}+\underline{\underline{Z}}\underline{\underline{Z}}^{*}>0.\label{eq:BoundXM}$$ Since $\underline{\underline{R}}$ is positive-definite, and $\underline{\underline{Z}}\underline{\underline{Z}}^{*}=-\underline{\underline{Z}}\underline{\underline{Z}}^{\dagger}$ is negative definite, we deduce that each element of $\underline{\underline{Z}}$ is bounded, which implies that $\sum_{j}\left|Z_{ij}\right|^{2}<1$ and $\left|Z_{ij}\right|<1$. The classical domain boundary of the real antisymmetric matrices $\underline{\underline{X}}$ is the same, except for the additional requirement that the elements are real. This restriction of the domain $\mathcal{R}_{III}$ corresponds to the hermitian Cartan symmetric space DIII [@Cartan:1926; @Cartan:1927Bull]. Altland and Zirnbauer [@Altland_Zirnbauer:1997] classified non-standard symmetry classes for fermionic systems. Here the classification is made by considering the Lie algebra which the symmetry class belongs to [@Altland_Zirnbauer:1997; @HeinznerZirnbauer:2005_SymmetryClasses_disorder]. In this notation, their symmetry class $D$ corresponds to $D_{N}\equiv so(2N)$ [@Altland_Zirnbauer:1997; @Caselle_Magnea:2004_ReviewRMT_SymSpaces; @HeinznerZirnbauer:2005_SymmetryClasses_disorder]. The Lie algebra is isomorphic to the real antisymmetric $2N\times2N$ matrices. The boundary condition can also be written in the real case as: $$\underline{\underline{X}}^{+}\underline{\underline{X}}^{-}>0,$$ where we define for convenience, $$\underline{\underline{X}}^{\pm}\equiv\underline{\underline{X}}\pm\underline{\underline{\mathcal{I}}}.\label{eq:Xpm}$$ For the unordered operator identities derived below, it is simpler to transform the resulting phase-space variable to a modified form such that: $$\underline{\underline{x}}=\underline{\underline{{\cal I}}}\underline{\underline{X}}^{T}\underline{\underline{{\cal I}}}.\label{eq:x}$$ We also introduce $\underline{\underline{x}}^{\pm}=\underline{\underline{x}}\pm\rmi\underline{\underline{I}}$, and one can prove that in this case the classical domain bounds are unchanged, with boundaries at: $$\underline{\underline{I}}+\underline{\underline{x}}^{2}=\underline{\underline{x}}^{+}\underline{\underline{x}}^{-}>0\,.$$ Resolution of identity ---------------------- The definition of a Q-function first requires one to have an expression for the resolution of the Hilbert space identity operator. In order to obtain the most general form of the resolution of identity in the fermion case, we define $\hat{\Lambda}^{N}\left(\underline{\underline{X}}\right)$ as the following Majorana Gaussian basis: $$\begin{aligned} \hat{\Lambda}^{N}\left(\underline{\underline{X}}\right) & = & \frac{1}{{\cal N}}\hat{\Lambda}\left(\underline{\underline{X}}\right)S\left(\underline{\underline{X}}^{2}\right),\label{eq:MajGBasis}\end{aligned}$$ with $\hat{\Lambda}\left(\underline{\underline{X}}\right)$ given in (\[eq:MajOpX\]), $S\left(\underline{\underline{X}}^{2}\right)$ is an even, positive scaling function, and $\mathcal{N}$ is a normalization constant that ensures that the identity expansion is normalized. We consider the following form for the scaling factor $S$, which vanishes at the phase-space boundaries: $$S=\det\left[\underline{\underline{I}}+\underline{\underline{X}}^{2}\right]^{k/2}.$$ Following the methods developed earlier [@ResUnityFGO:2013] for the complex variable Q-function, one can then prove that: $$\hat{I}=\int_{\mathcal{D}_{R}}\hat{\Lambda}^{N}\left(\underline{\underline{X}}\right)\rmd X\,,$$ where the integration is over the bounded real classical domain of antisymmetric matrices, and $dX=\prod_{1\text{\ensuremath{\le}}j<k\text{\ensuremath{\le}}2M}dX_{jk}.$ The value of the normalization constant is obtained by noting that: $${\cal N}=2^{-M}\int_{\mathcal{D}_{R}}S\left(\underline{\underline{X}^{2}}\right)\rmd X\,.$$ In the limit that $k\rightarrow0$, the constant $\mathcal{N}$ is related to the volume of the real classical Hua type $III$ domain [@Hua_Book_harmonic_analysis; @Mehta_RM_book; @Forrester_book:2010] using matrix polar coordinate methods, and is given by: $$\begin{aligned} \mathcal{N} & = & \frac{\pi^{M\left(M-1/2\right)}}{2^{M}}\prod_{j=1}^{M}\frac{\Gamma\left(k+j\right)}{\Gamma\left(k+M+j-\frac{1}{2}\right)}.\label{eq:normalization factor-1}\end{aligned}$$ Thus, for example, in the case of $M=1$, one obtains $\mathcal{N}=1$. We note that these results hold equally well when using the transformed variable $\underline{\underline{x}}$, which has an identical measure, determinant and integration domain. In the single mode case, the two forms are exactly the same. Majorana differential identities\[sec:Majorana-differential-properties\] ======================================================================== The main result of the paper is the derivation of differential identities using Majorana operators and antisymmetric matrices. It is useful to derive differential identities for the normalized Majorana Gaussian operators in terms of the matrices $\underline{\underline{X}}$, because they are related to the covariance matrices $\underline{\underline{\sigma}}$. The fermionic Q-function uses as a basis the Gaussian operators expressed in a symmetric form of these variables. Calculations using the Q-function require differential identities which will involve these variables. Thus, these identities allow one to perform simulations with fermionic phase space representations. We wish to write differential identities for the action of Majorana operators on the unit-trace Gaussian basis defined in (\[eq:MajOpX\]). There is a correspondence between Majorana operators and Fermi operators, hence we derive four differential identities which are written using normal, anti-normal ordering and mixed products of normal and anti-normal ordering. In order to derive the identities for the normalized Gaussian operators it is convenient to first derive the differential identities for the unnormalized Majorana operators. The next step is to perform a change of variables and use the corresponding normalization factor with its respectively derivative. The differential identities are given below and the details of this procedure are explained in \[sec:AppendixMajoranaDiffId\], \[sec:AppendixDiffIdNormGO\] and \[sec:AppendixDiffIdunorderedGO-1\]. Un-normalized Majorana differential identities \[subsec:Un-normalized-MajDifY\] ------------------------------------------------------------------------------- In order to derive the differential identities for the normalized Gaussian operators we need to derive first the corresponding differential identities for the un-normalized Gaussian operators given in (\[eq:GOY\]). These identities are needed in order to obtain the differential identities in terms of the antisymmetric matrix $\underline{\underline{X}}$. This matrix is related to the covariance matrix $\underline{\underline{\sigma}}$, through the expression given in (\[eq:XMat\]). We use the convention for matrix derivatives that: $$\left[\frac{\rmd}{\rmd\underline{\underline{Y}}}\right]_{\mu\nu}=\frac{\rmd}{\rmd Y_{\nu\mu}}\,.$$ These identities are given in terms of the different forms of the possible orderings between the Majorana variables and the Gaussian operators and are listed below. The corresponding proofs are shown in \[sec:AppendixMajoranaDiffId\]. - Mixed products: $$\Biggl\{\hat{\underline{\gamma}}:\hat{\underline{\gamma}}^{T}\hat{\Lambda}^{\left(u\right)}:\Biggr\}=\rmi\left[-\underline{\underline{{\cal I}}}\left(2\underline{\underline{Y}}+\underline{\underline{\mathcal{I}}}\right)\frac{\rmd}{\rmd\underline{\underline{Y}}}+2\underline{\underline{{\cal I}}}\right]\hat{\Lambda}^{\left(u\right)}.\label{eq:MajDiffIdY}$$ The explicit form of $\Bigl\{\hat{\underline{\gamma}}:\hat{\underline{\gamma}}^{T}\hat{\Lambda}:\Bigr\}$, in terms of the creation and annihilation operators is given in \[sec:AppendixFormOrdering\]. - Normally ordered products: $$:\hat{\underline{\gamma}}\hat{\underline{\gamma}}^{T}\hat{\Lambda}^{\left(u\right)}:=\rmi\frac{d}{d\underline{\underline{Y}}}\hat{\Lambda}^{\left(u\right)}.\label{eq:UnNormNormalDifId}$$ For clarity the detailed expression is given in \[sec:AppendixFormOrdering\]. - Anti-normally ordered products: $$\begin{aligned} \left\{ \underline{\widehat{\gamma}}\widehat{\underline{\gamma}}^{T}\hat{\Lambda}^{\left(u\right)}\right\} & = & \rmi\underline{\underline{\mathcal{I}}}\left\{ \left(2\underline{\underline{Y}}+\underline{\underline{\mathcal{I}}}\right)\frac{\rmd}{\rmd\underline{\underline{Y}}}\hat{\Lambda}^{u}-2\hat{\Lambda}^{u}\right\} \left(2\underline{\underline{Y}}+\underline{\underline{\mathcal{I}}}\right)\underline{\underline{\mathcal{I}}}.\label{eq:AntiNormDifIdUnGO}\end{aligned}$$ The full expression is given in \[sec:AppendixFormOrdering\]. Normalized Majorana differential identities\[subsec:Normalized-Majorana-DifId\] ------------------------------------------------------------------------------- In this section we will give the differential identities for the normalized Gaussian operators given in (\[eq:NormGOX\]). We note that in order to obtain these identities, we need to perform a change of variables from the matrices $\underline{\underline{Y}}$ to the $\underline{\underline{X}}$ using the relationship given in (\[eq:RelXY\]) hence: $$\underline{\underline{Y}}=\left(\underline{\underline{X}}-\underline{\underline{\mathcal{I}}}\right)^{-1}-\underline{\underline{\mathcal{I}}}.\label{eq:ChangeVarYX}$$ We also recall that $\hat{\Lambda}\left(\underline{\underline{X}}\right)=N\left(\underline{\underline{X}}\right)\hat{\Lambda}^{\left(u\right)}\left(\underline{\underline{X}}\right),$ where $N\left(\underline{\underline{X}}\right)$ is given in (\[eq:NormGOX\]), The normalized differential identities are given below and in \[sec:AppendixDiffIdNormGO\] we show the procedure in order to obtain these identities. - Mixed products: $$\Biggl\{\hat{\underline{\gamma}}:\hat{\underline{\gamma}}^{T}\hat{\Lambda}:\Biggr\}=\rmi\left[\underline{\underline{X}}^{+}\frac{\rmd\hat{\Lambda}}{\rmd\underline{\underline{X}}}-\hat{\Lambda}\right]\underline{\underline{X}}^{-}.\label{eq:MajMixProd}$$ $$:\hat{\underline{\gamma}}\Biggl\{\hat{\underline{\gamma}}^{T}\hat{\Lambda}\Biggr\}:=\rmi\left[\underline{\underline{X}}^{-}\frac{\rmd\hat{\Lambda}}{\rmd\underline{\underline{X}}}-\hat{\Lambda}\right]\underline{\underline{X}}^{+}.\label{eq:Majmix2}$$ - Normally ordered products: $$:\hat{\underline{\gamma}}\hat{\underline{\gamma}}^{T}\hat{\Lambda}:=-\rmi\left[\underline{\underline{X}}^{-}\frac{\rmd\hat{\Lambda}}{\rmd\underline{\underline{X}}}-\hat{\Lambda}\right]\underline{\underline{X}}^{-}.\label{eq:NormalNormMajId}$$ - Anti-normally ordered products: $$\begin{aligned} \left\{ \underline{\widehat{\gamma}}\widehat{\underline{\gamma}}^{T}\hat{\Lambda}\right\} & =-\rmi\left[\underline{\underline{X}}^{+}\frac{\rmd\hat{\Lambda}}{\rmd\underline{\underline{X}}}-\hat{\Lambda}\right]\underline{\underline{X}}^{+} & .\label{eq:AntiNormalDifId}\end{aligned}$$ Unordered Majorana differential identities\[subsec:Unordered-Majorana-differential\] ------------------------------------------------------------------------------------ In this section we obtain differential identities that do not use normal or anti-normal ordering as in the previous section. There is no natural normal or anti-normal ordering for Majorana operators, however, the differential identities for the unordered products of the Majorana operators and the Gaussian basis can be obtained by combining the four normalized differential identities given in Section \[subsec:Normalized-Majorana-DifId\]. The details of the derivation of the differential identities are given in \[sec:AppendixDiffIdunorderedGO-1\]. These results are simpler in the modified variables, $\underline{\underline{x}}=\underline{\underline{{\cal I}}}\underline{\underline{X}}^{T}\underline{\underline{{\cal I}}},$ together with $\underline{\underline{x}}^{\pm}=\underline{\underline{x}}\pm\rmi\underline{\underline{I}}$. The resulting identities are: - Unordered left product: $$\hat{\underline{\gamma}}\hat{\underline{\gamma}}^{T}\hat{\Lambda}=\rmi\left[\underline{\underline{x}}^{-}\frac{\rmd\hat{\Lambda}}{\rmd\underline{\underline{x}}}\underline{\underline{x}}^{+}-\hat{\Lambda}\underline{\underline{x}}^{+}\right].\label{eq:UnorderedDifId}$$ - Unordered right product: $$\hat{\Lambda}\hat{\underline{\gamma}}\hat{\underline{\gamma}}^{T}=\rmi\left[\underline{\underline{x}}^{+}\frac{\rmd\hat{\Lambda}}{\rmd\underline{\underline{x}}}\underline{\underline{x}}^{-}-\hat{\Lambda}\underline{\underline{x}}^{+}\right].\label{eq:UnorderedDifId-1}$$ - Unordered mixed product: $$\hat{\underline{\gamma}}\hat{\Lambda}\hat{\underline{\gamma}}^{T}=\rmi\left[-\underline{\underline{x}}^{-}\frac{\rmd\hat{\Lambda}}{\rmd\underline{\underline{x}}}\underline{\underline{x}}^{-}+\hat{\Lambda}\underline{\underline{x}}^{-}\right].\label{eq:unmixed}$$ Phase space representations\[sec:Phase-space-representations\] ============================================================== Phase space representations have been widely used in quantum optics and ultracold atoms. A fermionic $P$-representation can be introduced using Grassmann variables, which are variables that anti-commute [@Cahill_Glauber_fermions_1999], but these are exponentially complex and hence mostly useful for analytic calculations, although they can be used to obtain complex covariance type equations [@dalton2016grassmann]. Another approach is to define fermionic phase space using the Gaussian operators. For fermions there are two known complete representations, which are the fermionic P-representation [@Cahill_Glauber_fermions_1999; @Corney_PD_PRL2004_GQMC_ferm_bos; @Corney_PD_JPA_2006_GR_fermions; @Corney_PD_PRB_2006_GPSR_fermions; @Corboz_Chapter_PhaseSpaceMethodsFermions] and the fermionic Q-function [@FermiQ]. In this case the phase space variables correspond to the matrix of the Gaussian operator, which are the elements of the covariance matrix. Using the Majorana Gaussian representation it is also possible to define phase space representations in terms of the Majorana variables and an antisymmetric matrix. These are the analogous to the fermionic phase space representations. Majorana P-functions \[subsec:Majorana-P-representation\] --------------------------------------------------------- A Majorana fermionic P-function can be defined in analogy to the fermionic P-function using raising and lowering operators [@Cahill_Glauber_fermions_1999; @Corney_PD_PRL2004_GQMC_ferm_bos; @Corney_PD_JPA_2006_GR_fermions; @Corney_PD_PRB_2006_GPSR_fermions; @Corboz_Chapter_PhaseSpaceMethodsFermions]. In this case the density matrix $\hat{\rho}(t)$ is expanded in terms of the Gaussian basis given in (\[eq:MajOpX\]). The phase space variables are either the antisymmetric matrix $\underline{\underline{X}}$, or its cousin $\underline{\underline{x}}$. Each may be more suitable, depending on whether one wishes to use ordered or unordered Majorana identities. Here we choose the unordered form: $$\hat{\rho}(\tau)=\int P\left(\underline{\underline{x}},\tau\right)\hat{\Lambda}\left(\underline{\underline{x}}\right)\prod_{i}\rmd x_{i}.$$ The $P(\underline{\underline{x}},\tau)$ distribution has the same properties as the fermionic $P$-representation: 1. **Normalization**: It is normalized to one: $$\int_{{\cal D}_{C}}P\left(\underline{\underline{x}}\right)\prod_{i}\rmd x_{i}=1.$$ 2. **Moments**: Expectation values of observables $O$ are obtained as: $$\begin{aligned} \left\langle O\right\rangle & = & \int_{{\cal D}_{C}}P\left(\underline{\underline{x}},\tau\right)\Tr\left[O\hat{\Lambda}\left(\underline{\underline{x}}\right)\right]\prod_{i}\rmd x_{i}\equiv\left\langle O\right\rangle _{P}.\end{aligned}$$ Majorana Q-functions -------------------- A Majorana Q-function can be defined as: $$Q\left(\underline{\underline{x}}\right)=\Tr\left[\hat{\rho}\hat{\Lambda}^{N}\left(\underline{\underline{x}}\right)\right].\label{eq:MajQf}$$ This Majorana Q-function has the same properties as the fermionic Q-function which are: 1. It is defined for any quantum density-matrix. 2. It is a positive probability distribution. 3. Observables are moments of the distribution. These are obtained as: $$\begin{aligned} \left\langle \hat{O}_{n}\right\rangle & = & \int_{{\cal D}_{R}}Q\left(\underline{\underline{x}}\right)O_{n}\left(\underline{\underline{x}}\right)\prod_{i}\rmd x_{i}=\left\langle \hat{O}_{n}\left(\left(\underline{\underline{x}}\right)\right)\right\rangle _{Q}.\label{eq:moment}\end{aligned}$$ Single mode Majorana Q-function ------------------------------- It is instructive to have an expression for the Q function in the single mode case. Utilizing \[eq:MajOpX\], where there is only one real variable, $x\equiv x_{12}$, and: $$\underline{\underline{x}}=\left[\begin{array}{cc} 0 & x\\ -x & 0 \end{array}\right].\label{eq:U0-1}$$ We can begin with the expression for the single mode Gaussian operator in Majorana form, which can be re- expressed using the result that: $$\widehat{n}=\frac{1+\rmi\hat{\gamma}_{1}\hat{\gamma}_{2}}{2}$$ so that $$\hat{\Lambda}\Biggl(x\Biggr)=\frac{1}{2}+\frac{\rmi}{2}\hat{\gamma}_{1}\hat{\gamma}_{2}x.$$ All physical density operators are also Gaussian operators in single mode case[@Corney_PD_JPA_2006_GR_fermions], since if $n\equiv\left\langle \hat{n}\right\rangle $, the most general density matrix is $$\hat{\rho}=\hat{\Lambda}_{1}\left(n\right)=\left(1-n\right)\left|0\right\rangle \left\langle 0\right|+n\left|1\right\rangle \left\langle 1\right|.$$ We arrive at an expression for Majorana Q-function by using the above two equations in \[eq:MajQf\], to give: $$Q\left(x\right)=\frac{1}{{\cal N}}S\left(x\right)\left[\left(\frac{1-x}{2}\right)+nx\right],$$ where $S\left(x\right)=\left[1+x^{2}\right]^{2p}$. Just as with the unit trace Gaussian, this has a symmetry between particles and holes, since changing the occupation from $n$ to $1-n$ simply changes the sign of the argument $x$. We note that in the absence of the normalization factor $S$, the distribution $Q\left(x\right)$ would have a discontinuity at the boundary. Observables ----------- We can obtain observables in terms of the Majorana Q-function using the Majorana differential identities in (\[eq:moment\]). As an example of this, we consider the Majorana correlation function, $$\hat{X}_{\mu\nu}\equiv\frac{\rmi}{2}\left[\gamma_{\mu},\gamma_{\nu}\right].\label{eq:Xcap}$$ This is a general hermitian observable, which includes the occupation number operators since: $$\widehat{n}_{i}=\frac{1+\hat{X}_{i,M+i}}{2}\,.$$ The expectation is defined as: $$\Biggl\langle\hat{X}_{\mu\nu}\Biggr\rangle=Tr\left[\hat{\rho}\widehat{X}_{\mu\upsilon}\right].\label{eq:mom1}$$ Next, using the resolution of identity and the unordered product ordering, we can arrive at the expression: $$\Biggl\langle\hat{X}_{\mu\nu}\Biggr\rangle=\frac{\rmi}{2}\int\Tr\left[\widehat{\rho}\left[\widehat{\gamma}_{\mu}\widehat{\gamma}_{\upsilon}\hat{\Lambda}^{N}-\widehat{\gamma}_{\upsilon}\widehat{\gamma}_{\mu}\hat{\Lambda}^{N}\right]\right]\rmd\underline{\underline{x}}.\label{eq:mom2}$$ Now we can apply the differential identities corresponding to the unordered product given in (\[eq:UnorderedDifId\]) obtaining: $$\begin{aligned} \Bigl\langle\widehat{X}_{\mu\upsilon}\Bigr\rangle & = & \frac{1}{2}\int\Tr\left[\widehat{\rho}\frac{S}{{\cal N}}\left[-x_{\mu\alpha}^{-}\frac{\rmd\hat{\Lambda}}{\rmd x_{\beta\alpha}}x_{\beta\upsilon}^{+}+\hat{\Lambda}x_{\mu\upsilon}^{+}\right]\rmd\underline{\underline{x}}\right]\nonumber \\ & & -\frac{1}{2}\int\Tr\left[\widehat{\rho}\frac{S}{{\cal N}}\left[-x_{\upsilon\alpha}^{-}\frac{\rmd\hat{\Lambda}}{\rmd x_{\beta\alpha}}x_{\beta\mu}^{+}+\hat{\Lambda}x_{\upsilon\mu}^{+}\right]\rmd\underline{\underline{x}}\right].\end{aligned}$$ Considering the limit $S\rightarrow1$ and assuming that the boundary terms vanish and also using the definition of the Q-function, the above equation can be written as: $$\fl\Bigl\langle\widehat{X}_{\mu\upsilon}\Bigr\rangle=\frac{1}{2}\int\left[-x_{\mu\alpha}^{-}\frac{\rmd}{\rmd x_{\beta\alpha}}x_{\beta\upsilon}^{+}+x_{\mu\upsilon}^{+}\right]Q\rmd\underline{\underline{x}}-\frac{1}{2}\int\left[-x_{\upsilon\alpha}^{-}\frac{\rmd}{\rmd x_{\beta\alpha}}x_{\beta\mu}^{+}+x_{\upsilon\mu}^{+}\right]Q\rmd\underline{\underline{x}}.$$ Using the the chain rule the above equation is: $$\begin{aligned} \Bigl\langle\widehat{X}_{\mu\upsilon}\Bigr\rangle & = & -\frac{1}{2}\int\left[\frac{\rmd}{\rmd x_{\beta\alpha}}\left[x_{\mu\alpha}^{-}x_{\beta\upsilon}^{+}\right]-2x_{\mu\upsilon}\left(2M-1\right)-x_{\mu\upsilon}^{+}\right]Q\rmd\underline{\underline{x}}\nonumber \\ & & +\frac{1}{2}\int\left[\frac{\rmd}{\rmd x_{\beta\alpha}}\left[x_{\upsilon\alpha}^{-}x_{\beta\mu}^{+}\right]-2x_{\upsilon\mu}\left(2M-1\right)-x_{\upsilon\mu}^{-}\right]Q\rmd\underline{\underline{x}}\end{aligned}$$ Here we have assumed that the boundary terms for the normal components of $x_{\mu\alpha}^{-}x_{\beta\upsilon}^{+}Q$ vanish at the classical domain boundary, so that the corresponding total derivative integrates to zero: $$\int\frac{\rmd}{\rmd x_{\beta\alpha}}\left[x_{\mu\alpha}^{-}x_{\beta\upsilon}^{+}Q\right]\rmd\underline{\underline{x}}=0\,.$$ This will leads to the required observable equation: $$\Bigl\langle\widehat{\underline{\underline{X}}}\Bigr\rangle=\left(4M-1\right)\int\underline{\underline{x}}Q\left(\underline{\underline{x}}\right)\rmd\underline{\underline{x}}.$$ Similarly, we can extend this method to evaluate higher order correlations as well. The assumption of vanishing boundary terms is an important restriction on our results, and would need to be checked in individual cases. Hamiltonian and time evolution \[sec:Time-Evolution-MQf\] ========================================================= Majorana variables have been used to study decoherence effects in Fermi systems [@Prosen:2008] or fermionic entanglement [@Eisler:2015; @Meichanetzidis:2016Ent]. These effects are related to the dynamical evolution of the systems. The Majorana differential identities are useful in order to perform dynamical simulations. In order to illustrate this, we will consider the simplest case of quadratic Hamiltonians. While higher order terms can be treated, these result in diffusion-type differential operators, which are outside the scope of the present article. Quadratic Hamiltonians ---------------------- The symmetries of fermionic covariance matrices and quadratic Hamiltonians have identical properties. The general hermitian quadratic Hamiltonian has been widely investigated, and has the form: $$\hat{H}=\frac{1}{2}\left[\hat{\bm{a}}^{\dagger}\mathbf{h}\hat{\bm{a}}-\hat{\bm{a}}\bm{h}^{T}\hat{\bm{a}}^{\dagger}+\hat{\bm{a}}^{\dagger}\bm{\Delta}\hat{\bm{a}}^{\dagger}-\hat{\bm{a}}\bm{\Delta}^{*}\hat{\bm{a}}\right].\label{eq:BdGHam-2}$$ This Hamiltonian is also known as the Bogoliubov- de Gennes Hamiltonian, which can be written in matrix form using the extended ladder operators as: $$\hat{H}=\frac{1}{2}\hat{\underline{a}}^{\dagger}\underline{\underline{H}}\hat{\underline{a}},\label{eq:BdGHamMat-1}$$ where the matrix $\underline{\underline{H}}$ is defined as: $$\underline{\underline{H}}=\left(\begin{array}{cc} \mathbf{h} & \bm{\Delta}\\ -\bm{\Delta^{*}} & -\mathbf{h}^{T} \end{array}\right),\label{eq:H_Matrix-2}$$ with $\mathbf{h}=\mathbf{h}^{\dagger}$ and $\bm{\Delta}=-\bm{\Delta}^{T}$ as the only symmetry restrictions. Altland and Zirnbauer [@Altland_Zirnbauer:1997] consider this Hamiltonian in order to define the non-standard symmetry classes. This is done by considering how imposing time-reversal and/or spin-rotation symmetry on the Hamiltonian leads to different types of symmetry classes. Each one of these corresponds to a particular type of Lie group that has a corresponding mathematical symmetric space. Here we will focus on class D symmetry, which has neither time-reversal nor spin-rotation invariance, allowing us to treat an arbitrary Fermi system, which means that $\underline{\underline{H}}=\underline{\underline{H}}^{\dagger}.$ Using the relationship for the ladder operators and the Majorana operators given in (\[eq:OpMajUn\]) we can express the Hamiltonian of (\[eq:BdGHamMat-1\]) in the following form [@Kitaev:2009; @Alicea:2012]: $$\hat{H}=\frac{\rmi\hbar}{2}\hat{\underline{\gamma}}^{T}\underline{\underline{\Omega}}\hat{\underline{\gamma}}=\frac{\hbar}{2}\Omega_{\mu\nu}\hat{X}_{\mu\nu}.\label{eq:BdGHam_MajOp-2}$$ The expression of the matrix $\underline{\underline{\Omega}}$ in terms of matrices $\mathbf{h}$ and $\bm{\Delta}$ is: $$\underline{\underline{\Omega}}=\frac{1}{2\rmi\hbar}\left(\begin{array}{cc} \mathbf{h}_{-}+\bm{\Delta}_{-} & i\mathbf{h}_{+}-i\bm{\Delta}_{+}\\ -i\mathbf{h}_{+}-i\bm{\Delta}_{+} & \mathbf{h}_{-}-\bm{\Delta}_{-} \end{array}\right)\,.\label{eq:HamMatAntiSym-1}$$ Here we have defined $\mathbf{h}_{\pm}=\mathbf{h}\pm\mathbf{h}^{T}$ and $\bm{\Delta}_{\pm}=\bm{\Delta}\pm\bm{\Delta}^{*}$. We note that $\underline{\underline{\Omega}}$ is a real anti-symmetric matrix: $\underline{\underline{\Omega}}^{T}=-\underline{\underline{\Omega}}$, and therefore satisfies the group properties of these matrices, with a well-defined Haar measure. Time evolution -------------- The time evolution equation of the Majorana Q-function is obtained by considering that: $$\frac{\rmd Q\left(\underline{\underline{x}}\right)}{\rmd t}=\Tr\left[\frac{\rmd\hat{\rho}}{\rmd t}\hat{\Lambda}^{N}\left(\underline{\underline{x}}\right)\right].\label{eq:TE_MQf}$$ Here we have used the definition of the Majorana Q-function given in (\[eq:MajQf\]). Next we consider the time-evolution equation for the density operator: $$\begin{aligned} i\hbar\frac{\partial}{\partial t}\hat{\rho} & = & \left[\hat{H},\hat{\rho}\right].\label{eq:TimeEvolutionDM}\end{aligned}$$ On substituting this on the time-evolution equation for the Majorana Q-function and using the cyclic properties of the trace, we get: $$\frac{\rmd Q\left(\underline{\underline{x}}\right)}{\rmd t}=\frac{1}{\rmi\hbar}\Tr\left[\left[\hat{\Lambda}^{N}\left(\underline{\underline{x}}\right),\hat{H}\right]\hat{\rho}\right].\label{eq:TT}$$ Here we will consider the general Hamiltonian $\hat{H}$ given in (\[eq:BdGHam\_MajOp-2\]). Therefore we get that the time-evolution equation for the Majorana Q-function is: $$\frac{\rmd Q\left(\underline{\underline{x}}\right)}{\rmd t}=\frac{1}{2\rmi}\Tr\left[\hat{\Lambda}^{N}\Omega_{\mu\nu}\hat{X}_{\mu\nu}\hat{\rho}-\Omega_{\mu\nu}\hat{X}_{\mu\nu}\hat{\Lambda}^{N}\hat{\rho}\right].$$ We now use the definition of $\underline{\underline{\hat{X}}}$ given in (\[eq:Xcap\]), obtaining: $$\frac{\rmd Q\left(\underline{\underline{x}}\right)}{\rmd t}=-\frac{1}{4}\Tr\left[\Omega_{\mu\nu}\left[\gamma_{\mu}\gamma_{\nu}-\gamma_{\nu}\gamma_{\mu},\hat{\Lambda}^{N}\right]\hat{\rho}\right].\label{eq:TF}$$ The differential identity for $\left[\gamma_{\mu}\gamma_{\nu}-\gamma_{\nu}\gamma_{\mu},\widehat{\Lambda}\right]$ is derived from (\[eq:UnorderedDifId\]). This is given below: $$\left[\gamma_{\mu}\gamma_{\nu}-\gamma_{\nu}\gamma_{\mu},\widehat{\Lambda}\right]=4\left[x_{\kappa\upsilon}\frac{\rmd\hat{\Lambda}}{\rmd x_{\kappa\mu}}-x_{\mu\kappa}\frac{\rmd\hat{\Lambda}}{\rmd x_{\upsilon\kappa}}\right].\label{eq:COMMU}$$ On substituting (\[eq:COMMU\]) in (\[eq:TF\]), we get: $$\frac{\rmd Q\left(\underline{\underline{X}}\right)}{\rmd t}=-\Tr\left[\Omega_{\mu\nu}\frac{S}{{\cal N}}\left[-x_{\mu\kappa}\frac{\rmd\hat{\Lambda}}{\rmd x_{\upsilon\kappa}}+\frac{\rmd\hat{\Lambda}}{\rmd x_{\kappa\mu}}x_{\kappa\upsilon}\right]\hat{\rho}\right].\label{eq:dQ/dt}$$ Following the steps outlined in \[sec:AppendixTimeevolution\], we get: $$\frac{\rmd Q\left(\underline{\underline{X}}\right)}{\rmd t}=-\Omega_{\mu\nu}\left[-x_{\mu\kappa}\frac{\rmd Q}{\rmd x_{\upsilon\kappa}}+\frac{\rmd Q}{\rmd x_{\kappa\mu}}x_{\kappa\upsilon}\right]\label{eq:TEf}$$ Next, using the product rule, (\[eq:TEf\]) can be written as: $$\begin{aligned} \frac{\rmd Q\left(\underline{\underline{X}}\right)}{\rmd t} & = & \Omega_{\mu\nu}\left[\frac{\rmd}{\rmd x_{\upsilon\kappa}}\left(x_{\mu\kappa}Q\right)-\frac{\rmd}{\rmd x_{\kappa\mu}}\left(x_{\kappa\upsilon}Q\right)\right].\label{eq:TEf2}\end{aligned}$$ The method of characteristics allows us to solve the above equation as: $$\frac{d\underline{\underline{x}}}{dt}=\left[\underline{\underline{\Omega}},\underline{\underline{x}}\right].\label{eq:TEX}$$ We note that the Heisenberg equations of motion for the Majorana operators in this case are identical, and are given by: $$\frac{\rmd\underline{\underline{\hat{X}}}\left(t\right)}{\rmd t}=\left[\underline{\underline{\Omega}},\underline{\hat{\underline{X}}}\right].$$ This result is valid for completely arbitrary quadratic real antisymmetric matrices $\underline{\underline{\Omega}}$, even with anomalous terms or complex frequency matrices. Since the operator equations for $\underline{\underline{\hat{X}}}$ are simply proportional to the mean value equations, which in turn are proportional to the characteristic equations for $\underline{\underline{x}}$, this provides a verification of the phase-space results. The utility of the Q-function method in this case is that the general Q-function can be computed given an arbitrary initial distribution. One is not restricted to just calculating mean values of the lowest order correlation function. Bosonic Q-function ------------------ For comparison purposes, we can also write a time evolution for the bosonic Q-function [@Husimi1940] in a similar form that the one described in the above section for fermions. In this case we consider the following Hamiltonian of a non-interacting Bose gas: $$\hat{H}=\hbar\hat{\bm{a}}^{\dagger}\mathbf{\bm{\omega}}\hat{\bm{a}}.$$ Following the standard techniques for phase-space representations, where one uses the corresponding identities that map phase space variables into c-numbers [@Drummond:2014BookQT], we obtain the following time evolution equation for the bosonic Q function, $Q_{B}$: $$\frac{dQ_{B}(\boldsymbol{\alpha})}{dt}=i\bm{\bm{\omega}}\left[\frac{\partial}{\partial\boldsymbol{\alpha}}\boldsymbol{\alpha}-\frac{\partial}{\partial\boldsymbol{\alpha}^{*}}\boldsymbol{\alpha}^{*}\right]Q_{B}.\label{eq:btimeev}$$ We use the methods of characteristics in order to solve this differential equation obtaining: $$\frac{d\boldsymbol{\alpha}}{dt}=-i\bm{\bm{\omega}}\boldsymbol{\alpha}.\label{eq:char}$$ Next, we define the following two real vectors: $$\begin{aligned} \boldsymbol{\alpha}_{x} & = & \frac{1}{2}\left[\boldsymbol{\alpha}+\boldsymbol{\alpha}^{*}\right],\\ \boldsymbol{\alpha}_{y} & = & \frac{1}{2i}\left[\boldsymbol{\alpha}-\boldsymbol{\alpha}^{*}\right].\end{aligned}$$ Consequently we can introduce the following real vector: $$\underline{\boldsymbol{\alpha}}=\left[\begin{array}{c} \boldsymbol{\alpha}_{x}\\ \boldsymbol{\alpha}_{y} \end{array}\right],\label{eq:quad}$$ and a matrix of quadrature correlations, analogous to the Majorana phase space matrix, where $$\underline{\underline{x}}_{b}=\underline{x}\underline{x}^{T}.$$ Using these expressions in (\[eq:char\]), leads to a time evolution equation for $\underline{\underline{x}}_{b}$ of the form: $$\frac{d}{dt}\left(\underline{\underline{x}}_{b}\right)=\left[\underline{\underline{\Omega}}_{b},\underline{\underline{x}}_{b}\right],\label{eq:Xb}$$ where $\underline{\underline{\Omega}}_{b}=\left(\begin{array}{cc} 0 & \boldsymbol{\omega}\\ -\boldsymbol{\mathbf{\omega}} & 0 \end{array}\right).$ In summary, for the usual bosonic case, one also obtains a simple characteristic evolution, provided the Hamiltonian has no anomalous terms. More generally, the results are more complicated, and do not follow a simple first-order equation. In the Fermi Q-function case, a first-order characteristic evolution is obtained for any quadratic Hamiltonian. Time-Evolution of open quantum systems\[sec:TimeEvOpenSyst\] ============================================================ We now consider the following application of our method: the time evolution of an open quantum system. This will include the interaction of a system with the environment, which will show that our method can treat dissipative systems as well. The system is a small quantum dot coupled to a zero temperature reservoir. Here it is convenient to use the ordered identities, as the dissipative master-equation terms have normal ordering. Master Equation --------------- The time evolution of the density operator of this model is given by a master equation [@Corney_PD_PRB_2006_GPSR_fermions]: $$\frac{d\hat{\rho}}{dt}=-i\omega\hat{a}^{\dagger}\hat{a}\hat{\rho}+i\omega\hat{\rho}\hat{a}^{\dagger}\hat{a}+\gamma\left(\hat{a}\hat{\rho}\hat{a}^{\dagger}-\frac{1}{2}\hat{a}^{\dagger}\hat{a}\hat{\rho}-\frac{1}{2}\hat{\rho}\hat{a}^{\dagger}\hat{a}\right).\label{eq:DensityOpQD}$$ In the case of a multi-mode quantum system we get $$\frac{d\hat{\rho}}{dt}=-i\omega_{ji}\hat{a}_{i}^{\dagger}\hat{a}_{j}\hat{\rho}+i\omega_{ji}\hat{\rho}\hat{a}_{i}^{\dagger}\hat{a}_{j}+\gamma_{ij}\left(\hat{a}_{i}\hat{\rho}\hat{a}_{j}^{\dagger}-\frac{1}{2}\hat{a}_{j}^{\dagger}\hat{a}_{i}\hat{\rho}-\frac{1}{2}\hat{\rho}\hat{a}_{j}^{\dagger}\hat{a}_{i}\right),\label{eq:Denmul}$$ provided $\bm{\omega}=\bm{\omega}^{T}$ and $\bm{\gamma}=\bm{\gamma}^{T}$. On substituting this equation for the time evolution of the Q-function given in (\[eq:TE\_MQf\]) we obtain: $$\begin{aligned} \frac{dQ}{dt} & = & i\Tr\Biggl[\omega_{ji}\left[\widehat{a}_{i}^{\dagger}\widehat{a}_{j},\hat{\Lambda}^{N}\left(\underline{\underline{X}}\right)\right]\widehat{\rho}\Biggr]+\Tr\left[\hat{\Lambda}^{N}(\underline{\underline{X}})\gamma_{ij}\widehat{a}_{i}\widehat{\rho}\widehat{a}_{j}^{\dagger}\right]\nonumber \\ & & -\frac{1}{2}\Tr\left[\gamma_{ij}\left[\widehat{a}_{i}^{\dagger}\widehat{a}_{j},\hat{\Lambda}^{N}\left(\underline{\underline{X}}\right)\right]_{+}\widehat{\rho}\right].\label{eq:timeevo1}\end{aligned}$$ We now wish to express the above results in terms of the Majorana differential identities given in Section \[sec:Majorana-differential-properties\]. First, using the expressions of the different orderings of the Majorana variables and the Gaussian operators given in \[sec:AppendixDiffIdNormGO\] we rewrite (\[eq:timeevo1\]) as: $$\begin{aligned} \frac{dQ}{dt} & = & -\frac{i}{2}\frac{1}{{\cal N}}S\Tr\left[\tilde{\Omega}_{\kappa\nu}\Bigl\{\hat{\underline{\gamma}}:\hat{\underline{\gamma}}^{T}\hat{\Lambda}:\Bigr\}_{\nu\kappa}\widehat{\rho}\right]\nonumber \\ & & -\frac{1}{2i}\frac{1}{{\cal N}}S\Tr\left[\Upsilon_{\kappa\nu}\left(:\hat{\underline{\gamma}}\hat{\underline{\gamma}}^{T}\hat{\Lambda}:{}_{\nu\kappa}-\Bigl\{\hat{\underline{\gamma}}:\hat{\underline{\gamma}}^{T}\hat{\Lambda}:\Bigr\}_{\nu\kappa}\right)\widehat{\rho}\right]-\gamma_{ij}\delta_{ij}Q.\label{eq:timeevo2}\end{aligned}$$ Here we have defined $$\begin{aligned} \underline{\underline{\tilde{\Omega}}} & = & \left(\begin{array}{cc} \bm{\omega} & \mathbf{0}\\ \mathbf{0} & \bm{\omega} \end{array}\right),\qquad{\rm and}\label{eq:OmegaQDot}\\ \underline{\underline{\Upsilon}} & = & \left(\begin{array}{cc} \mathbf{0} & -\frac{\bm{\gamma}}{2}\\ \bm{\frac{\gamma}{2}} & \mathbf{0} \end{array}\right).\label{eq:Gamma}\end{aligned}$$ On using the Majorana differential identities given in (\[eq:MajMixProd\]) and (\[eq:NormalNormMajId\]) we get that the time evolution equation is: $$\begin{aligned} \frac{dQ}{dt} & = & \frac{1}{2}\tilde{\Omega}_{\kappa\nu}\frac{d}{dX_{p\ell}}\left(X_{\nu\ell}^{+}QX_{p\kappa}^{-}\right)+\Upsilon_{\kappa\nu}\frac{d}{dX_{pl}}\left(X_{\nu\ell}QX_{p\kappa}^{-}\right)\nonumber \\ & & -\left(4M-1\right)X_{\nu\kappa}\Upsilon_{\kappa\nu}Q+\left(2M-1\right)\Upsilon_{\kappa\nu}\mathcal{I}_{\nu\kappa}.\label{eq:finalEx}\end{aligned}$$ Details of the calculations are given in \[sec:AppTimeEvOpenQS\]. We will next illustrate the dynamic behavior for this dissipative system for the single mode case. Single mode case ---------------- We wish to show the dynamic behavior of the open quantum system. In order to do this, we will study the nature of trajectories for the single mode case, where $\underline{\underline{X}}=\underline{\underline{x}}$. In this case, defining $X=X_{12}$, (\[eq:finalEx\]) reduces to: $$\frac{dQ}{dt}=\gamma\left[\frac{d}{dX}\left[QX\left(X-1\right)\right]\right]+\gamma\left(1-3X\right)Q.\label{eq:TevQ_QDot_SingleMode}$$ We can solve the above differential equation using method of characteristics which leads to: $$\frac{dX}{dt}=\gamma X(1-X).\label{eq:dXdtQdotSM}$$ On integrating this equation we get: $$t-t_{0}=\ln\left[\frac{X}{1-X}\right].$$ This can be written as: $$e^{-(t-t_{0})}=\frac{1}{X}-1,$$ which gives two possible solutions: $$X=\frac{1}{1\pm e^{-(t-t_{0})}}.$$ In order to get this result we have used that since $t_{0}$ is arbitrary, it can be chosen as complex, so for $X<0$, one has: $t_{0}\rightarrow i\pi+t_{0}$, which gives the minus sign. Therefore, if $t\rightarrow\infty$, then, either with the plus sign, $X(0)>0$, leads to $X\rightarrow1$ or with the minus sign, $X(0)<0$, leads to $X\rightarrow-\infty.$ This means that for $X(0)<0$ the trajectory crosses the boundary after a finite time, and is lost. The solution of the differential equation (\[eq:TevQ\_QDot\_SingleMode\]) is depicted graphically in figure \[fig:DynQD\]. One usually has a first order Fokker Plank equation when dealing with a damped P-function for bosons. It decays down to a delta function. This means that for the bosonic P function, every trajectory decays to zero. Then the long-time solution in that case is a delta function with zero width. However, for bosons, the Q-function method would include a diffusion term, and give a finite width. In our case we have a first-order differential equation for a fermionic Q function. In this case we get a finite width for the ground state solution for the single mode case. We would normally expect to obtain this through a diffusion term in the Fokker Planck equation. However we have no diffusion term in our calculation, but we still get a finite width, which makes the dynamics unusual. In this case, the Q-function dissipative dynamics has both sources and sinks, with weighted trajectories being generated at the origin, and being lost at the boundary at $X=-1$. ![Dynamics of a single mode quantum dot coupled to a zero temperature reservoir.\[fig:DynQD\] ](Figure) Summary\[sec:Summary\_Conclusion\] ================================== In summary, we have introduced a formalism for a fermionic phase space representation in terms of Majorana operators and a phase-space of antisymmetric matrices defined within a Cartan bounded homogeneous domain. This phase space representation uses as a basis the Majorana Gaussian operators, whose symmetry group is associated with the same classical domain of real antisymmetric matrices. We have derived differential identities for the Majorana Gaussian operators. These identities are relevant to the important tasks of computing observables and carrying out dynamical simulations. As a simple illustration, we have derived time evolution equations of the Majorana Q-function for the number conserving Hamiltonian, in the case of a general quadratic Hamiltonian. We have also obtained time evolution equations for a quantum dot coupled to a zero temperature reservoir, which is an example of a dissipative system. The results obtained are quite different to those in the bosonic Q-function representation of Husimi. For example, we can treat all quadratic Hamiltonians as simple trajectory evolution - not just those with number-conserving Hamiltonians. These results show the usefulness of the Majorana differential identities. As an example, our results can also be used to study the dynamics of shock-wave formation in a one-dimensional Fermi gas at finite temperature. This study will be carried out elsewhere. Explicit form of the different orderings for the Majorana variables and Gaussian operators\[sec:AppendixFormOrdering\] ====================================================================================================================== In this section we write the explicit forms of the possible different orderings of the Majorana variables with the Majorana Gaussian operators. These expressions are given following the order in which the differential identities were given. First mixed product\[subsec:Mixed-product1\] -------------------------------------------- $$\begin{aligned} & & \fl\Bigl\{\hat{\underline{\gamma}}:\hat{\underline{\gamma}}^{T}\hat{\Lambda}:\Bigr\}=\nonumber \\ & & \fl\left(\begin{array}{cc} \bm{T}_{1}-\left(\hat{\Lambda}\hat{\bm{a}}\hat{\bm{a}}^{\dagger}\right)^{T}-\left(\hat{\bm{a}}^{\dagger T}\hat{\Lambda}\hat{\bm{a}}^{\dagger}\right)^{T} & -\rmi\left[\bm{T}_{2}-\left(\hat{\Lambda}\hat{\bm{a}}\hat{\bm{a}}^{\dagger}\right)^{T}+\left(\hat{\bm{a}}^{\dagger T}\hat{\Lambda}\hat{\bm{a}}^{\dagger}\right)^{T}\right]\\ -\rmi\left[\bm{T}_{1}+\left(\hat{\Lambda}\hat{\bm{a}}\hat{\bm{a}}^{\dagger}\right)^{T}+\left(\hat{\bm{a}}^{\dagger T}\hat{\Lambda}\hat{\bm{a}}^{\dagger}\right)^{T}\right] & -\left[\bm{T}_{2}+\left(\hat{\bm{a}}^{\dagger T}\hat{\Lambda}\hat{\bm{a}}^{\dagger}\right)^{T}-\left(\hat{\Lambda}\hat{\bm{a}}\hat{\bm{a}}^{\dagger}\right)^{T}\right] \end{array}\right),\label{eq:ExpFMP}\end{aligned}$$ where $\bm{T}_{1}=\hat{\bm{a}}\hat{\Lambda}\hat{\bm{a}}^{T}+\hat{\bm{a}}\hat{\bm{a}}^{\dagger}\hat{\Lambda}$ and $\bm{T}_{2}=\hat{\bm{a}}\hat{\Lambda}\hat{\bm{a}}^{T}-\hat{\bm{a}}\hat{\bm{a}}^{\dagger}\hat{\Lambda}$. Second mixed product\[subsec:Mixed-product-2\] ---------------------------------------------- $$\begin{aligned} & & \fl:\hat{\gamma}\Biggl\{\hat{\gamma}\hat{\Lambda}\Biggr\}:=\left(\begin{array}{cc} \bm{T}_{1}^{'}-\left(\hat{\Lambda}\hat{\bm{a}}^{\dagger T}\hat{\bm{a}}^{T}\right)^{T}-\left(\hat{\bm{a}}\hat{\Lambda}\hat{\bm{a}}^{T}\right)^{T} & \rmi\left[\bm{T}_{2}^{'}-\hat{\bm{a}}^{\dagger T}\hat{\bm{a}}^{T}\hat{\Lambda}+\hat{\bm{a}}^{\dagger T}\hat{\Lambda}\hat{\bm{a}}^{\dagger}\right]\\ \rmi\left[\bm{T}_{1}^{'}+\left(\hat{\Lambda}\hat{\bm{a}}^{\dagger T}\hat{\bm{a}}^{T}\right)^{T}+\left(\hat{\bm{a}}\hat{\Lambda}\hat{\bm{a}}^{T}\right)^{T}\right] & \bm{T}_{2}^{'}+\hat{\bm{a}}^{\dagger T}\hat{\bm{a}}^{T}\hat{\Lambda}-\hat{\bm{a}}^{\dagger T}\hat{\Lambda}\hat{\bm{a}}^{\dagger} \end{array}\right)\label{eq:ExpSMP}\end{aligned}$$ where $\bm{T}_{1}^{'}=+\hat{\bm{a}}^{\dagger T}\hat{\bm{a}}^{T}\hat{\Lambda}+\hat{\bm{a}}^{\dagger T}\hat{\Lambda}\hat{\bm{a}}^{\dagger}$ and $\bm{T}_{2}^{'}=-\left(\hat{\Lambda}\hat{\bm{a}}^{\dagger T}\hat{\bm{a}}^{T}\right)^{T}+\left(\hat{\bm{a}}\hat{\Lambda}\hat{\bm{a}}^{T}\right)^{T}$. Normally ordered products \[subsec:Normally-ordered-products\] -------------------------------------------------------------- $$\begin{aligned} & & \fl:\hat{\underline{\gamma}}\hat{\underline{\gamma}}^{T}\hat{\Lambda}:=\left(\begin{array}{cc} \bm{T}_{3}+\hat{\bm{a}}^{\dagger T}\hat{\Lambda}\hat{\bm{a}}^{T}+\hat{\bm{a}}^{\dagger T}\hat{\bm{a}}^{\dagger}\hat{\Lambda} & -\rmi\left[\bm{T}_{4}+\hat{\bm{a}}^{\dagger T}\hat{\Lambda}\hat{\bm{a}}^{T}-\hat{\bm{a}}^{\dagger T}\hat{\bm{a}}^{\dagger}\hat{\Lambda}\right]\\ -\rmi\left[\bm{T}_{3}-\hat{\bm{a}}^{\dagger T}\hat{\Lambda}\hat{\bm{a}}^{T}-\hat{\bm{a}}^{\dagger T}\hat{\bm{a}}^{\dagger}\hat{\Lambda}\right] & -\left[\bm{T}_{4}-\hat{\bm{a}}^{\dagger T}\hat{\Lambda}\hat{\bm{a}}^{T}+\hat{\bm{a}}^{\dagger T}\hat{\bm{a}}^{\dagger}\hat{\Lambda}\right] \end{array}\right),\label{eq:ENOP}\end{aligned}$$ where $\bm{T}_{3}=\hat{\Lambda}\hat{\bm{a}}\hat{\bm{a}}^{T}-\left(\hat{\bm{a}}^{\dagger T}\hat{\Lambda}\hat{\bm{a}}^{T}\right)^{T}$ and $\bm{T}_{4}=\hat{\Lambda}\hat{\bm{a}}\hat{\bm{a}}^{T}+\left(\hat{\bm{a}}^{\dagger T}\hat{\Lambda}\hat{\bm{a}}^{T}\right)^{T}$. Anti-normally ordered products \[subsec:Anti-normally-ordered-products\] ------------------------------------------------------------------------ $$\begin{aligned} & & \fl\left\{ \hat{\underline{\gamma}}\hat{\underline{\gamma}}^{T}\hat{\Lambda}\right\} =\left(\begin{array}{cc} \bm{T}_{5}-\left(\hat{\bm{a}}\hat{\Lambda}\hat{\bm{a}}^{\dagger}\right)^{T}+\hat{\Lambda}\hat{\bm{a}}^{\dagger T}\hat{\bm{a}}^{\dagger} & -\rmi\left[\bm{T}_{6}-\left(\hat{\bm{a}}\hat{\Lambda}\hat{\bm{a}}^{\dagger}\right)^{T}-\hat{\Lambda}\hat{\bm{a}}^{\dagger T}\hat{\bm{a}}^{\dagger}\right]\\ -\rmi\left[\bm{T}_{5}+\left(\hat{\bm{a}}\hat{\Lambda}\hat{\bm{a}}^{\dagger}\right)^{T}-\hat{\Lambda}\hat{\bm{a}}^{\dagger T}\hat{\bm{a}}^{\dagger}\right] & -\left[\bm{T}_{6}+\left(\hat{\bm{a}}\hat{\Lambda}\hat{\bm{a}}^{\dagger}\right)^{T}+\hat{\Lambda}\hat{\bm{a}}^{\dagger T}\hat{\bm{a}}^{\dagger}\right] \end{array}\right),\label{eq:ExpANOP}\end{aligned}$$ where $\bm{T}_{5}=\hat{\bm{a}}\hat{\bm{a}}^{T}\hat{\Lambda}+\hat{\bm{a}}\hat{\Lambda}\hat{\bm{a}}^{\dagger}$ and $\bm{T}_{6}=\hat{\bm{a}}\hat{\bm{a}}^{T}\hat{\Lambda}-\hat{\bm{a}}\hat{\Lambda}\hat{\bm{a}}^{\dagger}$. Unordered products \[subsec:UnOrderedProduct\] ---------------------------------------------- $$\begin{aligned} & & \fl\hat{\underline{\gamma}}\hat{\underline{\gamma}}^{T}\hat{\Lambda}=\left[\begin{array}{cc} \bm{T}_{7}+\hat{\bm{a}}^{\dagger T}\hat{\bm{a}}^{\dagger}\hat{\Lambda}+\hat{\bm{a}}^{\dagger}{}^{T}\hat{\bm{a}}^{T}\hat{\Lambda} & \rmi\left[\bm{T}_{8}+\hat{\bm{a}}^{\dagger T}\hat{\bm{a}}^{\dagger}\hat{\Lambda}-\hat{\bm{a}}^{\dagger}{}^{T}\hat{\bm{a}}^{T}\hat{\Lambda}\right]\\ \rmi\left[-\bm{T}_{7}+\hat{\bm{a}}^{\dagger T}\hat{\bm{a}}^{\dagger}\hat{\Lambda}+\hat{\bm{a}}^{\dagger}{}^{T}\hat{\bm{a}}^{T}\hat{\Lambda}\right] & \bm{T}_{8}-\hat{\bm{a}}^{\dagger T}\hat{\bm{a}}^{\dagger}\hat{\Lambda}+\hat{\bm{a}}^{\dagger}{}^{T}\hat{\bm{a}}^{T}\hat{\Lambda} \end{array}\right]\label{eq:ExpUn-Or}\end{aligned}$$ where $\bm{T}_{7}=\hat{\bm{a}}\hat{\bm{a}}^{\dagger}\hat{\Lambda}+\hat{\bm{a}}\hat{\bm{a}}^{T}\hat{\Lambda}$ and $\bm{T}_{8}=\hat{\bm{a}}\hat{\bm{a}}^{\dagger}\hat{\Lambda}-\hat{\bm{a}}\hat{\bm{a}}^{T}\hat{\Lambda}$. Majorana differential identities for un-normalized ordered products\[sec:AppendixMajoranaDiffId\] ================================================================================================= In this Section we give the detailed proofs of the un-normalized, ordered product differential identities shown in Section \[sec:Majorana-differential-properties\]. We first give the proof for the un-normalized Majorana differential identities and then in the next section we use these identities in order to obtain the normalized ones. The following differential identities are given considering the different types of ordering of the Majorana variables and Gaussian operators. The explicit form of these are given in \[sec:AppendixFormOrdering\]. These identities uses the un-normalized Gaussian operators given in (\[eq:GOY\]). Mixed products -------------- Here we are considering products of the Majorana variables and Gaussian operators of the form $\Biggl\{\hat{\underline{\gamma}}:\hat{\underline{\gamma}}^{T}\hat{\Lambda}^{\left(u\right)}:\Biggr\}$. In this case the differential identity is: $$\Biggl\{\hat{\underline{\gamma}}:\hat{\underline{\gamma}}^{T}\hat{\Lambda}^{\left(u\right)}:\Biggr\}=\rmi\left[-\underline{\underline{{\cal I}}}\left(2\underline{\underline{Y}}+\underline{\underline{\mathcal{I}}}\right)\frac{\rmd}{\rmd\underline{\underline{Y}}}+2\underline{\underline{{\cal I}}}\right]\hat{\Lambda}^{\left(u\right)}.$$ ***Proof.*** To prove this identity we proceed by using Grassmann variables and fermionic coherent states. We also use the following result [@Corney_PD_JPA_2006_GR_fermions] that is already known, but with more details included for clarity and completeness: $$\begin{aligned} \fl\Biggl\{\underline{\widehat{a}}:\widehat{\underline{a}}^{\dagger}\hat{\Lambda}^{\left(u\right)}:\Biggr\} & = & \int\rmd\underline{\gamma}\rmd\underline{\beta}\rmd\underline{\alpha}\rmd\underline{\epsilon}\Biggl|\bgamma\Biggr\rangle\Biggl\langle\bgamma\Biggr|\Biggl\{\underline{\widehat{a}}:\Biggl|\bbeta\Biggr\rangle\Biggl\langle\bbeta\Biggr|\widehat{\underline{a}}^{\dagger}\widehat{\Lambda}^{\left(u\right)}\Biggl|\balpha\Biggr\rangle\Biggl\langle\balpha\Biggr|:\Biggr\}\Biggl|\bepsilon\Biggr\rangle\Biggl\langle\bepsilon\Biggr|\nonumber \\ & = & \int\rmd\underline{\gamma}\rmd\underline{\beta}\rmd\underline{\alpha}\rmd\underline{\epsilon}\Biggl|\bgamma\Biggr\rangle\Biggl\langle\bepsilon\Biggr|\exp\left[-\bar{\balpha}\balpha-\bar{\bbeta}\bbeta-\frac{1}{2}\bar{\bgamma}\bgamma-\frac{1}{2}\bar{\bepsilon}\bepsilon+\bar{\bgamma}\bbeta+\bar{\balpha}\bepsilon\right]\nonumber \\ & \times & \left[\begin{array}{c} \bbeta\\ \bar{\balpha} \end{array}\right]\left[\begin{array}{cc} \bar{\bbeta} & \balpha\end{array}\right]\exp\left[-\frac{1}{2}\left[\begin{array}{cc} \bar{\bbeta} & \balpha\end{array}\right]\left(\underline{\underline{\mu}}+\underline{\underline{J}}\right)\left[\begin{array}{c} \balpha\\ \bar{\bbeta} \end{array}\right]\right]\nonumber \\ & = & \int\rmd\underline{\gamma}\rmd\underline{\beta}\rmd\underline{\alpha}\rmd\underline{\epsilon}\Biggl|\bgamma\Biggr\rangle\Biggl\langle\bepsilon\Biggr|\exp\left[-\bar{\balpha}\balpha-\bar{\bbeta}\bbeta-\frac{1}{2}\bar{\bgamma}\bgamma-\frac{1}{2}\bar{\bepsilon}\bepsilon+\bar{\bgamma}\bbeta+\bar{\balpha}\bepsilon\right]\nonumber \\ & \times & \left[\underline{\underline{J}}\left(\underline{\underline{\mu}}+\underline{\underline{J}}\right)\left[\begin{array}{c} \balpha\\ \bar{\bbeta} \end{array}\right]\left[\begin{array}{cc} \bar{\bbeta} & \balpha\end{array}\right]-\underline{\underline{J}}\right]\exp\left[-\left[\begin{array}{cc} \bar{\bbeta} & \balpha\end{array}\right]\frac{\left(\underline{\underline{\mu}}+\underline{\underline{J}}\right)}{2}\left[\begin{array}{c} \balpha\\ \bar{\bbeta} \end{array}\right]\right].\nonumber \\ \label{eq:Eq.B42}\end{aligned}$$ Here $\left|\balpha\right\rangle $, $\left|\bbeta\right\rangle $, $\left|\bgamma\right\rangle $ and $\left|\bepsilon\right\rangle $ denote fermionic coherent states, while $\balpha$, $\bbeta$, $\bgamma$ and $\bepsilon$ are the corresponding Grassmann variables, and $\rmd\underline{\gamma}$, $\rmd\underline{\beta},$ $\rmd\underline{\alpha}$ and $\rmd\underline{\epsilon}$ are the corresponding $2M$ integration measures. In order to obtain the identity given in (\[eq:Eq.B42\]) we have used an integration by parts and the properties of Grassmann calculus, as well as the eigenvalue properties for the fermionic coherent states in the form $\hat{\bm{a}}\left|\balpha\right\rangle =\balpha\left|\balpha\right\rangle $. We have also used the resolution of unity or completeness identity of the fermionic coherent states, together with the inner product property of coherent states which are given below: $$\begin{aligned} \int d\underline{\alpha}\left|\balpha\right\rangle \left\langle \balpha\right| & = & \mathbf{I},\label{eq:ResUnitCS}\\ \left\langle \bbeta\vert\balpha\right\rangle & = & \exp\left[\bar{\bbeta}\balpha-\left(\bar{\bbeta}\bbeta+\bar{\balpha}\balpha\right)/2\right].\label{eq:InnerProdCS}\end{aligned}$$ One important step is to express the variables $\left[\begin{array}{c} \bbeta\\ \bar{\balpha} \end{array}\right]$ in the form $\left[\begin{array}{c} \balpha\\ \bar{\bbeta} \end{array}\right],$ which is the one given in the exponential of the Gaussian operators. This is done by using the following identities, which use the derivative of the Gaussian operator and Grassmann calculus: $$\left[\begin{array}{c} \bbeta\\ \bar{\balpha} \end{array}\right]\exp\left[-\bar{\balpha}\balpha-\bar{\bbeta}\bbeta\right]=\left[\begin{array}{c} -\frac{\partial}{\partial\overline{\bbeta}}\\ \frac{\partial}{\partial\balpha} \end{array}\right]\exp\left[-\bar{\balpha}\balpha-\bar{\bbeta}\bbeta\right].$$ $$\begin{aligned} & & \fl\exp\left[-\bar{\balpha}\balpha-\bar{\bbeta}\bbeta\right]\left[\begin{array}{c} \frac{\partial}{\partial\overline{\bbeta}}\\ -\frac{\partial}{\partial\balpha} \end{array}\right]\exp\left[-\frac{1}{2}\left[\begin{array}{cc} \bar{\bbeta} & \balpha\end{array}\right]\left(\underline{\underline{\mu}}+\underline{\underline{J}}\right)\left[\begin{array}{c} \balpha\\ \bar{\bbeta} \end{array}\right]\right]\left[\begin{array}{cc} \bar{\bbeta} & \balpha\end{array}\right]\\ & & \fl=\exp\left[-\bar{\balpha}\balpha-\bar{\bbeta}\bbeta\right]\left[\underline{\underline{J}}\left(\underline{\underline{\mu}}+\underline{\underline{J}}\right)\left[\begin{array}{c} \balpha\\ \bar{\bbeta} \end{array}\right]\left[\begin{array}{cc} \bar{\bbeta} & \balpha\end{array}\right]-\underline{\underline{J}}\right]\\ & & \fl\times\exp\left[-\frac{1}{2}\left[\begin{array}{cc} \bar{\bbeta} & \balpha\end{array}\right]\left(\underline{\underline{\mu}}+\underline{\underline{J}}\right)\left[\begin{array}{c} \balpha\\ \bar{\bbeta} \end{array}\right]\right].\end{aligned}$$ Hence, using integration by parts and the above identities, (\[eq:Eq.B42\]) is obtained. Next, we wish to express the above identities in terms of Majorana operators. Thus without changing the order of the ladder operators, we use the identities that relate the Majorana operators with the ladder operators given in (\[eq:OpMajUn\]), as well as the expression given in (\[eq:YMat\]). We also multiply the right side of both sides of equation (\[eq:Eq.B42\]) by $2\underline{\underline{U_{0}}}^{-1}$, obtaining: $$\begin{aligned} \fl\Biggl\{\underline{\underline{U_{0}}}^{-1}\hat{\underline{\gamma}}:\hat{\underline{\gamma}}^{T}\hat{\Lambda}^{\left(u\right)}:\Biggr\} & = & \int\rmd\underline{\gamma}\rmd\underline{\beta}\rmd\underline{\alpha}\rmd\underline{\epsilon}\Biggl|\bgamma\Biggr\rangle\Biggl\langle\bepsilon\Biggr|\exp\left[-\bar{\balpha}\balpha-\bar{\bbeta}\bbeta-\frac{1}{2}\bar{\bgamma}\bgamma-\frac{1}{2}\bar{\bepsilon}\bepsilon+\bar{\bgamma}\bbeta+\bar{\balpha}\bepsilon\right]\nonumber \\ & & \times\left[-\rmi2\underline{\underline{J}}\underline{\underline{U_{0}}}^{-1}\left(2\underline{\underline{Y}}+\underline{\underline{\mathcal{I}}}\right)\underline{\underline{U_{0}}}\left[\begin{array}{c} \balpha\\ \bar{\bbeta} \end{array}\right]\left[\begin{array}{cc} \bar{\bbeta} & \balpha\end{array}\right]\underline{\underline{U_{0}}}^{-1}-2\underline{\underline{J}}\underline{\underline{U_{0}}}^{-1}\right]\nonumber \\ & & \times\exp\left[\frac{\rmi}{2}\left[\begin{array}{cc} \bar{\bbeta} & \balpha\end{array}\right]\underline{\underline{U_{0}}}^{-1}\left(2\underline{\underline{Y}}+\underline{\underline{\mathcal{I}}}\right)\underline{\underline{U_{0}}}\left[\begin{array}{c} \balpha\\ \bar{\bbeta} \end{array}\right]\right].\label{eq:MajOpT}\end{aligned}$$ We notice that the second and third lines of (\[eq:MajOpT\]) can be written in terms of the derivative of the un-normalized Gaussian operator itself with respect to $\underline{\underline{Y}}$. Hence we get the following result: $$\begin{aligned} \fl\Biggl\{\underline{\underline{U_{0}}}^{-1}\hat{\underline{\gamma}}:\hat{\underline{\gamma}}^{T}\hat{\Lambda}^{\left(u\right)}:\Biggr\} & = & \int\rmd\underline{\gamma}\rmd\underline{\beta}\rmd\underline{\alpha}\rmd\underline{\epsilon}\Biggl|\bgamma\Biggr\rangle\Biggl\langle\bepsilon\Biggr|\exp\left[-\bar{\balpha}\balpha-\bar{\bbeta}\bbeta-\frac{1}{2}\bar{\bgamma}\bgamma-\frac{1}{2}\bar{\bepsilon}\bepsilon+\bar{\bgamma}\bbeta+\bar{\balpha}\bepsilon\right]\nonumber \\ & \times & \left[\overline{\underline{\underline{I}}}\underline{\underline{U_{0}}}^{-1}\left(2\underline{\underline{Y}}+\underline{\underline{\mathcal{I}}}\right)\frac{\rmd}{\rmd\underline{\underline{Y}}}-2\underline{\underline{\overline{I}}}\underline{\underline{U_{0}}}^{-1}\right]\nonumber \\ & \times & \exp\left[\frac{\rmi}{2}\left[\begin{array}{cc} \bar{\bbeta} & \balpha\end{array}\right]\underline{\underline{U_{0}}}^{-1}\left(2\underline{\underline{Y}}+\underline{\underline{\mathcal{I}}}\right)\underline{\underline{U_{0}}}\left[\begin{array}{c} \balpha\\ \bar{\bbeta} \end{array}\right]\right]\\ & = & N\left[\overline{\underline{\underline{I}}}\underline{\underline{U_{0}}}^{-1}\left(2\underline{\underline{Y}}+\underline{\underline{\mathcal{I}}}\right)\frac{\rmd}{\rmd\underline{\underline{Y}}}-2\underline{\underline{\overline{I}}}\underline{\underline{U_{0}}}^{-1}\right]\hat{\Lambda}^{\left(u\right)}\left(\underline{\underline{Y}}\right).\label{eq:MajIdMPCS}\end{aligned}$$ In order to obtain the last line of (\[eq:MajIdMPCS\]) we have used the resolution of unity of coherent states given in (\[eq:ResUnitCS\]) and the inner product of fermionic coherent states given in (\[eq:InnerProdCS\]) in order to express the Majorana Gaussian operator as: $$\begin{aligned} \fl\hat{\Lambda}^{\left(u\right)}\left(\underline{\underline{Y}}\right) & = & \int\rmd\underline{\gamma}\rmd\underline{\beta}\rmd\underline{\alpha}\rmd\underline{\epsilon}\Biggl|\bgamma\Biggr\rangle\Biggl\langle\bepsilon\Biggr|\exp\left[-\bar{\balpha}\balpha-\bar{\bbeta}\bbeta-\frac{1}{2}\bar{\bgamma}\bgamma-\frac{1}{2}\bar{\bepsilon}\bepsilon+\bar{\bgamma}\bbeta+\bar{\balpha}\bepsilon\right]\nonumber \\ & \times & \exp\left[\frac{\rmi}{2}\left[\begin{array}{cc} \bar{\bbeta} & \balpha\end{array}\right]\underline{\underline{U_{0}}}^{-1}\left(2\underline{\underline{Y}}+\underline{\underline{\mathcal{I}}}\right)\underline{\underline{U_{0}}}\left[\begin{array}{c} \balpha\\ \bar{\bbeta} \end{array}\right]\right].\label{eq:MajGO_ResUnit}\end{aligned}$$ On multiplying on the left both sides of (\[eq:MajIdMPCS\]), by $\underline{\underline{U_{0}}}$, we get: $$\begin{aligned} \Biggl\{\hat{\underline{\gamma}}:\hat{\underline{\gamma}}^{T}\hat{\Lambda}^{\left(u\right)}:\Biggr\} & = & \left[\underline{\underline{U_{0}}}\overline{\underline{\underline{I}}}\underline{\underline{U_{0}}}^{-1}\left(2\underline{\underline{Y}}+\underline{\underline{\mathcal{I}}}\right)\frac{\rmd}{\rmd\underline{\underline{Y}}}-2\underline{\underline{U_{0}}}\underline{\underline{\overline{I}}}\underline{\underline{U_{0}}}^{-1}\right]\hat{\Lambda}^{\left(u\right)}\nonumber \\ & = & \rmi\left[-\underline{\underline{{\cal I}}}\left(2\underline{\underline{Y}}+\underline{\underline{\mathcal{I}}}\right)\frac{\rmd}{\rmd\underline{\underline{Y}}}+2\underline{\underline{{\cal I}}}\right]\hat{\Lambda}^{\left(u\right)}.\label{eq:DifIY}\end{aligned}$$ Here we have used that from the definition of $\underline{\underline{{\cal I}}}$ given in (\[eq:UnitFI\]) we get $-i\underline{\underline{{\cal I}}}=\underline{\underline{U_{0}}}\overline{\underline{\underline{I}}}\underline{\underline{U_{0}}}^{-1}$. Therefore we have proved the differential identity given in (\[eq:MajDiffIdY\]). Normally ordered products -------------------------- $$:\hat{\underline{\gamma}}\hat{\underline{\gamma}}^{T}\hat{\Lambda}^{\left(u\right)}:=\rmi\frac{d}{d\underline{\underline{Y}}}\hat{\Lambda}^{\left(u\right)}.$$ ***Proof***. Analogous to the previous case we make use of the eigenvalue property of the fermionic coherent states as well as the completeness identity and the inner product of coherent states. Thus, we get: $$\begin{aligned} \fl:\underline{\widehat{a}}\widehat{\underline{a}}^{\dagger}\hat{\Lambda}^{\left(u\right)}: & = & \int\rmd\underline{\gamma}\rmd\underline{\beta}\rmd\underline{\alpha}\rmd\underline{\epsilon}\Biggl|\bgamma\Biggr\rangle\Biggl\langle\bepsilon\Biggr|\exp\left[-\bar{\balpha}\balpha-\bar{\bbeta}\bbeta-\frac{1}{2}\bar{\bgamma}\bgamma-\frac{1}{2}\bar{\bepsilon}\bepsilon+\bar{\bgamma}\bbeta+\bar{\balpha}\bepsilon\right]\\ & & \times\left[\begin{array}{c} \balpha\\ \bar{\bbeta} \end{array}\right]\left[\begin{array}{cc} \bar{\bbeta} & \balpha\end{array}\right]\exp\left[-\frac{1}{2}\left[\begin{array}{cc} \bar{\bbeta} & \balpha\end{array}\right]\left(\underline{\underline{\mu}}+\underline{\underline{J}}\right)\left[\begin{array}{c} \balpha\\ \bar{\bbeta} \end{array}\right]\right].\end{aligned}$$ Next, without changing the order of the Grassmann variables, we can write the above expression in terms of Majorana operators using equations (\[eq:OpMajUn\]) and (\[eq:YMat\]). We also multiply both sides of the above equation on the left by $2\underline{\underline{U_{0}}}$ and on the right by $\underline{\underline{U_{0}}}^{-1}$, thus obtaining: $$\begin{aligned} \fl:\hat{\underline{\gamma}}\hat{\underline{\gamma}}^{T}\hat{\Lambda}^{\left(u\right)}: & = & \int\rmd\underline{\gamma}\rmd\underline{\beta}\rmd\underline{\alpha}\rmd\underline{\epsilon}\Biggl|\bgamma\Biggr\rangle\Biggl\langle\bepsilon\Biggr|\exp\left[-\bar{\balpha}\balpha-\bar{\bbeta}\bbeta-\frac{1}{2}\bar{\bgamma}\bgamma-\frac{1}{2}\bar{\bepsilon}\bepsilon+\bar{\bgamma}\bbeta+\bar{\balpha}\bepsilon\right]\\ & & \times2\underline{\underline{U_{0}}}\left[\begin{array}{c} \balpha\\ \bar{\bbeta} \end{array}\right]\left[\begin{array}{cc} \bar{\bbeta} & \balpha\end{array}\right]\underline{\underline{U_{0}}}^{-1}\exp\left[\frac{\rmi}{2}\left[\begin{array}{cc} \bar{\bbeta} & \balpha\end{array}\right]\underline{\underline{U_{0}}}^{-1}\left(2\underline{\underline{Y}}+\underline{\underline{\mathcal{I}}}\right)\underline{\underline{U_{0}}}\left[\begin{array}{c} \balpha\\ \bar{\bbeta} \end{array}\right]\right]\\ & = & \int\rmd\underline{\gamma}\rmd\underline{\beta}\rmd\underline{\alpha}\rmd\underline{\epsilon}\Biggl|\bgamma\Biggr\rangle\Biggl\langle\bepsilon\Biggr|\exp\left[-\bar{\balpha}\balpha-\bar{\bbeta}\bbeta-\frac{1}{2}\bar{\bgamma}\bgamma-\frac{1}{2}\bar{\bepsilon}\bepsilon+\bar{\bgamma}\bbeta+\bar{\balpha}\bepsilon\right]\\ & & \times\left(\frac{d}{d\underline{\underline{Y}}}\exp\left[\frac{\rmi}{2}\left[\begin{array}{cc} \bar{\bbeta} & \balpha\end{array}\right]\underline{\underline{U_{0}}}^{-1}\left(2\underline{\underline{Y}}+\underline{\underline{\mathcal{I}}}\right)\underline{\underline{U_{0}}}\left[\begin{array}{c} \balpha\\ \bar{\bbeta} \end{array}\right]\right]\right).\\ & = & \rmi\frac{d}{d\underline{\underline{Y}}}\hat{\Lambda}^{\left(u\right)}.\end{aligned}$$ Here we have expressed the results of the first two lines as the derivative of the Gaussian operator an we have used the expression of the Majorana Gaussian operator given in (\[eq:MajGO\_ResUnit\]). Therefore we have proved the differential identity given in (\[eq:UnNormNormalDifId\]). Anti-normally ordered products ------------------------------- $$\left\{ \underline{\widehat{\gamma}}\widehat{\underline{\gamma}}^{T}\hat{\Lambda}^{\left(u\right)}\right\} =\rmi\underline{\underline{\mathcal{I}}}\left\{ \left(2\underline{\underline{Y}}+\underline{\underline{\mathcal{I}}}\right)\frac{\rmd}{\rmd\underline{\underline{Y}}}\hat{\Lambda}^{u}-2\hat{\Lambda}^{u}\right\} \left(2\underline{\underline{Y}}+\underline{\underline{\mathcal{I}}}\right)\underline{\underline{\mathcal{I}}}.$$ ***Proof.*** As in the previous cases we use the eigenvalue property of the fermionic coherent states as well as the completeness identity and the inner product of coherent states in order to write $\left\{ \underline{\widehat{a}}\widehat{\underline{a}}^{\dagger}\hat{\Lambda}^{\left(u\right)}\right\} $ in terms of Grassmann variables, obtaining: $$\begin{aligned} \fl\left\{ \underline{\widehat{a}}\widehat{\underline{a}}^{\dagger}\hat{\Lambda}^{\left(u\right)}\right\} & = & \int\rmd\underline{\gamma}\rmd\underline{\beta}\rmd\underline{\alpha}\rmd\underline{\epsilon}\Biggl|\bgamma\Biggr\rangle\Biggl\langle\bgamma\Biggr|\Biggl\{\underline{\widehat{a}}\widehat{\underline{a}}^{\dagger}\Biggl|\bbeta\Biggr\rangle\Biggl\langle\bbeta\Biggr|\widehat{\Lambda}^{\left(u\right)}\Biggl|\balpha\Biggr\rangle\Biggl\langle\balpha\Biggr|:\Biggr\}\Biggl|\bepsilon\Biggr\rangle\Biggl\langle\bepsilon\Biggr|\nonumber \\ & = & \int\rmd\underline{\gamma}\rmd\underline{\beta}\rmd\underline{\alpha}\rmd\underline{\epsilon}\Biggl|\bgamma\Biggr\rangle\Biggl\langle\bepsilon\Biggr|\exp\left[-\bar{\balpha}\balpha-\bar{\bbeta}\bbeta-\frac{1}{2}\bar{\bgamma}\bgamma-\frac{1}{2}\bar{\bepsilon}\bepsilon+\bar{\bgamma}\bbeta+\bar{\balpha}\bepsilon\right]\nonumber \\ & & \times\exp\left[-\frac{1}{2}\left[\begin{array}{cc} \bar{\bbeta} & \balpha\end{array}\right]\left(\underline{\underline{\mu}}+\underline{\underline{J}}\right)\left[\begin{array}{c} \balpha\\ \bar{\bbeta} \end{array}\right]\right]\left[\begin{array}{c} \bbeta\\ \bar{\balpha} \end{array}\right]\left[\begin{array}{cc} \bar{\balpha} & \bbeta\end{array}\right].\label{eq:AntiNormalDPGv}\end{aligned}$$ Next we wish to express the variables $\left[\begin{array}{c} \bbeta\\ \bar{\balpha} \end{array}\right]$ and $\left[\begin{array}{cc} \bar{\balpha} & \bbeta\end{array}\right]$ in the form given in the exponential of the Gaussian operator. In this case we will also use integration by parts but it will be performed twice. To this end we will use the following identities, which make use of Grassmann calculus: $$\begin{aligned} & & \fl\exp\left[-\frac{1}{2}\left[\begin{array}{cc} \bar{\bbeta} & \balpha\end{array}\right]\left(\underline{\underline{\mu}}+\underline{\underline{J}}\right)\left[\begin{array}{c} \balpha\\ \bar{\bbeta} \end{array}\right]\right]\left[\begin{array}{c} \bbeta\\ \bar{\balpha} \end{array}\right]\left[\begin{array}{cc} \bar{\balpha} & \bbeta\end{array}\right]\exp\left[-\bar{\balpha}\balpha-\bar{\bbeta}\bbeta\right]=\\ & & \fl\exp\left[-\frac{1}{2}\left[\begin{array}{cc} \bar{\bbeta} & \balpha\end{array}\right]\left(\underline{\underline{\mu}}+\underline{\underline{J}}\right)\left[\begin{array}{c} \balpha\\ \bar{\bbeta} \end{array}\right]\right]\left[\begin{array}{c} -\frac{\partial}{\partial\overline{\bbeta}}\\ \frac{\partial}{\partial\balpha} \end{array}\right]\left[\begin{array}{cc} \frac{\partial}{\partial\balpha} & -\frac{\partial}{\partial\overline{\bbeta}}\end{array}\right]\exp\left[-\bar{\balpha}\balpha-\bar{\bbeta}\bbeta\right]\end{aligned}$$ $$\begin{aligned} & & \fl\exp\left[-\bar{\balpha}\balpha-\bar{\bbeta}\bbeta\right]\left[\begin{array}{c} -\frac{\partial}{\partial\overline{\bbeta}}\\ \frac{\partial}{\partial\balpha} \end{array}\right]\left[\begin{array}{cc} \frac{\partial}{\partial\balpha} & -\frac{\partial}{\partial\overline{\bbeta}}\end{array}\right]\exp\left[-\frac{1}{2}\left[\begin{array}{cc} \bar{\bbeta} & \balpha\end{array}\right]\left(\underline{\underline{\mu}}+\underline{\underline{J}}\right)\left[\begin{array}{c} \balpha\\ \bar{\bbeta} \end{array}\right]\right]\\ & & \fl=\exp\left[-\bar{\balpha}\balpha-\bar{\bbeta}\bbeta\right]\left[\underline{\underline{J}}\left(\underline{\underline{\mu}}+\underline{\underline{J}}\right)\left[\begin{array}{c} \balpha\\ \bar{\bbeta} \end{array}\right]\left[\begin{array}{cc} \bar{\bbeta} & \balpha\end{array}\right]-\underline{\underline{J}}\right]\left(\underline{\underline{\mu}}+\underline{\underline{J}}\right)\underline{\underline{J}}\\ & & \fl\times\exp\left[-\frac{1}{2}\left[\begin{array}{cc} \bar{\bbeta} & \balpha\end{array}\right]\left(\underline{\underline{\mu}}+\underline{\underline{J}}\right)\left[\begin{array}{c} \balpha\\ \bar{\bbeta} \end{array}\right]\right].\end{aligned}$$ The above expressions allow us to perform an integration by parts twice in (\[eq:AntiNormalDPGv\]) obtaining: $$\begin{aligned} \fl\left\{ \underline{\widehat{a}}\widehat{\underline{a}}^{\dagger}\hat{\Lambda}^{\left(u\right)}\right\} & = & \int\rmd\underline{\gamma}\rmd\underline{\beta}\rmd\underline{\alpha}\rmd\underline{\epsilon}\Biggl|\bgamma\Biggr\rangle\Biggl\langle\bepsilon\Biggr|\exp\left[-\bar{\balpha}\balpha-\bar{\bbeta}\bbeta-\frac{1}{2}\bar{\bgamma}\bgamma-\frac{1}{2}\bar{\bepsilon}\bepsilon+\bar{\bgamma}\bbeta+\bar{\balpha}\bepsilon\right]\\ & & \times\left[\underline{\underline{J}}\left(\underline{\underline{\mu}}+\underline{\underline{J}}\right)\left[\begin{array}{c} \balpha\\ \bar{\bbeta} \end{array}\right]\left[\begin{array}{cc} \bar{\bbeta} & \balpha\end{array}\right]-\underline{\underline{J}}\right]\left(\underline{\underline{\mu}}+\underline{\underline{J}}\right)\underline{\underline{J}}\\ & & \times\exp\left[-\frac{1}{2}\left[\begin{array}{cc} \bar{\bbeta} & \balpha\end{array}\right]\left(\underline{\underline{\mu}}+\underline{\underline{J}}\right)\left[\begin{array}{c} \balpha\\ \bar{\bbeta} \end{array}\right]\right].\end{aligned}$$ Next since we wish to express the identity in terms of Majorana operators and an anti-symmetric matrix, without changing the order of the ladder operators, we use the identities that relate the ladder operators with the Majorana operators given in (\[eq:OpMajUn\]), as well the relation given in (\[eq:YMat\]) and (\[eq:UnitFI\]). We also multiply both sides of the above equation on the left by $2\underline{\underline{U_{0}}}$ and on the right by $\underline{\underline{U_{0}}}^{-1}$, thus obtaining: $$\begin{aligned} \fl\left\{ \underline{\widehat{\gamma}}\widehat{\underline{\gamma}}^{T}\hat{\Lambda}^{\left(u\right)}\right\} & = & \int\rmd\underline{\gamma}\rmd\underline{\beta}\rmd\underline{\alpha}\rmd\underline{\epsilon}\Biggl|\bgamma\Biggr\rangle\Biggl\langle\bepsilon\Biggr|\exp\left[-\bar{\balpha}\balpha-\bar{\bbeta}\bbeta-\frac{1}{2}\bar{\bgamma}\bgamma-\frac{1}{2}\bar{\bepsilon}\bepsilon+\bar{\bgamma}\bbeta+\bar{\balpha}\bepsilon\right]\\ & & \times2\left[\underline{\underline{\mathcal{I}}}\left(2\underline{\underline{Y}}+\underline{\underline{\mathcal{I}}}\right)\underline{\underline{U_{0}}}\left[\begin{array}{c} \balpha\\ \bar{\bbeta} \end{array}\right]\left[\begin{array}{cc} \bar{\bbeta} & \balpha\end{array}\right]\underline{\underline{U_{0}}}^{-1}-\rmi\underline{\underline{\mathcal{I}}}\right]\left(2\underline{\underline{Y}}+\underline{\underline{\mathcal{I}}}\right)\underline{\underline{\mathcal{I}}}\\ & & \times\exp\left[\frac{\rmi}{2}\left[\begin{array}{cc} \bar{\bbeta} & \balpha\end{array}\right]\underline{\underline{U_{0}}}^{-1}\left(2\underline{\underline{Y}}+\underline{\underline{\mathcal{I}}}\right)\underline{\underline{U_{0}}}\left[\begin{array}{c} \balpha\\ \bar{\bbeta} \end{array}\right]\right].\end{aligned}$$ As in the previous cases we notice that the second and third line of the above expressions can be expressed in terms of the derivative of the un-normalized Gaussian operator with respect to $\underline{\underline{Y}}$. We also use the expression for the un-normalized Gaussian operators given in (\[eq:MajGO\_ResUnit\]). Therefore we get: $$\begin{aligned} \left\{ \underline{\widehat{\gamma}}\widehat{\underline{\gamma}}^{T}\hat{\Lambda}^{\left(u\right)}\right\} & = & \rmi\underline{\underline{\mathcal{I}}}\left\{ \left(2\underline{\underline{Y}}+\underline{\underline{\mathcal{I}}}\right)\frac{\rmd}{\rmd\underline{\underline{Y}}}\hat{\Lambda}^{u}-2\hat{\Lambda}^{u}\right\} \left(2\underline{\underline{Y}}+\underline{\underline{\mathcal{I}}}\right)\underline{\underline{\mathcal{I}}}.\end{aligned}$$ Thus we have proved the differential identity given in (\[eq:AntiNormDifIdUnGO\]). Normalized Majorana differential identities\[sec:AppendixDiffIdNormGO\] ======================================================================= Here we consider the normalized Gaussian operators given in (\[eq:NormGOX\]). We also used the change of variables given in (\[eq:ChangeVarYX\]). First mixed product ------------------- $$\Biggl\{\hat{\underline{\gamma}}:\hat{\underline{\gamma}}^{T}\hat{\Lambda}:\Biggr\}=\rmi\left(\underline{\underline{X}}+\underline{\underline{{\cal I}}}\right)\frac{\rmd\hat{\Lambda}\left(\underline{\underline{X}}\right)}{\rmd\underline{\underline{X}}}\left(\underline{\underline{X}}-\underline{\underline{\mathcal{I}}}\right)-\rmi\left(\underline{\underline{X}}-\underline{\underline{{\cal I}}}\right)\hat{\Lambda}\left(\underline{\underline{X}}\right).$$ ***Proof.*** We make a change of variables using (\[eq:ChangeVarYX\]) on the differential identities given in (\[eq:MajDiffIdY\]) obtaining: $$\fl\Biggl\{\hat{\underline{\gamma}}:\hat{\underline{\gamma}}^{T}\hat{\Lambda}:\Biggr\}=\rmi N\left(\underline{\underline{X}}\right)\left[-\underline{\underline{{\cal I}}}\left(2\left(\underline{\underline{X}}-\underline{\underline{\mathcal{I}}}\right)^{-1}-\underline{\underline{\mathcal{I}}}\right)\frac{\rmd}{\rmd\underline{\underline{Y}}}+2\underline{\underline{{\cal I}}}\right]\hat{\Lambda}^{\left(u\right)}\left(\underline{\underline{X}}\right).\label{eq:DifIndYX}$$ Next, we use the chain rule in order to change the derivative, which is: $$\fl\frac{\rmd\hat{\Lambda}^{\left(u\right)}}{\rmd\underline{\underline{Y}}}=\frac{\rmd\hat{\Lambda}^{\left(u\right)}}{\rmd\underline{\underline{X}}}\frac{\rmd\underline{\underline{X}}}{\rmd\underline{\underline{Y}}}=-\left(\underline{\underline{X}}-\underline{\underline{\mathcal{I}}}\right)\frac{\rmd\hat{\Lambda}^{\left(u\right)}}{\rmd\underline{\underline{X}}}\left(\underline{\underline{X}}-\underline{\underline{\mathcal{I}}}\right).\label{eq:RelDerGOYX}$$ On substituting the above identities in (\[eq:DifIndYX\]) and on simplifying terms we get: $$\begin{aligned} \fl\Biggl\{\hat{\underline{\gamma}}:\hat{\underline{\gamma}}^{T}\hat{\Lambda}:\Biggr\} & = & \rmi N\left(\underline{\underline{X}}\right)\left[2\underline{\underline{{\cal I}}}\hat{\Lambda}^{\left(u\right)}\left(\underline{\underline{X}}\right)+\left(\underline{\underline{X}}+\underline{\underline{{\cal I}}}\right)\frac{d\hat{\Lambda}^{\left(u\right)}\left(\underline{\underline{X}}\right)}{d\underline{\underline{X}}}\left(\underline{\underline{X}}-\underline{\underline{\mathcal{I}}}\right)\right].\label{eq:DifIdXUnGO}\end{aligned}$$ We now wish to relate the derivative of the un-normalized Gaussian operator with the derivative of the normalized one. Hence we use the following relation: $$\begin{aligned} \fl\frac{\rmd\hat{\Lambda}\left(\underline{\underline{X}}\right)}{\rmd\underline{\underline{X}}} & = & \frac{\rmd}{\rmd\underline{\underline{X}}}N\left(\underline{\underline{X}}\right)\hat{\Lambda}^{\left(u\right)}\left(\underline{\underline{X}}\right)=N\left(\underline{\underline{X}}\right)\hat{\Lambda}^{\left(u\right)}\left(\underline{\underline{X}}\right)\left(\underline{\underline{X}}-\underline{\underline{\mathcal{I}}}\right)^{-1}+N\left(\underline{\underline{X}}\right)\frac{\rmd\hat{\Lambda}^{\left(u\right)}\left(\underline{\underline{X}}\right)}{\rmd\underline{\underline{X}}}.\label{eq:RelDer}\end{aligned}$$ Here we have used the following result for the derivative of the square root of the determinant of an antisymmetric matrix: $$\fl\frac{\rmd N\left(\underline{\underline{X}}\right)}{\rmd\underline{\underline{X}}}=\frac{\rmd}{\rmd\underline{\underline{X}}}\frac{1}{2^{M}}\sqrt{\det\left[\underline{\underline{X}}\right]}=\frac{1}{2^{M}}\sqrt{\det\left[\underline{\underline{X}}\right]}\left(\underline{\underline{X}}-\underline{\underline{\mathcal{I}}}\right)^{-1}=N\left(\underline{\underline{X}}\right)\left(\underline{\underline{X}}-\underline{\underline{\mathcal{I}}}\right)^{-1}.$$ From (\[eq:RelDer\]) we get: $$\frac{\rmd\hat{\Lambda}^{\left(u\right)}\left(\underline{\underline{X}}\right)}{\rmd\underline{\underline{X}}}=\frac{1}{N\left(\underline{\underline{X}}\right)}\frac{\rmd\hat{\Lambda}\left(\underline{\underline{X}}\right)}{\rmd\underline{\underline{X}}}-\hat{\Lambda}^{\left(u\right)}\left(\underline{\underline{X}}\right)\left(\underline{\underline{X}}-\underline{\underline{\mathcal{I}}}\right)^{-1}.\label{eq:RelDersNUn}$$ On substituting this result on (\[eq:DifIdXUnGO\]) and simplifying we get: $$\begin{aligned} \Biggl\{\hat{\underline{\gamma}}:\hat{\underline{\gamma}}^{T}\hat{\Lambda}:\Biggr\} & = & \rmi\left(\underline{\underline{X}}+\underline{\underline{{\cal I}}}\right)\frac{\rmd\hat{\Lambda}\left(\underline{\underline{X}}\right)}{\rmd\underline{\underline{X}}}\left(\underline{\underline{X}}-\underline{\underline{\mathcal{I}}}\right)-\rmi\left(\underline{\underline{X}}-\underline{\underline{{\cal I}}}\right)\hat{\Lambda}\left(\underline{\underline{X}}\right).\end{aligned}$$ Therefore we have proved the differential identity given in (\[eq:MajMixProd\]). Second mixed product -------------------- $$:\hat{\underline{\gamma}}\Biggl\{\hat{\underline{\gamma}}^{T}\hat{\Lambda}\Biggr\}:=\rmi\left[\underline{\underline{X}}^{-}\frac{\rmd\hat{\Lambda}}{\rmd\underline{\underline{X}}}-\hat{\Lambda}\right]\underline{\underline{X}}^{+}$$ ***Proof.*** We can derive this fourth mixed identity from the third one. We know the orderings of both identities as given in (\[eq:ExpFMP\] and \[eq:ExpSMP\]). Utilizing those orderings and Fermi commutation relation in (\[eq:FerComm\]) we get: $$:\hat{\underline{\gamma}}\Biggl\{\hat{\underline{\gamma}}^{T}\hat{\Lambda}\Biggr\}:=-\Bigl\{\hat{\underline{\gamma}}:\hat{\underline{\gamma}}^{T}\hat{\Lambda}:\Bigr\}^{T}-2i\hat{\Lambda}\underline{\underline{{\cal I}}}$$ This will give us the new fourth identity as in (\[eq:Majmix2\]) Normally ordered products -------------------------- $$:\hat{\underline{\gamma}}\hat{\underline{\gamma}}^{T}\hat{\Lambda}:=\rmi\left(\underline{\underline{X}}-\underline{\underline{\mathcal{I}}}\right)\hat{\Lambda}\left(\underline{\underline{X}}\right)-\rmi\left(\underline{\underline{X}}-\underline{\underline{\mathcal{I}}}\right)\frac{\rmd\hat{\Lambda}\left(\underline{\underline{X}}\right)}{\rmd\underline{\underline{X}}}\left(\underline{\underline{X}}-\underline{\underline{\mathcal{I}}}\right).$$ **Proof**. We use $\hat{\Lambda}\left(\underline{\underline{X}}\right)=N\left(\underline{\underline{X}}\right)\hat{\Lambda}^{\left(u\right)}\left(\underline{\underline{X}}\right)$ and we also use the expressions given in (\[eq:RelDerGOYX\]) and (\[eq:RelDersNUn\]) in order to express the derivative in terms of the normalized Gaussian operators which is $$\begin{aligned} \fl\frac{\rmd}{\rmd\underline{\underline{Y}}}\hat{\Lambda}^{\left(u\right)} & = & -\frac{1}{N\left(\underline{\underline{X}}\right)}\left(\underline{\underline{X}}-\underline{\underline{\mathcal{I}}}\right)\frac{\rmd\hat{\Lambda}\left(\underline{\underline{X}}\right)}{\rmd\underline{\underline{X}}}\left(\underline{\underline{X}}-\underline{\underline{\mathcal{I}}}\right)+\left(\underline{\underline{X}}-\underline{\underline{\mathcal{I}}}\right)\hat{\Lambda}^{\left(u\right)}\left(\underline{\underline{X}}\right).\label{eq:derUnNGOXY}\end{aligned}$$ On substituting the above result in (\[eq:UnNormNormalDifId\]) we get: $$\begin{aligned} :\hat{\underline{\gamma}}\hat{\underline{\gamma}}^{T}\hat{\Lambda}: & = & \rmi\left(\underline{\underline{X}}-\underline{\underline{\mathcal{I}}}\right)\hat{\Lambda}\left(\underline{\underline{X}}\right)-\rmi\left(\underline{\underline{X}}-\underline{\underline{\mathcal{I}}}\right)\frac{\rmd\hat{\Lambda}\left(\underline{\underline{X}}\right)}{\rmd\underline{\underline{X}}}\left(\underline{\underline{X}}-\underline{\underline{\mathcal{I}}}\right).\end{aligned}$$ Thus we have proved the differential identity given in (\[eq:NormalNormMajId\]). Anti-normally ordered products ------------------------------- $$\left\{ \underline{\widehat{\gamma}}\widehat{\underline{\gamma}}^{T}\hat{\Lambda}\right\} =\rmi\left[-\left(\underline{\underline{\mathcal{I}}}+\underline{\underline{X}}\right)\frac{\rmd\hat{\Lambda}\left(\underline{\underline{X}}\right)}{\rmd\underline{\underline{X}}}\left(\underline{\underline{\mathcal{I}}}+\underline{\underline{X}}\right)+\hat{\Lambda}\left(\underline{\underline{X}}\right)\left(\underline{\underline{\mathcal{I}}}+\underline{\underline{X}}\right)\right].$$ **Proof**. As in the previous cases we use $\hat{\Lambda}\left(\underline{\underline{X}}\right)=N\left(\underline{\underline{X}}\right)\hat{\Lambda}^{\left(u\right)}\left(\underline{\underline{X}}\right)$ and we also make a change of variables using (\[eq:ChangeVarYX\]) on the differential identities given in (\[eq:AntiNormDifIdUnGO\]) obtaining: $$\fl\left\{ \underline{\widehat{\gamma}}\widehat{\underline{\gamma}}^{T}\hat{\Lambda}\right\} =\rmi N\left(\underline{\underline{X}}\right)\underline{\underline{\mathcal{I}}}\left\{ \left(2\left(\underline{\underline{X}}-\underline{\underline{\mathcal{I}}}\right)^{-1}-\underline{\underline{\mathcal{I}}}\right)\frac{\rmd}{\rmd\underline{\underline{Y}}}-2\right\} \hat{\Lambda}^{\left(u\right)}\left(\underline{\underline{X}}\right)\left(2\left(\underline{\underline{X}}-\underline{\underline{\mathcal{I}}}\right)^{-1}-\underline{\underline{\mathcal{I}}}\right)\underline{\underline{\mathcal{I}}}.$$ Next, we express the derivative in terms of the Gaussian operators and the $\underline{\underline{X}}$ matrix using the identity given in (\[eq:derUnNGOXY\]) obtaining: $$\begin{aligned} & & \fl\left\{ \underline{\widehat{\gamma}}\widehat{\underline{\gamma}}^{T}\hat{\Lambda}\right\} =\rmi\underline{\underline{\mathcal{I}}}\left\{ \left(2\left(\underline{\underline{X}}-\underline{\underline{\mathcal{I}}}\right)^{-1}-\underline{\underline{\mathcal{I}}}\right)\left[-\left(\underline{\underline{X}}-\underline{\underline{\mathcal{I}}}\right)\frac{\rmd\hat{\Lambda}\left(\underline{\underline{X}}\right)}{\rmd\underline{\underline{X}}}\left(\underline{\underline{X}}-\underline{\underline{\mathcal{I}}}\right)+\left(\underline{\underline{X}}-\underline{\underline{\mathcal{I}}}\right)\hat{\Lambda}\left(\underline{\underline{X}}\right)\right]\right.\\ & & \left.-2\hat{\Lambda}\left(\underline{\underline{X}}\right)\right\} \left(2\left(\underline{\underline{X}}-\underline{\underline{\mathcal{I}}}\right)^{-1}-\underline{\underline{\mathcal{I}}}\right)\underline{\underline{\mathcal{I}}}.\end{aligned}$$ On simplifying terms and using that $\underline{\underline{\mathcal{I}}}\mathbf{\underline{\underline{\mathcal{I}}}}=-\underline{\underline{I}}$ we get: $$\begin{aligned} \left\{ \underline{\widehat{\gamma}}\widehat{\underline{\gamma}}^{T}\hat{\Lambda}\right\} & = & \rmi\left[-\left(\underline{\underline{\mathcal{I}}}+\underline{\underline{X}}\right)\frac{\rmd\hat{\Lambda}\left(\underline{\underline{X}}\right)}{\rmd\underline{\underline{X}}}\left(\underline{\underline{\mathcal{I}}}+\underline{\underline{X}}\right)+\hat{\Lambda}\left(\underline{\underline{X}}\right)\left(\underline{\underline{\mathcal{I}}}+\underline{\underline{X}}\right)\right].\end{aligned}$$ Therefore we have proved the differential identity given in (\[eq:AntiNormalDifId\]). Majorana differential identities for unordered products\[sec:AppendixDiffIdunorderedGO-1\] ========================================================================================== In this Section we give the detailed calculations for unordered Majorana differential identities. These are derived using the ordered identities given in \[sec:AppendixDiffIdNormGO\]. Unordered left and right products --------------------------------- $$\hat{\underline{\gamma}}\hat{\underline{\gamma}}^{T}\hat{\Lambda}=\rmi\left[\left(1+\rmi\underline{\underline{x}}\right)\frac{d\hat{\Lambda}}{d\underline{\underline{x}}}\left(1-\rmi\underline{\underline{x}}\right)-\hat{\Lambda}\left(\underline{\underline{x}}+\rmi\right)\right].\label{eq:Un-OrdLP}$$ **Proof**. The explicit expression for $\hat{\underline{\gamma}}\hat{\underline{\gamma}}^{T}\hat{\Lambda}$ in terms of the Fermi operators is given in (\[eq:ExpUn-Or\]). The idea is to find the appropriate combination between the four different expressions of the ordered products of Majorana operators and the Gaussian operator that gives the expression given in (\[eq:ExpUn-Or\]). Each ordered product will be used in order to obtain the products of only Fermi creation operators with the Gaussian operators, or Fermi annihilation operators and the Gaussian operators or the two different product of Fermi creation and annihilation operator with the Gaussian operator in the left side. In order to do this we will use the real antisymmetric matrix $\underline{\underline{{\cal I}}}$. We first obtain the combination of the anti-normal product that will give the matrix form with only product of Fermi annihilation operators and the Gaussian one. Hence on pre and post multiplying (\[eq:ExpANOP\]) by $\underline{\underline{{\cal I}}}$, we get: $$\begin{aligned} & & \fl\underline{\underline{{\cal I}}}\left\{ \hat{\gamma}_{\mu}\hat{\gamma}_{\upsilon}\hat{\Lambda}\right\} \underline{\underline{{\cal I}}}=\nonumber \\ & & \fl\left(\begin{array}{cc} \widehat{a}_{i}\widehat{a}_{j}\hat{\Lambda}+\hat{\Lambda}\widehat{a}_{i}^{\dagger}\widehat{a}_{j}^{\dagger}-\widehat{a}_{i}\hat{\Lambda}\widehat{a}_{j}^{\dagger}+\widehat{a}_{j}\hat{\Lambda}\widehat{a}_{i}^{\dagger} & -i\widehat{a}_{i}\widehat{a}_{j}\hat{\Lambda}+i\hat{\Lambda}\widehat{a}_{i}^{\dagger}\widehat{a}_{j}^{\dagger}-i\widehat{a}_{i}\hat{\Lambda}\widehat{a}_{j}^{\dagger}-i\widehat{a}_{j}\hat{\Lambda}\widehat{a}_{i}^{\dagger}\\ -i\widehat{a}_{i}\widehat{a}_{j}\hat{\Lambda}+i\hat{\Lambda}\widehat{a}_{i}^{\dagger}\widehat{a}_{j}^{\dagger}+i\widehat{a}_{i}\hat{\Lambda}\widehat{a}_{j}^{\dagger}+i\widehat{a}_{j}\hat{\Lambda}\widehat{a}_{i}^{\dagger} & -\widehat{a}_{i}\widehat{a}_{j}\hat{\Lambda}-\hat{\Lambda}\widehat{a}_{i}^{\dagger}\widehat{a}_{j}^{\dagger}-\widehat{a}_{i}\hat{\Lambda}\widehat{a}_{j}^{\dagger}+\widehat{a}_{j}\hat{\Lambda}\widehat{a}_{i}^{\dagger} \end{array}\right).\label{eq:IpreIpost}\end{aligned}$$ Now adding (\[eq:ExpANOP\]) and (\[eq:IpreIpost\]) leads to: $$\begin{aligned} & & B_{1}=\left(\begin{array}{cc} 2\widehat{a}_{i}\widehat{a}_{j}\hat{\Lambda}+2\hat{\Lambda}\widehat{a}_{i}^{\dagger}\widehat{a}_{j}^{\dagger} & -2\rmi\widehat{a}_{i}\widehat{a}_{j}\hat{\Lambda}+2\rmi\hat{\Lambda}\widehat{a}_{i}^{\dagger}\widehat{a}_{j}^{\dagger}\\ -2\rmi\widehat{a}_{i}\widehat{a}_{j}\hat{\Lambda}+2\rmi\hat{\Lambda}\widehat{a}_{i}^{\dagger}\widehat{a}_{j}^{\dagger} & -2\widehat{a}_{i}\widehat{a}_{j}\hat{\Lambda}-2\hat{\Lambda}\widehat{a}_{i}^{\dagger}\widehat{a}_{j}^{\dagger} \end{array}\right),\label{eq:B1}\end{aligned}$$ where $B_{1}=\left\{ \hat{\gamma}_{\mu}\hat{\gamma}_{\upsilon}\hat{\Lambda}\right\} +\underline{\underline{{\cal I}}}\left\{ \hat{\gamma}_{\mu}\hat{\gamma}_{\upsilon}\hat{\Lambda}\right\} \underline{\underline{{\cal I}}}.$ We notice that we can obtain the required product of Fermi annihilation operators and $\hat{\Lambda}$ from the following expression: $$\begin{aligned} \frac{B_{1}-\rmi B_{1}\underline{\underline{{\cal I}}}}{4} & = & \left(\begin{array}{cc} \widehat{a}_{i}\widehat{a}_{j}\hat{\Lambda} & -i\widehat{a}_{i}\widehat{a}_{j}\hat{\Lambda}\\ -i\widehat{a}_{i}\widehat{a}_{j}\hat{\Lambda} & -\widehat{a}_{i}\widehat{a}_{j}\hat{\Lambda} \end{array}\right).\label{eq:antinormal_part}\end{aligned}$$ In a similar way we wish to obtain a matrix of the product of Fermi creation operators with $\hat{\Lambda}$. In this case we consider (\[eq:ENOP\]) and perform the same procedures as described above obtaining: $$B_{2}=\left(\begin{array}{cc} 2\widehat{a}_{i}^{\dagger}\widehat{a}_{j}^{\dagger}\hat{\Lambda}+2\hat{\Lambda}\widehat{a}_{i}\widehat{a}_{j} & -\rmi\left[-2\widehat{a}_{i}^{\dagger}\widehat{a}_{j}^{\dagger}\hat{\Lambda}+2\hat{\Lambda}\widehat{a}_{i}\widehat{a}_{j}\right]\\ -\rmi\left[-2\widehat{a}_{i}^{\dagger}\widehat{a}_{j}^{\dagger}\hat{\Lambda}+2\hat{\Lambda}\widehat{a}_{i}\widehat{a}_{j}\right] & -2\widehat{a}_{i}^{\dagger}\widehat{a}_{j}^{\dagger}\hat{\Lambda}-2\hat{\Lambda}\widehat{a}_{i}\widehat{a}_{j} \end{array}\right),$$ where $B_{2}=:\hat{\gamma}_{\mu}\hat{\gamma}_{\upsilon}\hat{\Lambda}:+\underline{\underline{{\cal I}}}:\hat{\gamma}_{\mu}\hat{\gamma}_{\upsilon}\hat{\Lambda}:\underline{\underline{{\cal I}}}$. Then the required expression is: $$\begin{aligned} \frac{B_{2}+\rmi B_{2}\underline{\underline{{\cal I}}}}{4} & = & \left(\begin{array}{cc} \widehat{a}_{i}^{\dagger}\widehat{a}_{j}^{\dagger}\hat{\Lambda} & \rmi\widehat{a}_{i}^{\dagger}\widehat{a}_{j}^{\dagger}\hat{\Lambda}\\ \rmi\widehat{a}_{i}^{\dagger}\widehat{a}_{j}^{\dagger}\hat{\Lambda} & -\widehat{a}_{i}^{\dagger}\widehat{a}_{j}^{\dagger}\hat{\Lambda} \end{array}\right).\label{eq:Normal_part}\end{aligned}$$ Similarly, we use the mixed product given in (\[eq:ExpFMP\]) in order to obtain: $$C_{1}=\left(\begin{array}{cc} 2\widehat{a}_{i}\widehat{a}_{j}^{\dagger}\hat{\Lambda}-2\hat{\Lambda}\widehat{a}_{j}\widehat{a}_{i}^{\dagger} & \rmi\left[2\widehat{a}_{i}\widehat{a}_{j}^{\dagger}\hat{\Lambda}+2\hat{\Lambda}\widehat{a}_{j}\widehat{a}_{i}^{\dagger}\right]\\ \rmi\left[-2\widehat{a}_{i}\widehat{a}_{j}^{\dagger}\hat{\Lambda}-2\hat{\Lambda}\widehat{a}_{j}\widehat{a}_{i}^{\dagger}\right] & 2\widehat{a}_{i}\widehat{a}_{j}^{\dagger}\widehat{\Lambda}-2\widehat{\Lambda}\widehat{a}_{j}\widehat{a}_{i}^{\dagger} \end{array}\right),$$ where $C_{1}=\Bigl\{\hat{\gamma}_{\mu}:\hat{\gamma}_{\upsilon}\hat{\Lambda}:\Bigr\}-\underline{\underline{{\cal I}}}\Bigl\{\hat{\gamma}_{\mu}:\hat{\gamma}_{\upsilon}\hat{\Lambda}:\Bigr\}\underline{\underline{{\cal I}}}$ and from this expression we get a matrix in terms of product of creation and annihilation operators and $\hat{\Lambda}$: $$\begin{aligned} \frac{C_{1}+\rmi C_{1}\underline{\underline{{\cal I}}}}{4} & = & \left(\begin{array}{cc} \widehat{a}_{i}\widehat{a}_{j}^{\dagger}\hat{\Lambda} & \rmi\widehat{a}_{i}\widehat{a}_{j}^{\dagger}\hat{\Lambda}\\ -\rmi\widehat{a}_{i}\widehat{a}_{j}^{\dagger}\hat{\Lambda} & \widehat{a}_{i}\widehat{a}_{j}^{\dagger}\widehat{\Lambda} \end{array}\right).\label{eq:mixedpart1}\end{aligned}$$ Following the same steps as described above from (\[eq:ExpSMP\]) we get: $$C_{2}=\left(\begin{array}{cc} 2\widehat{a}_{i}^{\dagger}\widehat{a}_{j}\hat{\Lambda}-2\hat{\Lambda}\widehat{a}_{j}^{\dagger}\widehat{a}_{i} & \rmi\left[-2\widehat{a}_{i}^{\dagger}\widehat{a}_{j}\hat{\Lambda}-2\hat{\Lambda}\widehat{a}_{j}^{\dagger}\widehat{a}_{i}\right]\\ \rmi\left[+2\widehat{a}_{i}^{\dagger}\widehat{a}_{j}\hat{\Lambda}+2\hat{\Lambda}\widehat{a}_{j}^{\dagger}\widehat{a}_{i}\right] & 2\widehat{a}_{i}^{\dagger}\widehat{a}_{j}\hat{\Lambda}-2\hat{\Lambda}\widehat{a}_{j}^{\dagger}\widehat{a}_{i} \end{array}\right),$$ where: $C_{2}=:\hat{\gamma}_{\mu}\Biggl\{\hat{\gamma}_{\upsilon}\hat{\Lambda}\Biggr\}:-\underline{\underline{{\cal I}}}:\hat{\gamma}_{\mu}\Biggl\{\hat{\gamma}_{\upsilon}\hat{\Lambda}\Biggr\}:\underline{\underline{{\cal I}}}$ and we also obtain: $$\begin{aligned} \frac{C_{2}-\rmi C_{2}\underline{\underline{{\cal I}}}}{4} & = & \left(\begin{array}{cc} \widehat{a}_{i}^{\dagger}\widehat{a}_{j}\hat{\Lambda} & -i\widehat{a}_{i}^{\dagger}\widehat{a}_{j}\hat{\Lambda}\\ i\widehat{a}_{i}^{\dagger}\widehat{a}_{j}\hat{\Lambda} & \widehat{a}_{i}^{\dagger}\widehat{a}_{j}\hat{\Lambda} \end{array}\right).\label{eq:mixed part2}\end{aligned}$$ On adding the identities given in (\[eq:Normal\_part\]), (\[eq:antinormal\_part\]), (\[eq:mixedpart1\]), and (\[eq:mixed part2\]) we obtain the combination of Fermi operators and the Gaussian operators corresponding to the unordered product of Majorana operators and $\hat{\Lambda}$ given in (\[eq:ExpUn-Or\]): $$\hat{\underline{\gamma}}\hat{\underline{\gamma}}^{T}\hat{\Lambda}=\frac{C_{1}+\rmi C_{1}\underline{\underline{{\cal I}}}}{4}+\frac{C_{2}-\rmi C_{2}\underline{\underline{{\cal I}}}}{4}+\frac{B_{1}-\rmi B_{1}\underline{\underline{{\cal I}}}}{4}+\frac{B_{2}+\rmi B_{2}\underline{\underline{{\cal I}}}}{4}.$$ In order to obtain the differential identity we notice that $B_{1},$ $B_{2}$, $C_{1}$ and $C_{2}$ and given in terms of the normalized ordered differential identities derived in \[sec:AppendixDiffIdNormGO\]. Thus using the expression in the right hand side of the four normalized differential identities and on simplifying terms we get: $$\hat{\underline{\gamma}}\hat{\underline{\gamma}}^{T}\hat{\Lambda}=-\rmi\underline{\underline{{\cal I}}}\frac{d\hat{\Lambda}}{d\underline{\underline{X}}}\underline{\underline{{\cal I}}}-\rmi\underline{\underline{{\cal I}}}\underline{\underline{X}}\frac{d\hat{\Lambda}}{d\underline{\underline{X}}}\underline{\underline{X}}\underline{\underline{{\cal I}}}+\rmi\underline{\underline{{\cal I}}}\hat{\Lambda}\underline{\underline{X}}\underline{\underline{{\cal I}}}-\underline{\underline{{\cal I}}}\frac{d\hat{\Lambda}}{d\underline{\underline{X}}}\underline{\underline{X}}\underline{\underline{{\cal I}}}+\underline{\underline{{\cal I}}}\underline{\underline{X}}\frac{d\hat{\Lambda}}{d\underline{\underline{X}}}\underline{\underline{{\cal I}}}+\hat{\Lambda}\underline{\underline{I}}.$$ The above equation can be written in a simple form as: $$\hat{\underline{\gamma}}\hat{\underline{\gamma}}^{T}\hat{\Lambda}=-\rmi\underline{\underline{{\cal I}}}\left[\left(1+i\underline{\underline{X}}\right)\frac{d\hat{\Lambda}}{d\underline{\underline{X}}}\left(1-\rmi\underline{\underline{X}}\right)-\rmi\hat{\Lambda}\left(1-\rmi\underline{\underline{X}}\right)\right]\underline{\underline{{\cal I}}}.\label{eq:id}$$ In terms of the alternate antisymmetric form, $\underline{\underline{x}}=\underline{\underline{{\cal I}}}\underline{\underline{X}}^{T}\underline{\underline{{\cal I}}}$ , we can define: $$\frac{d\hat{\Lambda}}{d\underline{\underline{x}}^{T}}=\underline{\underline{{\cal I}}}\frac{d\hat{\Lambda}}{d\underline{\underline{X}}}\underline{\underline{{\cal I}}}\,.\label{eq:dLdxT}$$ In this way we can introduce $\underline{\underline{x}}^{\pm}=\underline{\underline{x}}\pm\rmi\underline{\underline{I}},$ and simplify (\[eq:id\]) to give: $$\hat{\underline{\gamma}}\hat{\underline{\gamma}}^{T}\hat{\Lambda}=\rmi\left[\underline{\underline{x}}^{-}\frac{\rmd\hat{\Lambda}}{\rmd\underline{\underline{x}}}\underline{\underline{x}}^{+}-\hat{\Lambda}\underline{\underline{x}}^{+}\right].$$ On conjugating this expression, we obtain the right product identity: $$\hat{\Lambda}\hat{\underline{\gamma}}\hat{\underline{\gamma}}^{T}=\rmi\left[\underline{\underline{x}}^{+}\frac{\rmd\hat{\Lambda}}{\rmd\underline{\underline{x}}}\underline{\underline{x}}^{-}-\hat{\Lambda}\underline{\underline{x}}^{-}\right].\label{eq:UnorderedDifId-1-1}$$ Unordered mixed products ------------------------ $$\hat{\underline{\gamma}}\hat{\Lambda}\hat{\underline{\gamma}}^{T}=\rmi\left[\left(i\underline{\underline{x}}+\underline{\underline{I}}\right)\frac{d\hat{\Lambda}}{d\underline{\underline{x}}}\left(\underline{\underline{I}}+\rmi\underline{\underline{x}}\right)-i\hat{\Lambda}\left(\underline{\underline{I}}+\rmi\underline{\underline{x}}\right)\right].$$ **Proof**. Analogous to the previous differential identity, the method is to use the four different orderings of products of Majorana operators and Gaussian operators to give an explicit expression for $\hat{\underline{\gamma}}\hat{\Lambda}\hat{\underline{\gamma}}^{T}$, in terms of the ordered Fermi operators and Gaussian operators. Next we transform this into the corresponding differential identities. First, we obtain matrices of products of the Fermi creation and annihilation operators where the Gaussian operator is in the middle. Thus, pre and post multiplying (\[eq:ExpANOP\]) by the real antisymmetric matrix $\underline{\underline{{\cal I}}}$, we get: $$\begin{aligned} & & \fl\underline{\underline{{\cal I}}}\left\{ \hat{\gamma}_{\mu}\hat{\gamma}_{\upsilon}\hat{\Lambda}\right\} \underline{\underline{{\cal I}}}=\nonumber \\ & & \fl\left(\begin{array}{cc} \widehat{a}_{i}\widehat{a}_{j}\hat{\Lambda}+\hat{\Lambda}\widehat{a}_{i}^{\dagger}\widehat{a}_{j}^{\dagger}-\widehat{a}_{i}\hat{\Lambda}\widehat{a}_{j}^{\dagger}+\widehat{a}_{j}\hat{\Lambda}\widehat{a}_{i}^{\dagger} & -i\widehat{a}_{i}\widehat{a}_{j}\hat{\Lambda}+i\hat{\Lambda}\widehat{a}_{i}^{\dagger}\widehat{a}_{j}^{\dagger}-i\widehat{a}_{i}\hat{\Lambda}\widehat{a}_{j}^{\dagger}-i\widehat{a}_{j}\hat{\Lambda}\widehat{a}_{i}^{\dagger}\\ -i\widehat{a}_{i}\widehat{a}_{j}\hat{\Lambda}+i\hat{\Lambda}\widehat{a}_{i}^{\dagger}\widehat{a}_{j}^{\dagger}+i\widehat{a}_{i}\hat{\Lambda}\widehat{a}_{j}^{\dagger}+i\widehat{a}_{j}\hat{\Lambda}\widehat{a}_{i}^{\dagger} & -\widehat{a}_{i}\widehat{a}_{j}\hat{\Lambda}-\hat{\Lambda}\widehat{a}_{i}^{\dagger}\widehat{a}_{j}^{\dagger}-\widehat{a}_{i}\hat{\Lambda}\widehat{a}_{j}^{\dagger}+\widehat{a}_{j}\hat{\Lambda}\widehat{a}_{i}^{\dagger} \end{array}\right),\label{eq:IpreIpost-1}\end{aligned}$$ Adding (\[eq:ExpANOP\]) and (\[eq:IpreIpost-1\]) leads to: $$\begin{aligned} D_{1} & = & \left(\begin{array}{cc} 2\widehat{a}_{i}\hat{\Lambda}\widehat{a}_{j}^{\dagger}-2\widehat{a}_{j}\hat{\Lambda}\widehat{a}_{i}^{\dagger} & 2i\widehat{a}_{i}\hat{\Lambda}\widehat{a}_{j}^{\dagger}+2i\widehat{a}_{j}\hat{\Lambda}\widehat{a}_{i}^{\dagger}\\ -2i\widehat{a}_{i}\hat{\Lambda}\widehat{a}_{j}^{\dagger}-2i\widehat{a}_{j}\hat{\Lambda}\widehat{a}_{i}^{\dagger} & 2\widehat{a}_{i}\hat{\Lambda}\widehat{a}_{j}^{\dagger}-2\widehat{a}_{j}\hat{\Lambda}\widehat{a}_{i}^{\dagger} \end{array}\right),\label{eq:B1-1}\end{aligned}$$ where $D_{1}=\left\{ \hat{\gamma}_{\mu}\hat{\gamma}_{\upsilon}\hat{\Lambda}\right\} -\underline{\underline{{\cal I}}}\left\{ \hat{\gamma}_{\mu}\hat{\gamma}_{\upsilon}\hat{\Lambda}\right\} \underline{\underline{{\cal I}}}$. Using this expression we obtain: $$\begin{aligned} \frac{D_{1}+\rmi D_{1}\underline{\underline{{\cal I}}}}{4} & = & \left(\begin{array}{cc} \widehat{a}_{i}\hat{\Lambda}\widehat{a}_{j}^{\dagger} & \rmi\widehat{a}_{i}\hat{\Lambda}\widehat{a}_{j}^{\dagger}\\ -\rmi\widehat{a}_{i}\hat{\Lambda}\widehat{a}_{j}^{\dagger} & \widehat{a}_{i}\hat{\Lambda}\widehat{a}_{j}^{\dagger} \end{array}\right).\label{eq:antinormal_part-1}\end{aligned}$$ In this case, using the anti-normal ordering identity we obtain a matrix with products of the form $\widehat{a}_{i}\hat{\Lambda}\widehat{a}_{j}^{\dagger}$. We now use the the same procedure as described above for the normal ordering product, (\[eq:ENOP\]), obtaining: $$D_{2}=\left(\begin{array}{cc} -2\widehat{a}_{j}^{\dagger}\hat{\Lambda}\widehat{a}_{i}+2\widehat{a}_{i}^{\dagger}\hat{\Lambda}\widehat{a}_{j} & -i2\widehat{a}_{j}^{\dagger}\hat{\Lambda}\widehat{a}_{i}-i2\widehat{a}_{i}^{\dagger}\hat{\Lambda}\widehat{a}_{j}\\ i2\widehat{a}_{j}^{\dagger}\hat{\Lambda}\widehat{a}_{i}+i2\widehat{a}_{i}^{\dagger}\hat{\Lambda}\widehat{a}_{j} & -2\widehat{a}_{j}^{\dagger}\hat{\Lambda}\widehat{a}_{i}+2\widehat{a}_{i}^{\dagger}\hat{\Lambda}\widehat{a}_{j} \end{array}\right),$$ where $D_{2}=:\hat{\gamma}_{\mu}\hat{\gamma}_{\upsilon}\hat{\Lambda}:-\underline{\underline{{\cal I}}}:\hat{\gamma}_{\mu}\hat{\gamma}_{\upsilon}\hat{\Lambda}:\underline{\underline{{\cal I}}}$ . From this we get a matrix with products of the form $\widehat{a}_{i}^{\dagger}\hat{\Lambda}\widehat{a}_{j}$: $$\begin{aligned} \frac{D_{2}-\rmi D_{2}\underline{\underline{{\cal I}}}}{4} & = & \left(\begin{array}{cc} \widehat{a}_{i}^{\dagger}\hat{\Lambda}\widehat{a}_{j} & -\rmi\widehat{a}_{i}^{\dagger}\hat{\Lambda}\widehat{a}_{j}\\ \rmi\widehat{a}_{i}^{\dagger}\hat{\Lambda}\widehat{a}_{j} & \widehat{a}_{i}^{\dagger}\hat{\Lambda}\widehat{a}_{j} \end{array}\right).\label{eq:Normal_part-1}\end{aligned}$$ Next, using (\[eq:ExpFMP\]) we arrive at the following expression: $$\begin{aligned} \frac{E_{1}-\rmi E_{1}\underline{\underline{{\cal I}}}}{4} & = & \left(\begin{array}{cc} \widehat{a}_{i}\hat{\Lambda}\widehat{a}_{j} & -\rmi\widehat{a}_{i}\hat{\Lambda}\widehat{a}_{j}\\ -\rmi\widehat{a}_{i}\hat{\Lambda}\widehat{a}_{j} & -\widehat{a}_{i}\widehat{\Lambda}\widehat{a}_{j} \end{array}\right),\label{eq:mixedpart1-1}\end{aligned}$$ where $E_{1}=\Bigl\{\hat{\gamma}_{\mu}:\hat{\gamma}_{\upsilon}\hat{\Lambda}:\Bigr\}+\underline{\underline{{\cal I}}}\Bigl\{\hat{\gamma}_{\mu}:\hat{\gamma}_{\upsilon}\hat{\Lambda}:\Bigr\}\underline{\underline{{\cal I}}}$. Using (\[eq:ExpSMP\]) we obtain a matrix with products of the form $\widehat{a}_{i}^{\dagger}\hat{\Lambda}\widehat{a}_{j}^{\dagger}$: $$\begin{aligned} \frac{E_{2}+\rmi E_{2}\underline{\underline{{\cal I}}}}{4} & = & \left(\begin{array}{cc} \widehat{a}_{i}^{\dagger}\hat{\Lambda}\widehat{a}_{j}^{\dagger} & \rmi\widehat{a}_{i}^{\dagger}\hat{\Lambda}\widehat{a}_{j}^{\dagger}\\ \rmi\widehat{a}_{i}^{\dagger}\hat{\Lambda}\widehat{a}_{j}^{\dagger} & -\widehat{a}_{i}^{\dagger}\hat{\Lambda}\widehat{a}_{j}^{\dagger} \end{array}\right),\label{eq:mixed part2-1}\end{aligned}$$ where: $E_{2}=:\hat{\gamma}_{\mu}\Biggl\{\hat{\gamma}_{\upsilon}\hat{\Lambda}\Biggr\}:+\underline{\underline{{\cal I}}}:\hat{\gamma}_{\mu}\Biggl\{\hat{\gamma}_{\upsilon}\hat{\Lambda}\Biggr\}:\underline{\underline{{\cal I}}}$. On adding the expressions given in (\[eq:Normal\_part-1\]), (\[eq:antinormal\_part-1\]), (\[eq:mixedpart1-1\]), and (\[eq:mixed part2-1\]) we get the appropriate combination of products of Fermi operators and the Gaussian operator: $$\hat{\underline{\gamma}}\hat{\Lambda}\hat{\underline{\gamma}}^{T}=\frac{E_{1}-\rmi E_{1}\underline{\underline{{\cal I}}}}{4}+\frac{E_{2}+\rmi E_{2}\underline{\underline{{\cal I}}}}{4}+\frac{D_{1}+\rmi D_{1}\underline{\underline{{\cal I}}}}{4}+\frac{D_{2}-\rmi D_{2}\underline{\underline{{\cal I}}}}{4}.$$ Now utilizing the right hand side of the four normalized differential identities an on simplifying terms we arrive at the following expression: $$\hat{\underline{\gamma}}\hat{\Lambda}\hat{\underline{\gamma}}^{T}=-i\underline{\underline{{\cal I}}}\frac{d\hat{\Lambda}}{d\underline{\underline{X}}}\underline{\underline{{\cal I}}}+i\underline{\underline{{\cal I}}}\underline{\underline{X}}\frac{d\hat{\Lambda}}{d\underline{\underline{X}}}\underline{\underline{X}}\underline{\underline{{\cal I}}}-i\underline{\underline{{\cal I}}}\hat{\Lambda}\underline{\underline{X}}\underline{\underline{{\cal I}}}+\underline{\underline{{\cal I}}}\frac{d\hat{\Lambda}}{d\underline{\underline{X}}}\underline{\underline{X}}\underline{\underline{{\cal I}}}+\underline{\underline{{\cal I}}}\underline{\underline{X}}\frac{d\hat{\Lambda}}{d\underline{\underline{X}}}\underline{\underline{{\cal I}}}+\hat{\Lambda}\underline{\underline{I}}.$$ Taking common terms outside we get: $$\hat{\underline{\gamma}}\hat{\Lambda}\hat{\underline{\gamma}}^{T}=-i\underline{\underline{{\cal I}}}\left[\left(i\underline{\underline{X}}+1\right)\frac{d\hat{\Lambda}}{d\underline{\underline{X}}}\left(1+i\underline{\underline{X}}\right)-i\hat{\Lambda}\left(1+i\underline{\underline{X}}\right)\right]\underline{\underline{{\cal I}}}.\label{eq:id-1}$$ This leads to our final result, which can also be written as: $$\hat{\underline{\gamma}}\hat{\Lambda}\hat{\underline{\gamma}}^{T}=\rmi\left[-\underline{\underline{x}}^{-}\frac{\rmd\hat{\Lambda}}{\rmd\underline{\underline{x}}}\underline{\underline{x}}^{-}+\hat{\Lambda}\underline{\underline{x}}^{-}\right].$$ Time evolution of the Majorana Q-function\[sec:AppendixTimeevolution\] ====================================================================== Here we provide the details of the calculations in order to derive (\[eq:TEf\]), (\[eq:TEf2\]) and (\[eq:TEX\]) of Section \[sec:Time-Evolution-MQf\]. The chain rule allow us to obtain the relation between the Gaussian basis $\hat{\Lambda}^{N}\left(\underline{\underline{x}}\right)$ and the Gaussian operator $\hat{\Lambda}\left(\underline{\underline{x}}\right),$ which is: $$\frac{1}{{\cal N}}S\left(\left[\underline{\underline{x}}\right]^{2}\right)\frac{\rmd\hat{\Lambda}\left(\underline{\underline{x}}\right)}{\rmd\underline{\underline{x}}}=\frac{\rmd\hat{\Lambda}^{N}\left(\underline{\underline{x}}\right)}{\rmd\underline{\underline{x}}}-\frac{\rmd\ln S\left(\left[\underline{\underline{x}}\right]^{2}\right)}{\rmd\underline{\underline{x}}}\hat{\Lambda}^{N}\left(\underline{\underline{x}}\right).\label{eq:chainrule}$$ Using the above identity in (\[eq:dQ/dt\]) and the definition of the Majorana Q-function we get: $$\begin{aligned} \frac{\rmd Q\left(\underline{\underline{x}}\right)}{\rmd t} & = & -\Omega_{\mu\nu}\left[-x_{\mu\kappa}\frac{\rmd Q}{\rmd x_{\upsilon\kappa}}+x_{\mu\kappa}\frac{\rmd\ln S\left(\left[\underline{\underline{x}}\right]^{2}\right)}{\rmd x_{\upsilon\kappa}}\right]\nonumber \\ & & -\Omega_{\mu\nu}\left[\frac{\rmd Q}{\rmd x_{\kappa\mu}}x_{\kappa\upsilon}-\frac{\rmd\ln S\left(\left[\underline{\underline{x}}\right]^{2}\right)}{\rmd x_{\kappa\mu}}x_{\kappa\upsilon}\right].\label{eq:TEv_QFGB}\end{aligned}$$ Here we have used the following convention for calculation matrix derivatives $\left[\rmd/\rmd x\right]_{\kappa\upsilon}=\rmd/\rmd x_{\upsilon\kappa}$ [@Corney_PD_JPA_2006_GR_fermions]. In the limit $S\left(\left[\underline{\underline{x}}\right]^{2}\right)\rightarrow1$, the above expression reduces to (\[eq:TEf\]). Next, using the chain rule we obtain the following expressions: $$\begin{aligned} -x_{\mu\kappa}\frac{\rmd Q}{\rmd x_{\upsilon\kappa}} & = & -\frac{\rmd}{\rmd x_{\upsilon\kappa}}\left(x_{\mu\kappa}Q\right)+\left(\frac{\rmd}{\rmd x_{\upsilon\kappa}}x_{\mu\kappa}\right)Q\nonumber \\ & = & -\frac{\rmd}{\rmd x_{\upsilon\kappa}}\left(x_{\mu\kappa}Q\right)+\left(2M-1\right)\delta_{\mu\upsilon}Q,\label{eq:CR}\end{aligned}$$ and $$\begin{aligned} \frac{\rmd Q}{\rmd x_{\kappa\mu}}x_{\kappa\upsilon} & = & \frac{\rmd}{\rmd x_{\kappa\mu}}\left(Qx_{\kappa\upsilon}\right)-\left(\frac{\rmd}{\rmd x_{\kappa\mu}}x_{\kappa\upsilon}\right)Q\nonumber \\ & = & \frac{\rmd}{\rmd x_{\kappa\mu}}\left(Qx_{\kappa\upsilon}\right)-\left(2M-1\right)\delta_{\upsilon\mu}Q.\label{eq:CR-1}\end{aligned}$$ On substituting (\[eq:CR\] and \[eq:CR-1\]) in (\[eq:TEf\]) we get (\[eq:TEf2\]). The method of characteristics allow us to solve the above equation as: $$\frac{\rmd x_{\upsilon\kappa}}{\rmd t}=\Omega_{\nu\mu}x_{\mu\kappa}-x_{\upsilon\mu}\Omega_{\mu\kappa},$$ where we used that $\underline{\underline{\Omega}}$ is an antisymmetric matrix. In matrix form the above equation is: $$\frac{\rmd\underline{\underline{x}}}{\rmd t}=\left[\underline{\underline{\Omega}},\underline{\underline{x}}\right].\label{eq:dxdtT-1}$$ Here (\[eq:dxdtT-1\]) corresponds to (\[eq:TEX\]). Time evolution for the open quantum system \[sec:AppTimeEvOpenQS\] ================================================================== In this section we show the details of the calculations in order to obtain the time evolution equation of the small quantum dot coupled to a zero temperature reservoir, for the multimode case. This equation is given in (\[eq:finalEx\]). The time evolution equation for the Q-function written in terms of products of Majorana variables and Majorana Gaussian operators is given in (\[eq:timeevo2\]). This equation is: $$\begin{aligned} \frac{\rmd Q}{\rmd t} & = & -\frac{i}{2}\frac{1}{{\cal N}}S\Tr\left[\tilde{\Omega}_{\kappa\nu}\Bigl\{\hat{\underline{\gamma}}:\hat{\underline{\gamma}}^{T}\hat{\Lambda}:\Bigr\}_{\nu\kappa}\widehat{\rho}\right]\\ & & -\frac{1}{2i}\frac{1}{{\cal N}}S\Tr\left[\Upsilon_{\kappa\nu}\left(:\hat{\underline{\gamma}}\hat{\underline{\gamma}}^{T}\hat{\Lambda}:{}_{\nu\kappa}-\Bigl\{\hat{\underline{\gamma}}:\hat{\underline{\gamma}}^{T}\hat{\Lambda}:\Bigr\}_{\nu\kappa}\right)\widehat{\rho}\right]-\gamma_{ij}\delta_{ij}Q.\end{aligned}$$ We now use the Majorana differential identities given in (\[eq:MajMixProd\]) and (\[eq:NormalNormMajId\]) respectively. On simplifying terms and using identities corresponding identities we obtain: $$\begin{aligned} \frac{\rmd Q}{\rmd t} & = & \frac{1}{2}\frac{1}{{\cal N}}S\Tr\left[\tilde{\Omega}_{\kappa\nu}\left[\underline{\underline{X}}^{+}\frac{d\hat{\Lambda}\left(\underline{\underline{X}}\right)}{d\underline{\underline{X}}}\underline{\underline{X}}^{-}-\underline{\underline{X}}^{-}\hat{\Lambda}\left(\underline{\underline{X}}\right)\right]_{\nu\kappa}\widehat{\rho}\right]\nonumber \\ & + & \frac{1}{{\cal N}}S\Tr\left[\Upsilon_{\kappa\nu}\left[\underline{\underline{X}}\frac{d\hat{\Lambda}\left(\underline{\underline{X}}\right)}{d\underline{\underline{X}}}\underline{\underline{X}}^{-}\right]_{\nu\kappa}\widehat{\rho}\right]-\Upsilon_{\kappa\nu}X_{\nu\kappa}Q\label{eq:timeevo3}\end{aligned}$$ provided $\gamma_{ij}\delta_{ij}=\Upsilon_{\kappa\nu}\mathcal{I}_{\nu\kappa}.$ Due to the same reason as stated in the previous time evolution example, $\tilde{\Omega}_{\kappa\nu}X_{\nu\kappa}^{-}$ will be zero. Now utilizing the chain rule (\[eq:chainrule\]), the definition of the Q-function and considering the limit $S\left(\left[\underline{\underline{X}}\right]^{2}\right)\rightarrow1,$ we get: $$\frac{\rmd Q}{\rmd t}=\frac{1}{2}\tilde{\Omega}_{\kappa\nu}\left[\underline{\underline{X}}^{+}\frac{\rmd Q}{\rmd\underline{\underline{X}}}\underline{\underline{X}}^{-}\right]_{\nu\kappa}+\Upsilon_{\kappa\nu}\left[\underline{\underline{X}}\frac{\rmd Q}{\rmd\underline{\underline{X}}}\underline{\underline{X}}^{-}\right]_{\nu\kappa}-\Upsilon_{\kappa\nu}X_{\nu\kappa}Q\label{eq:timeevo4}$$ Next, using the following result of the product rule: $$\fl X_{\nu\ell}\frac{\rmd Q}{\rmd X_{p\ell}}X_{p\kappa}^{-}=\frac{\rmd}{\rmd X_{p\ell}}X_{\nu\ell}X_{p\kappa}^{-}Q-\left(\delta_{\nu p}\delta_{\ell\ell}-\delta_{\nu\ell}\delta_{\ell p}\right)X_{p\kappa}^{-}Q-\left(\delta_{pp}\delta_{\kappa\ell}-\delta_{p\ell}\delta_{\kappa p}\right)X_{\nu\ell}Q,\label{eq:CR2}$$ on the time evolution equation we finally obtain: $$\begin{aligned} \frac{\rmd Q}{\rmd t} & = & \frac{1}{2}\tilde{\Omega}_{\kappa\nu}\frac{\rmd}{\rmd X_{p\ell}}\left(X_{\nu\ell}^{+}QX_{p\kappa}^{-}\right)+\Upsilon_{\kappa\nu}\frac{\rmd}{\rmd X_{p\ell}}\left(X_{\nu\ell}X_{p\kappa}^{-}Q\right)-\Upsilon_{\kappa\nu}X_{\nu\kappa}Q\\ & & -\Upsilon_{\kappa\nu}\left(\delta_{\nu p}\delta_{\ell\ell}-\delta_{\nu\ell}\delta_{\ell p}\right)X_{p\kappa}^{-}Q-\Upsilon_{\kappa\nu}\left(\delta_{pp}\delta_{\kappa\ell}-\delta_{p\ell}\delta_{\kappa p}\right)X_{\nu\ell}Q.\end{aligned}$$ Upon summation by considering all the $2M$ modes, $\sum\delta_{\ell\ell}=2M$, and on simplifying we finally get: $$\begin{aligned} \frac{\rmd Q}{\rmd t} & = & \frac{1}{2}\tilde{\Omega}_{\kappa\nu}\frac{\rmd}{\rmd X_{p\ell}}\left(X_{\nu\ell}^{+}QX_{p\kappa}^{-}\right)+\Upsilon_{\kappa\nu}\frac{\rmd}{\rmd X_{pl}}\left(X_{\nu\ell}QX_{p\kappa}^{-}\right)\nonumber \\ & & -\left(4M-1\right)X_{\nu\kappa}\Upsilon_{\kappa\nu}Q+\left(2M-1\right)\Upsilon_{\kappa\nu}\mathcal{I}_{\nu\kappa}.\label{eq:finalEx-1}\end{aligned}$$ This corresponds to (\[eq:finalEx\]). 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--- abstract: | In this paper we consider a single-server cyclic polling system consisting of two queues. Between visits to successive queues, the server is delayed by a random switch-over time. Two types of customers arrive at the first queue: high and low priority customers. For this situation the following service disciplines are considered: gated, globally gated, and exhaustive. We study the cycle time distribution, the waiting times for each customer type, the joint queue length distribution at polling epochs, and the steady-state marginal queue length distributions for each customer type. **Keywords:** Polling, priority levels, queue lengths, waiting times author: - | M.A.A. Boon[^1]\ <marko@win.tue.nl> - | I.J.B.F. Adan\ <iadan@win.tue.nl> - | O.J. Boxma\ <boxma@win.tue.nl> date: 'May, 2008' title: 'A Two-Queue Polling Model with Two Priority Levels in the First Queue[^2][^3]' --- Introduction {#intro} ============ A polling model is a single-server system in which the server visits $n$ queues $Q_1, \dots, Q_n$ in cyclic order. Customers that arrive at $Q_i$ are referred to as type $i$ customers. The special feature of the model considered in the present paper is that, within a customer type, we distinguish high and low priority customers. More specifically, we study a polling system which consists of two queues, $Q_1$ and $Q_2$. The first of these queues contains customers of two priority classes, high ($H$) and low ($L$). The exhaustive, gated and globally gated service disciplines are studied. Our motivation to study a polling model with priorities is that the performance of a polling system can be improved through the introduction of priorities. In production environments, e.g., one could give highest priority to jobs with a service requirement below a certain threshold level. This might decrease the mean waiting time of an arbitrary customer without having to purchase additional resources [@wierman07]. Priority polling models also can be used to study traffic intersections where conflicting traffic flows face a green light simultaneously; e.g. traffic which takes a left turn may have to give right of way to conflicting traffic that moves straight on, even if the traffic light is green for both traffic flows. Another application is discussed in [@cicin2001], where a priority polling model is used to study scheduling of surgery procedures in medical emergency rooms. In the computer science community the Bluetooth and 802.11 protocols are frequently modelled as polling systems, cf. [@ieee802.11-1; @bluetooth1; @ieee802.11-2; @bluetooth2]. Many scheduling policies that have been considered or implemented in these protocols involve different priority levels in order to improve Quality-of-Service (QoS) for traffic that is very sensitive to delays or loss of data, such as Voice over Wireless IP. The 802.11e amendment defines a set of QoS enhancements for wireless LAN applications by differentiating between high priority traffic, like streaming multimedia, and low priority traffic, like web browsing and email traffic. Although there is quite an extensive amount of literature available on polling systems, only very few papers treat priorities in polling models. Most of these papers only provide approximations or focus on pseudo-conservation laws. In [@wierman07] exact mean waiting time results are obtained using the Mean Value Analysis (MVA) framework for polling systems, developed in [@winands06]. The MVA framework can only be used to find the first moment of the waiting time distribution for each customer type, and the mean residual cycle time. The main contribution of the present paper is the derivation of Laplace Stieltjes Transforms (LSTs) of the distributions of the marginal waiting times for each customer type; in particular it turns out to be possible to obtain exact expressions for the waiting time distributions of both high and low priority customers at a queue of a polling system. Probability Generating Functions (PGFs) are derived for the joint queue length distribution at polling epochs, and for the steady-state marginal queue length distribution of the number of customers at an arbitrary epoch. The present paper is structured as follows: Section \[general\] gathers known results of nonpriority polling models which are relevant for the present study. Sections \[gated\] (gated), \[globallygated\] (globally gated), and \[exhaustive\] (exhaustive) give new results on the priority polling model. In each of the sections we successively discuss the joint queue length distribution at polling epochs, the cycle time distribution, the marginal queue length distributions and waiting time distributions. The mean waiting times are given at the end of each section. A numerical example is presented in Section \[numericalexample\] to illustrate some of the improvements that can be obtained by introducing prioritisation in a polling system. Notation and description of the nonpriority polling model {#general} ========================================================= The model that is considered in this section, is a nonpriority polling model with two queues ($Q_1$ and $Q_2$). We consider three service disciplines: gated, globally gated, and exhaustive. The gated service discipline states that during a visit to $Q_i$, the server serves only those type $i$ customers who are present at the polling epoch. All type $i$ customers that arrive during this visit will be served in the next cycle. In this respect, a cycle is the time between two successive visit beginnings to a queue. The exhaustive service discipline states that when the server arrives at $Q_i$, all type $i$ customers are served until no type $i$ customer is present in the system. We also consider the globally gated service discipline, which means that during a cycle only those customers will be served that were present at the beginning of that cycle. Customers of type $i$ arrive at $Q_i$ according to a Poisson process with arrival rate $\lambda_i$ $(i = 1,2)$. Service times can follow any distribution, and we assume that a customer’s service time is independent of other service times and independent of the arrival processes. The LST of the distribution of the generic service time $B_i$ of type $i$ customers is denoted by $\beta_i(\cdot)$. The fraction of time that the server is serving customers of type $i$ equals $\rho_i := \lambda_i E(B_i)$. Switches of the server from $Q_i$ to $Q_{i+1}$ (all indices modulo 2), require a switch-over time $S_i$. The LST of this switch-over time distribution is denoted by $\sigma_i(\cdot)$. The fraction of time that the server is working (i.e., not switching) is $\rho := \rho_1+\rho_2$. We assume that $\rho < 1$, which is a necessary and sufficient condition for the steady state distributions of cycle times, queue lengths and waiting times to exist. [@takacs68] studied this model, but without switch-over times and only with the exhaustive service discipline. [@coopermurray69] analysed this polling system for any number of queues, and for both gated and exhaustive service disciplines. [@eisenberg72] obtained results for a polling system with switch-over times (but only exhaustive service) by relating the PGFs of the joint queue length distributions at visit beginnings, visit endings, service beginnings and service endings. [@resing93] was the first to point out the relation between polling systems and Multitype Branching Processes with immigration in each state. His results can be applied to polling models in which each queue satisfies the following property: \[resingproperty\] If the server arrives at $Q_i$ to find $k_i$ customers there, then during the course of the server’s visit, each of these $k_i$ customers will effectively be replaced in an i.i.d. manner by a random population having probability generating function $h_i(z_1,\dots,z_n)$, which can be any $n$-dimensional probability generating function. We use this property, and the relation to Multitype Branching Processes, to find results for our polling system with two queues, two priorities in the first queue, and gated, globally gated, and exhaustive service discipline. Notice that, unlike the gated and exhaustive service disciplines, the globally gated service discipline does not satisfy Property \[resingproperty\]. But the results obtained by Resing also hold for a more general class of polling systems, namely those which satisfy the following (weaker) property that is formulated in [@semphd]: \[borstproperty\] If there are $k_i$ customers present at $Q_i$ at the beginning (or the end) of a visit to $Q_{\pi(i)}$, with $\pi(i) \in \{1, \dots, n\}$, then during the course of the visit to $Q_i$, each of these $k_i$ customers will effectively be replaced in an i.i.d. manner by a random population having probability generating function $h_i(z_1,\dots,z_n)$, which can be any $n$-dimensional probability generating function. Globally gated and gated are special cases of the synchronised gated service discipline, which states that only customers in $Q_i$ will be served that were present at the moment that the server reaches the “parent queue” of $Q_i$: $Q_{\pi(i)}$. For gated service, $\pi(i) = i$, for globally gated service, $\pi(i) = 1$. The synchronised gated service discipline is discussed in [@khamisy92], but no observation is made that this discipline is a member of the class of polling systems satisfying Property \[borstproperty\] which means that results as obtained in [@resing93] can be extended to this model. [@borst97] combined the results of [@resing93] and [@eisenberg72] to find a relation between the PGFs of the marginal queue length distribution for polling systems with and without switch-over times, expressed in the Fuhrmann-Cooper queue length decomposition form [@fuhrmanncooper85]. Joint queue length distribution at polling epochs ------------------------------------------------- The probability generating function $h_i(z_1,\dots,z_n)$ which is mentioned in Property \[resingproperty\] depends on the service discipline. In a polling system with two queues and gated service we have $h_i(z_1, z_2) = \beta_i({\lambda_1(1-z_1)+\lambda_2(1-z_2)})$. For exhaustive service this PGF becomes $h_i(z_1, z_2) = \pi_i(\sum_{j\neq i} \lambda_j(1-z_j))$, where $\pi_i(\cdot)$ is the LST of a busy period (BP) distribution in an $M/G/1$ system with only type $i$ customers, so it is the root of the equation $\pi_i(\omega) = \beta_i(\omega + \lambda_i(1 - \pi_i(\omega)))$. We choose the beginning of a visit to $Q_1$ as start of a cycle. In order to find the joint queue length distribution at the beginning of a cycle, we relate the numbers of customers in each queue at the beginning of a cycle to those at the beginning of the previous cycle. Customers always enter the system during a switch-over time, or during a visit period. The first group is called *immigration*, whereas a customer from the second group is called *offspring* of the customer that is served at the moment of his arrival. We define the immigration PGF for each switch-over time and the offspring PGF for each visit period analogous to [@resing93]. The immigration PGFs are: $$\begin{aligned} g^{(2)}(z_1, z_2) &= \sigma_2({\lambda_1(1-z_1)+\lambda_2(1-z_2)}), \\ g^{(1)}(z_1, z_2) &= \sigma_1(\lambda_1(1-z_1)+\lambda_2(1-h_2(z_1, z_2))).\end{aligned}$$ $g^{(2)}(z_1, z_2)$ is the PGF of the joint distribution of type $1$ and $2$ customers that arrive during $S_2$. For $S_1$ things are slightly more complicated, since type $2$ customers arriving during $S_1$ may be served before the end of the cycle, and generate offspring. $g^{(1)}(z_1, z_2)$ is the joint PGF of the type $1$ and $2$ customers present at the end of the cycle that either arrived during $S_1$, or are offspring of type 2 customers that arrived during $S_1$. The total immigration PGF is the product of these two PGFs: $$g(z_1, z_2) = \prod_{i=1}^2 g^{(i)}(z_1,z_2) = g^{(1)}(z_1,z_2)g^{(2)}(z_1,z_2).$$ We define the offspring PGFs for each visit period in a similar manner:$$\begin{aligned} f^{(2)}(z_1, z_2) &= h_2(z_1, z_2),\\ f^{(1)}(z_1, z_2) &= h_1(z_1, h_2(z_1, z_2)).\end{aligned}$$ The term for $Q_1$ is again slightly more complicated than the term for $Q_2$, since type 2 customers arriving during a server visit to $Q_1$ may be served before the end of the cycle, and generate offspring. [@resing93] shows that the following recursive expression holds for the joint queue length PGF at the beginning of a cycle (starting with a visit to $Q_1$): $$P_1(z_1, z_2) = g(z_1, z_2)P_1\left(f^{(1)}(z_1, z_2), f^{(2)}(z_1, z_2)\right).\label{P1recursive}$$ This expression can be used to compute moments of the joint queue length distribution. Alternatively, iteration of this expression yields the following closed form expression for $P_1(z_1, z_2)$: $$P_1(z_1, z_2) = \prod_{n=0}^\infty g(f_n(z_1, z_2)),\label{p1twoqueues}$$ where we use the following recursive definition for $f_n(z_1, z_2)$, $n=0,1,2,\dots$: $$\begin{aligned} f_n(z_1, z_2) &= (f^{(1)}(f_{n-1}(z_1, z_2)), f^{(2)}(f_{n-1}(z_1, z_2))), \\ f_0(z_1, z_2) &= (z_1, z_2).\end{aligned}$$ [@resing93] proves that this infinite product converges if and only if $\rho < 1$. We can relate the joint queue length distribution at other polling epochs to $P_1(z_1, z_2)$. We denote the PGF of the joint queue length distribution at a visit beginning to $Q_i$ by $V_{b_i}(\cdot)$, so $P_1(\cdot) = V_{b_1}(\cdot)$. The PGF of the joint queue length distribution at a visit completion to $Q_i$ is denoted by $V_{c_i}(\cdot)$. The following relations hold: $$\begin{aligned} V_{b_1}(z_1, z_2) &= V_{c_2}(z_1, z_2)\sigma_2({\lambda_1(1-z_1)+\lambda_2(1-z_2)}) \nonumber\\ &= V_{b_2}(z_1, h_2(z_1, z_2)) \sigma_2({\lambda_1(1-z_1)+\lambda_2(1-z_2)}) \nonumber\\ &= V_{b_2}(z_1, f^{(2)}(z_1, z_2)) g^{(2)}(z_1, z_2), \label{vb1twoqueues}\\ V_{b_2}(z_1, z_2) &= V_{c_1}(z_1, z_2)\sigma_1({\lambda_1(1-z_1)+\lambda_2(1-z_2)}) \nonumber\\ &= V_{b_1}(h_1(z_1, z_2), z_2) \sigma_1({\lambda_1(1-z_1)+\lambda_2(1-z_2)}).\label{vb2twoqueues}\end{aligned}$$ Cycle time ---------- The cycle time, starting at a visit *beginning* to $Q_1$, is the sum of the visit times to $Q_1$ and $Q_2$, and the two switch-over times which are independent of the visit times. Since type 2 customers who arrive during the visit to $Q_1$ or the switch from $Q_1$ to $Q_2$ will be served during the visit to $Q_2$, it can be shown that the LST of the distribution of the cycle time $C_1$, $\gamma_1(\cdot)$, is related to $P_1(\cdot)$ as follows: $$\gamma_1(\omega) = \sigma_1(\omega + \lambda_2(1-\phi_2(\omega))) \, \sigma_2(\omega) \, P_1(\phi_1(\omega + \lambda_2(1-\phi_2(\omega))), \phi_2(\omega)),\label{cycletimelst}$$ where $\phi_i(\cdot)$ is the LST of the distribution of the time that the server spends at $Q_i$ due to the presence of one type $i$ customer there. For gated service $\phi_i(\cdot) = \beta_i(\cdot)$, for exhaustive service $\phi_i(\cdot) = \pi_i(\cdot)$. A proof of can be found in [@boxmafralixbruin08]. In some cases it is convenient to choose a different starting point for a cycle, for example when analysing a polling system with exhaustive service. If we define $C_1^*$ to be the time between two successive visit *completions* to $Q_1$, the LST of its distribution, $\gamma^*_1(\cdot)$, is: $$\begin{aligned} \gamma^*_1(\omega) =& \sigma_1(\omega + \lambda_1(1-\phi_1(\omega)) + \lambda_2(1-\phi_2(\omega + \lambda_1(1-\phi_1(\omega)))))\nonumber\\ &\cdot \sigma_2(\omega + \lambda_1(1-\phi_1(\omega))) \, V_{c_1}(\phi_1(\omega), \phi_2(\omega+\lambda_1(1-\phi_1(\omega)))),\label{cycletimlstVc}\end{aligned}$$ with $V_{c_1}(z_1,z_2) = P_1(h_1(z_1, z_2), z_2)$. Marginal queue lengths and waiting times ---------------------------------------- We denote the PGF of the steady-state marginal queue length distribution of $Q_1$ at the visit beginning by $\widetilde{V}_{b_1}(z) = V_{b_1}(z, 1)$. Analogously we define $\widetilde{V}_{b_2}(\cdot), \widetilde{V}_{c_1}(\cdot)$, and $\widetilde{V}_{c_2}(\cdot)$. It is shown in [@borst97] that the steady-state marginal queue length of $Q_i$ can be decomposed into two parts: the queue length of the corresponding $M/G/1$ queue with only type $i$ customers, and the queue length at an arbitrary epoch during the intervisit period of $Q_i$, denoted by ${N_{i|I}}$. [@borst97] show that by virtue of PASTA, ${N_{i|I}}$ has the same distribution as the number of type $i$ customers seen by an arbitrary type $i$ customer arriving during an intervisit period, which equals $$E(z^{N_{i|I}}) = \frac{E(z^{N_{i|I_{\textit{begin}}}}) - E(z^{N_{i|I_{\textit{end}}}})}{(1-z)(E(N_{i|I_{\textit{end}}}) - E(N_{i|I_{\textit{begin}}}))},$$ where $N_{i|I_{\textit{begin}}}$ is the number of type $i$ customers at the beginning of an intervisit period $I_i$, and $N_{i|I_{\textit{end}}}$ is the number of type $i$ customers at the end of $I_i$. Since the beginning of an intervisit period coincides with the completion of a visit to $Q_i$, and the end of an intervisit period coincides with the beginning of a visit, we know the PGFs for the distributions of these random variables: $\widetilde V_{c_i}(\cdot)$ and $\widetilde V_{b_i}(\cdot)$. This leads to the following expression for the PGF of the steady-state queue length distribution of $Q_i$ at an arbitrary epoch, $E[z^{N_i}]$: $$E[z^{N_i}] = \frac{(1-\rho_i)(1-z)\beta_i(\lambda_i(1-z))}{\beta_i(\lambda_i(1-z))-z} \cdot\frac{\widetilde{V}_{c_i}(z) - \widetilde{V}_{b_i}(z)}{(1-z)(E(N_{i|I_{\textit{end}}}) - E(N_{i|I_{\textit{begin}}}))}. \label{queuelengthdecomposition}$$ [@keilsonservi90] show that the distributional form of Little’s law can be used to find the LST of the marginal waiting time distribution: $E(z^{N_i}) = E({\textrm{e}}^{-\lambda_i(1-z)(W_i + B_i)})$, hence $E({\textrm{e}}^{-\omega W_i}) = E[(1-\frac{\omega}{\lambda_i})^{N_i}]/\beta_i(\omega)$. This can be substituted into : $$\begin{aligned} E[{\textrm{e}}^{-\omega W_i}] =& \frac{(1-\rho_i)\omega}{\omega-\lambda_i(1-\beta_i(\omega))}\cdot \frac{\widetilde{V}_{c_i}\left(1-\frac{\omega}{\lambda_i}\right) - \widetilde{V}_{b_i}\left(1-\frac{\omega}{\lambda_i}\right)}{(E(N_{i|I_{\textit{end}}}) - E(N_{i|I_{\textit{begin}}}))\omega/\lambda_i} \nonumber\\ =& E[{\textrm{e}}^{-\omega W_{i|M/G/1}}]E\left[\left(1-\frac\omega{\lambda_i}\right)^{N_{i|I}}\right].\label{waitingtimedecomposition}\end{aligned}$$ The interpretation of this formula is that the waiting time of a type $i$ customer in a polling model is the sum of two independent random variables: the waiting time of a customer in an $M/G/1$ queue with only type $i$ customers, $W_{i|M/G/1}$, and the remaining intervisit time for a customer that arrives at an arbitrary epoch during the intervisit time of $Q_i$. For *gated* service, the number of type $i$ customers at the beginning of a visit to $Q_i$ is exactly the number of type $i$ customers that arrived during the previous cycle, starting at $Q_i$. In terms of PGFs: $\widetilde V_{b_i}(z) = \gamma_i(\lambda_i(1-z))$. The number of type $i$ customers at the end of a visit to $Q_i$ are exactly those type $i$ customers that arrived during this visit. In terms of PGFs: $\widetilde V_{c_i}(z) = \gamma_i(\lambda_i(1-\beta_i(\lambda_i(1-z))))$. We can rewrite $E(N_{i|I_{\textit{end}}}) - E(N_{i|I_{\textit{begin}}})$ as $\lambda_i E(I_i)$, because this is the number of type $i$ customers that arrive during an intervisit time. In Section \[momentsgeneral\] we show that $\lambda_i E(I_i) = \lambda_i(1-\rho_i)E(C)$. Using these expressions we can rewrite Equation for gated service to: $$E[{\textrm{e}}^{-\omega W_i}] = \frac{(1-\rho_i)\omega}{\omega-\lambda_i(1-\beta_i(\omega))}\cdot \frac{\gamma_i(\lambda_i(1-\beta_i(\omega))) - \gamma_i(\omega)}{(1-\rho_i)\omega E(C)}.\label{lstwgated}$$ For *exhaustive* service, $\widetilde V_{c_i}(z) = 1$, because $Q_i$ is empty at the end of a visit to $Q_i$. The number of type $i$ customers at the beginning of a visit to $Q_i$ in an exhaustive polling system is equal to the number of type $i$ customers that arrived during the previous intervisit time of $Q_i$. Hence, $\widetilde{V}_{b_i}(z) = \widetilde{I}_i(\lambda_i(1-z))$, where $\widetilde{I}_i(\cdot)$ is the LST of the intervisit time distribution for $Q_i$. Substitution of $\widetilde{I}_i(\omega) = \widetilde{V}_{b_i}(1-\frac{\omega}{\lambda_i})$ in leads to the following expression for the LST of the steady-state waiting time distribution of a type $i$ customer in an exhaustive polling system: $$E[{\textrm{e}}^{-\omega W_i}] = \frac{(1-\rho_i)\omega}{\omega-\lambda_i(1-\beta_i(\omega))}\cdot \frac{1-\widetilde{I}_i(\omega)}{\omega E(I_i)}.\label{lstwexhaustive}$$ To the best of our knowledge, the following result is new. Let the cycle time $C^*_i$ be the time between two successive visit *completions* to $Q_i$. The LST of the cycle time distribution is given by . An equivalent expression for $E[{\textrm{e}}^{-\omega W_i}]$ if $Q_i$ is served exhaustively, is: $$\begin{aligned} E[{\textrm{e}}^{-\omega W_i}] &= \frac{1-\gamma^*_i(\omega - \lambda_i(1-\beta_i(\omega)))}{(\omega-\lambda_i(1-\beta_i(\omega)))E(C)}\label{lstwexhaustiveC}\\ &= E[{\textrm{e}}^{-(\omega - \lambda_i(1-\beta_i(\omega))) C^*_{i,\textit{res}}}],\nonumber\end{aligned}$$ where $C^*_{i,\textit{res}}$ is the residual length of $C^*_i$. The cycle time is the length of an intervisit period $I_i$ plus the length of a visit $V_i$, which is the time required to serve all type $i$ customers that have arrived during $I_i$, and their type $i$ descendants. Hence, the following equation holds: $$\gamma^*_i(\omega) = \widetilde{I}_i(\omega + \lambda_i(1-\pi_i(\omega))).\label{lstCexhaustiveI}$$ We use this equation to find the inverse relation: $$\begin{aligned} \widetilde{I}_i(\omega + \lambda_i(1-\pi_i(\omega))) &= \gamma^*_i(\omega) \\ &= \gamma^*_i(\omega + \lambda_i(1-\pi_i(\omega)) - \lambda_i(1-\pi_i(\omega)))\\ &= \gamma^*_i(\omega + \lambda_i(1-\pi_i(\omega)) - \lambda_i(1-\beta_i(\omega+\lambda_i(1-\pi_i(\omega))))).\end{aligned}$$ If we substitute $s := \omega + \lambda_i(1-\pi_i(\omega))$, we find $$\widetilde{I}_i(s) = \gamma^*_i(s - \lambda_i(1-\beta_i(s))).\label{intervisitC}$$ Substitution of into gives . We can write and as follows: $$\gamma^*_i(\omega) = \widetilde{I}_i(\psi(\omega)), \qquad \widetilde{I}_i(s) = \gamma^*_i(\phi(s)),$$ where $\phi(\cdot)$ equals the Laplace exponent of the Lévy process $\sum_{j=1}^{N(t)}B_{i,j}-t$, with $N(t)$ a Poisson process with intensity $\lambda_i$, and with $\psi(\omega) = \omega+\lambda_i(1-\pi_i(\omega))$, which is known to be the inverse of $\phi(\cdot)$. Moments {#momentsgeneral} ------- The focus of this paper is on LST and PGF of distribution functions, not on their moments. Moments can be obtained by differentiation, and are also discussed in [@wierman07]. In this subsection we will only mention some results that will be used later. First we will derive the mean cycle time $E(C)$. Unlike higher moments of the cycle time, the mean does not depend on where the cycle starts: $E(C) = \frac{E(S_1) + E(S_2)}{1-\rho}$. This can easily be seen, because $1-\rho$ is the fraction of time that the server is not working, but switching. The total switch-over time is $E(S_1) + E(S_2)$. The expected length of a visit to $Q_i$ is $E(V_i) = \rho_i E(C)$. The mean length of an intervisit period for $Q_i$ is $E(I_i) = (1-\rho_i)E(C)$. Notice that these expectations do not depend on the service discipline used. The expected number of type $i$ customers at polling moments does depend on the service discipline. For gated service the expected number of type $i$ customers at the beginning of a visit to $Q_i$ is $\lambda_i E(C)$. For exhaustive service this is $\lambda_i E(I_i)$. The expected number of type $i$ customers at the beginning of a visit to $Q_{i+1}$ is $\lambda_i (E(V_i) + E(S_i))$ for gated service, and $\lambda_i E(S_i)$ for exhaustive service. Moments of the waiting time distribution for a type $i$ customer at an arbitrary epoch can be derived from the LSTs given by , and . We only present the first moment: $$\begin{aligned} &\textrm{Gated: } & E(W_i) &= (1+\rho_i)\frac{E(C_i^2)}{2E(C)},\label{ewgated}\\ &\textrm{Exhaustive: } & E(W_i) &= \frac{E(I_i^2)}{2E(I_i)}+\frac{\rho_i}{1-\rho_i}\frac{E(B_i^2)}{2E(B_i)},\nonumber\\ & & &= (1-\rho_i)\frac{E({C^*_i}^2)}{2E(C)}.\label{ewexhaustiveC}\end{aligned}$$ Notice that the start of $C_i$ is the *beginning* of a visit to $Q_i$, whereas the start of $C^*_i$ is the *end* of a visit. Equations and are in agreement with Equations (4.1) and (4.2) in [@boxmaworkloadsandwaitingtimes89]. Although at first sight these might seem nice, closed formulas, it should be noted that the expected residual cycle time and the expected residual intervisit time are not easy to determine, requiring the solution of a large set of equations. MVA is an efficient technique to compute mean waiting times, the mean residual cycle time, and also the mean residual intervisit time. We refer to [@winands06] for an MVA framework for polling models. Gated service {#gated} ============= In this section we study the gated service discipline for a polling system with two queues and two priority classes in the first queue: high ($H$) and low ($L$) priority customers. All type $H$ and $L$ customers that are present at the moment when the server arrives at $Q_1$, will be served during the server’s visit to $Q_1$. First all type $H$ customers will be served, then all type $L$ customers. Type $H$ customers arrive at $Q_1$ according to a Poisson process with intensity $\lambda_H$, and have a service requirement $B_H$ with LST $\beta_H(\cdot)$. Type $L$ customers arrive at $Q_1$ with intensity $\lambda_L$, and have a service requirement $B_L$ with LST $\beta_L(\cdot)$. If we do not distinguish between high and low priority customers, we can still use the results from Section \[general\] if we regard the system as a polling system with two queues where customers in $Q_1$ arrive according to a Poisson process with intensity $\lambda_1 := \lambda_H + \lambda_L$ and have service requirement $B_1$ with LST $\beta_1(\cdot) = \frac{\lambda_H}{\lambda_1} \beta_H(\cdot) + \frac{\lambda_L}{\lambda_1} \beta_L(\cdot)$. We follow the same approach as in Section \[general\]. First we study the joint queue length distribution at polling epochs, then the cycle time distribution, followed by the marginal queue length distribution and waiting time distribution. The last subsection provides the first moment of these distributions. Joint queue length distribution at polling epochs ------------------------------------------------- Equations and give the PGFs of the joint queue length distribution at visit beginnings, $V_{b_i}(z_1, z_2)$. A type 1 customer entering the system is a type $H$ customer with probability $\lambda_H/\lambda_1$, and a type $L$ customer with probability $\lambda_L/\lambda_1$. We can express the PGF of the joint queue length distribution in the polling system with priorities, $V_{b_i}(\cdot,\cdot,\cdot)$, in terms of the PGF of the joint queue length distribution in the polling system without priorities, $V_{b_i}(\cdot,\cdot)$. \[p1\_3qs\] $$V_{b_i}(z_H, z_L, z_2) = V_{b_i}\left(\frac{\lambda_H z_H+\lambda_L z_L}{\lambda_1}, z_2\right).$$ Let $X_H$ be the number of high priority customers present in $Q_1$ at the beginning of a visit to $Q_i$, $i=1,2$. Similarly define $X_L$ to be the number of low priority customers present in $Q_1$ at the beginning of a visit to $Q_i$. Let $X_1 = X_H + X_L$. Since the type $H$/$L$ customers in $Q_1$ are exactly those $H$/$L$ customers that arrived since the previous visit beginning at $Q_i$, we know that $$P(X_H=i,X_L=k-i|X_1=k) = \binom ki \left(\frac{\lambda_H}{\lambda_1}\right)^i \left(\frac{\lambda_L}{\lambda_1}\right)^{k-i}.$$ Hence $$\begin{aligned} E[z_H^{X_H}z_L^{X_L}|X_1 = k] &= \sum_{i=0}^\infty\sum_{j=0}^\infty z_H^iz_L^j P(X_H=i,X_L=j|X_1=k) \\ &=\left(\frac{\lambda_H z_H + \lambda_L z_L}{\lambda_1}\right)^k.\end{aligned}$$ Finally, $$\begin{aligned} V_{b_i}(z_H, z_L, z_2) &= \sum_{i=0}^\infty\sum_{j=0}^\infty \left(\frac{\lambda_H z_H + \lambda_L z_L}{\lambda_1}\right)^i z_2^j P(X_1=i, X_2=j) \\ &= V_{b_i}\left(\frac{1}{\lambda_1}(\lambda_H z_H+\lambda_L z_L), z_2\right).\end{aligned}$$ Cycle time ---------- The LST of the cycle time distribution is still given by if we define $\lambda_1 := \lambda_H + \lambda_L$ and $\beta_1(\cdot) := \frac{\lambda_H}{\lambda_1} \beta_H(\cdot) + \frac{\lambda_L}{\lambda_1} \beta_L(\cdot)$, because the cycle time does not depend on the order of service. Equation is valid for polling systems with queues having any branching type service discipline. In the present section we can derive an alternative, shorter expression for $\gamma_1(\cdot)$ by explicitly using the fact that $Q_1$ receives gated service. The type 1 (i.e. both $H$ and $L$) customers present at the visit beginning to $Q_1$ are those that arrived during the previous cycle: $P_1(z, 1) = \gamma_1(\lambda_1(1-z))$. By setting $\omega = \lambda_1(1-z)$, this leads to the following expression for the LST of the distribution of $C_1$ if service in $Q_1$ is gated: $$\gamma_1(\omega) = P_1(1-\frac{\omega}{\lambda_1}, 1).\label{lstCgatedShort}$$ Marginal queue lengths and waiting times {#gatedmarginalw} ---------------------------------------- We first determine the LST of the waiting time distribution for a type $L$ customer, using the fact that this customer will not be served until the next cycle (starting at $Q_1$). The time from the start of the cycle until the arrival will be called “past cycle time”, denoted by $C_{1P}$. The residual cycle time will be denoted by $C_{1R}$. The waiting time of a type $L$ customer is composed of $C_{1R}$, the service times of all high priority customers that arrived during $C_{1P}+C_{1R}$, and the service times of all low priority customers that have arrived during $C_{1P}$. Let $N_H(T)$ be the number of high priority customers that have arrived during time interval $T$, and equivalently define $N_L(T)$. $$\begin{aligned} E\left[{\textrm{e}}^{-\omega W_L}\right] =& \frac{\gamma_1(\lambda_H(1-\beta_H(\omega))+\lambda_L(1-\beta_L(\omega))) - \gamma_1(\omega+\lambda_H(1-\beta_H(\omega)))}{(\omega-\lambda_L(1-\beta_L(\omega)))E(C)}.\end{aligned}$$ $$\begin{aligned} E\left[{\textrm{e}}^{-\omega W_L}\right] &= E\left[{\textrm{e}}^{-\omega (C_{1R}+\sum_{i=1}^{N_H(C_{1P}+C_{1R})}B_{H,i}+\sum_{i=1}^{N_L(C_{1P})}B_{L,i})}\right] \nonumber\\ &= \int_{t=0}^\infty \int_{u=0}^\infty \sum_{m=0}^\infty\sum_{n=0}^\infty E\left[{\textrm{e}}^{-\omega\sum_{i=1}^{m}B_{H,i}}\right]E\left[{\textrm{e}}^{-\omega\sum_{i=1}^{n}B_{L,i}}\right] \nonumber\\ &\quad\cdot{\textrm{e}}^{-\omega u} \frac{(\lambda_H(t+u))^m}{m!}{\textrm{e}}^{-\lambda_H(t+u)}\frac{(\lambda_L t)^n}{n!}{\textrm{e}}^{-\lambda_Lt} {\,\textrm{d}}P(C_{1P}<t, C_{1R}<u)\nonumber\\ \nonumber\\ &= \int_{t=0}^\infty \int_{u=0}^\infty {\textrm{e}}^{-t(\lambda_H(1-\beta_H(\omega))+\lambda_L(1-\beta_L(\omega)))}{\textrm{e}}^{-u(\omega + \lambda_H(1-\beta_H(\omega)))} {\,\textrm{d}}P(C_{1P}<t, C_{1R}<u)\nonumber\\ &=\frac{\gamma_1(\lambda_H(1-\beta_H(\omega))+\lambda_L(1-\beta_L(\omega))) - \gamma_1(\omega+\lambda_H(1-\beta_H(\omega)))}{(\omega-\lambda_L(1-\beta_L(\omega)))E(C)}. \label{lstwlgated}\end{aligned}$$ For the last step in the derivation of we used $$E[{\textrm{e}}^{-\omega_P C_{1P}-\omega_R C_{1R}}] = \frac{E[{\textrm{e}}^{-\omega_P C_1}]-E[{\textrm{e}}^{-\omega_R C_1}]}{(\omega_R-\omega_P)E(C)},$$ which is obtained in [@boxmalevyyechiali92]. The Fuhrmann-Cooper decomposition [@fuhrmanncooper85] still holds for the waiting time of type $L$ customers, because can be rewritten into $$\begin{aligned} E\left[{\textrm{e}}^{-\omega W_L}\right] =& \frac{(1-\rho_L)\omega}{\omega-\lambda_L(1-\beta_L(\omega))} \nonumber\\ &\cdot \frac{\gamma_1(\lambda_H(1-\beta_H(\omega))+\lambda_L(1-\beta_L(\omega))) - \gamma_1(\omega+\lambda_H(1-\beta_H(\omega)))}{(1-\rho_L)\omega E(C)}.\label{lstwlgateddecomposition}\end{aligned}$$ We recognise the first term on the right-hand side of as the LST of the waiting time distribution of an $M/G/1$ queue with only type $L$ customers. An interpretation of the other two terms on the right-hand side can be found when regarding the polling system as a polling system with *three* queues $(Q_H, Q_L, Q_2)$ and no switch-over time between $Q_H$ and $Q_L$. The service discipline of this equivalent system is synchronised gated, which is a more general version of gated. The gates for queues $Q_H$ and $Q_L$ are set simultaneously when the server arrives at $Q_H$, but the gate for $Q_2$ is still set when the server arrives at $Q_2$. In the following paragraphs we show that the second and third term on the right-hand side of together can be interpreted as $E[\left(1-\frac\omega{\lambda_L}\right)^{N_{L|I}}]$, where $N_{L|I}$ is the number of type $L$ customers at a random epoch during the intervisit period of $Q_L$. The expression for the LST of the distribution of the number of type $L$ customers at an arbitrary epoch is determined by first converting the waiting time LST to sojourn time LST, i.e., multiplying expression with $\beta_L(\omega)$. Second, we apply the distributional form of Little’s law [@keilsonservi90] to . This law can be applied because the required conditions are fulfilled for each customer class (H, L, and 2): the customers enter the system in a Poisson stream, every customer enters the system and leaves the system one at a time in order of arrival, and for any time $t$ the entry process into the system of customers after time $t$ and the time spent in the system by any customer arriving before time $t$ are independent. The result is: $$E\left[z^{N_L}\right] = \frac{(1-\rho_L)(1-z)\beta_L(\lambda_L(1-z))}{\beta_L(\lambda_L(1-z))-z} \cdot \frac{\widetilde{V}_{c_L}(z) - \widetilde{V}_{b_L}(z)}{(1-z)(E(N_{L|I_{\textit{end}}}) - E(N_{L|I_{\textit{begin}}}))}.\label{gfzlgateddecomposition}$$ In this equation $\widetilde{V}_{b_L}(z)$ denotes the PGF of the distribution of the number of type $L$ customers at the beginning of a visit to $Q_L$, and $\widetilde{V}_{c_L}(z)$ denotes the PGF at the completion of a visit to $Q_L$: $$\begin{aligned} \widetilde{V}_{b_L}(z) &= V_{b_1}(\beta_H(\lambda_L(1-z)), z, 1)\\ &= \gamma_1(\lambda_H(1-\beta_H(\lambda_L(1-z))) + \lambda_L(1-z)), \\ \widetilde{V}_{c_L}(z) &= V_{b_1}(\beta_H(\lambda_L(1-z)), \beta_L(\lambda_L(1-z)), 1)\\ &= \gamma_1(\lambda_H(1-\beta_H(\lambda_L(1-z))) + \lambda_L(1-\beta_L(\lambda_L(1-z)))).\end{aligned}$$ The last term in is the PGF of the distribution of the number of type $L$ customers at an arbitrary epoch during the intervisit period of $Q_L$, $E[z^{N_{L|I}}]$. Substitution of $\omega := \lambda_L(1-z)$ in , and using $(E(N_{L|I_{\textit{end}}}) - E(N_{L|I_{\textit{begin}}})) = \lambda_L E(I_L)$, shows that the second and third term at the right-hand side of together indeed equal $E[\left(1-\frac\omega{\lambda_L}\right)^{N_{L|I}}]$. The derivation of the LSTs of $W_H$ and $W_2$ is similar and leads to the following expressions: $$\begin{aligned} E\left[{\textrm{e}}^{-\omega W_H}\right] =& \frac{(1-\rho_H)\omega}{\omega-\lambda_H(1-\beta_H(\omega))} \cdot \frac{\gamma_1(\lambda_H(1-\beta_H(\omega))) - \gamma_1(\omega)}{(1-\rho_H)\omega E(C)},\label{lstwhgated}\\ E\left[{\textrm{e}}^{-\omega W_2}\right] =& \frac{(1-\rho_2)\omega}{\omega-\lambda_2(1-\beta_2(\omega))} \cdot \frac{\gamma_2(\lambda_2(1-\beta_2(\omega))) - \gamma_2(\omega)}{(1-\rho_2)\omega E(C)}.\label{lstw2gated}\end{aligned}$$ Equations and are equivalent to the LST of $W_i$ in a nonpriority polling system , which illustrates that the Fuhrmann-Cooper decomposition also holds for the waiting time distributions of high priority customers in $Q_1$ and type 2 customers in a polling system with gated service. Application of the distributional form of Little’s law to these expressions results in: $$\begin{aligned} E\left[z^{N_H}\right] &= \frac{(1-\rho_H)(1-z)\beta_H(\lambda_H(1-z))}{\beta_H(\lambda_H(1-z))-z} \cdot \frac{\gamma_1(\lambda_H(1-\beta_H(\lambda_H(1-z)))) - \gamma_1(\lambda_H(1-z))}{\lambda_H(1-\rho_H)(1-z)E(C)},\\ E\left[z^{N_2}\right] &= \frac{(1-\rho_2)(1-z)\beta_2(\lambda_2(1-z))}{\beta_2(\lambda_2(1-z))-z} \cdot \frac{\gamma_2(\lambda_2(1-\beta_2(\lambda_2(1-z)))) - \gamma_2(\lambda_2(1-z))}{\lambda_2(1-\rho_2)(1-z)E(C)}.\end{aligned}$$ If the service discipline in $Q_2$ is not gated, but another branching type service discipline that satisfies Property \[resingproperty\], should be replaced by the more general expression . Moments {#momentsgated} ------- As mentioned in Section \[momentsgeneral\], we do not focus on moments in this paper, and we only mention the mean waiting times of type $H$ and $L$ customers. For a type $H$ customer, it is immediately clear that $E(W_H) = (1+\rho_H)E(C_{1,\textit{res}})$. The mean waiting time for a type $L$ customer can be obtained by differentiating . This results in: $$E(W_L) = (1+2\rho_H+\rho_L)E(C_{1,\textit{res}}).$$ These formulas can also be obtained using MVA, as shown in [@wierman07]. Globally gated service {#globallygated} ====================== In this section we discuss a polling model with two queues $(Q_1, Q_2)$ and two priority classes ($H$ and $L$) in $Q_1$ with globally gated service. For this service discipline, only customers that were present when the server started its visit to $Q_1$ are served. This feature makes the model exactly the same as a nonpriority polling model with three queues $(Q_H, Q_L, Q_2)$. Although this system does not satisfy Property \[resingproperty\], it does satisfy Property \[borstproperty\] which implies that we can still follow the same approach as in the previous sections. Joint queue length distribution at polling epochs ------------------------------------------------- We define the beginning of a visit to $Q_1$ as the start of a cycle, since this is the moment that determines which customers will be served during the next visits to the queues. Arriving customers will always be served in the next cycle, so the three $(i=H,L,2)$ offspring PGFs are: $$\begin{aligned} f^{(i)}(z_H, z_L, z_2) &= h_i(z_H, z_L, z_2) \\ &= \beta_i(\lambda_H(1-z_H)+\lambda_L(1-z_L)+\lambda_2(1-z_2)),\end{aligned}$$ The two $(i=1,2)$ immigration functions are: $$g^{(i)}(z_H, z_L, z_2) = \sigma_i(\lambda_H(1-z_H)+\lambda_L(1-z_L)+\lambda_2(1-z_2)),$$ Using these definitions, the formula for the PGF of the joint queue length distribution at the beginning of a cycle is similar to the one found in Section \[general\]: $$P_1(z_H, z_L, z_2) = \prod_{n=0}^\infty g(f_n(z_H, z_L, z_2)).$$ Notice that in a system with globally gated service it is possible to express the joint queue length distribution at the beginning of a cycle in terms of the cycle time LST, since all customers that are present at the beginning of a cycle are exactly all of the customers that have arrived during the previous cycle: $$P_1(z_H, z_L, z_2) = \gamma_1(\lambda_H(1-z_H)+\lambda_L(1-z_L)+\lambda_2(1-z_2)).\label{p1globallygated}$$ Cycle time ---------- Since only those customers that are present at the start of a cycle, starting at $Q_1$, will be served during this cycle, the LST of the cycle time distribution is $$\gamma_1(\omega) = \sigma_1(\omega)\sigma_2(\omega)P_1(\beta_H(\omega), \beta_L(\omega), \beta_2(\omega)).\label{lstcgloballygated}$$ Substitution of into this expression gives us the following relation: $$\begin{aligned} &\gamma_1(\omega) = \sigma_1(\omega)\sigma_2(\omega)\\ &\cdot\gamma_1(\lambda_H(1-\beta_H(\omega))+\lambda_L(1-\beta_L(\omega))+\lambda_2(1-\beta_2(\omega))).\end{aligned}$$ [@boxmalevyyechiali92] show that this relation leads to the following expression for the cycle time LST: $$\gamma_1(\omega) = \prod_{i=0}^\infty \sigma(\delta^{(i)}(\omega)),$$ where $\sigma(\cdot) = \sigma_1(\cdot)\sigma_2(\cdot)$, and $\delta^{(i)}(\omega)$ is recursively defined as follows: $$\begin{aligned} \delta^{(0)}(\omega) &= \omega,\\ \delta^{(i)}(\omega) &= \delta(\delta^{(i-1)}(\omega)), \qquad\qquad i=1,2,3,\dots,\\ \delta(\omega) &= \lambda_H(1-\beta_H(\omega)) + \lambda_L(1-\beta_L(\omega)) + \lambda_2(1-\beta_2(\omega)).\end{aligned}$$ Marginal queue lengths and waiting times {#marginal-queue-lengths-and-waiting-times-1} ---------------------------------------- For type $H$ and $L$ customers, the expressions for $E({\textrm{e}}^{-\omega W_H})$ and $E({\textrm{e}}^{-\omega W_L})$ are exactly the same as the ones found in Section \[gatedmarginalw\], but with $\gamma_1(\cdot)$ as defined in . The expression for $E({\textrm{e}}^{-\omega W_2})$ can be obtained with the method used in Section \[gatedmarginalw\]: $$\begin{aligned} E\left[{\textrm{e}}^{-\omega W_2}\right] =&\sigma_1(\omega)\cdot\frac{\gamma_1(\sum_{i=H,L,2}\lambda_i(1-\beta_i(\omega))) - \gamma_1(\omega+\sum_{i=H,L}\lambda_i(1-\beta_i(\omega)))}{(\omega-\lambda_2(1-\beta_2(\omega)))E(C)}\\ =& \sigma_1(\omega)\cdot\frac{(1-\rho_2)\omega}{\omega-\lambda_2(1-\beta_2(\omega))} \\ &\cdot \frac{\gamma_1(\sum_{i=H,L,2}\lambda_i(1-\beta_i(\omega))) - \gamma_1(\omega+\sum_{i=H,L}\lambda_i(1-\beta_i(\omega)))}{(1-\rho_2)\omega E(C)}.\end{aligned}$$ We can use the distributional form of Little’s law to determine the LST of the marginal queue length distribution of $Q_2$: $$\begin{aligned} E\left[z^{N_2}\right] =& \sigma_1(\lambda_2(1-z))\frac{(1-\rho_2)(1-z)\beta_2(\lambda_2(1-z))}{\beta_2(\lambda_2(1-z))-z} \\ &\cdot \frac{\gamma_1\left(\sum_{i=H,L,2}\lambda_i(1-\beta_i(\lambda_2(1-z)))\right) - \gamma_1\left(\lambda_2(1-z)+\sum_{i=H,L}\lambda_i(1-\beta_i(\lambda_2(1-z)))\right)}{ \lambda_2(1-\rho_2)(1-z)E(C)}.\end{aligned}$$ The Fuhrmann-Cooper queue length decomposition also holds for all customer classes in a polling system with globally gated service. Moments {#moments} ------- The expressions for $E(W_H)$ and $E(W_L)$ from Section \[momentsgated\] also hold in a globally gated polling system, but with a different mean residual cycle time. We only provide the mean waiting time of type 2 customers: $$E(W_2) = E(S_1) + (1 + 2\rho_H + 2\rho_L + \rho_2) E(C_{1,\textit{res}}).$$ Exhaustive service {#exhaustive} ================== In this section we study the same polling model as in the previous two sections, but the two queues are served exhaustively. The section has the same structure as the other sections, so we start with the derivation of the LST of the joint queue length distribution at polling epochs, followed by the LST of the cycle time distribution. LSTs of the marginal queue length distributions and waiting time distributions are provided in the next subsection. In the last part of the section the mean waiting time of each customer type is studied. It should be noted that, although we assume that both $Q_1$ and $Q_2$ are served exhaustively, a model in which $Q_2$ is served according to another branching type service discipline, requires only minor adaptations. Joint queue length distribution at polling epochs ------------------------------------------------- We can derive the joint queue length distribution at the beginning of a cycle for a polling system with two queues and two priority classes in $Q_1$, $P_1(z_H, z_L, z_2)$, directly from for $P_1(z_1, z_2)$. Similar to the proof of Lemma \[p1\_3qs\], we can prove that $$P_1(z_H, z_L, z_2) = P_1\left(\frac{1}{\lambda_1}(\lambda_H z_H+\lambda_L z_L), z_2\right).$$ The same holds for $V_{b_2}(\cdot, \cdot, \cdot)$ and visit completion epochs $V_{c_i}(\cdot, \cdot, \cdot)$, for $i = 1, 2$. Cycle time ---------- For the cycle time starting with a visit to $Q_1$, is still valid. However, when studying the waiting time of a specific customer type in an exhaustively served queue, it is convenient to consider the *completion* of a visit to $Q_1$ as the start of a cycle. Hence, in this section the notation $C^*_1$, or the LST of its distribution, $\gamma^*_1(\cdot)$, refers to the cycle time starting at the completion of a visit to $Q_1$. Equation gives the LST of the distribution of $C^*_1$. Using the fact that customers in $Q_1$ are served exhaustively, we can find an alternative, compact expression for $\gamma_1^*(\cdot)$. The type 1 (i.e. both type $H$ and $L$ customers) customers at the beginning of a visit to $Q_1$ are exactly those type 1 customers that have arrived during the previous intervisit time: $P_1(z, 1) = \widetilde{I}_1(\lambda_1(1-z))$. Hence, by setting $\omega=\lambda_1(1-z)$, we get $\widetilde{I}_1(\omega) = P_1(1-\frac{\omega}{\lambda_1}, 1)$, and thus by , $$\gamma_1^*(\omega) = P_1(\pi_1(\omega)-\frac{\omega}{\lambda_1}, 1).\label{lstCexhaustiveShort}$$ Marginal queue lengths and waiting times {#marginal-queue-lengths-and-waiting-times-2} ---------------------------------------- Analysis of the model with exhaustive service requires a different approach. The key observation, made by [@fuhrmanncooper85], is that a nonpriority polling system from the viewpoint of a type $i$ customer is an $M/G/1$ queue with multiple server vacations. This implies that the Fuhrmann-Cooper decomposition can be used, even though the intervisit times are strongly dependent on the visit times. The $M/G/1$ queue with priorities and vacations can be analysed by modelling the system as a special version of the *nonpriority* $M/G/1$ queue with multiple server vacations, and then applying the results from Fuhrmann and Cooper. This approach has been used by [@kellayechiali88] who used the concept of *delay cycles*, and also by [@Shanthikumar89] who used *level crossing analysis*; see also [@takagi91]. We apply Kella and Yechiali’s approach to the polling model under consideration to find the waiting time LST for type $H$ and $L$ customers. In [@kellayechiali88] systems with single and multiple vacations, preemptive resume and nonpreemptive service are considered. In the present paper we do not consider preemptive resume, so we only use results from the case labelled as NPMV (nonpreemptive, multiple vacations) in [@kellayechiali88]. We consider the system from the viewpoint of a type $H$ and type $L$ customer separately to derive $E[{\textrm{e}}^{-\omega W_H}]$ and $E[{\textrm{e}}^{-\omega W_L}]$. From the viewpoint of a type $H$ customer and as far as waiting times are concerned, a polling system is a *nonpriority* single server system with multiple vacations. The vacation can either be the intervisit period $I_1$, or the service of a type $L$ customer. The LSTs of these two types of vacations are: $$\begin{aligned} E[{\textrm{e}}^{-\omega I_1}] &= P_1(1-\omega/\lambda_1, 1), \label{intervisitexhaustive}\\ E[{\textrm{e}}^{-\omega B_L}] &= \beta_L(\omega). \nonumber\end{aligned}$$ Equation follows immediately from the fact that the number of type 1 (i.e. both H and L) customers at the beginning of a visit to $Q_1$ is the number of type 1 customers that have arrived during the previous intervisit period: $P_1(z, 1) = E[{\textrm{e}}^{-(\lambda_1(1-z)) I_1}]$. We now use the concept of delay cycles, introduced in [@kellayechiali88], to find the waiting time LST of a type $H$ customer. The key observation is that an arrival of a tagged type $H$ customer will always take place within either an $I_H$ cycle, or an $L_H$ cycle. An $I_H$ cycle is a cycle that starts with an intervisit period for $Q_1$, followed by the service of all type $H$ customers that have arrived during the intervisit period, and ends at the moment that no type $H$ customers are left in the system. Notice that at the start of the intervisit period, no type $H$ customers were present in the system either. An $L_H$ cycle is a similar cycle, but starts with the service of a type $L$ customer. This cycle also ends at the moment that no type $H$ customers are left in the system. The fraction of time that the system is in an $L_H$ cycle is $\frac{\rho_L}{1-\rho_H}$, because type $L$ customers arrive with intensity $\lambda_L$. Each of these customers will start an $L_H$ cycle and the length of an $L_H$ cycle equals $\frac{E(B_L)}{1-\rho_H}$: $$\begin{aligned} E(L_H\textrm{ cycle}) &= E(B_L) + \lambda_H E(B_L) E(\textit{BP}_H) \\ &= E(B_L) + \lambda_H E(B_L) \frac{E(B_H)}{1-\rho_H} \\ &= (1+\frac{\rho_H}{1-\rho_H})E(B_L) = \frac{E(B_L)}{1-\rho_H},\end{aligned}$$ where $E(\textit{BP}_H)$ is the mean length of a busy period of type $H$ customers. The fraction of time that the system is in an $I_H$ cycle, is $1-\frac{\rho_L}{1-\rho_H} = \frac{1-\rho_1}{1-\rho_H}$. This result can also be obtained by using the argument that the fraction of time that the system is in an intervisit period is the fraction of time that the server is not serving $Q_1$, which is equal to $1-\rho_1$. A cycle which starts with such an intervisit period and stops when all type $H$ customers that arrived during the intervisit period and their type $H$ descendants have been served, has mean length $E(I_1) + \lambda_H E(I_1) E(\textit{BP}_H) = \frac{E(I_1)}{1-\rho_H}$. This also leads to the conclusion that $\frac{1-\rho_1}{1-\rho_H}$ is the fraction of time that the system is in an $I_H$ cycle. A customer arriving during an $I_H$ cycle views the system as a nonpriority $M/G/1$ queue with multiple server vacations $I_1$; a customer arriving during an $L_H$ cycle views the system as a nonpriority $M/G/1$ queue with multiple server vacations $B_L$. [@fuhrmanncooper85] showed that the waiting time of a customer in an $M/G/1$ queue with server vacations is the sum of two independent quantities: the waiting time of a customer in a corresponding $M/G/1$ queue without vacations, and the residual vacation time. Hence, the LST of the waiting time distribution of a type $H$ customer is: $$E[{\textrm{e}}^{-\omega W_H}] = \frac{(1-\rho_H)\omega}{\omega-\lambda_H(1-\beta_H(\omega))}\cdot \left[\frac{1-\rho_1}{1-\rho_H} \cdot \frac{1-\widetilde{I}_1(\omega)}{\omega E(I_1)}+ \frac{\rho_L}{1-\rho_H}\cdot\frac{1-\beta_L(\omega)}{\omega E(B_L)}\right].\label{lstwhexhaustive}$$ Equation is in accordance with the more general equation in Section 4.1 in [@kellayechiali88]. The LST of the distribution of the waiting time of a high priority customer in a two priority $M/G/1$ queue without vacations is $$E[{\textrm{e}}^{-\omega W_{H|M/G/1}}] =\frac{(1-\rho_1)\omega+\lambda_L(1-\beta_L(\omega))}{\omega-\lambda_H(1-\beta_H(\omega))}\label{lstwhmg1},\\$$ see, e.g., Equation (3.85) in [@cohen82], Chapter III.3. Equation can be rewritten to , with $\frac{1-\widetilde{I}_1(\omega)}{\omega E(I_1)}$ replaced by 1. Hence, the waiting time distribution of a high priority customer in a two priority $M/G/1$ queue equals the waiting time distribution of a customer in a nonpriority $M/G/1$ queue with only type $H$ customers, where the server goes on a vacation $B_L$ with probability $\frac{\rho_L}{1-\rho_H}$. Substitution of in expresses $E[{\textrm{e}}^{-\omega W_H}]$ in terms of the LST of the cycle time distribution starting at a visit *completion* to $Q_1$, $\gamma_1^*(\cdot)$: $$E[{\textrm{e}}^{-\omega W_H}] = \frac{1-\gamma_1^*(\omega - \lambda_H(1-\beta_H(\omega)) - \lambda_L(1-\beta_L(\omega))) +\lambda_L (1-\beta_L(\omega))E(C)}{(\omega-\lambda_H(1-\beta_H(\omega)))E(C)}.\label{lstwhexhaustiveC}$$ The concept of cycles is not really needed to model the system from the perspective of a type $L$ customer, because for a type $L$ customer the system merely consists of $I_{HL}$ cycles. An $I_{HL}$ cycle is the same as an $I_H$ cycle, discussed in the previous paragraphs, except that it ends when no type $H$ *or $L$* customers are left in the system. So the system can be modelled as a nonpriority $M/G/1$ queue with server vacations. The vacation is the intervisit time $I_1$, plus the service times of all type $H$ customers that have arrived during that intervisit time and their type $H$ descendants. We will denote this extended intervisit time by $I_1^*$ with LST $$\widetilde{I}_1^*(\omega) = \widetilde{I}_1(\omega+\lambda_H(1-\pi_H(\omega))).$$ The mean length of $I_1^*$ equals $E(I_1^*) = \frac{E(I_1)}{1-\rho_H}$. We also have to take into account that a busy period of type $L$ customers might be interrupted by the arrival of type $H$ customers. Therefore the alternative system that we are considering will not contain regular type $L$ customers, but customers still arriving with arrival rate $\lambda_L$, whose service time equals the service time of a type $L$ customer in the original model, plus the service times of all type $H$ customers that arrive during this service time, and all of their type $H$ descendants. The LST of the distribution of this extended service time $B_L^*$ is $$\beta_L^*(\omega) = \beta_L(\omega + \lambda_H(1-\pi_H(\omega))).$$ This extended service time is often called *completion time* in the literature. In this alternative system, the mean service time of these customers equals $E(B_L^*) = \frac{E(B_L)}{1-\rho_H}$. The fraction of time that the system is serving these customers is $\rho_L^* = \frac{\rho_L}{1-\rho_H} = 1 - \frac{1-\rho_1}{1-\rho_H}$. Now we use the results from the $M/G/1$ queue with server vacations (starting with the Fuhrmann-Cooper decomposition) to determine the LST of the waiting time distribution for type $L$ customers: $$\begin{aligned} E[{\textrm{e}}^{-\omega W_L}] =& \frac{(1-\rho_L^*)\omega}{\omega-\lambda_L(1-\beta_L^*(\omega))} \cdot \frac{1-\widetilde{I}_1^*(\omega)}{\omega E(I_1^*)} \nonumber\\ =& \frac{(1-\rho_1)(\omega+\lambda_H(1-\pi_H(\omega)))}{\omega-\lambda_L(1-\beta_L(\omega+\lambda_H(1-\pi_H(\omega))))} \cdot \frac{1-\widetilde{I}_1(\omega+\lambda_H(1-\pi_H(\omega)))}{(\omega + \lambda_H(1-\pi_H(\omega)))E(I_1)}. \label{lstwlexhaustive}\end{aligned}$$ The last term of is the LST of the distribution of the residual intervisit time, plus the time that it takes to serve all type $H$ customers and their type $H$ descendants that arrive during this residual intervisit time. The first term of is the LST of the waiting time distribution of a low-priority customer in an $M/G/1$ queue with two priorities, without vacations (see e.g. (3.76) in [@cohen82], Chapter III.3). The $M/G/1$ queue with two priorities can be viewed as a nonpriority $M/G/1$ queue with vacations, if we consider the waiting time of type $L$ customers. We only need to rewrite the first term of : $$\begin{aligned} E[{\textrm{e}}^{-\omega W_{L|M/G/1}}] =& \frac{(1-\rho_1)(\omega+\lambda_H(1-\pi_H(\omega)))}{\omega-\lambda_L(1-\beta_L(\omega+\lambda_H(1-\pi_H(\omega))))}\\ =&\frac{(1-\rho_L^*)\omega}{\omega-\lambda_L(1-\beta_L^*(\omega))} \cdot \frac{1-\rho_1}{1-\rho_L^*} \cdot \frac{\omega+\lambda_H(1-\pi_H(\omega))}{\omega} \\ =& E[e^{-\omega W_{L|M/G/1}^*}] \cdot\left[(1-\rho_H) + \rho_H \frac{1-\pi_H(\omega)}{\omega E(\textit{BP}_H)}\right],\end{aligned}$$ where $E[e^{-\omega W_{L|M/G/1}^*}]$ is the LST of the waiting time distribution of a customer in an $M/G/1$ queue where customers arrive at intensity $\lambda_L$ and have service requirement LST $\beta_L(\omega + \lambda_H(1-\pi_H(\omega)))$. So with probability $1-\rho_H$ the waiting time of a customer is the waiting time in an $M/G/1$ queue with no vacations, and with probability $\rho_H$ the waiting time of a customer is the sum of the waiting time in an $M/G/1$ queue and the residual length of a vacation, which is a busy period of type $H$ customers. Substitution of in leads to a different expression for $E[{\textrm{e}}^{-\omega W_L}]$: $$\begin{aligned} E[{\textrm{e}}^{-\omega W_L}] &=\frac{1-\gamma^*_1(\omega - \lambda_L(1-\beta_L(\omega+\lambda_H(1-\pi_H(\omega)))))}{(\omega-\lambda_L(1-\beta_L(\omega+\lambda_H(1-\pi_H(\omega)))))E(C)}\nonumber\\ &=E[{\textrm{e}}^{-(\omega - \lambda_L(1-\beta_L(\omega+\lambda_H(1-\pi_H(\omega)))))C^*_{1,\textit{res}}}].\label{lstwlexhaustiveC}\end{aligned}$$ The waiting time of type 2 customers is not affected at all by the fact that $Q_1$ contains multiple classes of customers, so is still valid for $E({\textrm{e}}^{-\omega W_2})$. We will refrain from mentioning the PGFs of the marginal queue length distributions here, because they can be obtained by applying the distributional form of Little’s law as we have done before. Moments {#moments-1} ------- The mean waiting times for high and low priority customers can be found by differentiation of and : $$\begin{aligned} E(W_H) &= \frac{\rho_H E(B_{H,\textit{res}})+\rho_L E(B_{L,\textit{res}})}{1-\rho_H}+\frac{1-\rho_1}{1-\rho_H}E(I_{1,\textit{res}}),\\ E(W_L) &= \frac{\rho_H E(B_{H,\textit{res}}) + \rho_L E(B_{L,\textit{res}})}{(1-\rho_H)(1-\rho_1)} + \frac{1}{1-\rho_H}E(I_{1,\textit{res}}).\end{aligned}$$ Differentiation of and leads to alternative expressions, that can also be found in [@wierman07]. $$\begin{aligned} E(W_H) &= \frac{(1-\rho_1)^2}{1-\rho_H}\frac{E({C^*_1}^2)}{2E(C)},\\ E(W_L) &= \frac{(1-\rho_1)^2}{(1-\rho_H)(1-\rho_1)}\frac{E({C^*_1}^2)}{2E(C)}\\ &= \left(1-\frac{\rho_L}{1-\rho_H}\right)\frac{E({C^*_1}^2)}{2E(C)}.\end{aligned}$$ Example {#numericalexample} ======= Consider a polling system with two queues, and assume exponential service times and switch-over times. Suppose that $\lambda_1 = \frac{6}{10}, \lambda_2 = \frac{2}{10}, E(B_1) = E(B_2) = 1, E(S_1) = E(S_2) = 1$. The workload of this polling system is $\rho = \frac{8}{10}$. This example is extensively discussed in [@winands06] where MVA was used to compute mean waiting times and mean residual cycle times for the gated and exhaustive service disciplines. In this example we show that the performance of this system can be improved by giving higher priority to jobs with smaller service times. We define a threshold $t$ and divide the jobs into two classes: jobs with a service time less than $t$ receive high priority, the other jobs receive low priority. In Figures \[fig:gated\] and \[fig:exhaustive\] the mean waiting times of customers in $Q_1$ are shown as a function of the threshold $t$. The following four cases are distinguished: - the mean waiting time of the low priority customers in $Q_1$ (indicated as “Type L”); - the mean waiting time of the high priority customers in $Q_1$ (indicated as “Type H”); - a weighted average of the above two mean waiting times: $\frac{\lambda_L}{\lambda_1}E(W_L) + \frac{\lambda_H}{\lambda_1}E(W_H)$ (indicated as “Type 1 with priorities”). This can be interpreted as the mean waiting time of an arbitrary customer in $Q_1$; - the mean waiting time of an arbitrary customer in $Q_1$ if no priority rules would be applied to this queue (indicated as “Type 1 no priorities”). In this situation there is no such thing as high and low priority customers, so the mean waiting time does not depend on $t$, and has already been computed in [@winands06]. The figures show that a unique optimal threshold exists that minimises the mean weighted waiting time for customers in $Q_1$. This value depends on the service discipline used and is discussed in [@wierman07]. In this example the optimal threshold is 1 for gated, and 1.38 for exhaustive. Figure \[fig:gated\] confirms that the mean waiting times for type $H$ and $L$ customers in the gated model only differ by a constant value: $E(W_L) - E(W_H) = \rho_1 E(C_{1,\textit{res}})$. For globally gated service no figure is included, because we again have $E(W_L) - E(W_H) = \rho_1 E(C_{1,\textit{res}})$. The mean residual cycle time is different from the one in the gated model, but this does not affect the optimal threshold which is still $t=1$. In the exhaustive model we have the following relation: $$E(W_L) - E(W_H) = \frac{\rho_1(1-\rho_1)}{1-\rho_H}E(C^*_{1,\textit{res}}).$$ If we increase threshold $t$, the fraction of customers in $Q_1$ that receive high priority grows, and so does their mean service time. This means that $\rho_H$ increases as $t$ increases, so $E(W_L) - E(W_H)$ gets bigger, which can be seen in Figure \[fig:exhaustive\]. Notice that $\frac{E(W_H)}{E(W_L)} = 1-\rho_1$, so it does not depend on $t$. ![Mean waiting time of customers in $Q_1$ in the gated polling system, versus threshold $t$.\[fig:gated\]](gatedMean){width="\linewidth"} ![Mean waiting time of customers in $Q_1$ in the exhaustive polling system, versus threshold $t$.\[fig:exhaustive\]](exhaustiveMean){width="\linewidth"} It is interesting to also consider the variance, or rather the standard deviation of the waiting time. Figures \[fig:gatedvar\] and \[fig:exhaustivevar\] show the standard deviation of the type $H$ and $L$ customers versus the threshold $t$. The figures also show the standard deviation of an arbitrary customer in $Q_1$, with and without priorities. The figures indicate that the waiting times in the gated system have smaller standard deviations than in the exhaustive case. In this example, the introduction of priorities affects the standard deviation of an arbitrary type 1 customer only slightly. However, it is interesting to zoom in to investigate the influence of threshold $t$. Figure \[fig:gatedexhaustivevarzoom\] contains zoomed versions of Figures \[fig:gatedvar\] and \[fig:exhaustivevar\] and indicates that the threshold $t$ that minimises the overall mean waiting time of type $1$ customers in the priority system does not minimise the standard deviation. In fact, changing threshold $t$ affects the entire service time distributions $B_H$ and $B_L$, which results in two local minima for the standard deviation as function of threshold $t$. ![Standard deviation of the waiting time of customers in $Q_1$ in the gated polling system, versus threshold $t$.\[fig:gatedvar\]](gatedVar){width="\linewidth"} ![Standard deviation of the waiting time of customers in $Q_1$ in the exhaustive polling system, versus threshold $t$.\[fig:exhaustivevar\]](exhaustiveVar){width="\linewidth"} ![Zoomed versions of Figures \[fig:gatedvar\] (left) and \[fig:exhaustivevar\] (right).\[fig:gatedexhaustivevarzoom\]](gatedVarZoom "fig:"){width="0.45\linewidth"} ![Zoomed versions of Figures \[fig:gatedvar\] (left) and \[fig:exhaustivevar\] (right).\[fig:gatedexhaustivevarzoom\]](exhaustiveVarZoom "fig:"){width="0.45\linewidth"} Possible extensions and future research {#extensions} ======================================= The polling system studied in the present paper leaves many possibilities for extensions or variations. In this section we discuss some of them. #### Multiple queues and priority levels. Probably the most obvious extension of the model under consideration, is a polling system with any number of queues and any number of priority levels in each queue. In recent research [@boonadanboxma2008], we have discovered that such a polling model can be analysed in detail. Each queue can have its own service discipline, either exhaustive or (synchronised) gated. #### Preemptive resume. In the present paper, the service of low priority customers is not interrupted by the arrival of a high priority customer. If we allow for service interruptions, these would only take place in a queue with exhaustive service, since (globally) gated service forces high priority customers to wait behind the gate. We note that allowing service interruptions does not affect the joint queue length distributions at polling instants, nor the cycle time. Also the waiting time of low priority customers is unaffected (but they might have a longer *sojourn time*). It only affects the waiting time of high priority customers, because they do not have to wait for a residual service time of a low priority customer. The LST of the waiting time distribution of a high priority customer if service is preemptive resume, is: $$E[{\textrm{e}}^{-\omega W_H}] = \frac{(1-\rho_H)\omega}{\omega-\lambda_H(1-\beta_H(\omega))}\cdot \left[\frac{1-\rho_1}{1-\rho_H} \cdot \frac{1-\widetilde{I}_1(\omega)}{\omega E(I_1)}+ \frac{\rho_L}{1-\rho_H}\right].$$ #### Mixed gated/exhaustive service. In the present paper, customers in $Q_1$ receive either exhaustive or (globally) gated service. One may consider serving each priority level according to a different service discipline. In [@boonadan2008], high priority customers receive exhaustive service, whereas low priority customers receive gated service. This gives high priority customers an additional advantage, but it turns out that for low priority customers this strategy may be better than, e.g., gated service for all priority levels. A mixture of globally gated service for low priority customers and exhaustive service for high priority customers can be analysed similarly. The “opposite” strategy, where low priority customers are served exhaustively and high priority customers are served according to the gated service discipline is easier to analyse, since we can model it as a nonpriority polling model with $Q_1$ replaced by two queues, $Q_{H}$ and $Q_L$, containing the type $H$ and type $L$ customers and having gated and exhaustive service respectively. #### Partially gated. A variant of the gated service discipline is partially gated service: every customer, type $H$ or $L$, standing in front of the gate is served during a visit with a fixed probability $p$, and is not served with probability $1-p$. The probability $p$ might even depend on the customer type. Whether a rejected customer is eligible for service in the next cycle, or leaves the system, does not matter. Both situations can be analysed. #### Different polling sequences. We assume that the server alternates between $Q_1$ and $Q_2$. A different way of introducing priorities to a polling system is by increasing the frequency of visits to a queue within a cycle. One can, e.g., decide to visit $Q_1$ two consecutive times if gated service is used. Or one can think of a system where the server switches to $Q_j$ after completing a visit to $Q_i$ with probability $p_{ij}$. #### Large setup times. [@winandsPhD] establishes fluid limits for polling systems with any branching type service discipline and *deterministic* switch-over times tending to infinity. The scaled waiting time distribution is shown to converge to a uniform distribution with bounds that can be computed explicitly. The results are relevant to applications in production systems, where large setup times are common. These fluid limits can also be computed for the polling model that is discussed in the present paper and give explicit insight in when each of the discussed service disciplines is optimal. [27]{} \[1\][\#1]{} \[1\][`#1`]{} urlstyle \[1\][doi: \#1]{} M. A. A. Boon and I. J. B. F. Adan. Mixed gated/exhaustive service in a polling model with priorities. <span style="font-variant:small-caps;">Eurandom</span> report 2008-045, submitted for publication, 2008. M. A. A. Boon, I. J. B. F. Adan, and O. J. Boxma. A polling model with multiple priority levels. <span style="font-variant:small-caps;">Eurandom</span> report 2008-029, submitted for publication, 2008. M. A. A. Boon, I. J. B. F. Adan, and O. J. Boxma. 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[^1]: <span style="font-variant:small-caps;">Eurandom</span> and Department of Mathematics and Computer Science, Eindhoven University of Technology, P.O. Box 513, 5600MB Eindhoven, The Netherlands [^2]: The research was done in the framework of the BSIK/BRICKS project, and of the European Network of Excellence Euro-FGI. [^3]: The present paper is an adapted and extended version of [@boonadanboxma2queues2008].

--- abstract: | We revisit a fundamental open problem in quantum information theory, namely whether it is possible to transmit quantum information at a rate exceeding the channel capacity if we allow for a non-vanishing probability of decoding error. Here we establish that the Rains information of any quantum channel is a strong converse rate for quantum communication: For any code with a rate exceeding the Rains information of the channel, we show that the fidelity vanishes exponentially fast as the number of channel uses increases. This remains true even if we consider codes that perform classical post-processing on the transmitted quantum data. Our result has several applications. Most importantly, for generalized dephasing channels we show that the Rains information is also achievable, and thereby establish the strong converse property for quantum communication over such channels. This for the first time conclusively settles the strong converse question for a class of quantum channels that have a non-trivial quantum capacity. Moreover, we show that the classical post-processing assisted quantum capacity of the quantum binary erasure channel satisfies the strong converse property. author: - 'Marco Tomamichel$^*$,' - 'Mark M. Wilde$^\dagger$,' - 'Andreas Winter$^\ddagger$ [^1] [^2][^3]' bibliography: - 'Ref.bib' title: Strong Converse Rates for Quantum Communication --- Introduction ============ The *quantum capacity* of a quantum channel ${\mathcal{N}}$, denoted $Q(\mathcal{N})$, is defined as the maximum rate (in qubits per channel use) at which it is possible to transmit quantum information over many memoryless uses of the channel, such that a receiver can recover the state with a fidelity that asymptotically converges to one as we increase the number of channel uses. (We will formally introduce capacity in Section \[sec:ent-gen-code\].) The question of determining the quantum capacity was set out by Shor in his seminal paper on quantum error correction [@S95]. Since then, a number of works established a multi-letter upper bound on the quantum channel capacity in terms of the coherent information [@SN96; @BKN98; @BNS98], and the coherent information lower bound on quantum capacity was demonstrated by a sequence of works [@L97; @capacity2002shor; @D05] which are often said to bear increasing standards of rigor.[^4] In more detail, the work in [@L97; @capacity2002shor; @D05] showed that the following *coherent information* of the channel$~\mathcal{N}$ is an *achievable rate for quantum communication*:$$I_{\rm c}\left( \mathcal{N}\right) := \max_{\phi_{RA}} I(R\rangle B)_{\rho}, \qquad \textrm{where} \qquad \rho_{RB} = \mathcal{N}_{A\rightarrow B}\left( \phi_{RA}\right) \label{eq:coh-info}$$ and the optimization is over all pure bipartite states $\phi_{RA}$. Here, the coherent information of a bipartite state $\rho_{RB}$ is defined as $I(R\rangle B)_{\rho}:= H(\rho_B)-H(\rho_{RB})$, with the von Neumann entropies $H(\rho) := -$Tr$\left\{ \rho \log\rho \right\}$.[^5] From the above result, we can also conclude that the rate $I_{\rm c}\left( \mathcal{N}^{\otimes l}\right) /l$ is achievable for any positive integer $l$, simply by applying the formula in (\[eq:coh-info\]) to the superchannel $\mathcal{N}^{\otimes l}$ and normalizing. By a limiting argument, we find that the regularized coherent information $\lim_{l\rightarrow\infty}I_{c}\left( \mathcal{N}^{\otimes l}\right) /l$ is also achievable, and Refs. [@SN96; @BKN98; @BNS98] established that this regularized coherent information is also an upper bound on quantum capacity. This establishes that $$\begin{aligned} Q(\mathcal{N}) = \lim_{l\rightarrow\infty} \frac{I_{\rm c}( \mathcal{N}^{\otimes l})}{l} . \label{eq:capaity-result}\end{aligned}$$ Clearly, the regularized coherent information is not a tractable characterization of quantum capacity. But the later work of Devetak and Shor proved that the single-letter coherent information formula in (\[eq:coh-info\]) is equal to the quantum capacity for the class of degradable quantum channels [@DS05]. Degradable channels are such that the receiver of the channel can simulate the channel to the environment by applying a degrading map to the channel output. All of the above works established an understanding of quantum capacity in the following sense: 1. (Achievability) If the rate of quantum communication is below the quantum capacity, then there exists a scheme for quantum communication such that the fidelity converges to one in the limit of many channel uses. 2. (Weak Converse) If the rate of quantum communication is above the quantum capacity, then there cannot exist an asymptotically error-free quantum communication scheme. It is crucial to note at this point that practical schemes for quantum communication are restricted to operate on finite block lengths, and it is thus in principle impossible to achieve exactly error-free communication (for most channels). Hence, the weak converse discussed above does not apply and we are left to wonder whether it is possible to transmit information at a larger rate than the regularized coherent information given in  if a non-vanishing error is permissible. Interestingly, it has been known since the early days of classical information theory that the capacity of a classical channel obeys the *strong converse property* [@Wolfowitz1964; @Arimoto73]: if the rate of communication exceeds the capacity, then the error probability necessarily converges to one in the limit of many channel uses. Furthermore, many works have now confirmed that the strong converse property holds for the classical capacity of several quantum channels [@ON1999; @W99; @KW09; @WWY13; @WW13] and also for the entanglement-assisted classical capacity of all quantum channels [@BDHSW12; @BCR09; @GW13]. Consequently, this work aims to sharpen the interpretation of quantum capacity in the same spirit. Overview of Results and Outline ------------------------------- In this paper, we define the *Rains information of a quantum channel* $\mathcal{N}_{A\rightarrow B}$ as follows:[^6]$$R\left( \mathcal{N}\right) := \max_{\rho_{RA}} \min_{\tau_{RB}\in{{\rm PPT}'}\left( R:B\right) }D\left( {\mathcal{N}}(\rho_{RA}) \Vert\tau_{RB}\right) , \label{eq:rains-bound}$$ where we maximize over input states $\rho_{RA}$ with $R$ a reference system, and minimize over a set of states (including all PPT states) first considered by Rains, namely the set $$\begin{aligned} {{\rm PPT}'}\left( R:B\right) & =\left\{ \tau_{RB}:\tau_{RB}\geq0\wedge\left\Vert T_{B}\left( \tau_{RB}\right) \right\Vert _{1}\leq1\right\} .$$ Here, $D(\rho\|\tau) := \operatorname{tr}\{ \rho (\log \rho - \log \tau) \}$ is the relative entropy and $T_{B}$ denotes the partial transpose. The quantity in (\[eq:rains-bound\]), often called the Rains bound, was first explored by Rains in the context of entanglement distillation [@R01] and later refined to the form above in [@AdMVW02]. Our main contribution is that the Rains information of a quantum channel is a *strong converse rate for quantum communication* (Theorem \[thm:strong-converse\]), even if we allow for classical pre- and post-processing. (The allowed codes are described in more detail in Section \[sec:ent-gen-code\].) That is, if the quantum communication rate of any protocol for a channel $\mathcal{N}_{A\rightarrow B}$ exceeds its Rains information, then the fidelity of the scheme decays exponentially fast to zero as the number $n$ of channel uses increases. (We also mention that our main contribution here sets some previous claims from [@SS08; @SSW08] on a firm foundation. Namely, in Refs. [@SS08; @SSW08], the authors claim that the Rains bound for entanglement distillation leads to a weak converse upper bound on the quantum capacity of any channel. However, these papers do not appear to support this claim in any way. Our strong converse result in the present paper thus provides a proof of these claims.) We establish this theorem by exploiting the generalized divergence framework of Sharma and Warsi [@SW12], which generalizes the related framework from the classical context [@PV10]. However, as discussed in Section \[sec:gen-div\], our main departing point from that work is to consider a different class of useless channels for quantum data transmission. That is, in [@SW12], the main idea was to exploit a generalized divergence to compare the output of the channel with an operator of the form $\pi_{R}\otimes\sigma_{B}$, where $\pi_R$ denotes the fully mixed state on the reference system $R$ and $\sigma_B$ is an arbitrary state on the output system. This state can be viewed as the output of a useless channel that replaces the input and reference system with the maximally mixed state and an uncorrelated state. The resulting information quantity is a generalization of the coherent information from (\[eq:coh-info\]), and one then invokes a data processing inequality to relate this quantity to communication rate and fidelity. Here, we instead compare the output of the channel with an operator in the set ${{\rm PPT}'}\left( R:B\right) $, which contains and is closely related to the positive partial transpose states, which in turn are well known to have no distillable entanglement (or equivalently, no quantum data transmission capabilities, so that they constitute another class of useless channels for quantum data transmission). It turns out that a Rényi-like version of the Rains information of a quantum channel appears to be easier to manipulate, so that we can show that it obeys a weak subadditivity property (Theorem \[thm:weak-subadditivity\] in Section \[sec:weaksub\]). From there, some standard arguments from [@ON1999] conclude the proof of our main result (Theorem \[thm:strong-converse\] in Section \[sec:main\]). The main application of this result is to establish the *strong converse property* for any generalized dephasing channel (Section \[sec:dephasing\]). The action of any channel in this class on an input state $\rho$ is as follows:$$\mathcal{N}\left( \rho\right) =\sum_{x,y=0}^{d-1}\left\langle x\right\vert _{A}\rho\left\vert y\right\rangle _{A}\ \left\langle \psi _{y}|\psi_{x}\right\rangle \ \left\vert x\right\rangle \left\langle y\right\vert _{B},$$ where $\{\left\vert x\right\rangle _{A}\}$ and $\{\left\vert x\right\rangle _{B}\}$ are orthonormal bases for the input and output systems, respectively, for some positive integer $d$, and $\{\left\vert \psi_{x}\right\rangle _{E}\}$ is a set of arbitrary pure quantum states. A particular example in this class is the qubit dephasing channel, whose action on a qubit density operator is$$\mathcal{N}\left( \rho\right) =\left( 1-p\right) \rho+pZ\rho Z,$$ where the dephasing parameter $p\in\left[ 0,1\right] $ and $Z$ is the Pauli $\sigma_Z$ operator. We prove this result by showing that the Rains information of a generalized dephasing channel is equal to its coherent information (Proposition \[prop:I\_c=R-for-H\]). Finally, in Section \[sec:erasure\], we establish the strong converse property for the classical post-processing assisted quantum capacity of the quantum erasure channel. Related Work ------------ A few papers have made some partial progress on or addressed the strong converse for quantum capacity question [@BDHSW12; @BCR09; @BBCW13; @MW13; @SW12; @WW14], with none however establishing that the strong converse holds for any nontrivial class of channels. Refs. [@BDHSW12; @BCR09] prove a strong converse theorem for the entanglement-assisted quantum capacity of a channel, which in turn establishes this as a strong converse rate for the unassisted quantum capacity. Ref. [@BBCW13] proves that the entanglement cost of a quantum channel is a strong converse rate for the quantum capacity assisted by unlimited forward and backward classical communication, which then demonstrates that this quantity is a strong converse rate for unassisted quantum capacity. Arguably the most important progress to date for establishing a strong converse for quantum capacity is from [@MW13]. These authors reduced the proof of the strong converse for the quantum capacity of degradable channels to that of establishing it for the simpler class of channels known as the symmetric channels (these are channels symmetric under the exchange of the receiver and the environment of the channel). Along the way, they also demonstrated that a pretty strong converse holds for degradable quantum channels, meaning that there is (at least) a jump in the quantum error from zero to $1/2$ as soon as the communication rate exceeds the quantum capacity. Sharma and Warsi established the generalized divergence framework for understanding quantum communication and reduced the task of establishing the strong converse to a purely mathematical additivity question [@SW12], which hitherto has remained unsolved. The later work in [@WW14] demonstrated that randomly selected codes with a communication rate exceeding the quantum capacity of the quantum erasure channel lead to a fidelity that decreases exponentially fast as the number of channel uses increases. (We stress that a strong converse would imply this behavior for *all* codes whose rate exceeds capacity.) The methods given in the present paper easily lead to the Rains relative entropy of a quantum state being a strong converse rate for entanglement distillation. After the completion of the present paper, we learned that this result was already established by Hayashi in [@H06 Section 8.6]. Preliminaries {#sec:notation} ============= States and Channels ------------------- #### States {#states .unnumbered} We denote different physical systems by capital letters (e.g. $A$, $B$) and we use these labels as subscripts to indicate with which physical system a mathematical object is associated. Let ${\mathcal{H}}_A$ denote the Hilbert space corresponding to the system $A$, where we restrict ourselves to finite-dimensional Hilbert spaces throughout this paper. We define $|A| = \dim \{ {\mathcal{H}}_A \}$. Moreover, let $\mathcal{B}(A)$ and $\mathcal{P}(A)$ denote the algebra of *bounded linear operators* acting on $\mathcal{H}_A$ and the subset of *positive semi-definite operators*, respectively. We also write $X_A \geq0$ if $X_A \in\mathcal{P}(A)$. An operator $\rho_A$ is in the set $\mathcal{S}( A) $ of *quantum states* if $\rho_A \geq 0$ and $\operatorname{tr}\left\{ \rho_A \right\} =1$. We say that a quantum state $\rho_A$ is *pure* if $\text{rank}(\rho_A) = 1$. We denote the *identity operator* in ${\mathcal{B}}(A)$ by $I_A$ and the *fully mixed* state by $\pi_A = I_A / |A|$. The fidelity between two density operators $\rho$ and $\sigma$ is defined as$$F\left( \rho,\sigma\right) :=\left\Vert \sqrt{\rho}\sqrt{\sigma }\right\Vert _{1}^{2} = \operatorname{tr}\Big\{ \sqrt{\sqrt{\rho}\,\sigma\sqrt{\rho}\,} \Big\}^2.$$ A bipartite physical system $AB$ is described by the tensor-product Hilbert space ${\mathcal{H}}_{AB} = \mathcal{H}_{A}\otimes\mathcal{H}_{B}$ with the sets $\mathcal{B}(AB)$, ${\mathcal{P}}(AB)$, and ${\mathcal{S}}(AB)$ defined accordingly. Given a bipartite quantum state $\rho_{AB} \in{\mathcal{S}}(AB)$, we unambiguously write $\rho_{A}=\operatorname{tr}_{B}\left\{ \rho_{AB}\right\}$ for the reduced state on system $A$. A state $\rho_{AB}$ is called *maximally entangled* if $\rho_{AB}$ is pure and $\rho_A = \pi_A$ or $\rho_B = \pi_B$. A product state is a state that can be written in the form $\rho_{AB} = \rho_A \otimes \rho_B$. A state $\rho_{AB}$ is called *separable* if it can be written in the form $$\begin{aligned} \rho_{AB} = \sum_{x \in {\mathcal{X}}} P(x) \rho_A^x \otimes \rho_B^x \,\end{aligned}$$ for a set ${\mathcal{X}}$, a probability mass function $P$ on ${\mathcal{X}}$ and $\rho_A^x \in {\mathcal{S}}(A)$, $\rho_B^x \in {\mathcal{S}}(B)$ for all $x \in {\mathcal{X}}$. We denote the set of all such states by ${{\rm SEP}}(A\!:\!B)$. Furthermore, we say that a bipartite state $\rho_{AB}$ is PPT if it has a positive partial transpose, namely if $T_{B}\left( \rho_{AB}\right) \geq0$, where $T_{B}$ indicates the partial transpose operation on system $B$. This is a necessary condition for a bipartite state to be separable [@P96].[^7] Let ${{\rm PPT}}( A\!:\!B) $ denote the set of all such states.[^8] For our purposes it is useful to further enlarge ${{\rm PPT}}(A\!:\!B)$ to the set ${{\rm PPT}'}( A:B)$, defined as $${{\rm PPT}'}\left( A:B\right) :=\left\{ \tau_{AB}:\tau_{AB}\geq 0\wedge\left\Vert T_{B}\left( \tau_{AB}\right) \right\Vert _{1}\leq1\right\} .$$ The set ${{\rm PPT}'}( A\!:\!B)$ includes all PPT states because $\left\Vert T_{B}\left( \tau_{AB}\right) \right\Vert _{1}=1$ if $\tau _{AB}\in\ {{\rm PPT}}\left( A\!:\!B\right) $. Furthermore, all operators $\tau_{AB}\in{{\rm PPT}'}( A\!:\!B) $ are subnormalized, in the sense that Tr$\left\{ \tau_{AB}\right\} \leq1$, because$$\operatorname{tr}\left\{ \tau_{AB}\right\} =\operatorname{tr}\left\{ T_{B}\left( \tau _{AB}\right) \right\} \leq\left\Vert T_{B}\left( \tau_{AB}\right) \right\Vert _{1}\leq1.$$ #### Channels {#channels .unnumbered} We denote linear maps from the set of bounded linear operators on one system to the bounded linear operators on another system by calligraphic letters. For example, ${\mathcal{N}}_{A\to B}$ denotes a map from ${\mathcal{B}}(A)$ to ${\mathcal{B}}(B)$, and we will drop the subscript $A \to B$ if it is clear from the context. Let id$_{A}$ denote the *identity map* acting on ${\mathcal{B}}(A)$. If a linear map is completely positive and trace-preserving (CPTP), we say that it is a *quantum channel*. We say that a CPTP map ${\mathcal{N}}_{AB \to A'B'}$ consists of local operations (LO) from $A\!:\!B$ to $A'\!:\!B'$ if it has the form ${\mathcal{N}}_{AB \to A'B'} = \mathcal{L}_{A \to A'} \otimes \mathcal{M}_{B \to B'}$ for CPTP maps $\mathcal{L}_{A \to A'}$ and $\mathcal{M}_{B \to B'}$. Similarly, we say that ${\mathcal{N}}$ is LOCC if it consists of local operations and classical communication. Furthermore, LOCC maps are contained in the set of separability preserving maps from $A\!:\!B$ to $A'\!:\!B'$ that take ${{\rm SEP}}(A\!:\!B)$ to ${{\rm SEP}}(A'\!:\!B')$. Finally, $\mathcal{N}$ is a PPT preserving operation from $A\!:\!B$ to $A'\!:\!B'$ if the map $T_{B^{\prime}}\circ\mathcal{N}_{AB\rightarrow A^{\prime}B^{\prime}}\circ T_{B}$ is CPTP [@R01]. (So we can conclude that any LOCC map is a PPT-preserving map.) Codes, Rates and Capacity {#sec:ent-gen-code} ------------------------- #### Codes {#codes .unnumbered} We consider a general class of classical pre- and post-processing (CPPP) assisted [entanglement generation (EG) codes]{}, for which the goal is for the sender (Alice) to use a quantum channel in order to share a state that is close to a maximally entangled state with the receiver (Bob).[^9] The [[CPPP ]{}]{}assisted codes allow classical pre- and post-processing before and after the quantum communication phase. In particular, the parties are allowed to prepare a separable resource state before the quantum communication commences. Note that any converse bounds for these classical communication assisted codes naturally also imply the same bounds for unassisted codes. Formally, we define a *[[CPPP ]{}]{}assisted EG code* for a channel $\mathcal{N}_{A \to B}$ as a triple $$\begin{aligned} {\mathcal{C}}= (M, {\mathcal{E}}_{\emptyset \to \tilde{A}A\tilde{B}}, {\mathcal{D}}_{\tilde{A}B\tilde{B} \to \hat{A}\hat{B}}) \,. \label{eq:code}\end{aligned}$$ Here, $\hat{A}$ and $\hat{B}$ are Hilbert spaces of dimension $M$, and $\tilde{A}$ and $\tilde{B}$ are auxiliary Hilbert spaces of arbitrary dimension. Moreover, ${\mathcal{E}}_{\emptyset \to \tilde{A}A\tilde{B}}$ is an LOCC quantum channel to $\tilde{A}A\!:\!\tilde{B}$, and ${\mathcal{D}}_{\tilde{A}B\tilde{B} \to \hat{A}\hat{B}}$ is an LOCC quantum channel from $\tilde{A}A\!:\!\tilde{B}$ to $\hat{A}\!:\!\hat{B}$. We write $|{\mathcal{C}}| = M$ for the *size* of the EG code. An *unassisted EG code* is defined in the same way, but ${\mathcal{E}}$ and ${\mathcal{D}}$ are restricted to be LO instead of LOCC. The corresponding coding schemes begin with Alice and Bob preparing a bipartite state $\rho_{A\tilde{A}\tilde{B}} \in {{\rm SEP}}(A\tilde{A}\!:\!\tilde{B})$ using the quantum channel ${\mathcal{E}}$. Alice then sends the system $A$ through the channel $\mathcal{N}$, resulting in the state $$\rho_{\tilde{A}B\tilde{B}} = \mathcal{N}_{A\rightarrow B}\left( \rho_{\tilde{A}A\tilde{B}}\right) . \label{eq:state-output-from-channel}$$ Finally, Alice and Bob perform a decoding $\mathcal{D}$, leading to the state $\omega_{\hat{A}\hat{B}} = \mathcal{D}_{\tilde{A}B\tilde{B}\rightarrow\hat{A}\hat{B}}( \rho_{\tilde{A}B\tilde{B}} )$. The *fidelity* of the above code ${\mathcal{C}}$ on the channel ${\mathcal{N}}$ is given by$$F({\mathcal{C}}, {\mathcal{N}}) := \left\langle \Phi\right\vert _{\hat{A}\hat{B}}\, \omega_{\hat{A}\hat{B}}\, \left\vert \Phi\right\rangle _{\hat{A}\hat{B}}, \label{eq:fidelity}$$ where $\left\vert \Phi\right\rangle _{\hat{A}\hat{B}}$ is a (fixed) maximally entangled state on $\hat{A}\hat{B}$. #### Rates And Capacity {#rates-and-capacity .unnumbered} The main focus of this paper is on EG codes for many parallel uses of a memoryless quantum channel. That is, we want to investigate product channels $\mathcal{N}^{\otimes n}$ for large $n$. Note that a [[CPPP ]{}]{}assisted EG code for ${\mathcal{N}}^{\otimes n}$ (as described above) allows for classical communication before and after the product channel is used, but does not allow for interactive schemes with classical communication between different channel uses. We say that a rate $r$ is an *achievable rate* for ([[CPPP ]{}]{}assisted) quantum communication over the channel ${\mathcal{N}}$ if there exist a sequence of codes $\{{\mathcal{C}}_n \}_{n \in \mathbb{N}}$ where ${\mathcal{C}}_n$ is a ([[CPPP ]{}]{}assisted) EG code for ${\mathcal{N}}^{\otimes n}$, such that $$\begin{aligned} \liminf_{n\to\infty} \frac{1}{n} \log |{\mathcal{C}}_n| \geq r \qquad \textrm{and} \qquad\lim_{n\to\infty} F({\mathcal{C}}_n, {\mathcal{N}}^{\otimes n}) = 1 .\end{aligned}$$ The *quantum capacity* of ${\mathcal{N}}$, denoted $Q({\mathcal{N}})$ is the supremum of all achievable rates. Analogously, the *[[CPPP ]{}]{}assisted quantum capacity* of ${\mathcal{N}}$, denoted $Q_{{\rm pp}}({\mathcal{N}})$, is the supremum of all [[CPPP ]{}]{}assisted achievable rates. On the other hand, $r$ is a *strong converse rate* for ([[CPPP ]{}]{}assisted) quantum communication if for every sequence of codes $\{ C_n \}_{n \in \mathbb{N}}$ as above, we have $$\begin{aligned} \liminf_{n\to\infty} \frac{1}{n} \log |{\mathcal{C}}_n| > r \qquad \implies \qquad\lim_{n\to\infty} F({\mathcal{C}}_n, {\mathcal{N}}^{\otimes n}) = 0 .\end{aligned}$$ The *strong converse quantum capacity*, denoted $Q^{\dagger}({\mathcal{N}})$, is the infimum of all strong converse rates. Analogously, the *[[CPPP ]{}]{}assisted strong converse quantum capacity* of ${\mathcal{N}}$, denoted $Q_{{\rm pp}}^{\dagger}({\mathcal{N}})$, is the infimum of all [[CPPP ]{}]{}assisted strong converse rates. Clearly, the following inequalities hold by definition: $$\begin{aligned} Q({\mathcal{N}}) \leq Q_{{\rm pp}}({\mathcal{N}}) \leq Q_{{\rm pp}}^{\dagger}({\mathcal{N}}) \qquad \textrm{and} \qquad Q({\mathcal{N}}) \leq Q^{\dagger}({\mathcal{N}}) \leq Q_{{\rm pp}}^{\dagger}({\mathcal{N}}) \,.\end{aligned}$$ Finally, we say that a channel ${\mathcal{N}}$ satisfies the *strong converse property for quantum communication* if $Q({\mathcal{N}}) = Q^{\dagger}({\mathcal{N}})$. Similarly, we say that a channel ${\mathcal{N}}$ satisfies the *strong converse property for [[CPPP ]{}]{}assisted quantum communication* if $Q_{{\rm pp}}({\mathcal{N}}) = Q_{{\rm pp}}^{\dagger}({\mathcal{N}})$. Generalized Divergence Framework {#sec:gen-div} ================================ A functional $\mathbf{D}:\mathcal{S} \times \mathcal{P} \rightarrow \mathbb{R}$ is a generalized divergence if it satisfies the monotonicity inequality$$\mathbf{D}\left( \rho \Vert\sigma \right) \geq\mathbf{D}\left( \mathcal{N}( \rho) \Vert\mathcal{N}( \sigma) \right) ,$$ where $\mathcal{N}$ is a CPTP map. It follows directly from monotonicity that any generalized divergence is invariant under unitaries, in the sense that $\mathbf{D}\left( \rho\Vert\sigma\right) =\mathbf{D}\mathcal{(}U\rho U^{\dag }\Vert U\sigma U^{\dag})$, where $U$ is a unitary operator, and that it is invariant under tensoring with another quantum state $\tau$, namely $\mathbf{D}\left( \rho\Vert\sigma\right) =\mathbf{D}\mathcal{(}\rho \otimes\tau\Vert\sigma\otimes\tau)$. Rains Relative Entropy and Rains Information -------------------------------------------- We now define some information measures which play a central role in this paper. They are inspired by the Rains bound on distillable entanglement [@R01] and the subsequent reformulation of it in [@AdMVW02]. We define the *generalized Rains relative entropy* of a bipartite state $\rho_{AB}$ as follows:$$R_{\mathbf{D}}(A\!:\!B)_{\rho} :=\min_{\tau_{AB}\in {{\rm PPT}'}( A:B) }\mathbf{D}\left( \rho_{AB}\Vert\tau _{AB}\right) ,$$ We sometimes abuse notation and write $R_{\mathbf{D}}(A\!:\!B)_{\rho} = R_{\mathbf{D}}(\rho_{AB})$ if the bipartition is obvious in context. One property of $R_{\mathbf{D}}$, critical for our application here, is that it is monotone under PPT preserving operations, in the sense that $$R_{\mathbf{D}}(A\!:\!B)_{\rho} \geq R_{\mathbf{D}}(A'\!:\!B')_{\tau} , \qquad \textrm{where} \qquad \tau_{A'B'} = \mathcal{P}_{AB\to A'B'}( \rho_{AB})$$ and $\mathcal{P}_{AB\to A'B'}$ is any PPT-preserving operation from $A\!:\!B$ to $A'\!:\!B'$. This is because PPT-preserving operations do not take operators $\tau_{AB}$ in ${{\rm PPT}'}\left( A\!:\!B\right) $ outside of this set, which follows from $$\Vert T_{B'} (\mathcal{P}_{AB\to A'B'}(\tau_{AB})) \Vert_{1} = \Vert T_{B'} (\mathcal{P}_{AB}(T_{B} (T_{B}(\tau_{AB})))) \Vert_{1} \leq\Vert T_{B}(\tau_{AB}) \Vert_{1} \leq1.$$ In the above, the first equality follows because the partial transpose $T_{B}$ is its own inverse, and the first inequality follows because the map $T_{B} \circ\mathcal{P}_{AB} \circ T_{B}$ is CPTP (as it is PPT preserving) and the fact that the trace norm is monotone decreasing under CPTP maps. Since the set of PPT preserving operations includes LOCC operations, $R_{\mathbf{D}}\left( A\!:\!B\right)_\rho $ is also monotone under LOCC operations. We also note that the Rains information is *subadditive*, namely for any two bipartite states $\rho_{AB}$ and $\sigma_{A'B'}$, we have $$\begin{aligned} \label{lm:sub-add} \widetilde{R}_{\mathbf{D}}(AA'\!:\!BB')_{\rho \otimes \sigma} \leq \widetilde{R}_{\mathbf{D}}(A\!:\!B)_{\rho} + \widetilde{R}_{\mathbf{D}}(A'\!:\!B')_{\sigma} .\end{aligned}$$ This follows directly from the definition, since we can always restrict the minimization to product states. Finally, we define the *generalized Rains information of a quantum channel* as $$R_{\mathbf{D}}\left( \mathcal{N}\right) := \max_{\rho_{RA}} R_{\mathbf{D}}(R\!:\!B)_{\omega}, \qquad \textrm{where} \qquad \omega_{RB} = \mathcal{N}_{A\rightarrow B}( \rho_{RA}) \, .$$ Another critical property of the above quantities has to do with the set ${{\rm PPT}'}\left( A:B\right) $ of operators over which we are optimizing. That is, all operators in this set satisfy the property given in Lemma 2 of [@R99], which we recall now: \[Lemma 2 of [@R99]\]\[lem:overlap\]Let $\tau_{AB}\in{{\rm PPT}'}( A\!:\!B)$. Then the overlap of $\tau_{AB}$ with any maximally entangled state $\Phi_{AB}$ of Schmidt rank $M$ is at most $1/M$, i.e.$\operatorname{Tr}\left\{ \Phi_{AB}\tau_{AB}\right\} \leq\frac{1}{M}.$ The same is true for $\sigma_{AB}\in\ \operatorname{PPT}( A\!:\!B) $ simply because $\operatorname{PPT}( A\!:\!B) \subseteq{{\rm PPT}'}( A\!:\!B) $. Covariance of Quantum Channels ------------------------------ Covariant quantum channels have symmetries which allow us to simplify the set of states over which we need to optimize their generalized Rains information. Let $G$ be a finite group, and for every $g\in G$, let $g\rightarrow U_{A}\left( g\right) $ and $g\rightarrow V_{B}\left( g\right) $ be unitary representations acting on the input and output spaces of the channel, respectively. Then a quantum channel $\mathcal{N}_{A\rightarrow B}$ is covariant with respect to these representations if the following relation holds for all input density operators $\rho_A \in {\mathcal{S}}(A)$ and group elements $g\in G$:$$\mathcal{N}_{A\rightarrow B}\left( U_{A}\left( g\right) \rho_A U_{A}^{\dag }\left( g\right) \right) =V_{B}\left( g\right) \mathcal{N}_{A\rightarrow B}\left( \rho_A \right) V_{B}^{\dag}\left( g\right) .$$ We then have the following proposition which allows us to restrict the form of the input states needed to optimize the generalized Rains information of a covariant channel: \[prop:covariance\] Let $\mathcal{N}_{A\rightarrow B}$ be a covariant channel with group $G$ as above and let $\rho_{A} \in {\mathcal{S}}(A)$, $\phi^{\rho}_{RA}$ a purification of $\rho_A$, and $\rho_{RB} = \mathcal{N}_{A\rightarrow B}(\phi^\rho_{RA})$. Let $\bar{\rho}_{A}$ be the group average of $\rho_{A}$, i.e.,$$\bar{\rho}_{A} = \frac{1}{\left\vert G\right\vert }\sum_{g}U_{A}\left( g\right) \rho_A U_{A}^{\dag}\left( g\right) ,$$ and let $\phi^{\bar{\rho}}_{RA}$ be a purification of $\bar{\rho}_A$ and $\bar{\rho}_{RB} = \mathcal{N}_{A\rightarrow B}(\phi^{\bar{\rho}}_{RA})$. Then, $R_{\mathbf{D}}(R\!:B)_{\bar{\rho}} \geq R_{\mathbf{D}}(R\!:B)_{{\rho}}$ Given the purification $\phi^\rho_{RA}$, consider the following state$$\left\vert \psi\right\rangle _{PRA}:=\sum_{g}\frac{1}{\sqrt{\left\vert G\right\vert }}\left\vert g\right\rangle _{P}\left[ I_{R}\otimes U_{A}\left( g\right) \right] \left\vert \phi^{\rho}\right\rangle _{RA}\text{.}$$ Observe that $\left\vert \psi\right\rangle _{PRA}$ is a purification of $\overline{\rho}_{A}$ with purifying systems $P$ and $R$. Let $\tau_{PRB}$ be an arbitrary operator in ${{\rm PPT}'}\left( PR:B\right) $. Then the following chain of inequalities holds:$$\begin{aligned} & \mathbf{D}\left( \mathcal{N}_{A\rightarrow B}(\psi_{PRA})\middle\Vert\tau _{PRB}\right) \nonumber\\ & \qquad \geq\mathbf{D}\left( \sum_{g}\frac{1}{\left\vert G\right\vert }\left\vert g\right\rangle \left\langle g\right\vert _{P}\otimes\mathcal{N}_{A\rightarrow B}(U_{A}\left( g\right) \phi_{RA}^{\rho}U_{A}^{\dag}\left( g\right) )\middle\Vert\sum_{g}p\left( g\right) \left\vert g\right\rangle \left\langle g\right\vert _{P}\otimes\tau_{RB}^{g}\right) \\ & \qquad =\mathbf{D}\left( \sum_{g}\frac{1}{\left\vert G\right\vert }\left\vert g\right\rangle \left\langle g\right\vert _{P}\otimes V_{B}\left( g\right) \mathcal{N}_{A\rightarrow B}(\phi_{RA}^{\rho})V_{B}^{\dag}\left( g\right) \middle\Vert\sum_{g}p\left( g\right) \left\vert g\right\rangle \left\langle g\right\vert _{P}\otimes\tau_{RB}^{g}\right) \label{eq:classical-on-P}\\ & \qquad =\mathbf{D}\left( \sum_{g}\frac{1}{\left\vert G\right\vert }\left\vert g\right\rangle \left\langle g\right\vert _{P}\otimes\mathcal{N}_{A\rightarrow B}(\phi_{RA}^{\rho})\middle\Vert\sum_{g}p\left( g\right) \left\vert g\right\rangle \left\langle g\right\vert _{P}\otimes V_{B}^{\dag}\left( g\right) \tau _{RB}^{g}V_{B}\left( g\right) \right) \\ & \qquad \geq\mathbf{D}\left( \mathcal{N}_{A\rightarrow B}(\phi_{RA}^{\rho})\middle\Vert\sum_{g}p\left( g\right) V_{B}^{\dag}\left( g\right) \tau_{RB}^{g}V_{B}\left( g\right) \right) \\ & \qquad \geq\min_{\tau_{RB}\in{{\rm PPT}'}\left( R:B\right) }\mathbf{D}\left( \mathcal{N}_{A\rightarrow B}(\phi_{RA}^{\rho})\middle\Vert\tau_{RB}\right) =R_{\mathbf{D}}(R\!:B)_{{\rho}} \,.\end{aligned}$$ The first inequality follows from monotonicity of the generalized divergence $\mathbf{D}$ under a dephasing of the $P$ register (where the dephasing operation is given by $\sum_{g}\left\vert g\right\rangle \left\langle g\right\vert \cdot\left\vert g\right\rangle \left\langle g\right\vert $). The first equality follows from the assumption of channel covariance. The second equality follows from invariance of the generalized divergence under unitaries, with the unitary chosen to be$$\sum_{g}\left\vert g\right\rangle \left\langle g\right\vert _{P}\otimes V_{B}^{\dag}\left( g\right) .$$ Furthermore, this unitary does not take the state out of the class ${{\rm PPT}'}$, i.e. $$\sum_{g}p\left( g\right) \left\vert g\right\rangle \left\langle g\right\vert _{P}\otimes V_{B}^{\dag}\left( g\right) \tau_{RB}^{g}V_{B}\left( g\right) \in{{\rm PPT}'}\left( PR\!:\!B\right) .$$ This is because, in this case, one could also implement this operation as a classically controlled LOCC operation, i.e., a von Neumann measurement $\left\{ \left\vert g\right\rangle \left\langle g\right\vert \right\} $ of the register $P$ followed by a rotation $V_{B}^{\dag}\left( g\right) $ of the $B$ register. One can do so here because both arguments to $\mathbf{D}$ in (\[eq:classical-on-P\]) are classical on $P$. The second inequality follows because the generalized divergence $\mathbf{D}$ is monotone under the discarding of the register $P$. The final inequality results from taking a minimization, and the final equality is by definition. Since $\tau_{PRB}$ is chosen to be an arbitrary operator in ${{\rm PPT}'}( PR\!:\!B) $, it follows that$$\min_{\tau_{PRB} \in {{\rm PPT}'}(PR:B)} \mathbf{D}\left( \mathcal{N}_{A\rightarrow B}(\psi_{PRA}) \middle\| \tau_{PRB} \right) \geq R_{\mathbf{D}}(R\!:\!B)_{{\rho}}$$ The conclusion then follows because all purifications are related by a unitary on the purifying system and the quantity $R_{\mathbf{D}}$ is invariant under unitaries on the purifying system. Specializing to Rényi Divergence -------------------------------- At this point we specialize our discussion to a particular type of Rényi divergence. For $\rho\in\mathcal{S}$, $\sigma\in\mathcal{P}$ and $\alpha \in (0,1) \cup (1,\infty)$, we define the *sandwiched Rényi relative entropy* of order $\alpha$ as [@MDSFT13; @WWY13]$$\widetilde{D}_{\alpha}\left( \rho\Vert\sigma\right) := \left\{ \begin{array} [c]{cc}\frac{1}{\alpha -1}\log \operatorname{tr}\left\{ \left( \sigma^{\left( 1-\alpha\right) /2\alpha}\rho \sigma^{\left( 1-\alpha\right) /2\alpha}\right) ^{\alpha}\right\} & \text{if supp}\left( \rho\right) \subseteq\text{supp}\left( \sigma\right) \text{ or } \alpha\in(0,1)\\ \infty & \text{else}\end{array} \right. .$$ The sandwiched Rényi relative entropy is defined for $\alpha\in\{0,1,\infty\}$ by taking the respective limit. In particular, $\lim_{\alpha \to 1} \widetilde{D}_{\alpha}(\rho\|\sigma) = D(\rho\|\sigma)$. In the following, we restrict our attention to the regime for which $\alpha \geq 1$. The sandwiched Rényi relative entropy is monotone under CPTP maps $\mathcal{N}$ for such values of $\alpha$ [@FL13; @B13monotone; @MO13] and thus constitutes a generalized divergence as discussed above. Moreover, the Rényi divergence is *quasi-convex* in the first argument, i.e. if $\rho$ decomposes as $\rho = \lambda \rho^1 + (1-\lambda) \rho^2$ for $\lambda \in [0,1]$ and $\rho^1, \rho^2 \in {\mathcal{S}}$, then $\max_{i \in \{0,1\}} \widetilde{D}_{\alpha}(\rho^i\|\sigma) \geq \widetilde{D}_{\alpha}(\rho\|\sigma)$. We denote the corresponding Rains relative entropy by $\widetilde{R}_{\alpha}(A\!:\!B)_{\rho}$ (or $\widetilde{R}_{\alpha}(\rho_{AB})$ if the bipartition is obvious). Consequently, the *Rényi Rains information of a quantum channel* ${\mathcal{N}}_{A\to B}$ is defined as $$\widetilde{R}_{\alpha}( \mathcal{N}) :=\max_{\rho_{RA}}\min_{\tau_{RB}\in{{\rm PPT}'}( R:B) }\widetilde{D}_{\alpha }\left( \mathcal{N}_{A\rightarrow B}( \rho_{RA}) \middle\Vert\tau _{RB}\right) .$$ It suffices to perform the maximization in $\widetilde{R}_{\alpha}\left( \mathcal{N}\right)$ over pure bipartite states $\rho_{RA}$, due to the quasi-convexity of $\widetilde{D}_\alpha$ whenever $\alpha > 1$. As a result, it suffices for the dimension of the reference system $R$ to be no larger than the dimension of the channel input $A$, due to the well known Schmidt decomposition. The Rényi Rains information converges to $R( \mathcal{N}) $ in the limit as $\alpha$ approaches one from above. This is shown in the following Lemma, whose proof is provided in Appendix \[sec:ogawa-nagaoka\]. \[lm:continuity\] For any quantum channel $\mathcal{N}$ and $\alpha > \beta > 1$, we have $$\widetilde{R}_{\alpha}( \mathcal{N}) \geq \widetilde{R}_{\beta}( \mathcal{N}) \geq R( \mathcal{N}) \qquad \textrm{and} \qquad \lim_{\alpha\rightarrow1^{+}}\widetilde{R}_{\alpha}( \mathcal{N}) =R\left( \mathcal{N}\right) .$$ Relating Fidelity of an Entanglement Generation Code to the Rényi Rains Information of a Channel ------------------------------------------------------------------------------------------------ The power of the generalized divergence framework is that it allows us to relate rate and fidelity to an information quantity. The usual approach is to compare the states resulting from any code to a set of states resulting from a useless channel. For the transmission of classical information, the only set of useless channels are those which trace out the input to the channel and replace it with an arbitrary density operator, effectively cutting the communication line. However, for the transmission of quantum information, there are more interesting classes of useless channels [@SS12]. For example, it is well known that a PPT entanglement binding channel has zero quantum capacity [@HHH00]. More generally, the bound in Lemma \[lem:overlap\] establishes that if both the input to the channel and the reference system are replaced with an operator $\tau_{RB}\in{{\rm PPT}'}(R\!:\!B)$, then the fidelity with a maximally entangled state can never be larger than $1/M$. Since for a memoryless channel we are taking $M=2^{nQ}$, this overlap will be exponentially small with the number of channel uses, so that channels that replace with $\tau_{RB}$ cannot send any quantum information reliably. The following proposition gives a one-shot bound on the fidelity of any CPPP assisted EG code: \[lem:one-shot\] Let ${\mathcal{N}}$ be a quantum channel. Any [[CPPP ]{}]{}assisted EG code ${\mathcal{C}}$ on ${\mathcal{N}}$ obeys the following bound. For all $\alpha>1$,$$F({\mathcal{C}}, {\mathcal{N}}) \leq2^{-\left( \frac{\alpha-1}{\alpha}\right) \left( \log |{\mathcal{C}}| - \widetilde {R}_{\alpha}\left( \mathcal{N} \right) \right) } \, . \label{eq:fidelity-bound-rains}$$ Let ${\mathcal{C}}= (M, {\mathcal{E}}_{\emptyset \to \tilde{A}A\tilde{B}}, {\mathcal{D}}_{\tilde{A}B\tilde{B} \to \hat{A}\hat{B}})$ as in  and recall the state $\rho_{\tilde{A}B\tilde{B}}$ in  prior to decoding and the state $\omega_{\hat{A}\hat{B}} = {\mathcal{D}}_{\tilde{A}B\tilde{B}\to\hat{A}\hat{B}}(\rho_{\tilde{A}B\tilde{B}})$. The binary test channel $\mathcal{B}_{\hat{A}\hat{B}\rightarrow Z}$ outputs a flag indicating whether a state is maximally entangled or not:$$\mathcal{B}_{\hat{A}\hat{B}\rightarrow Z}\left( \cdot\right) :=\operatorname{tr}\left\{ \Phi_{\hat{A}\hat{B}}\left( \cdot\right) \right\} \left\vert 1\right\rangle \left\langle 1\right\vert +\operatorname{tr}\left\{ \left( I_{\hat{A}\hat{B}}-\Phi_{\hat{A}\hat{B}}\right) \left( \cdot\right) \right\} \left\vert 0\right\rangle \left\langle 0\right\vert . \label{eq:entanglement-test}$$ Furthermore, consider an arbitrary state $\tau_{\tilde{A}B\tilde{B}} \in {{\rm PPT}'}(\tilde{A}\!:\!B\tilde{B})$ and observe that $\mathcal{D}_{\tilde{A}B\tilde{B}\to\hat{A}\hat{B}}(\tau_{\tilde{A}B\tilde{B}}) \in {{\rm PPT}'}(\hat{A}\!:\!\hat{B})$ because the decoding operator is restricted to be LOCC. For ease of presentation, we set $$p = \operatorname{tr}\left\{ \Phi_{\hat{A}\hat{B}}\, \mathcal{D}_{\tilde{A}B\tilde{B}\rightarrow\hat{A}\hat{B}} ( \tau_{\tilde{A}B\tilde{B}}) \right\} \qquad \textrm{and} \qquad F = F({\mathcal{C}},{\mathcal{N}}) = \operatorname{tr}\left\{ \Phi_{\hat{A}\hat{B}}\, \omega_{\hat{A}\hat{B}} \right\} . \label{eq:overlap-decoded-PPT}$$ Applying monotonicity of the divergence under the decoding map ${\mathcal{D}}$ and the test ${\mathcal{B}}$, we find $$\begin{aligned} \widetilde{D}_{\alpha}\left( \rho_{\tilde{A}B\tilde{B}}\middle\Vert\tau_{\tilde{A}B\tilde{B}}\right) & \geq\widetilde{D}_{\alpha}\left( \mathcal{B}_{\hat{A}\hat{B}\rightarrow Z}\left( \omega_{\hat{A}\hat{B}}\right) \middle\Vert\mathcal{B}_{\hat{A}\hat{B}\rightarrow Z}\left( \mathcal{D}_{\tilde{A}B\tilde{B}\rightarrow\hat{A}\hat{B}}\left( \tau_{\tilde{A}B\tilde{B}}\right) \right) \right) \label{eq:renyi-develop-1}\\ & =\frac{1}{\alpha-1}\log\left[ F^{\alpha}p^{1-\alpha}+\left( 1-F\right) ^{\alpha}\left[ \operatorname{tr}\left\{ \tau_{\tilde{A}B\tilde{B}} \right\} -p\right] ^{1-\alpha }\right] \\ & \geq\frac{1}{\alpha-1}\log\left[ F^{\alpha}p^{1-\alpha}\right] \\ & \geq\frac{1}{\alpha-1}\log\left[ F^{\alpha}\left( 1/M\right) ^{1-\alpha }\right] \\ & =\frac{\alpha}{\alpha-1}\log F+\log M \label{eq:renyi-develop-last} .$$ The second inequality follows by discarding the second term $\left( 1-F\right) ^{\alpha}\left[ \operatorname{tr}\left\{ \tau_{RB}\right\} -p\right] ^{1-\alpha}$ (recall that we are considering $\alpha>1$). The third inequality follows from (\[eq:overlap-decoded-PPT\]) and Lemma \[lem:overlap\]. Now, recall that the state $\rho_{\tilde{A}A\tilde{B}}$ is in ${{\rm SEP}}(\tilde{A}A\!:\!\tilde{B})$ and can thus be decomposed into products of pure states. Using the quasi-convexity of $\widetilde{D}_{\alpha}$ in the first argument, we find that there exist pure states $\sigma_{\tilde{A}A}$ and $\sigma_{\tilde{B}}$ such that for every $\tau_{\tilde{A}B} \in {{\rm PPT}'}(\tilde{A}:B)$, we have $$\begin{aligned} \widetilde{D}_{\alpha}\left( {\mathcal{N}}_{A\to B} \left( \sigma_{\tilde{A}A} \right) \middle\| \tau_{\tilde{A}B}\right) &\geq \widetilde{D}_{\alpha}\left( {\mathcal{N}}_{A\to B} \left( \sigma_{\tilde{A}A} \right) \otimes \sigma_{\tilde{B}} \middle\| \tau_{\tilde{A}B} \otimes \sigma_{\tilde{B}} \right) \\ &\geq \widetilde{D}_{\alpha}\left( {\mathcal{N}}_{A\to B} \left( \rho_{\tilde{A}A\tilde{B}} \right) \middle\| \tau_{\tilde{A}B} \otimes \sigma_{\tilde{B}} \right)\\ & \geq \frac{\alpha}{\alpha-1}\log F+\log M,\end{aligned}$$ where in the last line we apply the development in - given that $\tau_{\tilde{A}B} \otimes \sigma_{\tilde{B}} \in {{\rm PPT}'}( \tilde{A}\!:\!B\tilde{B}) $. Since the above bound holds for all $\tau_{\tilde{A}B} \in {{\rm PPT}'}(\tilde{A}:B)$, we can conclude that $$\min_{\tau_{\tilde{A}B} \in {{\rm PPT}'}(\tilde{A}:B) }\widetilde{D}_{\alpha}\left( {\mathcal{N}}_{A\to B} \left( \sigma_{\tilde{A}A} \right) \middle\| \tau_{\tilde{A}B}\right) \geq \frac{\alpha}{\alpha-1}\log F+\log M.$$ We can finally remove the dependence on any particular code by optimizing over all inputs to the channel. Identifying $\tilde{A}$ with $R$ to simplify notation, we find $$\widetilde{R}_{\alpha}\left( \mathcal{N}\right) =\max_{\rho_{RA}}\min _{\tau_{RB}\in{{\rm PPT}'}\left( R:B\right) }\widetilde{D}_{\alpha}\left( \mathcal{N}_{A\rightarrow B}\left( \rho_{RA}\right) \middle\Vert\tau_{RB}\right) \geq\frac{\alpha}{\alpha-1}\log F+\log M.$$ This bound is then equivalent to (\[eq:fidelity-bound-rains\]). Weak Subadditivity of the $\alpha$-Rains Information for Memoryless Channels {#sec:weaksub} ============================================================================ In this section, we prove an important theorem, which is critical for concluding that the Rains information of a channel is a strong converse rate for quantum communication. Before we commence, we need the following technical property. \[lem:mono-inc\] Let $\alpha > 1$, $\rho, \rho' \in \mathcal{S}$ and $\sigma \in \mathcal{P}$. If $\rho \leq \gamma \rho'$ for some $\gamma \geq 1$, then $$\widetilde{D}_{\alpha} \left( \rho\Vert\sigma\right) \leq \frac{\alpha}{\alpha-1} \log \gamma + \widetilde{D}_{\alpha}\left( \rho'\Vert\sigma\right) \,.$$ From the assumption that $\rho\leq \gamma \rho^{\prime}$, we get $\sigma^{\left( 1-\alpha\right) /2\alpha}\rho\sigma^{\left( 1-\alpha\right) /2\alpha}\leq \gamma \sigma^{\left( 1-\alpha\right) /2\alpha}\rho^{\prime}\sigma^{\left( 1-\alpha\right) /2\alpha}$. Then we have that $$\operatorname{tr}\{ (\sigma^{\left( 1-\alpha\right) /2\alpha}\rho\sigma^{\left( 1-\alpha\right) /2\alpha})^\alpha\} \leq \gamma^\alpha \operatorname{tr}\{ (\sigma^{\left( 1-\alpha\right) /2\alpha}\rho^{\prime}\sigma^{\left( 1-\alpha\right) /2\alpha})^\alpha \}$$ because Tr$\left\{ f\left( P\right) \right\} \leq\ $Tr$\left\{ f\left( Q\right) \right\} $ for $P\leq Q$ and $f$ a monotone increasing function (see, e.g., [@MO13 Lemma III.6]). Taking logarithms and dividing by $\alpha -1$ gives the statement of the lemma. Now we are ready to prove that the Rains information of the channel obeys a weak subadditivity property. \[thm:weak-subadditivity\] Let ${\mathcal{N}}_{A\to B}$ be a quantum channel. For all $\alpha>1$ and $n \in \mathbb{N}$, we have $$\widetilde{R}_{\alpha}\left( \mathcal{N}^{\otimes n}\right) \leq n\widetilde{R}_{\alpha}\left( \mathcal{N}\right) +\frac{\alpha\left\vert A\right\vert ^{2}}{\alpha-1}\log n \label{eq:weak-subadd} .$$ To begin with, we observe that a tensor-power channel is covariant with respect to permutations of the input and output systems, in the sense that$$\forall\pi\in S_{n}:W_{B^{n}}^{\pi}\mathcal{N}^{\otimes n}\left( \rho_{A^{n}}\right) \left( W_{B^{n}}^{\pi}\right) ^{\dag}=\mathcal{N}^{\otimes n}\left( W_{A^{n}}^{\pi}\rho_{A^{n}}\left( W_{A^{n}}^{\pi}\right) ^{\dag }\right) ,$$ where $W_{A^{n}}^{\pi}$ and $W_{B^{n}}^{\pi}$ are unitary representations of the permutation $\pi$, acting on the input space$~A^{n}$ and the output space $B^{n}$, respectively. So, letting $\left\vert \phi^{\rho}\right\rangle _{RA^{n}}$ denote a purification of $\rho_{A^{n}}$, we can apply Proposition \[prop:covariance\] to find that $$\widetilde{R}_{\alpha}\left( \mathcal{N}^{\otimes n}\left( \phi_{RA^{n}}^{\rho}\right) \right) \leq\widetilde{R}_{\alpha}\left( \mathcal{N}^{\otimes n}\left( \phi_{RA^{n}}^{\overline{\rho}}\right) \right) , \label{eq:1st-proof-line}$$ where $\phi_{RA^{n}}^{\overline{\rho}}$ is a purification of the permutation invariant state $\overline{\rho}_{A^{n}}$. Now, this purification $\phi_{RA^{n}}^{\overline{\rho}}$ is related by a unitary on the reference system $R$ to a state $\left\vert \psi\right\rangle _{\hat{A}^{n}A^{n}}\in\text{Sym}((\hat{A}\otimes A)^{\otimes n})$, where $\hat{A}\simeq A$ [@RennerThesis Lemma 4.3.1]. So it follows that$$\widetilde{R}_{\alpha}\left( \mathcal{N}^{\otimes n}\left( \phi_{RA^{n}}^{\rho}\right) \right) \leq\widetilde{R}_{\alpha}\left( \mathcal{N}^{\otimes n}\left( \phi_{RA^{n}}^{\overline{\rho}}\right) \right) =\widetilde{R}_{\alpha}\left( \mathcal{N}^{\otimes n}\left( \psi_{\hat{A}^{n}A^{n}}\right) \right) . \label{eq:2nd-proof-line}$$ For such a state in Sym$((\hat{A}\otimes A)^{\otimes n})$, we observe (see, e.g., [@CKR09]) that $$\psi_{\hat{A}^{n}A^{n}}\leq n^{\left\vert A\right\vert ^{2}}\omega_{\hat{A}^{n}A^{n}}^{(n) }, \quad \textrm{where} \quad \omega_{\hat{A}^{n}A^{n}}^{\left( n\right) } := \int d\mu\left( \varphi\right) \, \varphi_{\hat{A}A}^{\otimes n}, \label{eq:de-finetti-state} \,.$$ with $\mu\left( \varphi\right) $ denoting the uniform probability measure on the unit sphere consisting of pure bipartite states $\varphi _{\hat{A}A}$. Employing Lemma \[lem:mono-inc\], we find that$$\begin{aligned} \widetilde{R}_{\alpha}\left( \mathcal{N}^{\otimes n}\left( \psi_{\hat{A}^{n}A^{n}}\right) \right) \leq \frac{\alpha\left\vert A\right\vert ^{2}}{\alpha-1}\log n + \widetilde{R}_{\alpha}\left( \mathcal{N}^{\otimes n}\left( \omega_{\hat{A}^{n}A^{n} }\right) \right) .\end{aligned}$$ and since the right hand side does not depend on the state $\rho_{A^n}$ anymore, this yields directly $$\begin{aligned} \widetilde{R}_{\alpha}({\mathcal{N}}^{\otimes n}) \leq \frac{\alpha\left\vert A\right\vert ^{2}}{\alpha-1}\log n + \widetilde{R}_{\alpha}\left( \mathcal{N}^{\otimes n}\left( \omega_{\hat{A}^{n}A^{n} }\right) \right) \leq \frac{\alpha\left\vert A\right\vert ^{2}}{\alpha-1}\log n + \max_{\phi_{\hat{A}A}} \widetilde{R}_{\alpha}\left( \mathcal{N}^{\otimes n}\left( \phi_{\hat{A}A}^{\otimes n} \right) \right) \label{eq:use-this}\end{aligned}$$ where we used the quasi-convexity of $\widetilde{D}_{\alpha}$ and the definition of $\omega_{\hat{A}^{n}A^{n}}^{(n)}$ in  to establish the second inequality. Now simply note that $\widetilde{R}_{\alpha}$ is subadditive for product states, i.e.$\widetilde{R}_{\alpha}\big( \mathcal{N}^{\otimes n}( \phi_{\hat{A}A}^{\otimes n} ) \big) \leq n \widetilde{R}_{\alpha}( \mathcal{N}(\phi_{\hat{A}A}) )$. Combining this with  concludes the proof. A consequence of Theorem \[thm:weak-subadditivity\] is that the Rains information of a channel is weakly subadditive. This corollary is required in order to set some of the claims in [@SS08; @SSW08] on a firm foundation. We provide its proof in Appendix \[sec:appendix\]. \[cor:von-Neumann-subadd\] The Rains information of a quantum channel is weakly subadditive, in the sense that$$\limsup_{n \to\infty} \frac{1}{n}R\left( \mathcal{N}^{\otimes n}\right) \leq R\left( \mathcal{N}\right) .$$ The Rains Information is a Strong Converse Rate for Quantum Communication {#sec:main} ========================================================================= We are ready to state our main result, which states that the Rains information of the channel is an upper bound on the [[CPPP ]{}]{}assisted quantum capacity. \[thm:strong-converse\] For any quantum channel ${\mathcal{N}}$, we have $Q_{{\rm pp}}^{\dagger}({\mathcal{N}}) \leq \inf_{l \in \mathbb{N}} \frac{1}{l} R({\mathcal{N}}^{\otimes l})$. Before we prove this result, we first state a technical proposition which implies that the fidelity decreases exponentially fast as the number of channel uses increases, with the exponent bounded in  below. (However, we we do not know if the exponent in  is optimal.) \[pr:strong-converse\] Let ${\mathcal{N}}$ be a quantum channel. Consider any sequence of codes $\{ {\mathcal{C}}_n \}_{n \in \mathbb{N}}$, where ${\mathcal{C}}_n$ is a [[CPPP ]{}]{}assisted EG code for ${\mathcal{N}}^{\otimes n}$. Then the rate of this sequence, $r = \liminf_{n\to\infty} \frac{1}{n} \log |{\mathcal{C}}_n|$, satisfies $$\begin{aligned} \liminf_{n\to\infty} \left\{ - \frac{1}{n} \log F({\mathcal{C}}_n,{\mathcal{N}}^{\otimes n}) \right\} \geq \sup_{\alpha > 1} \left\{ \frac{\alpha-1}{\alpha} \big( r - \widetilde{R}_{\alpha}({\mathcal{N}}) \big) \right\}. \label{eq:exponent}\end{aligned}$$ The first statement is shown as follows. First, by definition of $r$, for any $\delta > 0$, there exists an $N_0 \in \mathbb{N}$ such that $\frac{1}{n} \log |{\mathcal{C}}_n| \geq r - \delta$ for all $n \geq N_0$. For such $n$ and any $\alpha > 1$, we employ Proposition \[lem:one-shot\] to find the following bound: $$\begin{aligned} F\left({\mathcal{C}}_n,{\mathcal{N}}^{\otimes n}\right) &\leq2^{-\left( \frac{\alpha-1}{\alpha}\right) \left( \log |{\mathcal{C}}_n|-\widetilde{R}_{\alpha}\left( \mathcal{N}^{\otimes n}\right) \right) }\\ &\leq 2^{-\left( \frac{\alpha-1}{\alpha}\right) \left( n(r-\delta) -\widetilde {R}_{\alpha}\left( \mathcal{N}^{\otimes n}\right) \right) }\\ & \leq2^{-\big( \frac{\alpha-1}{\alpha}\big) \big( n(r -\delta) - n \widetilde{R}_{\alpha}(\mathcal{N}) -\frac{\alpha\left\vert A\right\vert^{2}}{\alpha-1}\log n \big) } \label{eq:use-weak} = n^{\left\vert A\right\vert^{2}}2^{-n\left( \frac{\alpha-1}{\alpha }\right) \left( r-\delta-\widetilde{R}_{\alpha}( \mathcal{N}) \right) } .$$ Here, we used the weak subadditivity result from Theorem \[thm:weak-subadditivity\] to establish . Hence, we find $$\begin{aligned} \liminf_{n\to\infty} \left\{ - \frac{1}{n} \log F({\mathcal{C}}_n,{\mathcal{N}}^{\otimes n}) \right\} \geq \frac{\alpha-1}{\alpha} \big( r - \delta - \widetilde{R}_{\alpha}({\mathcal{N}}) \big) \end{aligned}$$ and the first statement of the theorem follows since we can choose $\delta$ and $\alpha$ arbitrarily. We briefly sketch a proof of the main theorem here for the case in which we want to show that $Q_{{\rm pp}}({\mathcal{N}}) \leq R({\mathcal{N}})$ and provide the full proof below. Consider any code with $r$ defined as in Proposition \[pr:strong-converse\]. If $r > R({\mathcal{N}})$, then by continuity of $\widetilde{R}_{\alpha}({\mathcal{N}})$ as $\alpha \to 1^+$, there always exists an $\alpha > 1$ such that $r > \widetilde{R}_{\alpha}({\mathcal{N}})$ as well. Thus, the right hand side of  is strictly positive and the fidelity thus vanishes. Consider any sequence of codes $\{ {\mathcal{C}}_n \}_{n \in \mathbb{N}}$, where ${\mathcal{C}}_n$ is a [[CPPP ]{}]{}assisted EG code for ${\mathcal{N}}^{\otimes n}$ such that $r = \liminf_{n\to\infty} \frac{1}{n} \log |{\mathcal{C}}_n| > \inf_{l \in \mathbb{N}} \frac{1}{l} R({\mathcal{N}}^{\otimes l})$. Then, by definition of the infimum there exists a value $l \in \mathbb{N}$ such that $l r > R({\mathcal{N}}^{\otimes l})$. Furthermore, by continuity of $\widetilde{R}_{\alpha}(\mathcal{N}^{\otimes l})$ as $\alpha \to 1^+$ (cf. Lemma \[lm:continuity\]), there exists an $\alpha > 1$ such that $l r > \widetilde{R}_{\alpha}({\mathcal{N}}^{\otimes l})$. We now consider the subsequence of codes $\{ {\mathcal{C}}_{kl} \}_{k \in \mathbb{N}}$ for the channels ${\mathcal{N}}^{\otimes kl} =\big({\mathcal{N}}^{\otimes l}\big)^{\otimes k}$. Then, Proposition \[pr:strong-converse\] applied to the channel ${\mathcal{N}}^{\otimes l}$ yields $$\begin{aligned} \liminf_{k\to\infty} \left\{ - \frac{1}{k} \log F({\mathcal{C}}_{kl},{\mathcal{N}}^{\otimes kl}) \right\} \geq \frac{\alpha-1}{\alpha } \big( lr - \widetilde{R}_{\alpha}({\mathcal{N}}^{\otimes l}) \big) > 0 . \label{eq:nonzero-lower} \end{aligned}$$ Here we used that $\liminf_{k\to\infty} \frac{1}{k} \log |{\mathcal{C}}_{kl}| \geq lr$ since the $\liminf$ of the subsequence $\{ \frac{1}{kl} \log |{\mathcal{C}}_{kl}| \}_{k}$ is lower bounded by the $\liminf$ of the sequence $\{ \frac{1}{n} \log |{\mathcal{C}}_n| \}_{n}$. On the other hand, we have $$\begin{aligned} \liminf_{n\to\infty} \left\{ - \frac{1}{n} \log F({\mathcal{C}}_n,{\mathcal{N}}^{\otimes n}) \right\} \geq \min_{j \in \{0,1, \dots l-1\}} \liminf_{k\to\infty} \left\{ - \frac{1}{k l+j} \log F({\mathcal{C}}_{kl+j},{\mathcal{N}}^{\otimes kl}) \right\} \,. \end{aligned}$$ Now note that  implies that the $\liminf$ for the subsequence with $j=0$ is strictly positive. In fact, an adaption of the argument leading to  reveals that the same is true for all $j \in \{0, 1, \ldots, l-1\}$. Hence, we conclude that the fidelity vanishes and that $\inf_{l \in \mathbb{N}} \frac{1}{l} R({\mathcal{N}}^{\otimes l})$ is a strong converse rate. Theorem \[thm:strong-converse\] establishes the Rains information of a quantum channel as an upper bound on the strong converse capacity $Q_{{\rm pp}}^{\dagger}({\mathcal{N}})$ for [[CPPP ]{}]{}assisted quantum communication over a channel ${\mathcal{N}}$. Thus, in summary, the following inequalities hold for all quantum channels: $$\begin{aligned} I_{\rm c}({\mathcal{N}}) \leq \lim_{l \to \infty} \frac{1}{l} I_{\rm c}({\mathcal{N}}^{\otimes l}) = Q({\mathcal{N}}) \leq Q^{\dagger}({\mathcal{N}}) \leq Q_{{\rm pp}}^{\dagger}({\mathcal{N}}) \leq \inf_{l\geq 1} \frac{1}{l} R({\mathcal{N}}^{\otimes l}) \leq R({\mathcal{N}}) . \label{eq:hier}\end{aligned}$$ Strong Converse Property for Quantum Communication over Dephasing Channels {#sec:dephasing} ========================================================================== In this section, we show that the Rains information of a generalized dephasing channel [@DS05; @YHD05MQAC; @itit2008hsieh; @BHTW10] (also known as Hadamard diagonal channels [@KMNR07] and Schur multiplier channels [@MW13]) is equal to the coherent information of this channel. Consequently, the hierarchy in  collapses for generalized dephasing channels. In particular, we establish that the quantum capacity of this class of channels obeys the strong converse property and also that classical pre- and post-processing does not increase the capacity for these channels. A generalized dephasing channel is any channel with an isometric extension of the form $$U_{A\rightarrow BE}^{\mathcal{N}}:=\sum_{x=0}^{d-1}\left\vert x\right\rangle _{B}\left\langle x\right\vert _{A}\otimes\left\vert \psi _{x}\right\rangle _{E}, \label{eq:deph}$$ where the states $\left\vert \psi_{x}\right\rangle $ are arbitrary (not necessarily orthonormal). \[prop:I\_c=R-for-H\]Let $\mathcal{N}$ be a generalized dephasing channel of the form . Then $I_{c}\left( \mathcal{N}\right) =R\left( \mathcal{N}\right)$. We have already seen in that $I_{c}\left( \mathcal{N}\right) \leq R\left( \mathcal{N}\right)$ holds for all channels. We now establish that the opposite inequality holds for a generalized dephasing channel $\mathcal{N}$. Consider that any generalized dephasing channel $\mathcal{N}$ obeys the following covariance property:[^10] $$\mathcal{N}\left( Z_{A}\left( z\right) \rho Z_{A}^{\dag}\left( z\right) \right) =Z_{B}\left( z\right) \mathcal{N}\left( \rho\right) Z_{B}^{\dag}\left( z\right) , \label{eq:covariance-dephasing}$$ for $z\in\left\{ 0,\cdots,d-1\right\} $, where$$\begin{aligned} Z_{A}\left( z\right) \left\vert x\right\rangle _{A} =\exp\left\{ 2\pi ixz/d\right\} \left\vert x\right\rangle _{A},\qquad Z_{B}\left( z\right) \left\vert x\right\rangle _{B} =\exp\left\{ 2\pi ixz/d\right\} \left\vert x\right\rangle _{B}. \label{eq:phase-op-B}$$ Furthermore, a uniform mixing of these operators is equivalent to a completely dephasing channel:$$\frac{1}{d}\sum_{i=0}^{d-1}Z_{A}\left( z\right) \left( \cdot\right) Z_{A}^{\dag}\left( z\right) =\sum_{i=0}^{d-1}\left\vert x\right\rangle \left\langle x\right\vert _{A}\left( \cdot\right) \left\vert x\right\rangle \left\langle x\right\vert _{A},$$ with the same true for the operators $\left\{ Z_{B}\left( z\right) \right\} _{z\in\left\{ 0,\ldots,d-1\right\} }$. Then we can apply Proposition \[prop:covariance\] to conclude that the Rains information of a generalized dephasing channel is maximized by a state with a Schmidt decomposition of the following form:$$\left\vert \varphi^{p}\right\rangle _{RA}:=\sum_{x}\sqrt{p_{X}\left( x\right) }\left\vert x\right\rangle _{R}\left\vert x\right\rangle _{A}, \label{eq:aligned-schmidt}$$ for some probability distribution $p_{X}\left( x\right) $ and some orthonormal basis $\left\{ \left\vert x\right\rangle _{R}\right\} $ for the reference system $R$ (with the key result being that the basis $\left\{ \left\vert x\right\rangle _{A}\right\} $ is aligned with the basis of the channel). That is,$$R\left( \mathcal{N}\right) =\max_{p_X}\min_{\tau_{RB}\in{{\rm PPT}'}\left( R:B\right) }D\left( \mathcal{N} \left( \varphi_{RA}^{p}\right) \middle\Vert\tau _{RB}\right) . \label{eq:rains-hadamard}$$ Let $\Delta_{P}$ be a CPTP map constructed as follows:$$\begin{aligned} \Delta_{P}\left( \cdot\right) & = P\left( \cdot\right) P+\left( I-P\right) \left( \cdot\right) \left( I-P\right) , \qquad \textrm{with} \qquad P =\sum_{x}\left\vert x\right\rangle \left\langle x\right\vert _{R}\otimes\left\vert x\right\rangle \left\langle x\right\vert _{B} \,.\end{aligned}$$ Then the following chain of inequalities holds:$$\begin{aligned} I_{c}\left( \mathcal{N}\right) &=\max_{\varphi_{RA}}\min_{\sigma_{B}}D\left( \mathcal{N} \left( \varphi_{RA}\right) \middle\Vert I_{R}\otimes\sigma_{B}\right) \\ & \geq\max_{p_X}\min_{\sigma_{B}}D\left( \mathcal{N} \left( \varphi_{RA}^{p}\right) \middle\Vert I_{R}\otimes\sigma_{B}\right) \\ & \geq\max_{p_X}\min_{\sigma_{B}}D\left( \Delta_{P}\left( \mathcal{N} \left( \varphi_{RA}^{p}\right) \right) \middle\Vert\Delta_{P}\left( I_{R}\otimes\sigma_{B}\right) \right) \\ & =\max_{p_X}\min_{\sigma_{B}} D\left( P\left( \mathcal{N} \left( \varphi_{RA}^{p}\right) \right) P\middle\Vert P\left( I_{R}\otimes\sigma_{B}\right) P\right) \\ & =\max_{p_X}\min_{q}D\left( \mathcal{N} \left( \varphi_{RA}^{p}\right) \middle\Vert\sum _{x}q\left( x\right) \left\vert x\right\rangle \left\langle x\right\vert _{R}\otimes\left\vert x\right\rangle \left\langle x\right\vert _{B}\right) \\ & \geq\max_{p_X}\min_{\tau_{RB}\in{{\rm PPT}'}\left( R:B\right) }D\left( \mathcal{N} \left( \varphi_{RA}^{p}\right) \middle\Vert\tau_{RB}\right) =R\left( \mathcal{N}\right) .\end{aligned}$$ The first inequality follows by restricting the maximization to be over pure bipartite vectors of the form in (\[eq:aligned-schmidt\]). The second inequality follows from monotonicity of the relative entropy under the CPTP map $\Delta_{P}$. The second equality follows because$$\begin{gathered} D\left( \Delta_{P}\left[ \left( \text{id}_{R}\otimes\mathcal{N}\right) \left( \varphi_{RA}^{p}\right) \right] \middle\Vert\Delta_{P}\left( I_{R}\otimes\sigma_{B}\right) \right) =D\left( P\left[ \left( \text{id}_{R}\otimes\mathcal{N}\right) \left( \varphi_{RA}^{p}\right) \right] P\middle\Vert P\left( I_{R}\otimes\sigma_{B}\right) P\right) \\ +D\left( \left( I-P\right) \left[ \left( \text{id}_{R}\otimes \mathcal{N}\right) \left( \varphi_{RA}^{p}\right) \right] \left( I-P\right) \middle\Vert\left( I-P\right) \left( I_{R}\otimes\sigma_{B}\right) \left( I-P\right) \right)\end{gathered}$$ since $P\perp I-P$, and because the state $\left( \text{id}_{R}\otimes\mathcal{N}\right) \left( \varphi_{RA}^{p}\right) $ has no support in the subspace onto which $I-P$ projects, so that$$D\left( \left( I-P\right) \left[ \left( \text{id}_{R}\otimes \mathcal{N}\right) \left( \varphi_{RA}^{p}\right) \right] \left( I-P\right) \middle\Vert\left( I-P\right) \left( I_{R}\otimes\sigma_{B}\right) \left( I-P\right) \right) =0.$$ The third equality follows because $$\begin{aligned} P\left[ \left( \text{id}_{R}\otimes\mathcal{N}\right) \left( \varphi_{RA}^{p}\right) \right] P & =\left( \text{id}_{R}\otimes \mathcal{N}\right) \left( \varphi_{RA}^{p}\right), \\ P\left( I_{R}\otimes\sigma_{B}\right) P & =\sum_{x}q\left( x\right) \left\vert x\right\rangle \left\langle x\right\vert _{R}\otimes\left\vert x\right\rangle \left\langle x\right\vert _{B},\end{aligned}$$ for some distribution $q\left( x\right) =\left\langle x\right\vert _{B}\sigma_{B}\left\vert x\right\rangle _{B}$. The last inequality follows because the state $\sum_{x}q\left( x\right) \left\vert x\right\rangle \left\langle x\right\vert _{R}\otimes\left\vert x\right\rangle \left\langle x\right\vert _{B}\in{{\rm PPT}'}\left( R\!:\!B\right)$, and the final equality follows from (\[eq:rains-hadamard\]). Strong Converse for Classical Communication\ Assisted Capacity of Erasure Channels {#sec:erasure} ============================================ In this section, we consider a class of erasure channels of the form $$\begin{aligned} {\mathcal{N}}_{A \to B} : \rho_A \mapsto (1-p) \rho_{B} + p | e \rangle\!\langle e|_{B}, \label{eq:erasure}\end{aligned}$$ where $p \in [0,1]$ is the erasure probability, $\rho_B$ is an isometric embedding of $\rho_A$ into $B$, and $\vert e\rangle$ is a quantum state orthogonal to $\rho_{B}$. Let ${\mathcal{N}}$ be an erasure channel of the form . Then, $Q_{{\rm pp}}({\mathcal{N}}) = Q_{{\rm pp}}^{\dagger}({\mathcal{N}}) = (1-p) \log |A|$. First note that erasure channels are covariant under the full unitary group acting on $A$, and thus Proposition \[prop:covariance\] implies that $$\begin{aligned} R({\mathcal{N}}) = R({\mathcal{N}}(\psi_{RA})) \leq D \big( {\mathcal{N}}(\psi_{RA}) \big\| \tau_{RB} \big) = H(\tau_{RB}) - H(\rho_{RB})\end{aligned}$$ where $\psi_{RA}$ is a maximally entangled state, $\rho_{RB} = {\mathcal{N}}(\psi_{RA})$, and $\tau_{RB}$ is chosen as follows: $$\begin{aligned} |\psi\rangle_{RA} = \sum_x \frac{1}{\sqrt{|A|}} |x\rangle_R \otimes |x\rangle_A , \qquad \textrm{and} \qquad \tau_{RB} = \sum_x \frac{1}{|A|} |x\rangle\!\langle x|_R \otimes {\mathcal{N}}\big(|x\rangle\!\langle x|_A\big) \,.\end{aligned}$$ Evaluating this, we find $$\begin{aligned} H(RB)_\tau & = H(R)_\tau + H(B|R)_\tau \\ & = \log |A| + \sum_x \frac{1}{|A|} H(B)_{{\mathcal{N}}(|x\rangle\!\langle x|_A)} \\ & = \log |A| + \sum_x \frac{1}{|A|} [H((1-p)|x\rangle\!\langle x|_A + p |e\rangle\langle e|)] \\ & = \log |A| + h_2(p) ,\\ H(RB)_\rho & = H( (1-p) \psi_{RA} + p \pi_R \otimes |e\rangle \langle e |) \\ & = h_2(p) + (1-p) H( \psi_{RA}) + p H(\pi_R \otimes |e\rangle \langle e |) \\ & = h_2(p) + p \log|A| .\end{aligned}$$ In the above, $H(B|R) := H(BR) - H(R)$ is the conditional entropy. Therefore, we conclude that $Q_{{\rm pp}}^{\dagger}({\mathcal{N}}) \leq R({\mathcal{N}}) \leq (1-p) \log |A|.$ Finally, it is well known that any rate $r < (1-p) \log |A|$ can be achieved with [[CPPP ]{}]{}assistance [@PhysRevLett.78.3217]. We review this argument briefly. Let Alice prepare and send $n$ maximally entangled states through ${\mathcal{N}}^{\otimes n}$. Bob then records which instances of the channel led to an erasure, and communicates this to Alice. They will then use the correctly transmitted states to distill a maximally entangled state of dimension $r n$. This will succeed whenever $r n / \log |A|$ is smaller than the number of correctly transmitted states, which will happen with probability 1 as $n \to \infty$ as long as $r < (1-p) \log |A|$. This establishes that $Q_{{\rm pp}}({\mathcal{N}}) \geq (1-p) \log |A|$ and concludes the proof. Finally, recall that the erasure channel is degradable for $p\leq 1/2$ [@DS05] and we can thus calculate its unassisted quantum capacity to be $Q({\mathcal{N}}) = \max\{(1-2p) \log |A|, 0\}$. Thus, we have an example of a channel where $Q({\mathcal{N}}) < R({\mathcal{N}})$, which means that our upper bound in terms of the Rains information of the channel is not sufficient to establish the strong converse property for general degradable channels. Conclusion {#sec:conclusion} ========== This paper has established that the Rains information of a quantum channel is a strong converse rate for quantum communication. The main application of the first result is to establish the strong converse property for the quantum capacity of all generalized dephasing channels. Going forward from here, there are several questions to consider. First, are there any other channels besides the generalized dephasing ones for which the Rains information is equal to the coherent information? If true, the theorems established here would establish the strong converse property for these channels. For example, can we prove a strong converse theorem for the quantum capacity of general Hadamard channels? Is it possible to show weak subadditivity of a Rényi coherent information quantity for some class of channels, addressing the original question posed in [@SW12]? To this end, the developments in [@TBH13] might be helpful. Next, is it possible to show that the Rains information of a quantum channel represents a strong converse rate for quantum communication assisted by interactive forward and backward classical communication? Similarly, can one show that the squashed entanglement of a channel [@TGW14] is a strong converse rate for this task? Recent work has proved that the squashed entanglement of a quantum channel is an upper bound on the quantum capacity with interactive forward and backward classical communication, and it could be that the quantities defined in [@BSW14] would be helpful for settling this question. Finally, now that the strong converse holds for the classical capacity, the quantum capacity, and the entanglement-assisted capacity of all generalized dephasing channels, can we establish that the strong converse holds for trade-off capacities of these channels, in the sense of [@BHTW10; @WH10b]? Can we establish second-order characterizations for this class of channels, in the sense of [@TH12; @TT13; @DL14; @DTW14]? **Acknowledgements.** We acknowledge many discussions with our colleagues regarding the strong converse for quantum capacity. This includes Mario Berta, Frederic Dupuis, Will Matthews, Ciara Morgan, Naresh Sharma, Stephanie Wehner, and Dong Yang. MT is funded by the Ministry of Education (MOE) and National Research Foundation Singapore, as well as MOE Tier 3 Grant Random numbers from quantum processes (MOE2012-T3-1-009). MMW acknowledges startup funds from the Department of Physics and Astronomy at LSU, support from the NSF through Award No. CCF-1350397, and support from the DARPA Quiness Program through US Army Research Office award W31P4Q-12-1-0019. MMW is also grateful to the quantum information theory group at the Universitat Autònoma de Barcelona and to Stephanie Wehner’s quantum information group at the Centre for Quantum Technologies, National University of Singapore, for hosting him for research visits during May-June 2014. AW acknowledges financial support by the Spanish MINECO, project FIS2008-01236 with the support of FEDER funds, the EC STREP RAQUEL, the ERC Advanced Grant IRQUAT, and the Philip Leverhulme Trust. Proof of Lemma \[lm:continuity\] {#sec:ogawa-nagaoka} ================================ The first statement follows because the underlying sandwiched relative entropy, $\widetilde{D}_{\alpha}(\rho\|\sigma)$, is monotonically increasing in $\alpha$ [@MDSFT13 Theorem 7] for all $\rho \in \mathcal{S}$ and $\sigma \in \mathcal{P}$, i.e.$$\widetilde{D}_{\alpha}\left( \rho\Vert\sigma\right) \geq\widetilde{D}_{\beta}\left( \rho\Vert\sigma\right)$$ for all $\alpha\geq\beta\geq 0$. This already establishes that the limit exists and satisfies $$\lim_{\alpha \to 1^+} \widetilde{R}_{\alpha}(\mathcal{N}) \geq R(\mathcal{N}) .$$ We would like to show the opposite inequality. Consider the following bound (from [@TCR09 Lemma 8] and [@T12 Lemma 6.3], see also [@WWY13 Eq. (19)]) $$\widetilde{D}_{1+\delta}\left( \rho\Vert\sigma\right) \leq D\left( \rho \Vert\sigma\right) +4 \delta \left[ \log\nu\left( \rho,\sigma\right) \right] ^{2},$$ which holds for $\delta\in\left( 0,\log3/\left( 4\log\nu\left( \rho,\sigma\right) \right) \right) $ where$$\nu\left( \rho,\sigma\right) := \operatorname{tr}(\rho^{3/2} \sigma^{- 1/2}) + \operatorname{tr}(\rho^{1/2} \sigma^{1/2}) +1.$$ Let us for the moment assume that $\sigma > 0$ with $\operatorname{tr}(\sigma) \leq 1$, and its smallest eigenvalue is denoted as $\lambda$. In this case, we can bound $ v(\rho,\sigma) \leq 2 + 1/{\sqrt{\lambda}}$. For any state $\rho_{RB}$ and $\delta > 0$ sufficiently small, we can apply the bound above to arrive at$$\begin{aligned} &\min_{\sigma_{RB}\in{{\rm PPT}'}\left( R:B\right) }\widetilde{D}_{1+\delta} \left( \rho_{RB} \Vert\sigma_{RB}\right) \\ &\qquad \leq \min_{\tau_{RB}\in{{\rm PPT}'}\left( R:B\right) }\widetilde{D}_{1+\delta} \left( \rho_{RB} \middle\Vert (1- \delta) \tau _{RB} + \delta \pi_{RB} \right) \\ &\qquad \leq \min_{\tau_{RB}\in{{\rm PPT}'}\left( R:B\right) } D\left( \rho_{RB} \middle\Vert (1-\delta) \tau _{RB} + \delta \pi_{RB} \right) + 4 \delta \left( \log \left( 2 + \frac{\sqrt{|A| |B|}}{\sqrt{\delta}} \right) \right)^2 \\ & \qquad \leq\min_{\tau_{RB}\in{{\rm PPT}'}\left( R:B\right) } D\left( \rho_{RB} \middle\Vert \tau _{RB} \right) + \log \frac1{1-\delta} + 4 \delta \left( \log \left( 2 + \frac{\sqrt{|A| |B|}}{\sqrt{\delta}} \right) \right)^2 \label{eq:uniform-bound}\end{aligned}$$ The first inequality follows by picking $\sigma_{RB}$ to be of the form $\sigma_{RB} = \left( 1- \delta \right) \tau_{RB} + \delta \pi_{RB}$ where $\pi_{RB}$ is the fully mixed state on $RB$. Also, note that $\left( 1-\delta\right) \tau_{RB} + \delta \pi_{RB}\in T\left( R:B\right)$. To verify the second inequality, note that the minimum eigenvalue of $\left( 1-\delta\right) \tau_{RB} + \delta \pi_{RB}$ is always larger than $\delta/(|A| |B|)$. (The system $R$ can be chosen to be of size $|A|$ without loss of generality, where $|A|$ is the dimension of the channel input.) Finally, recall that $D\left( \rho\Vert\sigma\right) \leq D\left( \rho \Vert\sigma^{\prime}\right)$ whenever $\sigma^{\prime}\leq\sigma$ to verify the last inequality. Since, crucially, the upper bound in  is uniform in $\rho_{RB}$, we can immediately conclude that $$\widetilde{R}_{1+\delta}(\mathcal{N}) \leq R(\mathcal{N}) + \log \frac1{1-\delta} + 4 \delta \left( \log \left( 2 + \frac{\sqrt{|A| |B|}}{\sqrt{\delta}} \right) \right)^2 \label{eq:nice-bound}$$ by maximizing over channel output states as in the definition of $\widetilde{R}_{1+\delta}$ and $R$. Thus, $\lim_{\alpha \to 1^+} \widetilde{R}_{\alpha}(\mathcal{N}) \leq R(\mathcal{N})$, concluding the proof. Proof of Corollary \[cor:von-Neumann-subadd\] {#sec:appendix} ============================================= We can focus only on operators $\tau_{RB}$ such that supp$\left( \rho _{RB}\right) \subseteq\ $supp$\left( \tau_{RB}\right) $, where $\rho _{RB}=\mathcal{N}_{A\rightarrow B}\left( \phi_{RA}\right) $. This is because the quantity of interest contains a minimization over all $\tau_{RB}$. For any sufficiently small $\delta>0$$$\begin{aligned} R\left( \mathcal{N}^{\otimes n}\right) \leq\widetilde{R}_{1+\delta }\left( \mathcal{N}^{\otimes n}\right) \leq\frac{1+\delta}{\delta}\left\vert A\right\vert ^{2}\log n+n\widetilde {R}_{1+\delta}\left( \mathcal{N}\right).\end{aligned}$$ The first inequality is from the monotonicity of the sandwiched Rényi relative entropy in the Rényi parameter [@MDSFT13 Theorem 7]. The second inequality follows from Theorem \[thm:weak-subadditivity\]. This invites an application of , which gives $$\frac{1}{n} R\left( \mathcal{N}^{\otimes n}\right) \leq R(\mathcal{N}) + \frac{1+\delta}{\delta}\left\vert A\right\vert ^{2} \frac{\log n}{n} + \log \frac1{1-\delta} + 4 \delta \left( \log \left( 2 + \frac{\sqrt{|A| |B|}}{\sqrt{\delta}} \right) \right)^2$$ Choosing $\delta = 1/\sqrt{n}$ and taking the limit $n \to \infty$ then immediately yields the desired result. [Marco Tomamichel]{} (M’13) received the M.Sc. in Electrical Engineering and Information Technology degree from ETH Zurich (Switzerland) in 2007. He then graduated with a Ph.D. in Physics at the Institute of Theoretical Physics at ETH Zurich in 2012. From 2012 to 2014 he was a Research Fellow and then Senior Research Fellow at the Centre for Quantum Technologies at the National University of Singapore. Currently he is a Lecturer in the School of Physics at the University of Sydney. His research interests include classical and quantum information theory, specifically finite resource analysis, as well as quantum cryptography. [Mark M. Wilde]{} (M’99-SM’13) was born in Metairie, Louisiana, USA. He received the Ph.D. degree in electrical engineering from the University of Southern California, Los Angeles, California, in 2008. He is an Assistant Professor in the Department of Physics and Astronomy and the Center for Computation and Technology at Louisiana State University. His current research interests are in quantum Shannon theory, quantum optical communication, quantum computational complexity theory, and quantum error correction. [Andreas Winter]{} received a Diploma degree in Mathematics from the Freie Universität Berlin, Berlin, Germany, in 1997, and a Ph.D. degree from the Fakultät für Mathematik, Universität Bielefeld, Bielefeld, Germany, in 1999. He was Research Associate at the University of Bielefeld until 2001, and then with the Department of Computer Science at the University of Bristol, Bristol, UK. In 2003, still with the University of Bristol, he was appointed Lecturer in Mathematics, and in 2006 Professor of Physics of Information. Since 2012 he has been ICREA Research Professor with the Universitat Autònoma de Barcelona, Barcelona, Spain. [^1]: $^*$ School of Physics, The University of Sydney, Sydney, NSW 2006, Australia. (Email: [marco.tomamichel@sydney.edu.sg]{}). [^2]: $^\dagger$ Hearne Institute for Theoretical Physics, Department of Physics and Astronomy, Center for Computation and Technology, Louisiana State University, Baton Rouge, Louisiana 70803, USA. [^3]: $^\ddagger$ ICREA & Física Teórica, Informació i Fenomens Quántics, Universitat Autònoma de Barcelona, ES-08193 Bellaterra (Barcelona), Spain. [^4]: However, see the later works [@qcap2008second] and [@qcap2008fourth], which set [@L97] and [@capacity2002shor], respectively, on a firm foundation. [^5]: All logarithms in this paper are taken base two. [^6]: It should be clear from the context whether $R$ refers to “Rains” or to a reference system. [^7]: It is also sufficient for $2\times2$ and $2\times3$ systems, but otherwise only necessary [@Horodecki19961]. [^8]: Note that there is no need to have an asymmetric notation here because a state is PPT with respect to transpose on system $A$ if and only if it is PPT with respect to a partial transpose on system $B$. [^9]: This generalizes the standard classical scenario in which we are interested in transmitting a uniformly distributed message. [^10]: The covariance in (\[eq:covariance-dephasing\]) in fact holds for any operators of the form $\sum_{x=0}^{d-1}\exp\left\{ i\varphi_{x}\right\} \left\vert x\right\rangle \left\langle x\right\vert _{A}$ and $\sum _{x=0}^{d-1}\exp\left\{ i\varphi_{x}\right\} \left\vert x\right\rangle \left\langle x\right\vert _{B}$ with $\varphi_{x}\in\mathbb{R}$, but it suffices to consider only the operators in (\[eq:phase-op-B\]) for our proof here.

--- abstract: 'Contextualized embeddings use unsupervised language model pretraining to compute word representations depending on their context. This is intuitively useful for generalization, especially in Named-Entity Recognition where it is crucial to detect mentions never seen during training. However, standard English benchmarks overestimate the importance of lexical over contextual features because of an unrealistic lexical overlap between train and test mentions. In this paper, we perform an empirical analysis of the generalization capabilities of state-of-the-art contextualized embeddings by separating mentions by novelty and with out-of-domain evaluation. We show that they are particularly beneficial for unseen mentions detection, especially out-of-domain. For models trained on CoNLL03, language model contextualization leads to a +1.2% maximal relative micro-F1 score increase in-domain against +13% out-of-domain on the WNUT dataset.' author: - Bruno Taillé - Vincent Guigue - Patrick Gallinari bibliography: - 'mendeley.bib' title: 'Contextualized Embeddings in Named-Entity Recognition: An Empirical Study on Generalization' --- Introduction ============ Named-Entity Recognition (NER) consists in detecting textual mentions of entities and classifying them into predefined types. It is modeled as sequence labeling, the standard neural architecture of which is BiLSTM-CRF [@Huang2015BidirectionalTagging]. Recent improvements mainly stem from using new types of representations: learned character-level word embeddings [@Lample2016NeuralRecognition] and contextualized embeddings derived from a language model (LM) [@Peters2018DeepRepresentations; @Akbik2018ContextualLabeling; @Devlin2019BERT:Understanding]. LM pretraining enables to obtain contextual word representations and reduce the dependency of neural networks on hand-labeled data specific to tasks or domains [@Howard2018UniversalClassification; @Radford2018ImprovingPre-Training]. This contextualization ability can particularly benefit to NER domain adaptation which is often limited to training a network on source data and either feeding its predictions to a new classifier or finetuning it on target data [@Lee2018TransferNetworks; @Rodriguez2018TransferClasses]. All the more as classical NER models have been shown to poorly generalize to unseen mentions or domains [@Augenstein2017GeneralisationAnalysis]. In this paper, we quantify the impact of ELMo [@Peters2018DeepRepresentations], Flair [@Akbik2018ContextualLabeling] and BERT [@Devlin2019BERT:Understanding] representations on generalization to unseen mentions and new domains in NER. To better understand their effectiveness, we propose a set of experiments to distinguish the effect of unsupervised LM contextualization ($C_{LM}$) from task supervised contextualization ($C_{NER}$). We show that the former mainly benefits unseen mentions detection, all the more out-of-domain where it is even more beneficial than the latter. Lexical Overlap =============== Neural NER models mainly rely on lexical features in the form of word embeddings, either learned at the character-level or not. Yet, standard NER benchmarks present a large lexical overlap between mentions in the train set and dev / test sets which leads to a poor evaluation of generalization to unseen mentions as shown by Augenstein et al. [@Augenstein2017GeneralisationAnalysis]. They separate seen from unseen mentions and evaluate out-of-domain to focus on generalization but only study models designed before 2011 and no longer in use. We propose to use a similar setting to analyze the impact of state-of-the-art LM pretraining methods on generalization in NER. We introduce a slightly more fine-grained novelty partition by separating unseen mentions in *partial match* and *new* categories. A mention is an *exact match* (EM) if it appears in the exact same case-sensitive form in the train set, tagged with the same type. It is a *partial match* (PM) if at least one of its non stop words appears in a mention of same type. Every other mentions are *new*. We study lexical overlap in CoNLL03 [@Sang2003IntroductionTask] and OntoNotes [@Weischedel2013OntoNotesLDC2013T19], the two main English NER datasets, as well as WNUT17 [@Derczynski2017ResultsRecognition] which is smaller, specific to user generated content (tweets, comments) and was designed without exact overlap. For out-of-domain evaluation, we train on CoNLL03 (news articles) and test on the larger and more diverse OntoNotes (see Table \[table:genres\] for genres) and the very specific WNUT. We remap OntoNotes and WNUT entity types to match CoNLL03’s and denote the obtained dataset with $^\ast$. [@ll\*[5]{}[Y]{}rYr\*[5]{}[Y]{}rYr\*[4]{}[Y]{}@]{} & & & & ON & & & & WNUT & &\ & & LOC & MISC & ORG & PER & ALL & & ALL& & LOC & MISC & ORG & PER & ALL & & ALL & & LOC & ORG & PER & ALL\ & EM & 82% & 67% & 54% & 14% & 52% & & 67% & & 87% & 93% & 54% & 49% & 69% & & - & & - & - &- &-\ & PM & 4% & 11% & 17% & 43% & 20% & & 24% & & 6% & 2% & 32% & 36% & 20% & & 12% & & 11% & 5% & 13% & 12%\ & New & 14% & 22% & 29% & 43% & 28% & & 9% & & 7% & 5% & 14% & 15% & 11% & & 88% & & 89% & 95% & 87% & 88%\ & EM &- & - & -& - &- & &- & & 70% & 78% & 18% & 16% & 42% & & - & & 26% & 8% & 1% & 7%\ & PM &- & - &- & -& - & & - & & 7% & 10% & 45% & 46% & 28% & & - & & 9% & 15% & 16% & 14%\ & New & - & - & - & - & - & & - & & 23% & 12% & 38% & 38% & 30%& & - & & 65% & 77% & 83% & 78%\ \[table:overlap\] As reported in Table \[table:overlap\], the two main benchmarks for English NER mainly evaluate performance on occurrences of mentions already seen during training, although they appear in different sentences. Such lexical overlap proportions are unrealistic in real-life where the model must process orders of magnitude more documents in the inference phase than it has been trained on, to amortize the annotation cost. On the contrary, WNUT proposes a particularly challenging low-resource setting with no exact overlap. Furthermore, the overlap depends on the entity types: Location and Miscellaneous are the most overlapping types, even out-of-domain, whereas Person and Organization present a more varied vocabulary, also more subject to evolve with time and domain. Word Representations ==================== ### Word embeddings map each word to a single vector which results in a lexical representation. We take **GloVe 840B** embeddings [@Pennington2014GloVe:Representation] trained on Common Crawl as the pretrained word embeddings baseline and fine-tune them as done in related work. ### Character-level word embeddings are learned by a word-level neural network from character embeddings to incorporate orthographic and morphological features. We reproduce the **Char-BiLSTM** from [@Lample2016NeuralRecognition]. It is trained jointly with the NER model and its outputs are concatenated to GloVe embeddings. We also experiment with the Char-CNN layer from ELMo to isolate the effect of LM contextualization and denote it **ELMo\[0\]**. ### Contextualized word embeddings take into account the context of a word in its representation, contrary to previous representations. A LM is pretrained and used to predict the representation of a word given its context. **ELMo** [@Peters2018DeepRepresentations] uses a Char-CNN to obtain a context-independent word embedding and the concatenation of a forward and backward two-layer LSTM LM for contextualization. These representations are summed with weights learned for each task as the LM is frozen after pretraining. **BERT** [@Devlin2019BERT:Understanding] uses WordPiece subword embeddings [@Wu2016GooglesTranslation] and learns a representation modeling both left and right contexts by training a Transformer encoder [@Vaswani2017AttentionNeed] for Masked LM and next sentence prediction. For a fairer comparison, we use the BERT~LARGE~ feature-based approach where the LM is not fine-tuned and its last four hidden layers are concatenated. **Flair** [@Akbik2018ContextualLabeling] uses a character-level LM for contextualization. As in ELMo, they train two opposite LSTM LMs, freeze them and concatenate the predicted states of the first and last characters of each word. Flair and ELMo are pretrained on the 1 Billion Word Benchmark [@Chelba2013OneModeling] while BERT uses Book Corpus [@Zhu2015AligningBooks] and English Wikipedia. Experiments =========== In order to compare the different embeddings, we feed them as input to a classifier. We first use the state-of-the-art BiLSTM-CRF [@Huang2015BidirectionalTagging] with hidden size 100 in each direction and present in-domain results on all datasets in Table \[table:in-domain\]. We then report out-of-domain performance in Table \[table:results\]. To better capture the intrinsic effect of LM contextualization, we introduce the Map-CRF baseline from [@Akbik2018ContextualLabeling] where the BiLSTM is replaced by a simple linear projection of each word embedding. We only consider domain adaptation from CoNLL03 to OntoNotes$^\ast$ and WNUT$^\ast$ assuming that labeled data is scarcer, less varied and more generic than target data in real use cases. We use the IOBES tagging scheme for NER and no preprocessing. We fix a batch size of 64, a learning rate of 0.001 and a 0.5 dropout rate at the embedding layer and after the BiLSTM or linear projection. The maximum number of epochs is set to 100 and we use early stopping with patience 5 on validation global micro-F1. For each configuration, we use the best performing optimization method between SGD and Adam with $\beta_1=0.9$ and $\beta_2=0.999$. We report the mean and standard deviation of five runs. [@lr@\*[4]{}[Y]{}r\*[4]{}[Y]{}r\*[3]{}[Y]{}@]{} & & & & & &\ Embedding & Dim & EM & PM & New & All & & EM & PM & New & All & & PM & New & All\ BERT & 4096 & 95.7$_{{\scalebox{\s}}{.1}}$ & 88.8$_{{\scalebox{\s}}{.3}}$ & 82.2$_{{\scalebox{\s}}{.3}}$ & 90.5$_{{\scalebox{\s}}{.1}}$ & & 96.9$_{{\scalebox{\s}}{.2}}$ & 88.6$_{{\scalebox{\s}}{.3}}$ & 81.1$_{{\scalebox{\s}}{.5}}$ & **93.5**$_{{\scalebox{\s}}{.2}}$ & & 77.0$_{{\scalebox{\s}}{4.6}}$ & 53.9$_{{\scalebox{\s}}{.9}}$ & **57.0**$_{{\scalebox{\s}}{1.0}}$\ ELMo & 1024 & 95.9$_{{\scalebox{\s}}{.1}}$ & 89.2$_{{\scalebox{\s}}{.5}}$ & 85.8$_{{\scalebox{\s}}{.7}}$ & **91.8**$_{{\scalebox{\s}}{.3 }}$ & & 97.1$_{{\scalebox{\s}}{.2}}$ & 88.0$_{{\scalebox{\s}}{.2}}$ & 79.9$_{{\scalebox{\s}}{.7}}$ & **93.4**$_{{\scalebox{\s}}{.2}}$ & & 67.7$_{{\scalebox{\s}}{3.2}}$ & 49.5$_{{\scalebox{\s}}{.9}}$ & 52.1$_{{\scalebox{\s}}{1.0}}$\ Flair & 4096 & 95.4$_{{\scalebox{\s}}{.1}}$ & 88.1$_{{\scalebox{\s}}{.6}}$ & 83.5$_{{\scalebox{\s}}{.5}}$ & 90.6$_{{\scalebox{\s}}{.2}}$ & & 96.7$_{{\scalebox{\s}}{.1}}$ & 85.8$_{{\scalebox{\s}}{.5}}$ & 75.0$_{{\scalebox{\s}}{.6}}$ & 92.1$_{{\scalebox{\s}}{.2}}$ & & 64.9$_{{\scalebox{\s}}{.7}}$ & 48.2$_{{\scalebox{\s}}{2.0}}$ & 50.4$_{{\scalebox{\s}}{1.8}}$\ ELMo\[0\] & 1024 & 95.8$_{{\scalebox{\s}}{.1}}$ & 87.2$_{{\scalebox{\s}}{.2}}$ & 83.5$_{{\scalebox{\s}}{.4}}$ & 90.7$_{{\scalebox{\s}}{.1}}$ & & 96.9$_{{\scalebox{\s}}{.1}}$ & 85.9$_{{\scalebox{\s}}{.3}}$ & 75.5$_{{\scalebox{\s}}{.6}}$ & 92.4$_{{\scalebox{\s}}{.1}}$ & & 72.8$_{{\scalebox{\s}}{1.3}}$ & 45.4$_{{\scalebox{\s}}{2.8}}$ & 49.1$_{{\scalebox{\s}}{2.3}}$\ GloVe + char & 350 & 95.3$_{{\scalebox{\s}}{.3}}$ & 85.5$_{{\scalebox{\s}}{.7}}$ & 83.1$_{{\scalebox{\s}}{.7}}$ & 89.9$_{{\scalebox{\s}}{.5}}$ & & 96.3$_{{\scalebox{\s}}{.1}}$ & 83.3$_{{\scalebox{\s}}{.2}}$ & 69.9$_{{\scalebox{\s}}{.6}}$ & 91.0$_{{\scalebox{\s}}{.1}}$ & & 63.2$_{{\scalebox{\s}}{4.6}}$ & 33.4$_{{\scalebox{\s}}{1.5}}$ & 38.0$_{{\scalebox{\s}}{1.7}}$\ GloVe & 300 & 95.1$_{{\scalebox{\s}}{.4}}$ & 85.3$_{{\scalebox{\s}}{.5}}$ & 81.1$_{{\scalebox{\s}}{.5}}$ & 89.3$_{{\scalebox{\s}}{.4}}$ & & 96.2$_{{\scalebox{\s}}{.2}}$ & 82.9$_{{\scalebox{\s}}{.2}}$ & 63.8$_{{\scalebox{\s}}{.5}}$ & 90.4$_{{\scalebox{\s}}{.2}}$ & & 59.1$_{{\scalebox{\s}}{2.9}}$ & 28.1$_{{\scalebox{\s}}{1.5}}$ & 32.9$_{{\scalebox{\s}}{1.2}}$\ \[table:in-domain\] [@l@lr\*[4]{}[Y]{}r\*[4]{}[Y]{}r\*[4]{}[Y]{}@]{} & & & & & & &\ & Emb & & EM & PM & New & All & & EM & PM & New & All & & EM & PM & New & All\ & BERT & & 95.7$_{{\scalebox{\s}}{.1}}$ & 88.8$_{{\scalebox{\s}}{.3}}$ & 82.2$_{{\scalebox{\s}}{.3}}$ & 90.5$_{{\scalebox{\s}}{.1}}$ & & 95.1$_{{\scalebox{\s}}{.1}}$ & 82.9$_{{\scalebox{\s}}{.5}}$ & 73.5$_{{\scalebox{\s}}{.4}}$ & **85.0**$_{{\scalebox{\s}}{.3}}$ & & 57.4$_{{\scalebox{\s}}{1.0}}$ & 56.3$_{{\scalebox{\s}}{1.2}}$ & 32.4$_{{\scalebox{\s}}{.8}}$ & 37.6$_{{\scalebox{\s}}{.8}}$\ & ELMo & & 95.9$_{{\scalebox{\s}}{.1}}$ & 89.2$_{{\scalebox{\s}}{.5}}$ & 85.8$_{{\scalebox{\s}}{.7}}$ & **91.8**$_{{\scalebox{\s}}{.3}}$ & & 94.3$_{{\scalebox{\s}}{.1}}$ & 79.2$_{{\scalebox{\s}}{.2}}$ & 72.4$_{{\scalebox{\s}}{.4}}$ & 83.4$_{{\scalebox{\s}}{.2}}$ & & 55.8$_{{\scalebox{\s}}{1.2}}$ & 52.7$_{{\scalebox{\s}}{1.1}}$ & 36.5$_{{\scalebox{\s}}{1.5}}$ & **41.0**$_{{\scalebox{\s}}{1.2}}$\ & Flair & & 95.4$_{{\scalebox{\s}}{.1}}$ & 88.1$_{{\scalebox{\s}}{.6}}$ & 83.5$_{{\scalebox{\s}}{.5}}$ & 90.6$_{{\scalebox{\s}}{.2}}$ & & 94.0$_{{\scalebox{\s}}{.3}}$ & 76.1$_{{\scalebox{\s}}{1.1}}$ & 62.1$_{{\scalebox{\s}}{.5}}$ & 79.0$_{{\scalebox{\s}}{.5}}$ & & 56.2$_{{\scalebox{\s}}{2.2}}$ & 49.4$_{{\scalebox{\s}}{3.4}}$ & 29.1$_{{\scalebox{\s}}{3.3}}$ & 34.9$_{{\scalebox{\s}}{2.9}}$\ & ELMo\[0\] & & 95.8$_{{\scalebox{\s}}{.1}}$ & 87.2$_{{\scalebox{\s}}{.2}}$ & 83.5$_{{\scalebox{\s}}{.4}}$ & 90.7$_{{\scalebox{\s}}{.1}}$ & & 93.6$_{{\scalebox{\s}}{.1}}$ & 76.8$_{{\scalebox{\s}}{.6}}$ & 66.1$_{{\scalebox{\s}}{.3}}$ & 80.5$_{{\scalebox{\s}}{.2}}$ & & 52.3$_{{\scalebox{\s}}{1.2}}$ & 50.8$_{{\scalebox{\s}}{1.5}}$ & 32.6$_{{\scalebox{\s}}{2.2}}$ & 37.6$_{{\scalebox{\s}}{1.8}}$\ & G + char & & 95.3$_{{\scalebox{\s}}{.3}}$ & 85.5$_{{\scalebox{\s}}{.7}}$ & 83.1$_{{\scalebox{\s}}{.7}}$ & 89.9$_{{\scalebox{\s}}{.5}}$ & & 93.9$_{{\scalebox{\s}}{.2}}$ & 73.9$_{{\scalebox{\s}}{1.1}}$ & 60.4$_{{\scalebox{\s}}{.7}}$ & 77.9$_{{\scalebox{\s}}{.5}}$ & & 55.9$_{{\scalebox{\s}}{.8}}$ & 46.8$_{{\scalebox{\s}}{1.8}}$ & 19.6$_{{\scalebox{\s}}{1.6}}$ & 27.2$_{{\scalebox{\s}}{1.3}}$\ & GloVe & & 95.1$_{{\scalebox{\s}}{.4}}$ & 85.3$_{{\scalebox{\s}}{.5}}$ & 81.1$_{{\scalebox{\s}}{.5}}$ & 89.3$_{{\scalebox{\s}}{.4}}$ & & 93.7$_{{\scalebox{\s}}{.2}}$ & 73.0$_{{\scalebox{\s}}{1.2}}$ & 57.4$_{{\scalebox{\s}}{1.8}}$ & 76.9$_{{\scalebox{\s}}{.9}}$ & & 53.9$_{{\scalebox{\s}}{1.2}}$ & 46.3$_{{\scalebox{\s}}{1.5}}$ & 13.3$_{{\scalebox{\s}}{1.4}}$ & 27.1$_{{\scalebox{\s}}{1.0}}$\ & BERT & & 93.2$_{{\scalebox{\s}}{.3}}$ & 85.8$_{{\scalebox{\s}}{.4}}$ & 73.7$_{{\scalebox{\s}}{.8}}$ & 86.2$_{{\scalebox{\s}}{.4}}$ & & 93.5$_{{\scalebox{\s}}{.2}}$ & 77.8$_{{\scalebox{\s}}{.5}}$ & 67.8$_{{\scalebox{\s}}{.9}}$ & 80.9$_{{\scalebox{\s}}{.4}}$ & & 57.4$_{{\scalebox{\s}}{.3}}$ & 53.5$_{{\scalebox{\s}}{2.6}}$ & 33.9$_{{\scalebox{\s}}{.6}}$ & 38.4$_{{\scalebox{\s}}{.4}}$\ & ELMo & & 93.7$_{{\scalebox{\s}}{.2}}$ & 87.2$_{{\scalebox{\s}}{.6}}$ & 80.1$_{{\scalebox{\s}}{.3}}$ & **88.7**$_{{\scalebox{\s}}{.2}}$ & & 93.6$_{{\scalebox{\s}}{.1}}$ & 79.1$_{{\scalebox{\s}}{.5}}$ & 69.5$_{{\scalebox{\s}}{.4}}$ & **82.2**$_{{\scalebox{\s}}{.3}}$ & & 61.1$_{{\scalebox{\s}}{.7}}$ & 53.0$_{{\scalebox{\s}}{.9}}$ & 37.5$_{{\scalebox{\s}}{.7}}$ & **42.4**$_{{\scalebox{\s}}{.6}}$\ & Flair & & 94.3$_{{\scalebox{\s}}{.1}}$ & 85.1$_{{\scalebox{\s}}{.3}}$ & 78.6$_{{\scalebox{\s}}{.3}}$ & 88.1$_{{\scalebox{\s}}{.03}}$ & & 93.2$_{{\scalebox{\s}}{.1}}$ & 74.0$_{{\scalebox{\s}}{.3}}$ & 59.6$_{{\scalebox{\s}}{.2}}$ & 77.5$_{{\scalebox{\s}}{.2}}$ & & 52.5$_{{\scalebox{\s}}{1.2}}$ & 50.6$_{{\scalebox{\s}}{.4}}$ & 28.8$_{{\scalebox{\s}}{.5}}$ & 33.7$_{{\scalebox{\s}}{.5}}$\ & ELMo\[0\] & & 92.2$_{{\scalebox{\s}}{.3}}$ & 80.5$_{{\scalebox{\s}}{1.0}}$ & 68.6$_{{\scalebox{\s}}{.4}}$ & 83.4$_{{\scalebox{\s}}{.4}}$ & & 91.6$_{{\scalebox{\s}}{.4}}$ & 69.6$_{{\scalebox{\s}}{1.0}}$ & 56.8$_{{\scalebox{\s}}{1.5}}$ & 75.0$_{{\scalebox{\s}}{1.0}}$ & & 51.9$_{{\scalebox{\s}}{1.1}}$ & 42.6$_{{\scalebox{\s}}{.9}}$ & 32.4$_{{\scalebox{\s}}{.3}}$ & 35.8$_{{\scalebox{\s}}{.4}}$\ & G + char & & 93.1$_{{\scalebox{\s}}{.3}}$ & 80.7$_{{\scalebox{\s}}{.9}}$ & 69.8$_{{\scalebox{\s}}{.7}}$ & 84.4$_{{\scalebox{\s}}{.4}}$ & & 91.8$_{{\scalebox{\s}}{.3}}$ & 69.3$_{{\scalebox{\s}}{.3}}$ & 55.6$_{{\scalebox{\s}}{1.1}}$ & 74.8$_{{\scalebox{\s}}{.5}}$ & & 50.6$_{{\scalebox{\s}}{.9}}$ & 42.5$_{{\scalebox{\s}}{1.4}}$ & 20.6$_{{\scalebox{\s}}{2.8}}$ & 28.7$_{{\scalebox{\s}}{2.5}}$\ & GloVe & & 92.2$_{{\scalebox{\s}}{.1}}$ & 77.0$_{{\scalebox{\s}}{.4}}$ & 61.7$_{{\scalebox{\s}}{.3}}$ & 81.5$_{{\scalebox{\s}}{.05}}$ & & 89.6$_{{\scalebox{\s}}{.3}}$ & 62.8$_{{\scalebox{\s}}{.6}}$ & 38.5$_{{\scalebox{\s}}{.4}}$ & 68.1$_{{\scalebox{\s}}{.4}}$ & & 46.8$_{{\scalebox{\s}}{.8}}$ & 41.3$_{{\scalebox{\s}}{.5}}$ & 3.2$_{{\scalebox{\s}}{.2}}$ & 18.9$_{{\scalebox{\s}}{.7}}$\ \[table:results\] General Observations {#general} -------------------- ### ELMo, BERT and Flair Drawing conclusions from the comparison of ELMo, BERT and Flair is difficult because there is no clear hierarchy accross datasets and they differ in dimensions, tokenization, contextualization levels and pretraining corpora. However, although BERT is particularly effective on the WNUT dataset in-domain, probably due to its subword tokenization, ELMo yields the most stable results in and out-of-domain. Furthermore, Flair globally underperforms ELMo and BERT, particularly for unseen mentions and out-of-domain. This suggests that LM pretraining at a lexical level (word or subword) is more robust for generalization than at a character level. In fact, Flair only beats the non contextual ELMo\[0\] baseline with Map-CRF which indicates that character-level contextualization is less beneficial than word-level contextualization with character-level representations. ### ELMo\[0\] vs GloVe+char Overall, ELMo\[0\] outperforms the GloVe+char baseline, particularly on unseen mentions, out-of-domain and on WNUT$^\ast$. The main difference is the incorporation of morphological features: in ELMo\[0\] they are learned jointly with the LM on a huge dataset whereas the char-BiLSTM is only trained on the source NER training set. Yet, morphology is crucial to represent words never encountered during pretraining and in WNUT$^\ast$ around 20% of words in test mentions are out of GloVe vocabulary against 5% in CoNLL03 and 3% in OntoNotes$^\ast$. This explains the poor performance of GloVe baselines on WNUT$^\ast$, all the more out-of-domain, and why a model trained on CoNLL03 with ELMo outperforms one trained on WNUT$^\ast$ with GloVe+char. Thus, ELMo’s improvement over previous state-of-the-art does not only stem from contextualization but also an effective non-contextual word representation. ### Seen Mentions Bias In every configuration, $F1_{exact} > F1_{partial} > F1_{new}$ with more than 10 points difference. This gap is wider out-of-domain where the context differs more from training data than in-domain. NER models thus poorly generalize to unseen mentions, and datasets with high lexical overlap only encourage this behavior. However, this generalization gap is reduced by two types of contextualization described hereafter. LM and NER Contextualizations {#analysis} ----------------------------- The ELMo\[0\] and Map-CRF baselines enable to strictly distinguish contextualization due to LM pretraining ($C_{LM}$: ELMo\[0\] to ELMo) from task supervised contextualization induced by the BiLSTM network ($C_{NER}$: Map to BiLSTM). In both cases, a BiLSTM incorporates syntactic information which improves generalization to unseen mentions for which context is decisive, as shown in Table \[table:results\]. ### Comparison However, because ${C_{NER}}$ is specific to the source dataset, it is more effective in-domain whereas $C_{LM}$ is particularly helpful out-of-domain. In the latter setting, the benefits from $C_{LM}$ even surpass those from ${C_{NER}}$, specifically on domains further from source data such as web text in OntoNotes$^\ast$ (see Table \[table:genres\]) or WNUT$^\ast$. This is again explained by the difference in quantity and quality of the corpora on which these contextualizations are learned. The much larger and more generic unlabeled corpora on which LM are pretrained lead to contextual representations more robust to domain adaptation than ${C_{NER}}$ learned on a small and specific NER corpus. Similar behaviors can be observed when comparing BERT and Flair to the GloVe baselines, although we cannot separate the effects of representation and contextualization. ### Complementarity Both in-domain and out-of-domain on OntoNotes$^\ast$, the two types of contextualization transfer complementary syntactic features leading to the best configuration. However, in the most difficult case of zero-shot domain adaptation from CoNLL03 to WNUT$^\ast$, ${C_{NER}}$ is detrimental with ELMo and BERT. This is probably due to the specificity of the target domain, excessively different from source data. [0.75]{}[@l\*[7]{}[Y]{}@]{} & bc & bn & nw & mz & tc & wb & All\ BERT & 87.2$_{{\scalebox{\s}}{.5}}$ & 88.4$_{{\scalebox{\s}}{.4}}$ & 84.7$_{{\scalebox{\s}}{.2}}$ & 82.4$_{{\scalebox{\s}}{1.2}}$ & 84.5$_{{\scalebox{\s}}{1.1}}$ & 79.5$_{{\scalebox{\s}}{1.0}}$ & **85.0**$_{{\scalebox{\s}}{.3}}$\ ELMo & 85.0$_{{\scalebox{\s}}{.6}}$ & 88.6$_{{\scalebox{\s}}{.3}}$ & 82.9$_{{\scalebox{\s}}{.3}}$ & 78.1$_{{\scalebox{\s}}{.7}}$ & 84.0$_{{\scalebox{\s}}{.8}}$ & 79.9$_{{\scalebox{\s}}{.5}}$ & 83.4$_{{\scalebox{\s}}{.2}}$\ Flair & 78.0$_{{\scalebox{\s}}{1.1}}$ & 86.5$_{{\scalebox{\s}}{.4}}$ & 80.4$_{{\scalebox{\s}}{.6}}$ & 71.1$_{{\scalebox{\s}}{.4}}$ & 73.5$_{{\scalebox{\s}}{1.8}}$ & 72.1$_{{\scalebox{\s}}{.8}}$ & 79.0$_{{\scalebox{\s}}{.5}}$\ ELMo\[0\] & 82.6$_{{\scalebox{\s}}{.5}}$ & 88.0$_{{\scalebox{\s}}{.3}}$ & 79.6$_{{\scalebox{\s}}{.5}}$ & 73.4$_{{\scalebox{\s}}{.6}}$ & 79.2$_{{\scalebox{\s}}{1.2}}$ & 75.1$_{{\scalebox{\s}}{.3}}$ & 80.5$_{{\scalebox{\s}}{.2}}$\ GloVe + char & 80.4$_{{\scalebox{\s}}{.8}}$ & 86.3$_{{\scalebox{\s}}{.4}}$ & 77.0$_{{\scalebox{\s}}{1.0}}$ & 70.7$_{{\scalebox{\s}}{.4}}$ & 79.7$_{{\scalebox{\s}}{1.8}}$ & 69.2$_{{\scalebox{\s}}{.8}}$ & 77.9$_{{\scalebox{\s}}{.5}}$\ \[table:genres\] Related Work ============ Augenstein et al. [@Augenstein2017GeneralisationAnalysis] perform a quantitative study of two CRF-based models and a CNN with classical word embeddings [@Collobert2011NaturalScratch] over seven NER datasets including CoNLL03 and OntoNotes. They separate performance on seen (*exact match*) and unseen mentions and show a drop in F1 on unseen mentions and out-of-domain. Although comprehensive in experiments, this analysis is limited to models dating back from 2005 to 2011. We use a similar experimental setting to draw new insights on state-of-the art architectures and word representations. We limit to the two main English NER benchmarks as well as WNUT which was specifically designed to tackle this generalization problem in the Twitter domain. These three datasets cover all the domains studied in [@Augenstein2017GeneralisationAnalysis]. Moosavi and Strube raise a similar lexical overlap issue in Coreference Resolution on the CoNLL2012 dataset. They first show that for out-of-domain evaluation the performance gap between Deep Learning models and a rule-based system fades away [@Moosavi2017LexicalCaution]. They then add linguistic features (such as gender, NER, POS...) to improve out-of-domain generalization [@Moosavi2018UsingResolvers]. Nevertheless, such features are obtained using models in turn based on lexical features and at least for NER the same lexical overlap issue arises. Finally, Pires et al. [@Pires2019HowBERT] concurrently evaluate the cross-lingual generalization capability of Multilingual BERT for NER and POS tagging. Our work on monolingual generalization to unseen mentions and domains naturally complements this study. Conclusion ========== NER benchmarks are biased towards seen mentions, at the opposite of real-life applications. Hence the necessity to disentangle performance on seen and unseen mentions and test out-of-domain. In such setting, we show that contextualization from LM pretraining is particularly beneficial for generalization to unseen mentions, all the more out-of-domain where it surpasses supervised contextualization. Despite this improvement, unseen mentions detection remains challenging and further work could explore attention or regularization mechanisms to better incorporate context and improve generalization. Furthermore, we can investigate how to best incorporate target data to improve this LM pretraining zero-shot domain adaptation baseline.

--- abstract: '[ The aim of this paper is to obtain a posteriori error bounds of optimal order in time and space for the linear second-order wave equation discretized by the Newmark scheme in time and the finite element method in space. Error estimate is derived in the $L^{\infty}$-in-time/energy-in-space norm. Numerical experiments are reported for several test cases and confirm equivalence of the proposed estimator and the true error. ]{} [ a posteriori error bounds in time and space, wave equation, Newmark scheme ]{}' author: - 'Olga Gorynina,[^1] Alexei Lozinski,[^2] and Marco Picasso[^3]' bibliography: - 'apost.bib' title: Time and space adaptivity of the wave equation discretized in time by a second order scheme --- Introduction ============ A posteriori error analysis of finite element approximations for partial differential equations plays an important role in mesh adaptivity techniques. The main aim of a posteriori error analysis is to obtain suitable error estimates computable using only the approximate solution given by the numerical method. The cases of elliptic and parabolic problems are well studied in the literature (for the parabolic case, we can cite, among many others [@ErikssonJohnson; @AMN; @LPP; @LakkisMakridakisPryer]). On the contrary, the a posteriori error analysis for hyperbolic equations of second order in time is much less developed. Some a posteriori bounds are proposed in [@BS; @Georgoulis13] for the wave equation using the Euler discretization in time, which is however known to be too diffusive and thus rarely used for the wave equation. More popular schemes, i.e. the leap-frog and cosine methods, are studied in [@Georgoulis16] but only the error caused by discretization in time is considered. On the other hand, error estimators for the space discretization only are proposed in [@Picasso10; @adjerid2002posteriori]. Goal-oriented error estimation and adaptivity for the wave equation were developed in [@bangerth2010adaptive; @bangerth2001adaptive; @bangerth1999finite]. The motivation of this work is to obtain a posteriori error estimates of optimal order in time and space for the fully discrete wave equation in energy norm discretized with the Newmark scheme in time (equivalent to a cosine method as presented in [@Georgoulis16]) and with finite elements in space. We adopt the particular choice for the parameters in the Newmark scheme, namely $\beta=1/4 $, $\gamma=1/2$. This choice of parameters is popular since it provides a conservative method with respect to the energy norm, cf. [@bathe1976numerical]. Another interesting feature of this variant of the method, which is in fact essential for our analysis, is the fact that the method can be reinterpreted as the Crank-Nicolson discretization of the reformulation of the governing equation in the first-order system, as in [@Baker76]. We are thus able to use the techniques stemming from a posteriori error analysis for the Crank-Nicolson discretization of the heat equation in [@LPP], based on a piecewise quadratic polynomial in time reconstruction of the numerical solution. This leads to optimal a posteriori error estimate in time and also allows us to easily recover the estimates in space. The resulting estimates are referred to as the $3$-point estimator since our quadratic reconstruction is drawn through the values of the discrete solution at 3 points in time. The reliability of 3-point estimator is proved theoretically for general regular meshes in space and non-uniform meshes in time. It is also illustrated by numerical experiments. We do not provide a proof of the optimality (efficiency) of our error estimators in space ans time. However, we are able to prove that the time estimator is of optimal order at least on sufficiently smooth solutions, quasi-uniform meshes in space and uniform meshes in time. The most interesting finding of this analysis is the crucial importance of the way in which the initial conditions are discretized (elliptic projections): a straightforward discretization, such as the nodal interpolation, may ruin the error estimators while providing quite acceptable numerical solution. Numerical experiments confirm these theoretical findings and demonstrate that our error estimators are of optimal order in space and time, even in situation not accessible to the current theory (non quasi-uniform meshes, not constant time steps). This gives us the hope that our estimators can be used to construct an adaptive algorithm in both time and space. The outline of the paper is as follows. We present the governing equations, the discretization and a priori error estimates in Section \[section2\]. In Section \[section3\], an a posteriori error estimate is derived and some considerations concerning the optimality of time estimators are given. Numerical results are analysed in Section \[section4\]. The Newmark scheme for the wave equation and a priori error analysis {#section2} ==================================================================== We consider initial boundary-value problem for the wave equation. Let $\Omega$ be a bounded domain in $\mathbb{R}^2$ with boundary $\partial \Omega$ and $T>0$ be a given final time. Let $u=u(x,t) : \Omega\times\left[0,T \right]\to\mathbb{R}$ be the solution to $$\begin{cases} \cfrac{\partial^2 u}{\partial t^2}-\Delta u=f,&\mbox{in}~ \Omega\times\left]0,T \right],\\ u=0,&\mbox{on}~ \partial\Omega\times\left]0,T \right],\\ u(\cdot,0)=u_0,&\mbox{in}~\Omega,\\ \cfrac{\partial u}{\partial t}(\cdot,0)=v_0,&\mbox{in}~\Omega, \end{cases} \label{wave}$$ where $f,u_0,v_0$ are given functions. Note that if we introduce the auxiliary unknown $v=\frac{\partial u}{\partial t}$ then model (\[wave\]) can be rewritten as the following first-order in time system $$\begin{cases} \cfrac{\partial u}{\partial t}-v=0, &\mbox{in}~ \Omega\times\left]0,T \right], \\ \cfrac{\partial v}{\partial t}-\Delta u=f, &\mbox{in}~ \Omega\times\left]0,T \right], \\ u=v=0,&\mbox{on}~ \partial\Omega\times\left]0,T \right] ,\\ u(\cdot,0)=u_0,~v(\cdot,0)=v_0,~&\mbox{in}~\Omega. \label{syst} \end{cases}$$ The above problem (\[wave\]) has the following weak formulation, cf. [@evans2010partial]: for given $f\in L^{2}(0,T;L^2(\Omega))$, $u_0\in H^1_0(\Omega)$ and $v_0\in L^2(\Omega)$ find a function $$u\in L^{2}\left(0,T;H^1_0(\Omega)\right),~\cfrac{\partial u}{\partial t}\in L^{2}\left(0,T;L^2(\Omega)\right),~\cfrac{\partial^2 u}{\partial t^2}\in L^{2}\left(0,T;H^{-1}(\Omega)\right)$$ such that $u(x,0)=u_0$ in $H^1_0(\Omega)$, $\cfrac{\partial u}{\partial t}(x,0)=v_0$ in $L^2(\Omega)$ and $$\left\langle\cfrac{\partial^2 u}{\partial t^2},\varphi\right\rangle+\left(\nabla u,\nabla \varphi\right)=\left(f,\varphi\right),~\forall\varphi \in H^1_0(\Omega), \label{weakwave}$$ where $\left\langle \cdot, \cdot \right\rangle$ denotes the duality pairing between $ H^{-1}(\Omega)$ and $ H^1_0(\Omega)$ and the parentheses $( \cdot, \cdot)$ stand for the inner product in $L^2 ( \Omega)$. Following Chap. 7, Sect. 2, Theorem 5 from [@evans2010partial], we observe that in fact $$u\in C^{0}\left(0,T;H^1_0(\Omega)\right),~\cfrac{\partial u}{\partial t}\in C^{0}\left(0,T;L^2(\Omega)\right),~\cfrac{\partial^2 u}{\partial t^2}\in C^{0}\left(0,T;H^{-1}(\Omega)\right).$$ Higher regularity results with more regular data are also available in [@evans2010partial]. Let us now discretize (\[wave\]) or, equivalently, (\[syst\]) in space using the finite element method and in time using an appropriate marching scheme. We thus introduce a regular mesh $\mathcal{T}_h$ on $\Omega$ with triangles $K$, $\mathrm{diam}~K=h_{K}$, $h=\max_{K\in{\mathcal{T}_h}}h_K$, internal edges $E\in\mathcal{E}_h$, where $\mathcal{E}_h$ represents the internal edges of the mesh $\mathcal{T}_h$ and the standard finite element space ${V}_h\subset H^1_0(\Omega ) $: $$V_h=\left\{v_h\in C(\bar\Omega):v_h|_K\in \mathbb{P}_1~\forall K\in\mathcal{T}_h \text{ and }v_h|_{\partial\Omega}=0\right\}.$$ Let us also introduce a subdivision of the time interval $[0,T]$ $$0=t_0<t_1<\dots<t_N=T$$ with time steps $\tau_n=t_{n+1}-t_n$ for $n=0,\ldots,N-1$ and $\tau=\displaystyle\max_{0 \leq n \leq N-1}\tau_n$ . Following [@Baker76], by applying Crank-Nicolson discretization to both equations in (\[syst\]) we get a second order in time scheme. The fully discretized method is as follows: taking $u^0_h,v^0_h\in V_h$ as some approximations to $u_0,v_0$ compute $u^n_h,v^n_h\in V_h$ for $n=0,\ldots,N-1$ from the system $$\begin{aligned} \label{CNh1} \frac{u_h^{n + 1} - u_h^n}{\tau_n} - \frac{v_h^n + v_h^{n+1}}{2} &= 0, \\ \label{CNh2} \left( \frac{v^{n + 1}_h - v^n_h}{\tau_n}, \varphi_h \right) + \left( \nabla \frac{u^{n + 1}_h + u^n_h}{2}, \nabla \varphi_h \right) &= \left( \frac{f^{n+1} + f^n}{2}, \varphi_h \right), \hspace{1em} \forall \varphi_h \in V_h.\end{aligned}$$ From here on, $f^n$ is an abbreviation for $f ( \cdot,t_n)$. Note that we can eliminate $v_h^n$ from (\[CNh1\])-(\[CNh2\]) and rewrite the scheme (\[CNh1\])-(\[CNh2\]) in terms of $u_h^n$ only. This results in the following method: given approximations $u^0_h, v^0_h \in V_h$ of $u_0, v_0$ compute $u^1_h \in V_h$ from $$\label{Newm1}\left( \frac{u^1_h - u^0_h}{\tau_0}, \varphi_h \right) + \left( \nabla\frac{\tau_0 (u^1_h + u^0_h)}{4}, \nabla \varphi_h \right) = \left( v_h^0 +\frac{\tau_0}{4} (f^1 + f^0), \varphi_h \right), \quad \forall\varphi_h \in V_h$$ and then compute $u^{n+1}_h \in V_h$ for $n = 1, \ldots, N-1$ from equation $$\begin{aligned} \left( \frac{u_h^{n + 1} - u_h^n}{\tau_n} - \frac{u_h^n - u_h^{n -1}}{\tau_{n - 1}}, \varphi_h \right) + \left( \nabla \frac{\tau_n (u_h^{n + 1}+ u_h^n) + \tau_{n - 1} (u_h^n + u_h^{n - 1})}{4}, \nabla \varphi_h \right)&\notag\\ =\left( \frac{\tau_n (f^{n + 1} + f^n) + \tau_{n - 1} (f^n + f^{n - 1})}{4},\varphi_h \right), \hspace{1em} \forall \varphi_h \in V_h.& \label{Newm2}\end{aligned}$$ This equation is derived by multiplying (\[CNh2\]) by $\tau_n/2$, doing the same at the previous time step, taking the sum of the two results and observing $$\frac{v_h^{n+1}-v_h^{n-1}}{2} =\frac{v_h^{n+1}-v_h^{n}}{2}+\frac{v_h^{n}-v_h^{n-1}}{2} =\frac{u_h^{n + 1} - u_h^n}{\tau_n} - \frac{u_h^n - u_h^{n -1}}{\tau_{n - 1}}$$ by (\[CNh1\]). We have thus recovered the Newmark scheme ([@newmark1959; @RT]) with coefficients $\beta = 1/4, \gamma = 1/2$ as applied to the wave equation (\[wave\]). Note that the presentation of this scheme in [[@newmark1959]]{} and in the subsequent literature on applications in structural mechanics is a little bit different, but the present form (\[Newm1\])-(\[Newm2\]) can be found, for example, in [[@RT]]{}. It is easy to see that for any $u^0_h, v^0_h \in V_h $, both schemes (\[CNh1\])-(\[CNh2\]) and (\[Newm1\])-(\[Newm2\]) provide the same unique solution $u^n_h, v^n_h \in V_h$ for $n = 1, \ldots, N$. In the case of scheme (\[Newm1\])-(\[Newm2\]), $v^n_h$ can be reconstructed from $u^n_h$ recursively with the formula $$\label{vhform} v_h^{n + 1} = 2 \frac{u_h^{n + 1} - u_h^n}{\tau_n} - v_h^n.$$ From now on, we shall use the following notations $$\begin{aligned} \label{notation} u_h^{n+1/2}& :=\frac{u_h^{n+1}+u_h^{n}}{2}, \quad \partial _{n+1/2}u_h:=\frac{u_h^{n+1}-u_h^{n}}{\tau_n}, \quad \partial _{n}u_h:=\frac{u_h^{n+1}-u_h^{n-1}}{\tau_n+\tau_{n-1}} \\ \notag \partial _{n}^{2}{u_h}&:=\frac{1}{\tau_{n-1/2}}\left(\frac{u_h^{n+1}-u_h^{n}}{\tau_n}-\frac{u_h^{n}-u_h^{n-1}}{\tau_{n-1}}\right) \text{ with } \tau_{n-1/2}:=\frac{\tau_n+\tau_{n-1}}{2}\end{aligned}$$ We apply this notations to all quantities indexed by a superscript, so that, for example, $f^{n+1/2}=({f^{n+1}+f^n})/{2}$. We also denote $u (x,t_n)$, $v(x,t_n)$ by $u^n$, $v^n$ so that, for example, $u^{n+1/2}=\left({u^{n+1}+u^n}\right)/{2}=\left(u(x,t_{n+1})+u(x,t_n)\right)/{2}$. We turn now to a priori error analysis for the scheme (\[CNh1\])-(\[CNh2\]). We shall measure the error in the following norm $$\label{EnNorm} u \mapsto \max_{t\in[0,T]}\left(\left\|\cfrac{\partial u}{\partial t} (t)\right\|_{L^2(\Omega)}^{2}+\left\vert u(t)\right\vert^{2}_{H^1(\Omega)}\right) ^{1/2}.$$ Here and in what follows, we use the notations $u(t)$ and $\cfrac{\partial u}{\partial t} (t)$ as a shorthand for, respectively, $u(\cdot,t)$ and $\cfrac{\partial u}{\partial t} (\cdot,t)$. The norms and semi-norms in Sobolev spaces $H^k(\Omega)$ are denoted, respectively, by $\|\cdot\|_{H^k(\Omega)}$ and $|\cdot|_{H^k(\Omega)}$. We call (\[EnNorm\]) the energy norm referring to the underlying physics of the studied phenomenon. Indeed, the first term in (\[EnNorm\]) may be assimilated to the kinetic energy and the second one to the potential energy. Note that a priori error estimates for scheme (\[CNh1\])-(\[CNh2\]) can be found in [@Baker76; @dupont19732; @RT]. We are going to construct a priori error estimates following the ideas of [@Baker76] but we measure the error in a different norm, namely the energy norm (\[EnNorm\]), and present the estimate in a slightly different manner, foreshadowing the upcoming a posteriori estimates. \[lemma\] Let $u$ be a smooth solution of the wave equation (\[wave\]) and $u_{h}^{n}$, $v_{h}^{n}$ be the discrete solution of the scheme (\[CNh1\])-(\[CNh2\]). If $u_0\in H^2(\Omega)$, $v_0\in H^1(\Omega)$ and the approximations to the initial conditions are chosen such that $\| v^0_h - v_0 \|_{L^2(\Omega)} \le Ch| v_0 |_{H^1(\Omega)} $ and $| u^0_h - u_0 |_{H^1(\Omega)} \le Ch | u_0 |_{H^2(\Omega)}$, then the following a priori error estimate holds $$\begin{gathered} \label{apriori} \max_{0\le n \le N}\left( \left\Vert v^n_h - \cfrac{\partial u}{\partial t} (t_n)\right\Vert_{L^2(\Omega)}^2 + | u^n_h - u (t_n) |^2_{H^1(\Omega)}\right)^{1/2} \\ \leq Ch\left(| v_0 |_{H^1(\Omega)} + | u_0 |_{H^2(\Omega)}\right) \\ + C\sum_{n = 0}^{N - 1} \tau^2_n \left( \int_{t_n}^{t_{n + 1}} \left\vert \cfrac{\partial^3 u }{\partial t^3} \right\vert_{H^1(\Omega)} {dt} + \int_{t_n}^{t_{n + 1}} \left\Vert \cfrac{\partial^4 u }{\partial t^4} \right\Vert_{L^2(\Omega)} {dt} \right) \\ + C h \left( \int_{t_0}^{t_N} \left\vert \cfrac{\partial^2 u }{\partial t^2} \right\vert_{H^1(\Omega)} {dt} + \sum_{n=0}^{N} \tau_n'\left\vert \cfrac{\partial u}{\partial t} (t_n)\right\vert_{H^2(\Omega)} + \left\vert \cfrac{\partial u }{\partial t}(t_N) \right\vert_{H^1(\Omega)} + | u ( t_N) |_{H^2(\Omega)} \right) \end{gathered}$$ with a constant $C>0$ depending only on the regularity of the mesh $\mathcal{T}_h$. We have set here $\tau_n'=\tau_{n-1/2}$ for $1<n<N-1$ and $\tau_0'=\tau_0$, $\tau_N'=\tau_N$. Let us introduce $e^n_u = u^n_h - \Pi_h u^n$ and $e^n_v = v^n_h - I_h v^n$ where $\Pi_h : H^1_0 (\Omega) \to V_h$ is the $H^1_0$-orthogonal projection operator, i.e. $$\label{Pih} \left(\nabla \Pi_h v, \nabla\varphi_h\right) = \left(\nabla v, \nabla\varphi_h\right), \hspace{1em} \forall v \in H^1_0 (\Omega),\hspace{1em}\forall \varphi_h \in V_h$$ and $\tilde I_h : H^1_0 (\Omega) \to V_h$ is a Cl[é]{}ment-type interpolation operator which is also a projection, i.e. $\tilde I_h=Id$ on $V_h$, cf. [@ErnGue; @ScoZh]. Let us recall, for future reference, the well known properties of these operators (see [@ErnGue]): for every sufficiently smooth function $v$ the following inequalities hold $$\label{Pinterp} | \Pi_h v |_{H^1(\Omega)}\leq | v |_{H^1(\Omega)}, \hspace{1em} | v - \Pi_h v |_{H^1(\Omega)} \leq Ch | v |_{H^2(\Omega)}$$ with a constant $C > 0$ which depends only on the regularity of the mesh. Moreover, for all $K\in{\mathcal{T}_h}$ and $E \in \mathcal{E}_h$ we have $$\label{Clement} \| v - \tilde I_h v \|_{L^2(K)} \leq Ch_K |v |_{H^1(\omega_K)} \text{ and } \| v - \tilde I_h v \|_{L^2(E)} \leq Ch_{E}^{1 / 2} | v |_{H^1(\omega_{E})}$$ Here $\omega_K$ (resp. $\omega_E$) represents the set of triangles of $\mathcal{T}_h$ having a common vertex with triangle $K$ (resp. edge $E$) and the constant $C > 0$ depends only on the regularity of the mesh. Observe that for $\varphi_h,\psi_h\in V_h$ the following equations hold $$\begin{aligned} \notag\left(\nabla\partial_{n + 1 / 2} e_u,\nabla\varphi_h\right) - \left(\nabla e_v^{n + 1 / 2},\nabla\varphi_h\right) &\\ = -&\left(\nabla \left( \partial_{n + 1 / 2} u - \tilde I_h v^{n + 1 / 2}\right),\nabla\varphi_h\right), \label{ErrEqApr1}\\ \left( \partial_{n + 1 / 2} e_v, \psi_h\right) + \left( \nabla e_u^{n + 1 / 2}, \nabla \psi_h\right) = & \left( \left( \cfrac{\partial^2 u}{\partial t^2}\right)^{n + 1 / 2} - \tilde I_h \left( \partial_{n + 1 / 2} v\right), \psi_h\right). \label{ErrEqApr2} \end{aligned}$$ The last equation is a direct consequence of (\[CNh2\]) together with the governing equation (\[wave\]) evaluated at times $t_n$ and $t_{n+1}$. In accordance with the conventions above, we have denoted here $$\left(\cfrac{\partial^2 u}{\partial t^2}\right)^{n + 1 / 2}:=\frac{1}{2}\left(\cfrac{\partial^2 u}{\partial t^2}(t_n)+\cfrac{\partial^2 u}{\partial t^2}(t_{n+1})\right)$$ Equation (\[ErrEqApr1\]) is obtained from (\[CNh1\]) taking the gradient of both sides, multiplying by $\nabla\varphi_h$ and integrating over $\Omega$. Putting $\varphi_h = e_u^{n + 1 / 2}$ and $\psi_h = e_v^{n + 1 / 2}$ and taking the sum of (\[ErrEqApr1\])–(\[ErrEqApr2\]) yields $$\begin{aligned} \label{ErrAprInterMed} \frac{| e^{n + 1}_u |_{H^1(\Omega)}^2 - | e^n_u |_{H^1(\Omega)}^2 + \| e^{n + 1}_v \|_{L^2(\Omega)}^2 - \| e^n_v \|_{L^2(\Omega)}^2}{2 \tau_n} = &- \left( \nabla R^n_1, \nabla e_u^{n + 1 / 2}\right) \\ \notag &+ \left(R^n_2, e_v^{n + 1 / 2}\right) \end{aligned}$$ with $$\begin{aligned} R^n_1 = \partial_{n + 1 / 2} u - \tilde I_hv^{n + 1 / 2} \text{ and } R^n_2 = \left( \cfrac{\partial^2 u}{\partial t^2}\right)^{n + 1 / 2} - \tilde I_h \left( \partial_{n + 1 / 2}v\right). \end{aligned}$$ Set $$E^n = \left(\left|e^n_u \right|_{H^1 (\Omega)}^2 +\left\|e^n_v \right\|_{L^2 (\Omega)}^2\right)^{1/2}$$ so that equality (\[ErrAprInterMed\]) with Cauchy-Schwarz inequality entails $$\frac{(E^{n + 1})^2 - (E^n)^2}{2 \tau_n} \leq \left(|R^n_1 |_{H^1 (\Omega)}^2 +\|R^n_2 \|_{L^2 (\Omega)}^2\right)^{1/2} \frac{E^{n + 1} + E^n}{2}$$ which implies $$E^{n + 1} - E^n \leq \tau_n\left(|R^n_1 |_{H^1 (\Omega)} +\|R^n_2 \|_{L^2 (\Omega)}\right).$$ Summing this over $n$ from 0 to $N - 1$ gives $$\begin{aligned} \label{sumeN} (|e^N_u |_{H^1 (\Omega)}^2 +\|e^N_v \|_{L^2 (\Omega)}^2)^{{1}/{2}} &\leq (|e^0_u |_{H^1 (\Omega)}^2 + \|e^0_v \|_{L^2 (\Omega)}^2)^{{1}/{2}}\\ \notag &+ \sum_{n = 0}^{N - 1} \tau_n (|R^n_1 |_{H^1 (\Omega)} +\|R^n_2 \|_{L^2 (\Omega)}) \end{aligned}$$ We have the following estimates for $R^n_1$ and $R^n_2$ $$\begin{aligned} \label{Rn1} |R^n_1 |_{H^1(\Omega)} &\leq C \tau_n \int_{t_n}^{t_{n + 1}} \left\vert \cfrac{\partial^3 u}{\partial t^3} \right\vert_{H^1(\Omega)} dt\\ \notag &+ Ch\left( \left\vert \cfrac{\partial u}{\partial t} \left(t^n\right) \right\vert_{H^2(\Omega)} + \left\vert \cfrac{\partial u}{\partial t} \left(t^{n+1}\right) \right\vert_{H^2(\Omega)} \right)\\ \label{Rn2}\|R^n_2 \|_{L^2(\Omega)} &\leq C \tau_n \int_{t_n}^{t_{n + 1}}\left\Vert \cfrac{\partial^4 u}{\partial t^4} \right\Vert_{L^2(\Omega)} dt + C \frac{h}{\tau_n} \int_{t_n}^{t_{n + 1}} \left\vert \cfrac{\partial^2 u}{\partial t^2} \right\vert_{H^1(\Omega)} dt\end{aligned}$$ The proof of (\[Rn1\])–(\[Rn2\]) is quite standard, but tedious. For brevity, we provide here only the proof of estimate (\[Rn2\]): we rewrite the definition of $R^n_2$ recalling that $v = \partial u/\partial t$ and using the Taylor expansion around $t=t_{n+1/2 }$ as follows $$\begin{aligned} R^n_2 &= \frac 12\left(\cfrac{\partial^2 u}{\partial t^2} ( t_{n+1}) + \cfrac{\partial^2 u}{\partial t^2} ( t_{n})\right) - \frac {1}{\tau_n}\left(\cfrac{\partial u}{\partial t} ( t_{n+1}) - \cfrac{\partial u}{\partial t} ( t_{n})\right)\\ & + \frac {1}{\tau_n}\left( I - \tilde I_h\right) \left(\cfrac{\partial u}{\partial t} ( t_{n+1}) - \cfrac{\partial u}{\partial t} ( t_{n})\right) = \int_{t_{n + 1 / 2}}^{t_{n + 1}} \left( \frac{t_{n + 1} - t}{2} - \frac{( t_{n + 1} - t)^2}{2 \tau_n} \right)\cfrac{\partial^4 u}{\partial t^4} dt \\ & - \int_{t_n}^{t_{n + 1 / 2}} \left( \frac{t_n - t}{2} + \frac{( t_n - t)^2}{2 \tau_n} \right) \cfrac{\partial^4 u}{\partial t^4}dt + \frac{1}{\tau_n} ( I - \tilde I_h) \int_{t_n}^{t_{n + 1}} \cfrac{\partial^2 u}{\partial t^2} dt.\end{aligned}$$ Taking the $L^2(\Omega)$ norm on both sides and applying the projection error estimate (\[Pinterp\]) in $L^2(\Omega)$ we obtain (\[Rn2\]). Substituting (\[Rn1\])–(\[Rn2\]) into (\[sumeN\]) yields $$\begin{aligned} \left(\left|e^N_u \right|_{H^1(\Omega)}^2 +\left\|e^N_v\right\|_{L^2(\Omega)}^2\right)&^{1/2} \leq \left(\left|e^0_u\right|_{H^1(\Omega)}^2 +\left\|e^0_v \right\|_{L^2(\Omega)}^2\right)^{1/2} \\ &+ C \sum_{n = 0}^{N - 1} \tau^2_n \left( \int_{t_n}^{t_{n + 1}} \left\vert \cfrac{\partial^3 u}{\partial t^3} \right\vert_{H^1(\Omega)} dt + \int_{t_n}^{t_{n + 1}} \left\Vert \cfrac{\partial^4 u}{\partial t^4} \right\Vert_{L^2(\Omega)} dt \right) \\ & + Ch \int_0^{t_N} \left\vert \cfrac{\partial^2 u}{\partial t^2} \right\vert_{H^1(\Omega)} dt + Ch \sum_{n=0}^{N} \tau_n'\left\vert \cfrac{\partial u}{\partial t} (t_n)\right\vert_{H^2(\Omega)}. \end{aligned}$$ Applying the triangle inequality and estimate (\[Pinterp\]) in the above inequality we get $$\begin{aligned} \left(\left\Vert v^N_h - \cfrac{\partial u}{\partial t} (t_N) \right\Vert_{L^2(\Omega)}^2 + \left|u^N_h - u (t_N) \right|^2_{H^1(\Omega)}\right)^{1/2} \leq \left(\left|e^N_u\right|_{H^1(\Omega)}^2 +\left\|e^N_v \right\|_{L^2(\Omega)}^2\right)^{1/2}& \notag\\+ \left(\left\Vert \left(I-\tilde I_h\right) \frac{\partial u}{\partial t} (t_N)\right\Vert_{L^2(\Omega)}^2 + \left\vert \left(I-\Pi_h\right) u (t_N) \right\vert^2_{H^1(\Omega)}\right)^{1/2}&\end{aligned}$$ which implies (\[apriori\]) since we can safely assume that the maximum of the error in (\[apriori\]) is attained at the final time $t_N$ (if not, it suffices to redeclare the time where the maximum is attained as $t_N$). Estimate (\[apriori\]) is of order $h$ in space which is due to the the presence of $H^1$ term in the norm in which we measure the error. One sees easily that essentially the proof above gives the estimate of order $h^2$, multiplied by the norms of the exact solution in more regular spaces, if the target norm is changed to $\displaystyle \max_{0 \leq n \leq N}\left\Vert v^n_h - \cfrac{\partial u}{\partial t} (t_n) \right\Vert_{L^2(\Omega)}$. One would rely then on the estimate $$\left\| v - \Pi_h v \right\|_{L^2(\Omega)} \leq Ch^2 | v |_{H^2(\Omega)}$$ for the orthogonal projection error and one would obtain $$\begin{aligned} \label{apriori2} \left\Vert v^N_h - \cfrac{\partial u}{\partial t} (t_N) \right\Vert_{L^2(\Omega)} & \leq \left\| v^0_h - v_0 \right\|_{L^2(\Omega)}^2 +Ch^2\left| v_0 \right|_{H^2(\Omega)}\\ \notag & + \sum_{n = 0}^{N - 1} \tau^2_n \left( \int_{t_n}^{t_{n + 1}} \left\vert \frac{\partial^3 u}{\partial t^3} \right\vert_{H^1(\Omega)} {dt} + \int_{t_n}^{t_{n + 1}} \left\Vert \frac{\partial^4 u}{\partial t^4} \right\Vert_{L^2(\Omega)} {dt} \right) \\ \notag & + C h^2 \left( \int_{t_0}^{t_N} \left\vert \frac{\partial^2 u}{\partial t^2} \right\vert_{H^2(\Omega)} {dt} + \left\vert \frac{\partial u}{\partial t}(t_N) \right\vert_{H^2(\Omega)} \right) \end{aligned}$$ A posteriori error estimates for the wave equation in the “energy” norm {#section3} ======================================================================= Our aim here is to derive a posteriori bounds in time and space for the error measured in the norm (\[EnNorm\]). We discuss some considerations about upper bound for $3$-point time estimator. A 3-point estimator: an upper bound for the error ------------------------------------------------- The basic technical tool in deriving time error estimator is the piecewise quadratic (in time) reconstruction of the discrete solution, already used in [@LPP] in a similar context. \[QuadRec\] Let $u^n_h$ be the discrete solution given by the scheme (\[Newm2\]). Then, the piecewise quadratic reconstruction $\tilde{u}_{h\tau} (t) : [0, T] \rightarrow V_h$ is constructed as the continuous in time function that is equal on $[t_n, t_{n + 1}]$, $n\ge 1$, to the quadratic polynomial in $t$ that coincides with $u^{n + 1}_h$ (respectively $u^n_h$, $u^{n - 1}_h$) at time $t_{n+1}$ (respectively $t_n$, $t_{n - 1}$). Moreover, $\tilde{u}_{h\tau} (t)$ is defined on $[t_0, t_{1}]$ as the quadratic polynomial in $t$ that coincides with $u^{2}_h$ (respectively $u^1_h$, $u^{0}_h$) at time $t_{2}$ (respectively $t_1$, $t_{0}$). Similarly, we introduce piecewise quadratic reconstruction $\tilde{v}_{h\tau} (t) : [0, T] \rightarrow V_h$ based on $v^n_h$ defined by (\[vhform\]) and $\tilde{f}_{\tau} (t) : [0, T] \rightarrow L^2(\Omega)$ based on $f(t_n,\cdot)$. Our quadratic reconstructions $\tilde{u}_{h\tau}$, $\tilde{v}_{h\tau}$ are thus based on three points in time (normally looking backwards in time, with the exemption of the initial time slab $[t_0,t_1]$). This is why the error estimator derived in the following theorem using Definition \[QuadRec\] will be referred to as the $3$-point estimator. \[lemest3\] The following a posteriori error estimate holds between the solution $u$ of the wave equation (\[wave\]) and the discrete solution $u_h^n$ given by (\[Newm1\])–(\[Newm2\]) for all $ t_n,~0\leq n\leq N$ with $v_h^n$ given by (\[vhform\]): $$\begin{gathered} \left(\left\Vert v^{n}_h- \cfrac{\partial u}{\partial t} (t_n)\right\Vert_{L^2(\Omega)} ^{2}+\left\vert u^{n}_h-u(t_{n})\right\vert ^{2}_{H^1(\Omega)}\right) ^{1/2}\\ \leq\left(\left\Vert v^{0}_h-v_0\right\Vert_{L^2(\Omega)} ^{2}+\left\vert u^{0}_h-u_0\right\vert ^{2}_{H^1(\Omega)}\right) ^{1/2} \\ +\eta _{S}(t_{N})+\sum_{k=0}^{N-1}\tau_k\eta _{T}(t_{k}) +\int_0^{t_n} \|f-\tilde{f}_\tau\|_{L^2(\Omega)}dt \label{estf}\end{gathered}$$ where the space indicator is defined by $$\begin{aligned} \eta_S (t_k) &= C_1 \max_{0 \leqslant t \leqslant t_k} \Biggl[ \sum_{K \in \mathcal{T}_h} h_K^2 \left\Vert \frac{\partial \tilde{v}_{h \tau}}{\partial t} - \Delta \tilde{u}_{h \tau}-f \right\Vert_{L^2(K)}^2 + \sum_{ E \in \mathcal{E}_h}h_{E} \left|\left[n \cdot \nabla \tilde{u}_{h \tau}\right]\right|_{L^2(E)}^2 \Biggl]^{1/2} \notag \\ \notag &+ C_2\sum_{m = 0}^{k-1} \int_{t_m}^{t_{m + 1}} \Biggl[ \sum_{K \in \mathcal{T}_h} h_K^2 \left\Vert \frac{\partial^2 \tilde{v}_{h \tau}}{\partial t^2} - \Delta \frac{\partial \tilde{u}_{h\tau}}{\partial t} -\frac{\partial{f}}{\partial{t}}\right\Vert_{L^2(K)}^2 + \sum_{ E \in \mathcal{E}_h}h_{E} \left\Vert\left[n \cdot \nabla \frac{\partial \tilde{u}_{h\tau}}{\partial t}\right]\right\Vert_{L^2(E)}^2 \Biggr]^{1/2}dt \notag \\ & + C_3\sum_{m = 1}^{k-1} {\tau_{m - 1}} \left[ \sum_{K \in \mathcal{T}_h} h_K^2 \left\Vert \partial_m^2 v_h - \partial_{m - 1}^2 v_h \right\Vert_{L^2(K)}^2 \right]^{1/2}\label{space}\end{aligned}$$ here $C_1,~C_2,~C_3$ are constants depending only on the mesh regularity, $[\cdot]$ stands for a jump on an edge $E\in\mathcal{E}_h$, and $\tilde{u}_{h\tau}$, $\tilde{v}_{h\tau}$ are given by Definition \[QuadRec\]. The error indicator in time for $k=1,\dots,N-1$ is $$\label{in} \eta_T (t_k)=\left(\frac{1}{12}\tau_{k}^2+\frac{1}{8}\tau_{k-1}\tau_{k}\right)\left(\left\vert\partial _{k}^{2}{v_h}\right\vert_{H^1(\Omega)} + \left\Vert \partial _{k}^{2}{f_h} - z_h^k\right\Vert_{L^2(\Omega)}^2\right)^{1/2}$$ where $z^k_h$ is such that $$\label{zh} \left(z_h^k, \varphi_h\right) =(\nabla \partial _{k}^{2}{u_h}, \nabla\varphi_h), \quad\forall\varphi_h\in V_h$$ and $$\label{in0step} \eta_T (t_0)=\left(\frac{5}{12}\tau_{0}^2+\frac{1}{2}\tau_{1}\tau_{0}\right)\left(\left\vert\partial _{1}^{2}{v_h}\right\vert_{H^1(\Omega)} + \left\Vert \partial _{1}^{2}{f_h} - z^1_h\right\Vert_{L^2(\Omega)}^2\right)^{1/2}$$ In the following, we adopt the vector notation $U (t, x) =\begin{pmatrix} u (t, x)\\ v (t, x) \end{pmatrix}$ where $v = {\partial u}/{\partial t}$. Note that the first equation in (\[syst\]) implies that $$\left(\nabla \cfrac{\partial u}{\partial t}, \nabla \varphi\right) - (\nabla v, \nabla \varphi) = 0, \quad \forall \varphi \in H^1_0 (\Omega)$$ by taking its gradient, multiplying it by $\nabla \varphi$ and integrating over $\Omega$. Thus, system (\[syst\]) can be rewritten in the vector notations as $${b} \left(\cfrac{\partial U}{\partial t}, \Phi\right) + \left(\mathcal{A} \nabla U, \nabla \Phi\right) = {b} (F,\Phi), \quad \forall \Phi \in (H^1_0 (\Omega))^2 \label{ODEf}$$ where $\mathcal{A} = \begin{pmatrix} 0 &- 1\\ 1&~0 \end{pmatrix}$, $F =\begin{pmatrix} 0\\ f \end{pmatrix}$ and $${b} ( U, \Phi) = {b}\left(\begin{pmatrix}u\\ v\end{pmatrix}, \begin{pmatrix} \varphi\\ \psi\end{pmatrix} \right) := (\nabla u, \nabla \varphi) + (v, \psi)$$ Similarly, Newmark scheme (\[CNh1\])–(\[CNh2\]) can be rewritten as $${b} \left( \frac{U_h^{n + 1} - U_h^n}{\tau_n}, \Phi_h \right) + \left( \mathcal{A} \nabla \frac{U_h^{n + 1} + U_h^n}{2}, \nabla \Phi_h \right) = {b} \left(F^{n+1/2}, \Phi_h\right), \hspace{1em} \forall \Phi_h \in V_h^2 \label{vectorScheme}$$ where $U_h^n = \begin{pmatrix} u_h^n\\ v_h^n \end{pmatrix}$ and $F^{n+1/2} = \begin{pmatrix} 0\\ f^{n + 1/2} \end{pmatrix}$. The a posteriori analysis relies on an appropriate residual equation for the quadratic reconstruction $\tilde{U}_{h\tau}=\begin{pmatrix} \tilde{u}_{h\tau}\\ \tilde{v}_{h\tau}\end{pmatrix}$. We have thus for $t \in [t_n,t_{n + 1}]$, $n = 1, \ldots, N-1$ $$\tilde{U}_{h\tau} (t) = U^{n + 1}_h + (t - t_{n + 1}) \partial_{n + 1/2} U_h + \frac{1}{2} (t - t_{n + 1}) (t - t_n) \partial_n^2 U_h$$ so that, after some simplifications, $$\begin{gathered} \label{Uode1} {b} \left( \frac{\partial \tilde{U}_{h\tau}}{\partial t}, \Phi_h \right) + (\mathcal{A} \nabla \tilde{U}_{h\tau}, \nabla \Phi_h) ={b} \left( (t - t_{n + 1/2}) \partial_n^2 U_h + F^{n+1/2}, \Phi_h \right) \\ + \left( (t - t_{n + 1/2}) \mathcal{A} \nabla \partial_{n + 1/2} U_h + \frac{1}{2} (t - t_{n + 1}) (t - t_n) \mathcal{A} \nabla \partial_n^2 U_h, \nabla \Phi_h \right)\end{gathered}$$ Consider now (\[vectorScheme\]) at time steps $n$ and $n - 1$. Subtracting one from another and dividing by $\tau_{n - 1/2}$ yields $${b} \left(\partial_n^2 U_h, \Phi_h\right) + \left(\mathcal{A} \nabla \partial_n U_h, \nabla \Phi_h\right) = {b}\left( \partial_n F, \Phi_h \right)$$ or $${b} \left(\partial_n^2 U_h, \Phi_h\right) + \left(\mathcal{A} \nabla \left( \partial_{n + 1/2} U_h - \frac{\tau_{n - 1}}{2} \partial_n^2 U_h \right), \nabla \Phi_h \right) = {b}\left(\partial_n F, \Phi_h \right)$$ so that (\[Uode1\]) simplifies to $$\begin{gathered} \label{Uode2} {b} \left(\frac{\partial \tilde{U}_{h\tau}}{\partial t}, \Phi_h \right) + \left(\mathcal{A} \nabla \tilde{U}_{h\tau}, \Phi_h\right) \\ = \left(p_n \mathcal{A} \nabla \partial_n^2 U_h, \nabla \Phi_h\right) + {b}\left( \left(t - t_{n + 1/2}\right) \partial_n F + F^{n+1/2}, \Phi_h \right) \\ = \left(p_n \mathcal{A} \nabla \partial_n^2 U_h, \nabla \Phi_h\right) + {b}\left( \tilde{F}_\tau - p_n \partial^2_n F, \Phi_h\right) \end{gathered}$$ where $$\begin{aligned} p_n &= \frac{\tau_{n - 1}}{2} (t - t_{n + 1/2}) + \frac{1}{2} (t - t_{n + 1}) (t - t_n), \\ \tilde{F}_{\tau} (t) &= F^{n + 1}_h + (t - t_{n + 1}) \partial_{n + 1/2} F + \frac{1}{2} (t - t_{n + 1}) (t - t_n) \partial_n^2 F.\end{aligned}$$ Introduce the error between reconstruction $\tilde{U}_{h\tau}$ and solution $U$ to problem (\[ODEf\]) : $$E = \tilde{U}_{h\tau} - U$$ or, component-wise $$E = \begin{pmatrix} E_u\\ E_v \end{pmatrix} = \begin{pmatrix} \tilde{u}_{h\tau} - u\\ \tilde{v}_{h\tau} - v \end{pmatrix}$$ Taking the difference between (\[Uode2\]) and (\[ODEf\]) we obtain the residual differential equation for the error valid for $t \in [t_n, t_{n + 1}]$, $n = 1, \ldots, N-1$ $$\begin{aligned} \label{errode}{b}({\partial}_{t}E,{\Phi})+(\mathcal{A}{\nabla} E,{\nabla} {\Phi}) &={b} \left(\cfrac{{\partial} \tilde{U}_{{\tau} h}}{\partial t}-F,{\Phi}-{\Phi}_{h}\right)+\left(\mathcal{A}{\nabla} \tilde{U}_{{\tau} h},{\nabla} ({\Phi}-{\Phi}_{h})\right)\\ \notag +\left(p_n \mathcal{A}{\nabla} {\partial}_{n}^{2} U_{h},{\nabla} {\Phi}_{h}\right) &+{b}\left( \tilde{F}_\tau -F - p_n \partial^2_n F, \Phi_h\right), \hspace{1em} \forall \Phi_h \in V_h^2 \end{aligned}$$ Now we take $\Phi = E$, $\Phi_h = \begin{pmatrix} \Pi_h E_u\\ \tilde I_h E_v \end{pmatrix}$ where $\Pi_h : H^1_0 (\Omega) \to V_h$ is the $H^1_0$-orthogonal projection operator (\[Pih\]) and $\tilde I_h : H^1_0 (\Omega) \to V_h$ is a Cl[é]{}ment-type interpolation operator satisfying $\tilde I_h=Id$ on $V_h$ and (\[Clement\]). Noting that $( \mathcal{A} \nabla E, \nabla E) = 0$ and $$\left( \nabla \cfrac{\partial \tilde{u}_{h\tau}}{\partial t}, \nabla ( E_u - \Pi_h E_u)\right) = \left( \nabla \tilde{v}_{h\tau}, \nabla \left( E_u - \Pi_h E_u\right)\right) = 0$$ Introducing operator $A_h:V_h\to V_h$ such that $$\label{Ah} \left(A_h w_h, \varphi_h\right) =(\nabla w_h, \nabla\varphi_h), \quad\forall\varphi_h\in V_h$$ we get $$\begin{aligned} \left(\cfrac{{\partial} E_{v}}{\partial t},E_{v}\right)+\left({\nabla} E_{u},{\nabla} \cfrac{{\partial} E_{u}}{\partial t}\right)=\left(\cfrac{{\partial}\tilde{v}_{{\tau} h}}{\partial t}-f,E_{v}-{\Pi}_{h} E_{v}\right)+\left({\nabla} \tilde{u}_{{\tau}h},{\nabla}\left(E_{v}-\tilde I_{h} E_{v}\right)\right)\\ + \left(p_n \left(A_h {\partial}_{n}^{2} u_{h}-{\partial}_{n}^{2}f_h\right),\tilde I_{h} E_{v}\right)-\left(p_n {\nabla}{\partial}_{n}^{2} v_{h},{\nabla}E_{u}\right) +\left(\tilde{f}_\tau-f,\tilde I_{h} E_{v}\right).\end{aligned}$$ Note that equation similar to (\[errode\]) also holds for $t \in [t_0, t_{1}]$ $$\begin{aligned} \label{errorode1st}{b}({\partial}_{t}E,{\Phi})+(\mathcal{A}{\nabla} E,{\nabla} {\Phi}) &={b} \left(\cfrac{{\partial} \tilde{U}_{{\tau} h}}{\partial t}-F,{\Phi}-{\Phi}_{h}\right)+\left(\mathcal{A}{\nabla} \tilde{U}_{{\tau} h},{\nabla} ({\Phi}-{\Phi}_{h})\right)\\ \notag &+\left(p_1 \mathcal{A}{\nabla} {\partial}_{1}^{2} U_{h},{\nabla} {\Phi}_{h}\right) + {b}\left( \tilde{F}_\tau -F - p_1 \partial^2_1 F, \Phi_h\right). \end{aligned}$$ That follows from the definition of the piecewise quadratic reconstruction $\tilde{u}_{h\tau} (t)$ for $t \in [t_0, t_{1}]$. Integrating (\[errode\]) and (\[errorode1st\]) in time from 0 to some $t^{\ast}\geq t_1$ yields $$\begin{aligned} \notag & {\frac{1}{2}} \left(|E_{u}|^{2}_{H^1(\Omega)}+\| E_{v}\|_{L^2(\Omega)}^{2}\right)(t^{{\ast}}) \\ &= {\frac{1}{2}} \left(|E_{u}|^{2}_{H^1(\Omega)}+\| E_{v}\|_{L^2(\Omega)}^{2}\right)(0) \notag\\ &+ \int_{0}^{t^{{\ast}}}\left(\cfrac{{\partial\tilde{v}_{{\tau} h}}}{\partial t}-f, E_{v}-\tilde I_{h} E_{v}\right) d t +\int_{0}^{t^{{\ast}}}\left({\nabla} \tilde{u}_{{\tau} h},{\nabla} ( E_{v}-\tilde I_{h} E_{v})\right) d t \notag \\ &+\int_{t_1}^{t^{{\ast}}}\left[\left(p_n \left(A_h {\partial}_{n}^{2} u_{h}-{\partial}_{n}^{2}f_h\right),\tilde I_{h} E_{v}\right)-\left(p_n {\nabla}{\partial}_{n}^{2} v_{h},{\nabla}E_{u}\right) +\left(\tilde{f}_\tau-f,\tilde I_{h} E_{v}\right)\right] d t \notag\\ &+\int_{0}^{t_1}\left[\left(p_1 \left(A_h {\partial}_{1}^{2} u_{h}-{\partial}_{1}^{2}f_h\right),\tilde I_{h} E_{v}\right)-\left(p_1 {\nabla}{\partial}_{1}^{2} v_{h},{\nabla}E_{u}\right) +\left(\tilde{f}_\tau-f,\tilde I_{h} E_{v}\right)\right] d t \notag\\ &\hspace{1cm}:=I+I I+I I I+IV .\notag \\ \label{4newterms} &\end{aligned}$$ Let $$Z (t) = \sqrt{| E_u |_{H^1(\Omega)}^2 + \| E_v \|_{L^2(\Omega)}^2}$$ and assume that $t^{\ast}$ is the point in time where $Z$ attains its maximum and $t^{\ast} \in (t_n, t_{n + 1}]$ for some $n$. Observe $$(I-\tilde I_h)E_v = (I-\tilde I_h)(\tilde{v}_{h\tau}-v ) = (I-\tilde I_h)\left(\cfrac{\partial\tilde{u}_{h\tau}}{\partial t}-\cfrac{\partial u}{\partial t} \right) = \cfrac{\partial}{\partial t}(I-\tilde I_h)E_{u}$$ since $( I - \tilde I_h) \varphi_h =0$ for any $\varphi_h\in V_h$. We thus get for the first and second terms in (\[4newterms\]) $$\begin{aligned} I + I I & =\int_{0}^{t^{{\ast}}}\left(\cfrac{{{\partial } \tilde{v}_{{\tau} h}}}{\partial t}-f,\cfrac{\partial}{\partial t} (E_{u}-\tilde I_{h} E_{u})\right) d t+\int_{0}^{t^{{\ast}}}\left({\nabla} \tilde{u}_{{\tau} h},\cfrac{\partial}{\partial t} {\nabla} (E_{u}-\tilde I_{h} E_{u})\right) dt.\end{aligned}$$ We now integrate by parts with respect to time in the two integrals above. Let us do it for the first term: $$\begin{aligned} \nonumber & \int_{0}^{t^{{\ast}}}\left(\cfrac{{{\partial } \tilde{v}_{{\tau} h}}}{\partial t}-f,\cfrac{\partial}{\partial t} (E_{u}-\tilde I_{h} E_{u})\right) d t \\ \nonumber & = \sum_{m = 0}^n \int_{t_m}^{\min (t_{m + 1}, t^{\ast})}\left(\cfrac{{{\partial } \tilde{v}_{{\tau} h}}}{\partial t}-f,\cfrac{\partial}{\partial t} (E_{u}-\tilde I_{h} E_{u})\right) d t \\ \nonumber & = \left( \cfrac{{{\partial } \tilde{v}_{{\tau} h}}}{\partial t}-f, E_u -\tilde I_h E_u \right) (t^{\ast}) - \sum_{m = 1}^n \left( \left[ \cfrac{{{\partial } \tilde{v}_{{\tau} h}}}{\partial t} \right]_{t_m}, ( E_u - \tilde I_h E_u) (t_n) \right) \\ & - \sum_{m = 0}^n \int_{t_m}^{\min (t_{m + 1}, t^{\ast})} \left( \cfrac{{{\partial^2 } \tilde{v}_{{\tau} h}}}{\partial t^2}-\cfrac{\partial {f}}{\partial t}, E_u -\tilde I_h E_u \right) dt .\end{aligned}$$ Here $[\cdot]_{t_n}$ denotes the jump with respect to time, i.e. $$[w]_{t_n} = \lim_{t \rightarrow t_n^+} w ( t) - \lim_{t \rightarrow t_n^-} w ( t).$$ Using the same trick in the other term we can finally write $$\begin{aligned} \nonumber I +I I & =\left( \cfrac{{{\partial } \tilde{v}_{{\tau} h}}}{\partial t}-f, E_u - \tilde I_h E_u \right) (t^{\ast})+\left({\nabla} \tilde{u}_{{\tau} h},{\nabla} ( E_{u}-\tilde I_{h} E_{u})\right)(t^{{\ast}})\\ \nonumber & -\sum_{m = 1}^n \left( \left[ \cfrac{{{\partial } \tilde{v}_{{\tau} h}}}{\partial t} \right]_{t_m}, ( E_u - \tilde I_h E_u) (t_n) \right)\\ \nonumber & - \sum_{m = 0}^n \int_{t_m}^{\min (t_{m + 1}, t^{\ast})} \left( \cfrac{{{\partial^2 } \tilde{v}_{{\tau} h}}}{\partial t^2}-\cfrac{\partial {f}}{\partial t}, E_u -\tilde I_h E_u \right) dt\\&-\sum_{m=0}^{n}\int_{t_{m}}^{min (t_{m+1},t^{{\ast}})}\left({\nabla} \cfrac{{{\partial } \tilde{u}_{{\tau} h}}}{\partial t},{\nabla} ( E_{u}-\tilde I_{h} E_{u})\right) d t.\end{aligned}$$ We have used here a simple expression for the jump of time of ${\partial \tilde{v}_{h\tau}}/\partial t$ $$\left[ \cfrac{\partial \tilde{v}_{h\tau}}{\partial t} \right]_{t_n} = {\tau_{n - 1}}{2} (\partial_n^2 v_h - \partial_{n - 1}^2 v_h)$$ and noted that $\tilde{u}_{h\tau}$ is continuous in time. Integration by parts element by element over $\Omega$ and interpolation estimates (\[Clement\]) yield $$\begin{aligned} I+II &\leq C_{1}\Biggl[\sum_{K{\in}\mathcal{T}_h}h_{K}^{2}\left\Vert\cfrac{{\partial}\tilde{v}_{h{\tau}}}{\partial t}-{\Delta}\tilde{u}_{h{\tau}}-f\right\Vert_{L^2( K)}^{2}\\ &~~~~~~~~~~~~~~~~~~~~~~~~~~~~+\sum_{ E \in \mathcal{E}_h}h_{E}\left\|[n{\cdot}{\nabla}\tilde{u}_{h{\tau}}]\right\|_{L^2(E)}^{2}\Biggr]^{1/2}(t^{{\ast}})|E_{u}|_{H^1(\Omega)}(t^{{\ast}}) \\ &+C_{1}\Biggl[\sum_{K{\in}\mathcal{T}_h}h_{K}^{2}\left\Vert\cfrac{{\partial}\tilde{v}_{h{\tau}}}{\partial t}-{\Delta}\tilde{u}_{h{\tau}}-f\right\Vert_{L^2(K)}^{2}\\ &~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~+\sum_{ E \in \mathcal{E}_h}h_{E}\left\|[n{\cdot}{\nabla}\tilde{u}_{h{\tau}}]\right\|_{L^2(E)}^{2}\Biggl]^{1/2}(0)|E_{u}|_{H^1(\Omega)}(0)\\ & +C_{2}\sum_{m=1}^{n}\frac{{\tau}_{m-1}}{2}\left[\sum_{K{\in}\mathcal{T}_h}h_{K}^{2}\left\|{\partial}_{m}^{2}v_{h}-{\partial}_{m-1}^{2}v_{h}\right\|_{L^2( K)}^{2}\right]^{1/2}|E_{u}|_{H^1(\Omega)}(t_{m})\\ & +C_{3}\sum_{m=0}^{n}\int_{t_{m}}^{min(t_{m+1},t^{{\ast}})}\Biggl[\sum_{K{\in}\mathcal{T}_h}h_{K}^{2}\left\Vert\cfrac{{\partial}^{2}\tilde{v}_{h{\tau}}}{\partial t^2}-{\Delta}\cfrac{{\partial}\tilde{u}_{{\tau}h}}{\partial t}-\cfrac{\partial f}{\partial t}\right\Vert_{L^2(K)}^{2}\\ &~~~~~~~~~~~~~~~~~~~~~~~~~~~~+\sum_{ E \in \mathcal{E}_h}h_{E}\left\Vert\left[n{\cdot}{\nabla}\cfrac{{\partial}\tilde{u}_{{\tau}h}}{\partial t}\right]\right\Vert_{L^2(E)}^{2}\Biggr]^{1/2 }(t)|E_{u}|_{H^1(\Omega)}(t)dt.\end{aligned}$$ We turn now to the third term in (\[4newterms\]) $$\begin{aligned} I I I &= \int_{t_1}^{t^{\ast}} \{(p_n (A_h {\partial}_{n}^{2} u_{h}-{\partial}_{n}^{2}f_h),\tilde I_{h} E_{v})-\left(p_n {\nabla}{\partial}_{n}^{2} v_{h},{\nabla}E_{u}\right) +(\tilde{f}_\tau-f,\tilde I_{h} E_{v}) \}dt \\ & \leq C \sum_{m = 1}^n \Biggl[ \left( \int_{t_m}^{t^{}_{m + 1}} |p_m|{dt} \right) \left( \left\Vert \partial_m^2 f_h - A_h \partial_m^2 u_h \right\Vert_{L^2(\Omega)} + \left\vert \partial_m^2 v_h\right\vert_{H^1(\Omega)}\right)\\ &+\int_{t_m}^{t^{}_{m + 1}} \left\Vert f-\tilde{f}_\tau \right\Vert_{L^2(\Omega)} {dt} \Biggr] Z ( t^{^{\ast}}) \\\end{aligned}$$ with $$\int_{t_m}^{t_{m + 1}} |p_m|{dt} \leq \frac{1}{12}{\tau}_{m}^{3}+\frac{1}{8}{\tau}_{m-1}{\tau}_{m}^2.$$ We have used here the bounds $|E_u |_{H^1(\Omega)} (t) \leqslant Z (t) \leqslant Z (t^{\ast})$ and $\|E_v \|_{L^2(\Omega)} \leqslant Z (t) \leqslant Z (t^{\ast})$ for all $t \in [0, t^{\ast}]$. Similar reasoning for the fourth term in (\[4newterms\]) give us $$\begin{aligned} I V &= \int_{t_0}^{t_1} \{(p_1 (A_h {\partial}_{1}^{2} u_{h}-{\partial}_{1}^{2}f_h),\tilde I_{h} E_{v})-\left(p_1 {\nabla}{\partial}_{1}^{2} v_{h},{\nabla}E_{u}\right) +(\tilde{f}_\tau-f,\tilde I_{h} E_{v}) \}dt \\ & \leq C \Biggl[ \left( \int_{t_0}^{t^{}_{ 1}} |p_1|{dt} \right) \left( \left\Vert \partial_1^2 f_h - A_h \partial_1^2 u_h \right\Vert_{L^2(\Omega)} + \left\vert \partial_1^2 v_h\right\vert_{H^1(\Omega)}\right)\\ &+\int_{t_0}^{t^{}_{1}} \left\Vert f-\tilde{f}_\tau \right\Vert_{L^2(\Omega)} {dt} \Biggr] Z ( t^{^{\ast}}) \\\end{aligned}$$ where $$\int_{t_0}^{t_{1}} |p_1|{dt} \leq \frac{5}{12}{\tau}_{0}^{3}+\frac{1}{2}{\tau}_{1}{\tau}_{0}^2.$$ Applying the same bounds for $|E_u |_{H^1(\Omega)} (t)$ and $\|E_v \|_{L^2(\Omega)} \leqslant Z (t)$ to the estimates for integrals $I+II$, inserting them into (\[4newterms\]) and noting that $A_h\partial^2_k u_h=z^k_h$ we obtain (\[estf\]). Comparing the a priori estimate (\[apriori\]) with the a posteriori one (\[estf\]) one sees that the time error indicator is essentially the same in both cases. Indeed, the term $\displaystyle\int_{t_n}^{t_{n + 1}} \left\Vert \cfrac{\partial^4 u }{\partial t^4} \right\Vert_{L^2(\Omega)} {dt}$ can be rewritten as $\displaystyle\int_{t_n}^{t_{n + 1}} \left\Vert \cfrac{\partial^2 f }{\partial t^2} + \Delta \cfrac{\partial^2 u }{\partial t^2} \right\Vert_{L^2(\Omega)} {dt}$ and it’s discrete counterpart is in \[in\] and \[in0step\]. Note also that the last term in (\[estf\]) is negligible, at least if $f$ the sufficiently smooth in time, since $\|f-\tilde{f}_\tau\|_{L^2(\Omega)}=O(\tau_n^3)$ for $t\in(t_n,t_{n+1})$. Moreover, in view of a posteriori estimate some of the terms are of higher order $\tau h^2$, so that neglecting the higher order terms, a posteriori space error estimator can be reduced to the two first lines in (\[space\]), i.e. $$\begin{aligned} \label{space1} \eta_S^{(1)} (t_k) &= C_1 \max_{0 \leqslant t \leqslant t_k} \Biggl[ \sum_{K \in \mathcal{T}_h} h_K^2 \left\Vert \frac{\partial \tilde{v}_{h \tau}}{\partial t} - \Delta \tilde{u}_{h \tau}-f \right\Vert_{L^2(K)}^2\\ \notag&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~+ \sum_{ E \in \mathcal{E}_h}h_{E} \|[n \cdot \nabla \tilde{u}_{h \tau}]\|_{L^2(E)}^2 \Biggr]^{1/2} (t),\\ \label{space2} \eta_S^{(2)} (t_k)&= C_2\sum_{m = 0}^k \int_{t_m}^{t_{m + 1}} \Biggl[ \sum_{K \in \mathcal{T}_h} h_K^2 \left\Vert \frac{\partial^2 \tilde{v}_{h \tau}}{\partial t^2} - \Delta \frac{\partial \tilde{u}_{h\tau}}{\partial t} -\frac{\partial{f}}{\partial{t}}\right\Vert_{L^2(K)}^2\\ \notag&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~+ \sum_{ E \in \mathcal{E}_h}h_{E} \left\Vert\left[n \cdot \nabla \frac{\partial \tilde{u}_{h\tau}}{\partial t}\right]\right\Vert_{L^2(E)}^2 \Biggr]^{1/2} (t) dt \end{aligned}$$ Optimality of the error estimators {#optimality} ---------------------------------- We do not have a lower bound for our error estimators in space and time. Note that such a bound is not available even in a simpler setting of Euler discretization in time, cf. [@BS]. We are going to prove a partial result in the direction of optimality, namely that the indicator of error in time provides the estimate of order $\tau^2$ at least on sufficiently smooth solutions and quasi-uniform meshes. For this, we should examine if the quantities $\partial_n^2 f_h - A_h \partial_n^2 u_h$ and $\partial^2_n v_h$ remain bounded in $L^2$ and $H^1$ norms respectively. This will be achieved in Lemma \[bound\_d4u\] assuming that the initial conditions are discretized in a specific way, via the $H^1_0$-orthogonal projection. We restrict ourselves to the constant time steps $\tau_n=\tau$ and introduce the notations $$\begin{aligned} {\partial}^{0}_{n} u_h&=u_h^{n+1},&&\quad {\partial}^{j+1}_{n} u_h={\frac{{\partial}^{j}_{n}u_h-{\partial}^{j}_{n-1}u}{{\tau}}}, &&\quad j=0,1,{\ldots},\quad n\ge j-1\\ \bar{\partial}^0_n {u}_h&=\frac{u_h^{n + 1} + u_h^n}{2},&&\quad\bar{\partial}^{j + 1}_n {u}_h= \frac{\bar{\partial}^j_n {u}_h - \bar{\partial}^j_{n - 1} {u}_h}{\tau}, &&\quad j=0,1,{\ldots},\quad n\ge j \end{aligned}$$ The Crank-Nicolson scheme for first-order system (\[CNh1\])-(\[CNh2\]) for $n\ge 0$ is written with these notations as $$\begin{aligned} \label{CNh1N} {\partial}^{1}_{n} u_h - \bar{\partial}^0_n {v}_h &= 0\\ \label{CNh2N} \left( {\partial}^{1}_{n} v_h, \varphi_h \right) + \left( \nabla \bar{\partial}^0_n {u}_h, \nabla \varphi_h \right) &= \left( \bar{\partial}^0_n {f}_h, \varphi_h \right), \hspace{1em} \forall \varphi_h \in V_h\end{aligned}$$ where $f^n_h$, $n\ge 0$, are the $L^2$-orthogonal projection of $f(t_n,\cdot)$ on $V_h$. The following lemma provides a higher regularity result on the discrete level, i.e. the boundedness of terms $\partial_n^j f_h - A_h \partial_n^j u_h$ and $\partial^j_n v_h$ for any $j\in\mathbb{N}^0$. \[bound\_d4u\_abst\] Let $u_h^n$ and $v_h^n$ be the solution to (\[CNh1\])-(\[CNh2\]) for $n\ge 0$. One has then for all $j\in\mathbb{N}^0$, $N\in\mathbb{N}$, $N\ge j$ $$\begin{gathered} \label{boundj} \left( \left\| \partial^{j}_N f_h-A_h\partial^{j}_N u_h \right\|_{L^2(\Omega)}^2 + \left| {\partial}^j_N v_h \right|_{H^1(\Omega)}^2\right)^{{1}/{2}} \\ \leq \left( \left\| \partial^{j}_jf_h-A_h\partial^{j}_j u_h \right\|_{L^2(\Omega)}^2 + \left|{\partial}^j_j v_h \right|_{H^1(\Omega)}^2\right)^{{1}/{2}} + \tau\sum_{n = j+1}^N\left\| {\partial}^{j+1}_n {f}\right\|_{L^2(\Omega)}\end{gathered}$$ Starting from (\[CNh1N\])-(\[CNh2N\]), taking the differences between steps $n$ and $n - 1$ and then making an induction on $j = 0, 1, \ldots$ one arrives at $$\begin{aligned} \label{CNh1NN} {\partial}^{j+1}_{n} u_h &= \bar{\partial}^j_n {v}_h, \\ \label{CNh2NN} {\partial}^{j+1}_{n} v_h &= \bar{\partial}^j_n {f}_h - A_h \bar{\partial}^j_n {u}_h. \end{aligned}$$ One can also prove that $\forall w^n_h\in V_h$ $$\label{propj} \bar{\partial}^j_n {w}_h = \frac{{\partial}^j_n w_h + {\partial}^j_{n - 1} w_h}{2}, \hspace{1em} j = 0, 1, \ldots$$ Indeed, this is obvious for $j = 0$ and then it follows for any $j$ by induction. Taking the inner product of (\[CNh2NN\]) with $\tau A_h{\partial}^{j + 1}_n u_h-\tau {\partial}^{j + 1}_n f_h$, using (\[propj\]) and definition of ${\partial}^{j+1}_n$ we obtain $$\begin{aligned} \biggl(\partial_n^{j+1} v_h , \tau A_h{\partial}^{j + 1}_n u_h&-\tau {\partial}^{j + 1}_n f_h \biggr)=\biggl(\bar{\partial}_n^{j} f_h - A_h\bar{\partial}_n^{j} u_h, \tau A_h{\partial}^{j + 1}_n u_h-\tau {\partial}^{j + 1}_n f_h \biggr)\\ &= -\frac{\left\| \partial^{j}_n f_h-A_h\partial^{j}_n u_h \right\|_{L^2(\Omega)}^2}{2} + \frac{\left\| \partial^{j}_{n-1} f_h-A_h\partial^{j}_{n-1} u_h \right\|_{L^2(\Omega)}^2}{2}.\end{aligned}$$ Now we apply (\[propj\]) and (\[CNh1NN\]) to the left-hand side above $$\begin{aligned} \biggl(\partial_n^{j+1} v_h , \tau A_h{\partial}^{j + 1}_n u_h&-\tau {\partial}^{j + 1}_n f_h \biggr) \\ &=\left({\partial_n^{j } v_h - \partial_{n-1}^{j} v_h} , A_h{\partial}^{j + 1}_n u_h\right) -\left(\partial_n^{j+1 } v_h , \tau{\partial}^{j + 1}_n f_h\right)\\ &= \frac{\left| {\partial}^j_n v_h \right|^2_{H^1(\Omega)} - \left| {\partial}^j_{n - 1} v_h \right|_{H^1(\Omega)}^2}{2} -\left(\partial_n^{j+1 } v_h , \tau{\partial}^{j + 1}_n f_h\right).\end{aligned}$$ Thus $$\begin{aligned} \frac{\left| {\partial}^j_n v_h \right|^2_{H^1(\Omega)} - \left| {\partial}^j_{n - 1} v_h \right|_{H^1(\Omega)}^2}{2} -\left(\partial_n^{j+1 } v_h , \tau{\partial}^{j + 1}_n f_h\right)&= -\frac{\left\| \partial^{j}_n f_h-A_h\partial^{j}_n u_h \right\|_{L^2(\Omega)}^2}{2}\\ &+ \frac{\left\| \partial^{j}_{n-1} f_h-A_h\partial^{j}_{n-1} u_h \right\|_{L^2(\Omega)}^2}{2}.\end{aligned}$$ We recall by (\[CNh2NN\]) $$\begin{aligned} \tau{\partial}^{j+1}_{n}v_h&=\tau\left( \bar{\partial}^{j}_{n}f_h-A_h\bar{\partial}^{j}_{n} u_h \right)\\ &= \frac{\tau}{2}\left({\partial}^{j}_{n}f_h+{\partial}^{j}_{n-1}f_h-A_h{\partial}^{j}_{n-1} u_h - A_h{\partial}^{j}_{n-1} u_h \right)\end{aligned}$$ and hence $$\begin{aligned} \left\| \partial^{j}_nf_h-A_h\partial^{j}_n u_h \right\|_{L^2(\Omega)}^2 + \left| {\partial}^j_{n} v_h \right|_{H^1(\Omega)}^2-\left\| \partial^{j}_{n-1}f_h-A_h\partial^{j}_{n-1} u_h \right\|_{L^2(\Omega)}^2 - \left| {\partial}^j_{n-1} v_h \right|_{H^1(\Omega)}^2 \\ \leq \tau\left\| {\partial}^{j+1}_n {f}_h \right\|_{L^2(\Omega)}\left(\left\| \partial^{j}_nf_h-A_h\partial^{j}_n u_h \right\|_{L^2(\Omega)}+ \left\| \partial^{j}_{n-1}f_h-A_h\partial^{j}_{n-1} u_h \right\|_{L^2(\Omega)}\right).\end{aligned}$$ Denoting $Z_n=\left(\left\| \partial^{j}_nf_h-A_h\partial^{j}_n u_h \right\|_{L^2(\Omega)}^2 + \left| {\partial}^j_{n} v_h \right|_{H^1(\Omega)}^2\right)^{{1}/{2}}$ the last inequality can be rewritten as $$\begin{aligned} Z_n^2-Z_{n-1}^2 &\leq \tau\left\| {\partial}^{j+1}_n {f}_h \right\|_{L^2(\Omega)}\left(\left\| \partial^{j}_nf_h-A_h\partial^{j}_n u_h \right\|_{L^2(\Omega)} \right.\\ & \left.+\left\| \partial^{j}_{n-1}f_h-A_h\partial^{j}_{n-1} u_h \right\|_{L^2(\Omega)}\right) \leq \tau\left\| {\partial}^{j+1}_n {f}_h \right\|_{L^2(\Omega)} (Z_n+Z_{n-1})\end{aligned}$$ so that $$Z_n-Z_{n-1} \leq \tau\left\| {\partial}^{j+1}_n {f}_h \right\|_{L^2(\Omega)}.$$ Summing this over $n$ we get (\[boundj\]). In order to take into account the initial conditions, we shall need the following auxiliary result about stability properties of operator $A_h$ defined by (\[Ah\]) and the $L^2$-orthogonal projection $P_h : L^2 (\Omega) \to V_h$ defined by $$\label{Ph} \forall v \in L^2 (\Omega) : \left( P_h v, \varphi_h\right) = \left( v, \varphi_h\right) \hspace{1em} \forall \varphi_h \in V_h$$ \[bound\_AhPhu\] Assuming the mesh $\mathcal{T}_h$ to be quasi-uniform, there exists $C>0$ depending only on the regularity of $\mathcal{T}_h$ such that $$\begin{aligned} \label{Phbound} \forall v \in H^1_0 (\Omega) & : & | P_h v |_{H^1(\Omega)} \leq C | v |_{H^1(\Omega)}, \\ \label{AhPhbound} \forall v \in H^2 (\Omega) \cap H^1_0 (\Omega) & : & \| A_hP_h v \|_{L^2(\Omega)} \leq C | v |_{H^2(\Omega)}\end{aligned}$$ Let $ v \in H^1_0 (\Omega)$. Using a Cl[é]{}ment-type interpolation operator $\tilde I_h$, satisfying $\tilde I_h=Id$ on $V_h$ and (\[Clement\]), together with an inverse inequality we observe $$| P_h v |_{H^1(\Omega)} \leq | P_h v -\tilde I_hv |_{H^1(\Omega)} + | \tilde I_h v |_{H^1(\Omega)} \leq \cfrac{C}{h}\| P_h v -\tilde I_hv \|_{L^2(\Omega)} + | v |_{H^1(\Omega)}$$ Then, from approximation properties (\[Clement\]) $$\| P_h v -v \|_{L^2(\Omega)} \leq \| \tilde I_h v -v \|_{L^2(\Omega)} \leq C h| v |_{H^1(\Omega)} \leq C h| v |_{H^1(\Omega)}$$ which entails (\[Phbound\]). We assume now $v\in H^2(\Omega) \cap H^1_0 (\Omega)$ and use a similar idea to prove (\[AhPhbound\]). For any $\varphi_h\in V_h$ $$\label{tr} \left( A_hP_h v,\varphi_h\right) = \left(\nabla\left(P_h-\tilde I_h\right) v ,\nabla\varphi_h\right)+\left( \nabla \tilde I_h v ,\nabla\varphi_h\right)$$ We can bound the first term in the right-hand side of (\[tr\]) using the inverse inequality and the approximation properties of $\tilde I_h$ $$\left(\nabla\left(P_h-\tilde I_h\right) v ,\nabla\varphi_h\right)\leq \cfrac{C}{h^2}\| P_h v -\tilde I_hv \|_{L^2(\Omega)}\| \varphi_h \|_{L^2(\Omega)}\leq C| v |_{H^2(\Omega)}\| \varphi_h \|_{L^2(\Omega)}$$ To deal with the second term in the right-hand side of (\[tr\]), we integrate by parts over all the triangles of the mesh and recall that $\Delta\varphi_h=0$ on any triangle, so that $$\left( \nabla \tilde I_h v ,\nabla\varphi_h \right)=\sum_{E \in \mathcal{E}_h}\int_{E}\left[\cfrac{\partial \tilde I_h v }{\partial n}\right]\varphi_h \leq \sum_{E \in \mathcal{E}_h}\left\Vert\left[\cfrac{\partial \tilde I_h v }{\partial n}\right]\right\Vert_{L^2(E)}\left\Vert\varphi_h\right\Vert_{L^2(E)}$$ Using the inverse trace inequality $\left\Vert\varphi_h\right\Vert_{L^2(E)}\leq \cfrac{C}{\sqrt{h}}\left\Vert\varphi_h\right\Vert_{L^2(\omega_E)}$ and the interpolation error bound $$\left\Vert\left[\cfrac{\partial \tilde I_h v }{\partial n}\right]\right\Vert_{L^2(E)} =\left\Vert\left[\cfrac{\partial }{\partial n}(v-\tilde I_h v)\right]\right\Vert_{L^2(E)} \leq C\sqrt{h}| v |_{H^2(\omega_{E})}$$ on all the edges $E\in \mathcal{E}_h$ leads, together with (\[tr\]), to $$\left( A_hP_h v,\varphi_h\right) \le C| v |_{H^2(\Omega)}\| \varphi_h \|_{L^2(\Omega)}$$ Taking here $\varphi_h=A_hP_h v$, we obtain desired result (\[AhPhbound\]). Our proof of Lemma \[bound\_AhPhu\] uses inverse inequalities and is thus restricted to the quasi-uniform meshes $\mathcal{T}_h$. The first estimate (\[Phbound\]) is actually established in [@bramble2002stability] under much milder hypotheses on the mesh compatible with usual mesh refinement techniques. We conjecture that the second estimate (\[AhPhbound\]) also holds under similar assumptions. Some numerical examples in this direction are given at the end of Subsection \[unstr\]. We are now able to complete the estimate of Lemma \[bound\_d4u\_abst\] in the case $j=2$ which is pertinent to our a posteriori analysis. \[bound\_d4u\] Let $u_h^n$ be the solution to (\[Newm1\])–(\[Newm2\]) on a quasi-uniform mesh with $$u^0_h = \Pi_h u^0,~v^0_h = \Pi_h v^0 \label{initialcond}$$ where $\Pi_h$ is the $H^1_0$-orthogonal projection on $V_h$. One has for all $N\ge 1$ $$\begin{aligned} \label{boundj4} \biggl(\bigl\| \partial^{2}_Nf_h-A_h&\partial^{2}_N u_h \bigr\|_{L^2(\Omega)}^2 + \left| {\partial}^2_{N} v_h \right|_{H^1(\Omega)}^2\biggr)^{{1}/{2}}\\ \notag &\leq C \left(\left| \cfrac{\partial^{3}u}{\partial t^{3}}(0) \right|_{H^1(\Omega) } + \left| \cfrac{\partial^{2}u}{\partial t^{2}}(0) \right|_{H^2(\Omega)} + \max_{t \in [0,2\tau]} \left\| \cfrac{\partial^{2}f}{\partial t^{2}}(t)\right\|_{L^2(\Omega)} \right)\\ \notag&+ \int_0^{t_N} \left\| \cfrac{\partial^{3}f}{\partial t^{3}}\right\|_{L^{2}(\Omega)}dt \end{aligned}$$ with a constant $C > 0$ independent of $h$, $\tau$, $N$. Denote $$Z = 2 \left( I + \frac{\tau^2}{4} A_h \right)^{- 1} \left( I - \frac{\tau^2}{4} A_h \right)$$ Then scheme (\[Newm2\]) for $n \geq 1$ can be rewritten as $$u^{n + 1}_h = Zu_h^n - u^{n - 1}_h + \tau^2 \left( I + \frac{\tau^2}{4} A_h \right)^{- 1} \bar{f}^n_h$$ Moreover, the initial step (\[Newm1\]) can be written as $$\frac{u^1_h - u^0_h - \tau v^0_h}{\tau^2} + A_h ^{} \frac{u^1_h + u^0_h}{4} = \bar{f}^0_h := \frac{f^1_h + f^0_h}{4} \label{Newmh0}$$ This gives the following expressions for $u_h^1, u_h^2$: $$\begin{aligned} \notag u_h^1 &= \tau^2 \left( I + \frac{\tau^2}{4} A_h \right)^{- 1} \left( \bar{f}^0_h + \frac{1}{\tau} v^0_h \right) + \frac{1}{2} Zu^0_h \\ \notag u_h^2 &= \tau^2 \left( I + \frac{\tau^2}{4} A_h \right)^{- 1} \left( Z \left( \bar{f}^0_h + \frac{1}{\tau} v^0_h \right) + \bar{f}^1_h \right) + \left( \frac{1}{2} Z^2_{} - I \right) u_h^0 \end{aligned}$$ Thus, $$\begin{aligned} \partial^{2}_1f_h-A_h\partial^{2}_1 u_h &= \partial^{2}_1f_h - \frac{A^2_hZ}{2\left(I + \frac{\tau^2}{4} A_h \right)}u^0_h \\ &- {A_h}\left( I + \frac{\tau^2}{4} A_h \right)^{- 1} \left( (Z-2I) \left( \bar{f}^0_h + \frac{1}{\tau} v^0_h \right) + \bar{f}^1_h \right)\end{aligned}$$ and $$\begin{aligned} {\partial}^2_1 v_h = &-A_h\frac{u^2_h-u^0_h}{2\tau}+\frac{f^2_h-f^0_h}{2\tau}=-\frac{A_h}{2 \tau} \left(\frac{1}{2}Z^2 - 2 I\right) u^0_h \\ &- \frac{A_h}{2 \tau} \tau^2 \left( I + \frac{\tau^2}{4} A_h \right)^{- 1} \left( Z \left( \bar{f}^0_h + \frac{1}{\tau} v^0_h \right) + \bar{f}^1_h \right) +\frac{f^2_h-f^0_h}{2\tau} \end{aligned}$$ After some tedious calculations, this can be rewritten as $$\begin{aligned} \notag \partial^{2}_1f_h-A_h\partial^{2}_1 u_h = -\frac{1}{2} \frac{Z} {\left( I + \frac{\tau^2}{4} A_h \right)^2} \left(A_h^2 u^0_h - A_h f^0_h\right) &+\frac{\tau A_h }{\left( I + \frac{\tau^2}{4} A_h \right)^2} \left(A_h v^0_h - \partial^1_0 f_h\right)\\ &+\left( I + \frac{\tau^2}{4} A_h \right)^{- 1}\partial^2_1 {f}_h \label{expAhd2u} \end{aligned}$$ and $$\begin{aligned} \label{expd2v}{\partial}^2_1 v_h = -\frac{\tau}{\left( I + \frac{\tau^2}{4} A_h \right)^2} \left(A_h^2 u^0_h - A_h f^0_h\right) &+ \frac{Z}{2 \left( I + \frac{\tau^2}{4} A_h \right)} \left(A_h v^0_h - \partial^1_0 f_h\right) \\ \notag &- \frac{\tau}{2 \left( I + \frac{\tau^2}{4} A_h \right)} \partial^2_1 f_h \end{aligned}$$ Since $A_h$ is a symmetric positive definite operator, we have $$\| R (\tau^2 A_h) v_h \|_{L^2(\Omega)} \leq C \| v_h \|_{L^2(\Omega)}$$ for any $v_h \in V_h$ and any rational function $R$ with the degree of nominator less or equal than that of the denominator and a constant $C$ depending only on $R$. Similarly, using the fact $| v_h |_{H^1(\Omega)} = (A_hv_h,v_h)^\frac{1}{2}=\left\|A_h^{{1}/{2}} v_h\right\|_{L^2(\Omega)}$ for any $v_h\in V_h$ one can observe $$\| \tau A_h R (\tau^2 A_h) v_h \|_{L^2(\Omega)} \le C \| A_h^{1/2} v_h \|_{L^2(\Omega)} = C | v_h |_{H^1(\Omega)}$$ for any rational function $R$ with the degree of nominator less than that of the denominator and a constant $C$ depending only on $R$. Applying these estimates to (\[expd2v\]) yields $$\begin{aligned} \| \partial^2_1 f_h - A_h \partial^2_1 u_h \|_{L^2 (\Omega)} & \leq C \left( \| A_h^2 u^0_h - A_h f^0_h \|_{L^2 (\Omega)} + \left| A_h v^0_h - \frac{\partial f_h}{\partial t} (0) \right|_{H^1 (\Omega)} \right.\\ & \left. + \left\| \frac{\tau A_h}{\left( I + \frac{\tau^2}{4} A_h \right)^2} \left( \frac{\partial f_h}{\partial t} (0) - \partial^1_0 f_h \right) \right\|_{L^2 (\Omega)} + \| \partial^2_1 f_h \|_{L^2 (\Omega)} \right) \end{aligned}$$ Since $$\partial^1_0 f_h = \frac{\partial f_h}{\partial t} (0) + \frac{1}{\tau} \int^{\tau}_0 (\tau - s) \frac{\partial^2 f}{\partial t^2} (s) ds$$ we have $$\begin{aligned} \left\| \frac{\tau A_h}{\left( I + \frac{\tau^2}{4} A_h \right)^2} \left( \frac{\partial f_h}{\partial t} (0) - \partial^1_0 f_h \right) \right\|_{L^2 (\Omega)} &\leq \max_{t \in [0, \tau]} \left\| \frac{\tau^2 A_h}{\left( I + \frac{\tau^2}{4} A_h \right)^2} \frac{\partial^2 f_h}{\partial t^2} (t) \right\|_{L^2 (\Omega)} \\ &\leq C\max_{t \in [0, \tau]} \left\| \frac{\partial^2 f_h}{\partial t^2} (t) \right\|_{L^2 (\Omega)} \end{aligned}$$ Noting finally that $\| \partial^2_1 f_h \|_{L^2(\Omega)}$ can be bounded by the maximum of $\left\| \cfrac{\partial^{2}f}{\partial t^{2}}(t)\right\|_{L^2(\Omega)}$ over time interval $[0,2\tau]$, we arrive at $$\begin{aligned} \|\partial^{2}_1f_h-A_h\partial^{2}_1 u_h\|_{L^2(\Omega)} &\leq C \left( \left\|A_h^2 u^0_h - A_h f^0_h \right\|_{L^2(\Omega)}+ \left|A_h v^0_h - \frac{\partial f_h}{\partial t}(0) \right|_{H^1(\Omega)}\right.\\ &\left. + \max_{t\in[0,2\tau]}\left\|\frac{\partial^2 f}{\partial t^2}(t)\right\|_{L^2(\Omega)}\right) \end{aligned}$$ By a similar reasoning we can also bound $\left| {\partial}^2_{1} v_h \right|_{H^1(\Omega)}$ by the same quantitity as in the right-hand side of the equation above. For this, we take the $H^1$ norm on both sides of (\[expd2v\]) and observe for the first term on the right hand side $$\begin{aligned} \left| \frac{\tau}{\left( I + \frac{\tau^2}{4} A_h \right)^2} \left(A_h^2 u^0_h - A_h f^0_h\right) \right|_{H^1(\Omega)} &= \left\| \frac{\tau A_h^{1/2}}{\left( I + \frac{\tau^2}{4} A_h \right)^2} \left(A_h^2 u^0_h - A_h f^0_h\right) \right\|_{L^2(\Omega)} \\ &\le C \left\| A_h^2 u^0_h - A_h f^0_h \right\|_{L^2(\Omega)}\end{aligned}$$ The other terms can be treated similarly so that, skipping some details, we obtain $$\begin{gathered} \label{boundedness} \left(\left\| \partial^{2}_1f_h-A_h\partial^{2}_1 u_h \right\|_{L^2(\Omega)}^2 + \left| {\partial}^2_{1} v_h \right|_{H^1(\Omega)}^2\right)^{{1}/{2}} \leq C \left( \left\|A_h^2 u^0_h - A_h f^0_h \right\|_{L^2(\Omega)}\right.\\ \left.+ \left|A_h v^0_h - \frac{\partial f_h}{\partial t}(0) \right|_{H^1(\Omega)} + \max_{t\in[0,2\tau]}\left\|\frac{\partial^2 f}{\partial t^2}(t)\right\|_{L^2(\Omega)} \right) \end{gathered}$$ We can now invoke the estimate of Lemma \[bound\_d4u\_abst\] with $j=2$ and combine it with (\[boundedness\]). This gives $$\begin{gathered} \label{boundj3} \left(\left\| \partial^{2}_Nf_h-A_h\partial^{2}_N u_h \right\|_{L^2(\Omega)}^2 + \left| {\partial}^2_{N} v_h \right|_{H^1(\Omega)}^2\right)^{{1}/{2}}\leq \sum_{n = 3}^N \tau \left\| {\partial}^3_n f\right\|_{L^2(\Omega)}\\ + C \left( \left\|A_h^2 u^0_h - A_h f^0_h \right\|_{L^2(\Omega)} + \left|A_h v^0_h - \frac{\partial f_h}{\partial t}(0) \right|_{H^1(\Omega)}\right.\\ + \max_{t\in[0,\tau]}\left\|\frac{\partial^2 f}{\partial t^2}(t)\right\|_{L^2(\Omega)} \Bigg).\end{gathered}$$ The first term in the right-hand side in (\[boundj3\]) can be easily bounded by $\displaystyle\int_0^{t_N} \left\| \cfrac{\partial^{3}f}{\partial t^{3}} \right\|_{L^2(\Omega)}dt$. The remaining terms in the middle line of (\[boundj3\]) are bounded using Lemma \[bound\_AhPhu\] and the relation $A_h\Pi_h=-P_h\Delta$ as follows $$\left\|A_h^2 u^0_h - A_h f^0_h \right\|_{L^2 (\Omega)} = \left\|A_h P_h (- \Delta u^0 - f^0) \right\|_{L^2(\Omega) } = \left\|A_h P_h \cfrac{\partial^{2}u}{\partial t^{2}} (0)\right \|_{L^2(\Omega) } \leq C \left| \cfrac{\partial^{2}u}{\partial t^{2}} (0) \right|_{H^2(\Omega) }$$ and $$\left|A_h v^0_h -\frac{\partial f_h}{\partial t}(0) \right|_{H^1(\Omega) } = \left|P_h \left(- \Delta v^0 - \frac{\partial f}{\partial t}(0)\right) \right|_{H^1 (\Omega)} \\ \leq \left|P_h \cfrac{\partial^{3}u}{\partial t^{3}} (0) \right|_{H^1(\Omega) } \leq C \left| \cfrac{\partial^{3}u}{\partial t^{3}} (0) \right|_{H^1(\Omega)}$$ This gives (\[boundj4\]). Note that in Lemma \[bound\_d4u\] the approximation of the initial conditions and of the right-hand side is crucial for boundedness of higher order discrete derivatives and consequently to optimality of our time and space error estimators. We illustrate this fact with some numerical examples in Subsection \[unstr\]. Let u be the solution of wave equation (\[wave\]) and $\displaystyle\cfrac{\partial^{3}u}{\partial t^{3}}(0)\in {H^1(\Omega)}$, $\displaystyle\cfrac{\partial^{2}u}{\partial t^{2}}(0) \in {H^2(\Omega)}$, $\displaystyle\cfrac{\partial^{2}f}{\partial t^{2}}(t)\in L^{\infty}(0,T;{L^2(\Omega)})$, $\displaystyle\cfrac{\partial^{3}f}{\partial t^{3}}(t)\in L^{2}(0,T;{L^2(\Omega)})$. Suppose that mesh ${\mathcal{T}_h}$ is quasi-uniform and the mesh in time is uniform ($t_k=k\tau$). Then, the 3-point time error estimator $\eta_T(t_k)$ defined by (\[in\],\[in0step\]) is of order $\tau^2$, i.e. $$\eta_T(t_k)\leq C \tau^2.\label{upperOpt}$$ with a positive constant $C$ depending only on $u$, $f$, and the mesh regularity. Follows immediately from Lemma \[bound\_d4u\]. Numerical results {#section4} ================= A toy model: a second order ordinary differential equation ---------------------------------------------------------- Let us consider first the following ordinary differential equation $$\begin{cases} \cfrac{d^{2}u(t)}{dt^{2}}+Au(t)=f(t) ,&t\in\left[ 0;T\right]\\ u(0)=u_0 ,&\\ \cfrac{du}{dt}(0)=v_0& \end{cases} \label{ODE}$$ with a constant $A>0$. This problem serves as simplification of the wave equation in which we get rid of the space variable. The Newmark scheme reduces in this case to $$\begin{aligned} \frac{u^{n+1}-u^{n}}{\tau_n}-\frac{u^{n}-u^{n-1}}{\tau_{n-1}}&+A\frac{\tau_n(u^{n+1}+u^{n})+\tau_{n-1}(u^{n}+u^{n-1})}{4}= \notag\\ &=\frac{\tau_n (f^{n+1}+f^n)+\tau_{n-1}(f^n+f^{n-1})}{4},~1\leq n\leq N-1 \label{schN}\\ \frac{u^1-u^0}{\tau_0}&=v_0-\frac{\tau_0}{4}A(u^1+u^0)+\frac{\tau_0}{4}(f^1+f^0), \notag\\ u^0&=u_0\notag\end{aligned}$$ the error becomes $e=\displaystyle\max_{0 \leq n \leq N}\left( \left\vert v^{n}-{u}'(t_{n})\right\vert ^{2}+A\left\vert u^{n}-u(t_{n})\right\vert ^{2}\right) ^{{1}/{2}}$, and the 3-point a posteriori error estimate $\forall n:~0\leq n \leq N$ simplifies to this form: $$\begin{aligned} \label{errest} e \leq \sum_{k=0}^{n-1}\tau_k\eta_{T}(t_k)&= \tau_0 \left(\frac{5}{12}\tau_{0}^2+\frac{1}{2}\tau_{0}\tau_{1}\right) \sqrt{A(\partial_1^2 v)^2 + (\partial_1^2f-A\partial_1^2 u)^2}\\ \notag &+\sum_{k=1}^{n-1}\tau_k \left(\frac{1}{12}\tau_{k}^2+\frac{1}{8}\tau_{k-1}\tau_{k}\right) \sqrt{A(\partial_k^2 v)^2 + (\partial_k^2f-A\partial_k^2 u)^2}.\end{aligned}$$ We define the following effectivity index in order to measure the quality of our estimators $\eta_T$: $$ei_T=\frac{\eta_T}{e}.$$ We present in Table \[tab:ode\] the results for equation (\[ODE\]) setting $f=0$, the exact solution $u=cos(\sqrt{A}t)$, final time $T=1$, and using constant time steps $\tau=\displaystyle {T}/{N}$. We observe that 3-point estimator is divided by about 100 when the time step $\tau$ is divided by 10. The true error $e$ also behaves as $O(\tau^2)$ and hence the time error estimator behaves as the true error. \[tab:ode\] In order to check behaviour of time error estimator for variable time step (see Table \[tab:ode2\]) we take the previous example with time step $\forall n:~0\leq n \leq N$ $$\label{tau10} \tau_n=\begin{cases} 0.1\tau_{\ast} ,&mod(n,2)=0\\ \tau_{\ast} ,&mod(n,2)=1 \end{cases}$$ where $\tau_{\ast}$ is a given fixed value. As in the case of constant time step we have the equivalence between the true error and the estimated error. We have plotted on Fig. \[fig:toyIndicators\] evolution in time of the value $\sum_{k=0}^{n-1}{\eta}_{T}(t_k)$ compared to $e$. The same conclusions hold when using even more non-uniform time step $\forall n:~0\leq n \leq N$ $$\label{tau100} \tau_n=\begin{cases} 0.01\tau_{\ast} ,&mod(n,2)=0\\ \tau_{\ast} ,&mod(n,2)=1 \end{cases}$$ on otherwise the same test case (see Table [\[tab:ode3\]]{}). Our conclusion is thus that for toy model classic and alternative a posteriori error estimators are sharp on both constant and variable time grids. ![Evolution in time of true error and 3-point error estimate for variable time step (\[tau10\]), $A=100$, $N=180$, $T=1$[]{data-label="fig:toyIndicators"}](timeEstimator.jpg){width=".85\textwidth"} \[tab:ode2\] \[tab:ode3\] The error estimator for the wave equation on structured mesh ------------------------------------------------------------ We now report numerical results for initial boundary-value problem for wave equation with uniform time steps when using 3-point time error estimator (\[in\], \[in0step\]). We compute space estimators (\[space1\]) and (\[space2\]) in practice as follows: $$\begin{aligned} \label{etas1} \eta_S^{(1)} (t_N) &= \max_{1 \leq n \leq N-1} \left[ \sum_{K \in \mathcal{T}_h} h_K^2 \left\Vert \partial_n v_h-{f}^n_{h} \right\Vert_{L^2(K)}^2\right. +\left.\sum_{ E \in \mathcal{E}_h}h_{E} \|[n \cdot \nabla {u}^n_{h}]\|_{L^2(E)}^2 \right]^{1/2}, \\ \label{etas2} \eta_S^{(2)} (t_N) &= \sum_{n = 1}^{N-1} \tau_n \left[ \sum_{K \in \mathcal{T}_h} h_K^2 \left\Vert \partial_n^2 v_h - \partial_n f_h\right\Vert_{L^2(K)}^2 + \sum_{ E \in \mathcal{E}_h}h_{E} \left\Vert\left[n \cdot \nabla \partial_n u_h\right]\right\Vert_{L^2(E)}^2 \right]^{1/2}.\end{aligned}$$ The quality of our error estimators in space and time is determined by following effectivity index: $$ei=\frac{\eta_T+\eta_S}{e}.$$ The true error is $$e=\max_{0 \leqslant n \leqslant N}\left(\left\Vert v^{n}_h-\cfrac{\partial u}{\partial t}(t_{n})\right\Vert_{L^2(\Omega)} ^{2}+\left\vert u^{n}_h-u(t_{n})\right\vert^{2}_{H^1(\Omega)}\right) ^{{1}/{2}}.$$ Consider the problem (\[wave\]) with $\Omega=(0,1)\times(0,1),~ T=1$ and the exact solution $u$ given by $$\begin{aligned} \text{case (a)}~~~ & u(x,y,t)=\cos(\pi t)\sin(\pi x)\sin(\pi y),\\ \text{case (b)}~~~ & u(x,y,t)=\cos(0.5\pi t)\sin(10\pi x)\sin(10\pi y),\\ \text{case (c)}~~~ & u(x,y,t)=\cos(15\pi t)\sin(\pi x)\sin(\pi y)\end{aligned}$$ We interpolate the initial conditions and the right-hand side with nodal interpolation. Structured meshes in space (see Fig. \[figura1\]) are used in all the experiments of this section. Numerical results are reported in Tables \[tab:wave1\]–\[tab:wave3\]. Note that these cases and the meshes in space in time in the following numerical experiments are chosen so that the error in case (a) should be due to both time and space discretization, that in case (b) comes mainly from the space discretization, and that in case (c) mainly from the time discretization. \[tab:wave1\] \[tab:wave2\] \[tab:wave3\] Referring to Table \[tab:wave1\], we observe from first three rows that setting $h = \tau^2$ the error is divided by 2 each time $h$ is divided by 2, consistent with $e\sim O(\tau^2+h)$. The space error estimator and the time error estimator behave similarly and thus provide a good representation of the true error. The effectivity index tends to a constant value. In rows 4-6, we choose $h = \tau$ in order to insure that the discretization in time gives an error of higher order than that in space, i.e. $O(h^2)$ vs. $O(h)$, respectively. Our estimators capture well this behaviour of the two parts of the error. In Table \[tab:wave2\], in order to illustrate the sharpness of the space estimator, we take case (b) where the error is mainly due to the space discretization. We can see from this table that the space error estimator $\eta_{S}$ behaves as the true error. Indeed, for a given space step, $\eta_{S}$ does not depend on the time step $\tau$, and for constant $\tau$, $\eta_{S}$ is divided by two when the space step $h$ is divided by two. Finally, we consider case (c), Table \[tab:wave3\]. We observe that the time error estimator $\eta_{T}$ behaves as the true error, when the error is mainly due to the time discretization. We therefore conclude that our time and space error estimators are sharp in the regime of constant time steps and structured space meshes. They separate well the two sources of the error and can be thus used for the mesh adaptation in space and time. As said already, the space estimator $\eta_S$ behaves as $O(h)$ in the numerical experiments reported in Tables \[tab:wave1\]-\[tab:wave2\]. The situation is slightly different in Table \[tab:wave3\]. Indeed, the first part of space error estimator $\eta^{(1)}_S$ behaves here as $O(\tau^2h)$. This can be explained by the fact that, as seen from the definitions (\[etas1\])–(\[etas2\]), both $\eta_S^{(1)}$ and $\eta_S^{(2)}$ are also influenced by discretization in time. In general, in the leading order in $h$ and $\tau$, one can conjecture $\eta_S^{(1,2)}=Ah+Bh\tau^2$ with case dependent $A$ and $B$. The second term $Bh\tau^2$ is asymptotically negligible but it can become visible in some situations where the solution is highly oscillating in time and the mesh in time is not sufficiently refined, as indeed observed with $\eta^{(1)}_S$ in Table \[tab:wave3\]. Fortunately, its value is small compared to the time error estimator and thus we can hope that this effect is not essential for mesh refinement. The error estimator for the wave equation on unstructured mesh {#unstr} -------------------------------------------------------------- We turn now to the numerical experiments on unstructured Delaunay meshes, cf. Fig. \[figura1\] (right). These experiments will reveal the dependence of the error estimators on approximation of initial conditions and of the right-hand side $f$. Indeed, as noted in Subsection \[optimality\], these approximations should be chosen carefully to ensure the optimality of our error estimators. \[tab:wave5\] \[tab:wave6\] We consider the test case from the previous subsection with the exact solution $u$ given by case (a). We test two different ways to approximate the initial conditions and the right-hand side: nodal interpolation $$\label{NodalInit} u^0_h = I_h u^0,~v^0_h = I_h v^0,~f^n_h= I_h f^n,~0\leq n \leq N$$ and orthogonal projections as in Lemma \[bound\_d4u\] $$\label{Projinit} u^0_h = \Pi_h u^0,~v^0_h = \Pi_h v^0,~f^n_h= P_h f^n,~0\leq n \leq N.$$ The results are reported in Tables \[tab:wave5\] and \[tab:wave6\]. The meshes, the time steps and other details of the numerical algorithm, are exactly the same in these two tables. We observe that the errors are very similar as well and conclude therefore that the accuracy of the method does not depend on the manner in which the initial conditions and $f$ are approximated, either (\[NodalInit\]) or (\[Projinit\]). On the contrary, the behaviour of error estimators is quite different in the two cases. From Table \[tab:wave5\] (nodal interpolation), we see that the time error estimator $\eta_T$ blows up with mesh refinement, while the second part of the space estimator $\eta^{(2)}_{S}$ behaves (non optimally) like $O(\tau+h)$. Only the first part of the space estimator $\eta^{(1)}_{S}$ behaves as the true error. Such a strange behaviour of our estimators indicates the unboundedness of higher order discrete derivatives in time. Indeed, the estimators $\eta_T$ and $\eta^{(2)}_{S}$ contain high order discrete derivatives ${\partial}^2_n f_h-A_h{\partial}^2_n u_h$ and ${\partial}^2_n v_h$ respectively. These error estimators can be of the optimal order only if all these derivatives are uniformly bounded. We recall that this property was examined in Lemma \[bound\_d4u\] and its proof hinges on the boundedness of $$\label{N0} N_0=\left\Vert A^2_hu^0_h-A_hf^0_h\right\Vert_{L^2(\Omega)}.$$ However, as reported in Table \[tab:wave5\], $N_0$ also blows up under the nodal interpolation of initial conditions and of the right-hand side. This is not surprising given that the boundedness of $N_0$ in Lemma \[bound\_d4u\] is a consequence of Lemma \[bound\_AhPhu\] and thus it is not guaranteed if one replaces projections (\[Projinit\]) by nodal interpolation (\[NodalInit\]). On the other hand, the results in Table \[tab:wave6\] corresponding to interpolation by projection (\[Projinit\]) confirm the order $O(\tau^2+h)$ for our error estimators, consistently with the theory developed in Lemmas \[bound\_d4u\] and \[bound\_AhPhu\]. \[tab:wave7\] The huge difference between the two data approximations can be also seen by looking directly at ${\partial}^4_4 u_h$. We report this quantity in Fig. \[figura2\] for the case (a) on a mesh with $h=0.0125$ and time step $\tau=0.025$ at $t=t_4=0.1$. On the left picture (nodal interpolation) we see that ${\partial}^4_4 u_h$ contains a lot of severe spurious oscillations, while the right picture (projection of initial conditions) contains a reasonable and quite smooth approximation of $\cfrac{\partial^4 u}{\partial t^4}$. This is another manifestation of the critical importance of the choice of an approximation of initial conditions and of the right-hand side for our error estimators. We note that such a phenomenon was not observed for the heat equation in [@LPP]. We also recall from Table \[tab:wave1\] that space and time error estimators provide a good representation of the true error on a structured mesh even under the nodal interpolation. Note that the quantity defined by (\[N0\]) remains also bounded on the structured mesh. We recall that the theory of Subsection \[optimality\], in particular Lemma \[bound\_AhPhu\], are established under the quasi-uniform meshe assumption. We conclude this article by a numerical test on non quasi-uniform meshes in order to asses the stability of operators $A_h$ and $P_h$. We apply our numerical method to (\[wave\]) with the exact solution $u$ from case (a) on meshes from Fig. \[figura3\]. The results are given in Table \[tab:wave7\]. We see that space and time error estimators provide a good representation of the true error, like in examples from Tables \[tab:wave1\] and \[tab:wave6\] with quasi-uniform meshes. Moreover, we observe stability for terms $\Vert A_hP_hu^0\Vert_{L^2(\Omega)}$, $\Vert P_hu^0\Vert_{H^1(\Omega)}$, and consequently $N_0$. This indicates that our error indicators may be useful for time and space adaptivity on rather general meshes. Conclusions =========== An a posteriori error estimate in the $L^{\infty}$-in-time/energy-in-space norm is proposed for the wave equation discretized by the Newmark scheme in time and the finite element method in space. Its reliability is proven theoretically in Theorem \[lemest3\]. Moreover, numerical experiments show its effectivity. Our estimators are designed to separate the error coming from discretization in space and that in time and should be therefore useful for time and space adaptivity. We have demonstrated, both theoretically and experimentally, the critical importance of the manner in which the initial conditions and the right-hand side are approximated. Indeed, under nodal interpolation the scheme in itself produces optimal results, bur certain quantities in a posteriori error estimates can blow up with mesh refinement. The remedy for this problem consists in using orthogonal pro1jections for initial conditions and the right-hand side, cf. Lemma \[bound\_d4u\]. [^1]: Laboratoire de Mathématiques de Besançon, Univ. Bourgogne Franche-Comté, 16 route de Gray, 25030 Besançon Cedex, France. Email: olga.gorynina@univ-fcomte.fr. [^2]: Laboratoire de Mathématiques de Besançon, Univ. Bourgogne Franche-Comté, 16 route de Gray, 25030 Besançon Cedex, France. Email: alexei.lozinski@univ-fcomte.fr. [^3]: Institute of Mathematics, Ecole Polytechnique Fédérale de Lausanne, Station 8, CH 1015, Lausanne, Switzerland. Email: marco.picasso@epfl.ch.

--- abstract: | On the night of Oct 31, 2015 two bright Southern Taurid fireballs occurred over Poland, being one of the most spectacular bolides of this shower in recent years. The first fireball - PF311015a Okonek - was detected by six video stations of Polish Fireball Network (PFN) and photographed by several bystanders, allowing for precise determination of the trajectory and orbit of the event. The PF311015a Okonek entered Earth’s atmosphere with the velocity of $33.2\pm0.1$ km/s and started to shine at height of $117.88 \pm 0.05$ km. The maximum brightness of $-16.0 \pm 0.4$ mag was reached at height of $82.5\pm0.1$ km. The trajectory of the fireball ended at height of $60.2\pm0.2$ km with terminal velocity of $30.2\pm1.0$ km/s. The second fireball - PF311015b Ostrowite - was detected by six video stations of PFN. It started with velocity of $33.2\pm0.1$ km/s at height of $108.05 \pm 0.02$ km. The peak brightness of $-14.8 \pm 0.5$ mag was recorded at height of $82.2\pm0.1$ km. The terminal velocity was $31.8\pm0.5$ km/s and was observed at height of $57.86\pm0.03$ km. The orbits of both fireballs are similar not only to orbits of Southern Taurids and comet 2P/Encke, but even closer resemblance was noticed for orbits of 2005 UR and 2005 TF50 asteroids. Especially the former object is interesting because of its close flyby during spectacular Taurid maximum in 2005. We carried out a further search to investigate the possible genetic relationship of Okonek and Ostrowite fireballs with both asteroids, that are considered to be associated with Taurid complex. Although, we could not have confirmed unequivocally the relation between fireballs and these objects, we showed that both asteroids could be associated, having the same origin in a disruption process that separates them. author: - | A. Olech$^{1}$[^1], P. Żo[ł]{}dek$^2$, M. Wiśniewski$^{2,3}$, R. Rudawska$^4$, M. Bben$^2$, T. Krzyżanowski$^2$, M. Myszkiewicz$^2$, M. Stolarz$^2$, M. Gawroński$^5$, M. Gozdalski$^2$, T. Suchodolski$^6$, W. Wgrzyk$^2$ and Z. Tymiński$^7$\ $^{1}$Nicolaus Copernicus Astronomical Center, Polish Academy of Sciences, ul. Bartycka 18, 00-716 Warszawa, Poland\ $^{2}$Comets and Meteors Workshop, ul. Bartycka 18, 00-716 Warszawa, Poland\ $^{3}$ Central Office of Measures, ul. Elektoralna 2, 00-139 Warsaw, Poland\ $^{4}$ ESA European Space Research and Technology Centre, Noordwijk, The Netherlands\ $^{5}$ Toruń Centre for Astronomy, Faculty of Physics, Astronomy and Applied Informatics, N. Copernicus University,\ ul. Grudziadzka 5, 87-100 Toruń, Poland\ $^{6}$ Space Research Centre, Polish Academy of Sciences, ul. Bartycka 18A, 00-716 Warszawa, Poland\ $^{7}$ Narodowe Centrum Badań Jdrowych, Ośrodek Radioizotopów POLATOM, ul. So[ł]{}tana 7, 05-400 Otwock, Poland date: 'Accepted 2016 March 15. Received 2016 February 29; in original form 2016 February 1' title: 2015 Southern Taurid fireballs and asteroids 2005 UR and 2005 TF50 --- \[firstpage\] meteorites, meteors, meteoroids, asteroids Introduction ============ The Taurids are an annual meteor shower active in October and November with maximum Zenithal Hourly Rates of about 5. The orbit of the shower has low inclination and thus, due to the gravitational perturbations of planets, swarm of particles is diffuse and separated into two main branches i.e. Northern Taurids (NTA) and Southern Taurids (STA). The parent body of Taurid complex is comet 2P/Encke (Whipple 1940), however both 2P/Encke and Taurids are believed to be remnants of a much larger object, which has disintegrated over the past 20000 to 30000 years (Asher et al. 1993, Babadzhanov et al. 2008). Recently, Porubčan et al. (2006) identified as many as 15 Taurid complex filaments and found possible association with 9 Near Earth Objects (NEOs). Most recently, Jopek (2011) identified as many as 14 parent bodies of the Taurids stream. It has been widely recognized that the Taurid complex, despite its moderate activity, produces a great number of bright fireballs. Asher (1991) suggested that a swarm of Taurids being in 7:2 resonance with Jupiter produces occasional enhanced activity. It was later confirmed by Asher and Izumi (1998) who predicted observed swarm encounters in 1998, 2005 and 2008. In period 2009-2014 activity of Taurid shower was typical with some fireballs were observed (Madiedo et al. 2011, 2014). The return in 2005 was spectacular with both enhanced global activity and maximum rich in fireballs (Dubietis and Arlt, 2006). In 2008 the activity of the shower was lower but still it may be considered as enhanced (Jenniskens et al. 2008, Shrbený and Spurný 2012). According to the Asher’s model, the next swarm encounter year was expected in 2015. Additionally, most recently, the Taurid shower was suspected to have ability to produce meteorites (Brown et al. 2013, Madiedo et al. 2014). On the other hand, Tubiana et al. (2015) found no spectroscopic evidence for a link between 2P/Encke, the Taurid complex NEOs and CM type carbonaceous chondrite meteorites which felt recently in Denmark and were suspected for origin from the Taurid-Encke complex. Moreover, there is still no consensus concerning origin of the complex while the spectral data of its largest objects do not support a common cometary origin (Popescu et al. 2014). In this paper we report the results of observations and data reduction of two very bright Taurid fireballs which were detected on 2015 October 31 over Poland. Afterwards we discuss their connection with the asteroids 2005 UR and 2005 TF50, and comet 2P/Encke. This is the pilot study preceding more comprehensive analysis devoted to an enhanced fireball activity of 2015 Taurid meteor complex and its comparison to 2005 Taurid return. Observations ============ The PFN is the project whose main goal is regularly monitoring the sky over Poland in order to detect bright fireballs occurring over the whole territory of the country (Olech et al. 2006, Żo[ł]{}dek et al. 2007, 2009, Wiśniewski et al. 2012). It is kept by amateur astronomers associated in Comets and Meteors Workshop (CMW) and coordinated by astronomers from Copernicus Astronomical Center in Warsaw, Poland. Currently, there are almost 30 fireball stations belonging to PFN that operate during each clear night. It total over 60 sensitive CCTV cameras with fast and wide angle lenses are used. Code Site Longitude \[$^\circ$\] Latitude \[$^\circ$\] Elev. \[m\] Camera Lens ------- ------------- ------------------------ ----------------------- ------------- --------------------- ------------------------ PFN38 Podgórzyn 15.6817 E 50.8328 N 360 Tayama C3102-01A4 Computar 4mm f/1.2 PFN38 Podgórzyn 15.6817 E 50.8328 N 360 KPF 131HR Panasonic 4.5mm f/0.75 PFN43 Siedlce 22.2833 E 52.2015 N 152 Mintron MTV-23X11C Ernitec 4mm f/1.2 PFN48 Rzeszów 21.9220 E 50.0451 N 230 Tayama C3102-01A4 Computar 4mm f/1.2 PFN52 Stary Sielc 21.2923 E 52.7914 N 90 DMK23GX236 Tamron 2.4-6mm f/1.2 PFN61 Piwnice 18.5603 E 53.0951 N 85 Tayama C3102-01A4 Ernitec 4mm f/1.2 PFN67 Nieznaszyn 18.1849 E 50.2373 N 200 Mintron MTV-23 X11E Panasonic 4.5mm f/0.75 During last ten years typical setup of the PFN station consisted of 2-3 Tayama C3102-01A4 cameras equipped with 4 mm f/1.2 Computar or Ernitec lenses. Tayama C3102-01A4 is cheap CCTV camera with 1/3" Sony SuperHAD CCD detector working in PAL interlaced resolution with 25 frames per second. The field of view of one camera with 4 mm lens is $69.8\times55.0$ deg with scale of $\sim 10'$/pixel. This setup allows detection of the atmospheric entries of debris (both natural and artificial) with accuracy of trajectory determination below 300 meters. Almost each station is equipped with a PC computer with Matrox Meteor II frame grabber. The signal from each camera is analyzed on-line. Our video stations use the [MetRec]{} software (Molau 1999) which automatically detects meteors in frames captured by Matrox Meteor II frame grabber. The frames containing meteors are stored into BMP files. Additionally, information about basic parameters of the event such as its time of appearance and $(x,y)$ coordinates on the frame are saved into [MetRec]{} INF files. In case of some new stations containing higher resolution cameras, [UFO Capture]{} software is used (SonotaCo 2009). On the evening of 2015 October 31 at 18:05 UT a very bright fireball appeared over northwestern Poland. The International Meteor Organization (IMO) received almost 70 visual reports concerning this event from Austria, Czech Republic, Denmark, Germany, Netherlands, Poland and Sweden[^2]. Good weather conditions that night in central Europe and the high brightness of the fireball were the main factors for receiving so many reports. However, there are also two other reasons for it as well. Firstly, the fireball appeared almost exactly at the moment of the close flyby of large 2015 TB145 asteroid. Such events attract not only the attention of astronomy amateurs but also general public, encouraging people for observations. What is more interesting, the asteroid was passing across the Ursa Major constellation at that time, which was exactly the same region of the sky where the fireball was visible for observers situated in central and southern Poland, Czech Republic and Slovakia. Finally, the date of the appearance of the fireball was the date of the All Saints’ Eve. This is a public holiday in Poland, when in the evening people gather in cemeteries lighting the candles at the graves of their relatives. The scenery of candles after dusk creates nice landscapes which many people try to photograph. It is thus no surprise then that one of the most beautiful images of this fireball was captures in the cementary in Czernice Borowe, Poland by Aleksander and Grzegorz Zieleniecki. This image, kindly shared by the authors, was used for analysis in this work. The fireball reached its maximum brightness over Okonek city, and therefore received designation PF311015a Okonek. It was observed by six regular PFN video stations, where from the PFN38 Podgórzyn station the fireball was recorded by two cameras. Basic properties of the stations that recorded the event are listed in Table 1. Figure 1 shows images of the fireball captured by the Podgórzyn and Rzeszów stations. Additionally, the bolide was accidentally photographed in Czernice Borowe (location of the former PFN22 station). Here the Nikon D3300 digital single-lens reflex camera with Nikkor AF-S 18-55 mm f/3.5-5.6 lens was used. The lens was set at 18 mm focal length with relative aperture of f/3.5. The exposure time was 20 seconds with ISO equal to 800. Five hours after the PF311015a Okonek appearance another very bright fireball appeared passed through the sky at 23:13 UT. It was only slightly fainter than Okonek fireball but due to the late hour (after midnight of local time) it was not observed by any bystanders. Fortunately, it was detected by six PFN stations which are listed in Table 2 and the images of the fireball captured by the Urzdów and Podgórzyn stations are shown in Figure 2. The maximum brightness of the meteor was observed over Ostrowite village and thus its designation is PF311015b Ostrowite. Code Site Longitude \[$^\circ$\] Latitude \[$^\circ$\] Elev. \[m\] Camera Lens ------- ------------- ------------------------ ----------------------- ------------- -------------------- ------------------------ PFN20 Urzdów 22.1456 E 50.9947 N 210 Tayama C3102-01A1 Ernitec 4mm f/1.2 PFN38 Podgórzyn 15.6817 E 50.8328 N 360 Tayama C3102-01A4 Computar 4mm f/1.2 PFN38 Podgórzyn 15.6817 E 50.8328 N 360 KPF 131HR Panasonic 4.5mm f/0.75 PFN43 Siedlce 22.2833 E 52.2015 N 152 Mintron MTV-23X11C Ernitec 4mm f/1.2 PFN48 Rzeszów 21.9220 E 50.0451 N 230 Tayama C3102-01A4 Computar 4mm f/1.2 PFN52 Stary Sielc 21.2923 E 52.7914 N 90 Watec 902B Computar 2.6mm f/1.0 PFN57 Krotoszyn 17.4416 E 51.7018 N 150 Tayama C3102-01A4 Computar 4mm f/1.2 All analog video cameras contributing to this paper work in PAL interlaced resolution of $768\times 576$ pixels, with 25 frames per sec offering 0.04 sec temporal resolution. While, the digital camera DMK23GX236 used in PFN52 station has resolution of $1920\times 1200$ pixels and works with 20 frames per sec. Data reduction ============== The data from all stations, after a previous conversion, were further reduced astrometricaly by the [UFO Analyzer]{} program (SonotaCo 2009). Initially only automatic data were taken into account. However, during the further processing it became obvious that significant overexposures, the presence of the wake and a possible fragmentation after the flare caused quite serious errors concerning the correct position of the points of the phenomenon. The measurement precision improved noticeably when the bolide’s position was determined using UFO Analyzer astrometric solution with manual centroid measurement [UFO Analyzer]{}. The trajectory and orbit of both fireballs was computed using [PyFN]{} software (Żo[ł]{}dek 2012). [PyFN]{} is written in Python with usage of SciPy module and CSPICE library. For the purpose of trajectory and orbit computation it uses the plane intersection method described by Ceplecha (1987). Moreover, [PyFN]{} accepts data in both [MetRec]{} (Molau 1999) and [UFOAnalyzer]{} (SonotaCo 2009) formats and allows for semi-automatic search for double-station meteors. In case of photographic image recorded in Czernice Borowe the astrometry was performed using [Astro Record 3.0]{} software (de Lignie 1997), with the accuracy of the meteor path determination of 3 arcmin. The image was not used for brightness estimate due to the fact that the meteor was visible through thin cirrus clouds. Results ======= Brightness determination of the fireballs ----------------------------------------- The photometry of both fireballs was not trivial because of the strong saturation observed. On every camera both fireballs looks like strongly overexposed bulbs of light. Video cameras in the northernmost stations were completly overexposed with whole image saturated. Only the most distant stations can be usable to any kind of photometric measurements. The best results has been obtained using PFN48 Rzeszów video recordings. Both fireballs were recorded completly and from the large distance. The Okonek fireball has been observed from the distance of 525 km (point of maximum brightness), the Ostrowite fireball from the distance of 380 km. From such large distance both fireballs appeared as overexposed objects with only slightly different brigtness. The same camera recorded also the Full Moon image and its brightness was used as a primary reference point. Two independent methods has been applied to estimate the real brightness of fireballs. The first one was based on the measurements of the sky brightness during the fireball flight. These measurements has been compared with the sky brigtness caused by the Full Moon in the same camera. As a result we had observed maximum magnitude for Okonek fireball reaching $-12.5$ and $-12.3$ magnitude for the second fireball. Due to low sensitivity of the cameras used and severe light pollution in the PFN48 site the resulted lightcurve is incomplete and contains only brightest part of the fireball. The second method used several comparison objects recorded by cameras of different types with the same optics as used in PFN48 and working in similar sky conditions. The only available comparison objects were bright planets like Jupiter and Venus (magnitudes in the range $-2.5$ to $-4.5$) and the Moon in the different phases (magnitudes from $-8$ do $-12.6$). We used two sets of reference objects - one set recorded by the same camera configuration as for both fireballs and one set recorded by camera with two magnitude higher sensitivity. This second set can be treated as a set of reference objects which are brighter by two magnitude and it is helpfull to fill the gap between -4.5 and -8 magnitude objects. Some trial and error photometric tests led us to choose the best measure method. Good results has been obtained using aperture photometry on the images with overexposed pixel only visible (pixel with value below 255 on eight bit image has been rejected). Further refinements lowered reject value to 230. From such measurements the $I(m)$ function has been derived (see Figure 3). It is an exponential function empiricaly defined in the form: $$I = A^{-0.45\cdot m}$$ where $I$ is intensity, $m$ magnitude and $A$ constant. In case of our meaurements this function is valid for magnitude range from $-4$ to $-12$ and is a bit different for higher magnitudes. This method has been used for both fireballs. The light curves has been determined. Measurements has been repeated using different sets of reference points, different results has been used to magnitude error determination. Maximum magnitudes measured using this method were $-12.4$ for Okonek fireball and $-11.9$ for Ostrowite fireball, respectively. These results are consistent with measuremets of the sky brightness mentioned before. Difference between two methods is less than 0.5 mag. Both fireballs observed from PFN48 Rzeszów, from the distance of hundreds kilometers, looked as very bright objects with brightness comparable to the Full Moon. Reduction to the standard absolute magnitude (fireball visible 100 km directly overhead) gives significantly higher brightness values. Resulting absolute brightness for Okonek fireball is $-16\pm0.4$ mag. The Ostrowite fireball was fainter and its absolute magnitude was $-14.8\pm0.5$. Both fireballs iluminated the southern parts of the country comparable to the Full Moon. In the north-western part of Poland these fireballs were observed as extremely bright objects which lit the sky with bright blue-greenish light. Observational properties of the fireballs ----------------------------------------- The PF311015a Okonek fireball appeared over Western Pomerania moving almost directly from east to west. The beginning of the meteor was recorded 117.88 km over the Radodzierz Lake. The entry velocity was $33.2 \pm 0.1$ km/s and was slightly higher than the mean velocity of Southern Taurids of 28 km/s[^3]. The peak brightness was observed at the height 82.5 km about 10 km east over Okonek city. The fireball travelled its 181.2 km luminous path in 5.62 seconds, ending at the height of 60.2 km over Z[ł]{}ocieniec. The basic characteristics of the PF311015a Okonek fireball are summarized in Table 3 and its luminous trajectory is shown in Figure 4. [lccc]{}\ \ & [**Beginning**]{} & [**Max. light**]{} & [**Terminal**]{}\ Vel. \[km/s\] & $33.2\pm0.1$ & $33.0\pm1.0$ & $30.2\pm1.0$\ Height \[km\] & $117.88\pm0.05$ & $82.5\pm0.1$ & $60.2\pm0.2$\ Long. \[$^\circ$E\] & $18.602\pm0.001$ & $17.04\pm0.01$ & $16.020\pm0.002$\ Lat. \[$^\circ$N\] & $53.6292\pm0.0004$ & $53.57\pm0.01$ & $53.526\pm0.001$\ Abs. magn. & $-0.8\pm0.5$ & $-16.0\pm0.4$ & $-2.3\pm0.3$\ Slope \[$^\circ$\] & $19.31\pm0.05$ & $18.39\pm0.05$ & $17.78\pm0.05$\ Duration &\ Length &\ Stations &\ \ & [**Observed**]{} & [**Geocentric**]{} & [**Heliocentric**]{}\ RA \[$^\circ$\] & $50.10\pm0.08$ & $51.06\pm0.07$ & -\ Decl. \[$^\circ$\] & $17.10\pm0.06$ & $15.11\pm0.07$ & -\ Vel. \[km/s\] & $33.2\pm0.1$ & $31.0\pm0.1$ & $37.2\pm0.1$\ The PF311015a Okonek fireball appeared as a meteor with absolute magnitude of $-0.8\pm0.5$. The brightness was increasing slowly by about 2 mag throughout the first second of the flight. While during the next 1.5 seconds much more steep increase of brightness was observed, ending with plateau lasting about one second when the peak brightness reaching $-16.0\pm0.4$ mag was recorded. The plateau finished abruptly at around 3.8 second of the flight with steep and almost linear decrease of brightness. At the terminal point the brightness of the meteor was $-2.3\pm0.5$ mag. At the end of the plateau phase a bright persistent train appeared. It started to shine with absolute magnitude of $-15$, and it slowly faded to $-13$ mag during almost two seconds. After that moment its brightness started to decrease much faster. Figure 5 shows the light curve of the PF311015a Okonek fireball and its persistent train. The PF311015b Ostrowite fireball appeared over the central Poland moving almost from south to north. It started its luminous path at the height of 108.05 km over place located about 5 km west of Konin. The entry velocity was $33.2 \pm 0.1$ km/s and was identical to the velocity of Okonek fireball. The peak brightness was observed at height of 82.2 km over the place located 5 km east of Ostrowite. The fireball travelled its 62.77 km luminous path in 2.0 seconds and ended at height 57.86 km with terminal velocity equal to $31.8 \pm 0.5$ km/s. The basic characteristics of the PF311015b Ostrowite fireball are summarized in Table 4, while its luminous trajectory is shown in Figure 4. [lccc]{}\ \ & [**Beginning**]{} & [**Max. light**]{} & [**Terminal**]{}\ Vel. \[km/s\] & $33.2\pm0.1$ & $32.2\pm0.5$ & $31.8\pm0.5$\ Height \[km\] & $108.05\pm0.02$ & $82.2\pm0.1$ & $57.86\pm0.03$\ Long. \[$^\circ$E\] & $18.1654\pm0.0003$ & $18.11\pm0.01$ & $18.0685\pm0.0007$\ Lat. \[$^\circ$N\] & $52.2005\pm0.0002$ & $52.37\pm0.01$ & $52.5313\pm0.0004$\ Abs. magn. & $-0.3\pm1.2$ & $-14.8\pm0.5$ & $-3.6\pm0.4$\ Slope \[$^\circ$\] & $52.29\pm0.05$ & $53.12\pm0.05$ & $52.96\pm0.05$\ Duration &\ Length &\ Stations &\ \ & [**Observed**]{} & [**Geocentric**]{} & [**Heliocentric**]{}\ RA \[$^\circ$\] & $52.09\pm0.06$ & $51.45\pm0.06$ & -\ Decl. \[$^\circ$\] & $15.78\pm0.07$ & $14.61\pm0.06$ & -\ Vel. \[km/s\] & $33.2\pm0.1$ & $31.23\pm0.109$ & $37.37\pm0.06$\ The PF311015b Ostrowite fireball appeared as object with absolute magnitude of $-0.3\pm1.2$. During the first second of the flight its brightness was increasing almost linearly to the value of $-9$ mag. After that, within a period of about 0.2 seconds, the brightness of the fireball increased by a factor of 100, reaching the plateau phase lasting for about 0.5 seconds. During this plateau the maximum absolute magnitude of $-14.8\pm0.5$ was recorded. While starting from 1.7 second of the flight the sudden drop of brightness was observed. So that the luminous path ended after about 2 seconds of the flight, with he terminal magnitude equal to $-3.6\pm0.4$. As in case of Okonek fireball, the bright persistent train was observed as well. It started to be visible just after the plateau phase with brightness of $-13$ mag, and within over one second it faded to $-3$ magnitude. The light curve of the PF311015b Ostrowite fireball and its persistent train is plotted in Figure 6. Orbits of the fireballs ----------------------- The orbital parameters of the Okonek and Ostrowite fireballs as computed from our observations are shown in Table 5. For comparison the orbital parameters of comet 2P/Encke are listed as well. The orbits of both fireballs are located almost in the ecliptic plane and have high eccentricity with perihelion distance slightly less than 0.3 a.u. The similarity of both orbits is evident with the Drummond criterion $D_D$ (Drummond 1981), and is equal to only 0.011. --------------------- ------------ ------------ ------------ ------------ ------------- ------------ ------------ $1/a$ $e$ $q$ $\omega$ $\Omega$ $i$ P \[1/AU\] \[AU\] \[deg\] \[deg\] \[deg\] \[years\] PF311015a Okonek 0.4440(55) 0.8690(21) 0.2948(17) 120.8(2) 37.77623(1) 4.73(12) 3.379(73) PF311015b Ostrowite 0.4408(50) 0.8720(17) 0.2903(15) 121.2(2) 37.98982(1) 5.636(51) 3.51(7) 2005 UR 0.4420(36) 0.8797(15) 0.2723(1) 141.03(14) 19.555(147) 6.972(26) 3.40(4) 2005 TF50 0.4401(5) 0.8689(3) 0.2978(3) 159.898(8) 0.666(23) 10.699(14) 3.425(6) 2P/Encke 0.45144(1) 0.84833(1) 0.33596(1) 186.546(1) 334.5682(1) 11.7815(1) 3.29698(1) --------------------- ------------ ------------ ------------ ------------ ------------- ------------ ------------ Object 2P/Encke 2005 UR 2005 TF50 PF311015a Okonek PF 311015b Ostrowite --------------------- ---------- --------- ----------- ------------------ ---------------------- 2P/Encke - 0.119 0.072 0.093 0.099 2005 UR 0.119 - 0.052 0.044 0.036 2005 TF50 0.072 0.052 - 0.045 0.042 PF311015a Okonek 0.093 0.044 0.045 - 0.011 PF311015b Ostrowite 0.099 0.036 0.042 0.011 - Comparison of the orbits of Okonek and Ostrowite fireballs to orbits of asteroids listed in Near Earth Objects - Dynamic Site (NEODyS-2)[^4] allowed us to select couple of asteroids with Drummond criterion $D_D<0.109$. Two of them are especially interesting. The Apollo type 2005 TF50 asteroid was discovered on 2005 October 10 by M. Block at the Steward Observatory, Kitt Peak. The absolute magnitude of the object is 20.3 mag which indicates the size of 260-590 meters. Its Tisserand parameter has value of 2.933. 2005 TF50 was listed in Porubčan et al. (2006) as one of NEOs connected with 2P/Encke and Taurid complex meteor showers. It is close to 7:2 resonance with Jupiter - the same resonance that was suggested as a possible source of Taurid enhanced activity by Asher (1991). The Drummond criterion describing the similarity of the orbit of 2005 TF50 to orbits of Okonek and Ostrowite fireballs is 0.045 and 0.042, respectively. The 2005 UR asteroid was discovered by Catalina Sky Survey on 2005 October 23. It belongs to the Apollo group and has a Tisserand parameter of 2.924. The absolute magnitude of the object is 21.6 mag which indicates the size of 140-320 meters. 2005 UR was not included in work of Porubčan et al. (2006) most probably due to the fact that paper was already written when the asteroid was discovered. On the other hand it was listed by Jopek (2011) as one of the parent bodies of the Taurid complex. The similarity of orbits of Okonek and Ostrowite fireballs to the orbit of 2005 UR is even more evident than in case of 2005 TF50 with Drummond criterion values 0.044 and 0.036, respectively. The interesting fact about 2005 UR asteroid is its close approach to Earth on 2005 October 30 at 13:11 UT with the distance of 0.041 a.u. The time of the close passage is at exactly the same moment as outburst of fireball activity of 2005 Taurids (Dubietis and Arlt, 2006). Table 6 lists the Drummond criterion $D_D$ values for PF311015a Okonek and PF311015b Ostrowite fireballs, 2005 UR and 2005 TF50 Near Earth Asteroids and comet 2P/Encke. Additionally, Figure 7 shows orbits of both fireballs in the inner Solar System together with the orbits of 2005 UR and 2005 TF50 asteroids and comet 2P/Encke. Modeling -------- A numerical integration of the orbital parameters backwards in time has been performed in order to test the link between the fireballs PF311015a Okonek and PF311015b Ostrowite, two NEOs: 2005 UR, 2005 TF50 and comet 2P/Encke. For the integrations of the asteroids and test particles representing fireball, the RADAU integrator in the Mercury software was used (Chambers 1999). The test particles means a series of clones of the radiant position and geocentric velocity of a fireball generated within the measurement uncertainties, that were later converted into orbital elements and propagated in the backward integration together with orbits of NEOs. We generated 100 massless clones of the fireballs individually. The model of the Solar System used in integrations included: 8 planets, four asteroids (Ceres, Pallas, Vesta, and Hygiea), and the Moon as a separate body. Additionally, we included the radiation pressure here as well. The positions and velocities of the perturbing planets and the Moon were taken from the DE406 (Standish 1998). The initial orbital elements of asteroids 2005 UR and 2005 TF50 and comet 2P/Encke were taken from JPL solar system dynamics web site.[^5] Together with initial orbital elements of asteroids and comet, the test particles were integrated to the same epoch of the beginning of the integration. Next, the backward integration was continued for 5000 yr. During the evolution the generated stream has been widely dispersed in longitude, therefore, we used Steel et al. (1991) criterion, $D_S$, instead of a conventional similarity functions (Southworth & Hawkins 1963, Drummond 1981, Jopek 1993). With the values being less than 0.15, the evolution of the $D_S$ criterion reveals a link between Okonek fireball and 2P/Encke, 2005 TF50, and 2005 UR, through 2300, 1600, and 1600 years, respectively (left panels in Figure 8). In case of Ostrowite fireball, the evolution of the $D_S$ criterion shows similarity through shorter period of times: 2000, 450, and 400 years with 2P/Encke, 2005 TF50, and 2005 UR, respectively (right panels of Figure 8). If there is a link between two bodies then the value of the dissimilarity criterion is very low at the moment of their separation, and increases with time. In theory, analysing results of the backward integration, we start in a moment when some time passed since the separation. Therefore, we start with a higher value of the dissimilarity criterion, then it decreases reaching a minimum value (at the possible moment of the separation). And then it increases again because in the integration two bodies are still treated as separate objects, as if the separation did not occur – unless we tell the program to stop integration when a given condition is fulfilled. We would see more complex image when involved in a study are objects which undergo a stronger perturbation and are in resonance with a planet (particularly Jupiter). The distance of an asteroid from the Jupiter’s orbit, characterised by the semimajor axis and aphelion distance, influences the amplitudes and rates of changes of the perihelion distance ($q$), eccentricity ($e$), and inclination ($i$). Moreover, all of used by us in the study asteroids are close to 7:2 resonance with Jupiter. All of this has its reflection is amplitudes and rates of changes of the dissimilarity criterion as well. Thus, instead of a stable, linear decreasing in going back in time to the possible separation moment we observe sinusoidal curve. Additional outcome of our work concerns the relation between asteroids themselves. Figure 9 shows the time evolution of semimajor axis ($a$), eccentricity ($e$), perihelion distance ($q$), inclination ($i$), argument of perihelion ($\omega$), and longitude of ascending node ($\Omega$) of NEAs and the comet. Our results show orbital similarity between 2005 TF50 and 2005 UR in the interval of almost 4000 years applying both $D_S$ and $D_{SH}$ (see Figure 10 for $D_S$ plot). The lower values are obtained around 2600 years, which corresponds with low values of $D_{S}$ and $D_{SH}$ when comparing asteroids’ orbit with orbit of 2P/Encke. This may suggest that around that time separation of both asteroids might have occurred. We generated 100 clones of asteroids 2005 UR and 2005 TF50, using their orbital covariance matrix taken from the JPL Horizon. Analysing results of the backward integration of those objects and their clones shows, as expected, that an orbit calculated from a short data-arc span (6 days for 2005 UR) would produce orbits with higher dispersion in time than for a longer data-arc (26 days for 2005 TF50). However, the amplitudes and rates of changes of orbital elements of clones have similar range and pattern, especially for 2005 TF50, for which orbital uncertainties of their nominal orbit are smaller. The current and past $D_S$ values for each NEO and 2P/Encke are not extremely low indicating that real separation of all these three bodies might took place 20000-30000 years ago in one catastrophic event which created the whole Taurid complex (Asher et al. 1993, Babadzhanov et al. 2008). Leaving for a while a connection with 2P/Encke comet, we can speculate about both NEOs and both fireballs origin. The orbits of 2005 UR and 2005 TF50 are very similar through a period of last 3600 years. Almost zero values of $D_S$ criterion are observed at moments of $-700$, $-1300 \div -1500$ and $-2700$ years (Figure 10). What is interesting is that the deep minimum of $D_S$ around the moment of $-1500$ years was obtained for each NEO-fireball orbit combination (see Figure 8). This epoch might be suspected as the time when larger body was disrupted creating both NEOs and meteoroids which caused the fireballs. Still we have take into account earlier minimum observed at epoch around $-400$ years. This is the first deep minimum of $D_S$ observed for all NEO-fireball combinations. In case of the disruption at that moment further backward integrations for earlier epochs have no physical sense. Summary ======= In this paper we presented an analysis of the multi-station observations of two bright Southern Taurid fireballs which occurred over Poland on 2015 October 31. Moreover, we investigated their connection with two NEOs and comet 2P/Encke. Our main conclusions are as follows: - both meteors are similar with many aspects including brightness higher than Full Moon, shape of the light curve, entry velocity, persistent train and orbital parameters ($D_D$ of only 0.011), - among over dozen of NEOs identified as possible parent bodies of Taurid complex two, namely 2005 UR and 2005 TF50, have orbits which are very similar to the orbits of observed fireballs (with $D_D<0.045$), - similarity of orbits of both fireballs and 2005 UR asteroid is especially interesting due to the fact that the close flyby of this NEO was observed exactly during last high maximum of Taurid complex shower in 2005, - the numerical backward integration of the orbital parameters of both fireballs and NEOs backwards in time, which has been performed in this work, indicates many similarities between orbits of these objects during past 5000 years. However, about 1500 years ago, $D_S$ criterion has close to zero values for each of NEO-fireball, NEO-NEO and fireball-fireball pair suggesting at that moment a disruption of a larger body might took place, - although, we could not have confirmed unequivocally the relation between fireballs and 2005 UR and 2005 TF50, we showed that at least both asteroids could be associated, having the same origin in a disruption process that separates them. The Taurid complex is certainly one of the most interesting objects in the Solar System. It is able to produce both impressive meteor maxima and extremely bright fireballs (Dubietis & Arlt 2006, Spurný 1994). Additionally, it can be connected with catastrophic events like Tunguska (Kresak 1978, Hartung 1993) and can affect the climate on Earth (Asher & Clube 1997). Accurate observations and analysis of all kind of bodies associated with the Taurid complex are then very a important task, demanding to continue and affecting the safety of our planet. Acknowledgments {#acknowledgments .unnumbered} =============== This work was supported by the NCN grant number 2013/09/B/ST9/02168. [99]{} Asher, D.J., 1991, The Taurid meteoroid complex, PhD thesis, New College, Oxford. Asher, D.J., Clube S.V.M., Steel D.I., 1993, MNRAS, 264, 93 Asher, D.J., Clube S.V.M., 1997, Celestial Mechanics and Dynamical Astronomy, v. 69, Issue 1/2, p. 149-170 Asher, D.J., Izumi, K., 1998, MNRAS, 297, 23 Babadzhanov, P.B., Williams, I.P., Kokhirova, G.I., 2008, MNRAS, 386, 1436 Brown, P., Marchenko, V., Moser, D.E.. 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J. Baggaley and V. Porubcan. Proceedings of the International Conference held at Tatranska Lomnica, Slovakia, August 17-21, 1998. Astronomical Institute of the Slovak Academy of Sciences, 1999., p.131 Olech, A. et. al, 2006, Proceedings of the International Meteor Conference, Oostmalle, Belgium, 15-18 September, Edt.: Bastiaens, L., Verbert, J., Wislez, J.-M., Verbeeck, C. International Meteor Organisation, p. 53 Popescu, M., Birlan, M., Nedelcu, D.A., Vaubaillon, J., Cristescu, C.P., 2014, A&A, 572, A106 Porubčan, V., Kronoş, L., Williams, I.P., 2006, Contributions of the Astronomical Observatory Skalnaté Pleso, 36, 103 Shrbený, L., Spurný, P., 2012, Asteroids, Comets, Meteors 2012, Proceedings of the conference held May 16-20, 2012 in Niigata, Japan. LPI Contribution No. 1667 SonotaCo, 2009, WGN, 37, 55 Southworth, R.B., Hawkins, G.S., 1963, Smithson. Contr. Astrophys, 7, 261 Spurný, P., 1994, Planetary and Space Science, vol. 42, no. 2, p. 157 Standish, E. M., 1998, JPL IOM, 312.F - 98 - 048 Steel, D.I., Asher, D.J., Clube, S.V.M., 1991, MNRAS, 251, 632 Tubiana, C., Snodgrass, C., Michelsen, R., Haack, H., Bohnhardt, H., Fitzsimmons, A., Williams, I.P., 2015, A&A, 584, A97 Whipple, F., 1940, Proc. Amer. Phil. Soc., 83, 711 Wiśniewski, M. et al., 2012, European Planetary Science Congress 2012, held 23-28 September, 2012 in Madrid, Spain. Żo[ł]{}dek, P., Olech, A., Wiśniewski, M., Kwinta, M., 2007, Earth, Moon and Planets, 100, 215 Żo[ł]{}dek, P. et al., 2009, WGN, 37, 161 Żo[ł]{}dek, P., 2012, Proceedings of the International Meteor Conference, Sibiu, Romania, 15-18 September, 2011 Eds.: Gyssens, M.; and Roggemans, P. p. 53 [^1]: e-mail: olech@camk.edu.pl [^2]: http://imo.net/node/1645 [^3]: Meteor Data Center:\ http://www.astro.amu.edu.pl/$\sim$jopek/MDC2007/ [^4]: http://newton.dm.unipi.it/neodys/index.php [^5]: http://ssd.jpl.nasa.gov

--- author: - | [^1]\ Institut de Physique Nucléaire d’Orsay, CNRS-IN2P3, Université Paris-Sud & Paris-Saclay, 91406 Orsay, France\ E-mail: title: 'Studying nucleon structure via Double Deeply Virtual Compton Scattering (DDVCS)' --- Introduction ============ There are essentially three experimental golden channels for direct measurements of GPDs: the electroproduction of a photon $eN\rightarrow eN\gamma$ which is sensitive to the deeply virtual Compton scattering (DVCS) amplitude, the photoproduction of a lepton pair $\gamma N\rightarrow l\bar{l}N$ which is sensitive to the timelike Compton scattering (TCS) amplitude, and the electroproduction of a lepton pair $eN\rightarrow eNl\bar{l}$ which is sensitive to the double deeply virtual Compton scattering (DDVCS) amplitude. Only the latter provides the framework necessary for an uncorrelated measurement of a GPD ($\xi',\xi,t$) as a function of both scaling variable $\xi'$ and $\xi$ [@ref1; @ref2]. The former two reactions cannot entirely serve the purpose of testing the angular momentum sum rule [@ref3] due to the reality of the final- or initial-state photons, which leads to the restriction $\xi'=\pm \xi$. For instance, the Compton form factors (CFF) $\mathcal{H}$ associated with the GPD $H$ and accessible in DVCS cross section or beam spin asymmetry experiments can be written $$\begin{aligned} \relax\mathcal{H}(\xi'=\xi,\xi,t)=\sum_{q}e_q^2&\bigg\{&\mathcal{P}\int_{-1}^1dx~H^q(x,\xi,t)\bigg[\frac{1}{x-\xi}+\frac{1}{x+\xi}\bigg] \nonumber\\ &&-i\pi\big[H^q(\xi,\xi,t)-H^q(-\xi,\xi,t)\big]\bigg\} \label{eq1}\end{aligned}$$ where the sum runs over all parton flavors with elementary electrical charge $e_q$, and $\mathcal{P}$ indicates the Cauchy principal value of the integral. While the imaginary part of the CFF accesses the GPD values at $\xi'=\pm \xi$, it is clear from Eq. \[eq1\] that the real part of the CFF is a more complex quantity involving the convolution of parton propagators and the GPD values out of the diagonals $\xi'=\pm \xi$, that is a domain that cannot be resolved unambiguously with DVCS experiments. Because of the virtuality of the final state photon, DDVCS provides a way to circumvent the DVCS limitation, allowing to vary independently $\xi'$ and $\xi$. Considering the same GPD $H$, the corresponding CFF for DDVCS process writes $$\begin{aligned} \mathcal{H}(\xi',\xi,t)=\sum_{q}e_q^2&\bigg\{&\mathcal{P}\int_{-1}^1dx~H^q(x,\xi,t)\bigg[\frac{1}{x-\xi'}+\frac{1}{x+\xi'}\bigg] \nonumber\\ &&-i\pi\big[H^q(\xi',\xi,t)-H^q(-\xi',\xi,t)\big]\bigg\} \label{eq2}\end{aligned}$$ providing access to the scaling variable $\xi' \neq \xi$. The DDVCS process is most challenging from the experimental point of view due to the small magnitude of the cross section and requires high luminosity and full exclusivity of the final state. Moreover, the difficult theoretical interpretation of electron-induced lepton pair production when detecting the $e^+ e^-$ pairs from the decay of the final virtual photon, hampers any reliable experimental study. Taking advantage of the energy upgrade of the CEBAF accelerator, it is proposed to investigate the electroproduction of $\mu^+ \mu^-$ di-muon pairs and measure the beam spin asymmetry of the exclusive $ep\rightarrow e'p'\gamma^* \rightarrow e'p'\mu^+\mu^-$ reaction in the hard scattering regime [@intent; @intent2; @ref6]. At sufficiently high virtuality of the initial space-like virtual photon and small enough four-momentum transfer to the nucleon with respect to the photon virtuality ($-t \ll Q^2$), DDVCS can be seen as the absorption of a space-like virtual photon by a parton of the nucleon, followed by the quasi-instantaneous emission of a time-like virtual photon by the same parton, which finally decays into a di-muon pair (Fig. \[fig1\]). $Q^2$ and $Q'^2$ represent the virtuality of the incoming space-like and outgoing time-like photons. The scaling variable $\xi'$ and $\xi$ write $$\begin{aligned} \xi' = \frac{Q^2-Q'^2+t/2}{2Q^2/x_\text{B}-Q^2-Q'^2+t}~~\text{and}~~ \xi = \frac{Q^2+Q'^2}{2Q^2/x_\text{B}-Q^2-Q'^2+t} \label{xipxi}\end{aligned}$$ from which one obtains $\xi' = \xi \frac{Q^2-Q'^2+t/2}{Q^2+Q'^2} .$ This relation indicates that $\xi'$, and consequently the CFF imaginary part, is changing sign about $Q^2=Q'^2$, which procures a strong testing ground of the universality of the GPD formalism. ![The handbag diagram symbolizing the DDVCS direct term with di-muon final states.[]{data-label="fig1"}](DDVCS.pdf){width=".4\textwidth"} In this proceeding, the feasibility of a DDVCS experiment at JLab 12 GeV is discussed. Section \[sec2\] describes DDVCS kinematics and experimental observables. Section \[sec3\] reports model-predicted experimental projection at a certain luminosity with ideal detectors. Preliminary conclusions about this study are drawn in the last section. Kinematics and experimental observables {#sec2} ======================================= The following kinematics cuts to ensure applicability of the GPD formalism have been applied: the center-of-mass energy $W>2$ GeV to ensure the deep inelastic scattering regime; $Q^2>1~($GeV$/c^2)^2$ to ensure the reaction at parton level; $t >-1~($GeV$/c^2)^2$ to support the factorization regime; and $Q'^2>(2m_\mu)^2$ to ensure the production of a di-muon pair. Fig. \[fig2\] shows the DDVCS allowed $(Q^2,~x_\text{B})$ phase space with kinematics cuts and one $(t,~Q'^{2})$ phase space at a specific $(Q^2,~x_\text{B})$ set. In the allowed region represented by the shaded area, kinematics bins with uniform widths have been chosen. $\Delta Q^2=0.5~($GeV$/c^2)^2$, $\Delta x_\text{B}=0.05$, $\Delta t=0.2~($GeV$/c^2)^2$ and $\Delta Q'^2=0.5~($GeV$/c^2)^2$, which is sufficiently small to allow a first-order estimation of the integral cross section. As a preliminary study of DDVCS, only the bins at $Q^2<5~($GeV$/c^2)^2$ have been studied, where the cross section is supposedly larger than at high $Q^2$. As a consequence, 664 four-dimensional bins have been considered. The bins boundary are shown in Fig. \[fig2\] as the dashed lines. ![Kinematics phase spaces: left panel shows the $(Q^2,~x_\text{B})$ phase space with physics inspired cuts. The shaded area represents the physics region of interest, and the point represents $(Q^2=1.25~($GeV$/c^2)^2,~x_\text{B}=0.1)$ whose correlated $(t,~Q'^{2})$ phase space is shown in the right panel together with the region of interest (shaded area).[]{data-label="fig2"}](bin1.pdf "fig:"){width=".49\textwidth"} ![Kinematics phase spaces: left panel shows the $(Q^2,~x_\text{B})$ phase space with physics inspired cuts. The shaded area represents the physics region of interest, and the point represents $(Q^2=1.25~($GeV$/c^2)^2,~x_\text{B}=0.1)$ whose correlated $(t,~Q'^{2})$ phase space is shown in the right panel together with the region of interest (shaded area).[]{data-label="fig2"}](bin2.pdf "fig:"){width=".49\textwidth"} The lepton-pair electroproduction process consists of three interfering elementary mechanisms, depicted in Fig. \[fig3\], with implied crossed contributions. The 7-fold differential cross section is proportional to the square of the total amplitude that is the coherent sum of the three processes, i.e. $d^7\sigma/(dQ^2dx_BdtdQ'^2d\phi d\Omega_\mu)$ $\propto |\mathcal{T}_\text{DDVCS}+\mathcal{T}_{\text{BH}_1}+\mathcal{T}_{\text{BH}_2}|^2$. We consider in this study the 5-fold cross section, integrating over the muon solid angle. The integration leads to the vanishing of the interference contributions originated from the BH$_2$ amplitude: $d^5\sigma/(dQ^2dx_BdtdQ'^2d\phi) \propto |\mathcal{T}_\text{DDVCS}+\mathcal{T}_{\text{BH}_1}|^2+|\mathcal{T}_{\text{BH}_2}|^2$ [@ref2; @ref7]. Though partial information is sacrificed, this simplification offers an easier understanding of this totally unexplored reaction. The cross section without target polarization can be described in terms of different contributions $$\sigma_{P}^{e}=\sigma_{\text{BH}_1}+\sigma_{\text{BH}_2} +\sigma_\text{DDVCS}+P\widetilde{\sigma}_\text{DDVCS}+(-e)\left( \sigma_{\text{INT}_1}+P\widetilde{\sigma}_{\text{INT}_1} \right) \label{eq3}$$ where, to simplify the notation, $\sigma$ stands for the 5-fold differential cross section, $e$ is the lepton beam electric charge, and $P$ is the polarization of the beam. The subscript INT$_1$ represents the interference terms between the DDVCS and the BH$_1$ amplitudes. BH terms are calculable since the nucleon form factors are well-known at small $t$. DDVCS terms are bi-linear in CFFs, while interference terms are linear. $\sigma_\text{DDVCS}$ and $\sigma_{\text{INT}_1}$ are sensitive to the real part of CFFs, while $\widetilde{\sigma}_\text{DDVCS}$ and $\widetilde{\sigma}_{\text{INT}_1}$ are sensitive to the imaginary part. ![Subprocesses contributing to electroproduction of a di-muon pair including DDVCS (left) and two kinds of Bethe-Heitler processes, i.e. BH$_1$ (middle) and BH$_2$ (right).[]{data-label="fig3"}](dd.pdf "fig:"){height=".2\textwidth"}          ![Subprocesses contributing to electroproduction of a di-muon pair including DDVCS (left) and two kinds of Bethe-Heitler processes, i.e. BH$_1$ (middle) and BH$_2$ (right).[]{data-label="fig3"}](dbh1.pdf "fig:"){height=".2\textwidth"}          ![Subprocesses contributing to electroproduction of a di-muon pair including DDVCS (left) and two kinds of Bethe-Heitler processes, i.e. BH$_1$ (middle) and BH$_2$ (right).[]{data-label="fig3"}](dbh2.pdf "fig:"){height=".2\textwidth"} Considering polarized positron and electron beams, single contributions can be separated from the three experimental observables: unpolarized cross section with electron beam ($\sigma_\text{UU}$), beam spin cross section difference with polarized electron and positron beam ($\Delta\sigma_\text{LU}$), and beam charge cross section difference ($\Delta\sigma^\text{C}$). From Eq. \[eq3\], $$\left\{ \begin{aligned} &\sigma_\text{UU}=\frac{1}{2}\left(\sigma_{+}^-+\sigma_{-}^-\right) &&=\sigma_{\text{BH}_1}+\sigma_{\text{BH}_2} +\sigma_{\text{DDVCS}} +\sigma_{\text{INT}_1}, \\ &\Delta\sigma_\text{LU}=\frac{1}{4}\left[\left(\sigma_{+}^--\sigma_{-}^-\right)-\left(\sigma_{+}^+-\sigma_{-}^+\right)\right] &&=\widetilde{\sigma}_{\text{INT}_1}, \\ &\Delta\sigma^\text{C}=\frac{1}{4}\left[\left(\sigma_{+}^-+\sigma_{-}^-\right)-\left(\sigma_{+}^++\sigma_{-}^+\right)\right] &&=\sigma_{\text{INT}_1}. \end{aligned} \right. \label{eq4}$$ It is difficult to extract CFFs from DDVCS term of bi-linear combination, $\Delta\sigma^\text{C}$ therefore provides pure interference term. The imaginary part of CFFs can be extracted from $\Delta\sigma_\text{LU}$, which provides directly the information for GPDs. In addition, we can also obtain pure $\sigma_\text{DDVCS}$ when combining $\sigma_\text{UU}$ and $\Delta\sigma^\text{C}$, and $\widetilde\sigma_\text{DDVCS}$ when combining $\Delta\sigma_\text{LU}$ and the electron beam spin cross section difference. The experimental projections of these observables have been performed, and is discussed in the next section. Projections {#sec3} =========== The projections have been performed in the ideal situation that all the particles of the final state can be detected with 100% efficiency. The count-rate calculation was done for a luminosity $\mathrsfso{L}=10^{37}~\text{cm}^{-2}\text{s}^{-1}$ considering 100 days running time equally distributed between each lepton beam charge. The number of events, for each five-dimensional bin ($Q^2,~x_\text{B},~t,~Q'^{2}~\text{and}~\phi$), was determined following $$\begin{aligned} N=\frac{d^5\sigma}{dQ^2dx_BdtdQ'^2d\phi} \cdot\Delta Q^2 \cdot\Delta x_B \cdot\Delta t \cdot\Delta Q'^2 \cdot\Delta\phi \cdot\mathrsfso{L} \cdot T, \label{eq5}\end{aligned}$$ where the differential cross section has been calculated with the VGG model [@refVGG] set at the central values of each four-dimensional bin at the beam energy of 11 GeV. Besides, 24 bins in $\phi$ 15$^\circ$-wide have been considered. Fig. \[fig4\] shows the observables with statistic errors at some different $Q'^2$ and a set of fixed ($Q^2,~x_\text{B}~\text{and}~t$) as a function of $\phi$ (upper half). The cross section decreases generally as $Q'^2$ increases, since the process at $Q'^2=0$ is equivalent to the DVCS process having one less electromagnetic vertex. $\Delta\sigma_\text{LU}$ in the $Q^2>Q'^2$ and $Q^2<Q'^2$ regions has opposite signs due to the antisymmetric property of GPD [@ref7]. The bottom half shows the ones at some different $t$ and a set of fixed ($Q^2,~x_\text{B}~\text{and}~Q'^2$). The cross section and precision decrease as $-t$ increases. ![The upper half of the figure shows the VGG projections at $Q^2 =1.25~(\text{GeV}/c^2)^2$, $x_\text{B}=0.1$, $-t=0.15~(\text{GeV}/c^2)^2$ and $Q'^2=0.3,~0.8,~1.3~\text{and}~1.8~(\text{GeV}/c^2)^2$ (left to right). The panels of the top row show the unpolarized cross section, the middle panels show the beam-spin cross section difference, and the bottom panels show the beam-charge cross section difference. Note that each panel has its own y-axis scale. The bottom half shows the ones at $Q^2 =1.75~(\text{GeV}/c^2)^2$, $x_\text{B}=0.125$, $Q'^2=0.8~(\text{GeV}/c^2)^2$ and $-t=0.15,~0.35,~0.55~\text{and}~0.75~(\text{GeV}/c^2)^2$ (left to right).[]{data-label="fig4"}](zcombin.pdf){width="100.00000%"} Fig. \[fig5\] shows the correlated location of all the four-dimensional bins in the CFFs phase space $(\xi',~\xi)$. Among the 664 bins, 82% have $\sigma_\text{UU}$ with a relative error less than 10% , 22% have $\Delta\sigma^\text{C}$ of the same quality, and only 7% for $\Delta\sigma_\text{LU}$ in the vicinity of the DVCS diagonal. ![CFFs phase space: the solid line indicates $\xi'=\xi$ or $Q'^2=0$ that is the DVCS correlation, the dashed line indicates $\xi'=-\xi$ or $Q^2=0$ that is the TCS correlation, and the colored markers represent the successful four-dimensional bins of DDVCS process (Eq. 1.2). The blue open circles indicate the bins where $\sigma_\text{UU}$ has the relative error less than 10%. Some of them, represented by the red open squares, have $\Delta\sigma^\text{C}$ with the relative error less than 10%, and a few of them, represented by green open triangles, have the $\Delta\sigma_\text{LU}$ at the same level of accuracy. The black crosses indicate the failed bins where all the three observables have the relative errors greater than 10%.[]{data-label="fig5"}](cff_uu_2d.pdf "fig:"){width=".32\textwidth"} ![CFFs phase space: the solid line indicates $\xi'=\xi$ or $Q'^2=0$ that is the DVCS correlation, the dashed line indicates $\xi'=-\xi$ or $Q^2=0$ that is the TCS correlation, and the colored markers represent the successful four-dimensional bins of DDVCS process (Eq. 1.2). The blue open circles indicate the bins where $\sigma_\text{UU}$ has the relative error less than 10%. Some of them, represented by the red open squares, have $\Delta\sigma^\text{C}$ with the relative error less than 10%, and a few of them, represented by green open triangles, have the $\Delta\sigma_\text{LU}$ at the same level of accuracy. The black crosses indicate the failed bins where all the three observables have the relative errors greater than 10%.[]{data-label="fig5"}](cff_uu_c_2d.pdf "fig:"){width=".32\textwidth"} ![CFFs phase space: the solid line indicates $\xi'=\xi$ or $Q'^2=0$ that is the DVCS correlation, the dashed line indicates $\xi'=-\xi$ or $Q^2=0$ that is the TCS correlation, and the colored markers represent the successful four-dimensional bins of DDVCS process (Eq. 1.2). The blue open circles indicate the bins where $\sigma_\text{UU}$ has the relative error less than 10%. Some of them, represented by the red open squares, have $\Delta\sigma^\text{C}$ with the relative error less than 10%, and a few of them, represented by green open triangles, have the $\Delta\sigma_\text{LU}$ at the same level of accuracy. The black crosses indicate the failed bins where all the three observables have the relative errors greater than 10%.[]{data-label="fig5"}](cff_uu_lu_2d.pdf "fig:"){width=".32\textwidth"} Conclusion {#sec4} ========== The model-predicted projections of a DDVCS experiment indicate a high degree of feasibility at a challenging luminosity with exclusive final states completely detected. The unpolarized cross section with very small statistics error can be obtained. Although the beam charge cross section difference has less precision, a better extraction of the real part of CFFs can be performed. The beam spin cross section difference can be obtained accurately only at a few specific kinematics, but it is the most powerful tools to directly access the totally unexplored GPDs phase space, otherwise inaccessible. An additional feature supporting the importance of this observable is the sign change of the beam spin cross section difference as $Q'^2$ becomes larger than $Q^2$. This behaviour is a strong prediction of the GPD formalism [@ref7], and consequently provides a stringent test for experimental investigations. Due to the strong sensitivity to $Q'^2$ and $t$ as well as the small cross section and insufficient statistics at high values, the binning strategy will be adapted in the next phase of DDVCS exploration covering the whole kinematic phase space, and finally extracting the CFFs. Acknowledgement {#acknowledgement .unnumbered} =============== I would like to express my appreciation to my thesis supervisor, E. Voutier, for the guidance of this work, also to M. Guidal, S. Niccolai, and M. Vanderheaghen for helpful discussions. [99]{} M. Guidal and M. Vanderhaeghen, *Double Deeply Virtual Compton Scattering off the Nucleon*, *Phys. Rev. Lett.* [**90**]{} (2003) 012001. A. V. Belitsky and D. Müller, *Exclusive electroproduction of lepton pairs as a probe of nucleon structure*, *Phys. Rev. Lett.* [**90**]{} (2003) 022001. X. Ji, *Gauge-Invariant Decomposition of Nucleon Spin*, *Phys. Rev. D* [**78**]{} (1997) 610. M. Boer, A. Camsonne, K. Gnanvo, E. Voutier, Z. Zhao, [*et al.*]{}, Jefferson Lab Experiment [**LOI12-15-005**]{} (2015). (CLAS Collaboration) S. Stepanyan, [*et al.*]{}, Jefferson Lab Experiment [**LOI12-16-004**]{} (2016). I. V. Anikin, [*et al.*]{}, *Nucleon and nuclear structure through dilepton production*, *Acta Phys. Pol. B* [**49**]{} (2018) 741 A. V. Belitsky and D. Müller, *Probing generalized parton distributions with electroproduction of lepton pairs off the nucleon*, *Phys. Rev. D* [**68**]{} (2003) 116005. M. Vanderhaeghen, P. A. M. Guichon and M. Guidal, *Deeply virtual electroproduction of photons and mesons on the nucleon: Leading order amplitudes and power corrections*, *Phys. Rev. D* [**60**]{} (1999) 094017. [^1]: Supported by the China Scholarship Council (CSC) and the French Centre National de la Recherche Scientifique (CNRS).

--- abstract: 'Using the Cosmic Origins Spectrograph onboard the [*Hubble Space Telescope*]{}, we have obtained high-resolution ultraviolet observations of GD 362 and PG 1225-079, two helium-dominated, externally-polluted white dwarfs. We determined or placed useful upper limits on the abundances of two key volatile elements, carbon and sulfur, in both stars; we also constrained the zinc abundance in PG 1225-079. In combination with previous optical data, we find strong evidence that each of these two white dwarfs has accreted a parent body that has evolved beyond primitive nebular condensation. The planetesimal accreted onto GD 362 had a bulk composition roughly similar to that of a mesosiderite meteorite based on a reduced chi-squared comparison with solar system objects; however, additional material is required to fully reproduce the observed mid-infrared spectrum for GD 362. No single meteorite can reproduce the unique abundance pattern observed in PG 1225-079; the best fit model requires a blend of ureilite and mesosiderite material. From a compiled sample of 9 well-studied polluted white dwarfs, we find evidence for both primitive planetesimals, which are a direct product from nebular condensation, as well as beyond-primitive planetesimals, whose final compositions were mainly determined by post-nebular processing.' author: - 'S. Xu(许偲艺), M. Jura, B. Klein, D. Koester, B. Zuckerman' bibliography: - 'apj-jour.bib' - 'Ref.bib' title: 'Two Beyond-Primitive Extrasolar Planetesimals' --- [UTF8]{}[gbsn]{} INTRODUCTION ============ Planetesimals are building blocks of planets and their formation is a key step towards planet formation. How do planetesimals form? What determines their bulk composition? To answer these questions, we start by examining our own solar system. The overall configuration of the solar system is that volatile-depleted, dry rocky objects are ubiquitous relatively close to the Sun while volatile-rich, icy objects are found beyond the snow line. This correlation between the volatile fraction and heliocentric distance can be explained by primitive nebular condensation: refractory elements condensed closer to the Sun while volatile elements can only be incorporated into the planetesimals where the temperature is low enough. Many solar system objects have experienced some additional processing that changed their initial compositions. For example, it has been argued that a collision between a large asteroid and proto-Mercury stripped off most of Mercury’s silicate mantle, leaving it $\sim$70% iron by mass [@Benz1988]. Also, the “late veneer" has delivered a large amount of water and volatiles onto Earth [@Chyba1990]. Post-nebular processing, such as collisions, melting and differentiation, is important in redistributing the elements among solar system objects. Currently, the best way to measure the elemental compositions of planetesimals in the solar system is from meteorites, which are fragments from collisions among asteroids. Following @ONeillPlame2008, we classify all meteorites into two categories in this paper. (i) “Chondritic" is used to refer to chondrites, which are a direct product of nebular processing. Objects in this category are described as “primitive" planetesimals. (ii) “Non-chondritic" objects consist of achondrites, stoney-iron meteorites and iron meteorites. Examples of their parent bodies include the Moon, Mars or asteroids that have experienced various amounts of post-nebular processing. Planetesimals in this category are considered to be “beyond-primitive". What about planetesimal formation in extrasolar planetary systems? High-resolution, high-sensitivity spectroscopic observations of externally-polluted white dwarfs are a powerful tool for determining the bulk elemental compositions of extrasolar planetesimals [@Jura2013]. Calculations show that minor planets can survive the red giant stage of a star and persist into the white dwarf phase with most of their internal water and volatiles intact [@Jura2008; @JuraXu2010]. Orbital perturbations from one or multiple planets can cause these planetesimals to stray into the tidal radius of the white dwarf and get tidally disrupted [@DebesSigurdsson2002; @Bonsor2011; @Debes2012a], sometimes producing a dust disk that emits mostly in the infrared [@Jura2003; @Kilic2006b; @VonHippel2007; @Farihi2009; @XuJura2012]. Eventually, all this planetary debris is accreted onto the central white dwarf and pollutes its otherwise pure hydrogen or helium atmosphere. The first comprehensive abundance measurement of an externally-polluted white dwarf was performed by @Zuckerman2007, who identified 15 elements heavier than helium in the atmosphere of GD 362, including Mg, Si and Fe, which are often called the “common elements" [@Larimer1988]. The disrupted object had a minimum mass $\sim$10$^{22}$ g, which is comparable to that of a massive solar system asteroid. Three years later, the abundances of eight heavy elements were determined in the atmosphere of GD 40, including all the major rock-forming elements – O, Mg, Si and Fe [@Klein2010]. Now there are many more high-resolution optical spectroscopic studies of externally-polluted white dwarfs \[e.g., @Klein2011 [@Melis2011; @Zuckerman2011; @Farihi2011a; @Dufour2012; @Vennes2010; @Vennes2011a]\]. However, optical spectroscopy of externally-polluted white dwarfs typically does not enable sensitive detection of highly-volatile elements, such as carbon, nitrogen and sulfur, which are key to understanding the thermal history of the system. Ultraviolet spectroscopy is complimentary to optical observations in determination of volatile abundances. To-date, there are four white dwarfs with both published high-resolution optical and ultraviolet measurements[^1]; we are beginning to accumulate an atlas of the compositions of extrasolar planetesimals. To zeroth order, we find that they are strikingly similar to meteorites in the solar system: (i) O, Mg, Si and Fe are always dominant and their sum is more than 85% of the accreted mass; (2) volatile elements, especially C, are typically depleted by more than a factor of 10 compared to solar abundances[^2]. In this paper, we report ultraviolet spectroscopic observations of GD 362 and PG 1225-079 with the Cosmic Origins Spectrograph (COS) onboard the [*Hubble Space Telescope*]{} ([*HST*]{}), complimentary to previous optical studies from the Keck High Resolution Echelle Spectrometer (HIRES) [@Zuckerman2007; @Klein2011]. PG 1225-079 has been observed with the low-resolution International Ultraviolet Explorer (IUE) [@Wolff2002]; there is no previous ultraviolet spectroscopy for GD 362. The rest of the paper is organized as follows. Data reduction is summarized in section 2 and atmospheric abundance determinations are reported in section 3. In section 4, we used a reduced chi-squared analysis to look for solar system analogs to the accreted parent bodies. The formation mechanisms of extrasolar planetesimals are assessed in section 5 and conclusions are given in section 6. In Appendix A, we report the [*Herschel*]{} Photodetecting Array Camera and Spectrometer (PACS) observation of GD 362. In Appendix B, we extend the reduced chi-squared analysis to two additional externally-polluted helium white dwarfs with both high-resolution optical and ultraviolet observations. OBSERVATIONS AND DATA REDUCTION =============================== GD 362 and PG 1225-079 were observed during [*HST*]{}/COS Cycle 18 under program 12290. These two white dwarfs are too cool to be observed effectively with the G130M grating centering around 1300 [Å]{}, as was employed by @Jura2012 and @Gaensicke2012 for other hotter white dwarfs. Instead, the G185M grating was used with a central wavelength of 1921 [Å]{} and wavelength coverage of 1800 – 1840 [Å]{}, 1903 – 1940 [Å]{} and 2008 – 2044 [Å]{}. The spectral resolution was $\sim$18,000. Total exposure times were 7411 and 1805 sec for GD 362 and PG 1225-079, respectively. The raw data were processed using the standard pipeline CALCOS 2.13.6. The fluxes at 2030 [Å]{} are 2.9 $\times$ 10$^{-15}$ erg s$^{-1}$ cm$^{-2}$ [Å]{}$^{-1}$ and 1.5 $\times$ 10$^{-14}$ erg s$^{-1}$ cm$^{-2}$ [Å]{}$^{-1}$ for GD 362 and PG 1225-079, respectively, in approximate agreement with broadband NUV fluxes from the [*GALEX*]{} satellite. The signal-to-noise ratio (SNR) in the original un-smoothed spectrum was 6 for PG 1225-079 and 4 for GD 362. Following previous data reduction procedures [@Klein2010; @Klein2011; @Jura2012], for PG 1225-079, equivalent widths (EWs) of each spectral line were measured in the un-smoothed spectra by fitting a Voigt profile with three different nearby continuum intervals in IRAF. The EW uncertainty is calculated by adding the standard deviation of the three EWs and the average uncertainty from the profile fitting in quadrature. The EW upper limit is obtained by artificially inserting a spectral line with different abundance into the model and comparing with the data. We adopt a different method to measure the EW for C I 1930.9 [Å]{} in GD 362, as described in section 3.1. The measured values are listed in Tables \[Tab: LinesGD\] and \[Tab: LinesPG\] for GD 362 and PG 1225-079, respectively. The average Doppler shift relative to the Sun for PG 1225-079 is 42 $\pm$ 13 km s$^{-1}$, in essential agreement with the value 49 $\pm$ 3 km s$^{-1}$ derived from optical studies [@Klein2011]. The large velocity dispersion in the ultraviolet is due to the low SNR of the spectrum and the $\sim$ 15 km s$^{-1}$ uncertainty of COS (COS Instrument Handbook). For GD 362, we marginally detected C I 1930.9 [Å]{} and it has a Doppler shift of 48 km s$^{-1}$, in agreement with 49.3 $\pm$ 1.0 km s$^{-1}$ from the optical study [@Zuckerman2007]. [lccccccc]{}\ Ion & $\lambda$ & E$_{low}$ & EW & log n(Z)/n(He)\ & (Å) & (eV) & (mÅ) &\ C I & 1930.905 & 1.26 & 560 $^{+230}_{-158}$ $^a$ & -6.70 $\pm$ 0.30\ \ S I & 1807.311 & 0 & $\lesssim$ 900 & $\lesssim$ -6.70\ S I & 1820.341 & 0.049 & $\lesssim$ 710 & $\lesssim$ -6.40\ S & & & & $\lesssim$ -6.70\ \[Tab: LinesGD\] $^a$ This is measured from the model spectra, as described in section 3.1. [lcccccc]{}\ \ Ion & $\lambda$ & E$_{low}$ & EW & log n(Z)/n(He)\ & (Å) & (eV) & (mÅ) &\ C I & 1930.905 & 1.26 & 1600 $\pm$ 200 & -7.80 $\pm$ 0.10\ \ S I & 1807.311 & 0 & $\lesssim$ 170 & $\lesssim$ -9.50\ S I & 1820.341 & 0.049 & $\lesssim$ 150 & $\lesssim$ -9.30\ S & & & & $\lesssim$ -9.50\ \ Mg I & 2026.477$^a$ & 0 & 288 $\pm$ 100$^b$ & $\lesssim$ -7.60\ \ Si II & 1808.013 & 0 & 936 $\pm$ 109 & -7.44 $\pm$ 0.10\ Si II & 1816.928 & 0.04 & 1232 $\pm$ 145 & -7.46 $\pm$ 0.10\ Si & & & & -7.45 $\pm$ 0.10\ \ Fe II & 1925.987 & 2.52 & 192 $\pm$ 72 & -7.62 $\pm$ 0.28\ Fe II & 2011.347 & 2.58 & 309 $\pm$ 97 & -7.35 $\pm$ 0.24\ Fe II & 2019.429 & 1.96 & 211 $\pm$ 67 & -7.56 $\pm$ 0.24\ Fe II & 2021.402 & 1.67 & 181 $\pm$ 68 & -7.71 $\pm$ 0.27\ Fe II & 2033.061 & 2.03 & 311 $\pm$ 67 & -7.24 $\pm$ 0.17\ Fe II & 2041.345 & 1.964 & 215 $\pm$ 50 & -7.24 $\pm$ 0.18\ Fe & & & & -7.45 $\pm$ 0.23\ \ Zn II & 2026.136 & 0 & 288 $\pm$ 47$^b$ & $\lesssim$ -11.30\ \[Tab: LinesPG\] $^a$ The atomic parameters for this line are taken from @KelleherPodobedova2008.\ $^b$ Mg I 2026.5 [Å]{} and Zn II 2026.1 [Å]{} are blended and the reported EW is for the entire feature. ATMOSPHERIC ABUNDANCE DETERMINATIONS ==================================== Because we are most interested in the abundance of an element relative to other heavy elements and these ratios are not strongly dependent upon the stellar temperature and surface gravity [@Klein2011], we only adopt one set of stellar parameters as listed in Table \[Tab: Properties\] and compute the model spectra following @Koester2010. Atomic data are mostly taken from the Vienna Atomic Line Database [@Kupka1999]. The computed model atmosphere spectra were convolved with the COS NUV line spread function[^3]. The abundance of each element was derived by comparing the EW of each spectral line with the value derived from the model atmosphere, as shown in Figures \[Fig: GD\_C\]-\[Fig: PG\_Zn\] and Tables \[Tab: LinesGD\] and \[Tab: LinesPG\]. The final abundances, combining ultraviolet with optical observations, are given in Tables \[Tab: AbundanceGD\] and \[Tab: AbundancePG\] for GD 362 and PG 1225-079, respectively. Our results mostly agree with previous reports but have a higher accuracy. For PG 1225-079, we newly derive the abundances of carbon and silicon and have tentative detections of sulfur and zinc. The magnesium abundance is updated while the iron abundance agrees with previous optical results. Because the data are noisier for GD 362, we are only able to crudely constrain the abundance of carbon and sulfur. --------------------- --------------- -------- ------------------- ------ ------------------------- ---------- -- -- star M$_*$ T log g D log M$_{cvz}$/M$_*$$^a$ Ref (M$_{\odot}$) (K) (cm$^2$ s$^{-1}$) (pc) GD 362 0.72 10,540 8.24 51 -6.71 \(1) (2) PG 1225-079 0.58 10,800 8.00 26 -5.02 \(3) (4) \[Tab: Properties\] --------------------- --------------- -------- ------------------- ------ ------------------------- ---------- -- -- : Adopted Stellar Properties $^a$ Newly-derived mass of the convective zone (see section 4).\ [**References.**]{}[(1) @Kilic2008b; (2) @Zuckerman2007; (3) @Klein2011; (4) @Farihi2005.]{}\ ------- ---------------------- --------------- -------------------------------- -- -- Z log n(Z)/n(He)$^a$ t$_{set}$$^b$ $\dot{M}$(Z$_i$)$^c$ (10$^5$ yr) (g s$^{-1})$ H -1.14 $\pm$ 0.10 ... ... C$^*$ -6.70 $\pm$ 0.30 2.1 2.5 $\times$ 10$^7$ N $<$ -4.14 2.2 $<$ 9.0 $\times$ 10$^9$ O $<$ -5.14 2.2 $<$ 1.1 $\times$ 10$^9$ Na -7.79 $\pm$ 0.20 2.2 3.7 $\times$ 10$^6$ Mg -5.98 $\pm$ 0.25 2.2 2.5 $\times$ 10$^8$ Al -6.40 $\pm$ 0.20 1.6 1.5 $\times$ 10$^8$ Si -5.84 $\pm$ 0.30 1.2 7.2 $\times$ 10$^8$ S$^*$ $\lesssim$ -6.70$^d$ 0.79 $\lesssim$ 1.7 $\times$ 10$^8$ Ca -6.24 $\pm$ 0.10 0.99 5.1 $\times$ 10$^8$ Sc -10.19 $\pm$ 0.30 0.93 6.8 $\times$ 10$^4$ Ti -7.95 $\pm$ 0.10 0.94 1.2 $\times$ 10$^7$ V -8.74 $\pm$ 0.30 0.95 2.1 $\times$ 10$^6$ Cr -7.41 $\pm$ 0.10 1.0 4.3 $\times$ 10$^7$ Mn -7.47 $\pm$ 0.10 1.0 4.0 $\times$ 10$^7$ Fe -5.65 $\pm$ 0.10 1.1 2.5 $\times$ 10$^9$ Co -8.50 $\pm$ 0.40 0.99 4.1 $\times$ 10$^6$ Ni -7.07 $\pm$ 0.15 1.0 1.1 $\times$ 10$^8$ Cu -9.20 $\pm$ 0.40 0.83 1.1 $\times$ 10$^6$ Sr -10.42 $\pm$ 0.30 0.56 1.3 $\times$ 10$^5$ Total 4.4 $\times$ 10$^9$ ------- ---------------------- --------------- -------------------------------- -- -- : Atmospheric Abundances for GD 362\[Tab: AbundanceGD\] $^*$ New measurements from this paper. The rest are from @Zuckerman2007 but we reference abundances relative to He, the dominant element in GD 362’s atmosphere, rather than H, as presented in @Zuckerman2007. Consequently, there is a possible systematic offset up to 0.1 dex in all entries derived from that paper.\ $^a$ The final abundance of an element combining optical and ultraviolet data.\ $^b$ Newly-derived settling times in the convective zone (see section 4); they are typically a factor of 2-3 longer than previously-derived values in @Koester2009a.\ $^c$ Accretion rates calculated from Equation (1).\ $^d$ The equality sign corresponds to the red model fit shown in figures. -------- ------------------- ------------- -------------------------------- Z log n(Z)/n(He) t$_{set}$ $\dot{M}$(Z$_i$) (10$^6$ yr) (g s$^{-1})$ H -4.05 $\pm$ 0.10 ... ... C$^*$ -7.80 $\pm$ 0.10 5.5 3.1 $\times$ 10$^6$ O $<$ -5.54 4.5 $<$ 9.1$\times$ 10$^8$ Na $<$ -8.26 4.4 $<$ 2.6 $\times$ 10$^6$ Mg$^*$ -7.50 $\pm$ 0.20 4.8 1.4 $\times$ 10$^7$ Al $<$ -7.84 3.6 $<$ 9.5 $\times$ 10$^6$ Si$^*$ -7.45 $\pm$ 0.10 3.0 3.0 $\times$ 10$^7$ S$^*$ $\lesssim$ -9.50 1.7 $\lesssim$ 5.2 $\times$ 10$^5$ Ca -8.06 $\pm$ 0.03 1.9 1.6 $\times$ 10$^7$ Sc -11.29 $\pm$ 0.07 1.8 1.1 $\times$ 10$^4$ Ti -9.45 $\pm$ 0.02 1.8 8.3 $\times$ 10$^5$ V -10.41 $\pm$ 0.10 1.8 9.6 $\times$ 10$^4$ Cr -9.27 $\pm$ 0.06 1.9 1.3 $\times$ 10$^6$ Mn -9.79 $\pm$ 0.14 2.0 4.0 $\times$ 10$^5$ Fe -7.42 $\pm$ 0.07 2.1 9.0 $\times$ 10$^7$ Ni -8.76 $\pm$ 0.14 2.3 4.0 $\times$ 10$^6$ Zn$^*$ $\lesssim$ -11.30 2.2 $\lesssim$ 1.3 $\times$ 10$^4$ Sr $<$ -11.65 1.2 $<$ 1.4 $\times$ 10$^4$ Total 1.6 $\times$ 10$^8$ -------- ------------------- ------------- -------------------------------- : Atmospheric Abundances for PG 1225-079\[Tab: AbundancePG\] $^*$ New results from this paper. The rest are from @Klein2011.\ [**Notes.**]{} The columns are defined the same as Table \[Tab: AbundanceGD\]. Carbon ------ There is only one useful carbon line in the observed wavelength interval, C I 1930.9 [Å]{}, as shown in Figures \[Fig: GD\_C\] and \[Fig: PG\_C\]. Because it arises from an excited level, it cannot be contaminated by interstellar absorption. However, this line can be blended with Mn II 1931.4 [Å]{}. Fortunately, accurate Mn abundances have been determined for both stars from optical data [@Zuckerman2007; @Klein2011] and the predicted EW for Mn II 1931.4 [Å]{} is less than 50 m[Å]{} in the model spectrum. Considering the measured EW of this feature is more than 500 m[Å]{} for both stars (see Tables \[Tab: LinesGD\] and \[Tab: LinesPG\]), we conclude that the line is dominated by C I 1930.9 [Å]{}. For PG 1225-079, our derived carbon abundance[^4] \[C\]/\[He\] = -7.80 $\pm$ 0.10 agrees with the IUE upper limit of -7.5 [@Wolff2002]. For GD 362, the largest uncertainty is from the low SNR of the data; the measured continuum flux is (3.1 $\pm$ 1.0) $\times$ 10$^{-15}$ erg s$^{-1}$ cm$^{-2}$ [Å]{}$^{-1}$. It is hard to measure the EW of C I 1930.9 [Å]{} directly from the noisy data. Instead, we computed model spectra with different carbon abundance to match the observed spectrum. In Figure \[Fig: GD\_C\], we present three best-fit models with \[C\]/\[He\] = - 6.4, \[C\]/\[He\] = -6.7, \[C\]/\[He\] = -7.0 and a continuum flux at 4.1 $\times$ 10$^{-15}$ erg s$^{-1}$ cm$^{-2}$ [Å]{}$^{-1}$, 3.1 $\times$ 10$^{-15}$ erg s$^{-1}$ cm$^{-2}$ [Å]{}$^{-1}$, 2.1 $\times$ 10$^{-15}$ erg s$^{-1}$ cm$^{-2}$ [Å]{}$^{-1}$, respectively. The final abundance is \[C\]/\[He\] = -6.7 $\pm$ 0.3 and the EW reported in Table \[Tab: LinesGD\] is measured from the model spectra. Sulfur ------ There are two useful sulfur lines, S I 1807.3 [Å]{} and S I 1820.3 [Å]{}. However, at best, we have only a tentative detection of sulfur in each star. S I 1807.3 [Å]{}, the stronger line, is adjacent to Si II 1808.0 [Å]{}. Fortunately, for GD 362, the silicon abundance is determined from previous optical data [@Zuckerman2007]; for PG 1225-079, other ultraviolet lines can be used to derive the silicon abundance (see section 3.4). The data and model atmosphere spectra for GD 362 and PG 1225-079 are presented in Figures \[Fig: GD\_S\] and \[Fig: PG\_S\], respectively. Considering the apparent match between the model and data for both S I lines, tentative sulfur abundances of -6.7 for GD 362 and -9.5 for PG 1225-079 can be assigned. Conservatively, these results are upper limits. Magnesium and Zinc ------------------ In PG 1225-079, Mg I 2026.4 [Å]{} and Zn II 2026.1 [Å]{} are heavily blended. As shown in Figure \[Fig: PG\_Zn\], our best fit model which matches the measured EW of the absorption feature requires \[Mg\]/\[He\] = -7.6 and \[Zn\]/\[He\] = -11.3. These values are individually taken as upper limits due to the blending. However, the reported magnesium abundance is -7.27 $\pm$ 0.06 from the optical data [@Klein2011], which is largely based on three Mg lines but the detections for two lines are only 2$\sigma$. @Wolff2002 reported \[Mg\]/\[He\] to be -7.6 $\pm$ 0.6 from the IUE data. Averaging these measurements, our final magnesium abundance is -7.50 $\pm$ 0.20. Because of the blending, the zinc abundance is only an upper limit. This provides the first stringent constraint on zinc in an extrasolar planetesimal. Silicon ------- In PG 1225-079, we measured two silicon lines, Si II 1808.0 [Å]{} and Si II 1816.9 [Å]{}, as shown in Figure \[Fig: PG\_S\]. Si II 1808.8 [Å]{} arises from the ground state and the photospheric line can be distorted by interstellar absorption. However, its measured EW is only 87 $\pm$ 11 m[Å]{} in $\zeta$ Oph, a star at a distance of 112 pc with a large amount of foreground interstellar gas [@Morton1975]. Considering PG 1225-079 is only 26 pc away, it has much less interstellar absorption. The measured EW is 936 $\pm$ 109 m[Å]{} and we conclude that Si II 1808.0 [Å]{} is largely photospheric and essentially free from interstellar absorption. The shape of Si II 1808.0 [Å]{} in the model does not quite fit the data; but the measured EW of the data, which is key in the abundance determination, has a good agreement with that in the model. Using these two Si II lines, we derive a final silicon abundance of -7.45 $\pm$ 0.10, in agreement with, but much better than the reported IUE abundance of -7.5 $\pm$ 0.5 [@Wolff2002] and the previous optical upper limit of -7.27 [@Klein2011]. Iron ---- In the COS data for PG 1225-079, there are six Fe II lines with EWs larger than 100 m[Å]{}. Four of them are shown in Figures \[Fig: PG\_C\] and \[Fig: PG\_Zn\]. We derived an iron abundance of -7.45 $\pm$ 0.23, in good agreement of the optical value of -7.42 $\pm$ 0.07, which is based on 28 high-SNR iron lines [@Klein2011]. Because the ultraviolet data are noisier, we adopt the optically-derived iron abundance. COMPARISON WITH SOLAR SYSTEM OBJECTS ==================================== Combined with previous data, we now have determined the abundances of 16 elements heavier than helium in the atmosphere of GD 362 and 11 heavy elements in PG 1225-079. However, the measured composition need not be identical to the composition of the accreted planetesimal because different elements gravitationally settle at different rates in a white dwarf atmosphere. Three major phases are proposed for a single accretion event: build-up, steady-state and decay [@Dupuis1993a; @Koester2009a]. Because an infrared excess is found for GD 362 and PG 1225-079 [@Becklin2005; @Kilic2005; @Farihi2010b], the accretion should be either in the build-up or steady-state phase. The timescale for build-up stage is comparable to the settling times [@Koester2009a]; it is $\sim$ 10$^5$ yr, for GD 362 and PG 1225-079 (see Tables \[Tab: AbundanceGD\] and \[Tab: AbundancePG\]). The rest of the disk-host stage should all be under the steady-state approximation. The dust disk lifetime has been under intensive studies for a few years but the values are still very uncertain, including 10$^5$ yr [@Farihi2009; @Rafikov2011b], 10$^6$ yr [@Rafikov2011a; @Girven2012; @Farihi2012b] and up to 10$^7$ yr [@Barber2012]. The true disk lifetime might have a range but it is likely to be longer than the settling times. Furthermore, @Zuckerman2010 suggested that steady-state approximation is the dominant situation for white dwarf accretion event based on a study of helium dominated stars; the settling times are only 0.1% of their cooling times but 30% of them show atmospheric pollution. GD 362 and PG 1225-079 are more likely to be under the steady-state approximation and that is the main focus of this paper. In the steady-state model, the observed concentration of an element is dependent on the time it takes to sink out of the convective envelope. To derive the theoretical settling times and obtain an improved understanding of the uncertainties, we formulated several numerical experiments with the code for the envelope structure and corrected two errors found in our previous calculations of diffusion timescales. In the course of changing the equations describing element diffusion from the version in @Paquette1986 (Equation 4) to the one in @Pelletier1986 (Equation 5), which is more accurate in the case of electron degeneracy, one of us (D.K.) discovered an error in the former paper. A factor of $\rho^{1/3}$ is missing in the second alternative of Equation 21, which we had not noticed before. A rederivation of all our equations uncovered another error in our implementation of the contribution of thermal diffusion. These errors have only a very small effect in stars with relatively shallow convection zones, like the hydrogen-dominated white dwarfs. However, for helium-dominated white dwarfs with T $<$ 15,000 K and a deep convection zone, the diffusion timescales can be slower by factors 2-3 relative to our earlier calculations[^5]. The accretion rate $\dot{M}(Z_i)$ of an element Z is calculated as [@Koester2009a] $$\dot{M}(Z_i) = \frac{M_{cvz} X(Z_i)}{t_{set}(Z_i)}$$ where M$_{cvz}$ is the mass of the convective envelope. X(Z$_i$) is the mass fraction of the element Z$_i$ relative to the dominant element in the atmosphere, either hydrogen or helium; t$_{set}$(Z$_i$) is the settling time. A longer settling time corresponds to a lower diffusion flux. Fortunately, the relative timescales for different elements, which are important for the determination of the abundances in the accreted matter, change much less. For GD 362 and PG 1225-079, compared to previously published values, the settling times listed in Tables \[Tab: AbundanceGD\] and \[Tab: AbundancePG\] typically increase by factors of 2-3 while the mass of the convective zone is 0.13 dex smaller for GD 362 and 0.05 dex larger for PG 1225-079 (Table \[Tab: Properties\]). These corrections lead to smaller total accretion rates by a factor of 3 for both stars. The next step is to compare the composition of the accreted parent body with those of solar system objects. We choose the summed mass of all the major elements as the normalization factor so that the analysis is independent of the chemical property and abundance uncertainty of each individual element. However, one complication is that no oxygen lines are detected in either GD 362 or PG 1225-079 due to their low photospheric temperatures relative to other helium-dominated white dwarfs; only upper limits were obtained for this major element. Therefore, our approach is to compare the mass fraction of an element relative to the summed mass of the common elements Mg, Si and Fe. For solar system objects, we include 80 representative and well-analyzed meteorite samples mostly from @Nittler2004. We also include the bulk composition of Earth from @Allegre2001 and an updated carbon abundance from @Marty2012. For our purpose, Earth appears to be chondritic and its bulk composition approaches CV chondrites even though Earth has experienced some post-nebular processing, such as differentiation and collisions. GD 362: Accretion from a Mesosiderite Analog? --------------------------------------------- In Figure \[Fig: GD362\], we compare the abundances of all 18 elements, including upper limits, of the accreted material in GD 362 with CI chondrites, which are the most primitive material in the solar system. The composition of CI chondrites is almost identical to the solar photosphere, with the exception of depletion of volatile elements C, N as well as H and noble gases. The parent body accreted onto GD 362 looks nothing like a CI chondrite, as first pointed out in @Zuckerman2007. For the volatile elements, the mass fraction of C and S are depleted by at least a factor of 7 and 3, respectively, relative to CI chondrites; refractory elements, such as V, Ca, Ti and Al, are all enhanced. Though oxygen is not detected in GD 362, its stringent upper limit can still provide useful insights. Following @Klein2010, we can calculate the required number of oxygen atoms to form oxides Z$_{p(Z)}$O$_{q(Z)}$ as $$n(O)=\sum_Z \frac{q(Z)}{p(Z)} n(Z)$$ Hydrogen is excluded here because GD 362 has an enormous amount and it might not be associated with the parent body or bodies currently in its atmosphere (see Appendix A). Under the steady-state approximation, \[O\]/\[He\] = -5.07 is required to form MgO, Al$_2$O$_3$, SiO$_2$ and CaO; this value is comparable to the observed oxygen upper limit of -5.14. However, Fe is the most abundant heavy element in the atmosphere of GD 362 and there is insufficient oxygen to tie it up in either FeO or Fe$_2$O$_3$. Thus, most, if not all the iron in the parent body is in metallic form, which is very different from CI chondrites where most iron is in oxides [@Nittler2004]. @ONeillPlame2008 suggested that \[Mn\]/\[Na\] can be used as an indicator of post-nebular processing. For example, \[Mn\]/\[Na\] is -0.79 for all chondrites as well as the solar photosphere while non-chondritic objects have a much higher value. Interestingly, \[Mn\]/\[Na\] is 0.65 $\pm$ 0.22 for GD 362, which is larger than -0.01 for Mars and 0.32 for the Moon [@ONeillPlame2008]. This suggests that the planetesimal accreted onto GD 362 is likely to be non-chondritic and have experienced some post-nebular processing. @Zuckerman2007 compared the \[Na\]/\[Ca\] ratio in GD 362 with solar system objects and reached a similar conclusion; the accreted planetesimal was non-chonridtic. The only other polluted white dwarf with both Mn and Na detections is WD J0738+1835 wherein \[Mn\]/\[Na\]= -0.54 $\pm$ 0.19 [@Dufour2012]; this agrees with the chondritic value within the uncertainties. To find the best solar system analog to the parent body accreted onto GD 362, we calculated a reduced chi-squared value for each object in our sample ($\chi^2_{red}$), defined as: $$\chi^2_{red}= \frac{1}{N} \sum ^N _{i=1} \frac{(M_{wd}(Z_i)-M_{mtr}(Z_i))^2}{\sigma_{wd}^2(Z_i)}$$ where N is the total number of elements considered in the analysis. M$_{wd}$(Z$_i$) and M$_{mtr}$(Z$_i$) represent the mass fraction of an element Z$_i$ relative to the summed mass of Mg, Si and Fe in the extrasolar planetesimal and solar system objects, respectively. $\sigma_{wd}$(Z$_i$) is the propagated uncertainty in mass fraction. For GD 362, we calculated $\chi^2_{red}$ for 11 heavy elements, C, Na, Mg, Al, Si, Ca, Ti, Cr, Mn, Fe and Ni, which have detections both in GD 362 and the meteorite sample[^6]. The results are shown in Figure \[Fig: GD\_chi\] for both steady-state and build-up approximations. There is no qualitative difference between these two models and mesosiderites provide the best fit considering all 11 elements. In particular, the mesosiderite ALH 77219 can match the overall abundance pattern to 95% confidence level. As shown in Figure \[Fig: GD362\], the abundance of individual elements agrees within 2$\sigma$ between mesosiderites and the planetesimal accreted onto GD 362. Mesosiderites are a rare type of stoney-iron meteorite with equal amounts of silicates and metallic iron and nickel. One mystery about mesosiderties is that the Si-rich crust and Fe, Ni-rich core materials are abundant but the olivine Mg-rich mantle seems to be missing. One model for the formation of mesosiderites is that a 200-400 km diameter asteroid with a molten core was nearly catastrophically disrupted by a 50-150 km diameter projectile at 4.42-4.52 Gyr ago [@Scott2001]. The collision mixed the target’s molten core with its crustal material but excluded the large and hot mantle fragments. The planetesimal accreted onto GD 362 may have been formed in a similar way. While mesosiderites may be a prototype for the accreted planetesimal onto GD 362, there are three major hurdles for this hypothesis to overcome. First, in the model of @Scott2001, only half of the original mass of a 200-400 km diameter asteroid was maintained after the collision and the final product only contains about 10% mesosiderite-like material by mass. This is equivalent to a 75-150 km diameter object. Mesosiderites that fall on Earth are only small fragments and the 180 kg NWA 2924 is among the largest (Meteorite Bulletin Database[^7]). However, the parent body accreted onto GD 362 has a minimum mass of 2.7 $\times$ 10$^{22}$ g, $\sim$260 km in diameter for an assumed density of 3 g cm$^{-3}$. It is unclear whether the same kind of collision can produce a mesosiderite parent body this big. Second, the mass fraction of hydrogen in mesosiderites is less than 0.2%; it cannot explain how there is 5 $\times$ 10$^{24}$ g hydrogen in the atmosphere of GD 362. Possibly, hydrogen was accreted during earlier events and it has been atop the atmosphere ever since (see Appendix A for more discussion). Third, GD 362 is currently accreting from its circumstellar disk and the disk material should also resemble the composition of mesosiderites. However, the shape of the mid-infrared spectrum for mesosiderite, which is dominated by a sharp peak at 9.13 $\mu$m and several other bands at 10.6 $\mu$m and 11.3 $\mu$m [@Morlok2012], cannot fully account for the broad 10 $\mu$m silicate emission feature observed for GD 362 [@Jura2007a]. This does not completely exclude the mesosiderite hypothesis but emission from some additional material is required to fully reproduce the observed infrared spectrum for GD 362. Mesosiderites are a good candidate for the parent body accreted onto GD 362 but there are remaining unresolved issues. PG 1225-079: Accretion from a Planetesimal with No Single Solar System Analog ----------------------------------------------------------------------------- In Figure \[Fig: PG1225\], we show a comparison of the mass fractions of 16 elements, including upper limits between PG 1225-079 and CI chondrites. Though the carbon abundance is approaching the chondritic value, the accreted planetesimal differs a lot from CI chondrites; the mass fraction of S is depleted by at least a factor of 40 while Zn is depleted by at least a factor of 8. In contrast, refractories, such as V, Ca, Ti and Sc are all enhanced. The overall pattern of relatively high carbon abundance and enhanced mass fractions of refractory elements does not follow a single condensation sequence and post-nebular processing is required. \(a) (b) As shown in Figure \[Fig: C-Si-S\], PG 1225-079 has a \[C\]/\[S\] value that is no smaller than the solar ratio, which is very different from other polluted white dwarfs and meteorites. Carbon and sulfur are among the most volatile elements that we can measure and their 50% condensation temperatures are 40 K and 655 K, respectively [@Lodders2003]. Most of the meteorites as well as polluted white dwarfs have a \[C\]/\[S\] ratio lower than the solar value, which can be explained by condensation at a temperature between 40 and 665 K though this is not necessarily true for all of them. The only solar system analog to PG 1225-079 with similar high carbon, low sulfur pattern is ureilites, a type of primitive achondrites. Ureilites are the second largest achondrite group and it is suggested that its high carbon abundance is derived from a carbon-rich parent body, but the exact formation mechanism is not well understood [@Goodrich1992]. However, as can be seen in Figure \[Fig: PG1225\](a), ureilites fail to match the overall composition of the parent body accreted onto PG 1225-079. We performed a $\chi^2_{red}$ analysis between solar system objects and the accreted planetesimal in PG 1225-079, comparing 9 elements, C, Mg, Si, Ca, Ti, Cr, Mn, Fe and Ni[^8]. The result is shown in Figure \[Fig: PG\_chi\]. There is no single solar system object that can match all nine elements; the closest is carbonaceous chondrite. Regardless, as shown in Figure \[Fig: PG1225\], the accreted abundance in PG 1225-079 is not at all identical to CI chondrites. The infrared excess around PG 1225-079 corresponds to $\sim$500 K dust [@Farihi2010b]; so far, only two white dwarfs are known to have such cool dust. The other 28 known disk-host stars all have $\sim$1000 K dust [@XuJura2012]. One hypothesis is that the inner disk region was recently impacted by another asteroid and all the material was dissipated [@Farihi2010b; @Jura2008]. If that is the case, PG 1225-079 can be accreting from a blend of two planetesimals, rather than one single parent body. After testing different combinations of the 80 meteorites in our database, the best fit model to the steady-state approximation consists of 30% ureilite North Haig and 70% mesosiderite Dyarrl Island by mass. This blend is also marked in Figure \[Fig: PG\_chi\]. Detailed abundance comparison is shown in Figure \[Fig: PG1225\](b); the abundances of S, Mn and Ca do not agree as well as the other elements but are all within 2$\sigma$. A possible scenario is that one extrasolar ureilite (mesosiderite) analog first got tidally disrupted and more recently, another mesosiderite (ureilite) analog impacted the disk and was blended with the previous material. ASSESSING THE FORMATION MECHANISMS OF EXTRASOLAR PLANETESIMALS ============================================================== Having established that the parent bodies accreted onto GD 362 and PG 1225-079 are beyond primitive, we now extend our analysis to other extrasolar planetesimals. We are most interested in understanding the formation mechanisms of extrasolar planetesimals and whether these are dominated by nebular or post-nebular processing. @JuraXu2013 suggested collisional rearrangement is important in determining the final composition of extrasolar planetesimals based on the scatter in \[Mg\]/\[Ca\] ratios in 60 externally-polluted white dwarfs. Here, we compile a sample of well-studied externally-polluted white dwarfs with abundance determinations of at least 9 elements. There are 9 stars in total, as listed in Table \[Tab: WDs\] and now we assess the formation mechanism for individual objects. star Dom. Dust Volatile Intermediate Refractory Process Ref --------------- ------ ------ -------------- ------------------- ------------------ --------------------- ------------ -- GD 40 He Y C,S:,O,Mn, P Cr,Si,Fe,Mg,Ni Ca,Ti,Al primitive 1,2 WD J0738+1835 He Y O,Na,Mn Cr,Si,Fe,Mg,Co,Ni V,Ca,Ti,Al,Sc primitive 3,4 PG 0843+517 H Y C,S,O,P Cr,Si,Fe,Mg,Ni Al beyond-primitive(?) 5 PG 1225-079 He Y C,Mn Cr,Si,Fe,Mg,Ni V,Ca,Ti,Sc beyond-primitive 6,7 NLTT 43806 H N Na Cr,Si,Fe,Mg,Ni Ca,Ti,Al beyond-primitive 8 GD 362 He Y C,Na,Cu,Mn Cr,Si,Fe,Mg,Co,Ni V,Sr,Ca,Ti,Al,Sc beyond-primitive 7,9 WD 1929+012 H Y C,S,O,Mn,P Cr,Si,Fe,Mg,Ni Ca,Al ??? 5,10,11,12 G241-6 He N S,O,Mn,P Cr,Si,Fe,Mg,Ni Ca,Ti primitive 2,6,13 HS 2253+8023 He N O,Mn Cr,Si,Fe,Mg,Ni: Ca,Ti primitive 6 \[Tab: WDs\] [**Note.**]{} This is a compiled sample of externally polluted white dwarfs with detections of at least 9 elements heavier than helium. Columns are defined as follows. “Dom" lists the dominant element in the atmosphere. “Dust" indicates whether a star has an infrared excess (“Y") or not (“N"). Following the classification scheme in @Lodders2003, “Volatile" lists the detected volatile elements, defined as having a 50% condensation temperature lower than 1290 K in a solar-system composition gas [@Lodders2003]; “Intermediate" lists the elements with a condensation temperature between 1290-1360 K –the same range as that of the common elements, Si, Fe and Mg; “Refractory" elements have a 50% condensation temperature higher than 1360 K. The elements are ordered with increasing condensation temperature. “Process" shows our proposed dominant mechanism that determines the final composition of the accreted extrasolar planetesimal (see section 5).\ [**References.**]{} (1) @Klein2010; (2) @Jura2012; (3) @Dufour2010; (4) @Dufour2012; (5) @Gaensicke2012; (6) @Klein2011; (7) this paper; (8) @Zuckerman2011; (9) @Zuckerman2007; (10) @Vennes2010; (11) @Vennes2011a; (12) @Melis2011; (13) @Zuckerman2010. [*GD 40*]{}: As discussed in @Jura2012 and Appendix B, the overall abundance pattern in GD 40 matches with carbonaceous chondrites and bulk Earth. Nebular condensation is sufficient to explain its observed composition. [*WD J0738+1835*]{}: @Dufour2012 found that there is a correlation between the abundance of an element and its condensation temperature: refractory elements are depleted while volatile elements are enhanced compared to bulk Earth. This indicates that the accreted planetesimal might be formed in a low temperature environment under nebular condensation. [*PG 0843+517*]{}: This star has the highest mass fraction of iron among all polluted white dwarfs. [@Gaensicke2012] found that all core elements, including Fe, Ni, S and Cr are enhanced relative to the values for bulk Earth while lithophile refractory Al is depleted. This star might be accreting from the core of a differentiated object. Nevertheless, considering the uncertainty for each element is at least 0.2 dex, the conclusion is still preliminary. [*PG 1225-079*]{}: As discussed in section 4.2, this star has a near chondritic carbon abundance but also enhanced mass fractions of refractory elements relative to CI chondrite; it cannot be formed solely under nebular processing. [*NLTT 43806*]{}: Compared to chondritic values, the accreted planetesimal is depleted in Fe and enhanced in Al. @Zuckerman2011 found that the best fit model corresponds to “30% crust 70% upper mantle". With detections of 9 elements, evidence is strong that NLTT 43806 has accreted the outer layer of a differentiated parent body. [*GD 362*]{}: As discussed in section 4.1, mesosiderite is the best solar system analog to the accreted parent body and post-nebular processing is required. [*WD 1929+012*]{}: @Gaensicke2012 showed that this star has a high iron content. However, the situation is perplexing in that different analyses yield different stellar parameters and atmospheric abundances. For example, both @Melis2011 and @Gaensicke2012 derived that \[Si\]/\[Fe\] is -0.25 but @Vennes2010 found that \[Si\]/\[Fe\] is 0.19. No final conclusion can be drawn before resolving such discrepancies. [*G241-6*]{}: This star is a near twin of GD 40 with a similar abundance pattern but without an infrared excess. One possible scenario is that G241-6 has accreted a planetesimal with a similar composition to GD 40 and now it is at the beginning of a decaying phase; all heavier elements appear to be depleted relative to GD 40 due to their short settling times [@Klein2011; @Jura2012]. As discussed in @Jura2012 and Appendix B, the overall abundances resemble those of chondrites and no post-nebular processing is required. [*HS 2253+8023*]{}: @Klein2011 showed that the composition of its parent body agrees with bulk Earth, except for the enhanced calcium abundance. Nebular processing can produce the observed abundance pattern. As summarized in Table \[Tab: WDs\], at least 4 out of the 9 white dwarfs have accreted planetesimals that can be formed under nebular processing while post-nebular processing is required for another 3 of them. It should be noted that some objects that we identify as primitive might still have undergone some post-nebular processing. For example, GD 40 has accreted from a planetesimal that has a similar composition as bulk Earth, whose overall abundance pattern is chondritic. However, it is still possible that the parent body was differentiated; when the entire object is accreted, the composition appears to be “chondritic". We can only put an upper limit on the number of objects formed under nebular condensation. From this sample of 9 stars, we see that post-nebular processing appears to play an important role in determining the final abundance of extrasolar planetesimals; beyond-primitive planetesimals might be as common as primitive planetesimals. In contrast, chondrites comprise more than 90% of all meteorites found on Earth by number (Meteorite Bulletin Database[^9]). Possibly, extrasolar planetesimals around white dwarfs have violent evolutionary histories with more collisions. This difference is not surprising since dynamical rearrangement of planetary systems at white dwarfs is expected to increase the frequency of collisions and produce more beyond-primitive extrasolar planetesimals. So far, 19 elements heavier than helium, including C, S, O, Na, Cu, Mn, P, Cr, Si, Mg, Fe, Co, Ni, V, Sr, Ca, Ti, Al and Sc, have been detected in the atmospheres of polluted white dwarfs, as shown in Table \[Tab: WDs\]. In terms of mass fraction in the accreted planetesimal, the lowest limit is $\sim$5 ppm, for Sc in WD J0738+1835 [@Dufour2012]. Studying externally-polluted white dwarfs proves to be a very sensitive probe of the bulk compositions of extrasolar planetesimals. CONCLUSIONS =========== We present [*HST*]{}/COS ultraviolet observations for GD 362 and PG 1225-079, two heavily polluted helium white dwarfs. In GD 362, the mass fractions of carbon and sulfur are depleted by at least a factor of 7 and 3 respectively, compared to CI chondrites. In PG 1225-079, a similar volatile depletion pattern is found: C by a factor of 2, S by at least a factor of 40 and Zn by at least a factor of 8. We provide good evidence for the presence of beyond-primitive extrasolar planetesimals: 1. Mesosiderites provide a good match to the composition of the parent body accreted onto GD 362. However, there are several unresolved issues for this hypothesis, especially the apparent difference between the mid-infrared spectrum of mesosiderites and the dust disk around GD 362. Additional material is required. 2. No single meteorite can reproduce the abundance pattern in PG 1225-079. A blend of 30% North Haig ureilite and 70% Dyarrl Island mesosiderite can provide a good fit to the overall composition. 3. Spectroscopic observations of externally-polluted white dwarfs enable sensitive measurement of the bulk compositions of extrasolar planetesimals, including 19 heavy elements down to a mass fraction of 5 ppm. Based on a sample of 9 well-studied white dwarfs, we find that post-nebular processing is as important as nebular condensation in determining the compositions of extrasolar planetesimals. Support for program \# 12290 was provided by NASA through a grant from the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 5-26555. This work also has been partly supported by NSF grants to UCLA to study polluted white dwarfs. [**APPENDIX**]{} The [*Herschel*]{}/PACS Observation of GD 362 ============================================= While hydrogen is detected in some helium-dominated white dwarfs [@Voss2007], GD 362 has an anomalously large amount. The helium-to-hydrogen number ratio is 14 in its convective zone, corresponding to 5 $\times$ 10$^{24}$ g of hydrogen; this is lower than 7 $\times$ 10$^{24}$ g reported in @Jura2009b because the mass of the convective zone for GD 362 is 0.13 dex lower in the updated calculation (Table \[Tab: Properties\]). The origin of the hydrogen is a mystery. Unlike heavy elements which have short settling times compared to the white dwarf cooling age, hydrogen never sinks and can be accumulated over the entire cooling history of the star [@Bergeron2011; @JuraXu2012]. If GD 362 has always been a helium-dominated white dwarf and all this hydrogen is from accretion of tidally disrupted objects, it can either be one Callisto-size object or $\sim$100 Ceres-like asteroids [@Jura2009b]. In the latter case, likely there would be many more asteroids orbiting the star and mutual collisions among them would generate a cloud of cold dust. We were awarded 1.1 hours of [*Herschel*]{}/PACS [@Poglitsch2010] observation time to look for cold dust around GD 362. The “mini-scan map" mode was used to observe in “blue" (85-125 $\mu$m) and “red" (125-210 $\mu$m) bands simultaneously with a medium scan speed of 20 s$^{-1}$ and a scan leg length of 4. The scan map size is 345 $\times$ 374 and the repetition number is 25. Two different scan angles, 45 degrees and 135 degrees were used and the total integration time was 1200 sec. Data reduction was performed using HIPE (Herschel Interactive Processing Environment) on a combined mosaic of level 2 products from pipeline SPG 7.1.0. The pixel scale is 1 pixel$^{-1}$ and 2 pixel$^{-1}$ for the blue and red band, respectively. Correcting for its proper motion, we expect GD 362 at $\alpha$ = 17:31:34.355, $\delta$ = +37:05:18.331 on the date of the observation. Because there is no detection, aperture photometry was performed at 25 locations within 5 pixels of the nominal position of GD 362. The aperture radius was 20 with a sky annulus between 61 and 70. The background intensity was estimated using the median sky estimation algorithm ([*Herschel*]{} Data Analysis Guide[^10]). Aperture correction factors are 0.949 for blue and 0.897 for red (PACS Observer’s Manual[^11]). Based on the dispersion of the 25 measurements, 3$\sigma$ upper limits are 5.1 mJy for blue and 5.6 mJy for red. What does this imply about dust mass? GD 362 has shrunk in mass from 3 M$_{\odot}$ on the main-sequence to its current mass of 0.72 M$_{\odot}$ [@Kilic2008b]. Consequently, asteroids initially at 3-5 AU are now orbiting at 13-21 AU. Currently, GD 362 has a stellar temperature of 10,540 K and cooling age $\sim$ 0.9 Gyr [@Farihi2009]. Extrapolating from white dwarf cooling models[^12] [@Bergeron2011], for GD 362, its stellar temperature is lower than 20,000 K for 90% of its cooling time. We approximate the stellar luminosity as a time-averaged luminosity of 0.01 L$_\odot$. Poynting-Robertson drag was able to remove particles smaller than 20 $\mu$m at a distance of 15 AU for a grain density of 3 g cm$^{-3}$. We therefore assume a dust particle radius of 20 $\mu$m in the putative asteroid belt orbiting GD 362. If the grains function as blackbodies with negligible albedo, then their temperature can be calculated as $$T_d = T_* \sqrt{\frac{R_*}{2D_{orb}}}$$ T$_*$, R$_*$ are the stellar temperature and radius; D$_{orb}$ is the orbital distance. The dust temperature is 14-11 K between 13-21 AU. The mass of the dust disk is $$M_{d}=\frac{ F_{\nu} D_*^2}{\chi B_{\nu}(T)}$$ where D$_*$ is the distance to GD 362, 51 pc [@Kilic2008b] and $\chi$ is the dust opacity. For a particle radius of 20 $\mu$m, $\chi$ = 100 cm$^2$ g$^{-1}$ in the geometric optics limit. As shown in Figure \[Fig: MD\], the upper limit of dust mass is between 10$^{25}$ g and 10$^{26}$ g at 13-21 AU; this mass is at least twice the hydrogen mass in the atmosphere of GD 362 and one order of magnitude larger than the mass of solar system’s asteroid belt [@Krasinsky2002]. The upper limit is not stringent enough to rule out the hypothesis that hydrogen in GD 362 is from accretion of multiple asteroids. So, the large hydrogen abundance in GD 362 remains an unsolved puzzle. Looking for Solar System Analogs to Extrasolar Planetesimals ============================================================ The $\chi^2_{red}$ analysis has proven to be an effective way to look for solar system analogs to the compositions of extrasolar planetesimals. Two other helium-dominated white dwarfs have reported volatile and refractory abundances from high-resolution optical and ultraviolet observations that are suitable for this kind of analysis[^13] – GD 40 and G241-6. Updated settling times and accretion rates are listed in Table \[Tab: GD40G241-6\] while the mass of the convective zone stays the same. Since all the major elements are determined, we compare the mass fraction of an element relative to the sum of O, Mg, Si and Fe. ------- --------------- ------------------------- ------------------------- -- t$_{set}$$^a$ $\dot{M}$(Z)$_{GD 40}$ $\dot{M}$(Z)$_{G241-6}$ Z (10$^6$ yr) (g s$^{-1}$) (g s$^{-1}$) C 1.1 2.2 $\times$ 10$^6$ $<$ 4.4 $\times$ 10$^5$ N 1.1 $<$ 2.6 $\times$ 10$^5$ $<$ 2.1 $\times$ 10$^5$ O 1.1 4.5 $\times$ 10$^8$ 4.3 $\times$ 10$^8$ Mg 1.2 1.7 $\times$ 10$^8$ 1.5 $\times$ 10$^8$ Al 1.2 1.4 $\times$ 10$^7$ $<$ 6.1 $\times$ 10$^6$ Si 1.0 1.3 $\times$ 10$^8$ 8.7 $\times$ 10$^7$ P 0.79 1.1 $\times$ 10$^6$ 4.7 $\times$ 10$^5$ S 0.64 1.0 $\times$ 10$^7$: 5.6 $\times$ 10$^7$ Cl 0.51 $<$ 8.0 $\times$ 10$^5$ $<$ 5.8 $\times$ 10$^5$ Ca 0.51 1.3 $\times$ 10$^8$ 5.1 $\times$ 10$^7$ Ti 0.49 3.2 $\times$ 10$^6$ 1.4 $\times$ 10$^6$ Cr 0.53 6.4 $\times$ 10$^6$ 4.5 $\times$ 10$^6$ Mn 0.53 3.1 $\times$ 10$^6$ 2.4 $\times$ 10$^6$ Fe 0.56 4.4 $\times$ 10$^8$ 2.0 $\times$ 10$^8$ Ni 0.61 1.8 $\times$ 10$^7$ 8.9 $\times$ 10$^6$ Cu 0.58 $<$ 1.8 $\times$ 10$^5$ $<$ 1.8 $\times$ 10$^5$ Ga 0.50 $<$ 2.9 $\times$ 10$^4$ $<$ 2.9 $\times$ 10$^4$ Ge 0.43 $<$ 1.4 $\times$ 10$^5$ $<$ 1.4 $\times$ 10$^5$ Total 1.4 $\times$ 10$^9$ 9.9 $\times$ 10$^8$ ------- --------------- ------------------------- ------------------------- -- : Updated Settling Times and Accretion Rates for GD 40 and G241-6 \[Tab: GD40G241-6\] $^a$ This column is for both GD 40 and G241-6 because their atmospheric conditions are similar. The total accretion rate for GD 40 is a factor of 2 lower than the value derived in @Klein2010, but the relative abundances change much less. The result of a $\chi^2_{red}$ analysis is presented in Figure \[Fig: GD40\_chi\]. When including all 13 detected elements, both carbonaceous chondrites and bulk Earth can match the composition to 95% confidence level for both steady-state and build-up approximations. The accreted planetesimal appears to be primitive and can be formed under nebular condensation, similar to what was concluded by @Jura2012. The newly-derived total accretion rate for G241-6 is about a factor of 2 lower than previously reported [@Zuckerman2010]. The non-detection of an infrared excess and the slight depletion of heavier elements suggest that it may be at the beginning of a decay phase [@XuJura2012; @Klein2011]. We assess both steady-state and decay phase for the $\chi^2_{red}$ analysis; in the latter case, we assume that accretion stopped 0.6 $\times$ 10$^6$ yr ago, approximately one settling time for Fe because its mass fraction is depleted by a factor of 2 relative to CI chondrites. The composition of the parent body is calculated following @Zuckerman2011 and Equation (5) in @Koester2009a. A fuller exploration of different time-varying models will be presented in the future in the spirit of @JuraXu2012. As shown in Figure \[Fig: G241-6\_chi\], both carbonaceous chondrites and ordinary chondrites provide good matches to all 11 elements, including O, Mg, Si, P, S, Ca, Ti, Cr, Mn, Fe and Ni. However, the carbon upper limit in G241-6, which is not included in the $\chi^2_{red}$ analysis, is at least one order of magnitude lower than most carbonaceous chondrites [@Jura2012]. Thus, ordinary chondrites are a more promising solar system analog to the parent body accreted onto G241-6 and nebular condensation is sufficient to produce the observed abundance pattern. The $\chi^2_{red}$ analysis for GD 40 and G241-6 confirms the previous results [@Jura2012]; the accreted extrasolar planetesimals can be formed under nebular condensation and their compositions resemble primitive chondrites in the solar system. [^1]: The four white dwarfs are: GD 61 [@Desharnais2008; @Farihi2011a]; GD 40, G241-6 [@Klein2010; @Klein2011; @Zuckerman2010; @Jura2012] and WD 1929+012 [@Vennes2010; @Vennes2011a; @Melis2011; @Gaensicke2012]. [^2]: Very recently, @Koester2012 reported several white dwarfs with solar carbon-to-silicon ratio. However, the source of this pollution is unclear and more analysis is forthcoming. [^3]: http://www.stsci.edu/hst/cos/performance/spectral\_resolution/nuv\_model\_lsf [^4]: Here, log n(X)/n(Y) is abbreviated as \[X\]/\[Y\]. [^5]: Updated diffusion timescales can be obtained at http://www.astrophysik.uni-kiel.de/ koester/astrophysics/ [^6]: For a couple of meteorites with no reported carbon abundance, we compute the $\chi^2_{red}$ for the other 10 elements. [^7]: http://www.lpi.usra.edu/meteor/ [^8]: Similar to the case of GD 362, for the meteorites with no reported carbon abundance, we only calculated $\chi^2_{red}$ for the other 8 elements. [^9]: http://www.lpi.usra.edu/meteor/ [^10]: http://herschel.esac.esa.int/hcss-doc-8.0/print/howtos/howtos.pdf [^11]: http://herschel.esac.esa.int/Docs/PACS/pdf/pacs\_om.pdf [^12]: http://www.astro.umontreal.ca/ bergeron/CoolingModels/ [^13]: GD 61 also has high-resolution optical and ultraviolet observations [@Desharnais2008; @Farihi2011a]. However, with a total of 5 detected elements, it is hard to make a comparison using the $\chi^2_{red}$ analysis.

--- abstract: 'Properties of disordered thin films are discussed based on the viewpoint that superconducting islands are formed in the system. These lead to superconducting correlations confined in space, which are known to form spontaneously in thin films. Application of a perpendicular magnetic field can drive the system from the superconducting state (characterized by phase-rigidity between the sample edges) to an insulating state in which there are no phase-correlations between the edges of the system. On the insulating side the existence of superconducting islands leads to a non-monotonic magnetoresistance. Several other features seen in experiment are explained.' address: - 'Physics Department, Ben-Gurion University, Beer Sheva 84105, Israel' - 'The Ilse Katz Center for Meso- and Nano-scale Science and Technology, Ben-Gurion University, Beer Sheva 84105, Israel ' - 'SPHT, CEA, F91191 Gif sur Yvette, France' author: - Yonatan Dubi - Yigal Meir - Yshai Avishai title: 'Superconducting islands, phase fluctuations and the superconductor-insulator transition ' --- , and Disordered superconductors,phase,fluctuations,superconductor,insulator,transition Introduction ============ Superconductivity (SC) in disordered thin films has been a subject of intense study for more than a decade [@Goldman_review]. Nevertheless, even elucidation of one of the most fundamental property of these systems, namely the superconductor-insulator transition (SIT), remains a puzzle. Specifically, it is still unclear whether this is a truely quantum phase-transition, what is the role of the magnetic field in the transition, and even its universality class remains undetermined. Another profound feature which is still in debate is the non-monotonic magneto-resistance (MR), which in some systems can reach several orders of magnitude [@Murty]. The non-monotonic MR is accompanied by a unique temperature, magnetic field and disorder dependence of the resistance [@Murty; @Kapitulnik] . On the theoretical front, the adequacy of treating the system in terms of strictly bosonic excitations (so called “dirty boson” models [@dirtybosons]) is still questionable. We adopt a perspective within which the system is composed of SC islands, a structure implying local SC correlations. The notion of SC islands appeared nearly a decade ago in two contexts. Galitski and Larkin [@Galitzki] suggested that in a strongly disordered system the SC order parameter ( and hence also $T_c$ and $H_c$) fluctuate in space. As a result, at a given temperature and magnetic field (close to $H_c$ of the clean system) there will be areas in the sample where the local critical field exceeds the external field. Therefore these domains still display SC correlations. Ghosal et.al. [@Ghosal] used a locally self-consistent solution of the Bogoliubov-de-Gennes equations to show that even in the presence of extremely strong disorder there remain regions in space where the SC order parameter is finite, surrounded by regions of vanishingly small order parameter (and hence dubbed SC islands). As it is shown below, the concept of SC islands might explain numerous experimental observations. In section 2 we describe the nature of the SIT as it emerges from the existence of SC islands with fluctuating phases. In section 3 we show how SC islands lead to non-monotonic MR and explain its dependence on temperature and magnetic field. Section 4 is devoted to a summary and outlook. From Superconducting islands to the superconductor – insulator transition ========================================================================= In this section we describe the nature of the SIT, based on our previous investigations [@us_Nature]. On the SC side of the transition, the system has a complex SC order parameter, and its amplitude strongly fluctuates in space. When a perpendicular magnetic field is turned on and then monotonically increased, it begins to penetrate the sample in the form of disordered vortices [@us_PRB]. As a consequence, there are regions in which the SC order parameter vanishes, and SC islands are formed. However, these islands are still interconnected via a Josephson coupling (JC), and hence their phases are well correlated. This means that Cooper-pairs can coherently traverse through the sample and carry super-current, i.e. the system is in a macroscopic SC state. When the JC between two islands becomes smaller than the temperature, thermal phase fluctuations overcome the phase-locking induced by the JC, and the islands become separate (in terms of SC wave functions). Now, as the magnetic field is increased the JC between the different islands decreases. However, as long as there is a phase-stiff path between the edges of the sample (that is a path of islands with JC larger than temperature) the system will maintain its SC nature. At a certain magnetic field $B_c$, the phase-stiff path is broken and the system is no longer in a SC state. Since it is a highly disordered two-dimensional system, one expects that it it will become insulating, as indeed observed. An immediate consequence of the above scenario is that SC correlations still survive on the insulating side of the transition. This idea is corroborated by a number of experiments. For instance, measurements of the AC conductance [@Crane] show that the superfluid stiffness remains finite well above the transition. Another example is magnetic-field induced conductance oscillations with $2 \phi_0$ period observed in a sample with a fabricated nano-hole lattice [@Valles], indicating the presence of Cooper pairs. To support the above picture we have performed numerical calculations of phase correlations between the two edges of a highly disordered superconducting thin film. We used a Monte-Carlo scheme [@Mayr] on-top of a self-consistent solution of the Bogoliubov-de-Gennes mean-field equations to obtain both the order parameter amplitude and phase, and calculated the correlation function $F_{LR}=\langle \cos (\delta \theta_i - \delta \theta_j) \rangle$ (where $\delta \theta_i$ is the shift of the order parameter phase from its mean-field value) as a function of magnetic field (in terms of the flux per plaquette, $\phi/\phi_0$). Here $i(j)$ are lattice sites on the left (right) edge of the sample. Details of the calculations are described elsewhere [@us_Nature]. In Fig. \[PhaseCorr\] we show $F_{LR}$ (diamonds) as a function of magnetic field (or flux per plaquette) for a single disorder realization of a system of size $20 \times 5$, with strong disorder $W/t=1.4$ (where $t$ is the hopping integral), attractive interaction strength $U/t=2$, average electron density (per site) $\langle n \rangle =0.92$ and temperature $T=0.04$. In addition we plot the average value of the SC order parameter amplitude $\bar{|\Delta|}$ (triangles), for the same magnetic field. As can be seen, at a certain (disorder and density dependent) magnetic field the phase correlation vanish, and hence this value corresponds to $B_c$. However, the order parameter amplitude is finite beyond this point and vanishes only at higher fields. This can also be seen from the density of states (DOS), plotted in the inset of Fig. \[PhaseCorr\] for two values of the magnetic field. The left inset displays the results for vanishing magnetic field, where a clear BCS-like DOS is observed. The right inset pertains to magnetic field of $\phi/\phi_0=0.06$, that is above the SIT. However, since there are still SC correlations a pseudogap develops. This may indicate that our picture is relevant also to the physics of high Tc superconductors [@Alvarez]. We point that similar results were obtained from tunnelig measurments [@valles], where the broadened BCS peak was accounted for by SC order parameter amplitude fluctuations. ![Phase correlations between the two sides of the system $F_{LR}$ (diamonds) and the SC order parameter amplitude $\bar{|\Delta|}$ (triangles) as a function of magnetic field. At a certain magnetic field $B_c$ phase-correlations vanish, indicating the SIT. However, $\bar{|\Delta|}$ is still finite after this point, indicating the existence of SC islands. Insets : the density of states for vanishing magnetic field (left inset) and just above the transition (right inset), demonstrating the appearance of a “pseudogap” above the transition. []{data-label="PhaseCorr"}](PhaseCorr.eps){width="8"} Although the transition is described in terms of thermal fluctuations, this description does not rule out the possibility that the transition is quantum in nature. All one needs to do is to replace thermal fluctuations with quantum fluctuations and temperature with an energy scale associated with quantum fluctuations. In fact, recent measurements [@Crane; @Aubin] indicate that the transition crosses smoothly from a thermal to a quantum phase transition, the phenomenology of the two classes being very similar. Superconducting islands and non-monotonic magneto-resistance ============================================================ In a recent set of experiments, Sambandamurthy et.al. [@Murty] showed that the magneto-resistance is non-monotonic above the SIT and develops a peak at a certain magnetic field $B_\mathrm{max}$. Surprisingly, the resistance at the peak can be several orders of magnitude larger than the resistance at the transition (which is found [ *not*]{} to be universal, see e.g. [@Hebard; @kapitulnik1]). Perhaps even more surprising is the fact that with increasing the magnetic field beyond $B_\mathrm{max}$ the resistance drops several orders of magnitude and comes close to its value at the SIT. While non-monotonicity in the MR was observed more than a decade ago [@Non-Mon], there it was a miniscule effect. Here, however, this effect is huge and cannot be overlooked. In addition, the resistance develops an activation dependence on temperature, which vanishes at low enough temperatures [@Murty]. An experimental study of MR for different values of disorder [@Kapitulnik] shows that with increasing disorder the value of the resistance at the MR peak increases, and at the same time the values of both $B_c$ and $B_\mathrm{max}$ decreases. How all these phenomena can be explained in terms of the formation of SC islands [@us_MR] is discussed below. The model is based on three assumptions. The first is that disorder induces the formation of SC islands due to fluctuations in the amplitude of the SC order parameter. The second assumption is that as the magnetic field is increased, the concentration and size of these SC islands decrease. This does not mean, however, that the physical size of the islands (that is, the spatial extent to which the order parameter amplitude is finite) decreases (although it might), but rather that islands become disconnected in the sense of phase-fluctuations described in the previous section. The third assumption is that the SC islands have a charging energy, and thus, a Cooper pair entering a SC island (via an Andreev tunneling process) has to overcome it. This charging energy is expected to be inversely proportional to the island size, and thus to increase with increasing magnetic field. All three assumptions have been corroborated by numerical calculations described elsewhere [@us_PRB]. Consider now such a system in the strong magnetic field regime, $B>>B_\mathrm{max}$. Due to the strong magnetic field the SC islands are small and have a large charging energy. Now the electrons can traverse through the system in two types of trajectories: those which follow normal areas of the sample and do not cross the SC islands (“normal paths”) and those in which an electron tunnels into a SC island via the Andreev channel (“island paths”). The resistance of the normal paths, $R_N$, has some value (which may depend on e.g. length, temperature, disorder etc.) and is assumed to be weakly affected by magnetic field. Due to Coulomb blockade, transport through the “island paths” is thermally activated, and hence the resistance of the island paths is of the form $R_I \sim \exp (E_{c}/T)$, where $E_c$ is the typical charging energy of the island. If $E_c$ is large then the main contribution to the conductance is due to transport along the normal paths. Consistent with experiment, the MR in this regime is small. As the magnetic field is decreased (but still in the regime $B>B_\mathrm{max}$), more SC islands are created and their size increases, but they are still small enough such that transport along normal paths is favourable. However, some paths which were normal at higher fields now become island paths and hence unavailable for electron transport. Thus, the effective density of normal electrons which contribute to the resistance diminishes, resulting in a negative MR. Eventually, at a certain magnetic field $B=B_\mathrm{max}$ some SC islands are large enough so that their charging energy is small and the resistance through them is comparable to the resistance through normal paths, i.e. $ R_N \approx R_I $. At this point the resistance reaches its maximum value, since as the magnetic field is further decreased the SC islands are so large that transport through them is always preferred over transport through normal paths. An increase in number and size of the SC islands will thus result in a decrease in the resistance. At the critical field $B_{c}$ a phase-stiff path percolates through the system, resulting in the SIT. The above model was encoded into a numerical calculation using a lattice network model of normal resistors, SC resistors and insulating resistors (to mimic the Coulomb blockade) [@us_MR]. Resistance of a normal resistor connecting two sites on the lattice is given by $ R_{ij} =R_0 \exp \left( \frac{2 r_{ij}}{ \xi_{loc}}+\frac{|\e_{i}|+|\e_{j}|+|\e_{i}-\e_{j}|}{2kT} \right), $ where $R_0$ is a constant, $r_{ij}$ is the distance between sites $i$ and $j$, $\xi_{loc}$ is the localization length, $\e_{i}$ is the energy of the $i$-th site measured from the chemical potential (taken from a uniform distribution $[-W/2,W/2] $) and $T$ is the temperature. The resistance between two (neighboring) SC sites is taken to be very small compared with that of a normal resistor, but still not zero and temperature dependent, in such a way that it vanishes as $T\rightarrow 0$ (distant SC sites are disconnected). The resistance between a normal site and a SC site (i.e. an insulating resistor) is taken to be $ R_{NS} \propto \exp (E_{c}/kT), $ where $E_c$ is the charging energy of the island. In Fig. \[MR\] we plot the resistance as a function of the concentration of the SC islands (which corresponds to the magnetic field) for different values of temperature. The quantitative resemblance to the experimental data (inset of Fig. \[MR\]) is self-evident. In addition, this simple phenomenological theory is capable of accounting for the breakdown of activation behaviour at low temperatures, the dependence of disorder and other experimental observations. ![Magneto-resistance as a function of concentration of SC islands for different temperatures (taken from Ref. [@us_MR]), showing good quantitative resemblance between the theory and experimental data (inset, taken from Ref. [@Murty]) is evident. []{data-label="MR"}](res1C.eps){width="8truecm"} Summary and outlook =================== In this contribution we have demonstrated that many of the properties of disorder SC thin films may be accounted for by the formation of SC islands. Specifically, we have demonstrated that phase-fluctuations between different islands lead to the SIT, and that above the transition formation of SC islands leads to a non-monotonic magneto-resistance. Still, there are many puzzles left in elucidating the physics of these systems. For instance, an unusual disorder and magnetic-field dependent anisotropy in the magneto-resistance was recently measured [@angular] and is up-to-date unexplained, although it seems that a percolation theory similar to that presented here may account for it [@Yigal]. Even more challenging is the appearance of a seemingly correlated [@Shahar2] and perhaps metallic state at high magnetic field [@Baturina]. A final example is the fact that some materials seem to exhibit a metallic intermediate state at the SIT while others do not [@Baturina2]. These and other questions are still waiting to be resolved. [99]{} For a review on the superconductor-insulator transition see, e.g. A. M. Goldman and N. Markovic, Phys. Today [**51**]{} (11), 39 (1998). M. P. A. Fisher, , 923 (1990); M. Cha, M. P. A. Fisher, S. M. Girvin, M. Wallin, and A. P. Young , 6883 (1991). G. Sambandamurthy, L. W. Engel, A. Johansson and D. Shahar, , 107005 (2004). M. Steiner and A. 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--- abstract: 'We have carried out a preliminary design and simulation of a single-electron resistive switch based on a system of two linear, parallel, electrostatically-coupled molecules: one implementing a single-electron transistor and another serving as a single-electron trap. To verify our design, we have performed a theoretical analysis of this “memristive” device, based on a combination of *ab-initio* calculations of the electronic structures of the molecules and the general theory of single-electron tunneling in systems with discrete energy spectra. Our results show that such molecular assemblies, with a length below 10 nm and a footprint area of about 5 nm$^2$, may combine sub-second switching times with multi-year retention times and high ($> 10^3$) ON/OFF current ratios, at room temperature. Moreover, Monte Carlo simulations of self-assembled monolayers (SAM) based on such molecular assemblies have shown that such monolayers may also be used as resistive switches, with comparable characteristics and, in addition, be highly tolerant to defects and stray offset charges.' author: - Nikita Simonian - 'Konstantin K. Likharev' - Andreas Mayr bibliography: - 'msbib.bib' title: 'Design and Simulation of Molecular Nonvolatile Single-Electron Resistive Switches' --- Introduction ============ Recently, a substantial progress was made in the fabrication of two-terminal “memristive” devices (including bistable “resistive” or “latching” switches) based on metal oxide thin-films, whose bistability is apparently based on the reversible formation/dissolution of conducting filaments — see, e.g., recent reviews [@likharev08; @waser09; @StrukovKohlstedt12]. However, scaling of resistive memories and hybrid CMOS/nano-crossbar integrated circuits [@likharev08], based on such switches, beyond the 10 nm frontier may still require more reproducible devices based on other physical principles. One possibility here is to use a molecular version of single-electron switches [@folling01]. Such a switch, schematically shown in Fig. 1a, is a combination of two electrostatically-coupled devices: a “single-electron trap” and a “single-electron transistor” [@likharev99] placed in parallel between two electrodes. (It will be convenient for us to call these electrodes the “drain/control” and the “source” — see Fig. 1a.) When the charge state of the trap island is electroneutral ($Q = 0$), the Coulomb blockade threshold $V_C$ of the transistor is large (Fig. 1b), so that at all applied voltages with $|V| < V_C$ the transistor carries virtually no current — the so-called OFF state of the switch. As soon as the voltage exceeds a certain threshold value $V_\leftarrow < V_C$, the rate of tunneling into the single-electron trap island increases sharply (Fig. 1c), and an additional elementary charge $q$ (either a hole or an electron) enters the trap island from the source electrode, charging it to $Q = q = \pm e$. The electrostatic field of this charge shifts the background electrostatic potential of transistor’s island and as a result reduces the Coulomb blockade threshold of the transistor to a lower value $V_C^\prime$. This is the ON state of the switch, with a substantial average current flowing through the transistor at $V > V_C^\prime$. The device may be switched back into the OFF state by applying a reverse voltage in excess of the trap-discharging threshold $|V_\rightarrow|$. As was experimentally demonstrated for metallic, low-temperature prototypes of the single-electron switch [@dresselhaus94], its retention time may be very long [^1]. However, for that the scale $e^2/2C$ of the single-electron charging energy of the trap island, with effective capacitance $C$, has to be much higher than the scale of thermal fluctuations, $k_{\mbox{\scriptsize B}}T$. For room temperature, this means the need for few-nm-sized islands [@likharev99], and so far the only way of reproducible fabrication of features so small has been the chemical synthesis of suitable molecules — see, e.g., [@tour03; @cuniberti05]. Transport properties of single molecules, captured between two metallic electrodes, have been repeatedly studied by several research groups for more than a decade — see, e.g., [@BummArnoldCygan96; @Reed97; @KerguerisBourgoinPalacin99; @ParkPasupathyGoldsmith02; @KubatkinDanilovHjort03; @DanilovKubatkinKafanov08; @WangBatsanovBryce09]. The practical use of such devices in VLSI circuits is still impeded by the unacceptably low yield of their fabrication and large device-to-device variability. The main reason of this problem is apparently the lack of atomic-scale control of the contacts between the molecules and the metallic electrodes. In addition, in three-terminal single-molecule devices that can work as transistors [@ParkPasupathyGoldsmith02; @KubatkinDanilovHjort03], there is an additional huge challenge of reproducible patterning of three-electrode geometries with the required sub-nm precision. These two challenges make single-molecule three-terminal stand-alone devices rather unlikely candidates for post-CMOS VLSI circuit technology. However, we believe that for resistive memories and CMOS/nano hybrids based on nano-crossbars (Fig. 1d [@likharev08]), these challenges may be met. Indeed, such crossbars use two-terminal crosspoint devices, so that their only critical dimension (the distance between the two electrodes) may be precisely controlled by layer thickness. In addition, if such devices are based on self-assembled monolayers (SAMs), the large number $N$ of molecules in a single device may mitigate the negative effects of interfacial and other uncertainties of single molecules [@akkerman07]. (Since the electrode footprint of a quasi-linear functional molecule stretched between the electrodes may be very small, $N$ may be as high as $\sim 10^2$ even for sub-10-nm-scale devices.) The goal of this paper is to describe the results of the design and *ab-initio* calculations of basic properties of a molecular resistive switch and SAMs based on such molecules. Our basic design is described, and its physics is discussed in the next section (Sec. II). In Sec. III we formulate a theoretical model which allows an approximate but reasonably accurate numerical simulation of electron transport properties of this device. The results of simulation of our most promising switch version are described in Sec. IV. In Sec. V we describe our approach to simulation of SAM layers consisting of resistive switch assemblies, and the results of these simulations. Finally, in Sec. VI we summarize our results and discuss the necessary further work towards the practical implementation of reproducible resistive switches. ![(a) The traditional version of the single-electron resistive switch, (b) its $I-V$ curve (schematically), and (c) the ON/OFF switching rates of the device, calculated using the orthodox theory of single-electron tunneling [@averin91], for $e^2/C = 20k_{\mbox{\scriptsize B}}T$. The inset in panel (c) schematically shows the switching between charge states of the trap resulting from repeated voltage sweeps with a rate $\Gamma_0 = \left|dV/dt\right|/(e/C) \ll \Gamma_r$. (d) Nano-crossbar with resistive switches as crosspoint devices.[]{data-label="fig1"}](fig1){width="21pc"} Resistive switch design ======================= Our initial design [@likharev03; @mayr07] of the molecular resistive switch was based on oligophenyleneethynylene (OPE) chains as tunnel barriers and diimide (namely, pyromellitdiimide, naphthalenediimide, and perylenediimide) groups as trap and transistor islands. However, already the first quantitative simulations have shown that the relatively narrow HOMO-LUMO bandgap of the OPE chains (of the order of 1.5 eV [@ke07]) cannot provide a tunnel barrier high enough to ensure sufficiently long electron retention times in traps with acceptable lengths. As a result, we have concluded that alkane chains (CH$_2$-CH$_2$-...), with a bandgap of $\sim 9$ eV [@wangreed03], are much better candidates for resistive switches. There is also substantial experience in the chemical synthesis of molecular electronic devices and SAMs using such chains as tunnel barriers [@tour03; @cuniberti05]. Figure 2 shows a possible realization of such a device, based on benzene-benzobisoxazole and naphthalenediimide acceptor groups (playing the role of single-electron islands), and alkane chains. In contrast with the usual (“orthodox”) design of the trap [@folling01; @dresselhaus94; @likharev89], where a long charge retention time is achieved by incorporation of several additional single-electron islands into the trap charging path (Fig. 1a), the long alkane chain used in the molecular trap has a band structure which enables its use simultaneously in two roles: as a tunnel barrier as well as a replacement of intermediate islands. ![Our final version of a molecular resistive switch featuring an alkane-naphthalenediimide single-electron trap electrostatically coupled to an alkane-benzobisoxazole-benzene single-electron transistor.[]{data-label="fig2"}](fig2){width="21pc"} In order to explain this novel approach, let us first review the role of intermediate islands in the conventional design of the trap (Fig. 1a). If a single-electron island is so large that the electron motion quantization inside it is negligible, its energy spectrum, at a fixed net charge $Q$, may be treated as a continuum. Elementary charging of the island with either an additional electron or an additional hole raises all energies in the spectrum by $e^2/2C$, where $C$ is the effective capacitance of the island [@likharev89; @likharev99]. As a result, the continua of the effective single-particle energies of the system for electrons and holes are separated by an effective energy gap $e^2/C$ — essentially, the “Coulomb gap” [@EfrosShklovskii75]. If this gap is much larger than $k_{\mbox{\scriptsize B}}T$ at applied voltages $V$ close to the “energy-equilibrating” voltage $V_e$ (see the middle panel of Fig. 3a), it may ensure a very low rate $\Gamma_r$ of single-charge tunneling in either direction and hence a sufficiently long retention time $t_r = 1/\Gamma_r$ of the trap. The energy gap may be suppressed by applying sufficiently high voltages $V \sim e/C$ of the proper polarity, enabling fast switching of the device into the counterpart charge state — see the left and right panels of Fig. 3a, and also Fig. 1c. ![Schematic single-particle energy diagrams of (a) the usual single-electron trap shown in Fig. 1a (for the sake of simplicity, with just one intermediate island) and (b) the molecular trap analyzed in this work (Fig. 2), each for three values of the applied voltage $V$. Occupied/unoccupied energy levels are shown in black/green. (The dotted green/black line denotes the energy level of the “working” orbital that is either empty or occupied during the device operation, defining its ON/OFF state.) Horizontal arrows show (elastic) tunneling transitions, while vertical arrows indicate inelastic relaxation transitions within an island, a molecule, or an electrode.[]{data-label="fig3"}](fig3){width="21pc"} In the molecular single-electron trap shown in Fig. 2, the “energy-equilibrating” voltage $V_e$ aligns the Fermi energy of the source electrode with the lowest unoccupied level of the acceptor group that is, by design, located in the middle of the HOMO-LUMO gap of the alkane chain. As a result, an electron from the source electrode may elastically tunnel into the group only with a very low rate $\Gamma_r$ — see the middle panel in Fig. 3b. The reciprocal process (at the same voltage) may be viewed as electron tunneling from the highest occupied molecular orbital of the singly-negatively charged molecule. (To simplify the terminology, in the reminder of the paper we call this molecular orbital the “working orbital” (indexed $W$), instead of HOMO or LUMO of the counterpart molecular ions, to make the name independent of the charge state of the device.) The energy-balance condition of both processes is similar, and may be expressed via the effective single-particle energy $\varepsilon_W$ of the working orbital [@simonian07]: $$\varepsilon_W=\Delta E(n) \equiv E_{gr}(n)-E_{gr}(n-1), \label{eq_epsilon_def_ground}$$ where $E_{gr}(n)$ is the ground-state energy of the molecular ion with $n$ electrons. (In the case of singly-negative ion we are discussing, $n = n_0 + 1$ [^2], where $n_0$ is the number of protons in the molecule.) In this notation, the energy-balance (level-alignment) condition, which defines the voltage $V_e$, is $$\varepsilon_W=W-e\gamma V_e, \label{eq_epsilon_balance_ground}$$ where $W$ is the workfunction of the source electrode material, and $\gamma$ is a constant factor imposed by the geometry of the junction; $0 < \gamma < 1$. (Its physical meaning is the fraction of the applied voltage, which drops between the trapping island and the source electrode.) At the charging threshold voltage $V_\leftarrow$, energy $\varepsilon_W$ becomes aligned with the valence band edge of the chain, allowing for a fast charging of the molecule — see the left panel in Fig. 3b. Similarly, as shown on the right panel in Fig. 3b, at $V_{\rightarrow}$ this energy becomes aligned with the conduction band edge of the chain, allowing for a fast discharging of the molecule. As an example, Fig. 4a shows the atomic self-interaction corrected (ASIC) [@pemmaraju07] Kohn-Sham electron eigenenergy spectrum $\varepsilon_{i}^{\mbox{\scriptsize ASIC}}(n)$ of the alkane-naphthalenediimide molecule used as our final trap design (Fig. 2), with the net charge $Q(n) = -e (n-n_0) = -e$, as a function of the applied voltage $V$. (Here $i$ is the spin-orbital index; see Sec. III below.) Point colors in Fig. 4a crudely represent the spatial localization of the orbitals: blue corresponds to their localization at the trapping (acceptor) group, while red marks the localization at the alkane chain’s part close to the source electrode. Figure 4b shows the probability density of the working orbital $\psi_W^{\mbox{\scriptsize ASIC}}=\psi_{n_0+1}^{\mbox{\scriptsize ASIC}}(n_0+1)$ of the molecular trap, integrated over the directions perpendicular to the molecule’s axis, with blue lines corresponding to the probability density at the most negative applied voltage. At the equilibrating voltage $V_e \approx 2.2 $ V, the working orbital is well localized at the acceptor group, and is isolated from the source electrode by a $\sim 4.5$-eV-high energy barrier. However, as Fig. 4b shows, the probability density of the orbital decays into the alkane group rather slowly, with the exponent coefficient $\beta \approx 0.4a_{\mbox{\scriptsize B}}^{-1}$, corresponding (in the parabolic approximation of the dispersion relation) to the effective electron mass $m_{ef} \approx 0.1 m_0$ [^3]. As a result, a long ($\sim 5$ nm) alkane chain is needed to ensure an acceptable retention time of the trap. (The 2-nm free-space separation between the other side of the molecule and the control/drain electrode, shown in Fig. 2, is quite sufficient for preventing electron escape in that direction.) ![ASIC density-functional-theory (DFT) results (corrected for level “freezing” at high positive and negative voltages — see Sec. III for details) for the singly-negatively charged alkane-naphthalenediimide trap molecule. (a) Kohn-Sham energy spectrum as a function of the applied voltage $V$, with colors representing the spatial localization of the corresponding orbitals — see the legend bar on the right. The vertical lines mark voltage values $V_\leftarrow$, $V_e$, and $V_\rightarrow$ corresponding to the left, middle, and right panels of Fig. 3b. The dashed lines labeled $E_{\mbox{\scriptsize F}}(s)$ and $E_{\mbox{\scriptsize F}}(c)$ show the Fermi levels of the source and control/drain electrodes whose workfunction was assumed to equal 5 eV. (b) The “working” orbital’s probability density, integrated over the directions perpendicular to molecule’s axis, for a series of applied voltages $V$ — see the legend bar on the right of the panel.[]{data-label="fig4"}](fig4){width="21pc"} At a sufficiently high forward/reverse bias voltage, the working orbital energy $\varepsilon_W$ crosses into the conduction/valence band of the alkane chain, so that the orbital partly hybridizes with the states localized near the source electrode interface, described by the rise of $|\psi_W|^2$ at larger values of $z$ — see Fig. \[fig4\]b. This rise enables fast electron tunneling to/from the source electrode, i.e. a fast switching of the device to the counterpart charge state, in a manner similar to that of the conventional single-electron trap, as shown schematically on the left and right panels of Fig. \[fig3\]b. Thus the long molecular chain, with a sufficiently large HOMO-LUMO gap, may indeed play the roles of both the tunnel junction and intermediate islands of the “orthodox” single-electron trap. For the design of the second component of the switch, the molecular single-electron transistor, the most important challenge is to satisfy the ON and OFF state current requirements. In particular, the ON current should not be too large to keep the power dissipation in the circuit at a manageable level, but simultaneously not too small, so that the device output does not vanish in the noise of the sense amplifier (for memory applications [@StrukovLikharevMemory05; @strukov07]) or the CMOS invertor (in hybrid logic circuits [@StrukovLikharevLogic05]). Also, the ON/OFF current ratio should be sufficiently high to suppress current “sneak paths” in large crossbar arrays [@franzon05; @strukov07]. In addition, the transistor molecule should be geometrically and chemically compatible with the trap molecule, enabling their chemical assembly as a unimolecular device, with their single-electron island groups sufficiently close to provide substantial electrostatic coupling. (Without it, the charge of the trap would not provide a substantial modulation of the transistor current.) At the same time, the molecules must not be too close, in order to prevent a parasitic discharge of the trap via electron cotunneling through the transistor into one of the electrodes. The chemical compatibility strongly favors the use of similar chains as the transistor’s tunnel junctions. We have analyzed several alkane-chain based transistor devices with naphthalenediimide, perylenediimide and benzobisoxazole acceptor groups as transistor islands. However, in all these cases the long alkane chains, needed to match the lengths of the transistor and trap molecules, make ON currents too low. Finally, we have decided to use an unusually long ($\sim 4.3$-nm) benzene-benzobisoxazole [@krausea88] island group — see Fig. \[fig2\] and Fig. \[fig5\]b. Figure \[fig5\]a shows the Kohn-Sham electron eigenenergy spectrum $\varepsilon_{i}^{\mbox{\scriptsize ASIC}}(l_0+1)$ of this molecule as a function of voltage $V$ (where $l_0$ is the number of protons in the transistor molecule). Blue/red colored points correspond to the orbitals localized at the left/right alkane chain, while green color points denote the orbitals extended over the whole acceptor group. This extension is clearly visible in Fig. 5b, which shows the probability density of the working orbital $\varepsilon_{W^\prime}^{\mbox{\scriptsize ASIC}}=\varepsilon_{l_0+1}^{\mbox{\scriptsize ASIC}}(l_0+1)$ of the transistor molecule. During transistor operation, the tunneling electron may populate any of several group-localized orbitals, resembling the operation of the usual (metallic) single-electron transistor. As a result of such island extension, alkane chains of the transistor could be substantially shortened, to $\sim 1.5$-nm-long C$_{11}$H$_{25}$, enabling low but still acceptable ON currents of the order of 0.1 pA, even if a small (0.25-nm) vacuum gap between the alkane chain and the source electrode is included into calculations to give a phenomenological description of the experimentally observed current reduction due to unknown interfacial chemistry [@simonian07]. ![ASIC results for the single-negatively charged benzene-benzobisoxazole transistor molecule. (a) Kohn-Sham energy spectrum as a function of the applied voltage $V$, with colors representing the spatial localization (within the junction) of the corresponding orbitals — see the legend bar on the right. The dashed lines labelled $E_{\mbox{\scriptsize F}}(s)$ and $E_{\mbox{\scriptsize F}}(c)$ show the Fermi levels of the source and control/drain electrodes whose workfunction was assumed to equal 5 eV. (b) Probability density of the working orbital, integrated over the directions perpendicular to the molecular axis, for a series of applied voltages — see the legend bar on the right.[]{data-label="fig5"}](fig5){width="21pc"} Theoretical model and approximations ==================================== Each molecule used in our device has a discrete set of possible excited states, and hence the electron transport is not limited to a single channel. In order to take into account all of these channels, we have used the “quasi-single-particle approximation” whose simplest version had been first formulated by Averin and Korotkov for semiconductor quantum dots [@AverinKorotkov90; @averin91] and which was recently generalized [@simonian07] to be more applicable to molecular structures. In this approximation, the energy of an arbitrary state $k=\{n,i\}$ of the molecule equals $$E_k=E_{gr}(n)+\sum_{i>n}\varepsilon_i(n)p_i-\sum_{i \leq n}\varepsilon_i(n)(1-p_i), \label{eq_eex_def}$$ where coefficients $\varepsilon_i(n)$ have the physical meaning of single-particle excitation energies of an $n$-electron ion, and numbers $p_i$ (equal to either 0 or 1) are the single-particle energy level occupancies. The condition of elastic tunneling, leading to a transition between states $k$ and $k^\prime$, is given by the natural generalization of Eqs. (\[eq\_epsilon\_def\_ground\]) and (\[eq\_epsilon\_balance\_ground\]): $$\varepsilon_{k \rightarrow k^\prime} = W - e\gamma V, \label{eq_epsilon_balance}$$ where the single-electron recharging/excitation energy is now defined as $$\varepsilon_{k \rightarrow k^\prime} \equiv E_k - E_{k^\prime}. \label{eq_epsilon_def}$$ Because of the large size and complexity of the molecules used in our design, the only practical way to calculate their electronic structure is to use a software package (such as SIESTA [@soler02]) [^4], based on the density-functional-theory (DFT) [@jones89], which may provide a reasonably accurate ground state energy $E_{gr}^{\mbox{\scriptsize DFT}}(n)$ and a single-particle spectrum $\varepsilon_{i}^{\mbox{\scriptsize DFT}}(n)$ at a fraction of the computational cost of more correct *ab-initio* methods. Unfortunately, for such strongly correlated electronic systems as molecules considered in this paper, results obtained using standard DFT software packages [^5] have significant self-interaction errors [@pedrew81]. We believe the source of such errors is that the approximate treatment of the exchange-correlation term in the Kohn-Sham Hamiltonian does not completely cancel the self-interaction energy present in the “Hartree term” of the Hamiltonian [^6]. Indeed, the standard DFT approach leads to errors, in the key energies (\[eq\_epsilon\_def\_ground\]) and (\[eq\_epsilon\_def\]), of the order of the single-electron charging energy $e^2/2C$, where $C$ is the effective capacitance of the island group — see Appendix A for details. This error may be rather substantial; for example in the naphthalenediimide-based trap molecule (Fig. 2), it is approximately equal to 1.8 eV. For this reason, the electron affinity $E_{gr}(n_0+1)-E_{gr}(n_0)$, calculated using the LSDA DFT for the singly-negatively charged ion of the molecular trap, is significantly (by $\sim 3.2$ eV) larger than the experimental value of similar molecules [@bhozale08; @singh06]. The LSDA energies may be readily corrected to yield a much better agreement with experiments (see Table 1 in Appendix A), however, it is not quite clear how such a theory may be used for a self-consistent calculation of the corresponding working orbital $\psi_W(\mathbf{r})$. We have found that a significant improvement may be achieved by using the recently proposed Atomic Self-Interaction Corrected DFT scheme (dubbed ASIC [@pemmaraju07]) implemented in a custom version of the SIESTA software package. For the molecules that we have considered here, this approach gives the Kohn-Sham energy $\varepsilon_{n_0+1}(n_0+1)$ very close to the experimental electron affinity. However, we have found that using even this advanced approach for our task faces two challenges. First, the algorithm gives (at least for our molecular trap states with $n = n_0+1$ and $n = n_0 + 2$ electrons) substantial deviations from the relation $\varepsilon_W = \varepsilon_{n+1}(n+1)$ for $n = n_0$ (which has to be satisfied in any exact theory [@janak78; @pemmaraju07]), with the ground energy difference (\[eq\_epsilon\_def\_ground\]) close to the LSDA DFT results. This means that Eq. (\[eq\_epsilon\_def\]) cannot be directly used with the ASIC results; instead, for the electron transfer energy between adjacent ions $n$ and $n - 1$ we have used the following expression: $$\varepsilon_{k \rightarrow k^\prime} = \varepsilon_{i^\prime}^{\mbox{\scriptsize ASIC}}(n). \label{eq_epsilon_asic}$$ This relation implies that the differences $\varepsilon_{i^\prime}^{\mbox{\scriptsize ASIC}}(n)-\varepsilon_{n}^{\mbox{\scriptsize ASIC}}(n)$ describe all possible single-particle excitations within the acceptor group, if the index $i^\prime$ is restricted to orbitals localized on the group. (Other orbitals, localized on the alkane chain are irrelevant for our calculations since they do not contribute to the elastic tunneling between the molecular group and the electrode.) In order to appreciate the second problem, look at Fig. 6 which shows the voltage-dependent Kohn-Sham spectra of the singly-negatively charged molecular trap, calculated using the ASIC SIESTA package for $T > 0$ K. Notice that above voltage $V_t \approx 13$V, and below voltage $V_t^\prime \approx -7$V, the eigenenergy spectrum is virtually “frozen”. (The LSDA SIESTA gives similar results.). As explained in Appendix B using a simple but reasonable model (similar to that used in Appendix A), at $V > V_t$ such “freezing” originates from the spurious self-interaction of an electron whose wavefunction cloud is gradually shifted from the top occupied orbital of the valence band of the chain, with energy $\varepsilon_v$, into the initially empty group-localized orbital with energy $\varepsilon_{W+1}$. (A similar freeze at voltages $V < V_t^\prime$, is due to the spurious gradual transfer of the electron wavefunction cloud from the working orbital, localized at the acceptor group, with energy $\varepsilon_W$, to the lowest orbital of the conduction band of the chain, with energy $\varepsilon_c$.) It is somewhat surprising that this spurious effect (which should not be present in any consistent quantum-mechanical approach — see Appendix B) is so strongly expressed in the ASIC version of the SIESTA code, which was purposely designed to get rid of the self-interaction in the first place. Being no SIESTA experts, we may only speculate that the nature of this artifact is related to the smoothing of the derivative discontinuity present in the ASIC method as the electron number passes through an integer value, which is mentioned in [@pemmaraju07] — see also Fig. 7 in that paper. ![The Kohn-Sham spectra of the singly-negatively charged molecular trap, calculated with the ASIC SIESTA at $T = 10$ K. At voltages below $V_t^\prime \approx -7$ V, the spectrum is virtually frozen due to a spurious gradual shift of the highest-energy electron from the “working” orbital (with energy $\varepsilon_W$, shown with a solid blue line) localized on the acceptor group, to the lowest orbital (with energy $\varepsilon_c$, shown with a solid red line) of the conduction band of the alkane chain. As a result, the calculated spectrum is virtually voltage-insensitive (“frozen”). In the voltage range $V_t^\prime < V < V_t \approx 13$ V, ASIC SIESTA gives apparently correct solutions, with the working orbital $\varepsilon_W$ fully occupied, and the next group-localized orbital (with energy $\varepsilon_{W+1}$, the dashed blue line) completely unoccupied. However, at $V > V_t$ the package describes a similar spurious gradual shift of the highest-energy electron from the highest level $\varepsilon_v$ of the valence band of the chain to orbital $\varepsilon_{W+1}$, resulting in a similar spectrum “freeze”. The spectrum evolution, calculated after the (approximate) correction of this spurious “freezing” effect, is shown in Fig. 4a above. []{data-label="fig6"}](fig6){width="21pc"} Fortunately, there is a way to correct this error very substantially by following the iterative process of self-consistent energy minimization within ASIC SIESTA. Indeed, for a fixed temperature $T > 0$ K (when the program automatically populates molecular orbitals in accordance with the single-particle Fermi-Dirac statistics) and voltages $V > V_t \approx 13$ V and $V < V_t^\prime \approx -7$ V, its iterative process converges to a wrong solution with the energy levels frozen at their $V_t$ and $V_t^\prime$ values, as is discussed above — see Fig. 6. However, if the temperature in that program is fixed at $T = 0$ K, its iterative process ends up in quasi-periodic oscillations between different solutions — most of them with frozen levels (just like in Fig. 6), but some of them with the group localized energies like the working orbital energies $\varepsilon_W$, $\varepsilon_{W+1}$ and the valence/conduction band edge energies $\varepsilon_v$, $\varepsilon_c$ close to their expected (unfrozen) values. (Those values were obtained by a linear extrapolation of their voltage behavior calculated at $V_t^\prime < V < V_t$.) Since such a solution is repeated almost exactly at each iterative cycle (see the vertical boxes in Fig. 7), we believe that it is close to the correct solution expected from the self-consistent quantum-mechanical theory — see Appendix B. These approximate solutions were used in our calculations both above $V_t$ and below $V_t^\prime$; they are illustrated in Fig. 4a, where we have substituted the incorrect “frozen” solutions for $T > 0$ K with solutions for $T = 0$ K, with $\varepsilon_W \approx \varepsilon_{W}^{\mbox{\scriptsize fit}}$, $\varepsilon_{W+1} \approx \varepsilon_{W+1}^{\mbox{\scriptsize fit}}$ and $\varepsilon_{c} \approx \varepsilon_{c}^{\mbox{\scriptsize fit}}$ at $V < V_t^\prime$ or $\varepsilon_{v} \approx \varepsilon_{v}^{\mbox{\scriptsize fit}}$ at $V > V_t$. Let us emphasize that the approximate nature of these solutions may have affected our calculations (we believe, rather insignificantly), only at $V > V_t \approx 13$ V and $V < V_t^\prime \approx -7$ V, i.e. only the device recharging time results, but not the most important retention time calculations for smaller voltages — see Fig. 9 below. ![The Kohn-Sham energy spectrum of our trap molecule, as calculated by successive iterations within ASIC SIESTA for $T = 0$ K and $V = 14.9$ V, i.e. above the threshold voltage $V_t \approx 13$ V. Vertical boxes mark the apparently correct solutions with energies of the working orbital ($\varepsilon_W^{\mbox{\scriptsize ASIC}}$), the next group-localized orbital ($\varepsilon_{W+1}^{\mbox{\scriptsize ASIC}}$), and the highest orbital of the valence band of the alkane chain ($\varepsilon_v^{\mbox{\scriptsize ASIC}}$) all close to their respective values $\varepsilon_W^{\mbox{\scriptsize fit}}$, $\varepsilon_{W+1}^{\mbox{\scriptsize fit}}$ and $\varepsilon_v^{\mbox{\scriptsize fit}}$ obtained by a linear extrapolation of their voltage dependence calculated at $V_t^\prime < V < V_t$. Just like in Figs. 4a, 5a and 6, point colors represent the spatial localization of the corresponding orbitals. Lines are only guides for the eye.[]{data-label="fig7"}](fig7){width="21pc"} With the electron orbitals and eigenenergies calculated, we have described dynamics of both the trap and the transistor, just as in our first work [@simonian07], by a set of master equations for state probabilities [@averin91], which are valid because of the incoherent character of single-electron tunneling to/from the continuum of electronic states in metallic electrodes [@likharev99]. Moreover, for the inelastic relaxation rates $\Gamma_{\mbox{\scriptsize inel}}$ and the rates $\Gamma_\leftarrow$ and $\Gamma_\rightarrow$ of the elastic tunneling between the molecular group and the metallic electrodes (see arrows in Fig. 3), the following strong inequality, $$\Gamma_{\mbox{\scriptsize inel}} \gg \Gamma_\leftarrow, \Gamma_\rightarrow \label{eq_gamma_ineq}$$ is well fulfilled. (Indeed, the rates $\Gamma_{\mbox{\scriptsize inel}}$ are crudely of the order of $10^{12}$ 1/s in both molecules and metals. On the other hand, our results, described in Sec. IV below, yield transistor currents $I \sim 10^{-13}$ A, meaning that $\Gamma_\leftarrow$ and $\Gamma_\rightarrow$ are of the order of $I/e \sim 10^6$ 1/s in the transistor; the rates are even much lower than that in the trap — see Fig. 9b.) Relation (\[eq\_gamma\_ineq\]) allows us to account only for the tunneling events starting from thermal equilibrium, and ensures that the rates $\Gamma_{\mbox{\scriptsize inel}}$ drop out of the calculations. In comparison with [@simonian07], one more new element of this work is the electrostatic coupling between the trap and the transistor which features similar but much more frequent single-charge transitions. This rate hierarchy allows the trap to be described by averaging rates $\Gamma$ of tunneling events in it over a time interval much longer than the average time period between tunneling events in the transistor, but still much shorter than $1/\Gamma$. These average rates may be calculated as $$\left<\Gamma_{\rightarrow}\right> = \sum_{l}\sigma_{n_0}(l)w_{l,+}(n_0), \label{eq_gammar}$$ for electron tunneling from the source into the trap molecule, and $$\left<\Gamma_{\leftarrow}\right> = \sum_{l}\sigma_{n_0+1}(l)w_{l,-}(n_0+1), \label{eq_gammal}$$ for the reciprocal event. Here $\sigma_n(l)$ are the conditional probabilities of certain charge states $l$ of the transistor island provided that the trap is in the $n$-electron charge state (with $n$ equal to either $n_0$ or $n_0+1$), while $w_{l,\pm}(n)$ are the total rates of single electron tunneling between the trap in its initial charge state $n$ and the source electrode. These rates have been calculated using Eq. (11) in [@simonian07], with an extra index $l$ added to account for the transistor’s state. The conditional probabilities $\sigma_n(l)$ satisfy the usual normalization condition: $$\sum_l \sigma_n(l)=1, \label{eq_sigmanorm}$$ and (together with the dc current $I$ flowing through the transistor) have been calculated as in [@simonian07], by combining the master equations of single-electronics [@AverinKorotkov90; @averin91] with *ab-initio* calculations of molecular orbitals and spectra, and the Bardeen formula [@bardeen61] for tunneling rates. ![A schematic view of charge densities participating in Eq. (\[eq\_coulomb\]).[]{data-label="fig8"}](fig8){width="21pc"} The electrostatic interaction between the two molecules is taken into account by an iterative incorporation of the numerically calculated Coulomb potential created by both molecules (as well as by the series of their charge images in the metallic electrodes of the system, which we have assumed to be plane, infinite surfaces — see Fig. \[fig8\]) into the Kohn-Sham potentials. From the elementary electrostatics, this potential may be expressed as $$\begin{gathered} \begin{split} \phi_s(\mathbf{r})=&\int\frac{\rho_{c}(\mathbf{r_0})}{\left|\mathbf{r}-\mathbf{r_0}\right|}d^3r_0\\ &+\sum_{j\neq 0}(-1)^{j} \int \frac{\rho_{c}(\mathbf{r_j})+\rho_{s}(\mathbf{r_j})}{\left|\mathbf{r}-\mathbf{r_j}\right|}d^3r_j \\ &-V\frac{z-d/2}{d}, \end{split} \nonumber \\ \mathbf{r}_j \equiv \mathbf{r}_0 + \mathbf{n}_z \times \left\{ \begin{split} jd, &\mbox{ for $j$ even,} \\ (j+1)d-2z, &\mbox{ for $j$ odd.} \end{split} \right. \label{eq_coulomb}\end{gathered}$$ where $\rho_{s}(\mathbf{r}_0)$, $\rho_{c}(\mathbf{r}_0)$ are the total charge distributions of the molecule under analysis and the complementary molecule, and $\rho_{s}(\mathbf{r}_j)$, $\rho_{c}(\mathbf{r}_j)$ are the corresponding charge images in the source $(j>0)$ and the drain $(j<0)$ electrodes — see Fig. \[fig8\]. The first term in Eq. (\[eq\_coulomb\]) is the potential created by the complement molecule, the second term describes the potential of the infinite set of charge images of both molecules in the source and control/drain electrodes, and the third term is the potential created by the applied source-drain voltage. At the 0-th iteration, the first two terms are taken equal to zero. Fortunately, the iterations give rapidly converging results, so that there was actually no need to go beyond the second iteration — see Fig. \[fig9\]. ![Differences between the energies of the working orbital of the molecular trap, calculated with ASIC SIESTA at the $k$-th and $(k-1)$-th iterations of the Coulomb interaction potential, given by Eq. (\[eq\_coulomb\]), as functions of the applied voltage.[]{data-label="fig9"}](fig9){width="21pc"} Another important change introduced into our calculations is that the charge transfer rates (see Fig. 4c in [@simonian07]) are calculated in a simpler way. Namely, one of the key conditions of validity of the Bardeen formula for the tunneling matrix elements $$T_{[s,c],i}=\frac{\hbar^{2}}{2m}\int_{S}\left(\psi_{s,c}^{\ast}\frac{\partial \psi_i}{\partial z} - \psi_i^\ast\frac{\partial \psi_{s,c}}{\partial z}\right)dS \label{eq_bardeen}$$ (where $\psi_i$ is the molecular orbital, $\psi_{s,c}$ are the wavefunctions of electrons located inside the source or control/drain electrodes, and $S$ is an arbitrary surface separating the single-electron island from the corresponding electrode) is that the result given by Eq. (\[eq\_bardeen\]) is independent of the position of the surface $S$. Due to electrostatic screening of the electric field by the electrode, the Kohn-Sham potential becomes very close to the vacuum potential at just a few Bohr radii $a_{\mbox{\scriptsize B}}$ away from the molecule’s last atom. Therefore, if the surface $S$ is selected inside the vacuum gap between the molecule’s end and the electrode surface, the effect of the molecule on wavefunctions $\psi_{s,c}$ is negligible with good accuracy (corresponding to a fraction of one order of magnitude in the resulting current). Hence, these wavefunctions may be calculated analytically to describe the usual exponential 1D decay into vacuum, instead of a numerical solution of a Schrödinger equation, as it had been done in [@simonian07]. On top of the *ab-initio* calculation scheme, we have performed the following check of the component molecule spacing. As was mentioned in Sec. II, the molecules have to be placed sufficiently far from each other to prevent a parasitic discharge of the trap via the elastic cotunneling through the transistor island, into one of the electrodes. This effect may be estimated using the following formula [@stoof96; @averin00]: $$\Gamma_\rightarrow^{\mbox{\scriptsize cot}}=\frac{\Delta^2 w_{\rightarrow}}{\hbar^2 w_{\rightarrow}^2 + 2\Delta^2+4\varepsilon^2}, \label{eq_averin}$$ where $\Delta$ is the matrix element of electron tunneling between the trap and the transistor islands (that exponentially depends on the distance $d_p$ between the molecules, see Fig. \[fig2\]), $\varepsilon$ is the difference between eigenenergies of these states, and $w_{\rightarrow}$ is the rate of tunneling between the transistor island and its electrodes. We have found that in order for the cotunneling rate $\Gamma_\rightarrow^{\mbox{\scriptsize cot}}$ to be below the retention rate of the trap $\Gamma_r$, the distance $d_p$ should not be lower than $\sim 1.5$ nm. Such relatively large separation justifies separate DFT calculations of the electronic structures of the trap and the transistor (with the molecular geometry of each component device initially relaxed, using LSDA SIESTA [^7]), related only by their electrostatic interaction described by Eq. (\[eq\_coulomb\]). Simulation results for a single resistive switch ================================================ Figure \[fig10\] shows our main results for the resistive switch shown in Fig. \[fig2\], for temperature $T = 300$ K. They include the dc $I-V$ curves of the transistor, plotted in Fig. \[fig10\]a for both charge states of the trap, and the rates of transitions between the neutral and single-negatively charged states of the trap, with and without the account of the transistor effect on the trap molecule (Fig. \[fig10\]b). The plots show that the resulting $I-V$ curves fit our initial specifications rather well, with a broad voltage window (from $\sim 2.0$ to $\sim 2.5$ V) for the trap state readout, a large ON/OFF current ratio within that window (inset in Fig. \[fig10\]a), and the ON current $I \sim 0.2$ pA. ![(a) Calculated dc $I-V$ curves of the transistor for two possible charge states of the trap molecule. The inset shows the ON/OFF current ratio of the transistor on a semi-log scale, within the most important voltage interval. (b) The trap switching rates, calculated with (solid lines) and without (dashed lines) taking into account the transistor’s back action, as functions of the applied voltage. Red dashed lines on panel (b) show the trap switching rates calculated without the level “freezing” correction (Sec. III and Appendix B) and without taking into the account the transistors’ back action.[]{data-label="fig10"}](fig10){width="21pc"} Figure \[fig10\]b shows that the trap features a high retention time, $\tau_r > 10^8$s, for both charge states, within a broad voltage range, $-2$V $< V <$ $+5$V. (It is somewhat surprising how little is the trap retention affected by the electrostatic “shot noise” generated by fast, quasi-periodic charging and discharging of the transistor island, which is taken into account by our theory.) The range includes point $V = 0$, so that the device may be considered a nonvolatile memory cell. At the same time, the device may be switched between its states relatively quickly by applied voltages outside of this window. The price being paid for using alkane chains with their large HOMO-LUMO gap is that the voltages necessary for fast switching are large — they must align the valence or conduction band of the alkane chain with the group-localized working orbital — see Fig. \[fig4\]a. SAMs of resistive switches ========================== Probably the largest problem of molecular electronics [@tour03; @cuniberti05] is the low reproducibility of interfaces between molecules and metallic electrodes. However, recent results [@akkerman06] indicate that this challenge may be met at least for self-assembled monolayers (SAMs) encapsulated using special organic counter-electrodes. This is why we have explored properties of SAMs consisting of square arrays of $N \times N$ resistive switches described above — see Fig. \[fig11\]. In order to increase the tolerance of the resulting SAM devices to self-assembly defects and charged impurities, it is beneficial to place the component molecular assemblies (Fig. 2) as close to each other as possible, say at distances comparable to that ($\sim 1.5$ nm) between the trap and transistor. In this case, the Coulomb interactions between the component molecules are very substantial, and properties of the system have to be calculated taking these interactions into account. ![Schematic view of a $5 \times 5$-switch SAM sandwiched between two electrodes.[]{data-label="fig11"}](fig11){width="21pc"} A system of $N \times N$ resistive switches has $2N \times 2N$ single-electron islands and hence at least $2^{2N \times 2N}$ possible charge states, which would require solving that many master equations for their exact description. Even for relatively small $N$, this approach is impracticable, and virtually the only way to explore the properties of the system is to perform its Monte Carlo simulations [@likharev89; @likharev99]. In this method a random number generator is used twice for each state change: first, to calculate the random time of some state change (which obeys Poissonian statistics), and second, to calculate the charge transition type (if several transitions are possible simultaneously). The procedure requires a prior calculation of rates of transitions between all pairs of charge configurations which differ by one single-electron tunneling event. ![Effect of a single charge of a trap molecule on the electron affinity of another molecule, located at distance $r$ without and with the account of the electric field screening by the common metallic electrodes.[]{data-label="fig12"}](fig12){width="21pc"} As was discussed above, the peculiarity of our particular system is that it features two very different time scales: the first one (for our devices, $\tau_t \approx e/I_{ON} \sim 10^{-4} - 10^{-6}$ s) characterizes fast charge tunneling through single-electron transistors, and the second one corresponds to the lifetimes of trap states ($\tau_r = 1/\Gamma \sim 10^8 - 10^{-2}$ s). In order to gather reasonable statistics of the switching rates, our data accumulation time, for each parameter set, corresponded to the physical times of up to 10 s, i.e., included up to a million transition tunneling events in the system’s transistors. ![Trap tunnel rates as functions of the applied voltage for two quasi-similar nearest-neighbor charge configurations shown in the insets.[]{data-label="fig13"}](fig13){width="21pc"} As a check of the validity of the procedure, the Monte Carlo algorithm was first applied to a single resistive switch, and it indeed gave virtually the same result as the master equation solution. We then used the approach for a direct simulation of SAM fragments with two and more coupled resistive switches. As the fragment is increased beyond a $2 \times 2$ switch array, even the Monte Carlo method runs into computer limitations, because of the exponentially growing number of the possible charge configurations. The calculations may be very significantly sped up by using the approximation in which each molecule’s state affects the potential of only its nearest neighbors. This approximation has turned out to be very reasonable (Fig. \[fig12\]) and may be justified by the fact that metallic electrodes of the system substantially screen the Coulomb potential of the charges of distant molecules: the distance between the acceptor group centers and the electrodes, $d/2 \approx 4$ nm, is of the same order as the 3-nm distance between the molecule and its next-next neighbors. In this nearest-neighbor approximation, each molecule (a trap or a transistor) is still affected by 8 other molecules. To limit the number of the charge configurations even further, we have treated all “essentially similar” of them (having charge pairs at equal distances, irrespective of their angular position) as identical — see Fig. \[fig13\]. ![Monte-Carlo simulated dc $I-V$ curves of a 25-switch SAM. The top inset shows the fraction $\beta$ of single-negatively charged traps, averaged over 40 sweeps of applied voltage between -8V and 13V. The bottom inset shows the ON/OFF current ratio averaged over the voltage sweeps, and its maximum sweep-to-sweep spread.[]{data-label="fig14"}](fig14){width="21pc"} Figure \[fig14\] shows the results of calculations, based on this approach, for a $5 \times 5$-switch SAM, of the total area close to $10 \times 10$ nm$^2$. The switching and state readout properties are very comparable with those of a single switch (Fig. \[fig10\]), despite a significant mutual repulsion between single electrons charging neighboring traps. In order to better understand why this repulsion does not have adverse effects on the operation of the SAM as a whole, we have calculated the correlation coefficients of charging of two molecules in the SAM as a function of the distance between them. At voltages above the transistor Coulomb blockade, transistor molecules switch their charge state fast and the correlation coefficient $K(r)$ between two transistor molecules may be calculated directly from their time evolution records at a constant $V$. On the other hand, trap molecules have quasi-stationary charge states, so that the correlation between two trap molecules has to be calculated from a set of snapshots of their charge states (at some voltage of interest) taken at repeated, slow sweeping of the applied voltage throughout the whole voltage range. ![The average correlation between two traps (green) and two transistors (blue) as a function of distance between them in a $5 \times 5$ device SAM.[]{data-label="fig15"}](fig15){width="21pc"} Figure \[fig15\] shows the resulting average correlation between molecules (and its fluctuations) as a function of the distance between them in the $5 \times 5$-switch SAM. The charge states of neighboring traps are significantly anticorrelated, while the next-next neighbor charge states are positively correlated. This means that the switching is due to a nearly-simultaneous entry of electrons into roughly every other trap [^8]. This explains why in the top inset in Fig. \[fig14\] the average fraction of charged traps is close to 1/2. Thus the only adverse effect of the Coulomb interaction between individual resistive switches is the approximately two-fold reduction of the average ON current per device. Figure \[fig16\] presents a summary characterization of the SAM operation as a function of its size (and hence its area). ![Summary of Monte Carlo simulations of SAMs of various area: (a) the average ON currents at voltages providing certain ON/OFF ratios; (b) the average fraction of negatively charged traps at the equilibrating voltage $V_e$.[]{data-label="fig16"}](fig16){width="21pc"} The fact that even the fractional charging of traps in SAMs is sufficient for a very good modulation of their net current suggests that these devices should have a high tolerance to defects and stray electric charges [@likharev99]. In order to verify this, we have carried out a preliminary evaluation of the defect tolerance by artificially fixing charge states of certain, randomly selected component molecules. The results, shown in Fig. \[fig17\], are rather encouraging, implying that the switches may provide the ON/OFF current ratios above 100 at defective switch fractions up to $\sim 10$%, and at a comparable concentration of random offset charges. ![Defect tolerance of the $5 \times 5$ SAM switch: ON current as a function of a number of molecules held artificially in a fixed, random charge state, at random locations, at the applied voltage values necessary to ensure a certain level of the ON/OFF current ratio. Error bars show the r.m.s. spread of results.[]{data-label="fig17"}](fig17){width="21pc"} Conclusion ========== Despite the problems with the description of single-electron charging in the density-functional theory, described in detail in Appendices A and B, we have managed to combine its advanced (ASIC) version to analyze the possibility of using single-electron tunneling effects in molecular assemblies for the implementation of bistable memristive devices (“resistive switches”). Our results indicate that chemically-plausible molecules and self-assembled monolayers of such molecules may indeed operate, at room temperature, as nonvolatile resistive switches which would combine multi-year retention times with sub-second switching times, and have ON/OFF current ratios in excess of $10^3$. Moreover, we have obtained strong evidence that operation of the SAM version of the device is tolerant to a rather high concentration of defects and randomly located charged impurities. The ON current of a single device ($\sim 0.1$ pA at $V \approx 2$ V) corresponds to a very reasonable density ($\sim 4$ W/cm$^2$) of the power dissipated in an open SAM switch, potentially enabling 3D integration of hybrid CMOS/nano circuits [@ChengStrukov12]. (Note that the average power density in a crossbar is at least 4 times lower because of the necessary crosspoint device spacing (Fig. \[fig1\]d); besides that, in all applications we are aware of, at least 50% of the switches (and frequently much more) are closed, decreasing the power even further.) However, even our best design (Fig. \[fig2\]) still requires additional work. First, proper spatial positions of the functional molecules have to be enforced by some additional molecular support groups which have not been taken into account in our analysis yet. If the spacer groups fixing the relative spatial arrangement of the islands can be constructed from saturated molecular units similar to the alkane chains used to separate the islands from the electrodes, then the calculations presented here should be applicable to complete devices, but this expectation still has to be verified. Second, we feel that there is room for improvement in the choice of molecular chains used as tunnel barriers and intermediate islands. For example, the low calculated effective mass, $m_{ef} \approx 0.1 m_0$, of electrons tunneling along alkane chains makes it necessary to use rather long chains, despite their large HOMO-LUMO gaps (which, in turn, require large switching voltages — see Figs. \[fig4\], \[fig10\]). The use of a molecular chain with a higher $m_{ef}$ and a narrower gap would decrease switching voltages (and hence energy dissipation at switching), and also reduce the total device length, resulting in shorter switching times (at the same charge retention). Third, the defect tolerance of SAM-based switches should be evaluated in more detail, for charged impurities located not only on the molecular acceptor groups, but also between them — say, inside the (still unspecified) support groups. Finally, an experimental verification of our predictions looks imperative for the further progress of work towards practicable molecular resistive switches. Acknowledgment {#acknowledgment .unnumbered} ============== This work was supported by the Air Force Office of Scientific Research. The supercomputer resources used in this work were provided by DOD’s HPCMP. Valuable comments by P. Allen, D. Averin and M. Fernandez-Serra are gratefully acknowledged. We would also like to thank C. Pammaraju and S. Sanvito for their generous help with the ASIC SIESTA software package. Single-electron charging correction =================================== Let us consider a simple but reasonable model of a well-conducting (say, metallic) island, of a size well above the Thomas-Fermi screening length, in which the single-electron addition energies are simply $$\Delta E(i) = K_i - e\phi_i, \label{eq_A1}$$ where $K_i$ is $i$-th electron’s kinetic energy (which, as well as the island capacitance $C$, is assumed to be independent of other electron state occupancies, but is an arbitrary function of $i$), and the second term describes the potential energy of that electron in the net electrostatic potential of all other charges, $$\phi_i=\phi_0-(i-1)\frac{e}{C}, \label{eq_A2}$$ where $\phi_0$ is the background potential of the nuclei, and the second term is due to the previously added electrons. In this model the total ground-state energy of an $n$-electron ion (besides the electron-independent contributions) is $$\begin{aligned} E_{gr}(n)&=&\sum_{i=1}^n\Delta E(i) \nonumber \\ &=&\sum_{i=1}^n K_i - en\phi_0+\frac{e^2}{2C}n(n-1), \label{eq_A3}\end{aligned}$$ so that the energy difference created by the last charging is $$\begin{aligned} \Delta E(n)&=&E_{gr}(n)-E_{gr}(n-1) \nonumber \\ &=&K_n-e\phi_0+\frac{e^2}{C}(n-1). \label{eq_A4}\end{aligned}$$ On the other hand, in a hypothetical naïve DFT theory, without the partial self-interaction corrections present in its LSDA, GGA and ASIC versions, the single-particle (Kohn-Sham) energies of ion $n$ of this model are written as $$\varepsilon_i^{\mbox{\scriptsize DFT}}(n)=K_i-e\phi_n,\mbox{ }\phi_n=\phi_0-\frac{e}{C}n. \label{eq_A5}$$ For the calculation of the full ground-state energy of ion $n$, such generic DFT sums up these energies from $i = 1$ to $i = n$, adding the “double-counting correction” term [@jones89], in the Gaussian units equal to $$E_{\mbox{\scriptsize corr}}=-\frac{1}{2}\int d^3r \int d^3r^\prime \frac{\rho(\textbf{r})\rho(\textbf{r}^\prime)}{\left|\textbf{r}-\textbf{r}^\prime \right|}, \label{eq_A6}$$ where $\rho(\textbf{r})$ is the total electron charge density at point **r**. For our simple model, this correction is just $–e^2n^2/2C$, so that $$\begin{aligned} E_{gr}^{\mbox{\scriptsize DFT}}&=&\sum_{i=1}^n\varepsilon_i^{\mbox{\scriptsize DFT}}(n)-\frac{e^2n^2}{2C} \nonumber \\ &=&\sum_{i=1}^nK_i-en\phi_0+\frac{e^2n^2}{2C}, \label{eq_A7}\end{aligned}$$ and $$\begin{aligned} \Delta E^{\mbox{\scriptsize DFT}}(n)& \equiv & E_{gr}^{\mbox{\scriptsize DFT}}(n)-E_{gr}^{\mbox{\scriptsize DFT}}(n-1) \nonumber \\ &=& K_n-e\phi_0+\frac{e^2}{C}\left(n-\frac{1}{2}\right). \label{eq_A8}\end{aligned}$$ Comparing this result with Eq. (\[eq\_A4\]), we obtain the following relation: $$\Delta E(n)=\Delta E^{\mbox{\scriptsize DFT}}(n)-\frac{e^2}{2C}. \label{eq_A9}$$ Thus in the naïve DFT theory, the single-electron addition energy differs from the correct expression (\[eq\_A4\]) by $e^2/2C$. Moreover, it does not satisfy the fundamental Eq. (\[eq\_epsilon\_def\_ground\]). Indeed, for $i = n$, Eq. (\[eq\_A9\]) gives the following result, $$\varepsilon_n^{\mbox{\scriptsize DFT}}(n)=K_n-e\phi_0+\frac{e^2}{C}n \label{eq_A10}$$ which, according to Eqs. (\[eq\_A4\]) and (\[eq\_A8\]) may be rewritten either as $$\Delta E(n) = \varepsilon_n^{\mbox{\scriptsize DFT}}(n)-\frac{e^2}{C}, \label{eq_A11}$$ or as $$\Delta E^{\mbox{\scriptsize DFT}}(n) = \varepsilon_n^{\mbox{\scriptsize DFT}}(n)-\frac{e^2}{2C}. \label{eq_A12}$$ This error is natural, because such DFT version ignores the fundamental physical fact that an electron does not interact with itself, even if it is quantum-mechanically spread over a finite volume. This difference can become quite substantial in small objects such as molecular groups. For example, Table 1 shows the results using LSDA SIESTA calculations for two different ions of our trap molecule (Fig. \[fig2\]), with $n = n_0 + 1$ and $n = n_0 +2$, where $n_0 = 330$ is the total number of protons in the molecule. The results show that the inconsistency described by Eq. (\[eq\_A12\]) is indeed very substantial and is independent (as it should be) of the applied voltage $V$ in the range keeping the working orbital’s energy inside the HOMO-LUMO gap of the alkane chain. The two last columns of the tables show the values of $e^2/2C$, calculated in two different ways: from the relation following from Eq. (\[eq\_A5\]): $$\frac{e^2}{2C}=\frac{\varepsilon_n^{\mbox{\scriptsize DFT}}(n)-\varepsilon_n^{\mbox{\scriptsize DFT}}(n-1)}{2}, \label{eq_A13}$$ and from the direct electrostatic expression $$\frac{e^2}{2C}=\frac{1}{2}\int \phi_n(\textbf{r})\left|\psi_n^{\mbox{\scriptsize DFT}}(\textbf{r})\right|^2d^3r, \label{eq_A14}$$ where $\phi_n(\textbf{r})$ is the part of the electrostatic potential, created by the electron of the $n$-th orbital of the $n$-th ion. The values are very close to each other and correspond to capacitance $C \approx 4.5 \times 10^{-20}$ F which a perfectly conducting sphere of diameter $d \approx 0.8$ nm would have. The last number is in a very reasonable correspondence with the size of the acceptor group of the molecule — see Fig. \[fig2\]. [ &gt;m[1.5cm]{} &gt;m[1.6cm]{} &gt;m[1.6cm]{} &gt;m[1.6cm]{} &gt;m[1.6cm]{}]{} & [ $\Delta E(n)$ from Eq. (\[eq\_A9\])]{} (eV) & [ $\Delta E(n)$ from Eq. (\[eq\_A11\]) (eV) ]{} & [ $e^2/2C$ from Eq. (\[eq\_A13\]) (eV)]{} & [ $e^2/2C$ from Eq. (\[eq\_A14\]) (eV)]{}\ \ (r)[1-5]{} -2.36 & -3.08 & -3.01 & 1.84 & 1.79\ -1.18 & -3.38 & -3.37 & 1.84 & 1.79\ 0.00 & -3.73$^{\mbox{\scriptsize (a)}}$ & -3.73$^{\mbox{\scriptsize (a)}}$ & 1.84 & 1.79\ 1.18 & -4.07 & -4.10 & 1.84 & 1.79\ 2.36 & -4.42 & -4.46 & 1.84 & 1.79\ 3.53 & -4.77 & -4.82 & 1.84 & 1.79\ (r)[1-5]{}\ (r)[1-5]{} 7.07 & -1.91 & -2.02 & 1.82 & 1.79\ 8.24 & -2.29 & -2.39 & 1.82 & 1.79\ 9.42 & -2.61 & -2.75 & 1.82 & 1.79\ 10.60 & -2.97 & -3.11 & 1.82 & 1.79\ 11.78 & -3.32 & -3.47 & 1.82 & 1.79\ \ $^{\mbox{\scriptsize (a)}}$ The numbers to be compared with experimental values of electron affinity: -3.31 eV Ref. [@bhozale08] and -3.57 eV Ref. [@singh06]. The second and third columns of the table present the genuine electron addition energies $\Delta E(n)$ calculated from, respectively, Eq. (\[eq\_A9\]) and (\[eq\_A11\]), using the average of the above values of $e^2/2C$. Not only do these values coincide very well; they are in a remarkable agreement with experimentally measured electron affinities [@singh06; @bhozale08] of molecules similar to our molecular trap. We believe that these results show that, first, LSDA SIESTA provides very small compensation of the self-interaction effects in the key energy $\Delta E(n)$ and, second, that (at least for the lowest negative ions of our trap molecules), an effective compensation may be provided using any of the simple relations (\[eq\_A9\]) and (\[eq\_A11\]). Level freezing in DFT ===================== For the analysis of the fictitious “level freezing” predicted by a naïve DFT at $V > V_t$ (see Fig. \[fig6\]), let us consider the following simple model: a molecule consisting of a small acceptor group with just one essential energy level, and a spatially separated chain with a quasi-continuous valence band. Figure \[fig\_B1\] shows the energy spectrum of the system at $V < V_t$. (As before, the occupied levels are shown in black, while the unoccupied ones are shown in green.) ![The schematic energy spectrum of our model at a voltage $V$ below voltage $V_t$ that aligns the group localized level $\varepsilon$ with the valence band edge $\varepsilon_v$.[]{data-label="fig_B1"}](figB1){width="21pc"} The edge $\varepsilon_v$ of the band is separated from the first unoccupied level in the group by energy $-e(V - V_t)$, where $V$ is the fraction of the voltage drop between the centers of the group and the tail of a molecule, and $V_t$ is its value which aligns the level with $\varepsilon_v$. Now let $V$ be close to $V_t$, so that the occupancy $p$ of the discrete level is noticeable. If the effect of group charging on the exchange-correlation energy is negligible, a generic DFT theory (e.g., LSDA) would describe the system energy as $$\begin{aligned} E& = & E_0-e(V-V_t)p \nonumber \\ &+&\frac{1}{2}\int d^3r \int d^3r^\prime \frac{\rho(\textbf{r})\rho(\textbf{r}^\prime)-\rho_0(\textbf{r})\rho_0(\textbf{r}^\prime)}{\left|\textbf{r}-\textbf{r}^\prime\right|}, \label{eq_B1}\end{aligned}$$ where index 0 marks the variable values at $p = 0$. Now let us simplify Eq. (\[eq\_B1\]) by assuming that due to a small size of the acceptor group, the Coulomb interaction of electrons localized on it is much larger than that on the chain, so that the latter may be neglected. (For the trap molecule shown in Fig. 2, this assumption is true within $\sim$5%.) Then Eq. (\[eq\_B1\]) is reduced to $$\begin{aligned} E &\approx& E_0 - e(V-V_t)p - \frac{e}{2}\int_{\mbox{\scriptsize group}} \phi(\textbf{r})\left|\psi(\textbf{r})\right|^2d^3r, \nonumber \\ p&=&\int_{\mbox{\scriptsize group}}\left|\psi(\textbf{r})\right|^2d^3r, \label{eq_B2}\end{aligned}$$ where $\phi(\textbf{r})$ is the electrostatic potential created by the part of the electronic wavefunction that resides on the group. In the simple capacitive model of the group (used in particular in Appendix A), $\phi(\textbf{r}) = -ep/C$, where $C$ is the effective capacitance of the group, so that $$E \approx E_0 - e(V-V_t)p + \frac{e^2p^2}{2C}. \label{eq_B3}$$ On the other hand, in accordance with Eq. (\[eq\_A7\]), the total energy in the DFT may also be presented in the form $$E= \sum_{i}p_i\varepsilon_i - \frac{1}{2}\int d^3r \int d^3r^\prime \frac{\rho(\textbf{r})\rho(\textbf{r}^\prime)}{\left|\textbf{r}-\textbf{r}^\prime\right|}, \label{eq_B4}$$ where $\varepsilon_i$ are all occupied (or partially occupied) single particle energies, so that in our simple model $$E \approx E_0 + (\varepsilon-\varepsilon_v)p - \frac{e^2p^2}{2C}. \label{eq_B5}$$ Comparing Eqs. (\[eq\_B3\]) and (\[eq\_B5\]), we arrive at the following expression: $$\varepsilon-\varepsilon_v \approx -e(V-V_t) + \frac{e^2p}{C}. \label{eq_B6}$$ In most DFT packages, level occupancies $p_i$ are calculated from the single-particle Fermi distribution, $$p_i=\frac{1}{\mbox{exp}\left\{(\varepsilon_i-\mu)/k_{\mbox{\scriptsize B}}T\right\}+1}; \label{eq_B7}$$ for our simple model, index $i$ may be dropped, and (due to the valence band multiplicity) $\mu \approx \varepsilon_v$. As is evident from the sketch of Eqs. (\[eq\_B6\]) and (\[eq\_B7\]), in Fig. \[fig\_B2\], if the thermal fluctuation scale $k_{\mbox{\scriptsize B}}T$ is much lower than the charging energy scale $e^2/C$, then almost within the whole range $V_t < V < V_t + e/C$, the approximate solution of the system of these equations is $$p \approx \frac{C}{e}(V-V_t),\mbox{ }\varepsilon \approx \varepsilon_v. \label{eq_B8}$$ ![A sketch of Eqs. (\[eq\_B6\]) and (\[eq\_B7\]).[]{data-label="fig_B2"}](figB2){width="21pc"} Panel (a) in Fig. \[fig\_B3\] shows (schematically) the resulting dependence of the energy spectrum of our model system on the applied voltage $V$, with level freezing in the range $V_t < V < V_t + e/C$. The dashed black-green line indicates the region with a partial occupancy $0 < p < 1$ of the group-localized orbital. In panel (b) in (Fig. \[fig\_B3\]) we show the evolution which should follow from the correct quantum-mechanical theory, in which electrons do not self-interact and as a result there is the usual anticrossing of energy levels $\varepsilon$ and $\varepsilon_v$ at $V = V_t$. (For clarity, Fig. \[fig\_B3\] strongly exaggerates the anticrossing width, which is less than $10^{-3}$ eV for our trap [^9].) ![(a) A sketch of the evolution of the energy spectrum from Fig. \[fig\_B1\] as a function of the applied voltage $V$, illustrating the self-interaction errors giving rise to a spurious level freezing in the $V_t < V < V_t + e/C$ voltage range. The dashed black-green line indicates the region with a partial occupancy $0 < p < 1$ of the group-localized orbital. (b) A sketch of the evolution of the same energy spectrum in a correct quantum-mechanical theory, in which electrons do not self-interact.[]{data-label="fig_B3"}](figB3){width="21pc"} The actual spectrum of our molecular trap is somewhat more complex than that of the simple model above — see Figs. \[fig4\]a and \[fig6\]. First, not only the valence energy band of the alkane chain, but also its conduction band is important for electron transfer in our voltage range. Second, the molecular group has not one, but a series of discrete energy levels, with the most important of them corresponding to the working orbital (energy $\varepsilon_W$), and one more group-localized orbital with energy $\varepsilon_{W+1} \approx \varepsilon_W + 0.7$ eV. ![(a) A sketch of the evolution of the molecular energy spectrum of our trap molecule as a function of the applied voltage $V$, illustrating the self-interaction errors giving rise to a spurious level freezing in voltage ranges $V_t^\prime - e/C < V < V_t^\prime$ and $V_t < V < V_t + e/C$. The dashed black-green line indicates the region with a partial occupancy $0 < p < 1$ of the group-localized orbitals with energies $\varepsilon_W$ or $\varepsilon_{W+1}$. (b) A sketch of the evolution of the molecular energy spectrum but in a correct quantum-mechanical theory, in which electrons do not self-interact.[]{data-label="fig_B4"}](figB4){width="21pc"} Nevertheless, the behavior of the spectrum, predicted by uncorrected versions of DFT (Fig. \[fig6\]) may still be well understood using our model. Just as was discussed above, for voltages $V$ above the threshold value $V_t$ (which now corresponds to the alignment of $\varepsilon_v$ with $\varepsilon_{W+1}$ rather than $\varepsilon_{W}$), it describes a gradual transfer of an electron between the top level of the valence band and the second group-localized orbital, with its occupation number $p_{W+1}$ gradually growing in accordance with Eq. (\[eq\_B8\]) — see panel (a) in Fig. \[fig\_B4\]. Similarly, at voltages $V$ below $V_t^\prime$ (which corresponds to the alignment of the working orbital’s energy $\varepsilon_W$ with the lowest level $\varepsilon_{c}$ of the chain’s conduction band), there is a similar spurious gradual transfer of an electron between the corresponding orbitals. In both voltage ranges, a spurious internal electrostatic potential is created; as is described by Eq. (\[eq\_B8\]), it closely compensates the changes of the applied external potential, thus “freezing” all orbital energies of the system at their levels reached at thresholds $V_t^\prime$ and $V_t$ — see panel (a) in Fig. \[fig\_B4\]. Figure \[fig\_B5\] shows that results of both the LSDA and ASIC DFT calculations at $V > V_t$ agree well with Eq. (\[eq\_B8\]), with a value $C = 4.5 \times 10^{-20}$ F calculated as discussed in Appendix A, indicating that the electron self-interaction effects remain almost uncompensated in these software packages, at least for complex molecules such as our trap. ![The DFT-calculated occupancy $p_{W+1}$ of the $(W+1)$’st orbital of the acceptor group of our trap molecule at voltages above the threshold voltage $V_t$ of the alignment of its energy $\varepsilon_{W+1}$ with alkane chain’s valence band edge $\varepsilon_v$. Black lines show results of two versions of DFT theory, for two ion states: the singly-negatively charged ion and the neutral molecule, while the red line shows the result given by Eq. (\[eq\_B8\]) with $C = 4.5\times 10^{-20}$ F.[]{data-label="fig_B5"}](figB5){width="21pc"} Again, in the correct quantum-mechanical theory, there should be a simple (and in our molecules, extremely narrow) anticrossing between the effective single-particle levels of the acceptor group and the alkane chain — see panel (b) in Fig. \[fig\_B4\]. As described in Sec. III of the main text, we have succeeded to describe this behavior rather closely, using the internal iteration dynamics of ASIC SIESTA with $T = 0$ K. [^1]: A resistive switch with a sufficiently long (a-few-year) retention time at $V = 0$ may be classified as a nonvolatile memory cell. [^2]: We use the notation in which the fundamental electric charge unit $e$ is positive, so that the electric charge of an ion with $n$ electrons is $Q(n) = -e(n-n_0)$. [^3]: Experiments (for a recent summary, see, e.g., Table 1 in [@mcdermott09]) give for the exponent coefficient $\beta$ a wide range $(0.26-0.53) a_{\mbox{\scriptsize B}}^{-1}$ corresponding to the effective mass range $(0.05-0.2) m_0$ (assuming a rectangular, 4.5-eV-high energy barrier). It has been suggested [@mcdermott09] that such a large variation is due to a complex dispersion law inside the alkane bandgap, making $\beta$ a strong function of the tunneling electron energy. [^4]: Initially, we made an attempt to use NRLMOL [@pederson00] which had been successfully employed in our previous study of single-electron tunneling through smaller molecules [@simonian07]. However, we have found the performance of SIESTA (with the “standard” double-Zeta polarized basis set) for our current problem to be substantially higher, though the results obtained from NRLMOL may be slightly more accurate. [^5]: this is valid not only for the DFT packages based on the local spin density approximation (LSDA), such as the standard version of SIESTA. Another popular DFT functional, the generalized gradient approximation (GGA) [@PerdewBurkeErnzerhof96], does not provide much improvement on these results [^6]: In contrast, in the Hartree-Fock theory the exchange energy is exact (of course, in the usual sense of the first approximation of the perturbation theory), and the self-interaction errors are absent [@pedrew81]. [^7]: The geometry relaxation was done for isolated neutral molecules only, at no applied bias voltage and without the account of the possible image charge effects; in the relaxed geometry all force components on the atoms are smaller than $0.05$ eV/$\AA$. To justify this procedure, we have verified that trap charging and discharging rates, calculated using the trap geometry relaxed in the presence of a high (8 V) bias, do not significantly differ from the rates (shown in Fig. \[fig10\]b) calculated for its relaxation at $V = 0$. [^8]: there is virtually no correlation between the transistor molecules, just as with their autocorrelation in time [@korotkov94], because at least two transition channels are open at any time. [^9]: Direct SIESTA calculation has shown that the anticrossing energy splitting is less than the calculation error (of the order of $10^{-3}$ eV). An indirect calculation using Eq. (\[eq\_bardeen\]), with $\psi_W$ and $\psi_v$ substituted instead of $\psi_{i}$ and $\psi_{s,c}$, suggests that this overlap is as small as $\sim 10^{-8}$ eV.

--- abstract: | This paper describes the recently developed mixed mimetic spectral element method for the Stokes problem in the vorticity-velocity-pressure formulation. This compatible discretization method relies on the construction of a conforming discrete Hodge decomposition, that is based on a bounded projection operator that commutes with the exterior derivative. The projection operator is the composition of a reduction and a reconstruction step. The reconstruction in terms of mimetic spectral element basis-functions are tensor-based constructions and therefore hold for curvilinear quadrilateral and hexahedral meshes. For compatible discretization methods that contain a conforming discrete Hodge decomposition, we derive optimal a priori error estimates which are valid for all admissible boundary conditions on both Cartesian and curvilinear meshes. These theoretical results are confirmed by numerical experiments. These clearly show that the mimetic spectral elements outperform the commonly used $H(\mathrm{div})$-compatible Raviart-Thomas elements. address: 'Delft University of Technology, Faculty of Aerospace Engineering, Kluyverweg 2, 2629 HT Delft, The Netherlands.' author: - Jasper Kreeft - Marc Gerritsma bibliography: - './literature.bib' title: A priori error estimates for compatible spectral discretization of the Stokes problem for all admissible boundary conditions --- [^1] [^2] Introduction ============ Let $\Omega\subset\mathbb{R}^n$, $n\geq2$, be a bounded contractible domain with boundary $\Gamma=\p\Omega$. On this domain we consider the Stokes problem, consisting of the equations for conservation of momentum and for conservation of mass, $$\begin{aligned} \nabla\cdot\sigma&=\vec{f}\quad\mathrm{on}\ \Omega,\\ \mathrm{div}\,\vec{u}&=g\quad\mathrm{on}\ \Omega,\end{aligned}$$ where the stress tensor $\sigma$ is given by $$\sigma=-\nu\nabla\vec{u}+pI,$$ with $\vec{u}$ the velocity vector, $p$ the pressure, $\vec{f}$ the forcing term, $g$ the mass source and $\nu$ the kinematic viscosity. For analysis purposes we choose $\nu=1$. This paper considers the recently developed mixed mimetic spectral element method (MMSEM) [@kreeft2012; @kreeftpalhagerritsma2011]. This compatible finite/spectral element method is based on the compatible discretization of the exterior derivative $\ud$ from differential geometry, which represents the vector operators, grad, curl and div. The Stokes problem expressed in terms of these vector operations is known as the *vorticity-velocity-pressure* (VVP) formulation, [@bochevgunzburger2009; @dubois2002]. For the VVP formulation, the Laplace operator is split using the vector identity, $-\Delta\vec{u}=\mathrm{curl}\,\mathrm{curl}^*\,\vec{u}-\mathrm{grad}^*\,\mathrm{div}\,\vec{u}$, and by introducing vorticity as auxiliary variable, $\vec{\omega}=\mathrm{curl}^*\,\vec{u}$. The VVP formulation of the Stokes problem becomes \[stokessinglevector\] $$\begin{aligned} \vec{\omega}-\mathrm{curl}^*\,\vec{u}&=0,\quad\mathrm{on}\ \Omega\label{stokessinglevector1}\\ \mathrm{curl}\,\vec{\omega}-\mathrm{grad}^*\,\mathrm{div}\,\vec{u}+\mathrm{grad}^*\,p&=\vec{f},\quad\mathrm{on}\ \Omega \label{stokessinglevector2}\\ %-\mathrm{grad}\,\mathrm{div}\,\vec{u} +\mathrm{grad}\,g \mathrm{div}\,\vec{u}&=g,\quad\mathrm{on}\ \Omega.\label{stokessinglevector3}\end{aligned}$$ Following [@bochevgunzburger2009; @kreeft2012] we make a distinction between the operators grad, curl and div, that correspond to the classical Newton-Leibnitz, Stokes circulation and Gauss divergence theorems, and the operators -grad$^*$, curl$^*$ and -div$^*$ that are their formal Hilbert adjoints, $$\big(\vec{a},-\mathrm{grad}^*\,b\big):=\big(\mathrm{div}\,\vec{a},b\big),\quad \big(\vec{a},\mathrm{curl}^*\,\vec{b}\big):=\big(\mathrm{curl}\,\vec{a},\vec{b}\big),\quad \big(a,-\mathrm{div}^*\,\vec{b}\big):=\big(\mathrm{grad}\,a,\vec{b}\big).$$ The distinction between the two types of differential operators is made explicitly, because the construction of our conforming finite element spaces relies on the three mentioned integration theorems, while the mixed formulation relies on the formal Hilbert adjoint relations. While in vector calculus this distinction is not common, in differential geometry these structures naturally appear since they make a clear distinction between metric-free (topological) and metric-dependent operations. The MMSEM is a compatible discretization method that relies on the construction of a conforming discrete Hodge-decomposition, which implies a discrete Poincaré inequality. It requires the development of a bounded projection operator that commutes with the exterior derivative. The bounded projection is a composition of a reduction by means of integration and mimetic spectral element basisfunctions as reconstruction. The reduction onto $k$-dimensional submanifolds result in the discrete unknowns representing *integral quantities*. This is one of the major differences with related methods as the Marker and Cell scheme [@harlowwelch1965] and the lowest-order Raviart-Thomas and Nédélec compatible finite elements [@nedelec1980; @raviartthomas1977], where use is made of averaged quantities. The basis functions, used for the reconstruction, are constructed using tensor products of one dimensional nodal and edge interpolation basis functions [@gerritsma2011], and therefore hold for quadrilateral and hexahedral meshes. They belong to the class of compatible finite elements, and were constructed based on the mimetic framework first described in [@hymanscovel1988] and later extended in [@bochevhyman2006]. The mimetic framework, including the mimetic spectral elements, were extensively described in [@kreeftpalhagerritsma2011]. This mimetic framework relies on the languages of differential geometry instead of vector calculus, and algebraic topology as its discrete counterpart. The use of differential geometry and algebraic topology enjoys increasing popularity for the development of compatible schemes, [@arnoldfalkwinther2006; @arnoldfalkwinther2010; @bochevhyman2006; @bossavit1998; @bossavit9900; @desbrun2005c; @hiptmair2001; @hiptmair2002]. Compatible discretizations are often combined with mixed formulations. Mixed formulations are described extensively in among others [@brezzifortin; @giraultraviart] and in terms of differential forms in [@arnoldfalkwinther2006; @arnoldfalkwinther2010] for the Hodge-Laplacian and in [@kreeft2012] for the VVP formulation of the Stokes problem. The MMSEM contains compatible finite elements that are compatible with all admissible types of boundary conditions for the Stokes problem in VVP formulation. We will show that the method obtains optimal rates of convergence for all variables on curvilinear meshes and for all admissible boundary conditions, i.e. standard and nonstandard. It is therefore extending the error estimates found in literature, which are often specifically constructed for certain types of boundary conditions, [@abboud2011; @arnold2011; @bochevgunzburger; @bramble1994; @dubois2003b; @girault1988]. To show optimal convergence a priori error estimates are derived. This is an improvement with respect to the well-known Raviart-Thomas compatible finite elements. These are not compatible in case of Dirichlet boundary conditions and therefore lead to suboptimal convergence behavior, as was shown in [@arnold2011; @dubois2003b]. This non-compatibility results in a decrease in rate of convergence of maximal $\tfrac{3}{2}$ order. From a physical/fluid dynamics point-of-view the new method is relevant because it combines optimal convergence with a pointwise divergence-free discretization (in absence of any mass source) of arbitrary order on curvilinear meshes, valid for all allowable types of boundary conditions, among which the no-slip condition. The derived rates of convergence are confirmed using simple manufactured solution problems, discretized on both Cartesian and curvilinear meshes. The fact that the analysis holds for all admissible boundary conditions is also reflected in the numerical results. This paper is organized as follows: First an introduction into differential geometry is given and the Stokes problem is reformulated in terms of differential forms. In the mixed formulation is given and well-posedness is proven. In the key properties of the mimetic discretization are explained that lead to compatible function spaces. This includes a discussion on the relevant properties of algebraic topology, the definitions op mimetic operators, the introduction of mimetic spectral element basisfunctions and finally the proof of discrete well-posedness. Having formulated the conforming/compatible finite element spaces, the error estimates are developed in and the numerical results are shown in . Notation and preliminaries {#sec:differentialgeometry} ========================== Differential forms ------------------ Differential forms offer significant benefits in the construction of structure-preserving spatial discretizations. For example, the coordinate-free action of the exterior derivative and generalized Stokes theorem give rise to commuting properties with respect to mappings between different manifolds. Acknowledging and respecting these kind of commuting properties are essential for the structure preserving behavior of the mimetic method. Only those concepts from differential geometry which play a role in the remainder of this paper will be explained. More can be found in [@abrahammarsdenratiu; @flanders; @frankel; @kreeftpalhagerritsma2011]. Let $\Lambda^k(\Omega)$ denote a space of *differential $k$-forms* or *$k$-forms*, on a sufficiently smooth bounded $n$-dimensional oriented manifold $\Omega\subset\mathbb{R}^n$ with boundary $\Gamma=\p\Omega$. Every element $a\in\Lambda^k(\Omega)$ has a unique representation of the form $$\label{differentialform} a=\sum_If_I(\mathbf{x})\ud x^{i_1}\wedge\ud x^{i_2}\wedge\cdots\wedge\ud x^{i_k},$$ where $I=i_1,\hdots,i_k$ with $1\leq i_1<\hdots<i_k\leq n$ and where $f_I(\mathbf{x})$ is a continuously differentiable scalar function, $f_I(\mathbf{x})\in \mathcal{C}^\infty(\Omega)$. Differential $k$-forms are naturally integrated over $k$-dimensional manifolds, i.e. for $a\in \Lambda^k(\Omega)$ and $\Omega_k\subset\mathbb{R}^n$, with $k=\mathrm{dim}(\Omega_k)$, $$\label{integration} \int_{\Omega_k}a\in\mathbb{R}\quad\Leftrightarrow\quad\langle a,\Omega_k\rangle\in\mathbb{R},$$ where $\langle\cdot,\cdot\rangle$ indicates a duality pairing between the differential form and the geometry. Note that the $n$-dimensional computational domain is indicated as $\Omega$, so without subscript. The differential forms live on manifolds and transform under the action of mappings. Let $\Phi:\widehat{\Omega}\rightarrow\Omega$ be a mapping between two manifolds. Then we can define the pullback operator, $\Phi^\star:\Lambda^k(\Omega)\rightarrow\Lambda^k(\widehat{\Omega})$, expressing the $k$-form on the $n$-dimensional reference manifold, $\widehat{\Omega}$. The mapping, $\Phi$, and the pullback, $\Phi^\star$, are each others formal adjoints with respect to a duality pairing , $$\int_{\Phi(\widehat{\Omega}_l)}a=\int_{\widehat{\Omega}_l}\Phi^\star a\quad\Leftrightarrow\quad\langle a,\Phi(\widehat{\Omega}_l)\rangle=\langle\Phi^\star a,\widehat{\Omega}_l\rangle,$$ where $\widehat{\Omega}_l$ is an $l$-dimensional submanifold of $\widehat{\Omega}$ and $\Omega_k=\Phi(\widehat{\Omega}_l)$ a $k$-dimensional submanifold of $\Omega$. A special case of the pullback operator is the trace operator. The trace of $k$-forms to the boundary, $\tr:\Lambda^k(\Omega)\rightarrow\Lambda^k(\p\Omega)$, is the pullback of the inclusion of the boundary of a manifold, $\p\Omega\hookrightarrow\Omega$, see [@kreeftpalhagerritsma2011]. The wedge product, $\wedge$, of two differential forms $a\in\Lambda^k(\Omega)$ and $b\in\Lambda^l(\Omega)$ is a mapping: $\wedge:\Lambda^k(\Omega)\times\Lambda^l(\Omega)\rightarrow\Lambda^{k+l}(\Omega),\ k+l\leq n$. The wedge product is a skew-symmetric operator, i.e. $a\wedge b=(-1)^{kl}b\wedge a$. An important operator in differential geometry is the exterior derivative, $\ud:\Lambda^k(\Omega)\rightarrow\Lambda^{k+1}(\Omega)$. It is induced by the *generalized Stokes’ theorem*, combining the classical Newton-Leibnitz, Stokes circulation and Gauss divergence theorems. Let $\Omega_{k+1}$ be a $(k+1)$-dimensional manifold and $a\in\Lambda^k(\Omega)$, then $$\label{stokestheorem} \int_{\p\Omega_{k+1}}a=\int_{\Omega_{k+1}}\ud a\quad\Leftrightarrow\quad \langle a,\partial\Omega_{k+1}\rangle=\langle\ud a,\Omega_{k+1}\rangle,$$ where $\partial\Omega_{k+1}$ is a $k$-dimensional manifold being the boundary of $\Omega_{k+1}$. The duality pairing in shows that the exterior derivative is the formal adjoint of the *boundary operator* $\p:\Omega_{k+1}\rightarrow\Omega_k$. The exterior derivative is independent of any metric and coordinate system. Applying the exterior derivative twice always leads to the null $(k+2)$-form, $\ud(\ud a)=0^{(k+2)}$ for all $a\in\Lambda^k(\Omega)$. As a consequence, on contractible domains the exterior derivative gives rise to an exact sequence, called *De Rham complex* [@frankel], and indicated by $(\Lambda,\ud)$, $$\mathbb{R}\hookrightarrow\Lambda^0(\Omega)\stackrel{\ud}{\longrightarrow}\Lambda^1(\Omega)\stackrel{\ud}{\longrightarrow}\cdots\stackrel{\ud}{\longrightarrow}\Lambda^n(\Omega)\stackrel{\ud}{\longrightarrow}0. \label{derhamcomplex}$$ In vector calculus a similar sequence exists, where, from left to right for $\mathbb{R}^3$, the $\ud$’s denote the vector operators grad, curl and div. The exterior derivative and wedge product are related according to Leibnitz’s rule as: for all $a\in \Lambda^k(\Omega)$ and $b\in\Lambda^l(\Omega)$, $$\ud\big(a\wedge b\big)=\ud a\wedge b+(-1)^ka\wedge\ud b,\quad\mathrm{for}\ k+l<n. \label{leibnitz}$$ The pullback operator and exterior derivative possess the following commuting property, $$\Phi^\star\ud a=\ud\Phi^\star a,\quad\forall a\in\Lambda^k(\Omega).$$ In this paper we will consider Hilbert spaces $L^2\Lambda^k(\Omega)\supset\Lambda^k(\Omega)$, where in the functions $f_I(\mathbf{x})\in L^2(\Omega)$. The pointwise inner-product of $k$-forms, $(\cdot,\cdot)$, is constructed using inner products of one-forms, that is based on the inner product on vector spaces, see [@flanders; @frankel]. The wedge product and inner product induce the Hodge-$\star$ operator, $\star:L^2\Lambda^k(\Omega)\rightarrow L^2\Lambda^{n-k}(\Omega)$, a metric operator that includes orientation. Let $a,b\in L^2\Lambda^k(\Omega)$, then $$a\wedge\star b:=\big(a,b\big)\sigma, \label{hodgestar}$$ where $\sigma\in\Lambda^n(\Omega)$ is a unit volume form, $\sigma=\star1$. In geometric physics the Hodge-$\star$ switches between an inner-oriented description of physical variables and an outer-oriented description. See [@kreeft2012; @kreeftpalhagerritsma2011; @mattiussi2000; @tonti1] for a thorough discussion on the concepts of inner and outer orientation. The space of square integrable $k$-forms on $\Omega$ can be equipped with a $L^2$ inner product, $\big(\cdot,\cdot\big)_\Omega:L^2\Lambda^k(\Omega)\times L^2\Lambda^k(\Omega)\rightarrow\mathbb{R}$, given by, $$\big(a,b\big)_\Omega:=\int_\Omega\big(a,b\big)\kdifform{\sigma}{n}=\int_\Omega a\wedge\star b. \label{L2innerproduct}$$ The norm corresponding to the space $L^2\Lambda^k(\Omega)$ is $\norm{a}_{L^2\Lambda^k}=\sqrt{\big(a,a\big)_\Omega}$. Higher degree Sobolev spaces, $H^m\Lambda^k$, consists of all $k$-forms as in where $f_I(\mathbf{x})\in H^m(\Omega)$, with corresponding norms $|a|_{H^m\Lambda^k}$ and $\Vert a\Vert_{H^m\Lambda^k}$. The Hilbert space associated to the exterior derivative $H\Lambda^k(\Omega)$ is defined as $$H\Lambda^k(\Omega)=\{a\in L^2\Lambda^k(\Omega)\;|\;\ederiv a\in L^2\Lambda^{k+1}(\Omega)\}.$$ and the norm corresponding to $H\Lambda^k(\Omega)$ is defined as $\Vert a\Vert^2_{H\Lambda^k}:=\Vert a\Vert^2_{L^2\Lambda^k}+\Vert\ederiv a\Vert^2_{L^2\Lambda^{k+1}}$. The $H\Lambda^k$-semi-norm is the $L^2$-norm of the exterior derivative, $\vert a\vert_{H\Lambda^k}=\Vert\ederiv a\Vert_{L^2\Lambda^{k+1}}$. Note that $H^1\Lambda^k(\Omega)\subseteq H\Lambda^k(\Omega)\subseteq L^2\Lambda^k(\Omega)$, where the left equality holds for $k=0$ and the right for $k=n$. The $L^2$-de Rham complex, also called *Hilbert complex* [@bruninglesch1992], $(H\Lambda,\ud)$, is the exact sequence of maps and spaces given by $$\mathbb{R}\hookrightarrow H\Lambda^0(\Omega)\stackrel{\ederiv}{\longrightarrow} H\Lambda^1(\Omega)\stackrel{\ederiv}{\longrightarrow}\cdots\stackrel{\ederiv}{\longrightarrow} H\Lambda^n(\Omega)\stackrel{\ud}{\longrightarrow}0.$$ In terms of vector operations the Hilbert complex becomes for $\Omega\subset\mathbb{R}^3$, $$H^1(\Omega)\stackrel{\rm grad}{\longrightarrow} H(\mathrm{curl},\Omega)\stackrel{\rm curl}{\longrightarrow}H(\mathrm{div},\Omega)\stackrel{\rm div}{\longrightarrow} L^2(\Omega),$$ and for $\Omega\subset\mathbb{R}^2$, either $$H^1(\Omega)\stackrel{\rm grad}{\longrightarrow} H(\mathrm{rot},\Omega)\stackrel{\rm rot}{\longrightarrow}L^2(\Omega),\quad\mathrm{or}\quad H^1(\Omega)\stackrel{\rm curl}{\longrightarrow} H(\mathrm{curl},\Omega)\stackrel{\rm div}{\longrightarrow}L^2(\Omega).$$ The two are related by the Hodge-$\star$ operator , see [@palha2010], $$\label{doublehilbertcomplex} \!\!\!\!\!\!\! \begin{matrix} H\Lambda^{0}(\Omega)\!\!&\!\!\stackrel{\ederiv}{\longrightarrow}\!\!&\!\! H\Lambda^{1}(\Omega)\!\!&\!\!\stackrel{\ederiv}{\longrightarrow}\!\!&\!\!L^2\Lambda^{2}(\Omega)\\ \star\updownarrow & & \star\updownarrow & &\star\updownarrow \\ L^2\Lambda^{2}(\Omega)\!\!&\!\!\stackrel{\ederiv}{\longleftarrow}\!\!&\!\!H\Lambda^{1}(\Omega)\!\!&\!\!\stackrel{\ederiv}{\longleftarrow}\!\!&\!\!H\Lambda^{0}(\Omega) \end{matrix} \quad\Leftrightarrow\quad \begin{matrix} H^1(\Omega)\!\!&\!\!\stackrel{\mathrm{curl}}{\longrightarrow}\!\!&\!\!H(\mathrm{curl},\Omega)\!\!&\!\!\stackrel{\mathrm{div}}{\longrightarrow}\!\!&\!\!L^2(\Omega)\\ \star\updownarrow & & \star\updownarrow & &\star\updownarrow \\ L^2(\Omega)\!\!&\!\!\stackrel{\mathrm{rot}}{\longleftarrow}\!\!&\!\!H(\mathrm{rot},\Omega)\!\!&\!\!\stackrel{\mathrm{grad}}{\longleftarrow}\!\!&\!\!H^1(\Omega). \end{matrix}$$ The upper complex is associated with outer-oriented $k$-forms, i.e. $k$-forms that are associated with outer-oriented manifolds, and the lower complex is associated with inner-oriented $k$-forms. In this paper we mainly consider the upper complex and circumvent the lower complex by means of integration by parts. Only the pressure and tangential velocity boundary conditions are given on the lower complex, as we will see in the following sections. A similar double Hilbert complex can be constructed in $\mathbb{R}^3$. Since the exterior derivative is nilpotent, it ensures that the range, $\mathcal{B}^k:=\ud\, H\Lambda^{k-1}(\Omega)$, of the exterior derivative on $(k-1)$-forms is contained in the nullspace, $\mathcal{Z}^k:=\{\;a\in H\Lambda^k(\Omega)\;|\;\ud a=0\;\}$, of the exterior derivative on $k$-forms, $\mathcal{B}^k\subseteq\mathcal{Z}^k$. Every space of $k$-forms in the complex $(H\Lambda,\ud)$ can be decomposed into the nullspace of $\ud$, and its orthogonal complement, $H\Lambda^k(\Omega)=\mathcal{Z}^k\oplus\mathcal{Z}^{k,\perp}$. This is the *Hodge decomposition*, where on contractible domains $\mathcal{Z}^k=\mathcal{B}^k$. By the Hodge decomposition it follows that the exterior derivative is an isomorphism $\ud:\mathcal{Z}^{k,\perp}\rightarrow\mathcal{B}^{k+1}$. The inner product gives rise to the formal Hilbert adjoint of the exterior derivative, the codifferential operator, $\ud^*:H^*\Lambda^k(\Omega)\rightarrow L^2\Lambda^{k-1}(\Omega)$. Let $H^*\Lambda^k(\Omega)=\{\,a\in L^2\Lambda^k(\Omega)\,|\,\ud^*a\in L^2\Lambda^{k-1}(\Omega)\}$, then $$\big(\ud a,b\big)_\Omega=(a,\ud^* b)_\Omega,\quad\forall\, a\in H\Lambda^{k-1}(\Omega),\ b\in H^*\Lambda^k.$$ In case of non-zero trace, and by combining , and , we obtain integration by parts, $$\big(a,\ud^*b\big)_\Omega=\big(\ud a,b\big)_\Omega-\int_{\p\Omega} \tr a\wedge \tr\star b. \label{integrationbyparts}$$ Also the codifferential operator is nilpotent, $\ud^*(\ud^*\kdifform{a}{k})=0$, i.e., its range is contained in its nullspace, $\mathcal{B}^{*,k}\subseteq\mathcal{Z}^{*,k}$, where $\mathcal{B}^{*,k}:=\ud^*H^*\Lambda^{k+1}(\Omega)$ and $\mathcal{Z}^{*,k}:=\{\; a\in H^*\Lambda^k(\Omega)\;|\;\ud^*a=0\;\}$. In fact the codifferential is an isomorphism $\ud^*:\mathcal{Z}^{*,k,\perp}\rightarrow\mathcal{B}^{*,k+1}$, where $\mathcal{Z}^{*,k,\perp}$ follows from the following Hodge decomposition, $\Lambda^k(\Omega)=\mathcal{Z}^{*,k}\oplus\mathcal{Z}^{*,k,\perp}$. On contractible manifolds this gives rise to the following exact sequence, $$0\stackrel{\ud^*}{\longleftarrow}H^*\Lambda^0(\Omega)\stackrel{\ud^*}{\longleftarrow}H^*\Lambda^1(\Omega)\stackrel{\ud^*}{\longleftarrow}\cdots\stackrel{\ud^*}{\longleftarrow}H^*\Lambda^n(\Omega)\hookleftarrow\mathbb{R}.$$ In vector notation from right to left the $\ud^*$’s denote the -grad$^*$, curl$^*$ and -div$^*$ operators in $\mathbb{R}^3$, as were also mentioned in the introduction. However, whereas the exterior derivative is a metric-free operator, the codifferential operator is metric-dependent. The Hodge-Laplace operator, $\Delta:H^2\Lambda^k(\Omega)\rightarrow L^2\Lambda^k(\Omega)$, is constructed as a composition of the exterior derivative and the codifferential operator, $$\label{laplace} -\Delta\, a:=(\ud^*\ud+\ud\ud^*)\,a.$$ An important inequality in stability analysis, relating the $L^2\Lambda^k$-norm and the $H\Lambda^k$-norm, is Poincaré inequality. \[lem:poicareinequality\][@arnoldfalkwinther2010] Consider the Hilbert complex $(H\Lambda,\ederiv)$, then the exterior derivative is a bounded bijection from $\mathcal{Z}^{k,\perp}$ to $\mathcal{B}^{k+1}$, and hence, by Banach’s bounded inverse theorem, there exists a constant $c_P$ such that $$\Vert a\Vert_{H\Lambda^k}\leq c_P\Vert\ederiv a\Vert_{L^2\Lambda^{k+1}},\quad\forall a\in \mathcal{Z}^{k,\perp}.$$ Finally, for Hilbert spaces with essential boundary conditions we write, $H_0\Lambda^k(\Omega):=\{\,a\in H\Lambda^k(\Omega)\,|\,\tr a=0\,\}$, and for natural boundary conditions we consider the following trace map, $\tr\star:H\Lambda^k(\Omega)\rightarrow H^{\frac{1}{2}}\Lambda^{n-k}(\p\Omega)$. Stokes problem in differential form notation -------------------------------------------- Consider again a bounded contractible domain $\Omega\subset\mathbb{R}^n$. Because we require exact conservation of mass and because we can perform exact discretization of the exterior derivative, see , we use the following formulation for the Stokes problem: let $(\omega,u,p)\in\{\Lambda^{n-2}(\Omega)\times\Lambda^{n-1}(\Omega)\times\Lambda^{n}(\Omega)\}$, then the VVP formulation is given by \[stokeseq\] $$\begin{aligned} \omega-\ud^* u&=0,\quad\mathrm{in}\ \Lambda^{n-2}(\Omega),\label{stokeseq1}\\ \ud^*\ud u+\ud\omega+\ud^*p&=f,\quad\mathrm{in}\ \Lambda^{n-1}(\Omega),\label{stokeseq2}\\ \ud u&=g,\quad\mathrm{in}\ \Lambda^n(\Omega).\label{stokeseq3}\end{aligned}$$ In the VVP formulation the pressure in acts as a Lagrange multiplier for the constraint on velocity, , whereas velocity in acts as a Lagrange multiplier for the constraint on vorticity in . Let $\Gamma=\p\Omega$ be the boundary of $\Omega$, where $$\Gamma=\Gamma_\omega\cup\Gamma_t,\quad \Gamma_\omega\cap\Gamma_t=\emptyset,\quad\mathrm{and}\quad \Gamma=\Gamma_n\cup\Gamma_\pi,\quad\Gamma_n\cap\Gamma_\pi=\emptyset.$$ We will impose the tangential vorticity and normal velocity as essential boundary conditions, and the tangential velocity and the pressure plus divergence of velocity as the natural boundary conditions: \[boundaryconditions\] $$\begin{aligned} \tr\omega&=0\quad\quad\ \mathrm{on}\ \Gamma_\omega,\\ \tr u&=0\quad\quad\ \mathrm{on}\ \Gamma_n,\\ \quad\tr\star u&=u_{b,t}\quad\ \mathrm{on}\ \Gamma_t,\quad\; \mathrm{with}\ u_{b,t}\in H^{\frac{1}{2}}\Lambda^{1}(\Gamma_t),\\ \tr\star(\ud u+p)&=\Pi_b\quad\ \ \, \mathrm{on}\ \Gamma_\pi,\quad \mathrm{with}\ \Pi_b\in H^{\frac{1}{2}}\Lambda^0(\Gamma_\pi).\end{aligned}$$ Then the boundary $\Gamma$ can be partitioned into four sections, $\Gamma=\bigcup_{i=1}^4\Gamma_i$, with $\Gamma_i\cap\Gamma_j=\emptyset$ for $i\neq j$, where $$\label{boundarydecomposition} \Gamma_1:=\Gamma_t=\Gamma_n,\quad\Gamma_2:=\Gamma_t=\Gamma_\pi,\quad\Gamma_3:=\Gamma_\omega=\Gamma_n,\quad\Gamma_4:=\Gamma_\omega=\Gamma_\pi.$$ This decomposition, introduced before in [@dubois2002; @hughesfranca1987; @kreeft2012], shows all admissible boundary conditions. It will also follow directly from the mixed formulation, see , Section \[sec:mixedformulation\]. In case, $\Gamma=\Gamma_1\cap\Gamma_3$, $\Gamma_2\cup\Gamma_4=\emptyset$, no pressure boundary conditions are prescribed, and so the pressure is only determined up to an element $\hat{p}\in\mathcal{Z}^{*,n}$, i.e. up to a constant. As a post processing step either the pressure in a point in $\Omega$ can be set, or a zero average pressure can be imposed; i.e. $\int_\Omega \hat{p}=0$. In case $\Gamma=\Gamma_4$, no velocity boundary conditions are prescribed, and so the solution of velocity is determined modulo a curl$^*$-free element, i.e. modulo $\hat{u}\in\mathcal{Z}^{*,n-1}$. Mixed formulation {#sec:mixedformulation} ================= Mixed formulation of Stokes problem ----------------------------------- The use of a mixed formulation is based on the following reasoning; We know how to discretize exactly the metric-free exterior derivative $\ud$, but it is less obvious how to treat the codifferential operator $\ud^*$. ### Generalized Poisson problem Take for example the generalized Poisson problem using the Hodge-Laplacian acting on $k$-forms, $(\ud\ud^*+\ud^*\ud)u=f$, on $\Omega$ with boundary $\Gamma=\p\Omega$. A standard Galerkin approach, using integration by parts , would give; find $u\in H\Lambda^k(\Omega)\cap H^*\Lambda^k(\Omega)$ with $\ud u\in H^*\Lambda^{k+1}(\Omega)$ and $\ud^*u\in H\Lambda^{k-1}(\Omega)$, given $f\in L^2\Lambda^k(\Omega)$, such that $$\label{standardgalerkin} \big(\ud^* v,\ud^* u\big)_\Omega+\big(\ud v,\ud u\big)_\Omega=\big(v,f\big)_\Omega,\quad\quad\forall v\in H\Lambda^k(\Omega)\cap H^*\Lambda^k(\Omega).$$ It has a corresponding minimization problem for an energy functional over the space $H\Lambda^k(\Omega)\cap H^*\Lambda^k(\Omega)$. The standard Galerkin formulation is coercive, which immediately implies stability. Corresponding to this standard Galerkin formulation one usually chooses a $H^1\Lambda^k(\Omega)$-conforming approximation space. This could be a standard continuous piecewise polynomial vector space based on nodal interpolation. However, in case of a nonconvex polyhedral or curvilinear or noncontractible domain $\Omega$, for allmost all $f$, $H\Lambda^k(\Omega)\cap H^*\Lambda^k(\Omega)\not\subset H^1\Lambda^k(\Omega)$. Consequently, the solution will be stable but inconsistent in general, [@costabel1991]. In other words, the solution converges to the wrong solution. Unfortunately, it seems not possible to construct $H\Lambda^k(\Omega)\cap H^*\Lambda^k(\Omega)$ conforming finite element spaces. Alternatively, one proposed to use *mixed formulations*, [@brezzifortin]. In contrast to standard Galerkin, the mixed formulation uses integration by parts to express each codifferential in terms of an exterior derivative and suitable boundary conditions. Consequently, mixed formulations require only $H\Lambda^k(\Omega)$-conforming finite element spaces, which are much easier to construct. Therefore, in all cases mixed formulations do converge to the true solution. Mixed formulations correspond to saddle point problems instead of minimization problems. The derivation of the mixed formulation of the Poisson problem consists of three steps: 1. Introduce an auxiliary variable $\omega=\ud^*u$ in $H\Lambda^{k-1}$, 2. multiply both equations by test functions $\big(\tau,v\big)\in\{H\Lambda^{k-1}\times H\Lambda^{k}\}$ using $L^2$-inner products, 3. use integration by parts, as in , to express the remaining codifferentials in terms of the exterior derivatives and boundary integrals. Again the boundary may constitute up to four different types of boundary conditions, \[boundaryconditions2\] $$\begin{aligned} \tr\omega&=0\quad\quad\ \mathrm{on}\ \Gamma_\omega,\\ \tr u&=0\quad\quad\ \mathrm{on}\ \Gamma_n,\\ \quad\tr\star u&=u_{b,t}\quad\ \mathrm{on}\ \Gamma_t,\quad\; \mathrm{with}\ u_{b,t}\in H^{\frac{1}{2}}\Lambda^{n-k}(\Gamma_t),\\ \tr\star\ud u&=g_b\quad\ \ \, \mathrm{on}\ \Gamma_\pi,\quad \mathrm{with}\ g_b\in H^{\frac{1}{2}}\Lambda^{n-k-1}(\Gamma_\pi).\end{aligned}$$ Then also for the generalized Poisson problem the boundary $\Gamma$ consists up to four sections as defined in . To obtain a unique solution for the corresponding mized formulation, we define the following Hilbert spaces, $$\begin{aligned} W:=&\{\,\tau\in H\Lambda^{k-1}(\Omega)\,|\,\tr\tau=0\ \mathrm{on}\ \Gamma_\omega\,\},\\ V:=&\left\{ \begin{aligned} &\{\, v\in H\Lambda^{k}(\Omega)\,|\,\tr v=0\ \mathrm{on}\ \Gamma_n\,\}\quad\mathrm{if}\ \Gamma_4=\emptyset,\\ &\{\, v\in H\Lambda^{k}(\Omega)\backslash\mathcal{Z}^{*,k}(\Omega)\,|\,\tr v=0\ \mathrm{on}\ \Gamma_n\,\}\quad\mathrm{if}\ \Gamma_4\neq\emptyset, \end{aligned} \right.\end{aligned}$$ with corresponding norms, $\norm{\cdot}_W,\ \norm{\cdot}_V$, respectively. The resulting mixed formulation for the Poisson problem for all $0\leq k\leq n$ becomes: find $(\omega,u)\in\{W\times V\}$, given $f\in L^2\Lambda^k$, for all $(\tau,v)\in\{W\times V\}$, such that \[poisson\] $$\begin{aligned} \big(\tau,\omega\big)_\Omega-\big(\ud\tau,u\big)_\Omega&=-\int_{\Gamma_1\cup\Gamma_2}\tr\tau\wedge u_{b,t},\label{poisson1}\\ %&&\forall\tau\in H\Lambda^{k-1}(\Omega), \big(v,\ud\omega\big)_\Omega+\big(\ud v,\ud u\big)_\Omega&=\big(v,f\big)_\Omega+\int_{\Gamma_2\cup\Gamma_4}\tr v\wedge g_b.\label{poisson2} %,&&\forall \kdifform{v}{k}\in H\Lambda^k(\Omega)\end{aligned}$$ Note that, for a scalar Poisson, it is not a choice whether to use Galerkin or mixed formulation, but it depends on whether the scalar is a 0-form or an $n$-form. This is determined by the physics. ### Stokes problem In a similar way the mixed formulation of the VVP formulation of the Stokes problem is obtained. Consider the Hilbert spaces $W$ and $V$ defined in the previous section, where $k=n-1$, and define the following Hilbert space $$Q:=\left\{ \begin{aligned} &q\in L^2\Lambda^n(\Omega),\quad\quad\quad\ \,\mathrm{if}\ \Gamma_\pi\neq\emptyset,\\ &q\in L^2\Lambda^n(\Omega)\backslash\mathcal{Z}^{*,n},\quad\mathrm{if}\ \Gamma_\pi=\emptyset, \end{aligned} \right.$$ with corresponding norm $\norm{\cdot}_Q$ and where $\mathcal{Z}^{*,n}=\mathbb{R}$. Then the mixed formulation of the VVP formulation reads: find $(\omega,u,p)\in\{W\times V\times Q\}$, for the given data $f\in L^2\Lambda^{n-1}(\Omega)$, $g\in L^2\Lambda^{n}(\Omega)$ and natural boundary conditions $u_{b,t}\in H^{\frac{1}{2}}\Lambda^{n-1}(\Gamma_t)$, $\Pi_b\in H^{\frac{1}{2}}\Lambda^n(\Gamma_\pi)$, for all $\big(\tau,v,q\big)\in\{W\times V\times Q\}$, such that \[mixedstokes\] $$\begin{aligned} \big(\tau,\omega\big)_\Omega-\big(\ud\tau,u\big)_\Omega&=-\int_{\Gamma_1\cup\Gamma_2}\tr\tau\wedge u_{b,t},\label{mixedstokes1}\\%,&&\forall\kdifform{\tau}{n-2}\in W \big(v,\ud\omega\big)_\Omega+\big(\ud v,\ud u\big)_\Omega+\big(\ud v,p\big)_\Omega&=\big(v,f\big)_\Omega+\int_{\Gamma_2\cup\Gamma_4}\tr v\wedge\Pi_b,\label{mixedstokes2}\\%,&&\forall \kdifform{v}{n-1}\in V \big(q,\ud u\big)_\Omega&=\big(q,g\big)_\Omega.\label{mixedstokes3}%\big(\kdifform{q}{n},\kdifform{g}{n}\big)_\Omega &&\forall \kdifform{q}{n}\in Q.\end{aligned}$$ Again use is made of integration by parts, . [@bernardi2006] Problems - and are equivalent, in the sense that any triple $\big(\omega,u,p\big)\in\{W\times V\times Q\}$ is a solution of problem - if and only if it is a solution of problem . Well-posedness of mixed formulation ----------------------------------- Before we continue first define the following nullspaces of $W$, $$\begin{aligned} Z_W&:=\{\;\tau\in W\;|\;\ud\tau=0\;\},\\ Z_W^*&:=\{\;\tau\in W\;|\;\ud^*\tau=0\;\},\end{aligned}$$ and consider the following decompositions, $W=Z_W\oplus Z_W^\perp$ and $W=Z_W^*\oplus Z_W^{*,\perp}$. Since vorticity is defined as $\omega=\ud^*u$, we have $\omega\in Z_W^*$, and because we consider contractible domains only, it follows that $\omega\in Z_W^\perp$. Note that for $n=2$, $Z^*_W\equiv W$. A similar decomposition can be made for $V$. Define $$Z_V:=\{\;v\in V\;|\;\ud v=0\;\},$$ then $V=Z_V\oplus Z_V^\perp$. The velocity is decomposed as $u=u_\mathcal{Z}+u_\perp$, where $u_\mathcal{Z}\in Z_V$ and $u_\perp\in Z_V^\perp$.\ We can write the mixed formulation of in a more general representation, using four continuous bilinear forms, $$\begin{aligned} \sfa(\cdot,\cdot)&:=(\cdot,\cdot)_\Omega\ :W\times W\rightarrow\mathbb{R},\quad\; \sfb(\cdot,\cdot):=(\ud\cdot,\cdot)_\Omega\ :V\times Q\rightarrow\mathbb{R},\\ \sfc(\cdot,\cdot)&:=(\ud\cdot,\cdot)_\Omega\ :W\times V\rightarrow\mathbb{R},\quad \sfe(\cdot,\cdot):=(\ud\cdot,\ud\cdot)_\Omega\ :V\times V\rightarrow\mathbb{R},\end{aligned}$$ and three continuous linear forms $$\begin{gathered} \sff(\cdot):=(\cdot,f)_\Omega+\int_{\Gamma_2\cup\Gamma_4}\tr\cdot\wedge\Pi_b\ :V\rightarrow\mathbb{R},\\ \sfg(\cdot):=(\cdot,g)_\Omega\ :Q\rightarrow\mathbb{R},\quad\quad \sfh(\cdot):=-\int_{\Gamma_1\cup\Gamma_2}\tr\cdot\wedge u_{b,t}\ :W\rightarrow\mathbb{R}.\end{gathered}$$ The mixed formulation becomes \[doublesaddlepoint\] $$\begin{aligned} \sfa(\tau,\omega)-\sfc(\tau,u)&=\sfh(\tau),\quad\quad\!\forall\tau\in W, \label{saddlepoint1}\\ \sfe(v,u)+\sfc(\omega,v)+\sfb(v,p)&=\sff(v),\quad\quad\forall v\in V, \label{saddlepoint2}\\ \sfb(u,q)&=\sfg(q).\quad\quad\forall q\in Q. \label{saddlepoint3}\end{aligned}$$ There exists continuity constants $0<c_\sfa,c_\sfb,c_\sfc,c_\sfe<\infty$ such that $$\sfa(\tau,\kappa)\leq c_\sfa\norm{\tau}_W\norm{\kappa}_W,\quad \sfb(v,q)\leq c_\sfb\norm{v}_V\norm{q}_Q,\quad \sfc(\tau,v)\leq c_\sfc\norm{\tau}_W\norm{v}_V,\quad\sfe(v,w)\leq c_\sfe\norm{v}_V\norm{w}_V.$$ By Cauchy-Schwarz we know that $c_\sfa=1$, however we write $c_\sfa$ for generality purpose. The continuous linear forms are bounded such that $$\sff(v)\leq \norm{f}\norm{v}_V,\quad \sfg(v)\leq \norm{g}\norm{v}_V,\quad\sfh(\tau)\leq\norm{h}\norm{\tau}_W.$$ At first restrict to all $v=v_\mathcal{Z}\in Z_V$. This gives the vorticity-velocity subproblem, which is a saddle point problem: \[vorticityvelocityproblem\] $$\begin{aligned} \sfa(\tau,\omega)-\sfc(\tau,u_\mathcal{Z})&=\sfh(\tau),\quad\quad\ \,\forall\tau\in W,\\ \sfc(v_\mathcal{Z},\omega)&=\sff(v_\mathcal{Z}),\quad\quad\forall v_\mathcal{Z}\in Z_V.\end{aligned}$$ [@dubois2002]\[prop:infsupac\] System has a unique solution $(\omega,u_\mathcal{Z})\in\{W\times Z_V\}$ if there exists positive constants $\alpha,\ \gamma$, such that we have coercivity in the kernel of $W$, $$\label{infsupa} \inf_{\tau_\mathcal{Z}\in Z_W}\sup_{\kappa_\mathcal{Z}\in Z_W}\frac{\sfa(\tau_\mathcal{Z},\kappa_\mathcal{Z})}{\norm{\tau_\mathcal{Z}}_W\norm{\kappa_\mathcal{Z}}_W}\geq\alpha,\quad\quad \inf_{\kappa_\mathcal{Z}\in Z_W}\sup_{\tau_\mathcal{Z}\in Z_W}\frac{\sfa(\tau_\mathcal{Z},\kappa_\mathcal{Z})}{\norm{\tau_\mathcal{Z}}_W\norm{\kappa_\mathcal{Z}}_W}\geq\alpha,$$ and satisfies the following inf-sup condition for $\sfc(\tau,v_\mathcal{Z})$, $$\label{infsupc} \inf_{v_\mathcal{Z}\in Z_V}\sup_{\tau\in W}\frac{\sfc(\tau,v_\mathcal{Z})}{\norm{\tau}_W\norm{v_\mathcal{Z}}_V}\geq\gamma,$$ The proof of is straightforward, see e.g. [@brezzifortin]. For , we have $\sfc(\tau,v_\mathcal{Z})=(\ud\tau,v_\mathcal{Z})_\Omega$, where $\ud:Z_W^\perp\rightarrow Z_V$. Thus, given $v_\mathcal{Z}\in Z_V$ there exists a unique $\tau_v\in Z_W^\perp$ such that $\ud\tau_v=v_\mathcal{Z}$ and $\norm{\tau_v}_W\leq c_P\norm{v_\mathcal{Z}}_V$ by . Therefore $$\sup_{\tau\in W}\frac{\sfc(\tau,v_\mathcal{Z})}{\norm{\tau}_W}\geq\frac{\sfc(\tau_v,v_\mathcal{Z})}{\norm{\tau_v}_W}=\frac{\norm{v_\mathcal{Z}}_V^2}{\norm{\tau_v}_W}\geq\frac{1}{c_P}\norm{v_\mathcal{Z}}_V.$$ \[prop:infsupeb\] The full problem has a unique solution $(\omega,u,p)\in\{W,V,Q\}$ if conditions and from Proposition \[prop:infsupac\] are satisfied and additionally is there exists positive constants $\ve,\beta$, such that we have coercivity in the range of $V$, $$\label{infsupe} \inf_{v_\perp\in Z_V^\perp}\sup_{w_\perp\in Z_V^\perp}\frac{\sfe(v_\perp,w_\perp)}{\norm{v_\perp}_V\norm{w_\perp}_V}\geq\ve,\quad\quad \inf_{w_\perp\in Z_V^\perp}\sup_{v_\perp\in Z_V^\perp}\frac{\sfe(v_\perp,w_\perp)}{\norm{v_\perp}_V\norm{w_\perp}_V}\geq\ve,$$ and satisfies the following inf-sup condition for $\sfb(v,q)$, $$\label{infsupb} \inf_{q\in Q}\sup_{v\in V}\frac{\sfb(v,q)}{\norm{v}_V\norm{q}_Q}\geq\beta>0.$$ The proof is similar to that of . See also [@bochev2003], Section 7.1. So well-posedness of the Stokes problem relies only on the Hodge decomposition and the Poincaré inequality. \[cor:wellposedness\][@bernardi2006; @dubois2002] Problem is well-posed according to Propositions \[prop:infsupac\] and \[prop:infsupeb\]. That is, for any given data $f\in L^2\Lambda^{n-1}(\Omega)$ and $g\in L^2\Lambda^n(\Omega)$ and natural boundary conditions $u_{b,t}\in H^{\frac{1}{2}}\Lambda^{n-1}(\Gamma_t)$ and $\Pi_b\in H^{\frac{1}{2}}\Lambda^n(\Gamma_\pi)$, there exists a unique solution $(\omega,u,p)\in W\times V\times Q$ satisfying . Moreover, this solution satisfies: $$\norm{\omega}_{W}+\norm{u}_{V}+\norm{p}_{Q}\leq C\left(\norm{f}_{L^2\Lambda^{n-1}}+\norm{g}_{L^2\Lambda^n}+\norm{u_{b,t}}_{H^{\frac{1}{2}}\Lambda^{n-1}}+\norm{\Pi_b}_{H^{\frac{1}{2}}\Lambda^n}\right),$$ where $C$ is a constant depending only on the Poincaré constant $c_P$ and the continuity constants. Compatible spectral discretization {#sec:discretization} ================================== Well-posedness of the Stokes problem in VVP formulation relies solely on the Hodge decomposition and the Poincaré inequality. For a compatible discretization, these properties need to be respected as well in the finite dimensional spaces. Key ingredient to obtain a discrete Hodge decomposition and discrete Poincaré inequality is the construction of a bounded projection operator that commutes with the exterior derivative. The compatible spectral discretization consists of three parts. First, the discrete structure is described in terms of chains and cochains from algebraic topology, the discrete counterpart of differential geometry. This discrete structure mimics many of the properties from differential geometry. Secondly, mimetic operators are introduced that relate the continuous formulation in terms of differential forms to the discrete representation based on cochains and finite dimensional differential forms. Thirdly, mimetic spectral element basis functions are described following the definitions of the mimetic operators. In this paper we address these topics only briefly. More details of the mimetic spectral element method can be found in [@kreeft2012; @kreeftpalhagerritsma2011]. Finally, well-posedness of the discrete numerical formulation is proven and interpolation error estimates are given. Algebraic Topology ------------------ Let $D$ be an *oriented cell-complex* covering the manifold $\Omega$, describing the topology of the mesh, and consisting of $k$-cells $\tau_{(k)}$, $k=0,\hdots,n$. The two most popular classes of $k$-cells in literature to describe the topology of a manifold are either in terms of *simplices*, see for instance [@munkres1984; @singerthorpe; @whitney], or in terms of *cubes*, see [@Massey2; @tonti1]. From a topological point of view both descriptions are equivalent, see [@dieudonne]. Despite this equivalence of simplicial complexes and cubical complexes, the reconstruction maps in terms of basis functions, to be discussed in , differ significantly. For mimetic methods based on simplices see [@arnoldfalkwinther2006; @arnoldfalkwinther2010; @desbrun2005c; @rapetti2009], whereas for mimetic methods based on singular cubes see [@arnoldboffifalk2005; @HymanShashkovSteinberg2002; @hymansteinberg2004; @RobidouxSteinberg2011]. We restrict ourselves to $k$-cubes, although we will keep calling them $k$-cells. The ordered collection of all $k$-cells in $D$ generate a basis for the space of $k$-chains, $C_k(D)$. Then a $k$-chain, $\kchain{c}{k}\in C_k(D)$, is a formal linear combination of $k$-cells, $\tau_{(k),i}\in D$, $$\label{kchain} \kchain{c}{k}=\sum_ic_i\tau_{(k),i}.$$ The boundary operator on $k$-chains, $\partial:\kchainspacedomain{k}{D}\spacemap\kchainspacedomain{k-1}{D}$, is an homomorphism defined by [@hatcher; @munkres1984], $$\label{algebraic::boundary_operator} \partial \kchain{c}{k} = \partial \sum_{i}c_{i}\tau_{(k),i} := \sum_{i}c_{i} \partial \left ( \tau_{(k),i} \right ) \;.$$ The boundary of a $k$-cell $\tau_{(k)}$ will then be a $(k-1)$-chain formed by the oriented faces of $\tau_{(k)}$. Like the exterior derivative, applying the boundary operator twice on a $k$-chain gives the null $(k-2)$-chain, $\p\p\kchain{c}{k}=\kchain{0}{k-2}$ for all $\kchain{c}{k}\in C_k(D)$. The set of $k$-chains and boundary operators gives rise to an exact sequence, the chain complex $(C_k(D),\p)$, $$\label{chaincomplex} \begin{CD} \cdots @<\p<< C_{k-1}(D) @<\p<< C_k(D) @<\p<< C_{k+1}(D) @<\p<< \cdots. \end{CD}$$ Let $B_k$ be the range and $Z_k$ be the nullspace of $\p$ in $C_k$. Then the topological Hodge decomposition of the space of $k$-chains is given by $C_k=Z_k\oplus Z_k^\perp$, where $Z_k=B_k$ on contractible domains[^3]. The boundary operator on chains in is a bijection that maps $\p:Z_k^\perp\rightarrow B_{k-1}$. Dual to the space of $k$-chains, $C_k(D)$, is the space of *$k$-cochains*, $C^k(D)$, defined as the set of all linear functionals, $\kcochain{c}{k}:C_k(D)\rightarrow\mathbb{R}$. The duality is expressed using the duality pairing $\langle\kcochain{c}{k},\kchain{c}{k}\rangle:=\kcochain{c}{k}(\kchain{c}{k})$. Note the resemblance between this duality pairing and the integration of differential forms . Let $\{\tau_{(k),i}\}$ form a basis of $C_k(D)$, then there is a dual basis $\{\tau^{(k),i}\}$ of $C^k(D)$, such that $\tau^{(k),i}(\tau_{(k),i})=\delta^i_j$ and all $k$-cochains can be represented as linear combinations of the basis elements, $$\kcochain{c}{k}=\sum_ic^i\tau^{(k),i}.$$ With the duality relation between chains and cochains, we can define the formal adjoint of the boundary operator which constitutes an exact sequence on the spaces of $k$-cochains in the cell complex. This formal adjoint is called the *coboundary operator*, $\delta:\kcochainspacedomain{k}{D}\rightarrow\kcochainspacedomain{k+1}{D}$, and is defined analogous to as $$\duality{\delta\kcochain{c}{k}}{\kchain{c}{k+1}} := \duality{\kcochain{c}{k}}{\partial\kchain{c}{k+1}}, \quad\forall\kcochain{c}{k}\in\kcochainspacedomain{k}{D} \text{ and } \,\forall\kchain{c}{k+1}\in\kchainspacedomain{k+1}{D} \;. \label{coboundary}$$ Note that expression is nothing but a discrete Stokes’ theorem and that the coboundary operator is nothing but a discrete exterior derivative. Also the coboundary operator satisfies $\delta\delta\kcochain{c}{k}=\kcochain{0}{k+2}$, for all $\kcochain{c}{k}\in C^k(D)$, and gives rise to an exact sequence, called the *cochain complex* $(C^k(D),\delta)$, $$\label{cochaincomplex} \begin{CD} \cdots @>\delta>> C^{k-1}(D)@>\delta>> C^k(D)@>\delta>>C^{k+1}(D)@>\delta>>\cdots\;. \end{CD}$$ Let $B^k$ be the range and $Z^k$ be the nullspace of $\delta$ in $C^k$, then a Hodge decomposition of the space of $k$-cochains is given by $C^k=Z^k\oplus Z^{k,\perp}$, where $Z^k=B^k$ on contractible domains. The coboundary operator in is a bijection that maps $\delta:Z^{k,\perp}\rightarrow B^{k+1}$. Note the similarity between this map, that of the boundary operator on $k$-chains and that of the exterior derivative on $k$-forms. Mimetic Operators {#mimeticoperators} ----------------- The discretization of the flow variables involves a bounded projection operator, $\pi_h$, from the complete space $H\Lambda^k(\Omega)$ to a conforming subspace $\Lambda^k_h(\Omega;C_k)\subset H\Lambda^k(\Omega)$. The projection operation consists of two steps, a *reduction operator*, $\reduction:H\Lambda^k(\Omega)\rightarrow C^k(D)$, that integrates the $k$-forms on $k$-chains to get $k$-cochains, and a *reconstruction operator*, $\reconstruction:C^k(D)\rightarrow\Lambda^k_h(\Omega;C_k)$, to reconstruct $k$-forms from $k$-cochains using appropriate basis-functions. These mimetic operators were already introduced before in [@bochevhyman2006; @hymanscovel1988]. A composition of the two gives the projection operator $\projection=\reconstruction\circ\reduction$ as is illustrated below. H\^k()&\^& \^k\_h(;C\_k)\ \^ & \_ &\ C\^k(D) & & These three operators together constitute the mimetic framework. An extensive discussion on mimetic operators can be found in [@kreeft2012; @kreeftpalhagerritsma2011]. The reduction $\reduction$ and reconstruction $\reconstruction$ operators are defined below. The fundamental property of $\reduction$ and $\reconstruction$ is the commutation with differentiation in terms of exterior derivative and coboundary operator. The reduction operator $\reduction:H\kformspacedomain{k}{\Omega}\rightarrow \kcochainspacedomain{k}{D}$ is a homomorphism that maps differential forms to cochains. This map is defined by integration as $$\duality{\reduction a}{\tau_{(k)}}:=\int_{\tau_{(k)}}a,\quad \forall a\in H\Lambda^k(\Omega),\ \tau_{(k)}\in C_k(D). \label{reduction}$$ Then for all $\kchain{c}{k}\in C_k(D)$, the reduction of the $k$-form, $a\in H\Lambda^{k}(\Omega)$, to the $k$-cochain, $\kcochain{a}{k}\in C^k(D)$, is given by $$\kcochain{a}{k}(\kchain{c}{k}):=\duality{\reduction a}{\kchain{c}{k}}\stackrel{\eqref{kchain}}{=}\sum_ic^i\duality{\reduction a}{\tau_{(k),i}}\stackrel{\eqref{reduction}}{=}\sum_ic^i\int_{\tau_{(k),i}}a=\int_{\kchain{c}{k}}a.$$ The reduction maps has a commuting property with respect to differentiation in terms of exterior derivative and coboundary operator, $$\reduction\ud=\delta\reduction,\quad\mathrm{on}\ H\Lambda^k(\Omega).\label{cdp1}$$ Since $\reduction$ is defined by integration, follows directly from Stokes theorem and the duality property . Next by definition also the reconstruction map $\reconstruction:C^k(D)\rightarrow\Lambda^k_h(\Omega;C_k)$ needs to have a commuting property with respect to differentiation in terms of exterior derivative and coboundary operator, $$\ud\reconstruction=\reconstruction\delta,\quad\mathrm{on}\ C^k(D).\label{cdp2}$$ The reconstruction $\reconstruction$ must be the right inverse of $\reduction$, so $\mathcal{RI}=Id$ on $C^k(D)$, and we want it to be an approximate left inverse of $\mathcal{R}$, so $\mathcal{IR}=Id+\mathcal{O}(h^p)$ on $H\Lambda^k(\Omega)$. This composition is defined as the projection operator. \[def:projection\] The composition $\reconstruction\circ\reduction$ will denote the projection operator, $\projection\define\reconstruction\reduction:H\kformspace{k}(\Omega)\rightarrow\kformspace{k}_h(\Omega;C_k)$, allowing for a finite dimensional representation of a $k$-form, $$\projection a:=\reconstruction\reduction a, \quad \pi_ha\in\kformspace{k}_h(\Omega;C_k)\subset H\kformspace{k}(\Omega). \label{projection}$$ where $\reconstruction\reduction a$ is expressed as a combination of $k$-cochains and interpolating $k$-forms. The projection operator $\pi_h$ is a bounded operator if for $C<\infty$ and for all $a\in\Lambda^k(\Omega)$ we have $\norm{\pi_ha}_{H\Lambda^k}\leq C\norm{a}_{H\Lambda^k}$. A proof that $\pi_h$ is indeed a projection operator is given in [@kreeftpalhagerritsma2011]. In also boundedness is proven. \[Lem:projectionextder\] There exists a commuting property for the projection and the exterior derivative, such that $$\label{projectionextder} \ederiv\projection=\projection\ederiv\quad\mathrm{on}\ H\Lambda^k(\Omega).$$ Express the projection in terms of the reduction and reconstruction operator, then $$\ederiv\projection\difform{a}\stackrel{\eqref{projection}}{=}\ederiv\reconstruction\reduction\difform{a}\stackrel{\eqref{cdp2}}{=}\reconstruction\delta\reduction\difform{a}\stackrel{\eqref{cdp1}}{=}\reconstruction\reduction\ederiv\difform{a}\stackrel{\eqref{projection}}{=}\projection\ederiv\difform{a},\quad\forall a\in H\Lambda^k(\Omega).$$ Note that it is the intermediate step $\mathcal{I}\delta\mathcal{R}a$ that is used in practice for the discretization, see [@kreeft2012], and on page . \[cor:projectionextder\] From it follows that $\mathcal{B}^k_h:=\pi_h\mathcal{B}^k\subset\mathcal{B}^k$, $\mathcal{Z}^k_h:=\pi_h\mathcal{Z}^k\subset\mathcal{Z}^k$ and that on contractible domains, $\mathcal{Z}^{k,\perp}_h:=\pi_h\mathcal{Z}^{k,\perp}\subset\mathcal{Z}^{k,\perp}$. Then the discrete Hodge decomposition is given by $\Lambda^k=\mathcal{Z}^k_h\oplus\mathcal{Z}^{k,\perp}_h$. As a consequence of and the discrete Hodge decomposition we have $Z_{W_h}\subset Z_W$, $Z_{V_h}\subset Z_V$, $\ud W_h \subset V_h$ and $\ud V_h= Q_h$, which shows that the discretization method is *compatible*. Finally, we do not restrict ourselves to affine mappings only, as is required in many other compatible finite elements, like Nédélec and Raviart-Thomas elements and their generalizations [@arnoldfalkwinther2006; @nedelec1980; @raviartthomas1977], but also allow non-affine maps such as curvilinear transfinite or isoparametric mappings of quadrilaterals or hexahedrals, [@gordonhall1973], where $\Phi$ and its inverse are piecewise sufficiently smooth, i.e. 1. $\Phi$ is a $\mathcal{C}^{p+1}$-diffeomorphism, 2. $|\Phi|_{W_\infty^l}\leq Ch^l,\quad\quad l\leq p+1$, 3. $|\Phi^{-1}|_{W_\infty^l}\leq Ch^{-l},\quad l\leq p+1$. This allows for better approximations in complex domains with curved boundaries, without the need for excessive refinement, while maintaining design convergence rates, [@ciarletraviart1972]. This is possible since the projection operator $\pi_h$ commutes with the pullback $\pullback$, $$\label{pullbackprojection} \pullback\projection=\projection\pullback\quad\mathrm{on}\ H\Lambda^k(\Omega).$$ An extensive proof is given in [@kreeftpalhagerritsma2011]. Numerical stability ------------------- Essential ingredients in proving numerical stability are the discrete Hodge decomposition and the discrete Poincaré inequality. Because the complexes $(H\Lambda,\ud)$ and $(\Lambda_h,\ud)$ are each others supercomplex and subcomplex, respectively, the discrete Poincaré inequality is directly related to the Poincaré inequality in and the bounded projection in . \[lem:discretepoincareinequality\]Let $(H\Lambda,\ud)$ be a bounded closed Hilbert complex, $(\Lambda_h,\ud)$ a subcomplex, and $\pi_h$ a bounded projection. Then $$\left\{ \begin{aligned} &\norm{a_h}_{H\Lambda^k}\leq c_{Ph}\norm{\ud a_h}_{L^2\Lambda^k},\quad a_h\in\mathcal{Z}^{k,\perp}_h,\\ &1\leq c_{Ph}\leq c_P. \end{aligned} \right.$$ Given $a_h\in\mathcal{Z}_h^{k,\perp}$. From the Hodge decomposition and the bounded projection it follows that $\mathcal{B}_h^k\subset\mathcal{B}^k$ and $\mathcal{Z}^k_h\subset\mathcal{Z}^k$ and from the commutation relation it follows that $\mathcal{Z}^{k,\perp}_h\subset\mathcal{Z}^{k,\perp}$. Since we consider a proper subspace, is still valid, with $c_{Ph}\leq c_P$. \[th:discretewellposedness\] Let $(\Lambda_h,\ud)$ be a subcomplex of the closed Hilbert complex $(H\Lambda,\ud)$. Then there exists constants $\alpha_h,\beta_h,\gamma_h$, depending only on $c_{Ph}$, such that for any $(\tau_h,v_h,q_h)\in W_h\times V_h\times Q_h$, there exists a stable finite dimensional solution $(\omega_h,u_h,p_h)\in W_h\times V_h\times Q_h$ of the Stokes problem , with $$\alpha_h>\alpha>0,\quad\beta_h>\beta>0,\quad\gamma_h>\gamma>0.$$ This is just Propositions \[prop:infsupac\] and \[prop:infsupeb\] applied to the complex $(\Lambda_h,\ud)$, combined with the fact that the constant in the Poincaré inequality for $\Lambda^k_h$ is $c_{Ph}\leq c_P$ by . Mimetic spectral element basis-functions {#sec:mimeticsem} ---------------------------------------- The finite dimensional differential forms used in this paper are polynomials, based on the idea of spectral element methods, [@canuto1]. The mimetic spectral elements used here were derived independently in [@gerritsma2011; @robidoux2008], and are more extensively discussed in [@kreeftpalhagerritsma2011]. Only the most important properties of the mimetic spectral element method are presented here. In spectral element methods the domain $\Omega$ is decomposed into $M$ non-overlapping, in this case curvilinear quadrilateral or hexahedral, closed sub-domains $Q_m$, $$\Omega=\bigcup_{m=1}^MQ_m,\quad Q_m\cap Q_l=\p Q_m\cap\p Q_l,\ m\neq l,$$ where in each sub-domain a Gauss-Lobatto grid is constructed. The complete mesh is indicated by $\mathcal{Q}:=\sum_{m=1}^MQ_m$. The collection of Gauss-Lobatto meshes in all elements $Q_m\in\mathcal{Q}$ constitutes the cell complex $D$. For each element $Q_m$ there exists a sub cell complex, $D_m$. Note that $D_m\cap D_l,\ m\neq l$, is not an empty set in case they are neighboring elements, but contains all $k$-cells, $k<n$, of the common boundary. Each sub-domain $Q_m\in\mathcal{Q}$ is mapped from the reference element, $\widehat{Q}=[-1,1]^n$, using the mapping $\Phi_m:\widehat{Q}\rightarrow Q_m$. Then all flow variables defined on $Q_m$ are pulled back onto this reference element using the following pullback operation, $\Phi^\star_m:\Lambda^k_h(Q_m;C_k)\rightarrow\Lambda^k_h(\widehat{Q};C_k)$. The basis-functions that interpolate the cochains on the quadrilateral or hexahedral elements are constructed using tensor products. It is therefore sufficient to derive interpolation functions in one dimension and use tensor products afterwards to construct $n$-dimensional basis functions. A similar approach was taken in [@buffa2011b]. Because projection operator and pullback operator commute , the interpolation functions are discussed for the reference element only. Since the mappings $\Phi_m$ and their inverse are assumed to be sufficiently smooth, the rates of convergence for interpolation estimates on the physical elements are equal to that of the reference element. Only the constants $C$ that will appear below will depend on the mappings $\Phi_m$, but will be independent of the meshsize and polynomial order.\ Consider a 0-form $a\in H\Lambda^0(\widehat{Q})$ on $\widehat{Q}:=\xi\in[-1,1]$, on which a cell complex $D$ is defined that consists of $N+1$ nodes, $\xi_i$, where $-1\leq \xi_0<\hdots<\xi_N\leq 1$, and $N$ edges, $\tau_{(1),i}=[\xi_{i-1},\xi_i]$, of which the nodes are their boundaries. Corresponding to this set of nodes (0-chains) there exists a projection using $N^{\rm th}$ order *Lagrange polynomials*, $l_i(\xi)$, to approximate a $0$-form, as $$\projection a=\sum_{i=0}^N a_il_i(\xi). \label{nodalapprox}$$ The property of Lagrange polynomials is that they interpolate nodal values. They are therefore suitable to reconstruct a 0-form form the 0-cochain $\kcochain{a}{0}=\reduction a$, $a\in\Lambda^0(\Omega)$, containing the set $a_i=a(\xi_i)$ for $i=0,\hdots,N$. Lagrange polynomials are in fact 0-forms, $l_i(\xi)\in\Lambda^0_h(\widehat{Q};C_0)$. Lagrange polynomials are constructed such that their value is one in the corresponding point and zero in all other grid points, $$\reduction l_i(\xi)=l_i(\xi_p)=\left\{ \begin{aligned} &1&{\rm if}\ i=p\\ &0&{\rm if}\ i\neq p \end{aligned} \right.. \label{nodalproperty}$$ In [@gerritsma2011; @robidoux2008] a similar basis for projection of 1-forms was derived, consisting of $1$-cochains and $1$-form polynomials, that is called the *edge polynomial*, $e_i(\xi)\in\Lambda^1_h(\widehat{Q})$. Let $b\in L^2\Lambda^1(\widehat{Q})$, then the projected 1-form is given by $$\projection b(\xi)=\sum_{i=1}^Nb_i e_i(\xi),$$ where the edge polynomial is defined as $$\begin{aligned} \label{edge} e_i(\xi)&=-\sum_{k=0}^{i-1}\ederiv l_k(\xi)=\sum_{k=i}^{N}\ederiv l_k(\xi)=\tfrac{1}{2}\sum_{k=i}^{N}\ederiv l_k(\xi)-\tfrac{1}{2}\sum_{k=0}^{i-1}\ederiv l_k(\xi).\end{aligned}$$ Let $a_h(\xi)\in\Lambda^0_h(\widehat{Q};C_0)$ be expressed as in , then $b_h=\ud a_h\in\Lambda^1_h(\widehat{Q};C_1)$ is expressed as $$\label{edgediscretization} \ud\sum_{i=0}^Na_il_i(\xi)=\sum_{i=1}^N(a_i-a_{i-1})e_i(\xi)=\sum_{i=1}^N\big(\delta\kcochain{a}{0}\big)_ie_i(\xi)=\sum_{i=1}^Nb_ie_i(\xi),$$ where $\delta$ is the coboundary operator , applied to the 0-cochain $\kcochain{a}{0}$. It therefore satisfies . For derivations and proofs see [@gerritsma2011; @kreeftpalhagerritsma2011; @robidoux2008]. Similar to , the edge basis-functions are constructed such that when integrating $e_i(\xi)$ over a line segment it gives one for the corresponding element and zero for any other line segment, so $$\reduction e_i(\xi)=\int_{\xi_{p-1}}^{\xi_p}e_i(\xi)=\left\{ \begin{aligned} &1&{\rm if}\ i=p\\ &0&{\rm if}\ i\neq p \end{aligned} \right.. \label{intedge}$$ Equations and show that indeed we have $\mathcal{RI}=Id$. The fourth-order Lagrange and third-order edge polynomials, corresponding to a Gauss-Lobatto grid with $N=4$, are shown in Figures \[fig:lagrange\] and \[fig:edgepoly\]. ![Edge polynomials on Gauss-Lobatto-Legendre grid.[]{data-label="fig:edgepoly"}](gll.pdf){width="0.9\linewidth"} ![Edge polynomials on Gauss-Lobatto-Legendre grid.[]{data-label="fig:edgepoly"}](edge.pdf){width="0.9\linewidth"} Bounded projections and interpolation estimates {#sec:boundedprojection} ----------------------------------------------- The mimetic framework uses Lagrange, $l_i(\xi)\in H\Lambda^0(\widehat{Q})$, and edge functions, $e_i(\xi)\in L^2\Lambda^1(\widehat{Q})$, for the reconstruction, $\mathcal{I}$. Because we consider tensor products to construct higher-dimensional interpolation, it is sufficient to show that the projection operator is bounded in one dimension. A similar approach was used in [@buffa2011b]. Due to the way the edge functions are constructed, there exists a commuting diagram property between projection and exterior derivative, $$\begin{CD} \mathbb{R} @>>> H\Lambda^0 @>\ederiv>> L^2\Lambda^1 @>>>0 \\ @. @VV \pi_h V @VV\pi_hV @. \\ \mathbb{R} @>>> \Lambda_h^0 @>\ederiv>> \Lambda_h^1 @>>>0, \end{CD}$$ which gives, for $a\in H\Lambda^0(\widehat{Q})$, the one form $\ederiv\pi_ha=\pi_h\ederiv a$ in $L^2\Lambda^1(\widehat{Q})$. Lagrange interpolation by itself does not guarantee a convergent approximation [@erdos1980], but it requires a suitably chosen set of points, $-1\leq\xi_0<\xi_1<\hdots<\xi_N\leq1$. Here, the Gauss-Lobatto distribution is proposed, because of its superior convergence behaviour. For $a\in H^m\Lambda^0(\Omega)$, the a priori error estimate in the $H\Lambda^0$-norm is given by [@canuto1], $$\label{hpconvergenceestimate} \Vert a-\pi_ha\Vert_{H\Lambda^0}\leq Ch^l|a|_{H^{l+1}\Lambda^0},\quad l=\mathrm{min}(N,m-1).$$ Equation also implies that the projection of zero-forms is stable in the $H\Lambda^0(\widehat{Q})$, as is shown in the following proposition. \[boundedh\][@kreeftpalhagerritsma2011] For $a\in H\Lambda^0(\widehat{Q})$ and the projection $\pi_h:H\Lambda^0\rightarrow \Lambda^0_h$, there exists the following two stability estimates in $H\Lambda^0$-norm and $H\Lambda^0$-semi-norm: $$\begin{aligned} \Vert\pi_ha\Vert_{H\Lambda^0}&\leq C\Vert a\Vert_{H\Lambda^0},\label{H1normstability}\\ |\pi_ha|_{H\Lambda^0}&\leq C|a|_{H\Lambda^0}.\label{H1seminormstability}\end{aligned}$$ Now that we have a bounded linear projection of zero forms in one dimension, we can also proof boundedness of the projection of one-forms. \[boundede\] Let $a\in H\Lambda^0$ and $b=\ederiv a\in L^2\Lambda^1$, then there exists a bounded linear projection $\pi_h:L^2\Lambda^1\rightarrow\Lambda^1_h$, such that $$\Vert\pi_hb\Vert_{L^2\Lambda^1}\leq C\Vert b\Vert_{L^2\Lambda^1}.$$ The proof is based on the result of the previous proposition and the commutation between the bounded projection operator and the exterior derivative, , $$\Vert\pi_hb\Vert_{L^2\Lambda^1}=|\pi_h\ederiv a|_{L^2\Lambda^1}=|\ederiv\pi_ha|_{L^2\Lambda^1}=|\pi_ha|_{H\Lambda^0}\leq C|a|_{H\Lambda^0}=C\Vert\ederiv a\Vert_{L^2\Lambda^1}=C\Vert b\Vert_{L^2\Lambda^1}.$$ Propositions \[boundedh\] and \[boundede\] show that the projection $\pi_h$ is a *bounded projection operator*, based on Lagrange functions and edge functions. As for zero forms using Lagrange interpolation, we can also give an estimate for the interpolation error of one forms, interpolated using edge functions. [@kreeftpalhagerritsma2011]\[prop:convratee\] Let $a\in H\Lambda^0$ and $b=\ederiv a\in L^2\Lambda^1$, the interpolation error $b-\pi_hb\in L^2\Lambda^1$ is given by $$\label{edgeinterpolationerror} \Vert b-\pi_hb\Vert_{L^2\Lambda^1}\leq Ch^{l}|b|_{H^{l}\Lambda^1},\quad l=\mathrm{min}(N,m-1).$$ The one dimensional results can be extended to the multidimensional framework by means of tensor products. This allows for the interpolation of integral quantities defined on $k$-dimensional cubes. Consider a reference element in $\mathbb{R}^2$, $\widehat{Q}=[-1,1]^2$. Then the interpolation functions for points, lines, surfaces (2D volumes) are given by, $$\begin{aligned} &\mathrm{point}:&&P^{(0)}_{i,j}(\xi,\eta)=l_i(\xi)\otimes l_j(\eta),\\ &\mathrm{line}:&&L^{(1)}_{i,j}(\xi,\eta)=\{e_i(\xi)\otimes l_j(\eta),\ l_i(\xi)\otimes e_j(\eta)\},\\ &\mathrm{surface}:&&S^{(2)}_{i,j}(\xi,\eta)=e_i(\xi)\otimes e_j(\eta). \end{aligned}$$ The approximation spaces are spanned by combinations of Lagrange and edge basis functions, $$\begin{aligned} H^1\Lambda^0(\Omega)\supset\Lambda^0_h(\mathcal{Q};C_0)&:=\mathrm{span}\left\{P^{(0)}_{i,j}\right\}_{i=0,j=0}^{N,N},\\ H\Lambda^1(\Omega)\supset\Lambda^1_h(\mathcal{Q};C_1)&:=\mathrm{span}\left\{\big(L^{(1)}_{i,j}\big)_1\right\}_{i=1,j=0}^{N,N}\times \mathrm{span}\left\{\big(L^{(1)}_{i,j}\big)_2\right\}_{i=0,j=1}^{N,N},\\ L^2\Lambda^2(\Omega)\supset\Lambda^2_h(\mathcal{Q};C_2)&:=\mathrm{span}\left\{S^{(2)}_{i,j}\right\}_{i=1,j=1}^{N,N}.\end{aligned}$$ For the variables vorticity, velocity and pressure in the VVP formulation of the Stokes problem, the $h$-convergence rates of the interpolation errors in $L^2\Lambda^k$-norm become, $$\norm{\omega-\pi_h\omega}_{L^2\Lambda^{n-2}}=\mathcal{O}(h^{N+s}),\quad\norm{u-\pi_hu}_{L^2\Lambda^{n-1}}=\mathcal{O}(h^N),\quad\norm{p-\pi_hp}_{L^2\Lambda^n}=\mathcal{O}(h^N), \label{L2interpolation}$$ in case the functions $(\omega,u,p)$ are sufficiently smooth, where $s=1$ for $n=2$ and $s=0$ for $n>2$. The interpolation errors in $H\Lambda^k$-norm become, $$\norm{\omega-\pi_h\omega}_{H\Lambda^{n-2}}=\mathcal{O}(h^N),\quad\norm{u-\pi_hu}_{H\Lambda^{n-1}}=\mathcal{O}(h^N), \label{Hinterpolation}$$ with $N$ defined as in . Error estimates {#sec:errorestimates} =============== Next consider the finite dimensional problem: find $(\omega_h,u_h,p_h)\in\{W_h\times V_h\times Q_h\}$, given $f\in L^2\Lambda^{n-1}(\Omega)$ and $g\in L^2\Lambda^{n}$ and boundary conditions in , for all $(\tau_h,v_h,q_h)\in\{W_h\times V_h\times Q_h\}$, such that \[discretedouble\] $$\begin{aligned} \sfa(\tau_h,\omega_h)-\sfc(\tau_h,u_h)&=\sfh(\tau_h),\quad\quad\forall\tau_h\in W_h, \label{discretedouble1}\\ \sfe(v_h,u_h)+\sfc(\omega_h,v_h)+\sfb(v_h,p_h)&=\sff(v_h),\quad\quad\forall v_h\in V_h, \label{discretedouble2}\\ \sfb(u_h,q_h)&=\sfg(q_h).\quad\quad\forall q_h\in Q_h. \label{discretedouble3}\end{aligned}$$ The following theorem gives the a priori error estimates of this problem when using the compatible spectral discretization method described in the previous section. Corollary \[cor:projectionextder\] showed that we have $Z_{W_h}\subset Z_W$ and $Z_{V_h}\subset Z_V$. From this it follows that we have compatible finite dimensional subspaces: $W_h\subset W$, $V_h=\ud W_h\oplus\ud^*Q_h\subset V$ and $Q_h=\ud V_h\subset Q$. The derivations of the error estimates are based on the methodology of [@brezzifortin]. The proofs are given in the subsequent propositions. \[th:errorestimates\] Let $(\omega,u,p)$ be the solution of the continuous problem given in or and $(\omega_h,u_h,p_h)$ the solution of the finite dimensional problem in . The continuous problem is well-posed by Propositions \[prop:infsupac\] and \[prop:infsupeb\] and the finite dimensional problem is well-posed by Theorem \[th:discretewellposedness\] and Propositions \[boundedh\] and \[boundede\]. Furthermore, from Corollary \[cor:projectionextder\] we have that for the compatible spectral discretization method, $Z_{W_h}\subset Z_W$ and $Z_{V_h}\subset Z_V$. Then the following a priori error estimates for the VVP formulation of the Stokes problem hold: $$\begin{gathered} \norm{\omega-\omega_h}_W\leq\left(1+\frac{c_\sfa}{\alpha_h}\right)\left(1+\frac{c_\sfc}{\gamma_h}\right)\inf_{\tau_h\in W_h}\norm{\omega-\tau_h}_W,\\ \norm{u-u_h}_V\leq\left(1+\frac{c_\sfc}{\gamma_h}\right)\left(1+\frac{c_\sfb}{\beta_h}\right)\inf_{v_h\in V_h}\norm{u-v_h}_V+\frac{c_\sfa}{\gamma_h}\left(1+\frac{c_\sfa}{\alpha_h}\right)\left(1+\frac{c_\sfc}{\gamma_h}\right)\inf_{\tau_h\in W_h}\norm{\omega-\tau_h}_W,\\ \norm{p-p_h}_Q\leq\left(1+\frac{c_\sfb}{\beta_h}\right)\inf_{q_h\in Q_h}\norm{p-q_h}_Q+\frac{c_\sfe}{\beta_h}\left(1+\frac{c_\sfc}{\gamma_h}\right)\left(1+\frac{c_\sfb}{\beta_h}\right)\inf_{v_h\in V_h}\norm{u-v_h}_V\\ +\left(\frac{c_\sfa}{\gamma_h}+\frac{c_\sfc}{\beta_h}\right)\left(1+\frac{c_\sfa}{\alpha_h}\right)\left(1+\frac{c_\sfc}{\gamma_h}\right)\inf_{\tau_h\in W_h}\norm{\omega-\tau_h}_W.\nonumber\end{gathered}$$ The proof of this Theorem will be given in a series of Propositions \[prop:vorticityerrorbound\] to \[prop:boundsigmas\]. \[prop:vorticityerrorbound\] Let $\sigma_h\in Z_{W_h}^\perp$, the error for vorticity is bounded by $$\label{vorticityerrorbound} \norm{\omega-\omega_h}_W\leq\left(1+\frac{c_\sfa}{\alpha_h}\right)\inf_{\sigma_h\in Z_{W_h}^\perp}\norm{\omega-\sigma_h}_W.$$ Subtract the velocity-vorticity relation in the finite dimensional problem from that of the continuous problem , we get $$\sfa(\tau_h,\omega-\omega_h)-\sfc(\tau_h,u-u_h)=0,\quad \forall\tau_h\in Z_{W_h}\subset Z_W.$$ Bound $\sigma_h-\omega_h\in Z_{W_h}$ using inf-sup condition , we get $$\begin{aligned} \alpha_h\norm{\sigma_h-\omega_h}_W&\leq\sup_{\tau_h\in Z_{W_h}}\frac{\sfa(\tau_h,\sigma_h-\omega_h)}{\norm{\tau_h}_W}\\ &=\sup_{\tau_h\in Z_{W_h}}\frac{\sfa(\tau_h,\sigma_h-\omega)+\sfa(\tau_h,\omega-\omega_h)}{\norm{\tau_h}_W}\\ &=\sup_{\tau_h\in Z_{W_h}}\frac{\sfa(\tau_h,\sigma_h-\omega)+\sfc(\tau_h,u-u_h)}{\norm{\tau_h}_W}. \end{aligned}$$ The last term vanishes since $\tau_h\in Z_{W_h}$ and $Z_{W_h}\subset Z_W$, hence $\alpha_h\norm{\sigma_h-\omega_h}_W\leq c_\sfa\norm{\omega-\sigma_h}_W$. By the triangle inequality and the infimum over all $\sigma_h\in Z_{W_h}^\perp$ we obtain . \[prop:velocityerrorbound\] Let $s_h\in Z_{V_h}^\perp$, the error for velocity is bounded by $$\label{velocityerrorbound} \norm{u-u_h}_V\leq\left(1+\frac{c_\sfc}{\gamma_h}\right)\inf_{s_h\in Z_{V_h}^\perp}\norm{u-s_h}_V+\frac{c_\sfa}{\gamma_h}\left(1+\frac{c_\sfa}{\alpha_h}\right)\inf_{\sigma_h\in Z_{W_h}^\perp}\norm{\omega-\sigma_h}_W.$$ Use the inf-sup condition to bound $s_h-u_h\in Z_V$, $$\begin{aligned} \gamma_h\norm{s_h-u_h}_V&\leq\sup_{\tau_h\in Z_{W_h}}\frac{\sfc(\tau_h,s_h-u_h)}{\norm{\tau_h}_W}\\ &=\sup_{\tau_h\in Z_{W_h}}\frac{\sfc(\tau_h,s_h-u)+\sfc(\tau_h,u-u_h)}{\norm{\tau_h}_W}\\ &=\sup_{\tau_h\in Z_{W_h}}\frac{\sfc(\tau_h,s_h-u)+\sfa(\tau_h,\omega-\omega_h)}{\norm{\tau_h}_W}\\ &\leq c_\sfc\norm{u-s_h}_V+c_\sfa\norm{\omega-\omega_h}_W. \end{aligned}$$ By triangle inequality, estimate and the infimum over all $s_h\in Z_{V_h}^\perp$, we obtain . \[prop:pressureerrorbound\] The error for pressure is bounded by $$\label{pressureerrorbound} \begin{aligned} \norm{p-p_h}_Q\leq&\left(1+\frac{c_\sfb}{\beta_h}\right)\inf_{q_h\in Q_h}\norm{p-q_h}_Q+\frac{c_\sfe}{\beta_h}\left(1+\frac{c_\sfc}{\gamma_h}\right)\inf_{s_h\in Z_{V_h}^\perp}\norm{u-s_h}_V\\ &+\left(\frac{c_\sfa}{\gamma_h}+\frac{c_\sfc}{\beta_h}\right)\left(1+\frac{c_\sfa}{\alpha_h}\right)\inf_{\sigma_h\in Z_{W_h}^\perp}\norm{\omega-\sigma_h}_W. \end{aligned}$$ Subtract from , we get $$\sfc(\omega-\omega_h,v_h)+\sfe(v_h,u-u_h)+\sfb(v_h,p-p_h)=0,\quad\forall v_h\in V_h.$$ So for $q_h\in Q_h$ we have $$\sfb(v_h,q_h-p_h)=-\sfc(\omega-\omega_h,v_h)-\sfe(v_h,u-u_h)-\sfb(v_h,p-q_h).$$ Use this and the inf-sup condition to bound $q_h-p_h\in Q_h$, $$\begin{aligned} \beta_h\norm{q_h-p_h}_Q&\leq\sup_{v_h\in V_h}\frac{\sfb(v_h,q_h-p_h)}{\norm{v_h}_V}\\ &=\sup_{v_h\in V_h}\frac{-\sfc(\omega-\omega_h,v_h)-\sfe(v_h,u-u_h)-\sfb(v_h,p-q_h)}{\norm{v_h}_V}\\ &\leq c_\sfc\norm{\omega-\omega_h}_W+c_\sfe\norm{u-u_h}_V+c_\sfb\norm{p-q_h}_Q. \end{aligned}$$ By triangle inequality, estimates and , and the infimum over all $q_h\in Q_h$, we obtain . Next we replace the infimums over $\sigma_h\in Z_{W_h}^\perp$ and $s_h\in Z_{V_h}^\perp$ by best approximation errors. \[prop:boundsigmas\] The terms $\inf_{\sigma_h\in Z_{W_h}^\perp}\norm{\omega-\sigma_h}_W$ and $\inf_{s_h\in Z_{V_h}^\perp}\norm{u-s_h}_V$ are bounded by the best approximation estimates $\inf_{\tau_h\in W_h}\norm{\omega-\tau_h}_W$ and $\inf_{v_h\in V_h}\norm{u-v_h}_V$, using the inf-sup conditions and , as $$\label{boundsigma} \inf_{\sigma_h\in Z_{W_h}^\perp}\norm{\omega-\sigma_h}_W\leq\left(1+\frac{c_\sfc}{\gamma_h}\right)\inf_{\tau_h\in W_h}\norm{\omega-\tau_h}_W,$$ $$\label{bounds} \inf_{s_h\in Z_{V_h}^\perp}\norm{u-s_h}_V\leq\left(1+\frac{c_\sfb}{\beta_h}\right)\inf_{v_h\in V_h}\norm{u-v_h}_V.$$ Take $\tau_h\in W_h$, then there exists a $\kappa_h\in W_h$ such that $$\sfc(\kappa_h,v_{\mathcal{Z}_h})=\sfc(\omega-\tau_h,v_{\mathcal{Z}_h}),\quad\forall v_{\mathcal{Z}_h}\in Z_{V_h}.$$ This is equivalent to $$\sfc(\kappa_h+\tau_h,v_{\mathcal{Z}_h})=\sfc(\omega,v_{\mathcal{Z}_h})=\sff(v_{\mathcal{Z}_h}),\quad\forall v_{\mathcal{Z}_h}\in Z_{V_h},$$ which shows that $\sigma_h=\kappa_h+\tau_h\in Z_{W_h}^\perp$. We can bound $\norm{\kappa_h}_W$ using the discrete inf-sup condition as follows $$\gamma_h\norm{\kappa_h}_W\leq\sup_{v\in Z_{V_h}}\frac{\sfc(\kappa_h,v)}{\norm{v}_V}=\sup_{v\in Z_{V_h}}\frac{\sfc(\omega-\tau_h,v)}{\norm{v}_V}\leq c_\sfc\norm{\omega-\tau_h}_V.$$ By triangle inequality and since $\tau_h\in W_h$ was arbitrary, we find . A similar proof holds for (see also [@brezzifortin], Proposition 2.5). Additionally, following section 7.7.6 in [@bochevgunzburger], we have the following $L^2\Lambda^{k}(\Omega)$ error estimates for the curl of vorticity and divergence of velocity, The errors of the curl of vorticity and divergence of velocity are bounded by their best approximation estimates, $$\begin{aligned} \norm{\ud(\omega-\omega_h)}_{L^2\Lambda^{n-1}}&\leq\inf_{\tau_h\in W_h}\norm{\ud(\omega-\tau_h)}_{L^2\Lambda^{n-1}},\label{curlvorticity}\\ \norm{\ud(u-u_h)}_{L^2\Lambda^{n}}&\leq\inf_{v_h\in V_h}\norm{\ud(u-v_h)}_{L^2\Lambda^{n}}.\label{divvelocity}\end{aligned}$$ Choose $v=v_{\mathcal{Z}_h}\in Z_{V_h}$ in and and subtract these. Set $v_{\mathcal{Z}_h}=\ud\tau_h$, this gives the orthogonality relation $$(\ud(\omega-\omega_h),\ud\tau_h))_\Omega=0,\quad\forall\tau_h\in W_h.$$ Substitute this into the following Cauchy-Schwarz inequality, $$\begin{aligned} \norm{\ud(\omega-\omega_h)}^2_{L^2\Lambda^{n-1}}&=(\ud(\omega-\omega_h),\ud(\omega-\tau_h))_\Omega\\ &\leq\norm{\ud(\omega-\omega_h)}_{L^2\Lambda^{n-1}}\norm{\ud(\omega-\tau_h)}_{L^2\Lambda^{n-1}},\quad\forall\tau_h\in W_h, \end{aligned}$$ and follows. Next choose $q_h=\ud v_h\in Q_h$ in and and subtract these. Then follows again from the Cauchy-Schwarz inequality. Because the projections of respectively $\omega,\ u,$ and $p$, belong to the finite dimensional subspaces $W_h\subset W$, $V_h\subset V$ and $Q_h\subset Q$, the best approximation errors can be bounded using the interpolation errors, $$\inf_{\tau_h\in W_h}\norm{\omega-\tau_h}_W\leq\norm{\omega-\pi_h\omega}_W,\ \inf_{v_h\in V_h}\norm{u-v_h}_V\leq\norm{u-\pi_h u}_V,\ \inf_{q_h\in Q_h}\norm{p-q_h}_Q\leq\norm{p-\pi_hp}_Q,$$ and therefore we obtain the following optimal a priori error estimates, $$\norm{\omega-\omega_h}_W=\mathcal{O}(h^{N}),\quad \norm{u-u_h}_V=\mathcal{O}(h^N), \quad \norm{p-p_h}_Q=\mathcal{O}(h^N).$$ So the convergence rates for the approximations are equal to those of the interpolations, , , thus we obtained optimal convergence. The error estimates were obtained independent of the chosen types of boundary conditions. In contrast to [@arnold2011; @dubois2003b] and [@bochevgunzburger], Table 7.5, where Raviart-Thomas elements were used, the proposed compatible method has provably optimal convergence also with standard velocity boundary conditions and with non-affine mappings. Numerical Results {#sec:numericalresults} ================= Now that the compatible spectral discretization method and its a priori error estimates are derived, we perform a series of test problems to show optimal convergence behavior. Purpose of the testcases is to show convergence behavior in case of various boundary conditions and in case of curvilinear meshes. In all cases we show optimal convergence. The first three testcases originate from a recent paper by Arnold et al [@arnold2011], where suboptimal convergence is shown for normal velocity - tangential boundary conditions in vector Poisson and Stokes problems, when using Raviart-Thomas elements [@raviartthomas1977]. Since Raviart-Thomas elements are the most popular $H(\mathrm{div,\Omega})$ conforming elements, we compare our method to these results. Vector Poisson problems ----------------------- shows the result of the vector Poisson problem on $\Omega=[0,1]^2$ with coordinates $\mathbf{x}:=(x,y)$, for a 1-form $u\in H\Lambda^1(\Omega)$, where $\Gamma=\Gamma_2$, i.e. with tangential velocity - divergence-free boundary conditions ($\tr\star u=0,\ \tr\star\ud u=0$). The corresponding solution is given by $$\begin{aligned} u^{(1)}&=-v(\mathbf{x})\,\ud x+u(\mathbf{x})\,\ud y\nonumber\\ &=-(2\sin\pi x\cos\pi y)\,\ud x+ (\cos\pi x\sin\pi y)\,\ud y.\end{aligned}$$ Both Raviart-Thomas and mimetic spectral element methods show optimal convergence rates. All results of this and the following two problems where obtained on the same quadrilateral mesh of $2^n\times 2^n$ subsquares, $n=1,2,3,4,\hdots$ just like the reference solutions from [@arnold2011]. ![Comparison of the $h$-convergence between Raviart-Thomas and Mimetic spectral elements for the 2D 1-form Poisson problem with tangential velocity - divergence-free boundary conditions.[]{data-label="fig:afgcase1"}](afgcase1.pdf){width=".8\textwidth"} shows again results for the vector Poisson problem for a 1-form, but now in combination with normal velocity - tangential velocity boundary conditions ($\tr u=0,\ \tr\star u=0$), so $\Gamma=\Gamma_1$. The corresponding manufactured solution is $$\begin{aligned} u^{(1)}&=-v(\mathbf{x})\,\ud x+u(\mathbf{x})\,\ud y\nonumber\\ &=-(\sin\pi x\sin\pi y)\,\ud x+ (\sin\pi x\sin\pi y)\,\ud y.\end{aligned}$$ The compatible spectral discretization method again shows optimal convergence, as was expected from the above analysis. The Raviart-Thomas elements only show suboptimal convergence in case of velocity boundary conditions. This suboptimality was proven in [@arnold2011]. Especially for $\omega_h$ and $\ud\omega_h$ the current method outperforms the Raviart-Thomas elements, with a difference in rate of convergence of $\tfrac{3}{2}$. ![Comparison of the $h$-convergence between Raviart-Thomas and Mimetic spectral elements for the 2D 1-form Poisson problem with tangential velocity - normal velocity boundary conditions.[]{data-label="fig:afgcase2"}](afgcase2.pdf){width=".8\textwidth"} Stokes problems --------------- The same difference in convergence behavior is found for the Stokes problem, where $\Gamma=\Gamma_1$, i.e. with normal velocity - tangential velocity boundary conditions, see . Again $\Omega$ is the unit square, and the velocity and pressure fields are given by $$\begin{aligned} u^{(1)}&=-v(\mathbf{x})\,\ud x+u(\mathbf{x})\,\ud y\nonumber\\ &=-\left(2y^2(y-1)^2x(2x-1)(x-1)\right)\,\ud x+ \left(-2x^2(x-1)^2y(2y-1)(y-1)\right)\,\ud y,\\ p^{(2)}&=p(\mathbf{x})\,\ud x\wedge\ud y=\left((x-\tfrac{1}{2})^5+(y-\tfrac{1}{2})^5\right)\,\ud x\wedge\ud y.\end{aligned}$$ While for velocity both methods show optimal convergence, for pressure a difference of $\tfrac{1}{2}$ is noticed in the rate of convergence and for vorticity and the curl of vorticity again a difference in rate of convergence of $\tfrac{3}{2}$ is revealed. ![Comparison of the $h$-convergence between Raviart-Thomas and Mimetic spectral element projections for the 2D Stokes problem with normal velocity - tangential velocity boundary conditions.[]{data-label="fig:afgcase3"}](afgcase3.pdf){width=".8\textwidth"} The error in divergence of velocity is not shown here for the Stokes problem, because the method is pointwise divergence-free up to machine precision. Special attention to this property is given in [@kreeft2012]. We would like to remark is that the results shown in are independent of the kind of boundary conditions used. Table \[tab:bcresults\] shows the results of vorticity for all types of admissible boundary conditions. --------------------- --------------------- ----------------- --------------- ------------- normal velocity tangential velocity vorticity   vorticity   convergence tangential velocity pressure normal velocity pressure rate 1.0280e-04 1.0109e-04 1.0030e-04 1.0035e-04 3.14 1.2445e-05 1.2410e-05 1.2364e-05 1.2375e-05 3.05 1.5424e-06 1.5426e-06 1.5399e-06 1.5416e-06 3.01 1.9238e-07 1.9247e-07 1.9230e-07 1.9255e-07 3.00 2.4035e-08 2.4042e-08 2.4032e-08 2.4065e-08 3.00 --------------------- --------------------- ----------------- --------------- ------------- : This table shows the vorticity error $\norm{\omega-\omega_h}_{L^2\Lambda^{0}}$ obtained using the four types of boundary conditions given in . The results are obtained on an uniform Cartesian mesh with $N=2$ and $h=\tfrac{1}{8},\tfrac{1}{16},\tfrac{1}{32},\tfrac{1}{64},\tfrac{1}{128}$. All four cases show third order convergence.[]{data-label="tab:bcresults"} The next testcase reveals the optimal convergence in case of higher-order approximation on curvilinear quadrilateral meshes for all admissible types of boundary conditions. The manufactured solution Stokes problem is given on a curvilinear domain, defined by the mapping $(x,y)=\Phi(\xi,\eta)$, $$\begin{aligned} x(\xi,\eta)&=\tfrac{1}{2}+\tfrac{1}{2}\left(\xi+\tfrac{1}{10}\cos(2\pi\xi)\sin(2\pi\eta)\right),\\ y(\xi,\eta)&=\tfrac{1}{2}+\tfrac{1}{2}\left(\eta+\tfrac{1}{10}\sin(2\pi\xi)\cos(2\pi\eta)\right).\end{aligned}$$ A $6\times6$ element $N=6$ mesh is show in . Each side of the domain has a different type of boundary condition, so $\Gamma=\Gamma_1\cup\Gamma_2\cup\Gamma_3\cup\Gamma_4$, as shown in the same figure and listed in . The solutions of vorticity $\omega\in\Lambda^0(\Omega)$, velocity $u\in\Lambda^1(\Omega)$ and pressure $p\in\Lambda^2(\Omega)$ are given by \[testcase4\] $$\begin{aligned} \omega^{(0)}&=\tfrac{3}{2}\pi\sin(\tfrac{3}{2}\pi x)\sin(\tfrac{3}{2}\pi y),\\ u^{(1)}&=-\left(\cos(\tfrac{3}{2}\pi x)\sin(\tfrac{3}{2}\pi y)\right)\,\ud x+\left(2\sin(\tfrac{3}{2}\pi x)\cos(\tfrac{3}{2}\pi y)\right)\,\ud y,\\ p^{(2)}&=\left(\sin(\pi x)\sin(\pi y)\right)\,\ud x\wedge\ud y.\end{aligned}$$ They lead to nonzero body force $f\in\Lambda^1(\Omega)$ and mass source $g\in\Lambda^2(\Omega)$. shows the convergence of the vorticity $\omega_h\in\Lambda_h^{0}(\Omega;C_0)$, velocity $u_h\in\Lambda^{1}_h(\Omega;C_1)$ and pressure $p_h\in\Lambda^2_h(\Omega;C_2)$. The errors for the vorticity and velocity are measured in the $H\Lambda^k$-norm, i.e. $\Vert\omega-\omega_h\Vert_{H\Lambda^{0}}$, and $\Vert u-u_h\Vert_{H\Lambda^{1}}$, respectively, and the error of the pressure is given in the $L^2\Lambda^2$-norm. In convergence rates are added which show the *optimal* $h$-convergence behavior of the Stokes problem on a curvilinear domain with curvilinear grid and all four types of boundary conditions. ![Upper left figure show the computational domain with a $6\times 6$ element mesh of $N=6$. Furthermore the velocity, vorticity and pressure $h$-convergence results are shown of Stokes problem . All variables are tested on grids with $N=2,4,6$ and 8.[]{data-label="fig:allinonestokes"}](stokesplots.pdf){width="80.00000%"} Concluding remark ================= Optimal approximation of the Stokes problem for all admissible boundary conditions essentially hinges on the construction of a conforming discrete Hodge decomposition, $\Lambda^k_h=\mathcal{Z}^k_h\oplus\mathcal{Z}^{k,\perp}_h$ and a discrete Poicaré inequality, that are based on the bijection of the exterior derivative on the conforming subspace, $\ud:\mathcal{Z}_h^{k,\perp}\rightarrow\mathcal{B}_h^{k+1}$. Ensuring these properties result in a compatible discretization method, and relied on the construction of a bounded projection operator, $\pi_h:\Lambda^k(\Omega)\rightarrow\Lambda_h^k(\Omega;C_k)$, that commutes with the exterior derivative, $\pi_h\ud=\reconstruction\delta\reduction=\ud\pi_h$. So the compatibility is based on the bijection of the coboundary operator, $\delta:Z^{k,\perp}\rightarrow B^{k+1}$, and the construction of interpolatory basis functions. From this it follows that, $\mathcal{B}_h^{k+1}\subset\mathcal{B}^{k+1}$, $\mathcal{Z}_h^k\subset\mathcal{Z}^k$ and $\mathcal{Z}_h^{k,\perp}\subset\mathcal{Z}^{k,\perp}$. From these properties the rest follows. For piecewise sufficiently smooth mappings, the optimal conference rates hold on curvilinear grids as well, since the pullback operator of the map from a curvilinear domain to the Cartesian frame commutes with the projection operator. Any projection (discretization) with these properties will yield similar results as described in this paper. Acknowledgments {#acknowledgments .unnumbered} =============== We would like to thank Pavel Bochev for the fruitful discussions on mimetic schemes, boundary conditions and error estimates. [^1]: Jasper Kreeft is funded by STW Grant 10113 [^2]: This paper is in final form and no version of it will be submitted for publication elsewhere. [^3]: Although ‘perpendicular’ in a topological space is not well defined, we refer to $Z^\perp_k$ as the complement space of $Z_k$ in $C_k$.

--- author: - 'A.A. Kaas' - 'G. Olofsson' - 'S. Bontemps' - 'P. André' - 'L. Nordh' - 'M. Huldtgren' - 'T. Prusti' - 'P. Persi' - 'A.J. Delgado' - 'F. Motte' - 'A. Abergel' - 'F. Boulanger' - 'M. Burgdorf' - 'M.M. Casali' - 'C.J. Cesarsky' - 'J. Davies' - 'E. Falgarone' - 'T. Montmerle' - 'M. Perault' - 'J.L. Puget' - 'F. Sibille' date: 'Received date; Accepted date' title: | The young stellar population in the Serpens Cloud Core:\ An ISOCAM survey[^1] [^2] --- Introduction {#intro} ============ The youngest stellar clusters are found deeply embedded in the molecular clouds from which they form. There are several reasons why very young clusters are particularly interesting for statistical studies such as mass functions and spatial distributions. Because mass segregation and loss of low mass members due to dynamical evolution has not had time to develop significantly for ages $\la 10^8$ yrs [@sca98], the stellar IMF can in principle be found for the complete sample, at least for sufficiently rich clusters. For ages $\la 10^5$ yrs the spatial distribution should in gross reflect the distribution at birth, which gives important input to the studies of cloud fragmentation and cluster formation. Only in the youngest regions of low mass star formation do we find the co-existence of newly born stars and pre-stellar clumps, which allows one to compare the mass functions of the different evolutionary stages. Low mass stars are more luminous when they are young, being either in their protostellar phase or contracting down the Hayashi track, which permits probing lower limiting masses. Severe cloud extinction, however, requires sensitive IR mapping at high spatial resolution to sample the stellar population of embedded clusters. ISOCAM, the camera aboard the ISO satellite [@kes96], provided sensitivity and relatively high spatial resolution in the mid-IR [@ces96]. The two broad band filters LW2 (5-8.5 $\mu$m) and LW3 (12-18 $\mu$m), designed to avoid the silicate features at 10 and 20 $\mu$m, were selected to sample the mid-IR Spectral Energy Distribution (SED) of Young Stellar Objects (YSOs) in different evolutionary phases. According to the current empirical picture for the early evolution of low mass stars [@ada87; @lad87; @and93; @and94], newborn YSOs can be observationally classified into 4 main evolutionary classes. Class0 objects are in the deeply embedded main accretion phase ($\ga$ 10$^4$ yrs), and have measured circumstellar envelope masses larger than their estimated central stellar masses, with overall SEDs resembling cold blackbodies and peaking in the far-IR. ClassI sources ($\sim 10^5$ yrs) are observationally characterised by a broad SED with a rising spectral index[^3] towards longer wavelengths ($\alpha_{\rm IR} > 0$) in the mid-IR. The ClassII sources spend some 10$^6$ yrs in a phase where most of the circumstellar matter is distributed in an optically thick disk, displaying broad SEDs with $-1.6 < \alpha_{\rm IR} < 0$. At $\alpha_{\rm IR} \approx -1.6$ the disk turns optically thin, and the sources evolve into the ($\sim 10^7$ yrs) ClassIII stage where the mid-IR imprints of a disk eventually disappear. A normal stellar photosphere has $\alpha_{\rm IR} = -3$. Thus, while Class0 objects are not favourably traced by mid-IR photometry, they are expected to be rare. At the other extreme, ClassIII sources cannot generally be distinguished using mid-IR photometry since most of them have SEDs similar to normal stellar photospheres. But mid-IR photometry from two broad bands, as obtained in this study with ISOCAM, is highly efficient when it comes to detection and classification of ClassI and ClassII sources. Thus, considering the fact that these latter objects constitute the major fraction of the youngest YSOs, the ISOCAM surveys provide a better defined sample for statistical studies than e.g. near-IR surveys for regions with very recent star formation [see @pru99]. This paper presents the results from an ISOCAM survey of $\sim $ 0.13 square degrees around the Serpens Cloud Core in two broad bands centred at 6.7 and 14.3 $\mu$m. This cloud, located at $b^{\rm II} = 5^{\circ}$ and $l^{\rm II} = 32^{\circ}$ at a distance of $259 \pm 37$ pc [@str96; @fes98], comprises a deeply embedded, very young cluster with large and spatially inhomogeneous cloud extinction, exceeding 50 magnitudes of visual extinction. Only a few sources are detected in the optical [@har85; @gom88; @gio98]. Serpens contains one of the richest known collection of Class0 objects [@cas93; @hur96; @wol98; @dav99], an indication that this cluster is young and active. On-going star formation is also evident from the presence of several molecular outflows [@bal83; @whi95; @hua97; @her97; @dav99], pre-stellar condensations seen as sub-mm sources [@cas93; @mcm94; @tes98; @wil00], a far-IR source (FIRS1) possibly associated with a non-thermal triple radio continuum source [@rod89; @eir89; @cur93], a FUor-like object [@hod96], and jets and knots in the 2.1 $\mu$m H$_2$ line [@eir97; @her97]. Investigations of the stellar content have been made with near-IR surveys [@str76; @chu86; @eir92; @sog97; @gio98; @kaa99a], identifying YSOs using different criteria, such as e.g. near-IR excesses, association with nebulosities, and variability. In this paper we identify new cluster members, characterize the YSOs into ClassI, flat-spectrum, and ClassII sources, estimate a stellar luminosity function for the ClassII sample and search for a compatible IMF and age, and finally describe the spatial distribution of both the protostars and the pre-main sequence population in this cluster. $$\begin{tabular}{lcccrrcrrrrrr} \hline \noalign{\smallskip} & $\alpha$(2000) & $\delta$(2000) & Size$^1$ & T$_{\rm int}$ & pfov & $<$n$_{\rm ro}>$ & N$_{\rm det}$ & N$_{\rm 6.7}$ & N$_{\rm 14.3}$ & N$_{\rm both}$ & 1$\sigma_{6.7}$ & 1$\sigma_{14.3}$ \\ & & & (\arcmin $\times$ \arcmin) & (sec) & (\arcsec) & & & & & & (mJy) & (mJy) \\ (1) & (2) & (3) & (4) & (5) & (6) & (7) & (8) & (9) & (10) & (11) & (12) & (13) \\ \hline CE & 18 29 48.3 & 01 16 04.5 &13$\times$13 & 0.28& 3/6 &4$\times$13& 133 & 113 & 56 & 51 & 2 &3 \\ CW & 18 29 06.3 & 01 16 01.4 &13$\times$13 & 2.1 & 6 &7 & 152 & 139 & 43 & 38 &0.8&2 \\ CS & 18 29 52.5 & 01 02 37.9 &13$\times$16.5& 0.28& 6 &4$\times$13& 165 & 152 & 41 & 34 &1.2&4 \\ D1 & 18 29 48.7 & 01 15 20.5 &1.8$\times$4.6& 2.1 & 3 &44 & 20 & 15 & 9 & 9 & 1 &2 \\ D2 & 18 29 52.3 & 01 15 20.8 &1.8$\times$4.6& 2.1 & 3 &44 & 24 & 19 & 11 & 10 & 1 &4.5 \\ D3 & 18 29 57.8 & 01 12 55.4 &4.6$\times$1.8& 2.1 & 3 &44 & 30 & 21 & 15 & 13 & 1 &3 \\ Tot$^2$ & & & & & & & 421 & 392 &140 &124 & & \\ \noalign{\smallskip} \hline \end{tabular}$$ $^1$ Approximate size since each raster has tagged edges.\ $^2$ The total number is corrected for multiply observed sources in overlapped regions.\ Observations and reductions {#obs} =========================== ISOCAM ------ This work presents data from the two ISOCAM star formation surveys LNORDH.SURVEY\_1 and GOLOFSSO.D\_SURMC. These were surveys within the ISO central programme mapping selected parts of the major nearby star formation regions in the two broad band filters LW2 (5-8.5 $\mu$m) and LW3 (12-18 $\mu$m). Other results on young stellar populations based on these surveys comprise the Chamaeleon I, II and III regions [@nor96; @olo98; @per00], the $\rho$ Ophiuchi star formation region [@bon01], the R Corona Australis region [@olo99], and the L1551 Taurus region [@gaa04], all observations performed basically in the same way. Data reduction methods were generally the same for all regions, using a combination of the CIA[^4] (CAM Interactive Analysis) package and own dedicated software. For overviews of survey results, see @nor98, @kaa00, and @olo00. In the Serpens Core region about 0.13 square degrees (deg$2$) were covered at 6.7 and 14.3 $\mu$m in 3 separate, but overlapping main rasters named CE, CW and CS (see Fig. \[figmap\]). In addition, 3 smaller fields within CE (named D1, D2, and D3) were observed at a higher sensitivity. CE covers the well known Serpens Cloud Core, CW is a reference region to the west of the Core which is substantially opaque in the optical but without appreciable 60 $\mu$m IRAS emission [@zha88a], and CS is a region directly to the south of the Cloud Core which has a peak in the 60 $\mu$m flux. The rasters were always made along the right ascension, with about half a frame (90) overlap in $\alpha$ and 24$\arcsec$ overlap in $\delta$. For the larger rasters the pixel field of view (PFOV) was set to the nominal survey value of 6$\arcsec$ except for the CE region in LW2, where a PFOV of 3$\arcsec$ was selected because of the risk of otherwise saturating the detector. Also in order to avoid saturation, the intrinsic integration time was set to $T_{\rm int} = 0.28$ s except for field CW where the nominal survey value of $T_{\rm int} = 2.1$ s was used. Each position in the sky was observed during about 15 s. For the deeper imaging within CE the overlap was 72$\arcsec$ in $\alpha$ and $\delta$, and each position was observed during about 92 s. The individual integration time was $T_{\rm int} = 2.1$ s and a PFOV of 3$\arcsec$ was used for better spatial resolution. See Table \[tbl-1\] for an overview. ### Image reduction Each raster consists of a cube of frames which is reduced individually, and in total 9112 individual frames were analysed. The dark current was subtracted using the CAL-G dark from the ISOCAM calibration library and, if necessary, further improved by a second order dark correction using a FFT thresholding method [@sta96a]. Cosmic ray hits were detected and masked by the multiresolution median transform (MMT) method [@sta96b]. The transients in the time history of each pixel due to the slow response of the LW detector were treated with the IAS inversion method v.1.0 [@abe96a; @abe96b]. Flat field images were constructed from the observations themselves. See @sta99 for a general description of ISOCAM data processing. ### Point source detection and photometry {#obs-phot} Bright point sources, even well below the saturation level, produce strong memory effects which are not entirely taken out by the transient correction. Due to the large PFOV which undersamples the point spread function (PSF), also remnants of cosmic ray glitches may be mistaken for faint sources. By looking at the time history of the candidate source fluxes, however, it is easy to distinguish real sources from memory effects, glitches or noise. This is done for each individual sky coverage, the redundancy being 2-6. Both the source detection and the photometry was made with the interactive software developed by our team. See also previous papers on this survey [@nor96; @olo99; @per00; @bon01]. The fluxes of each verified source were measured by aperture photometry, with nominal aperture radii 1.5 and 3 pixels, for the 6$\arcsec$ and 3$\arcsec$ PFOV, respectively, and aperture correction applied using the empirical PSF. For each redundant observation (2-6 overlaps) the flux is the median flux from the $N_{\rm ro}$ readouts per sky position. The sky level was estimated from the median image to reduce the noise. Uncertainties are estimated both from the standard deviation around this median (i.e. the temporal noise and also a measure of the efficiency of the transient correction method) and from the $\sigma$ of the sky background (i.e. the spatial noise). The quoted flux uncertainties are these two contributions added in quadrature. The photometric scatter between the overlaps is estimated and deviating measurements are discarded. If a source is affected by the dead column (no 24 was disconnected) or is hit by a serious glitch or memory effect in one of the redundant overlaps, this one is skipped, and the remaining ones are used to estimate the flux. If the redundancy is not sufficient (i.e. along raster edges), then the fluxes are flagged if affected by the dead column, detector edges, glitches, memory effects, close neighbours etc. Source positions were calculated from that of the redundant images where the source is closest to the centre, in order to diminish the effects of field distortion. The source centre is taken to be the peak pixel, and the default ISO pointing was used as a first source position value. The ISO position was then compared to near-IR positions of known sources in the literature [@eir92; @sog97; @gio98], and ISO positions of bright optical sources to the digital sky survey. Bulk offsets found for each raster registration were then corrected. The maximum bulk offset found was 5$\arcsec$, and we estimated an average uncertainty of $\pm 3\arcsec$ in RA and DEC. When 2MASS became available we checked the positions of 61 sources in Table \[tbl-2\] which have 2MASS counterparts and are not multiples unresolved by ISOCAM. The median deviation between ISOCAM and 2MASS positions is $2.2\arcsec$. One star deviates by as much as $9.2\arcsec$ (ISO-356), and 5 more sources by more than $4.5\arcsec$ (these are ISO-29, 207, 272, 357 and 367). ### Photometric Calibration The two broad band filters LW2 and LW3 have defined reference wavelengths at 6.7 and 14.3 $\mu$m, respectively. The fluxes in ADU/s are converted to mJy through the relations 2.32 and 1.96 ADU/gain/s/mJy for the bands LW2 and LW3, respectively (from in-orbit latest calibration by @blo00). These flux conversions are strictly valid only for sources with F$_{\nu} \propto \nu^{-1}$, however, and therefore a small colour correction must be applied to the blue sources (cf. Sect. \[excess\] for blue vs. red). This correction is obtained by dividing the above conversion factors by 1.05 and 1.02 for LW2 and LW3, respectively, yielding the effective factors 2.21 and 1.92 ADU/gain/s/mJy for blue sources. Conversion from flux density to magnitude is defined as m$_{\rm 6.7} = -2.5 \log (F_{\nu}(6.7 \mu m)/82.8)$ and m$_{\rm 14.3 \mu m} = -2.5 \log (F_{\nu}(14.3)/18.9)$, where $F_{\nu}$ is given in Jy. The $\sim 5$ % responsivity decrease throughout orbit has not been corrected for. Nordic Optical Telescope near-IR imaging ---------------------------------------- A $6\arcmin \times 8\arcmin$ region inside Serp-CE, the area which is usually referred to as the Serpens Cloud Core and has been covered to a smaller or larger extent by several studies in the near-IR [@eir92; @sog97; @gio98; @kaa99a], was mapped deeply in $J$ (1.25 $\mu$m), $H$ (1.65 $\mu$m) and $K$ (2.2 $\mu$m) in August 1996, only 4 months after the ISOCAM observations. See Fig. \[figmap\] for the location of the different maps. Also, a region to the NW of the $JHK$ field has been mapped in the $K$ band, see Fig. \[fig-nw\], but with a total coadded integration time from only 30 sec to 1 min. These observations were made with the ARcetri Near Infrared CAmera (Arnica) at the 2.56m Nordic Optical Telescope, La Palma. See @kaa99a for details on this near-IR dataset. IRAM 30m Telescope Observations ------------------------------- A 1.3 mm dust continuum mosaic of the Serpens main cloud core was taken with the IRAM 30-m telescope equipped with the MPIfR 37-channel bolometer array MAMBO-I [@kre98] during four night observing sessions in March 1998. The passband of the MAMBO bolometer array has an equivalent width $\approx 70$ GHz and is centered at $\nu_{eff} \approx 240$ GHz. The $\sim 19' \times 6'$ mosaic consists of eleven individual on-the-fly maps which were obtained in the dual-beam raster mode with a scanning velocity of 8$\arcsec$/sec and a spatial sampling of 4$\arcsec$ in elevation. In this mode, the telescope is scanning continuously in azimuth along each mapped row while the secondary mirror is wobbling in azimuth at frequency of 2 Hz. A wobbler throw of 45$\arcsec$ or 60$\arcsec$ was used. The typical azimuthal size of individual maps was 4$\arcmin$. The size of the main beam was measured to be $\sim$ 11$\arcsec$ (HPBW) on Uranus and other strong point-like sources such as quasars. The pointing of the telescope was checked every $\sim 1$ hr using the VLA position of the strong, compact Class 0 source FIRS1 (good to $\sim 0.1\arcsec $ – @cur93); it was found to be accurate to better than $\sim $ 3$\arcsec$. The zenith atmospheric optical depth, monitored by ‘skydips’ every $\sim 2$ hr, was between $\sim 0.2$ and $\sim 0.4$. Calibration was achieved through on-the-fly mapping and on-off observations of the primary calibrator Uranus [e.g. @gri93 and references therein]. In addition, the Serpens secondary calibrator FIRS1, which has a 1.3 mm peak flux density $\sim 2.4$ Jy in an 11$\arcsec$ beam was observed before and after each map. The relative calibration was found to be good to within $\sim 10\%$ by comparing the individual coverages of each field, while the overall absolute calibration uncertainty is estimated to be $\sim 20\%$. The dual-beam maps were reduced and combined with the IRAM software for bolometer-array data (“NIC”; cf. @bro95) which uses the EKH restoration algorithm [@eme79]. ISOCAM results {#results} ============== Source statistics, sensitivity and completeness {#stat} ----------------------------------------------- Table \[tbl-1\] gives an overview of the observational parameters and the results of the point source photometry for each of the six rasters named in col. 1. Columns 2, 3 and 4 give the centre position and size of the raster fields. Columns 5, 6 and 7 give the unit integration times, the PFOV and the average number of readouts per sky position. Column 8 gives the total number of source detections. Columns 9 and 10 give the number of sources for which photometry was obtained at 6.7 and 14.3 $\mu$m, respectively. Column 11 gives the number of sources with flux determinations in both of the photometric bands. Columns 12 and 13 give the measured 1$\sigma$ photometric limits which are about 4 and 6 times the read-out-noise for the large and the deep fields, respectively. Practically all sources detected at 14.3 $\mu$m are also detected at 6.7 $\mu$m, but there are 15 cases of 14.3 $\mu$m detections without 6.7 $\mu$m counterparts. On the average, only 30% of the 6.7 $\mu$m detections are also detected at 14.3 $\mu$m, but this value is clearly larger for the CE field. The sensitivity, which depends on the PFOV and T$_{\rm int}$ as well as the amount of nebulosity and bright point sources, is lowered by memory effects from bright sources in the CE field and by the presence of a nebula in the CS field. For the deep fields (D1, D2, D3) the main limiting factor in terms of sensitivity is the memory effect and the fact that the flat fields are based on fewer frames. The deep fields within the CE area are independently repeated observations (observed within the same satellite orbit), and therefore permit a check of the photometric repeatability for a number of sources. The median values of the individual direct scatter between the measured fluxes were found to be: about 30% for the 6.7 $\mu$m band measured from 28 sources between 4 mJy and 2 Jy, and about 15% for the 14.3 $\mu$m band, measured from 14 sources between 30 mJy and 5.7 Jy (note different pixel scale). This corresponds to a median error of 0.13 dex in the $[14.3/6.7]$ colour index, which is in good agreement with the scatter (of sources without mid-IR excesses) around the value expected for normal photospheres (see Fig. \[fig-323\]) at a flux density of 10 mJy in the 14.3 $\mu$m band. To estimate how many field stars to expect at a given sensitivity within the observed fields, the stellar content in a cone along the line-of-sight was integrated in steps of 0.02 kpc out to a distance of 20 kpc. The Galaxy was represented by an exponential disk, a bulge, a halo and a molecular ring, following @wai92. Absolute magnitudes at 2.2 $\mu$m (M$_{\rm K}$) and in the 12 $\mu$m IRAS band (M$_{\rm 12}$), local number densities and scale heights of the different stellar populations, as well as their contribution to the different components of the Galaxy were provided by @wai92 in their model of the mid-IR point source sky. Absolute magnitudes at 7 $\mu$m, M$_{\rm 7}$, are estimated by linear interpolation between M$_K$ and M$_{\rm 12}$. The upper panels of Fig. \[fig1\] show the histograms of the 6.7 $\mu$m and 14.3 $\mu$m sources in the CW field, which is free of nebulosity and practically free of IR excess sources. The bin size of 0.2 in $\log F_{\nu}$ corresponds to 0.5 magnitudes. Inserted are the model counts of galactic sources at 7 and 12 $\mu$m, scaled to the CW field size. The expected source number per bin is calculated assuming no cloud extinction (solid line), an average extinction of 0.2 magnitudes in the 6.7 $\mu$m band (dotted line), and A$_{\rm 6.7}$ = 0.4 magnitudes (dashed line), corresponding to roughly A$_V \sim 5$ and A$_V \sim 10$, respectively (cf. Sects. \[nir\] and \[char\] about extinction). For the CW field an average cloud extinction of A$_V \sim 7-8$ magnitudes is in agreement with the location of the majority of the sources in a DENIS $I-J/J-K'$ diagram, as well as an extinction map based on R star counts [@cam99]. Comparing the number of observed sources at 14.3 $\mu$m with the model expectation at 12 $\mu$m indicates completeness at $\sim$ 6 mJy or m$_{\rm 14.3}$ = 8.7 mag. The observations at 6.7 $\mu$m are estimated to be complete to $\sim $ 5 mJy or m$_{\rm 6.7}$ = 10.6 mag. The lower panels of Fig. \[fig1\] show the histograms for the total sample. The boldface steps show the contribution of the mid-IR excess sources. The dashed line represents the model expectation assuming an average cloud extinction of about 10 magnitudes of visual extinction, i.e. A$_{\rm 6.7}$ = 0.41 mag and A$_{\rm 14.3}$ = 0.36 mag (cf. Sects. \[nir\] and \[char\]). Thus, the whole sample taken together, and allowing for a subtraction of the mid-IR excess sources, indicates an overall completeness at $\sim$ 6 mJy for the 6.7 $\mu$m band and at 8 mJy for the 14.3 $\mu$m band (i.e. at m$_{\rm 6.7}$ = 10.4 mag and m$_{\rm 14.3}$ = 8.5 mag). Sources with mid-IR excesses {#excess} ---------------------------- For the 124 sources with fluxes in both bands we present a colour magnitude diagram in Fig. \[fig-323\]. The colour index $[14.3/6.7]$, defined as $ \log (F_{\nu}^{14.3}/F_{\nu}^{6.7})$, is plotted against $F_{\nu}^{14.3}$. This colour index can be converted to the commonly used index of the Spectral Energy Distribution (SED): $$\alpha_{\rm IR}^{7-14} = \frac{\log (\lambda_{14} F_{\lambda_{14}}) - \log (\lambda_7 F_{\lambda_7})} {\log \lambda_{14} - \log \lambda_7},$$ calculated between 6.7 and 14.3 $\mu$m and indicated on the right hand y-axis. The completeness limit is given with solid lines as the combined effect of the completeness in each of the two bands. The sources tend to separate into two distinct groups, with some few intermediate objects. The “red” sources (circles) are interpreted as pre-main-sequence (PMS) stars surrounded by circumstellar dust. This sample is not believed to be contaminated with galaxies. At the completeness flux level of 8 mJy at 14.3 $\mu$m the expected extracalagtic contamination is about half a source within our map coverage and at the level of 3 mJy (the faintest source in our sample at 14.3 $\mu$m) the contamination is below two sources within the mapped region [@hon03]. No correction for extinction has been made on these numbers, so that they should be considered as upper limits to the extragalactic contamination. We have decided to set the division line between mid-IR excess objects and “blue” objects (triangles) at $[14.3/6.7] = -0.2$ or $\alpha_{\rm IR}^{7-14} = -1.6$ (dotted line), which corresponds to the classical border between Class II and Class III objects (cf Sect \[char\]). $$\begin{tabular}{rccrrrrrrrrrrl} \hline \noalign{\smallskip} {ISO} & $\alpha_{2000}$ & $\delta_{2000}$ & {$J$} & {$\sigma_{J}$} & {$H$} & $\sigma_{H}$ & {$K$} & $\sigma_{K}$ & $F_{\nu}^{6.7}$ & $\sigma_{6.7}$ & $F_{\nu}^{14.3}$ & $\sigma_{14.3}$ & {Other ID} \\ \# & 18$^h$ & & mag & & mag & & mag & & mJy & mJy & mJy & mJy & \\ \hline {\null}29{&}28$^m$52$^s_.$5{&}1$^{\circ}$12$\arcmin$47$\arcsec${&}{&}{&}{&}{&}{&}{&}1$^g${&}0.6{&}3{&}2{&}{\\}{\null}150{&}29$^m$30$^s_.$6{&}1$^{\circ}$01$\arcmin$09$\arcsec${&}{&}{&}{&}{&}{&}{&}19{&}3{&}21{&}5{&}{\\}{\null}158{&}29$^m$31$^s_.$7{&}1$^{\circ}$08$\arcmin$22$\arcsec${&}{&}{&}{&}{&}{&}{&}10{&}1{&}16$^m${&}7{&}BD+01 3687{\\}{\null}159{&}29$^m$32$^s_.$0{&}1$^{\circ}$18$\arcmin$42$\arcsec${&}{&}{&}{&}{&}8.41{&}$<$0.01{&}1162{&}28{&}2351{&}26{&}IRAS 18269+0116{\\}{\null}160{&}29$^m$32$^s_.$0{&}1$^{\circ}$18$\arcmin$34$\arcsec${&}{&}{&}{&}{&}9.82{&}$<$0.01{&}114{&}2{&}-{&}-{&}IRAS 18269+0116{\\}{\null}173{&}29$^m$33$^s_.$4{&}1$^{\circ}$08$\arcmin$28$\arcsec${&}{&}{&}{&}{&}{&}{&}64{&}2{&}110{&}20{&}[CDF88] 6{\\}{\null}202{&}29$^m$39$^s_.$9{&}1$^{\circ}$17$\arcmin$55$\arcsec${&}{&}{&}{&}{&}11.80{&}$<$0.01{&}8{&}2{&}12$^c${&}3{&}{\\}{\null}207{&}29$^m$41$^s_.$1{&}1$^{\circ}$07$\arcmin$40$\arcsec${&}{&}{&}{&}{&}{&}{&}56{&}1{&}68{&}3{&}STGM3{\\}{\null}216{&}29$^m$42$^s_.$3{&}1$^{\circ}$20$\arcmin$19$\arcsec${&}{&}{&}{&}{&}12.02{&}$<$0.01{&}6{&}3{&}14$^c${&}2{&}{\\}{\null}219{&}29$^m$43$^s_.$7{&}1$^{\circ}$07$\arcmin$22$\arcsec${&}{&}{&}{&}{&}{&}{&}7{&}2{&}-{&}-{&}STGM2{\\}{\null}221{&}29$^m$44$^s_.$3{&}1$^{\circ}$04$\arcmin$55$\arcsec${&}{&}{&}{&}{&}{&}{&}2652{&}37{&}3664{&}23{&}IRAS 18271+0102{\\}{\null}224{&}29$^m$44$^s_.$8{&}1$^{\circ}$13$\arcmin$10$\arcsec${&}13.22{&}$<$0.01{&}12.26 {&}$<$0.01{&}11.82{&}$<$0.01{&}7{&}2{&}-{&}-{&}EC11{\\}{\null}226{&}29$^m$44$^s_.$8{&}1$^{\circ}$15$\arcmin$44$\arcsec${&}-{&}-{&}15.45 {&}0.02{&}13.51{&}$<$0.01{&}3{&}1{&}-{&}-{&}EC13{\\}{\null}231{&}29$^m$46$^s_.$0{&}1$^{\circ}$16$\arcmin$23$\arcsec${&}18.12{&}0.09{&}13.86 {&}$<$0.01{&}11.81{&}$<$0.01{&}11{&}4{&}-{&}-{&}EC21{\\}{\null}232{&}29$^m$46$^s_.$3{&}1$^{\circ}$12$\arcmin$14$\arcsec${&}13.22{&}$<$0.01{&}11.66 {&}$<$0.01{&}10.99{&}$<$0.01{&}6{&}4{&}-{&}-{&}EC23{\\}{\null}234{&}29$^m$46$^s_.$9{&}1$^{\circ}$16$\arcmin$10$\arcsec${&}-{&}-{&}18.73 {&}0.14{&}14.47{&}0.02{&}7{&}3{&}-{&}-{&}EC26{\\}{\null}237{&}29$^m$47$^s_.$2{&}1$^{\circ}$16$\arcmin$26$\arcsec${&}-{&}-{&}16.51 {&}0.03{&}13.19{&}0.01{&}33{&}3{&}27{&}5{&}EC28{\\}{\null}241{&}29$^m$48$^s_.$1{&}1$^{\circ}$16$\arcmin$43$\arcsec${&}-{&}-{&}- {&}-{&}16.11{&}0.05{&}28{&}4{&}41{&}8{&}{\\}{\null}242{&}29$^m$48$^s_.$6{&}1$^{\circ}$13$\arcmin$42$\arcsec${&}18.14{&}0.10{&}14.94 {&}$<$0.01{&}13.14{&}0.01{&}5{&}2{&}4{&}1{&}K8,EC33{\\}{\null}249{&}29$^m$49$^s_.$1{&}1$^{\circ}$16$\arcmin$32$\arcsec${&}-{&}-{&}17.09 {&}0.05{&}13.62{&}0.01{&}143$^f${&}6{&}637$^f${&}9{&}EC37{\\}{\null}250{&}29$^m$49$^s_.$3{&}1$^{\circ}$16$\arcmin$19$\arcsec${&}-{&}-{&}- {&}-{&}11.67{&}$<$0.01{&}2242$^f${&}54{&}4464$^f${&}35{&}DEOS{\\}{\null}252{&}29$^m$49$^s_.$9{&}0$^{\circ}$56$\arcmin$12$\arcsec${&}{&}{&}{&}{&}{&}{&}21{&}1{&}31{&}3{&}{\\}{\null}253{&}29$^m$49$^s_.$6{&}1$^{\circ}$14$\arcmin$57$\arcsec${&}-{&}-{&}- {&}-{&}14.90{&}0.03{&}49{&}13{&}49{&}5{&}EC40{\\}{\null}254{&}29$^m$49$^s_.$5{&}1$^{\circ}$17$\arcmin$07$\arcsec${&}{&}{&}{&}{&}12.46{&}0.01{&}156{&}12{&}214{&}5{&}EC38{\\}{\null}258a{&}29$^m$49$^s_.$6{&}1$^{\circ}$15$\arcmin$28$\arcsec${&}-{&}-{&}17.77 {&}0.08{&}14.80{&}0.03{&}19$^f${&}2{&}24$^f${&}4{&}EC41,GCNM23{\\}{\null}258b{&}29$^m$50$^s_.$3{&}1$^{\circ}$15$\arcmin$21$\arcsec${&}-{&}-{&}-{&}-{&}-{&}-{&}-{&}-{&}13$^f${&}4{&}{\\}{\null}259{&}29$^m$50$^s_.$6{&}1$^{\circ}$01$\arcmin$35$\arcsec${&}{&}{&}{&}{&}{&}{&}29{&}13{&}-{&}-{&}{\\}{\null}260{&}29$^m$50$^s_.$5{&}1$^{\circ}$14$\arcmin$17$\arcsec${&}-{&}-{&}- {&}-{&}16.76{&}0.15{&}3{&}1{&}-{&}-{&}{\\}{\null}265{&}29$^m$51$^s_.$2{&}1$^{\circ}$16$\arcmin$42$\arcsec${&}15.35{&}0.04{&}12.90 {&}0.01{&}11.32{&}$<$0.01{&}766{&}13{&}813{&}16{&}EC53{\\}{\null}266{&}29$^m$51$^s_.$3{&}1$^{\circ}$13$\arcmin$17$\arcsec${&}15.90{&}0.02{&}14.27 {&}0.01{&}13.02{&}$<$0.01{&}2{&}1{&}-{&}-{&}GCNM35,EC51{\\}{\null}269{&}29$^m$52$^s_.$2{&}1$^{\circ}$13$\arcmin$21$\arcsec${&}-{&}-{&}14.79 {&}0.01{&}13.31{&}0.01{&}2{&}1{&}-{&}-{&}K16,EC56{\\}{\null}270{&}29$^m$52$^s_.$2{&}1$^{\circ}$15$\arcmin$49$\arcsec${&}-{&}-{&}- {&}-{&}17.70{&}0.20{&}9{&}2{&}44{&}8{&}{\\}{\null}272{&}29$^m$52$^s_.$5{&}1$^{\circ}$12$\arcmin$54$\arcsec${&}19.01{&}0.22{&}14.63 {&}0.01{&}12.54{&}$<$0.01{&}3{&}1{&}-{&}-{&}EC59{\\}{\null}276{&}29$^m$52$^s_.$9{&}1$^{\circ}$14$\arcmin$56$\arcsec${&}-{&}-{&}- {&}-{&}14.80{&}0.03{&}74{&}12{&}231{&}4{&}GCNM53{\\}{\null}277{&}29$^m$53$^s_.$2{&}1$^{\circ}$15$\arcmin$43$\arcsec${&}-{&}-{&}- {&}-{&}14.88{&}0.03{&}9{&}2{&}-{&}-{&}EC63{\\}{\null}279{&}29$^m$53$^s_.$4{&}1$^{\circ}$13$\arcmin$13$\arcsec${&}-{&}-{&}14.15 {&}0.01{&}13.60{&}0.01{&}1{&}2{&}-{&}-{&}STGM14,EC66{\\}{\null}283{&}29$^m$53$^s_.$6{&}1$^{\circ}$17$\arcmin$00$\arcsec${&}-{&}-{&}11.58$^a$ {&}$<$0.01{&}10.63$^a${&}$<$0.01{&}30{&}4{&}29{&}6{&}EC67{\\}{\null}285{&}29$^m$53$^s_.$9{&}1$^{\circ}$13$\arcmin$32$\arcsec${&}16.58{&}0.04{&}14.21 {&}0.01{&}12.87{&}$<$0.01{&}6$^d${&}1{&}6{&}3{&}GCNM63,EC68{\\}{\null}287{&}29$^m$54$^s_.$1{&}1$^{\circ}$07$\arcmin$14$\arcsec${&}{&}{&}{&}{&}{&}{&}3{&}1{&}4{&}3{&}{\\}{\null}289{&}29$^m$54$^s_.$4{&}1$^{\circ}$15$\arcmin$03$\arcsec${&}16.15{&}0.06{&}12.66 {&}$<$0.01{&}10.86{&}$<$0.01{&}19{&}9{&}-{&}-{&}EC69,CK10{\\}{\null}291{&}29$^m$54$^s_.$4{&}1$^{\circ}$14$\arcmin$44$\arcsec${&}14.26{&}0.01{&}13.32 {&}$<$0.01{&}12.74{&}$<$0.01{&}2{&}2{&}-{&}-{&}GCNM70,EC70{\\}{\null}294{&}29$^m$55$^s_.$1{&}1$^{\circ}$13$\arcmin$22$\arcsec${&}16.32{&}0.03{&}13.84 {&}$<$0.01{&}12.39{&}$<$0.01{&}19{&}3{&}31{&}12{&}EC73,GEL3{\\}{\null}298{&}29$^m$55$^s_.$6{&}1$^{\circ}$14$\arcmin$31$\arcsec${&}14.78{&}0.02{&}12.21 {&}$<$0.01{&}10.44{&}$<$0.01{&}100{&}27{&}119$^d${&}16{&}EC74,CK9,GEL4{\\}{\null}304{&}29$^m$56$^s_.$6{&}1$^{\circ}$13$\arcmin$01$\arcsec${&}14.86{&}0.01{&}12.55 {&}$<$0.01{&}11.38{&}$<$0.01{&}33{&}5{&}38{&}3{&}EC79,GEL5{\\}{\null}306{&}29$^m$56$^s_.$6{&}1$^{\circ}$12$\arcmin$40$\arcsec${&}-{&}-{&}17.90 {&}0.08{&}13.75{&}0.02{&}43{&}4{&}32{&}4{&}EC80{\\}{\null}307{&}29$^m$56$^s_.$8{&}1$^{\circ}$14$\arcmin$46$\arcsec${&}12.01{&}$<$0.01{&}10.20 {&}$<$0.01{&}8.63{&}$<$0.01{&}680$^{e,d}${&}10{&}1762$^{e,d}${&}43{&}SVS2,CK3,EC82{\\}{\null}308{&}29$^m$56$^s_.$8{&}1$^{\circ}$13$\arcmin$18$\arcsec${&}-{&}-{&}- {&}-{&}16.62{&}0.19{&}11.0{&}2{&}-{&}-{&}K32,HCE170/171{\\}{\null}309{&}29$^m$56$^s_.$8{&}1$^{\circ}$12$\arcmin$49$\arcsec${&}15.15{&}0.02{&}12.44 {&}$<$0.01{&}11.00{&}$<$0.01{&}45{&}4{&}70{&}3{&}EC84,GEL7{\\}{\null}312{&}29$^m$57$^s_.$5{&}1$^{\circ}$12$\arcmin$59$\arcsec${&}15.56$^b${&}0.03{&}13.61$^b$ {&}0.01{&}11.24$^b${&}$<$0.01{&}453{&}8{&}697{&}10{&}EC88+EC89{\\}{\null}313{&}29$^m$57$^s_.$5{&}1$^{\circ}$13$\arcmin$49$\arcsec${&}17.81{&}0.12{&}15.47 {&}0.04{&}13.46{&}0.03{&}32{&}3{&}-{&}-{&}GCNM94,EC87{\\}{\null}314{&}29$^m$57$^s_.$5{&}1$^{\circ}$14$\arcmin$07$\arcsec${&}12.10{&}$<$0.01{&}9.16 {&}$<$0.01{&}7.03{&}$<$0.01{&}4479{&}97{&}6388{&}50{&}SVS20,CK1,EC90{\\}{\null}317{&}29$^m$57$^s_.$8{&}1$^{\circ}$12$\arcmin$52$\arcsec${&}15.30$^b${&}0.02{&}11.47$^b$ {&}$<$0.01{&}9.45$^b${&}$<$0.01{&}332{&}5{&}1039$^d${&}35{&}EC92+EC95{\\}{\null}318{&}29$^m$57$^s_.$8{&}1$^{\circ}$12$\arcmin$37$\arcsec${&}-{&}-{&}14.22 {&}0.01{&}11.47{&}$<$0.01{&}100$^f${&}9{&}105$^f${&}20{&}EC94{\\}{\null}319{&}29$^m$57$^s_.$7{&}1$^{\circ}$15$\arcmin$31$\arcsec${&}15.95{&}0.03{&}12.67 {&}$<$0.01{&}10.59{&}$<$0.01{&}45{&}4{&}-{&}-{&}EC93,CK13{\\}{\null}320{&}29$^m$57$^s_.$8{&}1$^{\circ}$12$\arcmin$29$\arcsec${&}-{&}-{&}15.61 {&}0.02{&}12.91{&}$<$0.01{&}23{&}7{&}19{&}5{&}EC91{\\}{\null}321{&}29$^m$58$^s_.$2{&}1$^{\circ}$15$\arcmin$22$\arcsec${&}12.97{&}$<$0.01{&}10.92 {&}$<$0.01{&}9.48{&}$<$0.01{&}204{&}9{&}327$^d${&}5{&}CK4,GEL12,EC97{\\}{\null}322{&}29$^m$58$^s_.$3{&}1$^{\circ}$12$\arcmin$49$\arcsec${&}-{&}-{&}14.61 {&}0.01{&}12.25{&}$<$0.01{&}32{&}3{&}-{&}-{&}EC98{\\}{\null}326{&}29$^m$58$^s_.$7{&}1$^{\circ}$14$\arcmin$26$\arcsec${&}16.62{&}0.06{&}13.82 {&}0.01{&}11.56{&}$<$0.01{&}202{&}39{&}335$^f${&}8{&}EC103{\\}\noalign{\smallskip} \hline \end{tabular}$$ [(continued on next page)]{}\ $$\begin{tabular}{rccrrrrrrrrrrl} \hline \noalign{\smallskip} {ISO} & $\alpha_{2000}$ & $\delta_{2000}$ & {$J$} & {$\sigma_{J}$} & {$H$} & $\sigma_{H}$ & {$K$} & $\sigma_{K}$ & $F_{\nu}^{6.7}$ & $\sigma_{6.7}$ & $F_{\nu}^{14.3}$ & $\sigma_{14.3}$ & {Other ID} \\ \# & 18$^h$ & & mag & & mag & & mag & & mJy & mJy & mJy & mJy & \\ \hline {\null}327{&}29$^m$58$^s_.$9{&}1$^{\circ}$12$\arcmin$31$\arcsec${&}-{&}-{&}- {&}-{&}15.41{&}0.04{&}9.7{&}2.0{&}8.1{&}4.3{&}HCE175{\\}{\null}328{&}29$^m$59$^s_.$2{&}1$^{\circ}$14$\arcmin$06$\arcsec${&}11.78{&}$<$0.01{&}10.36 {&}$<$0.01{&}9.46{&}$<$0.01{&}223{&}23{&}186$^f${&}10{&}EC105,CK8,GEL13{\\}{\null}330{&}29$^m$59$^s_.$5{&}1$^{\circ}$11$\arcmin$59$\arcsec${&}-{&}-{&}- {&}-{&}14.37{&}0.02{&}423{&}7{&}643{&}9{&}HB1{\\}{\null}331{&}29$^m$59$^s_.$7{&}1$^{\circ}$13$\arcmin$13$\arcsec${&}-{&}-{&}- {&}-{&}15.37{&}0.04{&}145{&}9{&}346{&}12{&}{\\}{\null}338{&}30$^m$00$^s_.$7{&}1$^{\circ}$13$\arcmin$38$\arcsec${&}13.40{&}$<$0.01{&}11.15 {&}$<$0.01{&}10.07{&}$<$0.01{&}16{&}4{&}-{&}-{&}EC117,CK6,GEL15{\\}{\null}341{&}30$^m$01$^s_.$1{&}1$^{\circ}$13$\arcmin$26$\arcsec${&}-{&}-{&}15.39 {&}0.02{&}12.90{&}$<$0.01{&}20{&}3{&}24{&}11{&}EC121{\\}{\null}345{&}30$^m$02$^s_.$1{&}1$^{\circ}$14$\arcmin$00$\arcsec${&}17.16{&}0.06{&}14.74 {&}0.01{&}12.88{&}$<$0.01{&}24{&}4{&}50{&}7{&}EC125,CK7{\\}{\null}347{&}30$^m$02$^s_.$8{&}1$^{\circ}$12$\arcmin$28$\arcsec${&}15.11{&}0.02{&}11.72 {&}$<$0.01{&}9.69{&}$<$0.01{&}499$^c${&}12{&}658{&}13{&}EC129{\\}{\null}348{&}30$^m$03$^s_.$2{&}1$^{\circ}$16$\arcmin$17$\arcsec${&}12.14{&}$<$0.01{&}10.89 {&}$<$0.01{&}10.15{&}$<$0.01{&}41$^n${&}11{&}59$^d${&}3{&}EC135,GGD29{\\}{\null}351{&}30$^m$04$^s_.$0{&}1$^{\circ}$12$\arcmin$38$\arcsec${&}16.51{&}0.03{&}13.54 {&}$<$0.01{&}12.13{&}$<$0.01{&}8.4{&}1.9{&}-{&}-{&}EC141{\\}{\null}356{&}30$^m$05$^s_.$2{&}1$^{\circ}$12$\arcmin$44$\arcsec${&}16.08{&}0.03{&}14.51 {&}0.01{&}13.53{&}0.01{&}2.5{&}1.6{&}-{&}-{&}K40,EC152{\\}{\null}357{&}30$^m$05$^s_.$8{&}1$^{\circ}$06$\arcmin$22$\arcsec${&}{&}{&}{&}{&}{&}{&}10{&}1.7{&}18{&}2.3{&}{\\}{\null}359{&}30$^m$06$^s_.$4{&}1$^{\circ}$01$\arcmin$09$\arcsec${&}{&}{&}{&}{&}{&}{&}32{&}4{&}47{&}4{&}{\\}{\null}366{&}30$^m$07$^s_.$6{&}1$^{\circ}$12$\arcmin$04$\arcsec${&}12.20{&}$<$0.01{&}10.74 {&}$<$0.01{&}9.92{&}$<$0.01{&}53{&}7{&}64{&}5{&}STGM8{\\}{\null}367{&}30$^m$08$^s_.$1{&}1$^{\circ}$01$\arcmin$42$\arcsec${&}{&}{&}{&}{&}{&}{&}12$^n${&}5{&}19$^n${&}4{&}{\\}{\null}370{&}30$^m$08$^s_.$4{&}0$^{\circ}$58$\arcmin$48$\arcsec${&}{&}{&}{&}{&}{&}{&}61{&}3{&}83{&}4{&}double in I{\\}{\null}379{&}30$^m$09$^s_.$3{&}1$^{\circ}$02$\arcmin$47$\arcsec${&}{&}{&}{&}{&}{&}{&}99{&}5{&}67{&}6{&}{\\}{\null}393{&}30$^m$11$^s_.$2{&}1$^{\circ}$12$\arcmin$40$\arcsec${&}12.97$^b${&}$<$0.01{&}11.67$^b$ {&}$<$0.01{&}11.06$^b${&}$<$0.01{&}6.3{&}4.1{&}11.2{&}4.5{&}{\\}{\null}407{&}30$^m$13$^s_.$9{&}1$^{\circ}$08$\arcmin$55$\arcsec${&}{&}{&}{&}{&}{&}{&}2.2{&}1.3{&}4.9{&}5.3{&}{\\}\noalign{\smallskip} \hline \end{tabular}$$ $^{a}$ $H$ and $K$ band data from the Arnica 1995 map, which is slightly displaced from the 1996 map and therefore includes this object.\ $^{b}$ ISOCAM source is resolved into two sources in the near-IR, and the near-IR fluxes are added.\ $^{c}$ Flux measurement affected by proximity to the dead column. If the dead column cannot be avoided by any of the redundant observations, this flagging. If source is located on the dead column, a flux measurement is not attempted at all.\ $^{d}$ Source close to the detector edge in all redundant observations.\ $^{e}$ Extended source.\ $^{f}$ ISOCAM source is not quite resolved from a bright neighbour.\ $^{g}$ Galaxy contamination? For fluxes $\sim$ 3 mJy at 14.3 $\mu$m the “red” sample is expected to contain less than two galaxies in our field according to @hon03.\ $^{m}$ Flux might be affected by memory effects from other sources.\ $^{n}$ Nebulous sky background.\ Empty space means no measurement available in this work, while a hyphen means no detection.\ Identifier acronyms are related to the following references: SVS (Strom et al. 1976); GGD (Gyulbudaghian et al. 1978); HL (Hartigan & Lada 1985); CK (Churchwell & Koornneef 1986); GEL (Gómez de Castro et al. 1988); CDF (Chavarria-K et al. 1988); EC (Eiroa & Casali 1992); SMM (Casali et al. 1993); MMW (McMullin et al. 1994); HHR (Hodapp et al. 1996); HB (Hurt & Barsony 1996); STGM (Sogawa et al. 1997); HCE (Horrobin et al. 1997); GCNM (Giovannetti et al. 1998); K (Kaas 1999) $$\begin{tabular}{rccrrrrrrrrrrl} \hline \noalign{\smallskip} {ISO} & $\alpha_{2000}$ & $\delta_{2000}$ & {$J$} & {$\sigma_{J}$} & {$H$} & $\sigma_{H}$ & {$K$} & $\sigma_{K}$ & $F_{\nu}^{6.7}$ & $\sigma_{6.7}$ & $F_{\nu}^{14.3}$ & $\sigma_{14.3}$ & {Other ID} \\ \# & 18$^h$ & & mag & & mag & & mag & & mJy & mJy & mJy & mJy & \\ \hline {\null}2\ {&}28$^m$41.2{&}1$^{\circ}$15$\arcmin$15$\arcsec${&}{&}{&}{&}{&}{&}{&}30{&}5{&}12{&}2{&}{\\}{\null}4\ {&}28$^m$41.9{&}1$^{\circ}$14$\arcmin$06$\arcsec${&}{&}{&}{&}{&}{&}{&}35{&}5{&}6.4{&}2.9{&}{\\}{\null}7\ {&}28$^m$42.1{&}1$^{\circ}$14$\arcmin$31$\arcsec${&}{&}{&}{&}{&}{&}{&}26{&}3{&}7.9{&}3.0{&}{\\}{\null}9\ {&}28$^m$43.6{&}1$^{\circ}$16$\arcmin$10$\arcsec${&}{&}{&}{&}{&}{&}{&}19{&}3{&}8.0{&}2.2{&}{\\}{\null}11\ {&}28$^m$45.1{&}1$^{\circ}$10$\arcmin$19$\arcsec${&}{&}{&}{&}{&}{&}{&}10{&}2{&}3.3{&}3.1{&}{\\}{\null}12\ {&}28$^m$45.2{&}1$^{\circ}$16$\arcmin$50$\arcsec${&}{&}{&}{&}{&}{&}{&}57{&}8{&}11{&}2{&}{\\}{\null}16\ {&}28$^m$47.0{&}1$^{\circ}$10$\arcmin$36$\arcsec${&}{&}{&}{&}{&}{&}{&}53{&}6{&}13{&}2{&}{\\}{\null}20\ {&}28$^m$48.4{&}1$^{\circ}$13$\arcmin$14$\arcsec${&}{&}{&}{&}{&}{&}{&}37{&}6{&}9{&}2{&}{\\}{\null}24\ {&}28$^m$49.7{&}1$^{\circ}$19$\arcmin$09$\arcsec${&}{&}{&}{&}{&}{&}{&}60{&}7{&}15{&}2{&}{\\}{\null}31\ {&}28$^m$52.6{&}1$^{\circ}$10$\arcmin$10$\arcsec${&}{&}{&}{&}{&}{&}{&}5$^d${&}1{&}1.2{&}2.8{&}{\\}{\null}42\ {&}28$^m$58.3{&}1$^{\circ}$10$\arcmin$55$\arcsec${&}{&}{&}{&}{&}{&}{&}20{&}2{&}3.5{&}1.7{&}[SCB96] 41{\\}{\null}46\ {&}28$^m$59.8{&}1$^{\circ}$09$\arcmin$58$\arcsec${&}{&}{&}{&}{&}{&}{&}27{&}4{&}4.8{&}2.5{&}{\\}{\null}49\ {&}29$^m$ 0.3{&}1$^{\circ}$18$\arcmin$17$\arcsec${&}{&}{&}{&}{&}{&}{&}114{&}14{&}32$^f${&}4{&}{\\}{\null}61\ {&}29$^m$ 3.4{&}1$^{\circ}$18$\arcmin$34$\arcsec${&}{&}{&}{&}{&}{&}{&}28{&}4{&}8{&}2{&}{\\}{\null}63\ {&}29$^m$ 3.8{&}1$^{\circ}$10$\arcmin$39$\arcsec${&}{&}{&}{&}{&}{&}{&}26{&}4{&}4.7{&}2.5{&}{\\}{\null}65\ {&}29$^m$ 4.6{&}1$^{\circ}$20$\arcmin$02$\arcsec${&}{&}{&}{&}{&}{&}{&}33{&}3{&}8{&}2{&}{\\}{\null}66\ {&}29$^m$ 4.7{&}1$^{\circ}$16$\arcmin$15$\arcsec${&}{&}{&}{&}{&}{&}{&}56{&}6{&}15{&}2{&}{\\}{\null}69\ {&}29$^m$ 5.0{&}1$^{\circ}$22$\arcmin$04$\arcsec${&}{&}{&}{&}{&}{&}{&}28{&}3{&}7{&}2{&}{\\}{\null}74\ {&}29$^m$ 8.1{&}1$^{\circ}$12$\arcmin$35$\arcsec${&}{&}{&}{&}{&}{&}{&}13{&}2{&}2.4{&}1.2{&}{\\}{\null}80\ {&}29$^m$ 9.6{&}1$^{\circ}$09$\arcmin$29$\arcsec${&}{&}{&}{&}{&}{&}{&}12{&}18{&}4.1$^d${&}2.0{&}{\\}{\null}82\ {&}29$^m$10.6{&}1$^{\circ}$18$\arcmin$30$\arcsec${&}{&}{&}{&}{&}{&}{&}27{&}3{&}6.3{&}1.8{&}{\\}{\null}84\ {&}29$^m$10.6{&}1$^{\circ}$18$\arcmin$06$\arcsec${&}{&}{&}{&}{&}{&}{&}12{&}2{&}3.8{&}2.5{&}{\\}{\null}91\ {&}29$^m$12.4{&}1$^{\circ}$11$\arcmin$20$\arcsec${&}{&}{&}{&}{&}{&}{&}23{&}3{&}5.3{&}1.9{&}{\\}{\null}94\ {&}29$^m$13.4{&}1$^{\circ}$11$\arcmin$59$\arcsec${&}{&}{&}{&}{&}{&}{&}118{&}12{&}33{&}4{&}{\\}{\null}96\ {&}29$^m$14.2{&}1$^{\circ}$21$\arcmin$30$\arcsec${&}{&}{&}{&}{&}{&}{&}31{&}4{&}8.3{&}2.3{&}{\\}{\null}98\ {&}29$^m$14.9{&}1$^{\circ}$21$\arcmin$55$\arcsec${&}{&}{&}{&}{&}{&}{&}79{&}13{&}19{&}4{&}{\\}{\null}102\ {&}29$^m$15.8{&}1$^{\circ}$ 9$\arcmin$32$\arcsec${&}{&}{&}{&}{&}{&}{&}128{&}22{&}34{&}5{&}{\\}{\null}104\ {&}29$^m$16.9{&}1$^{\circ}$18$\arcmin$37$\arcsec${&}{&}{&}{&}{&}{&}{&}28{&}2{&}7.0{&}2.0{&}{\\}{\null}112\ {&}29$^m$19.9{&}1$^{\circ}$20$\arcmin$59$\arcsec${&}{&}{&}{&}{&}{&}{&}65{&}8{&}38{&}5{&}{\\}{\null}126\ {&}29$^m$25.6{&}1$^{\circ}$03$\arcmin$50$\arcsec${&}{&}{&}{&}{&}{&}{&}59{&}2{&}12{&}4{&}{\\}{\null}132\ {&}29$^m$26.8{&}1$^{\circ}$11$\arcmin$56$\arcsec${&}{&}{&}{&}{&}{&}{&}21{&}2{&}5.0{&}2.7{&}{\\}{\null}134\ {&}29$^m$27.3{&}1$^{\circ}$12$\arcmin$58$\arcsec${&}{&}{&}{&}{&}{&}{&}15{&}2{&}1.4{&}2.5{&}BD+01 3686{\\}{\null}140\ {&}29$^m$28.6{&}1$^{\circ}$10$\arcmin$24$\arcsec${&}{&}{&}{&}{&}{&}{&}51{&}2{&}15{&}2{&}{\\}{\null}144\ {&}29$^m$29.4{&}1$^{\circ}$14$\arcmin$03$\arcsec${&}{&}{&}{&}{&}{&}{&}49{&}7{&}12{&}2{&}{\\}{\null}145\ {&}29$^m$29.7{&}1$^{\circ}$13$\arcmin$09$\arcsec${&}{&}{&}{&}{&}{&}{&}37{&}6{&}9.0{&}2.5{&}{\\}{\null}153\ {&}29$^m$30.9{&}1$^{\circ}$15$\arcmin$18$\arcsec${&}{&}{&}{&}{&}{&}{&}10{&}2{&}0.9{&}2.1{&}{\\}{\null}166\ {&}29$^m$32.6{&}1$^{\circ}$09$\arcmin$46$\arcsec${&}{&}{&}{&}{&}{&}{&}20{&}2{&}7.0{&}2.6{&}{\\}{\null}184\ {&}29$^m$37.2{&}1$^{\circ}$09$\arcmin$15$\arcsec${&}{&}{&}{&}{&}{&}{&}125{&}3{&}37{&}6{&}{\\}{\null}189\ {&}29$^m$37.7{&}1$^{\circ}$11$\arcmin$18$\arcsec${&}{&}{&}{&}{&}{&}{&}67{&}2{&}31{&}2{&}{\\}{\null}190\ {&}29$^m$37.6{&}1$^{\circ}$11$\arcmin$30$\arcsec${&}{&}{&}{&}{&}{&}{&}68{&}10{&}32{&}3{&}{\\}{\null}191\ {&}29$^m$37.9{&}1$^{\circ}$00$\arcmin$36$\arcsec${&}{&}{&}{&}{&}{&}{&}44$^c${&}2{&}10{&}3{&}{\\}{\null}208\ {&}29$^m$41.0{&}1$^{\circ}$12$\arcmin$40$\arcsec${&}{&}{&}{&}{&}{&}{&}36{&}6{&}9{&}4{&}SVS18{\\}{\null}209\ {&}29$^m$41.1{&}1$^{\circ}$20$\arcmin$56$\arcsec${&}{&}{&}{&}{&}8.69{&}$<$0.01{&}44{&}5{&}12{&}2{&}{\\}{\null}210\ {&}29$^m$41.5{&}1$^{\circ}$10$\arcmin$00$\arcsec${&}{&}{&}{&}{&}{&}{&}12{&}2{&}6{&}4{&}{\\}{\null}228\ {&}29$^m$45.2{&}1$^{\circ}$18$\arcmin$46$\arcsec${&}{&}{&}10.09 {&}{&}8.67{&}$<$0.01{&}91{&}9{&}19{&}3{&}SVS 9{\\}{\null}238\ {&}29$^m$47.6{&}1$^{\circ}$05$\arcmin$12$\arcsec${&}{&}{&}{&}{&}{&}{&}1414{&}19{&}285{&}7{&}StRS 208{\\}{\null}243\ {&}29$^m$48.9{&}1$^{\circ}$11$\arcmin$43$\arcsec${&}11.18{&}$<$0.01{&}9.53 {&}$<$0.01{&}8.84{&}$<$0.01{&}35{&}4{&}8{&}3{&}SVS3{\\}{\null}244\ {&}29$^m$49.2{&}0$^{\circ}$58$\arcmin$54$\arcsec${&}{&}{&}{&}{&}{&}{&}20{&}3{&}7$^e${&}3{&}{\\}{\null}255\ {&}29$^m$49.9{&}1$^{\circ}$09$\arcmin$24$\arcsec${&}{&}{&}{&}{&}{&}{&}91{&}2{&}24{&}4{&}{\\}{\null}264\ {&}29$^m$51.6{&}0$^{\circ}$56$\arcmin$21$\arcsec${&}{&}{&}{&}{&}{&}{&}14{&}2{&}5{&}1{&}{\\}{\null}267\ {&}29$^m$51.6{&}1$^{\circ}$10$\arcmin$32$\arcsec${&}{&}{&}{&}{&}{&}{&}48{&}3{&}13{&}6{&}SVS 5{\\}{\null}271\ {&}29$^m$52.7{&}1$^{\circ}$04$\arcmin$38$\arcsec${&}{&}{&}{&}{&}{&}{&}24{&}6{&}4{&}6{&}{\\}{\null}290\ {&}29$^m$54.3{&}1$^{\circ}$17$\arcmin$47$\arcsec${&}{&}{&}{&}{&}{&}{&}27{&}5{&}4{&}2{&}{\\}{\null}296\ {&}29$^m$55.8{&}1$^{\circ}$04$\arcmin$14$\arcsec${&}{&}{&}{&}{&}{&}{&}118{&}5{&}31{&}4{&}{\\}{\null}300\ {&}29$^m$56.1{&}1$^{\circ}$00$\arcmin$24$\arcsec${&}{&}{&}{&}{&}{&}{&}53{&}19{&}12{&}13{&}{\\}{\null}311\ {&}29$^m$57.6{&}1$^{\circ}$10$\arcmin$46$\arcsec${&}{&}{&}{&}{&}{&}{&}26{&}2{&}15{&}3{&}BD+01 3689B{\\}\noalign{\smallskip} \hline \end{tabular}$$ [(continued on next page)]{}\ $$\begin{tabular}{rccrrrrrrrrrrl} \hline \noalign{\smallskip} {ISO} & $\alpha_{2000}$ & $\delta_{2000}$ & {$J$} & {$\sigma_{J}$} & {$H$} & $\sigma_{H}$ & {$K$} & $\sigma_{K}$ & $F_{\nu}^{6.7}$ & $\sigma_{6.7}$ & $F_{\nu}^{14.3}$ & $\sigma_{14.3}$ & {Other ID} \\ \# & 18$^h$ & & mag & & mag & & mag & & mJy & mJy & mJy & mJy & \\ \hline {\null}324 {&}29$^m$58.4{&}1$^{\circ}$20$\arcmin$26$\arcsec${&}{&}{&}{&}{&}{&}{&}15{&}3{&}4{&}3{&}SVS 16{\\}{\null}332 {&}30$^m$ 0.2{&}1$^{\circ}$09$\arcmin$47$\arcsec${&}{&}{&}{&}{&}{&}{&}62{&}2{&}36{&}4{&}{\\}{\null}334 {&}30$^m$ 0.0{&}1$^{\circ}$21$\arcmin$55$\arcsec${&}{&}{&}{&}{&}{&}{&}160{&}10{&}40{&}3{&}SVS 15{\\}{\null}337 {&}30$^m$ 0.6{&}1$^{\circ}$15$\arcmin$19$\arcsec${&}19.13{&}0.279{&}12.61 {&}0.004{&}8.92{&}0.001{&}340{&}8{&}95{&}6{&}CK2,EC118{\\}{\null}339 {&}30$^m$ 0.8{&}1$^{\circ}$19$\arcmin$41$\arcsec${&}{&}{&}{&}{&}{&}{&}70{&}7{&}15{&}2{&}SVS 13{\\}{\null}342 {&}30$^m$ 1.6{&}0$^{\circ}$59$\arcmin$30$\arcsec${&}{&}{&}{&}{&}{&}{&}40{&}9{&}8{&}6{&}{\\}{\null}349 {&}30$^m$ 3.7{&}1$^{\circ}$ 4$\arcmin$52$\arcsec${&}{&}{&}{&}{&}{&}{&}97{&}6{&}21{&}9{&}{\\}{\null}372 {&}30$^m$ 8.4{&}0$^{\circ}$55$\arcmin$29$\arcsec${&}{&}{&}{&}{&}{&}{&}80{&}2{&}19{&}4{&}{\\}{\null}375 {&}30$^m$ 8.7{&}0$^{\circ}$55$\arcmin$16$\arcsec${&}{&}{&}{&}{&}{&}{&}98{&}3{&}28{&}5{&}{\\}{\null}378 {&}30$^m$ 9.0{&}1$^{\circ}$14$\arcmin$44$\arcsec${&}11.81{&}0.003{&}9.68 {&}0.001{&}8.67{&}0.001{&}70{&}7{&}14{&}4{&}SVS7{\\}{\null}382 {&}30$^m$ 9.2{&}1$^{\circ}$17$\arcmin$53$\arcsec${&}{&}{&}{&}{&}{&}{&}7{&}2{&}3{&}4{&}{\\}{\null}389 {&}30$^m$10.4{&}1$^{\circ}$19$\arcmin$36$\arcsec${&}{&}{&}{&}{&}{&}{&}18{&}2{&}5{&}4{&}BD+01 3693{\\}{\null}401 {&}30$^m$12.1{&}1$^{\circ}$16$\arcmin$41$\arcsec${&}11.39{&}0.003{&}9.59 {&}0.001{&}8.64{&}0.001{&}66{&}11{&}13{&}5{&}SVS6{\\}{\null}409 {&}30$^m$14.7{&}1$^{\circ}$05$\arcmin$24$\arcsec${&}{&}{&}{&}{&}{&}{&}15{&}3{&}5{&}3{&}{\\}{\null}410 {&}30$^m$14.7{&}1$^{\circ}$06$\arcmin$17$\arcsec${&}{&}{&}{&}{&}{&}{&}22{&}2{&}7{&}3{&}{\\}\noalign{\smallskip} \hline \end{tabular}$$ $^{b}$ Binary or close companion not quite resolved by ISOCAM.\ $^{c}$ Flux measurement affected by proximity to the dead column. If the dead column cannot be avoided by any of the redundant observations, this flagging. If source is located on the dead column, a flux measurement is not attempted at all.\ $^{d}$ Source close to the detector edge in all redundant observations.\ $^{e}$ Extended source.\ Identifier acronyms are related to the following references: SVS (Strom et al. 1976); CK (Churchwell & Koornneef 1986); StRS (Stephenson 1992); EC (Eiroa & Casali 1992), SCB (Straizys et al. 1996). Except for a few transition objects, the “blue” group (triangles) is mainly located at the colour index of normal photospheres, i.e. $[14.3/6.7] = -0.66$ or $\alpha_{\rm IR}^{7-14} = -3$ (dashed line). The spread around this value indicates the increasing photometric uncertainty with decreasing flux, although some of the scatter might be real. M giants have intrinsic excesses in the colour index $[14.3/6.7]$ of less than 0.1, while late M giants are expected to have excesses of up to $\sim 0.2$ above normal photospheres. Although late M giants are few, a number of M giants are expected in the observed sample of field stars. Thus, the slight displacement of this group of sources above the colour of normal photospheres could be due to a combination of extinction and intrinsic colours of M giants. The effect of extinction is small, however. A reddening vector of size corresponding to A$_K = 3$ is indicated in the figure. (More about extinction in Sect. \[nir\].) This same dichotomy in colour was found also in the Chamaeleon dark clouds [@nor96], in RCrA [@olo99], and in $\rho$ Ophiuchi [@bon01]. According to the SED index, $\alpha_{\rm IR}^{7-14}$, the mid-IR excess sources are Class II and Class I types of Young Stellar Objects (YSOs). In Chamaeleon I about 40% of the mid-IR excess sources had been previously classified as Classical T Tauri stars (CTTS), and in $\rho$ Ophiuchi $\sim$ 50% were previously known as Class II and Class Is. In Serpens, very few sources are optically visible and the IRAS data suffer badly from source confusion. The known YSOs are thus a sample of sources classified from near-IR excesses, association with nebulosity, emission-line stars, ice features, clustering properties, and variability [@eir92; @hod96; @hor97; @sog97; @gio98; @kaa99a]. Previously known YSOs are not found in the “blue” group, although ClassIII sources, young stars with marginal or no IR excess, have their locus there. A few objects in the transition phase between Class II and Class III are evident in the “blue” group. ClassIII sources without IR excess cannot be distinguished from field stars purely on the basis of broad band infrared photometry. They can, however, be efficiently identified through X-ray observations. The fact that intrinsic IR excess is so easily distinguished from reddening at these wavelengths, enables an unambiguous classification of the youngest YSOs on the basis of one single colour index $[14.3/6.7]$. In this way ISOCAM found $53$ Class I and Class II YSOs, of which only $28$ had been previously suggested as YSO candidates. Positions and photometry of the 53 Class I and Class II YSOs are listed among other members of the Serpens Core cluster in Table \[tbl-2\]. The 71 “blue” sources are listed in Table \[tbl-3\]. Many of these are field stars, but some are likely to be young cluster members of Class III type. Fig. \[fig1\] shows that the CW region contains as many as 20 (10) sources detected at 6.7 $\mu$m (14.3 $\mu$m) above the expected field star contribution according to the galactic model by @wai92. Because there are only two IR excess sources in the CW field, of which one is very faint, these are practically all belonging to the “blue” sample. Since the total region surveyed is about 3 times as large as the CW field, perhaps more than 20 “blue” sources from Table \[tbl-3\] could be ClassIIIs belonging to the cluster. Near-IR and mid-IR excess YSOs {#nir} ============================== When it comes to detecting IR excesses, the $J$ band usually limits the number of sources in a $J-H/H-K$ diagram because of its sensitivity to extinction. For the ISOCAM observations the 14.3 $\mu$m band is the less sensitive, unless the sources are extremely red (see limits in Fig. \[fig-323\]). Using the $H-K/K-m_{6.7}$ diagram as in @olo99, however, we should be able to detect more IR excess objects. Fig. \[fig-hk2\] shows that practically all sources with excesses in the colour index $[14.3/6.7]$ are also found to have excesses in their $K - m_{6.7}$ index. Approximate intrinsic colours of main-sequence, giant and supergiant stars, taken from the source table of @wai92 and interpolated between 2.2 and 12 $\mu$m, are given as boldface curves (solid, dotted and dashed, respectively). Sources with ISOCAM mid-IR excesses (circles) separate well from those without mid-IR excesses (triangles). A reddening vector has been calculated by fitting a line from origin through the 4 sources without mid-IR excesses. The slope (1.23) is mainly constrained by the star CK2, which is believed to be a background supergiant, see e.g. @cas96. On the basis of a number of background stars in the $J-H/H-K$ diagram, @kaa99a found a reddening law of the form A$_{\lambda} \sim \lambda^{-1.9}$ to fit the Serpens data. Extrapolation of this law to 6.7 $\mu$m would give a slope of 0.83 in the $H-K/K-m_{\rm 6.7}$ diagram, which is in disagreement with all the 4 sources without mid-IR excess. The A$_{\lambda} \sim \lambda^{-1.7}$ law [@whi88] gives an even shallower slope. By assuming, however, the A$_{\lambda} \sim \lambda^{-1.9}$ law to hold for $H$ and $K$, and using the empirical slope found in Fig. \[fig-hk2\], the extinction in the 6.7 $\mu$m band is estimated to be A$_{\rm 6.7} = 0.41 $ A$_{K}$. This is in good agreement with the values 0.37, 0.35 and 0.47 found by @jia03 from the ISOGAL survey, whose three values depend on the actual form of the near-IR extinction curve applied. As shown in @kaa00 only about 50% of the IR excess sources which display clear excesses in the single ISOCAM colour index $[14.3/6.7]$ show up as IR excess objects in the $J-H/H-K$ diagram. This was demonstrated by plotting the same sources in the two diagrams, using a statistically significant sample of several hundred sources in Serpens, $\rho$ Ophiuchi, Chamaeleon I, and RCrA. This result implies that a sole use of the $J-H/H-K$ diagram will severely underestimate the IR-excess population of YSOs in active star formation regions. @bon01 showed that in $\rho$ Ophiuchi only 50% of the 123 Class II sources have near-IR excesses large enough to be recognizable in the $J-H/H-K$ diagram. Similarly we confirm here specifically for Serpens that the mid-IR excess sources cluster along the reddening line when plotted in a $J-H/H-K$ diagram, and that only 50% would be recognized as having IR excesses from JHK data alone. The $H-K/K-m_{7}$ diagram is more efficient than the $J-H/H-K$ diagram in distinguishing between intrinsic circumstellar excess (circles) and reddening (triangles). Thus, by sampling the SED a bit more into the mid-IR, the use of two colour indices to select sources with intrinsic IR excesses becomes substantially less prone to the influence from cloud extinction. Fig. \[fig-hk2\] also shows that the mid-IR excesses apparent from the $[14.3/6.7]$ index, always appear already at 6.7 $\mu$m, but only in half of the cases at 2.2 $\mu$m. In addition to the $53$ IR excess sources obtained from Fig \[fig-323\], another eight sources were found to have IR excess from the $H-K/K-m_{7}$ diagram, i.e. displaced by more than 1$\sigma$ to the right of the reddening band (dotted line). This means we find 25% more IR excess sources within the centre region than by using the $[14.3/6.7]$ index alone. ClassII sources are most likely Classical T Tauri stars (CTTS), but we can not exclude that some of them are weak-lined T Tauri stars (WTTS). Among the ClassII sources detected with ISOCAM in ChamaeleonI about 1/3 had been previously classified as WTTS [@nor96]. For WTTS one could attribute the presence of mid-IR excess but a lack of near-IR excess to an inner hole in the circumstellar disk [@mon99]. CTTS, on the other hand, are believed to have inner disks, since they show strong H$\alpha$ emission, which is commonly interpreted as a signature of the accretion process onto the surface of the object, but could also arise in stellar winds. Optical information is scarce in the Serpens Cloud Core because of the large cloud extinction, and we do not know what fraction of the ClassIIs are strong H$\alpha$ emitters. It is well known from studies of CTTS locations in the $J-H/H-K$ diagram [@lad92; @mey97], that many CTTS lack detectable near-IR excesses; about 40% in Taurus-Auriga according to @str93. For typical CTTS the near-IR wavelength region is strongly dominated by the photospheric emission, and rather large amounts of dust hot enough to produce a strong excess at 2 $\mu$m are therefore needed in order to distinguish intrinsic IR excess from the effects of scattering and extinction in the $J-H/H-K$ diagram. In a region such as the Serpens Cloud Core, where 7 of the ClassIIs in question here were originally proposed to be YSOs due to their association with nebulosity [@eir92], it is likely that scattered light in the J and H bands adds to the lack of detectable near-IR excesses as it gives a bluer $H-K$ colour index. From an ISOCAM sample of CTTS with mid-IR excesses in ChamaeleonI, @com00 found evidence for the presence of near-IR excesses to be correlated with luminosity, suggesting an incapability of objects with very low temperatures and luminosities (young brown dwarfs and very low mass CTTS) to raise the temperature needed at the inner part of the circumstellar disk to produce a detectable excess at 2.2 $\mu$m, in agreement with model predictions [@mey97]. The majority of the Serpens sources are substantially more luminous. As pointed out by @hil98, the larger the stellar radius (i.e. the younger the star is), the more difficult it is to separate near-IR excesses from the stellar flux. This could perhaps be part of the explanation for the youngest sources (e.g. two flat-spectrum sources without near-IR excess). But we found all the ClassI sources to possess near-IR excesses, such that the role of larger radii in these cases seems to be well compensated for, probably by their larger disk accretion rates. A statistical interpretation based on broad band photometry is likely over-simplified, however, and there are many properties intrinsic to the star-disk system (such as disk inclination angle, disk accretion rate, stellar mass and radius) which contribute to the complexity of the individual YSOs. From the results presented here for Serpens, in @bon01 for $\rho$ Ophiuchi, and in @kaa00 for the ISOCAM star formation surveys in general, it is evident that one should be careful in estimating disk frequencies among YSOs based on near-IR excesses only, see also @hai01. Since our study samples only IR excess objects, we cannot say anything about the disk fraction among YSOs in Serpens. Characterization of the YSO population {#char} ====================================== Large fraction of ClassI sources {#classI} -------------------------------- We have combined the photometry from ISOCAM with the available $K$ band photometry from 1996[^5] [@kaa99a] and calculated the two SED indices: $\alpha_{\rm IR}^{2-14}$ and $\alpha_{\rm IR}^{2-7}$, which are plotted against each other in Fig. \[fig-k2k3\] for the 39 mid-IR excess sources and the 6 sources without mid-IR excesses. The index $\alpha_{\rm IR}^{2-14}$ is close to the index originally used to define the three classes I, II and III from the shape of their SED between 2.2 and 10 (or 25) $\mu$m by @lad84 [@lad87]. Figure \[fig-k2k3\] indicates the loci of these classes in addition to a transitional class referred to as “flat spectrum” sources. According to the updated IR spectral classification scheme [@and94; @gre94] we tentatively define YSOs with $\alpha_{\rm IR}^{2-14} > 0.3$ as ClassI sources, those with $-0.3 < \alpha_{\rm IR}^{2-14} < 0.3$ as flat-spectrum sources, and objects with $-1.6 < \alpha_{\rm IR}^{2-14} < -0.3$ as ClassII sources. There is a marginal hint of a gap in the distribution of sources along the $[14.3/6.7] = -0.2$ axis at $\alpha_{\rm IR}^{2-14} \approx 0.3$, which was seen also in the $\rho$ Ophiuchi sample [@bon01], though at $\approx 0.5$. It is apparent that we cannot distinguish between Class III sources and field stars. (The transition sources in the “blue” group (see Fig. \[fig-323\]) with some mid-IR excesses are all located outside the field for which we have K-band observations.) IR excess sources without detection at 14.3 $\mu$m (see Sec. 4.2) can be classified from the approximately linear relation between $\alpha_{\rm IR}^{2-14}$ and $\alpha_{\rm IR}^{2-7}$ found in Fig. \[fig-k2k3\]. In this case ClassIs are those which have $\alpha_{\rm IR}^{2-7} > 1.2$, corresponding to $K-m_{7} > 4.8$, ClassIIs are those which have $\alpha_{\rm IR}^{2-7} < -0.25$, corresponding to $K-m_{7} < 3.2$, and flat-spectrum sources are those in between. Thus, the total number of sources in each category is larger than apparent from Fig. \[fig-k2k3\]. From Fig. \[fig-hk2\] we have 1 more Class I, 2 more flat-spectrum sources, and 5 more Class IIs. Three sources (260, 277 and 308) have only $K$ and 6.7 $\mu$m detections, but their very red colours strongly suggest membership in the Class I category. The number of ClassI sources ($19$) is thus about equal to the number of ClassII sources ($18$) in the central part of the Serpens cluster. This is very unusual since ClassII sources outnumber ClassI sources by typically 10 to 1 in star formation regions. If we include the $13$ “flat spectrum” sources in the ClassII group, the ClassI/ClassII number ratio is still as high as $19/31$. This is exceptional, also compared to the results obtained from ISOCAM surveys in other regions; in Chamaeleon I this number ratio is $5/{\bf 42}$ [@kaa99b] and in $\rho$ Ophiuchi: $16/123$ [@bon01]. Such a large population of ClassI sources indicates a recent burst of star formation in this region and would be in line with the rich collection of Class0 objects found in this cluster by @cas93 and @hur96. Reddening effect on the $\alpha_{\rm IR}^{2-14}$ index? {#redden} ------------------------------------------------------- While a spectral index between two fixed wavelengths, such as 2.2 and 14.3 $\mu$m, is practical and provides a preliminary classification for a large number of sources, a truly reddening-independent classification is obtainable only when complete SEDs are available over the $\sim$ 2 to 100 $\mu$m range (or to 1.3 mm if Class0 sources are involved). In the following we discuss the possibility that sources defined as ClassIs from the $\alpha_{\rm IR}^{2-14}$ index could be heavily extinguished ClassII or flat-spectrum sources. The source CK2 (probably a background supergiant) is seen through more than 50 magnitudes of visual extinction [@kaa99a], and indicates an empirical slope (=1.6) of the extinction law in Fig. \[fig-k2k3\], as do five other sources without mid-IR excesses. This reddening line translates to a slope of $0.92$ in the $K-m_{7}/K-m_{14}$ diagram, which gives the relation A$_{\rm 14}$ = 0.88 A$_{\rm 7}$ in terms of magnitudes[^6], by using the relation $A_{7} = 0.41 A_K$ obtained in Sect. \[nir\]. It is remarkable that the group of mid-IR excess sources are lined up along the same slope as the one outlined by the reddened field stars. In this sense, some of the sources classified as ClassI objects on the basis of Fig. \[fig-k2k3\] could be interpreted as highly extinguished ClassII objects, either deeply embedded in the cloud or seen edge-on. Flat-spectrum sources and ClassIIs are generally expected to be optically visible objects. In Serpens, however, only one (ISO-Ser-159) of the 13 flat-spectrum sources and only three of the 18 ClassIIs are optically visible. This shows the extreme degree of embeddedness, and is in agreement with the C$^{18}$O map of @whi95, which suggests that the visual cloud extinction is larger than 30 magnitudes everywhere in the NW-SE ridge and may reach values as high as $A_V \sim 200$ in some places. The fact that the slopes are so similar suggests that ClassIs are either merely more embedded ClassIIs, or that the extinction behaves roughly in the same way for ClassI envelopes as for the cloud in general at these wavelengths. This would be in line with @pad99 who observed three ClassI SED YSOs with HST/NICMOS and found edge-on disks in all three cases. As shown in Fig. \[fig-323\] the $\alpha_{\rm IR}^{7-14}$ index is rather insensitive to extinction. As many as 15 YSOs in our sample have $\alpha_{\rm IR}^{2-14} > 0$, but $\alpha_{\rm IR}^{7-14} < 0$, among them the well known DEOS. For some of these we have more spectral information than the three flux points at 2.2, 6.7 and 14.3 $\mu$m. The two sources HB1 and EC129 are relatively isolated YSOs both in the near-IR and mid-IR images and should not suffer so much from the source confusion that sets a limit to the usefulness of IRAS data for the other sources. We use HIRES IRAS fluxes and upper limits from @hur96 and extend our mid-IR SEDs to longer wavelengths. Figure \[fig-sed\] shows that both HB1 and EC129 (HB2) have rising spectra beyond a clear dip in the SED at $\sim$ 14 $\mu$m. These sources have clearly ClassI SEDs, but the dip at $\sim$ 14 $\mu$m will produce a blue $\alpha_{\rm IR}^{7-14}$ index. @hur96 divided the flux from their HIRES map of IRAS 18272+0114 between the three sources DEOS, EC53 and S68N, while ISOCAM resolves six mid-IR excess sources (cf Fig. \[fig-iram\]), of which only EC37 has $\alpha_{\rm IR}^{7-14} > 0$. It is possible that a broad silicate absorption feature and/or the presence of H$_2$O and CO$_2$ ices in the 14.3 $\mu$m band [@whi96; @deg96] may cause absorption effects that correspond in magnitude to the observed dips in the SEDs at $\sim$ 14 $\mu$m, noting that the effect is there for the most deeply embedded sources. Recently, ISOCAM-CVF spectroscopy of 42 Class I and Class II YSOs in Serpens, $\rho$ Ophiuchi, Chamaeleon and RCrA by @ale03 reveals a number of absorption features from 5 to 18 $\mu$m. They find that the majority of Class I sources fall in their category [**a**]{}, having deep absorption features of ices and silicates, and conclude that in the cases of large extinction the continuum spectral index between 2 and 14 $\mu$m provides a truer value of the shape of the underlying continuum than the observed mid-IR spectrum. For the sub-sample of 20 sources that were studied by @ale03 in Serpens we found: four Class Is, six flat-spectrum sources, four Class II sources, one “blue” source, and we did not detect five of their sources. All four Class Is, three flat-spectrum sources, and one Class II fall in their group [**a**]{}, and two flat-spectrum sources and three Class IIs in their category [**b**]{} (only SVS2 is of type [**c**]{}). This comparison shows overall good agreement, and indicates that we have not overestimated the number of Class Is in our study. Also, the presence of shocked molecular H$_2$ line emission in the 6.7 $\mu$m band [e.g. @cab98; @lar02] included in the measured flux of some of the youngest sources could contribute to give a bluer $\alpha_{\rm IR}^{7-14}$ index than expected from a dust continuum. Recent 2-D models of Class I source geometries by @whi03 which include flared disk and bipolar cavity produce mid-IR SEDs with very broad dips around 10 $\mu$m, and overall much bluer mid-IR colours than produced by simple 1D or simplified 2-D models. Based on the previous discussion we trust that the $\alpha_{\rm IR}^{2-14}$ index gives a more reliable measure of the SED than the $\alpha_{\rm IR}^{7-14}$ index. We have also looked at the mid-IR fluxes of Class Is versus Class IIs. If all the 15 YSOs with $\alpha_{\rm IR}^{2-14} > 0$ and $\alpha_{\rm IR}^{7-14} < 0$ were reddened ClassII sources, one would expect them to be on the average fainter than (or at most as bright as) the YSOs with $\alpha_{\rm IR}^{2-14} < 0$. We note, however, that there is a slight tendency that sources with large $\alpha_{\rm IR}^{2-14}$ indices also are, on the average, the brightest ones in the mid-infrared. Figure \[fig-fav\] shows the average flux at 6.7 and 14.3 $\mu$m versus $\alpha_{\rm IR}^{2-14}$, with a reddening vector of A$_K = 3$ magnitudes inserted. The filled circles in the figure mark the 15 sources that have $\alpha_{\rm IR}^{2-14} > 0$ and $\alpha_{\rm IR}^{7-14} < 0$. Excluding the flat-spectrum sources, the median fluxes are $0.18$ and $0.029$ Jy for [**15**]{} ClassIs and [**27**]{} ClassIIs (including all those from Table \[tbl-6\] that have mid-IR fluxes), respectively. For comparison, the median mid-IR fluxes of 16 ClassIs and 76 ClassIIs in $\rho$ Ophiuchi are $1.17$ and $0.095$ Jy, respectively [@bon01]. In Chamaeleon I for 5 ClassIs and 42 ClassIIs these numbers are $0.28$ and $0.077$ Jy, respectively. While the statistics is low for the Chamaeleon I ClassI sources, there is an indication of luminosity evolution from the ClassI to the ClassII phase, the case being strongest for $\rho$ Ophiuchi. We conclude that on the basis of this statistical luminosity argument, there seems to be an intrinsic difference between the ClassI and the ClassII populations. Noting that ClassIs are on the average more luminous in the mid-IR than ClassIIs [cf. @bon01], we caution that it is not entirely excluded that the large number fraction of ClassI sources in Serpens could be a direct effect of a lower sensitivity because of a larger distance. There are about 80 faint YSO candidates in the Serpens Cloud Core below the sensitivity limit of ISOCAM [@kaa99a]. Although the exact ClassI/ClassII number ratio might become subject to modification when future observations at high spatial resolution and sensitivity towards longer wavelengths are available (SIRTF-Spitzer at 24 $\mu$m and FIRST-Herschel at 90-250 $\mu$m), we believe that the main conclusion of an unusually large fraction of ClassI sources in the Serpens Cloud Core will be maintained. The protostar sample in Serpens ------------------------------- Based on the discussion in the previous paragraph, we have arrived at a sample of 20 Class I SED sources which are listed in Table \[tbl-4\] with the two SED indices $\alpha_{\rm IR}^{2-14}$ and $\alpha_{\rm IR}^{2-7}$. These are all protostar candidates. Although Class0 sources normally are not expected to be detectable at shorter wavelengths than about 25 $\mu$m, we detect two ClassI sources, ISO-241 and ISO-308, within the $\pm$ 3$\arcsec$ positional uncertainties of S68N and SMM4, respectively. [^7] @mot01 found that about 60% of the Class I sources in Taurus-Auriga are “true protostars”, using the criterion that the envelope mass to stellar mass fraction is $\ga 0.1$. A similar method has been applied to the Serpens data. But first we exclude 3 sources in order to account for the possible confusion with Class0 sources (in the case of EC41) or contribution of Class0 sources (in the cases: ISO-241, ISO-308) in the sample of 19 ClassIs in the region mapped by IRAM. The fraction of bona-fide protostars in the Serpens Core is then estimated to be 9 out of 16 ClassIs or 56%, with an uncertainty of $\pm$ 10% because of source confusion problems with the strong clustering. [rlrrcc]{} [ISO]{} & [Other ID]{} & $\alpha_{IR}^{2-14}$ & $\alpha_{IR}^{2-7}$ & [fwhm]{} & $\Delta K$\ \# & & & & $\arcsec$ & mag\ 270[&]{}[&]{}2.56[&]{}3.54[&]{}2.31[&]{}[\ ]{}331[&]{}[&]{}2.51[&]{}4.12[&]{}4.57[&]{}[\ ]{}330[&]{}HB1[&]{}2.35[&]{}4.26[&]{}1.47[&]{}0.32[\ ]{}250[&]{}DEOS[&]{}2.06[&]{}3.52[&]{}7.52[&]{}0.92[\ ]{}276[&]{}GCNM53[&]{}2.02[&]{}3.05[&]{}1.89[&]{}[\ ]{}249[&]{}EC37[&]{}1.98[&]{}2.66[&]{}1.25[&]{}0.26[\ ]{}241[&]{}[&]{}1.74[&]{}3.26[&]{}1.49[&]{}[\ ]{}308[&]{}HCE170/171$^1$[&]{}-[&]{}2.84[&]{}4.48[&]{}0.64[\ ]{}253[&]{}EC40[&]{}1.23[&]{}2.76[&]{}1.21[&]{}[\ ]{}265[&]{}EC53[&]{}0.98[&]{}2.27[&]{}2.27[&]{}0.59[\ ]{}259[&]{}[&]{}-[&]{}1.94$^2$[&]{}-[&]{}[\ ]{}260[&]{}[&]{}-[&]{}1.88[&]{}1.83[&]{}[\ ]{}258a[&]{}EC41[&]{}0.96[&]{}1.81[&]{}1.33[&]{}[\ ]{}312[&]{}EC88/89$^3$[&]{}0.85[&]{}1.73[&]{}9.53[&]{}0.11[\ ]{}254[&]{}EC38[&]{}0.82[&]{}1.78[&]{}1.00$^4$[&]{}-[\ ]{}326[&]{}EC103[&]{}0.62[&]{}1.27[&]{}1.55[&]{}0.22[\ ]{}327[&]{}HCE175[&]{}0.53[&]{}1.73[&]{}1.42[&]{}[\ ]{}306[&]{}EC80[&]{}0.44[&]{}1.70[&]{}1.32[&]{}[\ ]{}277[&]{}EC63[&]{}-[&]{}1.21[&]{}1.26[&]{}[\ ]{}313[&]{}GCNM94 [&]{}-[&]{}1.19[&]{}1.32[&]{}0.43[\ ]{} $^1$ both are extended, HCE171=K32 is variable\ $^2$ Ks = 14.48 from 2MASS PSC, 16” south-west of extended IRAS 18273+0059, possible K extendedness not investigated\ $^3$ both are extended, EC88 is variable\ $^4$ from image serp-nw which has better seeing\ Identifier acronyms as in Table \[tbl-2\]. Extended, elongated and polarized near-IR emission is predicted by models of infalling envelopes which have developed cavities owing to bipolar outflows, where near-IR radiation from the central object may escape and scatter off dust in the cavity and the outer envelope, see e.g. @whi97. @par02 find that 70% of the bona-fide protostars in Taurus are extended in the near-IR, and that less than 10% of the sources that are extended in the near-IR show no extension in mm continuum. Our near-IR data has a spatial resolution about 10 times better than the IRAM map, and we have included in Table \[tbl-4\] the full width at half maximum (fwhm) of a gaussian fit to the source profiles in the $K$ band images. The median fwhm of 160 isolated and relatively bright sources over the field is $1.\arcsec27 \pm 0.\arcsec10$. Sources with a fwhm greater than $1.\arcsec6$, i.e. above 3$\sigma$, are here defined as extended in the K-band. For the Class I sources this concerns 8 of 19 sources or 42%, highly coinciding with continuum sources except for ISO-331 and ISO-260. There are no extended sources in the flat-spectrum and ClassII samples. Substantial K-band variability was found in 8 of the above Class I sources, 5 of the flat-spectrum sources and 5 of the Class II sources [@kaa99a]. The brightness variations of the Class I sources are given in Table \[tbl-4\] as $\Delta K$. The median value of the amplitude is $0.38 \pm 0.27$ mag, larger than the variations found for Class Is in Taurus [@par02]. [rlrrcr]{} [ISO]{} & [Other ID]{} & $\alpha_{IR}^{2-14}$ & $\alpha_{IR}^{2-7}$ & [fwhm]{} & $\Delta K$\ \# & & & & $\arcsec$ & mag\ 234[&]{}EC26[&]{}-[&]{}0.63[&]{}1.31[&]{}[\ ]{}322[&]{}EC98[&]{}-[&]{}0.17[&]{}1.31[&]{}[\ ]{}345[&]{}EC125[&]{}0.26[&]{}0.44[&]{}1.30[&]{}[\ ]{}317[&]{}EC92/95[&]{}0.19[&]{}-0.03[&]{}1.35[&]{}[\ ]{}159[&]{}IRAS 18269+0116[&]{}0.11[&]{}0.23[&]{}1.00$^1$[&]{}-[\ ]{}237[&]{}EC28[&]{}0.07[&]{}0.99[&]{}1.23[&]{}[\ ]{}307[&]{}SVS2[&]{}0.06[&]{}-0.06[&]{}1.38[&]{}0.19[\ ]{}347[&]{}EC129[&]{}0.06[&]{}0.53[&]{}1.34[&]{}[\ ]{}314[&]{}SVS20 (double)[&]{}-0.03[&]{}0.30[&]{}2.74$^2$[&]{}0.14[\ ]{}318[&]{}EC94[&]{}-0.05[&]{}0.56[&]{}1.35[&]{}[\ ]{}341[&]{}EC121[&]{}-0.14[&]{}0.30[&]{}1.27[&]{}0.21[\ ]{}294[&]{}EC73[&]{}-0.24[&]{}-0.18[&]{}1.31[&]{}0.66[\ ]{}320[&]{}EC91[&]{}-0.24[&]{}0.45[&]{}1.33[&]{}0.32[\ ]{} $^1$ From image serp-nw which has better seeing.\ $^2$ Double source, not extended.\ Identifier acronyms as in Table \[tbl-2\]. The flat-spectrum sources are believed to be in a transition phase between the Class I and the Class II stage, but a few of these might be found to be true protostars. We have listed them in Table \[tbl-5\] with SED indices, $K$-band fwhm and variability amplitude. Both SVS2 and SVS20 have centrosymmetric polarization patterns, indicating that evacuated bipolar cavities surround them, and both are double sources [@hua97]. Among the flat-spectrum sources ISO-159, EC129, SVS20, and ISO-237 (i.e. 30%) are detected as mm continuum sources in our IRAM map. [rlccrrl]{} [ISO]{} & [Other ID]{} & $\alpha_{IR}^{2-14}$ & $\alpha_{IR}^{2-7}$ & $A_K$ & $L_{\star}$ & [Additional information]{}\ \# & & & & (mag) & ($L_{\odot}$) &\ 29[&]{}[&]{}-2.13[&]{}-3.96[&]{}0.32 [&]{}0.009 [&]{}Mid-IR excess, $Ks = 11.17$ from 2MASS [\ ]{}150[&]{}[&]{}-1.30[&]{}-1.59[&]{}0.30 [&]{}0.19 [&]{}15” north of radio source S68 3, $Ks = 10.69$ from 2MASS [\ ]{}158[&]{}BD+01 3687[&]{}-1.88[&]{}-2.92 [&]{}0.0 [&]{}0.086 [&]{}$K = 9.8$ from @cha88. Transition object II-III?[\ ]{}160[&]{}[&]{}-[&]{}-0.69 [&]{}2.28 [&]{}2.4 [&]{}See Appendix \[ind\] [\ ]{}173[&]{}\[CDF88\] 6[&]{}-1.58 [&]{}-2.46 [&]{}0.0 [&]{}0.57 [&]{}$Ks = 9.21$ from 2MASS [\ ]{}202[&]{}[&]{}-1.05[&]{}-1.42 [&]{}0.77 [&]{}0.096 [&]{}Located 30” south-east of HH460[\ ]{}207[&]{}STGM 3[&]{}-1.18 [&]{}-1.48 [&]{}0.35 [&]{}0.57 [&]{}$Ks = 9.64$ from 2MASS, NIR excess [@sog97] [\ ]{}216[&]{}[&]{}-0.86[&]{}-1.47 [&]{}0.0 [&]{}0.057 [&]{}Located 1’ east of HH477[\ ]{}219[&]{}STGM 2[&]{}-[&]{}-1.70 [&]{}0.58 [&]{}0.077 [&]{}$Ks = 11.60$ from 2MASS, NIR excess [@sog97] [\ ]{}221[&]{}IRAS 18271+0102[&]{}-1.14[&]{}-1.53[&]{}1.09 [&]{}36 [&]{}$Ks = 5.38$ from 2MASS, see also @cla91 [\ ]{}224[&]{}EC11[&]{}-[&]{}-1.58 [&]{}0.11 [&]{}0.062 [&]{}Clear IR excess in the $H-K/K-m_7$ diagram [\ ]{}226[&]{}EC13[&]{}-[&]{}-0.84 [&]{}$\sim$ 2.0 [&]{}0.061 [&]{}Clear IR excess in the $H-K/K-m_7$ diagram[\ ]{}231[&]{}EC21[&]{}-[&]{}-1.18 [&]{}3.30 [&]{}0.33 [&]{}Uncertain, see Appendix \[ind\][\ ]{}232[&]{}EC23[&]{}-[&]{}-2.33 [&]{}0.69 [&]{}0.072 [&]{}Uncertain, See Appendix \[ind\] [\ ]{}242[&]{}K8,EC33[&]{}-1.02[&]{}-0.80 [&]{}2.28 [&]{}0.099 [&]{}Variable, $\Delta K=0.21$ [\ ]{}252[&]{}[&]{}-0.87 [&]{}-1.15[&]{}0.70 [&]{}0.24 [&]{}$Ks = 11.12$ from 2MASS [\ ]{}266[&]{}GCNM35,EC51[&]{}-[&]{}-1.77 [&]{}0.76 [&]{}0.021 [&]{}NIR excess[\ ]{}269[&]{}K16,EC56[&]{}-[&]{}-1.58 [&]{}$\sim$ 1.7 [&]{}0.029 [&]{}Variable, $\Delta K=0.21$ [\ ]{}272[&]{}EC59[&]{}-[&]{}-1.82 [&]{}3.42 [&]{}0.086 [&]{}Uncertain, see App.\[ind\] [\ ]{}279[&]{}STGM14,EC66[&]{}-[&]{}-1.56 [&]{}$\sim$ 1.4 [&]{}0.020 [&]{}Variable, $\Delta K=0.21$ + NIR excess[\ ]{}283[&]{}EC67[&]{}-1.14[&]{}-1.22 [&]{}$\sim$ 0.51 [&]{}0.32 [&]{}[\ ]{}285[&]{}GCNM63,EC68[&]{}-0.89[&]{}-0.77 [&]{}1.47 [&]{}0.097 [&]{}[\ ]{}287[&]{}[&]{}-0.80[&]{}-1.04[&]{}0.75 [&]{}0.030 [&]{}Mid-IR excess + optically visible, likely ClassII [\ ]{}289[&]{}EC69,CK10[&]{}-[&]{}-1.42 [&]{}2.56 [&]{}0.45 [&]{}See Appendix \[ind\] [\ ]{}291[&]{}GCNM70,EC70[&]{}-[&]{}-2.05 [&]{}0.09 [&]{}0.016 [&]{}NIR excess[\ ]{}298[&]{}EC74,CK9,GEL4[&]{}-0.48[&]{}-0.29 [&]{}1.67 [&]{}1.7 [&]{}Variable, $\Delta K=0.15$ [\ ]{}304[&]{}EC79,GEL5[&]{}-0.64[&]{}-0.52 [&]{}1.42 [&]{}0.50 [&]{}[\ ]{}309[&]{}EC84,GEL7[&]{}-0.49[&]{}-0.55 [&]{}1.80 [&]{}0.79 [&]{}[\ ]{}319[&]{}EC93,CK13[&]{}-[&]{}-0.89 [&]{}2.35 [&]{}0.97 [&]{}Clear IR excess in the $H-K/K-m_7$ diagram [\ ]{}321[&]{}CK4,GEL12,EC97[&]{}-0.42[&]{}-0.44 [&]{}1.16 [&]{}2.8 [&]{}[\ ]{}328[&]{}EC105,CK8,GEL13[&]{}-0.73[&]{}-0.38 [&]{}0.55 [&]{}2.4 [&]{}Variable, $\Delta K=0.63$[\ ]{}338[&]{}EC117,CK6,GEL15[&]{}-[&]{}-2.21 [&]{}1.36 [&]{}0.25 [&]{}No IR excess, but ass. with nebulosity [\ ]{}348[&]{}EC135,GGD29[&]{}-1.01[&]{}-1.34 [&]{}0.39 [&]{}0.42 [&]{}[\ ]{}351[&]{}EC141[&]{}-[&]{}-1.12 [&]{}2.05 [&]{}0.16 [&]{}Clear IR excess in the $H-K/K-m_7$ diagram[\ ]{}356[&]{}K40,EC152[&]{}-[&]{}-1.03 [&]{}0.70 [&]{}0.030 [&]{}Clear IR excess in the $H-K/K-m_7$ diagram [\ ]{}357[&]{}[&]{}-1.11[&]{}-1.73[&]{}0.25 [&]{}0.096 [&]{}$Ks = 11.23$ from 2MASS [\ ]{}359[&]{}[&]{}-0.92[&]{}-1.20[&]{}0.38 [&]{}0.33 [&]{}$Ks = 10.56$ from 2MASS, strong H$\alpha$ emission (to be publ.)[\ ]{}366[&]{}STGM8[&]{}-1.08[&]{}-1.29 [&]{}0.59 [&]{}0.59 [&]{}[\ ]{}367[&]{}[&]{}-0.87[&]{}-1.16[&]{}0.34 [&]{}0.12 [&]{}$Ks = 11.67$ from 2MASS, strong H$\alpha$ em.[\ ]{}370[&]{}[&]{}-0.83[&]{}-1.00[&]{}0.02 [&]{}0.54 [&]{}$Ks = 10.13$ from 2MASS, dbl in the optical, both H$\alpha$ em. (to be publ.) [\ ]{}379[&]{}[&]{}-1.95[&]{}-2.24[&]{}0.81 [&]{}1.2 [&]{}$Ks = 8.09$ from 2MASS [\ ]{}393[&]{}[&]{}-1.44[&]{}-2.27 [&]{}0.44 [&]{}0.066 [&]{}double in $K$ [\ ]{}407[&]{}[&]{}-1.24[&]{}-2.13[&]{}0.19 [&]{}0.021 [&]{}$Ks = 12.36$ from 2MASS [\ ]{} Identifier acronyms as in Tab \[tbl-2\]. The pre-main sequence sample in Serpens --------------------------------------- Based on the ISOCAM YSO sample found in Table \[tbl-1\] we have used a combination of criteria to arrive at a tentative sample of ClassII sources in Serpens. These are listed in Table \[tbl-6\] with SED indices, whenever available, and additional criteria used to argue for a ClassII status. Since our $K$ imaging from 1996 only covers about 10% of the ISOCAM survey, we have also used published $K$ band photometry whenever available to calculate the SED indices. Also, we have assumed that if the source shows strong mid-IR excess and is optically visible, it is more likely to be a ClassII than a flat-spectrum or ClassI type of YSO. As a supporting argument for ClassII designation we have found for a few mid-IR excess sources without JHK data, strong H$\alpha$ in emission (to be published in a future paper). When the 2MASS All-Sky Data Release became available while we were finalizing our investigation, we used the near-IR photometry from their point source catalogue for the ClassII source candidates where such data was lacking, and found spectral indices in agreement with our above reasoning. The pre-main sequence population also includes the ClassIII type of YSOs, objects which show no IR signatures of an optically thick disk. Since the basic criterion for selecting YSOs in our study is IR excess, we do not sample these sources. With the existing photometric data alone we cannot distinguish ClassIIIs from field stars, except for a few objects in the transition zone between the ClassII and ClassIII phases. The clear gap between the two groups formed in the ISOCAM colour-magnitude diagram in Fig. \[fig-323\] demonstrates that there are few objects undergoing this transition, i.e. that the transition must be rapid. This is evident in all the star formation regions surveyed by ISOCAM. In Serpens at least five sources from Table \[tbl-3\] belong to this transition group, cf. Fig. \[fig-323\], and we list them as candidate ClassIIIs in Table \[tbl-7\]. None of these are located in the central Cloud Core region covered by Arnica deep near-IR imaging. [rlrrr]{} [ISO]{} & [Other ID]{} & $\alpha_{IR}^{7-14}$ & $\alpha_{IR}^{2-14}$ & $\alpha_{IR}^{2-7}$\ \# & & & &\ 112[&]{}[&]{}-1.67[&]{}[&]{}[\ ]{}190[&]{}[&]{}-1.98[&]{}-1.83[&]{}-1.73[\ ]{}210[&]{}[&]{}-2.00[&]{}[&]{}[\ ]{}311[&]{}BD+01 3689B[&]{}-1.73[&]{}-2.21[&]{}-2.54[\ ]{}332[&]{}[&]{}-1.72[&]{}[&]{}[\ ]{} Identifier acronyms as in Table \[tbl-3\]. Luminosity distribution {#lf} ======================= Only about 10% of the Serpens area surveyed by ISOCAM was covered by deep JHK photometry. The total sample is therefore not as homogeneous as our surveys in $\rho$ Ophiuchi [@bon01], Chamaeleon [@per00] and RCrA [@olo99]. Also, since Serpens is at a larger distance the overall sensitivity is lower, and source confusion is worse, especially in the very clustered active centre. With our deep RIJHK photometric coverage over the whole ISOCAM survey, which will be published in a future paper, we expect to extend the population of IR excess sources, and reveal YSOs which are below the ISO sensitivity. With near-IR photometry for the whole sample we can also improve the luminosity estimate of the ClassII sources, by dereddening the J-band fluxes, analogously to what was done for $\rho$ Ophiuchi [@bon01]. At the moment we estimate the luminosities from the mid-IR 6.7 $\mu$m band only, because this is the most homogeneous measurement we have. The 6.7 $\mu$m luminosity function is shown in Fig. \[fig-lf\] for the ClassI and flat-spectrum sources (lower) and ClassII sources (upper). Protostars are expected to radiate most of their luminosity at longer wavelengths. To derive their bolometric luminosities we need to integrate the observed SEDs. As shown in Sect. \[clustering\], however, the four IRAS sources found in the CE region correspond each to a number of protostars. For reasons of source confusion we have not attempted to make calorimetric luminosity estimates of the ClassIs in Serpens, as was done for the ClassI sample in $\rho$ Ophiuchi, see @bon01 who found that the typical fraction of the luminosity radiated between 6.7 and 14.3 $\mu$m for a ClassI source is $\sim 10 \%$. As shown in Fig. \[fig-lf\] the protostars span a range in mid-IR luminosities about equal to that of the ClassIIs. It is clear that the protostars are on the average more luminous at 6.7 $\mu$m than the ClassIIs. This is expected as a large fraction of the luminosity of a protostar is the accretion luminosity. The accretion can be partly continuous and partly happen in bursts, so that the luminosity of a protostar is far from being a simple function of its age and mass. ClassII sources are found to radiate the bulk of their flux in the near-IR. The 6.7 $\mu$m band is expected to be dominated by disk emission, e.g. for a simple black-body at T = 3700 K only about 8% of the luminosity is radiated in this band [@bon01]. Thus we can relate the mid-IR flux to the total stellar luminosity by assuming that what we see is dust emission from a passive reprocessing circumstellar disk. For most ClassII sources any contribution to the luminosity from an active accretion disk is assumed to be negligible. This is supported by measurements of the disk accretion rates of CTTS in Taurus, which are found to have a median value of only $10^{-8}$ M$_{\odot}$ yr$^{-1}$ [e.g. @gul98]. If the dust in the disk is distributed roughly as in the ideal case of an infinite, spatially flat disk, 25 % of the central source luminosity is absorbed and reradiated by the dust [@ada86]. If the disk is flared, the percentage increases, and if it has an inner hole, it decreases. Furthermore, the disk inclination angle $i$ (to the line of sight) determines what fraction of the disk luminosity we observe. In addition, we have to correct for cloud extinction. Since many of our ClassII sources are not detected at 14.3$\mu$m we select to use the 6.7$\mu$m flux only. @olo99 calibrated an empirical relation between the observed 6.7$\mu$m flux and the stellar luminosity by selecting sources with known spectral class and very small extinction in RCrA and Chamaeleon I. Correcting this relation to the larger distance of Serpens we get here that $\log (L_{\star}/L_{\odot}) = \log F_{6.7} (Jy) + 0.95$. Before we can apply this relation, however, we have to correct $F_{6.7}$ for cloud extinction. For the 21 ClassIIs which have Arnica JHK photometry, we assume an intrinsic colour of $(J-H)_0 = 0.85$, which is a median value with small dispersion found for CTTS [@str93; @mey97], and estimate the extinction in the K-band, $A_K = 0.97 \times [(J-H) - (J-H)_0]$, applying the $\lambda^{-1.9}$ extinction law for the near-IR found for Serpens [@kaa99a]. Sources without J-band photometry are either not detected in $J$ - probably owing to extinction - or they are located outside the area observed in the near-IR. In the first case (ISO-Ser-226, 269, 279) we interpolate extinction values of their neighbouring ClassII sources, and in the second case we use the recently available 2MASS PSC and a dereddening as above. We do not transform from 2MASS $Ks$ to Arnica $K$, since the difference is less than or about equal to the estimated errors in the photometry (0.01-0.02 mag). All extinction values are listed in Table \[tbl-6\] expressed as $A_K$, but only approximate values are given for ISO-Ser-226, 269 and 279, since it cannot be known at which depth in the cloud these objects reside. In Sect. \[nir\] we found the relation $A_{6.7} = 0.41 A_K$, which is used to correct the 6.7 $\mu$m flux. The derived stellar luminosity $L_{\star}$ for each ClassII source is also listed in Table \[tbl-6\]. The uncertainty in the luminosity estimate is a function of the uncertainties in: 1) the extinction estimate, 2) the distance, 3) the $L_{\star}$ vs. $F_{6.7}$ relation itself, of which the last two will be systematic errors for the whole sample. Comparison with previous luminosity estimates by @eir92, but scaled to the distance we use, shows that for the 13 objects that overlap in these two samples, there is a large scatter; the fraction $L/L_{EC92}$ varies from 0.5 to 3.7 but the median value is 1.1. The completeness estimate we found in Sect. \[stat\] for detections in the 6.7 $\mu$m band is 6 mJy. Because of the variable extinction this cannot be directly translated to a completeness for ClassII sources in terms of stellar luminosities. We have used the average measured extinction of $A_K = 1.0$ for our ClassII sample and calculated that the completeness limit is at $L = 0.08 L_{\odot}$. Implications for the IMF in Serpens {#imf} =================================== @eir92 first estimated a luminosity function (LF) for the young cluster in the Serpens Cloud Core, including their 51 identified cluster members. The stellar luminosities were obtained from a trapezoidal integration over the detected wavelengths, extrapolated to longer wavelengths and extinction-corrected. This LF showed a pronounced peak around 1 $L_{\odot}$ and a turnover below 0.2 $L_{\odot}$. @gio98 evaluated synthetic K-band luminosity functions (KLFs) and found a best fit to their observed KLF with two bursts of star formation, one 0.1 Myrs ago and the other around 3 Myrs ago, and an underlying Miller & Scalo IMF. This was partly based on their finding of a turnover in the KLF above $K = 14$ mag. A later study expanded the number of Serpens members and found no evidence for a turnover of the KLF down to a limit of $K = 16$ mag [@kaa99a]. The weakness of KLFs is that differential extinction is not taken into account, and for Serpens this is especially important as values as high as $A_V \sim 50$ mag have been observed for background stars, and $A_V \sim 100$ mag is expected for the densest regions of the NW-SE ridge [@whi95]. By selecting sources in the ClassII phase only, we here constrain the age spread somewhat [@bon01]. This phase is thought to last of the order of a few Myr. This means that ClassIIs can have a similar age spread, but we cannot rule out that the ClassIIs once formed in a burst like we see now for the protostars (cf. Sect. \[clustering\]). We have here made the simple assumption of coeval star formation. In Fig. \[fig-lfimf\] we show our observed LF for the sample of 43 ClassII sources (shaded histograms). The bin width of the histogram is $d \log L = 0.325$, based on a factor of two uncertainty in the luminosity estimate, and the histogram has been shifted and the bin size slightly varied to check that the presented distribution remains stable. The observed LF shows a pronounced peak at $L \sim 0.09 L_{\odot}$, but this corresponds roughly to the completeness limit we have estimated for this sample (dotted vertical line). The error bars are the statistical counting errors ($\sqrt N$). The many luminous objects of around $ 1 L_{\odot}$ found by @eir92 have disappeared as the sample has been restricted to ClassII sources. We have assumed coeval populations and three different underlying IMFs: the Salpeter IMF [@sal55], the Kroupa, Tout and Gilmore three-segment power-law IMF [@kro93], and the Scalo three-segment power-law IMF [@sca98], hereafter S55, KTG93 and S98, respectively. With the IMF in the form: $d N/d \log M_{\star} \propto M_{\star}^{\alpha}$, the Salpeter IMF has one single index: $\alpha = -1.35$, originally determined for the mass interval 0.4 to 10 $M_{\odot}$ but in Fig. \[fig-lfimf\] extrapolated to lower masses for reference. Both the KTG93 and S98 IMFs have three different values of $\alpha$ for three (differently divided) segments. @bon01 found for a large sample of ClassII objects in $\rho$ Ophiuchi that the high mass end of the mass function was well fitted with the index $\alpha = -1.7$. Also, they found that the IMF starts to flatten at $M_{flat} = 0.55 M_{\odot}$ and stays “flat” down to $0.055 M_{\odot}$ with a power-law index $\alpha_{flat} = -0.35$. These best fits of the two free parameters $M_{flat}$ and $\alpha_{flat}$ result in a mass function close to both the KTG93 and S98 IMFs for $\rho$ Ophiuchi. On the basis of the pre-main sequence evolutionary models of @dan94 [@dan98], with the 1998 upgrade for low mass stars, we have computed synthetic LFs for each IMF and for the four different ages: 0.5, 1, 2 and 3 Myr of a coeval cluster. Each window in Fig. \[fig-lfimf\] shows the result for one age, and the computed LFs are overplotted on the observed LF. The peak in the LF which wanders towards lower luminosities with increasing age arises because of the deuterium burning phase [@zin93]. Deuterium burning acts like a thermostat and hampers somewhat the contraction down the Hayashi track, causing a build-up of sources in a given luminosity bin for the case of coeval star formation. The peak in the observed LF, however, is approximately at the completeness limit of the sample, and we cannot put too much confidence in it. Nevertheless, it is obvious from the figure that the observed LF excludes ages of less than about 1 Myr for this ClassII population for any of three underlying IMFs, which is in agreement with current understanding of YSO evolution. It is not possible to distinguish between the S98 IMF (solid line) and the KTG93 IMF (dashed), but both of them are plausible for a coeval population with an age of about 2 Myr, all the way down to the estimated completeness limit. Also the Salpeter IMF (dashed-dotted line) is in agreement with the observed LF for ages of 2-3 Myr at the high luminosity end. Its extrapolation to lower masses is given in the plots just for reference. Note, however, that if star formation proceeds in a more continuous fashion, the structures in the model LFs will smooth out. We have checked the model LFs for the case of continuous star formation over several intervals from 0.5 to 3 Myrs. Although the statistics in the observed LF is poor, it seems we can exclude the combination of the above tested IMFs and continuous SF over periods of less than 2 Myrs. We also tested the case of a single star formation burst taking place 2 Myr ago but having various durations from 0.2 to 1.0 Myr. The peak in the observed LF is not secure enough to discriminate between the various burst durations. For the tested IMFs and SF histories, all we can conclude is that an age of less than 2 Myr for the Class II population seems implausible. Corrections for binarity have not been applied, but such a correction would increase the populations in the lowest luminosity bins. We conclude that with the currently obtained LF the ClassII mass function in Serpens is similar to the one in $\rho$ Ophiuchi [@bon01], even though flat-spectrum sources were included in the $\rho$ Ophiuchi sample. We note that the typical stellar mass found for the Serpens ClassII sample (i.e. the median) is 0.17 $M_{\odot}$, the same as in $\rho$ Ophiuchi [@bon01]. The total mass of the ClassII sample in Serpens is $16.3 M_{\odot}$. We have also used the @bar98 evolutionary tracks and calculated synthetic LFs with the same underlying IMFs as above and for the ages: 1, 2, 3, and 4 Myr. This model gives basically the same result as that of @dan98 for the degree of confidence we can put in our observed LF, taking the large error bars in the histogram into account. Adopting the above deduced age of 2 Myr for the Serpens ClassII population, our study is estimated to be complete to about $0.17 M_{\odot}$, but to reach down to $0.03-0.04 M_{\odot}$. Taking $0.08 M_{\odot}$ as the border between low mass stars and brown dwarfs, we find 9 sub-stellar size objects in the Serpens ClassII sample (those with $L_{\star} \le 0.04 L_{\odot}$ in Tab. \[tbl-6\]). These are young [*brown dwarfs*]{} (BDs) apparently going through a ClassII phase just like low mass stars. They all have IR excesses with no apparent deviations from other ClassIIs in any aspect, which is consistent with the idea that they formed in the same way as low-mass stars do. Our sample of BDs is not complete, however, and we cannot discuss the frequency of occurrence of disks among BDs or set strong constraints on BD formation models. Based on the age estimate above, we find that $\sim$ 21% of the ClassII sample studied is made up of free-floating brown dwarfs. This is similar to the percentage found in $\rho$ Ophiuchi [@bon01]. The percentage in terms of mass is, however, only about 3% of the total mass for the whole ClassII sample. Spatial distributions {#clustering} ===================== The spatial distribution of the ISOCAM sources is shown in Fig. \[figmap\] together with the location of the three main rasters CE, CW and CS. The filled circles indicate sources with mid-IR excesses (cf. Table \[tbl-2\]), the open circles are the sources without mid-IR excesses (cf. Table \[tbl-3\]), the plus signs are sources with 6.7 $\mu$m fluxes only, and crosses sources with 14.3 $\mu$m fluxes only. The small box outlines the $8\arcmin \times 6\arcmin$ field for which deep JHK imaging is obtained. The overall spatial distribution of the mid-IR excess YSOs shows a very clear concentration along a SE-NW oriented ridge within the CE field, mainly coinciding with what is known as the Serpens Cloud Core. In the CW field there is practically no sign of activity, while the distribution of IR-excess YSOs in the CS field is quite scattered. The majority of the mid-IR excess sources seem to be distributed along a curved filament pointing towards a centre of curvature around 18:29:10, +01:05:00 (J2000). Nothing particular was found at this location searching SIMBAD, however, and whether star formation along this arc may be externally triggered remains speculative. The IRAS Point Source Catalogue contains six sources in the area surveyed by ISOCAM (cf. Appendix \[ind\] for details). Clustering scales ----------------- The concentration of young sources found along the NW-SE ridge, corresponds well to the pronounced density enhancement seen in the IRAM 1300 $\mu$m map in Fig. \[fig-iram\]. It is evident that the ClassI and the flat-spectrum sources are spatially more confined to the NW-SE ridge than the ClassII sources, which have a more scattered distribution. In particular, the protostars seem to form a set of sub-clusters lined up along the ridge. These sub-clusters are in good positional agreement with the kinematically separated N$_2$H$^+$ cores A,B,C,D found by @tes00, especially with B,C, and D. The SVS4 sub-cluster lies between cores A and B, however, and is also displaced to the west of the SMM4 mm continuum peak. The spatial distribution of all the classes of YSOs within the central 6’ $\times$ 8’ of the Cloud Core can be seen overlaid on a $K$ band image in Fig. \[fig-kim\]. To quantify the scale of sub-clustering of the two populations: protostar candidates (ClassI and flat-spectrum sources) and ClassII sources, we show for each population the distribution of separations between the sources. Figure \[fig-clust\] shows the result as the number of pairs vs. separation. The bin size was chosen to be 6 times the minimum separation. The difference between the two populations is clear. The protostar candidates (circles) have a minimum scale of clustering of about 0.12 pc (95$\arcsec$), a secondary peak at 0.36 pc, and practically no scattered distribution, while the ClassII sources have a minimum clustering scale of about 0.25 pc, a secondary peak at 0.59 pc and a much more distributed population, in agreement with the maps. The clustering scale of 0.12 pc for the protostars agrees well with the range of radii of the N$_2$H$^+$ cores: 0.055 to 0.115 pc [@olm02], and the second peak at about 0.4 pc reflects well the size scale of the most active region in the Serpens Core. The correspondence between the high density gas and the compactness of the protostar clusters indicates that these sources are found close to their birth place. Assuming a typical velocity dispersion in the range from 0.1 to 1 kms$^{-1}$, YSOs younger than $10^5$ yrs can be expected to have moved at most $0.1$ pc from their birth place. The typical ages of ClassI sources are estimated to be of this order in other regions such as $\rho$ Ophiuchi [@gre94] and Taurus [@ken90b]. Our results indicate that the protostar candidates in each cluster must have formed from the same core at about the same time, which may put constraints on models of cloud fragmentation and core formation within clumps. Each sub-cluster contains between 6 and 12 protostar candidates, and this is just a lower limit because of the spatial resolution of ISOCAM. Adopting the Serpens distance of $260$ pc yields a [*protostar*]{} surface density in the sub-clusters in the range from 500-1100 pc$^{-2}$. To our knowledge, such a high spatial density of protostars has not been found in any other nearby star formation region, although we note that the Class0 surface density in Orion-OMC3 is comparable to that in the Serpens Core [@chi97]. The sub-clustering in the observed 2D projection of the cluster does not necessarily correspond to the true spatial distribution of the objects, but it seems highly unlikely that these concentrations are due to elongated cloud structures seen end-on. Sub-clustering was also found for the ClassII sources in $\rho$ Ophiuchi [@bon01], with a similar size scale as for the ClassII population in Serpens. It seems likely that the strong clustering of Serpens protostars will evolve into looser clusters of ClassIIs over a few 10$^6$ yrs. The expansion of the clustering scales from 0.12 to 0.25 pc and the assumed ages of these two populations suggest that the velocity dispersion of the young stars is of the order of only 0.05 kms$^{-1}$. Star formation rate and efficiency ================================== Assuming that a recent burst of star formation took place $\sim 2 \times$ 10$^5$ yrs ago, producing the current population of protostars, we can derive a rough estimate of the star formation rate (SFR) and the star formation efficiency (SFE) in this burst. We use our sample of protostar candidates, the flat-spectrum and ClassI sources found clustered along the NW-SE oriented ridge. We are not able to estimate their masses, but we hypothesize that they follow the same IMF as the ClassII sample. Comparing the total mass of the ClassII sample and correcting for the number of sources, we estimate a total mass of 12.1 $M_{\odot}$ for the protostar population. If these sources were formed gradually over $2 \times$ 10$^5$ yrs, the SFR in this microburst would be 6.1 $\times$ 10$^{-5}$ M$_{\odot}$/yr, or for typical masses of 0.17 M$_{\odot}$, one newborn star every $\sim$ 2800 yr. This SFR is significantly higher than the one found for the whole YSO population in $\rho$ Ophiuchi [@bon01]. Comparing the protostellar masses with the gaseous mass in these cores will give us the star formation efficiency, defined as $SFE = M_{\rm star}/(M_{\rm star}+M_{\rm gas})$. According to @olm02 the two sub-clumps they name NW and SE, which contain our four sub-clusters of protostars, are virialised with masses of 60 $M_{\odot}$ each. For our protostar candidates this yields a SFE of about 9 %. This is the [*local*]{} SFE for the sub-clusters along the NW-SE ridge. The global SFE calculated for the ClassIIs and protostars over the entire cluster (i.e. 28.8 $M_{\odot}$ of stellar mass) is much lower. In the literature we find estimates of the total gas mass from 300 $M_{\odot}$ [@mcm00] to 1500 $M_{\odot}$ [@whi95] for surveyed areas of 5 to 10 arc minutes, which gives only upper limits on the SFE of the order of 2-9 %. The above estimates are based on the hypothesis that the protostars follow the same IMF as the one found for the ClassII sources. We have not attempted to derive the protostar mass function. Future high resolution far-IR observations (e.g. ESA’s Herschel Space Observatory) would be needed to yield bolometric luminosities of these clustered protostars, and with some assumptions on the accretion rates one could estimate their masses. Summary and conclusions ======================= We have used ISOCAM to survey 0.13 square degrees of the Serpens Cloud Core in two broad bands centred at 6.7 and 14.3 $\mu$m. In combination with our ground based deep $JHK$ imaging of the 48 square arcminute central region as well as additional $K$ band imaging of about 30 square arcminutes to the NW, and a 1.3 mm IRAM map of the central dense filament, we have investigated the mid-IR properties of the young stellar population. The following results are found: - The number of point sources with reliable flux measurements are 392 at 6.7 $\mu$m and 139 at 14.3 $\mu$m. Of these, 124 are detected in both bands. - On the basis of one single colour index, $[14.3/6.7]$, we found that 53 of the 124 objects possess strong mid-IR excesses. Only 28 of these were previously suggested YSO candidates. The large scale spatial distribution of these mid-IR excess sources is strongly concentrated towards the Cloud Core, where it is elongated along NW-SE. - The near-IR $J-H/H-K$ diagram is found to have an efficiency of less than 50% in detecting IR excess sources. This efficieny is comparable to that one found in star formation regions in general from ISOCAM data [@kaa00] and $\rho$ Ophiuchi in particular [@bon01] and should be kept in mind when interpreting $JHK$ based data in terms of disk fractions. - The $H-K/K-m_7$ diagram separates well intrinsic IR excess from the effects of reddening. From this diagram we were able to increase the number of mid-IR excess sources to 70. This means a fractional increase in the investigated region by 25%. - Combination of near-IR and mid-IR photometry for reddened stars without IR excesses enables us to estimate the extinction at 6.7 and 14.3 $\mu$m relative to that in the $K$ band in the Serpens direction. We find A$_{7}$ = 0.41 A$_K$ and A$_{14}$ = 0.36 A$_K$. Our results agree with the extinction law measured towards the Galactic centre by @lut96, as well as with the results from the ISOGAL survey [@jia03]. - Classification of the Serpens YSOs in terms of the SED indices gives 20 ClassI sources, 13 flat-spectrum sources, and 43 ClassII sources. The number of ClassI sources appears to be exceptionally large along the NW-SE oriented ridge, and the number fraction ClassI/ClassII is almost 10 times higher than normal, an indication that this part of the cluster is extremely young and active. - The mid-IR luminosities of the ClassI sources are on the average larger than those of the ClassII sources. Since ClassII sources are expected to have SEDs which peak approximately in the near and mid-IR, in contrast to ClassIs which radiate most of their luminosity in the far-IR, we conclude that there is a weak indication of luminosity evolution in the Serpens Cloud Core. - We have estimated extinction and stellar luminosities for the 43 ClassII sources found in our survey. The ClassII luminosity function is found to be compatible with co-eval formation about 2 Myrs ago and an underlying IMF of the three-segment power-law type [@kro93; @sca98], similar to the mass function found in $\rho$ Ophiuchi [@bon01]. With this assumption on age, every fifth ClassII is a young brown dwarf. - Except for one case, the ClassI sources are exclusively found in sub-clusters of sizes $\sim$ 0.12 pc distributed along the NW-SE oriented ridge. The sub-clusters also contain several flat-spectrum sources. In total, each core has formed between 6 and 12 protostars (lower limit) within a very short time. The spatial distribution of the ClassII sources, on the other hand, is in general much more dispersed. - On the assumption that the protostar candidates follow roughly the same IMF as the ClassIIs, we derive a SFR of 6.1 $\times$ 10$^{-5}$ M$_{\odot}$/yr and a local SFE of 9% in the recent microburst of star formation forming the dense sub-clusters of protostars. The results presented in this study show evidence that the sub-clusters in the central part of the Serpens Cloud Core were formed by a recent microburst of star formation. The extreme youth of this burst, deduced from the compact clusters of protostar candidates, is supported by independent investigations, such as the rich collection of Class0 sources found by @cas93 and @hur96 in the same regions. In addition to the clustered protostar population, we also find a more distributed population of ClassII sources, for which we have deduced an age more or less typical of ClassIIs found in other regions (2 Myr). In addition to these YSO generations, there is probably also a population of ClassIII sources, which is undistinguishable from the field star population in our study, but which should be looked for with proper search tools, such as e.g. X-ray mapping [@gro00]. @cas93 suggested that their submm sources without near-IR counterparts represented a second phase of active star formation in Serpens. Here we show the co-existence of sources in the various evolutionary stages from ClassII and flat-spectrum to ClassI and Class0 sources. While we find an age estimate of 2 Myr for the ClassII population, it is highly unlikely that the clustered ClassI and flat-spectrum sources are older than a few 10$^5$ yrs. Our results therefore support the conclusions of @cas93 that star formation has proceeded in several phases in Serpens. The authors wish to thank Carlos Eiroa for stimulating discussions and helpful comments. We also thank an anonymous referee for comments that helped us improve the paper. The ISOCAM data presented in this paper were reduced using “CIA”, a joint development by the ESA Astrophysics Division and the ISOCAM Consortium led by the ISOCAM PI, C. Cesarsky, Direction des Sciences de la Matière, C.E.A., France. The near-IR data presented in this paper were obtained with the ARcetri Near InfraRed CAmera ([Arnica]{}) at the Nordic Optical Telescope in 1996, and Carlo Baffa, Mauro Sozzi, Ruggero Stanga and Leonardo Testi from the [Arnica]{} team are acknowledged for the instrument support. The Nordic Optical Telescope is operated on the island of La Palma jointly by Denmark, Finland, Iceland, Norway, and Sweden, in the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrofisica de Canarias. This publication made use of the SIMBAD database, operated at CDS, Strasbourg, France, and data products from the Two Micron All Sky Survey, which is a joint project of the University of Massachusetts and the Infrared Processing and Analysis Center/California Institute of Technology, funded by the National Aeronautics and Space Administration and the National Science Foundation. Financial support from the Swedish National Space Board is acknowledged. 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Mannings, V., Boss, A.P., Russel, S.S., University of Arizona Press, Tucson, p. 429 Notes on some individual sources {#ind} ================================ IRAS 18269+0116 (ISO-159, 160) ------------------------------ At the location of IRAS 18269+0116 we find ISO-159 and ISO-160, of which the latter is not detected at 14.3 $\mu$m. A $K$-band image shows that ISO-159 has a bow-shaped nebulosity 7$\arcsec$ to WNW (at a level of about 3-4 $\sigma$, see Fig. \[fig-nw\]). The SED index classifies ISO-159 as a flat-spectrum source, while ISO-160 is either an extremely extinguished background source (analogous to CK2) or most likely a ClassII object (cf Fig. \[fig-k2k3\]). IRAS 18271+0102 (ISO-221) ------------------------- Identified as a YSO by @cla91 on the basis of its steeply rising SED towards longer wavelengths. With ISOCAM we see two very bright point sources (221 and 238), of which only ISO-221 has IR-excess. EC21 (ISO-231) -------------- This source was suggested as a YSO on the basis of its location in the $K$ vs $H-K$ diagram by @eir92. Deeper near-IR imaging detected the source also in the $J$ band, but found no near-IR excess [@kaa99a]. It is located in the very dense and active S68N region, but whether it has intrinsic IR-excess at 7 $\mu$m or is only reddened (in that case, by 3 magnitudes in the $K$ band, which translates to $A_V \sim 30$ magnitudes), is not clear. EC23 (ISO-232) -------------- Proposed as a YSO on the basis of its $H-K$ colour and 3.08 $\mu$m ice feature [@eir92], but not found to have near-IR excess in a deeper JHK study [@kaa99a], nor mid-IR excess in this study. IRAS 18272+0114 (DEOS, EC37, EC38, EC53, 234, 237, 241) ------------------------------------------------------- IRAS 18272+0114 is resolved into a cluster of protostar candidates: the Deeply Embedded Outburst Source [@hod96], EC37, EC38 and EC53 [@eir92], and three new ISOCAM sources (234, 237, 241). ISO-241 is within the positional uncertainties associated with S68N, a radio source detected by @mcm94 and found to satisfy the Class0 criteria. We use the position at its 450 $\mu$m emission peak [@wol98]. ISO-241 is a ClassI, however, from the shape of its SED from 2 to 14 $\mu$m. @hod99 denotes the $K$-band source SMM9-IR, since it is located close to the SMM9 positions quoted in the literature [@whi95; @wol98; @tes98; @dav99], and shows that it has a very faint $K$ companion about 2$\arcsec$ to the NE. EC41 (ISO-258a) and FIRS1 (ISO-258b) ------------------------------------ IRAS 18273+0113, also called FIRS1 and SMM1 [@rod89; @eir89; @cas93], is associated with 5 mid-IR excess sources (EC41, 253, 270, 276, 277) plus one 14.3 $\mu$m source (258b). ISO-Ser-258a is in agreement with the near-IR source positions of EC41 [@eir92] and GCNM23 [@gio98], and these are most likely the same source, a ClassI YSO according to the $\alpha_{\rm IR}^{2-14}$ index. The ISOCAM source appeared slightly extended towards the SE both at 6.7 and 14.3 $\mu$m in most of our data. @lar00 found evidence for two separate sources, however, tracing pixel by pixel the 5 to 15 $\mu$m SED from CVF observations. This prompted us to take a more careful look, and we identified two clearly separated sources at 14.3 $\mu$m (rather than one extended source) in one of the deeper (and higher resolution) survey maps, D2 (cf. Fig. \[fig-D2\]). The new source (called 258b), is located about 13$\arcsec$ to the SE of EC41. Since the radio position at RA(2000) = 18 29 49.73, DEC(2000) = 01 15 20.8 [@cur93] is 8.6$\arcsec$ to the west of 258b we find it unlikely that this source is the mid-IR counterpart of the far-IR source FIRS1, but it could be scattered light we see in the mid-IR. It is not entirely clear if this source is just a knot of the extended emission which appears to begin somewhat south of DEOS and is seen in both filters in Fig. \[fig-D2\]. The northernmost part of this extended emission is, however, disturbed by a memory feature from the strong DEOS source in the raster scanning. IRAS 18273+0059 (ISO-259) ------------------------- A prominent extended emission seen in the ISOCAM images corresponds in position to IRAS 18273+0059 [@cla91]. With ISOCAM, however, we see no apparent bright point source embedded in the nebulosity, but a moderately faint object is detected at 6.7 $\mu$m only. On optical plates the SE part of this extended emission is recognised as an optical reflection nebula. IRAS 18274+0112 --------------- IRAS 18274+0112 is also composed of multiple mid-IR excess sources centred on SVS20 (cf. Fig \[fig-kim\]). Higher spatial 25 $\mu$m resolution was obtained by @hur96, and their sources PS1 and PS2 (EC129) are among the bright mid-IR excess sources resolved with ISOCAM. EC59 (ISO-272) -------------- Included as a YSO candidate solely on the basis of its location in the $K$ vs $H-K$ diagram [@eir92], and no near-IR excess found in a deeper JHK study [@kaa99a]. Since we have no indication of mid-IR excess its status remains inconclusive. EC69 [ISO-289]{} ---------------- This source (also called CK10) was suggested as a YSO on the basis of its location in the $K$ vs $H-K$ diagram [@eir92], and while near-IR excess was found by @sog97, no near-IR excess was found in a deeper JHK study [@kaa99a]. The J-band magnitude has varied though: 15.7 in 1989, 14.96 in May 1992, 15.6 in Nov 1992, and 16.15 in Aug 1996 [@eir92; @sog97; @gio98; @kaa99a]. We conclude that it is a YSO, and with an $\alpha_{\rm IR}^{2-7} = -1.42$ it belongs in the ClassII group. EC95 & EC92 (ISO-317) --------------------- ISO-317 is located at the position of EC92 according to the coordinates given by @eir92, but the positional uncertainty is roughly $\pm 3\arcsec$. Also, the spatial resolution of ISOCAM is not sufficient to resolve the two neighbours EC92 and EC95, and it is likely that both are included in the ISOCAM fluxes. The mid-IR SED suggests a flat-spectrum source. Slightly closer to the position of EC95, but also with some positional uncertainty, @pre98 found an extremely strong X-ray source (Ser-X3). ISO-331 ------- This bright mid-IR source is detectable in $K$ as a nebulous spot only. Probably the $K$ flux is dominated by scattered light, and it is therefore not entirely correct to place ISO-331 in colour diagrams together with continuum sources. Nevertheless, the $K$ measurement gives a lower limit to the colour index of the object, which is classified as a ClassI source. Its mid-IR position is about 25to the NNW of SMM2 [@cas93], and coincides roughly with the VLA 3.6 cm continuum source \#11 found by @bon96. CK2 (ISO-337) ------------- According to ISOCAM data the source CK2, which is extremely red in the near-IR, has no mid-IR excess and can be interpreted as a background source, consistent with the suggestions of @chu86, @chi94, @cas96 and @sog97. The extinction towards CK2 is high, A$_V >$ 50 mag. [^1]: Based on observations with ISO, an ESA project with instruments funded by ESA Member States (especially the PI countries: France, Germany, the Netherlands and the United Kingdom) and with participation of ISAS and NASA. [^2]: Tables \[tbl-2\] and \[tbl-3\] are only available in electronic form at the CDS via anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or via http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/ [^3]: The spectral index is defined as $\alpha_{\rm IR} = d \log (\lambda F_{\lambda})/(d \log \lambda)$ and is usually calculated between 2.2 $\mu$m and 10 or 25 $\mu$m. [^4]: A joint development by the ESA Astrophysics Division and the ISOCAM Consortium led by the ISOCAM PI, C. Cesarsky, Direction des Sciences de la Matière, C.E.A., France [^5]: This means the central 8$\arcmin \times$ 6$\arcmin$ JHK field and the NW field shown in Fig. \[fig-nw\]. [^6]: The same was found in Cha I (unpublished) as well as in $\rho$ Ophiuchi [@bon01]. This implies that, for all these star formation regions, the extinction is slightly larger at 6.7 $\mu$m than at 14.3 $\mu$m, see also @olo99. Hence, we see no minimum in the extinction law in the 4-8 $\mu$m interval, as expected for standard graphite-silicate mixtures [@mat77; @dra84]. But our results agree with the extinction law found towards the Galactic centre by @lut96 and the ISOGAL results in @jia03 who find A$_{\rm 14}$ = 0.87 A$_{\rm 7}$ when using the Rieke & Lebofsky law updates[@rie89] for the near-IR extinction. [^7]: According to @and00 the following sources satisfy the Class0 criteria: FIRS1, S68N, SMM3, and SMM4, all of which are bright sources in the 1.3 mm IRAM map in Fig. \[fig-iram\]. In addition, SMM2 is a candidate Class0. ISO-258b, which may be a knot in the extended emission (cf. Fig \[fig-D2\]), is 8” away from FIRS1, but could result from scattered light (see Sect. \[ind\]). SMM2 and SMM3 have no ISOCAM detections within the positional uncertainties, that is to a $3\sigma$ upper limit of 6 and 9 mJy for 6.7 and 14.3 $\mu$m, respectively (cf. Table \[tbl-1\]).

--- abstract: 'We give upper bounds on the essential dimension of (quasi-)simple algebraic groups over an algebraically closed field that hold in all characteristics. The results depend on showing that certain representations are generically free. In particular, aside from the cases of spin and half-spin groups, we prove that the essential dimension of a simple algebraic group $G$ of rank at least two is at most $\dim G - 2(\operatorname{rank}G) - 1$. It is known that the essential dimension of spin and half-spin groups grows exponentially in the rank. In most cases, our bounds are as good or better than those known in characteristic zero and the proofs are shorter. We also compute the generic stabilizer of an adjoint group on its Lie algebra.' address: - 'Garibaldi: Center for Communications Research, San Diego, California 92121' - 'Guralnick: Department of Mathematics, University of Southern California, Los Angeles, CA 90089-2532' author: - Skip Garibaldi - 'Robert M. Guralnick' bibliography: - 'skip\_master.bib' title: 'Essential dimension of algebraic groups, including bad characteristic' --- [^1] Introduction ============ The essential dimension of an algebraic group $G$ is the minimal transcendence degree of the field of definition of a versal $G$-torsor. (Although inaccurate, one can think of it as the number of parameters needed to specify a $G$-torsor.) This invariant captures deep information about algebraic structures with automorphism group $G$, and it is difficult to calculate. For example, the fact that $\operatorname{ed}(\operatorname{PGL}_2) = \operatorname{ed}(\operatorname{PGL}_3) = 2$ corresponds to the classical fact that division algebras of dimension $2^2$ or $3^2$ over their center are cyclic, and it is an open problem whether the essential dimension of $\operatorname{PGL}_p$ is 2 for primes $p \ge 5$ [@ketura Problem 6.2], although it is known that $\operatorname{ed}(\operatorname{PGL}_n)$ is not $O(n)$ [@M:lower]. Therefore, the bulk of known results on essential dimension provide upper or lower bounds, as in [@ChSe], [@GiRei], [@BaekM], [@MR3039772], etc. (See [@M:bbk], [@M:ed], or [@Rei:ICM] for a survey of the current state of the art.) In this paper, we provide upper bounds on $\operatorname{ed}(G)$ for every simple algebraic group $G$ over an algebraically closed field $k$, regardless of the characteristic of $k$. Our bounds are in some cases as good as (Theorem \[A\]) or better (Theorems \[big.O\] and \[PSp\]) than the bounds known in characteristic zero, and have shorter proofs. One summary consequence of our results is the following. \[big.O\] Let $G$ be a simple algebraic group over an algebraically closed field. Then $$\operatorname{ed}(G) \le \dim G - 2 (\operatorname{rank}G) - 1$$ or $G \cong \operatorname{PGL}_2$, or $G \cong \operatorname{Spin}_n$ or $\operatorname{HSpin}_n$ for some $n$. For the excluded cases, $\operatorname{ed}(\operatorname{PGL}_2) = 2$ (Example \[A1.ed\]). For spin and half-spin groups, essential dimension grows exponentially in $n$ [@BRV] whereas the dimension, $\binom{n}{2}$, is quadratic in $n$. Specifically, $\operatorname{ed}(\operatorname{Spin}_n) > \dim \operatorname{Spin}_n$ for all $n \ge 19$ and $\operatorname{ed}(\operatorname{HSpin}_n) > \dim \operatorname{HSpin}_n$ for $n$ divisible by 4 and $\ge 20$. Adjoint groups {#adjoint-groups .unnumbered} -------------- Under the additional hypotheses that $G$ is adjoint and $\operatorname{char}k = 0$, it is well known that an adjoint semisimple group $G$ acts generically freely on $\operatorname{Lie}(G) \oplus \operatorname{Lie}(G)$[^2] and consequently $\operatorname{ed}(G) \le \dim G$, as was pointed out in [@BRV Remark 3-11]. In this setting, the stronger bound in Theorem \[big.O\] was proved in [@Lemire]. Dropping the hypothesis on $\operatorname{char}k$ but still assuming $G$ is adjoint, the bound $\operatorname{ed}(G) \le \dim G - 2(\operatorname{rank}G)$ was recently proved in [@BGS Cor. 10]. Theorem \[big.O\] for adjoint groups includes the following bounds, where we write $T^{{\mathrm{adj}}}_n$ for an adjoint group of type $T_n$: $$\begin{gathered} \label{BGS.1} \operatorname{ed}(E^{{\mathrm{adj}}}_6) \le 65, \quad \operatorname{ed}(E^{{\mathrm{adj}}}_7) \le 118, \quad \operatorname{ed}(E_8) \le 231, \\ \text{and} \quad \operatorname{ed}(D_n^{{\mathrm{adj}}}) \le \text{$2n^2 - 3n - 1$ for $n \ge 4$.} \notag\end{gathered}$$ (The adjoint group $D_n^{{\mathrm{adj}}}$ is sometimes denoted $\operatorname{PSO}_{2n}$.) These bounds agree with those in [@Lemire] for characteristic 0. (The number 112 given there for $E_7^{{\mathrm{adj}}}$ was a typo.) We remark that the essential dimension of ${\mathrm{SO}}_{2n+1}$ (adjoint of type $B_n$) is $2n$ if $\operatorname{char}k \ne 2$ [@Rei:ed] and is $n+1$ if $\operatorname{char}k =2$ [@BabicCh]. For ${\mathrm{SO}}_{2n}$ (of type $D_n$), the essential dimension is $2n-1$ if $\operatorname{char}k \ne 2$ and is $n$ or $n+1$ if $\operatorname{char}k = 2$. Groups of type $C$ {#groups-of-type-c .unnumbered} ------------------ We give the following upper bound for adjoint groups of type $C_n$, which improves on the bound $2n^2 - 3n -1$ given in [@Lemire] in characteristic zero. \[PSp\] Over an algebraically closed field $k$ and for $n \ge 4$: $$\operatorname{ed}(\operatorname{GL}_{2n}/\mu_2) \le \operatorname{ed}(\operatorname{PSp}_{2n}) \le \begin{cases} 2n^2 - 3n - 4 & \text{if $\operatorname{char}k \!\not\vert\, n$ or $n=4$} \\ 2n^2 - 3n - 6 & \text{if $\operatorname{char}k \mid n$ and $n > 4$.} \end{cases}$$ The interesting case of Theorem \[PSp\] is when $n$ is even; in the special case where $n$ is odd, the natural map $\operatorname{PGL}_2 \times {\mathrm{SO}}_n \hookrightarrow \operatorname{PSp}_{2n}$ gives a surjection $H^1(k, \operatorname{PGL}_2) \times H^1(k, {\mathrm{SO}}_n) \to H^1(k, \operatorname{PSp}_{2n})$, and $\operatorname{ed}(\operatorname{PSp}_{2n}) = n + 1$ for $n$ odd, cf. [@MacD:CIodd p. 302]. It is known that $\operatorname{ed}(\operatorname{GL}_8/\mu_2) = 8$ if $\operatorname{char}k \ne 2$ [@BaekM Cor. 1.4] and is $\le 10$ if $\operatorname{char}k = 2$ [@Baek:edp Cor. 1.4], somewhat better than the bound $\le 16$ provided by Theorem \[PSp\]. Groups of type $A$ {#groups-of-type-a .unnumbered} ------------------ The essential $p$-dimension of $\operatorname{GL}_n / \mu_m$ and of $\operatorname{SL}_n / \mu_m$ have been studied in [@BaekM], [@Baek:edp], and [@ChM:edpA]. Here and in we give upper bounds for the essential dimension (without the $p$). \[A\] Over an algebraically closed field and for $m$ dividing $n \ge 4$, we have: $$\operatorname{ed}(\operatorname{PGL}_n) \le n^2 - 3n + 1 \quad \text{and} \quad \operatorname{ed}(\operatorname{SL}_n / \mu_m) \le n^2 - 3n + n/m + 1.$$ If $m = 1$, then $\operatorname{ed}(\operatorname{SL}_n) = 0$. If $m = n$, then $\operatorname{SL}_n / \mu_m = \operatorname{PGL}_n$. Our bound for $\operatorname{PGL}_n$ agrees with the one given by [@Lemire] in $\operatorname{char}k = 0$; we remove this hypothesis. A better bound on $\operatorname{ed}(\operatorname{PGL}_n)$ is known for $n$ odd [@LRRS]. See [@M:ed §10] or [@Rei:ICM §7.6] for discussions of the many more results on upper bounds for $\operatorname{PGL}_n$. If $m = 2$, then (applying Lemma \[GLSL\]) the bound in Theorem \[PSp\] is better by about a factor of 2. Exceptional groups {#exceptional-groups .unnumbered} ------------------ Concerning exceptional groups, a series of papers [@Lemire], [@MacD:F4], [@MacD:E7], [@LoetscherMacD] have led to the following upper bounds for exceptional groups: $$\operatorname{ed}(F_4) \le 7, \quad \operatorname{ed}(E_6^{{\mathrm{sc}}}) \le 8, \quad \text{and} \quad \operatorname{ed}(E_7^{{\mathrm{sc}}}) \le 11 \quad \text{ if $\operatorname{char}k \ne 2, 3$.}$$ (Here $F_4$, $E_6^{{\mathrm{sc}}}$ and $E_7^{{\mathrm{sc}}}$ denote simple and simply connected groups of types $F_4$, $E_6$, and $E_7$; the displayed upper bounds are meant to be compared with the dimensions of 52, 78, and 133 respectively. These upper bounds are close to the known lower bounds of 5, 4, and 8 for $\operatorname{char}k \ne 2, 3$.) The proofs of these upper bounds for $F_4$ and $E_7^{{\mathrm{sc}}}$ are technical and detailed calculations. The following weaker bounds have the advantage of simple proofs and holding for fields of characteristic 2 and 3. \[ed.thm\] Over an algebraically closed field, we have: $$\operatorname{ed}(F_4) \le 19, \qquad \operatorname{ed}(E_6^{{\mathrm{sc}}}) \le 20, \qquad \text{and} \qquad \operatorname{ed}(E_7^{{\mathrm{sc}}}) \le 49.$$ The proofs of most of the theorems above rely on computations of the (scheme-theoretic) stabilizer of a generic element in a representation of $N_G(T)$ for $T$ a maximal torus in $G$. The proof of Theorem \[PSp\] uses the computation of a generic stabilizer in the action of $\operatorname{Sp}$ on $L(\lambda_2)$. Using the same technique, we prove calculate the generic stabilizer of an adjoint group $G$ acting on its Lie algebra. In particular, this stabilizer is connected unless $\operatorname{char}k = 2$. In the final section, we give upper bounds on essential dimension for small spin and half-spin groups, completing the list of upper bounds on $\operatorname{ed}(G)$ for $G$ simple and connected over an algebraically closed field. Generically free actions ======================== Let $G$ be an affine group scheme of finite type over a field $k$, which we assume is algebraically closed. (If $G$ is additionally smooth, then we say that $G$ is an *algebraic group*.) We put $G^\circ$ for the identity component of $G$. If $G$ acts on a variety $X$, the stabilizer $G_x$ of an element $x \in X(k)$ is a sub-group-scheme of $G$ with points $$G_x(R) = \{ g \in G(R) \mid gx = x \}$$ for every $k$-algebra $R$. Statements “for generic $x$” means that there is a dense open subset $U$ of $X$ such that the property holds for all $x \in U$. Suppose $G$ acts on a variety $X$ in the sense that there is a map of $k$-schemes $G \times X \to X$ satisfying the axioms of a group action. We say that $G$ acts *generically freely* on $X$ if there is a nonempty open subset $U$ of $X$ such that for every $u \in U$ the stabilizer $G_u$ is the trivial group scheme 1. It is equivalent to require that $G_u(k) = 1$ and $\operatorname{Lie}(G_u) = 0$. Indeed, if $\operatorname{Lie}(G_u) = 0$, then $G_u$ is finite étale and (since $k$ is algebraically closed) it follows that $G_u = 1$. \[torus\] For $T$ a diagonalizable group scheme (e.g., a split torus) acting linearly on a vector space $V$, the stabilizer $T_v$ of a generic vector $v \in V$ is $\cap_{\omega \in \Omega} \ker \omega$ where $\Omega \subset T^*$ is the set of weights of $V$, i.e., $T_v$ is the kernel of the action. (By the duality between diagonalizable group schemes and finitely generated abelian groups, this is a statement on the level of group schemes.) In particular, $T$ acts generically freely on $V$ if and only if $\Omega$ spans $T^*$. Similarly, the stabilizer $T_{[v]}$ of a generic element $[v] \in {\mathbb{P}}(V)$ is $\cap_{\omega, \omega'} \ker(\omega- \omega')$, so $T$ acts generically freely on ${\mathbb{P}}(V)$ iff the set of differences $\omega - \omega'$ span $T^*$. For other groups $G$, we have the following well known lemma, see for example [@GG:simple Lemma 2.2]. \[point\] Suppose $G$ is connected and $X$ is irreducible. If there is a field $K \supseteq k$ and an element $x_0 \in X(K)$ such that $G_{x_0}$ is finite étale, then there is an $n \ge 1$ and a nonempty open $U \subseteq X$ such that, for every algebraically closed field $E \supseteq k$ and every $u \in U(E)$, $G_u$ is finite étale and $|G_u(E)| = n$. $\hfill\qed$ Note that finding some $x_0$ with $G_{x_0} = 1$ does not imply that $G$ acts generically freely on $X$; it is common that such an $x_0$ will exist in cases where $G_x$ is finite étale but $\ne 1$ for generic $x$. This was pointed out already in [@AndreevPopov]; see [@GurLawther] for more discussion and examples. Nonetheless, Lemma \[point\] may be used to prove that an action is generically free as follows. Suppose $G$, $X$ and the action of $G$ on $X$ can be defined over a countable algebraically closed field $F$ and that $X$ is unirational, i.e., there is an $F$-defined dominant rational map $\phi \!: {\mathbb{A}}^d \dashrightarrow X$ for some $d$. Adjoin $d$ indeterminates $a_1, \ldots, a_d$ to $F$ and calculate $G_{x_0}$ for $x_0 = \phi(a_1, \ldots, a_d)$. As $F$ is countable, for $K$ an uncountable algebraically closed field containing $F$, the elements of ${\mathbb{A}}^d(K)$ with algebraically independent coordinates are the complement of countably many closed subsets so are dense. Therefore, modifying $\phi$ by an $F$-automorphism of ${\mathbb{A}}^d$, the calculation of $G_{x_0}$ implicitly also calculates $G_x$ for $x$ in a dense subset. In particular, if $G_{x_0} = 1$, then the lemma gives that $G$ acts generically freely on $X$. Groups whose identity component is a torus {#groups-whose-identity-component-is-a-torus .unnumbered} ------------------------------------------ Suppose that $G$ is an algebraic group whose identity component is a torus $T$. As $k$ is assumed algebraically closed, the component group $G/T$ is a finite constant group. We are interested in representations $V$ of $G$ such that $G$ acts generically freely on $V$ or ${\mathbb{P}}(V)$. Evidently it is necessary that $T$ acts faithfully on $V$ or ${\mathbb{P}}(V)$, respectively. \[proj\] Let $G$ be an algebraic group with identity component a torus $T$. Suppose that $G$ acts linearly on a vector space $V$ such that: 1. \[proj.mult\] every weight of $V$ has multiplicity $1$, and 2. \[proj.ker\] for $\Omega$ the set of weights of $V$, $G/T$ acts faithfully on the kernel of the map $\psi \!:\!~\!\oplus_{\omega \in \Omega} {\mathbb{Z}}\mapsto T^*$ given by $(n_\omega) \mapsto \sum_\omega n_\omega \omega$. If $T$ acts faithfully on $V$ (resp., ${\mathbb{P}}(V)$), then $G$ acts generically freely on $V$ (resp., ${\mathbb{P}}(V)$). We give a concrete proof. Alternatively one could adapt the proof of [@MeyerRei:PGL Lemma 3.3]. As $G$ is the extension of a finite constant group by a torus, it and the representation $V$ are defined over the algebraic closure of the prime field in $k$. Put $K$ for the algebraic closure of the field obtained by adjoining independent indeterminates $c_\chi$ to $k$ for each weight $\chi$ of $V$. Fix elements $v_\chi \in V$ generating the $\chi$ weight space for each $\chi$ and put $v := \sum_\chi c_\chi v_\chi \in V {\otimes}K$. We have a homomorphism $\delta \!: G_{[v]} \to {\mathbb{G}_m}$ given by $g \mapsto gv/v$. Thinking now of the function $\pi \!: V \to {\mathbb{A}}^1$ defined by $\pi(\sum \alpha_\chi v_\chi) = \prod \alpha_\chi$, we see that $\pi(gv) = \pi(\delta(g)v) = \delta(g)^{\dim V} \pi(v)$. On the other hand, for $\delta'$ denoting the composition $\delta \!: G \to \operatorname{GL}(V) \xrightarrow{\det} {\mathbb{G}_m}$, as the image of $G$ in $\operatorname{GL}(V)$ consists of monomial matrices, we find that $\pi(gv) = \pm \delta'(g) \pi(v)$. That is, $\delta(g)^{\dim V} = \pm \delta'(g)$, so the image of $\delta$ in ${\mathbb{G}_m}$ is the group scheme $\mu_a$ of $a$-th roots of unity for some $a$. For sake of contradiction, suppose there exists a $g \in (G_{[v]})(K)$ mapping to a non-identity element $w$ in $(G/T)(K)$. Pick $n \in G(k)$ with the same image $w$, so $g = nt$ for some $t \in T(K)$. Now $nv_\chi = m_\chi v_{w\chi}$ for some $m_\chi \in k^\times$, and we have an equation $$\delta(g) v = ntv = \sum_{\chi} c_\chi \chi(t) m_\chi v_{w\chi},$$ hence $ \chi(t) = \delta(g) c_{w\chi} / (c_\chi m_\chi)$ for all $\chi$. By hypothesis, there exist $\chi_1, \ldots, \chi_r \in \Omega$ and nonzero $z_1, \ldots, z_r \in {\mathbb{Z}}$ such that $\sum z_i \chi_i = 0$ in $T^*$, yet the tuple $(z_\chi) \in \oplus_{\chi \in \Omega} {\mathbb{Z}}$ is not fixed by $w$ where $z_\chi = z_i$ if $\chi = \chi_i$ and $z_\chi = 0$ otherwise. As $\sum z_i \chi_i = 0$, we have $$\label{proj.1} 1 = \prod_i \left( \frac{\delta(g)c_{w\chi_i} }{c_{\chi_i} m_{\chi_i}} \right)^{z_i},$$ an equation in $K$, where $\delta(g)$ and the $m_{\chi_i}$ belong to $k^\times$. But the indeterminates appearing in the numerator correspond to the tuple $(z_{w \cdot \chi})$ whereas those in the denominator correspond to $(z_\chi)$, so the equality is impossible. That is, the image of $G_{[v]}$ in the constant group $G/T$ is trivial, and $G_{[v]}$ is contained in $T$. Thus $G_v$ or $G_{[v]}$ is trivial by Example \[torus\]. Since we have proved that an element with algebraically independent coordinates has trivial stabilizer, $G$ acts generically freely by the discussion following Lemma \[point\]. \[proj.m1\] Suppose there is an element $-1 \in (G/T)(k)$ that acts by $-1$ on $T^*$. Then we may partition $\Omega \setminus \{ 0 \}$ as $P \coprod -P$ for some set $P$. *If $|P| > \dim T$, then $-1$ acts nontrivially on $\ker \psi$.* Indeed, there are $n_\pi \in {\mathbb{Z}}$ for $\pi \in P$, not all zero, so that $\sum n_\pi \pi = 0$ in $T^*$, which provides an element of $\ker \psi$ that is moved by $-1$. The group $\operatorname{AGL}_1$ {#the-group-operatornameagl_1 .unnumbered} -------------------------------- The following result will be used for groups of type $C$. Let $k$ be an algebraically closed field of characteristic $p \ge 0$. Let $X$ be the variety of monic polynomials of degree $n$ over $k$. Of course, $X$ is isomorphic to affine space $k^n$ and can also be identified with $k^n/S_n$ (where the coordinates are just the roots of the polynomial). Let $X_0$ be the subvariety of $X$ such that the coefficient of $x^{n-1}$ is $0$ (i.e., the sum of the roots of $f$ is $0$). Let $G = \operatorname{AGL}_1$, the group with $k$-points $\{ {\left( \begin{smallmatrix} c&b \\ 0&1 \end{smallmatrix} \right) } \mid c \in k^\times, b \in k \}$, so $G$ is a semidirect product ${\mathbb{G}_m}\ltimes {\mathbb{G}_a}$ and is isomorphic to a Borel subgroup of $\operatorname{PGL}_2$. An element $g \in G$ acts on $k$ by $y \mapsto cy + b$ and we can extend this to an action on $X$ (by acting on each root of $f$). Note that $G$ preserves $X_0$ if and only if $p$ divides $n$. In any case ${\mathbb{G}_m}$ does act on $X_0$. \[lem:polys\] If $p$ does not divide $n > 2$, then ${\mathbb{G}_m}$ acts generically freely on $X_0$. If $p$ divides $n$ and $n > 4$, then $G$ acts generically freely on $X_0$. We just give the proof of the group of $k$-points. The proof for the Lie algebra is identical. Note that if $c = (0,c) \in k^\times$ and $f$ has distinct roots, then $c$ fixes $f$ implies that $c^{n(n-1)} =1 $ since $c$ preserves the discriminant of $f$. In particular, there are only finitely many possibilities for $c$. Note that the dimension of the fixed point space of multiplication by $c$ on $X$ has dimension at most $n/2$ and so the fixed point space on $X_0$ is a proper subvariety (because $n > 2$) and has codimension at least $2$ if $n > 4$. If $p$ does not divide $n$, then we see that there are only finitely many elements of $k^\times$ which have a fixed space which intersects the open subvariety of $X_0$ consisting of elements with nonzero discriminant. Thus, the finite union of these fixed spaces is contained in a proper subvariety of $X_0$ whence for a generic point the stabilizer is trivial. Now suppose that $p$ divides $n$. Then translation by $b$ has a fixed space of dimension $n/p \le n/2$ on $X$ and so similarly the fixed space on $X_0$ has codimension at least $2$ for $n > 4$. There is precisely one conjugacy class of nontrivial unipotent elements in $G$ and this class has dimension $1$. Thus the union of all fixed spaces of nontrivial unipotent elements of $G$ is contained in a hypersurface for $n > 4$. Any semisimple element of $G$ is conjugate to an element of $k^\times$ (i.e., to an element of the form $(0,c)$) and so there are only finitely many such conjugacy classes which have fixed points on the locus of polynomials with nonzero discriminant. Again, since each class is $1$-dimensional and each fixed space has codimension greater than $1$, we see that the union of all fixed spaces is contained in a hypersurface of $X_0$ for $n > 4$. If $n=4$ and $p=2$, then any $f \in X_0$ is fixed by a translation and so the action is not generically free. Essential dimension =================== The essential dimension of an affine group scheme $G$ over a field $k$ can be defined as follows. For each extension $K$ of $k$, write $H^1(K, G)$ for the cohomology set relative to the fppf (= faithfully flat and finitely presented) site. For each $x \in H^1(K, G)$, we define $\operatorname{ed}(x)$ to be the minimum transcendence degree of $K_0$ over $k$ for $k \supseteq K_0 \supseteq K$ such that $x$ is in the image of $H^1(K_0, G) \to H^1(K, G)$. The *essential dimension* of $G$, denoted $\operatorname{ed}(G)$, is defined to be $\max \operatorname{ed}(x)$ as $x$ varies over all extensions $K$ of $k$ and all $x \in H^1(K, G)$. If $V$ is a representation of $G$ on which $G$ acts generically freely, then $\operatorname{ed}(G) \le \dim V - \dim G$, see, e.g., [@M:ed Prop. 3.13]. We can decrease this bound somewhat by employing the following. \[ed.proj\] Suppose $V$ is a representation of an algebraic group $G$. If there is a $G$-equivariant dominant rational map $V \dashrightarrow X$ for a $G$-variety $X$ on which $G$ acts generically freely, then $\operatorname{ed}(G) \le \dim X - \dim G$. Certainly, $G$ must act generically freely on $V$. In the language of [@DuncanRei] or [@M:ed p. 424], then, $V$ is a versal and generically free $G$-variety and the natural map $V \dashrightarrow X$ is a $G$-compression. Therefore, referring to [@M:ed Prop. 3.11], we find that $\operatorname{ed}(G) \le \dim X - \dim G$. The following lemma was pointed out in [@BRV:arxiv]. \[BRV.lem\] Let $V$ be a faithful representation of a (connected) reductive group $G$. Then $\operatorname{ed}(G) \le \dim V$. Fix a maximal torus $T$ in $G$. By [@ChGiRei:semi] there is a finite sub-$k$-group-scheme $S$ of $N_G(T)$ so that the natural map of fppf cohomology sets $H^1(K, S) \to H^1(K, G)$ is surjective for every extension $K$ of $k$, so $\operatorname{ed}(G) \le \operatorname{ed}(S)$. For generic $v$ in $V$, the stabilizer $S_v$ of $v$ in $S$ has $\operatorname{Lie}(S_v) \subseteq \operatorname{Lie}(S)_v \subseteq \operatorname{Lie}(T)_v = 0$, so $S_v$ is smooth. As the finite group $S(k)$ acts faithfully on $V$, $S(k)$ acts generically freely, so $\operatorname{ed}(S) \le \dim V$. While the lemma gives cheap upper bounds on $\operatorname{ed}(G)$, it is not sufficient to deduce Theorem \[big.O\] even in the coarse sense of big-$O$ notation: the minimal faithful representations of $\operatorname{SL}_n / \mu_m$ are too big for $3 \le m < n$, being at least cubic in $n$ whereas the dimension of the group is $n^2 - 1$. The short root representation {#short.sec} ============================= Let $G$ be an adjoint simple algebraic group and put $V$ for the Weyl module with highest weight the highest short root. Fixing a maximal torus $T$ in $G$, the weights of this representation are 0 (with some multiplicity) and the short roots $\Omega$ (each with multiplicity 1), and we put ${\overline{V}}$ for $V$ modulo the zero weight space. It is a module for $N_G(T)$. \[short\] Suppose $k$ is algebraically closed. If $G$ is of type $A_n$ ($n \ge 2$), $C_n$ ($n \ge 3)$, $D_n$ ($n \ge 4$), $E_6$, $E_7$, $E_8$, or $F_4$, then $N_G(T)$ acts generically freely on ${\mathbb{P}}({\overline{V}})$ and $$\operatorname{ed}(N_G(T)) \le |\Omega| - \dim T - 1.$$ The inequality in the proposition is reminiscent of the one in [@Lemire Th. 1.3]. \[A1.eg\] The group $\operatorname{PGL}_n$ is adjoint of type $A_{n-1}$ and we identify it with the quotient of $\operatorname{GL}_n$ by the invertible scalar matrices. We may choose $T \subset \operatorname{PGL}_n$ to be the image of the diagonal matrices and $N_G(T)$ is the image of the monomial matrices. The representation ${\overline{V}}$ is the space of matrices with zeros on the diagonal, on which $N_G(T)$ acts by conjugation. With this notation, for type $A_1$, the stabilizer in $N_G(T)$ of a generic element $v := {\left( \begin{smallmatrix} 0&x \\ y&0 \end{smallmatrix} \right) }$ of ${\overline{V}}$ is ${\mathbb{Z}}/2$, with nontrivial element the image of $v$ itself. So type $A_1$ is a genuine exclusion from the proposition. It suffices to prove that $N_G(T)$ acts generically freely, for then the inequality follows by Lemma \[ed.proj\]. We apply Lemma \[proj\]. For every short root $\alpha$, there is a short root $\beta$ such that ${{\left\langle{\beta, \alpha}\right\rangle}} = \pm 1$. (If $\alpha$ is simple, take $\beta$ to be simple and adjacent to $\alpha$ in the Dynkin diagram. Otherwise, $\alpha$ is in the Weyl orbit of a simple root.) Thus, the kernel of $T \to \operatorname{PGL}({\overline{V}})$ is contained in the kernel of $\beta - s_\alpha(\beta) = \pm\alpha$. As the lattice generated by the short roots $\alpha$ is the root lattice $T^*$, it follows that $T$ acts generically freely on ${\mathbb{P}}({\overline{V}})$. So it suffices to verify \[proj\]. Fix $w \ne 1$ in the Weyl group; we find short roots $\chi_1, \ldots, \chi_r$ such that $\sum \chi_i = 0$ and the set $\{ \chi_i \}$ is not $w$-invariant. If $w = -1$, then take $\chi_1$, $\chi_2$ to be non-orthogonal short simple roots. They generate an $A_2$ subsystem and we set $\chi_3 := -\chi_1 - \chi_2$. (Alternatively, apply Example \[proj.m1\].) This proves the claim for type $F_4$: the kernel of the $G/T$-action on $\ker \psi$ is a normal subgroup of the Weyl group not containing $-1$, and therefore it is trivial. If $G$ has type $A$, $D$, or $E$, then all roots are short. As $w \ne \pm 1$, there is a short simple root $\chi_1$ such that $w(\chi_1) \ne \pm \chi_1$. (Indeed, otherwise there would be simple roots $\alpha, \alpha'$ such that $w(\alpha) = \alpha$, $w(\alpha') = -\alpha'$, yet $\alpha$ and $\alpha'$ are adjacent in the Dynkin diagram.) Take $\chi_2 = -\chi_1$. For type $C_n$ ($n \ge 3$), as in [@Bou:g4] we may view the root lattice ${\mathbb{Z}}[\Phi]$ as contained in a copy of ${\mathbb{Z}}^n$ with basis ${\varepsilon}_1, \ldots, {\varepsilon}_n$. If the kernel of the $G/T$-action on $\ker \psi$ does not contain $-1$, it contains the group $H$ isomorphic to $({\mathbb{Z}}/2)^{n-1}$ consisting of those elements that send ${\varepsilon}_i \mapsto -{\varepsilon}_i$ for an even number of indexes $i$ and fix the others. Taking $\chi_1 = {\varepsilon}_1 - {\varepsilon}_2$, $\chi_2 = {\varepsilon}_2 - {\varepsilon}_3$, and $\chi_3 = -\chi_1 -\chi_2$ gives a set $\{ \chi_i \}$ not stabilized by $H$. Groups of type $C$: Proof of Theorem \[PSp\] ============================================ Let $G$ be the adjoint group of type $C_n$ for $n > 3$ over an algebraically closed field $k$ of characteristic $p$. Let $W:=L(\omega_2)$ be the irreducible module for $G$ with highest weight $\omega_2$ where $\omega_2$ is the the second fundamental dominant weight (as numbered in [@Bou:g4]). We view $W$ as the unique irreducible nontrivial $G$-composition factor of $Y:=\wedge^2(V)$ where $V$ is the natural module for $\operatorname{Sp}_{2n}$. We recall that $Y = W \oplus k$ if $p$ does not divide $n$. If $p$ divides $n$, then $Y$ is uniserial of length $3$ with 1-dimensional socle and radical. Any element in $Y$ has characteristic polynomial $f^2$ where $f$ has degree $n$, and the radical $Y_0$ of $Y$ is the set of elements with the roots of $f$ summing to $0$. (Note that aside from characteristic $2$, $Y_0$ are the elements of trace $0$ in $Y$.) In particular, $\dim W = 2n^2-n - 1$ if $p$ does not divide $n$ and $\dim W = 2n^2-n-2$ if $p$ does divide $n$. As in [@GoGu], we view $Y$ as the set of skew adjoint operators on $V$ with respect to the alternating form defining $\operatorname{Sp}_{2n}$ with $G$ acting as conjugation on $Y$. \[Sp\] If $n > 3$ and $(n,p) \ne (4,2)$, then $G$ acts generically freely on $W \oplus W$ and on ${\mathbb{P}}(W) \times {\mathbb{P}}(W)$. Any element $y \in Y$ is conjugate to an element of the form $\operatorname{diag}(A, A^{\top})$ acting on a direct sum of totally singular subspaces. A generic element of $Y$ is thus an element where $A$ is semisimple regular. Writing $V = V_1 \perp \ldots \perp V_n$ where the $V_i$ are 2-dimensional nonsingular spaces on which $y$ acts as a scalar, we see that a generic point of $Y$ has stabilizer (as a group scheme) $\operatorname{Sp}_2^{\times n} = \operatorname{SL}_2^{\times n}$ in $\operatorname{Sp}_{2n}$ (and by [@GoGu], this precisely the intersection of two generic conjugates of $\operatorname{Sp}_{2n}$ in $\operatorname{SL}_{2n}$). The same argument shows that this is true for a generic point of $Y_0$. In particular if $p$ does not divide $n$, the same is true for $W=Y_0$. It follows by Lemma \[lem:polys\] that for generic $w \in W$, $gw =cw$ for $g \in G$ and $c \in k^x$ implies that $c=1$. Thus, the stabilizer of a generic point in ${\mathbb{P}}(W)$ still has stabilizer $\operatorname{Sp}_2^{\times n} = \operatorname{SL}_2^{\times n}$. If $p$ does divide $n$, then $W = Y_0/k$ where we identify $k$ with the scalar matrices in $Y$. We claim that (for $n > 4$) the generic stabilizer is still $\operatorname{Sp}_2^{\times n} = \operatorname{SL}_2^{\times n}$ on ${\mathbb{P}}(W)$. Again, this follows by Lemma \[lem:polys\] since if $gw = cw + b$ with $b,c \in k$ and $g \in G$, then for $w$ generic, $b=0$ and $c=1$. It is straightforward to see that the same is true for $W$, because for a generic point anything stabilizing $y$ modulo scalars must stabilize $y$ (see Lemma \[lem:polys\]). Thus, in all cases, the generic stabilizer of a point in ${\mathbb{P}}(W) \times {\mathbb{P}}(W)$ is the same as for $Y \oplus Y$. Consider $\operatorname{GL}_{2n}$ acting on $Y \oplus Y \oplus Y$. The stabilizer of a generic point of $Y$ is clearly a conjugate of $\operatorname{Sp}_{2n}$. It follows from [@GoGu] that the stabilizer of a generic element of $Y \oplus Y \oplus Y$ is central. (The result is only stated for the algebraic group but precisely the same proof holds for the group scheme.) Thus, the same holds for $\operatorname{Sp}_{2n}$ acting on $Y \oplus Y$ and so also on ${\mathbb{P}}(W) \times {\mathbb{P}}(W)$. The result follows. We can now improve and extend Lemire’s bound for $\operatorname{ed}(\operatorname{PSp}_{2n})$ from [@Lemire Cor. 1.4] both numerically and to fields of all characteristics. For the first inequality, the group $\operatorname{GL}_{2n}/\mu_2$ has an open orbit on $\wedge^2(k^{2n})$, and the stabilizer of a generic element is $\operatorname{PSp}_{2n}$. Consequently, the induced map $H^1(K, \operatorname{PSp}_{2n}) \to H^1(K, \operatorname{GL}_{2n}/\mu_2)$ is surjective for every field $K$, see [@G:lens Th. 9.3] or [@SeCG §[III.2.1]{}]. (Alternatively, the domain classifies pairs $(A, \sigma)$ where $A$ is a central simple algebra of degree $2n$ and exponent 2 and $\sigma$ is a symplectic involution [@KMRT 29.22], and the codomain classifies central simple algebras of degree $2n$ and exponent $2$. The map is the forgetful one $(A, \sigma) \mapsto A$.) Thus $\operatorname{ed}(\operatorname{GL}_{2n}/\mu_2) \le \operatorname{ed}(\operatorname{PSp}_{2n})$. For the second inequality, assume that $n \ge 4$ and if $n=4$, then $p \ne 2$. As $\operatorname{PSp}_n$ acts generically freely on ${\mathbb{P}}(W) \times {\mathbb{P}}(W)$, $\operatorname{ed}(G) \le 2(\dim {\mathbb{P}}(W)) - \dim G$ by Lemma \[ed.proj\]. Theorem \[PSp\] follows, because $\dim {\mathbb{P}}(W) = 2n^2 - n - \delta$ where $\delta = 3$ if $p$ divides $n$ and 2 otherwise. If $n=4$ and $p=2$, $G$ still acts generically freely on $Y_0 \oplus Y_0$. Indeed, arguing as above we see that $G$ acts generically freely on ${\mathbb{P}}(Y_0) \times {\mathbb{P}}(Y_0)$ and the result follows in this case. Groups of type $A$: proof of Theorem \[A\] {#A.sec} ========================================== For the proofs of Theorems \[big.O\], \[A\], and \[ed.thm\], we use the fact that $\operatorname{ed}(N_G(T)) \ge \operatorname{ed}(G)$, because for every field $K \supseteq k$, the natural map $H^1(K, N_G(T)) \to H^1(K, G)$ is surjective (which in turn holds because, for $K$ separably closed, all maximal $K$-tori in $G$ are $G(K)$-conjugate). Let $T$ be a maximal torus in $G := \operatorname{PGL}_n$ for some $n \ge 4$. The representation ${\overline{V}}$ of $N_G(T)$ from §\[short.sec\] may be identified with the space of $n$-by-$n$ matrices with zeros on the diagonal. It decomposes as ${\overline{V}}= \oplus_{i=1}^n W_i$, where $W_i$ is the subspace of matrices whose nonzero entries all lie in the $i$-th row; $N_G(T)$ permutes the $W_i$’s. \[projs\] If $n \ge 4$, then $N_G(T)$ acts generically freely on $X := {\mathbb{P}}(W_1) \times {\mathbb{P}}(W_2) \times \cdots \times {\mathbb{P}}(W_n)$. Each element of the maximal torus $T$ is the image of a diagonal matrix $t := \operatorname{diag}(t_1, \ldots, t_n)$ under the surjection $\operatorname{GL}_n \to \operatorname{PGL}_n$. The kernel of the action of $T$ on ${\mathbb{P}}(W_i)$ are the elements such that $t_i t_j^{-1}$ are equal for all $j \ne i$. Thus the kernel of the action on $X$ is the subgroup of elements with $t_i = t_j$ for all $i, j$, so $T$ acts faithfully on $X$. For generic $x \in X$, the identity component of $N_G(T)_x$ is contained in $T_x$, so $\operatorname{Lie}(N_G(T)_x) \subseteq \operatorname{Lie}(T)_x = 0$, i.e., $N_G(T)_x$ is finite étale. To show that the (concrete) group $S$ of $k$-points of $N_G(T)_x$ is trivial, it suffices to check $1 \ne s \in S$ that $$\label{projs.1} \dim s^T + \dim X^s < \dim X$$ (compare, for example, [@GG:simple 10.2, 10.5]). As $s \ne 1$, it permutes the $W_i$’s nontrivially. If $s$ moves more than two of the $W_i$’s, then $$\dim X - \dim X^s \ge 2 \dim {\mathbb{P}}(W_i) = 2(n-2).$$ But of course $\dim s^T \le n - 1$, verifying for $n \ge 4$. If $s$ interchanges only two of the $W_i$’s, i.e., it is a transposition, then $\dim X - \dim X^s = n-2$, but $\dim s^T = 1 < n - 2$, and again has been verified. \[A1.ed\] $\operatorname{ed}(\operatorname{PGL}_2) = 2$, regardless of $\operatorname{char}k$, so type $A_1$ is a genuine exception to Theorems \[big.O\] and \[A\] (as $\dim \operatorname{PGL}_2 - 2(\operatorname{rank}\operatorname{PGL}_2) - 1 = 0$). Indeed, $H^1(k, G)$ classifies quaternion algebras over $k$, i.e., the subgroup ${\mathbb{Z}}/2 \times \mu_2$ of $\operatorname{PGL}_2$ gives a surjection in flat cohomology $H^1(k, {\mathbb{Z}}/2) \times H^1(k, \mu_2) \to H^1(k, \operatorname{PGL}_2)$, so $\operatorname{ed}(\operatorname{PGL}_2) \le 2$. On the other hand, the connecting homomorphism $H^1(K, \operatorname{PGL}_2) \to H^2(K, \mu_2)$, which sends a quaternion algebra to its class in the 2-torsion of the Brauer group of $K$, is nonzero for some extension $K$, and therefore also $\operatorname{ed}(\operatorname{PGL}_2) \ge 2$. Entirely parallel comments apply to $\operatorname{PGL}_3$, in which case the surjectivity $H^1(k, {\mathbb{Z}}/3) \times H^1(k, \mu_3) \to H^1(k, \operatorname{PGL}_3)$ is due to Wedderburn [@KMRT 19.2]. Thus $\operatorname{ed}(\operatorname{PGL}_3) = 2$ and $\operatorname{PGL}_3$ is a genuine exception to Theorem \[A\]. The proof of Theorem \[A\] requires a couple more lemmas. \[GLSL\] Suppose $1 \to A \to B \to C \to 1$ is an exact sequence of group schemes over $k$. If $H^1(K, C) = 0$ for every $K \supseteq k$, then $\operatorname{ed}(B) \le \operatorname{ed}(A) \le \operatorname{ed}(B) + \dim C$. For every $K$, the sequence $$\label{GLSL.1} C(K) \to H^1(K, A) \to H^1(K, B) \to 1$$ is exact. From here the argument is standard. The surjectivity of the middle arrow gives the first inequality. For the second, take $\alpha \in H^1(K, A)$. There is a field $K_0$ lying between $k$ and $K$ such that $\operatorname{trdeg}_k K_0 \le \operatorname{ed}(B)$ and an element $\alpha_0 \in H^1(K_0, A)$ whose image in $H^1(K, B)$ agrees with that of $\alpha$. Thus, there is a $c \in C(K)$ such that $c \cdot \operatorname{res}_{K/K_0}(\alpha_0) = \alpha$. There is a field $K_1$ lying between $k$ and $K$ such that $\operatorname{trdeg}_k K_1 \le \dim C$ such that $c$ belongs to $C(K_1) \subseteq C(K)$. In summary, $$\operatorname{ed}(\alpha) \le \operatorname{trdeg}_k (K_1 K_0) \le \operatorname{trdeg}_k K_1 + \operatorname{trdeg}_k K_0 \le \operatorname{ed}(B) + \dim C.$$ As this holds for every $K$ and every $\alpha \in H^1(K, A)$, the conclusion follows. Lemma \[GLSL\] applies, for example, when $B$ is an extension of a group $A$ by a quasi-trivial torus $C$, such as when $A = \operatorname{SL}_n / \mu_m$ and $B = \operatorname{GL}_n / \mu_m$. In that case, one can tease out whether $\operatorname{ed}(\operatorname{SL}_n / \mu_m) = \operatorname{ed}(\operatorname{GL}_n/\mu_m)$ or $\operatorname{ed}(\operatorname{GL}_n/\mu_m) + 1$ by arguing as in [@ChM:edpA]. \[SL.xfer\] Suppose $m$ divides $n \ge 2$. Then $$\operatorname{ed}(\operatorname{GL}_n / \mu_m) \le \operatorname{ed}(\operatorname{PGL}_n) + n/m - 1.$$ We omit the proof, which is the same as that for [@BaekM Lemma 7.1] apart from cosmetic details. In view of Lemmas \[projs\] and \[ed.proj\], we find that $$\operatorname{ed}(\operatorname{PGL}_n) \le \operatorname{ed}(N_G(T)) \le \dim X - \dim N_G(T) = n^2 - 3n + 1.$$ Therefore Lemma \[SL.xfer\] gives $$\label{GLSL.bound} \operatorname{ed}(\operatorname{GL}_n / \mu_m) \le n^2 - 3n + n/m$$ and Lemma \[GLSL\] gives the required bound on $\operatorname{ed}(\operatorname{SL}_n / \mu_m)$. \[coprime\] Suppose $m$ divides $n$, and write $n = n'q$ where $n'$ and $m$ have the same prime factors and $\gcd(n',q) = 1$. Then $H^1(K, \operatorname{GL}_n / \mu_m) = H^1(K, \operatorname{GL}_{n'} / \mu_m)$ for every extension $K$ of $k$ and $\operatorname{ed}(\operatorname{GL}_n / \mu_m) = \operatorname{ed}(\operatorname{GL}_{n'}/\mu_m)$. The set $H^1(K, \operatorname{GL}_n / \mu_m)$ is in bijection with the isomorphism classes of central simple $K$-algebras $A$ of degree $n$ and exponent dividing $m$. As $n'$ and $q$ are coprime, every such algebra can be written uniquely as $A' {\otimes}B$ where $A'$ has degree $n'$ and $B$ has degree $q$ [@GilleSz 4.5.16]. However, $B$ is split as its exponent must divide $\gcd(q, \exp A)$, i.e., $A \cong M_q(A')$. That is, $H^1(K, \operatorname{GL}_n / \mu_m) = H^1(K, \operatorname{GL}_{n'}/\mu_m)$. As this holds for every extension $K$ of $k$, the claim on essential dimension follows. One can eliminate $m$ from the bound appearing in Theorem \[A\] to obtain $$\operatorname{ed}(\operatorname{SL}_n / \mu_m) \le n^2 - 3n + 1 + n/4 \quad \text{for $m$ dividing $n \ge 4$.}$$ To check this, assume $m < 4$. If $m = 1$, $\operatorname{ed}(\operatorname{SL}_n) = 0$. If $m = 2$, then Theorem \[PSp\] gives a stronger bound. If $m = 3$, then write $n = n'q$ for $n' = 3^a$ for some $a \ge 1$ as in Lemma \[coprime\]. If $a = 1$, then $n \ge 6$ and $\operatorname{ed}(\operatorname{GL}_n / \mu_3) = \operatorname{ed}(\operatorname{PGL}_3) = 2$ by Lemma \[coprime\], which is less than $n^2 - 3n + n/4$. If $a > 1$, then $\operatorname{ed}(\operatorname{GL}_n/\mu_3) \le \operatorname{ed}(\operatorname{PGL}_{n'}) + n'/3 - 1$; as $n'$ is odd and $\ge 9$, [@LRRS] gives $\operatorname{ed}(\operatorname{PGL}_{n'}) \le \frac12 (n'-1)(n'-2)$, whence the claim. Here is another way to obtain an upper bound on $\operatorname{ed}(\operatorname{SL}_n / \mu_m)$; it is amusing because it requires $\operatorname{char}k = p$ to be nonzero. Fix an integer $e \ge 1$ and ${\varepsilon}= \pm 1$, and set $m := \gcd(p^e + {\varepsilon}, n)$. We will show that $$\label{Frob.bd} \operatorname{ed}(\operatorname{SL}_n / \mu_m) \le n^2 - n + 1.$$ To see this, consider the $\operatorname{GL}_n$-module $V := W {\otimes}W^{[e]}$ or $W^* {\otimes}W^{[e]}$, where $W$ is the natural module $k^n$, $[e]$ denotes the $e$-th Frobenius twist, and where we take the first option if ${\varepsilon}= +1$ and the second option if ${\varepsilon}= -1$. A scalar matrix $x \in \operatorname{GL}_n$ acts on $V$ as $x^{p^e + {\varepsilon}}$, and therefore the action of $\operatorname{SL}_n$ on $V$ gives a faithful representation of $G := \operatorname{SL}_n / \mu_m$. We consider the action of $N_G(T)$ on $V$ for $T$ a maximal torus in $G$, and apply Lemma \[proj\] to see that $N_G(T)$ acts generically freely on $V$ and so obtain . Minuscule representations of $E_6^{{\mathrm{sc}}}$ and $E_7^{{\mathrm{sc}}}$: proof of Theorem \[ed.thm\] ========================================================================================================= Recall that $E_6^{{\mathrm{sc}}}$ and $E_7^{{\mathrm{sc}}}$ have minuscule representations, i.e., representations where all weights are nonzero and occur with multiplicity 1 and make up a single orbit $\Omega$ under the Weyl group. For $E_6$ there are two inequivalent choices, both of dimension 27, and for $E_7$ there is a unique one of dimension 56. \[minu\] Let $T$ be a maximal torus in a simply connected group $G$ of type $E_6^{{\mathrm{sc}}}$ or $E_7^{{\mathrm{sc}}}$ over an algebraically closed field $k$. Then $N_G(T)$ acts generically freely on $V$ for every minuscule representation $V$ of $G$. We apply Lemma \[proj\]. The map $G \to \operatorname{GL}(V)$ is injective, so $T$ acts faithfully on $V$. It suffices to verify \[proj\]. One can list explicitly the weights $\Omega$ of $V$ and find $X = \{ \chi_1, \ldots, \chi_6 \} \subset \Omega$ with $\sum \chi_i = 0$ and $\chi_i \ne \pm \chi_j$ for $i \ne j$. It suffices, therefore, to check for every minimal normal subgroup $H$ of the Weyl group not containing $-1$, that $HX \ne X$. For this, it is enough to observe that $H$ has no fixed lines on the vector space ${\mathbb{C}}[\Phi]$ generated by the roots $\Phi$ (because ${\mathbb{Z}}[\Omega] = {\mathbb{Z}}[\Phi]$, so $H$ fixes no element of $\Omega$) and that $H$ has no orbits of size $2, 3, \ldots, 6$ (because its maximal subgroups have index greater than 6). For $E_6$, $H$ has order 25920 with largest maximal subgroups of index 27. For $E_7$, $H$ is isomorphic to $\operatorname{Sp}_6({\mathbb{F}}_2)$ with largest maximal subgroups of index 28. The description of these Weyl groups from [@Bou:g4 Ch. IV, §4, Exercises 2 and 3] make it obvious that $H$ does not preserve any line in ${\mathbb{C}}[\Phi]$. The group $F_4$ has 24 short roots, so by Proposition \[short\], we have $$\operatorname{ed}(F_4) \le \operatorname{ed}(N_G(T)) \le 24 - 4 - 1 = 19.$$ For $E_7^{{\mathrm{sc}}}$, we apply instead Proposition \[minu\] to obtain the desired upper bound. The group $E_6^{{\mathrm{sc}}}$ has a subgroup $F_4 \times \mu_3$ such that the map in cohomology $H^1(K, F_4 \times \mu_3) \to H^1(K, E_6^{{\mathrm{sc}}})$ is surjective for every extension $K \supseteq k$, see [@G:lens 9.12], hence $\operatorname{ed}(E_6^{{\mathrm{sc}}}) \le \operatorname{ed}(F_4) + 1$. Proof of Theorem \[big.O\] {#A.sec} ========================== Suppose first that $G$ has type $A_{n-1}$, i.e., $G \cong \operatorname{SL}_n / \mu_m$. Assume $m > 1$ for otherwise $\operatorname{ed}(G) = 0$. It is claimed that $\operatorname{ed}(G) \le n^2 - 2n$. As $\operatorname{ed}(\operatorname{PGL}_3) = 2$, we may assume $n \ge 4$. Combining Theorem \[A\] with the fact that $1 + n/m \le n$ gives the claim. Now suppose that $G$ is adjoint. If $G$ is one of the types covered by Proposition \[short\], then we are done by combining that proposition with the inequality $\operatorname{ed}(G) \le \operatorname{ed}(N_G(T))$. Type $B$ was already addressed in the introduction. For type $G_2$, the essential dimension is 3 because $H^1(K, G_2)$ is in bijection with the set of 3-Pfister quadratic forms over $K$ for every field $K$ containing $k$ [@KMRT 26.19]. Now suppose that $G$ is neither type $A$ nor adjoint. If $G$ has type $B$, then $G$ is a spin group, so there is nothing to prove. If $G$ has type $C$, then $G=\operatorname{Sp}_{2n}$ and $\operatorname{ed}(G)=0$. If $G$ has type $D$, then the only remaining case to consider is $G={\mathrm{SO}}_{2n}$ for $n \ge 4$ and then $\operatorname{ed}(G) \le 2n - 1< 2n^2 - 3n - 1 = \dim G - 2(\operatorname{rank}G) - 1$. The two remaining cases are the simply connected groups of type $E_6$ and $E_7$ for which we refer to Theorem \[ed.thm\]. Generic stabilizer for the adjoint action {#adj.sec} ========================================= As a complement to the above results, we now calculate the stabilizer in a simple algebraic group $G$ of a generic element in $\operatorname{Lie}(\operatorname{Ad}(G))$. (Note that, in case $G = \operatorname{SL}_2$, we are discussing the action on $\operatorname{Lie}(\operatorname{PGL}_2)$, not on $\operatorname{Lie}(\operatorname{SL}_2)$, and the two Lie algebras are distinct if $\operatorname{char}k = 2$.) We include this calculation here because the methods are similar to the previous results. The results are complementary, in the sense that previously we considered $N_G(T)$ acting on representations with no zero weights, and in this section we consider $N_G(T)$ acting on $\operatorname{Lie}(\operatorname{Ad}(T))$, for which zero is the only weight. The main result, Proposition \[adjoint\], is used in [@GG:spin]. After a preliminary result, we will calculate the stabilizer of a generic element of the adjoint representation. Let $\Phi$ be an irreducible root system and put $W$ for its Weyl group and $Q$ for its root lattice. For each prime $p$, tensoring $Q$ with the finite field ${\mathbb{F}}_p$ gives a homomorphism $$\rho_p \!: {{\left\langle{W, -1}\right\rangle}} \to \operatorname{GL}_{\operatorname{rank}Q}({\mathbb{F}}_p).$$ \[weyl\] The kernel of $\rho_p$ is $({\mathbb{Z}}/2)^n$ if $\Phi$ has type $B_n$ for some $n \ge 2$ and $p = 2$. Otherwise, $\ker \rho_p = {{\left\langle{-1}\right\rangle}}$ if $p = 2$ and $\ker \rho_p = 1$ for $p \ne 2$. If $p \ne 2$, $\ker \rho_p =1$ by an old theorem of Minkowski (see [@Mi87] and also [@Se07 Lemma 1.1]). So we may assume that $p=2$. it also follows by a similar argument that $\ker \rho_2$ is a $2$-group [@Se07 Lemma 1.1’]. Clearly $-1 \in \ker \rho_2$. Thus, the result follows immediately for $G$ of type $A_n$ for $n \ne 3$, $G_2$, or $E_n$, since the only normal $2$-subgroups in these cases are the subgroup of order $2$ containing $-1$. It is straightforward to check the result for the groups $A_3=D_3$ and $C_3$. Note that the root lattice of $D_{n-1}$ is a direct summand of $D_n, n > 3$ and any normal $2$-subgroup of the Weyl group of $D_n$ of order greater than $2$ intersects the Weyl group of of $D_{n-1}$ in a subgroup of order greater than $2$. Thus, the result for $D_3$ implies the result for all $D_n$. Similarly, the result for $C_3$ implies the result for $C_n, n > 3$. Finally, suppose $\Phi$ has type $B_n$ for some $n \ge 2$ and $p = 2$. Viewing ${\mathbb{Z}}^n$ as having basis ${\varepsilon}_i$ for $1 \le i \le n$, we can embed $\Phi$ in ${\mathbb{Z}}^n$ by setting the simple roots to be $\alpha_i = {\varepsilon}_i - {\varepsilon}_{i+1}$ for $1 \le i < n$ and $\alpha_n = {\varepsilon}_n$ as in [@Bou:g4]. The Weyl group $W$ is isomorphic to $({\mathbb{Z}}/2)^n \rtimes S_n$, where $({\mathbb{Z}}/2)^n$ consists of all possible sign flips of the ${\varepsilon}_i$ and $S_n$ acts by permuting the ${\varepsilon}_i$. The subgroup $({\mathbb{Z}}/2)^n$ obviously acts trivially on $Q {\otimes}{\mathbb{F}}_2$ (since there is a basis of eigenvectors for $Q$ for this subgroup of exponent $2$). In fact, $({\mathbb{Z}}/2)^n$ is precisely the kernel of the action of $W$ on $Q {\otimes}{\mathbb{F}}_2$, as is easy to check for $n < 5$ and is clear for $n \ge 5$. \[adjoint\] Let $G$ be a simple algebraic group. The action of $G$ on $\operatorname{Lie}(\operatorname{Ad}(G))$ has stabilizer in general position $S$, with identity component $S^\circ$ a maximal torus in $G$. Moreover, $S = S^\circ$ unless $\operatorname{char}k = 2$ and: 1. $G$ has type $B_n$ for $n \ge 2$; in this case $S/S^\circ \cong ({\mathbb{Z}}/2)^n$. 2. $G$ has type $A_1$, $C_n$ for $n \ge 3$, $D_n$ for $n \ge 4$ even, $E_7$, $E_8$, $F_4$, or $G_2$; in this case $S/S^\circ \cong {\mathbb{Z}}/2$ and the nontrivial element acts on $S^\circ$ by inversion. Suppose first that $G = \operatorname{Ad}(G)$ and fix a maximal torus $T$ of $G$. As $G$ is adjoint, the Lie algebra $\operatorname{Lie}(T)$ is a Cartan subalgebra of $\operatorname{Lie}(G)$, and the natural map $G \times \operatorname{Lie}(T) \to \operatorname{Lie}(G)$ is dominant [@SGA3.2 XIII.5.1, XIV.3.18]. Therefore, it suffices to verify that the stabilizer $S$ in $G$ of a generic vector $t$ in $\operatorname{Lie}(T)$ is as claimed. The subgroup of $G$ transporting $t$ in $\operatorname{Lie}(T)$ is the normalizer $N_G(T)$ [@SGA3.2 XIII.6.1(d)(viii)], hence $S$ is the centralizer of $t$ in $N_G(T)$ and it follows that $S^\circ = T$ and $S/S^\circ$ is isomorphic to the group of elements $w$ of the Weyl group fixing $t$, compare [@St:tor Lemma 3.7]. As $G$ is adjoint, the element $t$ is determined by its action on $\operatorname{Lie}(G)$, i.e., by the values of the roots on $t$; in particular $w(t) = t$ if and only if $w$ acts trivially on $Q {\otimes}k$. Lemma \[weyl\] completes the proof for $G$ adjoint. In case $G$ is not adjoint, the representation factors through the central isogeny $G \to \operatorname{Ad}(G)$, and $G_t$ is the inverse image of the generic stabilizer in $\operatorname{Ad}(G)$. To summarize the proof, the identity component of $C_G(t)$ is $T$ by [@SGA3.2], so $C_G(t)$ is contained in $N_G(T)$ and is determined by its image in the Weyl group $N_G(T)/T$; this statement is included in [@St:tor]. What is added here is the calculation of the component group $C_G(t)/T$, and in particular that it need not be connected. One can also compute the generic stabilizer for the action of $G$ on the projective space ${\mathbb{P}}(\operatorname{Lie}(G))$ of $\operatorname{Lie}(G)$ by the same argument. If $p = 2$, since $\operatorname{PGL}_n({\mathbb{F}}_2) = \operatorname{GL}_n({\mathbb{F}}_2)$ we see that the generic stabilizers for ${\mathbb{P}}(\operatorname{Lie}(G))$ and $\operatorname{Lie}(G)$ are the same. If $p$ is odd, an easy argument shows that a generic stabilizer is a maximal torus if $-1$ is not in the Weyl group and is just a maximal torus extended by $-1$ if $-1$ is in the Weyl group. (Clearly $-1$ does act by $-1$ on $\operatorname{Lie}(T)$, $T$ a maximal torus.) In any case, the connected component of the stabilizer of a generic line in $\operatorname{Lie}(G)$ is contained in the normalizer of a maximal torus, as we know from [@SGA3.2]. Action of $G$ on $\operatorname{Lie}(G) \oplus \operatorname{Lie}(G)$ {#action-of-g-on-operatornamelieg-oplus-operatornamelieg .unnumbered} --------------------------------------------------------------------- In case $k = {\mathbb{C}}$, it is well known that an adjoint simple group $G$ acts generically freely on $\operatorname{Lie}(G) \oplus \operatorname{Lie}(G)$. However we have also the following: \[A1A1\] Maintaining the notation of Example \[A1.eg\], the Lie algebra ${\mathfrak{pgl}}_2$ of $\operatorname{PGL}_2$ may be identified with the Lie algebra ${\mathfrak{gl}}_2$ of 2-by-2 matrices, modulo the scalar matrices. Write $T$ for the (image of the) diagonal matrices in $\operatorname{PGL}_2$. A generic element $v \in {\mathfrak{pgl}}_2$ is the image of some ${\left( \begin{smallmatrix} x&y \\ z&w \end{smallmatrix} \right) }$. The normalizer of $[v] \in {\mathbb{P}}({\mathfrak{pgl}}_2)$ in $N_G(T)$ is ${\mathbb{Z}}/2$, with nontrivial element the image $g$ of ${\left( \begin{smallmatrix} 0&y \\ -z&0 \end{smallmatrix} \right) }$, which satisfies $gv = -v$. If $\operatorname{char}k = 2$, the same calculation shows that the normalizer of $v \in {\mathfrak{pgl}}_2$ is ${\mathbb{Z}}/2$. The subgroup of $\operatorname{PGL}_2$ mapping a generic element of $\operatorname{Lie}(T)$ into $\operatorname{Lie}(T)$ is $N_G(T)$, as was already used in the proof of Proposition \[adjoint\]. Therefore, the stabilizer in $G$ of a generic element of ${\mathbb{P}}({\mathfrak{pgl}}_2) \oplus {\mathbb{P}}({\mathfrak{pgl}}_2)$ equals the stabilizer in $N_G(T)$ of a generic element of ${\mathbb{P}}({\mathfrak{pgl}}_2)$, i.e., ${\mathbb{Z}}/2$. Moreover, if $\operatorname{char}k = 2$, *the stabilizer in $\operatorname{PGL}_2$ of a generic element in ${\mathfrak{pgl}}_2 \oplus {\mathfrak{pgl}}_2$ is ${\mathbb{Z}}/2$.* We note that this is the only such example. \[double\] Let $G$ be an adjoint simple group. Then $G$ acts generically freely on ${\mathbb{P}}(\operatorname{Lie}(G)) \times{\mathbb{P}}(\operatorname{Lie}(G))$ unless $G$ has type $A_1$. If $G$ has type $A_1$ and $\operatorname{char}k \ne 2$, then $G$ acts generically freely on $\operatorname{Lie}(G) \oplus \operatorname{Lie}(G)$. Pick a maximal torus $T$ in $G$. The stabilizer in $G$ of a generic element of ${\mathbb{P}}(\operatorname{Lie}(G)) \times {\mathbb{P}}(\operatorname{Lie}(G))$ is contained in the intersection of two generic conjugates of $N_G(T)$. If $G$ is not of type $A_1$, then this intersection is 1 as in the proof of [@BGS Cor. 10]. If $G$ is of type $A_1$ and $\operatorname{char}k \ne 2$, then we apply the preceding example. Note that if $p \ne 2$ and we consider the action of $G$ on $\operatorname{Lie}(G)$, then a generic stabilizer is a maximal torus and it is elementary to see that two generic conjugates of a maximal torus intersect trivially. Groups of type $B$ and $D$ {#spin.sec} ========================== We have not yet discussed upper bounds for the simply connected groups $\operatorname{Spin}_n$ for $n \ge 7$ of type $B_\ell$ for $\ell \ge 3$ or $D_\ell$ for $\ell \ge 4$. Also, for $\operatorname{Spin}_n$ with $n$ divisible by 4 and at least 12, there is a quotient $\operatorname{Spin}_n / \mu_2$ that is distinct from ${\mathrm{SO}}_n$; it is denoted $\operatorname{HSpin}_n$ and is known as a half-spin group. The group $G = \operatorname{Spin}_n$ with $n > 14$ or $\operatorname{HSpin}_n$ with $n > 16$ act generically freely on a (half) spin representation or the sum of a half spin representation and the vector representation $\operatorname{Spin}_n \to {\mathrm{SO}}_n$ by [@AndreevPopov] and [@APopov] if $\operatorname{char}k = 0$ and [@GG:spin] for all characteristics. This gives an upper bound on $\operatorname{ed}(G)$, which is an equality if $\operatorname{char}k \ne 2$, see [@BRV] and [@GG:spin]. We now give bounds for $\operatorname{HSpin}_{12}$ and $\operatorname{HSpin}_{16}$. For $T$ a maximal torus in $G := \operatorname{HSpin}_n$ for $n$ divisible by $4$ and $n \ge 12$, the group $N_G(T)$ acts generically freely on the half-spin representation of $G$. Apply Lemma \[proj\]. The representation $V$ is minuscule and $T$ acts faithfully because $G$ does so. The element $-1$ of the Weyl group acts nontrivially on $\ker \psi$ by Example \[proj.m1\] because $\frac12 \dim V = 2^{n/2-2} > n/2 = \dim T$. As $-1$ is contained in every nontrivial normal subgroup of the Weyl group, the proof is complete. Over every algebraically closed field, $$\operatorname{ed}(\operatorname{HSpin}_{12}) \le 26 \quad \text{and} \quad \operatorname{ed}(\operatorname{HSpin}_{16}) \le 120.$$ The remaining groups are $\operatorname{Spin}_n$ with $7 \le n \le 14$. In case $\operatorname{char}k \ne 2$, the precise essential dimension is known by Rost, see [@Rost:14.1], [@Rost:14.2], and [@G:lens]. The same methods, combined with the calculations of the generic stabilizers from [@GG:spin], will provide upper bounds for $\operatorname{ed}(\operatorname{Spin}_n)$ in case $\operatorname{char}k = 2$. But these methods require detailed arguments, so for our purposes we note simply that $\operatorname{Spin}_n$ acts faithfully on the spin representation for $n$ odd and on the direct sum of the vector representation and a half-spin representation for $n$ even; Lemma \[BRV.lem\] then provides an upper bound on $\operatorname{ed}(\operatorname{Spin}_n)$. This completes the task of giving an upper bound on $\operatorname{ed}(G)$ for every simple algebraic group $G$ over an algebraically closed field $k$. Acknowledgements {#acknowledgements .unnumbered} ---------------- We thank the referee, Zinovy Reichstein, and Mark MacDonald for their helpful comments, which greatly improved the paper. [^1]: Guralnick was partially supported by NSF grants DMS-1265297 and DMS-1302886. [^2]: By, e.g., [@Richardson:n Lemma 3.3(b)]. For analogous statements in prime characteristic, see §\[adj.sec\].

--- abstract: 'We introduce the concept of a quasiresiduated lattice and prove that every lattice effect algebra can be organized into a commutative quasiresiduated lattice with divisibility. Also conversely, every such lattice can be converted into a lattice effect algebra and every lattice effect algebra can be reconstructed form its assigned quasiresiduated lattice. We apply this method also for lattice pseudoeffect algebras introduced recently by Dvurečenskij and Vetterlein. We show that every good lattice pseudoeffect algebra can be organized into a (possibly non-commutative) quasiresiduated lattice with divisibility and conversely, every such lattice can be converted into a lattice pseudoeffect algebra. Moreover, also a good lattice pseudoeffect algebra can be reconstructed from the assigned quasiresiduated lattice.' author: - Ivan Chajda and Helmut Länger title: Residuation in lattice effect algebras --- [**AMS Subject Classification:**]{} 03G25,03G12,06D35 [**Keywords:**]{} lattice effect algebra, lattice pseudoeffect algebra, quasiresiduated lattice, quasiadjointness, divisibility In order to axiomatize quantum logic effects in a Hilbert space, Foulis and Bennett ([@FB]) introduced the so-called effect algebras. These are partial algebras with one partial binary operation which can be converted into bounded posets in general and into lattices in particular cases. It turns out that effect algebras form a successful axiomatization of the logic of quantum mechanics, but we suppose that there exists a connection with a kind of substructural logics whose algebraic semantics is based on residuated lattices. An attempt in this direction was already done in [@CH] where the so-called conditional residuation was introduced. A disadvantage of this approach is that the axioms of residuated structures are reflected only in the case when the terms used in adjointness are defined. This is an essential restriction which prevents the development of this theory. The aim of the present paper is to introduce the more general concept of quasiresiduation and to show that lattice effect algebras and lattice pseudoeffect algebras satisfy this concept. Pseudoeffect algebras were introduced recently by Dvurečenski and Vetterlein ([@DV]). We start with the following definition. An [*effect algebra*]{} is a partial algebra $\mathbf E=(E,+,{}',0,1)$ of type $(2,1,0,0)$ where $(E,{}',0,1)$ is an algebra and $+$ is a partial operation satisfying the following conditions for all $x,y,z\in E$: 1. $x+y$ is defined if and only if so is $y+x$ and in this case $x+y=y+x$, 2. $(x+y)+z$ is defined if and only if so is $x+(y+z)$ and in this case $(x+y)+z=x+(y+z)$, 3. $x'$ is the unique $u\in E$ with $x+u=1$, 4. if $1+x$ is defined then $x=0$. On $E$ a binary relation $\leq$ can be defined by $$x\leq y\text{ if there exists some }z\in E\text{ with }x+z=y$$ ($x,y\in E$). Then $(E,\leq,0,1)$ is a bounded poset and $\leq$ is called the [*induced order*]{} of $\mathbf E$. If $(E,\leq)$ is a lattice then $\mathbf E$ is called a [*lattice effect algebra*]{}. In the sequel we will use the properties of effect algebras listed in the following lemma. \[lem1\] [(]{}see [[@DP],[@DV])]{} If $\mathbf E=(E,+,{}',0,1)$ is an effect algebra, $\leq$ its induced order and $a,b,c,d\in E$ then the following hold: 1. $a''=a$, 2. $a\leq b$ implies $b'\leq a'$, 3. $a+b$ is defined if and only if $a\leq b'$, 4. if $a\leq b$ and $b+c$ is defined then $a+c$ is defined and $a+c\leq b+c$, 5. if $a\leq b$ then $a+(a+b')'=b$, 6. $a+0=0+a=a$, 7. $0'=1$ and $1'=0$. A [*partial monoid*]{} is a partial algebra $\mathbf A=(A,\odot,1)$ of type $(2,0)$ where $1\in A$ and $\odot$ is a partial operation satisfying the following conditions for all $x,y,z\in A$: 1. $(x\odot y)\odot z$ is defined if and only if so is $x\odot(y\odot z)$ and in this case $(x\odot y)\odot z=x\odot(y\odot z)$, 2. $x\odot1=1\odot x=x$. The [*partial monoid*]{} $\mathbf A$ is called [*commutative*]{} if it satisfies the following condition for all $x,y\in A$: 1. $x\odot y$ is defined if and only if so is $y\odot x$ and in this case $x\odot y=y\odot x$, The authors already introduced a certain modification of residuation for sectionally pseudocomplemented lattices, see [@CL]. For lattice effect algebras, we introduce another version of residuation called quasiresiduation. \[def1\] A [*commutative quasiresiduated lattice*]{} is a partial algebra $\mathbf C=(C,\vee,\wedge,$ $\odot,\rightarrow,0,1)$ of type $(2,2,2,2,0,0)$ where $(C,\vee,\wedge,0,1)$ is a bounded lattice, $\odot$ is a partial and $\rightarrow$ a full operation satisfying the following conditions for all $x,y,z\in C$: 1. $(C,\odot,1)$ is a partial commutative monoid where $x\odot y$ is defined if and only if $x'\leq y$, 2. $x''=x$, and $x\leq y$ implies $y'\leq x'$, 3. $(x\vee y')\odot y\leq y\wedge z$ if and only if $x\vee y'\leq y\rightarrow z$. Here $x'$ is an abbreviation for $x\rightarrow0$. The [*commutative quasiresiduated lattice*]{} $\mathbf C$ is called [*divisible*]{} if $$x\leq y\text{ implies }y\odot(y\rightarrow x)=x$$ for all $x,y\in C$. Note that the terms in (C3) are everywhere defined. In case $y'\leq x$ and $z\leq y$ condition (C3) reduces to $$x\odot y\leq z\text{ if and only if }x\leq y\rightarrow z$$ which is usual adjointness. Therefore condition (C3) will be called [*commutative quasiadjointness*]{}. Hence, contrary to the similar concept in [@CH], in commutative quasiadjointness we have only everywhere defined terms in $\mathbf C$ although $\mathbf C$ is a partial algebra. Our aim is to show that every lattice effect algebra can be organized into a commutative quasiresiduated lattice. \[th1\] Let $\mathbf E=(E,+,{}',0,1)$ be a lattice effect algebra with lattice operations $\vee$ and $\wedge$ and put $$\begin{aligned} x\odot y & :=(x'+y')'\text{ if and only if }x'\leq y, \\ x\rightarrow y & :=(x\wedge y)+x'\end{aligned}$$ [(]{}$x,y\in E$[)]{}. Then $\mathbb C(\mathbf E):=(E,\vee,\wedge,\odot,\rightarrow,0,1)$ is a divisible commutative quasiresiduated lattice. Let $a,b,c\in E$. Obviously, $(E,\vee,\wedge,0,1)$ is a bounded lattice and (C1) and (C2) hold. If $(a\vee b')\odot b\leq b\wedge c$ then $((a\vee b')'+b')'\leq b\wedge c$ and $b'\leq a\vee b'$ and hence $$a\vee b'=b'+((a\vee b')'+b')'\leq b'+(b\wedge c)=b\rightarrow c.$$ If, conversely, $a\vee b'\leq b\rightarrow c$ then $a\vee b'\leq(b\wedge c)+b'$ and $b'\leq(b\wedge c)'$ and hence $$(a\vee b')\odot b=(b'+(a\vee b')')'\leq(b'+((b\wedge c)+b')')'=(b\wedge c)''=b\wedge c$$ proving (C3). If $a\leq b$ then $b'\leq a'$ and hence $$b\odot(b\rightarrow a)=(b'+(a+b')')'=a''=a$$ proving divisibility. Let us mention that Definition \[def1\] can be modified in such a way that it contains only everywhere defined operations. Namely, if we put $$x\otimes y:=(x\vee y')\odot y$$ for all $x,y\in C$ then $\otimes$ is everywhere defined and satisfies the identities $x\otimes1\approx1\otimes x\approx x$, and commutative quasiadjointness can then be expressed in the form $$x\otimes y\leq y\wedge z\text{ if and only if }x\vee y'\leq y\rightarrow z.$$ This means that our definition of commutative quasiresiduation differs from that of usual residuation only in the point that $y$ occurs on the right-hand side of $x\otimes y\leq y\wedge z$ and $y'$ on the left-hand side of $x\vee y'\leq y\rightarrow z$. On the other hand, using this version, divisibility cannot be easily defined. Moreover, since in lattice effect algebras we have $$y\rightarrow z=(y\wedge z)+y'=y\rightarrow(y\wedge z),$$ commutative quasiadjointness can be rewritten in the form $$(x\vee y')\odot y\leq y\wedge z\text{ if and only if }x\vee y'\leq y\rightarrow(y\wedge z)$$ which corresponds to usual adjointness if we abbreviate $x\vee y'$ by $X$ and $y\wedge z$ by $Z$, i.e. $$X\odot y\leq Z\text{ if and only if }X\leq y\rightarrow Z.$$ We can prove also the converse. Let $\mathbf C=(C,\vee,\wedge,\odot,\rightarrow,0,1)$ be a commutative quasiresiduated lattice and put $$\begin{aligned} x+y & :=(x'\odot y')'\text{ if and only if }x\leq y', \\ x' & :=x\rightarrow0\end{aligned}$$ [(]{}$x,y\in C$[)]{}. Then $\mathbb E(\mathbf C):=(C,+,{}',0,1)$ is a lattice effect algebra whose order coincides with that in $\mathbf C$. Let $a,b\in C$. It is easy to see that (E1), (E2) and (E4) hold. Since $$0\vee a'=a'\leq a'=a\rightarrow0$$ we have $$a\odot a'=a\odot(0\vee a')\leq a\wedge0=0,$$ i.e. $a\odot a'=0$. If, conversely, $a\odot b=0$ then $a'\leq b$ and hence $$a\odot(b\vee a')=a\odot b=0\leq a\wedge0$$ whence $$b=b\vee a'\leq a\rightarrow0=a'$$ showing $b=a'$. Hence $a\odot b=0$ if and only if $b=a'$. Now the following are equivalent: $$\begin{aligned} a+b & =1, \\ a'\odot b' & =0, \\ a & =b', \\ b & =a'.\end{aligned}$$ This shows (E3). Moreover, the following are equivalent: $$\begin{aligned} & a\leq b\text{ holds in }\mathbb E(\mathbf R), \\ & a+b'\text{ is defined}, \\ & a'\odot b\text{ is defined}, \\ & a\leq b\text{ holds in }\mathbf R.\end{aligned}$$ Since $(C,\vee,\wedge)$ is a lattice and the partial order relations in $\mathbf C$ and $\mathbb E(\mathbf C)$ coincide, $\mathbb E(\mathbf C)$ is a lattice effect algebra. Moreover, every lattice effect algebra can be reconstructed from the assigned quasiresiduated lattice as shown in the following result. Let $\mathbf E$ be a lattice effect algebra. Then $\mathbb E(\mathbb C(\mathbf E))=\mathbf E$. Let $$\begin{aligned} \mathbf E & =(E,+,{}',0,1)\text{ with lattice operations }\vee\text{ and }\wedge, \\ \mathbb C(\mathbf E) & =(E,\vee,\wedge,\odot,\rightarrow,0,1), \\ \mathbb E(\mathbb C(\mathbf E)) & =(E,\oplus,{}^*,0,1)\end{aligned}$$ and $a,b\in E$. Then $$a^*=a\rightarrow0=(a\wedge0)+a'=0+a'=a'.$$ Moreover, the following are equivalent: $$\begin{aligned} & a\oplus b\text{ is defined}, \\ & a\leq b'\text{ in }\mathbb C(\mathbf E), \\ & a\leq b'\text{ in }\mathbf E\end{aligned}$$ and in this case $$a\oplus b=(a^*\odot b^*)^*=(a'\odot b')'=(a''+b'')''=a+b.$$ Now we turn our attention to a more general case. The following concept was introduced by Dvurečenskij and Vetterlein ([@DV]). A [*pseudoeffect algebra*]{} is a partial algebra $\mathbf P=(P,+,\,\bar{}\,,\,\tilde{}\,,0,1)$ of type $(2,1,1,0,0)$ where $(P,\,\bar{}\,,\,\tilde{}\,,0,1)$ is an algebra and $+$ is a partial operation satisfying the following conditions for all $x,y,z\in P$: 1. If $x+y$ is defined then there exist $u,w\in P$ with $u+x=y+w=x+y$, 2. $(x+y)+z$ is defined if and only if $x+(y+z)$ is defined, and in this case $(x+y)+z=x+(y+z)$, 3. $\bar x$ is the unique $u\in P$ with $u+x=1$, and $\tilde x$ is the unique $w\in P$ with $x+w=1$, 4. if $1+x$ or $x+1$ is defined then $x=0$. The [*pseudoeffect algebra*]{} $\mathbf P$ is called [*good*]{} if $\widetilde{\bar x+\bar y}=\overline{\tilde x+\tilde y}$ for all $x,y\in P$ with $\tilde x\leq y$. On $P$ a binary relation $\leq$ can be defined by $$x\leq y\text{ if there exists some }z\in E\text{ with }x+z=y$$ ($x,y\in P$). Then $(P,\leq,0,1)$ is a bounded poset and $\leq$ is called the [*induced order*]{} of $\mathbf P$. If $(P,\leq)$ is a lattice then $\mathbf P$ is called a [*lattice pseudoeffect algebra*]{}. For our investigation we need the following results taken from [@DV]. If $\mathbf P=(P,+,\,\bar{}\,,\,\tilde{}\,,0,1)$ is a pseudoeffect algebra, $\leq$ its induced order and $a,b,c\in P$ then 1. $\bar{\tilde a}=\tilde{\bar a}=a$, 2. the following are equivalent: $a\leq b$, $\bar b\leq\bar a$, $\tilde b\leq\tilde a$, 3. $a+b$ is defined if and only if $a\leq\bar b$, 4. if $a\leq b$ and $b+c$ is defined then $a+c$ is defined and $a+c\leq b+c$, 5. if $a\leq b$ and $c+b$ is defined then $c+a$ is defined and $c+a\leq c+b$, 6. if $a\leq b$ then $a+\widetilde{\bar b+a}=\overline{a+\tilde b}+a=b$, 7. $a+0=0+a=a$, 8. $\bar0=\tilde0=1$ and $\bar1=\tilde1=0$, 9. the following are equivalent: $a\leq b$, there exists some $d\in P$ with $a+d=b$, there exists some $e\in P$ with $e+a=b$. Since pseudoeffect algebras are more general than effect algebras, we must define quasiresiduated lattice for the case when the partial operation $\odot$ is not commutative and the mapping $x\mapsto\bar x$ is not an involution. A [*quasiresiduated lattice*]{} is a partial algebra $\mathbf Q=(Q,\vee,\wedge,\odot,\rightarrow,\leadsto,0,1)$ of type $(2,2,2,2,2,0,0)$ where $(Q,\vee,\wedge,0,1)$ is a bounded lattice, $\odot$ is a partial and $\rightarrow$ and $\leadsto$ are full operations satisfying the following conditions for all $x,y,z\in Q$: 1. $(Q,\odot,1)$ is a partial monoid where $x\odot y$ is defined if and only if $\tilde x\leq y$, 2. $\tilde{\bar x}=\bar{\tilde x}=x$, and $x\leq y$ implies $\bar y\leq\bar x$ and $\tilde y\leq\tilde x$, 3. $(x\vee\bar y)\odot y\leq y\wedge z$ if and only if $x\vee\bar y\leq y\rightarrow z$, 4. $y\odot(x\vee\tilde y)\leq y\wedge z$ if and only if $x\vee\tilde y\leq y\leadsto z$, 5. $\widetilde{\bar x\odot\bar y}=\overline{\tilde x\odot\tilde y}$. Here $\bar x$ and $\tilde x$ are abbreviations for $x\rightarrow0$ and $x\leadsto0$, respectively. The quasiresiduated lattice $\mathbf Q$ is called [*divisible*]{} if $$x\leq y\text{ implies }(y\rightarrow x)\odot y=y\odot(y\leadsto x)=x$$ for all $x,y\in Q$. Note that the terms in (Q3) and (Q4) are everywhere defined. In case $\bar y\leq x$ and $z\leq y$ condition (Q3) reduces to $$x\odot y\leq z\text{ if and only if }x\leq y\rightarrow z$$ which is usual adjointness. Analogously, in case $\tilde y\leq x$ and $z\leq y$ condition (Q4) reduces to $$y\odot x\leq z\text{ if and only if }x\leq y\rightarrow z$$ which is usual adjointness if $\odot$ is commutative. Therefore conditions (Q3) and (Q4) will be called [*quasiadjointness*]{}. Hence, contrary to the similar concept in [@CH], in quasiadjointness we have only everywhere defined terms in $\mathbf Q$ although $\mathbf Q$ is a partial algebra. Similarly as for effect algebras, we prove that every good lattice pseudoeffect algebra can be organized into a quasiresiduated lattice which, however, need not be commutative. Let $\mathbf P=(P,+,\,\bar{}\,,\,\tilde{}\,,0,1)$ be a good lattice pseudoeffect algebra with lattice operations $\vee$ and $\wedge$ and put $$\begin{aligned} x\odot y & :=\widetilde{\bar x+\bar y}\text{ if and only if }\tilde x\leq y, \\ x\rightarrow y & :=\bar x+(x\wedge y),\;x\leadsto y:=(x\wedge y)+\tilde x\end{aligned}$$ [(]{}$x,y\in P$[)]{}. Then $\mathbb Q(\mathbf P):=(P,\vee,\wedge,\odot,\rightarrow,\leadsto,0,1)$ is a divisible quasiresiduated lattice. Let $a,b,c\in E$. Obviously, $(P,\vee,\wedge,0,1)$ is a bounded lattice and (Q1), (Q2) and (Q5) hold. If $(a\vee\bar b)\odot b\leq b\wedge c$ then $\widetilde{\overline{a\vee\bar b}+\bar b}\leq b\wedge c$ and $\bar b\leq a\vee\bar b$ and hence $$a\vee\bar b=\bar b+\widetilde{\overline{a\vee\bar b}+\bar b}\leq\bar b+(b\wedge c)=b\rightarrow c.$$ If, conversely, $a\vee\bar b\leq b\rightarrow c$ then $a\vee\bar b\leq\bar b+(b\wedge c)$ and $\bar b\leq\overline{b\wedge c}$ and hence $$(a\vee\bar b)\odot b=\widetilde{\overline{a\vee\bar b}+\bar b}\leq\widetilde{\overline{\bar b+(b\wedge c)}+\bar b}=\widetilde{\overline{b\wedge c}}=b\wedge c$$ roving (Q3). If $b\odot(a\vee\tilde b)\leq b\wedge c$ then $\overline{\tilde b+\widetilde{a\vee\tilde b}}\leq b\wedge c$ and $\tilde b\leq a\vee\tilde b$ and hence $$a\vee\tilde b=\overline{\tilde b+\widetilde{a\vee\tilde b}}+\tilde b\leq(b\wedge c)+\tilde b=b\leadsto c.$$ If, conversely, $a\vee\tilde b\leq b\leadsto c$ then $a\vee\tilde b\leq(b\wedge c)+\tilde b$ and $\tilde b\leq\widetilde{b\wedge c}$ and hence $$b\odot(a\vee\tilde b)=\overline{\tilde b+\widetilde{a\vee\tilde b}}\leq\overline{\tilde b+\widetilde{(b\wedge c)+\tilde b}}=\overline{\widetilde{b\wedge c}}=b\wedge c$$ proving (Q4). If $a\leq b$ then $\bar b\leq\bar a$ and $\tilde b\leq\tilde a$ and hence $$\begin{aligned} (b\rightarrow a)\odot b & =\widetilde{\mathit{\overline{\bar b+a}+\bar b}}=\tilde{\bar a}=a, \\ b\odot(b\leadsto a) & =\overline{\tilde b+\widetilde{a+\tilde b}}=\bar{\tilde a}=a\end{aligned}$$ proving divisibility. We can prove also the converse. Let $\mathbf Q=(Q,\vee,\wedge,\odot,\rightarrow,\leadsto,0,1)$ be a quasiresiduated lattice and put $$\begin{aligned} x+y & :=\widetilde{\bar x\odot\bar y}\text{ if and only if }x\leq\bar y, \\ \bar x & :=x\rightarrow0,\;\tilde x:=x\leadsto0\end{aligned}$$ [(]{}$x,y\in Q$[)]{}. Then $\mathbb P(\mathbf Q):=(Q,+,\,\bar{}\,,\,\tilde{}\,,0,1)$ is a good lattice pseudoeffect algebra whose order coincides with that in $\mathbf Q$. Let $a,b,c\in Q$. It is easy to see that (E2) and (E4) hold. Since $$0\vee\tilde a=\tilde a\leq\tilde a=a\leadsto0$$ we have $$a\odot\tilde a=a\odot(0\vee\tilde a)\leq a\wedge0=0,$$ i.e. $a\odot\tilde a=0$. If, conversely, $a\odot b=0$ then $\tilde a\leq b$ and hence $$a\odot(b\vee\tilde a)=a\odot b=0\leq a\wedge0$$ whence $$b=b\vee\tilde a\leq a\leadsto0=\tilde a$$ showing $b=\tilde a$. Hence $a\odot b=0$ if and only if $b=\tilde a$. Since $$0\vee\bar a=\bar a\leq\bar a=a\rightarrow0$$ we have $$\bar a\odot a=(0\vee\bar a)\odot a\leq a\wedge0=0,$$ i.e. $\bar a\odot a=0$. If, conversely, $b\odot a=0$ then $\tilde b\leq a$, i.e. $\bar a\leq b$, and hence $$(b\vee\bar a)\odot a=b\odot a=0\leq a\wedge0$$ whence $$b=b\vee\bar a\leq a\rightarrow0=\bar a$$ showing $b=\bar a$. Hence $b\odot a=0$ if and only if $b=\bar a$. Now the following are equivalent: $$\begin{aligned} a+b & =1, \\ \bar a\odot\bar b & =0, \\ a & =\bar b, \\ b & =\tilde a.\end{aligned}$$ This shows (P3). Since $$a\odot(1\vee\tilde a)=a\odot1=a\leq a\wedge a$$ we have $$1=1\vee\tilde a\leq a\leadsto a,$$ i.e. $a\leadsto a=1$. If $a\leq\bar b$ then because of $\bar b\vee a\leq\bar a\leadsto\bar a$ we have $$\bar a\odot\bar b=\bar a\odot(\bar b\vee a)\leq\bar a\wedge\bar a=\bar a$$ whence $$a=\tilde{\bar a}\leq\widetilde{\bar a\odot\bar b}=a+b$$ showing that $a+\widetilde{a+b}$ is defined. Now in case $a\leq\bar b$ the following are equivalent: $$\begin{aligned} c & =\overline{a+\widetilde{a+b}}, \\ c+(a+\widetilde{a+b}) & =1, \\ (c+a)+\widetilde{a+b} & =1, \\ c+a & =a+b.\end{aligned}$$ Since $$(1\vee\bar a)\odot a=1\odot a=a\leq a\wedge a$$ we have $$1=1\vee\bar a\leq a\rightarrow a,$$ i.e. $a\rightarrow a=1$. If $b\leq\tilde a$ then because of $\tilde a\vee b\leq\tilde b\rightarrow\tilde b$ we have $$\tilde a\odot\tilde b=(\tilde a\vee b)\odot\tilde b\leq\tilde b\wedge\tilde b=\tilde b$$ whence $$b=\bar{\tilde b}\leq\overline{\tilde a\odot\tilde b}=\widetilde{\bar a\odot\bar b}=a+b$$ showing that $\overline{a+b}+b$ is defined. Now in case $b\leq\tilde a$, i.e. $a\leq\bar b$ the following are equivalent: $$\begin{aligned} c & =\widetilde{\overline{a+b}+b}, \\ (\overline{a+b}+b)+c & =1, \\ \overline{a+b}+(b+c) & =1, \\ b+c & =a+b.\end{aligned}$$ This shows (P1). Now the following are equivalent: $$\begin{aligned} & a\leq b\text{ holds in }\mathbb P(\mathbf Q), \\ & a+\tilde b\text{ is defined}, \\ & \bar a\odot b\text{ is defined}, \\ & a\leq b\text{ holds in }\mathbf Q.\end{aligned}$$ Since $(Q,\vee,\wedge)$ is a lattice and the partial order relations in $\mathbf Q$ and $\mathbb P(\mathbf Q)$ coincide, $\mathbb P(\mathbf Q)$ is a lattice pseudoeffect algebra. As in the case of effect algebras, also every good lattice pseudoeffect algebra can be reconstructed from its assigned quasiresiduated lattice. Let $\mathbf P$ be a good lattice pseudoeffect algebra. Then $\mathbb P(\mathbb Q(\mathbf P))=\mathbf P$. Let $$\begin{aligned} \mathbf P & =(P,+,\,\bar{}\,,\,\tilde{}\,,0,1)\text{ with lattice operations }\vee\text{ and }\wedge, \\ \mathbb Q(\mathbf P) & =(P,\vee,\wedge,\odot,\rightarrow,\leadsto,0,1), \\ \mathbb P(\mathbb Q(\mathbf P)) & =(P,\oplus,{}^*,{}^+,0,1)\end{aligned}$$ and $a,b\in E$. Then $$\begin{aligned} a^* & =a\rightarrow0=\bar a+(a\wedge0)=\bar a+0=\bar a, \\ a^+ & =a\leadsto0=(a\wedge0)+\tilde a=0+\tilde a=\tilde a.\end{aligned}$$ Moreover, the following are equivalent: $$\begin{aligned} & a\oplus b\text{ is defined}, \\ & a\leq\bar b\text{ in }\mathbb Q(\mathbf P), \\ & a\leq\bar b\text{ in }\mathbf P\end{aligned}$$ and in this case $$a\oplus b=(a^*\odot b^*)^+=\widetilde{\bar a\odot\bar b}=\overline{\tilde a\odot\tilde b}=a+b.$$ 9 I. Chajda and R. Halaš, Effect algebras are conditionally residuated structures. Soft Comput. [**15**]{} (2011), 1383–1387. I. Chajda and H. Länger, Relatively residuated lattices and posets. Math. Slovaca (submitted). A. Dvurečenskij and S. Pulmannová, New Trends in Quantum Structures. Kluwer, Dordrecht 2000. ISBN 0-7923-6471-6. A. Dvurečenskij and T. Vetterlein, Pseudoeffect algebras. I. Basic properties. Internat. J. Theoret. Phys. [**40**]{} (2001), 685–701. D. J. Foulis and M. K. Bennett, Effect algebras and unsharp quantum logics. Found. Phys. [**24**]{} (1994), 1331–1352. Authors’ addresses: Ivan Chajda\ Palacký University Olomouc\ Faculty of Science\ Department of Algebra and Geometry\ 17. listopadu 12\ 771 46 Olomouc\ Czech Republic\ ivan.chajda@upol.cz Helmut Länger\ TU Wien\ Faculty of Mathematics and Geoinformation\ Institute of Discrete Mathematics and Geometry\ Wiedner Hauptstraße 8-10\ 1040 Vienna\ Austria, and\ Palacký University Olomouc\ Faculty of Science\ Department of Algebra and Geometry\ 17. listopadu 12\ 771 46 Olomouc\ Czech Republic\ helmut.laenger@tuwien.ac.at

--- abstract: | In this paper we develop an algorithm to calculate the prices and Greeks of barrier options in a hyper-exponential additive model with piecewise constant parameters. We obtain an explicit semi-analytical expression for the first-passage probability. The solution rests on a randomization and an explicit matrix Wiener-Hopf factorization. Employing this result we derive explicit expressions for the Laplace-Fourier transforms of the prices and Greeks of barrier options. As a numerical illustration, the prices and Greeks of down-and-in digital and down-and-in call options are calculated for a set of parameters obtained by a simultaneous calibration to Stoxx50E call options across strikes and four different maturities. By comparing the results with Monte-Carlo simulations, we show that the method is fast, accurate, and stable. *Keywords:* Hyper-exponential additive processes, matrix Wiener-Hopf factorization, first passage times, barrier options, multi-dimensional Laplace transform, Fourier transform, sensitivities. *Acknowledgements:* We would like to thank P. Howard and S. Obraztsov for their support, and also D. Madan for useful conversations. This research was supported by EPSRC grant EP/D039053, and was partly carried out while the authors were based at King’s College London. author: - | Marc Jeannin$^{\dagger, \star}$ and Martijn Pistorius$^{\star}$\ \ $\phantom{|}^{\dagger}$[Models and Methodology Group, Risk Management Department]{}\ [Nomura International plc]{}\ [Nomura House 1 St Martin’s-le-Grand, London EC1A 4NP, UK]{}\ [E-mail: marc.jeannin@nomura.com]{}\ \ $\phantom{|}^{\star}$[Department of Mathematics, Imperial College London]{}\ [South Kensington Campus, London SW7 2AZ, UK]{}\ [E-mail: martijn.pistorius@imperial.ac.uk ]{}\ \ title: | Pricing and hedging barrier options\ in a hyper-exponential additive model --- Introduction ============ Barrier options are contracts whose pay-offs are activated or de-activated when the underlying process crosses a pre-specified level. These contracts are among the most popular path-dependent options. To value barrier options, a model needs to be sufficiently flexible to calibrate call option prices at different strikes and maturities. However, it is desirable to maintain a degree of analytical tractability to facilitate the calculations, especially for the Greeks or the sensitivities. These sensitivities describe the change in the model price with respect to a change in the underlying parameter, and are important for an appreciation of the robustness of the model’s results. It is well known that the accurate evaluation of the Greeks is a challenging numerical problem, since standard PDE or Monte-Carlo methods are generally slow and unstable. It is well established that the geometric Brownian motion model lacks the flexibility to capture features in financial asset return data such as the skewness and the excess kurtosis. It cannot calibrate simultaneously to a set of call option prices. To address these limitations, one of the approaches consists of introducing jumps in the price process by replacing the Brownian motion by a Lévy process. Lévy models, such as the VG, CGMY, NIG, KoBoL, generalised hyperbolic, and Kou’s double exponential model, have been successfully applied to the valuation of European-type options. We refer to Cont and Tankov [@ContTankov], Boyarchenko and Levendorskii [@BLbook], and Schoutens [@Schoutens] for background and references on the application of Lévy models in option pricing. As observed by many authors, such as Eberlein and Kluge [@EberleinKluge], or Carr and Wu [@carr03], Lévy models are generally not capable of calibrating option prices simultaneously across strikes and maturities. Empirical studies of S&P500 index data by Carr and Wu [@carr03], and Pan [@pan02], show that the implied jump intensities and the implied jump size distributions vary greatly over time. The prices of short-dated options exhibit a significantly larger risk-premium than that of long-dated options. This is reflected in the thicker tails of the implied marginal risk-neutral distributions, especially at short maturities. For example, in the equity markets, short-dated out-of-the money put options are relatively expensive since the risk of a large negative jump in the share is priced. Because of the stationarity and independence of the increments of a Lévy process, the moments exhibit a rigid term structure that is different from what is observed in market data. This lack of flexibility can be overcome by considering models driven by additive processes, which have independent and time-inhomogeneous increments. Additive models have been used for equity option pricing by Carr et al. [@carr06], Galloway and Nolder [@galloway04], and by Eberlein and Kluge [@EberleinKluge] for interest rate option pricing. Motivated by modelling considerations, Carr [@carr06] proposed a self-similar additive model for the log-price, and reported good calibration results across time. Galloway and Nolder [@galloway04] carried out a calibration study for various related models. Eberlein and Kluge [@EberleinKluge] constructed an HJM model driven by an additive process with continuous characteristics, and they obtained a good fit for swaptions by using piecewise constant parameters. In this paper we follow a similar approach: we model the share price by an additive process with hyper-exponential jumps. Hyper-exponential distributions are finite mean-mixtures of exponential distributions which can approximate monotone distribution arbitrarily closely. As first observed by Asmussen et al. [@AMP], most of the popular Lévy models used in mathematical finance possess completely monotone Lévy densities and can therefore be approximated well by hyper-exponential Lévy models. A hyper-exponential additive model is sufficiently flexible to allow for an accurate calibration to European option prices across strikes and multiple maturities. In addition, if the parameters are piecewise constant, the model admits semi-analytical expressions for prices and Greeks of barrier options. There currently is a body of literature devoted to various aspects of pricing barrier options. In the setting of Lévy models, a transform-based approach to price barrier options has been developed in a number of papers, including Geman and Yor [@GemanYor], Kou and Wang [@KouWang], Davydov and Linetsky [@DavydovLinetsky], Boyarchenko and Levendorskii [@BoyLev]. In particular, Kou and Wang [@KouWang], Kou et al. [@KouPW], Sepp [@Sepp], Lipton [@Lipton], and Jeannin and Pistorius [@JeanninP] considered the cases of Lévy processes with double-exponential and hyper-exponential jumps. In this paper, the transform algorithm that we develop is based on a so-called matrix Wiener-Hopf factorization. Such matrix factorizations were first studied by London et al. [@london80] and Rogers [@rogers94] for (noisy) fluid models. Jiang and Pistorius [@JiangP] developed matrix-Wiener factorization results for regime-switching models with jumps. We show that by suitably randomizing the parameters the distributions of the infimum and supremum of the randomized hyper-exponential additive process can be explicitly expressed in terms of a matrix Wiener-Hopf factorization. We use these results to derive semi-analytical expressions for the first-passage time probabilities, for the prices, and for the Greeks of barrier options, up to a multi-dimensional transform. The actual prices are subsequently obtained by inverting this transform. As a numerical illustration, we calibrate the hyper-exponential additive model to Eurostoxx prices quoted on 27 February 2007 at four different maturities. We calculate in this setting down-and-in digital and down-and-in call option prices and Greeks (delta and gamma). To invert the transform, we use a contour deformation algorithm and a fractional Fast Fourier Transform algorithm, developed by Talbot [@talbot79], Bailey and Swarztrauber [@Bailey94], and Chourdakis [@choudhury94], [@chourdakis04]. We also compare it to Monte-Carlo Euler scheme simulations. We find that the algorithm is accurate and stable, and much faster than Monte-Carlo simulations (especially for the Greeks). This method is suitable for applications in which the number of periods is not too large (up to four). When a larger number of periods is required, the direct inversion method used here is no longer feasible. The subject still needs to be further investigated and is left for future research. The remainder of the paper is organized as follows. In Section \[sec:add\] we define the hyper-exponential additive model and present its application to European call option pricing. In Sections \[sec:WH\] and \[sec:price\] we derive semi-analytical expressions for the first-passage probabilities of a hyper-exponential additive process in terms of a matrix Wiener-Hopf factorisation, and for the prices and Greeks of barrier options. In Section \[sec:num\] we present numerical results. The model {#sec:add} ========= Additive processes ------------------ We consider an asset price process $S$ modelled as the exponential $$S_t = S_0 \te{X_t}$$ of an additive process $X$. Informally, an additive process can be described as a Lévy process with time-dependent characteristics or, equivalently, as a process with independent but non-stationary increments. We briefly review below some key properties of additive processes. For further background on additive processes and their applications in finance, we refer to Sato [@Sato], and to Cont and Tankov [@ContTankov]. An additive process can be defined more formally as follows. \[def:locallevy\] For a given $T>0$, $X = \{X_t, t\in[0,T]\}$ is an additive process if - $X_0=0$, - For any finite partition $0\leq t_0<t_1 \cdots <t_k\leq T$, the random variables $X_{t_k}-X_{t_{k-1}}, \cdots, X_{t_0}$ are independent, - The sample paths $t\mapsto X_t$ have càdlàg modifications almost surely. If $X$ is an additive process, then, for every $t\in[0,T]$, $X_t$ has an infinitely divisible distribution with Lévy triplet $(M_t, \Sigma^2_t, \Lambda_t)$; that is, the characteristic function of $X_t$ is given by $\Phi_t(u) = \exp [\Psi_{t}(u)\,]$. According to the Lévy-Khintchine formula, $\Psi_{t}$ is the characteristic exponent given by $$\Psi_{t}(u) = \mathbf i u M_t - \frac{\Sigma^2_t}{2} u^2 + \int_{-\i}^\i \le\{\te{\mathbf iux} - 1 - \mathbf iux1_{\{|x|<1\}}\ri\}\Lambda_t(\td x),$$ with $M_t$, $\Sigma_t\in\mathbb R$, and where $\Lambda_t$ the Lévy measure satisfies the integrability constraint $$\int (1\wedge x^2)\Lambda_t(dx)<\i.$$ The law of the additive process $\{X_t, t\in[0,T]\}$ is determined by the collection of Lévy triplets $\{(M_{t},\Sigma^2_{t}, \Lambda_{t})\ \text{for}\ t\in[0,T]\}$. If the Lévy triplets are time-independent, $X$ is a Lévy process. If the additive process has absolutely continuous characteristics, the Lévy triplets take the explicit form $$\begin{aligned} M(t) &=& \int_0^t \mu(s)\td s, \qquad \Sigma(t) = \int_0^t\sigma^2(s)\td s,\\ \Lambda_t(B) &=& \int_0^t\int_B g(s,x)\td x\td s \qquad \text{for Borel sets $B$},\end{aligned}$$ where $\mu,\sigma^2:[0,T]\to\mathbb R$ and $g:[0,T]\times\mathbb R\to\mathbb R$ are integrable functions, with $g$ and $\sigma^2$ non-negative. We call the functions $(\mu,\sigma^2, g)$ the local triplet of $X$.\ \ We assume that we have been given deterministic integrable functions $r(t)$ and $d(t)$ representing the short rate and the dividend yield, and that the characteristic exponent of $X_t$ satisfies $$\label{eq:psi1} \Psi_t(-\mathbf i) = \int_0^t[r(s)-d(s)]\td s,\qquad $$ or equivalently $$\label{eq:psi2} \mu(t) + \frac{\sigma^2(t)}{2} + \int_{-\infty}^\infty [\te{x} - 1 - 1_{\{|x|<1\}}x]g(t,x)\td x = r(t) - d(t).$$ It follows that the discounted process $$S_{0}\exp(\int_0^t [r(s)-d(s)]\td s)$$ is a martingale if and only if (or, equivalently, ) holds. Hyper-exponential additive processes ------------------------------------ In what follows we restrict the discussion to a hyper-exponential additive process $X$ which is specified by its local triplet $(\mu,\sigma^2,g)$ where $g$ is given by $$g(t,x) = \sum_{k=1}^{n^+}\pi^+_k(t)\alpha^+_k(t)\te{-\alpha^+_k(t)x}1_{\{x>0\}} + \sum_{j=1}^{n^-}\pi^-_j(t)\alpha_j^-(t)\te{-\alpha^-_j(t)|x|}1_{\{x<0\}},$$ where $\pi_k^\pm(t)$ and $\alpha^\pm_k(t)$ are non-negative. The continuous part of $X$ consists of a diffusion with time-dependent drift $\mu(t)$ and volatility $\s(t)$. The jump part of $X$ is of finite activity and forms an inhomogeneous compound Poisson process where positive and negative jumps occur at the rates $$\lambda^+(t):=\sum_{k=1}^{n^+}\pi^+_k(t) \q\text{ and }\q \lambda^-(t):= \sum_{j=1}^{n^-}\pi^-_j(t),$$ and jump sizes are distributed according to a hyper-exponential distribution.\ Small random price movements are intuitively modelled by the diffusion part, whereas sudden changes of the price are captured by the jump-part of $X$. If we take $n^\pm = 1$, the jump-sizes are exponentially distributed, and this model reduces to an extension of the Kou model with time-dependent parameters. Piecewise constant parameters ----------------------------- To reduce the dimension of the available parameter set, we take the functions $\mu(t),\sigma(t)$ and $g(t,\cdot)$ to be piecewise constant. Given that we have a finite set of European call options with different maturities $T_1, \ldots, T_N$, we take the local parameters to be constant between the different maturities $T_i$. Then for all $t \in (T_{i-1},T_i]$, (with $T_0=0$) we set $$\label{eq:para} \mu(t) = \mu^{(i)},\quad \sigma^2(t) = \sigma^{2(i)},\quad g(t,x) = g^{(i)}(x), \quad\text{$i=1, \ldots, N$}.$$ For $t\in(T_{i-1}, T_i]$ the characteristic exponent of $X_t-X_{T_{i-1}}$ is given by $$\Psi_{T_{i-1},t}(u) =: \Psi^{(i)}(u),$$ where $$\label{eq:Psi} \Psi^{(i)}(u) = \mu^{(i)} u\mathbf i - \frac{\sigma^{2(i)}}{2}u^2 + \sum_{k=1}^{n^+} \pi_k^{+(i)}\le(\frac{u\mathbf i}{\alpha_k^{+(i)} - u\mathbf i}\ri) - \sum_{j=1}^{n^-} \pi_j^{-(i)}\le(\frac{u\mathbf i}{\alpha_j^{-(i)} + u\mathbf i}\ri).$$ First passage probabilities {#sec:WH} =========================== The value of a digital barrier option can be expressed in terms of the distribution $$F^{(+)}(x;T) = P(\ovl X_T \leq x)$$ of the running supremum $$\ovl X_T = \sup_{s\leq T} X_s$$ of $X$, or equivalently, the distribution of the first-passage time $$T^+(x)=\inf\{t\ge0: X_t>x\}$$ which is related to $F^{(+)}$ by $$P(T^+(x)\leq T) = 1-F^{(+)}(x;T).$$ Whereas for a Lévy process the distributions of the infimum and supremum are linked to the characteristic exponent by the so-called Wiener-Hopf factorization, such a result does not exist for general additive processes, because of the time-dependence of the parameters. However, in the case of piecewise constant parameters, the triplet changes only at deterministic times, so that as a consequence the distribution function $F^{(+)}(x) = F^{(+)}(x; T^{(1)}, \ldots, T^{(N)})$ only depends on the inter-jump times $T^{(i)} = T_i - T_{i-1}$ (with $T_0=0$). In this case, as we show below, the $N$-dimensional Laplace transform $G^{(+)}(x,\mbf q)$ of $F^{(+)}$, given by $$G^{(+)}(x,\mbf q) = \int \te{- (q_1 u_1 + \cdots\, + q_Nu_N)} F^{(+)}(x; u_1, \ldots, u_N)\td u_1 \cdots \td u_N,$$ where $\mbf q = (q_1, \ldots, q_N)$, is expressed explicitly in terms of a matrix Wiener-Hopf factorization. To state this result we need to introduce some further notation. For any vector $\mbf v=(v_1, \ldots, v_n)$, we denote by $\Delta_{{{\mbox{\scriptsize\boldmath$v$}}}}$ the diagonal matrix $\Delta_{{{\mbox{\scriptsize\boldmath$v$}}}} = (v_i, i=1, \ldots, n)_{\mrm{diag}}$. Let $Q$ be the $N(1+n^++n^-)\times N(1+n^++n^-)$ matrix given in block notation by $$\label{eq:Q} Q = \begin{pmatrix} H^+ & D^-\\ C^- & T^- \end{pmatrix},$$ where $$\label{eq:H} H^+ = \begin{pmatrix} G - \Delta_{{{\mbox{\scriptsize\boldmath$\lambda$}}}} & b^+\\ t^+ & T^+ \end{pmatrix}.$$ Here ${{\mbox{\boldmath$\lambda$}}}=(\lambda_i^+ + \lambda_i^-, i=1, \ldots, N)$, and $G$ and $b^+$ are the $N\times N$ and $N\times Nn^+$ matrices in block notation given by $$\label{eq:b+} G = \begin{pmatrix} -q_1 &q_1 &&&\\ & -q_2&q_2&&\\ &&\ddots& \\&&&-q_{N-1} & q_{N-1}\\ &&&& -q_N \end{pmatrix}, \ b^+ = \begin{pmatrix} \mbf\pi^{+{(1)}} &&&\\ & \mbf\pi^{+{(2)}} &&\\ &&\ddots&\\ &&& \mbf\pi^{+{(N)}} \end{pmatrix}.$$ Here $\mbf\pi^{+(i)}$ is the row-vector ${{\mbox{\boldmath$\pi$}}}^{+(i)}= (\pi^{+(i)}_l, l=1, \ldots, n)$, and where $t^+$, $T^+$ are given by $$\begin{aligned} \label{eq:t+} T^+ &= \begin{pmatrix} -\Delta_{{{\mbox{\scriptsize\boldmath$\a$}}}^+} &&&\\ & -\Delta_{{{\mbox{\scriptsize\boldmath$\a$}}}^+} &&\\ &&\ddots&\\ &&& -\Delta_{{{\mbox{\scriptsize\boldmath$\a$}}}^+} \end{pmatrix},\quad t^+ = \begin{pmatrix} \mbf\a^+ &&&\\ & \mbf\a^+ &&\\ &&\ddots&\\ &&& \mbf\a^+ \end{pmatrix},\end{aligned}$$ where $\mbf\a^+$ is the column-vector $\mbf\a^+ = (\alpha_i^+, i=1, \ldots, n^+)'$, and $$\label{eq:C-D-} C^-= \begin{pmatrix} t^-& O^- \end{pmatrix}, \quad\quad\quad D^- = \begin{pmatrix} b^- \\ O^+ \end{pmatrix},$$ where $O^\pm$ are $n^\pm N\times n^\pm N$ zero matrices, and $b^-$, $T^-$, and $t^-$ are given by and with ${{\mbox{\boldmath$\pi$}}}^{+(i)}$ and ${{\mbox{\boldmath$\alpha$}}}^+$ replaced by ${{\mbox{\boldmath$\pi$}}}^{-(i)}$ and ${{\mbox{\boldmath$\alpha$}}}^-$. The matrix $Q$ is a generator matrix, that is, a square matrix with non-negative off-diagonal elements and non-positive row sums, and defines a Markov chain. This Markov chain is associated to a randomization and embedding of the additive process $X$ (which will be illustrated with a concrete example below). We recall that a sub-probability matrix is a matrix with non-negative elements and row sums not larger than one. By applying the matrix Wiener-Hopf factorization results of Jiang and Pistorius [@JiangP] to the current setting we arrive at the following conclusion. \[thm:wh\] It holds that $$\label{eq:GX} G^{(+)}(x,{{\mbox{\boldmath$q$}}}) = \frac{1}{q_1 \ldots q_N} \times \left[ 1 - e_1' \te{Q_+ x}\mathbf 1\right],$$ where $$\mbf e_1'=(1, 0, \ldots, 0)\q\text{ and }\q\mathbf 1=(1,\ldots, 1)'.$$ $Q_+$ is an $N(1+n^+)\times N(1+n^+)$ generator matrix that together with $\eta^+$ an $Nn^-\times N(1+n^+)$ sub-probability matrix, solves the system of matrix equations $$\label{eq:WH+} \begin{cases} \frac{1}{2}S^2 Q_+^2 - V^+ Q_+ + H^+ + D^-\eta^+ = O,\\ \\ -\eta^+ Q_+ + C^- + T^-\eta^+ = O. \end{cases}$$ Here the $O$’s are zero matrices of appropriate sizes, and in block notation we have, $$\label{eq:TSV} S^2 = \begin{pmatrix} \Delta_{{{\mbox{\scriptsize\boldmath$\s$}}}^2} & \\ & O^+ \end{pmatrix}, \qquad V^+ = \begin{pmatrix} + \Delta_{{{\mbox{\scriptsize\boldmath$\m$}}}} & \\ & I^+ \end{pmatrix},$$ with $${{\mbox{\boldmath$\s$}}}^2=(\s_i^2, i=1,\ldots, N),\qquad {{\mbox{\boldmath$\mu$}}}=(\mu_i, i=1, \ldots,N).$$ $O^+$ and $I^+$ represent $n^+N\times n^+N$ zero and identity matrices, respectively. By applying Theorem \[thm:wh\] to $-X$, we find the corresponding pair of matrices $(Q_-,\eta_-)$. The quadruple $(Q_+,\eta_+,Q_-, \eta_-)$ is called a matrix Wiener-Hopf factorization of $Q$.\ \ [**Example.**]{} To illustrate this approach, we consider a hyper-exponential additive process $X$ on $[0,T_2]$ whose parameters are constant during the periods $[0,T_1]$ and $[T_1, T_2]$. In the first period $X$ evolves as a jump-diffusion with positive and negative exponential jumps with means and jump rates $1/\alpha^+, \lambda^+$ and $1/\alpha^-, \lambda^-$. In the second period $X$ is a Brownian motion with drift. The idea is to randomize the times between maturities by replacing $T^{(1)}=T_1$ and $T^{(2)}=T_2-T_1$ with independent exponential random variables having means $q_1^{-1}$ and $q_2^{-1}$. This results in a regime-switching jump-diffusion with the regime only jumping from state 1 to state 2, according to the generator matrix $$G = \begin{pmatrix} -q_1 & q_1\\ 0 & -q_2 \end{pmatrix}.$$ We associate to the regime-switching process a continuous Markov additive process, which can be informally obtained by replacing positive and negative jumps with stretched slopes of $+1$ and $-1$ (see Asmussen [@asm00] for background on this embedding). As described in [@JiangP], in this case the generator of the modulating Markov process is given by $$Q = \begin{pmatrix} H^+ & D^-\\ C^- & T^- \end{pmatrix} = \le(\begin{array}{ccc|c} -q_1-\lambda^+-\lambda^- & q_1 & \lambda^+ & \lambda^-\\ 0 & -q_2 & 0 & 0\\ \alpha^+ & 0 & -\alpha^+ & 0\\ \hline \alpha^- & 0 & 0 & -\alpha^- \end{array}\ri),$$ with the matrices $S^2$ and $V^+$ in Theorem \[thm:wh\] given by $$S^2 = \begin{pmatrix} \sigma_1^2 \\ & \sigma_2^2\\ && 0 \end{pmatrix}, \qquad V^+ = \begin{pmatrix} \mu_1 \\ & \mu_2\\ && 1 \end{pmatrix}.$$ ![Paths of the various processes related to log-price process $X$ are illustrated. Here $X$ is a hyper-exponential additive process on the period $[0,T_2]$ whose parameters are constant during the periods $[0,T_1]$ and $[T_1, T_2]$. During the first period $[0,T_1]$, the process $X$ evolves as a jump-diffusion with volatility $\sigma_1$, drift $\mu_1=0$ and exponentially-distributed jumps. The positive and negative jumps are exponentially-distributed with means and jump rates $1/\alpha^+, \lambda^+$ and $1/\alpha^-, \lambda^-$, respectively. During the second period $[T_1,T_2]$, $X$ evolves as a linear Brownian motion with volatility $\sigma_2$ and drift $\mu_2=0$. Associated to $X$ is a continuous Markov additive process $A$, which can be obtained from $X$ by replacing the positive and negative jumps with linear stretches of path of slopes $+1$ and $-1$, and replacing the fixed times $T_1$, $T_2-T_1$ by independent exponential random times $\mbf e_1$ and $\mbf e_2$ with parameters $q_1, q_2$. The process $Y$ that records the current state or regime of $A$ is a Markov process with generating matrix $Q$. When $Y$ takes values 1 and 2, $A$ evolves as a linear Brownian motion with zero drift and volatility $\sigma_1$ and $\sigma_2$ respectively; and when $Y$ is 3 and 4, $A$ is a positive or negative unit drift. These linear stretches of paths of $A$ originate from the jumps of $X$. A jump of $Y$ from one state to another is induced either by a jump of $X$ or by a switch of the set of parameters that determine the dynamics of $X$. By time-changing $A$ by the time $T_0(t)$ up to time $t$ that $Y$ was equal to $1$ and $2$, we recover a regime-switching jump-diffusion; that is, the process $\{A(T_0(t)), t\ge 0\}$ is in law equal to a regime-switching jump-diffusion. Finally, replacement of the times at which a regime switch occurs by $T_1$ and $T_2$ results in a process that has the same distribution as $X$.[]{data-label="Fig:emb"}](Paths.eps){height="10.4cm" width="7.55cm"} Solution of the matrix equation ------------------------------- To solve the system , which is a Ricatti-type matrix equation, we follow a spectral approach and determine the spectral decomposition of $Q^+$. Denoting by $h(\rho)$ a (column) eigenvector of $Q^+$ corresponding to the eigenvalue $\rho$, one finds that it is a matter of algebra to verify that the system can be equivalently rewritten as $$\frac{1}{2} \T S^2\begin{pmatrix}I\\\eta^+\end{pmatrix}Q_+^2 - \T V\begin{pmatrix}I\\\eta^+\end{pmatrix}Q_+ + Q\begin{pmatrix}I\\\eta^+\end{pmatrix} = O.$$ Here $O$ is a $N(1+n^+ + n^-)$ square zero matrix, $I$ is an $N(1+n^+)$ identity matrix, and $$\T S^2 = \begin{pmatrix} \Delta_{{{\mbox{\scriptsize\boldmath$\s$}}}^2} & \\ & O^+\\ && O^- \end{pmatrix}, \qquad \T V = \begin{pmatrix} \Delta_{{{\mbox{\scriptsize\boldmath$\m$}}}} & \\ & I^+ \\ && -I^- \end{pmatrix}.$$ Defining the matrix $K(s)$ by $$\label{eq:K} K(s) = \frac{s^2}{2}\T S^2 + s\T V + Q,$$ we find that $h(\rho)$ solves the linear system $$K[-\rho]\begin{pmatrix}I\\\eta^+\end{pmatrix}h(\rho)=\mbf 0,$$ which implies that $\rho$ is a root of the equation $\det K(s) = 0$. The following result characterizes the eigenvalues of $Q^+$ (see Appendix A): \[lem:eigen\] - It holds that $$\label{eq:detK} |\det (K(s))| = \prod_{i=1}^N \le\{|\Psi^{(i)}(-\mathbf i s) - q_i| \prod_{k=1}^{n^+} |s-\alpha^+_k| \prod_{l=1}^{n^-} |s+\alpha^-_l|\ri\},$$ where $\Psi^{(i)}$ is given in . - The equation $$\label{eq:detK2} \det K(s)=0$$ has $N(1+n^+)$ positive roots and $N(1+n^-)$ negative roots. Since $-Q^+$ is the negative of a generator matrix, it is non-negative definite, so its eigenvalues are non-negative and are given by the positive roots of . In particular, if the positive roots $${{\mbox{\boldmath$\rho$}}}_+=\le(\rho^+_i: i=1, \ldots, N(n^++1)\ri)$$ of equation are distinct, it follows from Lemma \[lem:eigen\] and Theorem \[thm:wh\] that $$G^{(+)}(x,\mbf q) = (q_1\cdots q_N)^{-1} \times [1-\mbf e_i' U_+ \te{-\Delta_{{{\mbox{\scriptsize\boldmath$\rho$}}}_+} x} U_+^{-1}\mathbf 1],$$ where $U_+ = (h(\rho^+_i), i=1, \ldots, N(n^++1))$. The final position and the first exit time ------------------------------------------ The valuation of barrier options involves the joint distribution of the final position at maturity $T$ and the first exit time. We will extend the results in the previous section by considering the following: $$\begin{aligned} \ovl F^{(+)}(x,s) &:=& E[\te{sX_T}\mathbf 1_{\{\ovl X_T > x\}}]\\ &=& E[\te{sX_T}\mathbf 1_{\{ T^+(x) < T\}}].\end{aligned}$$ $\ovl F^{(+)}(x,s)$ depends on time only through the inter-maturity times $(T^{(1)},\ldots, T^{(N)})$. The Laplace transform $\ovl G^{(+)}(x,s;q)$ of $\ovl F^{(+)}(x,s)$ in $(T^{(1)},\ldots, T^{(N)})$, can be expressed in terms of $Q^+$ and $K(s)$ as follows: \[prop:XS\] It holds that $$\label{eq:GXS} \ovl G^{(+)}(x,s,\mbf q) = \frac{\te{sx}}{q_1\ldots q_N} \times \mbf e_1' \te{Q^+ x}K(s)^{-1} K(0)\mathbf 1$$ for all $s\in\mathbb C$ with $\mathrm{Re}(s)\in (-\min_{j=1,\ldots, n^-} \a_j^-,\min_{k=1,\ldots,n^+} \a^+_k)$. A proof is given in Appendix A. First passage to a lower level {#ssec:low} ------------------------------ The form of the analogous distributions concerning the infimum $$F^{(-)}(x) = P(- \unl X_T\leq x), \qquad \ovl F^{(-)}(x,s) = E[\te{sX_T}\mathbf 1_{\{- \unl X_T > x\}}],\quad x>0,$$ can be found by applying the results in the previous section to the process $-X$. More specifically, it is straightforward to check that the $N$-dimensional Laplace transforms $G^{(-)}(x,q)$ and $\ovl G^{(-)}(x,s,q)$ are given by and replacing $Q^+$ by $Q^-$. $(Q^-,\eta^-)$ satisfies the system of matrix equations with $V^+, H^+, D^-, C^-, T^-$ replaced by $V^-, H^-, D^+, C^+, T^+$, where the latter set is defined by interchanging $+$ and $-$ in equations , , and . It is straightforward to verify that *(i)* an eigenvector $h(\rho)$ of $Q^-$ corresponding to eigenvalue $\rho$ satisfies $$K[\rho]\begin{pmatrix}\eta^-\\ I\end{pmatrix}h(\rho)=0,$$ where $I$ is an $N(1+n^-)$ identity matrix, and *(ii)* that, in view of Lemma \[lem:eigen\], the eigenvalues of $Q^-$ are given by the negative roots $${{\mbox{\boldmath$\rho$}}}^- = \le(\rho^-_j, j=1, \ldots, N(n^-+1)\ri)$$ of $\det K(\rho) = 0$. Prices and Greeks of digital and barrier options {#sec:price} ================================================ Using the first-passage results from the previous section we derive semi-analytical expressions for the prices and sensitivities of down-and-in digital and knock-in call options. A down-and-in digital option at level $H<S_0$ is a contract that pays out one unit at maturity $T$ if the price $S$ has down-crossed the level $H$ before $T$. Similarly, a down-and-in call option at level $H<S_0$ and with strike $K$ is a call option whose pay-off is activated once $S$ down-crosses $H$. Taking the risk-free rate $r$ and the dividend rate $d$ to be constant, the arbitrage free prices of a down-and-in digital $(DID)$ and a call option $(DIC)$ are given respectively by $$DID(T,H,S_0) = \te{-(r-d)T} E\le[\mathbf 1_{\{\inf_{s\leq T} S_s < H\}}\ri] = \te{-(r-d)T} P(\unl X_T < h),$$ where $h=\log(H/S_0)$ is the log-barrier, and $$DIC(T,H,K,S_0) = \te{-(r-d)T} S_0 E\le[(\te{X_T} - \te{k})^+\mathbf 1_{\{\unl X_T < h\}}\ri],$$ where $k=\log(K/S_0)$ denotes the log-strike. Let $\WH{DID}(\mbf q)$ denote the joint Laplace transform of $DID$ in the inter-maturity times $(T^{(1)}, \ldots, T^{(N)})$ (with $T^{(N)}=T$), and denote by $\WH{DIC}^*(\mbf q,s)$ the Laplace-Fourier transform in $(T^{(1)}, \ldots, T^{(N)})$ and in the log-strike $k$. Then we have the following result: \[prop:price\] For $h=\log(H/S_0)<0$ it holds that $$\begin{aligned} \label{eq:DID} \WH{DID}(\mbf q) &=& \frac{1}{c(q)} \times \mbf e_1' \te{Q^- h}\mathbf 1,\\ \WH{DIC}^*(\mbf q,s) &=& \frac{S_0\te{bk}}{c(\mbf q) b(b-1)} \times \mbf e_1' \te{Q^- h}K(s)^{-1}K(0)\mathbf 1, \label{eq:DIC}\end{aligned}$$ where $k=\log(K/S_0)$, $\mbf q=(q_1,\ldots, q_N)$, and $b=\alpha + \mathbf i s + 1, c(\mbf q) = (q_1 + r) \cdots (q_N + r).$ Before we give the proof we observe that from the explicit expressions and semi-analytical formulas can be obtained for the delta and gamma of the down-and-in digital and call options (i.e. the first and second derivatives of the option value with respect to the spot $S_0$). Indeed, the derivatives of the expressions and with respect to $S_0$ are equal to the Laplace-Fourier transforms of the derivatives of the option, as integration and differentiation are interchangeable in this case. In the case of a down-and-in digital option we find that the Laplace transforms $\WH{\Delta}_{DID}$ and $\WH{\Gamma}_{DID}$ of the delta $\Delta_{DID}$ and gamma $\Gamma_{DID}$ are given by $$\WH{\Delta}_{DID}(\mbf q) = -\frac{1}{c(q)S_0}\times e_1' Q^-\te{Q^- h}\mathbf 1, \ \quad \WH{\Gamma}_{DID}(\mbf q) = \frac{1}{c(q)S_0^2}\times e_1' [(Q^-)^2 + Q^-]\te{Q^- h}\mathbf 1.$$ The expression is a direct consequence of Theorem \[thm:wh\] (see also Section \[ssec:low\]). To verify we start by taking the Fourier transform in $k$ and find as in that the Fourier transform $DIC^*$ is given by $$\label{eq:DIC*} DIC^*(\alpha+\mathbf i s) = \frac{\ovl F^{(-)}(-h,b)}{b(b-1)},$$ where $b = \alpha+\mathbf i s + 1$ and $$\ovl F^{(-)}(x,b) = E[\te{bX_T}\textbf{1}_{\{-\unl X_T \geq x\}}].$$ From Proposition \[prop:XS\] we deduce that the form the joint Laplace transform $\ovl G(x,b,\mbf q)$ of $\ovl F^{(-)}(x,b)$ in $(T^{(1)},\ldots, T^{(N)})$ is given by $$\label{eq:ovlG} \ovl G(x,b,\mbf q) = \frac{\te{bx}}{c(\mbf q)} \times \mbf e_1' \te{Q^- h}K(s)^{-1}K(0)\mathbf 1.$$ Combining and completes the proof.[ ]{} Numerical results {#sec:num} ================= Calibration ----------- To determine a parameter set to test the method, we calibrate the hyper-exponential additive model to Eurostoxx call options at four different maturities, observed in the market on 20 February 2007. The spot price is EUR $4150$, the risk-free rate is assumed to be fixed at $r=0.03$, and the dividend rate is taken to be zero. As we find that inclusion of positive jumps does not substantially improve the calibration results, we only consider negative jumps, and we specify the jump size parameters to be $(\alpha^-_1, \alpha^-_2)=(3,10)$. The jump arrival rates and the volatility are piecewise constant in time, and are estimated by minimizing the root-mean-square error between model and observed market call prices. Using the well known Fourier transform method (briefly recalled in Appendix \[app:euro\]) the calibration is carried out maturity by maturity under constraints through a bootstrapping method with well-defined local triplets: 1. Calibrate call prices at $T_1$ to obtain the parameters $(\sigma(T_1),\pi_1^{\pm}(T_1),\pi_2^{\pm}(T_1))$. 2. For $j=2, \ldots, N$ calibrate call prices at $T_j$ to obtain $(\sigma(T_j),\pi_1^{\pm}(T_j),\pi_2^{\pm}(T_j))$. In Figure 2 the calibration results are presented with plots of the market and model implied volatility surfaces corresponding to the four maturities $6$m, $1$Y, $3$Y and $5$Y. The root-mean-square error (RMSE) and the average relative percentage error (ARPE) are equal to $5.30$ and $1.1\%$. We compare it to a price process that follows a Lévy process with hyper-exponential jumps (i.e. with constant parameters over time), and find that the calibration of the four maturities in that case give a RMSE of $9.82$ and an ARPE of $2.9\%$. In Table 1 the resulting parameter sets are displayed under the hyper-exponential additive and Lévy models. In the case of the hyper-exponential model we observe a high jump intensity of small jumps for short maturities that decrease substantially over time. This is consistent with the finding of Carr and Wu [@carr03], and Pan [@pan02]. \[fig:cal\] Results for the barrier option prices and Greeks ------------------------------------------------ Using the parameter set found in the calibration of the Eurostoxx call options, we value barrier and digital options on the Eurostoxx index, modelling its price process as the exponential of a hyper-exponential additive process. We use the semi-analytical results in Proposition \[prop:price\]. To invert the multi-dimensional Laplace transforms we choose Talbot’s method [@talbot79] (see also [@choudhury94]) for down-and-in digital options. We combine it with the fractional FFT algorithm of Bailey and Swarztrauber [@Bailey94], and Chourdakis [@chourdakis04] for down-and-in call options. See Appendix \[app:trans\] for a detailed description of the implementation of these transform algorithms. We compare it to the same quantities calculated by Monte-Carlo simulations, using a standard Euler scheme. ### Down-and-in digital options We price down-and-in digital options with a maturity of five years for different spot levels. We evaluate the required 4-dimensional Laplace transform over two time increments of six months, and two of two years, that is, $T^{(1)}=0.5$, $T^{(2)}=0.5$, $T^{(3)}=2$, $T^{(4)}=2$. We use Talbot’s algorithm with $M=6$ (see Appendix \[app:trans\] for an explanation of this parameter). Using Mathematica to run the algorithm, the computation time was five minutes on a 3189 Mhz computer to calculate prices and Greeks for fourteen different spot levels. The calculation of first passage probabilities using Monte-Carlo simulations requires a large number of time steps and paths. We use one million paths with $\delta t= 5 \times 10^{-5}$ and it takes several hours to obtain stable Greeks in C++. Error bounds cannot be obtained analytically, but we observe in Table \[table:VGDD\] that the results of the transform method agree with Monte-Carlo simulation results. Figures 3 and 4 report prices and Greeks for down-and-in digital options. The options are expressed as a percentage of the spot price. The values of the sensitivities are expressed as fractions of the spot price $S_0$. This transform algorithm is particulary efficient at a book level, since once the generating matrices $Q^-$ of the infimum have been calculated for different values of the vector $\textbf{q}$ the calculation of prices and Greeks of any digital barrier product is just a matter of summation. ### Down-and-in call options We value down-and-in call options with a maturity of one year for different strike levels. In this case, a two-dimensional Laplace inversion is required over time increments $T^{(1)}=0.5$ and $T^{(2)}=0.5$. For the inversions of the Laplace transform and the Fourier transform, we set $M=7$ and $N=1024$ (refer to Appendix \[app:trans\] for an explanation of these parameters). For Monte-Carlo simulations, we use one million paths with time step $\delta t= 2.5 \times 10^{-5}$. The option prices and Greeks obtained by the two methods are reported in table \[table:VGDIC\] and figures \[fig:DID5YG\] and \[fig:DIC1Y\]. We observe that the results of the transform method agree with the Monte-Carlo simulation results. Using Mathematica again to run the algorithm, the computation time is ten minutes to calculate the option prices, delta and gamma for eleven different levels of the strike. Since the option prices and Greeks of a down-and-in call option are obtained via a Fourier-Laplace transform, it takes more time than in the case of a digital option (approximately twice as long), which is still much faster than a Monte-Carlo Euler scheme. We note that the transform algorithm is particulary efficient for the pricing of options with different strikes, as we obtain by FrFFT inversion the prices and Greeks of any down-and-in call options on a log-strike grid. \ \ **APPENDIX** Proofs ====== [*Proof of Lemma \[lem:eigen\]:*]{} - It is straightforward to verify that $K^\#(s)$ can be obtained from $K(s)$ by interchanging some columns and rows, where $K^\#(s)$ is given by $$K^\#(s) =\left( \begin{array}{ccccc} K_1(s) & D_{1} &&&\\ & K_2(s) &D_{2}& &\\ &&.&.&\\ &&& K_{N-1}(s) &D_{N-1} \\ && & &K_N(s)\\ \end{array} \right),$$ where $K_w(s)$ and $D_w$ are square matrices of dimension $n^++n^-+1$. There are defined respectively by $$K_w(s) = \left( \begin{array}{ccccccc} \frac{\sigma_w^2s^2}{2} + \mu_ws-q_w-\lambda_w^+-\lambda_w^- &\pi_1^{-}(w)&. &\pi_m^{-}(w)&\pi_1^{+}(w)&.&\pi_n^{+}(w)\\ \alpha_1^-&-s-\alpha_1^-&0&0&0&0&0\\ .&0&.&0&0&0&0\\ \alpha_m^-& 0 &0&-s-\alpha_m^-&0&0&0\\ \alpha_1^+&0&0&0&s-\alpha_1^+&0&0\\ .&0&0&0&0&.&0\\ \alpha_n^+&0&0&0&0&0&s-\alpha_n^+ \end{array} \right),$$ where $\lambda_w^\pm=\sum_i \pi^\pm_i(w)$, and $$(D_w)_{ij} = \begin{cases} q_w & \text{if $i=j=1$}\\ 0 & \text{otherwise} \end{cases}.$$ Therefore $|\det(K(s))|$ is equal to $|\det(K^\#(s)|$. To proceed we recall an identity from matrix algebra. Let $M$ be a matrix of the form $$M = \begin{pmatrix} M_{11} & M_{12} \\ M_{21} & M_{22} \end{pmatrix}$$ in block notation, where $M_{22}$ is invertible. Then $$\det(M) = \det(M_{22}) \det (M_{11} + M_{12} M_{22}^{-1} M_{21}).$$ Using this identity, it is a matter of algebra to verify by induction that $$\det(K^\#(s)) = \det(K_1(s)) \cdots det(K_N(s)).$$ As a consequence we find, by applying this matrix identity, that $$\begin{gathered} \hspace{-0.5cm}\det(K_w(s)) = \le(\frac{\sigma^2_w}{2} s^2 + \mu_w s + \sum_{i=1}^{n^+} \pi^+_i(w) \frac{\alpha_i^+}{s-\alpha_i^+} + \sum_{j=1}^{n^-} \pi^-_j(w) \frac{\alpha_j^-}{-s-\alpha_j^-} -\lambda_w^+ - \lambda_w^- - q_w\ri)\times \\ \times\prod_{i=1}^{n^+} (s-\alpha^+_i) \prod_{j=1}^{n^-} (-s-\alpha^-_j),\end{gathered}$$ and the assertion follows in view of . - Using the intermediate value theorem and the specific form of $\Psi$, it is straightforward to check that the equation $\Psi(-u\mathbf i)=q$, $q>0$ has $n+1$ positive roots $\rho^+_i$ and $m+1$ negative roots $\rho^-_j$, satisfying $$\rho^-_{m+1} < -\alpha_m^- < \rho_m^- < \ldots < -\alpha_1^- < \rho^-_1<0 < \rho_i^+ < \alpha_1^+ < \ldots < \rho_n^+ < \alpha_n^+ < \rho^+_{n+1}.$$ Since $\det(K_w(s))$ is a polynomial of degree $n+m+2$, it follows that all the roots of $\det(K_w(s))=0$ are given by $(\rho_i^+, i=1, \ldots, n+1)$ and $(\rho_j^-, j=1,\ldots, m+1)$. In view of the form of $\det(K(s))$ derived in (i) the assertion follows. [ ]{}\ \ [*Proof of Proposition \[prop:XS\]:*]{} Consider the following randomization of $X$ obtained by randomizing the inter-maturity times $T^{(i)}$ by replacing them by independent exponential random times with means $q_i^{-1}$, and call this process $\WT X$. The process $\WT X$ is a regime-switching jump-diffusion, where the only regime switches that can occur are from $i$ to $i+1$ at rate $q_i$ ($i=1,\ldots, N-1$), and from the final state $N$ to an absorbing ’graveyard state’ $\partial$. As shown in [@JiangP], the process $\WT X$ is equal to a time-changed continuous process $A$, say. Denoting by $\zeta$ the epoch at which $A$ is sent to $\partial$, by $Y$ the modulating Markov chain, and by $\tau=\inf\{t\ge0: A_\tau=x\}$, we have $$\begin{aligned} q_1\cdots q_N \ovl G(x,s,q) &=& E[\te{s A_{\zeta-}} \mathbf 1_{\{\ovl A_{\zeta-} > x\}}]\\ &=& E[\te{s A_{\zeta-}} \mathbf 1_{\{\tau < \zeta\}}]\\ &=& E[\te{s A_{\tau}} \mathbf 1_{\{\tau < \zeta\}}f(Y_\tau)]\\ &=& \te{s x} E[\mathbf 1_{\{\tau < \zeta\}}f(Y_\tau)],\end{aligned}$$ with $$f(y) = E[\te{s A_{\zeta-}}|A_0=0, Y_0=y],$$ where the last two lines follow by the Markov property of $(A,Y)$ and the fact that $A$ is continuous. To guarantee that all the expressions are well defined in this calculation $s$ has to be such that $E[\te{s X_1}]<\i$, which corresponds to the restriction that $$Re(s)\in(-\min_j \a_j^-,\min_i \a^+_i).$$ In [@JiangP] it was shown that the vector $\mbf f = (f(y), y\in N)$, where $N$ denotes the state space of $Y$, is given by $$\mbf f = K(s)^{-1} Q \mathbf 1,$$ where the matrix $K(s)$ is given in (\[eq:K\]). Combining these results with Theorem \[thm:wh\] we find that $$q_1\cdots q_N \ovl G(x,s,q) = e_1' \te{Q^+ x} K(s)^{-1} Q \mathbf 1,$$ and the proof is complete. [ ]{} European call options {#app:euro} ===================== Under the hyper-exponential additive model with piecewise constant parameters , the characteristic function at time $T$ is explicitly given by $$\Phi^{(i)}(u) = \exp\le(\sum_{j=1}^i\Psi^{(j)}(u)\ri),$$ with $\Psi^{(j)}$ as given in . The price of a European call with maturity $T_i$ can thus be efficiently calculated using a well-established Fourier transform method, which we briefly recall. The Fourier transform $C^*_{T_i}$ over $k$ of $C_{T_i}(k)$, the price of a call option with log-strike $k=\log(K/S_0)$ and maturity $T_i$, can be explicitly expressed in terms of the characteristic function $\Phi^{(i)}(u)$ as follows: $$\begin{aligned} \nn C^*_{T_i}(v-\mathbf i \alpha)&=&S_0 \te{-rT_i} \int_{-\infty}^{\infty}\te{\mathbf i vk}E[\te{\alpha k}(\te{X_{T_i}}-\te k)^+] \td k\\ &=&S_0\te{-rT_i}\frac{\Phi^{(i)}(v-(\alpha+1)\mathbf i)}{(\alpha+\mathbf iv)(\alpha+1+\mathbf iv)}.\label{eq:FFTCall}\end{aligned}$$ Since the call pay-off function itself is not square-integrable in the log-strike, the axis of integration is here shifted over $\mathbf i\alpha$ which corresponds to exponentially dampening the pay-off function at a rate $\alpha$, which is usually taken to be $\alpha=0.75$ (see Carr and Madan [@carr98]). The call option prices are then determined by inverting the Fourier transform: $$\label{eq:CallTi} C_{T_i}(k)=\frac{S_0 \te{-\alpha k}\te{-rT_i}}{\pi}\int_0^{\infty} \te{-\mathbf ikv}\frac{\Phi^{(i)}(v-(\alpha+1)\mathbf i)}{(\alpha+\mathbf iv)(\alpha+1+\mathbf iv)}\td v.$$ Transform inversion algorithms {#app:trans} ============================== Multi-dimensional Laplace inversion ----------------------------------- To evaluate down-and-in digital option prices (DID), we invert the multi-dimensional Laplace transform to obtain $$\label{eq:lDID} DID(S_0,h,\textbf{T})=\frac{1}{(2\pi \mathbf i)^N}\int_{C_N}\cdots\int_{C_1}\te{q_1 T_1+\ldots+q_NT_N} \WH{DID}(S_0,h,\textbf{q})\td \textbf{q}.$$ where $\mbf T=(T_1, \ldots, T_N)$ and $C_n$ are vertical lines in the complex plane defined by $q_n=r_n+\mathbf iy_n$ for $n=1, \ldots N$ with $-\infty<y_n<\infty$ and fixed values of $r_n$, chosen such that all the singularities of the transform $\WH{DID}(S_0,h,\texttt{q})$ are coordinate-wise on the left of the lines $C_n$. Many algorithms approximate the integrals in by a finite linear combination of the transform at some specific nodes with certain weights. Three approaches have been studied by Abate et al. [@abate06], based on Fourier series expansion, combinations of Gaver functionals, and deformation of the integral contour. Here we concentrate on the last method developed by Talbot [@talbot79], since reports in the literature (e.g. [@abate06]) suggest that this approach offers high performance for a short time of execution, which our numerical results confirm. We write $$\label{eq:DIDLT} DID(S_0,h,\textbf{T})=\frac{1}{(2\pi \mathbf i)^N}\int_{-\pi}^{\pi}\cdots\int_{-\pi}^{\pi} \beta_1(\theta) \cdots \beta_N(\theta) \WH{DID}(S_0,h,\textbf{q}(\theta)) \td \mbox{\boldmath $\theta$},$$ with $n=1, \ldots N$, $\beta_n(\theta)=w_n \te{\mathbf i r_n w_n T_n}$, $ q_n(\theta)=\mathbf i r_n w_n$, and $$w_n=-1+\mathbf i \theta+\mathbf i (\theta \cot \theta-1) \cot \theta.$$ Since $DID$ is a real valued function, $DID$ is also equal to the real part of the integral on the right-hand side of , which can be used to reduce the calculation by a factor of two. To illustrate the evaluation of the integrals , we present concrete expressions for the approximating sums when $N=4$ (which is the setting that will be implemented later on). Defining $$\theta^k_n =k\pi/M\ \text{ and }\ r_n=\frac{2M}{5T_n},$$ we obtain [$$\begin{aligned} &&\hspace{-0.8cm}DID(S_0,h,\textbf{T})\approx\frac{2}{5^4 T_1 T_2 T_3 T_4 }\sum_{k_1=0}^{M-1} \sum_{k_2=0}^{M-1} \sum_{k_3=0}^{M-1} \sum_{k_4=0}^{M-1}\\ && \beta_{k_1} \beta_{k_2} \beta_{k_3}\beta_{k_4} f(q_{k_1}/T_1,q_{k_2}/T_2,q_{k_3}/T_3,q_{k_4}/T_4)+\overline{\beta}_{k_1} \beta_{k_2} \beta_{k_3}\beta_{k_4} f(\overline{q}_{k_1}/T_1,q_{k_2}/T_2,q_{k_3}/T_3,q_{k_4}/T_4)\\ &+&\beta_{k_1} \overline{\beta}_{k_2} \beta_{k_3}\beta_{k_4} f(q_{k_1}/T_1,\overline{q}_{k_2}/T_2,q_{k_3}/T_3,q_{k_4}/T_4)+\beta_{k_1} \beta_{k_2} \overline{\beta}_{k_3}\beta_{k_4} f(q_{k_1}/T_1,q_{k_2}/T_2,\overline{q}_{k_3}/T_3,q_{k_4}/T_4)\\ &+&\beta_{k_1} \beta_{k_2} \beta_{k_3}\overline{\beta}_{k_4} f(q_{k_1}/T_1,q_{k_2}/T_2,q_{k_3}/T_3,\overline{q}_{k_4}/T_4)+\overline{\beta}_{k_1} \overline{\beta}_{k_2} \beta_{k_3}\beta_{k_4} f(\overline{q}_{k_1}/T_1,\overline{q}_{k_2}/T_2,q_{k_3}/T_3,q_{k_4}/T_4)\\ &+&\beta_{k_1} \overline{\beta}_{k_2} \beta_{k_3}\overline{\beta}_{k_4} f(q_{k_1}/T_1,\overline{q}_{k_2}/T_2,q_{k_3}/T_3,\overline{q}_{k_4}/T_4)+\overline{\beta}_{k_1} \overline{\beta}_{k_2} \overline{\beta}_{k_3}\beta_{k_4} f(\overline{q}_{k_1}/T_1,\overline{q}_{k_2}/T_2,\overline{q}_{k_3}/T_3,q_{k_4}/T_4),\end{aligned}$$]{} where $f$ is equal to $\WH{DID}$. The weights and the nodes are given by $$\begin{aligned} q_0&=&\frac{2M}{5},\quad q_k=\frac{2k\pi}{5}(\cot(k\pi/M)+\mathbf i), \quad \quad 0<k<M,\\ \beta_0&=&0.5 e^{q_0}, \quad \beta_k=(1+\mathbf i(k\pi/M)(1+[\cot(k \pi/M)]^2)-\mathbf i \cot(k\pi/M))e^{q_k}.\end{aligned}$$ Since the weights and nodes are independent of the transform, the calculation time of the algorithm can be reduced by pre computing and storing weights and nodes. The speed of convergence and the accuracy of the Talbot algorithm will depend on the regularity of the Laplace transform $f$. Although universal error bounds are not known, Abate et al. [@abate04] showed numerically that the single parameter $M$ can be used to control the error and can be seen as a measure for the precision. They found after extensive numerical experiments that for a large class of Laplace transforms the relative error is approximately $10^{-0.6M}$. For high dimensional inversion, extra accuracy in the inner sums may be needed to obtain a sufficient degree of precision for the outer sums, which can be achieved by increasing $M$. Fractional Fourier Transform ---------------------------- To evaluate down-and-in call option prices (DIC), we invert the Fourier-Laplace transform over log-strike and time periods. For the inversion of the Laplace transform we again apply the Talbot algorithm. In the case of two time periods, with $$f_v(q_1,q_2)=\WH{DIC}^*(S_0,h,v,\textbf{q}),$$ we find that the Fourier transform $DIC^*$ can be approximated by the following sums: $$\begin{aligned} \lefteqn{DIC^*(S_0,h,v,\textbf{T})}\\ &\approx&\frac{1}{5^2 T_1 T_2 }\sum_{k_1=0}^{M-1} \sum_{k_2=0}^{M-1} \le\{\beta_{k_1} \beta_{k_2}f_v(q_{k_1}/T_1,q_{k_2}/T_2)+\overline{\beta}_{k_1} \overline{\beta}_{k_2} f_v(\overline{q}_{k_1}/T_1,\overline{q}_{k_2}/T_2)\ri.\\ &\phantom{=}&\qquad\qquad\qquad\quad +\ \le.\overline{\beta}_{k_1} \beta_{k_2}f_v(\overline{q}_{k_1}/T_1,q_{k_2}/T_2)+\beta_{k_1} \overline{\beta}_{k_2}f_v(q_{k_1}/T_1,\overline{q}_{k_2}/T_2)\ri\}.\end{aligned}$$ Unlike the case of the inversion of $\WH{DID}$, we cannot reduce the calculation time by two by using complex conjugates, since the function $DIC^*$ is not real valued. Down-and-in call prices are then obtained by inverting the Fourier transform over strike: $$DIC(S_0,h,k, \textbf{T})=\frac{\te{-\alpha k}}{\pi}\int_0^{\infty} \te{-\mathbf ivk}DIC^*(v)\td v,$$ where $\alpha$ is the rate of exponential dampening. This integral is approximated for a set of log-strikes between $(-x_0,x_0)$ as a summation: $$\label{eq:DICsum} DIC(S_0,h,k, \textbf{T}) \approx \frac{S \te{-\alpha k}\te{-rT}}{\pi}\sum_{j=0}^{N-1}w_j\te{-\mathbf i\delta j(-x_0+k \lambda)}DIC^*(\delta j) \delta, \quad \quad k=1\cdots N-1,$$ where $(w_j)_{j=0}^{N-1}$ are the integration weights defined by the trapezoidal rule with $w_0=w_{N-1}=0.5$ and $1$ otherwise, $\lambda = 2x_0/N$ is the log-strike grid step-size and $\delta$ is the $v$-grid step-size. Carr and Madan [@carr98] and Chourdakis [@chourdakis04] set $\delta=0.25$. To have accurate prices for any strike, the log-strike grid spacing $\lambda$ needs to be sufficiently small. A common approach is to apply directly the Fast Fourier Transform (FFT) and to compute the summation on a fixed log-strike range $(-x_0,x_0)$ with $x_0=\pi/\delta$ using many points $N$. Bailey and Swarztrauber [@Bailey91], [@Bailey94] propose an alternative approach, and define the Fractional Fast Fourier transform (FrFFT), which uses an arbitrary range. Chourdakis [@chourdakis04] showed that the FrFFT can be used to calculate option prices with less points without losing accuracy. He reported that the FrFFT is $45$ times faster than the FFT for the calculation of European option prices. Since in our case the Fourier transform $DIC^*$ is obtained numerically, we chose to employ the FrFFT. We now briefly specify the form of this algorithm in our setting, and refer for further details to [@Bailey91], [@Bailey94], [@chourdakis04]. The resulting sum is then given by $$\begin{aligned} DIC(S_0,h,k,\textbf{T}) &\approx&\frac{S_0 \te{-(\alpha k+\mathbf i \pi k^2 \nu)}\te{-rT}}{\pi}\sum_{j=0}^{N-1}\widetilde{w}_j\te{-\pi \mathbf i j^2 \nu}\te{-\pi \mathbf i(k-j)^2 \nu}DIC^*(\delta j) \delta,\end{aligned}$$ where $k=1\cdots N-1$, $\widetilde{w}_j=w_je^{\mathbf i x_0 \delta j}$ and $\nu=\delta x_0/N \pi$. Extending this summation into a circular convolution over $2N$ yields $$\begin{aligned} DIC(S_0,h,k,\textbf{T})&\approx& \frac{S_0 \te{-(\alpha k+\mathbf i\pi k^2 \nu)}\te{-rT}}{\pi}\sum_{j=0}^{2N-1} y_j z_{k-j}, \quad \quad k=1\cdots N-1,\end{aligned}$$ where $$y_j=\widetilde{w}_j \te{-\pi \mathbf i j^2 \nu} DIC^*(\delta j)\delta, \quad z_{j}= \te{-\pi \mathbf i j^2 \nu}, \quad j<N,\\$$ and $$y_j=0, \quad z_{j}= \te{-\pi \mathbf i (j-2N)^2 \nu}, \quad j \geq N.$$ This equation can be rewritten in terms of three discrete Fourier transforms: $$\begin{aligned} DIC(k) &\approx&\frac{S_0\te{-(\alpha k+\mathbf i\pi k^2 \nu)}\te{-rT}}{\pi}F_k^{-1}(F_k(y)F_k(z)), \quad \quad k=1\cdots N-1,\end{aligned}$$ with $$F(x)=\sum_{j=0}^{N-1}x_j \te{-2\pi \mathbf i j k/N},\quad \quad \quad F^{-1}(x)=\sum_{j=0}^{N-1}x_j \te{2\pi \mathbf i j k/N}.$$ Although the latter sum is computed by invoking two Fourier transforms and one inverse Fourier transform, this approach has the advantage of computing the option prices on a specific log-strike window $(-x_0,x_0)$ with independent grids $\delta$ and $\lambda$ and requires less points. 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--- abstract: 'An antimagic labeling of a directed graph $D$ with $n$ vertices and $m$ arcs is a bijection from the set of arcs of $D$ to the integers $\{1, \cdots, m\}$ such that all $n$ oriented vertex sums are pairwise distinct, where an oriented vertex sum is the sum of labels of all arcs entering that vertex minus the sum of labels of all arcs leaving it. An undirected graph $G$ is said to have an antimagic orientation if $G$ has an orientation which admits an antimagic labeling. Hefetz, M[ü]{}tze, and Schwartz conjectured that every connected undirected graph admits an antimagic orientation. In this paper, we support this conjecture by proving that every biregular bipartite graph admits an antimagic orientation.' author: - | Songling Shan$^a$[^1],  Xiaowei Yu$^{a,b}$[^2]\ [$^a$ Department of Mathematics, Vanderbilt University,]{}\ [Nashville, TN 37240, U.S.A.]{}\ [$^b$ School of Mathematics, Shandong University,]{}\ [Jinan, Shandong 250100, P. R. China]{}\ bibliography: - 'SSL-BIB.bib' title: 'Antimagic orientation of biregular bipartite graphs[^3]' --- 0.65cm [**Keywords:**]{} Labeling; Antimagic labeling; Antimagic orientation Introduction ============ Unless otherwise stated explicitly, all graphs considered are simple and finite. A [*labeling*]{} of a graph $G$ with $m$ edges is a bijection from $E(G)$ to a set $S$ of $m$ integers, and the [*vertex sum*]{} at a vertex $v\in V(G)$ is the sum of labels on the edges incident to $v$. If there are two vertices have same vertex sums in $G$, then we call them *conflict*. A labeling of $E(G)$ with no conflicting vertex is called a [*vertex distinguishable labeling*]{}. A labeling is [*antimagic*]{} if it is vertex distinguishable and $S=\{1,2,\cdots, m\}$. A graph is [*antimagic*]{} if it has an antimagic labeling. Hartsfield and Ringel [@MR1282717] introduced antimagic labelings in 1990 and conjectured that every connected graph other than $K_2$ is antimagic. There have been significant progresses toward this conjecture. Let $G$ be a graph with $n$ vertices other than $K_2$. In 2004, Alon, Kaplan, Lev, Roditty, and Yuster [@MR2096791] showed that there exists a constant $c$ such that if $G$ has minimum degree at least $ c \cdot log n$, then $G$ is antimagic. They also proved that $G$ is antimagic when the maximum degree of $G$ is at least $n-2$, and they proved that all complete multipartite graphs (other than $K_2$) are antimagic. The latter result of Alon et al. was improved by Yilma [@MR3021347] in 2013. Apart from the above results on dense graphs, the antimagic labeling conjecture has been also verified for regular graphs. Started with Cranston [@MR2478227] showing that every bipartite regular graph is antimagic, regular graphs of odd degree [@MR3372337], and finally all regular graphs [@MR3515572] were shown to be antimatic sequentially. For more results on the antimagic labeling conjecture for other classes of graphs, see [@MR3527991; @MR2174213; @MR2682515; @MR2510327]. Hefetz, M[ü]{}tze, and Schwartz [@MR2674494] introduced the variation of antimagic labelings, i.e., antimagic labelings on directed graphs. An [*antimagic*]{} labeling of a directed graph with $m$ arcs is a bijection from the set of arcs to the integers $\{1,...,m\}$ such that any two oriented vertex sums are pairwise distinct, where an [*oriented vertex sum*]{} is the sum of labels of all arcs entering that vertex minus the sum of labels of all arcs leaving it. A digraph is called [*antimagic*]{} if it admits an antimagic labeling. For an undirected graph $G$, if it has an orientation such that the orientation is antimagic, then we say $G$ admits an [*antimagic orientation*]{}. Hefetz et al. in the same paper posted the following problems. \[question1\] Is every connected directed graph with at least 4 vertices antimagic? \[antimagic-orientation\] Every connected graph admits an antimagic orientation. Parallel to the results the on antimagic labelling conjecture, Hefetz, M[ü]{}tze, and Schwartz [@MR2674494] showed that every orientation of a dense graph is antimagic and almost all regular graphs have an antimagic orientation. Particulary, they showed that every orientation of stars(other than $K_{1,2}$), wheels, and complete graphs(other than $K_3$) is antimagic. Observe that if a bipartite graph is antimagic, then it has an antimagic orientation obtained by directing all edges from one partite set to the other. Thus by the result of Cranston [@MR2478227], regular bipartite graphs have an antimagic orientation. A bipartite graph is [*biregular*]{} if vertices in each of the same partite set have the same degree. In this paper, by supporting Conjecture \[antimagic-orientation\], we obtain the result below. \[th1\] Every biregular bipartite graph admits an antimagic orientation. Notation and Lemmas =================== Let $G$ be a graph. If $G$ is bipartite with partite sets $X$ and $Y$, we denote $G$ by $G[X,Y]$. Given an orientation of $G$ and a labeling on $E(G)$, for a vertex $v\in V(G)$ and a subgraph $H$ of $G$, we use $\omega_H(v)$ to denote the [*oriented sum at $v$*]{} in $H$, which is the sum of labels of all arcs entering $v$ minus the sum of labels of all arcs leaving it in the graph $H$. If $v$ is of degree 2 in $G$, we say the labels at edges incident to $v$ the [*label at $v$*]{} and write it as a pair in $\{(a,b),(-a,b),(a,-b),(-a,-b)\}$, where $a,b$ are the labels on the two edges incident to $v$, and $-a$ is used if the edge with label $a$ is leaving $v$ and $a$ is used otherwise; similar situation for the value $-b$ or $b$. A *trail* is an alternating sequence of vertices and edges $v_0,e_1,v_1,\ldots,e_t,v_t$ such that $v_{i-1}$ and $v_i$ are the endvertices of $e_i$, for each $i$ with $1\leq i\leq t$, and the edges are all distinct (but there might be repetitions among the vertices). A trail is *open* if $v_0\neq v_t$. The *length* of a trail is the number of edges in it. Occasionally, a trail $T$ is also treated as a graph whose vertex set is the set of distinct vertices in $T$ and edge set is the set of edges in $T$. We use the terminology “trail” without distinguishing if it is a sequence or a graph, but the meaning will be clear from the context. For two integers $a,b$ with $a<b$, let $[a,b]:=\{a,a+1,\cdots, b\}$. We need the result below which guarantees a matching in a bipartite graph. A simple proof of this result can be found in [@local-CE]. \[[@local-CE]\]\[matching\] Let $H$ be a bipartite graph with partite sets $X$ and $Y$. If there is no isolated vertex in $X$ and $d_H(x)\ge d_H(y)$ holds for every edge $xy$ with $x\in X$ and $y\in Y$, then $H$ has a matching which saturates $X$. For even regular graphs, Petersen proved that a 2-factor always exists. \[petersen\] Every regular (multi)graph with positive even degree has a $2$-factor. Also we need the following result on decomposing edges in a graph into trails. \[trail\] Given a connected graph $G$, and let $T=\{v\in V: d_G(v)$ is odd$\}$. If $T\neq \emptyset$, then $E(G)$ can be partitioned into $\frac{|T|}{2}$ open trails. \[cycle1\] Every simple $2$-regular graph $G$ admits a vertex distinguishable labeling with labels in $[a,b]$, where $a, b$ are two positive integers with $b-a=|E(G)|-1$. Moreover, the vertex sums belong to $[2a+1,2b-1]$. [**Proof**.]{}Note that $G$ is antimagic by Corollary 3 in [@MR2478227]. Assume that $\phi: E(G)\rightarrow [1, |E(G)|]$ is an antimagic labeling of $G$. Define another labeling $\varphi: E(G)\rightarrow [a,b]$ based on $\phi$ as follows. $$\begin{aligned} \varphi(e)=\phi(e)+a-1, \quad \forall \, e\in E(G).\nonumber\end{aligned}$$ Since $G$ is regular and $\phi$ is antimagic, it is clear that $\varphi$ is a vertex distinguishable labeling of $G$. Furthermore, the sums fall into the interval $[2a+1,2b-1]$. \[paths\] Let $T[X,Y]$ be an open trail with all vertices in $Y$ having degree 2 except precisely two having degree 1. Suppose $T$ has $2m$ edges. Let $y_1$ and $y_{m+1}$ be the two degree 1 vertices in $Y$ such that $T$ starts at $y_1$ and ends at $y_{m+1}$. Let $a, b$ be two integers with $a=b-2m+1$. Then there exists a bijection from $ E(T)$ to $[a,a+m-1]\cup [b-m+1,b]=[a,b]$ such that each of the following holds. (i) $\omega_T(x)=\frac{d_T(x)(a+b)}{2}$ for any $x\in X$; and $\omega_T(y)\ne \omega_T(z)$ for any distinct $y,z\in Y-\{y_1, y_{m+1}\}$. (ii) If $m\equiv0 \,(\operatorname{mod}2)$, then $\omega_T(y_1)=b$, $\omega_T(y_{m+1})=b-m+1$, and $\omega_T(y)$ is an odd number in $[2a+1,2a+2m-3]\cup [2b-2m+5,2b-3]$ for any $y\in Y-\{y_1, y_{m+1}\}$. (iii) If $m\equiv 1 \,(\operatorname{mod}2)$, then $\omega_T(y_1)=a$, $\omega_T(y_{m+1})=b-m+1$, and $\omega_T(y)$ is an odd number in $[2a+3,2a+2m-3]\cup [2b-2m+5,2b-1]$ for any $y\in Y-\{y_1, y_{m+1}\}$. [**Proof**.]{}Since $|E(T)|=2m$, and except precisely two degree 1 vertices, all other vertices in $Y$ have degree 2, we conclude that $|Y|=m+1$. Let $Y=\{y_1,y_2,\cdots, y_{m+1}\}$. Then there are precisely $m$ edges of $T$ incident to vertices in $Y$ with even indices, and $m$ edges of $T$ incident to vertices in $Y$ with odd indices. We treat $T$ as an alternating sequence of vertices and edges starting at $y_1$ and ending at $y_{m+1}$. If $m\equiv0 \,(\operatorname{mod}2)$, following the order of the appearances of edges in $T$, assign edges incident to vertices in $Y$ of even indices with labels $$a, a+1, \cdots, a+m-1,$$ and assign edges incident to vertices in $Y$ of odd indices with labels $$b, b-1, \cdots, b-m+1.$$ That is, the label at $y_i$ is $(a+i-2, a+i-1)$ if $i$ is even; and $(b-i+2, b-i+1)$ if $i$ is odd and not equal to $1$ or $m+1$. If $m\equiv1 \,(\operatorname{mod}2)$, following the order of the appearances of edges in $T$, assign edges incident to vertices in $Y$ of odd indices with labels $$a, a+1, \cdots, a+m-1,$$ and assign edges incident to vertices in $Y$ of even indices with labels $$b, b-1, \cdots, b-m+1.$$ That is, the label at $y_i$ is $(a+i-2, a+i-1)$ if $i$ is odd and not equal to $1$ or $m+1$; and $(b-i+2, b-i+1)$ if $i$ is even. If $m\equiv0 \,(\operatorname{mod}2)$, for each $y_i\in Y$ with $1\le i\le m+1$, by the assignment of labels, we have that $$\omega_T(y_i)= \left\{ \begin{array}{ll} b , & \hbox{if $i=1$;} \\ b-m+1 , & \hbox{if $i=m+1$;} \\ 2a+2i-3, & \hbox{if $i$ is even and $2\le i\le m$;} \\ 2b-2i+3, & \hbox{if $i$ is odd and $3\le i\le m-1$.} \end{array} \right.$$ If $m\equiv1 \,(\operatorname{mod}2)$, for each $y_i\in Y$ with $1\le i\le m+1$, by the assignment of labels, we have that $$\omega_T(y_i)= \left\{ \begin{array}{ll} a , & \hbox{if $i=1$;} \\ b-m+1 , & \hbox{if $i=m+1$;} \\ 2a+2i-3, & \hbox{if $i$ is odd and $3\le i\le m$;} \\ 2b-2i+3, & \hbox{if $i$ is even and $2\le i\le m-1$.} \end{array} \right.$$ The sum on each vertex $y_i$ with $y_i\in Y-\{y_1, y_{m+1}\}$ is expressed as either $2a+2i-3$ or $2b-2i+3$, which is an odd number. Furthermore, the sums on $y_2, y_4,\cdots, y_{m}$, starting at $2a+1$, strictly increase to $2a+2m-3$ if $m\equiv0 \,(\operatorname{mod}2)$, and the sums on $y_3, y_5,\cdots, y_{m-1}$, starting at $2a+3$, strictly increase to $2a+2m-3$ if $m\equiv1 \,(\operatorname{mod}2)$. The sums on $y_3, y_5,\cdots, y_{m-1}$, starting at $2b-3$, strictly decrease to $2b-2m+5$ if $m\equiv0 \,(\operatorname{mod}2)$, and the sums on $y_2, y_4,\cdots, y_{m}$, starting at $2b-1$, strictly decrease to $2b-2m+5$ if $m\equiv1 \,(\operatorname{mod}2)$. So these sums are all distinct. Since $a=b-2m+1$, it holds that $2a+2m-3<2b-2m+5$. Thus all $\omega_T(y)$ are distinct for $y\in Y$ with $y\in Y-\{y_1, y_{m+1}\}$. Let $x$ be a vertex in $X$. Suppose that one appearance of $x$ is adjacent to $y_i$ and $y_{i+1}$ in the sequence $T$. If $m\equiv0 \,(\operatorname{mod}2)$, for even $i$ with $2\le i\le m$, the labels on the two edges $xy_i$ and $xy_{i+1}$ contribute a value of $(a+i-1)+(b-(i+1)+2)=a+b$ to $\omega_T(x)$; for odd $i$ with $1\le i\le m-1$, the labels on the two edges $xy_i$ and $xy_{i+1}$ contribute a value of $(b-i+1)+(a+(i+1)-2)=a+b$ to $\omega_T(x)$. Since $x$ appears $d_T(x)/2$ times in $T$, $\omega_T(x)=\frac{d_T(x)(a+b)}{2}$. If $m\equiv1 \,(\operatorname{mod}2)$, for even $i$ with $2\le i\le m$, the labels on the two edges $xy_i$ and $xy_{i+1}$ contribute a value of $(b-i+1)+(a+(i+1)-2)=a+b$ to $\omega_T(x)$; for odd $i$ with $1\le i\le m-1$, the labels on the two edges $xy_i$ and $xy_{i+1}$ contribute a value of $(a+i-1)+(b-(i+1)+2)=a+b$ to $\omega_T(x)$. Since $x$ appears $d_T(x)/2$ times in $T$, $\omega_T(x)=\frac{d_T(x)(a+b)}{2}$. \[cycle2\] Let $C[X,Y]$ be a cycle of length $2m$ with $m\equiv 0\,(\operatorname{mod}2)$, and let $a, b$ be two integers with $a=b-2m+1$. Then there exists a bijection from $ E(C)$ to $[a,a+m-1]\cup [b-m+1,b]=[a,b]$ such that each of the following holds. (i) $\omega_C(x)=a+b$ for any $x\in X$. (ii) $\omega_C(y)\ne \omega_C(z)$ for any distinct $y,z\in Y$. (iii) $\omega_C(y)\in [2a+1,2a+2m-3]\cup [2b-2m+5,2b-2]$ for all $y\in Y$, and the sums in $[2a+1,2a+2m-3]$ are odd. [**Proof**.]{}Denote by $C=x_1y_1x_2y_2\cdots x_my_mx_1$ with $x_i\in X$ and $y_i\in Y$. Following the order of the appearances of edges in $C$, assign edges incident to vertices in $\{y_1, y_3,\cdots, y_{m-1}\}$ with labels $$a+1, a, a+2, a+3,\cdots, a+m-2, a+m-1.$$ Note that the labels are increasing consecutive integers after exchanging the positions of the first two; assign edges incident to vertices in $\{y_{2}, y_{4},\cdots, y_{m}\}$ with labels $$b, b-2, b-3, \cdots, b-m+2, b-m+1, b-1.$$ Note that the labels are decreasing consecutive integers after inserting the last number between the first two labels. For each $y_i\in Y$ with $1\le i\le m$, following the appearances of the edges in the sequence of $C$, we denote the labels on the edges incident to $y_i$ by an ordered pair. Particularly, by the assignment of the labels, we have that $$\mbox{label at $y_i$}= \left\{ \begin{array}{ll} (a+1, a) , & \hbox{if $i=1$;} \\ (b,b-2), & \hbox{if $i=2$;} \\ (b-m+1, b-1) , & \hbox{if $i=m$;} \\ (a+i-1,a+i), & \hbox{if $i$ is odd with $3\le i\le m-1$;}\\ (b-i+1,b-i), & \hbox{if $i$ is even with $4\le i\le m-2$.} \end{array} \right.$$ The sums on $y_1, y_3,\cdots, y_{m-1}$ starting at $2a+1$ strictly increase to $2a+2m-3$ and all of them are odd; and the sums on $y_4, y_6,\cdots, y_{m-2}$ starting at $2b-7$ strictly decrease to $2b-2m+5$ and all of them are odd; the sums on $y_2, y_m$ are even numbers $2b-2$ and $2b-m$, respectively. Since $a= b-2m+1$, $2a+2m-3<2b-2m+5$. Hence all $\omega_C(y)$ are distinct for $y\in Y$. This shows both (ii) and (iii). Let $x_i$ be a vertex in $X$ for $1\leq i\leq m$. If $i=1$, the labels on the two edges $x_1y_1$ and $x_1y_{m}$ are $a+1$ and $b-1$, respectively; if $i=2$, the labels on the two edges $x_2y_1$ and $x_2y_{2}$ are $a$ and $b$, respectively. Thus $\omega_C(x_i)=a+b$ if $i=1,2$. Suppose $i\ge 3$. If $i$ is even, then the labels on the two edges $x_iy_{i-1}$ and $x_iy_i$ are $a+(i-1)$ and $b-i+1$, respectively; if $i$ is odd, the labels on the two edges $xy_{i-1}$ and $xy_{i}$ are $b-(i-1)$ and $a+i-1$, respectively. Thus $\omega_C(x_i)=a+b$. This proves (i). \[pathcycle\] Let $G$ be a bipartite graph, and $H[X,Y]$ a subgraph of $G$. Suppose that $E(H)$ can be decomposed into edge-disjoint $p+q$ open trails $T_1, \cdots, T_p$, $T_{p+1}, \cdots, T_{p+q}$, and $\ell$ cycles $C_{p+q+1}, \cdots, C_{p+q+\ell}$. Suppose further that these $p+q+\ell$ subgraphs have no common vertex in $Y$, and for each of the trail, its vertices in $Y$ are all distinct and its endvertices are contained in $Y$. Let $2m:=|E(H)|$, and $c, d$ be two integers with $c=d-2m+1$. If the length of $T_1,\cdots, T_p$ are congruent to 2 modulo 4, and the length of each of the remaining trails and cycles is congruent to 0 modulo 4, then there exists a bijection from $ E(H)$ to $[c,c+m-1]\cup [d-m+1,d]=[c,d]$ such that each of the following holds. (i) $\omega_H(x)=\frac{d_H(x)(c+d)}{2}$ for any $x\in X$. (ii) For each $i$ with $1\le i\le p+q$, let $T_i$ start at $y_{1i}$ and end at $y_{(m_i+1)i}$, where $y_{1i}, y_{(m_i+1)i}\in Y$. Suppose that $$\begin{cases} \omega_G(y_{1i})=\omega_H(y_{1i})+(c-2p+i-1), & \hbox{if $1\le i\le p$}, \\ \omega_G(y_{(m_i+1)i})=\omega_H(y_{(m_i+1)i})+(c-i), & \hbox{if $1\le i\le p$},\\ \omega_G(y_{1i})=\omega_H(y_{1i})+(d+2q-2(i-p-1)) ,& \hbox{if $p+1\le i\le p+q$},\\ \omega_G(y_{(m_i+1)i})=\omega_H(y_{(m_i+1)i})+(d+2q-2(i-p-1)-1), & \hbox{if $p+1\le i\le p+q$},\\ \omega_G(y)=\omega_H(y), & \text{if $y\ne y_{1i},y_{(m_i+1)i} $}. \end{cases}$$ Then $\omega_G(y)\ne \omega_G(z)$ for any distinct $y,z\in Y$, and $\omega_G(y)\in [2c-2p,\max\{2d+2q-\sum\limits_{i=1}^pm_i, 2d\}]$ for all $y\in Y$. [**Proof**.]{}Since $T_1, \cdots, T_{p+q}$ and $C_{p+q+1}, \cdots, C_{p+q+\ell}$ are edge-disjoint and pairwise have no common vertex in $Y$, and for each of the trail, its vertices in $Y$ are all distinct and its endvertices are contained in $Y$, we conclude that in the graph $H$, all vertices in $X$ have even degree, all the endvertices of $T_i$ are precisely the degree 1 vertices in $Y$, and all other vertices in $Y$ have degree 2. For each $i$ with $1\le i\le p+q$ and each $j$ with $p+q+1\le j\le p+q+\ell$, let $$|E(T_i)|=2m_i, \quad |E(C_j)|=2m_j.$$ Since $V(T_i)\cap Y$ contains two degree 1 vertices and $|V(T_i)\cap Y|-2$ degree 2 vertices in $Y$, we conclude that $|V(T_i)\cap Y|=m_i+1$. Apply Lemma \[paths\] on each $T_i$, $1\le i\le p+q$, with $$a:=a_i:=c+\sum\limits_{j=1}^{i-1}m_j, \quad b:=b_i:=d-\sum\limits_{j=1}^{i-1}m_j;$$ and apply Lemma \[cycle2\] on each $C_i$, $p+q+1\le i\le p+q+\ell$, with $$a:=a_i:=c+\sum\limits_{j=1}^{i-1}m_j, \quad b:=b_i:=d-\sum\limits_{j=1}^{i-1}m_j.$$ Note that $a_i+b_i=c+d$, $1\le i\le p+q+\ell$. By Lemma \[paths\] and Lemma \[cycle2\], we get that $$\omega_H(x)=\frac{d_H(x)(c+d)}{2} \quad \mbox{for any $x\in X$.}$$ By Lemma \[paths\], the sums at $y_{1i},y_{m_ii}$, respectively, are \_H(y\_[1i]{})=c+\_[j=1]{}\^[i-1]{}m\_j, \_H(y\_[(m\_i+1)i]{})=d-\_[j=1]{}\^[i]{}m\_j+1, & ;\[sumy1\]\ \_H(y\_[1i]{})=d-\_[j=1]{}\^[i-1]{}m\_j, \_H(y\_[(m\_i+1)i]{})=d-\_[j=1]{}\^[i]{}m\_j+1,& ;\[sumy2\] and for each $i$ with $1\le i\le p+q$, the sums at vertices in $V(T_i)\cap Y-\{y_{1i}, y_{(m_i+1)i}\}$ fall into the intervals $$\begin{cases} [2a_i+3, 2a_i+2m_i-3]\cup [2b_i-2m_i+5, 2b_i-1], & \text{if $1\le i\le p$ }, \\ [2a_i+1, 2a_i+2m_i-3]\cup [2b_i-2m_i+5, 2b_i-3],& \text{if $p+1\le i\le p+q$}, \end{cases}$$ and all these sums are distinct and odd. By Lemma \[cycle2\], for each $i$ with $p+q+1\le i\le p+q+\ell$, the sums at vertices in $V(C_i)\cap Y$ are all distinct and fall into the intervals $$[2a_i+1, 2a_i+2m_i-3]\cup [2b_i-2m_i+5, 2b_i-2],$$ and all the sums in $[2a_i+1, 2a_i+2m_i-3]$ are odd. Since for each $i,j$ with $1\le i< j\le p+q+\ell$, $$\left\{ \begin{array}{ll} 2a_j+1>2a_i+2m_i-3;\\ 2b_i-2m_i+5>2b_j-1, \end{array} \right.$$ we see that $2a_{p+q+\ell}+2m_{p+q+\ell}-3$ is the largest value in the set $$\left(\bigcup_{i=1}^{p}[2a_i+3, 2a_i+2m_i-3]\right)\bigcup \left(\bigcup_{i=p+1}^{p+q+\ell}[2a_i+1, 2a_i+2m_i-3]\right),$$ and $2b_{p+q+\ell}-2m_{p+q+\ell}+5$ is the smallest value in the set $$\begin{aligned} \left(\bigcup_{i=1}^{p}[2b_i-2m_i+5, 2b_i-1]\right)&\bigcup& \left(\bigcup_{i=p+1}^{p+q}[2b_i-2m_i+5, 2b_i-3]\right)\\ & \bigcup& \left(\bigcup_{i=p+q+1}^{p+q+\ell}[2b_i-2m_i+5, 2b_i-2]\right).\end{aligned}$$ Furthermore, $$\begin{aligned} \begin{tabular}{lll} &$2b_{p+q+\ell}-2m_{p+q+\ell}+5-(2a_{p+q+\ell}+2m_{p+q+\ell}-3)$ &\\ =& $2d-\sum\limits_{j=1}^{p+q+\ell}2m_i+5-(2c+\sum\limits_{j=1}^{p+q+\ell}2m_i-3)$&\\ $=$& $2(c+2m-1)-2m+5-(2c+2m-3)$\quad \mbox{($d= c+2m-1$ and $\sum\limits_{j=1}^{p+q+\ell}2m_i=2m$)}& \\ $=$& $6$. & \end{tabular}\end{aligned}$$ Hence, all the vertex sums at $Y-\{y_{1i}, y_{(m_i+1)i}\,|\, 1\le i\le p+q\}$ are pairwise distinct. Let $$k_1:=\sum\limits_{i=1}^p m_i, \quad \mbox{and}\quad k_2:=\sum\limits_{i=p+1}^{p+q} m_i.$$ By Equalities (\[sumy1\]) and (\[sumy2\]), and the assumptions on the parity of each $m_i$, if $1\le i\le p$, \_G(y\_[1i]{})=\_H(y\_[1i]{})+(c-2p+i-1),\ =c+\_[j=1]{}\^[i-1]{}m\_j+c-2p+i-1=2c-2p+\_[j=1]{}\^[i-1]{}(m\_j+1)\ 0(2) &\ , \[y1ip\]\ \_G(y\_[(m\_i+1)i]{})=\_H(y\_[(m\_i+1)i]{})+(c-i)\ =d-\_[j=1]{}\^[i]{}m\_j+1+c-i\ =c+2m-1-\_[j=1]{}\^[i]{}m\_j+1+c-i\ =2c+2m-\_[j=1]{}\^[i]{}(m\_j-1)0(2)&\ , \[ym1p\] and if $p+1\le i\le p+q$, \_G(y\_[1i]{})=\_H(y\_[1i]{})+(d+2q-2(i-p-1)) \[y1iq\]\ = d-\_[j=1]{}\^[i-1]{}m\_j+d+2q-2(i-p-1)0(2),&\ \_G(y\_[(m\_i+1)i]{})=\_H(y\_[(m\_i+1)i]{})+(d+2q-2(i-p-1)-1) \[ym1q\]\ = d-\_[j=1]{}\^[i]{}m\_j+1+d+2q-2(i-p-1)-1\ 0(2),& By the above analysis, for each $i$ with $1\le i\le p+q$, both $\omega_G(y_{1i})$ and $\omega_G(y_{(m_i+1)i})$ are even. As all the sums at vertices in $\bigcup_{i=1}^{p+q} V(T_i)\cap Y-\{y_{1i}, y_{(m_i+1)i}\,|\, 1\le i\le p+q\}$ are odd by Lemma \[paths\], and by Lemma \[cycle2\] all the sums on vertices in $\bigcup_{i=p+q+1}^{p+q+\ell}V(C_i)\cap Y$ which fall into the set $\bigcup_{i=p+q+1}^{p+q+\ell}[2a_i+1, 2a_i+2m_i-3]$ are odd, all of them are distinct from these $2(p+q)$ $\omega_G$ sums on vertices in $\{y_{1i}, y_{(m_i+1)i}\,|\, 1\le i\le p+q\}$. Hence, to show that all the vertex sums at vertices in $Y$ are distinct, we are left to check that all these $2(p+q)$ $\omega_G$ sums are distinct with the sums on vertices in $\bigcup_{i=p+q+1}^{p+q+\ell}V(C_i)\cap Y$ which fall into the set $\bigcup_{i=p+q+1}^{p+q+\ell}[2b_i-2m_i+5, 2b_i-2]$, and all these $2(p+q)$ $\omega_G$ sums at vertices in $\{y_{1i}, y_{(m_i+1)i}\,|\, 1\le i\le p+q\}$ are pairwise distinct. If $1\le i<j\le p$, by (\[y1ip\]) and (\[ym1p\]), $\omega_G(y_{1i})<\omega_G(y_{1j})$ and $\omega_G(y_{(m_i+1)i})>\omega_G(y_{(m_i+1)j})$. Thus, all the sums at vertices either in $\{ y_{1i}\,|\, 1\le i\le p\}$ or in $\{ y_{(m_i+1)i}\,|\, 1\le i\le p\}$ are all distinct, and $$\omega_G(y_{1p})=\max\{\omega_G(y_{1i})\,|\, 1\le i\le p\}, \quad \omega_G(y_{(m_{p}+1)p})=\min\{\omega_G(y_{(m_i+1)i})\,|\, 1\le i\le p\}.$$ Furthermore, by (\[y1ip\]) and (\[ym1p\]), $$\begin{aligned} \begin{tabular}{lll} &$\omega_G(y_{(m_{p}+1)p})-\omega_G(y_{1p})$&\\ =& $2c+2m-k_1-p-(2c-p-1+k_1-m_p)$&\\ =& $2m-2k_1+m_p+1\ge 2$\quad \mbox{($2m\ge 2k_1, m_p\ge 1$).}& \end{tabular}\end{aligned}$$ Thus, the $\omega_G$ sums on vertices in $\{ y_{1i}, y_{(m_i+1)i}\,|\, 1\le i\le p\}$ are all distinct. By (\[y1iq\]) and (\[ym1q\]), $\omega_G(y_{1i})>\omega_G(y_{(m_i+1)i})$ for all $i$ with $p+1\le i\le p+q$, and $\omega_G(y_{(m_i+1)i})>\omega_G(y_{1j})$ if $p+1\le i<j\le p+q$. Thus, all the $\omega_G$ sums on vertices in $\{ y_{1i}, y_{(m_i+1)i}\,|\, p+1\le i\le p+q\}$ are all distinct, and $$\omega_G(y_{(m_{p+q}+1)(p+q)})=\min\{\omega_G(y_{1i}),\omega_G(y_{(m_i+1)i})\,|\, p+1\le i\le p+q\}.$$ Furthermore, $$\omega_G(y_{(m_{1}+1)1})=\max\{\omega_G(y_{1i}),\omega_G(y_{(m_i+1)i})\,|\, 1\le i\le p\},$$ and by (\[ym1q\]) and (\[ym1p\]), $$\begin{aligned} \begin{tabular}{lll} &$\omega_G(y_{(m_{p+q}+1)(p+q)})-\omega_G(y_{(m_1+1)1})$&\\ =& $2d-k_1-k_2+2-(2c+2m-m_1-1)$&\\ =& $2m-k_1-k_2+m_1+1\ge 2$\quad \mbox{($d=c+2m-1, 2m\ge k_1+k_2, m_1\ge 1$).}& \end{tabular}\end{aligned}$$ Thus, all the $\omega_G$ sums on vertices in $\{ y_{1i}, y_{(m_i+1)i}\,|\, 1\le i\le p+q\}$ are all distinct. We may assume that $\ell\ge 1$. Otherwise, we are done. By the definition of the parameters $b_i$ and easy calculations, $$\bigcup_{i=p+q+1}^{p+q+\ell}[2b_i-2m_i+5, 2b_i-2]\subseteq [2b_{p+q+\ell}-2m_{p+q+\ell}+5, 2b_{p+q+1}-2].$$ By (\[y1ip\]), (\[ym1p\]),(\[y1iq\]), and (\[ym1q\]), the $\omega_G$ sums at vertices in $\{ y_{1i}, y_{(m_i+1)i}\,|\, 1\le i\le p+q\}$ fall into the intervals $$[2c-2p, 2c+2m-m_1+1]\cup [2d-k_1-k_2+2, 2d+2q-k_1].$$ Since $$\begin{aligned} \begin{tabular}{lll} &$(2b_{p+q+\ell}-2m_{p+q+\ell}+5)-(2c+2m-m_1-1)$ &\\ =& $2d-2k_1-2k_2+5-(2(d-2m+1)+2m-m_1-1)$&\\ =& $2m-2k_1-2k_2+m_1+4$& \\ $\ge $& $5$ \quad \mbox{($2m\ge 2k_1+2k_2$ and $m_1\ge 1$)},& \end{tabular}\end{aligned}$$ and $$\begin{aligned} \begin{tabular}{lll} &$(2d-k_1-k_2+2)-(2b_{p+q+1}-2)$ &\\ =& $2d-2k_1-k_2+2-(2d-2k_1-2k_2-2)$&\\ =& $k_1+k_2+4 \ge 4$,& \end{tabular}\end{aligned}$$ we then conclude that these $2(p+q)$ $\omega_G$ sums at vertices in $\{y_{1i}, y_{(m_i+1)i}\,|\, 1\le i\le p+q\}$ are all distinct with the $\omega_G$ sums at vertices in $\bigcup_{i=p+q+1}^{p+q+\ell}V(C_i)\cap Y$ which fall into the set $\bigcup_{i=p+q+1}^{p+q+\ell}[2b_i-2m_i+5, 2b_i-2]$. By all the arguments above, we have shown that $\omega_G(y)\ne \omega_G(z)$ for any distinct $y,z\in Y$. Since the sums on vertices in $Y-\{y_{1i}, y_{(m_i+1)i}\,|\, 1\le i\le p+q\}$ fall into the interval $[2c, 2d]$ and the sums on vertices in $\{y_{1i}, y_{(m_i+1)i}\,|\, 1\le i\le p+q\}$ fall into the interval $[2c-2p,2d+2q-k_1]$, where the value $2d+2q-k_1$ is attained at $\omega_G(y_{1(p+1)})$, we have that $\omega_G(y)\in [2c-2p,\max\{2d+2q-k_1, 2d\}]$ for all $y\in Y$. Proof of Theorem \[th1\] ======================== Let $G=[X,Y]$ be a biregular bipartite graph. Assume that $|X|=m$, $|Y|=n$, $d_G(x)=s\geq d_G(y)=t$, where $x\in X, y\in Y$. Consequently $m\leq n$ and $|E(G)|=ms=nt$. Given an orientation of $G$, we will denote the orientation by $\overrightarrow{G}$. If $t=1$, then $G$ is the union of vertex-disjoint stars with centers in $X$. Denote $$X=\{x_1,x_2,\cdots,x_m\} \quad \mbox{and} \quad Y=\{y_1,y_2,\cdots, y_n\}.$$ For each $x_i$, $1\le i\le m$, we assign arbitrarily edges incident to $x_i$ with labels $$s(i-1)+1, s(i-1)+2,\cdots, s(i-1)+s.$$ Orient edges of $G$ from $X$ to $Y$. Thus, the oriented vertex sums for vertices in $X$ are negative, and the oriented vertex sums for vertices in $Y$ are positive. Hence, no two vertices $x$ and $y$ conflict if $x\in X$ and $y\in Y$. Also, it is routine to check that no two vertices in $X$ conflicting and no two vertices in $Y$ conflicting. Hence the labeling of $\overrightarrow{G}$ is antimagic. Thus we assume $t\ge 2$. We distinguish three cases for finishing the proof. Orient edges of $G$ from $X$ to $Y$, and denote the orientation by $\overrightarrow{G}$. By the orientation of $G$, the sums of vertices in $X$ are negative while the sums at vertices in $Y$ are positive. Hence in the following, we just need to find a labeling of $\overrightarrow{G}$ using labels in $[1,sm]$, which guarantees that the sums at vertices in $X$ are all distinct and the sums at vertices in $Y$ are all distinct. By Lemma \[matching\], $G$ has a matching $M$ saturating $X$. Assume, w.l.o.g, that $M=\{x_1y_1, x_2y_2,\ldots,x_my_m\}$. Let $H=G-M$. Note that $d_H(y_i)=t-1$ for $1\leq i\leq m$ and $d_H(y_i)=t$ for $m+1\leq i\leq n$. Reserve labels in $[1,m]$ for edges in $M$, and use labels in $[m+1, tn=sm]$ for edges in $H$. For each $y_i$ with $1\le i\le m$, assign arbitrarily the edges incident to $y_i$ with labels $$m+i, 3m-i+1, 3m+i, 5m-i+1, 5m+i, \cdots, (t-2)m-i+1, (t-2)m+i, tm;$$ and for each $y_i$ with $m+1\le i\le n$, assign arbitrarily the edges incident to $y_i$ with labels $$t(i-m-1)+tm+1, t(i-m-1)+tm+2, \cdots, t(i-m-1)+tm+t.$$ Assume, w.l.o.g., that under the above assignment of labels, $\omega_H(x_1)\le \omega_H(x_2)\le \cdots \le \omega_H(x_m)$. Now for each edge $x_iy_i\in M$, $1\le i\le m$, assign the edge $x_iy_i$ with the label $i$. We verify now that the labeling of $\overrightarrow{G}$ given above is antimagic. For each $x_i, x_j\in X$ with $i<j$, since $\omega_H(x_i)\le \omega_H(x_j)$, it holds that $\omega_G(x_i)=\omega_H(x_i)+i<\omega_H(x_j)+j=\omega_G(x_j)$. Next for each $y_i, y_j\in \{y_1,y_2,\cdots, y_m\}$ with $i<j$, since $\omega_H(y_i)=\omega_H(y_j)=\frac{t-1}{2}((t+1)m+1)$, we have that $\omega_G(y_i)=\omega_H(y_i)+i<\omega_H(y_j)+j=\omega_G(y_j)$. By the assignment of labels on edges incident to $y_i$ with $m+1\le i\le n$, the sums at $y_i$ are pairwise distinct. The smallest vertex sum among these values is $\omega_G(y_{m+1})=t^2m+\sum\limits_{i=1}^{t}i$. The largest vertex sum among values in $\{\omega_G(y_1),\cdots, \omega_G(y_m)\}$ is $\omega_G(y_m)=\frac{t-1}{2}((t+1)m+1)+m$. It is easy to check that $\omega_G(y_m)<\omega_G(y_{m+1})$. Hence, all the sums at vertices in $Y$ are distinct. Reserve labels in $\{2,4,\cdots, 2m\}$ for edges in $M$, and use the labels in $\{1,3,\cdots, 2m-1\}\cup \{2m+1,\cdots, tn=sm\}$ for edges in $H$. For each $y_i$ with $1\le i\le m$, assign arbitrarily the edges incident to $y_i$ with labels $$2i-1, 3m-i+1, 4m-i+1, 4m+i, 6m-i+1, \cdots, (t-2)m+i, tm-i+1;$$ and for each $y_i$ with $m+1\le i\le n$, assign arbitrarily the edges incident to $y_i$ with labels $$t(i-m-1)+tm+1, t(i-m-1)+tm+2, \cdots, t(i-m-1)+tm+t.$$ Assume, w.l.o.g., that under the above assignment of labels, $\omega_H(x_1)\le \omega_H(x_2)\le \cdots \le \omega_H(x_m)$. Now for each edge $x_iy_i\in M$, $1\le i\le m$, assign the edge $x_iy_i$ with the label $2i$. We verify now that the labeling of $\overrightarrow{G}$ given above is antimagic. Obviously, for each $x_i, x_j\in X$ with $1\leq i<j\leq m$, because $\omega_H(x_i)\le \omega_H(x_j)$, it holds that $\omega_G(x_i)=\omega_H(x_i)+2i<\omega_H(x_j)+2j=\omega_G(x_j)$. Next for each $y_i, y_j$ with $1\leq i<j\leq m$, since $\omega_H(y_i)=\omega_H(y_j)=(t^2-9)m+t-3$, we have that $\omega_G(y_i)=\omega_H(y_i)+2i<\omega_H(y_j)+2j=\omega_G(y_j)$. By the assignment of labels on edges incident to $y_i$ with $m+1\le i\le n$, the sums at $y_i$ are pairwise distinct. The smallest sum among these values is $\omega_G(y_{m+1})=t^2m+\sum\limits_{i=1}^{t}i$. The largest sum among values in $\{\omega_G(y_1),\cdots, \omega_G(y_m)\}$ is $\omega_G(y_m)=(t^2-9)m+t-3+2m$. It is easy to check that $\omega_G(y_m)<\omega_G(y_{m+1})$. Hence, all the sums at vertices in $Y$ are distinct. [**Case 2: $t=2$ and $s$ is odd**]{} By Lemma \[matching\], there exists a matching $M$ saturating vertices in $X$. In each component of $G-M$, the vertices contained in $X$ are all of even degree $s-1$, and all vertices contained in $Y$ are of degree 2 or 1. Thus, the number of vertices with degree 1 in the component is even. Since there are in total $m$ vertices of degree 1 in $Y$, by Lemma \[trail\], we can decompose $E(G-M)$ into $m/2$ open trials with endvertices in $Y$. Denote the trails by $T_1, \cdots, T_{m/2}$. Since the endvertices of each $T_i$ are in $Y$, all the vertices in $V(T_i)\cap X$ have even degree. Consequently, $T_i$ has even length. For each $i$ with $1\le i\le m/2$, let $$2m_i:=|E(T_i)| \quad \mbox{and}\quad T_i=y_{1i}x_{1i}\cdots x_{m_ii}y_{(m_i+1)i},$$ where $x_{1i},x_{2i},\cdots, x_{m_ii}\in X$ and $y_{1i},y_{2i},\cdots, y_{m_i+1i}\in Y$. Note that $x_{1i},x_{2i},\cdots, x_{m_ii}$ maynot be distinct vertices in $X$, but $y_{1i},y_{2i},\cdots, y_{m_i+1i}$ are distinct vertices in $Y$ because all vertices in $Y$ have degree 2 in $G$. Assume further, w.l.o.g., that there are $p$ trails $T_1,\cdots, T_p$ of length congruent to 2 modulo 4, and $q$ trails $T_{p+1},\cdots, T_{p+q}$ of length congruent to 0 modulo 4. Set $$c:=2p+1,\quad d:=sm-2q, \quad \text{and} \quad H=\bigcup_{i=1}^{p+q} T_i.$$ The endvertices of the $m/2$ open trails are exactly the set of Y-endvertices of the $m$ matching edges. Thus, for each $i$ with $1\le i\le m/2$, and for each edge $e\in M$, $e$ is incident to either $y_{1i}$ or $y_{(m_i+1)i}$. We assign labels in $[1,2p]\cup [sm-2q+1, sm]$ on $e$ as below. If $1\le i\le p$, i, & \[m1\]\ 2p-i+1, & \[m2\] if $p+1\le i\le p+q$, sm-2(i-p-1), & \[m3\]\ sm-2(i-p-1)-1, & \[m4\] Thus, $$\begin{cases} \omega_G(y_{1i})=\omega_H(y_{1i})+i, & \text{if $1\le i\le p$}, \\ \hphantom{\omega_G(y_{1i})}=\omega_H(y_{1i})+(c-2p+i-1),\\ \omega_G(y_{(m_i+1)i})=\omega_H(y_{(m_i+1)i})+(2p-i+1), & \text{if $1\le i\le p$},\\ \hphantom{\omega_G(y_{(m_i+1)i})}=\omega_H(y_{(m_i+1)i})+(c-i),\\ \omega_G(y_{1i})=\omega_H(y_{1i})+(sm-2(i-p-1)) ,& \text{if $p+1\le i\le p+q$},\\ \hphantom{\omega_G(y_{1i})}=\omega_H(y_{1i})+(d+2q-2(i-p-1)),\\ \omega_G(y_{(m_i+1)i})=\omega_H(y_{(m_i+1)i})+(sm-2(i-p-1)-1), & \text{if $p+1\le i\le p+q$},\\ \hphantom{\omega_G(y_{(m_i+1)i})}=\omega_H(y_{(m_i+1)i})+(d+2q-2(i-p-1)-1),\\ \omega_G(y)=\omega_H(y), & \text{if $y\in Y, y\ne y_{1i},y_{(m_i+1)i} $}. \end{cases}$$ Applying Lemma \[pathcycle\] on $H$ with $c=2p+1$ and $d=sm-2q$ defined as above, we get an assignment of labels on $E(H)$ such that (i) For any $x\in V(H)\cap X$, $\omega_H(x)=\frac{d_H(x)(c+d)}{2}=\frac{(s-1)(sm-2q+2p+1)}{2}$; and (ii) For any distinct $y, z\in V(H)\cap Y$, $\omega_{G}(y)\ne \omega_G(z)$. Orient all the edges of $G$ from $X$ to $Y$, and denote the orientation by $\overrightarrow{G}$. The labeling of $\overrightarrow{G}$ given above is antimagic. [**Proof**.]{}We first show that all the set of labels used is the set $[1,sm]$. The set of labels used on edges in $M$ is the set $[1,2p]\cup [sm-2q+1,sm]$, and the set of labels used on edges in $H$ is the set $[c,d]=[2p+1, sm-2q]$. The union of these two sets is the set $[1,sm]$. We then show that all the oriented sums on vertices in $\overrightarrow{G}$ are pairwise distinct. We first examine the sums on vertices in $Y$. For any $y\in Y$, since $Y=V(H)\cap Y$, we know that all the sums on vertices in $Y$ are pairwise distinct by (ii) preceding Claim 1. Next we show that in $\overrightarrow{G}$, the oriented sums on vertices in $X$ are all distinct. According to (i) preceding Claim 1 and the orientation of $G$, we see that for any $x\in X$, $$\omega_H(x)=\frac{-(s-1)(sm-2q+2p+1)}{2}.$$ Since all the labels on the edges in $M$ are distinct, and for any $x\in X$, $$\omega_G(x)=\omega_H(x)-\text{the label on $e\in M$ which is incident to $x$},$$ we know that the oriented sums on vertices in $X$ are all distinct. Finally, we show that for any $x\in X$ and $y\in Y$ the oriented sums at $x$ and $y$ are distinct in $G$. This is clear since all the oriented sums at vertices in $Y$ are positive while that at vertices in $X$ are negative. [**Case 3: $t=2$ and $s$ is even**]{} We may assume that $s\ge 4$. Otherwise $G$ is 2-regular and $|E(G)|=2m$. By Lemma \[cycle1\], $G$ has an antimatic labeling by taking $a:=1$ and $b:=2m$, and the labeling is also an antimagic labeling of $\overrightarrow{G}$ obtained by orienting all edges from $X$ to $Y$. The graph $G[X,Y]$ contains a subgraph $F$ such that - $F$ is a set of vertex disjoint cycles; and - $V(F)\cap X=X$. [**Proof**.]{}Suppressing all degree 2 vertices in $Y$, we obtain an $s$-regular (multi)graph $G'$. Since $s$ is even, by applying Lemma \[petersen\], we find a 2-factor of $G'$. Subdivide each edge in the 2-factor of $G'$, we get the desired graph $F$. Now $G-E(F)$ is a graph with all vertices having even degree. So $G-E(F)$ can be decomposed into edge-disjoint cycles. Assume that there are in total $\ell$ edge-disjoint cycles in $G-E(F)$ such that each of them has length congruent to 0 modulo 4, and there are in total $h$ edge-disjoint cycles in $G-E(F)$ such that each of them has length congruent to 2 modulo 4. For each $i$ with $1\le i\le h$, denote by $$C_i=x_{1i}y_{1i}\cdots x_{m_i i}y_{m_i i}x_{1i}, \quad \mbox{where}~ x_{1i},x_{2i}, \cdots, x_{m_i i}\in X,\quad y_{1i},y_{2i}, \cdots, y_{m_i i}\in X.$$ the $i$-th cycle of length congruent to 2 modulo 4. We pre-label edges in $\{x_{1i}y_{1i}, x_{1i}y_{m_ii}, y_{1i}x_{2i}, y_{m_ii}x_{m_ii}, x_{2i}y_{2i}, x_{m_ii}y_{(m_i-1)i}\,|\, 1 \le i\le h\}$. In doing so, we distinguish if $s=4$ or $s\ge 6$. If $s=4$, for each $i$ with $1\le i\le h$, use the labels in $[1,3h]\cup [2m-3h+1,2m]$ to label each edge $e$ indicated below.  i, & \[s43\]\  2m-(i-1), & \[s44\]\  h+2i-1, & \[s45\]\  h+2i, &\[s46\]\  2m+1-(h+2i-1), & \[s41\]\  2m+1-(h+2i), & If $s\ge 6$, for each $i$ with $1\le i\le h$, use the labels in $[1,4h]\cup [(s-2)m-2h+1,(s-2)m]$ to label each edge $e$ indicated below. i, & \[s63\]\ h+i, & \[s64\]\ 2h+2i-1, & \[s65\]\ 2h+2i, & \[s66\]\ (s-2)m+2h+1-(2h+2i-1), & \[s61\]\ (s-2)m+2h+1-(2h+2i), & \[s62\] Assume that there are $q$ paths with positive length after deleting the vertices $x_{1i}, x_{2i}, x_{m_ii}$, $y_{1i}, y_{m_ii}$ in each $C_i$ for $1\leq i\leq h$. Assume, w.l.o.g., that these paths are obtained from $C_{h-q+1}, \cdots, C_h$. For each $1\le i\le q $, denote these paths by $$P_i=C_{h-q+i}-\{x_{1(h-q+i)}, x_{2(h-q+i)}, x_{m_{(h-q+i)}(h-q+i)}, y_{1(h-q+i)}, y_{m_{(h-q+i)}(h-q+i)}\},$$ and assume that $P_i$ starts at $y_{2(h-q+i)}$ and ends at $y_{(m_{h-q+i}-1)(h-q+i)}$. Then each $P_i$ has length $2(m_{h-q+i}-3)\equiv 0(\operatorname{mod}4)$. Under this assumption, we know that $C_1, C_2, \cdots, C_{h-q} $ are 6-cycles. Denote the $\ell$ edge-disjoint cycles in $G-E(F)$ such that each of them has length congruent to 0 modulo 4 by $D_{q+1}, D_{q+2}, \cdots, D_{q+\ell}$. Let $$H=\left(\bigcup_{i=1}^q P_i\right)\bigcup \left(\bigcup_{i=q+1}^{q+\ell} D_i\right).$$ If $s=4$, then let $$c:=3h+1 \quad \mbox{and} \quad d:=2m-3h.$$ It is clear that $d-c=2m-6h-1=|E(H)|-1$. For $h-q+1\le j\le h$, let $i=j-h+q$. Then by Equations (\[s41\]) and (\[s42\]), the labels on the other edges not in $H$ incident to $y_{2j}$ and $y_{(m_j-1)j}$, respectively, are $$\begin{aligned} 2m+1-(h+2j-1)&=& (s-2)m-h-2(i+h-q)+2\\ &=&(s-2)m-3h+2q-2i+2=d+2q-2(i-1),\\ 2m+1-(h+2j)&=&(s-2)m-h-2(i+h-k)+1\\ &=&(s-2)m-3h+2q-2i+1=d+2q-2(i-1)-1.\end{aligned}$$ If $s\ge 6$, then let $$c:=4h+1 \quad \mbox{and} \quad d:=(s-2)m-2h.$$ Again $d-c=(s-2)m-6h-1=|E(H)|-1$. For $h-q+1\le j\le p$, let $i=j-h+q$. Then by Equations (\[s61\]) and (\[s62\]), the labels on the other edges not in $H$ incident to $y_{2j}$ and $y_{(m_j-1)j}$, respectively, are $$\begin{aligned} (s-2)m+2h+1-(2h+2j-1) &=& (s-2)m-2(i+h-q)+2\\ &=&(s-2)m-2h+2q-2i+2=d+2q-2(i-1),\\ (s-2)m+2h+1-(2h+2j) &=& (s-2)m-2(i+h-q)+1\\ &=&(s-2)m-2h+2q-2i+1=d+2q-2(i-1)-1.\end{aligned}$$ Thus, $$\omega_G(y_{2j})=\omega_H(y_{2j})+(d+2q-2(i-p-1)) , \quad \omega_G(y_{(m_j-1)j})=\omega_H(y_{(m_j-1)j})+(d+2q-2(i-p-1)-1).$$ Apply Lemma \[pathcycle\] on $H$ with $c$ and $d$ defined above(according to if $s=4$ or $s\ge 6$) and with $p=0$, we get an assignment of labels on $E(H)$ such that (i) For any $x\in V(H)\cap X$, $\omega_H(x)=\frac{d_H(x)(c+d)}{2}$; and (ii) For any distinct $y,z\in V(H)\cap Y$, $\omega_{G}(y)\ne \omega_G(z)$, and $\omega_{G}(y)\in [2c,2d+2q]$. Apply Lemma \[cycle1\] on $F$ with $$a:=(s-2)m+1 \quad \mbox{and} \quad b:=sm,$$ we get an antimagic labeling on $F$. If $s=4$, orient the edges in $\{x_{1i}y_{1i}, x_{1i}y_{m_ii}\,|\, 1\le i \le h\}$ from $Y$ to $X$, and orient all the remaining edges from $X$ to $Y$. If $s\ge 6$, orient the edges in $\{x_{1i}y_{1i}|\, 1\le i \le h\}$ from $Y$ to $X$, and orient all the remaining edges from $X$ to $Y$. Denote the orientation of $G$ by $\overrightarrow{G}$. The labeling of $\overrightarrow{G}$ given above is antimagic. [**Proof**.]{}We first show that the set of labels used is the set $[1,sm]$. The labels used on edges in $F$ are exactly numbers in the set $[(s-2)m+1, sm]$. If $s=4$, then the set of labels used on edges in $\{x_{1i}y_{1i}, x_{1i}y_{m_ii}, x_{2i}y_{1i}, x_{m_ii}y_{m_ii}, x_{2i}y_{2i}, x_{m_ii}y_{(m_i-1)i}\,|\, 1\le i\le h\}$ is $[1,3h]\cup [2m-3h+1,2m]$, and the set of labels used on $E(H)$ is $[3h+1,2m-3h]$. If $s\ge 6$, then the set of labels used on edges in $\{x_{1i}y_{1i}, x_{1i}y_{m_ii}, x_{2i}y_{1i}, x_{m_ii}y_{m_ii}, x_{2i}y_{2i}, x_{m_ii}y_{(m_i-1)i}\,|\, 1\le i\le h\}$ is $[1,4h]\cup [(s-2)m-2h+1,(s-2)m]$, and the set of labels used on $E(H)$ is $[4h+1,(s-2)m-2h]$. The union of these sets is the set $[1,sm]$. We then show that the oriented sums on vertices in $\overrightarrow{G}$ are all distinct. We separate the proof according to if $s=4$ or $s\ge 6$. [**Case $s=4$:**]{} For each $i$ with $1\le i\le h$, by (\[s43\])-(\[s46\]) and the orientation of $G$, the labels at $y_{1i}$, $y_{m_ii}$, respectively, are $$(-i, h+2i-1)\quad \mbox{and} \quad (-2m+(i-1), h+2i).$$ Thus, $$\omega_G(y_{1i})=h+i-1\in [h, 2h-1] \quad \mbox{and} \quad \omega_G(y_{m_ii})=-2m+h+3i-1\in[-2m+h+2,-2m+4h-1].$$ All these $2h$ values are pairwise distinct and fall into the interval $[-2m+h+2, 2h-1]$. For each $i$ with $1\le i\le h-q$, by (\[s41\]) and (\[s42\]), the label at $y_{2i}$ is, $$(2m+1-(h+2i-1), 2m+1-(h+2i).$$ Thus, $$\omega_G(y_{2i})=4m+3-2h-4i\in [4m+3-6h+4q, 4m-1-2h].$$ All these $h-q$ values are pairwise distinct. The sums at vertices in $V(H)\cap Y$ are all distinct and fall into the interval $[2c,2d+2q]=[6h+2,4m-6h+2q]\subseteq [6h+2,4m]$($q\le h$) by Lemma \[pathcycle\]. The sums at vertices in $V(F)\cap Y$ are all distinct and fall into the interval $[4m+3, 8m-1]$ by Lemma \[cycle1\]. Since these sets $[-2m+h+2, 2h-1], [6h+2,4m-6h+2q], [4m+3-6h+4q, 4m-1-2h]$, and $[4m+3, 8m-1]$ are pairwise disjoint, we see that the oriented sums on vertices in $Y$ are all distinct. Next we show that in $\overrightarrow{G}$, the oriented sums on vertices in $X$ are all distinct. For each $i$ with $1 \le i\le h$, $$\begin{cases} \omega_{G-E(F)}(x_{1i})=2m+1 & \text{by (\ref{s43}) and (\ref{s44}) }, \\ \omega_{G-E(F)}(x_{2i})=\omega_{G-E(F)}(x_{m_ii})=-2m-1& \text{by (\ref{s45}) (\ref{s41}), and (\ref{s46}) (\ref{s42})},\\ \omega_{G-E(F)}(x)=-\frac{d_H(x)(c+d)}{2}=-(c+d)=-2m-1 & \text{if $x\in V(H)\cap X$}. \end{cases}$$ Hence, $$\label{abdiffx} |\omega_{G-E(F)}(u)-\omega_{G-E(F)}(v)|=0~ \mbox{or}~ 4m+2, \mbox{for any $ u,v\in X$}.$$ For the graph $F$, by Lemma \[cycle1\], the sums on vertices in $V(F)\cap X$ are pairwise distinct. Since the set of labels used on $E(F)$ is $[2m+1, 4m]$ and $F$ is 2-regular, it follows that in $F$, $\omega_{F}(x)\in [-8m+1,-4m-3]$ and any two of the sums at vertices in $V(F)\cap X$ differ an absolute value of at most $4m-4$. Because of $\omega_{G}(x)=\omega_{G-E(F)}(x)+\omega_{F}(x)$ for $x\in X$ and the fact in (\[abdiffx\]), we conclude that the total oriented vertex sums at vertices in $X$ are all distinct. Finally, we show that for any $x\in X$ and $y\in Y$ the oriented vertex sums at $x$ and $y$ are distinct in $\overrightarrow{G}$. By the analysis above, $\omega_G(y)\in [-2m+h+2, 8m-1]$ for any $y\in Y$. And $\omega_G(x)\in [-10m, -2m-2]$ for any $x\in X$, which follows by the facts that $\omega_{G-E(F)}(x)=-2m-1$ or $2m+1$, $\omega_{F}(x)\in [-8m+1,-4m-3]$, and $\omega_G(x)=\omega_{G-E(F)}(x)+\omega_{F}(x)$. Thus, $\omega_G(x)\ne \omega_G(y)$. [**Case $s\ge 6$**]{}:For each $i$ with $1\le i\le h$, by (\[s63\])-(\[s66\]) and the orientation of $G$, the labels at $y_{1i}$, $y_{m_ii}$, respectively, are $$(-i, 2h+2i+1)\quad \mbox{and} \quad (h+i, 2h+2i).$$ Thus, $$\omega_G(y_{1i})=2h+i+1\in [2h+2,3h+1] \quad \mbox{and} \quad \omega_G(y_{m_ii})=3h+3i\in[3h+3,6h].$$ All these $2h$ values are pairwise distinct and fall into the interval $[2h+2, 6h]$. For each $i$ with $1\le i\le h-q$, by (\[s61\]) and (\[s62\]), the label at $y_{2i}$ is, $$((s-2)m+2h+1-(h+2i-1), (s-2)m+2h+1-(h+2i).$$ Thus, $$\omega_G(y_{2i})=2(s-2)m+3+2h-4i\in [ 2(s-2)m+3-2h+4q, 2(s-2)m+2h-1].$$ All these $h-q$ values are pairwise distinct. The sums at vertices in $V(H)\cap Y$ are all distinct and fall into the interval $[2c,2d+2q]=[8h+2,2(s-2)m-4h+2q]$ by Lemma \[pathcycle\]. The sums at vertices in $V(F)\cap Y$ are all distinct and fall into the interval $[2(s-2)m+3, 2sm-1]$ by Lemma \[cycle1\]. Since these sets $[2h+2, 6h], [8h+2,2(s-2)m-4h+2q]$, $[ 2(s-2)m+3-2h+4q, 2(s-2)m+2h-1]$, and $[2(s-2)m+3, 2sm-1]$ are pairwise disjoint, we see that the oriented vertex sums on vertices in $Y$ are all distinct. Next we show that in $\overrightarrow{G}$, the oriented sums at vertices in $X$ are all distinct. Assume that for each $x\in X$, $x$ appears $\alpha_x$ times in $\{x_{1i}\,|\, 1\le i\le h\}$, and $\beta_x$ times in $\{x_{2i}, x_{m_ii}\,|\, 1\le i\le h\}$. Since each of $x_{1i},x_{2i}$, and $x_{m_ii}$ has two distinct neighbors in $Y$, $x$ has degree $s-2-2\alpha_x-2\beta_x$ in $H$. In addtion, each appearance of $x$ in $\{x_{1i}\,|\, 1\le i\le h\}$ contributes a value of $-h$ to the oriented sum at $x$ by (\[s63\]) and (\[s64\]), and each appearance of $x$ in $\{x_{2i}, x_{m_ii}\,|\, 1\le i\le h\}$ contributes a value of $-(c+d)$ to the oriented sum at $x$ by (\[s65\]) (\[s61\]), and (\[s66\]) (\[s62\]). By (i) preceeding Claim 3, $\omega_{H}(x)=-\frac{d_H(x)(c+d)}{2}=-\frac{(s-2-2\alpha_x-2\beta_x)(c+d)}{2}$. Hence, $$\omega_{G-E(F)}(x)=-\frac{(s-2-2\alpha_x-2\beta_x)(c+d)}{2}-\alpha_xh-\beta_x(c+d)=-\frac{(s-2)(c+d)}{2}+\alpha_x(c+d-h).$$ Thus, for any $ u,v\in X$, $$\begin{aligned} |\omega_{G-E(F)}(u)-\omega_{G-E(F)}(v)| &=&|\alpha_u-\alpha_v|(c+d-h) \nonumber \\ &=&|\alpha_u-\alpha_v|((s-2)m+h+1)=0 ~\mbox{or}~ >4m. \label{abdiffx2}\end{aligned}$$ For the graph $F$, by Lemma \[cycle1\], the sums on vertices in $V(F)\cap X$ are pairwise distinct, any two of the sums at vertices in $V(F)\cap X$ differ an absolute value of at most $4m-4$. Because of $\omega_{G}(x)=\omega_{G-E(F)}(x)+\omega_{F}(x)$ for $x\in X$ and the fact in (\[abdiffx2\]), we conclude that the total oriented sums at vertices in $X$ are all distinct. Finally, we show that for any $x\in X$ and $y\in Y$ the oriented sums at $x$ and $y$ are distinct in $\overrightarrow{G}$. By the analysis above, for any $y\in Y$, $\omega_G(y)\in [2h+2, 2sm-1]$ is a positive integer. For any $x\in X$, $\omega_{G-F}(x)$ is negative and $\omega_F(x)$ is negative, so $\omega_G(x)=\omega_{G-E(F)}(x)+\omega_{F}(x)$ is negative. Hence $\omega_G(x)\ne \omega_G(y)$. The proof of Theorem \[th1\] is now complete. [^1]: Email: songling.shan@vanderbilt.edu [^2]: Corresponding author. Email: xwyu2013@163.com, yux6@vanderbilt.edu [^3]: This work was supported by the National Natural Science Foundation of China (11371355, 11471193, 11271006, 11631014), the Foundation for Distinguished Young Scholars of Shandong Province (JQ201501).

--- abstract: 'For any Coxeter system $(W,S)$ of rank $n$, we introduce an abstract boolean complex (simplicial poset) of dimension $2n-1$ that contains the Coxeter complex as a relative subcomplex. Faces are indexed by triples $(I,w,J)$, where $I$ and $J$ are subsets of the set $S$ of simple generators, and $w$ is a minimal length representative for the parabolic double coset $W_I w W_J$. There is exactly one maximal face for each element of the group $W$. The complex is shellable and thin, which implies the complex is a sphere for the finite Coxeter groups. In this case, a natural refinement of the $h$-polynomial is given by the “two-sided" $W$-Eulerian polynomial, i.e., the generating function for the joint distribution of left and right descents in $W$.' address: 'Department of Mathematical Sciences, DePaul University, Chicago, IL, USA' author: - 'T. Kyle Petersen' title: 'A two-sided analogue of the Coxeter complex' --- Introduction ============ Coxeter groups were developed to study symmetries of regular polytopes, and they play a major role in the study of Lie algebras (the Weyl group of a root system is a Coxeter group). The Coxeter complex is a simplicial complex associated with the reflection representation of the group, but which can also be defined abstractly via cosets of parabolic subgroups. The goal of this note is to provide a “two-sided" analogue of the Coxeter complex by considering double cosets of parabolic subgroups. Before turning to the new construction, let us recall some definitions and important properties of the usual Coxeter complex. We assume the reader has some familiarity with the study of Coxeter groups. See Humphreys’ book [@Humphreys] or Björner and Brenti’s book [@BjB] for background. Fix a finitely generated Coxeter system $(W,S)$, and let $W_J$ denote the standard parabolic subgroup generated by a subset of simple generators $J \subseteq S$. It is well known that the set of cosets of parabolic subgroups forms an abstract simplicial complex known as the *Coxeter complex*, and denoted by $$\Sigma = \Sigma(W,S) = \{ wW_J : w \in W, J\subseteq S\}.$$ The faces of $\Sigma$ are ordered by reverse inclusion of cosets, i.e., $$wW_J \leq_{\Sigma} w'W_{J'} \quad \mbox{ if and only if } \quad wW_J \supseteq w'W_{J'}.$$ Note that this means maximal elements are singleton sets: $w W_{\emptyset} = \{w\}$, and there is a unique minimal element: $w W_S = W$. Some well-known features of the Coxeter complex are highlighted in the following result. Most of these statements can be found in work of Björner [@Bjorner2] (see also Abramenko and Brown [@AbramenkoBrown Chapter 3]), though some of these facts were known earlier. See, e.g., Bourbaki [@Bourbaki]. \[thm:Coxeter\] For any Coxeter system $(W,S)$ with $|S|=n<\infty$ we have the following. 1. The Coxeter complex $\Sigma$ is a balanced simplicial complex of dimension $n-1$. 2. The facets (maximal faces) of $\Sigma$ are in bijection with the elements of $W$. 3. The Coxeter complex is shellable and any linear extension of the weak order on $W$ gives a shelling order for $\Sigma$. 4. If $W$ is infinite then $\Sigma$ is contractible. 5. If $W$ is finite, 1. the geometric realization of $\Sigma$ is a sphere, and 2. the $h$-polynomial of $\Sigma$ is the $W$-Eulerian polynomial, $$h(\Sigma;t) = \sum_{w \in W} t^{\operatorname{des}(w)},$$ where $\operatorname{des}(w)$ denotes the number of descents of the element $w$. We will try to emulate all these properties for a “two-sided" version of the Coxeter complex, denoted $\Xi = \Xi(W,S)$. While we will defer the definition of $\Xi$ to Section \[sec:defn\], let us comment on one matter. Although its faces are related to parabolic double cosets $W_IwW_J$, $\Xi$ is *not* merely the set of such cosets ordered by inclusion. (See Remark \[rem:distinct\].) Our main results are summarized as follows. \[thm:main\] For any Coxeter system $(W,S)$ with $|S|=n<\infty$, we have the following. 1. The complex $\Xi$ is a balanced boolean complex of dimension $2n-1$. 2. The facets (maximal faces) of $\Xi$ are in bijection with the elements of $W$, and the Coxeter complex $\Sigma$ is a relative subcomplex of $\Xi$. 3. The complex $\Xi$ is shellable and any linear extension of the two-sided weak order on $W$ gives a shelling order for $\Xi$. 4. If $W$ is infinite then $\Xi$ is contractible. 5. If $W$ is finite, 1. the geometric realization of $\Xi$ is a sphere, and 2. a refined $h$-polynomial of $\Xi$ is the two-sided $W$-Eulerian polynomial, $$h(\Xi;s,t) = \sum_{w \in W} s^{\operatorname{des}_L(w)}t^{\operatorname{des}_R(w)},$$ where $\operatorname{des}_L(w)$ denotes the number of left descents of $w$ and $\operatorname{des}_R(w)$ denotes the number of right descents of the element $w$. The main contrasts between $\Xi$ and $\Sigma$ lie in the fact that $\Xi$ is roughly twice the dimension of $\Sigma$ and in the fact that $\Xi$ is not a simplicial complex. While all the faces of $\Xi$ are simplices, many of these simplices share the same vertex set. Even for the rank one Coxeter group $A_1$, $\Xi$ is realized by two edges whose endpoints are paired off to form a circle: (0,0) circle (5pt); (-.2,0) node\[circle,fill=black,inner sep=1\] ; (.2,0) node\[circle,fill=black,inner sep=1\] ; . We remark that our approach in this work is combinatorial, not geometric. There are two different approaches to proving the topological results for the Coxeter complex listed in Theorem \[thm:Coxeter\]. One way (following Bourbaki [@Bourbaki]) is to study the reflection hyperplane arrangement for the Coxeter group. For example, in the finite case, intersecting this arrangement with a sphere realizes the Coxeter complex. Thus in this situation the topology of the Coxeter complex is manifest in the ambient space. On the other hand, Björner showed in [@Bjorner2] how to use poset-theoretic tools to study the topology of the complex with only the abstract definition of the face poset. The approach of this paper mirrors that of Björner. We define the face poset of $\Xi$ abstractly, and use Björner’s techniques to deduce Theorem \[thm:main\]. We hope to uncover a more geometric description of $\Xi$ in the future. The paper is structured as follows. The first few sections introduce $\Xi$ and establish the various parts of our main theorem. In Section \[sec:defn\] we provide the definition of $\Xi$ and the proof of parts (1) and (2) of Theorem \[thm:main\]. In Section \[sec:topology\] we prove parts (3), (4), and (5a) of Theorem \[thm:main\]. Section \[sec:faces\] discusses face enumeration in the case of finite groups $W$, and establishes part (5b) of Theorem \[thm:main\]. In Section \[sec:poly\] we define, for any finite Coxeter group $W$, the “two-sided" Eulerian polynomials $$W(s,t) := \sum_{w \in W} s^{\operatorname{des}_L(w)} t^{\operatorname{des}_R(w)}.$$ These polynomials have pleasant properties and we offer a generalization of a conjecture of Gessel that asserts that these polynomials expand positively in the basis $$\{ (st)^a(s+t)^b(1+st)^{n-2a-b} \}_{0\leq 2a+b \leq n},$$ where $n$ is the rank of the group. See Conjecture \[conj:gamma\]. Finally in Section \[sec:contingency\] we discuss a combinatorial model for faces of $\Xi$ in the case of the Coxeter group of type $A_{n-1}$, i.e., the symmetric group $S_n$. Here the faces of $\Xi$ can be encoded by two-way *contingency tables*. These tables are nonnegative integer arrays whose entries sum to $n$ and whose row and column sums are positive. The partial order on faces in this case is simply refinement ordering on contingency tables. Maximal tables are permutation matrices and the minimal element is the unique one-by-one array. Such arrays were studied by Diaconis and Gangolli [@DG], but not this partial ordering on the arrays. The author would like to thank John Stembridge for helpful conversations that inspired the idea for this paper, and Vic Reiner for comments on an earlier draft. A two-sided Coxeter complex {#sec:defn} =========================== Throughout this section we assume familiarity with basic Coxeter group concepts and terminology. We mostly follow the definitions and notational conventions of [@BjB] and [@Humphreys]. Fix a Coxeter system $(W,S)$ with $|S|=n$. We call the elements $s \in S$ the *simple generators* of $W$. Every element $w \in W$ can be written as a product of elements in $S$, $w=s_1\cdots s_k$, and if this expression is minimal, we say the *length* of $w$ is $k$, denoted $\ell(w) = k$. An expression of minimal length is called a *reduced expression*. Recall that a *cover relation* in a partially ordered set (“poset" for short) is a pair $x < y$ such that if $x\leq z \leq y$, then $x=z$ or $z=y$. A partial ordering of a set can be defined as the transitive closure of its cover relations. One important partial order on $W$ is known as the *weak order*. The weak order comes in two equivalent types: “left" and “right" weak order. We will also have reason to mention the ordering obtained from the union of the covers in left weak order and right right weak order, which we call the “two-sided" weak order. We now describe these orderings in terms of their cover relations. - The *left weak order* on $W$ says $v$ covers $u$ if and only if $\ell(v) = \ell(u)+1$ and $u^{-1}v \in S$. - The *right weak order* on $W$ says $v$ covers $u$ if and only if $\ell(v) = \ell(u)+1$ and $vu^{-1} \in S$. - The *two-sided weak order* on $W$ says $v$ covers $u$ if and only if $\ell(v) = \ell(u)+1$ and $vu^{-1}$ or $u^{-1}v$ is in $S$. The left and the right weak order are obviously subposets of the two-sided weak order. We write $u \leq_L v$ if $u$ is below $v$ in the left weak order, we write $u\leq_R v$ if $u$ is below $v$ in the right weak order, and we write $u \leq_{LR} v$ if $u$ is below $v$ in the two-sided weak order. The identity is the unique minimum in these partial orderings. When $W$ is finite, there is also a unique maximal element denoted $w_0$, and each poset is self-dual, i.e., isomorphic to its reverse ordering. Though will do not use the fact, we mention that all three of these posets are subposets of the strong Bruhat order on $W$, whose covers have $u^{-1}v$ or $vu^{-1}$ equal to a conjugate of an element of $S$. The *left (resp. right) descent set* of an element $w$ is the set of all simple generators that take us down in left (resp. right) weak order when multiplied on the left (resp. right). We denote the left and right descent sets by $\operatorname{Des}_L(w)$ and $\operatorname{Des}_R(w)$, respectively, i.e., $$\operatorname{Des}_L(w) = \{ s \in S : \ell(sw) < \ell(w) \} \mbox{ and } \operatorname{Des}_R(w) = \{ s \in S : \ell(ws) < \ell(w) \}.$$ We define the corresponding *ascent sets* as the complements of the descent sets in $S$: $$\operatorname{Asc}_L(w) = S-\operatorname{Des}_L(w)=\{ s \in S : \ell(sw) > \ell(w) \},$$ and $$\operatorname{Asc}_R(w) = S- \operatorname{Des}_R(w) = \{ s \in S : \ell(ws) > \ell(w)\}.$$ Intuitively, we move up and down in left (resp. right) weak order by multiplying elements on the left (resp. right) by simple generators. We move up and down in the two-sided weak order by multiplying on either side by simple generators. For Bruhat order, we move up and down by inserting simple generators anywhere in a given reduced expression. Suppose $J$ is a subset of simple generators, $J \subseteq S$, and let $W_J$ denote the group generated by the elements of $J$, i.e., $W_J = \langle s : s \in J\rangle$. This group is a Coxeter group in its own right, and we call such a subgroup a *standard parabolic subgroup*. The Coxeter complex arises when considering the quotients of the form $W/W_J$. That is, the faces of the Coxeter complex are identified with left cosets of parabolic subgroups, $wW_J$. To be precise, let $$\Sigma = \bigcup_{J \subseteq S} W/W_J = \{ wW_J : w \in W, J \subseteq S \}.$$ We partially order the elements of $\Sigma$ by reverse containment of sets, i.e., by declaring $$wW_J \leq_{\Sigma} w'W_{J'},$$ if and only if $$wW_J \supseteq w'W_{J'}.$$ The dimension of a face $wW_J$ is given by $\dim(wW_J) = |S-J|-1$, so that vertices correspond to cosets of the form $wW_{S-\{s\}}$, and maximal faces are singleton cosets of the form $wW_{\emptyset} = \{w\}$. For our two-sided analogue, we consider elements from all double quotients $W_I\backslash W /W_J$, so the faces will be related to double cosets of parabolic subgroups $W_I w W_J$, where $I$ and $J$ are subsets of $S$. However, the faces of $\Xi$ are *not* simply the double cosets of this form. See Remark \[rem:distinct\]. An essential fact about cosets of parabolic subgroups is that each coset $wW_J$ has a unique element of minimal length, call it $u$, such that $J \subseteq \operatorname{Asc}_R(u)$, or $\operatorname{Des}_R(u) \subseteq S-J$. In fact, the same is true for double cosets, and we record this in the following lemma, which can be found in [@Bourbaki Chapter 4, Exercise 1.3]. \[lem:minrep\] Each double coset $W_I w W_J$ has a unique element of minimal length, call it $u$, such that $$\operatorname{Des}_L(u) \subseteq S - I \quad \mbox{ and } \quad \operatorname{Des}_R(u) \subseteq S-J,$$ or $$I \subseteq \operatorname{Asc}_L(u) \quad \mbox{ and } \quad J \subseteq \operatorname{Asc}_R(u).$$ Moreover, for each $v \in W_I w W_J$, $u$ is below $v$ in the two-sided weak order: $u \leq_{LR} v$. Let ${}^I W^J$ denote the set of minimal representatives for $W_I\backslash W/W_J$, i.e., $${}^I W^J =\{ w \in W : I \subseteq \operatorname{Asc}_L(w) \mbox{ and } J \subseteq \operatorname{Asc}_R(w) \}.$$ If $I = \emptyset$ we have ${}^{\emptyset} W^J = W^J$ is the set of left coset representatives. With the lemma in mind, we could just as easily replace the cosets $wW_J$ in the definition of $\Sigma$ with pairs $(w,J)$ such that $w \in W^J$, i.e., $$\Sigma \cong \{ (w,J) : J \subseteq S, w \in W^J\}.$$ Extending this idea, we make the following definition. Let $$\Xi = \{ (I, w, J) : I, J \subseteq S \mbox{ and } w \in {}^I W^J \}.$$ We partially order the elements of $\Xi$ by reverse inclusion of the index sets $I$ and $J$ as well as the corresponding double coset, i.e., $$(I, w, J) \leq_{\Xi} (I',w',J') \quad \mbox{ if and only if } \quad \begin{cases} I \supseteq I',\\ J \supseteq J', \mbox{ and}\\ W_IwW_J \supseteq W_{I'}wW_{J'}. \end{cases}$$ We will refer to the $\Xi$ as the *two-sided Coxeter complex*. In Figure \[fig:A2\] we see the poset of faces of the two-sided Coxeter complex $\Xi(A_2)$. Faces are written as triples $(I,w,J)$, where $I, J \subseteq \{s_1,s_2\}$. We write only the subscripts for brevity, e.g., $(\{s_1\},e,\{s_1,s_2\})$ is written $(1,e,12)$. [$$\begin{tikzpicture}[xscale=.6,yscale=2.5] \tikzstyle{state}=[fill=white,inner sep = 2,scale=.6]; \tikzstyle{linet}=[line width=5, cap=round, color=white!80!black]; \draw (0,0) node[state] (12e12) {$(12,e,12)$}; \draw (-6,1) node[state] (1e12) {$(1,e,12)$}; \draw (-2,1) node[state] (2e12) {$(2,e,12)$}; \draw (6,1) node[state] (12e2) {$(12,e,2)$}; \draw (2,1) node[state] (12e1) {$(12,e,1)$}; \draw (-9,2) node[state] (e12) {$(\emptyset,e,12)$}; \draw (-7,2) node[state] (1e2) {$(1,e,2)$}; \draw (-5,2) node[state] (1e1) {$(1,e,1)$}; \draw (-3,2) node[state] (2e2) {$(2,e,2)$}; \draw (-1,2) node[state] (2e1) {$(2,e,1)$}; \draw (1,2) node[state] (12e) {$(12,e,\emptyset)$}; \draw (3,2) node[state] (2s12) {$(2,s_1,2)$}; \draw (5,2) node[state] (1s21) {$(1,s_2,1)$}; \draw (7.25,2) node[state] (2s1s21) {$(2,s_1s_2,1)$}; \draw (9.5,2) node[state] (1s2s12) {$(1,s_2s_1,2)$}; \draw (-11,3) node[state] (e2) {$(\emptyset,e,2)$}; \draw (-9,3) node[state] (e1) {$(\emptyset,e,1)$}; \draw (-7,3) node[state] (1e) {$(1,e,\emptyset)$}; \draw (-5,3) node[state] (2e) {$(2,e,\emptyset)$}; \draw (-3,3) node[state] (s12) {$(\emptyset,s_1,2)$}; \draw (-1,3) node[state] (2s1) {$(2,s_1,\emptyset)$}; \draw (1,3) node[state] (s21) {$(\emptyset,s_2,1)$}; \draw (3,3) node[state] (1s2) {$(1,s_2,\emptyset)$}; \draw (5.25,3) node[state] (s1s21) {$(\emptyset,s_1s_2,1)$}; \draw (7.5,3) node[state] (2s1s2) {$(2,s_1s_2,\emptyset)$}; \draw (9.75,3) node[state] (s2s12) {$(\emptyset,s_2s_1,2)$}; \draw (12,3) node[state] (1s2s1) {$(1,s_2s_1,\emptyset)$}; \draw (-10,4) node[state] (e) {$(\emptyset,e,\emptyset)$}; \draw (-6,4) node[state] (s1) {$(\emptyset,s_1,\emptyset)$}; \draw (-2,4) node[state] (s2) {$(\emptyset,s_2,\emptyset)$}; \draw (2,4) node[state] (s1s2) {$(\emptyset,s_1s_2,\emptyset)$}; \draw (6,4) node[state] (s2s1) {$(\emptyset,s_2s_1,\emptyset)$}; \draw (10,4) node[state] (s1s2s1) {$(\emptyset,s_1s_2s_1,\emptyset)$}; \draw[linet] (e)--(e2); \draw[linet] (e)--(e1); \draw[linet] (e)--(1e); \draw[linet] (e)--(2e); \draw[linet] (e2)--(e12); 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\draw (2e)--(2e1); \draw (2e)--(12e); \draw (e12)--(1e12); \draw (e12)--(2e12); \draw (1e2)--(1e12); \draw (1e2)--(12e2); \draw (1e1)--(1e12); \draw (1e1)--(12e1); \draw (2e2)--(2e12); \draw (2e2)--(12e2); \draw (2e1)--(2e12); \draw (2e1)--(12e1); \draw (12e)--(12e2); \draw (12e)--(12e1); \draw (12e12)--(1e12); \draw (12e12)--(2e12); \draw (12e12)--(12e2); \draw (12e12)--(12e1); \draw[linet] (s1)--(s12); \draw[linet] (s1)--(2s1); \draw[linet] (s12)--(2s12); \draw[linet] (2s1)--(2s12); \draw (s1)--(e1); \draw (s1)--(1e); \draw (s1)--(s12); \draw (s1)--(2s1); \draw (s12)--(e12); \draw (s12)--(1e2); \draw (s12)--(2s12); \draw (2s1)--(2e1); \draw (2s1)--(12e); \draw (2s1)--(2s12); \draw (2s12)--(2e12); \draw (2s12)--(12e2); \draw[linet] (s2)--(s21); \draw[linet] (s2)--(1s2); \draw[linet] (1s2)--(1s21); \draw[linet] (s21)--(1s21); \draw (s2)--(e2); \draw (s2)--(2e); \draw (s2)--(s21); \draw (s2)--(1s2); \draw (s21)--(e12); \draw (s21)--(2e1); \draw (s21)--(1s21); \draw (1s2)--(1e2); \draw (1s2)--(12e); \draw (1s2)--(1s21); \draw (1s21)--(1e12); \draw (1s21)--(12e1); \draw[linet] (s1s2)--(s1s21); \draw[linet] (s1s2)--(2s1s2); \draw[linet] (2s1s2)--(2s1s21); \draw[linet] (s1s21)--(2s1s21); \draw (s1s2)--(s12); \draw (s1s2)--(1s2); \draw (s1s2)--(s1s21); \draw (s1s2)--(2s1s2); \draw (s1s21)--(e12); \draw (s1s21)--(1s21); \draw (s1s21)--(2s1s21); \draw (2s1s2)--(12e); \draw (2s1s2)--(2s12); \draw (2s1s2)--(2s1s21); \draw (2s1s21)--(2e12); \draw (2s1s21)--(12e1); \draw[linet] (s2s1)--(s2s12); \draw[linet] (s2s1)--(1s2s1); \draw[linet] (s2s12)--(1s2s12); \draw[linet] (1s2s1)--(1s2s12); \draw (s2s1)--(s21); \draw (s2s1)--(2s1); \draw (s2s1)--(s2s12); \draw (s2s1)--(1s2s1); \draw (s2s12)--(e12); \draw (s2s12)--(2s12); \draw (s2s12)--(1s2s12); \draw (1s2s1)--(12e); \draw (1s2s1)--(1s21); \draw (1s2s1)--(1s2s12); \draw (1s2s12)--(1e12); \draw (1s2s12)--(12e2); \draw (s1s2s1)--(s1s21); \draw (s1s2s1)--(2s1s2); \draw (s1s2s1)--(s2s12); \draw (s1s2s1)--(1s2s1); \end{tikzpicture}$$ ]{} Before we move on to prove the various properties of $\Xi$ given in Theorem \[thm:main\], we include some remarks. \[rem:distinct\] A first guess to define a two-sided Coxeter complex is to consider the set of all double cosets $W_I wW_J$, ordered by reverse inclusion. Such a poset does indeed exist, but it is difficult to analyze. It is not even obvious when this poset is ranked. For one thing, there are many subtle equalities of cosets, e.g., with $w$ fixed, we might have $W_I w W_J = W_{I'}wW_{J'}$ and yet $I\neq I'$ or $J \neq J'$. For an extreme case, notice that for any $I \subseteq J$, we have $W_I e W_J = W_J$. Enumeration of the number of distinct parabolic double cosets is the topic of work of Billey, Konvalinka, Petersen, Slofstra, and Tenner [@BKPST]. \[rem:DGS\] If we fix a choice of $I$ and $J$, we can restrict the Bruhat order on $W$ to give a partial ordering on the elements ${}^IW^J$, or on the double quotient $W_I\backslash W /W_J$. Stembridge gives a geometric construction of this partial order in terms of root systems [@Stembridge]. Diaconis and Gangolli did the same in the case of the symmetric group, realized as a partial order on contingency tables with prescribed row and column sums [@DG]. $\Xi$ is boolean ---------------- The maximal elements in $\Xi$ are those of the form $(\emptyset, w, \emptyset)$, and there is a unique minimum, $(S, e, S)$. The rank one elements are those of the form $(S-\{i\}, e, S)$ and $(S,e,S-\{j\})$, i.e., those obtained by omitting a single element from $S$ on either the left or on the right. We will now prove that lower intervals in the poset $\Xi$ are isomorphic to boolean algebras. Since the face poset of a simplex is the boolean algebra on its vertex set, a poset with this property is known as a *simplicial poset*, or as a *boolean complex*. \[thm:boolean\] The poset $\Xi$ is a simplicial poset. In particular, the interval below the element $(I,w,J) \in \Xi$ isomorphic to the set of all subsets of $(S-I)\times (S-J)$. Fix an element $F=(I,w,J)$ of $\Xi$ and consider any element below $F$ in the partial order, i.e., suppose we have an element $(I',w',J') \leq_{\Xi} F$. Then by definition, $S \supseteq I' \supseteq I$ and $S \supseteq J' \supseteq J$, so $(I'-I, J'-J)$ is an element of $(S-I)\times (S-J)$. To finish the proof we must show that every pair of subsets $(I'-I, J'-J)$ in $(S-I)\times (S-J)$ corresponds to a unique element below $F$. Suppose $(I'-I, J'-J)$ is a pair of subsets in $(S-I)\times (S-J)$, i.e., $S \supseteq I' \supseteq I$ and $S \supseteq J' \supseteq J$. If $C = W_{I'} v W_{J'}$ is a coset that contains $W_I w W_J$, then in particular $w \in C$ and we can write $C=W_{I'} w W_{J'}$. Thus for fixed $I'$ and $J'$, there is one such coset. By Lemma \[lem:minrep\] there exists a unique element $w' \in C$ such that $\operatorname{Des}_L(w') \subseteq S-I'$ and $\operatorname{Des}_R(w') \subseteq S-J'$. This identifies the unique triple $G=(I',w',J')$ such that $G \leq_{\Xi} F$, completing the proof. Theorem \[thm:boolean\] means that each element of $\Xi$ can be thought of as an abstract simplex. As such, we will refer to the elements as *faces*. We say a face $(I,w,J)$ is *represented by $w$*. $\Xi$ is balanced ----------------- While each face of $\Xi$ is a simplex, it is not a simplicial complex, since distinct faces may share the same vertex set. In fact, we will see that for any $(W,S)$, $\Xi$ has the property that every *facet* (maximal face) has the same vertex set. The dimension of a face is given by one less than its rank in the poset, i.e., if $F = (I,w,J)$, $$\dim F = |S-I|+|S-J| - 1.$$ In particular, if $|S| = n$, then $\Xi$ has $2n$ vertices, each of the form $(S-\{i\},e,S)$ or $(S,e,S-\{j\})$. The dimension of $\Xi$ is the dimension of a maximal face, i.e., $\dim \Xi = \dim (\emptyset,w,\emptyset) = 2n-1$. A $(n-1)$-dimensional simplicial complex is *balanced* if there is an assignment of colors from the set $\{1,2,\ldots,n\}$ to its vertices such that each face has distinctly colored vertices. An important feature of the Coxeter complex $\Sigma$ is that it is balanced; a balanced coloring is given by declaring the *color* of the pair $F=(w,J)$ is $\operatorname{col}(F) = S-J$, where we fix an identification between $S$ and the set $\{1,2,\ldots,n\}$. To show $\Xi$ is balanced we will assign each vertex a color via $$\operatorname{col}( (S-\{i\},e,S)) = (\{i\}, \emptyset) \mbox{ and } \operatorname{col}( (S,e,S-\{j\})) = (\emptyset, \{j\}),$$ and for a general face $F$, $\operatorname{col}(F)$ is the union of the colors of its vertices, i.e., if $F=(I,w,J)$, then $$\operatorname{col}(F) = (S-I,S-J).$$ Since there are $2n$ colors and only $2n$ vertices, we see that $\Xi$ is trivially balanced, i.e., no face has two vertices of the same color since every vertex has a unique color. We have now established part (1) of Theorem \[thm:main\]. $\Sigma$ is a relative subcomplex of $\Xi$ ------------------------------------------ We have already mentioned that maximal faces of $\Xi$ are in bijection with elements of $W$. Let us denote the facet corresponding to an element $w$ by $F_w=(\emptyset, w, \emptyset)$. If we consider only adding elements to the right we get a subposet of $\Xi$ that corresponds to a facet of the usual Coxeter complex. That is, consider the interval $$[ (\emptyset,e,S), F_w ] = \{ G \in \Xi : (\emptyset,e,S) \leq_{\Xi} G \leq_{\Xi} (\emptyset, w, \emptyset) \}.$$ We can represent elements $G \in [ (\emptyset,e,S), F_w ]$ as $G= (\emptyset,u,J)$, such that $J \subseteq S$, $u \in {}^{\emptyset}W^J$, and $w \in uW_J$. Similarly, a facet of $\Sigma$ can be represented as an interval $$\begin{aligned} [(e,S), (w,\emptyset)] &= \{ G \in \Sigma : (e,S) \leq_{\Sigma} G \leq_{\Sigma} (w,\emptyset \},\\ &= \{ (u,J) : J \subseteq S, u \in W^J, w \in uW_J\}.\end{aligned}$$ Thus as posets $$[(\emptyset,e,S),F_w] \cong [ (e,S), (w,\emptyset)] \in \Sigma.$$ (Of course the same idea would work with right cosets, so we could also identify facets of $\Sigma$ with intervals of the form $[(S,e,\emptyset),F_w]$ if we wish.) Taking the union of all such intervals we get a full copy of $\Sigma$ as an upper order ideal (also known as an order filter) inside of $\Xi$. $$\begin{aligned} \Sigma = \{ (w,J) : J \subseteq S, w \in W^J\} &\cong \{ (\emptyset, w, J) : J \subseteq S, w \in {}^{\emptyset}W^J \},\\ &=\{ F \in \Xi : (\emptyset, e, S) \leq_{\Xi} F \}.\end{aligned}$$ To phrase this result another way, we say that $\Sigma$ is a *relative subcomplex* of $\Xi$. This establishes part (2) of Theorem \[thm:main\]. $\Xi$ is partitionable ---------------------- We can notice that the faces represented by a given element $w$ form an upper interval in $\Xi$, i.e., they form an interval whose maximal element has maximal rank in the face poset. To be specific, let $R_w = (\operatorname{Asc}_L(w), w, \operatorname{Asc}_R(w))$, which we call the *restriction* of $w$. Then the interval $[R_w, F_w]$ in $\Xi$ consists of all faces represented by $w$, and moreover this interval is boolean: $$\begin{aligned} [R_w, F_w] &= \{ (I,w,J) : I \subseteq \operatorname{Asc}_L(w), J\subseteq \operatorname{Asc}_R(w) \},\\ & \cong \operatorname{Asc}_L(w)\times \operatorname{Asc}_R(w). \end{aligned}$$ The union of all such intervals partitions the faces of $\Xi$, i.e., $$\label{eq:partition} \Xi = \bigcup_{w \in W} [R_w, F_w],$$ and this union is disjoint. Moreover, since each interval in the partition is an upper ideal isomorphic to a boolean algebra, $\Xi$ is *partitionable* in the topological sense as well. This property foreshadows the shellability result of the next section. See [@Stanley Section III.2] for the relevant definitions. Topology {#sec:topology} ======== In this section we will prove parts (3) and (4) of Theorem \[thm:main\]. $\Xi$ is shellable {#sec:shelling} ------------------ We first make the following simple observation. If $(I,u,J)$ is a face of $\Xi$ below the face $(I',v,J')$, then in particular $W_{I'}vW_{J'} \subseteq W_IuW_J$, and $v \in W_I uW_J$. But by Lemma \[lem:minrep\] this means $u$ is below $v$ in the two-sided weak order. \[obs:weak\] If $(I,u,J) \leq_{\Xi} (I',v,J')$, then $u\leq_{LR} v$. From this simple observation it follows that any choice of linear extension of the two-sided weak order for $W$ is a shelling order for $\Xi$. First recall the definition of a *shelling* of a boolean complex. This is an ordering of the facets $F_1, F_2,\ldots$ such that the intersection of the boundary of each new facet with the union of the boundaries of the prior facets is a pure codimension one complex. That is, for each $k$, we must show $$\partial F_k \cap \left( \bigcup_{i = 1}^{k-1} \partial F_i \right)$$ is a pure codimension one complex. Here $\partial F_k$ denotes the boundary of $F_k$, i.e., all proper faces of $F_k$. Consider all the codimension one faces of the facet $F_w=(\emptyset, w, \emptyset)$. These come in four types: - $(\{s\},sw,\emptyset)$ if $s \in \operatorname{Des}_L(w)$, - $(\emptyset,ws,\{s\})$ if $s \in \operatorname{Des}_R(w)$, - $(\{s\},w,\emptyset)$ if $s \in \operatorname{Asc}_L(w)$, - $(\emptyset,w,\{s\})$ if $s \in \operatorname{Asc}_R(w)$. In the first two cases, the elements $sw$ and $ws$ are below $w$ in the two-sided weak order. If we order the facets of $\Xi$ according to a linear extension of the two-sided weak order: $$F_{w_1}, F_{w_2}, \ldots, F_{w_k}, F_w, \ldots,$$ then the intersection of the boundary of $F_w$ with the union of the prior facets is given by those faces below $F_w$ in $\Xi$ that are not represented by $w$, i.e., $$\partial F_w \cap \left(\bigcup_{i=1}^k \partial F_{w_i} \right) = \bigcup_{\substack{ s \in \operatorname{Des}_L(w) \\ t \in \operatorname{Des}_R(w)}} [(S,e,S), (\{s\}, sw, \emptyset)] \cup [(S,e,S), (\emptyset, wt, \{t\})].$$ Because all maximal faces have codimension one, we have proved the following proposition. Any linear extension of the two-sided weak order on $W$ is a shelling order for $\Xi$. In particular, any linear extension of the Bruhat order is a shelling order. This proves part (3) of Theorem \[thm:main\]. A shelling of $\Xi(A_2)$ is indicated in Figure \[fig:A2\]. The highlighted edges represent the intervals $[R_w,F_w]$, and with facets taken left to right, we have a linear extension of the two-sided weak order. Consequences of shelling ------------------------ A simplicial complex is a *psuedomanifold* if every codimension one face is contained in exactly two maximal faces. A result of Björner tells us about shellable pseudomanifolds. Suppose $\Delta$ is a shellable pseudomanifold. If $\Delta$ is infinite, it is contractible. If $\Delta$ is finite it is a sphere. While $\Xi$ is not a simplicial complex, its barycentric subdivision is. Let $\Xi'$ denote this simplicial complex, whose faces are chains $$F'=\emptyset <_{\Xi} F_1 <_{\Xi} F_2 <_{\Xi} \cdots <_{\Xi} F_k, \qquad F_i \in \Xi.$$ The dimension of such a face is $k-1$, and inclusion of faces in $\Xi'$ is given by inclusion of the sets of faces, i.e., $F' \leq_{\Xi'} G'=\emptyset <_{\Xi} G_1 <_{\Xi} \cdots <_{\Xi} G_l$ if and only if $$\{ F_1,\ldots,F_k\} \subseteq \{ G_1,\ldots,G_l\}.$$ A poset is called *thin* if every interval of length two has exactly four elements. Since $\Xi$ is a simplicial poset, every interval is boolean, and $\Xi$ is clearly thin. The nice thing about being thin is that the barycentric subdivision $\Xi'$ is a pseudomanifold. Indeed if $F'$ is a codimension one face of $\Xi'$ it has the form $$F'= \emptyset <_{\Xi} F_1 <_{\Xi} F_2 <_{\Xi} \cdots <_{\Xi} F_{j-1} <_{\Xi} F_{j+1} <_{\Xi} \cdots <_{\Xi} F_d,$$ where $\dim(F_i) = i-1$ and $d=2n$. Since $\Xi$ is thin the interval $[F_{j-1},F_{j+1}] = \{ F_{j-1}, H, H', F_{j+1}\}$ has exactly four elements, so there are exactly two choices for how to fill the gap in $F'$ to create a facet of $\Xi'$; either $F_{j-1} <_{\Xi} H <_{\Xi} F_{j+1}$ or $F_{j-1} <_{\Xi} H' <_{\Xi} F_{j+1}$. Here we are tacitly assuming $j=1,\ldots,d-1$, but we also need to consider the $j=d$ case, i.e., faces in $\Xi'$ of the form $$F'= \emptyset <_{\Xi} F_1 <_{\Xi} F_2 <_{\Xi} \cdots <_{\Xi} F_{d-1}.$$ But if $F_{d-1}$ is a codimension one face of $\Xi$, we saw from Section \[sec:shelling\] it has the form $(\{s\},w,\emptyset)$ or $(\emptyset,w,\{s\})$, whose corresponding double cosets are $W_{\{s\}}wW_{\emptyset} = \{ w, sw\}$ or $W_{\emptyset}wW_{\{s\}} = \{ w, ws\}$. In either case, the coset has exactly two elements, so the face $(\{s\},w,\emptyset)$ is only contained in the facets $(\emptyset, w, \emptyset)$ and $(\emptyset, sw, \emptyset)$, while $(\emptyset,w,\{s\})$ is only contained in $(\emptyset,w,\emptyset)$ and $(\emptyset,ws,\emptyset)$. Thus we have shown that every codimension one face $F'$ of $\Xi'$ is contained in exactly two maximal faces, i.e., $\Xi'$ is a pseudomanifold. Having established that $\Xi'$ is a pseudomanifold, we also claim that $\Xi'$ inherits shellability from $\Xi$. This is well-known for finite posets, see, e.g., [@Bjorner Proposition 4.4(a)], and is easily generalized to arbitrary simplicial posets whose facets all have the same dimension. To summarize, the barycentric subdivision of $\Xi$ is a shellable pseudomanifold. Since barycentric subdivision respects topology, we obtain the following corollary, establishing parts (4) and (5a) of Theorem \[thm:main\]. \[cor:sphere\] The barycentric subdivision of $\Xi$ is a shellable pseudomanifold, and hence: - $\Xi$ is contractible when $W$ is infinite, - $\Xi$ is a sphere when $W$ is finite. Let $\hat\Xi = \Xi \cup \{\hat 1\}$ be the poset obtained by adding a unique maximal element $\hat 1$ to the poset $\Xi$. Our argument for showing that $\Xi'$ is a pseudomanifold is essentially the argument that the poset $\hat\Xi$ is thin. The fact that $\Xi$ is a sphere in the finite case thus follows from [@Bjorner Proposition 4.5]. Face enumeration for finite $W$ {#sec:faces} =============================== Throughout this section we assume $W$ is finite and fix an ordering on the generating set, $S=\{s_1,\ldots,s_n\}$. In this way we can identify subsets of $S$ with subsets of $[n]:=\{1,2,\ldots,n\}$. Let $x_1,\ldots,x_n$ and $y_1,\ldots,y_n$ be indeterminates. If $I \subseteq [n]$, let $x_I = \prod_{i \in I} x_i$, and similarly for $y_I$. For a face $F = (I, w, J)$ in $\Xi$, the *face monomial* for $F$ is $$m(F) = x_{[n]-I} y_{[n]-J}= \prod_{i \in [n]-I} x_i \prod_{j \in [n]-J} y_j.$$ Notice this encodes the color of the face $F$; the $x$ variables encode the left sided vertices, the $y$ variables encode the right sided vertices. Let $f(\mathbf{x},\mathbf{y})= f(x_1,\ldots,x_n,y_1,\ldots,y_n)$ denote the generating function for colors of faces, i.e., $$f(\mathbf{x},\mathbf{y}) = \sum_{F \in \Xi} m(F) = \sum_{I, J} f_{I,J} x_I y_J.$$ Notice that the coefficient $f_{I,J}$ is the number of faces $(S-I,w,S-J)$, i.e., it counts the cardinality of the corresponding double quotient: $$\label{eq:double} f_{I,J} = |{}^{S-I}W^{S-J}|=|W_{S-I}\backslash W /W_{S-J}|.$$ By Lemma \[lem:minrep\] this is $$f_{I,J} = |\{ w \in W: \operatorname{Des}_L(w) \subseteq I, \operatorname{Des}_R(w) \subseteq J\}|.$$ Now define the quantities $$\begin{aligned} h_{I,J} &= \sum_{\substack{ K\subseteq I \\ L \subseteq J}} (-1)^{|I-K|+|J-L|} f_{K,L},\\ &=|\{ w \in W: \operatorname{Des}_L(w) = I, \operatorname{Des}_R(w) = J\}|,\end{aligned}$$ and the corresponding generating function $$\begin{aligned} h(x_1,\ldots,x_n,y_1,\ldots,y_n) &= \sum_{I,J} h_{I,J}x_Iy_J,\\ &= \sum_{w \in W} x_{\operatorname{Des}_L(w)} y_{\operatorname{Des}_R(w)}.\end{aligned}$$ Recall that for a fixed element $w \in W$, the interval $[R_w,F_w]$ contains all the faces represented by $w$, and this interval is isomorphic to the boolean interval $\operatorname{Asc}_L(w)\times \operatorname{Asc}_R(w)$. This means the generating function for faces in this interval has the following form: $$\begin{aligned} \sum_{R_w \leq F \leq F_w} m(F) &= m(R_w)\cdot \prod_{i \in \operatorname{Asc}_L(w)} (1+x_i)\cdot \prod_{ j \in \operatorname{Asc}_R(w) } (1+y_j),\\ &= x_{\operatorname{Des}_L(w)}y_{\operatorname{Des}_R(w)} \cdot \prod_{i \in \operatorname{Asc}_L(w)} (1+x_i)\cdot \prod_{ j \in \operatorname{Asc}_R(w) } (1+y_j),\\ &= \left( \prod_{i=1}^n (1+x_i)(1+y_i) \right)\prod_{ j \in \operatorname{Des}_L(w)} \frac{x_j}{1+x_j} \prod_{k \in \operatorname{Des}_R(w)} \frac{y_k}{1+y_k},\end{aligned}$$ where the final equality comes from the fact that ascent sets and descent sets are complementary. Now using the partitioning of faces of $\Xi$ given in , we get$$\begin{aligned} f(\mathbf{x},\mathbf{y}) &= \sum_{F \in \Xi} m(F),\nonumber \\ &= \sum_{w \in W} \sum_{R_w \leq F \leq F_w} m(F),\nonumber \\ &= \prod_{i=1}^n (1+x_i)(1+y_i) \sum_{w \in W} \prod_{ j \in \operatorname{Des}_L(w)} \frac{x_j}{1+x_j} \prod_{k \in \operatorname{Des}_R(w)} \frac{y_k}{1+y_k},\nonumber \\ &= \prod_{i =1}^n (1+x_i)(1+y_i) h\left( \frac{x_1}{1+x_1}, \ldots,\frac{x_n}{1+x_n}, \frac{y_1}{1+y_1}, \ldots, \frac{y_n}{1+y_n} \right). \label{eq:ftoh}\end{aligned}$$ That is, we obtain the $f$-polynomial as a multiple of a certain specialization of the $h$-polynomial. Putting identity the other way around, we can write $$\label{eq:htof} h(\mathbf{x},\mathbf{y}) = \prod_{i=1}^n (1-x_i)(1-y_i) f\left( \frac{x_1}{1-x_1}, \ldots,\frac{x_n}{1-x_n}, \frac{y_1}{1-y_1}, \ldots, \frac{y_n}{1-y_n} \right).$$ Setting $x_j=x$ and $y_k=y$, we have $$f(x,y) = \sum_{F \in \Xi} x^{l(F)}y^{r(F)},$$ where if $F=(J,w,K)$, $l(F) = |S-J|$ and $r(F) = |S-K|$, which counts faces according to the number of “left" and “right" vertices. The $h$-polynomial specializes to $$h(x,y) = \sum_{w \in W} x^{\operatorname{des}_L(w)} y^{\operatorname{des}_R(w)}.$$ In other words, the polynomial $h(x,y)$ is a “two-sided" Eulerian polynomial. This establishes the claim in part (5b) of Theorem \[thm:main\]. The usual $f$- and $h$-polynomials of $\Xi$ can be obtained by the further specialization of $x=y$: $$f(x) = \sum_{F \in \Xi} x^{|F|}, \qquad h(x) = \sum_{w \in W} x^{\operatorname{des}_L(w) + \operatorname{des}_R(w)}.$$ We can see in Figure \[fig:A2\] that $$\begin{aligned} f(A_2;\mathbf{x},\mathbf{y}) &= 1 + (x_1 + x_2 + y_1 + y_2) \\ &+ (x_1x_2 + 2x_1y_1 + 2x_1y_2 + 2x_2y_1 + 2x_2y_2 + y_1y_2) \\ &+ (3x_1x_2y_1 + 3x_1x_2y_2 + 3x_1y_1y_2 + 3x_2y_1y_2) + 6x_1x_2y_1y_2,\end{aligned}$$ Which after a bit of rearranging equals $$\begin{aligned} &(1+x_1)(1+x_2)(1+y_1)(1+y_2) \\ &+ x_1y_1(1+x_2)(1+y_2) + x_1y_2(1+x_2)(1+y_1) \\ &+ x_2y_1(1+x_1)(1+y_2) + x_2y_2(1+x_1)(1+y_1)\\ &+ x_1x_2y_1y_2.\end{aligned}$$ The elements of $A_2$ have the following descent sets, $$\begin{array}{c | c | c} w & \operatorname{Des}_L(w) & \operatorname{Des}_R(w) \\ \hline e & \emptyset & \emptyset\\ s_1 & \{1\} & \{1\} \\ s_2 & \{2\} & \{2\} \\ s_1s_2 & \{1\} & \{2\}\\ s_2s_1 & \{2\} & \{1\} \\ s_1s_2s_1 = s_2 s_1s_2 & \{1,2\} & \{1,2\} \end{array}$$ so we can see that $$\begin{aligned} f(A_2; \mathbf{x},\mathbf{y}) &= \sum_{w \in A_2} x_{\operatorname{Des}_L(w)}y_{\operatorname{Des}_R(w)}\prod_{i \in \operatorname{Asc}_L(w)} (1+x_i) \cdot \prod_{j \in \operatorname{Asc}_R(w)} (1+y_j),\\ &= h\left(A_2; \frac{x_1}{1+x_1}, \frac{x_2}{1+x_2},\frac{y_1}{1+y_1},\frac{y_2}{1+y_2}\right).\end{aligned}$$ The coarser polynomials are then $$f(A_2; x, y) = 1 + 2(x+y) + x^2 + 8xy + y^2 + 6(x^2y + xy^2) + 6x^2y^2,$$ and $$h(A_2; x,y) = 1 + 4xy + x^2y^2.$$ Two-sided Eulerian polynomials {#sec:poly} ============================== With finite $W$, we can define the *two-sided $W$-Eulerian polynomial*, denoted $W(x,y)$, as the joint distribution of left and right descents: $$W(x,y) =\sum_{w \in W} x^{\operatorname{des}_L(w)}y^{\operatorname{des}_R(w)} = \sum_{0\leq i,j\leq n} {\genfrac{\langle}{\rangle}{0pt}{}{W}{i,j}} x^i y^j,$$ where ${\genfrac{\langle}{\rangle}{0pt}{}{W}{i,j}}$ denotes the number of elements in $W$ with $i$ left descents and $j$ right descents. We call ${\genfrac{\langle}{\rangle}{0pt}{}{W}{i,j}}$ a *two-sided $W$-Eulerian number*. In Tables \[tab:WpolyABD\] and \[tab:WpolyEF\] we have the arrays of coefficients $$\left[ {\genfrac{\langle}{\rangle}{0pt}{}{W}{i,j}} \right]_{0\leq i,j \leq n},$$ for some finite Coxeter groups of small rank. $$\begin{array}{| c c |} \hline & \\ W & \left[ {\genfrac{\langle}{\rangle}{0pt}{}{W}{i,j}} \right]_{0\leq i,j \leq n} \\ & \\ \hline \hline & \\ A_n\, (n\geq 1): & {\small \left[ \begin{array}{r r} 1 & 0 \\ 0 & 1\end{array} \right], \left[ \begin{array}{r rr} 1 & 0 &0 \\ 0 & 4 & 0 \\ 0 & 0 & 1 \end{array} \right], \left[ \begin{array}{r rrr} 1 & 0 &0 &0 \\ 0 & 10 & 1 & 0 \\ 0 & 1 & 10 & 0 \\ 0 & 0 & 0 & 1 \end{array} \right], \left[ \begin{array}{r rrrr} 1 & 0 &0 &0& 0\\ 0 & 20 & 6 & 0 & 0\\ 0 & 6 & 54 & 6 & 0 \\ 0 & 0 & 6 & 20& 0 \\ 0 & 0 & 0 & 0 &1 \end{array} \right]} \\ & \\ \hline & \\ B_n\, (n\geq 2): & {\small \left[ \begin{array}{r rr} 1 & 0 &0 \\ 0 & 6 & 0 \\ 0 & 0 & 1 \end{array} \right], \left[ \begin{array}{r rrr} 1 & 0 &0 &0 \\ 0 & 19 & 4 & 0 \\ 0 & 4 & 19 & 0 \\ 0 & 0 & 0 & 1 \end{array} \right], \left[ \begin{array}{r rrrr} 1 & 0 &0 &0& 0\\ 0 & 45 & 30 & 1 & 0\\ 0 & 30 & 170 & 30 & 0 \\ 0 & 1 & 30 & 45& 0 \\ 0 & 0 & 0 & 0 &1 \end{array} \right] }\\ & \\ \hline & \\ D_n\, (n\geq 4): & {\small \left[ \begin{array}{r rrrr} 1 & 0 &0 &0& 0\\ 0 & 30 & 12 & 2 & 0\\ 0 & 12 & 78 & 12 & 0 \\ 0 & 2 & 12 & 30 & 0 \\ 0 & 0 & 0 & 0 &1 \end{array} \right], \left[ \begin{array}{r rrrrr} 1 & 0 &0 &0& 0 & 0\\ 0 & 69 & 69 & 18 & 1 & 0\\ 0 & 69 & 486 & 229 & 18 & 0 \\ 0 & 18 & 229 & 486 & 69 & 0 \\ 0 & 1 & 18 & 69 & 69 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{array} \right]},\\ & \\ & {\small\left[ \begin{array}{rrrrrrr} 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 135 & 262 & 117 & 16 & 0 & 0 \\ 0 & 262 & 2433 & 2330 & 510 & 16 & 0 \\ 0 & 117 & 2330 & 5982 & 2330 & 117 & 0 \\ 0 & 16 & 510 & 2330 & 2433 & 262 & 0 \\ 0 & 0 & 16 & 117 & 262 & 135 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \end{array} \right]}\\ & \\ \hline \end{array}$$ $$\begin{array}{| c c |} \hline & \\ W & \left[ \gamma^{W}_{a,b} \right]_{0\leq 2a+b \leq n} \\ & \\ \hline \hline & \\ A_n\, (n\geq 1): & {\small \left[ \begin{array}{r r} 1 \end{array} \right], \left[ \begin{array}{r rr} 1 & 0 \\ 0 & 2 \end{array} \right], \left[ \begin{array}{r rrr} 1 & 0 \\ 0 & 7 \\ 0 & 1 \end{array} \right], \left[ \begin{array}{r rrrr} 1 & 0 &0 \\ 0 & 16 & 0 \\ 0 & 6 & 16 \end{array} \right]} \\ & \\ \hline & \\ B_n\, (n\geq 2): & {\small \left[ \begin{array}{r rr} 1 & 0 \\ 0 & 4 \end{array} \right], \left[ \begin{array}{r rrr} 1 & 0 \\ 0 & 16 \\ 0 & 4 \end{array} \right], \left[ \begin{array}{r rrrr} 1 & 0 & 0 \\ 0 & 41 & 0 \\ 0 & 30 & 80 \\ 0 & 1 & 0 \end{array} \right] }\\ & \\ \hline & \\ D_n\, (n\geq 4): & {\small \left[ \begin{array}{r rrrr} 1 & 0 & 0 \\ 0 & 26 & 0 \\ 0 & 12 & 16 \\ 0 & 2 & 0 \end{array} \right], \left[ \begin{array}{r rrrrr} 1 & 0 & 0 \\ 0 & 64 & 0 \\ 0 & 69 & 248 \\ 0 & 18 & 88 \\ 0 & 1 & 0 \end{array} \right]},{\small\left[ \begin{array}{rrrrrrr} 1 & 0 & 0 & 0 \\ 0 & 129 & 0 & 0 \\ 0 & 262 & 1668 & 0 \\ 0 & 117 & 1496 & 832 \\ 0 & 16 & 276 & 0 \end{array} \right]}\\ & \\ \hline \end{array}$$ $$\begin{array}{| c c |} \hline & \\ W & \left[ {\genfrac{\langle}{\rangle}{0pt}{}{W}{i,j}} \right]_{0\leq i,j \leq n} \\ & \\ \hline \hline & \\ E_6: & {\small \left[ \begin{array}{rrr rrrr} 1 & 0 &0 &0& 0& 0 &0\\ 0 & 232 & 584 & 389 & 64& 3 & 0\\ 0 & 584 & 4785 & 5440 & 1310 & 64 & 0 \\ 0 & 389 & 5440 & 13270 & 5440 & 389 &0 \\ 0 & 64 & 1310 & 5440 & 4785 & 584 & 0 \\ 0 & 3 & 64 & 389 & 584 & 232 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \end{array} \right]}\\ & \\ \hline & \\ E_7: & {\small \left[ \begin{array}{rrrrrrrr} 1& 0& 0& 0& 0& 0& 0& 0 \\ 0& 945& 5414& 7693& 3208& 367& 8& 0 \\ 0& 5414& 64905& 143036& 83491& 12756& 367& 0 \\ 0& 7693& 143036& 484551& 401936& 83491& 3208& 0 \\ 0& 3208& 83491& 401936& 484551& 143036& 7693& 0 \\ 0& 367& 12756& 83491& 143036& 64905& 5414& 0 \\ 0& 8& 367& 3208& 7693& 5414& 945& 0 \\ 0& 0& 0& 0& 0& 0& 0& 1 \end{array} \right]}\\ & \\ \hline & \\ E_8: & {\tiny \left[ \begin{array}{rrrrr rrrr} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 8460 & 113241 & 338944 & 318372 & 94540 & 8103 & 92 & 0 \\ 0 & 113241 & 2348364 & 9509809 & 11520216 & 4360423 & 476192 & 8103 & 0 \\ 0 & 338944 & 9509809 & 48819660 & 72638788 & 33260660 & 4360423 & 94540 & 0 \\ 0 & 318372 & 11520216 & 72638788 & 131292998 & 72638788 & 11520216 & 318372 & 0 \\ 0 & 94540 & 4360423 & 33260660 & 72638788 & 48819660 & 9509809 & 338944 & 0 \\ 0 & 8103 & 476192 & 4360423 & 11520216 & 9509809 & 2348364 & 113241 & 0 \\ 0 & 92 & 8103 & 94540 & 318372 & 338944 & 113241 & 8460 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \end{array} \right]} \\ & \\ \hline & \\ F_4: & \left[ \begin{array}{rrr rrrr} 1 & 0 & 0 & 0 & 0 \\ 0 & 108 & 112 & 16 & 0 \\ 0 & 112 & 454 & 112 & 0 \\ 0 & 16 & 112 & 108 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{array} \right]\\ & \\ \hline \end{array}$$ $$\begin{array}{| c c |} \hline & \\ W & \left[ \gamma^{W}_{a,b} \right]_{0\leq 2a+b \leq n} \\ & \\ \hline \hline & \\ E_6: & {\small \left[ \begin{array}{rrr rrrr} 1 & 0 & 0 & 0 \\ 0 & 226 & 0 & 0 \\ 0 & 584 & 3088 & 0 \\ 0 & 389 & 3496 & 3104 \\ 0 & 64 & 520 & 0 \\ 0 & 3 & 0 & 0 \end{array} \right]}\\ & \\ \hline & \\ E_7: & {\small \left[ \begin{array}{rrrrrrrr} 1 & 0 & 0 & 0 \\ 0 & 938 & 0 & 0 \\ 0 & 5414 & 44808 & 0 \\ 0 & 7693 & 111756 & 174464 \\ 0 & 3208 & 58944 & 107712 \\ 0 & 367 & 6300 & 0 \\ 0 & 8 & 0 & 0 \end{array} \right]}\\ & \\ \hline & \\ E_8: & {\small \left[ \begin{array}{rrrrr rrrr} 1 & 0 & 0 & 0 & 0 \\ 0 & 8452 & 0 & 0 & 0 \\ 0 & 113241 & 1619736 & 0 & 0 \\ 0 & 338944 & 7988488 & 19362528 & 0 \\ 0 & 318372 & 9786280 & 34500160 & 17111296 \\ 0 & 94540 & 3364792 & 9750496 & 0 \\ 0 & 8103 & 286560 & 0 & 0 \\ 0 & 92 & 0 & 0 & 0 \end{array} \right]} \\ & \\ \hline & \\ F_4: & \left[ \begin{array}{rrr rrrr} 1 & 0 & 0 & 0 \\ 0 & 104 & 0 & 0 \\ 0 & 112 & 208 & 0 \\ 0 & 16 & 0 & 0 \end{array} \right]\\ & \\ \hline \end{array}$$ In type $A_n$, these numbers were first studied by Carlitz et al. [@Carlitz], but have been recently revisited by the author [@Petersen] and Visontai [@Visontai] (who also discussed type $B_n$ Coxeter groups). The recent interest in these polynomials stems from a conjecture of Gessel that we will now describe and generalize from the symmetric group to all finite Coxeter groups. To state Gessel’s conjecture, one must first make note of certain symmetries in the two-sided Eulerian numbers. Notice that the map $w \mapsto w^{-1}$ swaps left and right descents, $\operatorname{Des}_L(w) = \operatorname{Des}_R(w^{-1})$, so we get symmetry in $i$ and $j$: $$\label{eq:ijsym} {\genfrac{\langle}{\rangle}{0pt}{}{W}{i,j}} = {\genfrac{\langle}{\rangle}{0pt}{}{W}{j,i}}.$$ Also recall that left multiplication by the long element $w_0$ complements the right descent set: $$\operatorname{Des}_R(w_0 w) = S- \operatorname{Des}_R(w),$$ while conjugation by $w_0$ conjugates the elements of the right descent set: $$\operatorname{Des}_R(w_0 w w_0) = \{ w_0sw_0 : s \in \operatorname{Des}_R(w)\} = w_0\operatorname{Des}_R(w) w_0.$$ These facts follow, e.g., from [@BjB Section 2.3]. Taken together, we see that left multiplication by $w_0$ complements the conjugate of the left descent set: $$\begin{aligned} \operatorname{Des}_L(w_0w) &=\operatorname{Des}_R(w^{-1}w_0),\\ &=\operatorname{Des}_R(w_0(w_0w^{-1}w_0)),\\ &=S-\operatorname{Des}_R(w_0w^{-1}w_0),\\ &=S-w_0\operatorname{Des}_R(w^{-1})w_0,\\ &=S-w_0\operatorname{Des}_L(w) w_0.\end{aligned}$$ Hence we have $\operatorname{des}_L(w_0w) = n-\operatorname{des}_L(w)$ and $\operatorname{des}_R(w_0w) = n- \operatorname{des}_R(w)$, implying the following symmetry: $$\label{eq:n-isym} {\genfrac{\langle}{\rangle}{0pt}{}{W}{i,j}} = {\genfrac{\langle}{\rangle}{0pt}{}{W}{n-i,n-j}}.$$ Phrasing symmetries and in terms of generating functions, we have the following observation about the two-sided $W$-Eulerian polynomials. For any finite Coxeter group $W$ of rank $n$, 1. $W(x,y) = W(y,x)$, and 2. $W(x,y)=x^ny^nW(1/x,1/y)$. Integer polynomials that possess symmetries (1) and (2) have an expansion in the following basis: $$\Gamma_n=\{ (x y)^a (x+y)^b (1+xy)^{n-2a-b} \}_{0\leq 2a + b \leq n}.$$ The generalized Gessel conjecture is that the two-sided Eulerian polynomials expand positively in this basis. \[conj:gamma\] For any finite Coxeter group $W$ of rank $n$, there exist nonnegative integers $\gamma_{a,b}^W$ such that $$W(x,y) = \sum_{0\leq 2a+b \leq n} \gamma_{a,b}^W (xy)^a(x+y)^b(1+xy)^{n-2a-b}.$$ The integers $\gamma_{a,b}^W$ for $W$ of small rank are shown in Tables \[tab:WpolyABD2\] and \[tab:WpolyEF2\]. In practice, traversing the group $W$ to compute the polynomial $W(x,y)$ is not very efficient, as the order of the group is roughly factorial in the rank. From Equation we know $f_{S-I,S-J}$ is the cardinality of the double quotient $|W_I\backslash W/W_J|$ and from [@ec2 Exercise 7.77a] we can compute this cardinality with an inner product of trivial characters on the parabolic subgroups induced up to $W$. That is, $$|W_I\backslash W/W_J| = \left\langle \mathrm{ind}_{W_{I}}^{W}1_{W_{I}}, \mathrm{ind}_{W_{J}}^{W}1_{W_{J}} \right\rangle,$$ where $1_{W_J}$ denotes the trivial character on $W_J$. Stembridge has a nice implementation of this character computation in Maple [@StM]. Having computed the numbers $f_{I,J}$ for all pairs of subsets $I, J \subseteq S$, we obtain the polynomial $f(\mathbf{x},\mathbf{y})$ and we can use Equation to compute the polynomial $h(\mathbf{x},\mathbf{y})$, which then specializes to $W(x,y)$. To put it succinctly, we have $$W(x,y) = \sum_{I,J \subseteq S} f_{I,J}x^{|I|}y^{|J|}(1-x)^{n-|I|}(1-y)^{n-|J|}.$$ Roughly speaking, this method reduces the problem of computing $W(x,y)$ from that of traversing the $|W|$ elements of $W$ to one of traversing $4^n$ pairs of subsets. The two-sided Eulerian numbers for $E_8$ were computed in about half an hour on a standard desktop machine in this manner. Very recently, the author was informed that Gessel’s original conjecture (for $W=A_n = S_{n+1}$) was proved by Lin [@Lin]. The method of proof seems to be a careful induction argument using a recurrence for the $\gamma_{a,b}^{A_n}$ given by Visontai [@Visontai]. The other cases have been verified for small rank $(n\leq 10)$. Type $B_n$ is governed by similar combinatorics, so perhaps a similar induction proof can be found. In all cases, it would be nice to know what the $\gamma_{a,b}^W$ count. Contingency tables {#sec:contingency} ================== Throughout this section we consider the special case where $W=S_n$ is the symmetric group. The generating set is $S=\{s_1,s_2,\ldots,s_{n-1}\}$, where $s_i$ is the $i$th adjacent transposition. As shown in Diaconis and Gangolli [@DG], for fixed $I$ and $J$ the double cosets $W_I w W_J$ are in bijection with arrays of nonnegative integers. (They attribute the idea to N. Bergeron.) To see how this connection is made, we draw double cosets as diagrams of “balls in boxes." First, we draw permutations as two-dimensional arrays, with a ball in column $i$ (left to right), row $j$ (bottom to top), if $w(i)=j$, then we insert some vertical and horizontal bars in gaps between balls. The group $S_n$ acts on the left by permuting rows; it acts on the right by permuting columns. For example, $w = 7142536$ is drawn in Figure \[fig:double\]. To indicate a parabolic double coset $W_IwW_J$, we draw solid horizontal bars in gaps that correspond to $S-I$ and solid vertical bars in gaps that correspond to $S-J$. In Figure \[fig:double\], $I=\{s_1,s_2,s_3, s_5\}$ and $J=\{s_2,s_3,s_6\}$. We can get all elements of $W_IwW_J$ by swapping columns and rows that are not separated by a solid bar. Notice that the balls cannot leave the boxes formed by the bars. The minimal representative for the double coset corresponds to the permutation obtained by sorting the balls in increasing order from left to right and from bottom to top. The minimal representative for the coset illustrated in Figure \[fig:double\] would then be $u = 7123546$. Notice that both the right descents and left descents of $u$ occur in barred positions. $$\begin{tikzpicture} \draw (0,0) node (a) { \begin{tikzpicture}[scale=.25] \draw[dashed] (0,0) grid[step=2] (14,14); \draw (1,13) node[circle,fill=black,inner sep =2] {}; \draw (3,1) node[circle,fill=black,inner sep =2] {}; \draw (5,7) node[circle,fill=black,inner sep =2] {}; \draw (7,3) node[circle,fill=black,inner sep =2] {}; \draw (9,9) node[circle,fill=black,inner sep =2] {}; \draw (11,5) node[circle,fill=black,inner sep =2] {}; \draw (13,11) node[circle,fill=black,inner sep =2] {}; \draw[thick] (2,-1)--(2,15); \draw[thick] (8,-1)--(8,15); \draw[thick] (10,-1)--(10,15); \draw[thick] (-1,8)--(15,8); \draw[thick] (-1,12)--(15,12); \draw (4,-1) node {$s_2$}; \draw (6,-1) node {$s_3$}; \draw (12,-1) node {$s_6$}; \draw (-1,2) node {$s_1$}; \draw (-1,4) node {$s_2$}; \draw (-1,6) node {$s_3$}; \draw (-1,10) node {$s_5$}; \end{tikzpicture} }; \draw (6,0) node (b) { $\left[ \begin{array}{ r r r r} 1 & 0 & 0 &0\\ 0 & 0 & 1 & 1 \\ 0 & 3 & 0 & 1 \end{array}\right]$ }; \draw[->] (a)--(b); \end{tikzpicture}$$ Given the diagram for a double coset as in Figure \[fig:double\], we can map the diagram to an array of nonnegative integers by merely counting the number of balls in each box. Let $\Xi(n)$ denote the set of all such arrays, which are known as *two-way contingency tables*. More precisely, define $\Xi(n)$ to be the set of all nonnegative integer arrays whose entries sum to $n$ and whose row sums and column sums are positive. To move up in the partial order, we refine our balls and boxes picture by inserting more bars. On the contingency table side, this means our arrays get more rows and columns. Each cover relation corresponds to adding or deleting a single bar, so rank is given by the total number of bars. A balls-in-boxes picture with $k$ horizontal bars and $l$ vertical bars will correspond to a $(k+1)\times(l+1)$ contingency table. [$$\begin{tikzpicture}[xscale=.6,yscale=3] \tikzstyle{state}=[fill=white,inner sep = 2,scale=.6]; \tikzstyle{linet}=[line width=5, cap=round, color=white!80!black]; \draw (0,.5) node[state] (12e12) {$[3]$}; \draw (-6,1) node[state] (1e12) {$\left[ \begin{array}{c} 1 \\ 2 \end{array} \right]$}; \draw (-2,1) node[state] (2e12) {$\left[ \begin{array}{c} 2 \\ 1 \end{array} \right]$}; \draw (6,1) node[state] (12e2) {$\left[ \begin{array}{cc} 1 & 2 \end{array} \right]$}; \draw (2,1) node[state] (12e1) {$\left[ \begin{array}{cc} 2 & 1 \end{array} \right]$}; \draw (-9,1.75) node[state] (e12) {$\left[ \begin{array}{c} 1 \\ 1 \\ 1 \end{array} \right]$}; \draw (-7,1.75) node[state] (1e2) {$\left[ \begin{array}{cc} 0 & 1 \\ 1 & 1\end{array} \right]$}; \draw (-5,1.75) node[state] (1e1) {$\left[ \begin{array}{cc} 0 & 1 \\ 2 & 0 \end{array} \right]$}; \draw (-3,1.75) node[state] (2e2) {$\left[ \begin{array}{cc} 0 & 2 \\ 1 & 0\end{array} \right]$}; \draw (-1,1.75) node[state] (2e1) {$\left[ \begin{array}{cc} 1 & 1 \\ 1 & 0 \end{array} \right]$}; \draw (1,1.75) node[state] (12e) {$\left[ \begin{array}{ccc} 1 & 1 & 1 \end{array} \right]$}; \draw (3,1.75) node[state] (2s12) {$\left[ \begin{array}{cc} 1 & 1 \\ 0 & 1 \end{array} \right]$}; \draw (5,1.75) node[state] (1s21) {$\left[ \begin{array}{cc} 1 & 0 \\ 1 & 1 \end{array} \right]$}; \draw (7.25,1.75) node[state] (2s1s21) {$\left[ \begin{array}{cc} 2 & 0 \\ 0 & 1 \end{array} \right]$}; \draw (9.5,1.75) node[state] (1s2s12) {$\left[ \begin{array}{cc} 1 & 0 \\ 0 & 2 \end{array} \right]$}; \draw (-11,3) node[state] (e2) {$\left[ \begin{array}{cc} 0 & 1 \\ 0 & 1 \\ 1 & 0 \end{array} \right]$}; \draw (-9,3) node[state] (e1) {$\left[ \begin{array}{cc} 0 & 1 \\ 1 & 0 \\ 1 & 0 \end{array} \right]$}; \draw (-7,3) node[state] (1e) {$\left[ \begin{array}{ccc} 0 & 0 & 1 \\ 1 & 1 & 0 \end{array} \right]$}; \draw (-5,3) node[state] (2e) {$\left[ \begin{array}{ccc} 0 & 1 & 1 \\ 1 & 0 & 0 \end{array} \right]$}; \draw (-3,3) node[state] (s12) {$\left[ \begin{array}{cc} 0 & 1 \\ 1 & 0 \\ 0 & 1 \end{array} \right]$}; \draw (-1,3) node[state] (2s1) {$\left[ \begin{array}{ccc} 1 & 0 & 1 \\ 0 & 1 & 0 \end{array} \right]$}; \draw (1,3) node[state] (s21) {$\left[ \begin{array}{cc} 1 & 0 \\ 0 & 1 \\ 1 & 0 \end{array} \right]$}; \draw (3,3) node[state] (1s2) {$\left[ \begin{array}{ccc} 0 & 1 & 0 \\ 1 & 0 & 1 \end{array} \right]$}; \draw (5.25,3) node[state] (s1s21) {$\left[ \begin{array}{cc} 1 & 0 \\ 1 & 0 \\ 0 & 1 \end{array} \right]$}; \draw (7.5,3) node[state] (2s1s2) {$\left[ \begin{array}{ccc} 1 & 1 & 0 \\ 0 & 0 & 1 \end{array} \right]$}; \draw (9.75,3) node[state] (s2s12) {$\left[ \begin{array}{cc} 1 & 0 \\ 0 & 1 \\ 0 & 1 \end{array} \right]$}; \draw (12,3) node[state] (1s2s1) {$\left[ \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 1 \end{array} \right]$}; \draw (-10,4) node[state] (e) {$\left[ \begin{array}{ccc} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right]$}; \draw (-6,4) node[state] (s1) {$\left[ \begin{array}{ccc} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{array} \right]$}; \draw (-2,4) node[state] (s2) {$\left[ \begin{array}{ccc} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{array} \right]$}; \draw (2,4) node[state] (s1s2) {$\left[ \begin{array}{ccc} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{array} \right]$}; \draw (6,4) node[state] (s2s1) {$\left[ \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{array} \right]$}; \draw (10,4) node[state] (s1s2s1) {$\left[ \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array} \right]$}; \draw[linet] (e)--(e2); \draw[linet] (e)--(e1); \draw[linet] (e2)--(e12); \draw[linet] (e1)--(e12); \draw[linet] (e12)--(s12); \draw[linet] (s1)--(e1); \draw[linet] (s2)--(e2); \draw[linet] (s12)--(s1s2); \draw[linet] (e12)--(s21); \draw[linet] (s21)--(s2s1); \draw[linet] (e12)--(s2s12); \draw[linet] (e12)--(s1s21); \draw (e)--(e2); \draw (e)--(e1); \draw (e)--(1e); \draw (e)--(2e); \draw (e2)--(e12); \draw (e2)--(1e2); \draw (e2)--(2e2); \draw (e1)--(e12); \draw (e1)--(1e1); \draw (e1)--(2e1); \draw (1e)--(1e2); \draw (1e)--(1e1); \draw (1e)--(12e); \draw (2e)--(2e2); \draw (2e)--(2e1); \draw (2e)--(12e); \draw (e12)--(1e12); \draw (e12)--(2e12); \draw (1e2)--(1e12); \draw (1e2)--(12e2); \draw (1e1)--(1e12); \draw (1e1)--(12e1); \draw (2e2)--(2e12); \draw (2e2)--(12e2); \draw (2e1)--(2e12); \draw (2e1)--(12e1); \draw (12e)--(12e2); \draw (12e)--(12e1); \draw (12e12)--(1e12); \draw (12e12)--(2e12); \draw (12e12)--(12e2); \draw (12e12)--(12e1); \draw[linet] (s1)--(s12); \draw (s1)--(e1); \draw (s1)--(1e); \draw (s1)--(s12); \draw (s1)--(2s1); \draw (s12)--(e12); \draw (s12)--(1e2); \draw (s12)--(2s12); \draw (2s1)--(2e1); \draw (2s1)--(12e); \draw (2s1)--(2s12); \draw (2s12)--(2e12); \draw (2s12)--(12e2); \draw[linet] (s2)--(s21); \draw (s2)--(e2); \draw (s2)--(2e); \draw (s2)--(s21); \draw (s2)--(1s2); \draw (s21)--(e12); \draw (s21)--(2e1); \draw (s21)--(1s21); \draw (1s2)--(1e2); \draw (1s2)--(12e); \draw (1s2)--(1s21); \draw (1s21)--(1e12); \draw (1s21)--(12e1); \draw[linet] (s1s2)--(s1s21); \draw (s1s2)--(s12); \draw (s1s2)--(1s2); \draw (s1s2)--(s1s21); \draw (s1s2)--(2s1s2); \draw (s1s21)--(e12); \draw (s1s21)--(1s21); \draw (s1s21)--(2s1s21); \draw (2s1s2)--(12e); \draw (2s1s2)--(2s12); \draw (2s1s2)--(2s1s21); \draw (2s1s21)--(2e12); \draw (2s1s21)--(12e1); \draw[linet] (s2s1)--(s2s12); \draw (s2s1)--(s21); \draw (s2s1)--(2s1); \draw (s2s1)--(s2s12); \draw (s2s1)--(1s2s1); \draw (s2s12)--(e12); \draw (s2s12)--(2s12); \draw (s2s12)--(1s2s12); \draw (1s2s1)--(12e); \draw (1s2s1)--(1s21); \draw (1s2s1)--(1s2s12); \draw (1s2s12)--(1e12); \draw (1s2s12)--(12e2); \draw (s1s2s1)--(s1s21); \draw (s1s2s1)--(2s1s2); \draw (s1s2s1)--(s2s12); \draw (s1s2s1)--(1s2s1); \end{tikzpicture}$$ ]{} Notice that we can permute the balls before insertion, so more than one cover relation can arise from inserting the same bar. For example, using the balls and boxes diagram of Figure \[fig:double\], there are two covers that come from inserting a horizontal bar in the gap corresponding to $s_5$: $$\begin{tikzpicture}[scale=.25, baseline =1.5cm] \draw[dashed] (0,0) grid[step=2] (14,14); \draw (1,13) node[circle,fill=black,inner sep =2] {}; \draw (3,1) node[circle,fill=black,inner sep =2] {}; \draw (5,7) node[circle,fill=black,inner sep =2] {}; \draw (7,3) node[circle,fill=black,inner sep =2] {}; \draw (9,9) node[circle,fill=black,inner sep =2] {}; \draw (11,5) node[circle,fill=black,inner sep =2] {}; \draw (13,11) node[circle,fill=black,inner sep =2] {}; \draw[thick] (2,-1)--(2,15); \draw[thick] (8,-1)--(8,15); \draw[thick] (10,-1)--(10,15); \draw[thick] (-1,8)--(15,8); \draw[thick] (-1,10)--(15,10); \draw[thick] (-1,12)--(15,12); \draw (4,-1) node {$s_2$}; \draw (6,-1) node {$s_3$}; \draw (12,-1) node {$s_6$}; \draw (-1,2) node {$s_1$}; \draw (-1,4) node {$s_2$}; \draw (-1,6) node {$s_3$}; \end{tikzpicture} \quad\mbox{ and }\quad \begin{tikzpicture}[scale=.25,baseline=1.5cm] \draw[dashed] (0,0) grid[step=2] (14,14); \draw (1,13) node[circle,fill=black,inner sep =2] {}; \draw (3,1) node[circle,fill=black,inner sep =2] {}; \draw (5,7) node[circle,fill=black,inner sep =2] {}; \draw (7,3) node[circle,fill=black,inner sep =2] {}; \draw (9,11) node[circle,fill=black,inner sep =2] {}; \draw (11,5) node[circle,fill=black,inner sep =2] {}; \draw (13,9) node[circle,fill=black,inner sep =2] {}; \draw[thick] (2,-1)--(2,15); \draw[thick] (8,-1)--(8,15); \draw[thick] (10,-1)--(10,15); \draw[thick] (-1,8)--(15,8); \draw[thick] (-1,10)--(15,10); \draw[thick] (-1,12)--(15,12); \draw (4,-1) node {$s_2$}; \draw (6,-1) node {$s_3$}; \draw (12,-1) node {$s_6$}; \draw (-1,2) node {$s_1$}; \draw (-1,4) node {$s_2$}; \draw (-1,6) node {$s_3$}; \end{tikzpicture},$$ corresponding to $$\left[ \begin{array}{rrrr} 1& 0 & 0 & 0 \\ 0 & 0 & 0 & 1\\ 0 & 0 & 1 & 0 \\ 0 & 3 & 0 & 1 \end{array} \right] \quad \mbox{ and } \quad \left[ \begin{array}{rrrr} 1& 0 & 0 & 0 \\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1 \\ 0 & 3 & 0 & 1 \end{array} \right],$$ respectively. Downward covers in the partial order correspond to removing a single bar from the balls in boxes picture, which therefore adds all the entries in two adjacent rows or two adjacent columns of the corresponding contingency tables. In Figure \[fig:covers\] we see all the upper and lower covers of the table from Figure \[fig:double\]. The reader might like to translate these arrays into pictures of balls in boxes. In Figure \[fig:S3\] we see the full refinement order on $\Xi(3)$. We finish by stating what should be clear at this point. The two-sided Coxeter complex of the symmetric group $S_n$ is isomorphic to $\Xi(n)$ under refinement order. $$\begin{tikzpicture}[rotate=-90, xscale=.65,yscale=4] \coordinate (u1) at (-11,1); \coordinate (u2) at (-9,1); \coordinate (u3) at (-7,1); \coordinate (u4) at (-5,1); \coordinate (u5) at (-3,1); \coordinate (u6) at (-1,1); \coordinate (u7) at (1,1); \coordinate (u8) at (3,1); \coordinate (u9) at (5,1); \coordinate (u10) at (7,1); \coordinate (u11) at (9,1); \coordinate (u12) at (11,1); \coordinate (d1) at (-6,-1); \coordinate (d2) at (-3,-1); \coordinate (d3) at (0,-1); \coordinate (d4) at (3,-1); \coordinate (d5) at (6,-1); \coordinate (o) at (0,0); \draw (o)--(u1); \draw (o)--(u2); \draw (o)--(u3); \draw (o)--(u4); \draw (o)--(u5); \draw (o)--(u6); \draw (o)--(u7); \draw (o)--(u8); \draw (o)--(u9); \draw (o)--(u10); \draw (o)--(u11); \draw (o)--(u12); \draw (o)--(d1); \draw (o)--(d2); \draw (o)--(d3); \draw (o)--(d4); \draw (o)--(d5); \draw (o) node[scale=.5,fill=white,inner sep = 0] { $\left[ \begin{array}{ r r r r} 1 & 0 & 0 &0\\ 0 & 0 & 1 & 1 \\ 0 & 3 & 0 & 1 \end{array}\right]$}; \draw (u1) node[scale=.5,fill=white,inner sep = 0, right] { $\left[ \begin{array}{ r r r r} 1 & 0 & 0 &0\\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 3 & 0 & 1 \end{array}\right]$}; \draw (u2) node[scale=.5,fill=white,inner sep = 0, right] { $\left[ \begin{array}{ r r r r} 1 & 0 & 0 &0\\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 3 & 0 & 1 \end{array}\right]$}; \draw (u3) node[scale=.5,fill=white,inner sep = 0, right] { $\left[ \begin{array}{ r r r r} 1 & 0 & 0 &0\\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 1 \\ 0 & 3 & 0 & 0 \end{array}\right]$}; \draw (u4) node[scale=.5,fill=white,inner sep = 0, right] { $\left[ \begin{array}{ r r r r} 1 & 0 & 0 &0\\ 0 & 0 & 1 & 1 \\ 0 & 1 & 0 & 0 \\ 0 & 2 & 0 & 1 \end{array}\right]$}; \draw (u5) node[scale=.5,fill=white,inner sep = 0, right] { $\left[ \begin{array}{ r r r r} 1 & 0 & 0 &0\\ 0 & 0 & 1 & 1 \\ 0 & 1 & 0 & 1 \\ 0 & 2 & 0 & 0 \end{array}\right]$}; \draw (u6) node[scale=.5,fill=white,inner sep = 0, right] { $\left[ \begin{array}{ r r r r} 1 & 0 & 0 &0\\ 0 & 0 & 1 & 1 \\ 0 & 2 & 0 & 0 \\ 0 & 1 & 0 & 1 \end{array}\right]$}; \draw (u7) node[scale=.5,fill=white,inner sep = 0, right] { $\left[ \begin{array}{ r r r r} 1 & 0 & 0 &0\\ 0 & 0 & 1 & 1 \\ 0 & 2 & 0 & 1 \\ 0 & 1 & 0 & 0 \end{array}\right]$}; \draw (u8) node[scale=.5,fill=white,inner sep = 0, right] { $\left[ \begin{array}{ r r r r} 1 & 0 & 0 &0\\ 0 & 0 & 1 & 1 \\ 0 & 3 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right]$}; \draw (u9) node[scale=.5,fill=white,inner sep = 0, right] { $\left[ \begin{array}{ r r r r r} 1 & 0 & 0 & 0 &0\\ 0 & 0 & 0 & 1 & 1 \\ 0 & 1 & 2 & 0 & 1 \end{array}\right]$}; \draw (u10) node[scale=.5,fill=white,inner sep = 0, right] { $\left[ \begin{array}{ r r r r r} 1 & 0 & 0 & 0 &0\\ 0 & 0 & 0 & 1 & 1 \\ 0 & 2 & 1 & 0 & 1 \end{array}\right]$}; \draw (u11) node[scale=.5,fill=white,inner sep = 0, right] { $\left[ \begin{array}{ r r r r r} 1 & 0 & 0 & 0 &0\\ 0 & 0 & 1 & 0 & 1 \\ 0 & 3 & 0 & 1 & 0 \end{array}\right]$}; \draw (u12) node[scale=.5,fill=white,inner sep = 0, right] { $\left[ \begin{array}{ r r r r r} 1 & 0 & 0 & 0 &0\\ 0 & 0 & 1 & 1 & 0 \\ 0 & 3 & 0 & 0 & 1 \end{array}\right]$}; \draw (d1) node[scale=.5,fill=white,inner sep = 0, left] { $\left[ \begin{array}{ r r r r } 1 & 0 & 1& 1\\ 0 & 3 & 0 & 1 \end{array}\right]$}; \draw (d2) node[scale=.5,fill=white,inner sep = 0, left] { $\left[ \begin{array}{ r r r r } 1 & 0 & 0& 0 \\ 0 & 3 & 1 & 2 \end{array}\right]$}; \draw (d3) node[scale=.5,fill=white,inner sep = 0, left] { $\left[ \begin{array}{ r r r } 1 & 0 & 0 \\ 0 & 1 & 1 \\ 3 & 0 & 1 \end{array}\right]$}; \draw (d4) node[scale=.5,fill=white,inner sep = 0, left] { $\left[ \begin{array}{ r r r } 1 & 0 & 0 \\ 0 & 1 & 1 \\ 0 & 3 & 1 \end{array}\right]$}; \draw (d5) node[scale=.5,fill=white,inner sep = 0, left] { $\left[ \begin{array}{ r r r } 1 & 0 & 0 \\ 0 & 0 & 2 \\ 0 & 3 & 1 \end{array}\right]$}; \end{tikzpicture}$$ It is well-known that the faces of the Coxeter complex for the symmetric group are modeled by ordered set partitions of $[n]$. Ordered set partitions are in bijection with contingency tables that have $n$ rows (or by those with $n$ columns). To see the correspondence, we simply record, from left to right in each column, the rows that have nonzero entries (counting from bottom to top). For example, the following array corresponds to the ordered set partition $(\{4,5\},\{3,6\},\{1\},\{2\})$: $$\left[\begin{array}{rrrr} 0 & 1 & 0 & 0 \\ 1& 0 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{array} \right].$$ The dual of the type $A_n$ Coxeter complex is the *permutahedron*, which plays an interesting role in the study of combinatorial Hopf algebras, such as the Malvenuto-Reutenauer algebra and the algebra of quasisymmetric functions. See work of Aguiar and Sottile, for example [@AS]. Suggestively, two-way contingency tables provide an indexing set for a bialgebra known as the set of *matrix quasisymmetric functions*, which contains many well-known combinatorial bialgebras as subalgebras or quotients. See work of Duchamp, Hivert, and Thibon [@DHT Section 5]. It would be interesting to explore whether $\Xi(n)$ might play a role for the matrix quasisymmetric functions similar to the role the permutahedron plays for the Malvenuto-Reutenauer algebra. We finish this article by remarking that refinement ordering on contingency tables makes sense not only for two-way tables. A *$k$-way contingency table* of $n$ objects is an array of nonnegative integers $$A = [ a_{i_1,\ldots,i_k} ],$$ such that the sum of the entries is $n$ and all *marginal sums* $$m_r = \sum_{i_1,\ldots,i_{j-1},i_{j+1},\ldots,i_k} a_{i_1,\ldots,i_{j-1},r,i_{j+1},\ldots,i_k},$$ are positive. In practical terms, a contingency table involves the study of a population according to several criteria that partition the population, say gender versus age versus income. Requiring the marginal sums to be positive means each criterion is satisfied by at least one member of the population. This seems reasonable, for otherwise the criterion gives no information. We can inductively define $k$-way contingency tables for $k>2$ by considering $(k-1)$-way tables whose entries are nonnegative integer vectors of the same size, such that when all the vectors with nonzero entries are put into the columns of an array they form a $2$-way table. Refinement order on $k$-way contingency tables whose entries sum to $n$ has maximal elements given by arrays whose marginal sums all equal to $1$. By induction we see there are $(n!)^{k-1}$ maximal tables. For any $k$, let the set of $k$-way contingency tables whose entries sum to $n$ be denoted by $\Xi(k;n)$. It is not hard to check the partial ordering given by refinement is ranked and boolean, just as in the $2$-way case. (Downward covers are given by adding adjacent entries in some coordinate.) It seems reasonable to expect that we get a shelling order from any linear extension of some sort of natural analogue of two-sided weak order on the facets. If so, refinement ordering on the set of $k$-way contingency tables of $[n]$ defines a thin, shellable simplicial poset and the geometric realization of $\Xi(k;n)$ is a sphere. [99]{} P. Abramenko and K. S. Brown, “Buildings. Theory and applications." Graduate Texts in Mathematics, 248. Springer, New York, 2008. xxii+747 pp. M. Aguiar and F. Sottile, *Structure of the Malvenuto-Reutenauer Hopf algebra of permutations.* Adv. Math. 191 (2005), no. 2, 225–275. S. Billey, M. Konvalinka, T. K. Petersen, W. Slofstra, and B. E. Tenner, *Parabolic double cosets in Coxeter groups*, (2016) preprint. A. Bjorner, *Posets, regular CW complexes and Bruhat order.* European J. Combin. 5 (1984), no. 1, 7–16. A. Björner, *Some combinatorial and algebraic properties of Coxeter complexes and Tits buildings.* Adv. in Math. 52 (1984), no. 3, 173–212. A. Björner and F. Brenti, “Combinatorics of Coxeter groups." Graduate Texts in Mathematics, 231. Springer, New York, 2005. xiv+363 pp. N. Bourbaki, “Lie groups and Lie algebras. Chapters 4Ð6." Translated from the 1968 French original by Andrew Pressley. Elements of Mathematics (Berlin). Springer-Verlag, Berlin, 2002. xii+300 pp. L. Carlitz, D. P. Roselle, and R. A. Scoville, *Permutations and sequences with repetitions by number of increases.* J. Combinatorial Theory 1 1966 350–374. P. Diaconis and A. Gangolli, *Rectangular arrays with fixed margins.* Discrete probability and algorithms (Minneapolis, MN, 1993), 15–41, IMA Vol. Math. Appl., 72, Springer, New York, 1995. G. Duchamp, F. Hivert, J.-Y. Thibon, *Noncommutative symmetric functions. VI. Free quasi-symmetric functions and related algebras.* Internat. J. Algebra Comput. 12 (2002), no. 5, 671–717. J. Humphreys, *Reflection groups and Coxeter groups.* Cambridge Studies in Advanced Mathematics, 29. Cambridge University Press, Cambridge, 1990. xii+204 pp. Z. Lin, *Proof of Gessel’s $\gamma$-positivity conjecture*, (2015) preprint. T. K. Petersen, *Two-sided Eulerian numbers via balls in boxes.* Math. Mag. 86 (2013), no. 3, 159–176. R. P. Stanley, “Combinatorics and commutative algebra." Second edition. Progress in Mathematics, 41. BirkhŠuser Boston, Inc., Boston, MA, 1996. x+164 pp. R. P. Stanley, “Enumerative combinatorics. Vol. 2." With a foreword by Gian-Carlo Rota and appendix 1 by Sergey Fomin. Cambridge Studies in Advanced Mathematics, 62. Cambridge University Press, Cambridge, 1999. xii+581 pp. J. R. Stembridge, *Tight quotients and double quotients in the Bruhat order.* Electron. J. Combin. 11 (2004/06), no. 2, Research Paper 14, 41 pp. J. R. Stembridge, John Stembridge’s Maple packages for symmetric functions, posets, root systems, and finite Coxeter groups, `http://www.math.lsa.umich.edu/~jrs/maple.html` M. Visontai, *Some remarks on the joint distribution of descents and inverse descents.* Electron. J. Combin. 20 (2013), no. 1, Paper 52, 12 pp.

--- author: - 'P. Ranalli' - 'A. Comastri' - 'G. Zamorani' - 'N. Cappelluti' - 'F. Civano' - 'I. Georgantopoulos' - 'R. Gilli' - 'E. Schinnerer' - 'V. Smol[č]{}i[ć]{}' - 'C. Vignali' bibliography: - '../fullbiblio.bib' date: 'Received 2011-12-22; accepted 2012-4-20' title: | X-ray properties of radio-selected star forming galaxies\ in the [[*Chandra*]{}]{}-COSMOS survey --- Introduction ============ Radio and far-infrared observations have been widely accepted as unbiased estimators of star formation (SF) in spiral galaxies for decades [see the @cond92; @kenn98 reviews]. The X-ray domain has also been recognized as a SF tracer in non-active galaxies (hereafter just “galaxies”) thanks to a number of works highlighting the presence of X-ray vs. radio/infrared correlations (@djf92 [@grimm02]; @rcs03, hereafter RCS03; @gilfanov04a). Strong absorption (i.e. with column densities $\gtrsim 10^{22}$ cm$^{-2}$) is also rare among galaxies, making the X-ray domain scarcely sensitive to extinction. Thus, an X-ray based Star Formation Rate (SFR) indicator can be considered not biased by absorption (RCS03). An interpretation framework, whose main idea is the dominance of High-Mass X-ray Binaries among the contributors to the X-ray luminosity of galaxies, has also been developed [@gilfanov04b; @pr07] and is currently the subject of further investigation. The observations of deep fields, especially with [[*Chandra*]{}]{}, have prompted the search for galaxies at high redshifts [@alexander02; @bauer02; @hornsch03; @ranalli03AN]. The galaxies X-ray luminosity function and its evolution has been investigated both in the local universe and at high redshift (@ioannis99 [@norman04; @georgantopoulos05]; @rcs05 [hereafter RCS05]; @georgakakis07 [@ptak07; @lehmer08]) with also the goals of obtaining an absorption-free estimate of the cosmic star formation history, and deriving the contribution by galaxies to the X-ray background. However, any work involving galaxies in X-ray surveys has to deal with the fundamental fact that AGN are preferentially selected in flux-limited X-ray surveys. A careful and efficient classification of the detected objects is necessary to identify the galaxies among the dominant AGN population. In early studies [@maccacaro88] AGN were found to populate a well defined region of the X-ray/optical vs. X-ray flux plane, bounded by an X-ray/optical flux ratio $X/O=-1$ (see definition in Sect. \[sec:X/O\]). This threshold has been often adopted as an approximate line dividing AGN and X-ray emitting galaxies. A more robust separation between AGN and star forming galaxies is obtained [@xue-cdfs4Ms; @vattakunnel12] by considering several different criteria (X-ray hardness ratio, X-ray luminosity, optical spectroscopy, X-ray/infrared or X-ray/radio flux ratios). An analysis of the relative merits of the different criteria when taken separately, and of the most effective trade-offs to identify star-forming galaxies is one of the aims of this paper. A similar need of a careful object classification has arisen in deep radio observations. It has been known since long that a population of faint radio sources associated with faint blue galaxies was emerging at radio fluxes below $\sim 1$ mJy [@windhorst85; @fomalont91; @vla98; @vla00]. In recent years, it has been shown that a sizable fraction (about 50%) of this sub-mJy population is actually made up of AGN [@gruppioni99; @ciliegi03; @seymour08; @smolcic08; @vla-cdfs-padovani09; @strazzullo10]. This means that an accurate screening is needed also for radio-selected faint galaxies. This screening has been the subject of the work by @smolcic08 [hereafter S08], who made use of the extensive data sets of the COSMOS survey to analyze $\sim 2400$ sub-mJy radio sources and classified them according to a newly developed, photometry-based method to separate SF galaxies and AGN. Their method is based on optical rest-frame synthetic colours, which are the result of a principal component analysis of many combinations of narrow-band colours, and which correlate with the position of the objects in the classical BPT diagram (@baldwin81; see Sect. \[sec:selezione\] for details). Here we build on this work, and use the S08 samples as the starting point for our classification of the X-ray galaxies in COSMOS. We intend to test the X-ray based selection criteria against the S08 method, and eventually refine the selection. The Cosmological Evolution Survey (COSMOS) is an all-wavelength survey, from radio to X-ray, designed to probe the formation and evolution of astronomical objects as a function of cosmic time and large scale structure environment in a field of 2 deg$^2$ area [@cosmos-overview]. In this paper, we build mainly on the radio (VLA-COSMOS, @schinnerer07), X-ray ([[*Chandra*]{}]{}-COSMOS, or C-COSMOS, @ccosmos-cat), and optical spectroscopic (Z-COSMOS, @zcosmos) observations. The radio data were taken at 1.4 GHz and have a RMS noise 7–10 $\mu$Jy (with the faintest sources discussed in this paper having fluxes around 60$\mu$Jy), while the X-ray data have a flux limit of $1.9{\cdot 10^{-16}}$ [erg s$^{-1}$ cm$^{-2}$]{} in the 0.5-—2 keV band. The area considered here is that covered by [[*Chandra*]{}]{}, which is a fraction (0.9 deg$^2$) of the whole COSMOS field. The [XMM-[*Newton*]{}]{} observations (XMM-COSMOS, @xmm-cosmos) covered the whole field but with a brighter flux limit ($1.7{\cdot 10^{-15}}$ [erg s$^{-1}$ cm$^{-2}$]{} in the 0.5–2 keV band). We focus on C-COSMOS here because its combination of area and limiting flux offers the best trade off for the subject of our study. The structure of this paper is as follows. In Sect. \[sec:selezione\] we define a sample of galaxies based on radio and optical selection, and subsequent X-ray detection. An AGN sample is selected with the same method to allow for comparisons. In Sect. \[sec:optical-radio\] we characterize the sample, in terms of magnitudes, redshifts, and optical spectra. In Sect. \[sec:characterization\] we review some commonly used X-ray-based indicators of star formation vs. AGN activity, and test them on our sample of galaxies; on this basis, a refined sample is then defined. In Sect. \[sec:stacking\] we investigate the average properties of galaxies with the same radio-optical selection but without a detection by [[*Chandra*]{}]{}. In Sect. \[sec:misclassif\] we discuss the number of composite and mis-classified sources. In Sect. \[sec:flussi\] we consider if the COSMOS data can further constrain the radio/X-ray correlation. Sect. \[sec:lognlogs\] is devoted to an analysis of the X-ray number counts of the radio-selected COSMOS galaxies. Finally, in Sect. \[sec:conclusions\] we review our conclusions. The cosmological parameters assumed in this paper are $H_0=70$ km s Mpc$^{-1}$, $\Omega_\Lambda=0.7$ and $\Omega_{\mathrm M}=0.3$. Selection criteria {#sec:selezione} ================== The catalogue of the COSMOS radio sources was published in @schinnerer07; the objects in this catalogue were then classified by S08, on the basis of their photometry-based method. Only objects with redshift $\le 1.2$ were considered by S08, because the errors on the classification would be larger beyond this threshold. AGN can be broadly divided in two classes: objects where the AGN dominates the entire Spectral Energy Distribution (SED), i.e. mainly QSO, and objects where it does not, such as type-II QSO, low luminosity AGN (Seyfert and LINER galaxies), and absorption-line AGN. The fraction of type-I QSO in the VLA-COSMOS catalogue is small ($\sim 5\%$). They have higher X-ray luminosities than the SF and the other kinds of AGN, making the few type-I objects easier to separate from other classes of objects[^1]; since we are mainly interested in the properties of galaxies, we will not discuss them further. Hereafter with the term “AGN” we will only refer to the other kind of AGN (type-II and low-luminosity). These AGN have broad-band properties similar to those of SF galaxies. The main tools to disentangle SF galaxies and low-luminosity AGN are spectroscopic diagnostic diagrams [BPT diagrams @baldwin81; @vo87; @kewley], which rely on the \[ 5007\]/H$\beta$ and \[ 6584\]/H$\alpha$ line ratios. Their main drawback is, however, the long telescope time needed to obtain good-quality spectra. Alternative methods which only build on photometric data can therefore be useful. Based on the observation of a tight correlation between rest-frame colours of emission-line galaxies and their position in the BPT diagram, @smolcic06 used the Sloan Digital Sky Survey (SDSS) photometry (a modified Strömgren system) to i) calculate rest-frame colours, and ii) use the Principal Component Analysis (PCA) technique to identify, among all linear combinations of colours, those which correlate best with the position in the BPT diagram. One of these combinations, named $P1$, was found to correlate strong enough with the emission line properties of SF galaxies and AGN, to be used alone for the classification. By applying this method to the COSMOS multi-band photometric data, S08 produced a list of 340 ‘star forming’ (hereafter SF) and 601 type II/dusty/low luminosity AGN candidates. The [[*Chandra*]{}]{} field, which is smaller than the radio-surveyed area, contains 242 SF and 398 AGN. Some mis-classifications are inherent in any colour- or line-based methods, and a fraction of objects may also exhibit composite or intermediate properties. Thus, S08 estimate that SF samples actually may contain $\sim 20\%$ of AGN and $\sim 10\%$ of composite objects. Conversely, AGN samples contain $\sim 5\%$ SF and $\sim 15\%$ composite. We matched the radio positions of the SF and AGN sources with those of the C-COSMOS catalogue [@ccosmos-cat; @puccetti09]. In Fig. \[fig:crosscorr\] we show the number of matched SF sources for different matching radii: the number of matches rises steeply from $0.1\arcsec$ to $0.5\arcsec$, and flattens for larger radii. To adopt a threshold for the maximum separation between the radio and X-ray coordinates, we considered that in the C-COSMOS survey some areas have been observed by [[*Chandra*]{}]{} only at large off-axis angles. For these areas, the point spread function (PSF) is much broader than the on-axis value ($0.5\arcsec$ FWHM), and this can also introduce errors in the determination of the source position. The position errors reported in the C-COSMOS catalogue are in fact larger than $1\arcsec$ for 221 sources out of 1761. Thus we considered all matches within $3\arcsec$, and visually inspected every match to check that the X-ray PSFs and the errors on the X-ray positions were wide enough to justify the larger threshold. Following this criterion, one match was excluded because the PSF in that position was much narrower than the distance between the radio and X-ray positions. ![Number of matched sources in cross-correlating the C-COSMOS catalogue [@ccosmos-cat] with the S08 sample of radio-selected star forming galaxies.[]{data-label="fig:crosscorr"}](crosscorr_result.ps){width=".9\columnwidth"} The samples of radio selected, optically classified, X-ray detected sources consist thus of 33 SF ($\sim 14\%$ of the SF-classified radio sources) and 82 AGN-type objects ($\sim 21\%$ of the AGN-classified radio sources). The breakdown of the sources according to their detection in the 0.5–2 keV, 2–7 keV and 0.5–7 keV bands is shown in Table \[tab:num\_src\]. Note the presence of 10 SF candidates lacking a detection in the soft band: this hints for the presence of AGN-type objects in the SF sample, which will be discussed in detail in the next sections. Band SF AGN ------- ---- ----- F+S+H 10 46 F+S 11 16 F+H 10 18 F — 1 S 2 1 H — — : Number of X-ray detected radio sources, according to their optical classification and X-ray band of detection (F: full, 0.5–7 keV; S: soft, 0.5–2 keV; H: hard, 2-7 keV).[]{data-label="tab:num_src"} Optical and radio properties {#sec:optical-radio} ============================ The samples of X-ray detected vs. undetected sources exhibit different properties, as shown both in the following tests and in the cumulative distributions in Fig. \[fig:samplehistograms\]. For the SF sample we find that: - detected sources have brighter (observer-frame) R magnitudes than undetected ones, at a confidence level of 99.96% (according to a Wilcoxon-Mann-Whitney test; the median magnitudes are $R=20.53$ and 21.66 for the detected and undetected, respectively); - detected sources have brighter radio fluxes than undetected, at a confidence level of 98.5% (median fluxes: ${{S_{1.4 \mathrm GHz}}}=0.148$ and 0.124 mJy, respectively); - detected sources have lower redshifts than undetected, at a confidence level of 99.8% (median redshifts: $z=0.36$ and 0.61, respectively); - detected sources have lower radio luminosities than undetected, at a confidence level of 97.8% (median luminosities: $S=8.7{\cdot 10^{29}}$ and $5.0{\cdot 10^{30}}$ [erg s$^{-1}$ Hz$^{-1}$]{}, respectively). This is in line with the expectation of the detected sources being closer to us, and the undetected ones probing a larger volume where more luminous yet less common objects can be found. The X-ray detected AGN sources also have brighter R magnitudes and radio fluxes than the undetected ones, but do not show any significant difference in the redshift distribution. The behaviour of the radio luminosity is reversed: the undetected AGN have lower radio luminosities than the detected ones. Optical spectra of X-ray detected sources {#sec:spectra .unnumbered} ----------------------------------------- Optical spectra are available for most of the sources with an X-ray detection from several spectroscopic campaigns: the ZCOSMOS project [@zcosmos; @zcosmos10k], Magellan/IMACS surveys, the Sloan Digital Sky Survey (SDSS), [@trump07; @trump09] and deeper observations[^2] with Keck/DEIMOS and VIMOS/VLT. A simple classification based on diagnostic diagrams [@bongiorno10], further checked by visual inspection, has been used to the determine the optical classifications [@civano12]. Since the signal/noise ratio varies a lot in the sample, some objects have noisy spectra which can only be classified tentatively. In the SF sample, 21 objects (out of 33) are classified as emission line galaxies; 9 are classified as AGN, and 3 have no spectral information or have spectra with low signal/noise ratios. In the AGN sample, 25 objects (out of 82) are classified as AGN; 30 as emission line galaxies; 7 as absorption line galaxies, and 20 have no spectral information or spectra with low signal/noise ratio. This partial overlap in the optical classification between the two catalogues is expected (see also Sect. \[sec:selezione\]), because [*i)*]{} the two phenomena of accretion and star formation are often present together in the same object, [*ii)*]{} the overlap of the areas covered by different populations (SF and AGN) in the diagnostic diagrams used by S08, [*iii)*]{} low-luminosity, narrow-line AGN and actively star forming galaxies can be difficult to distinguish in noisy spectra. The last point is particularly true at $z\gtrsim 0.4$, where the H$\alpha$ line is not sampled by optical spectra and, therefore, the standard BPT diagram (\[O <span style="font-variant:small-caps;">III</span>\]/H$\beta$ vs. \[H <span style="font-variant:small-caps;">II</span>\]/H$\alpha$; @baldwin81) cannot be used in the optical spectral classification. A comparison of the observed fraction of mis-classifications and composite objects with the expectations will be presented in Sect. \[sec:misclassif\]. In the following Section, we will try and characterize further the two samples on the basis of the X-ray properties of the sources, with the aim of improving the classification. X-ray characterization of the selected sources {#sec:characterization} ============================================== X-ray spectra of SF galaxies are rather complex, as they include emission from hot gas, supernova remnants (thermal spectra) and X-ray binaries (non-thermal, power-law spectrum), with the thermal components being usually softer than the non-thermal ones. A detailed description of the expected spectra of the different components may be found in @pr02. The relative importance of the spectral components may vary; however, in most cases the average flux ratio between the 0.5–2.0 keV and the 2.0–10 keV bands is the same that would be obtained if the spectrum was a power-law spectrum with spectral index $\Gamma=2.1$ and negligible absorption (RCS03; @lehmer08). This does not imply the lack of X-ray absorption of X-rays in SF galaxies: M82 and NGC3256 are notable examples (RCS03; @m82centok). However, the spectral analysis of 23 SF galaxies in RCS03 did not find heavy absorption to be a general property of that sample. The fluxes of candidate SF objects used in this paper have been recomputed from the counts by assuming the $\Gamma=2.1$ spectrum, instead of the $\Gamma=1.4$ used in @ccosmos-cat. Conversely, for AGN we have used the latter, harder spectrum. In the following, we review a few common indicators of SF vs. AGN activity, and apply them to the SF sample for further screening. Hardness ratio {#sec:HR} -------------- {width=".995\columnwidth"} {width=".995\columnwidth"} The hardness ratio (HR) is a simple measure of the spectral shape, defined as $HR=(H-S)/(H+S)$, where $H$ and $S$ are the net (i.e.background-subtracted) counts in the hard and soft bands respectively. It is most useful when the sources are too faint for a proper spectral analysis. The error on the HR is usually calculated by taking the uncertainties on source and background counts according to the @gehrels86 approximation, and applying the error propagation for Gaussian distributions. However, it has been shown that for faint sources this approach largely overestimates the errors on the HR [@behr]. To obtain a more realistic estimate of the HR uncertainties, we use the Bayesian approach described in @behr. The source and local background counts have been extracted from the [[*Chandra*]{}]{} event files in soft (0.5–2 keV) and hard (2–7 keV) observer-frame energy bands. For the sources in the SF and AGN samples we show in Fig. \[fig:behr\] the median value of the HR posterior distributions, and the 68% and 90% highest probability density (HPD) intervals of the posterior distributions, vs. the redshifts. The HR corresponding to eight model spectra (absorbed power-laws) are also superimposed.These model HR have been calculated with XSPEC, using average effective areas (see Sect. \[sec:stacking\]) and assuming no background. About half of the SF sources detected in C-COSMOS have a HR which is consistent with very flat or absorbed spectra ($N_{\rm H}\gtrsim 10^{21.5}$ cm$^{-2}$, or equivalently $\Gamma\lesssim 1.2$ without absorption). Conversely, about half of the AGN sources have a HR consistent with spectra steeper or less absorbed than the above thresholds. The average hardness ratio of galaxies is the one given by a non-absorbed $\Gamma=2.1$ spectrum (RCS03). Harder spectra are sometimes indeed found in galaxies (e.g., see the slopes for single power-law fits in @dahlem98) where they can result from bright X-ray binaries [@pr02]. However, @swartz04 found that the average slopes of X-ray binaries in nearby galaxies are $\Gamma=1.88\pm 0.06$ ($1.97\pm 0.11$) for binaries with luminosity larger (smaller) than $10^{39}$ [erg s$^{-1}$]{}, respectively. Only 7% (8%) of the binaries studied by @swartz04 have slopes $\Gamma\le 1.4$. The large number of hard objects in the SF sample is probably due to a selection effect. Because of the C-COSMOS flux limit, the [[*Chandra*]{}]{}-detected SF sample only contains the brightest 14% of all the SF objects in the field of view. In addition, 30% of the S08 SF sources are expected to be composite or misclassified, as is inherent in any selection method based on diagnostic diagrams. Since AGN are on average brighter than galaxies, the composite/misclassified objects should mingle with the brightest galaxies, hence with the [[*Chandra*]{}]{}-detected sample rather than with the [[*Chandra*]{}]{}-undetected one. X-ray/optical flux ratio {#sec:X/O} ------------------------ A fast and widely used, yet coarse method to classify X-ray objects is to look at their X-ray/optical flux ratio $X/O$, defined as $$\label{eq:rapp_xottico} X/O = {\mbox{Log}}\left( F_{\rm X} \right) + 0.4 R + 5.71$$ where $F_{\rm X}$ is the 0.5–2 keV flux, and $R$ is the optical apparent magnitude in the R filter. On average, AGN have higher $X/O$ values than SF galaxies. Given the intrinsic dispersion in the $X/O$ values for both AGN and SF galaxies, no single $X/O$ value can be used to unambiguously separate AGN from SF galaxies. However, it has been shown [@schmidt98; @bauer04; @alexander02] that the value $X/O=-1$ can be taken as a rough boundary between objects powered by star formation ($X/O<-1$) and by nuclear activity ($X/O>-1$). In the SF sample, 22 objects have $X/O<-1$ (2/3 of the total sample), while 11 have $X/O>-1$. Out of this latter 11 objects, which this criterion would classify as AGN, 10 have a HR whose 68% HPD interval is consistent with a column density $N_{\rm H}\gtrsim 10^{21.5}$ cm$^{-2}$. For comparison, the AGN sample has only 1/3 of the objects with $X/O<-1$ (27 objects out of 82). X-ray luminosity {#sec:lum42} ---------------- An X-ray luminosity of $10^{42}$ erg/s is also often used as another rough boundary between SF galaxies and AGN, with the band in which the luminosity is measured varying among different authors. While this criterion is, on its own, so unrefined that it ignores the existence of low luminosity AGN, it may still be of use when considered along with other criteria. In the SF sample, 17 objects have a 2–10 keV luminosity greater than this limit. However, it is expected that the X-ray luminosity of SF galaxies evolves with the redshift; RCS05 found that a pure luminosity evolution of the form $L_{\rm X} \propto (1+z)^{2.7}$ is a good description of the available data [see also @norman04]. Thus one should consider as AGN candidates only the objects with $L_{\rm X}> 10^{42}(1+z)^{2.7}$ [erg s$^{-1}$]{}: 12 objects in the SF sample satisfy this criterion. All these 12 objects have a HR compatible with a column density larger than $10^{21.5}$ cm$^{-2}$. Off-nuclear sources {#sec:offnuclear} ------------------- If the X-ray position is not coincident with the galaxy centre, but is still within the area covered by the galaxy in the optical band, then the X-ray emission is probably due to an off-nuclear X-ray binary. Thus any contribution from nuclear accretion is unlikely to be significant. An off-nuclear flag is present in the catalogue of optical identifications (@civano12; see also @mainieri10). The only SF source classified as off-nuclear is CXOCJ100058.6+021139. No source from the AGN sample is classified as off-nuclear. While in principle this is an important selection criterion, in practice it applies to only one object in our samples, and therefore we will not consider it any further. Classification and catalogue of sources {#sec:cat} --------------------------------------- From the considerations made above, it seems that only about half of the objects in the SF sample have properties compatible with SF-powered X-ray emission. This hints to the presence of several objects in the sample which show intermediate, rather than SF or AGN properties. Some refinement of the S08 SF selection criteria seems thus possible by inspecting the X-ray properties of the sources. We consider the following conditions as indicators of SF origin of the X-ray luminosity: - $L_{2-10}\le 10^{42}(1+z)^{2.7}$ [erg s$^{-1}$]{}, where $L_{2-10}$ is the hard X-ray (2.0-10 keV) luminosity; - HR lower (softer) than what expected for an absorbed power-law with $\Gamma=1.4$ and $N_\mathrm{H}= 10^{22}$ cm$^{-2}$; - $X/O\le -1$; - classification as galaxy from optical spectroscopy. For the purposes of classification, we only consider optical spectra with clear AGN-like emission line ratios as non-matching. This avoids that low signal/noise spectra can influence the classification. Among many possibilities to combine to above criteria, we use the following method: the number of matched criteria is counted, and an object is classified accordingly. An object is assigned to class 1, if it fulfils all the conditions; to class 2, if it fulfils all conditions but one; to class 3, if there are at least two conditions not satisfied. Objects for which one condition can not be checked (e.g., a missing optical spectrum), are classified as if that condition had been matched. The idea is that an object status is affected only by conditions which have been checked and not matched. Thus we recognize two samples of SF galaxies: one more conservative (class 1 objects), and another less conservative (objects in classes 1 or 2). Class 3 objects probably do not have the majority of their X-ray emission powered by star formation related processes. There are 8 class-1 objects; 8 class-2; and 17 class-3 objects in the SF sample. If the same selection is applied to the AGN sample, we find 14 class-1; 6 class-2; and 62 class-3. The classifications for both the SF and AGN sample are reported in Tables \[tab:cat\] and \[tab:catAGN\], along with the other parameters of interest: X-ray fluxes, luminosities, medians of the HR posterior probability distributions, $X/O$ ratios, X-ray/radio ratios (see Sect. \[sec:flussi\]), classification from optical spectroscopy. Average properties of undetected objects {#sec:stacking} ======================================== -------------------------------------------------------- ------------ ------------ ---------- ------------------- ---------- ------------ ----------- ------------ -------------------- ----------------- ---------- Selection No. of Avg. radio Avg. Band Exposure Net X-ray Lumi- $1-p_{\rm detect}$ $1-p_{\rm cts}$ $\Gamma$ candidates flux (mJy) redshift (Ms) counts Flux nosity ${{S_{1.4 \mathrm GHz}}}\le 0.20$ 156 0.113 0.67 soft 16.9 $176\pm20$ 4.8 0.94 $99.99\%$ $99.99\%$ \[1.5– “ & ” “ & ” hard 17.7 $105\pm25$ 4.8 0.94 $99.98\%$ $>99.99\%$ 2.5\] $0.20<{{S_{1.4 \mathrm GHz}}}\le 0.63\!\!\!\!\!\!\!\!$ 43 0.292 0.58 soft 4.3 $80\pm11$ 8.2 1.1 $99.96\%$ $99.92\%$ \[1.7– “ & ” “ & ” hard 4.5 $16^{+4.7}_{-10}$ 8.2 1.1 — — 3.8\] -------------------------------------------------------- ------------ ------------ ---------- ------------------- ---------- ------------ ----------- ------------ -------------------- ----------------- ---------- The method of stacking analysis allows to determine the average properties of objects which are not individually detected; it can be briefly described as follows. Candidate objects for stacking are selected from the list of SF sources in S08 which are not detected in C-COSMOS, with the additional criterion that no detected C-COSMOS source should be present within 7 arcsec from the position of the candidate. The reason is to avoid contamination from X-ray brighter sources. This does not introduce any bias in the selection of sources, because very few sources are excluded in this way (only 6 out of 209). Because of the radio–X-ray correlation, it is expected that the average X-ray properties are dominated by the brightest radio sources. Thus it may be advisable to include in the stack only sources with a narrow spread in their radio flux, to avoid biases due to the brightest sources. We split the sample on the basis of the radio flux, dividing the candidates in two lists as follows: 1. 156 SF sources with ${{S_{1.4 \mathrm GHz}}}\le 0.20 $mJy; 2. 43 SF sources with $0.20< {{S_{1.4 \mathrm GHz}}}\le 0.63$ mJy. Each sub-sample is 0.5 dex wide in flux; one starts from the lowest radio flux, while the other one follows continuously. For each list of candidates, postage-stamp size images measuring $20\times 20$ pixels (each one $0.491\arcsec$ wide) around the position of every candidate, are extracted and summed; the latter sum is hereafter called “stacked image”. If most candidates contribute some X-ray photons, then a “stacked source” appears on top of the background in the centre of the stacked image (similar results can be obtained using the software described in @miyaji_stack). The [wavdetect]{} tool is then used to determine the net counts of the stacked source. This analysis was done for both the 0.5–2.0 keV and 2.0–7.0 keV bands. The stacked source was successfully detected by [wavdetect]{} for the low-radio flux sub-sample in both bands, and for the high-radio flux in the soft band only. This is in line with the expectation that a lower number of counts should be present in the high- than in the low-radio-flux subsample: given the average fluxes and the number of positions (Table \[tab:stacking\]), $\sim 40\%$ more counts are expected in the latter than in the former. The significance of the detection of a stacked source is best estimated with simulations: we draw as many random positions as the number of sources in the list, reproducing the same spatial distribution of the VLA-COSMOS sources, and build a stacked image from these positions. Total counts ($c_{\rm sim}$) within a radius of 3.5 from the centre are extracted; the [wavdetect]{} software is run on the stacked image; this cycle is repeated 10000 times for each sub-sample and each band. We then define the following $p$-values: - $p_{\rm detect}$ as the fraction of times that the [wavdetect]{} software finds a source within 1.1 arcsec of the centre of the stacked image; - $p_{\rm cts}$ as the fraction of times that $c_{\rm sim} \ge c_{\rm stack}$, where $c_{\rm stack}$ are the total counts of the ‘real’ stacked source. We identify the $p$-values as two estimates of the probability that the stacked source was actually a background fluctuation. The $p$-values are shown in Table \[tab:stacking\] (actually, $1-p$ is shown, i.e. the probability that the source is not a fluctuation). Using the source and background regions defined above, and a power-law average spectrum with $\Gamma=2.1$, we extracted the net counts and derived the fluxes and luminosities shown in Table \[tab:stacking\]. Stacked X-ray spectra have also been extracted for the two subsamples, using CIAO 4.0. Background spectra have been extracted around the source positions, by taking 4 circular background regions for each source, each background region having the same radius of the source region, and being placed $10\arcsec$ east, north, west, or south of the source. This ensures that the background is the most accurate, given the actual sky positions of the sources. Then, we removed background positions which fell within $7\arcsec$ from any [[*Chandra*]{}]{}-detected source. Response matrices have been calculated considering the stacked source like it was an extended source consisting of many small pieces scattered around the detector area, weighted by the photons actually present in each position. ![Confidence contours for relevant parameters of simple models of stacked spectra of SF sources and (for comparison) AGN sources not detected by [[*Chandra*]{}]{}  (sources with radio flux $F<0.20$ mJy). The contours are shown at levels of $\Delta \mathrm{C}$: +2.92, +5.63, +8.34 above minima. Solid lines: SF sources; dotted lines: AGN.[]{data-label="fig:stackedspectra"}](contorni.ps){width=".49\textwidth"} The stacked spectra were fitted with a model which is the weighted sum of many absorbed power-laws; the number of power-law components is the number of sources in the stack. Each absorbed power-law is redshifted to the $z$ of the corresponding source. The slope and absorption are free parameters, but are assumed to be the same for all sources. The weights are proportional to the radio fluxes. Finally, observer-frame absorption due to the Galaxy is added. This method allows to fully account for the redshift distribution of the sources. We used XSPEC version 11.3.2 for this analysis. The confidence contours for the parameters are shown in Fig. \[fig:stackedspectra\] (solid curves) for the bin with sources with ${{S_{1.4 \mathrm GHz}}}\le 0.2$ mJy. The spectrum is consistent with moderately steep photon indices ($1.5\lesssim \Gamma \lesssim 2.5$). This behaviour is expected for star forming galaxies (see Sect. \[sec:characterization\]). The bin with sources with $0.2<{{S_{1.4 \mathrm GHz}}}\le 0.63$ mJy (not shown) has a lower number of X-ray photons in the spectrum and thus a wider range of slopes allowed, yet the spectrum is still consistent with the other bin. ![Radio vs. X-ray (0.5–2 keV) fluxes for the VLA-COSMOS SF sources detected in C-COSMOS. Detected sources are marked with squares and error bars. The black squares with an attached arrow mark the sources which are not detected in the band to which the panel refers, but are detected in any other of the C-COSMOS bands. Conversely, the grey upper limits show the sources without a detection in any X-ray band, hence without an entry in the C-COSMOS catalogue. The solid line shows the RCS03 relationship, K-corrected to the average redshift of the detected sources ($z=0.46$), while the dashed lines show the 1$\times$ and 3$\times$ standard deviation of the relationship. []{data-label="fig:radiox"}](radiox-soft.ps){width="\columnwidth"} For comparison, in Fig. \[fig:stackedspectra\] (dotted curves) we show also the confidence contours for stacked spectra of 207 AGN, selected from the S08 sample in the same radio flux intervals of the SF galaxies, and also not detected by [[*Chandra*]{}]{}. The average redshifts of these AGN are not significantly different from the SF ones. The AGN X-ray spectra are flatter; this is probably the result of absorption and of redshift (summing absorbed spectra of sources at different redshift leads to flat or inverted spectra, like in the case of the cosmic X-ray background). While a single power-law model is probably simplistic, a more detailed modeling would be beyond the scope of this paper. The average X-ray fluxes for the AGN, calculated with the best-fit slopes, are $7.2{\cdot 10^{-18}}$ ($4.8{\cdot 10^{-17}}$) [erg s$^{-1}$ cm$^{-2}$]{} for the bin with ${{S_{1.4 \mathrm GHz}}}\le 0.2$ mJy and $3.0{\cdot 10^{-18}}$ ($1.0{\cdot 10^{-17}}$) [erg s$^{-1}$ cm$^{-2}$]{} for the bin with $0.2<{{S_{1.4 \mathrm GHz}}}\le 0.63$ mJy in the 0.5–2 (2–10) keV band. The allowed ranges for the AGN spectral slopes are \[0.4–0.9\] and \[0.6–2.2\] for the ${{S_{1.4 \mathrm GHz}}}\le 0.2$ mJy and $0.2<{{S_{1.4 \mathrm GHz}}}\le 0.63$ mJy bins, respectively. Mis-classified and composite objects {#sec:misclassif} ==================================== The fractions of X-ray detected objects whose classification based on optical spectral line ratios (where available) is different from that based on the synthetic $P1$ colour in S08 are $9/33\sim 27\%$ (SF sample) and $37/82\sim 45\%$ (AGN sample). These numbers are of the same magnitude of those quoted in Sects. \[sec:HR\],\[sec:X/O\], \[sec:lum42\], though it is important to stress that different criteria yield different mis-classified and composite (hereafter MCC) objects. These fractions should be compared with the number of class 3 objects in the SF sample ($17/33\sim 51\%$) and of classes 1-2 in the AGN sample ($20/82\sim 24\%$). However, these fractions do not take into account the large number of X-ray undetected objects and should probably be considered as upper limits to the true fractions of MCC objects. In fact, the stark difference found between the average X-ray spectra of undetected SF and AGN sources (Sect.\[sec:stacking\]) would rather suggest much lower fractions of MCC. A lower limit to the true fractions of MCC can be derived by assuming that the MCC are only present among the X-ray detected sources. In this case, the fractions would be $17/242\sim 7\%$ ($20/398\sim 5\%$) for the class 3 (classes 1-2) in the SF (AGN) sample. The rationale for this assumption would be, for the SF sample, that galaxies with an AGN contribution are on average brighter in X-rays than the galaxies without and thus are more likely to be X-ray detected. For the AGN sample, it would be that intense star formation could cause X-ray emission at a level similar to that of a low-luminosity or absorbed active nucleus. The fractions reported by S08 (30% of MCC in the SF and 20% in the AGN sample; see Sect. \[sec:selezione\]) are intermediate between our upper and lower limits’ estimates, and therefore we regard them as in agreement with our findings. In the following, we use the method described in Sect. \[sec:cat\] to identify the MCC candidates. The same method cannot be applied to X-ray undetected objects, and therefore the optical colour-based classification is used for them in the remaining of this paper. The X-ray/radio flux ratio {#sec:flussi} ========================== Correlations between X-ray and radio luminosities of star forming galaxies, and between the X-ray and far infrared (FIR) ones, are well established for the local universe and have been tested for objects up to $z\sim 1$ (@bauer02; RCS03; @gilfanov04a [@pr07]). Both the radio and FIR luminosity are tracers of the star formation rate (SFR). These correlations are linear, and imply that in absence of any contribution from an AGN, the X-ray emission is powered by star-formation related processes. High-Mass X-ray Binaries (HMXB) seem to have the same luminosity function in all galaxies, only normalised according to the actual SFR. Thus, if HMXB are the dominant contributors to the X-ray emission, the X-ray luminosity is a tracer of the SFR [@grimm02; @gilfanov04b; @pr07]. The other possibly dominant contributors to the X-ray emission are the Low-Mass X-ray Binaries (LMXB), whose number scales with the galaxy stellar mass. The actual balance of the two populations thus depends on the ratio between SFR and mass, thus it is possible that there is some evolution of the correlation due to the mass build-up by the galaxies with the cosmic time. The VLA- and C-COSMOS surveys contain a sizable number of objects at medium-deep redshifts on which the correlation might be tested. However, many objects have radio fluxes close to the VLA-COSMOS flux limit. This is clearly visible in Fig. \[fig:radiox\], where we show the radio and X-ray fluxes and upper limits for all the SF galaxies in the C-COSMOS field. Since we account for the X-ray upper-limits, the main source of potential bias is the radio flux limit; the results may therefore be biased towards radio-brighter-than-X-rays objects. To partially mitigate this effect, we split the sample in two redshift bins, defined as follows. The knee of the radio LF of galaxies is found at a luminosity $L_{\mathrm k}\sim 1.5{\cdot 10^{29}}$ [erg s$^{-1}$ Hz$^{-1}$]{} at 1.4 GHz (see, e.g., Fig.2 in RCS05). We define the first bin as the redshift interval in which all objects with luminosity $\ge \frac{1}{2} L_{\mathrm k}$ can be observed. This corresponds to $z\le 0.2$. In other words, this redshift threshold guarantees that the luminosities around $L_{\mathrm k}$ are included in the sampled volume. While this bin does not contain a strictly volume-limited subsample, here the selection bias should be mitigated as much as possible, and still the bin includes a sizable number of objects. The second bin contains the remaining objects, i.e. those with $z>0.2$. Finally, we refer to calculations made on all objects as a third bin named “any-$z$”. For most of the objects, the 0.5–2 keV flux limit is a few times larger than what expected from the radio flux limits and the radio/X-ray correlation. Many objects therefore only have X-ray upper limits, which need to be properly accounted for. The 2–10 keV limit is about one order of magnitude larger and thus this band is not considered in this Section. The hypothesis we would like to test is if the COSMOS data are consistent with the extrapolation of the RCS03 correlation, or if they require substantially different parameters. This kind of question is usually what Bayesian methods are most suited to answer. The ratio $$\label{eq:q} q = {\mbox{Log}}( F_{0.5-2 \mathrm{keV}} / {{S_{1.4 \mathrm GHz}}})$$ may be defined in the same way of the analogous $q$ parameter often employed in testing the radio/FIR correlation of spiral galaxies [@cond92]. Non-detections in X-rays therefore lead to upper limits for the $q$’s and can be properly accounted for. To include the K-correction in the $q$’s, we consider the individual redshifts of the sources and assume average power-law spectra with average slopes. Using X-rays (0.5–2.0 keV luminosity less than $10^{42}$ [erg s$^{-1}$]{}) to remove bright AGN from the VLA-CDFS survey [@vla-cdfs-cat; @vla-cdfs-tozzi09], a radio spectral energy index $\alpha\sim 0.69$ can be assumed for the galaxies. The radio/X-ray correlation is described by its slope (here assumed unity), its mean $\bar q$, and its standard deviation $\sigma$. The two latter parameters are the subject of the present statistical analysis, and need prior probability distributions, which we take as follows: - $\bar q_\mathrm{prior}$ is assumed to follow a normal distribution with mean $\bar q_0$ and standard deviation $\sigma_{0}$ taken equal to the values found by RCS03 in the local universe ($\bar q_0=11.10$ and $\sigma_{0}=0.24$); - the standard deviation $\sigma_{\mathrm{prior}}$ is assumed to be uniformly distributed. In Fig. \[fig:qbayes\], the prior distribution is represented as the grey shade. The Bayesian posterior probabilities were calculated with the Montecarlo method [@meekerescobar]; the credible contours[^3] for the mean and standard deviations of $q$ are also shown in Fig. \[fig:qbayes\], along with the RCS03 estimate. The posterior distribution for $q$ and $\sigma$ is consistent with the RCS03 values, within the 68.3% HPD area for the $z\le 0.2$ and any-$z$ bins, and within the 90% Highest Posterior Density (HPD) area for the $z>0.2$ bin. However, the centres of the posterior distributions of the $z>0.2$ and the any-$z$ bins appear to be shifted to lower values of $q$. The $z>0.2$ and the any-$z$ bins allow a larger $\sigma$ than RCS03 (by a factor $\lesssim 2$), while the $z\le 0.2$ bin also allows smaller $\sigma$. The reasons for the larger dispersion may include the large number of upper limits, uncertainties on the K-correction, and a residual contamination by AGN of the objects with X-ray upper limits. One possible explanation of a smaller $q$ at high redshift is that the radio luminosity is evolving in redshift at a faster pace than the X-ray one. The possibilities for an increased efficiency of the radio emission might include different details of cosmic ray acceleration and propagation, or larger magnetic fields at high redshift. However, it has been shown that magnetic fields of a strength similar to those found in local galaxies are in place at $z\sim 1$ [@bernet08]. The two redshift bins can also be seen, approximately, as two luminosity bins. Thus, a different interpretation may be that $q$ is luminosity dependent, as suggested by @symeonidis2011 that the X-ray/far-infrared ratio may be lower in UltraLuminous InfraRed Galaxies (ULIRGs) than in normal galaxies with lower luminosities. If the analysis presented in this Section is repeated by dividing the sample in two luminosity bins (radio luminosities lower and higher than $4{\cdot 10^{29}}$ [erg s$^{-1}$ Hz$^{-1}$]{}, which is the median luminosity of the SF sample), then a picture quite similar to Fig. \[fig:qbayes\] is obtained: the high-luminosity bin favours a smaller $q$ than the low-luminosity bin, yet there is a sizable fraction of the parameter space which is allowed by both bins, and which contains the RCS03 value. However, the redshift bins whose data prompt for the evolution only sample the high-luminosity tail of the radio luminosity function, and it is possible that a selection bias is part of the explanation: it may be that radio-luminous objects have lower X-ray luminosity than average. In fact, it has been shown [@vattakunnel12] that galaxies in the [[*Chandra*]{}]{} Deep Fields, whose average X-ray luminosity is about one order of magnitude lower than those presented here, follow the same X-ray/radio correlation of galaxies in the local universe. An estimate of the bias due to the radio selection may be done with the method employed by @sargent2010 for the infrared/radio correlation (see also @kellerman64 [@condon84; @francis93; @lauer07]), in which the bias for a flux limited survey (where the luminosity function of the sources is not fully sampled) depends only on the scatter of the correlation $\sigma$ and on the slope of the differential number counts $\beta$: $$\label{eq:qbias} \Delta q_{\mathrm bias}=\mathrm{ln}(10)\, (\beta-1)\, \sigma^2\sim 0.18$$ where $\beta\sim 2.35$ (see Sect. \[sec:lognlogs\]) and $\sigma$=0.24. The amount of correction is shown in Fig. \[fig:qbayes\] as the green dashed arrow. Applying this correction to the $z>0.2$ and any-$z$ bins would mostly remove the redshift (or luminosity) evolution of $q$. This correction needs not to be applied to the $z\le 0.2$ bin because here the luminosity function should be almost correctly sampled. However, this correction should be only taken as a first-order approximation, because one of its hypotheses is that the scatter of the X-ray/radio correlation is described by a Gaussian function. Because the $\Delta q_{\mathrm bias}$ is sensitive to the shape of wings of the function, any deviation of the correlation from a Gaussian would require Eq. (\[eq:qbias\]) to be modified accordingly [@lauer07]. For these reasons, a further investigation of this issue with deeper observations in both the radio and X-ray bands, and including a proper statistical treatment of truncated data[^4] could be an interesting subject for a follow-up analysis. Demographics of star forming galaxies ===================================== {width=".7\textwidth"} \[sec:lognlogs\] An important test for the selection criteria described so far is to check for the size of the population of candidate SF X-ray galaxies. In this Section, we describe four different alternatives, which are plotted in Fig. \[fig:lognlogs\]. The galaxy X-ray number counts from the [[*Chandra*]{}]{} Deep Fields [CDFS @xue-cdfs4Ms] are also shown for reference. First, we consider the 22 class 1 objects from Tables \[tab:cat\] and \[tab:catAGN\]. This is the strictest selection that we discuss, since it requires that an object is radio-detected, and that all the X-ray based criteria are fulfilled. It should thus be considered as a lower limit. We keep the assumption that a criterion is considered fulfilled if the relevant data are lacking, as done in Sect. \[sec:cat\]. The number counts for this selection are plotted in Fig. \[fig:lognlogs\] as the dotted red histogram. The addition of the 14 class 2 objects to the above selection gives the solid red histogram, with the errors shown as the grey area. We only plot the errors for this selection, in order not to clutter the figure; however, they can be taken as representative of the other alternatives. The ratio between the class-1-only and the class$\le2$ histograms is about a factor of 2 at high X-ray fluxes and less than that at lower fluxes. A different approach is to discard the radio selection criterion, and to apply the X-ray based criteria to the whole C-COSMOS catalogue[@ccosmos-cat; @civano12]. This opens the possibility to include a larger number of composite SF/AGN and of low luminosity AGN in the selection. A number of 63 class-1 and 192 class-2 objects are selected in this way. The dotted and solid black histograms show the result for the class-1 and class$\le2$ objects, respectively. While the class-1 histogram lies a factor of $\sim 2$-3 above the red ones and is still within the 1–2$\sigma$ errors for the above determinations, a much larger difference (a factor of $\sim 7$) is observed for the class$\le2$ histogram. The latter [Log $N$–Log $S$]{} should be regarded as a likely overestimate for the X-ray galaxy number counts. In fact, if this histogram were extrapolated at fainter fluxes, it would predict a number of galaxy which, summed to expected number of AGN from synthesis models [@gilli07; @treister09], would be incompatible with the observed total [Log $N$–Log $S$]{}. A further hint comes from the integration of the galaxy X-ray luminosity function: the theoretical [Log $N$–Log $S$]{} relations in RCS05 would lie on average a factor of 3 below the histogram. (The same [Log $N$–Log $S$]{} are however consistent with the other three determinations). The galaxy number counts were derived by @xue-cdfs4Ms for the 4Ms CDFS by considering the following criteria: X-ray luminosity, photon index, X/O, optical spectroscopic classification, and X-ray/radio ratio. The criteria were joined is a similar manner to what done here for the class-1 sources. It is thus not surprising that, although each threshold is somewhat different from what used in this paper, the final result is very similar to the counts of radio-selected galaxies presented here. A power-law with the form ${\mbox{Log}}\ N(>S)=-1.35\, {\mbox{Log}}\ S -19.15$ may thus be considered a useful description of both @xue-cdfs4Ms and our determinations of the galaxy X-ray number counts. Conclusions {#sec:conclusions} =========== We have presented the X-ray properties of a sample of 242 SF galaxies in the C-COSMOS field, selected in the radio band and classified according to the optical colours with the method described in S08. This method builds on the definition of a synthetic rest-frame colour, which can be calculated from narrow-band photometry in several bands, and which has been shown to correlate with the position in the BPT diagram [@smolcic06]. It has a similar power to the BPT diagram, with the advantage of not requiring expensive spectral observations. In [[*Chandra*]{}]{} observations, 33 objects were detected. A comparison sample of 398 candidate type-II AGN (with 82 detections) is also presented. We have reviewed some X-ray based selection criteria commonly used in the literature, and analyzed how they affect the composition of the SF and AGN samples. We have thus refined the SF sample, and recovered some objects from the AGN one, on the basis of the following parameters: - hardness ratio; - X-ray luminosity; - X-ray/optical flux ratio; - classification from optical spectroscopy. This is a small yet effective set of indicators based only on X-ray and optical properties. We also mention that a similar method has been applied by @xue-cdfs4Ms in the [[*Chandra*]{}]{} Deep Field South. We have proposed two refined subsamples of C-COSMOS SF galaxies, based on the absence of AGN-like properties, one (class-1) being more strict in its criteria than the other (class$\le 2$), containing 8 and 16 objects respectively. If the same method is applied to the AGN sample, 14 objects may be recovered as SF under the stricter method, and 20 under the more liberal one; these objects may be composite, or may have been misclassified by the optical colour method. Of 33 detections in the SF sample, 17 exhibit AGN-like properties in terms of hardness ratio, non-detection in the 0.5-2.0 keV band, optical spectrum, X-ray/optical flux ratio and absolute X-ray luminosity. Among 82 detections in the AGN sample, 20 have SF-like properties. Thus the fraction of composite/misclassified objects is 50% for the SF sample and 25% for the AGN, while S08 reported fractions of 30% and 20%, respectively. The larger fractions observed here are likely to be explained as a selection effect, due to the fact that AGN are on average brighter than galaxies in the X-rays. Conversely, the stacked spectra of X-ray undetected SF and AGN are significantly different: the SF one can be fit with power-law spectra with $1.5\lesssim \Gamma\lesssim 2.5$, while the AGN one is flatter ($0.4\lesssim \Gamma\lesssim 0.9$). This suggests that the two samples do have different physical properties and that the fractions of composite/misclassified are actually lower for X-ray-undetected objects. Thus we regard the fraction of mis-classified and/or composite objects to be in line with the expectations from S08. We have investigated if the radio/X-ray luminosities correlation (RCS03) applies to our data, and if there is any evidence for redshift evolution of the correlation parameters. A subsample of SF objects at $z\le 0.2$ yields an X-ray/radio ratio fully consistent with the local RCS03 estimate. Data at larger redshifts are still consistent with the local value. Some evolution towards lower X-ray/radio ratios is possible, but at least part of the evolution may be explained by selection biases arising in flux-limited surveys. Further analysis of deeper data, or the use of statistical techniques appropriate to truncated data may be necessary. We have presented the number counts of the C-COSMOS SF galaxies according to different selection criteria, and compared them to the number counts of the CDFS galaxies [@xue-cdfs4Ms]. Considering only the radio-selected class-1, or the radio-selected class$\le 2$, or dropping the radio selection and only considering class-1, all lead to estimates which are consistent to each other within the 1–2$\sigma$ errors. Dropping the radio selection and considering class$\le 2$ objects gives an overestimate of the galaxy number counts. Further observations of the COSMOS field with [[*Chandra*]{}]{} would allow a much better determination of the X-ray demographics of the SF galaxies at the redshifts probed in this paper. By extending the coverage to the full 2 deg$^2$, with a uniform exposure of 180 ks over 1.7deg$^2$ (the HST-observed area), the final size of the X-ray detected COSMOS SF galaxy sample should be of the order of 200. The AEGIS survey [@nandra05-aegis] is similar in methodology to COSMOS, and currently has an observed area and flux limit similar to C-COSMOS, and recent observations have added deeper coverage (uniform exposure of 800 ks) to a 0.6 deg$^2$ sub-field. Even deeper is the exposure (4 Ms) in the [[*Chandra*]{}]{} Deep Field South (CDFS) field in the GOODS survey. Though the probed area is smaller (0.2 deg$^2$; @xue-cdfs4Ms [@vattakunnel12]), the CDFS already provides 179 objects classified as galaxies (24% of the total). This latter field also has a 3 Ms coverage with [XMM-[*Newton*]{}]{}, which is providing good quality spectroscopy [@comastri11-cdfs]. The surveys described above have also extensive optical spectroscopy, and have been observed in the far infrared by [*Spitzer*]{} and [ *Herschel*]{}. The inclusion of infrared data could provide a further improvement in the object classification, and together with optical photometry could allow to break down the AGN and host galaxy contributions for the composite objects. Finally, these data sets and the classifications done insofar could be used as testbeds for innovative statistical methods in object recognition and classification. This would be especially useful in light of the future large surveys, both in X-rays (e.g., [ *eROSITA*]{}) and in optical (LSST, Pan-STARRS, SNAP). We thank an anonymous referee whose comments have contributed to improve the presentation of this paper. This research has made use of the Perl Data Language (PDL) which provides a high-level numerical functionality for the Perl programming language [@pdl http://pdl.perl.org]. We acknowledge financial contribution from the agreement ASI-INAF I/009/10/0, and a grant from the Greek General Secretariat of Research and Technology in the framework of the program Support of Postdoctoral Researchers. The research leading to these results has received funding from the European Union’s Seventh Framework programme under grant agreement 229517. [^1]: The number of Chandra-detected type-I QSO is 11, out of 33 which are in the C-COSMOS field of view. The minimum 2-10 keV luminosity of these type-I QSO is $2.6{\cdot 10^{43}}$ [erg s$^{-1}$]{}, therefore all type-I QSO would be selected as QSO/AGN (as opposed to galaxies) by the absolute luminosity criterion of Sect. \[sec:lum42\]. More in general the type-I and type-II classes have a different distribution of luminosities: the median 2–10 keV luminosities are $6{\cdot 10^{43}}$ and $7{\cdot 10^{42}}$ [erg s$^{-1}$]{}, respectively. Thus, we regard a simple X-ray luminosity argument to be sufficient to differentiate the S08 type-I QSO from the star forming galaxy and the type-II populations. [^2]: PIs: Capak, Kartalpepe, Salvato, Sanders, Scoville. [^3]: It may be worth reminding that in the Bayesian framework one speaks of *credible* contours and intervals, leaving the word *confidence* for frequentist statistics in order to avoid confusion. [^4]: In statistical nomenclature, upper limits to the fluxes of objects otherwise detected at a different wavelength are an example of *censored data*, while a flux-limited survey for which no information about the existence of objects at fluxes lower than the limit is an example of *truncated data*. Truncated data cannot be treated with the same tools valid for censored data [see, e.g., @lawless].

--- abstract: 'The Erdös-Hajnal conjecture states that for every graph $H$, there exists a constant $\delta(H) > 0$ such that every graph $G$ with no induced subgraph isomorphic to $H$ has either a clique or a stable set of size at least $|V(G)|^{\delta(H)}$. This paper is a survey of some of the known results on this conjecture.' author: - | Maria Chudnovsky [^1]\ Columbia University, New York NY 10027 title: 'The Erdös-Hajnal Conjecture—A Survey' --- Introduction ============ All graphs in this paper are finite and simple. Let $G$ be a graph. We denote by $V(G)$ the vertex set of $G$. The [*complement*]{} $G^c$ of $G$ is the graph with vertex set $V(G)$, such that two vertices are adjacent in $G$ if and only if they are non-adjacent in $G^c$. A [*clique*]{} in $G$ is a set of vertices all pairwise adjacent. A [*stable set*]{} in $G$ is a set of vertices all pairwise non-adjacent (thus a stable set in $G$ is a clique in $G^c$). Given a graph $H$, we say that $G$ is [*$H$-free*]{} if $G$ has no induced subgraph isomorphic to $H$. For a family $\mathcal{F}$ of graphs, we say that $G$ is $\mathcal{F}$-free if $G$ is $F$-free for every $F \in \mathcal{F}$. It is a well-known theorem of Erdös [@Erdos1] that there exist graphs on $n$ vertices, with no clique or stable set of size larger than $O(\log n)$. However, in 1989 Erdös and Hajnal [@EH] made a conjecture suggesting that the situation is dramatically different for graphs that are $H$-free for some fixed graph $H$, the following: \[EHconj\] For every graph $H$, there exists a constant $\delta(H) > 0$ such that every $H$-free graph $G$ has either a clique or a stable set of size at least $|V(G)|^{\delta(H)}$. This is the [*Erdös-Hajnal conjecture*]{}. The same paper [@EH] also contains a partial result toward Conjecture \[EHconj\], showing that $H$-free graphs behave very differently from general graphs: \[EHthm\] For every graph $H$, there exists a constant $c(H) > 0$ such that every $H$-free graph $G$ has either a clique or a stable set of size at least $e^{c(H) \sqrt{log |V(G)|}}$. However, obtaining the polynomial bound of Conjecture \[EHconj\] seems to be a lot harder, and Conjecture \[EHconj\] is still open. The goal of this paper is to survey some recent results on Conjecture \[EHconj\]. We start with some definitions. We say that a graph $H$ has the [*Erdös-Hajnal property*]{} if there exists a constant $\delta(H) > 0$ such that every $H$-free graph $G$ has either a clique or a stable set of size at least $|V(G)|^{\delta(H)}$. Clearly, $H$ has the Erdös-Hajnal property if and only if $H^c$ does. Let $G$ be a graph. For $X \subseteq V(G)$, we denote by $G|X$ the subgraph of $G$ induced by $X$. We write $G \setminus X$ for $G|(V(G) \setminus X)$, and $G \setminus v$ for $G \setminus \{v\}$, where $v \in V(G)$. We denote by $\omega(G)$ the maximum size of a clique in $G$, by $\alpha(G)$ the maximum size of a stable set in $G$, and by $\chi(G)$ the chromatic number of $G$. The graph $G$ is [*perfect*]{} if $\chi(H)=\omega(H)$ for every induced subgraph $H$ of $G$. The Strong Perfect Graph Theorem [@CRST] characterizes perfect graphs by forbidden induced subgraphs: \[SPGT\] A graph $G$ is perfect if and only if no induced subgraph of $G$ or $G^c$ is an odd cycle of length at least five. Thus, in a perfect graph $G$ $$|V(G)| \leq \chi(G) \alpha(G) = \omega(G) \alpha(G),$$ and so \[perfect\] If $G$ is perfect, then either $\omega(G) \geq \sqrt{|V(G)|}$, or $\alpha(G) \geq \sqrt{|V(G)|}$. This turns out to be a useful observation in the study of Conjecture \[EHconj\]; in fact it is often convenient to work with the following equivalent version of Conjecture \[EHconj\]: \[EHconj3\] For every graph $H$, there exists a constant $\psi(H) > 0$, such that every $H$-free graph $G$ has a perfect induced subgraph with at least $|V(G)|^{\psi(H)}$ vertices. The equivalent of Conjecture \[EHconj\] and Conjecture \[EHconj3\] follows from Theorem \[perfect\]. The main advantage of Conjecture \[EHconj3\] is that instead of having two outcomes: a large clique or a large stable set, it only has one, namely a large perfect induced subgraph, thus making inductive proofs easier. This paper is organized as follows. In Section 2 we discuss graphs and families of graphs that are known to have the Erdös-Hajnal property. Section 3 deals with weakenings of Conjecture \[EHconj\] that are known to be true for all graphs. Is Section 4 we state an analogue of Conjecture \[EHconj\] for tournaments, and discuss related results and techniques. Sections 5 and 6 deal with special cases of Conjecture \[EHconj\] and its tournament analogue when we restrict our attention to graphs $H$ with a certain value of $\delta(H)$. Graphs with the Erdös-Hajnal property ===================================== Obviously, Conjecture \[EHconj\] can be restated as follows: \[EHconj2\] Every graph has the Erdös-Hajnal property. However, at the moment, only very few graphs have been shown to have the Erdös-Hajnal property. The goal of this section is to describe all such graphs. It is clear that graphs on at most two vertices have the property. Complete graphs and their complements have the property; this follows from the famous Ramsey theorem. If $H$ is the two-edge path, then every $H$-free graph $G$ is the disjoint union of cliques, and thus has either a clique or a stable set of size $\sqrt{|V(G)|}$, so the two-edge path has the property. By taking complements, this shows that all three-vertex graphs have the property. Another graph for which the Erdös-Hajnal property is easily established is the three-edge path. It follows immediately from Theorem \[SPGT\] that all graphs with no induced subgraph isomorphic to the three-edge path are perfect; this fact can also be obtained by an easy induction from the following theorem of Seinsche [@Seinsche]: \[3edgepath\] If $G$ is a graph with at least two vertices, and no induced subgraph of $G$ is isomorphic to the three-edge path, then either $G$ or $G^c$ is not connected. Let us now define the [*substitution*]{} operation. Given graphs $H_1$ and $H_2$, on disjoint vertex sets, each with at least two vertices, and $v \in V(H_1)$, we say that $H$ is [*obtained from $H_1$ by substituting $H_2$ for $v$*]{}, or [*obtained from $H_1$ and $H_2$ by substitution*]{} (when the details are not important) if: - $V(H)=(V(H_1) \cup V(H_2)) \setminus \{v\}$, - $H|V(H_2)=H_2$, - $H|(V(H_1) \setminus \{v\})=H_1 \setminus v$, and - $u \in V(H_1)$ is adjacent in $H$ to $w \in V(H_2)$ if and only if $u$ is adjacent in $H_1$ to $v$. A graph is [*prime*]{} if it is not obtained from smaller graphs by substitution. In [@APS] Alon, Pach, and Solymosi proved that the Erdös-Hajnal property is preserved under substitution: \[substitution\] If $H_1$ and $H_2$ are graphs with the Erdös-Hajnal property, and $H$ is obtained from $H_1$ and $H_2$ by substitution, then $H$ has the Erdös-Hajnal property. This is the only operation known today that allows us to build bigger graphs with the Erdös-Hajnal property from smaller ones. The idea of the proof of Theorem \[substitution\] is to notice the following: since both $H_1$ and $H_2$ have the Erdös-Hajnal property, it follows that if an $H$-free graph $G$ does not contain a “large” clique or stable set, then every induced subgraph of $G$ with at least $|V(G)|^{\epsilon}$ vertices (where $\epsilon$ depends on the precise definition of “large”) contains an induced copy of $H_1$ and an induced copy of $H_2$. Let $v \in V(H_1)$ be such that $H$ is obtained from $H_1$ by substituting $H_2$ for $v$. Then counting shows that some copy of $H_1\setminus v$ in $G$ can be extended to $H_1$ in at least $n^{\epsilon}$ ways. But this guarantees that there is a copy of $H_2$ among the possible extensions, contrary to the fact that $G$ is $H$-free, and Theorem \[substitution\] follows. Since the only prime graph on four vertices is the three-edge path, using Theorem \[substitution\] and the fact that the three-edge-path has the Erdös-Hajnal property, it is easy to check that all graphs on at most four vertices have the Erdös-Hajnal property. Moreover, there are only four prime graphs on five vertices: - the cycle of length five - the four-edge path - the complement of the four-edge path - the bull (the graph with vertex set $\{a_1,a_2,a_3,b_1,b_2\}$ and edge set $\{a_1a_2,a_2a_3,a_1a_3, a_1b_1, a_2b_2\}$), and Theorem \[substitution\] implies that all the other graphs on five vertices have the Erdös-Hajnal property. A much harder argument [@CSafra] shows that \[EHbull\] The bull has the the Erdös-Hajnal property. Moreover, every bull-free graph $G$ has a clique or a stable set of size at least $|V(G)|^{\frac{1}{4}}$. The exponent $\frac{1}{4}$ is in fact best possible because of the following construction. Take a triangle-free graph $T$ with $m$ vertices and no stable set of size larger than $\sqrt{m \log m}$ (such graphs exist by an old theorem of Kim [@Kim]). Let $G$ be obtained from $T$ by substituting a copy of $T^c$ for every vertex of $T$. Then $|V(G)|=m^2$, and it is easy to check that $G$ has no clique or stable set of size larger than $2\sqrt{m \log m}$. The proof of Theorem \[EHbull\] uses structural methods to show that every prime bull-free graph belongs to a certain subclass where a clique or a stable set of the appropriate size can be shown to exist. For graphs that are not prime, the result follows by induction. Let us say that a function $f: V(G) \rightarrow [0,1]$ is [*good*]{} if for every perfect induced subgraph $P$ of $G$ $$\Sigma_{v \in V(P)}f(v) \leq 1.$$ For $\alpha \geq 1$, the graph $G$ is [*$\alpha$-narrow*]{} if for every good function $f$ $$\Sigma_{v \in V(G)}f(v)^{\alpha} \leq 1.$$ Thus perfect graphs are $1$-narrow. Let $G$ be an $\alpha$-narrow graph for some $\alpha \geq 1$, and let $K=max |V(P)|$ where the maximum is taken over all perfect induced subgraphs of $G$. Then the function $f(v)=\frac{1}{K}$ (for all $v \in V(G)$) is good, and so, since $G$ is $\alpha$-narrow, $\frac{|V(G)|}{K^{\alpha}} \leq 1$. Thus $K \geq |V(G)|^{\frac{1}{\alpha}}$. By Theorem \[perfect\], this implies that in order to prove that a certain graph $H$ has the Erdös-Hajnal property, it is enough to show that there exists $\alpha \geq 1$ such that all $H$-free graphs are $\alpha$-narrow. This conjecture was formally stated in [@CZwols]: \[narrow\] For every graph $H$, there exists a constant $\alpha(H) \geq 1$ such that every $H$-free graph $G$ is $\alpha(H)$-narrow. Fox [@Fox] (see [@CSeymour2] for details) proved that Conjecture \[narrow\] is in fact equivalent to Conjecture \[EHconj\]. More specifically, he showed that for every graph $H$ with the Erdös-Hajnal property, there exists a constant $\alpha(H)$ such that every $H$-free graph is $\alpha(H)$-narrow. However, at least for the purely structural approach, Conjecture \[narrow\] seems to be more convenient to work with than \[EHconj\]. In fact, what is really proved in [@CSafra] is the following: \[narrowbull\] Every bull-free graph is $2$-narrow. And the inductive step proving Theorem \[narrowbull\] for bull-free graphs that are not prime is that $\alpha$-narrowness is preserved under substitutions: \[subnarrow\] If $H_1$ and $H_2$ are $\alpha$-narrow for $\alpha \geq 1$, and $H$ is obtained from $H_1$ and $H_2$ by substitution, then $H$ is also $\alpha$-narrow. Conjecture \[EHconj\] is still open for the four-edge path, its complement, and the five-cycle; and no prime graph on at least six vertices is known to have the Erdös-Hajnal property. We remark that the bull is a self-complementary graph, and one might think that to be the reason for its better behavior. This philosophy is supported by the following result: \[P4P4C\] Every graph with no induced subgraph isomorphic to the four-edge-path or the complement of the four-edge-path is $2$-narrow. This follows from Theorem \[subnarrow\] and from (a restatement of) a theorem of Fouquet [@Fouquet]: \[Fouquet\] Every prime graph with no induced subgraph isomorphic to the four-edge-path or the complement of the four-edge-path is either perfect or isomorphic to the five-cycle. On the other hand, the five-cycle is another self-complementary graph, and yet it seems to be completely intractable. We thus propose the following conjecture that may be slightly easier than the full Conjecture \[EHconj\] in special cases: \[EHconj4\] For every graph $H$, there exists a constant $\epsilon(H) > 0$, such that every $\{H,H^c\}$-free graph $G$ has either a clique or a stable set of size at least $|V(G)|^{\epsilon(H)}$. In [@CSeymour2] Conjecture \[EHconj4\] was proved in the case when $H$ is the five-edge path. This is a strengthening of a result of [@CZwols]. Approximate results =================== The previous section listed a few graphs for which Conjecture \[EHconj2\] is known to hold. The goal of this section is to list facts that are true for all graphs, but that do not achieve the full strength conjectured in Conjecture \[EHconj2\]. The first such statement is Theorem \[EHthm\] which we already mentioned in the Introduction. For two disjoint subsets $A$ and $B$ of $V(G)$, we say that $A$ is [*complete*]{} to $B$ if every vertex of $A$ is adjacent to every vertex of $B$, and that $A$ is [*anticomplete*]{} to $B$ if every vertex of $A$ is non-adjacent to every vertex of $B$. If $A=\{a\}$ for some $a \in V(G)$, we write “$a$ is complete (anticomplete) to $B$” instead of “$\{a\}$ is complete (anticomplete) to $B$”. Here is another theorem similar to Theorem \[EHthm\], due to Erdös, Hajnal and Pach [@EHP]. \[twosets\] For every graph $H$, there exists a constant $\delta(H)>0$ such that for every $H$-free graph $G$ there exist two disjoint subsets $A,B \subseteq V(G)$ with the following properties: 1. $|A|,|B| \geq |V(G)|^{\delta(H)}$, and 2. either $A$ is complete to $B$, or $A$ is anticomplete to $B$. The idea of the proof here is to partition $V(G)$ into $|V(H)|$ equal subsets (which we call “sets of candidates”), and then try to build an induced copy of $H$ in $G$, one vertex at a time, where each vertex of $H$ is chosen from the corresponding set of candidates. In this process, sets of candidates shrink at every step, and since $G$ is $H$-free, eventually we reach a situation where there do not exist enough vertices in one of the sets of candidates with the right adjacencies to another. At this stage we obtain the sets $A$ and $B$, as required in Theorem \[twosets\]. Theorem \[twosets\] was recently strengthened by Fox and Sudakov in [@FS]: \[halfway\] For every graph $H$, there exists a constant $\delta(H)>0$ such that for every $H$-free graph $G$ with $\omega(G) < {{|V(G)|}^{\delta(H)}}$ there exist two disjoint subsets $A,B \subseteq V(G)$, with the following properties: 1. $|A|,|B| \geq |V(G)|^{\delta(H)}$, and 2. $A$ is anticomplete to $B$. In [@LRSTT] another weakening of Conjecture \[EHconj2\] is considered. It is shown that for every $H$, the proportion of $H$-free graphs with $n$ vertices and no “large” cliques or stable sets tends to zero as $n \rightarrow \infty$. Let $\mathcal{F}_H^n$ be the class of all $H$-free graphs on $n$ vertices. Let $\mathcal{Q}_H^{n,\epsilon}$ be the subclass of $\mathcal{F}_H^n$, consisting of all graphs $G$ that have either a clique or a stable set of size at least $n^{\epsilon}$. The main result of [@LRSTT] is the following: \[almostall\] For every graph $H$, there exists a constant $\epsilon(H)>0$ such that ${\frac {|\mathcal{Q}_H^{n,\epsilon(H)}|}{|\mathcal{F}_H^n|}} \rightarrow 1$ as $n \rightarrow \infty$. The proof of Theorem \[almostall\] involves an application of Szemerédi’s Regularity Lemma [@Szemeredi]. Also, somewhat surprisingly, it uses Theorem \[EHbull\]. Tournaments =========== A [*tournament*]{} is a directed graph, where for every two vertices $u,v$ exactly one of the (ordered) pairs $uv$ and $vu$ is an edge. A tournament is [*transitive*]{} if it has no directed cycles (or, equivalently, no cyclic triangles). For a tournament $T$, we denote by $\alpha(T)$ the maximum number of vertices of a transitive subtournament of $T$. Transitive subtournaments seem to be a good analogue of both cliques and stable sets in graphs; furthermore, like induced perfect subgraphs in Conjecture \[EHconj3\], transitive tournaments have the advantage of being one object instead of two. For tournaments $S$ and $T$, we say that $T$ is [*$S$-free*]{} if no subtournament of $T$ is isomorphic to $S$. As with graphs, if $\mathcal{S}$ is a family of tournaments, then $T$ is [*$\mathcal{S}$-free*]{} if $T$ is $S$-free for every $S \in \mathcal{S}$. In [@APS] the following conjecture was formulated, and shown to be equivalent to Conjecture \[EHconj\]: \[EHtourn\] For every tournament $S$, there exists a constant $\delta(S) > 0$ such that every $S$-free tournament $T$ satisfies $\alpha(T) \geq |V(T)|^{\delta(H)}$. As with graphs, let us say that a tournament $S$ has the [*Erdös-Hajnal property*]{} if there exists $\delta(S) > 0$ such that every $S$-free tournament $T$ satisfies $\alpha(T) \geq |V(T)|^{\delta(H)}$. We remark that, like in graphs, the maximum number of vertices of a transitive subtournament in a random $n$-vertex tournament is $O(\log n)$ [@Erdos1]. A [*substitution*]{} operation can be defined for tournaments as follows. Given tournaments $S_1$ and $S_2$, with disjoint vertex sets and each with at least two vertices, and a vertex $v \in V(S_1)$, we say that $S$ is [*obtained from $S_1$ by substituting $S_2$ for $v$*]{} (or just [*obtained by substitution from $S_1$ and $S_2$*]{}) if $V(S)=(V(S_1) \cup V(S_2)) \setminus \{v\}$ and $uw$ is an edge of $S$ if and only if one of the following holds: - $u,w \in V(S_1)$ and $uw$ is an edge of $S_1$ - $u,w \in V(S_2)$ and $uw$ is an edge of $S_2$ - $u \in S_1, w \in S_2$ and $uv$ is an edge of $S_1$ - $u \in S_2$, $w \in S_1$, and $vw$ is an edge of $S_1$. A tournament is [*prime*]{} if it is not obtained by substitution from smaller tournaments. Repeating the proof of Theorem \[substitution\] in the setting of tournaments instead of graphs, it is easy to show that \[subsitutiontourn\] If $S_1$ and $S_2$ are tournaments with the Erdös-Hajnal property, and $S$ is obtained from $S_1$ and $S_2$ by substitution, then $S$ has the Erdös-Hajnal property. Clearly, all tournaments on at most three vertices have the Erdös-Hajnal property, and it is easy to check that there are no prime four-vertex tournaments. Consequently, all four-vertex tournaments have the Erdös-Hajnal property. So far this is very similar to the state of affairs in graphs, but here is a fact to which we do not have a graph analogue: we can define an infinite family of prime tournaments all with the Erdös-Hajnal property (recall that the largest prime graph known to have the property is the bull). Let us describe this family. First we need some definitions. Let $T$ be a tournament, and let $(v_1, \ldots, v_{|T|})$ be an ordering of its vertices; denote it by $\theta$. We say that an edge $v_{j}v_{i}$ of $T$ is a [*backward edge*]{} under $\theta$ if $i<j$. The [*graph of backward edges*]{} under $\theta$, denoted by $B(T, \theta)$, has vertex set $V(T)$, and $v_i v_j \in E(B(T, \theta))$ if and only if $v_iv_j$ or $v_jv_i$ is a backward edge of $T$ under the ordering $\theta$. For an integer $t>0$, we call the graph $K_{1,t}$ a [*star*]{}. Let $S$ be a star with vertex set $\{c, l_1, \ldots, l_t\}$, where $c$ is adjacent to $l_1, \ldots, l_t$ and $\{l_1, \ldots, l_t\}$ is a stable set. We call $c$ the [*center of the star*]{}, and $l_1, \ldots, l_t$ [*the leaves of the star*]{}. A [*right star*]{} in $B(T, \theta)$ is an induced subgraph with vertex set $\{v_{i_0}, \ldots, v_{i_t}\}$, such that $B(T,\theta)|\{v_{i_0}, \ldots, v_{i_t}\}$ is a star with center $v_{i_t}$, and $i_t > i_0, \ldots, i_{t-1}$. A [*left star*]{} in $B(T, \theta)$ is an induced subgraph with vertex set $\{v_{i_0}, \ldots, v_{i_t}\}$, such that $B(T,\theta)|\{v_{i_0}, \ldots, v_{i_t}\}$ is a star with center $v_{i_0}$, and $i_0 < i_1, \ldots, i_t$. Finally, a [*star*]{} in $B(T,\theta)$, is a left star or a right star. A tournament $T$ is a [*galaxy*]{} if there exists an ordering $\theta$ of its vertices such that every component of $B(T, \theta)$ is a star or a singleton, and - no center of a star appears in $\theta$ between two leaves of another star. In [@BCC] the following is proved: \[galaxy\] Every galaxy has the Erdös-Hajnal property. The proof uses the directed version of Szemerédi’s Regularity Lemma formulated in [@AS], and extensions of ideas from the proof of Theorem \[twosets\]. Instead of starting with arbitrary sets of candidates, the way it is done in Theorem \[twosets\], we get a head start by using sets given by a regular partition. Let us describe the proof in a little more detail. Let $T$ be a tournament. For disjoint subsets $A,B$ of $V(T)$, we say that $A$ is [*complete to*]{} $B$ if every vertex of $A$ is adjacent to every vertex of $B$. We say that $A$ is [*complete from*]{} $B$ if $B$ is complete to $A$. Denote by $e_{A,B}$ the number of directed edges $ab$, where $a \in A$ and $b \in B$. We define the [*directed density from A to B*]{} to be $d(A,B)=\frac{e_{A,B}}{|A||B|}.$ Given $\epsilon>0$ we call a pair $(X,Y)$ of disjoint subsets of $V(T)$ $\epsilon$-[*regular*]{} if all $A \subseteq X$ and $B \subseteq Y$ with $|A| \geq \epsilon |X|$ and $|B| \geq \epsilon |Y|$ satisfy: $|d(A,B)-d(X,Y)| \leq \epsilon$ and $|d(B,A)-d(Y,X)| \leq \epsilon.$ Consider a partition $\{V_{0},V_{1},...,V_{k}\}$ of $V(T)$ in which one set $V_{0}$ has been singled out as an [*exceptional*]{} set. (This exceptional set $V_{0}$ may be empty). Such a partition is called an [*$\epsilon$-regular partition of $T$*]{} if it satisfies the following three conditions: - $|V_{0}| \leq \epsilon |V|$ - $|V_{1}|=...=|V_{k}|$ - all but at most $\epsilon k^{2}$ of the pairs $(V_{i},V_{j})$ with $1 \leq i < j \leq k$ are $\epsilon$-regular. The following was proved in [@AS]: \[regularitylemma\] For every $\epsilon>0$ and every $m \geq 1$ there exists an integer $DM=DM(m,\epsilon)$ such that every tournament of order at least $m$ admits an $\epsilon$-regular partition $\{V_{0},V_{1},...,V_{k}\}$ with $m \leq k \leq DM$. The following is also a result from [@AS]; it is the directed analogue of a well-known lemma for undirected graphs [@BT]. \[universalgraphlemma\] Let $k \geq 1$ be an integer, and let $0 < \lambda < 1$. Then there exists a constant $\eta_0$ (depending on $k$ and $\lambda$) with the following property. Let H be a tournament with vertex set $\{x_{1},...,x_{k}\}$, and let $T$ be a tournament with vertex set $V(T)=\bigcup_{i=1}^{k} V_{i}$, where the $V_{i}$’s are disjoint sets, each of order at least one. Suppose that each pair $(V_{i},V_{j})$, $1 \leq i < j \leq k$ is $\eta$-regular, that $d(V_{i},V_{j}) \geq \lambda$ and $d(V_{j},V_{i}) \geq \lambda$. Then there exist vertices $v_{i} \in V_{i}$ for $i \in \{1, \ldots, k\}$, such that the map $x_{i} \rightarrow v_{i}$ gives an isomorphism between $H$ and the subtournament of $T$ induced by $\{v_{1},...,v_{k}\}$ provided that $\eta \leq \eta_{0}$. Now, given a galaxy $G$, we start with a regular partition of a $G$-free tournament given by Theorem \[regularitylemma\]. Using Theorem \[universalgraphlemma\] along with a few standard techniques which we will not describe here, we can find subsets $V_{i_1}, \ldots , V_{i_t}$ (for an appropriately chosen constant $t$), such that $d(V_{i_p},V_{i_q})>.999$ for every $1 \leq p <q \leq t$. This means that for every $1 \leq p <q \leq t$, vertices of $V_{i_p}$ tend to be adjacent to a substantial proportion of the vertices of $V_{i_q}$. On the other hand, if a substantial subset of $V_{i_p}$ is complete to a substantial subset of $V_{i_q}$, then we can apply induction to get a large transitive subtournament in $T$, and so we may assume that no such subsets exist. We now construct a copy of $G$ in $T$, choosing at most one vertex from each of $V_{i_1}, \ldots, V_{i_t}$, and using the fact that for $ 1 \leq p < q \leq t$ no substantial subset of $V_{i_p}$ is complete to a substantial subset of $V_{i_q}$ to obtain the backward edges in the galaxy ordering of $G$, thus obtaining the result of Theorem \[galaxy\]. Obviously, every tournament obtained from a transitive tournament by adding a vertex is a galaxy. It is not difficult to check that there is only one tournament on five vertices that is not a galaxy. Here it is: its vertex set is $\{v_1, \ldots, v_5\}$, and $v_iv_j$ is an edge if and only if $(j-i) \; mod \; 5 \in \{1,2\}$. We call this tournament $S_5$. We remark that $S_5$ is an example of a tournament that is obtained from a transitive tournament by adding two vertices, and that is not a galaxy. Another result of [@BCC] is that: \[S\_5\] The tournament $S_5$ has the Erdös-Hajnal property. The proof of Theorem \[S\_5\] uses similar ideas to the ones in the proof of Theorem \[galaxy\]. Theorem \[galaxy\] and Theorem \[S\_5\] together imply: \[smalltourn\] Every tournament on at most five vertices has the Erdös-Hajnal property. We finish this section with another curious corollary of Theorem \[galaxy\]. Let $P_{k}$ denote a tournament of order $k$ whose vertices can be ordered so that the graph of backward edges is a $k$-vertex path. \[path\] For every $k$, the tournament $P_{k}$ has the Erdös-Hajnal conjecture. Theorem \[path\] follows from the fact that, somewhat surprisingly, $P_k$ has a galaxy ordering. Linear-size cliques, stable sets and transitive subtournaments ============================================================== In Conjecture \[EHconj\] and Conjecture \[EHtourn\], every graph (or tournament) is conjectured to have a certain constant, lying in the $(0,1]$ interval, associated with it. A natural question is: when is this constant at its extreme? Excluding which graphs (or tournaments) guarantees a linear-size clique or stable set (or transitive subtournament)? For undirected graphs this question turns out not to be interesting, because if for some graph $H$ every $H$-free graph were to contain a linear size clique or stable set, then $H$ would need to have at most two vertices (we explain this later). However, for tournaments the answer is quite pretty. We say that a tournament $S$ is a [*celebrity*]{} if there exists a constant $0<c(S)\leq 1$ such that every $S$-free tournament $T$ contains a transitive subtournament on at least $c(S)|V(T)|$ vertices. So the question is to describe all celebrities. This was done in [@heroes], but before stating the result we need some definitions. For a tournament $T$ and $X \subseteq V(T)$, we denote by $T|X$ the subtournament of $T$ induced by $X$. Let $T_k$ denote the transitive tournament on $k$ vertices. If $T$ is a tournament and $X,Y$ are disjoint subsets of $V(T)$ such that $X$ is complete to $Y$, we write $X\Rightarrow Y$. We write $v\Rightarrow Y$ for $\{v\}\Rightarrow Y$, and $X\Rightarrow v$ for $X \Rightarrow \{v\}$. If $T$ is a tournament and $(X,Y,Z)$ is a partition of $V(T)$ into nonempty sets satisfying $X\Rightarrow Y$, $Y\Rightarrow Z$, and $Z\Rightarrow X$, we call $(X,Y,Z)$ a [*trisection*]{} of $T$. If $A,B,C,T$ are tournaments, and there is a trisection $(X,Y,Z)$ of $T$ such that $T|X,T|Y,T|Z$ are isomorphic to $A,B,C$ respectively, we write $T = \Delta(A,B,C)$. A [*strongly connected component*]{} of a tournament is a maximal subtournament that is strongly connected. One of the main results of [@heroes] is the following: \[celebrity\] A tournament is a celebrity if and only if all its strongly connected components are celebrities. A strongly connected tournament with more than one vertex is a celebrity if and only if it equals $\Delta(S,T_k,T_1)$ or $\Delta(S, T_1, T_k)$ for some celebrity $S$ and some integer $k\ge 1$. Following the analogy between stable sets in graphs and transitive subtournaments in tournaments, let us define the [*chromatic number*]{} of a tournament $T$ to be the smallest integer $k$, for which $V(T)$ can be covered by $k$ transitive subtournaments of $T$. We denote the chromatic number of $T$ by $\chi(T)$. Here is a related concept: let us say that a tournament $S$ is a [*hero*]{} if there exists a constant $d(S)>0$ such that every $S$-free tournament $T$ satisfies $\chi(T) \leq d(S)$. Clearly every hero is a celebrity. Moreover, the following turns out to be true (see [@heroes]) \[hero\] A tournament is a hero if and only if it is a celebrity. Another result of [@heroes] is a complete list of all minimal non-heroes (there are five such tournaments). Let us now get back to undirected graphs. What if instead of asking for excluding a single graph $H$ to guarantee a linear size clique or stable set, we ask the same question for a family of graphs? Let us say that a family $\mathcal{H}$ of graphs is [*celebrated*]{} if there exists a constant $0<c(\mathcal{H}) \leq 1$ such that every $\mathcal{H}$-free graph $G$ contains either a clique or a stable set of size at least $c(\mathcal{H})|V(G)|$. The [*cochromatic number*]{} of a graph $G$ is the minimum number of stable sets and cliques with union $V(G)$. We denote the cochromatic number of $G$ by $co\chi(G)$. Let us say that a family $\mathcal{H}$ is [*heroic*]{} if there exists a constant $d(\mathcal{H})>0$ such that $co\chi(G) < d(\mathcal{H})$ for every every $\mathcal{H}$-free graph $G$. Clearly, if $\mathcal{H}$ is heroic, then it is celebrated. Heroic families were studied in [@CSeymour]. Let $G$ be a complete multipartite graph with $m$ parts, each of size $m$. Then $G$ has $m^2$ vertices, and no clique or stable set of size larger than $m$; and the same is true for $G^c$. Thus every celebrated family contains a complete multipartite graph and the complement of one. Recall that for every positive integer $g$ there exist graphs with girth at least $g$ and no linear-size stable set (this is a theorem of Erdös [@Erdos2]). Consequently, every celebrated family must also contain a graph of girth at least $g$, and, by taking complements, a graph whose complement has girth at least $g$. Thus, for a finite family of graphs to be celebrated, it must contain a forest and the complement of one. In particular, if a celebrated family only contains one graph $H$, then $|V(H)| \leq 2$. The following conjecture is proposed in [@CSeymour], stating that these necessary conditions for a finite family of graphs to be celebrated are in fact sufficient for being heroic. \[heroicconj\] A finite family of graphs is heroic if and only if it contains a complete multipartite graph, the complement of a complete multipartite graph, a forest, and the complement of a forest. We remark that this is an extension of a well-known conjecture made independently by Gyárfás [@Gyarfas] and Sumner [@Sumner], that can be restated as follows in the language of heroic families: \[Gyarfas\] For every complete graph $K$ and every tree $T$, the family $\{K,T\}$ is heroic. The main result of [@CSeymour] is that Conjecture \[heroicconj\] and Conjecture \[Gyarfas\] are in fact equivalent. Since a complete graph is a multipartite graph, the complement of one, and the complement of a forest, we deduce that Conjecture \[heroicconj\] implies Conjecture  \[Gyarfas\]. The converse is a consequence of the following theorem of [@CSeymour]: \[ksplit\] Let $K$ and $J$ be graphs, such that both $K$ and $J^c$ are complete multipartite. Then there exists a constant $c(K,J)$ such that for every $\{K,J\}$-free graph $G$, $V(G)=X \cup Y$, where - $\omega(X) \leq c(K,J)$, and - $\alpha(Y) \leq c(K,J)$. The situation for infinite heroic families is more complicated. Another open conjecture of Gyárfás [@Gyarfas2] can be restated to say that a certain infinite family of graphs is heroic: \[hole\] For every complete graph $K$, and every integer $t>0$, the family consisting of $K$ and all cycles of length at least $t$ is heroic. If Conjecture \[hole\] is true, this is an example of a heroic set that does not include a minimal heroic set. Near-linear transitive subtournaments ===================================== In this section we discuss an extension of the property of being a hero studied in [@pseudo]. Let us say that $\epsilon \ge 0$ is an [*EH-coefficient*]{} for a tournament $S$ if there exists $c>0$ such that every $S$-free tournament $T$ satisfies $\alpha(T)\ge c|V(T)|^{\epsilon}$. (We introduce $c$ in the definition of the Erdös-Hajnal coefficient to eliminate the effect of tournaments $T$ with bounded number of vertices; now, whether $\epsilon$ is an EH-coefficient for $S$ depends only on arbitrarily large $S$-free tournaments.) Thus, Conjecture \[EHconj\] is equivalent to: \[EHcoeff\] Every tournament has a positive EH-coefficient. If $\epsilon$ is an EH-coefficient for $S$, then so is every smaller non-negative number; and thus a natural invariant is the supremum of the set of all EH-coefficients for $S$. We call this the [*EH-supremum*]{} for $S$, and denote it by $\xi(S)$. We remark that the EH-supremum for $S$ is [*not*]{} necessarily itself an EH-coefficient for $S$ (we will see an example later). One of the results of [@pseudo] is a characterization of all tournaments with EH-supremum $1$; and not all of these tournaments turn out to be celebrities (in this language, a celebrity is a tournament for which $1$ is its EH-coefficient, and not just its EH-supremum). The following theorem from [@Choromanski] suggests that EH-suprema tend to be quite small: \[upper\] Let $\mathcal{H}^{n,c}$ be the set of all $n$-vertex tournaments having EH-supremum at most $\frac{c}{n}$, where $c$ is an arbitrary constant such that $c > 4$, and let $\mathcal{H}^n$be the set of all $n$-vertex tournaments. Then $${\frac{|\mathcal{H}^{n,c}|}{|\mathcal{H}^{n}|}} \rightarrow 1$$ as $n \rightarrow \infty$. We say that a tournament $S$ is - a [*pseudo-hero*]{} if there exist constants $c(S),d(S) \ge 0$ such that every $S$-free tournament $T$ with $|V(T)|>1$ satisfies $\chi(T) \leq c(S) (\log(|V(T)|))^{d(S)}$; and - a [*pseudo-celebrity*]{} if there exist constants $c(S) >0$ and $d(S)\ge 0$ such that every $S$-free tournament $T$ with $|V(T)|>1$ satisfies $\alpha(T)\ge c(S)\frac{|V(T)|}{(\log(|V(T)|))^{d(S)}}$. In [@pseudo] all pseudo-celebrities and pseudo-heroes are described explicitly. \[pseudo\] The following statements hold: - A tournament is a pseudo-hero if and only if it is a pseudo-celebrity. - A tournament is a pseudo-hero if and only if all its strongly connected components are pseudo-heroes. - A strongly-connected tournament with more than one vertex is a pseudo-hero if and only if either - it equals $\Delta(T_2,T_k,T_l)$ for some $k,l \geq 2$, or - it equals $\Delta(S,T_1,T_k)$ or $\Delta(S,T_k,T_1)$ for some pseudo-hero $S$ and some integer $k>0$. We remind the reader that by Theorem \[celebrity\] the tournament $\Delta(T_2,T_2,T_2)$ is not a celebrity, and yet by Theorem \[pseudo\] it is a pseudo-celebrity. Thus it is an example of a tournament that does not attain its EH-supremum as an EH-coefficient. We conclude this section with another result from [@pseudo], that shows that after $1$, there is a gap in the set of EH-suprema. \[gap\] Every tournament $S$ with $\xi(S)>5/6$ is a pseudo-hero and hence satisfies $\xi(S) = 1$. The reason for Theorem \[gap\] is another theorem from [@pseudo] that states that a tournament is a pseudo-hero if and only if it is $\mathcal{S}$-free for a certain family $\mathcal{S}$ consisting of six tournaments ($S_5$ is one of them). Thus for any tournament $T$ that is not a pseudo-hero, $\xi(T) \leq \max_{s \in \mathcal{S}} \xi(S)$, and it is shown that $\max_{s \in \mathcal{S}} \xi(S) \leq \frac{5}{6}$. 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--- abstract: '5G network slicing is essential to providing flexible, scalable and on-demand solutions for the vast array of applications in 5G networks. Two key challenges of 5G network slicing are function isolation (intra-slice) and guaranteeing end-to-end delay for a slice. In this paper, we address the question of optimal allocation of a slice in 5G core networks by tackling these two challenges. We adopt and extend the work by D. Dietrich *et al.* [@7945385] to create a model that satisfies constraints on end-to-end delay as well as isolation between components of a slice for reliability.' author: - 'Danish Sattar and Ashraf Matrawy [^1] [^2] [^3]' bibliography: - '5G\_bib\_cletter.bib' title: Optimal Slice Allocation in 5G Core Networks --- 5G slicing, network slicing, 5G security, 5G reliability, 5G optimization, 5G isolation Introduction ============ The 5G network design and the standard are still in development, but it is envisioned to be an agile and elastic network. Network slicing has emerged as a key to realizing this vision. In 5G networks, an end-to-end network slice is a complete logical network that includes Radio Access Network (RAN) and Core Network (CN), and it has capabilities to provide different telecommunication services [@rfcns]. An end-to-end slice is created by pairing the RAN and core network slice, but the relationship between both slices could be 1-to-1 or 1-to-M. For instance, one RAN slice could be connected to multiple core slices and vice versa [@DBLP:journals/corr/LiWPU16; @5gamericas]. Fig. \[fig:Topology\] shows an example of the relationship between core and RAN slices as well as 5G network slicing use cases. In Fig. \[fig:Topology\], two different use cases for 5G network slicing are shown i.e. IoT, and Remote Health Services. ![An example of Network Slicing - A RAN slice can be connected with one or more core slice and vice versa. The slice pairing function is used to connect the RAN-core slice pair.[]{data-label="fig:Topology"}](Slicing6){width="45.00000%"} The first issue we consider in slice allocation is intra-slice isolation (physical isolation between Virtual Network Functions (VNF) of a slice). This might be required by the slice for more reliability because if the entire slice is hosted on the same server, and if the server is compromised or becomes unavailable, the entire slice would also be affected (compromised/unavailable). However, if the there is some level of intra-slice isolation, the slice operator might be able to recover from partial compromise/unavailability of the network slice. We note that our aim does not include inter-slice isolation where other aspects need to be taken into consideration including but not limited to physical isolation, hardware-based isolation, virtual machine based isolation [@8104638]. However, our focus is on providing on-demand physical isolation between different VNFs of a slice for added reliability and security. The second issue we consider is the end-to-end delay. 5G networks have strict requirements for the end-to-end delay. To support real-time applications (e.g., health services, autonomous driving, etc.) 5G network needs to guarantee end-to-end delay for a certain application across the network (only considering end-to-end delay for a core network slice). In this paper, we address the question of optimal slice allocation in 5G core networks (virtual Evolved Core (vEPC)). We do this by adopting and extending VNF placement in the LTE core network presented by D. Dietrich *et al.* [@7945385]. Our **contributions** are to (1) guarantee end-to-end delay, (2) provide intra-slice isolation for slice allocation and (3) find a minimum delay path between the slice components. We aim to provide an optimal solution for allocating a core network slice in 5G networks. The formulation we use for the optimization model is Mixed-Linear Integer Programming (MILP). We take into consideration some of the core requirements for allocating a 5G network slice. We consider the physical isolation requirement between different components of a slice for a variable degree of reliability. The optimization model also ensures the end-to-end delay required by a core network slice. A 5G network slice creation would be dynamic, and a slice could have a variable number of components that require on-demand service chaining (network slices might have different combination of VNFs). For instance, a slice could have several components e.g., Authentication Server Function (AUSF), Security Anchor Function (SEAF), Session Management Function (SMF), Application Function (AF) and several User Plane Functions (UPF) with on-demand service chaining between them. We are aware that there are several other requirements and properties that need to be addressed before a complete end-to-end 5G slice can be instantiated but those requirements are out-of-the-scope of this work. Related Work {#sec:relatedwork} ============ V. Sciancalepore et al. [@2018arXiv180103484S] have proposed a practical implementation of network slicing. The proposed model aims to provide an efficient network slicing solution by analyzing the past network slicing information. D. Dietrich *et al.* [@7945385] proposed linear programming formulation for the placement of VNFs in the LTE core network. In the proposed algorithm, they provided a balance between optimality and time complexity. R. Ford *et al.* [@DBLP:journals/corr/FordSMJR17] proposed optimal VNF placement for the SDN-based 5G mobile-edge cloud. Their optimization algorithm provides resilience by placing VNFs in distributed data centers. A. Baumgartner et al. [@7116162] have presented optimal VNF placement for the mobile virtual core. They used the cost of placement to allocate the VNFs. In their problem formulation, they considered physical network constraints for storage, processing, and switching capacity as well as service chains when allocating VNFs to the physical substrate network. S. Agarwal et al. [@Francesco2017JointVP] used a queuing model to perform VNF placement in 5G networks. Latency was used as the primary Key Performance Indicator (KPI) to formulate the optimization problem. MILP Formulation {#sec:mathmodel} ================ In this section, we will explain the optimization model we used in this paper. We are adopting and extending the work presented by D. Dietrich *et al.* in [@7945385]. The focus of their work was on the LTE cellular core and placement of network functions in an optimal manner while load balancing the resources. They transformed the optimization problem into Linear Programming (LP) problem by relaxing some MILP constraints to reduce time complexity. We adopt their model to achieve optimal slice allocation. Our objective is to allocate 5G core network slice VNFs optimally to provide intra-slice isolation for added reliability. We also fulfill one of the core 5G network requirements by guaranteeing the end-to-end core slice delay. In the following MILP formulation, we use the network model and variables from [@7945385]. In that model, each request is associated with a computing demand ($g^i$) and bandwidth requirement ($g^{ij}$). Additionally, for our slice request, we consider end-to-end delay ($d_{E2E}$) and intra-slice isolation (reliability) required between the VNFs ($K_{rel}$). We use the following objective function. $$\label{eq1} \begin{multlined} $Minimize$\\ \sum _{i\in{V}_{F}}\sum _{u\in {V}_{S}}\left( 1- {\frac{{r}_{u}}{r_{u,max}}}\right) g^i x^i_u \gamma^i_u \\ + \sum _{(i,j)\in{E}_{F}}\sum _{\substack{(u,v)\in {E}_{S} \\(u\neq v)}} L_{uv} f^{ij}_{uv} \end{multlined}$$ subject to: $$\begin{aligned} \label{eq9} \begin{multlined} \sum _{i \in V_F} x^i_u\leq K_{rel} \hspace{.5cm}\forall u \in V_S, K_{rel}=1,2,3... \end{multlined} \end{aligned}$$ $$\label{eq6} \begin{multlined} \sum _{\substack{(i,j)\in {E}_{F} }} \sum _{\substack{(u,v)\in {E}_{S}\\u\neq v}}\left(\dfrac{f^{ij}_{uv}}{g^{ij}} L_{uv}\right) + \sum_{i\in V_F} \alpha^i\leq d_{E2E} \end{multlined}$$ $$\label{eq7} \begin{multlined} \sum _{i \in V_F} g^i \leq \sum_{u \in V_S} r_u \end{multlined}$$ $$\label{eq8} \begin{multlined} \sum _{(i,j) \in E_F} g^{ij} \leq \sum_{(u,v) \in E_S} r_{uv} \end{multlined}$$ The objective function (\[eq1\]) will assign the incoming slice requests to the least utilized server and find a path with minimum delay. The first term is identical to the objective function in [@7945385] while the second one differs in the way we select paths between VNFs. The first term of the objective function assigns computing demands to the least utilized physical servers. The parameter $\gamma_u^i$ used to avoid infeasible mapping of the VNF/server combination. The second term takes into consideration the physical link delay ($L_{uv}$). Each time when a virtual link $(i,j)\in E_F$ is assigned to a physical link $(u,v)\in E_s$, it increases $L_{uv}$. $L_{uv}$ is a function of link utilization, and it is calculated using eq. (\[eq10\]), where $L_{uv,init}$ is the initial delay assigned to the link $(u,v)\in E_s$. Minimizing both terms will result in the assignment of a network slice to the least utilized servers, and it will find a path with least delay between the slice components (D. Dietrich *et al.* [@7945385] did not consider the minimum delay path). $$\label{eq10} \begin{multlined} L_{uv}=(1-\frac{r_{uv}}{r_{uv,max}})\:2.5\:ms + L_{uv,init}\hspace{.3cm} \forall (u,v) \in E_S \end{multlined}$$ The objective function is subjected to several MILP constraints that we will explain next. In our work, in addition to the constraints listed here, we use constraints (2-5) and (9-10) from [@7945385]. We are not listing/describing all the parameters and constraints due to the space limitation. If the slice has requested that each VNF needs to be assigned to different physical servers, constraint (\[eq9\]) will provide the desired degree of reliability (intra-slice isolation) for the slice ($K_{rel}$). The end-to-end delay for the 5G network is an important requirement. Constraint (\[eq6\]) enforces the end-to-end delay requirement for the core network slice[^4]. It includes the delay incurred along the entire path and the processing delay of each VNF ($\alpha^i$). Since the partial or incomplete assignment of the slice components serves no purpose, constraints (\[eq7\]) and (\[eq8\]) ensure that the remaining computing and bandwidth capacity of the entire data center is enough to accommodate the slice creation request. $x_u^i \in {0,1}$ and $f_{uv}^{ij} \geq 0$ are binary and real variables, respectively. Results and Discussion {#sec:resutls} ====================== \[tab:parameters\] -------------------------------------------------- -- **Parameter & **Value\ CPU capacity/server ($r_{u,max}$) & 12.0 GHz\ Total Servers ($V_S$) & 200\ Total Slice Requests & 200\ $K_{rel}$ & 1-10\ VNF/slice ($V_F$) & 10\ Bandwidth request/slice ($g^{ij}$) & 30-70 Mbps\ VNF CPU request/slice ($g^i$) & 0.5-2.0 GHz\ $\alpha^i$ & 0.3-2.0 ms\ $\epsilon$ & $10^{-10}$\ **** -------------------------------------------------- -- To test the optimization model, we used MATLAB to simulate 5G core network and slice requests. AMPL is used to model optimization algorithm and CPLEX 12.8.0.0 is used as MILP solver. The optimization algorithm is evaluated on Intel Core i7 3.2 GHz with 32 GB RAM. ![Simulation topology. S1-S200, E1-E10, A1-A4, and DC1-DC2 represent physical servers, Edge, Aggregation and Datacenter switches, respectively[]{data-label="fig:Topology1"}](topology){width="50.00000%"} We simulate 200 physical servers that can host different types of VNFs. Other parameters used for the evaluation are listed in Table \[tab:parameters\]. In our simulations, we vary the level of intra-slice isolation using the $K_{rel}$ parameter. This parameter provides the upper limit for how many VNFs can be placed on one physical server. The model guarantees the requested computing resources, bandwidth resources, and end-to-end delay for a slice in the current network state. After allocating each slice, we update the remaining computing and bandwidth resources. The flow link delay $L_{uv}$ can be dynamic. For instance, when the network is congested, this parameter can be updated to reflect the current state of the network, but we did not consider this case. In our simulations, we used two configurations for link bandwidth (Servers$\leftrightarrow$EdgeSwitches). In the first configuration as shown in fig. \[fig:Topology1\], the link bandwidth between servers and edge switches is set to $250$ Mbps. In this case, the overall system performance is limited by the available CPU capacity (CPU bound). Therefore our simulated slice requests, the CPU capacity of the physical servers becomes the limiting factor when allocation slices rather than the link bandwidth. In the second configuration, the link bandwidth (Servers$\leftrightarrow$EdgeSwitches) is set to $100$ Mbps (Bandwidth bound). In this case, the overall system performance is limited by the available link bandwidth between servers and Edge switches. Please note that in all the presented results, the simulation setup was “CPU bound” unless otherwise stated. Intra-slice isolation --------------------- [0.4]{} table\[x=K,y=CPU\]; table\[x=K,y=BW\]; table\[x=K,y=Requests\]; [0.4]{} table\[x=K,y=CPU\]; table\[x=K,y=BW\]; table\[x=K,y=Requests\]; [0.4]{} table\[x=K,y=solver\]; In the first part of the simulation, we fix the end-to-end delay ($d_{E2E}$) to a relatively high value ($500$ ms) to minimize its affect on the results and vary the levels of intra-slice isolation ($K_{rel}$). Fig. \[fig:utilization\] shows the overall average system utilization for CPU and bandwidth resources and accepted requests for different levels of intra-slice isolation. The system is CPU bound, so that overall system bandwidth is higher than total requested bandwidth, hence we see relatively low bandwidth utilization. When slices request intra-slice isolation where $K_{rel}<4$, the bandwidth utilization is higher because all VNFs would have to utilize physical links to communicate with each other. Whereas, when we relax the intra-slice isolation requirement, we get lower network utilization (i.e., $K_{rel}\ge4$). The reason is that as we can allocate more VNFs on the same physical server and the communication between the VNFs does not involve physical communication link, we see lower network activity in this case. However, there is a marginal difference in CPU utilization and requests accepted for variable levels of $K_{rel}\ge2$. We also simulated another topology where the system was bandwidth bound. Fig. \[fig:utilizationBW\] shows the overall system utilization for CPU, bandwidth and requests accepted. The performance of the system is worse compared to when the system is CPU bound (Fig. \[fig:utilization\]). table\[x=E2E,y=K2\] [E2ECPU.txt]{}; table\[x=E2E,y=K4\] [E2ECPU.txt]{}; table\[x=E2E,y=K6\] [E2ECPU.txt]{}; table\[x=E2E,y=K8\] [E2ECPU.txt]{}; table\[x=E2E,y=K10\] [E2ECPU.txt]{}; Fig. \[fig:sisolation\] provide some interesting results for the average solver runtime. Obviously, with stricter requirements for intra-slice isolation are, more time is required to find an optimal solution for allocation of slice components and to find an optimal path with least delay. A factor that impacts these values is that when the requirement for intra-isolation are flexible, the optimization algorithm can place more components on the same physical system and it would eliminate the need to find optimal paths between these components. We can see this behavior when $K_{rel}>4$ in Fig. \[fig:sisolation\]. However, as we can see, when a slice requests that no more than two or three VNFs can be placed on a single physical server, there is a significant variation in solver runtime. We ran these simulations multiple times and using multiple parameter value and each time we obtain almost identical results. We have not been able to identify the reason behind the anomalous behavior for $K_{rel}=2$ and $K_{rel}=3$. End-to-end delay ---------------- In the second part of the simulation, we use different end-to-end delay requirements. Please note that we ran simulations for $K_{rel}=1$ to $K_{rel}=10$ but results are only shown for a few values of $K_{rel}$ to present more readable graphs. The end-to-end delay parameter has a noticeable effect on CPU utilization because setting $K_{rel}\le2$ reduces the number of available solutions as shown in Fig. \[fig:e2ecpu\]. However, this effect becomes minimal when $d_{E2E}\ge150$. We note that the CPU utilization shows the same behaviour as the request acceptance rate (not shown here). table\[x=E2E,y=K2\] [E2EBW.txt]{}; table\[x=E2E,y=K4\] [E2EBW.txt]{}; table\[x=E2E,y=K6\] [E2EBW.txt]{}; table\[x=E2E,y=K8\] [E2EBW.txt]{}; table\[x=E2E,y=K10\] [E2EBW.txt]{}; + table\[y=K2\] [E2ETime.txt]{}; + table\[y=K4\] [E2ETime.txt]{}; + table\[y=K6\] [E2ETime.txt]{}; + table\[y=K8\] [E2ETime.txt]{}; + table\[y=K10\] [E2ETime.txt]{}; Fig. \[fig:e2ebw\] shows that different end-to-end delay requirements have minimal impact on overall bandwidth utilization for all levels of $K_{rel}$. Fig. \[fig:e2esrt\] shows the average solver runtime for different end-to-end delay requirements. We can see a consistent behavior for all levels of intra-slice isolation. Conclusion {#sec:con} ========== In this paper, we addressed the optimal allocation of 5G core network slices. The optimization model provides intra-slice isolation as well as ensures that the end-to-end delay meets the minimum requirement. We evaluated the optimization model by simulating a virtualized mobile core. Our evaluation shows that when there is little or no restriction on the intra-slice isolation ($K_{rel} > 2$), CPU utilization is increased and the demand for bandwidth is reduced due to the reduction between inter-machine communications. On the other hand, stricter intra-slice isolation ($K_{rel} \leq 2$) requires more bandwidth and leads to relatively lower CPU utilization. [^1]: D. Sattar, Department of Systems and Computer Engineering, Carleton University, Ottawa, ON K1S 5B6, Canada e-mail: danish.sattar@carleton.ca [^2]: A. Matrawy is with the School of Information Technology, Carleton University, Ottawa, ON K1S 5B6, Canada e-mail: ashraf.matrawy@carleton.ca [^3]: This work was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) through the NSERC Discovery Grant program. [^4]: We note that meeting end-to-end delay requirements would need traffic engineering which is outside the scope of this paper.

--- abstract: 'By Bernoulli’s law, an increase in the relative speed of a fluid around a body is accompanies by a decrease in the pressure. Therefore, a rotating body in a fluid stream experiences a force perpendicular to the motion of the fluid because of the unequal relative speed of the fluid across its surface. It is well known that light has a constant speed irrespective of the relative motion. Does a rotating body immersed in a stream of photons experience a Bernoulli-like force? We show that, indeed, a rotating dielectric cylinder experiences such a lateral force from an electromagnetic wave. In fact, the sign of the lateral force is the same as that of the fluid-mechanical analogue as long as the electric susceptibility is positive ($\epsilon>\epsilon_{0}$), but for negative-susceptibility materials (e.g. metals) we show that the lateral force is in the opposite direction. Because these results are derived from a classical electromagnetic scattering problem, Mie-resonance enhancements that occur in other scattering phenomena also enhance the lateral force.' author: - Ramis Movassagh - 'Steven G. Johnson' bibliography: - 'mybib.bib' title: 'Optical “Bernoulli” forces' --- *\[sec:Bernoulli\_intro\]Photonic Bernoulli’s Law?* When considering a rotating body in a fluid stream such as air, the body experiences a pressure gradient caused by the difference of the relative velocity of its motion to that of the fluid at various points on its boundary. For example, an idealized tornado such as a spinning cylinder moves perpendicular to the streamlines of the fluid. The direction of motion is along the direction connecting the center of the cylinder to the point of maximum relative velocity. In a famous experiment, Michelson and Morley [@Moller1957] showed that even if the earth were immersed in a fluid in motion, the speed of light would be constant relative to perpendicular directions. Later, the special theory of relativity established the constancy of the speed of light regardless of observer’s relative motion to the light source. Here, we ask to what extent can a stream of photons resembles a stream of massive fluids? In particular, if one considers a stream of photons (classically described by Maxwell’s equations) as a fluid in motion and places a rotating dielectric body in it, one might naively expect that no Bernoulli-type force would be experienced by the body since the relative speed of light is the same on both sides. Here we show that such a force *is* experienced by the rotating body, though the cause is the asymmetry of the scattered field [@Tai1964] from the dielectric, by which a net force is imparted to the rotating body.  Light scattering from a rotating dielectric cylinder.](E_incident_Cylinder.pdf) *\[sec:ScatteringField\] Cylindrically Rotating Dielectric.–* The exact electromagnetic constitutive equations in a medium moving at velocity ${\mathbf{v}}$, discovered by Minkowski [@Minkowski1907], are $$\begin{aligned} {\mathbf{D}}+{\mathbf{v}}\times{\mathbf{H}}/c & = & \epsilon\left({\mathbf{E}}+{\mathbf{v}}\times{\mathbf{B}}/c\right)\label{eq:FundamentalEquationsDielectricsExact}\\ {\mathbf{B}}+{\mathbf{E}}\times{\mathbf{v}}/c & = & \mu\left({\mathbf{H}}+{\mathbf{D}}\times{\mathbf{v}}/c\right),\label{eq:FundamentalEquationsDielectricsExact2}\end{aligned}$$ where ${\mathbf{E}}$, ${\mathbf{D}}$, ${\mathbf{B}}$ and ${\mathbf{H}}$ are the usual electromagnetic fields, $c$ is the speed of light in vacuum and $\epsilon$ is the electric permittivity in the rest frame and $\mu$ is the magnetic permeability in the rest frame. These equations presuppose uniform motion of the dielectric, where special relativity is sufficient. For *accelerated* dielectrics, the equations become more complicated; however, for rotating bodies with axial symmetry, the body in motion has the same shape as the one in the rest frame and it has been shown that the same equations would apply [@Sommerfeld1952; @Ridgely1996; @Tai1964]. This assertion has been successfully used in applications [@Tai1964] and was later proved rigorously by Ridgely [@Ridgely1996], who showed that the general relativistic treatment for *uniformly rotating* dielectrics with axial-symmetry, to first order in $v/c$, gives Minkowski’s results (Eqs. \[eq:FundamentalEquationsDielectricsExact\] and \[eq:FundamentalEquationsDielectricsExact2\]). In the limit where $v/c$ is small, Tai considered the scattered field of a plane wave incident upon a uniformly rotating dielectric cylinder with angular speed $\Omega$ [@Tai1964]. We begin by reviewing Tai’s derivation of the scattered field and then we use these fields to compute the force. As depicted in Fig. \[fig:CylindricalGeometry\], the velocity of the rotating body is ${\mathbf{v}}=\Omega r{\mathbf{\hat{{\phi}}}}$ at a radius $r$, the radius of the cylinder is denoted by $a$, and the ${\mathbf{E}}$ field of the incident wave is assumed to be polarized in the direction of the axis of the cylinder (which we take to be $\mathbf{\hat{z}}$). Derivations of key equations are provided in the appendix. We solve the scattering problem by standard technique of expanding the field in each region in basis of Bessel functions $J_{n}$ and then matching boundary conditions at the interface. In this basis, an incident $z$-polarized plane wave propagating in the $+x$ direction with amplitude $E_{0}$ (see Fig. \[fig:CylindricalGeometry\]) is given in polar $(r,\phi)$ coordinates by $$E_{i}=E_{0}\mbox{exp}\left\{ ik_{0}r\cos\phi\right\} =E_{0}\sum_{n=-\infty}^{+\infty}i^{n}J_{n}\left(k_{0}r\right)\mbox{exp}\left(in\phi\right),\label{eq:IncidentE_Tai}$$ where $E_{0}$ is the amplitude, $k_{0}=\omega/c$ is the wave number in vacuum, $\omega$ is the frequency in the time-harmonic oscillating field $e^{-i\omega t}$. The scattered and “transmitted” (interior) fields, respectively, can be written (using the Hankel function $H_{n}^{(1)}=J_{n}+iY_{n}$) with to-be-determined coefficients $\alpha_{n}$ and $\beta_{n}$: $$\begin{aligned} E_{s} & = & E_{0}\sum_{n=-\infty}^{+\infty}\alpha_{n}i^{n}H_{n}^{\left(1\right)}\left(k_{0}r\right)\mbox{exp}\left(in\phi\right).\label{eq:ScatteredE_Tai}\\ E_{t} & = & E_{0}\sum_{n=-\infty}^{+\infty}\beta_{n}i^{n}J_{n}\left(\gamma_{n}r\right)\mbox{exp}\left(in\phi\right),\label{eq:E_t_RAW}\end{aligned}$$ where $\gamma_{n}$ is defined by $$\begin{aligned} K & \equiv & \mu_{0}\left(\epsilon-\epsilon_{0}\right)\Omega\label{eq:K}\\ m & \equiv & 1-\frac{\epsilon_{0}}{\epsilon}\\ \gamma_{n}^{2} & = & k^{2}-2n\omega K=k^{2}\left(1-\frac{2nm\Omega}{\omega}\right).\label{eq:gamma_n}\end{aligned}$$ The total field is therefore $\mathbf{E}=E_{z}\hat{\mathbf{z}}=\left(E_{i}+E_{s}\right)\hat{\mathbf{z}}$ and the magnetic field in the vacuum regions is given by ${\mathbf{H}}=\frac{1}{i\omega\mu_{0}}\nabla\times{\mathbf{E}}$. The unknown coefficients $\beta_{n}$ and $\alpha_{n}$ are found by requiring continuity of $E_{z}$ and $H_{\phi}$ at $r=a$, yielding $$\begin{aligned} J_{n}\left(k_{0}a\right)+\alpha_{n}H_{n}^{\left(1\right)}\left(k_{0}a\right) & = & \beta_{n}J_{n}\left(\gamma_{n}a\right)\label{eq:BC_alpha_beta}\\ k_{0}\left[J_{n}'\left(k_{0}a\right)+\alpha_{n}H_{n}^{\left(1\right)}\left(k_{0}a\right)'\right] & = & \beta_{n}\gamma_{n}J_{n}'\left(\gamma_{n}a\right),\label{eq:BC_alpha_beta2}\end{aligned}$$ where the prime on $J$ and $H^{\left(1\right)}$ denotes derivative with respect to the entire argument. Solving for $\alpha_{n}$ gives $$\alpha_{n}=-\frac{J_{n}\left(\rho_{0}\right)\left[J_{n-1}\left(\rho_{n}\right)-J_{n+1}\left(\rho_{n}\right)\right]-\frac{k_{0}}{\gamma_{n}}J_{n}\left(\rho_{n}\right)\left[J_{n-1}\left(\rho_{0}\right)-J_{n+1}\left(\rho_{0}\right)\right]}{H_{n}^{\left(1\right)}\left(\rho_{0}\right)\left[J_{n-1}\left(\rho_{n}\right)-J_{n+1}\left(\rho_{n}\right)\right]-\frac{k_{0}}{\gamma_{n}}J_{n}\left(\rho_{n}\right)\left[H_{n-1}^{\left(1\right)}\left(\rho_{0}\right)-H_{n+1}^{\left(1\right)}\left(\rho_{0}\right)\right]}.\label{eq:alphan_simplified}$$ where $\rho_{0}=k_{0}a$ and $\rho_{n}=\gamma_{n}a$. For $\Omega\ne0$, the rotation breaks the $y=0$ mirror symmetry leading to asymmetrical scattering $\alpha_{n}\ne\alpha_{-n}$ as shown by Tai [@Tai1964]. If $\Omega=0$, then $\alpha_{n}=\alpha_{-n}$ and Eq. \[eq:ScatteredE\_Tai\] reduces to symmetrical scattering. \[sec:Force\]*Force imparted to the Rotating Cylinder.–* The asymmetry in the momentum transport by the scattered field should manifest itself as a lateral force on the dielectric. This force can be computed by integrating the Maxwell stress tensor over a closed surface around the object. Because we only evaluate the stress tensor in vacuum, we avoid the well-known difficulties that arise in defining the stress tensor inside the material [@LandauVol8(2ndEdition)], nor does the rotation affect the vacuum stress tensor. The stress tensor in SI units is $$\begin{aligned} \overleftrightarrow{{\mathbf{\sigma}}} & = & \epsilon_{0}{\mathbf{E}}\otimes{\mathbf{E}}+\mu_{0}{\mathbf{H}}\otimes{\mathbf{H}}\label{eq:StressTensorSI}\\ & - & \frac{1}{2}\left(\epsilon_{0}E^{2}+\mu_{0}H^{2}\right)\left({\mathbf{\hat{x}}}\otimes{\mathbf{\hat{x}}}+{\mathbf{\hat{y}}}\otimes{\mathbf{\hat{y}}}+{\mathbf{\hat{z}}}\otimes{\mathbf{\hat{z}}}\right)\nonumber \end{aligned}$$ where the hatted quantities are unit vectors. To calculate the force on the cylinder in any direction $\hat{\mathbf{n_{0}}}$ on the plane at a fixed radius $r_{0}$, we evaluate $$F_{{\mathbf{\hat{n_{0}}}}}=\frac{\omega}{2\pi}\int_{0}^{\frac{2\pi}{\omega}}dt\oint r\, dr\, d\phi\,\delta\left(r-r_{0}\right)\left\{ \hat{\mathbf{n_{0}}}\cdot\overleftrightarrow{{\mathbf{\sigma}}}\cdot\hat{\mathbf{r}}\right\} ,\label{eq:Force}$$ where $\hat{\mathbf{r}}=\cos\phi{\mathbf{\hat{x}}}+\sin\phi{\mathbf{\hat{y}}}$ and the time average is taken over a full period to obtain a real force. For our polarization, $E_{x}=E_{y}=H_{z}=0$. The force in $\hat{y}$ direction is $$\begin{aligned} F_{{\mathbf{\hat{y}}}} & = & r_{0}\oint d\phi\left\{ -\frac{\epsilon_{0}\left|E_{z}\right|^{2}+\mu_{0}\left|{\mathbf{H}}\right|^{2}}{4}\sin\phi\right.\label{eq:ForceY}\\ & + & \left.\frac{\mu_{0}}{2}\mbox{Re}\left(H_{x}^{*}H_{y}\right)\cos\phi+\frac{\mu_{0}\left|H_{y}\right|^{2}}{2}\sin\phi\right\} .\nonumber \end{aligned}$$ Using orthogonality conditions on $e^{in\phi}$, Eq. \[eq:ForceY\] can be integrated analytically. In the figures below we use the dimensionless force $\frac{F_{{\mathbf{\hat{n_{0}}}}}c}{P}$, where $P=2a\left|E_{0}\right|^{2}\sqrt{\frac{\epsilon_{0}}{\mu_{0}}}$ is the incident power on the scatterer’s geometric cross section. This is a convenient normalization because, for light incident on a perfectly absorbing flat surface the force is exactly $\frac{P}{c}$, so this normalization gives a measure of the lateral force relative to the incident photon pressure. Furthermore, we use the dimensionless angular frequency $\frac{a\Omega}{c}$, which is the ratio of speed of the cylinder boundary to that of light. Fig. \[fig:Force-vs.-rotational\] plots the force vs. angular frequency for $\epsilon/\epsilon_{0}=10$. As discussed below, whenever $\epsilon>\epsilon_{0}$, corresponding to a positive electric susceptibility $\chi^{(\mbox{e})}=\epsilon/\epsilon_{0}-1$, there is a force of sign analogous to Bernoulli’s law, where $\Omega>0$ (counterclockwise rotation) gives a force in the positive $y$ direction. Clearly, $\Omega<0$ gives the same magnitude of the force in the opposite direction as expected from the symmetry of the problem and in accordance with the fluid-mechanical analogy.  Normalized force vs. rotational frequency for $\lambda_{0}=0.01$ and $\epsilon/\epsilon_{0}=10$ where $P=2a\left|E_{0}\right|^{2}\sqrt{\frac{\epsilon_{0}}{\mu_{0}}}$.](Fy_wc_Apr22.pdf) In the case of a perfect conductor, $\beta_{n}$ is zero (see Eq. \[eq:E\_t\_RAW\]) and therefore Eqs. \[eq:BC\_alpha\_beta\] and \[eq:BC\_alpha\_beta2\] do not give an asymmetry with respect to $n$ for $\alpha_{n}$. Consequently, in this limit there is no lateral force. Intuitively, because a perfect conductor allows no penetration of the electromagnetic fields, the fields cannot “notice” that it is rotating or be “dragged” by the moving matter. However, for imperfect metals (finite $\epsilon<0$) there is some penetration of the radiation into the material which results in a lateral force. Interestingly, in the case of $\epsilon<0$, and in fact whenever $\epsilon<\epsilon_{0}$ (negative susceptibility), the force is in the opposite direction of the force for $\epsilon>\epsilon_{0}$ dielectrics (see Fig. \[fig:Metals\]). The reason is an immediate consequence of Eq. \[eq:K\]. For $\epsilon<\epsilon_{0}$ and $\Omega>0$, $K$ becomes negative and the phenomenology, looking at $\gamma_{n}$ in Eq. \[eq:gamma\_n\], become equivalent to the case of $\epsilon>\epsilon_{0}$ and $\Omega<0$. The same relationship between the sign of the force and the sign of ${\operatorname{Re}}\epsilon-\epsilon_{0}$ holds for complex $\epsilon$ as long as $|{\operatorname{Im}}\epsilon|\ll|{\operatorname{Re}}\epsilon|$, whereas for large $|{\operatorname{Im}}\epsilon|$ we observe a similar relationship with the sign of the imaginary part.  Normalized Force vs. Rotational frequency for $\lambda_{0}=0.01$ and various $\epsilon/\epsilon_{0}$ where $P=2a\left|E_{0}\right|^{2}\sqrt{\frac{\epsilon_{0}}{\mu_{0}}}$.](Negative_epsilon_Apr23.pdf)  Mie resonances: Normalized force vs. $\lambda_{0}$ for $a\Omega/c=3.34e-4$ and $a=0.01$.](f_vs_lam0-1.pdf) Lastly, we investigate the dependence of the normalized force on $2\pi\omega a/c=a/\lambda_{0}$, varying the vacuum wavelength $\lambda_{0}$ (see Fig. \[fig:Force-vs.-alam\]). For $\lambda_{0}\ll a$ the scattering approaches a ray-optics limit, while for $\lambda_{0}\gg a$ it is in the Rayleigh-scattering (dipole approximation) regime [@Jackson1998]. For $\lambda_{0}\sim a$, the force spectrum becomes more interesting due to the presence of Mie resonances [@Stratton1941]. *\[sec:Discussion-and-Future\]Discussion and Future Work.–* Given a finite amount of power, one would use a focused beam rather than a plane-wave, and an interesting question for future work is what beam width (and profile) maximizes the lateral force for a given total power; we conjecture that the optimal beam width should be comparable to the scattering cross-section. Furthermore, recent work has shown that an appropriate beam can form an optical tweezers [@ashkin1986] or tractor beam in which the sign of the longitudinal force on a non-spinning particle can reverse [@Chen2011; @lee2010; @Marston2006]. Applied to a spinning particle, the ability to change the sign of the longitudinal force implies that there should also be a zero point: a beam for which the force of is *purely lateral*. The forces obtained here are only a fraction of the incident radiation pressure and seem to require infeasible rotation rates, but we expect that they can be resonantly enhanced by techniques similar to those that have been used by other authors to enhance scattered power for a given particle diameter. Mie resonances are already visible in Fig. \[fig:Force-vs.-alam\], but much stronger resonant phenomena can be designed by using multilayer spheres that trap light using Bragg mirrors and/or specially designed surface plasmons, and one can even obtain “superscattering” by aligning multiple resonances at the same frequency [@Ruan11]. Material dispersion will contribute an additional source of lateral force: similar to the origin of quantum friction  [@Zhao12; @Manjavacas10; @Pendry10], the Doppler shift in the material dispersion should differ between the sides of the object moving toward and away from the light source, causing additional asymmetry in the scattered field and hence additional lateral force. Such enhancement mechanism, in combination with recent progress in generating rotating particles (of graphene) at near-GHz $\Omega$ [@kane2010], may permit the future experimental observation and exploitation of optical “Bernoulli” forces. SGJ was supported in part by the U.S. Army Research Office under contract W911NF-13-D-0001. Appendix ======== First let $\sigma,\mu,\epsilon$ be the conductivity, permeability and permittivity respectively and define $\mathbf{\Lambda}=\left(\epsilon\mu-\epsilon_{0}\mu_{0}\right){\mathbf{v}}$, where subscript $0$ corresponds to quantities in the vacuum. The electric and magnetic field vectors satisfy $$\begin{aligned} \nabla\times{\mathbf{E}} & = & -\frac{\partial}{\partial t}\left(\mu{\mathbf{H-\Lambda}}\times{\mathbf{E}}\right)\label{eq:EM_Tai-1}\\ \nabla\times{\mathbf{H}} & = & \sigma\left({\mathbf{E}}+\mu{\mathbf{v}}\times{\mathbf{H}}\right)+\frac{\partial}{\partial t}\left(\sigma{\mathbf{E+\Lambda}}\times{\mathbf{H}}\right).\label{eq:EMB_Tai-1}\end{aligned}$$ To determine the proper expression for the transmitted wave, we use harmonically oscillating fields to reduce Eqs. \[eq:EM\_Tai-1\] and \[eq:EMB\_Tai-1\] to (neglecting $\mathcal{O}\left(\left(v/c\right)^{2}\right)$ terms) $$\begin{aligned} \left(\nabla+i\omega{\mathbf{\Lambda}}\right)\times{\mathbf{E}} & = & i\omega\mu{\mathbf{H}}\label{eq:HarmonicReduction_1}\\ \left(\nabla+i\omega{\mathbf{\Lambda}}-\sigma\mu{\mathbf{v}}\right)\times{\mathbf{H}} & = & \left(\sigma-i\omega\epsilon\right){\mathbf{E}}\label{eq:HarmonicReduction_2}\end{aligned}$$ Looking at Figure \[fig:CylindricalGeometry\], the incident wave is given by $$E_{i}=E_{0}\mbox{exp}\left\{ ik_{0}r\cos\phi\right\} =E_{0}\sum_{n=-\infty}^{+\infty}i^{n}J_{n}\left(k_{0}r\right)\mbox{exp}\left(in\phi\right),\label{eq:IncidentE_Tai-1}$$ where $k_{0}=\omega/c$ is the wave number in vacuo; $\omega$ being the frequency in the time harmonic oscillating field $e^{i\omega t}$. The scattering field can be written in the form $$E_{s}=E_{0}\sum_{n=-\infty}^{+\infty}\alpha_{n}i^{n}H_{n}^{\left(1\right)}\left(k_{0}r\right)\mbox{exp}\left(in\phi\right).\label{eq:ScatteredE_Tai-1}$$ To solve for the transmitted field inside the dielectric we subject Eqs. \[eq:HarmonicReduction\_1\] and \[eq:HarmonicReduction\_2\] to the particular form of the velocity which is independent of $\mathbf{\hat{z}}$. The parameter ${\mathbf{\Lambda}}$ is then a function of $r$ alone, $${\mathbf{\Lambda=}}\mu_{0}\left(\epsilon-\epsilon_{0}\right){\mathbf{v}}=Kr\mathbf{\hat{{\mathbf{\phi}}}}\label{eq:Lambda-1}$$ where $K\equiv\mu_{0}\epsilon\left(1-\frac{\epsilon_{0}}{\epsilon}\right)\Omega=\frac{m\Omega}{c^{2}}$, with $m\equiv1-\frac{\epsilon_{0}}{\epsilon}$ and $c^{2}=1/\mu_{0}\epsilon$. Substituting Eq. \[eq:Lambda-1\] into Eqs. \[eq:HarmonicReduction\_1\] and \[eq:HarmonicReduction\_2\] and eliminating ${\mathbf{H}}$ we obtain a differential equation for $E_{z}$ which is the only component of the electric field inside the dielectric cylinder. $$\frac{1}{r}\frac{\partial}{\partial r}\left(r\frac{\partial E_{z}}{\partial r}\right)+\frac{1}{r^{2}}\frac{\partial^{2}E_{z}}{\partial\phi^{2}}+2i\omega K\frac{\partial E_{z}}{\partial\phi}+k^{2}E_{z}+\mathcal{O}\left(\left(\frac{v}{c}\right)^{2}\right)=0,\label{eq:DiffEq_Ez-1}$$ where $k^{2}=\left(\omega/c\right)^{2}$. To solve let us seek separable solutions for $E_{z}=F\left(r\right)e^{in\phi}$ and below we drop $\mathcal{O}\left(\left(\frac{v}{c}\right)^{2}\right)$ by understanding that the results are accurate to first order. The function $F\left(r\right)$ then satisfies $$\frac{1}{r}\frac{\partial}{\partial r}\left(r\frac{\partial F}{\partial r}\right)-\left(\frac{n^{2}}{r^{2}}+2n\omega K-k^{2}\right)F=0.$$ If we introduce $\gamma_{n}^{2}=k^{2}-2n\omega K=k^{2}\left(1-\frac{2nm\Omega}{\omega}\right),$ then the proper set of radial functions to describe the field inside the rotating cylinder is $$F\left(r\right)=J_{n}\left(\gamma_{n}r\right),\qquad n=0,\pm1,\pm2,\cdots.$$ The complete expression for the transmitted field can be written in the form $$E_{t}=E_{0}\sum_{n=-\infty}^{+\infty}\beta_{n}i^{n}J_{n}\left(\gamma_{n}r\right)\mbox{exp}\left(in\phi\right).\label{eq:E_t_RAW-1}$$ By matching $E_{z}$ as defined by Eqs. \[eq:IncidentE\_Tai-1\], \[eq:ScatteredE\_Tai-1\] and \[eq:E\_t\_RAW-1\], and the $\phi-$component of the magnetic field at the boundary $r=a$ one obtains the following two simultaneous equations: $$\begin{aligned} J_{n}\left(k_{0}a\right)+\alpha_{n}H_{n}^{\left(1\right)}\left(k_{0}a\right) & = & \beta_{n}J_{n}\left(\gamma_{n}a\right)\label{eq:BC_alpha_beta-1}\\ k_{0}\left[J_{n}'\left(k_{0}a\right)+\alpha_{n}H_{n}^{\left(1\right)}\left(k_{0}a\right)'\right] & = & \beta_{n}\gamma_{n}J_{n}'\left(\gamma_{n}a\right),\end{aligned}$$ where the prime on $J$ and $H^{\left(1\right)}$ denotes derivative with respect to the entire argument of these functions. The solutions for $\alpha_{n}$ and $\beta_{n}$ are $$\begin{aligned} \alpha_{n} & = & -\frac{J_{n}\left(\rho_{0}\right)J'_{n}\left(\rho_{n}\right)-\frac{k_{0}}{\gamma_{n}}J_{n}\left(\rho_{n}\right)J'_{n}\left(\rho_{0}\right)}{H_{n}^{\left(1\right)}\left(\rho_{0}\right)J'_{n}\left(\rho_{n}\right)-\frac{k_{0}}{\gamma_{n}}J_{n}\left(\rho_{n}\right)H_{n}^{\left(1\right)}\mbox{}'\left(\rho_{0}\right)}\label{eq:alpha_n-1}\\ \beta_{n} & = & -\frac{k_{0}}{\gamma_{n}}\frac{\left[J_{n}\left(\rho_{0}\right)H_{n}^{\left(1\right)}\mbox{}'\left(\rho_{0}\right)-J_{n}'\left(\rho_{0}\right)H_{n}^{\left(1\right)}\left(\rho_{0}\right)\right]}{H_{n}^{\left(1\right)}\left(\rho_{0}\right)J'_{n}\left(\rho_{n}\right)-\frac{k_{0}}{\gamma_{n}}J_{n}\left(\rho_{n}\right)H_{n}^{\left(1\right)}\mbox{}'\left(\rho_{0}\right)},\label{eq:beta_n-1}\end{aligned}$$ where $\rho_{0}=k_{0}a$ and $\rho_{n}=\gamma_{n}a$. Using identities $J'_{n}\left(x\right)=\frac{1}{2}\left[J_{n-1}\left(x\right)-J_{n+1}\left(x\right)\right]$, $J'_{0}\left(x\right)=-J_{1}\left(x\right)$ and $H_{n}^{\left(1\right)}\mbox{}'\left(x\right)=\frac{1}{2}\left[H_{n-1}^{\left(1\right)}\left(x\right)-H_{n+1}^{\left(1\right)}\left(x\right)\right]$, $H_{0}^{\left(1\right)}\mbox{}'\left(x\right)=-H_{1}^{\left(1\right)}\left(x\right)$, $\beta_{n}$ can be eliminated to give $$\alpha_{n}=-\frac{J_{n}\left(\rho_{0}\right)\left[J_{n-1}\left(\rho_{n}\right)-J_{n+1}\left(\rho_{n}\right)\right]-\frac{k_{0}}{\gamma_{n}}J_{n}\left(\rho_{n}\right)\left[J_{n-1}\left(\rho_{0}\right)-J_{n+1}\left(\rho_{0}\right)\right]}{H_{n}^{\left(1\right)}\left(\rho_{0}\right)\left[J_{n-1}\left(\rho_{n}\right)-J_{n+1}\left(\rho_{n}\right)\right]-\frac{k_{0}}{\gamma_{n}}J_{n}\left(\rho_{n}\right)\left[H_{n-1}^{\left(1\right)}\left(\rho_{0}\right)-H_{n+1}^{\left(1\right)}\left(\rho_{0}\right)\right]}.\label{eq:alphan_simplified-1}$$ The numerical value of $\alpha_{n}\ne\alpha_{-n}$ for $\Omega\ne0$ because in this case $\rho_{n}\ne\rho_{-n}$ and $\gamma_{n}\ne\gamma_{-n}$ hence the scattering field has an asymmetrical part with respect to the direction of incidence, $\phi=0$. When $\Omega=0\implies\alpha_{n}=\alpha_{-n}$ and Eq. \[eq:ScatteredE\_Tai-1\] reduces to the well known results. The total field therefore is $\mathbf{E}=E_{z}\hat{\mathbf{z}}=\left(E_{i}+E_{s}\right)\hat{\mathbf{z}}$. Below we suppress $k_{0}r$ as the argument of the Bessel functions unless stated otherwise. $$E_{z}=E_{0}\sum_{n=-\infty}^{+\infty}i^{n}\left\{ J_{n}+\alpha_{n}H_{n}^{\left(1\right)}\right\} e^{in\phi}\label{eq:Ez-1-2}$$ Further in free space we have ${\mathbf{H}}=\frac{1}{i\omega\mu_{0}}\nabla\times{\mathbf{E}}$ which gives $$\begin{aligned} {\mathbf{H}} & = & \frac{1}{i\omega\mu_{0}}\nabla\times E\hat{{\mathbf{z}}}=\frac{1}{i\omega\mu_{0}}\left(\frac{1}{r}\frac{\partial E_{z}}{\partial\phi}\hat{{\mathbf{r}}}-\frac{\partial E_{z}}{\partial r}\hat{{\mathbf{\phi}}}\right)\label{eq:H-1}\\ & = & \frac{1}{i\omega\mu_{0}}\left(\frac{1}{r}\frac{\partial E_{z}}{\partial\phi}\cos\phi+\frac{\partial E_{z}}{\partial r}\sin\phi\right)\hat{{\mathbf{x}}}\nonumber \\ & + & \frac{1}{i\omega\mu_{0}}\left(\frac{1}{r}\frac{\partial E_{z}}{\partial\phi}\sin\phi-\frac{\partial E_{z}}{\partial r}\cos\phi\right)\hat{{\mathbf{y}}}.\nonumber \end{aligned}$$ Let ${\mathbf{\hat{n_{0}}=}}\cos\psi{\mathbf{\hat{x}}}+\sin\psi{\mathbf{\hat{y}}}$ be any unit vector, then Eq. \[eq:StressTensorSI\], evaluated at the radius $r_{0}$, reads $$\begin{aligned} F_{{\mathbf{\hat{n_{0}}}}} & = & r_{0}\oint d\phi\left\{ -\frac{\epsilon_{0}\left|E\right|^{2}+\mu_{0}\left|H\right|^{2}}{4}\cos\left(\psi-\phi\right)\right.\label{eq:ForceGeneralEz}\\ & + & \frac{\mu_{0}}{2}\mbox{Re}\left(H_{x}^{*}H_{y}\right)\sin\left(\psi+\phi\right)\nonumber \\ & + & \left.\frac{\mu_{0}\left|H_{x}\right|^{2}}{2}\cos\phi\cos\psi+\frac{\mu_{0}\left|H_{y}\right|^{2}}{2}\sin\phi\sin\psi\right\} .\nonumber \end{aligned}$$ In particular we are interested in $\psi=\frac{\pi}{2}$ to calculate the transverse force $$\begin{aligned} F_{{\mathbf{\hat{y}}}} & = & r_{0}\oint d\phi\left\{ -\frac{\epsilon_{0}\left|E\right|^{2}+\mu_{0}\left|H\right|^{2}}{4}\sin\phi\right.\label{eq:ForceY-1}\\ & + & \left.\frac{\mu_{0}}{2}\mbox{Re}\left(H_{x}^{*}H_{y}\right)\cos\phi+\frac{\mu_{0}\left|H_{y}\right|^{2}}{2}\sin\phi\right\} .\end{aligned}$$ where $E=\left(E_{i}+E_{s}\right)$ is given by Eqs. \[eq:IncidentE\_Tai\] and \[eq:ScatteredE\_Tai\]. Evaluating the Integral ======================= Here we evaluate $\left|E\right|^{2}$, $\left|H\right|^{2}$, $\left|H_{y}^{2}\right|$ and $\mbox{Re}\left(H_{x}^{*}H_{y}\right)$ as they are useful for calculating the force below (Eqs. \[eq:ForceGeneralEz\] and \[eq:ForceY\]). The key equations are $$\begin{aligned} E_{z} & = & E_{0}\sum_{n=-\infty}^{+\infty}i^{n}\left\{ J_{n}+\alpha_{n}H_{n}^{\left(1\right)}\right\} e^{in\phi}\\ {\mathbf{H}} & \equiv & H_{x}\hat{{\mathbf{x}}}+H_{y}\hat{{\mathbf{y}}}\\ & = & \frac{1}{i\omega\mu_{0}}\left(\frac{1}{r}\frac{\partial E_{z}}{\partial\phi}\cos\phi+\frac{\partial E_{z}}{\partial r}\sin\phi\right)\hat{{\mathbf{x}}}\\ & + & \frac{1}{i\omega\mu_{0}}\left(\frac{1}{r}\frac{\partial E_{z}}{\partial\phi}\sin\phi-\frac{\partial E_{z}}{\partial r}\cos\phi\right)\hat{{\mathbf{y}}}.\end{aligned}$$ where $$\begin{aligned} H_{x} & = & \frac{E_{0}}{i\omega\mu_{0}}\sum_{n=-\infty}^{+\infty}e^{in\phi}i^{n}\left\{ \frac{in}{r}\left[J_{n}+\alpha_{n}H_{n}^{\left(1\right)}\right]\cos\phi+\frac{k_{0}}{2}\left[J_{n-1}-J_{n+1}+\alpha_{n}\left(H_{n-1}^{\left(1\right)}-H_{n+1}^{\left(1\right)}\right)\right]\sin\phi\right\} \\ H_{y} & = & \frac{E_{0}}{i\omega\mu_{0}}\sum_{n=-\infty}^{+\infty}e^{in\phi}i^{n}\left\{ \frac{in}{r}\left[J_{n}+\alpha_{n}H_{n}^{\left(1\right)}\right]\sin\phi-\frac{k_{0}}{2}\left[J_{n-1}-J_{n+1}+\alpha_{n}\left(H_{n-1}^{\left(1\right)}-H_{n+1}^{\left(1\right)}\right)\right]\cos\phi\right\} \end{aligned}$$ as well as the orthogonality relations $$\begin{aligned} \oint d\phi e^{i\left(n-m\right)\phi}\sin\phi & = & \left(-i\pi\right)\left(\delta_{n,m-1}-\delta_{n,m+1}\right)\\ \oint d\phi e^{i\left(n-m\right)\phi}\sin^{3}\phi & = & \left(\frac{i\pi}{4}\right)\left(\delta_{n,m-3}-\delta_{n,m+3}-3\delta_{n,m-1}+3\delta_{n,m+1}\right)\\ \oint d\phi e^{i\left(n-m\right)\phi}\sin^{2}\phi\cos\phi & = & \left(\frac{-\pi}{4}\right)\left(\delta_{n,m-3}+\delta_{n,m+3}-\delta_{n,m-1}-\delta_{n,m+1}\right)\\ \oint d\phi e^{i\left(n-m\right)\phi}\sin\phi\cos^{2}\phi & = & \left(\frac{-i\pi}{4}\right)\left(\delta_{n,m-3}-\delta_{n,m+3}+\delta_{n,m-1}-\delta_{n,m+1}\right)\\ \oint d\phi e^{i\left(n-m\right)\phi}\cos^{3}\phi & = & \left(\frac{\pi}{4}\right)\left(\delta_{n,m-3}+\delta_{n,m+3}+3\delta_{n,m-1}+3\delta_{n,m+1}\right).\end{aligned}$$ We proceed $$\begin{aligned} \oint d\phi\left|E_{z}\right|^{2}\sin\phi & = & E_{0}^{2}\sum_{n,m=-\infty}^{+\infty}\oint d\phi e^{i\left(n-m\right)\phi}\sin\phi\left(-1\right)^{m}i^{n+m}\left\{ J_{n}+\alpha_{n}H_{n}^{\left(1\right)}\right\} \left\{ J_{m}+\alpha_{m}^{*}H_{m}^{\left(2\right)}\right\} \\ & = & E_{0}^{2}\sum_{n,m=-\infty}^{+\infty}\left(-1\right)^{m}i^{n+m}\left\{ J_{n}+\alpha_{n}H_{n}^{\left(1\right)}\right\} \left\{ J_{m}+\alpha_{m}^{*}H_{m}^{\left(2\right)}\right\} \left(-i\pi\right)\left\{ \delta_{n,m-1}-\delta_{n,m+1}\right\} \end{aligned}$$ Further $\oint d\phi\left|H_{x}\right|^{2}\sin\phi=\oint d\phi\left(H_{x}^{*}H_{x}\right)\sin\phi$, similarly for $\oint d\phi\left|H_{y}\right|^{2}\sin\phi$ $$\begin{aligned} \oint d\phi\left|H_{x}\right|^{2}\sin\phi & = & \left(\frac{E_{0}}{\omega\mu_{0}}\right)^{2}\sum_{n,m=-\infty}^{+\infty}\left(-1\right)^{m}i^{n+m}\oint d\phi\sin\phi e^{i\left(n-m\right)\phi}\\ & & \left\{ \left(\frac{-im}{r}\left[J_{m}+\alpha_{m}^{*}H_{m}^{\left(2\right)}\right]\cos\phi+\frac{k_{0}}{2}\left[J_{m-1}-J_{m+1}+\alpha_{m}^{*}\left(H_{m-1}^{\left(2\right)}-H_{m+1}^{\left(2\right)}\right)\right]\sin\phi\right)\right.\\ & \times & \left.\left(\frac{in}{r}\left[J_{n}+\alpha_{n}H_{n}^{\left(1\right)}\right]\cos\phi+\frac{k_{0}}{2}\left[J_{n-1}-J_{n+1}+\alpha_{n}\left(H_{n-1}^{\left(1\right)}-H_{n+1}^{\left(1\right)}\right)\right]\sin\phi\right)\right\} \end{aligned}$$ which using the orthogonality relations yields $$\begin{aligned} \oint d\phi\left|H_{x}\right|^{2}\sin\phi & = & \left(\frac{E_{0}}{\omega\mu_{0}}\right)^{2}\sum_{n,m=-\infty}^{+\infty}\left(-1\right)^{m}i^{n+m}\\ & & \frac{nm}{r^{2}}\left(J_{m}+\alpha_{m}^{*}H_{m}^{\left(2\right)}\right)\left(J_{n}+\alpha_{n}H_{n}^{\left(1\right)}\right)\left(\frac{-i\pi}{4}\right)\left(\delta_{n,m-3}-\delta_{n,m+3}+\delta_{n,m-1}-\delta_{n,m+1}\right)\\ & + & \frac{k_{0}^{2}}{4}\left[J_{m-1}-J_{m+1}+\alpha_{m}^{*}\left(H_{m-1}^{\left(2\right)}-H_{m+1}^{\left(2\right)}\right)\right]\left[J_{n-1}-J_{n+1}+\alpha_{n}\left(H_{n-1}^{\left(1\right)}-H_{n+1}^{\left(1\right)}\right)\right]\\ & \times & \left(\frac{i\pi}{4}\right)\left(\delta_{n,m-3}-\delta_{n,m+3}-3\delta_{n,m-1}+3\delta_{n,m+1}\right)\\ & + & \frac{ik_{0}}{2r}\left[-m\left(J_{m}+\alpha_{m}^{*}H_{m}^{\left(2\right)}\right)\left(J_{n-1}-J_{n+1}+\alpha_{n}\left(H_{n-1}^{\left(1\right)}-H_{n+1}^{\left(1\right)}\right)\right)\right.\\ & & \;\left.+n\left(J_{n}+\alpha_{n}H_{n}^{\left(1\right)}\right)\left(J_{m-1}-J_{m+1}+\alpha_{m}^{*}\left(H_{m-1}^{\left(2\right)}-H_{m+1}^{\left(2\right)}\right)\right)\right]\\ & \times & \left(\frac{-\pi}{4}\right)\left(\delta_{n,m-3}+\delta_{n,m+3}-\delta_{n,m-1}-\delta_{n,m+1}\right)\end{aligned}$$ Similarly $$\begin{aligned} \oint d\phi\left|H_{y}\right|^{2}\sin\phi & = & \left(\frac{E_{0}}{\omega\mu_{0}}\right)^{2}\sum_{n,m=-\infty}^{+\infty}\left(-1\right)^{m}i^{n+m}\oint d\phi\sin\phi e^{i\left(n-m\right)\phi}\\ & & \left\{ \left(\frac{-im}{r}\left[J_{m}+\alpha_{m}^{*}H_{m}^{\left(2\right)}\right]\sin\phi-\frac{k_{0}}{2}\left[J_{m-1}-J_{m+1}+\alpha_{m}^{*}\left(H_{m-1}^{\left(2\right)}-H_{m+1}^{\left(2\right)}\right)\right]\cos\phi\right)\right.\\ & \times & \left.\left(\frac{in}{r}\left[J_{n}+\alpha_{n}H_{n}^{\left(1\right)}\right]\sin\phi-\frac{k_{0}}{2}\left[J_{n-1}-J_{n+1}+\alpha_{n}\left(H_{n-1}^{\left(1\right)}-H_{n+1}^{\left(1\right)}\right)\right]\cos\phi\right)\right\} \end{aligned}$$ which using the orthogonality relations yields $$\begin{aligned} \oint d\phi\left|H_{y}\right|^{2}\sin\phi & = & \left(\frac{E_{0}}{\omega\mu_{0}}\right)^{2}\sum_{n,m=-\infty}^{+\infty}\left(-1\right)^{m}i^{n+m}\\ & & \frac{mn}{r^{2}}\left(J_{n}+\alpha_{n}H_{n}^{\left(1\right)}\right)\left(J_{m}+\alpha_{m}^{*}H_{m}^{\left(2\right)}\right)\left(\frac{i\pi}{4}\right)\left(\delta_{n,m-3}-\delta_{n,m+3}-3\delta_{n,m-1}+3\delta_{n,m+1}\right)\\ & + & \frac{k_{0}^{2}}{4}\left(J_{m-1}-J_{m+1}+\alpha_{m}^{*}\left(H_{m-1}^{\left(2\right)}-H_{m+1}^{\left(2\right)}\right)\right)\left(J_{n-1}-J_{n+1}+\alpha_{n}\left(H_{n-1}^{\left(1\right)}-H_{n+1}^{\left(1\right)}\right)\right)\\ & \times & \left(\frac{-i\pi}{4}\right)\left(\delta_{n,m-3}-\delta_{n,m+3}+\delta_{n,m-1}-\delta_{n,m+1}\right)\\ & + & \frac{ik_{0}}{2r}\left[m\left(J_{m}+\alpha_{m}^{*}H_{m}^{\left(2\right)}\right)\left(J_{n-1}-J_{n+1}+\alpha_{n}\left(H_{n-1}^{\left(1\right)}-H_{n+1}^{\left(1\right)}\right)\right)\right.\\ & & \;\left.-n\left(J_{n}+\alpha_{n}H_{n}^{\left(1\right)}\right)\left(J_{m-1}-J_{m+1}+\alpha_{m}^{*}\left(H_{m-1}^{\left(2\right)}-H_{m+1}^{\left(2\right)}\right)\right)\right]\\ & \times & \left(\frac{-\pi}{4}\right)\left(\delta_{n,m-3}+\delta_{n,m+3}-\delta_{n,m-1}-\delta_{n,m+1}\right).\end{aligned}$$ Lastly we need $\oint d\phi\mbox{Re}\left(H_{x}^{*}H_{y}\right)\cos\phi=\frac{1}{2}\oint d\phi H_{x}^{*}H_{y}\cos\phi+c.c.$ $$\begin{aligned} \oint d\phi H_{x}^{*}H_{y}\cos\phi & = & \left(\frac{E_{0}}{\omega\mu_{0}}\right)^{2}\sum_{n,m=-\infty}^{+\infty}\left(-1\right)^{m}i^{n+m}\oint d\phi\cos\phi e^{i\left(n-m\right)\phi}\\ & & \left\{ \left(\frac{-im}{r}\left[J_{m}+\alpha_{m}^{*}H_{m}^{\left(2\right)}\right]\cos\phi+\frac{k_{0}}{2}\left[J_{m-1}-J_{m+1}+\alpha_{m}^{*}\left(H_{m-1}^{\left(2\right)}-H_{m+1}^{\left(2\right)}\right)\right]\sin\phi\right)\right.\\ & \times & \left.\left(\frac{in}{r}\left[J_{n}+\alpha_{n}H_{n}^{\left(1\right)}\right]\sin\phi-\frac{k_{0}}{2}\left[J_{n-1}-J_{n+1}+\alpha_{n}\left(H_{n-1}^{\left(1\right)}-H_{n+1}^{\left(1\right)}\right)\right]\cos\phi\right)\right\} \end{aligned}$$ which using the orthogonality relations yields $$\begin{aligned} \oint d\phi H_{x}^{*}H_{y}\cos\phi & = & \left(\frac{E_{0}}{\omega\mu_{0}}\right)^{2}\sum_{n,m=-\infty}^{+\infty}\left(-1\right)^{m}i^{n+m}\\ & & \left[\frac{mn}{r^{2}}\left(J_{m}+\alpha_{m}^{*}H_{m}^{\left(2\right)}\right)\left(J_{n}+\alpha_{n}H_{n}^{\left(1\right)}\right)-\frac{k_{0}^{2}}{4}\left(J_{m-1}-J_{m+1}+\alpha_{m}^{*}\left(H_{m-1}^{\left(2\right)}-H_{m+1}^{\left(2\right)}\right)\right)\right.\\ & & \left.\times\left(J_{n-1}-J_{n+1}+\alpha_{n}\left(H_{n-1}^{\left(1\right)}-H_{n+1}^{\left(1\right)}\right)\right)\left(-\frac{i\pi}{4}\right)\right]\left(\delta_{n,m-3}-\delta_{n,m+3}+\delta_{n,m-1}-\delta_{n,m+1}\right)\\ & + & \frac{imk_{0}}{2r}\left(J_{m}+\alpha_{m}^{*}H_{m}^{\left(2\right)}\right)\left(J_{n-1}-J_{n+1}+\alpha_{n}\left(H_{n-1}^{\left(1\right)}-H_{n+1}^{\left(1\right)}\right)\right)\\ & & \times\left(\frac{\pi}{4}\right)\left(\delta_{n,m-3}+\delta_{n,m+3}+3\delta_{n,m-1}+3\delta_{n,m+1}\right)\\ & + & \frac{ink_{0}}{2r}\left(J_{n}+\alpha_{n}H_{n}^{\left(1\right)}\right)\left(J_{m-1}-J_{m+1}+\alpha_{m}^{*}\left(H_{m-1}^{\left(2\right)}-H_{m+1}^{\left(2\right)}\right)\right)\\ & & \times\left(\frac{-\pi}{4}\right)\left(\delta_{n,m-3}+\delta_{n,m+3}-\delta_{n,m-1}-\delta_{n,m+1}\right).\end{aligned}$$ All of the above integrals were checked against numerics before calculating the cumulative effect that appears in the force Eq. \[eq:ForceY\]. The total force was also checked against numerical experiments. In all cases agreements were found with errors of order $\mathcal{O}\left(10^{-26}\right)$.

--- abstract: 'A tetragonal phase is predicted for $\mathrm{Hf_2O_3}$ and $\mathrm{Zr_2O_3}$ using density functional theory. Starting from atomic and unit cell relaxations of substoichiometric monoclinic $\mathrm{HfO_2}$ and $\mathrm{ZrO_2}$, such tetragonal structures are only reached at zero temperature by introducing the oxygen vacancy pair with the lowest formation energy. The tetragonal $\mathrm{Hf_2O_3}$ and $\mathrm{Zr_2O_3}$ structures belong to space group $P\mathrm{\bar{4}}m\mathrm{2}$ and are more stable than their corundum structure counterparts. These phases are semi-metallic, as confirmed through further $\mathrm{G_0W_0}$ calculations. The carrier concentrations are estimated to be $1.77\times{10^{21}}$$\mathrm{cm^{-3}}$ for both electrons and holes in tetragonal $\mathrm{Hf_2O_3}$, and $1.75\times{10^{21}}$$\mathrm{cm^{-3}}$ for both electrons and holes in tetragonal $\mathrm{Zr_2O_3}$. The tetragonal $\mathrm{Hf_2O_3}$ phase is probably related to the low resistivity state of hafnia-based resistive random access memory (RRAM).' author: - 'Kan-Hao Xue' - Philippe Blaise - 'Leonardo R. C. Fonseca' - Yoshio Nishi title: 'Prediction of semi-metallic tetragonal $\mathrm{Hf_2O_3}$ and $\mathrm{Zr_2O_3}$ from first-principles' --- Hafnia ($\mathrm{HfO_2}$) and zirconia ($\mathrm{ZrO_2}$) are found in a number of important technological applications.^^ In particular, hafnia has become a key component in sub-micrometer silicon MOS technology as the current choice of high permittivity dielectric layer.^^ In addition, it is also a promising candidate material for resistive random access memory (RRAM), which is one of the leading technologies for the next-generation non-volatile memory.^^ The core element of RRAM is a metal/insulator/metal capacitor which is subject to an electroforming process, where a high electric field (some MV/cm) is applied across the capacitor to create conduction paths in the insulating thin film, here named filaments. These filaments are of unknown composition or shape, and can be easily disturbed under electrical stress, leading to a memory effect. Knowing the composition of the filaments is crucial to the understanding of RRAM’s physics. Previous work on $\mathrm{TiO_2}$ RRAM reveals that the conductive filament is possibly due to $\mathrm{Ti_nO_{2n-1}}$ Magnéli phases, where the value of $n$ is mostly 4 or 5.^^ For hafnia RRAM, the structure of the conductive filaments has not been reported, though it is widely accepted that the filaments are associated to an oxygen-deficient phase.^^ Since the impact of electroforming is expected to occur in small and random patches of the capacitor, experimental investigation of the filaments suffers from great difficulty. In the present paper we employ first-principles density functional theory^^ (DFT) calculations to identify O-poor stable compositions of hafnium and zirconium oxides which may be reachable from the room temperature normal pressure monoclinic $\mathrm{Hf(Zr)O_2}$ \[$m$-$\mathrm{Hf(Zr)O_2}$\] with the assistance of an external source of energy, possibly an applied electric field. The processing and operation of oxide-based RRAM (OXRRAM) stimulated our search of a conductive phase in these materials. However, our predictions are quite general and may have broader implications. For DFT calculations, plane-wave based Vienna Ab initio Simulation Package (VASP) program^^ was implemented, using projector-augmented-wave (PAW) pseudopotentials^^ with Hf $5p$, $5d$ and $6s$ (Zr $4s$, $4p$, $4d$ and $5s$) electrons and O $2s$ and $2p$ electrons in the valence. Generalized gradient approximation (GGA) was used for the exchange-correlation energy, within the Perdew-Burke-Ernzerhof (PBE) functional.^^ The plane wave energy cutoff is chosen as 500eV which converges for all the involved compounds, and sufficiently dense Monkhorst-Pack^^ or $\Gamma$-centered $k$-mesh was utilized for sampling the Brillouin zone. Because DFT/GGA usually underestimates band gaps, and may even deem a material metallic rather than semiconductor as the case of bulk germanium,^^ for the metallic or semi-metallic candidates we calculated the first order energy shifts with $\mathrm{GW}$ approximation^^ ($\mathrm{G_0W_0}$) using the ABINIT^^ program. Hf (Zr) semi-core electrons were explicitly included through the $5s^25p^65d^26s^2$ ($4s^24p^64d^25s^2$) configuration while core electrons were replaced by Troullier-Martins pseudopotentials.^^ Convergence was achieved with 360 bands and a 15 Ha cutoff for the wave functions employed in the evaluation of the dielectric function and the $\Sigma$ function. A plasmon-pole approximation^^ was used. The $m$-$\mathrm{Hf(Zr)O_2}$ unit cells were fully relaxed until all Hellmann-Feynman forces were less than 0.01eV/Å and all stresses were less than 400MPa. The relaxed $m$-$\mathrm{Hf(Zr)O_2}$ unit cell parameters are a=5.146(5.219)Å, b/a=1.010(1.012), c/a=1.036(1.036) and $\beta=99.68^{\mathrm{o}}$($99.68^{\mathrm{o}}$), close to experimental values:^^ a=5.117(5.151)Å, b/a=1.011(1.010), c/a=1.034(1.032) and $\beta=99.22^{\mathrm{o}}$($99.20^{\mathrm{o}}$). The formation enthalpy of $m$-$\mathrm{Hf(Zr)O_2}$ calculated with respect to $h.c.p.$ Hf (Zr) and an isolated $\mathrm{O_2}$ molecule is -1166(-1106)kJ/mol, after adopting the 1.36eV energy correction for the $\mathrm{O_2}$ molecule provided by Wang *et al*.^^ These results are in accord with the experimental values, which are -1145kJ/mol for hafnia and -1101kJ/mol for zirconia.^^ {width="3.5in" height="1.98in"} For reference we first calculated the formation energies of single and double oxygen vacancies in a 96-atom $m$-$\mathrm{Hf(Zr)O_2}$ supercell ($2\times2\times2$). Aiming at energetically favorable metallic phases, only the neutral oxygen vacancy was considered. There are two inequivalent O sites regarding O coordination, namely the 3- and 4-coordinated O(A) and O(B) sites, respectively. The formation energy of a neutral oxygen vacancy is defined as $$E_{\mathrm{for}} = E_\mathrm{D} - E_\mathrm{0} + \mu_\mathrm{O}$$ where $E_\mathrm{D}$ is the energy of the defective supercell, $E_\mathrm{0}$ is the energy of the defect-free supercell and $\mu_\mathrm{O}$ is the chemical potential of oxygen. Under oxygen-rich condition $\mu_\mathrm{O}$ is usually set as one half of that of an $\mathrm{O_2}$ molecule. In this case the calculated $E_{\mathrm{for}}$ for a neutral O(A) vacancy in $m$-$\mathrm{Hf(Zr)O_2}$ is 7.13(6.62)eV, while for a neutral O(B) vacancy it is 7.00(6.53)eV. The difference between the two formation energies, 0.13(0.09)eV, is similar to Zheng *et al.*,^^ but larger than Foster *et al.* who reported a 0.02eV difference in both cases.^^ Next we calculated the formation energies of di-oxygen-vacancy pairs in $m$-$\mathrm{Hf(Zr)O_2}$. To this end, the 8 oxygen sites in a unit cell were named A1–A4 and B1–B4, as shown in Fig. 1. The distance between two di-oxygen-vacancy pairs is around 10Å, casting them as isolated pairs. Since $m$-$\mathrm{Hf(Zr)O_2}$ possesses the baddeleyite structure with space group $P\mathrm{2}_\mathrm{1}/c$, there are 10 inequivalent di-oxygen-vacancy pairs. The most energetically favorable pair is the (B1,B2) pair, whose formation energy per vacancy is the same as of a single O(B) vacancy. Under usual experimental conditions the dielectric is placed between two metal electrodes where oxygen can migrate as an interstitial. We thus calculated the incorporation energies of oxygen, starting from its molecular form, into bulk $h.c.p.$ Ti and $f.c.c.$ Pt, two commonly used electrodes. The results are -6.24 eV and 0.91 eV, respectively. The formation energies of single and double oxygen vacancy in all three environments are thus compared in Tab. [slowromancap1@]{}. Notice that the formation energy of a neutral O(B) vacancy in $m$-$\mathrm{Hf(Zr)O_2}$ plus an oxygen interstitial in $h.c.p.$ Ti is merely 0.76(0.29)eV, which is attributed to the strong Ti–O bonding. -------------------------- ---------- --------------------- ---------------- ------------- ------------ Chemical Vacancy Supercell formula site(s) units $\mathrm{O_2}$ Ti Pt $\mathrm{Me_{32}O_{63}}$ O(A) 2$\times$2$\times$2 7.13(6.62) 0.89(0.38) 8.04(7.53) $\mathrm{Me_{32}O_{63}}$ O(B) 2$\times$2$\times$2 7.00(6.53) 0.76(0.29) 7.91(7.44) $\mathrm{Me_{32}O_{62}}$ (B1, B2) 2$\times$2$\times$2 7.01(6.52) 0.77(0.28) 7.92(7.43) $\mathrm{Me_{32}O_{62}}$ (B2, B3) 2$\times$2$\times$2 7.02(6.60) 0.78(0.36) 7.93(7.51) $\mathrm{Me_{32}O_{62}}$ (A1, B1) 2$\times$2$\times$2 7.03(6.53) 0.79(0.29) 7.94(7.44) $\mathrm{Me_{4}O_{7}}$ O(A) 1$\times$1$\times$1 7.16(6.66) 0.92(0.42) 8.07(7.57) $\mathrm{Me_{4}O_{7}}$ O(B) 1$\times$1$\times$1 7.03(6.57) 0.79(0.33) 7.94(7.48) $\mathrm{Me_{4}O_{6}}$ (B1, B2) 1$\times$1$\times$1 6.55(5.66) 0.31(-0.58) 7.46(6.57) $\mathrm{Me_{4}O_{6}}$ (A1, B1) 1$\times$1$\times$1 6.92(6.40) 0.68(0.16) 7.83(7.31) $\mathrm{Me_{4}O_{6}}$ (B2, B3) 1$\times$1$\times$1 6.95(6.33) 0.71(0.09) 7.86(7.24) -------------------------- ---------- --------------------- ---------------- ------------- ------------ : Formation energies of isolated single oxygen vacancies and di-oxygen-vacancy pairs, of isolated oxygen vacancy chains, and of substoichiometric $\mathrm{Hf(Zr)_{4}O_{7}}$ and $\mathrm{Hf(Zr)_{2}O_{3}}$ in different structures derived from monoclinic $\mathrm{HfO_2}$ and $\mathrm{ZrO_2}$. In the table, Me stands for Hf or Zr; the parenthesed formation energy values are Zr-based while others are Hf-based. {width="3.5in" height="2.84in"} The search for metallic states began with the $\mathrm{Hf(Zr)_4O_7}$ models generated by introducing one oxygen vacancy per 12-atom $m$-$\mathrm{Hf(Zr)O_2}$ for two inequivalent cases, *i.e.*, O(A) and O(B). The defective unit cells were fully relaxed until all Hellmann-Feynman forces were less than 0.01 eV/Å. Atomic coordinates, cell dimensions and shape were subject to relaxation. The formation energies per vacancy of the two $\mathrm{Hf(Zr)_4O_7}$ phases are almost the same as in the single oxygen vacancy cases. The resulting DOS are shown in Figs. 2(b) and 2(c). In either case, a fully occupied defect-induced band emerges in the band gap, indicating a semiconductor. While the DOS of this band is broad and high, it cannot account for the measured metallic state in hafnia-based RRAM, because in that state the resistance is on the order of hundreds of Ohms for 10nm thin films.^^ However, the trend does hint that stronger off-stoichiometric hafnia or zirconia might undergo a phase transition from dielectric to metal. $a$ $c$ $\mathrm{Hf_z}$ $\mathrm{O(A)_z}$ Bulk modulus -------------------- -------- -------- ----------------- ------------------- -------------- $\mathrm{Hf_2O_3}$ 3.135Å 5.646Å 0.2553 0.1351 246GPa $\mathrm{Zr_2O_3}$ 3.174Å 5.763Å 0.2525 0.1367 228GPa : Calculated structural parameters of tetragonal $\mathrm{Hf_2O_3}$ and $\mathrm{Zr_2O_3}$. Hence, several $\mathrm{Hf(Zr)_2O_3}$ models were set up with two oxygen vacancies per 12-atom $m$-$\mathrm{Hf(Zr)O_2}$ unit cell. For all inequivalent cases (Fig. 1) the cells remain monoclinic during relaxation of the unit cell vectors, except for the (B1,B2) case which suffers from a monoclinic-to-tetragonal transition. The tetragonal $\mathrm{Hf(Zr)_2O_3}$ \[$t$-$\mathrm{Hf(Zr)_2O_3}$\] phase (Fig. 3; structural parameters in Tab. [slowromancap2@]{}) is the ground state of all ten $\mathrm{Hf(Zr)_2O_3}$ candidates. It belongs to the $D_{\mathrm{2}d}$ point group and $P\mathrm{\bar{4}}m\mathrm{2}$ (No. 115) space group^^. The Hf(Zr) coordination number is 7 as in $m$-$\mathrm{Hf(Zr)O_2}$, while 2/3 of the oxygen sites have coordination number 5 \[named O(A)\] and 1/3 have coordination number 4 \[named O(B)\]. Symmetry analysis indicates that Hf(Zr) and O(A) are at the $2g$ position while O(B) is at the $1c$ position^^. The average Hf(Zr)–O(A) and Hf(Zr)–O(B) bond lengths are 2.295(2.322)Å and 2.089(2.134)Å, respectively, compared with 2.084(2.117)Å and 2.209(2.240)Å in $m$-$\mathrm{Hf(Zr)O_2}$. Bader analysis reveals that the Hf (Zr) charge changes from 2.73e(2.57e) in $m$-$\mathrm{Hf(Zr)O_2}$ to 2.10e(2.02e) in $t$-$\mathrm{Hf(Zr)_2O_3}$; the O(A) charge changes from -1.34e(-1.25e) to -1.39e(-1.34e); and the O(B) charge changes from -1.39e(-1.31e) to -1.41e(-1.37e). {width="4in" height="3.65in"} {width="4in"} Figure 4 shows the band diagram and orbital-projected DOS of $t$-$\mathrm{Hf_2O_3}$. Results for $t$-$\mathrm{Zr_2O_3}$ (not shown) are similar. The high symmetry points in the Brillouin zone are named according to the Bilbao crystallographic server.^^ A semi-metallic behavior is revealed in the overlap of the partially occupied valence band top and conduction band bottom located at different high symmetry points, R and Z. Both band edges are mostly derived from Hf $5d$ states. The semi-metallic character of the two compounds was confirmed by $\mathrm{G_0W_0}$ calculations of the many-body correction to the DFT/GGA energy levels. Figure 4 implies that the energy shifts are of the order of a few tenths of eV, retaining the semi-metallic nature of $t$-$\mathrm{Hf_2O_3}$. The energy shifts obtained for $\mathrm{Zr_2O_3}$ (not shown) are similar. From the calculated band structures the densities of conduction electrons and holes can be obtained by integrating the electron occupation of the blue and orange bands in Fig. 4, respectively. The electron and hole densities are both $1.77\times{10^{21}}$$\mathrm{cm^{-3}}$ for $t$-$\mathrm{Hf_2O_3}$, and both $1.75\times{10^{21}}$$\mathrm{cm^{-3}}$ for $t$-$\mathrm{Zr_2O_3}$, typical of semi-metals. To evaluate the relative stability of this structure, we calculated the molar formation enthalpy of various $\mathrm{Hf_2O_3}$, $\mathrm{Zr_2O_3}$ and $\mathrm{Ti_2O_3}$ models with respect to their corresponding metals and $\mathrm{O_2}$. Still, the 1.36eV energy correction to $\mathrm{O_2}$ was applied to all cases. The formation enthalpy of $t$-$\mathrm{Hf(Zr)_2O_3}$ is -1700(-1666)kJ/mol, more favorable than fully relaxed corundum $\mathrm{Hf(Zr)_2O_3}$, -1586(-1580)kJ/mol. Nevertheless, a fully relaxed $\mathrm{Ti_2O_3}$ arranged in the $P\mathrm{\bar{4}}m\mathrm{2}$ tetragonal structure, possesses a formation enthalpy of -1576kJ/mol, less favorable than corundum $\mathrm{Ti_2O_3}$ whose formation enthalpy is -1598kJ/mol. These data confirm that for $\mathrm{Hf(Zr)_2O_3}$ the tetragonal $P\mathrm{\bar{4}}m\mathrm{2}$ structure is preferred, while for $\mathrm{Ti_2O_3}$ the corundum structure is preferred. To our best knowledge, the proposed $t$-$\mathrm{Hf_2O_3}$ and $t$-$\mathrm{Zr_2O_3}$ structures have not been reported before, though some published data may suggest their existance. Hildebrandt *et al.*^^ performed high-resolution transmission electron microscopy of a conducting $\mathrm{HfO_{2-x}}$ thin film where the enlarged inverse Fourier-transformed images show a similar structure as in Fig. 3(b). Manory *et al.*^^ discovered two unidentified X-ray diffraction (XRD) peaks at $2\theta=40^{\mathrm{o}}$ and $2\theta=52^{\mathrm{o}}$ in hafnia films grown by ion beam assisted deposition at a transport ratio of 5 and an ion energy of 20keV. They attributed these peaks to a new tetragonal structure and suggested the $\mathrm{Hf_2O_3}$ stoichiometry. However, they simulated their data with a $P\mathrm{4}/mmm$ phase with lattice parameters a=5.055Å and c=5.111Å, resulting in two small peaks around $40^{\mathrm{o}}$. We calculated powder XRD patterns^^ for $t$-$\mathrm{Hf_2O_3}$ \[Fig.  3(d)\] using Cu $K\alpha$ radiation ($\lambda=1.5418$Å), and found a (110) peak at $40.7^{\mathrm{o}}$ and a (112) peak at $52.5^{\mathrm{o}}$, similar to data. Similar calculation for $t$-$\mathrm{Zr_2O_3}$ yielded a (110) peak at $40.2^{\mathrm{o}}$ and a (112) peak at $51.6^{\mathrm{o}}$. In conclusion, we have predicted tetragonal semi-metallic $\mathrm{Hf_2O_3}$ and $\mathrm{Zr_2O_3}$ structures as the ground state highly oxygen deficient hafnia and zirconia which undergo a monoclinic-to-tetragonal phase transition. Their semi-metallic properties are characterized by an overlap of the valence band maximum and conduction band minimum at different points of the Brillouin zone, and by the low density of conduction electrons and holes. Also, $t$-$\mathrm{Hf_2O_3}$ may be the physical origin of the conductive state in hafnium-based RRAM. This work is financially supported by the Nanosciences Foundation of Grenoble (France) in the frame of the Chairs of Excellence awarded to L.R.C. Fonseca in 2008 and to Y. Nishi in 2010. LRCF also acknowledges CNPq for financial support. 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--- abstract: | In this paper, we present lower bounds on the rank of the split closure, the multi-branch closure and the lattice-free closure for packing sets as a function of the integrality gap. We also provide a similar lower bound on the split rank of covering polyhedra. These results indicate that whenever the integrality gap is high, these classes of cutting planes must necessarily be applied for many rounds in order to obtain the integer hull.\ \ **Keywords.** Integer programming, packing, covering, split rank, multi-branch split rank, lattice-free rank author: - 'Merve Bodur[^1]' - 'Alberto Del Pia[^2]' - 'Santanu S. Dey[^3]' - 'Marco Molinaro[^4]' bibliography: - 'LatticeFreeRank.bib' title: 'Lower bounds on the lattice-free rank for packing and covering integer programs' --- Introduction {#sec:intro} ============ *Split cuts* are a very important class of cutting planes in integer programming both from a theoretical and computational perspective (see for example [@balas:1979; @BalasS08; @cook:ka:sc:1990]). Recently, many generalizations of split cuts have been studied, such as the *multi-branch split cuts* [@dash2014lattice; @dash2013t; @li:ri:2008] and the *lattice-free cuts* [@andersen2007inequalities; @basu2015geometric; @borozan:2007; @RichardDey]. In order to study the strength of the cutting plane procedures, a very useful concept is the notion of *rank* which represents the minimum rounds of cuts needed to obtain the integer hull. The notion of rank was first studied in the context of Chvátal-Gomory (CG) cuts [@Schrijver80]. Many lower bounds on the rank of the above mentioned closures have been proven; see [@BasuCM12; @bodur2017cutting; @cook:ka:sc:1990; @dey:lowerbnd:2009; @DeyL11; @li:ri:2008] for the split rank, see [@dash2013t] for the multi-branch rank, and see [@averkov2017approximation] for the lattice-free rank. A standard notion describing the [difficulty]{} of an integer program is the *integrality gap* which in this paper refers to the ratio between the optimal objective function values of the integer program and its linear programming relaxation. While it is natural to expect that the rank of a cutting plane procedure should increase with the increase in the integrality gap, only a few results of this nature exist in the literature [@bodur2016aggregation; @PokuttaS11]. In this paper, we present lower bounds on the rank of the split closure, the multi-branch closure and the lattice-free closure for *packing sets* as a function of the integrality gap. We also provide a similar lower bound on the split rank of *covering polyhedra*. These results indicate that whenever the integrality gap is high, these classes of cutting planes must necessarily be applied for many rounds in order to obtain the integer hull. The rest of the paper is organized as follows. We provide all necessary definitions in Section \[sec:Prelim\]. We state all our main results in Section \[sec:Statements\]. Finally, in Section \[sec:Packing\] and Section \[sec:Covering\] we present the proofs for results concerning the packing and covering cases, respectively. Preliminaries {#sec:Prelim} ============= For an integer $t \geq 1$, we use $[t]$ to describe the set $\{1, \dots, t\}$. Also, we represent the $j^{\text{th}}$ unit vector, the vector of ones and the vector of zeros in appropriate dimension by $e_j$, ${\boldsymbol{1}}$ and ${\boldsymbol{0}}$, respectively. Given a set of vectors $v^1, \hdots, v^t$, we denote the linear subspace spanned by these given vectors as $\spann ( \{ v^j \}_{j \in [t]} )$. #### Sets. In this paper, we work with *covering [polyhedra]{}* and *packing [polyhedra]{}*, which are of the form $$P_C = \{x \in {\mathbb{R}}^n_+ \mid Ax \ge b \} \quad \text{and} \quad P_P = \{x \in {\mathbb{R}}^n_+ \mid Ax \le b \},$$ respectively, where all the data $(A,b) \in {\mathbb{Q}}_+^{m \times n} \times {\mathbb{Q}}_+^m$. [Thus, an inequality of packing type (respectively covering type) is one of the form $a^\top x \le b$ (respectively $a^\top \ge b$) for non-negative $(a,b) \in {\mathbb{Q}}_+^n \times {\mathbb{Q}}_+$.]{} If it is obvious from the context that the polyhedron is of covering (resp. packing) type, we may drop the subscript $C$ (resp. $P$). For the packing case, we also work with more general sets. We call $Q \ {{\color{black}}\subseteq} \ {\mathbb{R}}_+^n$ a *packing set* if $x \in Q$ and ${\boldsymbol{0}}\leq y \leq x$ imply that $y \in Q$. Throughout this paper, we make a technical assumption regarding the sets under consideration that we call as *well-behavedness*. The set $P_C$ is *well-behaved* if $A_{ij} \leq b_i$ for all $i \in [m], j \in [n]$. Notice that this is a natural assumption since if $A_{ij} > b_i$ for some $i \in [m], j \in [n]$, then we can replace the coefficient $A_{ij}$ by $b_i$ to obtain a tighter linear programming relaxation with the same set of feasible integer points. A packing set $Q$ is *well-behaved* if $e_j \in Q$ for all $j \in [n]$. This is not a restrictive assumption since if $e_j \notin Q$ for some $j \in [n]$, then we replace $Q$ with the packing set $\{ x \in Q \mid x_j = 0\}$, which provides a tighter linear relaxation with the same set of feasible integer points. Note that if $Q$ is the polytope $P_P$, then [the]{} well-behavedness definition is equivalent to $A_{ij} \leq b_i$ for all $i \in [m], j \in [n]$. [A *relaxation* of a set $P$ is any superset $\tilde{P} \supseteq P$.]{} Let $\alpha > 0$ be a scalar. If a given covering polyhedron $\tilde{P}_C$ is a relaxation of $P_C$ and satisfies $$\min\{c^\top x \mid x \in \tilde{P}_C\} \ge \frac{1}{\alpha} \cdot \min\{c^\top x \mid x \in P_C\}, \ \ \forall c \in {\mathbb{R}}_+^n,$$ then $\tilde{P}_C$ is an *$\alpha$-approximation* of $P_C$. Similarly, given a packing set $P_P$ and one of its relaxation $\tilde{P}_P$ of packing type, $\tilde{P}_P$ is an *$\alpha$-approximation* of $P_P$ if $$\max\{c^\top x \mid x \in \tilde{P}_P\} \le \alpha \cdot \max\{c^\top x \mid x \in P_P \}, \ \ \forall c \in {\mathbb{R}}_+^n.$$ For a set $P \subseteq {\mathbb{R}}^n$, we define $\alpha P:= \{\alpha x \mid x \in P\}$. The equivalent definitions of $\alpha$-approximation for covering and packing cases are provided in [@bodur2016aggregation] as $$\frac{1}{\alpha} \tilde{P}_C {\subseteq}P_C \quad \text{and} \quad \tilde{P}_P {\subseteq}\alpha P_P,$$ respectively. Given a polyhedron $P {\subseteq}{\mathbb{R}}^n$, we denote its integer hull by $P^I := \operatorname{conv}( \{x \mid x \in P \cap {\mathbb{Z}}^n\})$ where $\operatorname{conv}(\cdot)$ is the convex hull operator. We let ${z^{LP}(c)}$ and ${z^{I}(c)}$ to denote the optimal value of a given objective function $c^\top x$ over $P$ and $P^I$, respectively. For convenience, we will sometimes refer to ${z^{LP}(c)}$ and ${z^{I}(c)}$ as $z^{LP}$ and $z^{I}$, respectively. #### Closures. We call a set ${M}\in {\mathbb{R}}^n$ a *strict lattice-free set* if $M \cap {\mathbb{Z}}^n = \emptyset$. Note that the set ${M}$ need not to be convex. Given a set $Q$, one can obtain a relaxation of $Q^I$ as $$Q^{M}:= \operatorname{conv}(Q \setminus {M}).$$ Given a collection of strict lattice-free sets ${\mathcal{{M}}}$, we define the corresponding closure as $${\mathcal{{M}}}(Q) = \bigcap_{{M}\in {\mathcal{{M}}}} Q^{M}.$$ For convenience, we sometimes refer to ${\mathcal{{M}}}$ as the closure operator or just as closure. Next, we define three special cases of the strict lattice-free closures, namely the split closure, the multi-branch closure and the lattice-free closure. We denote the *split set* associated with $(\pi,\pi_0) \in {\mathbb{Z}}^n \times {\mathbb{Z}}$ by $$S(\pi,\pi_0) := \{ x \in {\mathbb{R}}^n \mid \pi_0 < \pi^\top x < \pi_0+1\}.$$ [Denoting]{} the collection of all split sets by $${{\color{black}}{\mathcal{S}}= \{S(\pi,\pi_0) \mid (\pi,\pi_0) \in {\mathbb{Z}}^n \times {\mathbb{Z}}\},}$$ the split closure of $Q$, denoted as ${\mathcal{S}}(Q)$, is defined to be $${\mathcal{S}}(Q) = \bigcap_{S \in {\mathcal{S}}} Q^S.$$ For convenience, we denote $Q^{S(\pi,\pi_0)}$ by $Q^{\pi,\pi_0} $ which is explicitly defined as $$Q^{\pi,\pi_0} = \operatorname{conv}(Q \setminus S(\pi,\pi_0)) = \operatorname{conv}\big( (Q \cap \{ \pi^\top x \leq \pi_0 \}) \cup (Q \cap \{ \pi^\top x \geq \pi_0+1 \}) \big).$$ A generalization of split closure, called the *$k$-branch split closure*, which is defined by [@li:ri:2008], is obtained by removing the union of *at most* $k$ split sets simultaneously. Letting $$\begin{aligned} Q^{\pi^1,\dots, \pi^k;\pi^1_0, \dots, \pi^k_0} :&= \operatorname{conv}\Big( Q \setminus \bigcup_{i \in [k]} S(\pi^i,\pi^i_0) \Big) \\ &= \operatorname{conv}\left( \bigcap_i (Q \cap \{ (\pi^i)^\top x \leq \pi^i_0 \}) \cup (Q \cap \{ (\pi^i)^\top x \geq \pi^i_0+1 \}) \right),\end{aligned}$$ the $k$-branch split closure of $Q$, denoted by ${{\mathcal{S}}^{k}}(Q)$, can be written as $${{\mathcal{S}}^{k}}(Q) = \bigcap_{(\pi^i,\pi^i_0) \in {\mathbb{Z}}^n \times {\mathbb{Z}},~i \in [k]} Q^{\pi^1,\dots, \pi^k;\pi^1_0, \dots, \pi^k_0}.$$ Note that the 1-branch split closure is equivalent to the split closure, i.e, ${\mathcal{S}}^{1}(Q)={\mathcal{S}}(Q)$. A further generalization of the split closure is the so-called *lattice-free closure*, which is obtained by considering convex sets having no integer point in their interior; see [@dash2012two; @dash2014lattice] for relations to the $k$-branch split closure. A set $L {\subseteq}{\mathbb{R}}^n$ is called a *lattice-free set* if $\int(L) \cap {\mathbb{Z}}^n = \emptyset$ where $\int(\cdot)$ is the interior operator. For each integer $k \geq 2$, we define ${\mathcal{L}}^k$ as the family of full-dimensional lattice-free polyhedra $L \subset {\mathbb{R}}^n$ defined by *at most* $k$ inequalities. (Note that it is not possible to have lattice-free sets defined by only one inequality.) We denote the *$k$-lattice-free closure* of $P$ by ${\mathcal{L}}^k(Q)$, i.e., $${\mathcal{L}}^k(Q) = \bigcap_{L \in {{\color{black}}{\mathcal{L}}^k}} Q^L,$$ where $$Q^L := \operatorname{conv}(Q \setminus \int(L)).$$ Given a closure operator ${\mathcal{{M}}}$ and a nonnegative objective function $c \in {\mathbb{R}}^n_+$, we use $z^{{\mathcal{{M}}}}$ to denote the optimal value of the minimization (or maximization) of $c^\top x$ over the closure ${\mathcal{{M}}}(Q)$. Lastly, we define the *rank* of the closure ${\mathcal{{M}}}$, denoted by $\operatorname{rank}_{{\mathcal{{M}}}}(Q)$, as the minimum number of iterative applications of ${\mathcal{{M}}}$ to obtain the integer hull of $Q$. We note that the split rank, thus the multi-branch rank and the lattice-free rank, are finite whenever $Q$ is a rational polyhedron or is a bounded set [@Schrijver80]. Main results {#sec:Statements} ============ Packing ------- The main proof strategy to prove lower bounds on ranks of various cutting plane closures is presented in the proposition below. \[thm:thm1\] Let ${\mathcal{{M}}}$ be a collection of strict lattice-free sets which satisfies the following two conditions: 1. Packing invariance: For any packing set $Q$, ${\mathcal{{M}}}(Q)$ is a packing set. 2. Constant approximation: There exists $\alpha_{\mathcal{{M}}}\geq 1$ such that $Q {\subseteq}\alpha_{\mathcal{{M}}}{\mathcal{{M}}}(Q)$ for every well-behaved packing set $Q$. Then, for any well-behaved packing set $Q$, $$\operatorname{rank}_{{\mathcal{{M}}}}(Q) \geq {{\color{black}}\sup_{c \in {\mathbb{R}}_+^n}} \left\lceil \frac {\log_2 \left( \frac{{z^{LP}(c)}}{{z^{I}(c)}}\right)}{\log_2 \alpha_{\mathcal{{M}}}}\right\rceil.$$ The proof of Proposition \[thm:thm1\] is based on a simple iterative argument, which is provided in Section \[subsec:4.1\]. ### Tools to prove the assumptions of Proposition \[thm:thm1\] In order to use Proposition \[thm:thm1\], we need to verify the packing invariance and constant approximation properties. The next tool is very helpful in proving packing invariance. \[thm:thm2\] Let ${\mathcal{{M}}}$ be a collection of strict lattice-free sets. For ${T}{\subseteq}[n]$, define $H[{T}] := \{ x \in {\mathbb{R}}^n \mid x_j = 0, \ \forall j \in T\}$. Given ${M}\in {\mathcal{{M}}}$, let $$M[{T}] := ( M \cap H[{T}] ) + \spann ( \{ e_{j} \}_{j \in T} ).$$ Suppose that ${\mathcal{{M}}}$ satisfies the following property: For any ${M}\in {\mathcal{{M}}}$ and $T {\subseteq}[n]$, $M[{T}] \neq \emptyset$ implies that $M[{T}] \in {\mathcal{{M}}}$. Then ${\mathcal{{M}}}$ is packing invariant. Note that it is straightforward to see that the set ${M}[T]$ in Theorem \[thm:thm2\] is guaranteed to be a strict lattice-free set by construction. The proof of Theorem \[thm:thm2\] is essentially based on the fact that a cut generated using a strict lattice-free set $M$ is dominated by a packing type inequality that is obtained using the strict lattice-free set $M[T]$ for a specifically chosen set $T$. The details of the proof of Theorem \[thm:thm2\] are given in Section \[subsec:4.2\]. We observe here that in order to use Proposition \[thm:thm1\], we must prove [the]{} constant approximation property for general well-behaved packing sets, rather than just for polyhedra. The reason is that the closures of some cutting plane families we consider are not known to be polyhedral. In order to prove [the]{} constant approximation property for general well-behaved packing sets, we will find it convenient to prove this property first for well-behaved packing polyhedra. It turns out that this is sufficient to prove constant approximation property for any well-behaved packing set as the next theorem states. \[thm:thm3\] Let ${\mathcal{{M}}}$ be a collection of strict lattice-free sets with the following property: There exist $\alpha_{\mathcal{{M}}}\geq 1$ such that $P_P {\subseteq}\alpha_{\mathcal{{M}}}{\mathcal{{M}}}(P_P)$ for every well-behaved packing polyhedron $P_P$. Then, $Q {\subseteq}\alpha_{\mathcal{{M}}}{\mathcal{{M}}}(Q)$ for every well-behaved packing set $Q$. Theorem \[thm:thm3\] is proven by first constructing a well-behaved packing polyhedron which is an inner approximation of $Q$ and is arbitrarily close to $Q$. We then show how to “transfer" the $\alpha_{\mathcal{{M}}}$ factor from this polyhedron to $Q$. The details of the proof of Theorem \[thm:thm3\] are provided in Section \[subsec:4.3\]. ### Applications of Proposition \[thm:thm1\] to split, multi-branch split and lattice-free closures We use Theorem \[thm:thm2\] to verify the following result. \[thm:thm4\] ${\mathcal{{M}}}$ is packing invariant for ${\mathcal{{M}}}\in \{ {\mathcal{S}}, {\mathcal{S}}^k, {\mathcal{L}}^k \}$. Theorem \[thm:thm4\] is proven in Section \[subsec:4.4\]. \[thm:thm5\] For ${\mathcal{{M}}}\in \{ {\mathcal{S}}, {\mathcal{S}}^k, {\mathcal{L}}^k \}$, ${\mathcal{{M}}}$ satisfies the constant approximation property, where [we can choose]{} $\alpha_{{\mathcal{S}}} = 2$, [$\alpha_{{\mathcal{S}}^k} = \min \{ 2^k,n \}+1$, and $\alpha_{{\mathcal{L}}^k} = \min \{ k,n \}+1$]{}. Moreover, the factor $\alpha_{{\mathcal{S}}}$ is tight, i.e., for every $\epsilon >0$, there exists a well-behaved packing polyhedron $\tilde{P}_P$ such that $\tilde{P}_P \not\subseteq (2-\epsilon) {\mathcal{S}}(\tilde{P}_P)$. Observe that the split cuts are a special case of multi-branch split cuts. However, we have stated their constant approximation result separately since the general factor for multi-branch split closure is not tight for the split closure. Indeed, proving the factor of 2 in the case of split cuts involves more careful analyses. Moreover, this factor of 2 for the split case is tight as stated in the theorem. The proof of Theorem \[thm:thm5\] for the split, multi-branch split and lattice-free cases are given in Sections \[subsubsec:4.5.1\], \[subsubsec:4.5.2\] and \[subsubsec:4.5.3\], respectively. Note that a factor of $2$ is proven in [@bodur2016aggregation] as an approximation factor of the *aggregation closure*, which is very similar to the result [for the]{} split closure in Theorem \[thm:thm5\]. However, the split closure result of Theorem \[thm:thm5\] is not implied by the result of [@bodur2016aggregation] since for packing polyhedra, split cuts are not [always]{} dominated by *aggregation cuts*, see the example given in Observation \[obs:PackingSplitAgg\] in Appendix \[sec:appendix\]. Proposition \[thm:thm1\], Theorem \[thm:thm4\] and Theorem \[thm:thm5\] lead us to the following lower bounds on the rank of [the]{} split closure, $k$-branch split closure and $k$-lattice-free closure of packing sets. As Corollary \[cor:cor1\] is a direct application of Proposition \[thm:thm1\], we omit its proof. \[cor:cor1\] Let $Q$ be a well-behaved packing set. Then 1. $\operatorname{rank}_{{\mathcal{S}}}(Q) \geq {{\color{black}}\sup_{c \in {\mathbb{R}}_+^n}} \left\lceil \log_2 \left( \frac{{z^{LP}(c)}}{{z^{I}(c)}}\right)\right\rceil$. 2. $\operatorname{rank}_{{\mathcal{S}}^k}(Q) \geq {{\color{black}}\sup_{c \in {\mathbb{R}}_+^n}} \left\lceil\frac{\log_2 \left( \frac{{z^{LP}(c)}}{{z^{I}(c)}}\right)}{\log_2 (\min \{ 2^k,n \}+1)}\right\rceil$ for any $k \in {\mathbb{Z}}_+, k \geq 1$. 3. $\operatorname{rank}_{{\mathcal{L}}^k}(Q) \geq {{\color{black}}\sup_{c \in {\mathbb{R}}_+^n}} \left\lceil\frac{\log_2 \left( \frac{{z^{LP}(c)}}{{z^{I}(c)}}\right)}{\log_2 (\min \{ k,n \}+1)}\right\rceil$ for any $k \in {\mathbb{Z}}_+, k \geq 2$. Corollary \[cor:cor1\] shows that if the integrality gap is high, then we cannot expect the split rank, the multi-branch split rank or the lattice-free rank of a well-behaved packing set to be low. To the best of our knowledge, the only other paper analyzing the rank of general lattice-free closures is [@averkov2017approximation], and the only papers presenting lower bounds on the rank of multi-branch split closure for very special [kinds]{} of polytopes are [@dash2013t] and [@li:ri:2008]. We note that none of these bounds are related to the [integrality]{} gap. There have been a number of papers giving lower bounds on split ranks such as [@BasuCM12; @bodur2017cutting; @cook:ka:sc:1990; @dey:lowerbnd:2009; @DeyL11] and bounds on a closely related concept, the [reverse]{} split rank [@ConfortiPSFSpli15]. To the best of our knowledge, this is the first work connecting the integrality gap to the split rank. We note that the first part of Corollary \[cor:cor1\] can be seen as a generalization of the result given in [@PokuttaS11] for the CG rank. The lower bound on the split rank given in Corollary \[cor:cor1\] is tight within a constant factor as formally stated below. \[prop:prop1\] There exists a well-behaved packing polyhedron $Q$ and a nonnegative objective function $c$ such that $\operatorname{rank}_{{\mathcal{S}}}(Q) \le O\left(\textup{log}_2\left( \frac{{z^{LP}(c)}}{{z^{I}(c)}}\right)\right)$. The proof of Proposition \[prop:prop1\] is given in Section \[subsubsec:4.5.4\]. Covering -------- We now state our results for covering [polyhedra]{}. All the proofs regarding the covering case are given in Section \[sec:Covering\]. \[thm:mainAppCovering\] Let $P_C$ be well-behaved. Then, the followings hold: - ${\mathcal{S}}(P_C)$ is a well-behaved covering polyhedron. - $\frac{1}{2} P_C {\subseteq}{\mathcal{S}}(P_C) $. Moreover, the bound given in (ii) is tight, i.e., for every $\epsilon >0$, there exists a well-behaved covering polyhedron $\tilde{P}_C$ such that $\frac{1}{2-\epsilon} \tilde{P}_C \not {\subseteq}{\mathcal{S}}(\tilde{P}_C)$. Regarding part (i) of Theorem \[thm:mainAppCovering\], ${\mathcal{S}}(P_C)$ is known to be a rational polyhedron since $P_C$ is assumed to be a rational polyhedron [[@cook:ka:sc:1990]]{}, and it is straightforward to show that the split closure is of covering type (Proposition \[prop:CoveringPoly\]); whereas its well-behavedness can be proven by showing that each split cut that violates the well-behavedness property is dominated by a well-behaved split cut (Proposition \[prop:CoveringSplitWell\]). Proof of part (ii) follows from a case analysis that gives the correct factor of [$\frac{1}{2}$]{} (Proposition \[prop:covering2approx\]). For the last statement in the theorem, we provide a tight example in Proposition \[prop:LBAC\]. Note that a [result similar]{} to Theorem \[thm:mainAppCovering\] is proven in [@bodur2016aggregation] with respect to the [aggregation closure]{}. However, Theorem \[thm:mainAppCovering\] is not implied by the result of [@bodur2016aggregation] since for covering polyhedra, split cuts are not dominated by [aggregation cuts]{}, see the example given in Observation \[obs:CoveringSplitAgg\] in Appendix \[sec:appendix\]. Similar to the proof of Proposition \[thm:thm1\] in the packing case, Theorem \[thm:mainAppCovering\] yields the following lower bound on the split rank of covering polyhedra. \[cor:SplitRankPolyCovering\] Let $P_C$ be well-behaved. Then, $$\operatorname{rank}_{{\mathcal{S}}}(P_C) \geq {{\color{black}}\sup_{c \in {\mathbb{R}}_+^n}} \left\lceil \textup{log}_2\left( \frac{{z^{I}(c)}}{{z^{LP}(c)}}\right)\right\rceil.$$ Unlike the packing case, we are unable to generalize the result of Corollary \[cor:SplitRankPolyCovering\] for the case of $k$-lattice-free rank. The key technical argument that is a roadblock is to prove the well-behavedness of the $k$-lattice-free closure of covering polyhedron. [We do not know if the $k$-lattice-free closure of covering polyhedron is well-behaved. Note that in contrast]{} in the packing case, if we start from a well-behaved set and the closure is of packing type, then trivially the closure is also well-behaved. Proofs for packing problems {#sec:Packing} =========================== We use the following observation, from [@bodur2016aggregation], in some of the proofs. \[obs:bijection\] Let $\phi:\mathbb{R}^n \rightarrow \mathbb{R}^n$ be a bijective map, let $\{S^i\}_{i \in I}$ be a collection of subsets of $\mathbb{R}^n$ and [for $S {\subseteq}\mathbb{R}^n$]{} let $\phi(S):= \{ \phi(x) \,|\, x \in S\}$. Then $\phi\left(\bigcap_{i \in I} S^i\right) = \bigcap_{i \in I} \phi(S^i)$. Proof of Proposition \[thm:thm1\] {#subsec:4.1} --------------------------------- Let ${\mathcal{{M}}}$ be a collection of strict lattice-free sets which satisfies the packing invariance and constant approximation properties. Let $Q {\subseteq}{\mathbb{R}}^n$ be a well-behaved packing set. Since $Q^I {\subseteq}{\mathcal{{M}}}(Q)$, we have that $e_j \in {\mathcal{{M}}}(Q)$ for all $j \in [n]$. Therefore, by the packing invariance property, ${\mathcal{{M}}}(Q)$ is also a well-behaved packing set. [Assume that the rank of the closure ${\mathcal{{M}}}$ is finite, as there is nothing to prove otherwise.]{} Let $t = \operatorname{rank}_{{\mathcal{{M}}}}(Q)$ and let $c \in {\mathbb{R}}_+^n$ be a given objective vector. Define $z^{i}$ to be the optimal objective function value of maximizing $c^\top x$ over the $i^{\text{th}}$ closure with respect to ${\mathcal{{M}}}$ of $Q$. Since, ${\mathcal{{M}}}(Q)$ is a well-behaved packing set, by induction, the $i^{\text{th}}$ closure with respect to ${\mathcal{{M}}}$ of $Q$ is a well-behaved packing set. Therefore, the constant approximation property guarantees that $z^i \le {{\color{black}}\alpha_{\mathcal{{M}}}} z^{i+1}$. Thus, $$\begin{aligned} \frac{{z^{LP}(c)}}{{z^{I}(c)}} = \frac{{z^{LP}(c)}}{z^1} \frac{z^1}{z^2}\dots\frac{z^{t-1}}{z^{t}} \leq {(\alpha_{\mathcal{{M}}})}^{t}.\end{aligned}$$ This implies the inequality $$\begin{aligned} t = \operatorname{rank}_{{\mathcal{{M}}}}(Q) \geq \left\lceil \frac{\textup{log}_2\left( \frac{{z^{LP}(c)}}{{z^{I}(c)}}\right)}{\textup{log}_2 \alpha_{\mathcal{{M}}}} \right\rceil,\end{aligned}$$ which is the required result. Proof of Theorem \[thm:thm2\] {#subsec:4.2} ----------------------------- Let ${\mathcal{{M}}}$ be a collection of strict lattice-free sets with the following property. For ${T}{\subseteq}[n]$, define $H[{T}] := \{ x \in {\mathbb{R}}^n | x_j = 0, \ \forall j \in T\}$. Given ${M}\in {\mathcal{{M}}}$, let $$M[{T}] := ( M \cap H[{T}] ) + \spann ( \{ e_{j} \}_{j \in T} ).$$ Assume that for any ${M}\in {\mathcal{{M}}}$ and $T {\subseteq}[n]$, if $M[{T}] \neq \emptyset$, then $M[{T}] \in {\mathcal{{M}}}$. We will show that ${\mathcal{{M}}}$ is packing invariant. Let $Q$ be a packing set. If $Q$ is empty, then there is nothing to prove. Therefore, assume that $Q$ is nonempty. Let ${M}\in {\mathcal{{M}}}$. Let ${\beta}^\top x \le \delta$ be a valid inequality for $Q^{M}$. We will show that this inequality is dominated by a packing type inequality valid for ${\mathcal{{M}}}(Q)$. Let $T = \{ j \in [n] : {\beta}_j < 0\}$. If $T = \emptyset$, there is nothing to prove. So, assume that $T \neq \emptyset$. For convenience, we define an operator $\breve{(\cdot)}$ as follows: For a given vector $u \in {\mathbb{R}}^n$, $\breve{u} \in {\mathbb{R}}^n$ is $$\begin{aligned} \breve{u}_j = \left \{ \begin{array}{rl} u_j, & \text{if} \ j \in [n] \setminus T \\ 0, & \text{if} \ j \in T \end{array} \right. . \label{eq:opBreve} \end{aligned}$$ We will show that $\breve {\beta}^\top x \le \delta$ is a valid inequality for ${\mathcal{{M}}}(Q)$. Since $\breve {\beta}\in {\mathbb{R}}^n_+$ and $\{x \in {\mathbb{R}}^n_+ : \breve {\beta}x \le \delta \} \subseteq \{x \in {\mathbb{R}}^n_+ : {\beta}^\top x \le \delta \}$, we obtain the required result. Let $\bar Q := Q \cap H[T]$. As $\bar Q \subseteq Q$, we have that ${\beta}^\top x \le \delta$ is a valid inequality for ${\bar Q}^{M}$. Since $\breve {\beta}^\top x = {\beta}^\top x$ for every $x \in H[T]$, we obtain that $$\label{betaineqValidForQbarM} \breve {\beta}^\top x \le \delta \ \text{is a valid inequality for} \ {\bar Q}^{M}.$$ Now, we [distinguish]{} two cases: - $H[{T}] \cap {M}= \emptyset$: In this case, we know that $\bar Q = {\bar Q}^{M}$, thus, using , we have that $\breve {\beta}x \leq \delta$ is valid for $\bar Q$. We show that $\breve {\beta}^\top x \le \delta$ is valid for $Q$, and therefore trivially for ${\mathcal{{M}}}(Q)$. Assume by contradiction that there is a point $x \in Q$ such that $\breve {\beta}^\top x > \delta$. We have $\breve {\beta}^\top \breve x = \breve {\beta}^\top x > \delta$. As $Q$ is a packing set, we have $\breve x \in Q$. Moreover, since $\breve x \in H[T]$, we have $\breve x \in \bar Q$. Thus $\breve x$ is a vector in $\bar Q$ with $\breve {\beta}^\top \breve x > \delta$, a contradiction since $\breve {\beta}x \leq \delta$ is valid for $\bar Q$. Therefore, in this case the statement is trivially satisfied. - $H[{T}] \cap {M}\neq \emptyset$: By the definition of $M[T]$, we have that ${M}[{T}] \neq \emptyset$ and $$H[{T}] \cap {M}= H[{T}] \cap {M}[T].$$ Therefore, $\bar Q \setminus {M}= \bar Q \setminus {M}[T]$, which together with imply that $$\label{factForContradictionMT} \breve {\beta}^\top x \le \delta \ \text{is a valid inequality for} \ {\bar Q}^{{\mathcal{{M}}}[T]}.$$ We now show that $\breve {\beta}^\top x \le \delta$ is a valid inequality for $Q^{{M}[T]}$. Assume by contradiction that there is a point $x \in Q \setminus {M}[T]$ such that $\breve {\beta}^\top x > \delta$. We have $\breve {\beta}^\top \breve x = \breve {\beta}^\top x > \delta$. As $Q$ is a packing set, we have $\breve x \in Q$. Moreover, since $\breve x \in H[T]$, we have $\breve x \in \bar Q$. Finally, since $x \notin {M}[T]$, we obtain that also $\breve x \notin {M}[T]$ by definition of ${M}[T]$. Thus $\breve x$ is a vector in $\bar Q \setminus {M}[T]$ with $\breve {\beta}^\top \breve x > \delta$, a contradiction to . Proof of Theorem \[thm:thm3\] {#subsec:4.3} ----------------------------- Let ${\mathcal{{M}}}$ be a collection of strict lattice-free sets with the following property: There exist $\alpha_{\mathcal{{M}}}\geq 1$ such that $P_P {\subseteq}\alpha_{\mathcal{{M}}}{\mathcal{{M}}}(P_P)$ for every well-behaved packing polyhedron $P_P$. Let $Q$ be a well-behaved packing set. We will show that $Q {\subseteq}\alpha_{\mathcal{{M}}}{\mathcal{{M}}}(Q)$. Our strategy to prove this statement is to first construct, in Lemma \[prop:SantanuLemma\], a well-behaved packing polyhedron which is an inner approximation of $Q$ and [can be chosen]{} arbitrarily close to $Q$.. Then, we apply the $\alpha_{\mathcal{{M}}}$ factor to this polyhedral approximation and “transfer" it to $Q$. \[prop:SantanuLemma\] Let $\epsilon > 0$. Then, there exists a well-behaved packing polyhedron $P_\epsilon$ such that $\frac{1}{1+\epsilon} Q \subseteq P_\epsilon \subseteq Q$. [First, consider the case that $Q$ is bounded.]{} Let $\sigma_Q$ be the support function of $Q$, i.e., $$\sigma_Q (u) = \sup \{ u^\top x | x \in Q \},$$ and $$C^n := \{ u \in {\mathbb{R}}^n_+ \, | \, \|u\|_2 = 1 \}.$$ Also, let $\tilde{Q} = \frac{1}{1+\epsilon} Q$. We first show that there exists $M > 0$ such that $$\label{eq:sigmaLB} \sigma_{\tilde{Q}}(u) \geq M \ \text{for all} \ u \in C^n.$$ Let $S = \{ x \in {\mathbb{R}}^n_+ ~|~ {\boldsymbol{1}}^\top x \leq 1 \}$ and $\tilde{S} = \frac{1}{1+\epsilon} S$. Since $Q$ is well-behaved, we have that $S \subseteq Q$, thus $\tilde{S} \subseteq \tilde{Q}$. Therefore, $\sigma_{\tilde{S}}(u) \leq \sigma_{\tilde{Q}}(u)$ for all $u \in C^n$. Since [$\sigma_{\tilde{S}}(u) \geq \frac{1}{\sqrt{n}(1+\epsilon)}$]{} for all $u \in C^n$, holds. Let $\bar{M} = \max \{ \|x\|_\infty ~|~ x \in \tilde{Q} \}$. It is well-known that $\sigma_{\tilde{Q}} (\cdot)$ is continuous since $\tilde{Q}$ is a compact convex set [@rockafellar:1970]. Moreover, as $\|\cdot\|_2$ is also continuous, [for any $u \in C^n$ and $\epsilon > 0$ there exists a neighborhood $N_u$ of $u$ (in the topology of the sphere)]{} such that for all $v \in N_u$ we have [$$\label{eq:contsigma} |\sigma_{\tilde{Q}}(u) - \sigma_{\tilde{Q}}(v)| \le \frac{\epsilon M}{4}$$ ]{} and $$\label{eq:cont2norm} \| u - v\|_2 \leq \frac{\epsilon M}{4 \bar{M}\sqrt{n}}.$$ Since $C^n$ is a compact set, there exists a finite list of vectors $v_1,\hdots,v_\ell $ such that $C^n = \cup_{i=1}^\ell N_{v_i}$. Define $$\label{eq:Psetdef} P^1_\epsilon := \{ x \in {\mathbb{R}}^n_+ ~|~ (v_i)^\top x \leq \sigma_{\tilde{Q}}(v_i), \forall i=1,\hdots,\ell, \ \text{and} \ x_i \leq \bar{M}, \forall i=1,\hdots,n \}$$ We now show that $$\label{eq:innerapprox} \tilde{Q} \subseteq P^1_\epsilon \subseteq (1+\frac{\epsilon}{2}) \tilde{Q} \subseteq Q.$$ Note that the first and the last containments are straightforward. In order to show the second containment, we [show]{} that $\sigma_{P^1_\epsilon}(u) / \sigma_{\tilde{Q}}(u) \leq 1+\frac{\epsilon}{2}$ for all $u \in C^n$. For a given $u \in C^n$, let $i \in \{ 1, \hdots,\ell\}$ such that $u \in N_{v_i}$. Observe that $$\begin{aligned} \sigma_{P^1_\epsilon}(u) & \leq \sigma_{P^1_\epsilon}(v_i) + \sigma_{P^1_\epsilon}(u-v_i) \nonumber \\ & \leq \sigma_{P^1_\epsilon}(v_i) + \| u - v_i \|_2 \cdot \max_{x \in P^1_\epsilon} \{ \| x \|_2 \} \nonumber \\ & \leq \sigma_{P^1_\epsilon}(v_i) + \frac{\epsilon M}{{{\color{black}}4} \sqrt{n} \bar{M}} \sqrt{n} \bar{M} \nonumber \\ & = \sigma_{P^1_\epsilon}(v_i) + \frac{\epsilon M}{4} \nonumber \\ & \leq \sigma_{\tilde{Q}}(v_i) + \frac{\epsilon M}{4} \nonumber \\ & \leq \sigma_{\tilde{Q}}(u) + \frac{\epsilon M}{4} + \frac{\epsilon M}{4} \nonumber \\ & = \sigma_{\tilde{Q}}(u) + \frac{\epsilon M}{2}, \label{eq:fifthineq} \end{aligned}$$ where the first inequality is due to the subadditivity property of the support functions [@rockafellar:1970], the second is due to the Cauchy-Schwartz inequality, the third follows from and , the fourth inequality is implied by the constraints defining $P_\epsilon$ in , and the last inequality is satisfied by . Inequality can be written as $$\frac{\sigma_{P^1_\epsilon}(u)}{\sigma_{\tilde{Q}}(u)} \leq 1+ \frac{\epsilon M}{2 \sigma_{\tilde{Q}}(u)} \leq 1+ \frac{\epsilon}{2},$$ [the second inequality]{} follows from . Due to , $P^1_\epsilon$ achieves almost all the required conditions except the fact that it may not be well-behaved. Therefore, let $$P_\epsilon = \operatorname{conv}{(P^1_\epsilon \cup S)}.$$ First, note that $\tilde{Q} \subseteq P^1_\epsilon \subseteq P_\epsilon \subseteq Q$ where the first two containments are straightforward, and the last containment follows from the fact that $S \subseteq Q$ and $P^1_\epsilon \subseteq Q$. It remains to verify that $P_\epsilon$ is a packing polyhedron which would imply that it is [a]{} well-behaved packing polyhedron since $S \subseteq P_\epsilon$. However, observe that $P_\epsilon$ is the convex hull of the union of two packing polyhedra, and therefore it is straightforward to verify that it is a packing polyhedron. Now suppose $Q$ is not bounded. Then we can decompose it as $Q = B + R$, where $B$ is a bounded packing set and $R$ is the recession cone of $Q$; explicitly, let $I = \{i \mid \operatorname{cone}(e_i) \subseteq Q\}$, so $B = Q \cap \{x \mid x_i \le 0~\forall i \in I\}$ and $R = \operatorname{cone}(\{e_i\}_{i \in I})$. Furthermore, let $\bar{B} = \operatorname{conv}(B \cup \bigcup_{i \in I} e_i)$, so that $\bar{B}$ is a *well-behaved* bounded packing set; notice $Q = B + R = \bar{B} + R$. Applying the proof above to the bounded set $\bar{B}$, we obtain a well-behaved packing *polyhedron* $P_\epsilon$ satisfying $\frac{1}{1+\epsilon} \bar{B} \subseteq P_\epsilon \subseteq \bar{B}$. Then $P_\epsilon + R$ is a well-behaved packing polyhedron (the polyhedrality follows from the fact $\bar{B}$ is a polytope and $R$ is finitely generated, see for example Theorem 3.13 of [@ConCorZam14b]). Finally, since $\alpha P_\epsilon + R = \alpha (P_\epsilon + R)$ for all $\alpha > 0$, $$\frac{1}{1+\epsilon} (\bar{B} + R) = \frac{1}{1 + \epsilon} \bar{B} + R \subseteq P_\epsilon + R \subseteq \bar{B} + R.$$ Since $\bar{B} + R = Q$, $P_\epsilon + R$ is the desired polyhedral approximation. This concludes the proof. Noting that $${\mathcal{{M}}}(Q) = \bigcap_{{M}\in {\mathcal{{M}}}} Q^{M},$$ it is sufficient to prove that $$\label{eq:Pkplusone} Q \subseteq (\alpha_{\mathcal{{M}}}) \, Q^{M},$$ [for an arbitrary $M \in {\mathcal{{M}}}$]{} (see Observation \[obs:bijection\]). Let $\epsilon > 0$ and $P_\epsilon$ be the well-behaved packing polyhedron satisfying the conditions of Lemma \[prop:SantanuLemma\]. Then, observe that $$\label{eq:generalContainment} \frac{1}{1+\epsilon} Q \subseteq P_\epsilon \subseteq (\alpha_{\mathcal{{M}}}) (P_\epsilon)^{M}\subseteq (\alpha_{\mathcal{{M}}}) \, Q^{M},$$ where the first and the last containments follow due to $\frac{1}{1+\epsilon} Q \subseteq P_\epsilon \subseteq Q$, whereas the second one holds by assumption and the fact that $P_\epsilon$ is well-behaved. Note that can be written as $Q \subseteq (1+\epsilon) (\alpha_{\mathcal{{M}}}) \, Q^{M}$. Since $\epsilon$ can be arbitrarily small, we obtain that $Q \subseteq (\alpha_{\mathcal{{M}}}) \, Q^{M}$. Proof of Theorem \[thm:thm4\] {#subsec:4.4} ----------------------------- Note that it is sufficient to prove the statement for ${\mathcal{{M}}}\in \{ {\mathcal{S}}^k, {\mathcal{L}}^k \}$ since ${\mathcal{S}}$ is a special case of ${\mathcal{S}}^k$. We will use Theorem \[thm:thm2\] to prove this statement. That is, letting $T {\subseteq}[n]$, we will show that for every ${M}\in {\mathcal{{M}}}$, we have ${M}[{T}] \in {\mathcal{{M}}}$ as well. [Recall the operator $\breve{(\cdot)}$ from equation .]{} - Consider an arbitrary element of ${\mathcal{S}}^k$ as $$M = \bigcup_{i \in [k]} S(\pi^i,\pi^i_0).$$ Observe that $$M[{T}] = \bigcup_{i \in [k]} S(\breve \pi^i,\pi^i_0).$$ If $\breve \pi^i = {\boldsymbol{0}}$, then $S(\breve \pi^i,\pi^i_0) = \emptyset$. Therefore, $M[T]$ is also a $k$-branch split set since $k$-branch split is defined to be the union of at most $k$ split sets. - Consider an arbitrary element of ${\mathcal{L}}^k$ as $$M = \{ x \in {\mathbb{R}}^n | (\pi^i)^\top x < \pi^i_0, \ i = 1,\hdots,k \} .$$ Observe that $$M[{T}] = \{ x \in {\mathbb{R}}^n | (\breve \pi^i)^\top x < \pi^i_0, \ i = 1,\hdots,k \} .$$ If $\breve \pi^i = {\boldsymbol{0}}$, then either the inequality $(\breve \pi^i)^\top x < \pi^i_0$ is trivially satisfied, or $M[{T}] = \emptyset$. Therefore, $M[T]$ is also a $k$-lattice-free set since $k$-lattice-free set is defined to be the union of at most $k$ lattice-free sets. Proof of Theorem \[thm:thm5\] ----------------------------- In order to make the proofs more self-contained, we record here standard bounds on the integrality gap of well-behaved packing polyhedra, which are essentially Proposition 6 of [@bodur2016aggregation]. \[prop:intGapPack\] Consider a well-behaved packing polyhedron $P_P = \{x \in {\mathbb{R}}^n_+ \mid (a^i)^{\top} x \le b_i,~\forall i \in [m]\}$. Then $P_P$ is a $(\min\{m,n\} + 1)$-approximation of the integer hull $P_P^I$. It is equivalent to show that $P_P$ is both an $(m+1)$- and an $(n+1)$-approximation of $P_P^I$. The former is precisely Proposition 6 of [@bodur2016aggregation], and the latter follows from similar arguments, reproduced here. Given a cost vector $c \in {\mathbb{R}}^n_+$, we need to show that $\max\{c^\top x \mid x \in P_P\} \le (n + 1) \max\{c^\top x \mid x \in P_P^I\} =: (n+1) z^I$. Let $x^{LP}$ be an optimal solution for the left-hand side, and let $\hat{x}$ be the integer solution obtained by rounding down each component of $x^{LP}$. Since each component of the difference $x^{LP} - \hat{x}$ is in $[0,1]$, we have $$\begin{aligned} z^I \ge c^\top \hat{x} \ge c^{\top} x^{LP} - \|c\|_{\infty}. \end{aligned}$$ Moreover, since $P_P$ is well-behaved, all canonical vectors $e_i$ belong to $P_P^I$ and hence $z^I \ge \|c\|_{\infty}$. Adding $n$ times this lower bound to the displayed equation, we obtain $(n+1)z^I \ge c^{\top} x^{LP}$, thus proving the desired result. This concludes the proof. ### Case of ${\mathcal{S}}$ {#subsubsec:4.5.1} We show that $\alpha_{\mathcal{S}}= 2$ in Proposition \[prop:packing2approx\]. The proof of Proposition \[prop:packing2approx\] involves a reduction to analyzing split closure of a packing polyhedron in ${\mathbb{R}}^2$, and a case analysis in ${\mathbb{R}}^2$ gives the correct factor of 2 (Lemma \[lem:pack\_2approx\]). For the last statement in the theorem, we provide a tight example in Proposition \[prop:pack\_TightEx\]. [We start with the proof of the result for the two-dimensional case.]{} \[lem:pack\_2approx\] Let $P_P \subseteq {\mathbb{R}}^2$ be well-behaved. Then $P_P {\subseteq}2 {\mathcal{S}}(P_P)$. [By [@cook:ka:sc:1990] and by Theorem \[thm:thm4\], the set $S(P_P)$ is a well-behaved packing polyhedron. To show that $P_P {\subseteq}2 {\mathcal{S}}(P_P)$, we just need to show that for all facet-defining inequalities ${\beta}^\top x \le \delta$ of $S(P_P)$, the inequality ${\beta}^\top x \le 2 \delta$ is valid for $P_P$. This is trivially satisfied for the facet-defining inequalities of $S(P_P)$ of the type $x_i \ge 0$, thus it remains to be shown for the other facet-defining inequalities of $S(P_P)$. Since $S(P_P)$ is of packing type, such facet-defining inequalities are of the form ${\beta}^\top x \le \delta$ with ${\beta}\in {\mathbb{R}}^2_+$. Since the split closure is finitely generated [@Ave12], each facet-defining inequality ${\beta}^\top x \le \delta$ of $S(P_P)$ defines a facet of a set $P_P^T := \operatorname{conv}(P_P \setminus \int T)$, where $T$ is a split set. (Note that the polyhedra $P_P^T$ are not necessarily of packing type.) Therefore, to complete the proof of the lemma, it suffices to show that for every split set $T$, and for every inequality ${\beta}^\top x \le \delta$ with ${\beta}\in {\mathbb{R}}^2_+$ valid for $P_P^T$, the inequality ${\beta}^\top x \le 2 \delta$ is valid for $P_P$. We show that for every split set $T$, and for every ${\beta}\in {\mathbb{R}}^2_+$, there exists $\hat x \in P_P^T$ that satisfies $\max\{{\beta}^\top x \mid x \in P_P\} \le 2{\beta}^\top \hat x$. This completes the proof because ${\beta}^\top \hat x \le \delta$ implies that every point in $P_P$ satisfies ${\beta}^\top x \le \max\{{\beta}^\top x \mid x \in P_P\} \le 2{\beta}^\top \hat x \le 2 \delta$.]{} [Now, fix a split set $T$ and a vector]{} ${\beta}\in {\mathbb{R}}^2_+$, and let $\bar x$ be a vector in $P_P$ that achieves $\max\{{\beta}^\top x \mid x \in P_P\}$. Since $P_P$ is a packing polyhedron, we have $\bar x \ge 0$. We divide the proof in three main cases based on the position of vector $\bar x$. 1\. In the first case we assume that $\bar x \ge (1,1)$, and we define $\hat x := \lfloor \bar x \rfloor$. Since $P_P$ is a packing polyhedron, we have that $\hat x \in P_P \cap {\mathbb{Z}}^2 \subseteq P_P^T$. As $\bar x \ge (1,1)$, we have $2 \hat x \ge \bar x$. Finally, ${\beta}\ge 0$ implies $2 {\beta}^\top \hat x \ge {\beta}^\top \bar x$ as desired. 2\. In the second case we assume that $\bar x \le (1,1)$. Since $P_P$ is well-behaved, we have that points $(1,0)$ and $(0,1)$ are in $P_P$ and therefore in $P_P^T$. If ${\beta}_1 \ge {\beta}_2$, we define $\hat x := (1,0)$. Then $2{\beta}^\top \hat x = 2 {\beta}_1 \ge {\beta}_1 + {\beta}_2$. Since ${\beta}\ge 0$ and $\bar x \le (1,1)$, we have ${\beta}_1 + {\beta}_2 \ge {\beta}^\top \bar x$, which implies $2{\beta}^\top \hat x \ge {\beta}^\top \bar x$ as desired. Symmetrically, if ${\beta}_2 \ge {\beta}_1$, we define $\hat x := (0,1)$, and obtain $2{\beta}^\top \hat x \ge {\beta}^\top \bar x$. 3\. In the third case we assume that $\bar x_1 < 1$ and $\bar x_2 > 1$. (The same argument works for the symmetric case $\bar x_2 < 1$ and $\bar x_1 > 1$.) Assume first that $T$ is *not* a vertical split set [$\{x \mid t \le x_1 \le t+1\}$]{} for some integer $t$. Define now $\hat x^1 := \lfloor \bar x \rfloor = (0,{\lfloor\bar x_2\rfloor})$. Since $P_P$ is [a packing polyhedron]{}, [the]{} vector $\hat x^1$ is in $P_P \cap {\mathbb{Z}}^2$, and therefore in $P_P^T$. If $2{\beta}^\top \hat x^1 = 2 {\beta}^\top (0,{\lfloor\bar x_2\rfloor}) \ge {\beta}^\top \bar x$, then we are done, thus we now assume $2 {\beta}^\top (0,{\lfloor\bar x_2\rfloor}) \le {\beta}^\top \bar x$. Let $\hat x^2 := (\bar x_1, \bar x_2-1)$. It can be shown that, since $T$ is not a vertical split set, the vector $\hat x^2$ is in $P_P^T$. We show $2 {\beta}^\top \hat x^2 = 2{\beta}^\top (\bar x_1,\bar x_2 -1) \ge {\beta}^\top \bar x$. Since ${\lfloor\bar x_2\rfloor} \ge 1$ and ${\beta}_2 \ge 0$, we have ${\beta}^\top \bar x \ge 2 {\beta}_2 {\lfloor\bar x_2\rfloor} \ge 2 {\beta}_2$. By adding ${\beta}^\top \bar x$ to both sides we obtain $2 {\beta}^\top \bar x - 2 {\beta}_2 \ge {\beta}^\top \bar x$, thus $2{\beta}^\top (\bar x_1,\bar x_2 -1) \ge {\beta}^\top \bar x$. Finally, assume that $T$ is a vertical split set [$\{x \mid t \le x_1 \le t+1\}$]{} for some integer $t$. Define now $\hat x^1 := (1,0)$. Since $P_P$ is well-behaved, [the]{} vector $\hat x^1$ is in $P_P \cap {\mathbb{Z}}^2$, and therefore in $P_P^T$. If $2{\beta}^\top \hat x^1 = 2 {\beta}^\top (1,0) \ge {\beta}^\top \bar x$, then we are done, thus we now assume $2 {\beta}^\top (1,0) \le {\beta}^\top \bar x$. Define $\hat x^2 := (0, \bar x_2)$ and note that $\hat x^2 \in P_P^T$ since $T$ is a vertical split set. We show $2{\beta}^\top\hat x^2 = 2{\beta}^\top (0, \bar x_2) \ge {\beta}^\top \bar x$. Since ${\beta}_1 \ge 0$ and $\bar x_1 < 1$, we have $2 {\beta}_1 \bar x_1 < 2 {\beta}_1$. By summing the latter with $2 {\beta}_1 \le {\beta}^\top \bar x$ we obtain $2 {\beta}_1 \bar x_1 \le {\beta}^\top \bar x$. By adding and subtracting $2{\beta}_2 \bar x_2$ to the left-hand, we get $2 {\beta}^\top \bar x - 2{\beta}_2 \bar x_2 \le {\beta}^\top \bar x$ which implies $2 {\beta}_2 \bar x_2 \ge {\beta}^\top \bar x$ as desired. \[prop:packing2approx\] Let $Q {\subseteq}{\mathbb{R}}^n$ be a well-behaved packing set. Then, $Q {\subseteq}2 {\mathcal{S}}(Q).$ It is sufficient to prove this proposition for a packing polyhedron, $P_P$, due to Theorem \[thm:thm3\]. Let $(\pi,\pi_0) \in {\mathbb{Z}}^n \times {\mathbb{Z}}$ and let ${\beta}^\top x \leq \delta$ be a valid inequality for $(P_P)^{\pi,\pi_0}$. Note that, due to Observation \[obs:bijection\], it is sufficient to show that ${\beta}^\top x \leq 2 \delta$ is valid for $P_P$. Due to Farkas’ Lemma [(e.g., Theorem 3.22 in [@ConCorZam14b])]{}, there exist ${{\color{black}}\lambda^1, \lambda^2} \in {\mathbb{R}}_+^m, \ {{\color{black}}\mu_1, \mu_2 \in {\mathbb{R}}_+}$ and ${{\color{black}}\sigma^1, \sigma^2} \in {\mathbb{R}}_+^n$ such that for any $j \in [n]$, we have $${\beta}_j = \sum_{i=1}^m \lambda^1_i A_{ij} + \mu_1 \pi_j - \sigma^1_j = \sum_{i=1}^m \lambda^2_i A_{ij} - \mu_2 \pi_j - \sigma^2_j.$$ Let $$Q := \{ x \mid (\lambda^1)^\top Ax \leq (\lambda^1)^\top b, ~ (\lambda^2)^\top Ax \leq (\lambda^2)^\top b, ~ x \geq 0\}.$$ Now, observe that $Q \supseteq P_P$. Therefore, it is sufficient to show that ${\beta}^\top x \leq 2 \delta$ is valid for $Q$. We will prove that the following holds: $$\label{eq:QclaimPacking} Q \subseteq 2 Q^{\pi,\pi_0}.$$ Since ${\beta}^\top x \leq \delta$ is valid for $Q^{\pi,\pi_0}$ by the definition of $Q$, this will imply that ${\beta}^\top x \leq 2 \delta$ is valid for $Q$. In order to show that holds, we verify that $$\label{eq:minPacking} \max \{ c^\top x \mid x \in Q\} \leq 2 \max \{ c^\top x \mid x \in Q^{\pi,\pi_0} \},$$ for any objective vector $c \in {\mathbb{R}}_+^n$. Let $x^*$ be a vertex of $Q$ that maximizes $c^\top x$ over $Q$. As $Q$ is defined by two linear inequalities, together with non-negativities, we know that at least $n-2$ components of $x^*$ are zero, say $x^*_j = 0$ for all $j=3,\hdots,n$. We will focus on the restriction of $Q$ to the first two variables, which we denote by $Q |_{{\mathbb{R}}^2}$. Observe that $$\label{eq:qc1} \max \{ c^\top x \mid x \in Q\} = \max \{ c_1 x_1 + c_2 x_2 | (x_1,x_2) \in Q |_{{\mathbb{R}}^2} \}.$$ Moreover, we have $$\label{eq:qc2} \max \{ c^\top x \mid x \in Q^{\pi,\pi_0} \} \geq \max \{ c_1 x_1 + c_2 x_2 | (x_1,x_2) \in (Q |_{{\mathbb{R}}^2})^{\pi,\pi_0} \}$$ because [$(Q |_{{\mathbb{R}}^2})^{\pi,\pi_0} \subseteq Q^{\pi,\pi_0} |_{{\mathbb{R}}^2}$.]{} Due to and , in order to prove , it is sufficient to only prove in ${\mathbb{R}}^2$. Since $Q |_{{\mathbb{R}}^2}$ is well-behaved, this immediately follows from Lemma \[lem:pack\_2approx\]. \[prop:pack\_TightEx\] For every $\epsilon >0$, there exists a well-behaved packing polyhedron $\tilde{P}_P$ such that $\tilde{P}_P \not\subseteq (2-\epsilon) {\mathcal{S}}(\tilde{P}_P)$. Let $\epsilon > 0$ and $M = \max\{1,\lceil \frac{2}{\epsilon}-1 \rceil \}$. Consider the instance $\textup{max} \{ x_1 + x_2 \,|\, x \in \tilde{P}_P\}$, where $$\begin{aligned} \tilde{P}_P = \{x \in {\mathbb{R}}^2_+ \,|\, x_1 + Mx_2 \leq M, \ Mx_1 + x_2 \leq M \}.\end{aligned}$$ Note that $\tilde{P}_P$ is well-behaved. It is sufficient to show that $\frac{z^{LP}}{z^{{\mathcal{S}}}} \ge 2 - \epsilon$ for this instance. 1. $z^{LP} \ge \frac{2M}{M + 1}$: It can be checked that the point $\bar x_1 = \bar x_2 = \frac{M}{M + 1}$ is in $\tilde{P}_P$. Thus, $z^{LP} \ge \frac{2M}{M + 1}$. 2. $z^{{\mathcal{S}}} \le1$: Adding the two constraints defining $\tilde{P}_P$ we obtain the valid inequality $$\begin{aligned} x_1 + x_2 \le \frac{2M}{M + 1}\end{aligned}$$ The corresponding CG cut is $x_1 + x_2 \leq 1$. Since each CG cut is also a split cut we obtain $z^{{\mathcal{S}}} \le 1$. Thus, $\frac{z^{LP}}{z^{{\mathcal{S}}}} \ge \frac{2M}{M + 1}$; and our choice of $M$ completes the proof. We note that the example given in Proposition \[prop:pack\_TightEx\] is the same as the one used in [@bodur2016aggregation] to show that the $2$-approximation bound for the CG closure of a well-behaved packing polyhedron is tight. ### Case of ${\mathcal{S}}^k$ {#subsubsec:4.5.2} We will show that [we can choose $\alpha_{{\mathcal{S}}^k} = \min \{ 2^k,n \}+1$]{}. It is sufficient to prove this proposition for a packing polyhedron, $P_P$, due to Theorem \[thm:thm3\]. Let $P_P = \{ x \in {\mathbb{R}}^n | Ax \leq b, x \geq 0\}$ and $\pi^i \in {\mathbb{Z}}^n, \pi^i_0 \in {\mathbb{Z}}\textup{ for all } i \in [k]$. It is sufficient to prove that $(\min\{2^k,n\}+1) \, (P_P)^{\pi^1,\dots, \pi^k;\pi^1_0, \dots, \pi^k_0} \supseteq P_P$. Let ${\beta}^\top x \leq \delta$ be a valid inequality for $(P_P)^{\pi^1,\dots, \pi^k;\pi^1_0, \dots, \pi^k_0}$. Since\ ${\boldsymbol{0}}\in (P_P)^{\pi^1,\dots, \pi^k;\pi^1_0, \dots, \pi^k_0}$, we have $\delta \geq 0$. Therefore, it is sufficient to prove that $$(\min\{2^k,n\}+1) \left( \{ x | {\beta}^\top x \leq \delta \} \right) \supseteq P_P.$$ Let $\mathcal{G} = \{ G {\subseteq}{{\color{black}}[k]} : (P_P)_G^{{\pi}^1,\dots, {\pi}^k;\pi^1_0, \dots, \pi^k_0} \neq \emptyset \}$, where $(P_P)_G^{{\pi}^1,\dots, {\pi}^k;\pi^1_0, \dots, \pi^k_0}$ is defined as $$(P_P)_G^{\pi^1,\dots, \pi^k;\pi^1_0, \dots, \pi^k_0} = P_P {\mathbin{\scalebox{1.5}{\ensuremath{\cap}}}}\left(\bigcap_{i \in G} \{ (\pi^i)^\top x \geq \pi^i_0 + 1)\} \right){\mathbin{\scalebox{1.5}{\ensuremath{\cap}}}}\left(\bigcap_{i \in {{\color{black}}[k]} \setminus G} \{ (\pi^i)^\top x \leq \pi^i_0)\}\right).$$ By Farkas’ Lemma, we know that ${\beta}^\top x \leq \delta$ is valid for $$\label{eq:aggPPG} \{ x \in {\mathbb{R}}_+^n | (\lambda^G)^\top A x \leq (\lambda^G)^\top b, (\pi^i)^\top x \geq \pi^i_0 + 1, \ \forall \ i \in G, \ (\pi^i)^\top x \leq \pi^i_0, \ \forall \ i \in {{\color{black}}[k]} \setminus G \},$$ for some $\lambda^G \in {\mathbb{R}}_+^m$. Let $$Q = \{ x \in {\mathbb{R}}_+^n | (\lambda^G)^\top A x \leq (\lambda^G)^\top b, \ \forall G \in \mathcal{G} \}$$ which is well-behaved since $P_P$ is assumed to be well-behaved. Now, observe that $$\begin{aligned} (\min\{2^k,n\}+1) \left( \{ x | {\beta}^\top x \leq \delta \} \right) & \supseteq (\min\{|\mathcal{G}|,n\}+1) \left( \{ x | {\beta}^\top x \leq \delta \} \right) \\ & \supseteq (\min\{|\mathcal{G}|,n\}+1) Q^I \supseteq Q \supseteq P_P,\end{aligned}$$ where the second containment follows from , the third one follows from [Proposition \[prop:intGapPack\]]{} since $Q$ is well-behaved, and the last one is straightforward. ### Case of ${\mathcal{L}}^k$ {#subsubsec:4.5.3} We show that [we can choose $\alpha_{{\mathcal{L}}^k} = \min \{ k,n \}+1$]{}. It is sufficient to prove this proposition for a packing polyhedron, $P_P$, due to Theorem \[thm:thm3\]. Let $P_P = \{ x \in {\mathbb{R}}^n | Ax \leq b, x \geq 0\}$ and [let]{} $$L = \{ x \in {\mathbb{R}}^n | (\pi^j)^\top x \leq \pi^j_0, \ j = 1,\hdots,k \}$$ [be lattice-free.]{} Then, observe that $$\label{eq:convPPL} (P_P)^L = \operatorname{conv}(P_P \setminus \int(L)) = \operatorname{conv}\left(\bigcup_{j=1}^k \left\{ x \in P_P ~|~ (\pi^j)^\top x \geq \pi^j_0 \right\} \right).$$ Without loss of generality, assume that the set $\{ x \in P_P ~|~ (\pi^j)^\top x \geq \pi^j_0 \}$ is non-empty if $j \leq r$, and empty otherwise, for some $r$ with $1 \leq r \leq k$. Let ${\beta}^\top x \leq \delta$ be a valid inequality for $(P_P)^L$. Since the origin is contained in $(P_P)^L$, we have $\delta \geq 0$. Therefore, it is sufficient to prove that $$(\min\{k,n\}+1) \left( \{ x | {\beta}^\top x \leq \delta \} \right) \supseteq P_P.$$ By equation and Farkas’ Lemma, we know that ${\beta}^\top x \leq \delta$ is valid for $$\label{eq:aggPP} \{ x \in {\mathbb{R}}_+^n | (\lambda^j)^\top A x \leq (\lambda^j)^\top b, (\pi^j)^\top x \geq \pi^j_0 \},$$ for $j = 1,\hdots,r$ where $\lambda^j \in {\mathbb{R}}_+^m$. Let $$Q = \{ x \in {\mathbb{R}}_+^n | (\lambda^j)^\top A x \leq (\lambda^j)^\top b, \ j = 1,\hdots,r \}$$ which is well-behaved since $P_P$ is assumed to be well-behaved. Now, observe that $$(\min\{k,n\}+1) \left( \{ x | {\beta}^\top x \leq \delta \} \right) \supseteq (\min\{k,n\}+1) Q^I \supseteq Q \supseteq P_P,$$ where the first containment follows from and $L$ being a lattice-free set, the second one follows from [Proposition \[prop:intGapPack\]]{} since $Q$ is well-behaved, and the last one is straightforward. ### Proof of Proposition \[prop:prop1\] {#subsubsec:4.5.4} Let $P_P$ be the standard relaxation of the stable set polytope: $$\begin{aligned} P_P = \{ x \in {\mathbb{R}}_+^{n} \,|\, x_i + x_j \le 1 \ \forall i,j \in [n], \ i < j\}.\end{aligned}$$ Corresponding to the clique inequality ${\boldsymbol{1}}^\top x \le 1$, we optimize the all ones vector over $P_P$ and $(P_P)^I$, and obtain $z^{LP}=n/2$ and $z^{I}=1$, respectively. The CG rank of the clique inequality is known to be $\lceil \log_2 (n-1) \rceil$ [@hartmann], therefore it also constitutes an upper bound on the split rank. Proofs for covering problems {#sec:Covering} ============================ \[prop:CoveringPoly\] ${\mathcal{S}}(P_C)$ is a covering polyhedron. If $P_C$ is empty, then there is nothing to prove. So, assume that $P_C$ is not empty. It is known that the split closure of a polyhedron is also a polyhedron [@cook:ka:sc:1990]. Let ${\beta}^\top x \geq \delta$ be a valid inequality for ${\mathcal{S}}(P_C)$. Then, there exists $(\pi,\pi_0) \in {\mathbb{Z}}^n \times {\mathbb{Z}}$ such that ${\beta}^\top x \geq \delta$ is valid for $P_C^{\pi,\pi_0}$. If one side of the disjunction is empty, then we already know that ${\beta}^\top x \geq \delta$ is of covering type as it is a CG cut. Now, assume that both sides are nonempty. Then, due to Farkas’ Lemma, there exist multipliers $\lambda^1, \lambda^2 \in {\mathbb{R}}_+^m, \ \mu_1, \mu_2 \in {\mathbb{R}}_+$ and $\sigma^1, \sigma^2 \in {\mathbb{R}}_+^n$ for the aggregation $$\begin{aligned} {4} & (\lambda^1) \quad && Ax \geq b \qquad \qquad \qquad && (\lambda^2) \quad && Ax \geq b \\ & (\mu_1) && - \pi^\top x \geq -\pi_0 && (\mu_2) && \pi^\top x \geq \pi_0 +1 \\ & (\sigma^1_j) && x_j \geq 0 && (\sigma^2_j) && x_j \geq 0 \qquad \qquad j=1,\hdots,n\end{aligned}$$ such that, for any $j=1,\hdots,n$, $$\label{eq:cSplit} {\beta}_j = \sum_{i=1}^m \lambda^1_i A_{ij} - \mu_1 \pi_j + \sigma^1_j = \sum_{i=1}^m \lambda^2_i A_{ij} + \mu_2 \pi_j + \sigma^2_j.$$ This implies that, for any $j=1,\hdots,n$, we have ${\beta}_j \geq 0$ (based on the sign of $\pi_j$, either the middle or the last expression [witnesses non-negativity]{}). Lastly, note that if $\delta < 0$, then ${\beta}^\top x \geq \delta$ is dominated by ${\beta}^\top x \geq 0$, which concludes the proof. \[prop:covering2approx\] Let $P_C$ be well-behaved, i.e., $A_{ij} \leq b_i$ for all $i \in [m], j \in [n]$. Then, $$z^{LP} \geq \frac{1}{2} z^{{\mathcal{S}}}.$$ Let ${\beta}^\top x \geq \delta$ be a facet-defining inequality for ${\mathcal{S}}(P_C)$. Note that, due to Observation \[obs:bijection\], it is sufficient to show that ${\beta}^\top x \geq \frac{\delta}{2}$ is valid for $P_C$. [If ${\beta}^\top x \geq \delta$ is valid for $P_C$, then there is nothing to show. So, assume that it is a non-trivial inequality.]{} If ${\beta}^\top x \geq \delta$ is a [non-trivial]{} CG cut, then [$\delta \geq 1$. We know that the strict inequality ${\beta}^\top x > \delta - 1$ is valid for $P_C$. If $\delta \geq 2$, then $\delta - 1 \geq \frac{\delta}{2}$, which implies that ${\beta}^\top x \geq \frac{\delta}{2}$ is valid for $P_C$. So, now assume that $\delta = 1$. Let $\lambda \in {\mathbb{R}}_+^m$ such that the CG cut is obtained by rounding up the coefficients and the right-hand-side of the base inequality $\lambda^\top A x \geq \lambda^\top b$, i.e., $\beta = {\lceil\lambda^\top A\rceil}$ and $\delta = {\lceil\lambda^\top b\rceil}$. As $\delta = 1$, we have $0 < \lambda^\top b \leq 1$. Thus, the scaled base inequality $(\lambda^\top A / \lambda^\top b) x \geq 1$ is valid for $P_C$. In addition, since $P_C$ is well-behaved, $\lambda^\top A_{\cdot j} \leq \lambda^\top b \ (\leq 1)$ for all $j \in [n]$ (where $A_{\cdot j}$ denotes the $j^\text{th}$ column of $A$. Then, for any $x \in P_C$, we have $$\beta^\top x = \sum_{j \in [n]} {\lceil\lambda^\top A_{\cdot j}\rceil} x_j \geq \sum_{j \in [n]} \frac{\lambda^\top A_{\cdot j}}{\lambda^\top b} x_j \geq 1 \geq \frac{{\lceil\lambda^\top b\rceil}}{2} = \frac{\delta}{2},$$ which implies that is $\beta^\top x \geq \frac{\delta}{2}$ valid for $P_C$. ]{} Otherwise, let $(\pi,\pi_0) \in {\mathbb{Z}}^n \times {\mathbb{Z}}$ be a corresponding vector such that ${\beta}^\top x \geq \delta$ is valid for $P_C^{\pi,\pi_0}$. Let $$Q := \{ x \mid (\lambda^1)^\top Ax \geq (\lambda^1)^\top b, ~ (\lambda^2)^\top Ax \geq (\lambda^2)^\top b, ~ x \geq 0\},$$ where $\lambda^1$ and $\lambda^2$ are the multipliers that satisfy . Now, observe that $Q \supseteq P_C$. Therefore, it is sufficient to show that ${\beta}^\top x \geq \frac{\delta}{2}$ is valid for $Q$. We will prove that the following holds: $$\label{eq:Qclaim} Q \subseteq \frac{1}{2} Q^{\pi,\pi_0}.$$ Since ${\beta}^\top x \geq \delta$ is valid for $Q^{\pi,\pi_0}$ by the definition of $Q$, this will imply that ${\beta}^\top x \geq \frac{\delta}{2}$ is valid for $Q$. In order to show that holds, we verify that $$\label{eq:min} \min \{ c^\top x \mid x \in Q\} \geq \frac{1}{2} \min \{ c^\top x \mid x \in Q^{\pi,\pi_0} \},$$ for any objective vector $c \in {\mathbb{R}}_+^n$. Let $x^*$ be a vertex of $Q$ that minimizes $c^\top x$ over $Q$. If $x^*$ belongs to $Q^{\pi,\pi_0}$, we are done. Thus, assume that $x^* \notin Q^{\pi,\pi_0}$. We will prove by showing that there exists a point $\hat{x} \in Q^{\pi,\pi_0}$ such that $c^\top \hat{x} \leq 2 c^\top x^*$. As $Q$ is defined by two linear inequalities, together with non-negativities, we know that at least $n-2$ components of $x^*$ are zero, say $x^*_j = 0$ for all $j=3,\hdots,n$. We will focus on this restriction of $Q$ in ${\mathbb{R}}_+^2$ in order to identify $\hat{x}$. Without loss of generality, assume that $c_1 \geq c_2$. A key observation that follows from the definition of split cuts is $$\label{eq:vee} (x^*_1+1,x^*_2,{\boldsymbol{0}}) \in Q^{\pi,\pi_0} \vee (x^*_1,x^*_2+1,{\boldsymbol{0}}) \in Q^{\pi,\pi_0}.$$ [Moreover, if $\pi \neq e_1$, then $(x^*_1,x^*_2+1,{\boldsymbol{0}}) \in Q^{\pi,\pi_0}.$\ ]{} Now, we will consider two cases to prove .\ \ *Case 1*. $x^*_1 \geq 1$: [Using (\[eq:vee\]), there are two subcases based on whether $(x^*_1+1,x^*_2,{\boldsymbol{0}}) \in Q^{\pi,\pi_0} $ or $ (x^*_1,x^*_2+1,{\boldsymbol{0}}) \in Q^{\pi,\pi_0}$. If $\hat{x} = (x^*_1+1,x^*_2,{\boldsymbol{0}}) \in Q^{\pi,\pi_0}$, then it is sufficient to show that $$c^\top x^* \geq \frac{1}{2} (c^\top x^* + c_1),$$ which is equivalent to $c^\top x^* \geq c_1$, which holds because $x^*_1 \geq 1$ and $c_1, c_2, x^*_2 \geq 0$. If $\hat{x} = (x^*_1,x^*_2 + 1,{\boldsymbol{0}}) \in Q^{\pi,\pi_0}$, then it is sufficient to show that $$c^\top x^* \geq \frac{1}{2} (c^\top x^* + c_2),$$ which is equivalent to $c^\top x^* \geq c_2$, which holds because $x^*_1 \geq 1$, $c_1 \geq c_2$ and $c_2, x^*_2 \geq 0$. ]{}\ \ *Case 2*. $0 \leq x^*_1 < 1$: Note that by construction, $Q$ is a well-behaved covering polyhedron. Now, consider the following two subcases:\ \ *Case 2a*. $(\pi,\pi_0) \neq (e_1,0)$: [In this case as discussed above]{}, [$\hat{x} = (x^*_1,x^*_2 + 1,{\boldsymbol{0}}) \in Q^{\pi,\pi_0}$]{}. It is sufficient to show that $$c^\top x^* \geq \frac{1}{2} (c^\top x^* + c_2),$$ which is equivalent to $c^\top x^* \geq c_2$. This holds as we have $x^*_1+x^*_2 \geq 1$ since $Q$ is well-behaved [and because $c_1 \geq c_2$]{}.\ \ *Case 2b*. $(\pi,\pi_0) = (e_1,0)$: Let $\hat{x} = (x^*_1,x^*_2 + x^*_1 x^*_2,{\boldsymbol{0}})$. We will first show that $\hat{x} \in Q^{\pi,\pi_0}$. Figure \[fig:1pic\] illustrates the restriction of $Q$ to the first two variables, which we denote by $Q |_{{\mathbb{R}}^2}$. Observe that $Q |_{{\mathbb{R}}^2}$ is a well-behaved covering polyhedron because $Q$ is a well-behaved covering polyhedron. Note that $x_1^* > 0$ due to the assumption $x^* \notin Q^{e_1,0}$. [Since $0 < x^*_1 < 1$, and $Q |_{{\mathbb{R}}^2}$ is a well-behaved covering polyhedron, we have that $x_1 \geq \gamma$ cannot be a valid inequality for $Q |_{{\mathbb{R}}^2}$ where $0< \gamma \leq x^*_1$. In other words, all non-trivial inequalities $\alpha_1 x_1 + \alpha_2 x_2 \geq \theta$ defining $Q |_{{\mathbb{R}}^2}$ must have $\alpha_2 > 0$. Therefore, there must exist a vertex of the form $(0,y)$ of $Q |_{{\mathbb{R}}^2}$.]{} Since $(x^*_1,x^*_2)$ is a vertex of $Q |_{{\mathbb{R}}^2}$ [with $x_1^* > 0$]{}, we know that $(0,y) \neq (x^*_1,x^*_2)$. Let $(h,0)$ be the intercept of the line passing through the vertices $(0,y) $ and $(x^*_1,x^*_2)$. [We first claim that $h \geq 1$. Note that the supporting hyperplane corresponding to non-trivial facet-defining inequality at $(0, y)$ (i.e., different from $x_1 \geq 0$) intersects the $x_1$-axis at a point $(\tilde{h}, 0)$ with $\tilde{h} \geq 1$ due to well-behavedness of $Q |_{{\mathbb{R}}^2}$. Since the line passing through the vertices $(0,y) $ and $(x^*_1,x^*_2)$ has an intercept at least as large as $\tilde{h}$, we obtain that $h \geq 1$.]{} Then, we have (0,0) – (0,4.5); (0,0) – (4.5,0); (0,3.5) – (1.5,1.5); (1.5,1.5) – (4,0); (0,4.5) – (0,3.5) – (1.5,1.5) – (4,0) – (4.5,0) – (4.5,4.5) – cycle; (1.5,1.5) – (2.635,0); at (4,0) ; at (1.5,1.5) ; (xstar) at (2.1,1.7) [$(x^*_1,x^*_2)$]{}; at (0,3.5) ; (xstar) at (0.58,3.65) [$(0,y)$]{}; (h) at (2.635,-0.3) [$(h,0)$]{}; at (2.635,0) [x]{}; $$\label{eq:heq} \frac{y}{h} = \frac{y - x^*_2}{x^*_1} \Rightarrow h = \frac{y x^*_1}{y-x^*_2} \ .$$ [Since $h \geq 1$, we have:]{} $$y x^*_1 \geq y - x^*_2 \iff \displaystyle y \leq \frac{x^*_2}{1-x^*_1} \Rightarrow \displaystyle (0, \frac{x^*_2}{1-x^*_1},{\boldsymbol{0}}) \in Q^{e_1,0}.$$ The last implication follows from the fact that $Q$ is a covering polyhedron, $(0,y,{\boldsymbol{0}}) \in Q$ and $(\pi,\pi_0) = (e_1,0)$. Similarly, we have $(1,x^*_2,{\boldsymbol{0}}) \in Q^{e_1,0}$. The following convex combination of these two points yields $\hat{x}$ as $$(1-x^*_1) (0, \frac{x^*_2}{1-x^*_1},{\boldsymbol{0}}) + x^*_1 (1,x^*_2,{\boldsymbol{0}}) = (x^*_1,x^*_2 + x^*_1 x^*_2,{\boldsymbol{0}}) = \hat{x}.$$ Finally, observe that [since $c_1 \geq 0$ and $0 \leq x^*_1 \leq 1$]{} $${{\color{black}}c^\top x^* \geq c_2 x^*_2 \geq c_2 x^*_1 x^*_2 \Rightarrow c^\top x^* \geq \frac{1}{2} (c^\top x^* + c_2 x^*_1 x^*_2) = \frac{1}{2}c^\top\hat{x}},$$ which completes the proof. We now show that Proposition \[prop:covering2approx\] is tight. In order to do so, we exhibit an instance of a well-behaved covering polyhedron and a nonnegative objective function such that LP is not better than a 2-approximation of ${\mathcal{S}}$. The construction given in the following example is the same that we used in [@bodur2016aggregation] to show that our $2$-approximation bounds for *1-row closure* and *1-row CG closure* are tight. \[prop:LBAC\] For every $\epsilon >0$, there exists a well-behaved covering polyhedron $\tilde{P}_C$ such that $\frac{1}{2-\epsilon} \tilde{P}_C \not {\subseteq}{\mathcal{S}}(\tilde{P}_C)$. Let $\epsilon >0$ and $n = \textup{max}\{2,\lceil \frac{1}{\epsilon}\rceil\}$. Consider the instance $\textup{min} \{ \sum_{j = 1}^n x_j \,|\, x \in \tilde{P}_C\}$, where $$\begin{aligned} \tilde{P}_C = \{x \in {\mathbb{R}}^n_+ \,|\, x_i + \displaystyle \sum_{j \in [n]\setminus \{i\}} 2x_j \geq 2, \ \forall i \in [n]\}.\end{aligned}$$ Note that $\tilde{P}_C$ is well-behaved. It is sufficient to show that $\frac{z^{{\mathcal{S}}}}{z^{LP}} \geq 2 - \epsilon$ for this instance. 1. $z^{LP} \le \frac{2n}{2n - 1}$: It can be checked that the point $\bar x_j = \frac{2}{2n - 1}$ for each $j \in [n]$ is in $\tilde{P}_C$. Thus, $z^{LP} \le \frac{2n}{2n - 1}$. 2. $z^{{\mathcal{S}}} \geq 2$: Adding all the constraints defining $\tilde{P}_C$ we obtain the valid inequality $$\begin{aligned} \sum_{j \in [n]} x_j \geq \frac{2n}{2n - 1}.\end{aligned}$$ The corresponding CG cut is $\sum_{j \in [n]} x_j \geq 2$. Since each CG cut is also a split cut we obtain $z^{{\mathcal{S}}} \geq 2$. Thus, $\frac{z^{{\mathcal{S}}}}{z^{LP}} \geq 2 - \frac{1}{n}$; and our choice of $n$ completes the proof. \[prop:CoveringSplitWell\] Let $P_C$ be well-behaved, i.e., $A_{ij} \leq b_i$ for all $i \in [m], j \in [n]$. Then, ${\mathcal{S}}(P_C)$ is well-behaved. Let ${\beta}^\top x \geq \delta$ be a facet-defining (i.e., nondominated) inequality for ${\mathcal{S}}(P_C)$. If it is a CG cut, then we are done. Thus, we assume that it is a non-CG cut. For a contradiction, suppose that ${\beta}_1 > \delta$. [This is because any CG cut can be written as $\sum_j \lceil \sum_{i = 1}^m \lambda_i A_{ij} \rceil x_j \geq \lceil \sum_{i = 1}^m \lambda_i b_i \rceil$, where $\lambda_i \geq 0 $ for $i \in [m]$. Therefore $\sum_{i = 1}^m \lambda_i A_{ij} \leq \sum_{i = 1}^m \lambda_i $ for all $j \in [n]$, which implies that $\lceil \sum_{i = 1}^m \lambda_i A_{ij} \rceil \leq \lceil \sum_{i = 1}^m \lambda_i b_i \rceil$ for all $j \in [n]$.]{} We know that ${\beta}^\top x \geq \delta$ is valid for $P_C^{\pi,\pi_0}$ for some $(\pi,\pi_0) \in {\mathbb{Z}}^n \times {\mathbb{Z}}$. Then, there exist multipliers $\lambda^1, \lambda^2 \in {\mathbb{R}}_+^m, \ \mu_1, \mu_2 \in {\mathbb{R}}_+$ and $\sigma^1, \sigma^2 \in {\mathbb{R}}_+^n$ such that $$\begin{aligned} \label{eqn:CovWell_SideOne} & ({\beta},\delta) = \lambda^1 (A,b) + \mu_1 (-\pi,-\pi_0) + \sigma^1 ({\boldsymbol{1}},0) \\ \label{eqn:CovWell_SideTwo} & ({\beta},\delta) = \lambda^2 (A,b) + \mu_2 (-\bar{\pi},-\bar{\pi}_0) + \sigma^2 ({\boldsymbol{1}},0)\end{aligned}$$ where $(\bar{\pi},\bar{\pi}_0) = (-\pi,-\pi_0-1)$. Note that if $\sigma^1_1 > 0$ and $\sigma^2_1 > 0$, then we can obtain another split cut by decreasing both $\sigma^1_1$ and $\sigma^2_1$ by $\min \{ \sigma^1_1,\sigma^2_1\}$, which dominates the given split cut ${\beta}^\top x \geq \delta$. Therefore, we assume, WLOG, that $\sigma^2_1 = 0$. Then, we make the following two cases:\ *Case 1*. $- \bar{\pi}_0 \geq - \bar{\pi}_1$: This implies that $\lambda^2 A_{\cdot 1} + \mu_2 (-\bar{\pi}_1) \leq \lambda^2 b + \mu_2 (-\bar{\pi}_0)$ (where $A_{\cdot 1}$ denotes the first column of $A$), equivalently ${\beta}_1 \leq \delta$, which is a contradiction.\ *Case 2*. $- \bar{\pi}_0 \ {{\color{black}}<} - \bar{\pi}_1$: This condition is equivalent to $1-\pi_1 < -\pi_0$. We first claim that $\sigma^1_1 > \mu_1$. From and ${\beta}_1 > \delta$, we have $$\lambda^1 A_{\cdot 1} - \mu_1 \pi_1 + \sigma^1_1 > \lambda^1 b + \mu_1 (-\pi_0) > \lambda^1 b + \mu_1 (1-\pi_1),$$ which implies that $$(\lambda^1 A_{\cdot 1}-\lambda^1 b) + \sigma^1_1 - \mu_1 > 0.$$ As $P_C$ is well-behaved, we have $\lambda^1 A_{\cdot 1}-\lambda^1 b \leq 0$, thus we get $\sigma^1_1 > \mu_1$. Next, we let $$\tilde{\pi} := \pi - e_1, \ \tilde{\sigma}^1 := \sigma^1 - \mu_1 e_1, \ \tilde{\sigma}^2 := \sigma^2 + \mu_2 e_1.$$ Note that $\tilde{\sigma}^1_1 > 0$. Also, due to and , we have $$\label{eq:CovWellBothSides} ({\beta},\delta) = \lambda^1 (A,b) + \mu_1 (-\tilde{\pi},-\pi_0) + \tilde{\sigma}^1 ({\boldsymbol{1}},0) = \lambda^2 (A,b) + \mu_2 (\tilde{\pi},-\bar{\pi}_0) + \tilde{\sigma}^2 ({\boldsymbol{1}},0).$$ Note that $\mu_2 > 0$ since otherwise, i.e., when $\mu_2 = 0$, the equation and $\sigma^2_1 = 0$ give the contradiction ${\beta}_1 = \lambda^2 A_{\cdot 1} \leq \lambda^2 b = \delta$. Therefore, we have $\tilde{\sigma}^2_1 > 0$ as well. If we reduce both $\tilde{\sigma}^1_1$ and $\tilde{\sigma}^2_1$ by a sufficiently small $\epsilon > 0$, so that they are still nonnegative, from we obtain another valid split cut $({\beta}- 2 \epsilon e_1)^\top x \geq \delta$ which dominates ${\beta}^\top x \geq \delta$, hence a contradiction. **Acknowledgements.** Santanu S. Dey would like to acknowledge the support of the NSF grant CMMI\#1149400. Marco Molinaro would like to acknowledge the support of CNPq grants Universal \#431480/2016-8 and Bolsa de Produtividade em Pesquisa \#310516/2017-0; this work was partially while the author was a Microsoft Research Fellow at the Simons Institute for the Theory of Computing. [We would like to thank the reviewers for their careful and constructive comments that have significantly improved the paper.]{} Additional proofs {#sec:appendix} ================= \[obs:PackingSplitAgg\] For packing polyhedra, split cuts are not necessarily aggregation cuts. An example, where there exists a split cut that cannot be obtained as an aggregation cut, is provided in Figure \[fig:PackingSplitAgg\]. (0,0) – (1.8,0) – (0.9,1.9) – (0,1.9) – cycle; (0,0) – (0,4.2); (0,0) – (4.2,0); (0,3.8) – (1.8,0); (0,1.9) – (4,1.9); at (0,2.7) ; at (1.8,1.4) ; (A) at (-0.2,3.8) [7]{}; (A) at (-0.2,2.7) [2]{}; (A) at (-0.35,1.9) [7/4]{}; (A) at (1.8,-0.2) [1]{}; (0,1.4) – (1.8,1.4); (A) at (-0.152,1.4) [1]{}; (1.8,3.8) – (1.8,0); (0,1.9) – (1.8,0); In the figure, the shaded region represents the packing polyhedron $P = \{ x \in {\mathbb{R}}_+^2 \mid 7 x_1 + x_2 \leq 7, 4 x_2 \leq 7\}$. It is easy to see that $7 x_1+4 x_2 \leq 7$ (the dashed line in the figure) is a split cut obtained by using the split set $S(e_1,0)$. Note that this cut separates both of the points $(0,2)$ and $(1,1)$. We next show that this cut is not an aggregation cut by proving that $(0,2)$ and $(1,1)$ are not separated at the same time by any aggregation cut. An inequality is an aggregation cut for $P$ if it is valid for the set $P(\alpha) := \operatorname{conv}( \{ x\in {\mathbb{R}}_+^2 \mid (7-7\alpha) x_1 + (3 \alpha +1) x_2 \leq 7 \})$ for some $\alpha \in [0,1]$. It can be easily verified that if $\alpha \leq 5/6$, then $(0,2) \in P(\alpha)$, and if $\alpha \geq 1/4$, then $(1,1) \in P(\alpha)$. \[obs:CoveringSplitAgg\] For covering polyhedra, split cuts are not necessarily aggregation cuts. An example, where there exists a split cut that cannot be obtained as an aggregation cut, is provided in Figure \[fig:CoveringSplitAgg\]. (0,3.8) – (0.7108,1.1) – (4,1.1) – (4,3.8) – cycle; (0,0) – (0,4.2); (0,0) – (4.2,0); (0,3.8) – (1,0); (0,1.1) – (4,1.1); at (0,2.7) ; at (1,0.7) ; (A) at (-0.2,3.8) [7]{}; (A) at (-0.2,2.7) [6]{}; (A) at (-0.35,1.1) [7/4]{}; (A) at (1,-0.2) [1]{}; (0,0.7) – (1,0.7); (A) at (-0.152,0.7) [1]{}; (1,3.8) – (1,0); (0,3.8) – (1.222,0.5); In the figure, the shaded region represents the covering polyhedron $P = \{ x \in {\mathbb{R}}_+^2 \mid 7 x_1 + x_2 \geq 7, 4 x_2 \geq 7\}$. It is easy to see that $21 x_1+4 x_2 \geq 28$ (the dashed line in the figure) is a split cut obtained by using the split set $S(e_1,0)$. Note that this cut separates both of the points $(0,6)$ and $(1,1)$. We next show that this cut is not an aggregation cut by proving that $(0,6)$ and $(1,1)$ are not separated at the same time by any aggregation cut. An inequality is an aggregation cut for $P$ if it is valid for the set $P(\alpha) := \operatorname{conv}( \{ x\in {\mathbb{R}}_+^2 \mid (7-7\alpha) x_1 + (3 \alpha +1) x_2 \geq 7 \})$ for some $\alpha \in [0,1]$. It can be easily verified that if $\alpha \geq 1/18$, then $(0,6) \in P(\alpha)$, and if $\alpha \leq 1/4$, then $(1,1) \in P(\alpha)$. [^1]: bodur@mie.utoronto.ca [^2]: delpia@wisc.edu [^3]: santanu.dey@isye.gatech.edu [^4]: mmolinaro@inf.puc-rio.br

--- abstract: 'It is well known that the Cox-Ingersoll-Ross (CIR) stochastic model to study the term structure of interest rates, as introduced in 1985, is inadequate for modelling the current market environment with negative short interest rates. Moreover, the diffusion term in the rate dynamics goes to zero when short rates are small; both volatility and long-run mean do not change with time; they do not fit with the skewed (fat tails) distribution of the interest rates, etc. The aim of the present work is to suggest a new framework, which we call the *CIR\# model*, that well fits the term structure of short interest rates so that the market volatility structure is preserved as well as the analytical tractability of the original CIR model.' address: - 'Dipartimento di Scienze Economiche e Metodi Matematici, Università degli Studi di Bari Aldo Moro, Italy' - 'Dipartimento di Matematica, Università degli Studi di Bari Aldo Moro,Italy' - 'Dipartmento di Metodi e Modelli per l’Economia, il Territorio e la Finanza, Università degli Studi di Roma “La Sapienza", Italy' author: - Giuseppe Orlando - 'Rosa Maria Mininni[^1]' - 'Michele Bufalo[@xdefthefnmark[\[note1\]]{}footnotemark]{}' title: 'On The Calibration of Short-Term Interest Rates Through a CIR Model' --- \#1 Interest rates forecasting, volatility, ARIMA models, simulation, jumps fitting, translation\ JEL Classification: G12, E43, E47 91G30, 91B84, 91G60, 91G70, 62M10 Introduction {#section1} ============ The aim of the present work is to provide a new numerical methodology for the CIR framework, which we call the *CIR\# model*, that well fits the term structure of short interest rates as observed in a real market. Our approach is based on a proper translation of interest rates so that the market volatility structure is preserved as well as the analytical tractability of the original CIR model. Cox, Ingersoll & Ross (1985) [@CIR_85] proposed a term structure model, well known as the CIR model, to describe the price of discount zero-coupon bonds with variuous maturities under no-arbitrage condition. This model generalizes the Vasicek model [@Vasicek1977] to the case of non constant volatility and assumes that the evolution of the underlying short term interest rate is a diffusion process, i.e. a continuous Markov process unique solution to the following stochastic differential equation (SDE) $$\label{CIRrate} dr(t) = [k(\theta-r(t)) - \lambda(t,r(t))]dt + \sigma \sqrt{r(t)}dW(t),$$ with initial condition $r(0)= r_{0} >0 $. $(W(t))_{t\ge 0} $ denotes a standard Brownian motion under the risk neutral probability measure, intended to model a random risk factor. The interest rate process $(r(t))_{t\ge 0}$ is usually known as the [*CIR process*]{} or [*square root process*]{}. The SDE is classified as a one-factor time-homogeneous model, because the parameters $k, \theta$ and $\sigma, $ are time-independent and the short interest rate dynamic is driven only by the market price of risk $ \lambda(t,r(t)) := \lambda r(t), $ where $\lambda$ is a constant. Therefore the SDE is composed of two parts: the “mean reverting" drift component $k[\theta-r(t)],$ which ensures the rate $r(t)$ is elastically pulled towards a long-run mean value $\theta >0$ at a speed of adjustment $k>0,$ and the random component $W(t),$ which is scaled by the standard deviation $\sigma \sqrt{r(t)}$. The volatility of the instantaneous short rate is denoted by $\sigma>0$. The paths of the CIR process never reach negative values and their behaviour depends on the relationship between the three constant positive parameters $k, \theta, \sigma$. Indeed, it can be shown (see, e.g., Jeanblanc, Yor & Chesney (2009) [@Jeanblanc2009 Ch. 3]) that if the condition $2k\theta \ge\sigma ^{2}$ is satisfied, then the interest rates $r(t)$ are strictly positive for all $t>0$, and, for small $r(t)$, the process rebounds as the random perturbation dampens with $r(t)\rightarrow 0$. Furthermore, the CIR process belongs to the class of processes satisfying the “affine property", i.e., the logarithm of the characteristic function of the transition distribution of such processes is an affine (linear plus constant) function with respect to their initial state (for more details the reader can refer to Duffie, Filipovi[ć]{} & Schachermayer (2003) [@Duffie_Filipovic Section 2]. As a consequence, the non-arbitrage price of a discount zero coupon bond with maturity $T>0$ and underlying interest rate dynamics described by a CIR process, is given by $$\label{CIRprice} P(T-t,r(t)) = A(T-t)e^{ -B(T-t)r(t)},\quad t\in [0,T],$$ where $A(\cdot)$ and $B(\cdot)$ are deterministic functions (see Cox, Ingersoll & Ross (1985) [@CIR_85]). The final condition is $P(T,r(T)) = 1,$ which corresponds to the nominal value of the bond conventionally set equal to 1 (monetary unit). Since bonds are commonly quoted in terms of yields rather than prices, the formula allows derivation of the yield-to-maturity curve $$\begin{aligned} Y(T-t,r(t)) &:= - \ln{P(t,T,r(t))}/(T-t) \\[2.\jot] &= [B(T-t)r(t) - \ln(A(T-t))]/(T-t),\quad t\in [0,T],\end{aligned}$$ which tends to the asymptotic value $\varUpsilon=2k\theta/(\gamma + k + \lambda)$ as $T\to\infty, $ where $\gamma := \sqrt{(k+\lambda)^2 + 2\sigma^2}.$ Thus the CIR process is characterized by the following properties: 1) short interest rates never become negative; 2) if the interest rate reaches the zero value (it may occur under the condition $2k\theta <\sigma ^{2}$), it is immediately reflected in positive values; 3) the diffusion term in increases when $r(t)$ increases; 4) the transition density of short rates is a noncentral chi-squared distribution, which converges to a gamma distribution as time grows towards infinity; 5) the trajectories describing the short rate dynamics cannot be explicitly derived, but exact simulation is still possible and bond prices are explicitly computable from it. The remainder of the paper is organized as follows. Section 2 summarizes the existing literature on the CIR model and the related extensions. Section 3 describes the principal steps of the proposed *CIR\# model*; Section 4 presents in more detail the numerical procedure and tests the goodness-of-fit of the new methodology to market data. Finally, Section 5 concludes. Literature review ================= The CIR model became very popular in finance among practitioners because it was perceived as an improvement on the Vasicek model, not allowing for negative rates and introducing a rate dependent volatility, as well as for its relatively handy implementation and analytical tractability. Other applications include stochastic volatility modelling in option pricing problems (see Orlando and Taglialatela (2017) [@Orlando_T], Heston (1993) [@Heston]), or default intensities in credit risk (see Duffie (2005) [@DuffieC]). Despite the previously listed properties, the CIR model fails as a satisfactory calibration to market data since it depends on a small number of constant parameters, $k,\theta$ and $\sigma.$ As proved by Keller-Ressel and Steiner (2008) [@Keller Theorem 3.9 and Section 4.2] the yield-to-maturity curve of any time-homogenous, affine one-factor model is either normal (i.e., a strictly increasing function of $T-t$), humped (i.e., with one local maximum and no minimum on $]0,\infty[$) or inverse (i.e., a strictly decreasing function of $T-t$) (see Figure \[Fig-KRS\]). For the CIR process, the yield is normal when $r(t) \le k\theta/(\gamma - 2(k + \lambda))$, while it is inverse when $r(t) \ge k\theta/(k + \lambda).$ For intermediate values the yield curve is humped. ![Inverse, humped or normal yield-to-maturity curves (Keller-Ressel & Steiner(2008) [@Keller Figure 1]).[]{data-label="Fig-KRS"}](KelleHumpInvNorm-TermStr){width="8cm"} However, Carmona and Tehranchi (2006), [@Carmona Section 2.3.5] explained that: “Tweaking the parameters can produce yield curves with one hump or one dip (a local minimum), but it is very difficult (if not impossible) to calibrate the parameters so that the hump/dip sits where desired. There are not enough parameters to calibrate the models to account for observed features contained in the prices quoted on the markets" (see Figure \[fig:EUR\_USD-Term\_Str\]). [0.65]{} ![EUR and USD term structure (1Y-50Y) as observed on monthly basis from January to December 2013.[]{data-label="fig:EUR_USD-Term_Str"}](EUR_IR_1Y-50Y "fig:"){width="120.00000%"} [0.65]{} ![EUR and USD term structure (1Y-50Y) as observed on monthly basis from January to December 2013.[]{data-label="fig:EUR_USD-Term_Str"}](USD_IR_1Y-50Y "fig:"){width="120.00000%"} Thus the need for more sophisticated models for an exact fit to the currently-observed yield curve, which could take into account multiple correlated sources of risk as well as shocks and/or structural changes of the market, led some years later to the development of extensions of the CIR model. Among the best known we mention: the Hull-White (1990) [@Hull1990] model based on the idea of considering time-dependent coefficients; the Chen (1996) [@Chen] three-factor model; the CIR++ model by Brigo & Mercurio (2001) [@Brigo2001] that considers short rates shifted by a deterministic function chosen to fit exactly the initial term structure of interest rates; the jump diffusion JCIR model (see Brigo & Mercurio (2006) [@BrigoMercurio2006]) and JCIR++ by Brigo & El-Bachir (2006)[@Brigo2006] where jumps are described by a time-homogeneous Poisson process; the CIR2 and CIR2++ two-factor models (see Brigo & Mercurio (2006) [@BrigoMercurio2006]). Very recently, Zhu (2014) [@Zhu], in order to incorporate the default clustering effects, proposed a CIR process with jumps modelled by a Hawkes process (which is a point process that has self-exciting property and the desired clustering effect), Moreno et al. (2015) [@Moreno] presented a cyclical square-root model, and Najafi et al. (2017) ([@Najafi1], [@Najafi2]) proposed some extensions of the CIR model where a mixed fractional Brownian motion applies to display the random part of the model. Note that all the above cited extensions to CIR model preserve the positivity of interest rates, in some cases through reasonable restrictions on the parameters. But the financial crisis of 2008 and the ensuing quantitative easing policies brought down interest rates, as a consequence of reduced growth of developed economies, and accustomed markets to unprecedented negative interest regimes under the so called “new normal". As observed in Engelen (2015) [@Engelen] and BIS (2015) [@BIS]: “Interest rates have been extraordinarily low for an exceptionally long time, in nominal and inflation-adjusted terms, against any benchmark"(see Figure \[Fig-BIS-2015-G1\]). “Between December 2014 and end-May 2015, on average around \$2 trillion in global long-term sovereign debt, much of it issued by euro area sovereigns, was trading at negative yields", “such yields are unprecedented. Policy rates are even lower than at the peak of the Great Financial Crisis in both nominal and real terms. And in real terms they have now been negative for even longer than during the Great Inflation of the 1970s. Yet, exceptional as this situation may be, many expect it to continue“. ”Such low rates are the most remarkable symptom of a broader malaise in the global economy: the economic expansion is unbalanced, debt burdens and financial risks are still too high, productivity growth too low, and the room for manoeuvre in macroeconomic policy too limited. The unthinkable risks becoming routine and being perceived as the new normal." ![$ $BIS 85th Annual Report 2015.[]{data-label="Fig-BIS-2015-G1"}](BIS-2015-G1){width="12cm"} Therefore, the need for adjusting short term interest rate models for negative rates has become an additional characteristic that a “good" model should possess. It is worth noting that the main drawback of the Vasicek model [@Vasicek1977], which allows for negative interest rates, is that the conditional volatility of changes in the interest rate is constant, independent on the level of it, and this may unrealistically affect the prices of bonds that can grow exponentially (see Rogers [@Rogers]). For this reason, the Vasicek model is unused by practitioners. The CIR\# model {#section2} =============== In the following we will illustrate our original approach, but first let us recap the main issues of the CIR model: 1. Negative interest rates are precluded; \[item1\] 2. The diffusion term in goes to zero when $r(t)$ is small (in contrast with market data); \[item2\] 3. The instantaneous volatility $\sigma$ is constant (in real life $\sigma$ is calibrated continuously from market data);\[item3\] 4. There are no jumps (e.g. caused by government fiscal and monetary policies, by release of corporate financial results, etc.);\[item4\] 5. Risk premia are linear with interest rates (false if credit worthiness of a counterparty and market volatility are considered); \[item5\] 6. The change in interest rates depends only on the market risk. \[item6\] The aim of the present work is to provide a new methodology that gives an answer to points [**i.- iv.**]{} by preserving the structure of the original CIR model to describe the dynamics of spot interest rates observed in financial markets. For this purpose the first step is partitioning the available market data sample into sub-samples - not necessarily of the same size - in order to capture all the statistically significant changes of variance in real spot rates and consequently, to give an account of jumps (see Section \[section3.1\]). This should allow to overcome the critical issue pointed out in [**iv.**]{}. After that, to overcome challenges [**i.- ii.**]{}, the real spot rates are properly translated to shift them away from zero or negative values and such that the diffusion term in is not dampened by the proximity to zero but fully reflects the same level of volatility present on the market (see Section \[section3.2\]). The second step consists in fitting an “optimal" - as explained in Section \[section3.4\] - ARIMA model to each sub-sample of market data. To ensure that the residuals of the chosen “optimal" ARIMA model in each sub-sample look like Gaussian white noise, the Johnson’s transformation (Johnson (1949)[@Johnson1949]) is applied to the standardized residuals (see Section \[section3.3\]). As a third step, the parameters $k, \theta, \sigma$ in are calibrated to the (eventually) shifted market interest rates by estimating them for each sub-sample of available data, as explained in Section \[section3.3.1\] (which allows to overcome the issue [**iii.**]{}). For this purpose, trajectories of the CIR process are simulated by a strong convergent discretization scheme, using the standardized residuals of the “optimal" ARIMA model selected for each sub-sample in place of realizations of a standard Brownian motion. As a result, exact CIR fitted values to real data are calculated and the computational cost of the numerical procedure is considerably reduced. Finally, the short-term interest rates estimated by the CIR model are shifted back and compared to real data. As a measure of goodness-of-fit to the available market data, we compute: - the statistics $R^2$ given by the following expression [@Kvalseth] $$\label{rsquare} R^2 = 1 - \frac{\sum\limits_{h=1}^{m} (e_h - \overline{e})^2}{\sum\limits_{h=1}^{m} (r_h - \overline{r})^2},$$ where $e_h = r_h - \widehat{r}_h$ denotes the residual between the observed market interest rate $r_{h}$ and the corresponding fitted value $\widehat{r}_h,$ evaluated on a data sample of size $m\ge 2.$ Furthermore, $\overline{e}$ and $\overline{r}$ denote the sample mean of $e_h$ and $r_{h}$, respectively; - the square root of the mean square error (RMSE) $$\varepsilon=\sqrt{\frac{1}{m}\sum_{h=1}^{m} {e^2_h}}.$$ Numerical Implementation and Empirical Analysis {#section3} =============================================== The Dataset {#sec:Dataset} ----------- In this section we give explicit numerical results for the *CIR\# model* described in Section \[section2\]. Our dataset records EUR and USD interest rates with maturities 1/360A–360/360A and 1Y–50Y (i.e. at 1 day (overnight), 30 days, 60 days,...., 360 days and 1 Year,...,50 Years) available from IBA [@IBA]. For each maturity, interest rates are recorded on monthly basis from 31 December 2010 to 29 July 2016 for a total number of 68 datapoints. In the book [@OMB1](2018) we performed a qualitative analysis on this dataset. We found that the most challenging task was to fit short-term interest rates with maturity 1/360A, 30/360A, 60/360A, 90/360A, 120/360A,..., 360/360A, respectively, due to the presence of next-to-zero and/or negative spot rate values. This led us to implement a novel numerical procedure, the CIR$\#$ model, that would allow description of the short-term structure by the original CIR model. The new methodology will be discussed in more details in the next subsections and we will give explicit numerical results for the CIR$\#$ model applied to the data sample consisted of $n=68$ EUR interest rates with 1 day (overnight) maturity. All computations have been executed using MATLAB^^ R$2017$a. ANOVA test with a fixed segmentation {#section3.1} ------------------------------------ As explained in Section \[section2\], our first objective is to overcome the issues pointed out in [**i., ii.**]{} and [**iv.**]{} for the CIR model. Thus we start to partition the whole data sample into sub-samples, which we call [*groups*]{}, by a one-way ANOVA analysis to highlight statistically significant changes of variance in real spot rates and so to give an account of possible jumps. The main difficulty concerns the choice of the optimal partition into groups to apply the ANOVA test; we had to take into account both the size (the smaller the group is, the more refined the analysis) and the ability to capture any jumps (the larger the group, the better in terms of statistical significance). After several tests, we decided to segment the whole sample into eight groups each of size $m=8$ or a multiple thereof (except for the last group, obviously). The results of the one-way ANOVA test are reported in Table \[tab:ANOVA\]. The *p-value* (Prob$>$F) of $8.00796\cdot 10^{-19}$ indicates a statistically significant difference between groups. ------------------------------------------------------- -- -- -- -- -- -- **Source&**SS&**df&**MS&**F&**Prob$>$F\ Groups & 10.8783 & 7 & 1.55405 & 34.71 & 8.00796e-19\ Error & 2.6862 & 60 & 0.04477\ Total & 13.5645 & 67\ ************ ------------------------------------------------------- -- -- -- -- -- -- : The ANOVA Table shows the between-groups (Groups) and the within-groups (Error) variation. “SS" is the sum of squares and “df" means degrees of freedom associated to SS. MS indicates the mean squared error, i.e. the estimate of the error variance. The value of the F-statistic is given by the ratio of the mean squared errors. \[tab:ANOVA\] Furthermore, the boxplot (Figure \[Fig:ANOVA\].a)) and a multiple comparison test performed on the eight groups (Figure \[Fig:ANOVA\].b)) have suggested partitioning the data sample into the following four groups of observations 1–8,  9–16,  17–56, 57–68. \ \ [![[**a).**]{} Boxplot; [**b).**]{} Multiple comparison test.[]{data-label="Fig:ANOVA"}](changeD4 "fig:"){width="80.00000%"}]{}\ \ Jumps fitting by translation {#section3.2} ---------------------------- Since the CIR model does not fit negative interest rates and normal/high volatility when the rate value is small, spot rates must be shifted away from zero and/or negative values. One chance is given by an affine type transformation, as follows $$r_{shift} (t) = \hat{\mu}_{r(t)} + \hat{\sigma}_{r(t)}\, r_{real}(t), \quad t\in [0,T]$$ where $\hat{\mu}_{r(t)}$ and $\hat{\sigma}_{r(t)}$ are the sample mean and the sample standard deviation of $r_{real}(t),$ respectively. But, in practice, this is not the best choice due to some potential issues such as persistence of negative values, worse fitting, changes in the short interest rates behaviour, etc. Among different options, a translation of type $$\label{t1} r_{shift}(t) = r_{real}(t) + \alpha,\quad t\in [0,T],$$ with $\alpha>0$ a constant term, is a reasonable choice because it leaves unchanged the stochastic dynamics of short interest rates, i.e. $dr_{shift}(t)=dr_{real}(t)$. The value of the parameter $\alpha$ can be arbitrarily selected but, in our opinion, the most appropriate choice must take into account the empirical distribution of interest rates. For our purpose, we set $\alpha$ equal to the approximate value, calculated from the available market data, corresponding to the $99th$-percentile of the conditional distribution of the process $\{r_{real}(t), t\in[0,T]\}.$ However, if further negative values are between the $99th$- and the $100th$-percentile, then $\alpha$ can be set equal to the approximate value corresponding to the $1st$-percentile of the conditional distribution of spot rates. Hence the translation becomes $r_{shift}(t)=r_{real}(t)-{\alpha}$. The translation is applied after a check carried out on each group partitioning the original data sample; the check consists in calculating the harmonic mean - because it is more robust than the arithmetic one in the presence of extreme values - and in verifying whether it is smaller than a constant value arbitrarily chosen small (e.g. $10^{-2}$). If this happens in at least one group, then the whole sample is translated. Sub-optimal ARIMA models {#section3.3} ------------------------ The second step consists in deriving the best fitting ARIMA$(p,i,q)$ model to each group of interest rates partitioning the original market data sample. Thus we start by selecting, for each group, a set of ARIMA $(p,i,q)$ models whose standardized residuals satisfy the following “sub-optimal" conditions: 1. Absence of both autocorrelation (AC) and partial autocorrelation (PAC) in the time series[^2]; 2. Absence of unit roots (stationarity of the time series); 3. Normally distributed standardized residuals; 4. $R^{2}_{ARIMA}>0.5$, where $R^{2}_{ARIMA}$ denotes the statistics $R^2$, defined in , computed for the ARIMA $(p,i,q)$ model. We look for only the indices $i\in\{0,1,2\}$ and $p,q\in\{1,2,3\}.$ As mentioned, to ensure that the residuals of the selected ARIMA $(p,i,q)$ models look like a Gaussian white noise, the Johnson’s transformation (Johnson (1949)[@Johnson1949]) is applied to the standardized residuals. The Johnson’s method consists in developing a flexible system of distributions, based on three families of transformations, that translates an observed, non-normal distribution to one conforming to the standard normal distribution. The transformation of a non-normal random variable $X$ to a standard normal variable $Z$ is written as $$\label{Jtransform} Z=\gamma +\delta f\left(\frac{X-\xi}{\lambda}\right),\quad \lambda,\; \delta>0$$ where $f$ is function of a simple form. In particular, $f((X-\xi)/\lambda)$ must be a monotonic function of $X$, and its range of values have to correspond to the actual range of possible values of $(X-\xi)/\lambda.$ The parameters $\delta$ and $\gamma$ reflect respectively the skewness and kurtosis of $f$, while $\xi $ and $\lambda $ are the mean and the standard deviation of $X.$ The algorithm to estimate the four parameters $\gamma$, $\delta$, $\lambda$ and $\xi$, and perform the appropriate transformation is available as a Matlab Toolbox written by Jones (2014) [@Jones]). In the sequel the normally distributed standardized ARIMA residuals applies to the random part of the CIR model to simulate trajectories of the interest rate process and calibrate the parameters $k, \theta, \sigma$ in to the (eventually) translated market interest rates. ### Calibration of CIR parameters {#section3.3.1} For the sake of simplicity, we rewrite the SDE as follows $$\begin{aligned} \label{CIRQ} dr(t) &= (\tilde{k}(\tilde{\theta} - r(t)) - \lambda r(t))\, dt + \sigma\sqrt{r(t)}\, dW(t) \nonumber \\ &= (k(\theta - r(t))\, dt + \sigma\sqrt{r(t)}\, dW(t),\end{aligned}$$ and we set $$k=\tilde{k} + \lambda >0, \quad \theta=\frac{\tilde{k}\tilde{\theta}}{k}>0.$$ Consider the $j{th}$-group partitioning the available market data sample, which we assume to be of length $n_j$. The calibration of the CIR parameters in the group is performed as follows 1. The volatility $\sigma$ is estimated by the group standard deviation, namely $\hat{\sigma}_j$; 2. The long-run mean parameter $\theta$ is estimated by the group mean, namely $\hat{\theta}_j$; 3. The speed of mean reversion $k$ is estimated by that value, say $\hat{k}_j,$ solving the following minimization problem: $$\label{7} \min\limits_{k>0}\, S_{j}(k) := \min\limits_{k>0}\, \sqrt{\frac{\sum\limits_{h=1}^{n_j} (u^{j}_{h}(k) - \overline{u}^j (k))^{2} }{n_j-1}}.$$ For any $k>0,$ we define $$\label{8} u^{j}_{h} (k) := r^{j}_{h}(k) - r^{j}_{shift, h}, \quad h=1,\cdots,n_j,$$ being $r^{j}_{shift, h}$ the real shifted interest rate value, and $r^{j}_{h}(k)$ the corresponding simulated CIR interest rate value expressed as a function of the unknown parameter $k$. The $r^{j}_{h}(k)$ are calculated by applying the strong convergent Milstein discretization scheme (1979) [@Milnstein] to the SDE . Brigo & Mercurio (2006) [@BrigoMercurio2006 Section 22.7] showed that the Milstein scheme converges in a much better way than other numerical schemes for the CIR process. It reads as $$\label{9} r^j_{h+1}(k) = r^j_h (k) + k(\hat{\theta}_j - r^j_h)\, \Delta + \hat{\sigma}_j\sqrt{r^j_h \Delta}\; Z^j_{h+1} + \frac{(\hat{\sigma}_j)^2}{4}\, [(\sqrt{\Delta}\; Z^j_{h+1})^{2} - \Delta],$$ where $\Delta$ is the time step - we set $\Delta = 1/30$ due to monthly observed data - and $Z^{j}_{h+1}$ are standard normally distributed random variables. Indeed, in this case, $Z^{j}_{h+1}$ are the normally distributed standardized residuals of each ARIMA $(p,i,q)$ model selected for the $jth$-group. They are computed by applying the Johnson’s transformation , with the ARIMA residuals as realizations of the random variable $X$. After calculation of the estimates $(\hat{k}_j,\, \hat{\theta}_j,\, \hat{\sigma}_j),$ the CIR fitted values to the shifted observed spot rates in the $jth$-group, $r^{j}_{shift, h}$, are computed by the simulation scheme as follows $$\label{CIRfitted} \hat{r}^j_{h+1} = \hat{r}^j_{h} + \hat{k}_j(\hat{\theta}_j - \hat{r}^j_{h})\, \Delta + \hat{\sigma}_j\sqrt{\hat{r}^j_{h}\Delta}\, Z^j_{h+1} + \frac{(\hat{\sigma}_j)^2}{4}\, [(\sqrt{\Delta}\, Z^j_{h+1})^{2} - \Delta],$$ where $\Delta$ and $Z^{j}_{h+1}$ are as before. To measure the goodness-of-fit, the statistics $R^{2}$ is computed. For sake of clarity, in the sequel we will denote by $R^{2}_{CIR}$ the statistics when referring to the CIR model. Optimal ARIMA-CIR model {#section3.4} ------------------------ For each group $j$, the “optimal" ARIMA$(p,i,q)$ model providing the best CIR fitting to real data will be chosen among the selected sub-optimal ARIMA $(p,i,q)$ models, as described in Section \[section3.3\], that satisfy the following additional conditions: 1. The ARIMA $(p,i,q)$ minimizes the Bayesian Information Criterion (BIC) matrix whose rows and columns are the possible $p$ and $q$ lags, respectively ([*$BIC$ condition*]{}); 2. $R^{2}_{CIR}>0.5$. Therefore we define the following sets of candidate ARIMA models: $$\mathcal{I}_{AC} = \left\lbrace (p,i,q)\, | \, \text{ARIMA$(p,i,q)$ satisfies conditions 1.-- 4. and 6.} \right\rbrace$$ and $$\mathcal{I}_{ACB} = \left\lbrace (p,i,q) |\, \text{ARIMA$(p,i,q)$ satisfies conditions 1.-- 6.} \right\rbrace.$$ Obviously, $\mathcal{I}_{ACB}\subset \mathcal{I}_{AC}$. Last but not least, the “optimal" ARIMA model is chosen in the above defined classes as the model minimizing the RMSE $\varepsilon_j$ $$\min\limits_{\hat{r}^j}\, \varepsilon_j := \min\limits_{\hat{r}^j}\; \sqrt{\frac{1}{n^j}\sum_{h=1}^{n^j}(r^j_{shift,h} - \hat{r}^j_h)^{2}},$$ where the minimum is computed with respect to all the CIR fitted values vectors, $\hat{r}^j,$ simulated for the $jth$-group. The algorithm proposed in Table \[tab:ARIMA-CIR\] finds, for each group $j,$ the “optimal" ARIMA $(p,i,q)$ model and returns as output: the matrix of indices $(p,i,q)$ belonging to the sets $\mathcal{I}_{AC}$ and $\mathcal{I}_{ACB},$ the corresponding CIR fitted values vector $\hat{r}^j$ computed by , and the associated values of the statistics $R^2_{CIR}$ and $\varepsilon_j$. The main steps of the algorithm can be summarized as follows: ------------------------------------------------------------------------------------------------ **Step 1:** verify if $check1=1$ for the $j{th}$-group; **Step 2:** if $check1=1$ verify if $check2=1$ for the current group; **Step 3:** if $check2=1$ print the output. Else, reduce the size $n^{j}$ of the current group to $(n^{j}-m)$ where $m=8.$ **Step 4:** repeat **Step 1**-**Step 3** for the remaining observations in the current group. **Step 5:** return to **Step 1** for the group $j+1.$ ------------------------------------------------------------------------------------------------ : ARIMA-CIR algorithm \[tab:ARIMA-CIR\] Note that $check1$ and $check2$ refer respectively to conditions 1.– 5. and 6. Their value is equal to 1 if those conditions are satisfied. It is worth mentioning that a test on the efficiency of the above algorithm could be done by verifying that $\varepsilon^j$ is small for all $j$ and that the weighted mean of the $\varepsilon^j$ (see formula (\[total\_error\])) is small for the whole data sample. We applied the ARIMA-CIR algorithm to the $n=68$ monthly observed EUR interest rates with 1 day ( overnight) maturity, mentioned in Section \[sec:Dataset\]. We recall that the ANOVA analysis suggested to partition the data sample into four groups of observations: 1–8,  9–16,  17–56, 57–68. Table \[tab:outputs\] shows in detail the outputs for this sample. The group containing the observations 17–56 has been futher segmented into three sub-groups of size $m=8$ or a multiple thereof: 17–32,  33–48 and 49–56. The triplets $(p,i,q)$ identified by a rectangle in Table \[tab:outputs\], indicates the “optimal" ARIMA model chosen for each group/sub-group (with the bigger $(R^{2}_{CIR})_j$ and the smaller $\varepsilon_j$ values). As it can be seen, none of these models fulfils the $BIC$ *condition*. ------------------------------------------------------- -- -- -- -- -- -- -- -- **j&**group/sub-groups&**ARIMA model&&&**BIC cond.**\ 1&1–8&&0.8166&\ &&(2,0,2)&0.5930&0.2414\ &&(3,0,2)&0.780&0.1865\ &&(1,1,1)&0.8166&0.1643\ &&(1,2,1)&0.7309&0.2026\ &&(1,2,2)&0.7805&0.1845&$\surd$\ &&(3,2,1)&0.7023&0.2104\ 2&9–16&(1,0,1)&0.6842&0.2090\ &&(2,0,2)&0.7799&0.2588\ &&(3,0,2)&0.6418&0.2661&$\surd$\ &&(1,1,1)&0.7378&0.2043\ &&&0.8472&\ &&(2,1,1)&0.6842&0.2169\ &&(2,1,2)&0.7799&0.2012\ &&(3,1,2)&0.6418&0.2333&\ 3&17–32&&0.9174&\ &&(3,0,1)&0.5485&0.0646\ 4&33–48&&0.6901&\ &&(3,1,2)&0.6332&0.1146\ &&(3,2,1)&0.6332& 0.1146\ ****** ------------------------------------------------------- -- -- -- -- -- -- -- -- : Outputs from the ARIMA-CIR algorithm for the 68 monthly EUR interest rates on overnight maturity[]{data-label="tab:outputs"} ------------------------------------------------------- -- -- -- -- -- -- -- -- **j&**group/sub-groups&**ARIMA model&&&**BIC cond.**\ 5&49–56&(1,0,1)&0.5076&0.0597\ &&(1,0,2)&0.6030&0.0526\ &&(2,0,1)&0.5702&0.0577\ &&(3,0,1)&0.7648&0.0483\ &&(3,0,2)&0.6240&0.0537\ &&(1,1,1)&0.5702&0.0577\ &&(2,1,1)&0.5393&0.0588\ &&&0.7715&\ &&(1,2,1)&0.5542&0.0582\ &&(3,2,1)&0.6987&0.0479\ &&(3,2,2)&0.6343&0.0536&$\surd$\ 6&57–68&(1,0,2)&0.5964&0.0752\ &&(1,0,3)&0.8080&0.0570\ &&(2,0,2)&0.8136&0.0559\ &&(3,0,2)&0.7899&0.0577\ &&(3,0,3)&0.6560&0.0704\ &&(1,1,2)&0.8136&0.0559\ &&(1,1,3)&0.8006&0.0614\ &&&0.8239&\ &&(2,1,2)&0.8542&0.0525\ &&(2,1,3)&0.8936&0.0551\ &&(3,1,2)&0.9023&0.0511\ &&(3,1,3)&0.8000&0.0580\ ****** ------------------------------------------------------- -- -- -- -- -- -- -- -- For a more exact comparison we use the numeric format long. Figure \[fig:subgroup\] below reports the qualitative statistical analysis carried out by applying the ARIMA $(1,1,2)$ model chosen as the “optimal" forecasting model for the group 9–16 (similar plots for the other group/sub-groups are reported in \[A-MainGraphs\]). \[!h\] ![Qualitative statistical analysis related to the group 9–16. [**Top line:**]{} ARIMA $(1,1,2)$ standardized residuals versus Johnson’s transformed residuals ([*left*]{}); Q-Q normal plot for the ARIMA $(1,1,2)$ standardized residuals ([*right*]{}). [**Middle line:**]{} AC plot ([*left*]{}) and PAC plot ([*right*]{}). [**Bottom line:**]{} real interest rates versus ARIMA $(1,1,2)$ fitted values ([*left*]{}); comparison among the cumulative distribution function (CDF) of the standard normal distribution, the empirical CDF of ARIMA $(1,1,2)$ standardized residuals and the empirical CDF of the residuals after the Johnson’s transformation ([*right*]{}).[]{data-label="fig:subgroup"}](QualitativeARIMA2 "fig:"){width="100.00000%"} Figure \[fig:CIRfittedvalues1\] compares the short-term interest rates structure of the analysed market data sample with the corresponding curve of CIR fitted values computed by the simulation scheme for each group partitioning the whole data sample. It is worth noting that the normally distributed standardized residuals of the “optimal" ARIMA-CIR model selected for each group/sub-group (see Table \[tab:outputs\]) replace the realizations of a standard Brownian motion in . This strategy allows us to get an exact trajectory of CIR fitted values instead of a curve averaged over 100,000 simulated trajectories. Consequently, the computational cost is considerably reduced. The real interest rates have been shifted by using a translation of type , where $\alpha$ corresponds to the $99^{th}$-percentile of the conditional distribution of the spot rates process, as described in Section \[section3.2\]. Finally, the CIR fitted values have been shifted back. ![Monthly EUR interest rates with T=1 day (overnight) maturity vs CIR fitted rates[]{data-label="fig:CIRfittedvalues1"}](changeD6){width="90.00000%"} The total values of the statistics $R^{2}_{CIR}$ and the RMSE $\varepsilon$ have been computed on the whole sample as a weighted mean of the $(R^{2}_{CIR})_j$ and $\varepsilon_{j}$ values corresponding to the “optimal" ARIMA model chosen for each group/sub-group, i.e. $$\widetilde{R}^{2}_{CIR}= \sum_{j=1}^{J} \frac{n_j}{n}(R^{2}_{CIR})_j,$$ $$\label{total_error} \varepsilon= \sqrt{\sum_{j=1}^{J} \frac{n_j}{n}\sum_{h=1}^{n_j} (r^j_{shift,h} - \hat{r}^j_h)^{2}}.$$ The Appendix \[B-Estimates\] reports the CIR parameters estimates and the plots of the function $S_j(k),$ defined in , for all groups/sub-groups. We would like to emphasize that the data sample segmentation does not imply a change in the bond’s pricing formula , but only affects the estimation of the CIR parameters $k,\theta, \sigma$ in that are locally calibrated to each group/sub-group of data. The change points detection problem ----------------------------------- As explained in Section \[section3.1\], where we have partitioned the available market data sample in groups with an arbitrary fixed length, say $m=8$, the main difficulty concerns the choice of the optimal segmentation to detect abrupt changes in the variance of the interest rates dynamics. In the literature there exist several approaches for detecting multiple changes in the probability distribution of a stochastic process or a time series such as sequential analysis (i.e., “online" methods), clustering based on maximum likelihood estimation (i.e. “offline" methods), minimax change detection, etc. (see, for example, Bai and Perron (2003) [@Bai], Lavielle (2005) [@Lavielle2005] and (2006) [@Lavielle2006], Hacker and Hatemi-J (2006)[@Hacker], Adams and MacKay (2007) [@Adams], Arlot and Cenisse (2011) [@Arlot]). ------------------------------------------------------------------------------------------- - compute $v(1:end)$ the array of change points detected in the real data array $x$    by the Lavielle method - set $l=v(1)$; - initialize $xstart=1$, $xend=l$;    ($xstart, \, xend$ denote the first and last component of a partitioning group     at each processing cycle) - initialize $j=1$; - set $smax=xstart$ +1 (each group must have a minimum length equal to $2$); - **while** $l<v(end)$ & $l\not=smax-1$ - compute $check \ 1$ and the matrix $L$ ($L$ is the matrix of possible ARIMA $(p,i,q)$ for $x(xstart:xend)$); - **if** $check1=1$ - compute $check2$; - **if** $check2=1$ - compute $\varepsilon_{j}$ and $R^{2}_{CIR};$ - let $ex(j)=l;$   ($ex$ is the array of the rescaled change points, see Figure \[fig:pointsdetection\] ); - set $xstart=ex(j)+1$; - set $j=j+1$; - set $l=v(j)$; - set $xend=l$; - **else** - set $l=l-1;$ - set $xend=l$; - **end** - **end** - **if** $l=smax$ - **if** $length(v)\leq j+1$ - set $l=v(j+1)$; - **else** $break$; - **end** - **end** - **end** ------------------------------------------------------------------------------------------- : Numerical scheme for change points detection by Lavielle’s algorithm[]{data-label="T-11"} Some statistical tests (involved in $check1$) require a minimum sample length equal to $7$, so one can also set $smax=xstart+6$. In this work we decided to implement the Matlab algorithm proposed by Lavielle (2005) [@Lavielle2005] for the detection of changes in the variance, which led to partitioning the real data sample into the following six groups: 1–13, 14–19, 20–30, 31–39, 40–52, 53–68. However, taking into account that our CIR\# model is based on the combination of an ARIMA model and the original CIR model, the above detected change points reported in Figure \[fig:pointsdetection\], namely 13, 19, 34, 39, 52, have to be adjusted according to the numerical scheme described in Table \[T-11\]. ![Scheme for change points detection from the algorithm in Table \[T-11\][]{data-label="fig:pointsdetection"}](Cattura3){width="100.00000%"} Table \[tab:outputs1\] lists the results from the ARIMA-CIR algorithm after application of the change point detection algorithm. Figure \[fig:CIRfittedvalues2\] plots the real interest rates versus the CIR\# fitted values according to results in Table \[tab:outputs1\]. We found that the total $R^{2}_{CIR}=0.7584$ and the total RMSE $\varepsilon=0.4159$. As before, for all $j$, the errors $\varepsilon_{j}$ are at most of the order of $10^{-1}$. --------------------------------------------- -- -- -- -- -- -- -- **j&**Groups&**ARIMA model&&&**BIC cond.**\ 1&1–13&&0.6223&\ &&(3,1,1)&0.6117&0.2258\ &&(3,1,2)&0.5985&0.2299\ 2&14–19&(1,0,1)&0.8842&0.0048\ &&(2,0,1)&0.7960&0.0062\ &&(2,0,2)&0.8814&0.0047\ &&&0.8795&\ &&(1,1,1)&0.8291&0.0058\ &&(1,1,2)&0.8821&0.0048\ &&(1,2,1)&0.7623&0.0068\ &&(2,1,1)&0.8421&0.0055\ ****** --------------------------------------------- -- -- -- -- -- -- -- : Outputs from the ARIMA-CIR algorithm applied to $n=68$ monthly EUR interest rates with T=1 day (overnight) maturity (after application of the Lavielle method) \[tab:outputs1\] --------------------------------------------- -- -- -- -- -- -- -- **j&**Groups&**ARIMA model&&&**BIC cond.**\ 3&20–30&(3,1,2)&0.6345&0.0087&$\surd$\ &&(3,1,3)&0.6193&0.0089\ &&(1,2,3)&0.6452&0.0099\ &&(2,2,2)&0.5178&0.0096&$\surd$\ &&(2,2,3)&0.5777&0.0089\ &&&0.6369&\ 4&31–39&(1,0,1)&0.7165&0.0444\ &&(1,0,2)&0.6895&0.0465\ &&(1,0,3)&0.6815&0.0469\ &&(1,1,1)&0.7168&0.0439\ &&(1,1,2)&0.7247&0.0415\ &&(1,1,3)&0.5409&0.0546\ &&(2,1,1)&0.6876&0.0458\ &&&0.7478&\ &&(2,1,3)&0.6778&0.0449\ &&(1,2,3)&0.5911&0.0507\ &&(2,2,3)&0.5817&0.0509\ &&(3,2,2)&0.6379&0.0473\ 5&40–52&&0.8841&\ &&(2,0,1)&0.8868&0.1196\ &&(2,0,2)&0.7985&0.1317&$\surd$\ &&(2,0,3)&0.7979&0.1315\ &&(3,0,1)&0.8944&0.1178\ &&(3,0,2)&0.8186&0.1230\ &&(3,0,3)&0.8786&0.1158\ &&(1,1,1)&0.6359&0.1653\ &&(1,1,2)&0.6004&0.1752&$\surd$\ &&(1,1,3)&0.6436&0.1595\ &&(2,1,1)&0.8459&0.1287\ &&(2,1,2)&0.5547&0.1783\ &&(2,2,1)&0.6694&0.1637\ &&(3,2,3)&0.7306&0.1630\ 6&53–68&&0.8111&\ ****** --------------------------------------------- -- -- -- -- -- -- -- ![Monthly EUR interest rates with T=1 day (overnight) maturity vs CIR\# fitted rates (after application of the Lavielle method)[]{data-label="fig:CIRfittedvalues2"}](changeD7){width="90.00000%"} Forecast of future interest rates ================================= In this section we will address briefly the CIR\# model’s progress on future interest rate forecasts from a window of observed market data. Ex-ante forecasts require a thorough analysis and will be extensively treated in a forthcoming research. It is worth noting that in this work we decided to impose the most challenging conditions by modelling the shortest part of the yield curve (e.g. the overnight rate) and using only a handful of number of observations. For instance, with monthly data we have found that $m=8$ observations are sufficient for a good calibration. Thus we start to consider a fixed size window of 8 real interest rates that is rolled through time, each month adding the new rate and taking off the oldest rate. The length of this window (8 months) is the historical period over which we forecast the next-month spot rate value. The numerical procedure described in Sections \[section3.2\]–\[section3.4\] has been applied to forecast future next-month interest rates based on monthly EUR interest rates with overnight maturity. The predicted curve is shown in Figure \[Fig:vsc2\], compared with the real observed term structure. ![[**Forecast of the next-month interest rate based on a rolling window of 8 real data:**]{} monthly EUR interest rates with maturity T=1 day (overnight) versus predicted next-month interest rates.[]{data-label="Fig:vsc2"}](changeD9){width=".90\textwidth"} It is evident that the predicted next-month spot rates computed by the CIR\# model follow the market trend. Moreover, the values of $R^2_{CIR}$ and RMSE $\varepsilon,$ computed to measure the goodness-of-fit of forecast interest rates to real data, are respectively $0.8045$ and $0.1392$. CIR\# forecasts versus CIR forecasts ------------------------------------ Here we are going to analyze the CIR\# improvements in forecast as compared to the original CIR model. It is worth noting that calibrating the CIR parameters $(k,\theta,\sigma)$ to real data, a hystorical window of $m>8$ observations is usually needed. To do that, herein we applied the martingale estimating functions method for diffusion processes proposed by Bibby et al. [@Bibby Example 5.4](2005), which shows a better performance with respect to the Maximum Likelihood approach, as empirically confirmed in [@OMB2]. In this case, to forecast a next-month rate by the standard CIR model the minimum length of a rolling window is $m \ge 14$, against the window size of $m=8$ observations, required by the CIR\#. Figure \[Fig:vs2\] illustrates the future next-month values predicted by the CIR\# compared with the ones forecasted by the classical $CIR$ model, showing a better fitting to the real observed term structure. ![[ **Forecast of future next-month expected interest rates:**]{} monthly EUR interest rates with overnight maturity compared with future next-month interest rates predicted by the CIR\# model based on a rolling window of $m=8$ real data, and future next-month interest rates predicted by the classical CIR model based on a rolling window of $m=14$ real data.[]{data-label="Fig:vs2"}](changeD12){width=".90\textwidth"} The values of $R^2$ and $\varepsilon$, listed in Tables \[tab:ForRsquare\] and \[tab:ForRMSE\] for ten term structures ranging from maturity $T= 1$ day to $T=270$ days, confirm the improvement of the proposed novel model compared to the original $CIR$ one applied to shifted data. -- -------------------------------------------------------- -- -- -- **A & **B & **C & **D\ **Maturity & $R^2$ & $R^2$ & Difference & Performance\ & CIR\# & CIR & **A-B & **C/A\ 1 d & 0.8082 & 0.6279 & 0.1803 & 22.38%\ 30 d & 0.9506 & 0.9232 & 0.0274 & 2.88%\ 60 d & 0.9676 & 0.9234 & 0.0445 & 4.59%\ 90 d & 0.9674 & 0.9343 & 0.0332 & 3.43%\ 120 d & 0.9705 & 0.9090 & 0.0615 & 6.33%\ 150 d & 0.9735 & 0.9533 & 0.0202 & 2.07%\ 180 d & 0.9752 & 0.9567 & 0.0185 & 1.89%\ 210 d & 0.9758 & 0.9588 & 0.0170 & 1.74%\ 240 d & 0.9782 & 0.9589 & 0.0193 & 1.97%\ 270 d & 0.9735 & 0.9589 & 0.0146 & 1.49%\ ************** -- -------------------------------------------------------- -- -- -- : CIR\# versus CIR forecasts of future next-months interest rates for ten different maturities. Each column reports the $R^2$ values for both the models.[]{data-label="tab:ForRsquare"} -- ------------------------------------------------------------------------ -- -- -- **A & **B & **C & **D\ **Maturity & $\varepsilon$ & $\varepsilon$ & Difference & Performance\ & CIR\# & CIR & **A-B & **C/A\ 1 d & 0.0963 & 0.1416 & -0.0453 & 47.04%\ 30 d & 0.0463 & 0.0576 & -0.0113 & 24.40%\ 60 d & 0.0406 & 0.0628 & -0.0222 & 54.67%\ 90 d & 0.0476 & 0.0675 & -0.0199 & 41.80%\ 120 d & 0.0484 & 0.0851 & -0.0367 & 75.82%\ 150 d & 0.0488 & 0.0646 & -0.0158 & 32.37%\ 180 d & 0.0500 & 0.0660 & -0.0160 & 32.00%\ 210 d & 0.0511 & 0.0667 & -0.0157 & 30.72%\ 240 d & 0.0503 & 0.0689 & -0.0186 & 36.97%\ 270 d & 0.0570 & 0.0709 & -0.0139 & 24.38%\ ************** -- ------------------------------------------------------------------------ -- -- -- : CIR\# versus CIR forecasts of future next-months interest rates for ten different maturities. Each column reports the RMSE $\varepsilon$ values for both the models.[]{data-label="tab:ForRMSE"} Conclusions =========== Several different extensions of the original model have been proposed to date, with the aim of overcoming the limitations of the CIR model: from one-factor models including time-varying coefficients or jump diffusions to multi-factor models. All these extensions preserve the positivity of interest rates but, in some cases, the analytical tractability of the basic model is violated. Our approach, instead, is based on a proper translation of interest rates such that the market volatility structure is maintained as well as the analytical tractability of the original CIR model. Thus the suggested *CIR\# model* is quite powerful for the following reasons. First, all the improvements are obtained within the CIR framework in order to preserve the single-factor property and the analytical tractability of the original model. Second, market interest rates are properly translated away from zero and/or negative values. The market data sample is partitioned into sub-groups in order to capture all the statistically significant changes of variance in real spot rates and therefore gives an account of jumps. Third, we have introduced a new way of calibration of the CIR model parameters to actual data. The standard Brownian motion process in the random part of the model is replaced with normally distributed standardized residuals of “optimal" ARIMA models suitably chosen. As a result, exact CIR fitted values to real data are calculated and the computational cost of the numerical procedure is considerably reduced. Fourth, we have shown that the *CIR\# model* is efficient and able to follow very closely the structure of market short-term interest rates (especially for short maturities that, notoriously, are very difficult to handle) and to predict future interest rates better then the original CIR model. As a measure of goodness-of-fit, we obtained high values of the statistics $R^{2}$ and small values of the square error $\varepsilon$ for each sub-group and the entire data sample. Future research will show the predictive power of the model by extending the dataset in terms of frequency and size. References {#references .unnumbered} ========== [9]{} R.P. Adams and D.J. MacKay, Bayesian Online Changepoint Detection, arXiv:0710.3742, 2007; S. Arlot and A. Celisse, Segmentation of the Mean of Heteroscedastic Data Via Cross-Validation, [*Statistics and Computing*]{} 21(4), 2011, 613-632 ; J. Bai and P. Perron, Computation and Analysis of Multiple Structural Change Models, [*Journal of Applied Econometrics*]{} 18(1), 2003, 1-22; B.M. Bibby, M. Jacobsen and M. Sørensen, Estimating Functions for Discretely Sampled Diffusion Type Models. In Y. A[ï]{}t-Sahalia and L. P. Hansen (Eds.) 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Mehrdoust, Bond Pricing Under Mixed Generalized CIR Model with Mixed Wishart Volatility, [*Journal of Computational and Applied Mathematics*]{} 319(C), 2017, 108-116; A.R. Najafi, F. Mehrdoust and S. Shirinpour, Pricing American Put Option on Zero-Coupon Bond under Fractional CIR Model with Transaction Cost, [*Communications in Statistics - Simulation and Computation*]{} 0(0), 2017, 1-7; G. Orlando, R. M. Mininni and M. Bufalo, A New Approach to CIR Short-Term Rates Modelling. In M. Mili, F. di Pietro and R. Samaniego Medina (Eds.), [*New Methods in Fixed Income Modeling*]{}, Springer International (USA),2018 (to appear); G. Orlando, R. M. Mininni and M. Bufalo, On the Forecast of Expected Short-Term Interest Rates in the CIR Model,Preprint 2018; G. Orlando and G. Taglialatela, A Review on Implied Volatility Calculation, [*Journal of Computational and Applied Mathematics*]{} 320, 2017, 202-220; L. Rogers, Gaussian errors, [*Risk*]{} 9(1), 1996, 42-45; O. Vasicek, An Equilibrium Characterization of the Term Structure, [*Journal of Financial Economics*]{} 5(2), 1977, 177-188; L. Zhu, Limit Theorems for a Cox-Ingersoll-Ross Process with Hawkes Jumps, [*Journal of Applied Probability*]{} 51(3), 2014, 699-712; Qualitative analysis related to Table \[tab:outputs\] {#A-MainGraphs} ===================================================== We report the qualitative statistical analysis carried out for each group/sub group according to the results reported in Table \[tab:outputs\]. The qualitative analysis related to the results in Table \[tab:outputs1\] is analogous and so it is omitted. ![Qualitative statistical analysis related to the sub-group 1–8. [**Top line:**]{} ARIMA $(2,0,1)$ standardized residuals versus Johnson’s transformed residuals ([*left*]{}); Q-Q normal plot for the ARIMA $(2,0,1)$ standardized residuals ([*right*]{}). [**Middle line:**]{} AC ([*left*]{}) and PAC ([*right*]{}) plots. [**Bottom line:**]{} real interest rates versus ARIMA $(2,0,1)$ fitted values ([*left*]{}); comparison of the standard normal cumulative distribution function (CDF) with the empirical CDF of ARIMA $(2,0,1)$ standardized residuals and of Johnson’s transformed residuals ([*right*]{}).](QualitativeARIMA1){width="100.00000%"} ![Qualitative statistical analysis related to the sub-group 17–32. [**Top line:**]{} ARIMA $(1,0,3)$ standardized residuals versus Johnson’s transformed residuals ([*left*]{}); Q-Q normal plot for the ARIMA $(1,0,3)$ standardized residuals ([*right*]{}). [**Middle line:**]{} AC ([*left*]{}) and PAC ([*right*]{}) plots. [**Bottom line:**]{} real interest rates versus ARIMA $(1,0,3)$ fitted values ([*left*]{}); comparison of the standard normal cumulative distribution function (CDF) with the empirical CDF of ARIMA $(1,0,3)$ standardized residuals and of Johnson’s transformed residuals ([*right*]{}).](QualitativeARIMA3){width="100.00000%"} ![Qualitative statistical analysis related to the sub-group 33–48. [**Top line:**]{} ARIMA $(3,0,1)$ standardized residuals versus Johnson’s transformed residuals ([*left*]{}); Q-Q normal plot for the ARIMA $(3,0,1)$ standardized residuals ([*right*]{}). [**Middle line:**]{} AC ([*left*]{}) and PAC ([*right*]{}) plots. [**Bottom line:**]{} real interest rates versus ARIMA $(3,0,1)$ fitted values ([*left*]{}); comparison of the standard normal cumulative distribution function (CDF) with the empirical CDF of ARIMA $(3,0,1)$ standardized residuals and of Johnson’s transformed residuals ([*right*]{}).](figA3.eps){width="100.00000%"} ![Qualitative statistical analysis related to the sub-group 49–56. [**Top line:**]{} ARIMA $(3,1,2)$ standardized residuals versus Johnson’s transformed residuals ([*left*]{}); Q-Q normal plot for the ARIMA $(3,1,2)$ standardized residuals ([*right*]{}). [**Middle line:**]{} AC ([*left*]{}) and PAC ([*right*]{}) plots. [**Bottom line:**]{} real interest rates versus ARIMA $(3,1,2)$ fitted values ([*left*]{}); comparison of the standard normal cumulative function (CDF) with the empirical CDF of ARIMA $(3,1,2)$ standardized residuals and of Johnson’s transformed residuals ([*right*]{}).](figA4.eps){width="100.00000%"} ![Qualitative statistical analysis related to the sub-group 57–68. [**Top line:**]{} ARIMA $(2,1,1)$ standardized residuals versus Johnson’s transformed residuals ([*left*]{}); Q-Q normal plot for the ARIMA $(2,1,1)$ standardized residuals ([*right*]{}). [**Middle line:**]{} AC ([*left*]{}) and PAC ([*right*]{}) plots. [**Bottom line:**]{} real interest rates versus ARIMA $(2,1,1)$ fitted values ([*left*]{}); comparison of the standard normal cumulative distribution function (CDF) with the empirical CDF of ARIMA $(2,1,1)$ standardized residuals and of Johnson’s transformed residuals ([*right*]{}).](figA5.eps){width="100.00000%"} CIR parameter estimates {#B-Estimates} ======================= We report the estimates of the CIR parameters $k,\theta,\sigma$ and, in particular, the plots of the function $S_j(k)$ defined in , corresponding to the selected “optimal" ARIMA models reported in Table \[tab:outputs\]. {width="100.00000%"} ------------------------------- -- -- -- -- **j&**group/sub-group& &&\ 1&1–8&20.6364&2.7699&0.4027\ 2&9–16&5.6621&2.3338&0.3663\ 3&17–32&5.1649&1.7924&0.0954\ 4&33–48&4.4462&1.9223&0.1546\ 5&49–56&1.9092&1.6637&0.0709\ 6&57–68&1.3555&1.4264&0.0958\ **** ------------------------------- -- -- -- -- : CIR parameter estimates based on 68 monthly observed 1-day (overnight) EUR interest rates \[tab:estimates\] [^1]: \[note1\]The contribution of both authors to this work is equivalent. [^2]: If this condition is not verified, we can require just the absence of autocorrelation.

--- abstract: | The Neumann boundary problem for the perturbed sine-Gordon equation describing the electrodynamics of Josephson junctions has been considered. The behavior of a viscous term, described by a higher-order derivative with small diffusion coefficient $ \varepsilon, $ is investigated. The Green function related to the linear third order operator is determined by means of Fourier series, and properties of rapid convergence are established. Furthermore, some classes of solutions of the hyperbolic equation have been determined, proving that there exists at least one solution whose derivatives are bounded. Results prove that diffusion effects are bounded and tend to zero when $ \varepsilon $ tends to zero. [**[Keywords]{}**]{}:[ Superconductivity; Junctions Initial- boundary problems for higher order parabolic equations ]{} **Mathematics Subject Classification (2000)**44A10 35A08,74K30, 35K35,35E05 **PACS**[ 74.50.+r 02.30.Jr ]{} author: - 'Monica De Angelis[^1]' title: 'On diffusion effects of the perturbed sine-Gordon equation with Neumann boundary conditions' --- Introduction {#intro} ============ Let us consider the sine-Gordon equation: $$\label{11} u_{xx}- u_{tt} = \gamma + \sin u.$$ This equation models the flux dynamics in the Josephson junction where two superconductors are separated by a thin insulating layer. Indeed, denoting by $ \lambda _L $ the London penetration depth of the superconducting electrodes, the spatial coordinate $ x$ is normalized to $ \lambda _L, $ while the time $ t $ is normalized to the inverse plasma frequency $ \omega _0 = \lambda _L / \tilde c$ (where $\tilde c$ is the maximum velocity of the electromagnetic waves in the junction). So, function $u(x,t)$ denotes the phase difference of the electrons between the top and the bottom superconductor, while constant $\gamma γ = j/j_0 $ (with $j_0 = $ maximum Josephson current) represents the normalized current bias. In this case superconductors are ideal, i.e. there are no quasi-particle currents and all the electrons form Cooper pairs. Conversely, when a real junction is considered [@sco], a term $\alpha u_t$ can denote the dissipative normal electron current flow across the junction, while the flowing of quasi-particles parallel to the junction can be represented by a third term such as $\varepsilon \,u_{xxt}$. So that the following perturbed sine Gordon equation holds: $$\label{12} u_{xx}- u_{tt} = \sin u + \gamma + \alpha u_t - \varepsilon \,u_{xxt}$$ where the value range for $\alpha $ and $\varepsilon$ depends on the material of the real junction. Indeed, denoting by $ C, R, L_p,$ respectively, the capacitance, the resistance and the inductance per unit length, it results $ \alpha =1/ \omega _0 RC$ and $ \beta = \omega _0 L_p /R.$ So, there are cases in which $0\leq\alpha ,\varepsilon \leq 1$ [@bp; @pfssu] but, when the resistance of the junction is so low to completely shorten the capacitance, the case $ \alpha $ large with respect to 1 arises [@ca; @tin]. Equation (\[12\]) characterizes rectangular or annular junctions, but other geometries can be considered such as window Josephson junctions (WJJ) ([@bc] and reference therein) or elliptic annular Josephson tunnel junctions (EAJTJs) [@mm], that reduce to circular annular junctions as soon as eccentricity vanishes. Moreover it is possible to consider also confocal annular Josephson tunnel junctions (CAJTJ) that are subtended by two ellipses with the same foci but do not have a constant annulus width [@rm]. Besides, if an exponentially shaped Josephson junction (ESJJ)[@bcs2]-[@32] is examined, the equation achieved is the following: $$\label {13} \varepsilon u_{xxt} +\, u_{xx} - u_{tt} - \varepsilon\lambda u_{xt} -\lambda u_x - \alpha u_t \, \, = \, \sin u - \gamma$$ where $ \lambda $ is a positive constant and the current due to the tapering is represented by terms $ \,\,\lambda \,u_x $ and $\,\lambda \, \varepsilon u_{xt} \,$. In particular, $ \lambda u _{x} $ characterizes the geometrical force driving the fluxons from the wide edge to the narrow edge. In others cases, such as a semiannular or an S-shaped Josephson junction, indicating by $ L $ the length of the junction and by $ b $ an applied magnetic field parallel to the plane of the dielectric barrier, the term $ b \cos (x \pi/\ell)$, with $ \ell = L/ \lambda _L, $ has to be considered, too. [@bp; @sk6]. Moreover, if a harmonically oscillating magnetic field applied parallel to the dielectric barrier, and a dc bias across the superconducting electrodes are considered, one has [@sk]: $$\label{14} u_{tt}- u_{xx} + \sin u = -\gamma - \alpha u_t + \varepsilon \,u_{xxt}- b \sin (\omega t) \cos ( x\pi/\ell)$$ where $\omega $ is the normalized frequency of the magnetic field normalized to the Josephson plasma frequency $\omega _0$. **•** There exist numerous applications for Josephson junctions. For example, by means of the superconducting quantum interference device (SQUID), it is possible using magnetocardiograms, to diagnose heart and/or blood circulation problems, while, through magnetoencephalography -MEG- magnetic fields generated by electric currents in the brain, can be evaluated [@bp]. In geophysics, on the other hand, they are used as gradiometers [@tin] or as gravitational wave detectors ([@car] and reference therein) and they play an important role in the study of the potential virtues of superconducting digital electronics, too [@f]. SQUIDs are also used in nondestructive testing as a convenient alternative to ultra sound or x-ray methods ([@bp; @tin][@bpagano]-[@cla2] and reference therein). Finally, SQUIDs can be used as fast, switchable meta-atoms [@jbm]. Mathematical considerations --------------------------- In all the previous equations (\[12\])-(\[14\]), the following linear operator appears: $$\label{15} {\cal L} u \equiv \,\,\partial_{xx} \,(\varepsilon u _{t}+u) - \partial_t( u_{t}+\alpha\,u).\,$$ This is a third order parabolic operator that, as it is well known, is involved in a vast number of realistic mathematical models concerning superconductivity, neurobiology, and viscoelasticity [@mda2010]-[@dannaf] where the evolution is often characterized by deep interactions between wave propagation and diffusion. So that, ${\cal L} $ can be also considered as a linear hyperbolic operator perturbed by viscous terms described by higher-order derivatives with small diffusion coefficients $ \varepsilon $. When $ \varepsilon \,\equiv 0, $ the parabolic operator turns into a hyperbolic one: $$\label{16} {\mathcal L}_0\,U \equiv \,U_{xx} - \partial_t(U_t+\alpha\, U),$$ and the influence of the dissipative terms, represented by $\, \varepsilon \, \partial _{xxt}$, on the wave behaviour has been estimated. Similar problems, for Dirichlet conditions, have already been studied in [@dr1; @dmr]. In particular, when $ \alpha=0, $ an asymptotic approximation is established by means of the two characteristic times: slow time $ \tau = \varepsilon \,t $ and fast time $ \theta =\, t/\varepsilon $. Moreover, for equation (\[13\]) in [@df213], an analytical analysis has proved that the surface damping has little influence on the behaviour of the oscillator, thus confirming numerical results already determined in [@bcs00]. Numerical investigations on influence of surface losses can be found in [@pskm],too Here the Neumann boundary conditions added with equation (\[15\]) are considered, and diffusion effects have been evaluated. In order to analyze the influence of the dissipative term on the wave behavior, a rigorous estimate of operator ${\mathcal L}$ has been achieved by means of the Green function determined by Fourier series, and an evaluation of the following difference: $$\label {17} d(x,t,\varepsilon)\,= u(x,t,\varepsilon) - U(x,t)$$ has been done. Hence, the solution of the non-linear problem related to $ d $ is determined. Furthermore, some classes of solutions of the hyperbolic equation have been obtained, proving that there exists at least one solution whose derivatives are bounded. Finally, since the hyperbolic equation admits solutions with limited derivatives, an estimate for the remainder term is achieved by proving that the diffusion effects are of the order of $ \varepsilon^h $ with $h <1 $ in each interval-time $ [0, T_\varepsilon]$, with $\displaystyle T_\varepsilon = \,\min\, \{\frac{1}{N}\,\lg \bigl( \,\, \frac{1}{\varepsilon ^{1-h}}\bigr)\,; \lg \bigl( \,\, \frac{1}{\varepsilon ^{1-h}}\bigr)\}$ and $ N>0 $ independent from $ \varepsilon $. The paper is organized as follows: Section II describes the mathematical problem, and attention is fixed on the Green function of the linear operator $ {\cal L} $ defined in (\[15\]). In section III properties of the Green Function $ G $ are pointed out in Theorem 1 whose proof can be found in appendix. Moreover, by means of the fixed point theorem, the solution of the problem related to the remainder term is showed. In section IV, some explicit solutions of the non linear hyperbolic equation (\[16\]) are determined. Finally, in section V an estimate for the remainder term is given in theorem 3. Statement of the problem ======================== Let $T$ be a prefixed positive value constant and $ \Omega =\{(x,t) : 0\leq x\leq \ell , \ \ 0 < t \leq T \} $. The Neumann boundary value problem for equation (\[12\]) refers to the phase gradient value and is proportional to the magnetic field [@for; @j05]. So that one has: $$\label{21} \left \{ \displaystyle \begin{array}{ll} \partial_{xx}(\varepsilon u _{t}+ u) - \partial_t(u_{t}+\alpha)= \sin u+\gamma , &\quad (x,t)\in \Omega, \vspace{2mm} \\ u(x,0)=h_0(x), \ \ \ u_t(x,0)=h_1(x), &\quad x\in [0,\ell], \vspace{2mm} \\ u_x(0,t)=\varphi_0(t), \ \ \ u_x(\ell,t)=\varphi_1(t), &\quad 0<t \leq T. \end{array} \right.$$ When $\varepsilon \equiv 0$, problem (\[21\]) turns into the Neumann problem related to parabolic operator ${\mathcal L}_0\,$, which has, of course, [*the same initial boundary conditions*]{}: $$\label{22} \left \{ \begin{array}{ll} \displaystyle U_{xx} - \partial_t(U_{t}+\alpha\,U)\,=\sin U\ +\gamma &\quad(x,t)\in \Omega,\vspace{2mm}\\ U(x,0)=h_0(x), \ \ \ U_t(x,0)=h_1(x), &\quad x\in [0,\ell],\vspace{2mm} \\ U_x(0,t)=\varphi_0(t), \ \ \ U_x(\ell,t)=\varphi_1(t), &\quad \ 0<t \leq T, \end{array} \right.$$ The influence of the dissipative term on the wave behavior of $\,U\,$ can be estimated when the difference $ d, $ defined in (\[17\]), is evaluated. So, let us consider the following problem related to the [*remainder*]{} term $ \, d:\,$ $$\label{23} \left \{ \begin{array}{ll} \partial_{xx} \, \,(\varepsilon \partial_t \,+1)\,d - \partial_t(\partial+\alpha\,)d\,=\,F(x,t,d),\ & (x,t)\in \Omega,\vspace{2mm} \\ d(x,0)=0, \ \ \ d_t(x,0)=0, \ \ & x\in [0,\ell], \vspace{2mm} \\ d_x(0,t)=0, \ \ d_x(l,t)=0, & 0<t \leq T \end{array} \right.$$ with $$\label{24} \,\,{F}(x,t,d)= \sin (d+U) - \sin U - \varepsilon \, U_{xxt}.\,$$ The operator $ {\cal L} $ has already been examined by means of convolutions of Bessel functions and a short review can be found in [@uffa], while for Dirichlet problem it has already been studied in [@mda; @mda12] and a recent approach can be found in [@cor]. Here, in order to deduce an exhaustive asymptotic analysis, it has been analyzed by means of the Green function determined by Fourier series. So, assuming $$\label{25} \left \{ \begin{array}{ll} &\gamma_n=\frac{n\pi}{\ell},\qquad h_n=\frac{1}{2}(\alpha+\varepsilon\gamma_n^2),\vspace{2mm} \\ \\ & \omega_n=\sqrt{h_n^2-\gamma_n^2} \\ \\ & H_{n}(t)=\,\, \frac{1}{\omega_n}\,\,e^{-h_nt}\,\, \sinh\,(\omega_nt), \end{array} \right.$$ by means of standard techniques, the Green function $ G=G(x,t) $ of problem (\[23\]) is given by $$\label{26} G(x,t,\xi)= \frac{1}{\ell}\,\, \frac{1- e^{-\alpha \, t }}{ \,\alpha}\,\, \,\,+ \,\frac{2}{\ell•}\,\,\sum_{n=1}^{\infty} H_{n}(t) \, \, \cos\gamma_n\xi\, \, \cos\gamma_nx.$$ Properties of the Green function and solution related to the remainder term ============================================================================= In order to achieve the explicit solution of problem (\[23\]), attention should be paid to function G. Some properties of the Green function have already been determined in [@mda; @mda12; @df13] proving among other things, that function G is exponentially vanishing as $t\rightarrow\infty$. Moreover, since (\[26\]), it results: $$\label{311} G(x,t,\xi) \leq \, \frac{2}{\ell} \,\, \sum_{n=0}^{\infty} \,\,H_n(t)\, \cos\gamma_n\xi\, \, \cos\gamma_nx,$$ and denoting by $$\begin{aligned} \label{31} \nonumber &\beta \equiv \min \,\,\biggl\{ \frac{1}{\varepsilon+\alpha(\ell/\pi)^2}, \ \frac{\alpha+\varepsilon (\pi/\ell)^2}{2}, \ \alpha/2 \biggr\},\\ \\ &\nonumber r = \frac{q-1}{q•} \,\, \,(q>1) $$ the following theorem, whose proof can be found in appendix, holds: **Theorem 1** \[thonG\] The function $G(x,\xi,t)$ defined in (\[26\]), and all its time derivatives are continuous functions and there exist some positive constants $ A_j \,\, (j \in {\sf N})$ depending on $\alpha ,\varepsilon$ and positive constants $ M, \, N $ depending on $ \alpha $ and independent from $\,\varepsilon $ such that: $$\label{32} | \, G(x,\xi,t)|\,\leq \,\, 1/2\,\, (\,M \,\,\varepsilon^{r} +N \,\, \,e^{- \beta\,\, t}\,)$$ $$\label{33} \left | \frac{\partial ^j G}{\partial t^j} \right |\, \leq \,A_j \, e^{-\beta t}, \qquad j \in {\sf N} $$ Furthermore, one has: $$\label{34} {\cal L} \,\,G \, =\partial_{xx}(\varepsilon G _{t}+ G) - \partial_t(G_{t}+\alpha G)=0.$$ ------------------------------------------------------------------------ Now, let us consider the nonlinear source (\[24\])$\,\displaystyle F(x,t,d)= \sin (d+U) - \sin U - \varepsilon \, U_{xxt}\,$. By means of standard methods related to integral equations and thanks to the fixed point theorem, owing to basic properties of the Green function $ G $ and the source $\,\, F,\,\, $ it is possible to prove that problem (\[23\])-(\[24\]) admits a unique regular solution in $ \Omega $ and it results: [@c]-[@dew13] $$\begin{aligned} \label{41} d(x,t)=\,\, -\,\frac{1}{\ell} \int_{0}^{t} \, d\tau\,\, \int_0^\ell \,\,\bigl[\frac{1- e^{-\alpha \, (t-\tau) }}{\alpha} \bigr]\,\, F(\xi,\tau,d(\xi,\tau)) \, d\xi\ \\ \nonumber \\ \nonumber - \,\,\frac{2}{\ell•}\,\int_0^ t d\tau\, \int_0^\ell\, H(x,\xi,t-\tau)\, F(\xi,\tau,d(\xi,\tau))\,\,d\xi. $$ where $$H= \sum_{n=1}^{\infty} H_{n}(t) \, \, \cos\gamma_n\xi\, \, \cos\gamma_nx.$$ Explicit solutions of the hyperbolic equation ============================================== Let us consider the semilinear second order equation: $$\label{51} U_{xx} \,- \,U_{tt} \, - \, \alpha \,U_{t} = \sin U \, \ + \gamma$$ When $ \alpha = \gamma = \, 0 $, (\[51\]) represents the sine - Gordon equation and there is plenty of literature about its classes of solution. [@jsb]-[@adc] Now, let $\, f \,$ be an arbitrary function, and let us consider the following function $ \,\,\Pi (f)\,\,$: $$\label{52} \Pi(f) = \, 2 \, \arctan\,\, e^f \,\,$$ so that $$\label{53} \sin \, \Pi (f) \,\, = \frac{1}{\cosh (f) }, \qquad \cos \, \Pi (f) \,\, = - \tanh (f) .$$ By means of function (\[52\]) it is possible to find a class of solutions of equation (\[51\]). Indeed, it is possible to verify that the following function: $$U \, = 2 \, \Pi [f(\xi)]\, \qquad \mbox{with}\qquad \xi =\, \frac{x-t}{\alpha}$$ is a solution of (\[51\]) provided that one has: $$\label{55} -\alpha\, U_{t}\, = \, \sin U \, +\, \gamma.$$ Moreover, since (\[53\]) and being $\,\, \dot \Pi = \frac{1}{\cosh f}, \,\,\, $ it results: $$\displaystyle -\alpha\, U_{t}\, = 2\frac{f'}{\cosh f}; \,\,\qquad \sin U \,\,= 2 \sin \Pi \cos \Pi \,\,= \,-\,\,2 \,\frac{\tanh f }{\cosh f}.$$ So, from (\[55\]), one deduces that function $ f $ must satisfy the following equation: $$\label{56} \frac{df}{-\tanh f\, +\gamma/2\, \cosh f•}\,= \,d\,\xi$$ When $ 0\leq \gamma \leq 1,$ we point our attention to those cases in which it results: $$U\,=\, 4 \arctan (\, y\,+\,\sqrt{y^2 \, +\, 1}\,).$$ So, let $ h $ be an arbitrary constant of integration, one obtains: $$y= \,h\, e^{-\, \xi\,}\,\,\,\,\quad\quad \qquad \qquad \mbox{when}\qquad \qquad \gamma = 0,$$ $$y= \,\frac{1-(\xi\,-h)}{1+ \xi -h}\qquad \qquad \mbox{when}\qquad \qquad \gamma = 1.$$ Moreover, assuming $ \gamma^2 <1, $ let $$A = \, \mp \sqrt{1-\gamma^2}\,\, \qquad \mbox{and} \qquad \delta = - 1+A.$$ For $ \gamma^2 \neq \delta^2, $ it results: $$\label{510} y= \,\frac{h\, \frac{\gamma}{\delta\,\,}\,\,e^{\xi A}\,\,-\frac{\delta}{\gamma\,\,}}{1-h \, e^{\,\,\xi \,\,A}}.$$ When $ \gamma > 1,$ it is possible to prove that $$U = 2 \,\,\arctan \biggl\{\frac{1}{\gamma}\,\, \biggl[\sqrt{\gamma^2-1}\,\, \tan \biggl(\frac{\sqrt{\gamma^2-1}}{2} \, \,\,\xi + h\,\,\biggr)\,\,-1 \,\,\biggr]\,\,\biggr\}$$ Physical cases show that generally $ \gamma \,$ is less than $ \,1, $ and in this case, indicating by $ \displaystyle \eta=\sqrt{1-\gamma^2},$ it also results: $$\label {49} U(x,t) \,\ = \, 2\,\, \arctan \,\biggl[ \,\, \frac{\eta}{\gamma} \,\, \biggl( \frac{1+ h\,\,e^{\, \xi}}{1- h\,\,e^{\,\, \xi}} \, - \,\, \frac{1}{\eta}\,\,\biggr) \biggr]$$ So, Indicating by $$z= \,\, \frac{\eta}{\gamma} \,\, \biggl( \frac{1+ h\,\,e^{\, \xi}}{1- h\,\,e^{\,\, \xi}} \, - \,\, \frac{1}{\eta}\,\,\biggr),$$ one has: $$\label{514} U_{xxt}(x,t)\,= \displaystyle{\frac{2 \,\,z_{xxt}}{ 1+z^2}\,}\,\,+ \,\frac{12 \, z \, z_x \,z_{xx}+ 4z_x^3}{ (1+z^2)^2}\,\,\,- \frac{ 16\, z^2 \, \,z^3_{x}}{ (1+z^2)^3}\, \,\,$$ which is bounded for all $ (x,t) \in \Omega_T, $ being $$\begin{aligned} \begin{split} &z_x = \ \frac{\eta}{\gamma \, \alpha}\,\,\biggl[\, \frac{he^\xi}{1-he^\xi}\, +\,\frac{h (1+h e^\xi)}{(1-he^\xi)^2}\biggr]; \\ \\ &z_{xx} = \frac{\eta}{\gamma \, \alpha^2}\,\,\biggl[\, \frac{he^\xi}{1-he^\xi}\, +\,\frac{h^2 e^\xi (1+ e^\xi)}{(1-he^\xi)^2} \,\,+ \,\frac{ 2 h^2 e^\xi (1-h^2 e^{2\xi})}{(1-he^\xi)^4} \,\biggr]\\ \nonumber \\ & z_{xxt} = - \frac{\eta}{\gamma \, \alpha^3}\,\,\biggl[\, \frac{he^\xi}{1-he^\xi}\, +\,\frac{3h^2 e^{2\xi}+ h^2e^\xi}{(1-he^\xi)^2} \,\,+ \,\frac{2 h^2 e^{\xi} -6h^4 e^{3\xi}}{(1-he^\xi)^4} \, + \frac{ 8 h^3 e^{2\xi} (1+h^2 e^{2\xi})}{(1-he^\xi)^5}\biggr]. \end{split}\end{aligned}$$ Estimates for the remainder term ================================ Let us assume $ \varepsilon =0 $ and let $ U $ be a solution of the reduced problem (\[22\]) In the following we will have to refer to a known inequality of Gronwall type due to S.M. Sardar’ly ([@mpf] p 359): **Theorem 2** Let $ x $ and $a_2 $ be continuous and $ a,\,a_1,\,\int_0^t b(t,s)ds \,$ Riemann integrable functions on $ J=[0,\beta], $ with $ a_1 $ and $a_2$ nonnegative on J. If $$x(t) \leq a(t) + \int_0^t b(t,s)ds + a_1(t)\int_0^t a_2(s) x(s) ds \qquad t\in J$$ then $$\begin{aligned} \nonumber x(t) \leq & a(t) & + \int_0^t b(t,s)ds + a_1(t)\int_0^t a(s)a_2(s) \exp\biggl( \int_s^t a_1(z) a_2(z) dz\biggr) ds \, + \\ \\ \nonumber & a_1(t)&\int_0^ta_2(s) \int_0^s b(t,z)dz \,\, \exp\biggl( \int_s^t a_1(z) a_2(z) dz\biggr) ds \qquad t\in J.\end{aligned}$$ ------------------------------------------------------------------------ According to this, it is possible to state: **Theorem 3** \[remainder\] Let us assume $$S(t)\, = \sup_{0\leq x \leq \ell }\, | d(x,t)|.$$ If there exists a positive constant $ k $ such that $$\label{61} |U_{xxt}\,(x,t)|\, \leq \,k,$$ then there exist two positive constants $\Gamma $ and $ h $ with $ \,\displaystyle \, h\,<\,\frac{q-1}{q}\, (1<q<\infty) \, $ such that, indicated by $ N $ the positive constant defined in (\[32\]) and $$\label{63} T_ \varepsilon \,: = \,\min \biggl\{ \frac{1}{N} \lg \biggl( \,\, \frac{1}{\varepsilon ^{1-h}}\biggr);\,\, \lg \biggl( \,\, \frac{1}{\varepsilon ^{1-h}}\biggr)\biggr\},$$ it results: $$\label{64} 0\,\leq \,S(t)\,\leq\, \Gamma \,\,\varepsilon ^h$$ for every $ t \leq T_ \varepsilon. $ : Let us consider function $ F $ defined in (\[24\]): $$\label{65} \, F \,= \, \sin \, (d+U) \, -\sin U \,- \, \varepsilon \,U_{xxt}.$$ Since (\[311\]) and (\[41\]), it results: $$\label{66} |d(x,t,\varepsilon)| \,\leq \,\frac{2}{\ell•}\int_0^ t d\tau\, \int_0^l\, |G(x,t-\tau \,,\xi)| \,\,\, | F(x,\tau,\xi)|\,\,d\xi\ $$ where, function $ F(x,t,u), $ according to (\[61\]) and (\[65\]), satisfies the following inequalities: $$\begin{aligned} &|F(x,t)| \leq \,\, |d(x,t)| +\varepsilon \,\,k \nonumber \\ \\\nonumber &|F(x,t)| \leq 2+\varepsilon \,\,k. $$ So that, by means of properties of the Green function $G, $ and in particular since $(\ref{32}), $ one obtains: $$\begin{aligned} \label{68} \nonumber & |d(x,t,\varepsilon)| \,\leq \frac{1}{\ell•}\int_0^ t d\tau \int_0^l\, M\,\varepsilon^r (2+\varepsilon k)d\xi\ + \\ \\ \nonumber \,\,&\ \qquad \qquad \qquad\frac{1}{\ell•}\int_0^ t d\tau\, \int_0^l\, N\, \,e^{-\beta(t-\tau)} \,[|d(\tau,\xi)|\,\, +\varepsilon \,\,k]\,\,d\xi $$ where $ \beta\,\, \mbox{and } r $ are defined in (\[31\]). Hence, it follows: $$\begin{aligned} \label{69} \nonumber &\,S(t)\,\leq\, \, \,(2+\varepsilon \,\,k) \,\, M \, \varepsilon^r\,\, t \,\,+\,k\,\,\varepsilon\,\, {N} \int_0^ t \, e^{\,-\,{\beta} \,(t-z)\,}\, dz \,+ \\ \\ \nonumber &\,N\,\, e^{\,-\,\beta \,t\,}\,\, \int_0^ t \, e^{\,\,{\beta} \,z\,}\, \,S(z)\, dz . $$ Applying theorem 2 it results: $$\begin{aligned} \label{610} \nonumber & S (t)\,\,\, \leq &\,M \,\,\varepsilon^r \,({2+\varepsilon \,\,k }) \,\, t \,\, + N \,k\,\,\frac{1-e^{-\beta \,t}}{\beta} \,\,\varepsilon \, + \\ \\ \nonumber && {N} \, e^{-\beta t} \int_0^t [ N\,k\,\,\varepsilon\,\frac{1-e^{-\beta \,s}}{ \, \beta} \,\,\, \, + M \,\,\varepsilon^r\,\, ({2+\varepsilon \,k})\,\, s] e^{\beta s} \,\, e^{N(t-s) }\,\,\, ds. $$ So, one has: $$\begin{aligned} \label{611} \nonumber & S(t) \,\leq \,&\biggl[ M \,(2+\varepsilon \,\,k ) \,\, t \, \,\, + N M (2+\varepsilon \,k) \biggl(\frac{t}{\beta-N} +\frac{e^{-t(\beta-N)}-1}{(\beta-N)^2•}\biggr)\,\,\biggr] \varepsilon^r\,+\\ \\ \nonumber \\ \nonumber &&\biggl[ N \,k\,\,\frac{1-e^{-\beta \,t}}{\beta} \,\,\, \,\, + \frac{N^2 k \,}{\beta•}\biggl(\frac{1}{\beta-N•}+\frac{\beta\, e^{-t(\beta-N)}}{N-\beta•}\,\, + \frac{{e^{-t\beta}}}{N•}\biggr)\,\biggr] \varepsilon $$ Now, since for all $ t\in \Re $ one has $t+1 \leq e^t, \, $ it results $$t \,\,\varepsilon ^r \leq \varepsilon^h; \qquad \varepsilon ^r \,e^{Nt}\leq \varepsilon^h$$ as soon as $ \displaystyle h < r= \frac{q-1}{q} \,\,(q>1)\,\, \mbox{and }\,\, t\leq T_\varepsilon. $ ------------------------------------------------------------------------ **Remark 6.1** Estimate (\[64\]) specifies the infinite time-intervals where the effects of diffusion are of the order $\,\displaystyle \varepsilon ^h \, \,$ with $\, 0<\,h\,<1. $ Indeed, the evolution of the superconductive model is characterized by diffusion effects which are of the order of $ \,\displaystyle \varepsilon ^h\,\,$ in each time-interval $ \, [\, 0, \, \, T_\varepsilon \,]$ with $ T_\varepsilon $ defined in (\[63\]). **Remark 6.2** Formula (\[514\]) shows that the class of functions satisfying hypotheses of Theorem 3 is not empty. **Acknowledgments** This work was supported by National Group of Mathematical Physics (GNFM-INDAM) Theorem 1 ========== Let us assume: $$\begin{aligned} \label{31} \nonumber &\beta \equiv \min \,\,\biggl\{ \frac{1}{\varepsilon+\alpha(\ell/\pi)^2}, \ \frac{\alpha+\varepsilon (\pi/\ell)^2}{2}, \ \alpha/2 \biggr\},\\ \\ &\nonumber r = \frac{q-1}{q•} \,\, \,(q>1) $$ The function $G(x,\xi,t)$ defined in (\[26\]) and all its time derivatives are continuous functions and there exist some positive constants $ A_j \,\, (j \in {\sf N})$ depending on $\alpha ,\varepsilon$ and positive constants $ M, \, N $ depending on $ \alpha $ and independent from $\,\varepsilon $ such that: $$\label{A1} | \, G(x,\xi,t)|\,\leq \,\, 1/2\,\, (\,M \,\,\varepsilon^{r} +N \,\, \,e^{- \beta\,\, t}\,)$$ $$\label{A2} \left | \frac{\partial ^j G}{\partial t^j} \right |\, \leq \,A_j \, e^{-\beta t}, \qquad j \in {\sf N} $$ Furthermore, one has: $$\label{A3} {\cal L} \,\,G \, =\partial_{xx}(\varepsilon G _{t}+ G) - \partial_t(G_{t}+\alpha G)=0.$$ Proof: Since $ \alpha \,\varepsilon <1, $ indicating by $ N_1, N_2 $ the integer part of $ \ell/(\,\pi \varepsilon ) (1\mp \sqrt{1-\alpha\varepsilon }), $ respectively, circular functions have to be distinguished from hyperbolic terms. In this case, since Taylor formula, it results $ \sqrt{1-\frac{b_n^2}{h_n^2}•}\,\,\ <1-\frac{b_n^2}{2h_n^2},$ and for all $ n\geq 1 $ one has: $$\displaystyle e^{-t(h_n-\omega_n)}\leq e^{-h_{n}t} \,\, e^{h_{n}\bigl(1-\frac{b_n^2}{2h_n^2•}\bigr)t} \leq e^{\,\frac{1}{\varepsilon+\alpha(\ell/\pi)^2}}$$ Moreover, indicating by $c$ an arbitrary constant less than $ 1, $ denoting by $ N_c $ the integer part of $ \displaystyle \ell/(\pi \varepsilon \sqrt{c}) (1+ \sqrt{1-\alpha\varepsilon \,c}), $ for all $ \displaystyle n \geq N_c,$ it results $ \, \displaystyle h_n > b_n $ and $ \displaystyle\frac{b_n}{\sqrt{c}•} < h_n.$ So that one has: $$\omega_n = h_n \sqrt{1- \frac{b_n^2}{h_n^2•}}\geq h_n \sqrt{1-c}$$ and hence $$\sum_{n=N_c}^\infty \frac{e^{-t(h_n-\omega_n)}}{\omega_n} \leq \frac{2 \ell^2\,\,\varepsilon }{\pi^2\sqrt{1-c}•} \,\, \xi (2)\,\, e^{\,\frac{1}{\varepsilon+\alpha(\ell/\pi)^2}}.$$ Besides, if $ \displaystyle \ell\geq 2\pi/a(1+\sqrt{1-\alpha \varepsilon)}, $ terms $\displaystyle \sum_{n=0}^{N_1-1} \frac{e^{-t(h_n-\omega_n)}}{\omega_n } $ have to be considered. Since $ \displaystyle\sqrt{1-\alpha \varepsilon } = 1- \alpha \varepsilon /2 - (\alpha \varepsilon)^2 \,\,(1-\theta \alpha \varepsilon )^{-3/2} /8\,\,(0<\theta <1) $ there exists a positive constant $ B $ such that: $$\sum_{n=0}^{N_1-1} \frac{e^{-t(h_n-\omega_n)}}{\omega_n}\leq B \, m \biggl(1+ e^{\,\frac{1}{\varepsilon+\alpha(\ell/\pi)^2}}\biggr) \varepsilon$$ being $ m $ the minimum value of $\displaystyle \frac{\omega_n•}{\varepsilon^2} $. Otherwise, when $ N_1 <1 $ attention must be paid to $\displaystyle \sum_{n=0}^{N_2-1} \frac{e^{-h_n t} \sinh(\omega_nt)}{\omega_n}. $ If $ \displaystyle 1<q<\infty $ and $1/p+1/q=1 $ Holder inequality can be considered: $$\sum_{n=0}^{N_2-1} \frac{e^{-h_n t} \sinh(\omega_nt)}{\omega_n}\leq \biggl(\sum_{n=0}^{N_2-1} |e^{-h_n t} \sinh(\omega_nt)|^p\biggr)^{1/p} \,\, \biggl(\sum_{n=0}^{N_2-1} \left |\frac{1}{\omega_n} \right |^q\biggr)^{1/q}.$$ So, having $$\displaystyle \left | \frac{1}{\omega_n} \right| ^q \,\,\leq \frac{\varepsilon^q}{ \sqrt{1-\alpha \varepsilon}}$$ it is possible to find a positive constant $D $ such that the following inequality holds: $$\sum_{n=0}^{N_2-1} \frac{e^{-h_n t} \sinh(\omega_nt)}{\omega_n}\leq D\,\, \varepsilon^{\frac{q-1}{q}} \frac{1}{\sqrt{1-\alpha \varepsilon}}.$$ In this way (\[A1\]) holds. As for the x-differentiation of Fourier series like (\[A3\]), attention must be given to convergence problems. 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--- abstract: | We consider effects of CP violating phases in $\mu$ and $A_t$ parameters in the effective supersymmetric standard model on $B\rightarrow X_s l^+ l^-$ and $\epsilon_K$. Scanning over the MSSM parameter space with experimental constraints including edm constraints from Chang-Keung-Pilaftsis (CKP) mechanism, we find that the ${\rm Br} (B\rightarrow X_s l^+ l^-)$ can be enhanced by upto $\sim 85 \%$ compared to the standard model (SM) prediction, and its correlation with ${\rm Br} ( B \rightarrow X_s \gamma)$ is distinctly different from the minimal supergravity scenario. Also we find $1 \lesssim \epsilon_K / \epsilon_K^{SM} \lesssim 1.4$, and fully supersymmetric CP violation in $K_L \rightarrow \pi \pi$ is not possible. Namely, $|\epsilon_K^{\rm SUSY}| \lesssim O(10^{-5})$ if the phases of $\mu$ and $A_t$ are the sole origin of CP violation. address: | Department of Physics, KAIST\ Taejon 305-701, Korea author: - 'Seungwon Baek[^1] and Pyungwon Ko[^2]' title: | Effects of supersymmetric CP violating phases\ on $B \rightarrow X_s l^+ l^-$ and $\epsilon_K$ --- [**1**]{}. In the minimal supersymmetric standard model (MSSM), there can be many new CP violating (CPV) phases beyond the KM phase in the standard model (SM) both in the flavor conserving and flavor violating sectors. The flavor conserving CPV phases in the MSSM are strongly constrained by electron/neutron electric dipole moment (edm) and believed to be very small ($\delta \lesssim 10^{-2}$ for $M_{\rm SUSY} \sim O(100)$ GeV ) [@susycp]. Or, one can imagine that the 1st/2nd generation scalar fermions are very heavy so that edm constraints are evaded via decoupling even for CPV phases of order $O(1)$ [@kaplan]. Also it is possible that various contributions to electron/neutron EDM cancel with each other in substantial parts of the MSSM parameter space even if SUSY CPV phases are $\sim O(1)$ and SUSY particles are relatively light [@nath] [@kane]. In the last two cases where SUSY CPV phases are of $\sim O(1)$, these phases may affect $B$ and $K$ physics in various manners. In the previous letter [@baek], we presented effects of these SUSY CPV phases on $B$ physics : the $B^0 - \overline{B^0}$ mixing and the direct asymmetry in $B\rightarrow X_s \gamma$, assuming that EDM constraints and SUSY FCNC problems are evaded by heavy 1st/2nd generation scalar fermions. In this letter, we extend our previous work to $B\rightarrow X_s l^+ l^-$ and $\epsilon_K$ (see also Ref. [@demir].) within the same assumptions. An important ingredient for large $\tan\beta$ in our model is the constraint on the $\mu$ and $A_t$ phases coming from electron/neutron edm’s through Chang-Keung-Pilaftsis (CKP) mechanism [@chang]. Two loop diagrams with CP-odd higgs and photon (gluon) exchanges between the fermion line and the sfermion loop (mainly stops and sbottoms) can contribute significantly to electron/neutron edm’s in the large $\tan\beta$ region. The authors of Ref. [@chang] find that $$( {d_f \over e } )_{\rm CKP} = Q_f {3 \alpha_{\rm em} \over 64 \pi^2} {R_f ~m_f \over M_A^2} \sum_{q=t,b}~\xi_q Q_q^2 \left[ F \left( { M_{\tilde{q}_1}^2 \over M_A^2 } \right) - F \left( { M_{\tilde{q}_2}^2 \over M_A^2 } \right) \right],$$ where $R_f = \cot\beta (\tan\beta)$ for $I_{3 f} = 1/2 ~(-1/2)$, and $$\xi_t = {\sin 2\theta_{\tilde{t}} m_t {\rm Im} (\mu e^{i \delta_t} ) \over \sin^2 \beta ~v^2}, ~~~~ \xi_b = {\sin 2\theta_{\tilde{b}} m_b {\rm Im} ( A_b e^{-i \delta_b} ) \over \sin \beta ~\cos\beta~v^2},$$ with $\delta_q = {\rm Arg} (A_q + R_q \mu^* )$, and $F(z)$ is a two-loop function given in Ref. [@chang]. This new contribution is independent of the 1st/2nd generation scalar fermion masses, so that it does not decouple for heavy 1st/2nd generation scalar fermions. Therefore it can be important for the electron or down quark edm for the large $\tan\beta$ case. This is in sharp contrast with the usual one-loop contributions to edm’s, for which [@chang] $$\left( {d_f \over e} \right) \sim 10^{-25} {\rm cm} \times { \left\{ {\rm Im} \mu, {\rm Im } A_f \right\} \over {\rm max} ( M_{\tilde{f}}, M_{\lambda} ) }~\left( { 1 ~{\rm TeV} \over {\rm max} ( M_{\tilde{f}}, M_{\lambda} ) } \right)^2~\left( {m_f \over 10~{\rm MeV}} \right),$$ and one can evade the edm constraints by having small phases for $\mu, A_{e,u,d}$, or heavy 1st/2nd generation scalar fermions. However, this would involve enlargement of our model parameter space, since one has to consider the sbottom sector as well as the stop sector. Therefore, more parameters have to be introduced in principle : $m_{\tilde{b}}^2$ and $A_b$ where $A_b$ may be complex like $A_t$. In order to avoid such enlargement, we will assume that there is no accidental cancellation between the stop and sbottom loop contributions. This CKP edm constraint has not been included in the recent paper by Demir [*et al.*]{} [@demir], who made claims that there could be a large new phase shift in the $B^0 - \overline{B^0}$ mixing and it is possible to have a fully supersymmetric $\epsilon_K$ from the phases of $\mu$ and $A_t$ only. However, if $\tan\beta$ is large ($\tan\beta \approx 60$) as in Ref.  [@demir], the CKP edm constraints via the CKP mechanism have to be properly included. This constraint reduces the possible new phase shift in the $B^0 - \overline{B^0}$ mixing to a very small number, $2 | \theta_d | \lesssim 1^{\circ}$, as demonstrated in Fig. 1 (a) of Ref. [@baek]. On the other hand, the CKP edm constraint does not affect too much the direct CP asymmetry in $B\rightarrow X_s \gamma$ [@baek]. In this work, we continue studying the effects of the phases of $\mu$ and $A_t$ on $B \rightarrow X_s l^+ l^-$ and $\epsilon_K$. We also reconsider a possibility of fully supersymmetric CP violation, namely generating $\epsilon_K$ entirely from the phases of $\mu$ and $A_t$ with vanishing KM phase ($\delta_{\rm KM} = 0$). Our conclusion is at variance with the claim made in Ref. [@demir]. [**2.**]{} As in Refs. [@baek] [@demir], we assume that the 1st and the 2nd family squarks are degenerate and very heavy in order to solve the SUSY FCNC/CP problems. Only the third family squarks can be light enough to affect $B$ and $K$ physics. We also ignore possible flavor changing squark mass matrix elements that could generate gluino-mediated flavor changing neutral current (FCNC) processes in addition to those effects we consider below, relegating the details to the existing literature [@randall]-[@kkl]. Therefore the only source of the FCNC in our case is the CKM matrix, whereas there are new CPV phases coming from the phases of $\mu$ and $A_t$ parameters (see below), in addition to the KM phase $\delta_{KM}$. Definitions for the chargino and stop mass matrices are the same as Ref.  [@baek]. There are two new flavor conserving CPV phases in our model, ${\rm Arg} (\mu)$ and ${\rm Arg} (A_t)$ in the basis where $M_2$ is real. We scan over the MSSM parameter space as in Ref. [@baek] indicated below (including that relevant to the EWBGEN scenario in the MSSM) : $$\begin{aligned} \label{eq:input} 80~{\rm GeV} < | \mu | < 1~{\rm TeV} , &~~& 80~{\rm GeV} < M_2 < 1~{\rm TeV} , \nonumber \\ 60~{\rm GeV} < M_A < 1~{\rm TeV} , &~~& 2 < \tan\beta < 70, \nonumber \\ (130~{\rm GeV})^2 < M_Q^2 & < & ( 1 ~{\rm TeV} )^2 , \nonumber \\ - ( 80~{\rm GeV})^2 < M_U^2 & < & (500~{\rm GeV})^2 , \nonumber \\ 0 < \phi_{\mu}, \phi_{A_t} < 2 \pi ,&~~& 0 < | A_t | < 1.5 ~{\rm TeV},\end{aligned}$$ with the following experimental constraints : $M_{\tilde{t}_1} > 80$ GeV independent of the mixing angle $\theta_{\tilde{t}}$, $M_{\tilde{\chi^{\pm}}} > 83$ GeV, and $0.77 \leq R_{\gamma} \leq 1.15$ [@alexander], where $R_{\gamma}$ is defined as $R_{\gamma}= BR(B \to X_s \gamma)^{expt}/ BR(B \to X_s \gamma)^{SM}$ and $BR(B \to X_s \gamma)^{SM} = (3.29 \pm 0.44) \times 10^{-4}$. We also impose ${\rm Br} (B\rightarrow X_{sg}) < 6.8 \%$ [@bsg], and vary $\tan\beta$ from 2 to 70 [^3]. This parameter space is larger than that in the constrained MSSM (CMSSM) where the universality of soft terms at the GUT scale is assumed. Especially, our parameter space includes the electroweak baryogenesis scenario in the MSSM [@cline]. In the numerical analysis, we used the following numbers for the input parameters (running masses in the $\overline{MS}$ scheme are used for the quark masses) : $\overline{m_c}(m_c(pole)) = 1.25$ GeV, $\overline{m_b}(m_b(pole)) = 4.3$ GeV, $\overline{m_t}(m_t(pole)) = 165$ GeV, and $| V_{cb} | = 0.0410, |V_{tb}| = 1, | V_{ts} | = 0.0400$ and $\gamma ( \phi_3 ) = 90^{\circ}$ in the CKM matrix elements. [**3.**]{} Let us first consider the branching ratio for $B\rightarrow X_s l^+ l^-$. The SM and the MSSM contributions to this decay were considered by several groups [@bsll] and [@bsll_susy], respectively. We use the standard notation for the effective Hamiltonian for this decay as described in Refs.  [@bsll] and [@bsll_susy]. The new CPV phases in $C_{7,9,10}$ can affect the branching ratio and other observables in $B\rightarrow X_s l^+ l^-$ as discussed in the first half of Ref. [@kkl] in a model independent way. In the second half of Ref. [@kkl], specific supersymmetric models were presented where new CPV phases reside in flavor changing squark mass matrices. In the present work, new CPV phases lie in flavor conserving sector, namely in $A_t$ and $\mu$ parameters. Although these new phases are flavor conserving, they affect the branching ratio of $B\rightarrow X_s l^+ l^-$ and its correlation with $Br (B\rightarrow X_s \gamma)$, as discussed in the first half of Ref. [@kkl]. Note that $C_{9,10}$ depend on the sneutrino mass, and we have scanned over $ 60 ~{\rm GeV} < m_{\tilde{\nu}} < 200~{\rm GeV}$. In the numerical evaluation for $R_{ll} \equiv {\rm Br} (B\rightarrow X_s l^+ l^-) / {\rm Br} (B\rightarrow X_s l^+ l^-)_{\rm SM}$, we considered the nonresonant contributions only for simplicity, neglecting the contributions from $J/\psi, \psi^{'}, etc$.. It would be straightforward to incorporate these resonance effects. In Figs. 1 (a) and (b), we plot the correlations of $R_{\mu\mu}$ with ${\rm Br} (B\rightarrow X_s \gamma)$ and $\tan\beta$, respectively. Those points that (do not) satisfy the CKP edm constraints are denoted by the squares (crosses). Some points are denoted by both the square and the cross. This means that there are two classes of points in the MSSM parameter space, and for one class the CKP edm constraints are satisfied but for another class the CKP edm constraints are not satisfied, and these two classes happen to lead to the same branching ratios for $B\rightarrow X_s \gamma$ and $R_{ll}$. In the presence of the new phases $\phi_{\mu}$ and $\phi_{A_t}$, $R_{\mu\mu}$ can be as large as 1.85, and the deviations from the SM prediction can be large, if $\tan\beta > 8$. As noticed in Ref. [@kkl], the correlation between the ${\rm Br} ( B\rightarrow X_s \gamma)$ and $R_{ll}$ is distinctly different from that in the minimal supergravisty case [@okada]. In the latter case, only the envelop of Fig. 1 (a) is allowed, whereas everywhere in between is allowed in the presence of new CPV phases in the MSSM. Even if one introduces the phases of $\mu$ and $A_0$ at GUT scale in the minimal supergravity scenario, this correlation does not change very much from the case of the minimal supergravity scenario with real $\mu$ and $A_0$, since the $A_0$ phase becomes very small at the electroweak scale because of the renormalization effects [@keum]. Only $\mu$ phase can affect the electroweak scale physics, but this phase is strongly constrained by the usual edm constraints so that $\mu$ should be essentially real parameter. Therefore the correlation between $B\rightarrow X_s \gamma$ and $R_{ll}$ can be a clean distinction between the minimal supergravity scenario and our model (or some other models with new CPV phases in the flavor changing [@kkl]). [**4.**]{} The new complex phases in $\mu$ and $A_t$ will also affect the $K^0 - \overline{K^0}$ mixing. The relevant $\Delta S = 2 $ effective Hamiltonian is given by $$H_{\rm eff}^{\Delta S = 2} = - {G_F^2 M_W^2 \over (2 \pi )^2}~ \sum_{i=1}^3 C_i Q_i,$$ where $$\begin{aligned} C_1 ( \mu_0 ) & = & \left( V_{td}^* V_{ts} \right)^2 \left[ F_V^W (3;3) + F_V^H (3;3) + A_V^C \right] \nonumber \\ & + & \left( V_{cd}^* V_{cs} \right)^2 \left[ F_V^W (2;2) + F_V^H (2;2) \right] \nonumber \\ & + & 2 \left( V_{td}^* V_{ts} V_{cd}^* V_{cs} \right) ~\left[ F_V^W (3;2) + F_V^H (3;2) \right], \nonumber \\ C_2 ( \mu_0 ) & = & \left( V_{td}^* V_{ts} \right)^2 F_S^H (3;3) %\nonumber \\ + \left( V_{cd}^* V_{cs} \right)^2 ~F_S^H (2;2) \nonumber \\ & + & 2 \left( V_{td}^* V_{ts} V_{cd}^* V_{cs} \right) ~F_S^H (3;2), \nonumber \\ C_3 ( \mu_0 ) & = & \left( V_{td}^* V_{ts} \right)^2 A_S^C,\end{aligned}$$ where the charm quark contributions have been kept. The superscripts $W,H,C$ denote the $W^{\pm}, H^{\pm}$ and chargino contributions respectively, and $$\begin{aligned} %F_V (i;j) & = & %\nonumber \\ A_V^C & = & \sum_{i,j=1}^2 \sum_{k,l=1}^2 ~{1\over 4}~G^{(3,k)i} G^{(3,k)j*} G^{(3,l)i*} G^{(3,l)j} Y_1 (r_k, r_l, s_i, s_j ), \nonumber \\ A_S^C & = & \sum_{i,j=1}^2 \sum_{k,l=1}^2 ~H^{(3,k)i} G^{(3,k)j*} G^{(3,l)i*} H^{(3,l)j} Y_2 (r_k, r_l, s_i, s_j ).\end{aligned}$$ Here $G^{(3,k)i}$ and $H^{(3,k)i}$ are the couplings of $k-$th stop and $i-$th chargino with left-handed and right-handed quarks, respectively : $$\begin{aligned} G^{(3,k)i} & = & \sqrt{2} C_{R 1i}^* S_{t k1} - { C_{R 2i}^* S_{t k2} \over \sin\beta } ~{m_t \over M_W}, \nonumber \\ H^{(3,k)i} & = & { C_{L 2 i}^* S_{tk1} \over \cos\beta } ~{m_s \over M_W},\end{aligned}$$ and $C_{L,R}$ and $S_t$ are unitary matrices that diagonalize the chargino and stop mass matrices [@branco]. : $C_R^{\dagger} M_{\chi}^- C_L = {\rm diag} ( M_{\tilde{\chi_1}},M_{\tilde{\chi_2}} )$ and $S_t M_{\tilde{t}}^2 S_t^{\dagger} = {\rm diag} ( M_{\tilde{t}_1}^2, M_{\tilde{t}_2}^2 )$. Explicit forms for functions $Y_{1,2}$ and $F$’s can be found in Ref. [@branco], and $r_k = M_{\tilde{t}_k}^2 / M_W^2$ and $s_i = M_{\tilde{\chi^{\pm}}_i} / M_W^2$. It should be noted that $C_2 ( \mu_0 )$ was misidentified as $C_3^H ( \mu_0 )$ in Ref. [@demir]. The gluino and neutralino contributions are negligible in our model. The Wilson coefficients at lower scales are obtained by renomalization group running. The relevant formulae with the NLO QCD corrections at $\mu = 2$ GeV are given in Ref. [@contino]. It is important to note that $C_1 ( \mu_0 )$ and $C_2 ( \mu_0 )$ are real relative to the SM contribution in our model. On the other hand, the chargino exchange contributions to $C_3 (\mu_0 )$ (namely $A_S^C $) are generically complex relative to the SM contributions, and can generate a new phase shift in the $K^0 - \overline{K^0}$ mixing relative to the SM value. This effect is in fact significant for large $\tan\beta (\simeq 1/\cos\beta)$ [@demir], since $C_3 (\mu_0)$ is proportional to $ (m_{s} / M_W \cos\beta )^2$. The CP violating parameter $\epsilon_K$ can be calculated from $$\epsilon_K \simeq {e^{i \pi / 4}~{\rm Im} M_{12} \over \sqrt{2} \Delta M_K},$$ where $M_{12}$ can be obtained from the $\Delta S = 2 $ effective Hamiltonian through $2 M_K M_{12} = \langle K^0 | H_{\rm eff}^{\Delta S = 2} | \overline{K^0} \rangle$. For $\Delta M_K$, we use the experimental value $\Delta M_K = (3.489 \pm 0.009) \times 10^{-12}$ MeV, instead of theoretical relation $\Delta M_K = 2 {\rm Re} M_{12}$, since the long distance contributions to $M_{12}$ is hard to calculate reliably unlike the $\Delta S = 2$ box diagrams. For the strange quark mass, we use the $\overline{\rm MS}$ mass at $\mu = 2$ GeV scale : $m_s (\mu = 2 {\rm GeV}) = 125$ MeV. In Figs. 2 (a) and (b), we plot the results of scanning the MSSM parameter space : the correlations between $\epsilon_K / \epsilon_K^{\rm SM}$ and (a) $\tan\beta$ and (b) the lighter stop mass. We note that $\epsilon_K / \epsilon_K^{\rm SM}$ can be as large as $1.4 $ for $\delta_{KM} = 90^{\circ}$ if $\tan\beta$ is small. This is a factor 2 larger deviation from the SM compared to the minimal supergravity case [@kek]. The dependence on the lighter stop is close to the case of the minimal supergravity case, but we can have a larger deviations. Such deviation is reasonably close to the experimental value, and will affect the CKM phenomenology at a certain level. In the MSSM with new CPV phases, there is an intriguing possibility that the observed CP violation in $K_L \rightarrow \pi\pi$ is fully due to the complex parameters $\mu$ and $A_t$ in the soft SUSY breaking terms which also break CP softly. This possibility was recently considered by Demir [*et al.*]{} [@demir]. Their claim was that it was possible to generate $\epsilon_K$ entirely from SUSY CPV phases for large $\tan\beta \approx 60$ with certain choice of soft parameters [^4]. In such a scenario, only ${\rm Im}~(A_S^C)$ in Eq. (6) can contribute to $\epsilon_K$, if we ignore a possible mixing between $C_2$ and $C_3$ under QCD renormalization. In actual numerical analysis we have included this effect using the results in Ref. [@contino]. We repeated their calculations using the same set of parameters, but could not confirm their claim. For $\delta_{KM} = 0^{\circ}$, we found that the supersymmetric $\epsilon_K$ is less than $\sim 2\times 10^{-5}$, which is too small compared to the observed value : $ | \epsilon_K | = (2.280 \pm 0.019) \times 10^{-3}$ determined from $K_{L,S} \rightarrow \pi^+ \pi^-$ [@pdg98]. Let us give a simple estimate for fully supersymmetric $\epsilon_K$, in which case only $C_3 ( \mu_0 )$ develops imaginary part and can contribute to $\epsilon_K$. For $m_{\tilde{t}_1} \sim m_{\chi^{\pm}} \sim M_W$, we would get $Y_2 \sim Y_2 (1,1,1,1) = 1/6$, and $$| G^{(3,k) i} | \lesssim O(1), ~~~{\rm and}~~~ | H^{(3,k) i} | \sim {m_s \tan\beta \over M_W},$$ because any components of unitary matrices $C_R$ and $S_t$ are $\lesssim O(1)$. Therefore ${\rm Im} ( A_S^C ) \lesssim O( 10^{-3} )$. Now using $${\rm Im} (M_{12}) = -{G_F^2 M_W^2 \over (2 \pi )^2} %~( V_{td}^* V_{ts} )^2 f_K^2 M_K \left( {M_K \over m_s } \right)^2~{1\over 24}~B_3(\mu)~{\rm Im} (C_3(\mu)),$$ and Eq. (9), we get $|\epsilon_K | \lesssim 2 \times 10^{-5}$. [**7.**]{} In conclusion, we extended our previous studies of SUSY CPV phases to $B\rightarrow X_s l^+ l^-$ and $\epsilon_K$. Our results can be summarized as follows : - The branching ratio for $B\rightarrow X_s l^+ l^-$ can be enhanced upto $\sim 85 \%$ compared to the SM prediction, and the correlation between ${\rm Br} (B\rightarrow X_s \gamma)$ and ${\rm Br} (B\rightarrow X_s l^+ l^-)$ is distinctly different from the minimal supergravity scenario (CMSSM) (even with new CP violating phases) [@okada] in the presence of new CP violating phases in $C_{7,8,9}$ as demonstared in model-independent analysis by Kim, Ko and Lee [@kkl]. - $\epsilon_K / \epsilon_{K}^{SM}$ can be as large as 1.4 for $\delta_{KM} = 90^{\circ} $. This is the extent to which the new phases in $\mu$ and $A_t$ can affect the construction of the unitarity triangle through $\epsilon_K$. - Fully supersymmetric CP violation is not possible even for large $\tan\beta \sim 60$ and light enough chargino and stop, contrary to the claim made in Ref. [@demir]. With real CKM matrix elements, we get very small $|\epsilon_K| \lesssim O(10^{-5})$, which is two orders of magnitude smaller than the experimental value. Before closing this paper, we’d like to emphasize that all of our results are based on the assumption that there are no new CPV phases in the flavor changing sector. Once this assumption is relaxed, then gluino-mediated FCNC with additional new CPV phases may play important roles, and many of our results may change [@kkl]. Within our assumption, the results presented here and in Ref. [@baek] are conservative since we did not impose any conditions on the soft SUSY breaking terms except that the resulting mass spectra for chargino, stop and other sparticles satisfy the current lower bounds from LEP and Tevatron. More detailed analysis of phenomenological implications of our works on $B_{d(s)}^0 - \overline{B_{d(s)}^0}$ mixing, $B\rightarrow X_{s(d)} \gamma, X_{s(d)} l^+ l^-, B_{s(d)}^0 \rightarrow l^+ l^-$ and their direct CP asymmetries will be presented elsewhere. The authors wich to thank G.C. Cho for clarifying $O_2$ and $O_3$ in Ref.  [@demir], and A. Ali, A. Grant, A. Pilaftsis and O. Vives for useful communications. A part of this work was done while one of the authors (PK) was visiting Harvard University under the Distinguished Scholar Exchange Program of Korea Research Foundation. This work is supported in part by KOSEF Contract No. 971-0201-002-2, KOSEF through Center for Theoretical Physics at Seoul National University, Korea Rsearch Foundation Program 1998-015-D00054 (PK), and by KOSEF Postdoctoral Fellowship Program (SB). See, for example, S.M. Barr and W.J. Marciano, in [*CP Violation*]{}, edited by C. Jarlskog (World Scientific, Singapore, 1989), p. 455 ; W. Bernreuther and M. 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Kizukuri and N. Oshimo, Phys. Lett. [**B 337**]{}, 316 (1994) ; Nucl. Phys. [**B 449**]{}, 483 (1995). R. Contino and I. Scimemi, hep-ph/9809437. T. Goto, T. Nihei and Y. Okada, Phys. Rev. [**D 53**]{}, 5233 (1996). Particle Data Group, Eur. Phys. J. [**C 3**]{}, 1 (1998). =9.3cm =9.3cm =9.3cm =9.3cm [^1]: swbaek@muon.kaist.ac.kr [^2]: pko@muon.kaist.ac.kr [^3]: This may be too large for perturbation theory to be valid, but we did extend to $\tan\beta \sim 70$ in order to check the claims made in Ref. [@demir]. [^4]: Their choice of parameters leads to $M_{\chi^{\pm}} = 80$ GeV and $M_{\tilde{t}} = 85$ GeV, which are very close to the recent lower limits set by LEP2 experiments.

--- abstract: 'We study excess noise in a quantum conductor in the presence of constant voltage and alternating external field. Due to a two particle interference effect caused by Fermi correlations the noise is sensitive to the phase of the time dependent transmission amplitude. We compute spectral density and show that at $T=0$ the noise has singular dependence on the [dc]{} voltage $V$ and the [ac]{} frequency $\Omega$ with cusplike singularities at integer $eV/\hbar\Omega$. For a metallic loop with an alternating flux the phase sensitivity leads to an oscillating dependence of the strengths of the cusps on the flux amplitude.' address: - | Universit[" a]{}t zu K[" o]{}ln,\ Institut f[" u]{}r Theoretische Physik,\ Z[" u]{}lpicher Str. 77, D-5000 K[" o]{}ln 41, Germany - | Massachusetts Institute of Technology,\ 12-112, Department of Physics,\ 77 Massachusetts Ave., Cambridge, MA 02139 author: - 'G.B. Lesovik[@*]' - 'L.S.Levitov[@**]' title: | Noise in an [ac]{} biased junction.\ Non-stationary Aharonov-Bohm effect. --- There is a variety of phenomena related with the quantum coherence of transport in small conductors[@1]: weak localization, Aharonov-Bohm effect with the flux quantum $hc/2e$, universal conductance fluctuations, etc. Each of these effects can also be seen in the spectrum of noise, equilibrium or non-equilibrium. The equilibrium noise is simply proportional to conductance according to the fluctuation-dissipation theorem. The non-equilibrium noise in coherent conductors is expressed through eigenvalues of the scattering matrix[@2; @3; @4], and, therefore, is also related with the conductance, though in a less trivial way. For that reason all the coherence phenomena are present in the non-equilibrium noise as well. However, for a better understanding of transport in small conductors it is interesting to analyze the converse line of thinking and to look for coherence effects that are present in the noise but are absent in the conductance. Such effects, if they exist, are genuinely many-particle[@5], otherwise they would show up in the conductance. As long as we are talking about non-interacting fermions it is only statistics that can produce such coherence. A purpose of this letter is to describe an effect caused by two-particle statistical correlations that leads to phase sensitivity of a two-particle observable, i.e., of electric noise, but does not affect one-particle observables, e.g., conductance. The phase sensitivity manifests itself in an oscillating dependence on the amplitude of an [ac]{} flux, in many aspects similar to the A-B effect. However, it will occur in a single-connected conducting loop, i.e., in the geometry where the normal A-B effect is absent. Let us specify which coherence effects we are going to study. In simple words, when an electron is scattered inside a conductor its wavepacket splits into two portions, forward and backward, presenting a choice to the electron to be either transmitted or reflected with the probabilities $D$ and $1-D$. Part of this picture of the wavepacket splitting, involving the relation of $D$ with the conductance[@6; @7] and of $D(1-D)$ with the noise[@2; @3; @4; @8] is well understood. However, there is another part, quite unusual, related with the behavior of current fluctuations in the time domain. Recently, we studied the distribution of the charge transmitted through a resistor during fixed interval of time[@9]. We found that the distribution is very close to the binomial, which means that the attempts to have electron transmitted are highly correlated in time. (Were the sequence of the attempts perfectly periodic the distribution would be exactly binomial.) The origin of the correlation is the Pauli principle that forbids passing of electrons through the resistor simultaneously. The attempts follow almost periodically, spaced by the interval $h/eV$. Because the periodicity is not perfect it does not affect the average current, but shows up in its second moment, i.e., noise, leading[@10] at zero temperature to a sharp edge of the spectral density of excess noise $S_\omega$ near $\omega_0= eV/\hbar$: $S_\omega = 2{e^2\over\pi }D(1-D)\hbar (\omega_0 - |\omega |)$ for $|\omega | < \omega_0$, $0$ otherwise. (Excess noise is the difference of the actual noise and the equilibrium $S_\omega=2{e^2\over\pi }D\hbar\omega\ {\rm coth}(\hbar\omega/2T)$.) The corresponding current-current correlation in time is ${\langle}{\langle}j(t)j(t+{\tau}){\rangle}{\rangle}= {2e^2 \over {\pi}^2}D(1-D) {\rm sin}^2({\omega}_0{\tau}/2)/{\tau}^2$, oscillating with the period $2\pi/\omega_0$ and decaying. Having realized that the frequency $eV/\hbar$ is characteristic for the time correlation of the attempts one has to think of a simple experimental situation were the presence of this frequency could be studied. It is natural to consider a system driven both [dc]{} and [ac]{}, and to look for the effects of commensurability of $\Omega$ and $eV/\hbar$, where $\Omega$ is the frequency of the [ac]{} bias and $V$ is the [dc]{} voltage. In this letter we study such a system and demonstrate that due to the [ac]{} bias the singularity at $\omega =\omega_0$ can be shifted down to zero frequency thus making it easier to observe. Below, we compute the noise in a model resistor in the presence of combined [dc]{}-[ac]{} bias and find that the low frequency noise power $S_0$ has singularities at $eV=n\hbar\Omega$, when the “internal” frequency of the problem $eV/\hbar$ is a multiple of the external frequency $\Omega$. We find that $\partial S_0/\partial V$ is a stepwise function of $V$ that rises in positive steps at $V_n=n\hbar\Omega/e$. Another interesting observation is that the heights of the steps of $\partial S_0/\partial V$ are [*phase sensitive*]{}, i.e., they depend on the phase of the transmission amplitude in an oscillating way resembling A-B effect. The phase sensitivity of the noise should be opposed to the pure [dc]{} situation where only the probabilities of transmission and reflection enter the expression for the noise, which makes the noise power insensitive to the phase picked by the wavefunction across the system. In the simplest situation when the [ac]{} bias is supplied by alternating flux threading the current loop, $\Phi (t)=\Phi_a\sin (\Omega t)$, the heights of the steps in $\partial S_0/\partial V$ are proportional to the squares of the Bessel functions $J_n^2(2\pi\Phi_a/\Phi_0)$, where $\Phi_0=h c/e$. Let us note that we are not talking about the trivial effect of the e.m.f. $-\partial\Phi /c \partial t$ induced in the circuit by the alternating flux. The effect in the noise will persist in the quasistatic limit $|\partial\Phi/c\partial t|\ll V$ when the [ac]{} component of the current vanishes. Let us start with recalling general facts about scattering off an oscillating potential. We consider a model one dimensional system where electrons are scattered by alternating scalar and vector potentials $U(x,t)$, $A(x,t)$ localized in the interval $[-d,d]$, $U(x,t)=A(x,t)=0$ for $|x|>d$. As a function of time they are periodic: $U(x,t)=\sum_{m=-\infty}^{\infty} U_m(x)\exp(-im\Omega t)$, where $U_0(x)$ is the static part of the potential, and the other harmonics $U_m(x)$, $m\ne0$ describe [ac]{} bias. (Expression for $A(x,t)$ is similar.) The [dc]{} bias is expressed in the framework of the Landauer model as the difference of the population of the right and the left scattering states. An important difference is that in our case the states describe [*inelastic*]{} scattering because an electron can gain several quanta $\hbar \Omega$ while passing through the region $[-d,d]$. It will be useful to have the states expressed through the amplitudes of transmission and reflection: $${\psi}_{L,k}(x,t)=\left\{ \begin{array}{ll} e^{-iEt+ikx} +\sum\limits_n B_{L,n}e^{-iE_nt-ik_nx} & x<-d \\ \sum\limits_n A_{L,n}e^{-iE_nt+ik_nx} & x>d \end{array} \right. ,$$ $${\psi}_{R,k}(x,t)=\left\{ \begin{array}{ll} \sum\limits_n A_{R,n}e^{-iE_nt-ik_nx} & x<-d \\ e^{-iEt-ikx}+\sum\limits_n B_{R,n}e^{-iE_nt+ik_nx} & x>d \end{array} \right. ,$$ where the amplitudes $A_{L(R),n},B_{L(R),n}$ are time independent. Here $E=\hbar^2k^2/2m$, $E_n=E+n\hbar\Omega$, and $k_n$ are defined by $\hbar^2k_n^2=2mE_n$. The states (1) are solutions of the Schr[ö]{}dinger equation\ $$E\psi (x,t)=[{1\over 2}(-i{\partial\over\partial x }-{e\over c}A(x,t))^2+U(x,t)]\psi (x,t)\ .$$ They can be used as a basis to study transport through the system the same way it is done for the static barrier. The amplitudes $A_{L(R),n}, B_{L(R),n}$ satisfy unitarity relation, $$\sum_{n,n',\alpha,\alpha'} \delta (E_n-E'_{n'}) {\big (} \bar A_{\alpha,n}(E) A_{\alpha',n'}(E')+ \bar B_{\alpha,n}(E) B_{\alpha',n'}(E'){\big )}= \delta(E-E') \delta _{\alpha \alpha'}$$ that one obtains by the standard reasoning about conservation of current. The operator of electric current, $\hat j(x,t)=-ie\hat \psi^+(x,t)\nabla\hat \psi(x,t)$ is written in terms of second-quantized electrons, ${\hat {\psi}}(x,t)= {\hat {\psi}}_L(x,t)+{\hat {\psi}}_R(x,t)$, $\hat \psi_L(x,t)= \sum_k \psi_{L,k}(x,t) {\hat a}_k$, $\hat \psi_R(x,t)= \sum_k \psi_{R,k}(x,t) {\hat b}_k$, where $a_k$ and $b_k$ are canonical Fermi operators corresponding to the states (1) coming out of the reservoirs, the left and the right respectively. It is straightforward to compute the mean value $I(t)=\langle\hat j(x,t)\rangle$, where the brackets $\langle...\rangle$ stand for averaging with the density matrix $\rho$ of the reservoirs. As usual, we assume absence of correlations in the reservoirs, $\hat \rho =\hat \rho_L \otimes \hat \rho_R$, which physically means that after having been scattered into a reservoir electrons have enough time to relax to the equilibrium before they return. Below we assume equilibrium Fermi distributions $\rho_{L,R}= n(E-E_F\pm eV/2)$. One obtains a generalized Landauer formula: $$I(t)=\sum_{m=-\infty}^{\infty} I_m\exp(-im\Omega t)\ ,$$ where it is straightforward to write the coefficients $I_m$ in terms of the scattering amplitudes $A_{L(R),n}$ and $B_{L(R),n}$. Expr.(2) describes both steady current and generation of harmonics in the presence of the [ac]{} bias. Quite similarly one can obtain an expression through $A_{L(R),n}$, $B_{L(R),n}$ for the noise. Noise is related with the correlation function $S(t_1,t_2)=\langle\!\langle [\hat j(x,t_1), \hat j(x,t_2)]_+ \rangle\!\rangle$. In the usual [dc]{} situation $S(t_1,t_2)=S(t_1-t_2)$ and its Fourier transform gives spectral density $S_\omega=\langle\!\langle j_\omega j_{-\omega}\rangle\!\rangle$ of the noise. In our [ac]{} case the situation is somewhat more complex because the spectral density $S_\omega$ does not provide a complete description of the noise. Indeed, $S(t_1,t_2)$ will depend now separately on $t_1$ and $t_2$, not only on the difference $t_1-t_2$. However, it satisfies $S(t_1,t_2)= S(t_1+2\pi/\Omega, t_2+ 2\pi/\Omega)$ resulting from the periodicity of the [ac]{} bias. For the Fourier components $\hat j_\omega$ it means that the average $\langle\!\langle j_\omega j_{\omega'} \rangle \! \rangle$ does not vanish whenever $\omega+\omega'=m\Omega$, where $m$ is any integer. Thus, in addition one gets generalized spectral densities $S_{\omega,m}=\langle\!\langle j_\omega j_{m \Omega -\omega} \rangle \! \rangle$, an integer parameter family of functions. Among them there is the ’ordinary’ $m=0$ spectral density $S_\omega = S_{\omega,0}=\langle\!\langle j_\omega j_{-\omega} \rangle \! \rangle$, the one easiest to access experimentally. In what follows we concentrate on it and do not study other $S_{\omega,m}$, $m\ne 0$. To compute the noise one has to average the product of two current operators over the distribution in the reservoirs. Evaluation of the average is similar to Refs.[@2; @3; @4], so we do not need to repeat it here. General expression simplifies quite substantially in the practically interesting limit of $t_f$, the time of flight through the barrier $U(x,t)$ being much shorter than $2\pi/\Omega$ and $\hbar/eV$. The point is that $\hbar /t_f$ defines the characteristic scale of energy dependence of the scattering amplitudes, so the condition $t_f\Omega \ll 1$, $t_feV \ll \hbar$ enables one to neglect the energy dependence of $A_{L(R),n}$, $B_{L(R),n}$ in the interesting energy domain $E_F \pm {\rm max}[eV, \hbar \Omega ]$. We also assume $E_F \gg {\rm max}[eV, \hbar \Omega]$, which allows to neglect the difference of $k_n/m$ and $k/m$, the velocities of scattered and incident states, and set $k_n/m=v_F$. It should be remarked that the physical picture we discuss below is not really dependent on any of these assumptions, they only make our expressions more compact. The more general case of arbitrary relation between $\hbar /t_f$, $E_F$, $eV$, $\hbar \Omega$ presents no difficulty. With the above assumptions made it becomes convenient to use Fourier transform of the amplitudes $A$ and $B$. Let us define $A_\alpha (t)=\sum_n A_{\alpha,n} \exp ( -i n \Omega t)$, $\alpha=L,R$. Similarly we introduce $B_\alpha (t)$, and rewrite Expr.(1) as $${\psi}_{L,k}(x,t)=\left\{ \begin{array}{ll} e^{ikx}+B_L(t+x/v_F)e^{-ikx} & x<-d \\ A_L(t-x/v_F)e^{ikx} & x>d \end{array} \right. ,$$ $${\psi}_{R,k}(x,t)=\left\{ \begin{array}{ll} A_R(t+x/v_F)e^{-ikx} & x<-d \\ e^{-ikx}+B_R(t-x/v_F)e^{ikx} & x>d \end{array}\right. .$$ (To obtain (3) from (1) we substitute $k_n=k+n\Omega /v_F$ in the phase shifts $e^{ik_nx}$ and then do the sum over $n$.) The amplitudes $A_{L(R)}(t)$, $B_{L(R)}(t)$ have clear meaning of the transmission and reflection amplitudes at given instant of time for a slowly varying potential. The retarded time $t-|x|/v_F$ in Expr.(3) accounts for the finite speed of motion after scattering. The unitarity relation now takes the form $$|A_{L(R)}(t)|^2+|B_{L(R)}(t)|^2=1,\ \bar A_L(t)B_R(t)+\bar B_L(t) A_R(t)=0\ .$$ To clarify the character of the simplification thus achieved let us remark that with Expr.(3) the formula (2) for the current $I(t)$ becomes just $I(t)=2{e^2\over h}|A(t)|^2eV$ which means that the current ’adiabatically’ follows time variation of the transparency of the barrier according to the Landauer formula. Now we shall compute noise and find that, unlike $I(t)$, it is not reduced to anything trivially related with the static limit. Let us write the average of two currents $\langle\!\langle\hat j(t_1)\hat j(t_2)\rangle\!\rangle=$ $${2e^2\over h^2}\sum_{E,E'}e^{-i(E_k-E_{k'})(t_1-t_2)} {\big [} |A(t_1)A(t_2)|^2{\big (}n_L(E')(1-n_L(E))+ n_R(E')(1-n_R(E)){\big )}$$ $$+\bar B(t_1)A(t_1) \bar A(t_2)B(t_2) n_R(E') (1-n_L(E))+ \bar A(t_1)B(t_1) \bar B(t_2)A(t_2) n_L(E') (1-n_R(E)){\big ]} .$$ To compute $S_\omega$ we have to do Fourier transform and substitute Fermi distributions $n_{L(R)}(E)$. Explicit calculation yields the result $$S_{\omega}= {2e^2\over \pi} \sum\limits_n 2 N_0(\omega-n\Omega) |(|A|^2)_n|^2 + N_1(\omega ,n\Omega+eV) |(A\bar B)_n|^2,$$ where $$N_0(x)=\int (n(E-x)+n(E+x))(1-n(E))dE= x{\rm coth}(x/2T)\ ,$$ $$N_1(x,y)=N_0(x+y)+N_0(x-y)={x \sinh (x/2T) - y \sinh (y/2T)\over \cosh (x/2T) - \cosh (y/2T)}\ ,$$ and $(...)_n$ denotes Fourier components, e.g., $(A\bar B)_n={\Omega \over 2\pi}\int A(t)\bar B(t) e^{in\Omega t}dt$. Expr.(4) describes the noise as function of $eV,\ \Omega,\ \omega$ and $T$. The behavior is simplest at $T=0$ when $N_0(x)=|x|$, $N_1(x,y)=|x+y|+|x-y|$. Given by Expr.(4) as a weighted sum of terms like $|n\Omega + eV \pm\omega|$, $|\omega-n\Omega|$ the noise $S_{\omega}$ will then depend on $V,\ \Omega,\ \omega$ in a piecewise linear way, changing from one slope to another when $n\Omega+eV\pm\omega$ or $\omega-n\Omega$ equals $0$. This condition defines the locations where $S_{\omega}$ has singularities. They are cusps, sharp at $T=0$ and rounded on the scale $T$ at $T>0$. With the general Expr.(4) one can explore the noise in all possible limiting situations that one obtains for different combinations of $eV,\ \Omega,\ \omega$ and $T$. Particularly interesting for us will be the case $T=0$, $\omega =0$ corresponding to the noise $S_0 =\langle\!\langle j_\omega j_{-\omega}\rangle\! \rangle_{\omega \rightarrow 0}$ measured at low frequency. Let us remark here that setting $\omega =0$ means only that $\omega$ is small compared to the parameters $eV$ and $\Omega$ that define the width of the frequency band of the excess noise. Such $\omega$ may still be much higher than the band width for other sources of noise, e.g., the $1/f$. Let us concentrate on the dependence of $S_0$ on $V$. It is a piecewise linear function which is easiest to characterize by its derivative, $$\partial S_0/\partial V={2e^3\over \pi }\sum\limits_n \lambda_n \theta (eV-n\hbar\Omega),$$ where $\lambda_n=|(\bar A_L B_R)_n|^2$ and $\theta (x)=1$ for $x>0$, $-1$ otherwise. The function $\partial S_0/\partial V$ rises in positive steps at all $V_n=\hbar \Omega n/e$ (see Fig. \[figure1\]), the property that can be alternatively formulated as convexity of $S_0(V)$ in $V$. The meaning of the singularities in $S_0(V)$ was clarified recently in a study of the statistics of transmitted charge [@IL]. The generating function of the charge distribution was expressed through the single-particle scattering matrix, and it was found that the distribution arises from Bernoulli statistics (i.e., it is a generalized binomial distribution). The frequencies of attempts were given as function of $V$ and $\Omega$. The probabilities of outcomes of a single attempt were found in terms of many-particle scattering amplitudes, and it was shown that they change at the thresholds $V_n=n\hbar\Omega/e$ in a discontinuous way due to statistical correlation in the outgoing channels of the scattering. The discontinuity manifests itself in the second moment of the distribution that corresponds to the noise $S_0(V)$ discussed above. There is an interesting and simple example where one can explicitly evaluate the heights of the steps. Let us consider a junction with ideal leads bent into a loop of length $L$ (see inset of Fig. \[figure1\]) and placed into an external magnetic field varying with time. In this problem the junction is the only source of scattering. For simplicity let us assume that only one scattering channel is involved and that the junction is symmetric, $A_L=A_R=A$, $B_L=B_R=B$. The [ac]{} bias is supplied by the alternating flux of the magnetic field through the loop, $\Phi (t)=\Phi_a\sin (\Omega t)$. Also let us suppose that the magnetic field is quasistatic, i.e., the time of flight through the system, $t_f=L/v_F$ is much shorter than $2\pi/\Omega$, that makes it possible to introduce the time dependent amplitudes $A_{L(R)}(t),\ B_{L(R)}(t)$ as it was discussed above. As is common, in such a situation the vector potential can be treated semiclassically, and one can write the wavefunction as $\psi (x,t)= \exp ({ie\over\hbar c}\int_{-\infty}^{x} A(x')dx') \psi_0 (x,t)$, where $x$ is the coordinate along the lead and $\psi_0 (x,t)$ is found by solving the Schr[ö]{}dinger equation in the absence of the magnetic field. Thus all the dependence on the magnetic flux can be accumulated in the phase of the transmission amplitude, $$A_{R(L)}(t)=\exp (\pm i\Phi (t)/\Phi_0) A$, $B_{R(L)}(t)=B,$$ where $\Phi_0=h c/e$ is single electron flux quantum. Since $|A(t)|^2=D=const$ the current is time independent: $I={2e^2\over h}DV$. According to Expr.(4) $S_{\omega}$ is written through the Fourier components of $\bar A_L(t) B_R(t)$ in this case given by the Bessel functions: $(\bar A_L B_R)_n=J_n(2\pi\Phi_a/\Phi_0)AB$. Thus we find $$S_{\omega}={2e^2\over \pi} {\big [} 2N_0(\omega )D^2+ \sum\limits_n N_1(\omega ,n\Omega+eV) D(1-D)J_n^2(2\pi\Phi_a/\Phi_0){\big ]}.$$ The heights $\lambda_n$ of the steps in $\partial S_0/\partial V$ are then given by $$\lambda_n= D(1-D) J_n^2(2\pi\Phi_a/\Phi_0).$$ They oscillate as function of $\Phi_a / \Phi_0$ and vanish at the nodes of Bessel functions. Exprs.(6),(7) illustrate one important feature of the noise in the [ac]{} biased system, the sensitivity to the [*phase*]{} of the transmission amplitude $A$. By varying the amplitude $\Phi_a$ of the alternating flux one can make $\lambda_n$ vanish separately for each harmonic $n\Omega$ of the [ac]{} frequency. This should be compared with the case of the [dc]{} bias where the noise is expressed only through $|A|^2$ and thus cannot be phase dependent. We call the oscillating dependence (6),(7) [*non-stationary Aharonov-Bohm effect*]{}. To compare it with the usual A-B effect let us recall that the latter is observed as an oscillation of the [dc]{} conductance under variation of flux in the situation when one has interference of transmission amplitudes corresponding to different classical trajectories of a quantum system, e.g., in a conductor with multiply connected leads forming one or several closed loops. The [dc]{} A-B effect cannot be observed in the single path geometry like Fig.1. Alternatively, the non-stationary A-B effect appears as a result of interference of the right and left scattering states travelling in the opposite directions along same path and having energies shifted by $n\Omega$. It is clear from our discussion that such interference does not contribute to the [ac]{} conductance but is important for the noise and, therefore, one obtains the non-stationary A-B effect in the noise even in the topologically trivial situation of Fig.1. One can derive a sum rule: $$\sum\limits_n \lambda_n={\Omega\over 2\pi} \int_0^{2\pi/\Omega} D(t)(1-D(t))dt \ ,$$ where $D(t)=|A(t)|^2$. For $\lambda_n$ given by Expr.(7) it follows from the definition of the Bessel functions. In the general case of Expr.(5) the sum rule is obtained by applying Plancherel’s formula to Fourier components of $A(t)\bar B(t)$. The sum rule clarifies the relation of our problem with the previous calculation[@2; @3; @4] of the noise in the pure [dc]{} case for which the result does not depend on the phase of $A(t)$. When the limit is taken $\Omega\rightarrow 0$, $V=const$, the steps in $\partial S_0/\partial V$ do not vanish but just move closer to zero, thus effectively condensing then all together in a single step at $V=0$. The height of this step is not phase sensitive and is simply given by the expression (8) for the [dc]{} noise averaged over the period $2\pi/\Omega$. It is worth mentioning that our results for $S_\omega$ are quite general. Indeed, it is clear after what have been said that the singularities at $V=n\hbar\Omega/e$ are only due to the sharp edge of the Fermi distribution, and not related with any specific geometry assumed for the junction. Because of that the phenomenon should be displayed by any coherent conductor, provided that the main source of inelastic scattering is the [ac]{} potential. The reason is that an elastic scattering, if any, can smear the Fermi distribution of momenta but it will not affect the sharpnes of the step in the energies distribution, and our effect is sensitive only to the latter. The same remark applies to the oscillating dependence of the singularities on the amplitude of the [ac]{} signal. Let us briefly discuss a generalization of the system shown in Fig.1 where the loop is not an ideal lead but a real metallic wire with disorder, i.e., instead of one scatterer there are now many of them uniformly distributed over the bulk of the wire. Most interesting is the case of a purely coherent conductor for which the energy relaxation time $\tau_E$ and the phase breaking time $\tau_\phi$ are much longer than the flight time $t_f$. (One can estimate $t_f\approx \hbar/E_c$, where $E_c$ is Thouless’ energy $\hbar D/L^2$.) In such a system transport is described by channels of the scattering matrix with transmission coefficients $T_m$ assigned to each channel[@7]. In the [dc]{} case the noise can be written[@4] in terms of $T_m$ as $S_0={2e^2\over \pi} \sum_m T_m(1-T_m)eV$. In the presence of the alternating flux the extension of our formalism can be carried out easily and one obtains expressions similar to (6) and (7), with $D^2$ and $D(1-D)$ replaced by $\sum_m T_m^2$ and $\sum_m T_m (1-T_m)$ respectively. However, the limitations under which the result is valid, $eV\ll\hbar/t_f$, $\Omega\ll 1/t_f$, are now slightly more stringent than for Exprs.(6),(7) because the flight time $t_f$ is longer. A more fundamental limitation to the general validity of our calculation is in the assumption that the flux threads only the phase coherent part of the conductor. It would certainly be of interest to better understand the opposite limit when the [ac]{} voltage increases smoothly over a distance much larger than the phase breaking length $L_\phi=\sqrt {\tau_\phi/D}$. To summarize, we studied current and noise in a conductor driven by [dc]{} and [ac]{} and we expressed them through time-dependent one particle scattering amplitudes. In the quasistatic limit of short time of flight through the conductor the current is given by the Landauer formula with time-dependent transmission coefficient, i.e., by a trivial generalization of the static case. The situation with the noise is quite different because of the two-particle interference. Spectral density of the noise $S_\omega$ depends on the scattering amplitudes in such a way that the phases do not drop out, and this leads to a non-stationary Aharonov-Bohm effect. Because of the way the Fermi statistics affects the two-particle interference the noise measured at $T=0$ is singular at $\omega =\pm eV/\hbar +m\Omega$, where $m$ is any integer. To illustrate the phase sensitivity of the noise we consider a conducting metallic loop in which the [ac]{} signal is supplied by an oscillating magnetic flux. Because of the sensitivity to the phase of transmission amplitude the strengths of the singularities in the noise display oscillatory dependence on the amplitude of the [ac]{} flux given by squares of the Bessel functions. We are indebted to D. E. Khmelnitskii for drawing our attention to the problem of harmonic generation in coherent conductors, and to J. Hajdu for illuminating discussions.\ Research of L. L. is partly supported by Alfred Sloan fellowship. The work of G. L. is performed within the research program of the Sonderforshungsbereich 341, Köln-Aachen-Jülich. also the Institute for Solid State Physics, Chernogolovka 142432, Moscow district, Russia also at the L. D. Landau Institute for Theoretical Physics, Moscow 117334, Russia P. A. Lee, T. V. Ramakrishnan, Rev. Mod. Phys. [**57**]{}, p.287 (1985),\ B. L. Altshuler, P. A. Lee, Phys. Today (December), p.2 (1988) G. B. Lesovik, JETP Letters, [**49**]{}, p.594 (1989) B. Yurke and G. P. Kochanski, Phys. Rev. [**41**]{}, p.8184 (1989) M. B[ü]{}ttiker, Phys. Rev. Lett., [**65**]{}, p.2901 (1990),\ M. B[ü]{}ttiker, in: Granular Nanoelectronics, D. K. Ferry (ed.), Plenum Press, New York (1991) M. Büttiker, Phys. Rev. Lett., [**68**]{}, p.843 (1992) R. Landauer, in: Localization, Interaction and Transport Phenomena, eds. B. Kramer, G. Bergmann and Y. Bruynsraede (Springer, Heidelberg, 1985) Vol.[**61**]{}, p.38;\ see also a review by R. Landauer in: W. van Haeringen and D. Lenstra (eds.), Analogies in Optics and Micro Electronics, pp.243-257, Kluwer Academic Publishers (1990) M. B[ü]{}ttiker, Phys. Rev. Lett., [**57**]{}, p.1761 (1986) Th. Martin and R. Landauer, Phys. Rev. [**B45**]{}, 1742 (1992) L. S. Levitov, G. B. Lesovik, JETP Letters [**58**]{} (3), p.230, (1993) S.-R. E. Yang, Solid State Commun. [**81**]{}, p. 375 (1992) D. Ivanov, L. S. Levitov, JETP Letters [**58**]{}(6), p.461 (1993)

--- abstract: 'We study the problem of recovering a structured signal $\mathbf{x}_0$ from high-dimensional data $\y_i=f(\mathbf{a}_i^T\mathbf{x}_0)$ for some nonlinear (and potentially unknown) link function $f$, when the regressors $\ab_i$ are iid Gaussian. Brillinger (1982) showed that ordinary least-squares estimates $\x_0$ up to a constant of proportionality $\mu_\ell$, which depends on $f$. Recently, Plan & Vershynin (2015) extended this result to the high-dimensional setting deriving sharp error bounds for the generalized Lasso. Unfortunately, both least-squares and the Lasso fail to recover $\mathbf{x}_0$ when $\mu_\ell=0$. For example, this includes all even link functions. We resolve this issue by proposing and analyzing an alternative convex recovery method. In a nutshell, our method treats such link functions as if they were linear in a lifted space of higher-dimension. Interestingly, our error analysis captures the effect of both the nonlinearity and the problem’s geometry in a few simple summary parameters.' author: - Christos Thrampoulidis - Ankit Singh Rawat bibliography: - 'compbib.bib' title: | Lifting high-dimensional nonlinear models\ with Gaussian regressors ---

--- abstract: 'Cold dark matter particles with an intrinsic matter-antimatter asymmetry do not annihilate after gravitational capture by the Sun and can affect its interior structure. The rate of capture is exponentially enhanced when such particles have self-interactions of the right order to explain structure formation on galactic scales. A ‘dark baryon’ of mass 5 GeV is a natural candidate and has the required relic abundance if its asymmetry is similar to that of ordinary baryons. We show that such particles can solve the ‘solar composition problem’. The predicted small decrease in the low energy neutrino fluxes may be measurable by the Borexino and SNO+ experiments.' author: - 'Mads T. Frandsen' - Subir Sarkar title: Asymmetric dark matter and the Sun --- We consider the capture by the Sun of asymmetric dark matter (ADM) particles which have a relic asymmetry just as do baryons, in contrast to the usual candidates for cold dark matter (CDM) such as supersymmetric neutralinos which have a relic thermal abundance determined by ‘freeze-out’ from chemical equilibrium. Hence ADM does not annihilate upon capture in astrophysical bodies such as the Sun, leading to a build up of its concentration. In particular, self-interactions can lead to an [*exponential*]{} increase of the ADM abundance in the Sun as it orbits around the Galaxy, accreting dark matter. ADM does not have the usual indirect signatures e.g. there will be no high energy neutrino signal from annihilations in the Sun. Instead ADM will alter heat transport in the solar interior thus affecting the low energy neutrino flux. This had been proposed as a solution to the ‘solar neutrino problem’ [@Spergel:1984re; @Faulkner:1985rm; @Gilliland:1986]. Although the solution is now understood to be neutrino oscillations [@Bahcall:2004mz], small changes induced by accreted CDM particles may account for the current discrepancy [@PenaGaray:2008qe] between helioseismological data and the revised ‘Standard Solar Model’ (SSM). An asymmetry in ADM similar to that in baryons naturally explains why their observed abundances are of the same order of magnitude. If ADM arises from a strongly coupled theory (like the baryon of QCD), then there is a conserved $U(1)$ global symmetry (like $B$ number in QCD) which guarantees stability of the lightest $U(1)$ charged object. Technicolour models of electroweak symmetry breaking [@Weinberg:1979bn] provide an example of ADM from strong dynamics in the form of the lightest neutral technibaryon [@Nussinov:1985xr]. Recently, new viable types of technibaryon dark matter (TIMPs) [@Gudnason:2006ug] as well as other particle candidates for ADM [@Kaplan:1991ah] have been suggested. While gravitational instability in collisionless CDM provides a good explanation for the large-scale structures of the universe, observations on galactic and smaller scales suggest that CDM may be self-interacting [@Spergel:1999mh; @Wandelt:2000ad]. If ADM arises as a composite TIMP or a ‘dark baryon’ from a strongly coupled model, then it would naturally have such self-interactions. We consider how the capture of self-interacting ADM by the Sun can alter helioseismology and low energy neutrino fluxes. Capture of self-interacting ADM =============================== We refer to earlier discussions of the capture by the Sun of heavy Dirac neutrinos having an asymmetry [@Griest:1986yu], and of symmetric CDM with self-interactions [@Zentner:2009is]. The capture rate for CDM particles $\chi$ with [*both*]{} an asymmetry and self-interactions is governed by the equation: $$\label{Nddm} \frac {\mathrm{d}N_\chi} {\mathrm{d}t} = C_\mathrm{\chi N} + C_{\chi\chi} N_\chi.$$ Here $C_\mathrm{\chi \cal{N}}$ is the usual rate of capture of CDM particles by scattering off nuclei (mainly protons) within the Sun, while $C_{\chi\chi}$ is the rate of self-capture through scattering off already captured $\chi$ particles. Hence the number of captured particles would have grown as $$\label{Nddm'} N_\chi (t) = \frac{C_\mathrm{\chi \cal{N}}}{C_{\chi\chi}} \left(\mathrm{e}^{C_{\chi\chi} t} -1 \right),$$ i.e. [*exponentially*]{} for $t \gtrsim C_{\chi\chi}^{-1}$. However the effective cross-section for self-captures cannot increase beyond $\pi r_\chi^2$ where $r_\chi$ is the scale-height of the region where they are gravitationally trapped [@Spergel:1984re]. The linear growth by contrast can continue up to the ‘black disk’ limit i.e. $\pi R_\odot^2$. In both cases there is an additional enhancement due to ‘gravitational focussing’ [@Spergel:1984re; @Gould:1987ju] as we quantify later. The ejection of captured ADM particles by recoil effects in the self-scattering can be neglected [@Zentner:2009is] and evaporation is negligible for a mass exceeding 3.7 GeV [@Gould:1987ju]. The ADM capture rate through spin-independent (SI) and spin-dependent (SD) interactions can be written [@Jungman:1995df]: $$\label{eq-CaptureRate} C_\mathrm{\chi \cal{N}}^\mathrm{SI,SD} = c_\mathrm{\chi \cal{N}}^\mathrm{SI,SD} \left(\frac{\rho_\mathrm{local}}{0.4~\mathrm{GeV\,cm}^{-3}}\right) \sum_i \mathcal{F}_i \left(\frac{\sigma_i^\mathrm{SI,SD}}{10^{-40}\mathrm{cm}^2} \right)$$ where $c_\mathrm{\chi \cal{N}}^\mathrm{SI} = 6.4 \times 10^{24}\mathrm{s}^{-1}$ and $c_\mathrm{\chi \cal{N}}^\mathrm{SD} = 1.7 \times 10^{25}\mathrm{s}^{-1}$, $\rho_\mathrm{local}$ is the estimated local CDM density, and $\mathcal{F}_i (m_\chi)$ encodes the form factors for different nuclei $i$ weighted by the solar chemical composition — the sum is over all nuclei (hydrogen only for SD interactions). Here $\sigma_\mathrm{\chi \cal{N}}^\mathrm{SI,SD}$ is the ADM-[*nucleus*]{} cross-section, which is related to $\sigma_\mathrm{\chi N}$, the ADM-nucleon cross-section [@Jungman:1995df]. For spin-independent interactions, $\sigma_\mathrm{\chi N}$ is constrained by direct detection experiments such as CDMS-II [@Ahmed:2009zw], XENON10 [@Angle:2009xb] and CoGeNT [@Aalseth:2010vx] to be $\lesssim 10^{-39}~\mathrm{cm}^2$ for $m_\chi = 5$ GeV. For spin-dependent interactions the constraints are considerably weaker, e.g. PICASSO [@Archambault:2009sm] sets the strongest bound of $\lesssim 10^{-36}~\mathrm{cm}^2$ for this mass. Next we consider the self-capture rate in the Sun [@Zentner:2009is]: $$\label{selfcapturerate} C_{\chi\chi} = \sqrt{\frac{3}{2}}\ \rho_\mathrm{local}\ s_\chi \ \frac{v_\mathrm{esc}^2(R_\odot)}{\bar{v}}\ \langle\phi\rangle\ \frac{\rm{erf(\eta)}}{\eta}$$ where $s_\chi \equiv \sigma_{\chi\chi}/m_\chi$ is the ADM self-interaction cross section divided by its mass, and $v_\mathrm{esc} (R_\odot) \sim 618~\mathrm{km} \mathrm{s}^{-1}$ is the escape velocity at the surface of the Sun, which is assumed to be moving at $v_\odot = 220~\mathrm{km} \mathrm{s}^{-1}$ through a Maxwell-Bolzmann distribution of CDM particles with velocity dispersion, $\bar{v} \sim 270~\mathrm{km} \mathrm{s}^{-1}$. Here $\langle\phi\rangle \sim 5.1$ is the average over $\phi (r) \equiv v_\mathrm{esc}^2(r)/v_\mathrm{esc}^2(R)$ and $\eta \equiv \sqrt{3/2}v_\odot/\bar{v}$. Self-interacting CDM was proposed [@Spergel:1999mh] to account for observations of galactic and subgalactic structure on scales $\lesssim$ a few Mpc which are not in accord with numerical simulations using collisionless cold particles. The discrepancy can be solved if CDM has a mean free path against self-interactions of $\lambda \sim 1~\mathrm{kpc} - 1~\mathrm{Mpc}$ corresponding to a self-scattering cross-section between $s_\chi \sim 8 \times 10^{-22}$ and $\sim 8 \times 10^{-25}~\mathrm{cm}^2 \mathrm{GeV}^{-1}$ [@Spergel:1999mh]. A detailed analysis sets an upper limit of $s_\chi \lesssim 10^{-23}~\mathrm{cm}^2 \mathrm{GeV}^{-1}$ [@Wandelt:2000ad], while a study [@Randall:2007ph] of the colliding ‘Bullet cluster’ of galaxies implies a stronger bound of $\sim 2 \times 10^{-24}~\mathrm{cm}^2 \mathrm{GeV}^{-1}$, which we adopt for our calculations below. A ‘dark baryon’ from a QCD-like strongly interacting sector but with a mass of about 5 GeV is a natural candidate for ADM. Its relic density is linked to the relic density of baryons via $\Omega_\chi \sim (m_\chi {\cal N}_\chi/m_\mathrm{B} {\cal N}_\mathrm{B})\Omega_B$ where ${\cal N}_{\mathrm{B}, \chi}$ are the respective asymmetries. If ${\cal N}_\mathrm{B} \sim {\cal N}_\chi$ (e.g. if both asymmetries are created by ‘leptogenesis’ [@Davidson:2008bu]) then the required CDM abundance is realised naturally. The self-interaction cross-section of such a neutral particle can be estimated by scaling up the neutron self-scattering cross-section $\sim 10^{-23}$ cm$^2$ [@Gardestig:2009zz] as: $\sigma_{\chi\chi} = (m_\mathrm{n}/m_\chi)^2 \sigma_\mathrm{nn}$ which is just of the required order. Note that the self-annihilation cross-section will be of the same order which ensures that the ADM thermal ([*symmetric*]{}) relic abundance is negligible, just as it is for baryons. Photon exchange, via a magnetic moment of the dark baryon, will give rise to both spin-independent and spin-dependent interactions of $\chi$ with nucleons. Recently this has been investigated in a model of a 5 GeV dark baryon in a ‘hidden sector’ interacting with the photon through mixing with a hidden photon magnetic moment [@An:2010kc]. From this model we infer that spin-independent cross-section with nuclei of ${\cal O}(10^{-39})$ cm$^2$ can be achieved. Moreover this will be accompanied by spin-dependent interactions which would aid further in the heat transport in the Sun as discussed below. Since the photon couples [*only*]{} to the proton in direct detection experiments, the limit on $\sigma_\mathrm{\chi N}$ is degraded for this model to $\sim 4 \times 10^{-39}$ cm$^2$ which we adopt as an example later. Helioseismology and Solar neutrinos =================================== Fig. \[Ncaptmax\] shows the growth of the number of captured ADM particles in ratio to the number of baryons in the Sun, for a scattering cross-section on nucleons as large as is experimentally allowed, including the ‘gravitational focussing’ factor of $(v_\mathrm{esc}(r)/\bar{v})^2$ [@Gould:1987ju] and setting $r=R_\odot$ or $r_\chi$ ($\simeq 0.07 R_\odot$ for $m_\chi = 5$ GeV) as appropriate. [![Growth of the relative abundance of 5 GeV mass ADM particles in the Sun until its present age (vertical line) assuming $s_{\chi\chi} = 2 \times 10^{-24} \mathrm{cm}^2 \mathrm{GeV}^{-1}$, and $\sigma_{\chi N} = 10^{-39}~\mathrm{cm}^2$ (solid line) and $10^{-36}~\mathrm{cm}^2$ (dashed line), these being the maximum experimentally allowed values for spin-independent and spin-dependent interactions respectively. Also shown is the ‘black disk’ limit (dotted line) for the Sun.[]{data-label="Ncaptmax"}](rightcapt2.pdf "fig:"){width="\linewidth"}]{} Note that due to the self-captures, the limiting abundance $N_\chi/N_\odot \sim 2 \times 10^{-11}$ is almost independent of the actual scattering cross-section. Such an ADM fraction in the Sun can affect the thermal conductivity and thereby solar neutrino fluxes [@Faulkner:1985rm; @Spergel:1984re]. The SSM [@Bahcall:2004pz] predicts 3 times the observed neutrino flux (the ‘Solar neutrino problem’) but this is now well understood taking into account neutrino oscillations [@Bahcall:2004mz]. Moreover until recently, the SSM with the ‘standard’ solar composition [@Grevesse:1998bj] agreed very well with helioseismology [@Serenelli:2009ww]. However the revision of the solar composition [@Asplund:2009fu] means that the SSM no longer reproduces the sound speed and density profile so there is now a ‘solar composition problem’ [@PenaGaray:2008qe]. We show that the presence of ADM in the Sun can resolve this problem and precision measurements of solar neutrino fluxes can constrain the properties of self-interacting ADM. A simple scaling argument gives for the luminosity carried by the ADM [@Spergel:1984re]: $$L_\chi \sim 4 \times 10^{12} L_{\odot} \frac{N_\chi}{N_\mathrm{\odot}} \frac{\sigma_\mathrm{\chi N}}{\sigma_\odot}\sqrt{\frac{m_\mathrm{N}}{m_\chi}}\ , \label{lumisimple}$$ where $L_{\odot}\sim 4 \times 10^{33} \rm{ergs}\, \rm{s}^{-1}$. When the ADM mean free path $\lambda_\chi$ is large compared to the scale-height $r_\chi$ then the energy transfer is [*non-local*]{}. This is the case when $\sigma_\mathrm{\chi N} \ll \sigma_{\rm{\odot}}$ where $\sigma_\odot \equiv (m_\mathrm{N}/M_\odot) R_\odot^2 \sim 4 \times 10^{-36}~\mathrm{cm}^2$ is a critical scattering cross-section. We consider the ADM trapped in the Sun as an isothermal gas at temperature $T_\chi$ [@Spergel:1984re], so the luminosity $L_\mathrm{\chi}$ carried by the particles is: $$L_\mathrm{\chi}(r) = \int^{r}_{0} \mathrm{d}r' \; 4 \pi r'\,^2\,\rho(r') \,\epsilon_\chi(r') \ ,$$ where $\epsilon_\chi (r') \propto (T(r') - T_{\chi} ) N_\chi \sigma_\mathrm{\chi N}$ is the energy transferred to the ADM per second per gram of nuclear matter and $\rho(r')$ is the density in the Sun [@Spergel:1984re]. The ADM temperature $T_{\chi}$ is fixed by requiring that the energy absorbed in the inner region ($T(r) > T_{\chi}$) is equal to that released in the outer region ($T(r) < T_{\chi}$), such that $L_{\rm \chi}(R_{\odot}) = 0$. This approximation overestimates the energy transfer by a small factor [@Gould:1989ez; @Dearborn:1990mm] but is sufficiently accurate for the present study. We adopt a simple polytropic model for the Sun’s temperature $T$, number density $n_\mathrm{p}$ and gravitational potential $V$ [@Spergel:1984re]. The resulting variation of the solar luminosity $\delta L(r) \equiv L_\chi(r)/L_{\odot}(r)$ is shown in Fig.\[luminosityfunction\] assuming $\sigma_\mathrm{\chi N} = 4 \times 10^{-39}~\mathrm{cm}^2$ (i.e. $10^{-3} \sigma_\mathrm{\odot}$) and $N_\chi = 2 \times 10^{-11} N_\mathrm{\odot}$ from Fig \[Ncaptmax\]. Note that the luminosity scales linearly with both $\sigma_{\mathrm{\chi N}}$ and $N_\chi/N_\odot$. [![The radial variation of $\delta L(r) \equiv L_\chi(r)/L_{\odot}(r)$ due to ADM of mass 5 GeV, using the approximation of Ref.[@Spergel:1984re] and $L_\odot (r)$ from the BS05 (OP) Standard Solar Model [@Bahcall:2004pz].[]{data-label="luminosityfunction"}](Luminosityfunctionnew2.pdf "fig:"){width="\linewidth"}]{} From the radiative transport equation it follows that a small variation of the solar luminosity is equivalent to an [*opposite*]{} small variation in the effective radiative opacity: $\delta L(r) \sim -\delta\kappa_\gamma(r) \equiv -\kappa_\chi(r)/\kappa_\gamma(r)$ [@Bottino:2002pd]. The effect of such a localised opacity variation in the region $r \lesssim 0.2 R_\odot$ has been studied by a Monte Carlo simulation [@Fiorentini:2001et] and results in excellent agreement obtained using a linear approximation to the solar structure equations [@Villante:2009xs]. Fig. \[luminosityfunction\] shows that the opacity modification due to a 5 GeV ADM with a relative concentration of $10^{-11}$ is roughly equivalent to the effect of a $~10\%$ opacity variation. The impact of this luminosity variation on neutrino fluxes can be estimated by evaluating $\delta L(r)$ at the scale height of the ADM distribution, $\delta L (r_\chi)$ [@Spergel:1999mh]. In general, to have an observable effect requires $\sigma_\mathrm{\chi N} N_\chi / \sigma_\odot N_\odot \gtrsim 10^{-14}$. It is possible through helioseismology to determine e.g. the mean variations of the sound speed profile $\langle \delta c/c \rangle$ and density $\langle\delta\rho/\rho\rangle$ of the Sun, as well as the boundary of the convective zone $R_\mathrm{CZ}$. In particular $R_\mathrm{CZ}$ is determined to be $(0.713 \pm 0.001) R_\odot$ while the SSM with the revised composition [@Asplund:2009fu] predicts values that are too high by up to 15$\sigma$ [@Serenelli:2009ww]. Lowering the opacity in the central region of the Sun with ADM also lowers the convective boundary. The $~10\%$ opacity variation shown in Fig. \[luminosityfunction\] leads to a $\sim 0.7\%$ reduction in $R_\mathrm{CZ}$ [@Villante:2009xs] and thus [*restores*]{} the agreement with helioseismology. The sound speed and density profiles which are presently underestimated in the region $0.2 R_\odot \lesssim r \lesssim R_\mathrm{CZ}$ would also be corrected by the opacity modification displayed in Fig. \[luminosityfunction\]. [![Neutrino producing regions in the solar interior[]{data-label="neutrinosBPS05"}](neutrinosBS05.pdf "fig:"){width="\linewidth"}]{} The modification of the luminosity profile extends into the neutrino producing region as displayed for the SSM in Fig. \[neutrinosBPS05\]. Comparing with Fig. \[luminosityfunction\] we see that precision measurements of different neutrino fluxes may be able to test the ADM model and determine its parameters. The ADM mass determines the scale height $r_\chi$, hence the relative modifications of individual neutrino fluxes, while the cross-section determines the capture rate and thereby the overall modification. Both Monte Carlo simulations [@Fiorentini:2001et] and the ‘linear solar model’ [@Villante:2009xs] show that the variation of neutrino fluxes with respect to localised opacity changes in the neutrino producing region ($r \lesssim 0.2 R_\odot$) scales approximately as $\delta \Phi_\mathrm{B} \sim 1.5 \delta\kappa$ and $\delta \Phi_\mathrm{Be} \sim 0.7 \delta\kappa$. The opacity variation in Fig. \[luminosityfunction\] leads to variations $\delta\Phi_\mathrm{B} = -17\% , \delta\Phi_\mathrm{Be} = -6.7\%$ and $\delta\Phi_\mathrm{N} = -10\% , \delta\Phi_\mathrm{O} = -14\%$ [@Villante:2009xs]. Measurements of the $^8$B flux by Super-Kamiokande [@:2008zn], SNO [@Aharmim:2008kc] and Borexino [@Collaboration:2008mr] are precise to 10% while the expectations vary by up to 20% depending on whether the old [@Grevesse:1998bj] or the new [@Asplund:2009fu] composition is used [@Serenelli:2009ww]. For the $^7$Be flux, the theoretical uncertainty is 10%, while Borexino aims to make a measurement precise to 3% [@Arpesella:2008mt]. SNO+ is expected to make a first measurement of the pep and CN-cycle fluxes [@Haxton:2008yv]. Thus the effects of metallicity and luminosity variations can be distinguished in principle. Conclusions =========== Asymmetric dark matter does not annihilate upon capture in the Sun and can therefore affect heat transport in the solar interior and consequently neutrino fluxes. This is particularly true for particles with self-interactions which would also explain the paucity of sub-galactic structure. We have shown that the presence of such particles in the Sun can solve the ‘solar composition problem’. Intriguingly a 5 GeV ‘dark baryon’ would naturally a) have the required relic abundance if it has an initial asymmetry similar to that of baryons, b) have a self-interaction cross-section of the right order to explain sub-galactic structure, c) modify the deep interior of the Sun, restoring agreement between the standard solar model and helioseismology, and d) be consistent with recent hints of signals in direct detection experiments [@Ahmed:2009zw; @Aalseth:2010vx]. Such a 5 GeV ADM particle would lower the solar neutrino fluxes which ought to be testable by the Borexino and (forthcoming) SNO+ experiments. [**Note added:**]{} After this paper was submitted to arXiv (1003.4505), another study appeared [@Cumberbatch:2010hh] with similar findings concerning the effect of ADM on helioseismology and neutrino fluxes. 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--- abstract: 'Processing-in-memory (PIM) turns out to be a promising solution to breakthrough the memory wall and the power wall. While prior PIM designs yield successful implementation of bitwise Boolean logic operations locally in memory, it is difficult to accomplish the multiplication (MUL) instruction in a fast and efficient manner. In this paper, we propose a new stochastic computing (SC) design to perform MUL with in-memory operations. Instead of using the stochastic number generators (SNGs), we harness the inherent stochasticity in the memory write behavior of the magnetic random access memory (MRAM). Each memory bit serves as an SC engine, performs MUL on operands in the form of write voltage pulses, and stores the MUL outcome in-situ. The proposed design provides up to 4x improvement in performance compared with conversational SC approaches, and achieves 18x speedup over implementing MUL with only in-memory bitwise Boolean logic operations.' author: - | Xin Ma$^{1,2}$, Liang Chang$^{1,3}$, Shuangchen Li$^{1}$, Lei Deng$^{1}$, Yufei Ding$^{2}$, Yuan Xie$^{1}$\ \ $^1$Department of Electrical and Computer Engineering, UCSB, California, USA\ $^2$Department of Computer Science, UCSB, California, USA\ $^3$School of Electronic and Information Engineering, Behang University, Beijing, China title: 'In-memory multiplication engine with SOT-MRAM based stochastic computing\' --- Stochastic computing, PIM, SOT-MRAM Introduction ============ The processing-in-memory (PIM) paradigm has been considered as a promising alternative to break the bottlenecks of conventional von-Neumann architecture. In the era of big data, data movement between the processor and the memory results in huge power consumption (power wall) and performance degradation (memory wall), known as the von-Neumann bottleneck[@koo2017summarizer]. By placing the processing units inside/near the memory, PIM remarkably reduces the energy and performance overhead induced by data transport[@zhang2014top][@ahn2016scalable]. In the recent advancement of PIM designs, it also allows fully leverage the large memory internal bandwidth and embrace massive parallelism by simultaneously activating multiple rows, subarrays, and banks in memory arrays for bit-wise operations [@li2016pinatubo]. These performance gains are all achieved at a minimal cost of slightly modifying the memory array peripheral circuits [@chi2016prime] [@li2017drisa]. Multiplication (MUL) is always a complex task to accomplish in efficient PIM designs, despite that MUL instructions are frequently used in Neuro Network (NN) algorithms and linear transforms (e.g. Discrete Fourier Transform). As shown by the recent developed DRISA [@li2017drisa], it takes 143 cycles to calculate an 8-bit multiplication, which deviates from its original motivation to achieve high performance with in-memory bitwise operations. The situation may be even worse with operands composing of more bits, as the cycle count can increase exponentially with the operand’s bit length. The challenge mainly lies in the fact that MUL can not be effectively decomposed into a small serial of bitwise Boolean logic operations which can be performed locally in memory. To tackle such challenge, prior efforts propose to either approximate the MUL or utilize the analog computing features of hardware devices. On one hand at the algorithm level, binary NN with approximate binary weights and activations has been developed [@courbariaux2016binarized]. As such, the MUL is simplified into bitwise XNOR operations that become PIM friendly [@angizi2018imce]. Unfortunately, such simplification comes at the cost of the undesired and significant degradation in the classification accuracy of NN. On the other hand at the hardware level, ReRAM is implemented to ease MUL in novel PIM designs, taking advantage of ReRAM’s analog storage. The analog resistance/conductance of ReRAM encodes the weights in NN. By activating one entire row/column simultaneously in a ReRAM crossbar, the dot product between a matrix and a vector in NN can be easily achieved using Ohm’s law[@chi2016prime]. Nevertheless, ReRAM itself suffers from the long write latency, high programming voltage and limited endurance, which hinders its application in high-speed and energy efficient architecture design. In this work, we propose a new stochastic computing (SC) design to effectively perform MUL with in-memory operations, in light of the simplicity to implement MUL with SC. In order to tightly couple SC with PIM, we embrace the inherent stochasticity of the memory bit in spin-orbit-torque magnetic random access memory (SOT-MRAM). Specifically, the stochastic number generation and massive AND operations in the conventional SC-based MUL are implemented with simple memory write operations in SOT-MRAM. Consequently, each bit serves as an SC engine, and the large supporting circuits for stochastic number generation and logic operations can be effectively saved. Finally, the MUL outcome is represented by the probability distribution of the binary storage states among MRAM bits, and can be converted back to its binary form with pop-count. The contributions of this paper are summarized as follows: - We propose the idea of employing the inherent stochastic write in SOT-MRAM to promote SC in the PIM design. - We develop an efficient approach to implement MUL in the way of memory write, by converting the binary multipliers to the write voltage pulse with varied duration. - We propose two strategies of pop-count to convert the MUL result back to its binary format, offering flexibility to further trade performance with area. - The proposed design provides up to 4x improvement in performance and significant reduction in area occupancy compared with conversational SC approaches, and achieves 18x speedup over implementing MUL with only in-memory bitwise Boolean logic operations. Preliminaries ============= This section introduces the motivation to combine SC with PIM and the preliminary design with the stochastic switching behavior of SOT-MRAM. SC and PIM ---------- SC provides an alternative approach to implement the MUL function. SC is an approximate computing method, which has been studied for decades and widely applied to image/signal processing, control systems and general purpose computing[@hayes2015introduction][@alaghi2013survey]. SC method essentially trades the data representation density for simpler logic design and lower power. For instance, SC represents a n-bit binary number with a stochastic bitstream ( $2^n$-bit). The value of the binary number $X$ equals to the probability of the appearance of “1”s in the bitstream $(\{x_i\})$: $$X=2/6(binary)\rightarrow \{x_i\}=\{1,0,0,0,1,0\}(stoch.),$$ Benefiting from such data representation, the MUL between two numbers can be converted to simple bitwise AND operations, which dramatically reduces the complexity of logic design. $$\begin{split} &Y=3/6(binary)\rightarrow \{x_i\}=\{0,1,0,1,1,0\}(stoch.),\\ &X \bullet Y=1/6 \rightarrow \{x_i\&y_i\}=\{0,0,0,0,1,0\}(stoch.). \end{split}$$ However, SC is not friendly to conventional von-Neumann architecture. The data explosion of SC aggravates data movement between processor and memory, which offsets the simplicity brought by SC. Instead, SC tightly couples with PIM from multi-fold aspects, leading to significant performance gain: First, the many bits of stochastic bitstream can be stored in off-chip memory with large capacity. Second, the logic operation with reduced complexity can be implemented by the processing units locally in memory. Finally, the stochastic feature of bitstream allows parallel computing on the individual bit, so that the internal memory bandwidth can be fully leveraged. Therefore, the MUL instruction can be significantly accelerated by combining SC with PIM. Several challenges still exist towards combine SC with PIM. The random bitstream still relies on stochastic number generators (SNGs), which incurs large area overhead for the supporting circuits. In addition, those stochastic bits can be hardly generated in parallel and with eliminated correlations, resulting in degradation of performance and accuracy in computing MUL. In our design, we overcome these drawbacks by utilizing the inherent stochasticity in MRAM bit. SOT-MRAM and its stochastic switching ------------------------------------- SOT-MRAM utilizes the spin-orbit torques to write the memory cell, overcoming the drawbacks of Spin Transfer Torque-MRAM (STT-MRAM) in terms of high write latency, and large write energy dissipation[@wang2018high] [@chang2017prescott]. Fig. \[MRAMbit\] compares the similarity and difference between SOT-MRAM and STT-MRAM cells. Similarly, both types of MRAM cells store the bit value in a magnetic tunnerling junction (MTJ). The bit value “0” or “1” is read out electrically as high or low tunneling magneto-resistance, which is controlled by the antiparallel (AP) or parallel (P) alignment of magnetization in the free layer (FL) and the reference layer (RL). Although the write of MRAM bit is always fulfilled by controlling the magnetization direction of the FL, the mechanisms used are different between STT-MRAM and SOT-MRAM. In STT-MRAM, the write current passes through MTJ and the spin polarized current exerts notable STT to switch the FL magnetization[@wang2018high] [@chang2016evaluation]. Differently in SOT-MRAM, SOTs are generated by transversing write current though an additional heavy metal layer (HML) to switch the magnetization in the adjunct FL. As a result, SOT-MRAM does not suffer from the asymmetric of write latency between “AP $\rightarrow$ P” and “P $\rightarrow$ AP” in STT-MRAM, speeding up the write procedure. Moreover, the energy efficiency of write is fundamentally higher in SOT-MRAM. That’s because each electron can be reused multiple times to exert SOTs after bounced back from HML and FL interface, while it can be used once at most in STT. ![Illustration of the memory cells and the write strategies of STT-MRAM (a) and SOT-MRAM (b).[]{data-label="MRAMbit"}](MRAMbit.jpg){width="3"} We harness the stochastic behavior within the memory write of SOT-MRAM to perform SC. The probability $P_{usw}$ of MRAM bit remains **not** switched under the appliance of electrical current $I$ is described [@seki2011switching] $$P_{usw}=exp(-\tau exp(-\Delta (1-I/I_c))). %\lable{Probability}$$ Here, $\tau$ denotes the pulse duration of the applied $I$ in nanosecond, $\Delta$ represents the thermal stability parameter of the MTJ, and $I_c$ is the critical current strength required to switch the FL magnetization. Fig. \[Probability\] plots the $P_{usw}$ as functions of $I$ and $\tau$, with $\Delta=60.9$ and $I_c=80\mu A$ estimated from previous micromagnetic simulations on SOT driven magnetization dynamics[@chang2017prescott]. By finely controlling the parameters in the write of SOT-MRAM, each memory bit can serve as a stochastic bit generator with the desired probability of holding either “0” or “1”. Utilizing this feature of SOT-MRAM, the large amount of stochastic bits in SC can be generated in parallel and in-situ stored in memory with a simple write operation. ![The probability of a MRAM bit remaining unswitched under different pulse duration and strength of the applied electrical current[]{data-label="Probability"}](uswP.jpg){width="3"} Data conversion and hardware design =================================== To implement the MUL operation with the stochastic switching of MRAM bit, the binary operands have to be translated into certain parameter of the write voltage pulse. The flow of our proposed sequential data conversion can be summarized as: $$\begin{split} &X(binary), Y(binary) \rightarrow ln(X)(binary), ln(Y)(binary)\\ &\rightarrow ln(X)(time), ln(Y)(time) \rightarrow X*Y (stoch.) \\ &\rightarrow X*Y (Binary) \end{split}$$ In this section, we will introduce them and the related hardware design step by step. Binary numbers to logarithmic timing signals -------------------------------------------- We first perform logarithmic operation on the digital numbers stored in memory, i.e. $X(binary), Y(binary) \rightarrow ln(X)(binary), ln(Y)(binary)$. The multiple bits of the operand $X$ are read out by sensing amplifiers (SAs) and decoded to find their logarithmic values using a lookup table (LUT) (Fig. \[LUT\]). The LUT method is usually used in logarithm multiplication, and has been demonstrated to be fast and accurate[@nandan201865]. This conversion step is necessary, since an exponential operation is inherently included in the following stochastic switching of the MRAM bit. Afterwards, we convert the $ln(X)$ to timing signals with a digital-to-time converter (DTC). The DTC outputs a voltage square pulse $V_{tX}$, where the pulse duration $\tau_{X}$ in Eq. 3 is proportional to the value of input $ln(X)$. The magnitude of the $V_{tX}$ pulse is normalized and fixed to drive SOT-MRAM bit in its non-deterministic switching region. ![Data conversion from binary to logarithmic timing signals[]{data-label="LUT"}](Circuits.jpg){width="3"} Logarithmic timing signals to stochastic bitstream -------------------------------------------------- The write voltage pulse $V_{tX}$ is subsequently applied onto the source lines (SLs) of multiple rows of SOT-MRAM bits, and drives their stochastic switching behaviors. The entire row of MRAM array can be written simultaneously with a cross-point design (Fig. \[Crosspoint\])[@chang2017prescott]. The MTJs in a row share a set of driving transistors, and are directly linked to the BLs and SLs without additional transistors for individual bit. As a result, minimal area and energy overhead are introduced to enable such simultaneous write. ![Schematics of the cross-point SOT-MRAM array, with stochastic bits in-situ stored under a pulsed voltage[]{data-label="Crosspoint"}](Memoryarray.jpg){width="3"} Fig. \[SCexample\] shows how the SC-based MUL is performed. **Initialization**: a preset operation is required to initialize all the bits to “1” with reversed current $I_c$. **Input first operand**: the converted write voltage pulse $V_{tX}$ is input onto the MRAM array, resulting in partial switching of the bits. The probability of remaining “1”s equals to $P_X$, where $P_X$ is proportional to the value of operand $X$. **MUL with the second operand input**: The MUL operation is performed by inputing a subsequent voltage pulse $V_{tY}$ (similarly converted from operand $Y$) onto the same MRAM array. As a result, the remaining “1”s survive from not switched by neither pulse $V_{tX}$ nor $V_{tY}$, and they are distributed among the MRAM arrays with a probability equaling $P_X*P_Y$ (proportional to $X*Y$). ![Illustration for the SC-based MUL in SOT-MRAM[]{data-label="SCexample"}](SCexample.jpg){width="3"} Stochastic bits to Binary numbers --------------------------------- At last, we perform bit counting to convert the outcome from the stochastic representation to its binary format. Either approximate pop-count (APC) [@kim2015approximate] or PIM-based ADD operations [@li2017drisa] can be employed to bit counting. APC method can be performed with one clock cycle, but introduces much area overhead. Alternatively, PIM-based ADD is area-efficient, but takes many clock cycles to perform the pop-count. Specifically, we can accelerate the PIM-based pop-count for the vectored multiply-and-accumulate (MAC) in NN. Fig. \[Popcount\] shows the two-step strategy, where the sum is performed after several MULs have been done. In the first step, we perform row-wise sum with a carry-save addition (CSA). Then in the second step, the intermediate sum results undergo a column-wise additions with full adder (FA). Our motivation here is to lessen the usage of FA for column-wised addition, since it takes more clock cycles than the lock step bitwise operations of CSA. As shown in Fig. \[Popcount\], the delay from FA can be averaged out, and the pop-count related cycle count converges to that of CSA after many MULs. ![The two steps of the PIM-based pop-count strategy.[]{data-label="Popcount"}](Bitcount.jpg){width="3.5"} Put them all together --------------------- After putting all the pieces together, we point out strategies to further improve the performance and accuracy, and explain certain considerations in the design. The sequential flow of data conversion can be separated and pipelined to improve the throughput and performance. For example, the LUT operation on the second operand can be performed simultaneously with the stochastic memory write for the first operand. Moreover, the bit counting can work in parallel with MUL operations for NN applications. There is no need for the relative slow pop-count to start until all the fast MULs between $w_i$ and $x_i$ have been finished in the computation of $\sum_iw_ix_i$. Furthermore, one could pre-convert certain frequently used data (e.g. weight $w_i$ in NN) into stochastic bits, which can be stored non-volatilely in MRAM arrays. Once other multipliers (e.g. inputs $x_i$) come, their converted timing signals can be directly input onto the corresponding MRAM arrays to perform MUL operations. There are several normalization units in the circuits that can be used to fine tune the accuracy and performance. For example, the pulse duration of $V_t$ can be scaled to a range where $P_{usw}\approx 0.5$. Through such scaling, the switching voltage pulse can not be longer than the usual time required to switch MRAM bit, avoiding unnecessary slowdown in computing. Moreover, the bitstream can be tuned neither sparse nor dense to guarantee the accuracy of MUL, so that more bits are effectively involved in SC. This is fundamentally similar to the improved classification accuracy of NN with more neurons involved. Multiple rows can be simultaneously activated and wrote to generate more stochastic bits in parallel. This situation happens when performing MUL on operands with more bits. In the cross-point MRAM design, we limit the number of memory cells in each row due to the concern of IR drop. The MTJs farther away from the driving transistors in the row would suffer from a lower switching voltage[@liang2010cross], and would likely undergo stochastic switching with undesired and incorrect probability. Finally, we note that the pulse duration $\tau$ is used here for computing, instead of the magnitude of switching voltage pulse $V_t$ (equivalent to $I$ in Eq. 3). That’s because the usage of the magnitude $I$ requires more complicated circuits design for data conversion, owing to the complex dependence of $P_{usw}$ on $I$. In addition, the two inputs $I_X$ and $I_Y$ has to be input simultaneously onto the MRAM arrays. This is not friendly to the pipeline strategies mentioned above, but will introduce large area overhead onto the driven transistors to enable higher current write instead. Monte Carlo simulations ======================= To estimate the accuracy and its dependence on hardware variance from statistics, we performe the Monte Carlo simulations on the stochastic switching of MRAM bits. In the following, $nbit$ denotes the number of stochastic bits per MUL, $P$ represents the probability $bitcount(B_i=1, i=1:nbit)/nbit$ that the bit remains not switched under certain input voltage pulse. For one MUL operation, we test the proposed SC with 1000 iterations and make statistics on the results among iterations. Accuracy -------- Fig. \[Statadd\](a) shows the distribution of the error $P_{XY}-P_X*P_Y$ among the 1000 iterations, where the the probability $P_{XY}$ is stochastically computed (with $\tau_{X}=0.3 ns, \tau_{Y}=0.4 ns$) and $P_X, P_Y$ are theoretically calculated from the two operands. The error distribution is centered to zero, indicating that there is no intrinsic bias in the SC arithmetic. The distribution can be well fitted with a Gaussian function (red line), with the standard deviation $\sigma \approx 1.6\%$. This indicates that the MUL is with about $3.2\%$ uncertainty for $nbit=1000$. ![(a) The probability distribution of the deviation from theoretical calculated MUL among 1000 iterations. (b) The MUL uncertainty as a function of stochastic bits number and input values.[]{data-label="Statadd"}](Statadd.jpg){width="3"} We further investigate the dependence of $\sigma$ on the inputs $\tau_Y$ and the number of stochastic bits $nbit$. As shown in Fig. \[Statadd\](b), $\sigma$ is almost independent on the inputs $\tau_Y$, but decreases with larger $nbit$. Therefore, we can improve the accuracy of SC by using more MRAM bits, despite that the improvement becomes more gradual with larger $nbit$. The impact of hardware variance ------------------------------- We also investigate the impact of hardware variance on the accuracy of MUL operation, by introducing random fluctuations on the devices’ parameters in Monte Carlo simulation. The critical currents of MRAM bits $I_c$ may be slightly varied, since the many MRAM bits can not be manufactured identically and they may also experience different thermal fluctuations when in use[@an2016current]. Therefore, we introduce 0% to 10% random fluctuations $\sigma(I_c)$ on the $I_c$. As shown in Fig. \[Faulttorr\](a), the accuracy of SC remains almost unchanged under different strength of fluctuations. We also compare the fault tolerance of our design with that of logarithm multiplication. To implement logarithm multiplication[@nandan201865], we replace the DTC and SOT-MRAMs with an antilogarithm amplifier. Then we introduce 4% to 10% random fluctuations $\sigma(Circuits)$ on DTC and antilogarithm respectively for the two cases. As shown in Fig. \[Faulttorr\](b), the accuracy of our SC+PIM design remains almost unchanged, while logarithm multiplication suffers from severe degradation in accuracy with stronger fluctuations. ![The dependence of MUL uncertainty on the variance of critical current among SOT-MRAM bits. The MUL uncertainties as functions of circuit variance in our SC+PIM design and logarithm multiplication[]{data-label="Faulttorr"}](Faulttorr.jpg){width="3"} Evaluation ========== In this section, we evaluate the performance, power and area overhead of the proposed SC+PIM design, and compare them with that of other approaches using either SC or PIM. Experimental setup ------------------ We adopt the cross-point design of SOT-MRAM arrays similar to PRESCOTT[@chang2017prescott], to enable the parallel memory write. The low-power DTC generates voltage pulses with 22 ps time resolution and occupies $75\mu m*25\mu m$ in area[@wang2015digital]. For the APC, we design one-cycle fully parallel circuit synthesized with 45nm FreePDK[@stine2009freepdk], integrating parameters from[@kim2015approximate]. Our evaluation is based on the multiplication between two 10-bit operands that represented by $2^{10}$ stochastic bits. In the following, different configurations have been compared: **SC+PIM (with APC)** denotes our SC+PIM design with pop-count conducted by APC. **SC+PIM (with CSA)** is our SC+PIM design with pop-count performed with CSA+FA. Specially, the evaluation is averaged onto each MUL for the situation of performing 100 MULs in a MAC. **SC** represents the usage of a built multiplier with the state-of-the-art SNG [@kim2016energy] and popcount with APC. **PIM** is the situation that we only use in-memory Boolean logic operations to implement MUL. Performance ----------- Fig. \[Performance\](a) compares the cycle count used to perform each MUL operation with different designs. Evidently, our SC+PIM approach outperforms prior approaches using either SC or PIM. The boost of performance in our design benefits from the parallel generation of stochastic bits. In contrast, prior SC approaches requires additional cycles to generate stochastic bitstreams or to shuffle the existing pseudo-stochastic or deterministic bitstreams [@kim2016energy]. In addition, we investigate the dependence of MUL cycle count on the operands’ bit length as shown in Fig. \[Performance\](b). The cycle count remains unchanged in our SC+PIM design, since different amount of stochastic bits ($2^n$ for n-bit operand) can be generated in parallel. As a comparison, the cycle count required for MUL increases exponentially for the operands’ bits length in prior PIM design. Therefore, the speedup of SC+PIM over PIM becomes more attractive for MUL between operands with more bits. ![(a) The cycle count to implement each MUL with different approaches. (b) The MUL latency as a function of multipliers’ bit length with SC+PIM and PIM approaches[]{data-label="Performance"}](Perfcomp2.jpg){width="3.5"} Energy consumption ------------------ Our SC+PIM design consumes 58% less energy compared with the SC method (Fig. \[Energy\]), thanks to the low write energy of SOT-MRAM[@jabeur2014spin]. In our design, most energy is spent through memory write, such as in the generation/computing of stochastic bits and the pop-count with bitwise addition (CSA). The situation is similar to prior SC approaches, where 88% of the energy is consumed in data buffering related operations. As shown by the breakdown of the energy consumption in Fig. \[Energy\], the initialization step costs more energy than the following steps performing SC for MUL. That’s because a write voltage pulse with a higher magnitude and a longer pulse duration needs to be applied to guarantee the initialization. Afterwards, the memory bits are mainly driven in a non-deterministic switching region which consumes less energy. ![The energy consumption for each MUL with different approaches and their breakdown.[]{data-label="Energy"}](Powercomp2.jpg){width="3"} Area overhead ------------- The area overhead of different designs is compared in Fig. \[Area\]. The area overhead is smaller by about one order of magnitude for our SC+PIM design than conventional SC. The improvement originates from the removal of the additional circuits for SNG, which occupies 95% of the area in the conventional SC approach. As shown by the breakdown of area overhead in Fig. \[Area\], the memory space required for the LUT table is comparable to the DTC and APC in our design, for the case of 10-bit multiplication. The LUT table size will shrink for regular 8-bit multiplication, since it depends exponentially on the bit length of the operands. ![The area overhead in different approaches with their breakdown. []{data-label="Area"}](Areacomp2.jpg){width="3"} Conclusion ========== In this paper, we propose a new SC design to perform MUL with in-memory operations. The stochastic random generation and AND operation in conventional SC are implemented by the simple write operations onto the SOT-MRAM. Such design is enabled by converting the binary multipliers to the varied pulse duration of the write voltage for SOT-MRAM. Consequently, the stochastic bits for the MUL outcome are in-situ stored. Two strategies of pop-count (APC or PIM-based ADD) have been proposed to convert the MUL result back to its binary format, offering flexibility to further trade off performance with area. Our approach improves the performance to compute MUL with PIM, in synergy with the mitigation of area overhead for supporting circuits of SC.

--- abstract: 'The optimization of large portfolios displays an inherent instability to estimation error. This poses a fundamental problem, because solutions that are not stable under sample fluctuations may look optimal for a given sample, but are, in effect, very far from optimal with respect to the average risk. In this paper, we approach the problem from the point of view of statistical learning theory. The occurrence of the instability is intimately related to over-fitting which can be avoided using known regularization methods. We show how regularized portfolio optimization with the expected shortfall as a risk measure is related to support vector regression. The budget constraint dictates a modification. We present the resulting optimization problem and discuss the solution. The L2 norm of the weight vector is used as a regularizer, which corresponds to a diversification “pressure". This means that diversification, besides counteracting downward fluctuations in some assets by upward fluctuations in others, is also crucial because it improves the stability of the solution. The approach we provide here allows for the simultaneous treatment of optimization and diversification in one framework that enables the investor to trade-off between the two, depending on the size of the available data set.' address: - 'Information and Computer Sciences, University of Hawaii at M$\bar{\rm a}$noa, Honolulu, Hawaii, USA' - 'Collegium Budapest–Institute for Advanced Study and Department of Physics of Complex Systems, E[ö]{}tv[ö]{}s University, Budapest, Hungary' author: - Susanne Still - Imre Kondor bibliography: - 'RPOsubmit.bib' title: 'Regularizing Portfolio Optimization.' --- Introduction ============ Markowitz’ portfolio selection theory [@Markowitz52; @Markowitz59] is one of the pillars of theoretical finance. It has greatly influenced the thinking and practice in investment, capital allocation, index tracking, and a number of other fields. Its two major ingredients are (i) seeking a trade-off between risk and reward, and (ii) exploiting the cancellation between fluctuations of (anti-)correlated assets. In the original formulation of the theory, the underlying process was assumed to be multivariate normal. Accordingly, reward was measured in terms of the expected return, risk in terms of the variance of the portfolio. The fundamental problem of this scheme (shared by all the other variants that have been introduced since) is that the characteristics of the underlying process generating the distribution of asset prices are not known in practice, and therefore averages are replaced by sums over the available sample. This procedure is well justified as long as the sample size, $T$ (i.e. the length of the available time series for each item), is sufficiently large compared to the size of the portfolio, $N$ (i.e. the number of items). In that limit, sample averages asymptotically converge to the true average due to the central limit theorem. Unfortunately, the nature of portfolio selection is not compatible with this limit. Institutional portfolios are large, with $N$’s in the range of hundreds or thousands, while considerations of transaction costs and non-stationarity limit the number of available data points to a couple of hundreds at most. Therefore, portfolio selection works in a region, where $N$ and $T$ are, at best, of the same order of magnitude. This, however, is not the realm of classical statistical methods. Portfolio optimization is rather closer to a situation which, by borrowing a term from statistical physics, might be termed the “thermodynamic limit", where $N$ and $T$ tend to infinity such that their ratio remains fixed. It is evident that portfolio theory struggles with the same fundamental difficulty that is underlying basically every complex modeling and optimization task: the high number of dimensions and the insufficient amount of information available about the system. This difficulty has been around in portfolio selection from the early days and a plethora of methods have been proposed to cope with it, e.g. single and multi-factor models [@eltongruber], Bayesian estimators [@jobson1979; @jorion1986; @frost_savarino; @safePO; @Jagannathan2003; @LedoitWolf2003; @LedoitWolf2004; @LedoitWolfHoney; @DeMiguel2007; @Garlappi2007; @Golosnoy2007; @Kan2007; @frahm_memmel; @DeMiguel2009], or, more recently, tools borrowed from random matrix theory [@Laloux1999; @Plerou1999; @Laloux2000; @Plerou2002; @burda; @Potters2005]. In the thermodynamic regime, estimation errors are large, sample to sample fluctuations are huge, results obtained from one sample do not generalize well and can be quite misleading concerning the true process. The same problem has received considerable attention in the area of machine learning. We discuss how the observed instabilities in portfolio optimization (elaborated in Section \[instab\]) can be understood and remedied by looking at portfolio theory from the point of view of machine learning. Portfolio optimization is a special case of regression, and therefore can be understood as a machine learning problem (see Section \[reasons\]). In machine learning, as well as in portfolio optimization, one wishes to minimize the [*actual risk*]{}, which is the risk (or error) evaluated by taking the ensemble average. This quantity, however, can not be computed from the data, only the [*empirical risk*]{} can. The difference between the two is not necessarily small in the thermodynamic limit, so that a small empirical risk does not automatically guarantee small actual risk [@VapnikCh71]. Statistical learning theory [@VapnikCh71; @Vapnik95; @Vapnik98] finds upper bounds on the generalization error that hold with a certain accuracy. These error bounds quantify the expected generalization performance of a model, and they decrease with decreasing [*capacity*]{} of the function class that is being fitted to the data. Lowering the capacity therefore lowers the error bound and thereby improves generalization. The resulting procedure is often referred to as regularization and essentially prevents over-fitting (see Section \[RPO\]). In the thermodynamic limit, portfolio optimization needs to be regularized. We show in Section \[RPO-ES\] how the above mentioned concepts, which find their practical application in support vector machines [@Boser92; @CortesVapnik95], can be used for portfolio optimization. Support vector machines constitute an extremely powerful class of learning algorithms which have met with considerable success. We show that regularized portfolio optimization, using the expected shortfall as a risk measure, is almost identical to support vector regression, apart from the budget constraint. We provide the modified optimization problem which can be solved by linear programming. In Section \[diversification\], we discuss the financial meaning of the regularizer: minimizing the L2 norm of the weight vector corresponds to a diversification pressure. We also discuss alternative constraints that could serve as regularizers in the context of portfolio optimization. Taking this machine learning angle allows one to organize a variety of ideas in the existing literature on portfolio optimization filtering methods into one systematic and well developed framework. There are basically two choices to be made: (i) which risk measure to use, and (ii) which regularizer. These choices result in different methods, because different optimization problems are being solved. While we focus here on the popular expected shortfall risk measure (in Section \[RPO-ES\]), the variance has a long history as an important risk measure in finance. Several existing filtering methods that use the variance risk measure essentially implement regularization, without necessarily stating so explicitly. The only work we found in this context [@safePO] that mentiones regularization in the context of portfolio optimization has not been noticed by the ensuing, closely related, literature. It is easy to show that when the L2 norm is used as a regularizer, then the resulting method is closely related to Bayesian ridge regression, which uses a Gaussian prior on the weights (with the difference of the additional budget constraint). The work on covariance shrinkage, such as [@Jagannathan2003; @LedoitWolf2003; @LedoitWolf2004; @LedoitWolfHoney], falls into the same category. Other priors can be used [@DeMiguel2009], which can be expected to lead to different results (for an insightful comparison see e.g. [@Tibshirani96]). Using the L1 norm has been popularized in statistics as the “LASSO" (least absolute shrinkage and selection operator) [@Tibshirani96], and methods that use any Lp norm are also known as the “bridge" [@Frank93]. Preliminaries – Instability of classical portfolio optimization. {#instab} ================================================================ Portfolio optimization in large institutions operates in what we called the thermodynamic limit, where both the number of assets and the number of data points are large, with their ratio a certain, typically not very small, number. The estimation problem for the mean is so serious [@chopraziemba1993; @merton1980] as to make the trade-off between risk and return largely illusory. Therefore, following a number of authors [@Jagannathan2003; @LedoitWolf2003; @okhrin2006; @kempf_memmel2006; @Frahm2008], we focus on the minimum variance portfolio and drop the usual constraint on the expected return. This is also in line with previous work (see [@kondor2007] and references therein), and makes the treatment simpler without compromising the main conclusions. An extension of the results to the more general case is straightforward. Nevertheless, even if we forget about the expected return constraint, the problem still remains that covariances have to be estimated from finite samples. It is an elementary fact from linear algebra that the rank of the empirical $N\times N$ covariance matrix is the smaller of $N$ and $T$. Therefore, if $T < N$, the covariance matrix is singular and the portfolio selection task becomes meaningless. The point $T = N$ thus separates two regions: for $T > N$ the portfolio problem has a solution, whereas for $T < N$, it does not. Even if $T$ is larger than $N$, but not [*much*]{} larger, the solution to the minimum variance problem is unstable under sample fluctuations, which means that it is not possible to find the optimal portfolio in this way. This instability of the estimated covariances, and hence of the optimal solutions, has been generally known in the community, however, the full depth of the problem has only been recognized recently, when it was pointed out that the average estimation error diverges at the critical point $N = T$ [@pafka2002; @pafka2003; @pafka2004]. In order to characterize the estimation error, Kondor and co-workers used the ratio $q_0^2$ between (i) the risk, evaluated at the optimal solution obtained by portfolio optimization using finite data and (ii) the true minimal risk. This quantity is a measure of generalization performance, with perfect performance when $q_0^2 = 1$, and increasingly bad performance as $q_0^2$ increases. As found numerically in [@pafka2003] and demonstrated analytically by random matrix theory techniques in [@burda2003], the quantity $q_0$ is proportional to $(1 - N/T)^{-1/2}$ and diverges when $T$ goes to $N$ from above. The identification of the point $N = T$ as a phase transition [@kondor2007; @ciliberti_1] allowed for the establishment of a link between portfolio optimization and the theory of phase transitions, which helped to organize a number of seemingly disparate phenomena into a single coherent picture with a rich conceptual content. For example, it has been shown that the divergence is not a special feature of the variance, but persists under all the other alternative risk measures that have been investigated so far: historical expected shortfall, maximal loss, mean absolute deviation, parametric VaR, expected shortfall, and semivariance [@kondor2007; @ciliberti_1; @ciliberti_2; @hasszan2008]. The critical value of the $N/T$ ratio, at which the divergence occurs, depends on the particular risk measure and on any parameter that the risk measure may depend on (such as the confidence level in expected shortfall). However, as a manifestation of universality, the power law governing the divergence of the estimation error is independent of the risk measure [@kondor2007; @ciliberti_1; @ciliberti_2], the covariance structure of the market [@pafka2004], and the statistical nature of the underlying process [@hasszan2007]. Ultimately, this line of thought led to the discovery of the instability of coherent risk measures [@kondor2008]. Statistical reasons for the observed instability in portfolio optimization {#reasons} ========================================================================== As mentioned above, for simplicity and clarity of the treatment we do not impose a constraint on the expected return, and only look for the global minimum risk portfolio. This task can be formalized as follows: Given a fixed budget, customarily taken to be unity, given $T$ past measurements of the returns of $N$ assets: $x_i^k$, $i=1, \dots,N$, $k = 1, \dots, T$, and given the risk functional $F({{\bf w} \cdot {\bf x}})$, find a weighted sum (the portfolio), ${{\bf w} \cdot {\bf x}}$,[^1] such that it minimizes the [*actual*]{} risk $$R ({{\bf w}}) = \langle F({{\bf w} \cdot {\bf x}}) \rangle_{p({{\bf x}})},$$ under the constraint that $\sum_i w_i = 1$. The central problem is that one does not know the distribution $p({{\bf x}})$, which is assumed to underly the generation of the data. In practice, one then minimizes the [*empirical*]{} risk, replacing ensemble averages by sample averages: $$R_{\rm emp} ({{\bf w}}) = {1 \over T} \sum_{k=1}^T F({{\bf w} \cdot {{\bf x}^{(k)}}}) \label{emprisk}$$ Now, let us interpret the weight vector as a linear model. The model class given by the linear functions has a [*capacity*]{} $h$, which is a concept that has been introduced by Vapnik and Chervonenkis in order to measure how powerful a learning machine is [@VapnikCh71; @Vapnik95; @Vapnik98]. (In the statistical learning literature, a learning machine is thought of as having a function class at its disposal, together with an induction principle and an algorithmic procedure for the implementation thereof [@Bernhard_thesis]). The capacity measures how powerful a function class is, and thereby also how easy it is to learn a model of that class. The rough idea is this: a learning machine has larger capacity if it can potentially fit more different types of data sets. Higher capacity comes, however, at the cost of potentially over-fitting the data. Capacity can be measured, for example, by the Vapnik-Chervonenkis (VC-) dimension [@VapnikCh71], which is a combinatoric measure that counts how many data points can be separated in all possible ways by any function of a given class. To make the idea tangible for linear models, focus on two dimensions ($N=2$). For each number of points, $n$, one can choose the geometrical arrangement of the points in the plane freely. Once it is chosen, points are labeled by one of two labels, say “red" and “blue". Can a line separate the red points from the blue points for [*any*]{} of the $2^n$ different ways in which the points could be colored? The VC-dimension is the largest number of points for which this can be done. Two points can trivially be separated by a line. Three points that are not arranged collinear can still be separate for any of the 8 possible labelings. However, for four points this is no longer the case, since there is no geometrical arrangement for which one could not find a labeling that can not be separated by a line. The VC-dimension is 3, and in general, for linear models in $N$ dimensions, it is $N+1$ [@Bernhard_thesis; @Bernhard_Book1]. In the regime in which the number of data points are much larger than the capacity of the learning machine, $h/T << 1$, a small empirical risk guarantees small actual risk [@VapnikCh71]. For linear functions through the origin that are otherwise unconstrained, the VC-dimension grows with $N$. In the thermodynamic regime, where $N/T$ is not very small, minimizing the empirical risk does not necessarily guarantee a small actual risk [@VapnikCh71]. Therefore it is not guaranteed to produce a solution that generalizes well to other data drawn from the same underlying distribution. In solving the optimizing problem that minimizes the [*empirical*]{} risk, Eq. (\[emprisk\]) in the regime in which $N/T$ is not very small, portfolio optimization [*over-fits*]{} the observed data. It thereby finds a solution that essentially pays attention to the seeming correlations in the data which come from estimation noise due to finite sample effects, rather than from real structure. The solution is thus different for different realizations of the data, and does not necessarily come close to the actual optimal portfolio. Overcoming the instability {#RPO} ========================== The generalization error can be bounded from above (with a certain probability) by the empirical error plus a confidence term that is monotonically increasing with some measure of the capacity, and depends on the probability with which the bound holds [@Vapnik79]. Several different bounds have been established, connected with different measures of capacity, see e.g. [@Bernhard_Book1]. Poor generalization and over-fitting can be improved upon by decreasing the capacity of the model [@Vapnik95; @Vapnik98], which helps to lower the generalization error. Support vector machines are a powerful class of algorithms that implement this idea. We suggest that if one wants to find a solution to the portfolio optimization problem in the thermodynamic regime, then one should not minimize the empirical risk alone, but also constrain the capacity of the portfolio optimizer (the linear model). How can portfolio optimization be regularized? Portfolio optimization is essentially a regression problem, and therefore we can apply statistical learning theory, in particular the work on support vector regression. Note first that the capacity of a linear model class for which the length of the weight vector is restricted to $\|w\|^2 \leq A$ has an upper bound which is smaller than the capacity of unconstrained linear models [@Vapnik95; @Vapnik98]. The capacity is minimized when the length of the weight vector is minimized [@Vapnik95; @Vapnik98]. Vapnik’s concept of [*structural risk minimization*]{} [@Vapnik79] results in the support vector algorithm [@Boser92; @CortesVapnik95] which finds the model with the smallest capacity that is consistent with the data, that is the model with smallest $\|w\|^2$. This leads to a convex constrained optimization problem [@Boser92; @CortesVapnik95] which can be solved using linear programming. Regularized portfolio optimization with the expected shortfall risk measure. {#RPO-ES} ============================================================================ While the original Markowitz’ formulation [@Markowitz52] measures risk by the variance, many other risk measures have been proposed since. Today, the most widely used risk measure, both in practice and in regulation, is Value at Risk (VaR) [@jorion; @riskmetrics]. VaR has, however, been criticized for its lack of convexity, see e.g. [@Artzner99; @embrechts; @acerbi1], and an axiomatic approach, leading to the introduction of the class of coherent risk measures, was put forward [@Artzner99]. Expected shortfall, essentially a conditional average measuring the average loss above a high threshold, has been demonstrated to belong to this class [@acerbi2; @acerbi3; @acerbi4]. Expected shortfall has been steadily gaining popularity in recent years. The regularization we propose here is intended to cure its weak point, the sensitivity to sample fluctuations, at least for reasonable values of the ratio $N/T$. Choose the risk functional $F(z) = z \theta(z-\alpha_\beta)$, where $\alpha_\beta$ is a threshold, such that a given fraction $\beta$ of the (empirical) loss-distribution over $z$ lies above $\alpha_\beta$. One now wishes to minimize the average over the remaining tail distribution, containing the fraction $\nu :=1 - \beta$, and defines the expected shortfall as $$ES = \min_{\epsilon}\left[\epsilon + \frac{1}{\nu T} \sum_{k=1}^T {1 \over 2} \left(-\epsilon -{{\bf w} \cdot {{\bf x}^{(k)}}}+ |-\epsilon -{{\bf w} \cdot {{\bf x}^{(k)}}}|\right)\right]. \label{ExpS}$$ The term in the sum implements the $\theta$-function, while $\nu$ in the denominator ensures normalization of the tail distribution. It has been pointed out [@Rockafellar] that this optimization problem maps onto solving the linear program: $$\begin{aligned} &&\min_{{{\bf w}}, {\bf \xi}, \epsilon} \left[ {1 \over T} \sum_{k=1}^{T} \xi_k + \nu \epsilon \right] \label{CVaR} \\ &{\rm s.t.}\;\;\; & {{\bf w} \cdot {{\bf x}^{(k)}}}+ \epsilon + \xi_k \geq 0; \;\;\; \xi_k; \geq 0 \label{ES-constr} \\ && \sum_i w_i = 1.\end{aligned}$$ We propose to implement regularization by including the minimization of $\|{{\bf w}}\|^2 $. This can be done using a Lagrange multiplier, $C$, to control the trade-off – as we relax the constraint on the length of the weight vector, we can, of course, make the empirical error go to zero and retrieve the solution to the minimal expected shortfall problem. The new optimization problem reads: $$\begin{aligned} &&\min_{{{\bf w}}, {\bf \xi}, \epsilon} \left[ {1 \over 2} \|{{\bf w}}\|^2 + C \left({1 \over T} \sum_{k=1}^{T} \xi_k + \nu \epsilon \right) \right] \label{newPO} \\ &{\rm s.t.}\;\;\; & - {{\bf w} \cdot {{\bf x}^{(k)}}}\leq \epsilon + \xi_k; \label{con1}\\ && \xi_k \geq 0; \;\;\; \epsilon \geq 0;\\ && \sum_i w_i = 1. \label{b}\end{aligned}$$ The problem is mathematically almost identical to a support vector regression (SVR) algorithm called $\nu$-SVR. There are two differences: (i) the budget constraint is added, and (ii) the loss function is asymmetric. Expected shortfall is an asymmetric version of the $\epsilon$-intensive loss, used in support vector regression, defined as the maximum of $\{0;| f({{\bf x}}) - y| - \epsilon \}$, where $f({{\bf x}})$ is the interpolant, and $y$ the measured value (response). In that sense $\epsilon$ measures an allowable error below which deviations are discarded.[^2] The use of asymmetric risk measures in finance is motivated by the consideration that investors are not afraid of upside fluctuations. However, to make the relationship to support vector regression as clear as possible, we will first solve the more general symmetrized problem, before restricting our treatment to the completely asymmetric case, corresponding to expected shortfall. In addition, one may argue that focusing exclusively on large negative fluctuations might not be advisable even from a financial point of view, especially when one does not have sufficiently large samples. In a relatively small sample it may happen that a particular item, or a certain combination of items, dominates the rest, i.e. produces a larger return than any other item in the portfolio at each time point, even though no such dominance exists on longer time scales. The probability of such an apparent arbitrage increases with the ratio $N/T$, and when it occurs it may encourage an investor acting on a lopsided risk measure to take up very large long positions in the dominating item(s), which may turn out to be detrimental on the long run. This is the essence of the argument that has led to the discovery of the instability of coherent and downside risk measures [@hasszan2008; @kondor2008]. According to the above, let us consider the general case where positive deviations are also penalized. The objective function, Eq. (\[newPO\]), then becomes $$\min_{{{\bf w}}, {\bf \xi}, \epsilon} \left[ {1 \over 2} \|{{\bf w}}\|^2 + C \left({1 \over T} \sum_{k=1}^{T} \left( \xi_k + \xi_k^*\right) + \nu \epsilon \right) \right] \label{gen-RES},$$ and additional constraints have to be added to Eqs. (\[con1\]) to (\[b\]): $$\begin{aligned} {{\bf w} \cdot {{\bf x}^{(k)}}}\leq \epsilon + \xi_k^*; \;\;\; \xi_k^* \geq 0.\label{con1sym}\end{aligned}$$ This problem corresponds to $\nu$-SVR, a well understood regression method [@Nu-SVM], with the only difference that the budget constraint, Eq. (\[b\]) is added here. In the finance context the associated loss might be called [*symmetric tail average*]{} (STA). Solving the regularized expected shortfall minimization problem, Eqs. (\[newPO\])–(\[b\]) is a special case of solving the regularized STA minimization problem, Eq. (\[gen-RES\]) with the constraints Eqs. (\[con1\])–(\[b\]) and (\[con1sym\]). Therefore, we solve the more general problem first (Section \[RSTA\]), before providing, in Section \[RES-par\], the solution to the regularized expected shortfall, Eqs. (\[newPO\])–(\[b\]). Regularized Symmetric Tail Average Minimization {#RSTA} ----------------------------------------------- The solution to the regularized symmetric tail average problem, Eq. (\[gen-RES\]) with the constraints Eqs. (\[con1\])–(\[b\]) and (\[con1sym\]), is found in analogy to support vector regression, following [@Nu-SVM], by writing down the Lagrangean, using Lagrange multipliers, $\{ {\bf \alpha}, {\bf \alpha^*}, \gamma, \lambda, {\bf \eta}, {\bf \eta^*} \}$, for the constraints. The solution is then a saddle point, i.e. minimum over primal and maximum over dual variables. The Lagrangean is different from the one that arises in $\nu$-SVR in that it is modified by the budget constraint: $$\begin{aligned} \!\!\!\!\! \!\!\!\!\! \!\!\!\!\! \!\!\!\!\! \!\!\!\!\! \!\!\!\!\! \!\!\!\!\! L[{{\bf w}}, {\bf \xi}, {\bf \xi^*}, \epsilon, {\bf \alpha}, {\bf \alpha^*}, \gamma, \lambda, {\bf \eta}, {\bf \eta^*}] &=&{1 \over 2} \|{{\bf w}}\|^2 + {C \over T} \sum_{k=1}^{T} (\xi_k + \xi_k^*) + C \nu \epsilon - \lambda \epsilon + \gamma \left( \sum_i w_i -1 \right)\nonumber \\ && + \sum_{k=1}^{T} \alpha_k^* ({{\bf w} \cdot {{\bf x}^{(k)}}}- \epsilon - \xi_k^*) - \sum_{k=1}^{T} \alpha_k ({{\bf w} \cdot {{\bf x}^{(k)}}}+ \epsilon + \xi_k) \nonumber \\ && - \sum_{k=1}^{T} (\eta_k \xi_k + \eta_k^* \xi_k^*) \label{L-sym} \\ &=& F[{{\bf w}}] + \epsilon \left( C \nu - \lambda - \sum_{k=1}^{T} (\alpha_k + \alpha_k^*) \right) -\gamma \label{Lagr-w-constr} \\ &&+ \sum_{k=1}^{T} \left[ \xi_k \left( {C \over T} -\alpha_k - \eta_k\right) + \xi_k^* \left( {C \over T} - \alpha_k^* - \eta_k^* \right) \right] \nonumber \label{L1}\end{aligned}$$ with $$\begin{aligned} F[{{\bf w}}] &=& {{\bf w}}\cdot \left({1 \over 2} {{\bf w}}- \left(\sum_{k=1}^{T} (\alpha_k - \alpha_k^*) {{\bf x}^{(k)}}- \gamma {\bf 1}\right) \right),\end{aligned}$$ where ${\bf 1}$ denotes the unit vector of length $N$. Setting the derivative of the Lagrangian w.r.t. ${{\bf w}}$ to zero gives: $$\begin{aligned} {{\bf w}}_{\rm opt} = \sum_{k=1}^{T} (\alpha_k - \alpha_k^*) {{\bf x}^{(k)}}- \gamma {\bf 1} \label{w-sym}\end{aligned}$$ This solution for the optimal portfolio is sparse in the sense that, due to the Karush-Kuhn-Tucker conditions (see e.g. [@Bertsekas95]), only those points contribute to the optimal portfolio weights, for which the inequality constraints in (\[con1\]), and the corresponding constraints in Eq. (\[con1sym\]), are met exactly. The solution of ${{\bf w}}_{\rm opt} $ contains only those points, and effectively ignores the rest. This sparsity contributes to the stability of the solution. Regularized portfolio optimization (RPO) operates, in contrast to general regression, with a fixed budget. As a consequence, the Lagrange multiplier $\gamma$ now appears in the optimal solution, Eq. (\[w-sym\]). Compared to the optimal solution in support vector (SV) regression, ${{\bf w}}_{\rm SV}$, the solution vector under the budget constraint, ${{\bf w}}_{\rm RPO}$, is shifted by $\gamma$: $${{\bf w}}_{\rm RPO} = {{\bf w}}_{\rm SV} - \gamma {\bf 1}.$$ Let us now consider the dual problem. The dual is, in general, a function of the dual variables, which are here $\{ {\bf \alpha}, {\bf \alpha^*}, \gamma, \lambda, {\bf \eta}, {\bf \eta^*} \}$, although we will see in the following that some of these variables drop out. The dual is defined as , and the dual problem is then to maximize $D$ over the dual variables. We can replace the minimization over ${{\bf w}}$ by evaluating the Lagrangian at ${{\bf w}}_{\rm opt}$. For that we have to evaluate $$\begin{aligned} F[{{\bf w}}_{\rm opt}] &=& - {1 \over 2} \| {{\bf w}}_{\rm opt} \|^2 \\ &=& \left[- {1 \over 2} \left(\sum_{k=1}^{T} (\alpha_k - \alpha_k^*) {{\bf x}^{(k)}}- \gamma {\bf 1}\right)^2 \right]. $$ For the other terms in the Lagrangian, we have to consider different cases: 1. If $\left( C \nu - \lambda - \sum_{k=1}^{T} (\alpha_k + \alpha_k^*) \right) < 0$, then $L$ can be minimized by letting $\epsilon \rightarrow \infty$, which means that $D = -\infty$. 2. If $\left( C \nu - \lambda - \sum_{k=1}^{T} (\alpha_k + \alpha_k^*) \right) \geq 0$: The term $\epsilon \left( C \nu - \lambda - \sum_{k=1}^{T} (\alpha_k + \alpha_k^*) \right)$ vanishes. Reason: if equality holds, this is trivially true, and if the inequality holds strictly then $L$ can be minimized by setting $\epsilon =0$. Similarly, for the other constraints (the notation $(*)$ means that this is true for variables with and without the asterisk): 1. If $\left( {C \over T} -\alpha_k^{(*)} - \eta_k^{(*)} \right) < 0$, then $L$ can be minimized by letting $\xi_k^{(*)} \rightarrow \infty$, which means that $D = -\infty$. 2. If $\left( {C \over T} -\alpha_k^{(*)} - \eta_k^{(*)} \right) \geq 0$, then $\xi_k \left( {C \over T} -\alpha_k^{(*)} - \eta_k^{(*)} \right) = 0$. Reason: If the inequality holds strictly then $L$ can be minimized by $\xi_k^{(*)} = 0$. If equality holds then it is trivially true. By a similar argument, the term $\gamma$ in Eq. (\[Lagr-w-constr\]) disappears in the Dual. Altogether we have that either $D = - \infty$, or $$\begin{aligned} && D({\bf \alpha}, {\bf \alpha^*}, \gamma) = \min_{{\bf \xi}, {\bf \xi^*}, \epsilon} F[{{\bf w}}_{\rm opt}({\bf \alpha}, {\bf \alpha^*}, \gamma)] = - {1 \over 2} \| {{\bf w}}_{\rm opt} \|^2 \\ {\rm and} &\;\;\;& \sum_{k=1}^{T} (\alpha_k^* + \alpha_k) \leq C \nu - \lambda \\ {\rm and} &\;\;\;& \alpha_k^{(*)} + \eta_k^{(*)} \leq {C \over T}.\end{aligned}$$ Note that the variables $\xi_k^{(*)}, \eta_k^{(*)}, \epsilon, \lambda$ do not appear in $F[{{\bf w}}_{\rm opt}({\bf \alpha}, {\bf \alpha^*}, \gamma)]$. The dual problem is therefore given by $$\begin{aligned} \max_{{\bf \alpha}, {\bf \alpha^*}, \gamma} && \left[- {1 \over 2} \left(\sum_{k=1}^{T} (\alpha_k - \alpha_k^*) {{\bf x}^{(k)}}- \gamma {\bf 1}\right)^2 \right]. \\ {\rm s.t.}&& \{ \alpha_k, \alpha_k^* \} \in \left[0,{C \over T}\right] \\ && \sum_{k=1}^{T} (\alpha_k^* + \alpha_k) \leq C\nu.\end{aligned}$$ We can analytically maximize over $\gamma$ and obtain for the optimal value $$\gamma = {1 \over N} \left(\sum_{k=1}^{T} (\alpha_k - \alpha_k^*) \sum_{i=1}^N {x^{(k)}_i}- 1 \right) \label{gamma}$$ The optimal projection (= optimal portfolio) is given by $$\!\!\!\!\!\!\!\! \!\!\!\! \!\!\!\! \!\!\!\! \!\!\!\! {{\bf w}}_{\rm opt} \cdot {{\bf x}}= \sum_{k=1}^{T} (\alpha_k - \alpha_k^*) {{\bf x}^{(k)}}\cdot {{\bf x}}- {1 \over N} \left(\sum_{k=1}^{T} (\alpha_k - \alpha_k^*) \sum_{i=1}^N {x^{(k)}_i}- 1 \right) {\bf1} \cdot {{\bf x}}. \label{RPO-sol-w}$$ For $N \rightarrow \infty$ the second term vanishes and the solution is the same as the the solution in support vector regression. Note that the kernel-trick (see e.g. [@Bernhard_Book1]), which is used in support vector machines to find nonlinear models hinges on the fact that only dot products of input vectors appear in the support vector expansion of the solution. As a consequence of the budget constraint, one can no longer use the kernel-trick (compare Eq. (\[RPO-sol-w\])). As long as we disregard derivatives, this is not a problem for portfolio optimization. Keep in mind, however, that the budget constraint introduces this otherwise undesirable property. Support vector algorithms typically solve the dual form of the problem (for a recent survey see [@leon2006]), which is in our case given by $$\begin{aligned} \!\!\!\!\!\!\!\! \max_{{\bf \alpha}, {\bf \alpha^*}, \gamma} && - {1 \over 2} \left[ \sum_{k=1}^{T} \sum_{l=1}^{T} (\alpha_k - \alpha_k^*)(\alpha_l - \alpha_l^*) \left( {{\bf x}^{(k)}}{{\bf x}^{(l)}}- {1\over N} \sum_{i=1}^{N} x^{(k)}_i \sum_{i=1}^{N} x^{(l)}_i \right) \right] \label{dual-sym-rpo} \\ \!\!\!\!\!\! {\rm s.t.}&& \{ \alpha_k, \alpha_k^* \} \in \left[0,{C \over T}\right]; \nonumber \\ && \sum_{k=1}^{T} (\alpha_k^* + \alpha_k) \leq C\nu. \nonumber\end{aligned}$$ For $N \rightarrow \infty$ the problem becomes [*identical*]{} to $\nu$-SVR, which can be solved by linear programming, for which software packages are available [@LOQO]. For finite $N$, it can still be solved with existing methods, because it is quadratic in the $\alpha_k$’s. Solvers such as the ones discussed in [@bordes-ertekin-weston-bottou-2005] and [@leon2006] can be used, but have to be adapted to this specific problem. The regularized symmetric tail average minimization problem (Eq. (\[gen-RES\]) with the constraints Eqs. (\[con1\])–(\[b\]) and (\[con1sym\])) is, as we have shown here, directly related to support vector regression which uses the $\epsilon$-insensitive loss function. The $\epsilon$-insensitive loss is stable to local changes for data points that fall outside the range specified by $\epsilon$. This point is elaborated in Section 3 in [@Nu-SVM], and relates this method to robust estimation of the mean. It can also be extended to robust estimation of quantiles [@Nu-SVM] by scaling of the slack variables $\xi_k$ by $\mu$ and $\xi_k^*$ by $1-\mu$, respectively. This scaling translates directly to the portfolio optimization problem, which is an extreme case: downside risk measures penalize only loss, not gain. The asymmetry in the loss function corresponds to $\mu =1$. Regularized expected shortfall. {#RES-par} ------------------------------- By this final change we arrive at the regularized portfolio optimization problem, Eqs. (\[newPO\])–(\[b\]), which we originally set out to solve. This is now easily solved in analogy to the previous paragraphs: the slack variables $\xi_k^*$ disappear, together with the respective Lagrange multipliers which enforce constraints, including $\alpha_k^*$. The optimal solution is now $$\begin{aligned} &{{\bf w}}_{\rm opt} = \sum_{k=1}^{T} \alpha_k {{\bf x}^{(k)}}- \gamma {\bf1},\end{aligned}$$ with $$\begin{aligned} \gamma &=& {1 \over N} \left(\sum_{k=1}^{T} \alpha_k \sum_{i=1}^{N} {x^{(k)}_i}- 1 \right).\end{aligned}$$ The dual problem is given by $$\begin{aligned} \max_{\alpha_k}&& - {1 \over 2} \left[ \sum_{k=1}^{T} \sum_{l=1}^{T} \alpha_k \alpha_l \left( {{\bf x}^{(k)}}{{\bf x}^{(l)}}- {1\over N} \sum_{i=1}^{N} x^{(k)}_i \sum_{i=1}^{N} x^{(l)}_i \right) \right] \nonumber \\ {\rm s.t.}&& \alpha_k \in \left[0,{C \over T}\right]; \;\; \sum_{k=1}^{T} \alpha_k \leq C\nu.\end{aligned}$$ which, like its symmetric counterpart, Eq. (\[dual-sym-rpo\]), can be solved by adjusting existing algorithms. The formalism provides a free parameter, $C$, to set the balance between the original risk function and the regularizer. Its choice may depend on a number of factors, such as the investors time horizon, the nature of the underlying data, and, crucially, on the ratio $N/T$. Intuitively, there must be a maximum allowable value $C_{\rm max}(N/T)$ for $C$, such that when one puts more emphasis on the data, $C > C_{\rm max}(N/T)$, then over fitting will occur with high probability. It would be desirable to know an analytic expression for (a bound on) $C_{\rm max}(N/T)$. In practice, cross-validation methods are often employed in machine learning to set the value of $C$. Those methods are not free of problems (see, for example, the treatment in [@bengio]), and the optimal choice of this parameter remains an open problem. Regularization corresponds to portfolio diversification. {#diversification} ======================================================== Above, we have controlled the capacity of the linear model by minimizing the L2 norm of the portfolio weight vector. In the finance context, minimizing $$\| {{\bf w}}\|^2 = \sum_i w_i^2 \simeq {1 \over N_{\rm eff}}$$ corresponds roughly to maximizing the effective number of assets, $N_{\rm eff}$, i.e. to exerting a [*pressure*]{} towards portfolio diversification [@Bouchaudpotters]. We conclude that diversification of the portfolio is crucial, because it serves to counteract the observed instability by acting as a regularizer. Other constraints that penalizes the length of the weight vector could alternatively be considered as a regularizer, in particular any Lp norm. The budget constraint [*alone*]{}, however, does not suffice as a regularizer, since it does not constrain the length of the weight vector. Adding a ban on short selling, $w_i \geq 0$, to the budget constraint, $\sum_i w_i = 1$, limits the allowable solutions to a finite volume in the space of weights and is equivalent to requiring that $\sum_i | w_i | \leq 1$.[^3] It thereby imposes a limit on the L1 norm, that is on the sum of the absolute amplitudes of long and short positions. One may argue that it may be a good idea to use the L1 norm instead of the L2 norm, because that may make the solution sparser. However, the L1 norm has a tendency to make some of the weights vanish. Indeed, it has been shown that in the orthonormal design case (using the variance as the risk measure) an L1 regularizer will set some of the weights to zero, while an L2 regularizer will scale all the weights [@Tibshirani96]. The spontaneous reduction of portfolio size has also been demonstrated in numerical simulations [@Kondor4]: as one goes deeper and deeper into the regime where $T$ is significantly smaller than $N$, under a ban on short selling, more and more of the weights will become zero. The same “freezing out" of the weights has been observed in portfolio optimization [@scherermartin] as an empirical fact. It is important to stress that the vanishing of some of the weights does not reflect any structural property of the objective function, it is just a random effect: as clearly demonstrated by simulations [@Kondor4], for a different sample a different set of weights vanishes. The angle of the weight vector fluctuates wildly from sample to sample. (The behavior of the solutions is similar for other limit systems as well.) This means that the solutions will be determined by the limit system and the random sample, rather than by the structure of the market. So the underlying instability is merely “masked", in that the solutions do not run away to infinity, but they are still unstable under sample fluctuations when $T$ is too small. As it is certainly not in the interest of the investor to obtain a portfolio solution which sets weights to zero on the basis of unreliable information from small samples, the above observations speak strongly in favor of using the L2 norm over the L1 norm. Conclusion ========== We have made the observation that the optimization of large portfolios minimizes the empirical risk in a regime where the data set size is similar to the size of the portfolio. In that regime, a small empirical risk does not necessarily guarantee a small actual risk [@VapnikCh71]. In this sense naive portfolio optimization over-fits the data. Regularization can overcome this problem by reducing the capacity of the considered model class. Regularized portfolio optimization has choices to make, not only about the risk function, but also about the regularizer. Here, we have focussed on the increasingly popular expected shortfall risk measure. Using the L2 norm as a regularizer leads to a convex optimization problem which can be solved with linear programming. We have shown that regularized portfolio optimization is then a variant of support vector regression. The differences are an asymmetry, due to the tolerance to large positive deviations, and the budget constraint, which is not present in regression. Our treatment provides a novel insight into why diversification is so important. The L2 regularizer implements a pressure towards portfolio diversification. Therefore, from a statistical point of view, diversification is important as it is one way to control the capacity of the portfolio optimizer and thereby to find a solution which is more stable, and hence meaningful. In summary, the method we have outlined in this paper allows for the unified treatment of optimization and diversification in one principled formalism. It shows how known methods from modern statistics can be used to improve the practice of portfolio optimization. Acknowledgements ================ We thank Leon Bottou for helpful discussions and comments on the manuscript. This work has been supported by the “Cooperative Center for Communication Networks Data Analysis", a NAP project sponsored by the National Office of Research and Technology under grant No. KCKHA005. SS thanks the Collegium Budapest for hosting her during this collaboration, and the community at the Collegium for providing a creative and inspiring atmosphere. [^1]: Notation: bold face symbols are understood to denote vectors. [^2]: The mathematical similarity between minimum expected shortfall [*without*]{} regularization and the E$\nu$-SVM algorithm [@EnuSVM] was pointed out, but incorrectly, in [@Takeda2008]. There is an important difference between the two optimization problems. In E$\nu$-SVM, the length of the weight vector, $\| {{\bf w}}\|$, is constrained, which implements capacity control. In the pure expected shortfall minimization, Eq. (\[CVaR\]), this is not done. Instead, the total budget $\sum_i w_i$ is fixed. This difference is not correctly identified in the proof of the central theorem (Theorem 1) in [@Takeda2008]. [^3]: This point has been made independently by [@DeMiguel2009].

--- author: - '[Sophia Yakoubov]{}' date: | [`sonka89@mit.edu`]{}\ [MIT]{} title: | \ [Pattern Avoidance in Extensions of Comb-Like Posets]{} ---

--- abstract: 'Long-lived alpha and beta emitters in the $^{222}$Rn decay chain on (and near) detector surfaces may be the limiting background in many experiments attempting to detect dark matter or neutrinoless double beta decay, and in screening detectors. In order to reduce backgrounds from radon-daughter plate-out onto the wires of the BetaCage during its assembly, an ultra-low-radon cleanroom is being commissioned at Syracuse University using a vacuum-swing-adsorption radon-mitigation system. The radon filter shows $\sim$20$\times$ reduction at its output, from $7.47\pm0.56$ to $0.37\pm0.12$Bq/m$^{3}$, and the cleanroom radon activity meets project requirements, with a lowest achieved value consistent with that of the filter, and levels consistently $<2$Bq/m$^{3}$.' author: - 'R.W. Schnee' - 'R. Bunker' - 'G. Ghulam' - 'D. Jardin' - 'M. Kos' - 'A.S. Tenney' bibliography: - 'schnee.bib' title: 'Construction and measurements of a vacuum-swing-adsorption radon-mitigation system' --- [ address=[Department of Physics, Syracuse University, Syracuse, NY 13244]{} ]{} [ address=[Department of Physics, Syracuse University, Syracuse, NY 13244]{} ]{} [ address=[Department of Physics, Syracuse University, Syracuse, NY 13244]{} ]{} [ address=[Department of Physics, Syracuse University, Syracuse, NY 13244]{} ]{} [ address=[Department of Physics, Syracuse University, Syracuse, NY 13244]{} ]{} [ address=[Department of Physics, Syracuse University, Syracuse, NY 13244]{} ]{} Introduction to radon mitigation {#sec:intro} ================================ A potentially dominant background for many rare-event searches or screening detectors is from radon daughters deposited from the atmosphere onto detector components. Examples include $^{214}$Po for SuperNEMO [@LRT2013SuperNEMO]; $^{210}$Pb for EDELWEISS [@LRT2013edelweiss], SuperCDMS [@LRT2013SuperCDMS] and the BetaCage [@LRT2013bunker]; $^{210}$Po for CUORE [@LRT2013cuore]; the $^{206}$Pb recoil nucleus from $^{210}$Po $\alpha$ decay for CRESST [@cresst2012], DEAP/CLEAN [@deap2011surface], and SuperCDMS [@LRT2013SuperCDMS]; and neutrons from ($\alpha,n$) reactions on Teflon for LUX, XENON1T, and DArKSIDE. To protect detector components, assembly within vacuum glove boxes and/or cleaning after assembly ( [@LRT2013schneeEP]) may be effective. However, vacuum glove boxes are impractical for large objects or for delicate assembly that could be jeopardized by reduced feel and range of motion. Similarly, cleaning after assembly may be difficult and risky for complicated structures. For these cases, providing radon-reduced air in a breathable atmosphere may be necessary . There are two basic types of radon-mitigation systems: those with continuous flow through a single filter (typically of activated charcoal), and “swing” systems with flow alternating through two or more different filters. Continuous systems ( [@nemoLRT2006]) are designed so that most radon decays before it exits the filter. For an ideal column, the final radon concentration $C_{\mathrm{final}} = C_{\mathrm{initial}} \exp(-t/t_{\mathrm{Rn}})$, where $C_{\mathrm{initial}}$ is the concentration of the input air, $t$ is the characteristic breakthrough time of the filter, and $t_{\mathrm{Rn}} = 5.5$days is the Rn lifetime. In order to have a sufficiently large breakthrough time to be effective, the carbon must be cooled. Continuous systems are relatively simple and robust, are available commercially, and typically achieve reduction factors of $\sim1000$, to $\sim$10–30mBq/m$^{3}$. In a swing system, one stops gas flow well before the breakthrough time $t$, and regenerates the first filter column while switching flow to a second column. For an ideal column, no radon reaches the output. Swing systems are more complicated than continuous systems (both in terms of their analysis and operation). Vacuum-swing systems ( [@LRT2004Pocar; @PocarThesis]) can potentially provide better performance than a continuous system at lower cost. Temperature-swing systems ( [@LRT2010HallinRadon]) should provide the best performance, albeit at the highest cost and complexity. Due to its potential and especially its low cost, we chose to build a vacuum-swing system in order to achieve the relatively modest radon reduction needed for construction of the BetaCage at Syracuse [@LRT2013bunker]. A vacuum-swing adsorption system takes advantage of the filtering medium’s greater adsorption capacity at high pressures. The carbon is regenerated by flowing a small fraction $f$ of filtered gas of mass flow $F$ back through the tank at low purge pressure $P_\mathrm{purge}$. The volume purge flow $$\phi_{\mathrm{purge}} = \frac{P_\mathrm{atm}} {P_\mathrm{purge}} f F = \frac{P_\mathrm{atm}} {P_\mathrm{purge}} f \phi_{\mathrm{feed}} .$$ On each cycle, the radon front is pushed back more than it moves forward if the volume flow gain $G \equiv \phi_\mathrm{purge} / \phi_\mathrm{feed}>1$, that is if $f P_\mathrm{atm} > P_\mathrm{purge}$ . ![[*Left*]{}: Photo of the cleanroom exterior, showing the externally housed HVAC system (before retrofitting) and the anteroom door. [*Right*]{}: The radon-mitigation system. Air ($\sim$10Bq/m${^3}$) enters through the top intake, then passes through a cooling coil, dehumidifier, and a second cooling coil in order to reduce the air’s dew point to $<-12$C, before flowing through Carbon Tank 1 or 2. A small fraction of the air passes back through the other tank to the vacuum pump, while most of it passes through a filter (where it is monitored with a RAD7) and to the output duct that supplies air to the cleanroom one floor below.[]{data-label="fig:setup"}](clean_room.jpg "fig:"){height="2.25in"} ![[*Left*]{}: Photo of the cleanroom exterior, showing the externally housed HVAC system (before retrofitting) and the anteroom door. [*Right*]{}: The radon-mitigation system. Air ($\sim$10Bq/m${^3}$) enters through the top intake, then passes through a cooling coil, dehumidifier, and a second cooling coil in order to reduce the air’s dew point to $<-12$C, before flowing through Carbon Tank 1 or 2. A small fraction of the air passes back through the other tank to the vacuum pump, while most of it passes through a filter (where it is monitored with a RAD7) and to the output duct that supplies air to the cleanroom one floor below.[]{data-label="fig:setup"}](MitigationSystemFrontSmall.pdf "fig:"){height="2.25in"} Design and Construction of the Syracuse radon-mitigation system {#sect:system} =============================================================== In order to increase the effectiveness of filtration on removal of radon daughters and to limit radon sources, the cleanroom itself (see left panel of Fig. \[fig:setup\]) was designed to be as small as practically possible. The room is 8ft by 12ft by 8ft high, with a 4ft by 8ft anteroom. To minimize emanation and permeation, it uses all aluminum panels and extrusions, with thick acrylic windows, and steps were taken to ensure that the room is very leak tight. The HVAC is outside the cleanroom and required retrofitting to make it sufficiently leak tight not to be a dominant source of radon. Aged water is used for humidification. The room was designed for 30cfm low-radon makeup air, with fast HEPA filtration recirculating at a rate of one air exchange per 43 seconds. ---------------------------------------------------------------------------------------- ------ -------- ---------------------------------------------- tanks 8k 9k specweld.com charcoal 6k 1.5k less and different carbon (see text) vacuum pumps 22k 10k roughing pump has lower capacity (see text) valves 4k 7k VAT 3“ (2” for purge and make-up air) dryer 3.5k 7.5k Munters HC-150 and Hack Air cooling coils blower 1.5k (none) filter + housing 1.5k 1k Clark Air ASHRAE filters at input and output PC and valve control boards 1.5k 6k including gauges other (fittings, tubing, *[etc.]{}) & 5k & 5k + 8k chiller & Pro Air Plus ACCPS015-2B\ total & 53k & 55k &\ * ---------------------------------------------------------------------------------------- ------ -------- ---------------------------------------------- : Comparison of costs of components for 2004 Princeton [@PocarThesis] and 2013 Syracuse VSA systems.[]{data-label="tab:a"} The Syracuse radon-mitigation system (see right panel of Fig. \[fig:setup\]) was based closely on the Princeton design [@LRT2004Pocar; @PocarThesis]. We tried to make some improvements focusing on ensuring that the radon reduction at the filter output was realized in the cleanroom, and we cut several corners in an effort to minimize costs. Table \[tab:a\] compares the costs of components for the Princeton [@PocarThesis] and Syracuse systems. Most notably, we use a roughing pump (Edwards E2M80) with significantly lower capacity at high pressures (and a significant cost savings). We also use about 60% as much carbon (possible due to the $\sim$2$\times$ lower airflow). Because the carbon used in the Princeton system is no longer available, we selected the most similar product available (Calgon Coconut Activated Carbon Product OVC Plus 4$\times$8 mesh). The carbon was multiply rinsed (with a final rinse in deionized water), then dried under high-flow fume hoods. Two identical stainless-steel vacuum vessels were filled with $\sim$150kg each and spring-loaded in order to maintain firmly packed columns during swing operation. As a check, we opened a tank after the first month of commissioning, finding that the carbon was still in good shape and well packed. As shown in the left panel of Fig. \[fig:predictions\], the lower capacity of the Syracuse pump at high pressures leads to a 5-min pump down to $\sim$10Torr (vs. Princeton $\sim$1min), so part of the purging cycle is inefficient. However, the lower achieved base pressure allows the Syracuse system to operate at a lower purge pressure for the same purge flow, as shown in the center panel of Fig. \[fig:predictions\], resulting in a high predicted volume flow gain $G$, as shown in the right panel of Fig. \[fig:predictions\]. Although system performance is best in principle when the swing period is short compared to the breakthrough time, the finite pump-down time limits the minimum swing period in practice. All results described here use a swing period of 90min (although further optimization may be possible); during each 45-min half-period, one column filters air for the cleanroom while the other is evacuated for $\sim$5min to $<$10Torr, regenerated for $\sim$39min, and finally repressurized to 1atm for 1min (which also acts as necessary dead time for recovery of the roughing pump). Measurements to determine the optimal output flow rate are ongoing, while the purge flow rate is typically set to $\sim$3cfm. ![[*Left*]{}: Pressure in a carbon tank on the pump side (light squares) and far side (dark stars) of the charcoal as a function of time since start of pump down, for the Princeton (small symbols) and Syracuse (large symbols) systems. The Syracuse system takes 4min longer to pump down to 10Torr, but also achieves a lower base pressure. [*Center*]{}: Purge pressure as a function of the purge flow for the Princeton (thin solid) and Syracuse (thick dash) systems. [*Right*]{}: Predicted volume flow gain $G$ (grayscale) as a function of the output and purge flows of the Syracuse system. Lower output flow results in higher $G$ but is less effective overcoming emanation or leaks in the cleanroom. []{data-label="fig:predictions"}](pumpdownPUSU.pdf "fig:"){height="2.2in"} ![[*Left*]{}: Pressure in a carbon tank on the pump side (light squares) and far side (dark stars) of the charcoal as a function of time since start of pump down, for the Princeton (small symbols) and Syracuse (large symbols) systems. The Syracuse system takes 4min longer to pump down to 10Torr, but also achieves a lower base pressure. [*Center*]{}: Purge pressure as a function of the purge flow for the Princeton (thin solid) and Syracuse (thick dash) systems. [*Right*]{}: Predicted volume flow gain $G$ (grayscale) as a function of the output and purge flows of the Syracuse system. Lower output flow results in higher $G$ but is less effective overcoming emanation or leaks in the cleanroom. []{data-label="fig:predictions"}](PUSUradonPurge.png "fig:"){height="2.2in"} ![[*Left*]{}: Pressure in a carbon tank on the pump side (light squares) and far side (dark stars) of the charcoal as a function of time since start of pump down, for the Princeton (small symbols) and Syracuse (large symbols) systems. The Syracuse system takes 4min longer to pump down to 10Torr, but also achieves a lower base pressure. [*Center*]{}: Purge pressure as a function of the purge flow for the Princeton (thin solid) and Syracuse (thick dash) systems. [*Right*]{}: Predicted volume flow gain $G$ (grayscale) as a function of the output and purge flows of the Syracuse system. Lower output flow results in higher $G$ but is less effective overcoming emanation or leaks in the cleanroom. []{data-label="fig:predictions"}](SUradonGgray.png "fig:"){height="2.2in"} Current radon-mitigation results and status =========================================== The radon-mitigation system was first turned on in December 2012. First results, shown in the left panel of Fig. \[fig:results\], indicated a radon reduction of $\sim$20$\times$, to $0.4\pm0.1$Bq/m$^{3}$. Assuming a dynamic adsorption coefficient of 3m$^{3}$/kg (STP), which is on the low end of the range for typical activated carbon, the predicted breakthrough time for each column is 4.2h for a volume flow rate $\phi_{\mathrm{feed}} = 60$cfm. By simply switching from the swing cycle to flowing air continuously through a single tank, a lower limit on the actual breakthrough time of about two hours was measured. First measurement of the radon level in the clean room, with the cleanroom air circulation off, was performed in April 2013, as shown in Fig. \[fig:results\]. The measured radon activity of 0.4Bq/m$^{3}$ was the same as that measured at the input duct to the cleanroom and indicates that emanation and leaks in the cleanroom itself are minor. Subsequent measurements with the air circulation on indicated significant leaks in the air circulation path which had to be fixed. Operation of the radon filter since this time with various configurations has led to a range of radon output levels (some consistent with zero), averaging to $0.33\pm0.13$Bq/m$^{3}$, while levels in the cleanroom have been consistently $<2$Bq/m$^{3}$. The 0.4Bq/m$^{3}$ output level of the radon filter is slightly worse than that of the Princeton Borexino system on which it was based. Optimization of the feed and purge flows and cycle times may yield improved performance. Moreover, we expect planned improvements to the leak tightness of the filter’s output duct will bring the cleanroom radon concentration (with circulation on) in line with the filter’s output. Even so, the cleanroom radon concentration already achieved is better than the level required for construction of the BetaCage [@LRT2013bunker]. The concentration is lower than that achieved for the Princeton system (by $\sim4\times$ with the circulation off), likely due to reduced emanation from the cleanroom itself. The results indicate that the VSA technique remains a viable lower-cost alternative to radon mitigation using cooled carbon filters. ![[*Left*]{}: Radon activities measured for the input air (light $\times$’s and upper band) and output air (dark $\times$’s and lower band) of the VSA filter, after subtraction of the RAD7’s intrinsic background (measured with boil-off nitrogen to be $0.19\pm0.03$Bq/m$^{3}$ and consistent with expectation). Average values (shaded bands) of the multi-hour periods (error bars) derived by rebinning RAD7 measurements of $^{214}$Po and $^{218}$Po alpha decays (originally taken in 1-hour intervals) indicate $\sim$20$\times$ reduction, from $7.47\pm0.56$ to $0.37\pm0.12$Bq/m$^{3}$. [*Right*]{}: Radon activities measured at the filter output (dark $\times$’s and narrow band around lower dashed line) and within the cleanroom (light $\times$’s and wide band around higher dashed line). Average values (shaded bands) of the 7-hour periods (error bars) indicate a cleanroom radon activity consistent with the level measured at the filter’s output. Uncertainties on the cleanroom activity are relatively large due to measurement with an older RAD7 with a larger intrinsic background (of $0.67\pm0.10$Bq/m$^{3}$). Note that the RAD7 uncertainties quoted here are known to be overly conservative. []{data-label="fig:results"}](VSAresult.png "fig:"){height="2.25in"} ![[*Left*]{}: Radon activities measured for the input air (light $\times$’s and upper band) and output air (dark $\times$’s and lower band) of the VSA filter, after subtraction of the RAD7’s intrinsic background (measured with boil-off nitrogen to be $0.19\pm0.03$Bq/m$^{3}$ and consistent with expectation). Average values (shaded bands) of the multi-hour periods (error bars) derived by rebinning RAD7 measurements of $^{214}$Po and $^{218}$Po alpha decays (originally taken in 1-hour intervals) indicate $\sim$20$\times$ reduction, from $7.47\pm0.56$ to $0.37\pm0.12$Bq/m$^{3}$. [*Right*]{}: Radon activities measured at the filter output (dark $\times$’s and narrow band around lower dashed line) and within the cleanroom (light $\times$’s and wide band around higher dashed line). Average values (shaded bands) of the 7-hour periods (error bars) indicate a cleanroom radon activity consistent with the level measured at the filter’s output. Uncertainties on the cleanroom activity are relatively large due to measurement with an older RAD7 with a larger intrinsic background (of $0.67\pm0.10$Bq/m$^{3}$). Note that the RAD7 uncertainties quoted here are known to be overly conservative. []{data-label="fig:results"}](CleanroomResult.png "fig:"){height="2.25in"} This work was supported in part by the National Science Foundation (Grant No. PHY-0855525). The authors thank T. Shutt, A. Hallin, A. Pocar, W. Rau, and V. Guiseppe for useful discussions and advice.

--- abstract: 'We establish a moduli space $\mathbb E$ of stationary vacuum metrics in a spacetime, and set up a well-defined boundary map $\Pi$ in $\mathbb E$, assigning a metric class with its Bartnik boundary data. Furthermore, we prove the boundary map $\Pi$ is Fredholm by showing that the stationary vacuum equations (combined with proper gauge terms) and the Bartnik boundary conditions form an elliptic boundary value problem. As an application, we show that the Bartnik boundary data near the standard flat boundary data admits a unique (up to diffeomorphism) stationary vacuum extension locally.' author: - Zhongshan An title: Ellipticity of Bartnik boundary data for stationary vacuum spacetimes --- Introduction ============ In general relativity, one of the most interesting and well-known notions of quasi-local mass is the Bartnik quasi-local mass. Let $\Omega$ be a bounded smooth 3-manifold with nonempty boundary $\Sigma$. Equip $\Omega$ with a Riemannian metric $g$ and a symmetric 2-tensor $K$, which is essentially the second fundamental form of $\Omega$ when it is embedded in some spacetime. The Bartnik quasi-local mass of the data set $(\Omega,g,K)$ is defined as (cf.\[B1\],\[B2\]), $$m_B[(\Omega,g,K)]=\text{inf}\{m_{ADM}[(M,g,K)]\},$$ where the infimum is taken over all asymptotically flat admissible initial data sets $(M,g,K)$ such that after gluing $M$ and $\Omega$ along the boundary $\partial M\cong\partial \Omega$, the manifold $M\cup\Omega$ forms a complete asymptotically flat spacetime initial data set. By analyzing the constraint equations across the boundary $\Sigma=\partial\Omega\cong\partial M$, Bartnik proposed a set of geometric boundary data for $(\Omega,g,K)$ given by, $$(g_{\Sigma}, H_{\Sigma}, tr_{\Sigma}K,\omega_{\mathbf n_{\Sigma}}).$$ Here $g_{\Sigma}$ is the induced metric on the boundary $\Sigma$ obtained from $(\Omega, g)$; $H_{\Sigma}$ is the mean curvature of $\Sigma\subset (\Omega,g)$; $tr_{\Sigma}K$ is the trace of the restriction $K|_{\Sigma}$ of the second fundamental form; and $\omega_{\mathbf n_{\Sigma}}$ is the connection 1-form of the spacetime normal bundle of $\Sigma$, which is defined as, $$\omega_{\mathbf n_{\Sigma}}(v)=K(\mathbf n_{\Sigma},v),~\forall v\in T\Sigma,$$ where $\mathbf n_{\Sigma}$ is the outward unit normal vector field on $\Sigma\subset (\Omega,g)$. Then definition (1.1) of the Bartnik quasi-local mass can be reduced to the infimum ADM mass taken over all asymptotically flat admissible initial data sets $(M,g,K)$ which satisfy the following boundary conditions: $$\begin{cases} g_{\partial M}=g_{\Sigma}\\ H_{\partial M}=H_{\Sigma}\\ tr_{\partial M}K=tr_{\Sigma}K\\ \omega_{\mathbf n_{\Sigma}}=\omega_{\mathbf n_{\partial M}}. \end{cases}$$ The tuple of geometric boundary data (1.2) is called the Bartnik boundary data. It also arises naturally from a Hamiltonian analysis of the vacuum Einstein equations. In fact, a regularization $\mathcal H$ of the Regge-Teitelboim Hamiltonian is constructed in \[B3\]. When the spacetime has empty boundary, by analyzing the functional $\mathcal H$ and following an approach initiated by Brill-Deser-Fadeev (cf.\[BDF\]), Bartnik proved that stationary metrics are critical points of the ADM energy functional on the constraint manifold. However, if the spacetime has non-empty boundary, boundary terms arise from the variation of $\mathcal H$; they were explicitly identified by Bartnik in \[B1\], and the boundary terms vanish if and only if the Bartnik boundary data (1.2) is preserved in the variation. It was conjectured in \[B1\] that the Bartnik quasi-local mass of a given data set $(\Omega,g,K)$ must be realized by an admissible extension $(M,g,K)$ which can be embedded as an initial data set into a stationary vacuum spacetime. To solve this conjecture, one of the well-known and fundamental open problems raised by Bartnik in \[B1\] is the following: $$\textit{Is the Bartnik boundary data elliptic for stationary vacuum metrics?}$$ In this paper, we give a positive answer to this question. A stationary spacetime $(V^{(4)},g^{(4)})$ is a 4-manifold $V^{(4)}$ with a smooth Lorentzian metric $g^{(4)}$ of signature $(-,+,+,+)$, which admits a time-like Killing vector field. In addition, a stationary spacetime is called $vacuum$ if it solves the vacuum Einstein equation $$Ric_{g^{(4)}}=0.$$ Throughout this paper, we assume that the stationary spacetime $(V^{(4)},g^{(4)})$ is *globally hyperbolic*, i.e. it admits a Cauchy surface $M$ and $V^{(4)}\cong\mathbb{R}\times M$. In this case, there exists a global time function $\tau$ on $V^{(4)}$ so that $M=\{\tau=0\}$ and every surface of constant $\tau$ is a Cauchy surface. Since the spacetime is stationary, one can choose local coordinates $\{\tau, x^i\}~(i=1,2,3)$ so that ${\partial_{\tau}}$ is the time-like Killing vector field. Then the metric $g^{(4)}$ can be written globally in the form $$g^{(4)}=-N^2d\tau^2+g_{ij}(dx^i+X^id\tau)(dx^j+X^jd\tau).$$ Notice that since $\partial_{\tau}$ is a Killing vector field, the stationary spacetime $(V^{(4)},g^{(4)})$ is vacuum if and only if the equation (1.5) holds on $M$. *Remark* In the expression of $g^{(4)}$ above, the scalar field $N$ and the vector field $X$ in $V^{(4)}$ are usually called the *lapse function* and the *shift vector* of $g^{(4)}$ in this 3+1 formalism of the spacetime. The tensor field $g$ is the induced (Riemannian) metric on the Cauchy surfaces $\{\tau=\text{constant}\}\subset (V^{(4)},g^{(4)})$. Since the spacetime is stationary, the fields $g, X$ and $N$ are all independent of the time variable $\tau$, so they can be regarded as tensor fields on the hypersurface $M$. Consequently, the vacuum equation (1.5) is an elliptic system (modulo gauge) of the fields $(g,X,N)$ on $M$. Let $K$ be the second fundamental form of $M\subset (V^{(4)},g^{(4)})$. The triple $(M,g,K)$ is called an $initial$ $data$ $set$ of the spacetime. In the case where the boundary $\partial M$ is nonempty, we can impose the Bartnik boundary condition (1.3) on this data set, coupling with the vacuum equation (1.5). So we obtain a boundary value problem (BVP) as, $$\begin{split} &Ric_{g^{(4)}}=0\quad\text{on }M,\\ &\begin{cases} g_{\partial M}=\gamma\\ H_{\partial M}=H\\ tr_{\partial M}K=k\\ \omega_{\mathbf n_{\partial M}}=\tau \end{cases}\quad\text{on }\partial M, \end{split}$$ where $\gamma,H,k$, and $\tau$ are prescribed tensor fields on $\partial M$. Now, the ellipticity question (1.4) is essentially asking whether this BVP is elliptic. Another way to formulate question (1.4) is to establish a boundary map. Let $\mathbf B(\partial M)$ denote the space of Bartnik boundary data, i.e. space of tuples $(\gamma,H,k,\tau)$ on $\partial M$. Let $\mathcal E$ be the space of stationary vacuum metrics on $V^{(4)}$. Then a natural boundary map $\Pi_1$ arises as, $$\begin{split} \Pi_1:\mathcal E&\rightarrow\mathbf B(\partial M),\\ \Pi_1(g^{(4)})&=(g_{\partial M},H_{\partial M}, tr_{\partial M} K,\omega_{\mathbf n}). \end{split}$$ The map $\Pi_1$ being Fredholm is essentially equivalent to that BVP (1.7) is elliptic. However, it is easy to observe that equation (1.5) is not elliptic, since it is invariant under diffeomorphisms, i.e., if $g^{(4)}$ is a stationary metric that solves (1.5), then the pull back metric $\Phi^*g^{(4)}$ of $g^{(4)}$ under an arbitrary time-independent diffeomorphism $\Phi$ of $V^{(4)}$ gives another stationary vacuum solution. This means that we need to add gauge terms to the BVP (1.7), and at the same time modify the domain space $\mathcal E$ in (1.8) to a moduli space. In this paper, we first analyze how to choose the right domain space for the boundary map to be well-defined. We conclude in §2 that the Bartnik boundary map should be established as, $$\begin{split} \Pi:\mathbb E&\rightarrow\mathbf B(\partial M),\\ \Pi([g^{(4)}])&=(g_{\partial M},H_{\partial M}, tr_{\partial M} K,\omega_{\mathbf n}). \end{split}$$ Here the moduli space $\mathbb E$ is the quotient of $\mathcal E$ by a particular diffeomorphism group $\mathcal D$. We refer to §2, cf.(2.14), for the exact definition of $\mathcal D$; roughly it is a natural intermediate group $\mathcal D_3\subset\mathcal D\subset\mathcal D_4$ between the groups of 3-dimensional diffeomorphisms on $M$ fixing the boundary $\partial M$ and 4-dimensional time-independent diffeomorphisms on $V^{(4)}$ fixing $\partial V^{(4)}$. In order to prove ellipticity of the map $\Pi$, we establish in §3 an associated BVP under a particular technical assumption (cf. Assumption 3.1). We prove this BVP is elliptic in §4, and from this derive the main theorem of this paper: The moduli space $\mathbb E$ is a $C^{\infty}$ smooth Banach manifold of infinite dimension and the boundary map $\Pi$ is Fredholm. We show in §5 that the theorem is still true without the technical assumption in §3, completing the proof of Theorem 1.1. To conclude, we apply this ellipticity result in §6 to show that the Bartnik boundary data near the standard flat (Minkowski) metric $\tilde g^{(4)}_0$ on $\mathbb R\times(\mathbb R^3\setminus B^3)$ can be locally uniquely realized by a stationary vacuum metric up to diffeomorphisms in $\mathcal D$. There is a neighborhood $\mathcal U\subset\mathbf B(S^2)$ of the standard flat boundary data $(g_0,2,0,0)$ such that for any $(\gamma,H,k,\tau)\in\mathcal U$, there is a unique stationary vacuum metric $g^{(4)}\in\mathcal E$ near $\tilde g^{(4)}_0$ up to diffeomorphisms in $\mathcal D$, for which $$\Pi_1(g^{(4)})=(\gamma,H,k.\tau).$$ Throughout, we assume the hypersurface $M\cong\mathbb R^3\setminus B^3$ (exterior problem), together with certain asymptotically flat conditions on the metric $g^{(4)}$. Meanwhile, all the methods and results here can be applied equally well in the case where $M\cong B^3$ (interior problem). This paper is a continuation of our previous paper \[Az\], in which we developed a general method to prove the ellipticity of boundary value problems for the stationary vacuum spacetime. Expanding this method, we study the ellipticity of the Bartnik boundary data here. Theorem 1.1 is a generalization of the results proved in \[AK\], where spacetimes are static. Theorem 1.2 generalizes the result in \[A2\] for static metrics. We refer to \[Mi\],\[J\],\[R\] for other existence results on stationary vacuum extensions of boundary data. The results we prove in this paper provide a firm foundation for future work on Bartnik’s conjecture about the quasi-local mass in spacetimes and the existence problem of stationary vacuum metrics that satisfy the Bartnik boundary conditions. To the author’s knowledge, this is the first ellipticity result of the Bartnik boundary data for general stationary vacuum metrics.\ \ **Acknowledgements** I would like to express great thanks to my advisor Michael Anderson for suggesting this problem and for valuable discussions and comments. background ========== Fix a 3-dimensional manifold $M\cong\mathbb R^3\setminus B^3$. Let $V^{(4)}\cong\mathbb R\times M$. By fixing a diffeomorphism between $V^{(4)}$ and $\mathbb R\times M$, we can equip $V^{(4)}$ with the natural coordinates $\{(t,p)\}$ where $t\in\mathbb R$ and $p\in M$. Fix the hypersurface $\{t=0\}\subset V^{(4)}$ and identify it with $M$. Let $\mathcal S$ denote the space of Lorenztian metrics $g^{(4)}$ on $V^{(4)}$ which satisfy the following conditions:\ \ *1. (globally hyperbolic) The metric $g^{(4)}$ can be expressed globally as $$g^{(4)}=-N^2dt^2+g_{ij}(dx^i+X^idt)(dx^j+X^jdt),$$ where $\{x^i\}(i=1,2,3)$ are local coordinates on $M$. There exists some time independent function $f\in C^{m,\alpha}_{\delta}(V^{(4)})$ (cf.§7.1) so that the transformation $\tau=t+f$ makes $\tau$ a time function of the spacetime $(V^{(4)},g^{(4)})$ in the sense of general relativity and every surface of constant $\tau$ is a Cauchy surface (cf. equation (1.6)). In particular, if $f\equiv0$ then expression (2.1) and (1.6) agree.\ 2. (stationary) The vector field $\partial_t$ is a time-like Killing vector field in $(V^{(4)},g^{(4)})$. So the triple $(g,X,N)$ is independent of $t$ and can be regarded as tensor fields on $M$ (cf. the remark below equation (1.6)). In addition, since $\langle\partial_t,\partial_t\rangle_{g^{(4)}}=-N^2+||X||_{g}^2<0$, one has $$N^2>||X||_{g}^2.$$ 3. (asymptotically flat) The metric $g^{(4)}$ decays to the flat (Minkowski) metric at infinity. Explicitly, $N$, $X$ and $g$ belong to the weighted Hölder spaces on $M$, given by, $$\begin{split} &g\in Met^{m,\alpha}_{\delta}(M),\\ &N-1\in C^{m,\alpha}_{\delta}(M),\\ &X\in T^{m,\alpha}_{\delta}(M), \end{split}$$ for some fixed number $m\geq 2,~ 0<\alpha<1,$ and $\frac{1}{2}<\delta<1$. We refer to the Appendix §7.1 for the precise definition of the weighted Hölder spaces.\ *$~~$ Throughout this paper, we will use $\langle X,Y \rangle_{g}$ to denote the inner product of two vector fields with respect to the metric $g$. The (square) norm of a vector field $X$ with respect to the metric $g$ is $\langle X,X\rangle_{g}=||X||^2_{g}$. We will omit the metric in the subcript when it is clear in the context what metric is being used. Based on the definition of the space $\mathcal S$, it is easy to observe that $\mathcal S$ is invariance under the action of the diffeomorphism group $\mathcal D_4$ (cf.(2.6) below). Consequently, the tensor field $g$ in (2.1), which can be taken as the induced metric on the hypersurface $M\subset (V^{(4)},g^{(4)})$, is not necessarily Riemannian. It is obvious that an element in $\mathcal S$ is uniquely determined by a triple of fields $(g,X,N)$ on $M$. Thus $\mathcal S$ is an open domain in a Banach space and so admits smooth Banach manifold structure. As is mentioned in the introduction, one can establish BVP (1.7) for $g^{(4)}\in\mathcal S$, but in order to make it elliptic, we need to add gauge terms. A standard choice is to use the Bianchi gauge, leading to a modified system with unknown $g^{(4)}\in\mathcal S$ as follows: $$\begin{split} &Ric_{g^{(4)}}+\delta^*_{g^{(4)}}\beta_{\tilde g^{(4)}}g^{(4)}=0 \quad\text{on}\quad M,\\ &\begin{cases} g_{\partial M}=\gamma\\ H_{\partial M}=H\\ tr_{\partial M}K=k\\ \omega_{\mathbf n}=\tau\\ \beta_{\tilde g^{(4)}}g^{(4)}=0. \end{cases}\quad\text{on}\quad\partial M \end{split}$$ In the system above, we add the term $\delta^*_{g^{(4)}}\beta_{\tilde g^{(4)}}g^{(4)}$ in the vacuum equation, and add the Dirichlet condition of the gauge term $\beta_{\tilde g^{(4)}}g^{(4)}$ on the boundary. Here the gauge term $\beta_{\tilde g^{(4)}}g^{(4)}$ is the Bianchi operator acting on the metric $g^{(4)}$ with respect to a fixed stationary vacuum metric $\tilde g^{(4)}$, i.e. $\beta_{\tilde g^{(4)}}g^{(4)}=\delta_{\tilde g^{(4)}}g^{(4)}+\frac{1}{2}dtr_{\tilde g^{(4)}}g^{(4)}$, where the reference metric $\tilde g^{(4)}\in\mathcal E$ (cf.(2.5) below). We use $\delta$ to denote the divergence operator $\delta =-tr\nabla $, and $\delta^*$ denotes the formal adjoint of the divergence operator, i.e. $\delta^*_{g^{(4)}}Y=\frac{1}{2}L_{Y}g^{(4)}$ for any $Y\in TV^{(4)}$. Among the Bartnik boundary conditions, here and throughout the following, we use $\omega_{\mathbf n}$ as the abbreviation of $\omega_{\mathbf n_{\partial M}}$. The effect of adding the gauge term to BVP (1.7) as above is to give a slice to the action on the solution space of (1.7) by the group $\mathcal D_4$ (cf.(2.6) below) of diffeomorphisms of the spacetime fixing the boundary $\partial V^{(4)}$. However, such a modification has two issues. First, it is easy to observe that $(2.4)$ is not well posed, because there are 10 interior equations on $M$ but 11 boundary conditions on $\partial M$ — notice that, the gauge term $\beta_{\tilde g^{(4)}}g^{(4)}$ defines a vector field in $V^{(4)}$, so it contributes $4$ extra boundary equations in $(2.4)$. Secondly, the associated boundary map to BVP (2.4) is not well-defined. Let $\mathcal E$ be the space of stationary vacuum metrics, i.e. $$\mathcal E=\{g^{(4)}\in\mathcal S:~Ric_{g^{(4)}}=0\}.$$ As is explained above, after adding the gauge term $\beta_{\tilde g^{(4)}}g^{(4)}$, the boundary map $\Pi_1$ defined in (1.8) should be modified to $\Pi_2$ as follows, $$\begin{split} \Pi_2:&\mathcal E/\mathcal D_4\rightarrow\mathbf B(\partial M),\\ \Pi_2([g^{(4)}])&=(g_{\partial M}, H_{\partial M}, tr_{\partial M}K,\omega_{\mathbf n}), \end{split}$$ where the target space $\mathbf B(\partial M)$ is given by $\mathbf B(\partial M)=Met^{m,\alpha}(\partial M)\times [C^{m-1,\alpha}(\partial M)]^2\times \wedge_1^{m-1,\alpha}(\partial M)$ (cf.§7.1 for the notations of various spaces of tensor fields). However, this map is not well defined, because elements in $\mathcal D_4$ do not always preserve the Bartnik boundary data (cf. Proposition 2.2 below), which means that the Bartnik boundary data is not well defined for an element $[g^{(4)}]$ — an equivalence class of metrics — in the moduli space $\mathcal E/\mathcal D_4$. Since we are working with stationary metrics, it is natural to require elements in $\mathcal D_4$ to be time-independent and preserve the Killing vector field $\partial_t$. Thus a general element in $\mathcal D_4$ can be decomposed into two parts — a diffeomorphism on the hypersurface $M$ and a translation of time, i.e. $\mathcal D_4$ can be defined as, $$\begin{split} \mathcal D_4=\{\Phi_{(\psi,f)}|~&\psi\in D^{m+1,\alpha}_{\delta}(M)\text{ and }\psi|_{\partial M}=Id_{\partial M};\\ &~f \in C^{m+1,\alpha}_{\delta}(M)\text{ and } f|_{\partial M}=0;\\ &\Phi_{(\psi,f)}:V^{(4)}\rightarrow V^{(4)},\\ &\Phi_{(\psi,f)}[t,p]=[t+f,\psi(p)],\quad\forall t\in\mathbb R,~p\in M.~\}, \end{split}$$ Here $D^{m+1,\alpha}_{\delta}(M)$ denotes the group of $C^{m+1,\alpha}$ diffeomorphisms of $M$ which are asymptotically $Id_{M}$ at the rate of $\delta$ (cf.§7.1). If an element $\Phi_{(\psi,f)}\in\mathcal D_4$ has a nontrivial time translation function $f$, then it does not preserve the Bartnik boundary data on $\partial M$. Equip $M=\{t=0\}$ with local coordinates $\{x^i\},~i=1,2,3.$ Choose a function $f\in C^{m+1,\alpha}_{\delta}(M)$, and take the diffeomorphism $\Phi_{(Id_M,f)}\in\mathcal D_4:$ $$\begin{split} \Phi_{(Id_M,f)}: V^{(4)}&\rightarrow V^{(4)}\\ \Phi_{(Id_M,f)}(t,x_1,x_2,x_3)&=(t+f,x_1,x_2,x_3). \end{split}$$ In the following, we will use $\Phi_f$ as the abbreviation of $\Phi_{(Id_M,f)}$. Take a spacetime metric $g^{(4)}\in\mathcal S$ expressed as, $$g^{(4)}=-N^2dt^2+g_{ij}(dx^i+X^idt)(dx^j+X^jdt).$$ Let $\hat g^{(4)}$ denotes the pull back metric, i.e. $\hat g^{(4)}=\Phi_f^*g^{(4)}$. Then we have, $$\begin{split} \hat g^{(4)}&=-N^2[d(t+f)]^2+g_{ij}[dx^i+X^id(t+f)][dx^j+X^jd(t+f)]\\ &=-u^2dt^2-u^2df\odot dt+X_idx^i\odot dt-u^2(df)^2+X_idx^i\odot df+g_{ij}dx^idx^j, \end{split}$$ where $u^2=N^2-|X|_g^2$. Here we use $\odot$ to denote the symmetrized product between two 1-forms $A,B$, i.e. $A\odot B=A\otimes B+B\otimes A$. From the expression above, one easily observes that the induced metric on $M\subset (V^{(4)}, \hat g^{(4)})$ is given by, $$\hat g=-u^2(df)^2+X_idx^i\odot df+g_{ij}dx^idx^j.$$ Plugging $f|_{\partial M}=0$ in the equation above, it is obvious that the first Bartnik boundary term $g_{\partial M}$ in (2.4) remains the same under such a time translation. However, this is not the case for the other data $H_{\partial M}$, $tr_{\partial M}K$ and $\omega_{\mathbf n_{\partial M}}$. Let $\mathbf{N}$ denote the future-pointing time-like unit normal vector to the slice $M\subset (V^{(4)},g^{(4)})$ and $\mathbf n$ denote the outward unit normal of $\partial M\subset (M, g)$. Define $(\mathbf{\hat N},\mathbf{\hat n})$ in the same way for $M\subset (V^{(4)},\hat g^{(4)})$. Then on the boundary $\partial M$, the pairs $(\mathbf N,\mathbf n)$ and $(\mathbf{\hat N},\mathbf{\hat n})$ are related in the following way, $$\begin{bmatrix} d\Phi_f(\mathbf{\hat N})\\ d\Phi_f(\mathbf{\hat n}) \end{bmatrix} = \begin{bmatrix} a&b\\ b&a \end{bmatrix} \begin{bmatrix} \bf N\\ \bf n \end{bmatrix},$$ where $a,b$ are scalar fields on $\partial M$ and $a^2-b^2=1$. We refer to §7.2 for the detailed proof. Let $\nabla$ be the Levi-Civita connection of the spacetime $(V^{(4)},g^{(4)})$, and $\hat{\nabla}$ denotes that of the spacetime $(V^{(4)},\hat g^{(4)})$, then $\hat{\nabla}=\Phi_f^*(\nabla)$. We use $H_{\partial M}$, $tr_{\partial M}K$ and $\omega_{\mathbf n}$ to denote the Bartnik boundary data of $(V^{(4)},g^{(4)})$ on $\partial M$; and use $\hat H_{\partial M}$, $tr_{\partial M}\hat K$ and $\hat\omega_{\hat{\mathbf n}}$ as that of $(V^{(4)},\hat g^{(4)})$. Then we have the following formula for the mean curvature: $$\begin{split} \hat H_{\partial M}&=tr_{\partial M}(\hat\nabla\mathbf{\hat n})\\ &=tr_{\partial M}[\nabla d\Phi_f(\mathbf{\hat n})]\\ &=tr_{\partial M}[\nabla(b\mathbf{N}+a\mathbf{n})]\\ &=btr_{\partial M}(\nabla \mathbf{N})+atr_{\partial M}(\nabla\mathbf{n})\\ &=btr_{\partial M}K+aH_{\partial M}. \end{split}$$ It is easy to show that $tr_{\partial M}K$ is transformed in a similar way as above, i.e. $$tr_{\partial M}\hat K=atr_{\partial M}K+bH_{\partial M}.$$ As for the last boundary term $\omega_{\mathbf n}$, one has $\forall v\in T(\partial M)$, $$\begin{split} \hat \omega_{\hat{\mathbf n}}(v)&=\hat K(\hat{\mathbf n},v)=\langle\hat\nabla_v\hat{\mathbf N},\hat{\mathbf n}\rangle_{\hat g^{(4)}}\\ &= \langle\Phi_f^*(\nabla)_{v}\mathbf{\hat N},\mathbf{\hat n}\rangle_{\Phi_f^*g^{(4)}}\\ &=\langle\nabla_{d\Phi_f(v)}(a\mathbf{N}+b\mathbf n),~b\mathbf{N}+a\mathbf n\rangle_{g^{(4)}}\\ &=-b\cdot \nabla_{d\Phi_f(v)}a+a\cdot \nabla_{d\Phi_f(v)}b+(a^2-b^2)\langle\nabla_{d\Phi_f(v)} \mathbf N,\mathbf n\rangle_{g^{(4)}}\\ &=a^2\nabla_{d\Phi_f(v)}(b/a)+K(\mathbf n,d\Phi_f(v)),\\ &=a^2v(b/a)+\omega_{\mathbf n}(v). \end{split}$$ Here the last equality is based on the observation that $d\Phi_f(v)=v~\forall v\in T(\partial M)$, since $\Phi_f|_{\partial M}=Id_{\partial M}$. From the formula above, we conclude that, $$\hat \omega_{\mathbf {\hat n}}=a^2d_{\partial M}(b/a)+\omega_{\mathbf n},$$ where $d_{\partial M}(b/a)$ denotes the exterior derivative of the scalar field on $\partial M$. Along the boundary $\partial M$, one has $$a=\frac{1+\langle X,\mathbf n\rangle\mathbf n(f)}{\sqrt{[1+\langle X,\mathbf n\rangle\mathbf n(f)]^2-N^2|\mathbf n(f)|^2}}.$$ We refer to the Appendix §7.2 for the detailed calculation of the scalar fields $a,b$. Therefore, if the function $f$ is nontrivial, in the sense that $\mathbf n(f)|_{\partial M}\neq0$ and $a\neq1$ in (2.11), then it is easy to observe from equations (2.8-10) that the Bartnik boundary conditions are not invariant under the diffeomorphism $\Phi_f$. In view of the fact above, one may suggest to reduce the diffeomorphism group $\mathcal D_4$ in the definition of the boundary map to a smaller one $\mathcal D_3$ consisting of only 3-dim diffeomorphism on the slice, i.e. $$\mathcal D_3=\{\Phi_{(\psi,f)}\in\mathcal D_4:~f\equiv 0 \text{ on }M\}.$$ However, this approach does not work either. Let $\Pi_3$ be the associated boundary map as follows, $$\Pi_3:\mathcal E/\mathcal D_3\rightarrow \mathbf B$$ $$\Pi_3([g^{(4)}])=(g_{\partial M},H_{\partial M},tr_{\partial M}K,\omega_{\mathbf n}).$$ Given a fixed boundary condition $(\gamma,H,k,\tau)$, and an element $g^{(4)}$ in the pre-image set $\Pi_3^{-1}[(\gamma,H,k,\tau)]$, we can take an arbitrary function $f\in C^{m+1,\alpha}_{\delta}(M)$ such that $f|_{\partial M}=\mathbf n(f)|_{\partial M}=0$, and make time translation $\Phi_f$ to obtain a new metric $\bar g^{(4)}=\Phi_f^*g^{(4)}$. Then, by the previous analysis, $\bar g^{(4)}$ also belongs to $\Pi_3^{-1}[(\gamma,H,k,\tau)]$. By taking a smooth curve (parametrized by $\tau$) of such time translations $\Phi_{f(\tau)}$ $(\tau\in (-1,1))$, we get a family of metrics $g^{(4)}(\tau)=\Phi^*_{f(\tau)}g^{(4)}$. Let $f_{\tau}=\frac{\partial}{\partial\tau}|_{\tau=0}f$, then the infinitesimal deformation of the spacetime metric at $\tau=0$ is of the form, $$(g^{(4)})'=L_{f_{\tau}\partial_t}g^{(4)}= df_{\tau}\odot (\partial t)^{\flat}=df_{\tau}\odot (-u^2dt+X_idx^i).$$ By construction, $g^{(4)}(\tau)\in\Pi_3^{-1}[(\gamma,H,k,\tau)]$ for all $\tau$, which implies that $(g^{(4)})'\in \text{Ker} D\Pi_3$. Such a kernel element is nontrivial if it is not tangent to any 3-dim diffeomorphism variation, i.e. the following equation is not solvable for $Z\in T^{m,\alpha}_{\delta}M$, $$df_{\tau}\odot (-u^2dt+X_idx^i)= L_Zg^{(4)}.$$ Since $(2.13)$ is an overdetermined system for $Z$, it is not solvable for generic choices of $f_{\tau}$. This means that the kernel of $D\Pi_3$ should be of infinite dimension, which indicates that $\Pi_3$ is not a Fredholm map. From all the previous analysis, we notice that the Neumann data $\mathbf n(f)$ of the time translation function plays an important role in choosing the right diffeomorphism group. This suggests defining a new group $\mathcal D$ as, $$\mathcal D=\{\Phi_{(\psi,f)}\in\mathcal D_4:\mathbf n_g(f)=0\text{ on }\partial M\}.$$ It is in fact an intermediate group in the sense that $\mathcal D_3\subset \mathcal D\subset\mathcal D_4$. The vector field $\mathbf n_g$ in (2.14) can be taken as the unit normal vector of $\partial M$ with respect to any Riemannian metric $g$ on $M$ — the group $\mathcal D$ does not depend on the choice of the metric $g$. In fact, it is easy to observe that $\mathcal D$ can be defined in an equivalent way: $$\mathcal D=\{\Phi_{(\psi,f)}\in\mathcal D_4: df=0\text{ at }\partial M\}.$$ Notice that $\mathbf n(f)=0$ in (2.14) yields $a=1$ in (2.11). This further implies that, geometrically, elements in the group $\mathcal D$ are diffeomorphisms of the spacetime $(V^{(4)},g^{(4)})$ which fix the boundary $\partial M$ and the time-like unit normal vector field $\mathbf N$ along $\partial M$. Now, define $\mathbb E$ to be the quotient space, $$\mathbb E=\mathcal E/\mathcal D.$$ Elements in $\mathbb E$ are equivalence classes $[g^{(4)}]$ given by, $$\begin{split} [g^{(4)}]=\{\Phi_{(\psi,f)}^*g^{(4)}:~ g^{(4)}\in\mathcal E,~ \Phi_{(\psi,f)}\in\mathcal D\}. \end{split}$$ Now we can consider the natural boundary map: $$\begin{split} \Pi:&\mathbb E\rightarrow\mathbf B\\ \Pi([g^{(4)}])&=(g_{\partial M},H_{\partial M},tr_{\partial M}K,\omega_{\mathbf n}). \end{split}$$ This map is well defined — the Bartnik boundary data is the same for all the metrics inside one equivalence class $[g^{(4)}]\in\mathbb E$, because the transformation formulas $(2.8-10)$ show that Bartnik boundary data is preserved under diffeomorphisms in $\mathcal D$. In the following sections we will prove this boundary map $\Pi$ is Fredholm. By the Remark 2.1, a general spacetime metric $g^{(4)}$ in an equivalence class $[g^{(4)}]\in\mathbb E$ does not necessarily induce Rimannian geometry on the fixed slice $M$, which might make the Bartnik boundary data not well-defined on the slice $M\subset (V^{(4)},g^{(4)})$. To solve this problem, we can restrict the boundary map $\Pi$ in (2.15) to the open subset $\mathbb E'\subset \mathbb E$, where every equivalent class $[g^{(4)}]\in\mathbb E'$ admits a representative $g^{(4)}$ which induces Riemannian metric on $M$. On the other hand, since the ellipticity of the Bartnik boundary data is only a local property, we can focus the study in a neighborhood of such metric $g^{(4)}\in\mathcal E$ and think of $\mathbb E$ locally as a slice of $\mathcal E$ under the action of $\mathcal D$. The boundary map $\Pi$ being Fredholm implies that the linearization $D\Pi_1$ of the map $\Pi_1$ in (1.8) at $g^{(4)}$ has finite dimensional kernel transverse to the action of $\mathcal D$ in the sense that $\text{Ker}D\Pi_1/\sim$ has finite dimension. Here two deformations $h^{(4)}_1,h^{(4)}_2\in \text{Ker}D\Pi_1$ are equivalent ($h^{(4)}_1\sim h^{(4)}_2$) if the difference $(h^{(4)}_1-h^{(4)}_2)=\delta^*_{g^{(4)}}Y$ for some $Y\in T\mathcal D$. In the following sections we always assume that the reference spacetime metric $\tilde g^{(4)}$ is chosen so that the slice $M$ is a Cauchy surface in $(V^{(4)},\tilde g^{(4)})$. The well-defined BVP ==================== Throughout this section, we take $\tilde g^{(4)}\in\mathcal E$ as a fixed reference metric and make the following assumption: The BVP with unknown $Y\in T^{m,\alpha}_{\delta}(V^{(4)})$, given by, $$\begin{cases} \beta_{g^{(4)}}\delta_{g^{(4)}}^*Y=0\quad\text{on}\quad M\\ Y=0\quad\text{on}\quad\partial M \end{cases}$$ has only the zero solution $Y=0$ when $g^{(4)}=\tilde g^{(4)}$. In the above, $T^{m,\alpha}_{\delta}(V^{(4)})$ denotes the space of $C^{m,\alpha}$ vector fields in $V^{(4)}$, which are asymptotically zero at the rate of $\delta$ and in addition time-independent(cf.§7.1). Throughout this paper, we say a tensor field $T$ in $V^{(4)}$ is *time-independent* if $L_{\partial_t}T=0$. In the following, we call the operator $\beta_{\tilde g^{(4)}}\delta^*_{\tilde g^{(4)}}$ *invertible* if Assumption 3.1 holds. Note that since the system (3.1) is elliptic and self-adjoint (cf.§5.1), it has trivial kernel if and only if the following map is an isomorphism: $$\beta_{\tilde g^{(4)}}\delta^*_{\tilde g^{(4)}}: \mathcal T\to T^{m-2,\alpha}_{\delta+2}(V^{(4)}),$$ where the domain space $\mathcal T=\{Y\in T^{m,\alpha}_{\delta}(V^{(4)}): Y=0\text{ on }\partial M\}$. This is an open condition. Thus if $\beta_{\tilde g^{(4)}}\delta^*_{\tilde g^{(4)}}$ is invertible, then so is the operator $\beta_{g^{(4)}}\delta^*_{ g^{(4)}}$ for $g^{(4)}$ near $\tilde g^{(4)}$ in the space $\mathcal S$. We refer to Remark 4.3 for more insights about the Assumption 3.1. Based on the discussions in §2, we modify the system (2.4) to a new BVP with unknowns $(g^{(4)},F)\in \mathcal S\times C^{m,\alpha}_{\delta}(M)$ as follows, $$\begin{split} &\quad\quad\begin{cases} Ric_{g^{(4)}}+\delta^*_{g^{(4)}}\beta_{\tilde g^{(4)}}g^{(4)}=0\\ \Delta F=0 \end{cases} \quad\text{on}\quad M,\\ &\quad\quad\begin{cases} g_{\partial M}=\gamma\\ aH_{\partial M}+btr_{\partial M}K=H\\ atr_{\partial M}K+bH_{\partial M}=k\\ \omega_{\mathbf n}+a^2d_{\partial M}(a/b)=\tau\\ \beta_{\tilde g^{(4)}}g^{(4)}=0 \end{cases}\quad\text{on}\quad\partial M, \end{split}$$ where $$a=\frac{1+\langle X,\mathbf n\rangle F}{\sqrt{(1+\langle X,\mathbf n\rangle F)^2-N^2F^2}},~\text{ and }~b=\sqrt{a^2-1},$$ with $N$ and $X$ denoting the lapse function and shift vector of $g^{(4)}$ as in (2.1). In the second equation above, $\Delta=-trHess$ denotes the Laplace operator with respect to the metric $g^{(4)}$. Here we think of $F\in C^{m,\alpha}_{\delta}(M)$ as a function in $V^{(4)}$ by expanding it time-independently. The argument to follow works in the same way if one sets $\Delta$ to be the Laplacian of the induced Riemannian metric $g$ on the slice $M$. But with the former choice, the principal symbol which we will compute in $\S 4$ is simpler. Applying the Bianchi operator to the first equation of $(3.2)$, one obtains, $$\beta_{g^{(4)}}\delta^*_{g^{(4)}}[\beta_{\tilde g^{(4)}}g^{(4)}]=0\quad\text{on }M.$$ In addition, the last boundary condition in (3.2) gives, $$\beta_{\tilde g^{(4)}}g^{(4)}=0\quad\text{on } \partial M.$$ Combining (3.4) and (3.5), together with the Assumption 3.1, it follows that, $$\begin{split} \beta_{\tilde g^{(4)}}g^{(4)}=0,\quad\forall \text{ solution } g^{(4)}\text{ of (3.2) near }\tilde g^{(4)}. \end{split}$$ Therefore, if $g^{(4)}$ is a solution to (3.2) near $\tilde g^{(4)}$, then it must be Ricci flat and Bianchi-free with respect to $\tilde g^{(4)}$. So we define a solution space $\mathcal C$ as follows: $$\begin{split} \mathcal C:=\{~(g^{(4)},F)\in\mathcal S\times C^{m,\alpha}_{\delta}(M):~&Ric_{g^{(4)}}=0,~\beta_{\tilde g^{(4)}}g^{(4)}=0,~\Delta F=0~\text{on}~M~\}. \end{split}$$ Obviously, $(\tilde g^{(4)},0)\in\mathcal C$. Let $\tilde \Pi$ be the boundary map: $$\begin{split} \tilde \Pi:~\mathcal C&\rightarrow\mathbf{B}\\ \tilde \Pi(g^{(4)},F)=(g_{\partial M},a H_{\partial M}+btr_{\partial M} K,&atr_{\partial M} K+b H_{\partial M},\omega_{\mathbf n}+a^2d_{\partial M}(b/a)). \end{split}$$ This map $\tilde\Pi$ is closely related to the boundary map $\Pi$ defined in (2.15) near the reference pair $(\tilde g^{(4)},0)\in\mathcal C$. In fact, we have the following theorem. There is a map $\mathcal P:\mathcal C\to\mathbb E$ which is locally a diffeomorphism near $(\tilde g^{(4)},0)$, and the boundary maps $\Pi$ and $\tilde\Pi$ are related by $$\tilde\Pi=\Pi\circ\mathcal P.$$ Given an element $(\hat g^{(4)},\hat F)\in\mathcal C$, one can take a function $f$ on $M$ such that $f|_{\partial M}=0$ and $\mathbf n(f)|_{\partial M}=\hat F|_{\partial M}$, and apply the diffeomorphism $\Phi_{(\psi,f)}\in\mathcal D_4$ to $\hat g^{(4)}$ where $\psi$ is an arbitrary diffeomorphism in $\mathcal D^{m,\alpha}_{\delta}(M)$ with $\psi|_{\partial M}=Id_{\partial M}$. Thus, any element $(\hat g^{(4)},\hat F)\in\mathcal C$ gives rise to a class of elements as follows, $$\{\Phi_{(\psi,f)}^*(\hat g^{(4)}):~\Phi_{(\psi,f)}\in\mathcal D_4,~\mathbf n( f)|_{\partial M}=\hat F|_{\partial M}\}.$$ It is easy to observe that the equivalence class above actually defines an element in $\mathbb E$. Henceforth we can define a map $\mathcal P$ as, $$\begin{split} \mathcal P:\mathcal C&\rightarrow\mathbb E,\\ \mathcal P(\hat g^{(4)},\hat F)&=[g^{(4)}], \end{split}$$ where $[g^{(4)}]$ is defined as the equivalence class $(3.6)$. On the other hand, consider the following map: $$\begin{split} &\mathcal G: \mathcal S\times \mathcal D_4\rightarrow (\wedge_1)^{m,\alpha}_{\delta} (V^{(4)})\\ &\mathcal G(g^{(4)}, \Phi)=\beta_{\tilde g^{(4)}}\Phi^*g^{(4)}, \end{split}$$ where $(\wedge_1)^{m,\alpha}_{\delta} (V^{(4)})$ denotes the space of $C^{m,\alpha}$ 1-forms in $V^{(4)}$ which are time-independent and asymptotically zero at the rate of $\delta$(cf.§7.1). The linearization of $\mathcal G$ at $(\tilde g^{(4)}, Id_{V^{(4)}})$ is given by, $$\begin{split} &D\mathcal G|_{(\tilde g^{(4)},Id_{V^{(4)}})}: T\mathcal S\times T\mathcal D_4\rightarrow (\wedge_1)^{m,\alpha}_{\delta}V^{(4)}\\ &D\mathcal G|_{(\tilde g^{(4)},Id_{V^{(4)}})}[(h^{(4)},Y)]=\beta_{\tilde g^{(4)}}\delta_{\tilde g^{(4)}}^*Y+\beta_{\tilde g^{(4)}}h^{(4)}. \end{split}$$ By the definition of $\mathcal D_4$, the vector field $Y\in T\mathcal D_4$ is time-independent, asymptotically zero and $Y=0$ on $\partial M$. So the operator $\beta_{\tilde g^{(4)}}\delta^*_{\tilde g^{(4)}}$ in the linearization above is invertible by the Assumption 3.1. Therefore, by the implicit function theorem, there is a neighborhood $U_{\tilde g^{(4)}}$ of $\tilde g^{(4)}$ in $\mathcal S$ such that for any $g^{(4)}\in U$, there is a unique element $\Phi_{(\psi,f)}\in\mathcal D_4$ near $Id_{V^{4}}$ such that the pull back metric $\Phi_{(\psi,f)}^*g^{(4)}$ is gauge-free, i.e. $\beta_{\tilde g^{(4)}}(\Phi_{(\psi,f)}^*g^{(4)})=0$ in $V^{(4)}$. Moreover, if $g^{(4)}$ is vacuum, i.e. $g^{(4)}\in U_{\tilde g^{(4)}}\cap\mathcal E$, then the gauge-free metric $\hat g^{(4)}=\Phi_{(\psi,f)}^*g^{(4)}$ is also vacuum. Trivially it follows, $$g^{(4)}=(\Phi^*_{(\psi,f)})^{-1}\hat g^{(4)}=\Phi_{(\psi^{-1},-f)}^*\hat g^{(4)}.$$ Let $\hat F\in C^{m,\alpha}_{\delta}(M)$ be the unique harmonic function (with respect to the metric $\hat g^{(4)}$) on $M$ satisfying the Dirichlet boundary condition $\hat F=\mathbf n(-f)$ on $\partial M$. Since the diffeomorphism $\Phi_{(\psi,f)}$ is near $Id_{V^{(4)}}$, $\mathbf n(f)$ is close to zero. Thus $F$ is also near the zero function on $M$. Pairing it with $\hat g^{(4)}$, we obtain an element $(\hat g^{(4)},\hat F)\in\mathcal C$ near $(\tilde g^{(4)},0)$. In addition, if two elements $ g_1^{(4)},~ g_2^{(4)}\in U_{\tilde g^{(4)}}\cap\mathcal E$ are equivalent under some diffeomorphism $\Phi_{(\psi_0,f_0)}\in\mathcal D_4$, then they correspond to the same gauge-free metric $\hat g^{(4)}$ because of the uniqueness shown above. In addition, since the time translation $f_0$ makes $\mathbf n(f_0)=0$ on $\partial M$, $g_1^{(4)}$ and $g_2^{(4)}$ also generate the same harmonic function $\hat F$ as described above. Therefore, in the neighborhood $U_{\tilde g^{(4)}}$ of $\tilde g^{(4)}$ in $\mathcal S$, all the metrics that belong to the same equivalence class $[g^{(4)}]\in\mathbb E$ give rise to a unique pair $( \hat g^{(4)},\hat F)\in\mathcal C$ near $(\tilde g^{(4)},0)$. Define $\mathbb U_{\tilde g^{(4)}}$ to be the corresponding neighborhood of $[\tilde g^{(4)}]$ in $\mathbb E$, i.e. $$\begin{split} \mathbb U_{\tilde g^{(4)}}=\{[g^{(4)}]\in\mathbb E:~\text{The equivalence class }[g^{(4)}]\text{ admits a representative } g_0^{(4)}&\\\text{ such that }g_0^{(4)}\in U_{\tilde g^{(4)}}\}&. \end{split}$$ Then for any $[g^{(4)}]\in\mathbb U_{\tilde g^{(4)}}$ we can take an arbitrary representative $g^{(4)}\in U_{\tilde g^{(4)}}$ and then obtain a pair $(\hat g^{(4)},\hat F)$ in the manner described above. Moreover, as is shown, the pair $(\hat g^{(4)},\hat F)$ does not depend on the representative we choose in $U_{\tilde g^{(4)}}$. In this way, one can establish a map $\mathcal {\tilde P}$ locally defined in $\mathbb E$ near $[\tilde g^{(4)}]$ and mapping $\mathbb U_{\tilde g^{(4)}}$ to a neighborhood of $(\tilde g^{(4)},0)$ in $\mathcal C$, given by, $$\begin{split} \mathcal{\tilde P}:\mathbb U_{\tilde g^{(4)}}&\rightarrow\mathcal C,\\ \mathcal{\tilde P}([g^{(4)}])&=(\hat g^{(4)},\hat F). \end{split}$$ It is easy to check that $\mathcal P$ and $\mathcal {\tilde P}$ are the inverse map of each other near $(\tilde g^{(4)},0)$. Thus, the spaces $\mathcal C$ and $\mathbb E$ are locally diffeomorphic via $\mathcal P$. Moreover, based on the formulas $(2.8-10)$, one can easily observe that if $[g^{(4)}]=\mathcal P(\hat g^{(4)},\hat F)$, then their Bartnik boundary data are related in the following way, $$\begin{split} (g_{\partial M}, &H_{\partial M}, tr_{\partial M} K, \omega_{\mathbf n})\\ &=(\hat g_{\partial M},a\hat H_{\partial M}+btr_{\partial M}\hat K,atr_{\partial M}\hat K+b\hat H_{\partial M},\hat\omega_{\mathbf n}+a^2d_{\partial M}(b/a)), \end{split}$$ where $a,b$ are given by the formulas in $(3.3)$ inside which $F=\hat F$ and $X,N$ are the lapse and shift of $\hat g^{(4)}$. Therefore, near $(\tilde g^{(4)},0)$ the boundary maps $\tilde \Pi$ and $\Pi$ are related by, $$\tilde \Pi= \Pi\circ\mathcal P.$$ From the analysis above, we see that locally the solution space $\mathcal C$ is a coordinate chart of the moduli space $\mathbb E$ near the reference metric class $[\tilde g^{(4)}]$, and the map $\tilde\Pi$ is the Bartnik boundary map $\Pi$ expressed in this local chart. In the following, we show that the space $\mathcal C$ admits Banach manifold structure. The space $\mathcal C$ admits smooth Banach manifold structure near $(\tilde g^{(4)},0)$. For any stationary vacuum metric $g^{(4)}$, define $\mathcal H_{g^{(4)}}$ as the space of harmonic functions on $M$: $$\mathcal H_{g^{(4)}}=\{f\in C^{m,\alpha}_{\delta}(M):\Delta_{g^{(4)}} f=0 \text{ on } M\}.$$ Since $\Delta_{g^{(4)}}$ is invertible when subjected to Dirichlet boundary conditions, it is easy to prove that, $$\mathcal H_{g^{(4)}}\cong C^{m,\alpha}(\partial M).$$ Thus $\mathcal H_{g^{(4)}}$ is a smooth Banach manifold. Let $P:\mathcal C\to \mathcal E$ be the projection $P(\hat g^{(4)},\hat F)=\hat g^{(4)}$. We observe that if $(\hat g^{(4)},\hat F)\in\mathcal C$ near $(\tilde g^{(4)},0)$, then $\hat g^{(4)}\in\mathcal E$ and it satisfies the gauge-free condition $\beta_{\tilde g^{(4)}}\hat g^{(4)}=0$. By the analysis in the proof of Theorem 3.2, any $g^{(4)}\in\mathcal E$ near $\tilde g^{(4)}$ is isometric via a diffeomorphism in $\mathcal D_4$ to a gauge free element $\hat g^{(4)}$. So the projection space $P(\mathcal C)$ is a slice for $\mathcal E$ under the action of $\mathcal D_4$. Therefore, locally the space $\mathcal C$ is a fiber bundle over $\mathcal E/\mathcal D_4$, with the fiber at $[g^{(4)}]$ being $\mathcal H_{g^{(4)}}$. Thus near the element $(\tilde g^{(4)},0)$, we have, $$\mathcal C\cong \mathcal E/\mathcal D_4\times\mathcal H_{\tilde g^{(4)}}.$$ It is proved in \[Az\] that the moduli space $\mathcal E/\mathcal D_4$ is a smooth Banach manifold, and hence it follows that $\mathcal C$ admits smooth Banach manifold structure. It then follows directly from Theorem 3.2 that the domain space $\mathbb E$ of the Bartnik boundary map $\Pi$ is a smooth Banach manifold. In the following sections, we will show that the boundary map $\tilde\Pi$ is Fredholm, which then implies so is $\Pi$. Ellipticity of BVP (3.2) ========================     In this section, we will prove the ellipticity of BVP (3.2), implementing the criterion developed by Agmon-Douglis-Nirenberg (cf.\[ADN\]). We use the following standard notation. Let $\xi$ denote a $1-$form on $M$, $\eta$ denote a nonzero $1-$form tangential to the boundary $\partial M$, and $\mu$ a unit $1-$form normal to the boundary $\partial M$. The index $0$ denotes the direction along $\partial t$ in $V^{(4)}$, and index $1,2,3$ denote the tangential direction on $M$. When restricted on the boundary, index $1$ denotes the (outward) normal direction to $\partial M\subset M$ and indices $2,3$ denote directions tangent to $\partial M$. We use greek letters when $0$ is included in the indices, and latin letters when there are only tangential components involved. Based on the system (3.2), we define a differential operator $\mathcal F=(\mathcal L,\mathcal B)$ with interior operator $\mathcal L$, mapping a pair $(g^{(4)},F)$ to the interior equations in (3.2): $$\begin{split} \mathcal L:\mathcal S\times C^{m,\alpha}_{\delta}(M)\rightarrow S^{m-2,\alpha}_{\delta+2}(V^{(4)})\times C^{m-2,\alpha}_{\delta+2}(M)\\ \mathcal L(g^{(4)},F)= (~2(Ric_{g^{(4)}}+\delta^*_{g^{(4)}}\beta_{\tilde g^{(4)}}g^{(4)}),~~\Delta F~~); \end{split}$$ and a boundary operator $\mathcal B$ mapping $(g^{(4)}, F)$ to the boundary equations in $(3.2)$: $$\begin{split} \mathcal B: \mathcal S\times C^{m,\alpha}_{\delta}(M)&\rightarrow \mathbb B\\ \mathcal B(g^{(4)},F)= (~~~ &g_{\partial M},\\ &aH_{\partial M}+btr_{\partial M}K,\\ &atr_{\partial M}K+bH_{\partial M},\\ &\omega_{\mathbf n}+a^2d_{\partial M}(b/a),\\ &\beta_{\tilde g^{(4)}}g^{(4)}~~~). \end{split}$$ In the above, $S^{m-2,\alpha}_{\delta+2}(V^{(4)})$ denotes the space of symmetric 2-tensors in $V^{(4)}$, which are time independent, $C^{m-2,\alpha}$ smooth and asymptotically zero at the rate of $(\delta+2)$; $\mathbb B$ is an abbreviation of the target space of $\mathcal B$, given by, $$\mathbb B=S^{m,\alpha}(\partial M)\times [C^{m-1,\alpha}(\partial M)]^2\times \wedge_1^{m-1,\alpha}(\partial M)\times C^{m-1,\alpha}(\partial M)\times \wedge_1^{m-1,\alpha}(\partial M).$$ We refer to §7.1 for these notations of tensor spaces. The linearization $D\mathcal F$ of $\mathcal F$ at $(\tilde g^{(4)},0)$ is elliptic. We use the characterization of ellipticity in \[ADN\] to prove the thoerem. We first show in §4.1 that $D\mathcal F$ is properly elliptic. Then in §4.2 we show that $D\mathcal F$ satisfies the complementing boundary condition. Properly elliptic condition ---------------------------     The linearization of the interior operator at $(\tilde g^{(4)},0)$ is given by (cf.\[Be\]) $$\begin{split} D\mathcal L: T_{\tilde g^{(4)}}\mathcal S\times C^{m,\alpha}_{\delta}(M)&\rightarrow S^{m-2,\alpha}_{\delta+2}(V^{(4)})\times C^{m-2,\alpha}_{\delta+2}(M)\\ D\mathcal L(h^{(4)},G)&= (~D_{\tilde g^{(4)}}^*D_{\tilde g^{(4)}} h^{(4)},~\Delta G~). \end{split}$$ Let $\{e_i\},~(i=1,2,3)$ be a local orthonormal basis of the tangent bundle on $M$. Recall that $\mathbf N$ denotes the future pointing time-like unit vector perpendicular to $M$ in the spacetime. Based on (2.1), $\mathbf N=N^{-1}(\partial_t-X)$, with $X,N$ being the shift vector and lapse function of $\tilde g^{(4)}$. Then the Laplacian $D_{\tilde g^{(4)}}^*D_{\tilde g^{(4)}} h^{(4)}_{\alpha\beta}$ in the above can be expressed in the $3+1$ slice formalism (2.1) of the spacetime as: $$\begin{split} D_{\tilde g^{(4)}}^*D_{\tilde g^{(4)}} h^{(4)}_{\alpha\beta}&=-D_{\mathbf N}D_{\mathbf N} h^{(4)}_{\alpha\beta}+\Sigma_{i=1}^3D_{e_i}D_{e_i} h^{(4)}_{\alpha\beta}+O_1(h^{(4)})\\ &=-D_{\frac{1}{N}(\partial t-X)}D_{\frac{1}{N}(\partial t-X)} h^{(4)}_{\alpha\beta}+\Sigma_{i=1}^3D_{e_i}D_{e_i} h^{(4)}_{\alpha\beta}+O_1(h^{(4)})\\ &=-\frac{1}{N^2}\partial_X\partial_X h^{(4)}_{\alpha\beta}+\Sigma_{i=1}^3\partial_{e_i}\partial_{e_i} h^{(4)}_{\alpha\beta}+O_1(h^{(4)}).\end{split}$$ Here $O_1(h^{(4)})$ denotes those terms with lower($\leq 1$) order derivatives. A similar formula holds for the term $\Delta G$, i.e. $$\begin{split} \Delta G =-\frac{1}{N^2}\partial_X\partial_X G+\Sigma_{i=1}^3\partial_{e_i}\partial_{e_i} G+O_1(G).\end{split}$$ Thus, the matrix of principal symbol for $D\mathcal L$ is given by, $$L(\xi)=a(\xi)I_{11\times 11}$$ with $$a(\xi)=\xi_1^2+\xi_2^2+\xi_3^2-\frac{1}{N^2}(X_i\xi^i)^2.$$ The determinant of this matrix is obviously $$\text{det}(L(\xi))=[a(\xi)]^{11}.$$ Notice that $\frac{||X||^2}{N^2}<1$ by $(2.2)$ and hence, $$a(\xi)=|\xi|^2-\langle\frac{X}{N},\xi\rangle^2\geq |\xi|^2-\frac{||X||^2}{N^2}|\xi|^2>0,$$ Therefore, the interior operator $L$ is properly elliptic.\ Complementing boundary condition --------------------------------     The complementing boundary condition is defined as (cf.\[ADN\]): *Let $L^*(\xi)$ be the adjoint matrix of $L(\xi)$ and set $\xi=\eta+z\mu$. The rows of the matrix $[B\cdot L^*](\eta+z\mu)$ are linearly independent modulo $l^+(z)=\prod (z-z_k)$, where $\{z_k\}$ are the roots of $detL(\eta+z\mu)=0$ having positive imaginary parts.*\ Since the principal symbol of $L$ is the identity matrix (up to a scalar) as shown in $(4.1)$, the complementing condition will hold as long as the boundary matrix $B(\eta+z\mu)$ is non-degenerate when $z$ is a root of $\text{det}L(\eta+z\mu)=0$ with positive imaginary part. The linearization of the boundary operator $\mathcal B$ at $(\tilde g^{(4)},0)$ is given by, $$\begin{split} \mathcal B:T\mathcal S\times C^{m,\alpha}_{\delta}(M)&\rightarrow \mathbb B\\ D\mathcal B(h^{(4)},G)= (~&h_{\partial M}\\ &(H_{\partial M})'_{h^{(4)}}+O_0(G)\\ &tr_{\partial M}K'_{h^{(4)}}+O_0(G)\\ &(\omega_{\mathbf n})'_{h^{(4)}}+Nd_{\partial M}G+O_0(G)\\ &\beta_{ g^{(4)}}h^{(4)}~). \end{split}$$ Here we use the notation $T'_{h^{(4)}}$ to denote the variation of the tensor $T$ with respect to the deformation $h^{(4)}$. Notice that at $(\tilde g^{(4)},0)$, $a=1,~b=0$. The formula $(3.3)$ of the scalar field $a$ involves only the $0-$order information of $F$. Thus the 2nd and 3rd lines in the expression of $D\mathcal B$ above, which represent the linearization of Bartnik data $(aH_{\partial M}+btr_{\partial M}K)$ and $(atr_{\partial M}K+bH_{\partial M})$ at $(a=1,b=0)$, do not contain high order ($\geq 1$) derivatives of $G$. It is easy to check at $(a=1,b=0)$, the variation $(a^2d_{\partial M}b/a\big)'_G=Nd_{\partial M}G+O_0(G),$ which contributes to the fourth line in $D\mathcal B$. Based on $(4.3)$, the principal symbol of $\mathcal B$ is of the form: $$B(\xi)= \begin{bmatrix} 0_{3\times 8}& \resizebox{.08\textwidth}{!}{$ \begin{matrix} 1&0&0\\ 0&1&0\\ 0&0&1 \end{matrix} $} \\ \tilde B_{8\times8}&* \end{bmatrix}.$$ Notice that $B(\xi)$ is a $11\times11$ matrix, since the boundary terms in (4.3) contain 11 equations in total and 11 (ordered) unknowns. Here for a simple expression of the boundary matrix, the unknown vector $(h^{(4)},G)$ is particularly arranged in the order of: $$(G,h^{(4)}_{00},h^{(4)}_{01},h^{(4)}_{02},h^{(4)}_{03},h^{(4)}_{11},h^{(4)}_{12},h^{(4)}_{13},h^{(4)}_{22},h^{(4)}_{23},h^{(4)}_{33}).$$ Obviously, the first boundary term $h_{\partial M}=h^{(4)}_{ij},~(2\leq i\leq j\leq3)$. Thus the first three rows of $B$ in (4.4) contain only zeros in the first eight columns and a $3\times 3$ identity matrix at the end. The remaining eight rows of $B$ represent the symbol of 2nd-5th boundary terms in (4.3), with $\tilde B_{8\times 8}$ denoting the first eight columns which are determined by the $G$ and $h^{(4)}_{\alpha\beta}~~(0\leq\alpha\leq1,~\alpha\leq\beta\leq 3)$ components of the corresponding boundary terms. Detailed calculation given in §7.3 shows that the matrix $\tilde B_{8\times 8}$ is given by $$\tilde B_{8\times 8}=-\frac{1}{2N^2}[(\hat B_1)_{8\times 1}~ (\hat B_2)_{8\times 4}~(\hat B_3)_{8\times 3}].$$ Here the scalar $-\frac{1}{2N^2}$ is for the purpose that the matrix following it can be written in a simpler way. Since $G$ only appears as $Nd_{\partial M}G$ in the fourth boundary term in (4.3), it is easy to derive that the first column of $\tilde B$ is given by $$\hat B_1= \begin{bmatrix} 0&0&-2N^3\xi_2&-2N^3\xi_3&0&0&0&0 \end{bmatrix}^t$$ In the above, we have the extra factor $-2N^2$, because $-\frac{1}{2N^2}$ has been factored out from these rows, which contributes to the factor in the front of the expression of $\tilde B$. Based on the symbol calculation in §7.3, we obtain the 2nd-5th columns of $\tilde B$ as, $$\hat B_2=\resizebox{.66\textwidth}{!}{$ \begin{bmatrix} 0&0&0&0\\ 0&0&2N\xi_2&2N\xi_3\\ 0&N\xi_2&N\xi_1&0\\ 0&N\xi_3&0&N\xi_1\\ \xi_1&2S-2\xi_1X^1&-2\xi_1X^2&-2\xi_1X^3~\\ \xi_2&-2\xi_2X^1&2S-2\xi_2X^2&-2\xi_2X^3~\\ \xi_3&-2\xi_3X^1&-2\xi_3X^2&2S-2\xi_3X^3~\\ 2S&2(N^2\xi_1-SX^1)&2(N^2\xi_2-SX^2)&2(N^2\xi_3-SX^3) \end{bmatrix}$},$$ and the last three columns are given by $$\hat B_2=\resizebox{.9\textwidth}{!}{$ \begin{bmatrix} 0&2N^2\xi_2&2N^2\xi_3\\ 0&-2N\xi_2X^1&-2N\xi_3X^1\\ -N\xi_2X^1&N\xi_3X^3&-N\xi_2X^3\\ -N\xi_3X^1&-N\xi_3X^2&N\xi_2X^2\\ \xi_1X^1X^1+N^2\xi_1-2S X^1&\xi_1X^1X^2-2S X^2+2N^2\xi_2&\xi_1X^1X^3-2S X^3+2N^2\xi_3\\ \xi_2X^1X^1-N^2\xi_2&\xi_2X^1X^2-2S X^1+2N^2\xi_1&\xi_2X^1X^3\\ \xi_3X^1X^1-N^2\xi_3&\xi_3X^1X^2&\xi_3X^1X^3-2S X^1+2N^2\xi_1\\ 0&0&0 \end{bmatrix}$},$$ inside which $S=\xi_1X^1+\xi_2X^2+\xi_3X^3$. Obviously, to prove the complementing boundary condition, it suffices to verify that $\tilde B(\eta+z\mu)$ is nonsingular when $z$ is a root of $a(\eta+z\mu)$ in (4.2) with positive imaginary part. We can simplify $\tilde B$ using elementary row and column operation of matrices (cf.$\S 7.4$ for the detailed calculations) and obtain an equivalent matrix $\hat B$ so that $\text{det}\tilde B=-\frac{1}{32N^{11}}\text{det}\hat B$. The matrix $\hat B(\xi)$ is given by, $$\resizebox{.90\textwidth}{!}{$ \begin{bmatrix} 0&0&0&0&0&0&-\xi_2&-\xi_3\\ 0&0&0&\xi_2&\xi_3&0&0&0\\ -2N^2\xi_2&0&0&\xi_1&0&0&\xi_1X^1+\xi_3X^3&-\xi_2X^3\\ -2N^2\xi_3&0&0&0&\xi_1&0&-\xi_3X^2&\xi_1X^1+\xi_2X^2\\ 0&\xi_1&2S&0&0&2N^2\xi_1&-2S X^2&-2S X^3\\ 0&\xi_2&0&2S&0&0&2N^2\xi_1&0\\ 0&\xi_3&0&0&2S&0&0&2N^2\xi_1\\ 0&S&N^2\xi_1+SX^1&SX^2&SX^3&N^2S+N^2\xi_1X^1&0&0 \end{bmatrix}$}.$$ Computing the determinant of the matrix above gives $$\text{det}(\hat B)(\xi)=8N^8(\xi_1^2-\frac{S^2}{N^2})^2(\xi_1^2+\xi_2^2)^2.$$ If $\xi=\eta+z\mu$, then $$\text{det}(\hat B)(\eta+z\mu)=8N^8(z^2-\frac{\langle X,\eta+z\mu\rangle^2}{N^2})^2|\eta|^4.$$ If $z$ is a complex root of $a(\eta+z\mu)=0$, then from (4.2) it follows, $$|\eta+z\mu|^2-\frac{1}{N^2}\langle X,\eta+z\mu\rangle^2=0,$$ i.e. $|\eta|^2+z^2=\frac{\langle X,\eta+z\mu\rangle^2}{N^2}$, and thus $$\begin{split} \text{det}(\tilde B)(\eta+z\mu)&=8N^8(z^2-\frac{\langle X,\eta+z\mu\rangle^2}{N^2})^2|\eta|^4\\ &=8N^8(z^2-z^2-|\eta|^2)^2|\eta|^4\\ &=8N^8|\eta|^8, \end{split}$$ which is obviously nonzero for $\eta\neq 0$. Thus the complementing boundary condition holds. This finishes the proof of Theorem 4.1. It then follows from Theorem 4.1 that the linearization of BVP (3.2) is elliptic, which further implies that the boundary map $\tilde \Pi$ defined in §3 is Fredholm – the linearization $D\tilde \Pi$ is a Fredholm operator at $(\tilde g^{(4)},0)$. Now according to the Theorem 3.2, we can conclude that Theorem 1.1 is true on condition of the Assumption 3.1, i.e. If $\tilde g^{(4)}$ is a stationary vacuum spacetime metric such that the Assumption 3.1 holds, then the moduli space $\mathbb E$ admits smooth Banach manifold structure near $[\tilde g^{(4)}]$, and the boundary map $\Pi$ is Fredholm at $[\tilde g^{(4)}]$. To conclude this section, recall that in §3 we show that $\mathcal C$ can be interpreted geometrically as a local coordinate chart of the moduli space $\mathbb E$, and the map $\tilde\Pi$ is exactly the map $\Pi$ expressed in this chart. However, such a local chart is effective only if Assumption 3.1 holds. In the following section, we will develop an alternative local chart at a reference metric $\tilde g^{(4)}\in\mathcal E$ where Assumption 3.1 fails. Furthermore, we show that the ellipticity result still holds in this case. The operator $\beta_{\tilde g^{(4)}}\delta^*_{\tilde g^{(4)}}$ with Dirichlet boundary condition is elliptic and self-adjoint. This is shown in §5.1 using the quotient formalism of stationary spacetimes. When expressed on the quotient manifold $(S,g_S)$ (cf.(5.9)), this operator is in the form of the Laplace operator plus lower order terms – especially nontrivial 0-order terms generated by the twist tensor of the metric. If the spacetime metric $\tilde g^{(4)}$ is static, then the twist tensor is zero and the operator $\beta_{\tilde g^{(4)}}\delta^*_{\tilde g^{(4)}}$ reduces to the Laplacian, which is invertible. However, if the metric is not static, the 0-order terms in the operator are not necessarily vanishing or positive, so they may result in a nontrivial kernel of the operator, in which case Assumption 3.1 might fail. On the other hand, because of ellipticity and self-adjointness, this operator must be invertible at least for generic metrics in the space $\mathcal E$. It would be interesting to understand whether invertibility holds for all $g^{(4)}\in\mathcal E$. Alternative charts ================== In this section, we assume that $\tilde g^{(4)}$ is a fixed stationary vacuum metric where Assumption 3.1 fails. Perturbation of the metric -------------------------- $~~$ We will use the projection formalism of stationary spacetimes (cf.\[Kr\],\[G\]) in this subsection. In a globally hyperbolic stationary spacetime $(V^{(4)},g^{(4)})$, the Killing vector field $\partial_t$ generates an isometric and proper $\mathbb R-$action on the spacetime. Let $S$ be the orbit space of this action, i.e. $S=V^{(4)}/\mathbb R$. Then $S$ is a smooth 3-manifold and inherits a Riemannian metric $g_S$, which is the restriction of the metric $g^{(4)}$ to the horizontal distribution — the orthogonal complement of $\text{span}\{\partial_t\}$ in $TV^{(4)}$. It is shown in \[G\] that there is a one-to-one correspondence between tensor fields $T^{'b...d}_{a...c}$ on $S$ and tensor fields $T^{b...d}_{a...c}$ on $V^{(4)}$ which satisfy $$(\partial_t)^aT^{b...d}_{a...c}=0,...,(\partial_t)_dT^{b...d}_{a...c}=0\text{ and }L_{\partial_t}T^{b...d}_{a...c}=0.$$ In the following we will identify tensor fields being on $S$ as tensor fields in $V^{(4)}$ satisfying the conditions above. For example, a vector field $X'\in TS$ corresponds to a vector field $X$ in $V^{(4)}$ such that $\langle X,\partial_t\rangle_{g^{(4)}}=0$ and $L_{\partial_t}X=0$. So we will drop the prime – identify $X'$ as $X$. In the projection formalism, any stationary spacetime metric $g^{(4)}$ is globally of the form $$g^{(4)}=-u^2(dt+\theta)^2+g_S.$$ Here $g_S$ is the metric on $S$, which can also be interpreted as a time-independent tensor field in $V^{(4)}$ as discussed above, and $\theta$ is a 1-form on $S$ (in the same sense) such that $-u^2(dt+\theta)$ is the dual of $\partial_t$ with respect to $g^{(4)}$. In our case $V^{(4)}\cong\mathbb R\times M$, there is a natural diffeomorphism between the quotient manifold $S$ and the hypersurface $M=\{t=0\}$. Thus we can pull back the radius function on $M$ to $S$ and define weighted Hölder spaces of tensor fields on $S$ similarly as in §7.1. So if a vector field $Y\in T^{m,\alpha}_{\delta}(V^{(4)})$ satisfies $\langle Y,\partial_t\rangle_{g^{(4)}}=0$ and $L_{\partial_t}Y=0$, then it can be identified with a tensor field $Y\in T^{m,\alpha}_{\delta}(S)$ Suppose that in the projection formalism, $\tilde g^{(4)}$ is expressed as, $$\tilde g^{(4)}=-u^2(dt+\theta)^2+g_S.$$ Take a smooth curve (parametrized by $\epsilon$) of perturbations of $\tilde g^{(4)}$ given by, $$g^{(4)}_{\epsilon}=\tilde g^{(4)}+\epsilon(dt+\theta)^2.$$ First we prove the following property of this family of metrics. The metric $g_{\epsilon}^{(4)}$ is Bianchi-free with respect to $\tilde g^{(4)}$, i.e. $$\beta_{\tilde g^{(4)}}g^{(4)}_{\epsilon}=0.$$ Clearly by (5.2), $$\beta_{\tilde g^{(4)}}g_{\epsilon}^{(4)}=\epsilon\beta_{\tilde g^{(4)}}(dt+\theta)^2.$$ Let $$\alpha=(dt+\theta),$$ then $\alpha=-\frac{1}{u^2}\xi$, where $\xi=-u^2(dt+\theta)$ is the dual of $\partial_t$. Obviously $\alpha(\partial_t)=1$, $\alpha(v)=0,~\forall v\in TS$, and hence $tr_{\tilde g^{(4)}}\alpha^2=-u^{-2}$. As a result, $$\begin{split} \beta_{\tilde g^{(4)}}(\alpha^2)&=\delta_{\tilde g^{(4)}}(\alpha^2)+\frac{1}{2}d(tr_{\tilde g^{(4)}}\alpha^2)=\delta_{\tilde g^{(4)}}(\alpha^2)+u^{-3}du. \end{split}$$ For the divergence term above, we have $$\begin{split} \delta_{\tilde g^{(4)}}(\alpha^2)&=-\frac{1}{u^2}\{-\nabla_{\partial_t}[\alpha^2(\partial_t)]+\alpha^2(\nabla_{\partial_t}\partial_t)\}\\ &=\frac{1}{u^2}\nabla_{\partial_t}\alpha=-\frac{1}{u^2}\nabla_{\partial_t}(\frac{1}{u^2}\xi)\\ &=-\frac{1}{u^4}\nabla_{\partial_t}\xi=-u^{-3}d u. \end{split}$$ In the first equality above, we have used the fact that $\nabla_{\partial_t}\partial_t=u\nabla u$ (cf.§7.5), which gives a vector field on $S$ and so $\alpha(\nabla_{\partial_t}\partial_t)=0$. In the last equality, trivially we have $\nabla_{\partial_t}\xi=udu$. Equations (5.4) and (5.5) now imply that $\alpha^2$ is Bianchi-free. In addition to Bianchi-free, the family of metrics $g_{\epsilon}^{(4)}$ possesses another property — for generic $\epsilon$ the operator $\beta_{\tilde g^{(4)}}\delta^*_{g_{\epsilon}^{(4)}}$ is invertible, in the following sense: In any neighborhood $I$ of $0$, there is an $\epsilon\in I$ such that the BVP with unknown $Y\in T^{m,\alpha}_{\delta}(V^{(4)})$ given by, $$\begin{cases} \beta_{\tilde g^{(4)}}\delta^*_{g_{\epsilon}^{(4)}}Y=0\quad\text{on }S\\ Y=0\quad\text{on }\partial S \end{cases}$$ has only the trivial solution $Y=0$. To prove this proposition, we state the following lemma first. The BVP (5.6) is elliptic (for $\epsilon$ small) and formally self-adjoint. Since $\delta^*_{g_{\epsilon}^{(4)}}Y=\frac{1}{2}L_{Y}g_{\epsilon}^{(4)}=\frac{1}{2}L_{Y}(\tilde g^{(4)}+\epsilon\alpha^2)=\delta^*_{\tilde g^{(4)}}Y+\frac{\epsilon}{2}L_Y\alpha^2$, one has, $$\beta_{\tilde g^{(4)}}\delta^*_{g_{\epsilon}^{(4)}}Y=\beta_{\tilde g^{(4)}}\delta^*_{\tilde g^{(4)}}Y+\frac{\epsilon}{2}\beta_{\tilde g^{(4)}}L_Y\alpha^2,$$ where $\alpha$ is as defined in (5.3). Notice that any time-independent vector field $Y$ in $V^{(4)}$ can be decomposed into a vector field on $S$ and another part proportional to $\partial_t$. Let $Y^{\perp}=u^{-1}\langle Y,\partial_t\rangle$ and $Y^T=Y+u^{-1}Y^{\perp}\partial_t$. Obviously, $Y^{\perp}$ is independent of $t$, so it can be taken as a function on $S$. It is also easy to verify that $\langle Y^T,\partial_t\rangle_{g^{(4)}}=0$ and $L_{\partial_t}Y^T=0$, i.e. $Y^T$ is a vector fields on $S$. Thus we have the decomposition $$Y=Y^T-u^{-1}Y^{\perp}\partial_t.$$ We decompose the vector $\beta_{\tilde g^{(4)}}\delta^*_{\tilde g^{(4)}}Y$ on the right side of equation (5.7) in the same way as described above. It is shown in §7.5 that the components are given by, $$\begin{cases} [\beta_{\tilde g^{(4)}}\delta^*_{\tilde g^{(4)}}Y]^T=(\nabla_{g_S})^*\nabla_{g_S}Y^T+u^{-2}Y^T(u)\nabla u -u^{-1}(\nabla_{g_S})_{\nabla u}Y^T\\ \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad+\frac{1}{2}u^2d\theta(d\theta(Y^T))-u^2d\theta(\nabla\frac{Y^{\perp}}{u})\\ ~~~\\ -[\beta_{\tilde g^{(4)}}\delta^*_{\tilde g^{(4)}}Y]^{\perp}=-u\Delta_{g_S}(\frac{Y^{\perp}}{u})+3\langle\nabla\frac{Y^{\perp}}{u},\nabla u\rangle-\frac{1}{4}u^2Y^{\perp}|d\theta|^2 \\ \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad+u\langle d\theta,\nabla_{g_S}Y^T\rangle-d\theta(\nabla u,Y^T), \end{cases}$$ where $\nabla_{g_S}$ (and $\Delta_{g_S}$) denotes connection (and Laplace operator) of $g_S$. Notice the leading terms of the operator in (5.9) are the Laplacian $[(\nabla_{g_S})^*\nabla_{g_S}Y^T]$ and $[-u\Delta_{g_S}(\frac{Y^{\perp}}{u})]$. Thus $\beta_{\tilde g^{(4)}}\delta^*_{\tilde g^{(4)}}$ is an elliptic operator with respect to the Dirichlet boundary condition, and so is the operator $\beta_{\tilde g^{(4)}}\delta^*_{g_{\epsilon}^{(4)}}$ (for $\epsilon$ small) according to the expression (5.7). Let $Y_1,Y_2\in T^{m,\alpha}_{\delta}(V^{(4)})$ be two asymptotically zero time-independent vector fields which are vanishing along $\partial V^{(4)}$. Then, $$\begin{split} &\int_{S}\langle\beta_{\tilde g^{(4)}}\delta^*_{\tilde g^{(4)}}Y_1,Y_2\rangle_{g^{(4)}}\cdot u\cdot dvol_{g_S}\\ =& \int_{S}\{\langle[\beta_{\tilde g^{(4)}}\delta^*_{\tilde g^{(4)}}Y_1]^T,Y_2^T\rangle_{g_S}+(-[\beta_{\tilde g^{(4)}}\delta^*_{\tilde g^{(4)}}Y_1]^{\perp})\cdot Y_2^{\perp}\}\cdot u\cdot dvol_{g_S} \end{split}$$ Substituting equations in (5.9) into the integral above and then integrating by parts gives, $$\begin{split} &\int_{S}\{\langle[\beta_{\tilde g^{(4)}}\delta^*_{\tilde g^{(4)}}Y_1]^T,Y_2^T\rangle_{g_S}+(-[\beta_{\tilde g^{(4)}}\delta^*_{\tilde g^{(4)}}Y_1]^{\perp})\cdot Y_2^{\perp}\}\cdot u\cdot dvol_{g_S}\\ &=\int_{S}\{\langle[\beta_{\tilde g^{(4)}}\delta^*_{\tilde g^{(4)}}Y_2]^T,Y_1^T\rangle_{g_S}+(-[\beta_{\tilde g^{(4)}}\delta^*_{\tilde g^{(4)}}Y_2]^{\perp})\cdot Y_1^{\perp}\}\cdot u\cdot dvol_{g_S}\\ &\quad+(\int_{\partial S}+\int_{\infty})[B(Y_2,Y_1)-B(Y_1,Y_2)], \end{split}$$ where $B(Y_2,Y_1)=u\langle\nabla_{\mathbf n}Y_2^T,Y_1^T\rangle]+2u^2d\theta(\mathbf n,Y_1^T)Y_2^{\perp}+u\mathbf n(Y_1^{\perp})Y_2^{\perp}$. It is obvious that the boundary integral on $\partial S$ is zero, since $Y_1,Y_2$ vanish on the boundary. The boundary term at infinity $\int_{\infty}=\lim_{r\to\infty}\int_{S_r}$, with $S_r$ denoting the sphere of radius $r$ on $S$. It is also zero because the decay rate of the bilinear form $B(Y_1,Y_2)$ is $2\delta+1>2$. (cf. Remark 5.1) Thus it follows that, the differential operator (5.9) is formally self-adjoint with respect to the measure $u\cdot dvol_{g_S}$ on $S$. $Remark.$ One has the following integration by parts formula in the spacetime $(V^{(4)},g^{(4)})$: $$\begin{split} \int_{V^{(4)}}\langle \nabla_{\tilde g^{(4)}}^*\nabla_{\tilde g_{(4)}}Y_1,Y_2\rangle_{\tilde g^{(4)}}&dvol_{\tilde g^{(4)}}= \int_{V^{(4)}}\langle \nabla_{\tilde g^{(4)}}^*\nabla_{\tilde g_{(4)}}Y_2,Y_1\rangle_{\tilde g^{(4)}}dvol_{\tilde g^{(4)}}\\ &+\int_{\partial V^{(4)}}\langle (\nabla_{\tilde g^{(4)}})_{\mathbf n}Y_2,Y_1\rangle_{\tilde g^{(4)}}-\langle (\nabla_{\tilde g^{(4)}})_{\mathbf n}Y_1,Y_2\rangle_{\tilde g^{(4)}}. \end{split}$$ When the spacetime $(V^{(4)},\tilde g^{(4)})$ is stationary, the equation above reduces to the equation (5.10) on the quotient manifold $(S,g_S)$. Using the same method as above, it is easy to check the following equality holds for any time-independent symmetric 2-tensor $h\in S^{m,\alpha}_{\delta+1}(V^{(4)})$ and vector field $Y\in T^{m,\alpha}_{\delta}(V^{(4)})$ with $Y|_{\partial V^{(4)}}=0$: $$\begin{split} \int_{S}\langle\beta_{\tilde g^{(4)}}h,Y\rangle_{\tilde g^{(4)}}\cdot u\cdot dvol_{g_S}= \int_{S}\langle h,\delta_{\tilde g^{(4)}}^*Y+\frac{1}{2}(\delta_{\tilde g^{(4)}}Y)\tilde g^{(4)}\rangle_{\tilde g^{(4)}} u\cdot dvol_{g_S}. \end{split}$$ Thus, as for the second term on the right side of equation (5.7), we have the following equality for all vector fields $Y_1,Y_2\in T^{m,\alpha}_{\delta}(V^{(4)})$ which vanish at $\partial V^{(4)}$: $$\begin{split} &\int_{S}\langle \beta_{\tilde g^{(4)}}L_{Y_1}\alpha^2,Y_2\rangle_{\tilde g^{(4)}} u\cdot d vol_{g_S}\\ &=\int_{S}\langle L_{Y_1}\alpha^2,\delta^*_{\tilde g^{(4)}}Y_2+\frac{1}{2}(\delta_{\tilde g^{(4)}}Y_2)\tilde g^{(4)}\rangle_{\tilde g^{(4)}} u\cdot d vol_{g_S}\\ &=\int_S-2u^{-2}\langle L_{Y_1}\alpha^2(\partial_t)^T,\delta^*_{\tilde g^{(4)}}Y_2(\partial_t)^T\rangle_{g_S} u\cdot dvol_{g_S}\\ &=\int_{S}-2u^{-2}\langle~ d\theta(Y_1^T)-d(\frac{Y_1^{\perp}}{u}),~-\frac{u^2}{2}[d\theta(Y^T_2)-d(\frac{Y_2^{\perp}}{u})]~\rangle_{g_S}u\cdot d vol_{g_S}\\ &=\int_{S}-2u^{-2}\langle~ -\frac{u^2}{2}[d\theta(Y_1^T)-d(\frac{Y_1^{\perp}}{u})],~d\theta(Y^T_2)-d(\frac{Y_2^{\perp}}{u})~\rangle_{g_S}u\cdot d vol_{g_S}\\ \end{split}$$ $$\begin{split} &=\int_S-2u^2\langle \delta^*_{\tilde g^{(4)}}Y_1(\partial_t)^T,L_{Y_2}\alpha^2(\partial_t)^T\rangle_{g_S}u\cdot dvol_{g_S}\\ &=\int_{S}\langle \delta^*Y_1+\frac{1}{2}(\delta_{\tilde g^{(4)}}Y_1)\tilde g^{(4)}, L_{Y_2}\alpha^2\rangle_{\tilde g^{(4)}} u\cdot d vol_{g_S} \\ &=\int_{S}\langle Y_1, \beta_{\tilde g^{(4)}}L_{Y_2}(dt+\theta)^2\rangle_{\tilde g^{(4)}} u\cdot d vol_{g_S}. \end{split}$$ In the calculation above, the first equality comes from formula (5.11). The second and third equalities are based on the decompositions (equations (5.12-13) below) of the time-independent vector fields $L_{Y_1}\alpha^2$ and $\delta^*_{\tilde g^{(4)}}Y_2$. Furthermore, the last three equalities above are carried out in the reversed way as the first three. The time-independent vector fields $L_{Y_1}\alpha^2$ and $\delta^*_{\tilde g^{(4)}}Y_2$ can be decomposed as, $$\begin{split} \begin{cases} [L_{Y_1}\alpha^2]^T=0\\ [L_{Y_1}\alpha^2](\partial t,\partial t)=0\\ \{[L_{Y_1}\alpha^2](\partial t)\}^T=d\theta( Y_1^T)-d(\frac{ Y_1^{\perp}}{u}), \end{cases} \end{split}$$ and $$\begin{split} \begin{cases} (\delta^*_{\tilde g^{(4)}}Y_2)^T=\delta^*_{g_S} Y_2^T\\ \delta^*_{\tilde g^{(4)}}Y_2(\partial_t,\partial_t)=-u Y_2^T(u)\\ [\delta^*_{\tilde g^{(4)}}Y_2(\partial_t)]^T=-\frac{1}{2}u^2d\theta( Y_2^T)+\frac{1}{2}u^2d(\frac{ Y_2^{\perp}}{u}). \end{cases} \end{split}$$ We refer to §7.4 for detailed proof of the equations (5.12-13). Summing up all the facts above, we conclude that the system (5.6) is formally self-adjoint. Now we give the proof for Proposition 5.3. We prove it by contradiction. Assume that Proposition 5.3 is not true, so there exists an interval $I$ which contains $0$ such that for any $\epsilon\in I$, the system (5.6) has a nonzero solution. From Lemma 5.4, we see that system $(5.6)$ gives rise to a analytic curve of elliptic self-adjoint operators parametrized by $\epsilon$. By the perturbation theory for self-adjoint operators (cf.\[K\],\[W\]), there exists a smooth curve of nontrivial solutions $Y(\epsilon)~(\epsilon\in I)$ solving the system $(5.6)$, i.e. $$\begin{cases} \beta_{\tilde g^{(4)}}\delta^*_{g_{\epsilon}^{(4)}}Y(\epsilon)=0\quad\text{on }S\\ Y(\epsilon)=0\quad\text{on }\partial S \end{cases} \quad\forall \epsilon\in I.$$ The proof of this is discussed in detail in §7.6. In particular, $Y(0)$ is a nontrivial solution to $(5.6)$ at $\epsilon=0$. In the following we will denote it as $\tilde Y=Y(0)$. Taking the linearization of the system above at $\epsilon=0$, we obtain: $$\begin{cases} \beta_{\tilde g^{(4)}}\delta_{\tilde g^{(4)}}^* Y'+\beta_{\tilde g^{(4)}}\delta^*_{g'}\tilde Y=0\quad\text{on }S\\ Y'=0\quad\text{on }\partial S, \end{cases}$$ where $$Y'=\frac{d}{d\epsilon}|_{\epsilon=0}Y(\epsilon), ~\delta^*_{g'}\tilde Y=\frac{d}{d\epsilon}|_{\epsilon=0}\delta^*_{g_{\epsilon}}\tilde Y =\frac{1}{2}\frac{d}{d\epsilon}|_{\epsilon=0}L_{\tilde Y}g_{\epsilon}=\frac{1}{2}L_{\tilde Y}\alpha^2 .$$ The first equation in (5.14) gives, $$-\beta_{\tilde g^{(4)}}\delta_{\tilde g^{(4)}}^* Y'=\beta_{\tilde g^{(4)}}\delta^*_{g'}\tilde Y.$$ Since $\beta_{\tilde g^{(4)}}\delta^*_{\tilde g^{(4)}}$ is self-adjoint, the equation above yields that, $$\begin{split} \int_{V^{(4)}}\langle \beta_{\tilde g^{(4)}}\delta^*_{g'}\tilde Y,\tilde Y\rangle dvol_{\tilde g^{(4)}}&=-\int_{V^{(4)}}\langle \beta_{\tilde g^{(4)}}\delta^*_{\tilde g^{(4)}}Y',\tilde Y\rangle dvol_{\tilde g^{(4)}}\\ &=-\int_{V^{(4)}}\langle Y',\beta_{\tilde g^{(4)}}\delta^*_{\tilde g^{(4)}}\tilde Y\rangle dvol_{\tilde g^{(4)}}\\ &=0. \end{split}$$ Apply integration by parts to the left side of $(5.15)$ and obtain, $$\int_{V^{(4)}}\langle\delta^*_{g'}\tilde Y,\delta^*_{\tilde g^{(4)}}\tilde Y+\frac{1}{2}(\delta \tilde Y)\tilde g^{(4)}\rangle dvol_{\tilde g^{(4)}}=0.$$ In the above, $\delta^*_{g'}\tilde Y=\frac{1}{2}L_{\tilde Y}\alpha^2$. Now apply the formulas (5.12-13) to $L_{\tilde Y}\alpha^2$ and $\delta^*_{\tilde g^{(4)}}\tilde Y$, and substitute them into $(5.16)$. It follows that, $$\int_S\frac{1}{2}||d\theta(\tilde Y^T)-d(\frac{\tilde Y^{\perp}}{u})||_{g_S}^2u\cdot dvol_{g_S}=0.$$ Therefore, we have $$d\theta(\tilde Y^T)=d(\frac{\tilde Y^{\perp}}{u}).$$ Recall that $\tilde Y$ is a nontrivial solution to system $(5.6)$ at $\epsilon=0$. By applying the decomposition equations in (5.9) to the vector field $\tilde Y$, we express the time-independent system (5.6) (at $\epsilon=0$) as an equivalent system: $$\begin{cases} \nabla_{g_S}^*\nabla_{g_S}\tilde Y^T-\frac{1}{u}(\nabla_{g_S})_{\nabla u}\tilde Y^T+\frac{1}{u^2}\tilde Y^T(u)\nabla u\\ \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad+\frac{1}{2}u^2d\theta(d\theta(\tilde Y^T))-u^2d\theta(\nabla\frac{\tilde Y^{\perp}}{u})=0\\ ~~\\ \Delta_{g_S}(\frac{\tilde Y^{\perp}}{u})-3\frac{1}{u}\langle\nabla u,\nabla\frac{\tilde Y^{\perp}}{u}\rangle+\frac{1}{4}uY^{\perp}|d\theta|^2\\ \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad-\langle d\theta,\nabla_{g_S}\tilde Y^T\rangle+\frac{1}{u}d\theta(\nabla u,Y^T)=0. \end{cases}$$ Observe that the last two terms in the first equation in $(5.18)$ can be manipulated as: $$\begin{split} &\frac{1}{2}u^2d\theta(d\theta(\tilde Y^T))-u^2d\theta(\nabla \frac{\tilde Y^{\perp}}{u})\\ =&\frac{1}{2}u^2d\theta[d\theta(\tilde Y^T)-2d(\frac{\tilde Y^{\perp}}{u})]\\ =&-\frac{1}{2}u^2d\theta(d\theta(Y^T)), \end{split}$$ where the last equality is based on $(5.17)$. Plugging this back to $(5.18)$, we obtain $$\nabla_{g_S}^*\nabla_{g_S}\tilde Y^T-\frac{1}{u}(\nabla_{g_S})_{\nabla u}\tilde Y^T+\frac{1}{u^2}\tilde Y^T(u)\nabla u-\frac{1}{2}u^2d\theta(d\theta(\tilde Y^T))=0$$ Pairing the equation above with $\tilde Y^T$ yields, $$\frac{1}{2}\Delta_{g_S}(||\tilde Y^T||^2)+||\nabla_{g_S}\tilde Y^T||^2-\frac{1}{2u}(\nabla_{g_S})_{\nabla u}||\tilde Y^T||^2+\frac{1}{u^2}||\tilde Y^T(u)||^2+\frac{1}{2}u^2||d\theta(\tilde Y^T)||^2=0$$ Based on this equation and the fact that $\tilde Y^T$ is asymptotically zero and equals to zero on $\partial S$, it is easy to derive by the maximum principle that $\tilde Y^T=0$, and consequently $\tilde Y^{\perp}=0$ according to the second equation in (5.18). This contradicts with the assumption that $\tilde Y$ is nontrivial. Combining Propositions 5.2 and 5.3, it is straightforward to derive that, In any neighborhood of $\tilde g^{(4)}\in\mathcal S$, there always exists a perturbation $g^{(4)}_0$ of $\tilde g^{(4)}$ such that $\beta_{\tilde g^{(4)}}g^{(4)}_0=0$ (Bianchi-free) and $\beta_{\tilde g^{(4)}}\delta^*_{g^{(4)}_0}$ is invertible. Alternative local charts ------------------------ Theorem 1.1 still holds without Assumption 3.1. In the case Assumption 3.1 fails, we take a perturbation $g^{(4)}_0$ near $\tilde g^{(4)}$ as described in Theorem 5.5. and modify (3.2) to a new BVP with unknowns $( g^{(4)},F)\in \mathcal S\times C^{m,\alpha}_{\delta}(M)$ as follows: $$\begin{split} &\quad\quad\begin{cases} Ric_{g^{(4)}}-\delta^*_{g^{(4)}_0}\beta_{g^{(4)}}g^{(4)}_0=0\\ \Delta F=0, \end{cases} \quad\text{on}\quad M\\ &\quad\quad\begin{cases} g_{\partial M}=\gamma\\ aH_{\partial M}+btr_{\partial M}K=H\\ atr_{\partial M}K+bH_{\partial M}=k\\ \omega_{\mathbf n}+a^2d_{\partial M}(a/b)=\tau\\ -\beta_{g^{(4)}} g_0^{(4)}=0.\\ \end{cases}\quad\text{on}\quad\partial M \end{split}$$ By applying Bianchi operator to the first equation above, one obtains, $$\begin{cases} \beta_{g^{(4)}}\delta^*_{g^{(4)}_0}\big(\beta_{g^{(4)}}g^{(4)}_0\big)=0\quad\text{ on }M,\\ -\beta_{g^{(4)}}g^{(4)}_0=0\quad\text{ on }\partial M. \end{cases}$$ Since the operator $\beta_{\tilde g^{(4)}}\delta^*_{g^{(4)}_0}$ is invertible, so is the operator $\beta_{g^{(4)}}\delta^*_{g^{(4)}_0}$ when $g^{(4)}$ is near $\tilde g^{(4)}$. Thus $(5.20)$ implies that $\beta_{g^{(4)}}g^{(4)}_0=0$ on $M$. So a solution $g^{(4)}$ of (5.19) near $\tilde g^{(4)}$ must be Ricci flat and gauge free with respect to $g_0^{(4)}$. As in §3, to associate the perturbed BVP (5.19) with a natural boundary map, we first construct a solution space $\mathcal C_0$ near $\tilde g^{(4)}$ given by, $$\mathcal C_0=\{(g^{(4)},F)\in \mathcal S\times C^{m,\alpha}_{\delta}(M):~Ric_{g^{(4)}}=0, \beta_{g^{(4)}}g^{(4)}_0=0,\Delta F=0~\text{on}~M~\}.$$ Obviously, $(\tilde g^{(4)},0)\in\mathcal C_0$ by construction. Next, as in the proof of Theorem 3.2, we need to prove that any stationary vacuum metric $g^{(4)}$ near $\tilde g^{(4)}$ can be transformed by a diffeomorphism in $\mathcal D_4$ so that it satisfies the gauge condition $\beta_{g^{(4)}}g^{(4)}_0=0$. Consider the following map: $$\begin{split} &\mathcal G: \mathcal S\times \mathcal D_4\rightarrow (\wedge_1)^{m.\alpha}_{\delta} (V^{(4)})\\ &\mathcal G(g^{(4)},\Phi)=\beta_{\Phi^*g^{(4)}}g^{(4)}_0. \end{split}$$ Notice that $$\beta_{\Phi^*g^{(4)}}g^{(4)}_0=\Phi^*\{\beta_{g^{(4)}}[(\Phi^*)^{-1}g^{(4)}_0]\}.$$ Thus the linearization of $\mathcal G$ at $(\tilde g^{(4)}, Id)$ is given by, $$\begin{split} &D\mathcal G|_{(\tilde g^{(4)},Id)}: T\mathcal S\times T\mathcal D_4\rightarrow (\wedge_1)^{m.\alpha}_{\delta} (V^{(4)})\\ &D\mathcal G|_{(\tilde g^{(4)},Id)}[(h^{(4)},Y)]=-\beta_{\tilde g^{(4)}}\delta_{g^{(4)}_0}^*Y+\beta'_{h^{(4)}}g^{(4)}_0. \end{split}$$ Since in the linearization above, the operator $[-\beta_{\tilde g^{(4)}}\delta_{g^{(4)}_0}^*]$ is invertible, it follows by the implicit function theorem that, for any $g^{(4)}$ near $\tilde g^{(4)}$, there is a unique element $\Phi\in\mathcal D_4$ near to $Id_{V^{(4)}}$ such that the gauge term $\beta_{\Phi^*g^{(4)}}g^{(4)}_0$ vanishes. Therefore the perturbed solution space $\mathcal C_0$ has similar structure as the space $\mathcal C$ in §3, i.e. near $(\tilde g^{(4)},0)$, $\mathcal C_0$ is locally a fiber bundle over the quotient space $\mathcal E/\mathcal D_4$ with fiber being the space of harmonic functions in $C^{m,\alpha}_{\delta}(M)$. Furthermore, based on the Theorems 3.2 and 3.3, we conclude there exists a local diffeomorphism $\mathcal P_0$ such that $\mathcal C_0\cong\mathbb E$ near $(\tilde g^{(4)},0)$ via $\mathcal P_0$ and $$\Pi_0=\Pi\circ\mathcal P_0,$$ where $\Pi_0$ is the natural boundary map defined on $\mathcal C_0$ given by, $$\begin{split} \Pi_0:\mathcal C_0&\rightarrow\mathbf B\\ \Pi_0(g^{(4)},F)=(g_{\partial M},a H_{\partial M}+btr_{\partial M} K,&atr_{\partial M} K+b H_{\partial M},\omega_{\mathbf n}+a^2d_{\partial M}(b/a)). \end{split}$$ As for ellipticity of the system (5.19), notice that linearization of the equality $\beta_{g^{(4)}}g^{(4)}=0$ yields, $$(\beta_{g^{(4)}})'_{h^{(4)}}g^{(4)}=-\beta_{g^{(4)}}{h^{(4)}}.$$ Thus the linearization of the gauge term in (5.19) at $(\tilde g^{(4)},0)$ is given by: $$\begin{split} [-\delta^*_{g^{(4)}_0}\beta_{g^{(4)}}g^{(4)}_0]'_{h^{(4)}}&=-\delta^*_{g^{(4)}_0}(\beta_{\tilde g^{(4)}})'_{h^{(4)}}g^{(4)}_0\\ &=-\delta^*_{g^{(4)}_0}(\beta_{\tilde g^{(4)}})'_{h^{(4)}}(\tilde g^{(4)}+g^{(4)}_0-\tilde g^{(4)})\\ &=\delta^*_{g^{(4)}_0}\beta_{\tilde g^{(4)}}{h^{(4)}}-\delta^*_{g^{(4)}_0}(\beta_{\tilde g^{(4)}})'_{h^{(4)}}(g^{(4)}_0-\tilde g^{(4)}) \end{split}$$ Comparing the system (5.19) with the previous one (3.2), it is easy to see that, at the reference metric $\tilde g^{(4)}$, the only differences between their linearizations are given by the term $$[\delta^*_{g^{(4)}_0}\beta_{\tilde g^{(4)}}-\delta^*_{\tilde g^{(4)}}\beta_{\tilde g^{(4)}}]({h^{(4)}})-\delta^*_{g^{(4)}_0}(\beta_{\tilde g^{(4)}})'_{h^{(4)}}(g^{(4)}_0-\tilde g^{(4)})$$ in the interior equations, and $$-(\beta)'_{h^{(4)}}(g_0^{(4)}-\tilde g^{(4)})$$ in the boundary equations. We can choose $g^{(4)}_0$ close enough to $\tilde g^{(4)}$ so that the terms in (5.22-23) are very small. Then the principal symbols of the perturbed system $(5.19)$ is close to that of the system (3.2). It has been proved that $(3.2)$ is elliptic. So (5.19) is also elliptic. As a consequence $\Pi_0$ is a Fredholm map and hence so is $\Pi$ because of the equivalence relation (5.21). This completes the proof. Local existence and uniqueness ============================== In this section we choose $V^{(4)}=\mathbb R\times(\mathbb R^3\setminus B^3)\subset \mathbb R^4$, equipped with the standard coordinates $\{t,x^i\}~(i=1,2,3)$ induced from $\mathbb R^4$. Let $M$ be the hypersurface $\{t=0\}$. Then $\partial M=S^2$, the unit sphere. Set the reference metric $\tilde g^{(4)}=\tilde g^{(4)}_0$, where $\tilde g^{(4)}_0$ is the standard flat (Minkowski) metric on $\mathbb R\times (\mathbb R^3\setminus B)$, i.e. $\tilde g_0^{(4)}=-dt^2+\Sigma_i (dx^i)^2$. Since it is static, i.e. its twist tensor in the quotient formalism is zero, it is easy to verify that Assumption 3.1 holds in this case (cf.§7.5). So we can use the local chart $(\mathcal C,\tilde\Pi)$ in §3 for the Bartnik boundary map at $[\tilde g^{(4)}_0]\in\mathbb E$. Obviously, the Bartnik data of this metic is $$\tilde\Pi(\tilde g^{(4)}_0, 0)=(g_{S^2}, 2, 0, 0),$$ where $g_{S^2}$ is the standard round metric on $S^2$. In this section we apply the ellipticity result proved in the previous sections to show that in a neighborhood of the standard flat boundary data $(g_{S^2}, 2, 0, 0)$, Bartnik boundary data admits unique stationary vacuum extensions up to diffeomorphisms. The kernel of $D\tilde\Pi_{(\tilde g_0^{(4)},0)}$ is trivial. Assume that $(h^{(4)},G)\in\text{Ker}(D\tilde\Pi_{(\tilde g_0^{(4)},0)})$. Since $(h^{(4)},G)\in T_{(\tilde g_0^{(4)},0)}\mathcal C$, it must be a vacuum deformation, in the sense that the following equations hold on $M$: $$\begin{cases} (Ric)'_{h^{(4)}}=0\\ \Delta G=0. \end{cases}$$ In addition, since elements in $\mathcal C$ satisfy the gauge condition $\beta_{\tilde g_0^{(4)}}g^{(4)}=0$, the same equation holds for the deformation $h^{(4)}$: $$\beta_{\tilde g_0^{(4)}}h^{(4)}=0\quad\text{on }M.$$ Since $(h^{(4)}, G)$ preserves the Bartnik boundary data, linearization of the boundary equations are zero, i.e. $$\begin{cases} h_{\partial M}=0\\ (H_{\partial M})'_{h^{(4)}}=0\\ (tr_{\partial M}K)'_{h^{(4)}}+2G=0\\ (\omega_{\mathbf n})'_{h^{(4)}}+\nabla_{\partial M}G=0. \end{cases}$$ As we know, a stationary spacetime metric is uniquely determined by the data set $(g,X,N)$ on the hypersurface $M$, where $g$ is the induced metric on $M$, $X$ is the shift vector and $N$ is the lapse function. For the standard metric $\tilde g_0^{(4)}$, the corresponding data is $(g_0,0,1)$ with $g_0$ being the flat (Euclidean) metric on $\mathbb R^3\setminus B$. Thus the deformation $h^{(4)}$ can be decomposed as $h^{(4)}=(h,Y,v)$, where $h$ is the deformation of the Riemannian metric $g_0$, $Y$ is the deformation of the shift vector and $v$ is that of the lapse function. The vacuum condition $Ric_{g^{(4)}}=0$ is equivalent to the following equations in terms of $(g,X,N)$ on $M$ (cf.\[Mo\]): $$\begin{cases} K=-\frac{1}{2N}L_Xg\\ Ric_g+(trK)K-2K^2-\frac{1}{N}D^2N-\frac{1}{N}L_XK=0\\ \frac{1}{N}\Delta N+|K|^2-\frac{1}{N}tr(L_XK)=0\\ \delta K+d(trK)=0. \end{cases}$$ It is easy to linearize the equations above at $(g_0,0,1)$ and obtain a system in terms of $(h,Y,v)$, which is equivalent to equation (6.2), given by, $$\begin{cases} Ric'_h-D^2v=0\\ \Delta_{g_0} v=0\\ \delta_{g_0}\delta_{g_0}^*Y-d\delta_{g_0} Y=0\\ \Delta G=0. \end{cases} \quad\text{on}~M.$$ The gauge equation (6.3) is equivalent to (cf.§7.7 for the proof) $$\begin{cases} \delta_{g_0}Y=0\\ \delta_{g_0}h+\frac{1}{2}d(tr_{g_0}h+2v)=0, \end{cases}\quad\text{on }M.$$ The boundary conditions (6.4) are equivalent to: $$\begin{cases} h_{\partial M}=0\\ H'_h=0\\ tr_{\partial M}\delta_{g_0}^*Y+2G=0\\ [\delta_{g_0}^*Y(\mathbf n)]^T+\nabla_{g_0^T}G=0. \end{cases} \quad\text{on}~\partial M.$$ In the last equation above, we use the superscript $'$$'$$^T$$'$$'$ to denote the restriction of tensors to the tangent bundle of $\partial M$. It is proved in \[A2\] that the first two equations in (6.5) combined with the first two boundary conditions in (6.7) imply that $v=0$ and $h=\delta_{g_0}^* Z$ for some vector field $Z\in C^{m+1,\alpha}_{\delta}(M)$ vanishing on $\partial M$. Additionally, $h$ must satisfy the gauge equation in (6.6). It follows that $\beta_{g_0}\delta_{g_0}^*Z=0$, which further implies that $Z=0$. So we obtain $h=0$ on $M$. It remains to prove $Y=0$ and $G=0$. The third equation in (6.5) and the first equation in (6.6) together imply: $$\delta_{g_0}\delta_{g_0}^*Y=0\quad\text{on}~M.$$ Pair the equation above with $Y$. Then integration by parts gives, $$\begin{split} 0&=\int_M \langle\delta_{g_0}\delta_{g_0}^*Y,Y\rangle_{g_0}dvol_{g_0}\\ &=\int_M|\delta_{g_0}^*Y|^2-\int_{\partial M}\delta_{g_0}^*Y(\mathbf n,Y)-\int_{\infty}\delta^*_{g_0}Y(\mathbf n,Y)\\ &=\int_M|\delta_{g_0}^*Y|^2-\int_{\partial M}\delta_{g_0}^*Y(\mathbf n,Y^T)-\int_{\partial M}\delta_{g_0}^*Y(\mathbf n,\mathbf n) Y^{\perp}. \end{split}$$ In the boundary integral, we have decomposed $Y$ as $Y=Y^T+Y^{\perp}\mathbf n$, with $Y^{\perp}=\langle Y,\mathbf n\rangle_{g_0}$. In the second line, the boundary term at infinity $\int_{\infty}=\lim_{r\to\infty}\int_{S_r}$ is zero because the decay rate of $[\delta^*_{g_0}Y(\mathbf n,Y)]$ is $2\delta+1>2$. For the second term in the last line of (6.8), one has, $$\begin{split} \delta_{g_0}^*Y(\mathbf n,Y^T)&=\langle[\delta_{g_0}^*Y(\mathbf n)]^T,Y^T\rangle =-\langle\nabla_{g_0^T}G,Y^T\rangle\\ &=-div_{g_0^T}(G\cdot Y^T)+G\cdot div_{g_0^T}Y^T\\ &=-div_{g_0^T}(G\cdot Y^T)-\frac{1}{2}(tr_{\partial M}\delta^*_{g_0}Y)\cdot div_{g_0^T}Y^T. \end{split}$$ Here the second equality comes from the last boundary equation in (6.7) and the last equality is based on the third boundary equation in (6.7). As for the last term in (6.8), notice that we have the following equality on the boundary: $$0=\delta_{g_0} Y=-\delta^*_{g_0}Y(\mathbf n,\mathbf n)-tr_{\partial M}\delta_{g_0}^*Y,$$ so that $\delta^*_{g_0}Y(\mathbf n,\mathbf n)=-tr_{\partial M}\delta_{g_0}^*Y$. In addition, $ tr_{\partial M}\delta^*_{g_0}Y =tr_{\partial M}\delta_{g_0}^*Y^T+tr_{\partial M}\delta^*_{g_0}(Y^{\perp}\mathbf n) =div_{g_0^T}Y^T+H_{g_0}Y^{\perp} =div_{g_0^T}Y^T+2Y^{\perp}. $ Substituting these computations into the integral equation (6.8) gives, $$\begin{split} 0&=\int_M|\delta_{g_0}^*Y|^2\\ &\quad\quad+\int_{\partial M}\frac{1}{2}(div_{g_0^T}Y^T+2Y^{\perp})\cdot div_{g_0^T}Y^T+\int_{\partial M}(div_{g_0^T}Y^T+2Y^{\perp}) Y^{\perp}\\ &=\int_M|\delta_{g_0}^*Y|^2+\frac{1}{2}\int_{\partial M}(div_{g_0^T}Y^T)^2+4Y^{\perp}\cdot div_{g_0^T}Y^T+4(Y^{\perp})^2\\ &=\int_M|\delta_{g_0}^*Y|^2+\frac{1}{2}\int_{\partial M}(div_{g_0^T}Y^T+2Y^{\perp})^2\\ &=\int_M|\delta_{g_0}^*Y|^2+\frac{1}{2}\int_{\partial M}(tr_{\partial M}\delta^*_{g_0}Y)^2. \end{split}$$ It immediately follows, $$\begin{split} &\delta_{g_0}^*Y=0\quad\text{on }M,\\ &tr_{\partial M}\delta_{g_0}^*Y=0\quad\text{on }\partial M. \end{split}$$ The first equation above implies that $Y$ is a Killing vector field of the flat metric $g_0$ on $\mathbb R^3\setminus B$. In addition $Y$ must be asymptotically zero since it comes from a deformation of the asymptotically flat metrics in $\mathcal C$. Thus it follows $Y=0$ on $M$. The boundary equation in (6.9) implies that $G=0$ on $\partial M$ according to (6.7). Furthermore, $G$ is asymptotically zero and harmonic according to (6.5). So $G=0$ on $M$. This completes the proof. Next, we prove that the Fredholm map $D\tilde\Pi_{(\tilde g_0^{(4)},0)}$ is of index 0 by showing the operator $D\mathcal F=(D\mathcal L,D\mathcal B)$ defined in §4 has index 0 at $(\tilde g_0^{(4)},0)$. Here we use the idea in \[A1\] — the boundary data in $D\mathcal B$ can be continuously deformed to a new collection of self-adjoint boundary data. Define the new boundary operator $D\mathcal{\tilde B}$ as follows: $$\begin{split} D\mathcal {\tilde B}: T_{(0,\tilde g_0^{(4)})}[\mathcal S\times C^{m,\alpha}_{\delta}(M)]\rightarrow \mathbb B\\ D\mathcal {\tilde B}(h^{(4)},G)= (\quad &h_{\partial M},\\ &\nabla_{\mathbf n}(h^{(4)}(\mathbf n,\mathbf n)),\\ &\mathbf n(G),\\ &-\frac{1}{2}\nabla_{\mathbf n}[h^{(4)}(\partial_t)]^T,\\ &-\frac{1}{2}\nabla_{\mathbf n}h^{(4)}(\partial_t,\partial_t),\\ &-\nabla_{\mathbf n}h^{(4)}(\mathbf n)^T,\\ &-\nabla_{\mathbf n}h^{(4)}(\partial_t,\mathbf n) \quad). \end{split}$$ Let $\mathcal N$ denote the space of deformations $(h^{(4)},G)$ of $(\tilde g_0^{(4)},0)$ in $\mathcal S\times C^{m,\alpha}_{\delta}(M)$ that are in the kernel of the boundary operator $D\mathcal{\tilde B}$, i.e. $$\begin{split} \mathcal N=\{~(h^{(4)},G)\in T_{(0,\tilde g_0^{(4)})}[ \mathcal S\times C_{\delta}^{m,\alpha}(M)]:~~D\mathcal{\tilde B}(h^{(4)},G)=0\quad\}. \end{split}$$ The operator $D\mathcal L: \mathcal N\rightarrow [(S_2)_{\delta+2}^{m-2,\alpha}\times C_{\delta+2}^{m-2,\alpha}](M)$, given by $$\begin{split} D\mathcal L(h^{(4)},G)=(D^*_{\tilde g_0^{(4)}}D_{\tilde g_0^{(4)}}h^{(4)},\Delta G), \end{split}$$ is formally self-adjoint. Let $(h^{(4)},G),(k^{(4)},J)$ denote two deformations in $\mathcal N$. Integration by parts yields: $$\begin{split} &\int_{M}\langle D\mathcal L(h^{(4)},G),(k^{(4)},J)\rangle_{\tilde g^{(4)}_0}dvol_{g_0} =\int_{M}\langle D\mathcal L(k^{(4)},J),(h^{(4)},G)\rangle_{\tilde g^{(4)}_0}\\ &\quad\quad\quad\quad\quad+\int_{\partial M}B[(k^{(4)},J),(h^{(4)},G)]-B[(h^{(4)},G),(k^{(4)},J)]. \end{split}$$ Here the boundary term at infinity is zero because of the decay behavior of the the deformations. The bilinear form $B$ is given by, $$B[(k^{(4)},J),(h^{(4)},G)]=\langle\nabla_{\mathbf n}k^{(4)},h^{(4)}\rangle_{\tilde g_0^{(4)}}+\mathbf n(J)G.$$ It is easy to verify that the terms above are zero because $(h^{(4)},G)$ and $(k^{(4)},J)$ make all the boundary terms listed in (6.10) vanish. Therefore $D\mathcal L$ is formally self-adjoint. Thus it follows that the new operator $(D\mathcal L,D\mathcal{\tilde B})$ is of index 0. Next we show that the boundary data in $D\mathcal B$ can be deformed continuously through elliptic boundary data to $D\mathcal{\tilde B}$. Define a family of boundary operator $D\mathcal B_t,~t\in[0,1]$ as follows, $$\begin{split} D\mathcal B_t: T_{(\tilde g_0^{(4)},0)}[&\mathcal S\times C^{m,\alpha}_{\delta}(M)]\rightarrow \mathbb B\\ D\mathcal B_t(h^{(4)},G)= (\quad &h_{\partial M},\\ &(1-t)(H_{\partial M})'_{h^{(4)}}+t\nabla_{\mathbf n}(h^{(4)}(\mathbf n,\mathbf n)),\\ &(1-t)(tr_{\partial M}K)'_{h^{(4)}}+t\mathbf n(G),\\ &-\frac{1}{2}[\nabla_{\mathbf n}[h^{(4)}(\partial_t)](e_i)+(1-t)\nabla_{e_i}[h^{(4)}(\partial_t)](\mathbf n)]+(1-t)e_i(G),\\ &-\frac{1}{2}\nabla_{\mathbf n}h^{(4)}(\partial_t,\partial_t)+(1-t)[\frac{1}{2}\nabla_{\mathbf n}tr_Mh+\delta h(\mathbf n)],\\ &-\nabla_{\mathbf n}h^{(4)}(\mathbf n)^T+(1-t)[-\nabla_{e_i}h^{(4)}(e_i)^T+\frac{1}{2}\nabla_{g_0^T}(trh^{(4)})],\\ &-\nabla_{\mathbf n}h^{(4)}(\mathbf n,\partial_t)-(1-t)\nabla_{e_i}h^{(4)}(e_i,\partial_t)\quad). \end{split}$$ Here $\{e_i\},~i=2,3$ denotes a local orthonormal basis of $T(\partial M)$. It is easy to check that $D\mathcal B_1=D\mathcal{\tilde B}$. When $t=0$, it is obvious that the first three lines of the boundary data above give the first three of the linearized Bartnik boundary data. It is also easy to check that the fourth line above (at $t=0$) has the same principal part as the linearized Bartnik boundary term $(\omega_{\mathbf n}+d_{\partial M}G)'$. Moreover, the last three lines above are respectively the $\mathbf n$, tangential ($\partial M$) and $\partial_t$ components of the gauge term $\beta_{\tilde g_0^{(4)}}h^{(4)}$ when $t=0$. Therefore, $D\mathcal B_0=D\mathcal B$. The operator $(D\mathcal L, D\mathcal B_t)$ is elliptic for $t\in[0,1]$. One can carry out the same proof as in §4. Since the shift vector and lapse function of $\tilde g_0^{(4)}$ are simply $X=0$ and $N=1$, the principal symbol of the interior operator is (cf.equation (4.2)), $$L(\xi)=(\xi_1^2+\xi_2^2+\xi_3^2)I_{11\times 11}.$$ Set $(X,N)=(0,1)$ in equation (4.4) to compute the principal matrix of boundary operator $D\mathcal B_0$. Make a linear combination with the matrix of $D\mathcal B_1$. One can derive the principal matrix of the boundary operator $D\mathcal B_t~(t\in[0,1])$ as, $$B_t(\xi)= \begin{bmatrix} 0_{3\times 8}& \resizebox{.08\textwidth}{!}{$ \begin{matrix} 1&0&0\\ 0&1&0\\ 0&0&1 \end{matrix} $} \\ (\tilde B_t)_{8\times8}&* \end{bmatrix},$$ where $\tilde B_t$ is given by (up to a factor of $-32^{-1}$), $$\resizebox{.99\textwidth}{!}{$ \begin{bmatrix} 0&0&0&0&0&t\xi_1&-(1-t)\xi_2&-(1-t)\xi_3\\ -t\xi_1&0&0&(1-t)\xi_2&(1-t)\xi_3&0&0&0\\ -2(1-t)\xi_2&0&(1-t)\xi_2&\xi_1&0&0&0&0\\ -2(1-t)\xi_3&0&(1-t)\xi_3&0&\xi_1&0&0&0\\ 0&\xi_1&0&0&0&(1-t)\xi_1&2(1-t)\xi_2&2(1-t)\xi_3\\ 0&(1-t)\xi_2&0&0&0&-(1-t)\xi_2&2\xi_1&0\\ 0&(1-t)\xi_3&0&0&0&-(1-t)\xi_3&0&2\xi_1\\ 0&0&\xi_1&(1-t)\xi_2&(1-t)\xi_3&0&0&0 \end{bmatrix}.$}$$ The determinant of $B_t(\xi)$ is $$\text{det}B_t(\xi)=\frac{1}{32}[t\xi_1^4-(2+t)(1-t)^2\xi_1^2(\xi_2^2+\xi_3^2)]\cdot[2(2+t)(1-t)^2(\xi_2^2+\xi_3^3)\xi_1^2-4t\xi_1^4] .$$ Let $\xi=z\mu+\eta$, where $z=i|\eta|^2$, the root of det$L(z\mu+\eta)=0$ with positive imaginary part. Then $$\text{det}(B_t(z\mu+\eta))=-\frac{1}{32}[t+(2+t)(1-t)^2]\cdot[2(2+t)(1-t)^2+4t]|\eta|^8,$$ which obviously never vanishes for $t\in[0,1]$, $\eta\neq 0$. Thus the complementing boundary condition holds for all $t\in[0,1]$, which completes the proof. To conclude, we have the following theorem: The boundary map $\tilde\Pi$ is locally a diffeomorphism near $(\tilde g_0^{(4)},0)$. From Lemma 6.2, 6.3 and the homotopy invariance of the index, it follows that the index of the boundary map $\tilde\Pi$ is 0 at $(\tilde g^{(4)}_0,0)$. In addition, it is proved in Theorem 6.1 that the kernel of $D\tilde\Pi_{(\tilde g_0^{(4)},0)}$ is trivial. Thus, the linearization $D\tilde\Pi_{(\tilde g_0^{(4)},0)}$ is an isomorphism. Then the inverse function theorem in Banach spaces gives the theorem. Now the equivalence relation between the maps $\tilde\Pi$ and $\Pi$ (cf.§3) gives Theorem 1.2. Appendix ======== Notations --------- $~~$ On a Riemannian manifold $M\cong\mathbb R^3\setminus B^3$, we can pull back the standard coordinates $\{x^i\} (i=1,2,3)$ and the radius function $r$ from $\mathbb R^3\setminus B^3$ to $M$ under a chosen diffeomorphism. Then given $m\in\mathbb{N}$, and $\alpha,\delta\in\mathbb{R}$, the weighted Hölder spaces on $M$ are defined as, $$\begin{split} &C^m_{\delta}(M)=\{\text{functions}~v~\text{on}~M: ||v||_{C^m_{\delta}}=\Sigma_{k=0}^msup~r^{k+\delta}|\nabla^kv|<\infty\},\\ &C^{m,\alpha}_{\delta}(M)=\{\text{functions}~v~\text{on}~M:\\ &\quad\quad\quad\quad\quad||v||_{C^m_{\delta}}+sup_{x,y}[\text{min}(r(x),r(y))^{m+\alpha+\delta} \frac{\nabla^mv(x)-\nabla^mv(y)}{|x-y|^{\alpha}}] <\infty\},\\ &Met^{m,\alpha}_{\delta}(M)=\{\text{metrics}~g~\text{on}~M: (g_{ij}-\delta_{ij})\in C^{m,\alpha}_{\delta}(M)\},\\ &T^{m,\alpha}_{\delta}(M)=\{\text{vector fields}~X~\text{on}~M: X^i\in C^{m,\alpha}_{\delta}(M)\},\\ &(\wedge_1)^{m,\alpha}_{\delta}(M)=\{1-\text{forms}~\sigma~\text{on}~M: \sigma_{i}\in C^{m,\alpha}_{\delta}(M)\},\\ &D^{m,\alpha}_{\delta}(M)=\{\text{diffeomorphisms }\Psi:M\to M, \Psi(x^1,x^2,x^3)=(y^1,y^2,y^3): \\ &\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad y^i-x^i\in C^{m,\alpha}_{\delta}(M)\}. \end{split}$$ A tensor field is called asymptotically trivial (or zero) at the rate of $\delta$, if it belongs to one of the spaces above. It is well-known that Laplace-type operators with Dirichlet boundary condition are Fredholm when acting on these weighted Hölder spaces (cf.\[LM\],\[Mc\],\[MV\]). On the compact manifold $\partial M$, we use the standard Hölder norm to define various Banach spaces of tensor fields as following, $$\begin{split} &C^m(\partial M)=\{\text{functions}~v~\text{on}~M: ||v||_{C^m}=\Sigma_{k=0}^msup~|\nabla^kv|<\infty\},\\ &C^{m,\alpha}(\partial M)=\{\text{functions}~v~\text{on}~M:~||v||_{C^m}+sup_{x,y}[ \frac{\nabla^mv(x)-\nabla^mv(y)}{|x-y|^{\alpha}}] <\infty\},\\ &Met^{m,\alpha}(\partial M)=\{\text{metrics}~g~\text{on}~\partial M: g_{ab}\in C^{m,\alpha}(\partial M)\},\\ &(\wedge_1)^{m,\alpha}(\partial M)=\{1-\text{forms}~\sigma~\text{on}~\partial M: \sigma_{a}\in C^{m,\alpha}(\partial M)\}. \end{split}$$ On the boundary $\partial M$ we use the index $1$ to denote the normal direction to $\partial M$ and $2,3$ the tangential direction. So in the definition above, the subscripts $a,b=2,3$. Let $(V^{(4)},g^{(4)})$ be a stationary spacetime such that $V^{(4)}\cong \mathbb R\times M$, with coordinates $\{t,x^i\}$ such that $\mathbb R$ is parametrized by $t$ and $M=\{t=0\}$. Assume $\partial_t$ is the time-like Killing vector field. In such a spacetime, a tensor field $\tau$ is called *time-independent* if $L_{\partial_t}\tau=0$. In this case, we can think of $\tau$ as a tensor field on $M$. Define the following spaces of time-independent tensor fields in $(V^{(4)},g^{(4)})$: $$\begin{split} &C^{m,\alpha}_{\delta}(V^{(4)})=\{\text{functions }f\text{ in }V^{(4)}:~\partial_t f=0,~f\in C^{m,\alpha}_{\delta}(M)\},\\ &T^{m,\alpha}_{\delta}(V^{(4)})=\{\text{vector fields }Y\text{ in }V^{(4)}:~L_{\partial_t}Y=0,~Y^{\gamma}\in C^{m,\alpha}_{\delta}(M)\},\\ &S^{m,\alpha}_{\delta}(V^{(4)})=\{\text{symmetric 2-tensor fields }h^{(4)}\text{ in }V^{(4)}:~L_{\partial_t}h^{(4)}=0,~h^{(4)}_{\gamma\beta}\in C^{m,\alpha}_{\delta}(M)\},\\ &(\wedge_1)^{m,\alpha}_{\delta}(V^{(4)})=\{\text{1-forms }\omega\text{ in }V^{(4)}:~L_{\partial_t}\omega=0,~\omega_{\gamma}\in C^{m,\alpha}_{\delta}(M)\}. \end{split}$$ In the definition above, the greek letters $\gamma,\beta=0,1,2,3$, where the index $0$ denotes the $\partial_t$ component of the tensor field and $1,2,3$ the tangential (to $M$) components. Scalar fields $a,b$ in the time translation ------------------------------------------- $~~$ As in proposition 2.1, set up the diffeomorphism $$\begin{split} &\Phi_f: V^{(4)}\to V^{(4)}\\ &\Phi_f(t,x_1,x_2,x^3)=(t+f,x^1,x^2,x^3). \end{split}$$ It maps the hypersurface $M=\{t=0\}$ to a new slice $\Phi_f(M)=\{t=f\}$ in $V^{(4)}$. Recall that $\mathbf N$ is the future-pointing time-like unit normal vector to the slice $M\subset (V^{(4)},g^{(4)})$ and $\mathbf n$ is the outward unit normal to $\partial M\subset (M,g)$. Their correspondence in the pull-back spacetime $(V^{(4)},\Phi_f^*(g^{(4)}))$ are $\hat{\mathbf N}$ and $\hat{\mathbf n}$. Thus $d\Phi_f(\hat{\mathbf N})$ and $d\Phi_f(\hat{\mathbf n})$ are the unit normal vector fields to the new slice $\Phi_f(M)$ and its boundary $\Phi_f(\partial M)$ in $(V^{(4)},g^{(4)})$ respectively. Since $f|_{\partial M}=0$, $\partial M=\Phi_f(\partial M)$ and $d\Phi_f|_{\partial M}=Id_{\partial M}$. So along the boundary $\partial M$ the unit normal vectors $(\mathbf{\hat N},\mathbf{\hat n})$ must be mapped to the same subspace as $(\mathbf N,\mathbf n)$ in $TV^{(4)}$, i.e. $$\begin{split} d\Phi_f(\mathbf{\hat N}),d\Phi_f(\mathbf{\hat n})\in \text{span}\{\mathbf N,\mathbf n\}. \end{split}$$ Therefore there exist scalar fields $a,b,c,d$ belongs to $C^{m,\alpha}(\partial M)$ so that $$\begin{cases} d\Phi_f(\mathbf{\hat N})=a\mathbf N+b\mathbf n,\\ d\Phi_f(\mathbf{\hat n})=c\mathbf N+d\mathbf n. \end{cases}$$ In addition, by the definition of pull-back metric we have $$\begin{split} &\langle d\Phi_f(\mathbf{\hat N}),d\Phi_f(\mathbf{\hat N})\rangle_{g{(4)}}=\langle\hat{\mathbf N},\hat{\mathbf N}\rangle_{\Phi_f^*g^{(4)}}=-1;\\ &\langle d\Phi_f(\mathbf{\hat n}),d\Phi_f(\mathbf{\hat n})\rangle_{g{(4)}}=\langle\hat{\mathbf n},\hat{\mathbf n}\rangle_{\Phi_f^*g^{(4)}}=1;\\ &\langle d\Phi_f(\mathbf{\hat N}),d\Phi_f(\mathbf{\hat n})\rangle_{g{(4)}}=\langle\hat{\mathbf N},\hat{\mathbf n}\rangle_{\Phi_f^*g^{(4)}}=0. \end{split}$$ So we obtain the following equations for $(a,b,c,d)$, $$\begin{cases} -a^2+b^2=-1,\\ -c^2+d^2=1,\\ -ac+bd=0. \end{cases}$$ It further implies that $a^2=d^2$ and $b^2=c^2$. Without loss of generality (up to the choice of directions), we can assume, $$a=d>0,~b=c>0.$$ Moreover, the vector field $d\Phi_f(\mathbf{\hat N})$ must be orthogonal to $d\Phi_f(\partial_{x^i})$ with respect to $g^{(4)}$ because $$\begin{split} \langle d\Phi_f(\mathbf{\hat N}),d\Phi_f(\partial_{x^i})\rangle_{g^{(4)}}=\langle \mathbf{\hat N},\partial_{x^i}\rangle_{\Phi^*g^{(4)}}=0,~\forall i=1,2,3. \end{split}$$ By the definition of $\Phi_f$, it follows that $d\Phi_f(\partial_{x^i})=(\partial_if)\partial_t+\partial_{x^i}$. On the other hand, it is easy to verify that, $$\langle \partial_t-X+N^2\nabla f,(\partial_if)\partial_t+\partial_{x^i}\rangle_{g^{(4)}}=0,~\forall i=1,2,3.$$ where $\nabla f$ denotes the gradient of $f$ with respect to the metric $g^{(4)}$. Thus, equations (7.3) and (7.4) imply that $d\Phi_f(\mathbf{\hat N})$ must be proportional to $\partial_t-X+N^2\nabla f$. We make the direction choice so that $d\Phi_f(\mathbf{\hat N})$ is future pointing, then $$\begin{split} d\Phi_f(\mathbf{\hat N})&=\frac{\partial_t-X+N^2\nabla f}{\sqrt{-||\partial_t-X+N^2\nabla f||^2_{g^{(4)}}}}\\ &=\frac{\partial_t-X+N^2\nabla f}{N\sqrt{1+2X(f)-||N^2\nabla f||^2_{g^{(4)}}}}\\ &=\frac{\partial_t-X+N^2\nabla_{g} f}{N\sqrt{1+2X(f)+X(f)^2-N^2||\nabla_{g} f||^2_g}}\\ &=\frac{\partial_t-X+N^2\nabla_{g} f}{N\sqrt{(1+X(f))^2-N^2||\nabla_{g} f||^2}_g}, \end{split}$$ where $\nabla_{g}f$ denotes the gradient of $f$ with respect to the induced metric $g$ on $M$. Therefore, according to the first equation in (7.1), we obtain $$\begin{split} a&=-\langle\mathbf{N},d\Phi_f(\mathbf{\hat N})\rangle_{g^{(4)}}\\ &=-\frac{g^{(4)}(\partial_t-X,\partial_t-X+N^2\nabla_g f)}{N^2\sqrt{(1+X(f))^2-N^2||\nabla_{g} f||^2}}\\ &=\frac{1+X(f)}{\sqrt{(1+X(f))^2-N^2||\nabla_g f||^2}}. \end{split}$$ In the above, the second equality is because $\mathbf N=(\partial_t-X)/N$ according to the 3+1 formalism (2.1). Moreover, since $f$ is chosen to be vanishing on $\partial M$, so $\nabla_gf=\mathbf n(f)\cdot\mathbf n$ on the boundary. Thus $||\nabla_g f||_g=\mathbf n(f)$ on the boundary $\partial M$, and $X(f)|_{\partial M}=\langle X,\mathbf n\rangle \mathbf n(f)$ and consequently, $$\begin{split} a=\frac{1+\langle X,\mathbf n\rangle \mathbf n(f)}{\sqrt{[1+\langle X,\mathbf n\rangle \mathbf n(f)]^2-N^2|\mathbf n(f)|^2}}\quad\text{on }\partial M, \end{split}$$ which is the formula (2.11). Based on (7.2), we easily derive the formula for $b$ as follows, $$\begin{split} b=\frac{N\mathbf n(f)}{\sqrt{[1+\langle X,\mathbf n\rangle \mathbf n(f)]^2-N^2|\mathbf n(f)|^2}}. \end{split}$$ Linearization of boundary operator $\mathcal B$ ----------------------------------------------- $~~$ For simplicity of notation, we will write $h$ instead of $h^{(4)}$ in this section. Subindex 0 denotes the $\partial_t$ direction in $V^{(4)}$, index 1 denotes the outward normal direction to $\partial M$ and $2,3$ denote the tangential directions on $\partial M$.\ \ 1.With respect to the deformation $h$, linearization of $g_{\partial M}$ is easily seen to be: $$(g_{\partial M})'_h=(h_{22},h_{23},h_{33}).$$\ 2.Linearization of $H_{\partial M}$: By the formula of the linearization of mean curvature (cf.§5.2 in \[Az\] for example), one has $$\begin{split} 2(H_{\partial M})'_h=-2\partial_2h_{12}-2\partial_3h_{13}+\partial_1(h_{22}+h_{33})+O_0(h). \end{split}$$\ 3.Linearization of the second fundamental form $K$: The defining equation for $K$ is $$K_{ij}=-\frac{1}{2N}L_{X}g_{ij},$$ where $g_{ij}$ denotes the Riemannian metric induced from $g^{(4)}$ on $M$ and $X$ is the shift vector on $M$. Notice that $X$ is the dual of the $1-$form $g^{(4)}(\partial_t)|_{M}$, i.e. $X^{\flat}_i=g^{(4)}_{0i}$. Variation of $K$ with respect to $h$ is given by, $$(K'_h)_{ij}=-\frac{1}{2N}(L_{(X)'}g_{ij}+L_{X}h_{ij})+O_0(h).$$ As for the variation $X'$, it is given by, $$\begin{split} X^i&=g^{ik}g^{(4)}_{0k},\\ (X')^i&=\tilde h^{ik}g^{(4)}_{0k}+g^{ik}h_{0k}, \end{split}$$ where $\tilde h$ is the variation of the inverse $g^{ij}$. It is easy to see that $$\tilde h^{ij}g_{jk}=-g^{ij}h_{jk}.$$ Therefore, $$\begin{split} L_{X'}g_{ij}&=g_{ik}\nabla^M_j (X')^{k}+g_{jk}\nabla^M_i (X')^{k}\\ &=\nabla^M_j\{g_{ik} (X')^{k}\}+\nabla^M_i\{g_{jk} (X')^{k}\}\\ &=\nabla^M_j\{g_{il} \tilde h^{lk}g^{(4)}_{0k}+g_{il}g^{lk}h_{0k}\}+\nabla^M_i\{g_{jl} \tilde h^{lk}g^{(4)}_{0k}+g_{jl}g^{lk}h_{0k}\}\\ &=\nabla^M_j\{-h_{il} g^{lk}g^{(4)}_{0k}+h_{0i}\}+\nabla^M_i\{-h_{jl} g^{lk}g^{(4)}_{0k}+h_{0j}\}\\ &=\nabla^M_j\{-h_{il} X^l+h_{0i}\}+\nabla^M_i\{-h_{jl}X^l+h_{0j}\}, \end{split}$$ and $$\begin{split} L_Xh_{ij}&=\nabla^M_{X}h_{ij}+h_{ik}\nabla^M_{j} X^{k}+h_{jk}\nabla^M_{i} X^{k}=\nabla^M_{X}h_{ij}+O_0(h). \end{split}$$ In the above the connection $\nabla^M$ denotes the covariant derivative with respect to the induced metric $g$ on $M$. Thus, $$\begin{split} (K'_h)_{ij}=-\frac{1}{2N}[\partial_i h_{0j}+\partial_j h_{0i}+\partial_{X}h_{ij}-X^l(\partial_ih_{jl}+\partial_jh_{il})]+O_0(h), \end{split}$$ and consequently, $$\begin{split} [tr_{\partial M}K]'_h&=tr_{\partial M}(K'_h)+O_0(h)\\ &=-\frac{1}{2N}[2\partial_2 h_{02}+2\partial_3 h_{03}+\partial_{X}(h_{22}+h_{33})-X^l(2\partial_2h_{2l}+2\partial_3h_{3l})]+O_0(h),\\ \big((\omega_{\mathbf n})'_h\big)_i&=[(K(\mathbf n)|_{\partial M})'_h]_i\\ &=[K'_h(\mathbf n)|_{\partial M}]_i+O_0(h)\\ &=-\frac{1}{2N}[\partial_1 h_{0i}+\partial_i h_{01}+\partial_{X}h_{1i}-X^l(\partial_1h_{il}+\partial_ih_{1l})]+O_0(h),\\ \text{with}~i=2,3. \end{split}$$\ 4. Linearization of the gauge term $\beta_{\tilde g^{(4)}}g^{(4)}$.\ Obviously $[\beta_{\tilde g^{(4)}}g^{(4)}]'_h=\beta_{\tilde g^{(4)}}h$. Take an arbitrary vector field $Y\in T^{m,\alpha}_{\delta}(V^{(4)})$. Then $\beta_{g^{(4)}}h(Y)=\delta_{g^{(4)}}h(Y)+\frac{1}{2}Y(trh),$ where the two terms on the right side are computed as, $$\begin{split} \delta_{g^{(4)}} h(Y)&=\nabla_{\mathbf N}h(\mathbf N, Y)-\Sigma_{k=1,2,3}\nabla_k h(Y)_k+O_0(h)\\ &=\frac{1}{N^2}\nabla_{\partial_t-X} h(\partial_t-X,Y)-\Sigma_{k=1,2,3}\nabla_k h(Y)_k+O_0(h)\\ &=-\frac{1}{N^2}\partial_{X} h(\partial_t-X,Y)-\Sigma_{k=1,2,3}\partial_k h(Y)_k+O_0(h),\\ trh&=-h(\mathbf N,\mathbf N)+h_{11}+h_{22}+h_{33}\\ &=-\frac{1}{N^2}h(\partial_t-X,\partial_t-X)+h_{11}+h_{22}+h_{33}\\ &=-\frac{1}{N^2}(h_{00}+X^iX^jh_{ij}-2X^lh_{0l})+h_{11}+h_{22}+h_{33}. \end{split}$$ Therefore, for $i=1,2,3$, the $i$th component of the linearized gauge term is given by, $$\begin{split} [\beta_{\tilde g^{(4)}}h^{(4)}]_i=&-\frac{1}{2N^2}[\partial_ih_{00}+X^kX^j\partial_ih_{kj}-2X^l\partial_ih_{0l}]+\frac{1}{2}\partial_i(h_{11}+h_{22}+h_{33})\\ &-\frac{1}{N^2}[\partial_{X} h_{0i}-X^k\partial_Xh_{ki}]-\Sigma_{k=1,2,3}\partial_k h_{ki}+O_0(h); \end{split}$$ and the $\partial_t$ component of the gauge term is given by, $$\begin{split} [\beta_{\tilde g^{(4)}}h^{(4)}]_0=-\frac{1}{N^2}[\partial_{X} h_{00}-X^k\partial_Xh_{k0}]-\Sigma_{k=1,2,3}\partial_k h_{k0}+O_0(h). \end{split}$$ Summing up all the computations above, we obtain the boundary symbol matrix $\tilde B$ in equation (4.4), given by, $$\begin{split} &\tilde B=\\ &\resizebox{.99\textwidth}{!}{$-\frac{1}{2N^2} \begin{bmatrix} 0&0&0&0&0~&0&2N^2\xi_2&2N^2\xi_3\\ 0&0&0&2N\xi_2&2N\xi_3~&0&-2N\xi_2X^1&-2N\xi_3X^1\\ -2N^3\xi_2&0&N\xi_2&N\xi_1&0~&-N\xi_2X^1&N\xi_3X^3&-N\xi_2X^3\\ -2N^3\xi_3&0&N\xi_3&0&N\xi_1~&-N\xi_3X^1&-N\xi_3X^2&N\xi_2X^2\\ 0&\xi_1&2S-2\xi_1X^1&-2\xi_1X^2&-2\xi_1X^3~&\xi_1X^1X^1+N^2\xi_1-2S X^1&\xi_1X^1X^2-2S X^2+2N^2\xi_2&\xi_1X^1X^3-2S X^3+2N^2\xi_3\\ 0&\xi_2&-2\xi_2X^1&2S-2\xi_2X^2&-2\xi_2X^3~&\xi_2X^1X^1-N^2\xi_2&\xi_2X^1X^2-2S X^1+2N^2\xi_1&\xi_2X^1X^3\\ 0&\xi_3&-2\xi_3X^1&-2\xi_3X^2&2S-2\xi_3X^3~&\xi_3X^1X^1-N^2\xi_3&\xi_3X^1X^2&\xi_3X^1X^3-2S X^1+2N^2\xi_1\\ 0&2S&2(N^2\xi_1-SX^1)&2(N^2\xi_2-SX^2)&2(N^2\xi_3-SX^3)&0&0&0 \end{bmatrix}$}, \end{split}$$ inside which $S=\xi_1X^1+\xi_2X^2+\xi_3X^3$. Calculation of the boundary symbol matrix ------------------------------------------ $~~~$ To compute the determinant of the matrix $\tilde B$, we can first simplify it by factoring out the common factors in every row: factor out $(-2N^2)$ from the first row, $2N$ from the second row, $N$ from the third and fourth row, and $2$ from the last row. Then we obtain that $\text{det}\tilde B=-\frac{1}{32N^{11}}\text{det}\hat B$, with $\hat B$ given by, $$\hat B=\resizebox{.97\textwidth}{!}{$ \begin{bmatrix} 0&0&0&0&0~&0&-\xi_2&-\xi_3\\ 0&0&0&\xi_2&\xi_3~&0&-\xi_2X^1&-\xi_3X^1\\ -2N^2\xi_2&0&\xi_2&\xi_1&0~&-\xi_2X^1&\xi_3X^3&-\xi_2X^3\\ -2N^2\xi_3&0&\xi_3&0&\xi_1~&-\xi_3X^1&-\xi_3X^2&\xi_2X^2\\ 0&\xi_1&2S-2\xi_1X^1&-2\xi_1X^2&-2\xi_1X^3~&\xi_1X^1X^1+N^2\xi_1-2S X^1&\xi_1X^1X^2-2S X^2+2N^2\xi_2&\xi_1X^1X^3-2S X^3+2N^2\xi_3\\ 0&\xi_2&-2\xi_2X^1&2S-2\xi_2X^2&-2\xi_2X^3~&\xi_2X^1X^1-N^2\xi_2&\xi_2X^1X^2-2S X^1+2N^2\xi_1&\xi_2X^1X^3\\ 0&\xi_3&-2\xi_3X^1&-2\xi_3X^2&2S-2\xi_3X^3~&\xi_3X^1X^1-N^2\xi_3&\xi_3X^1X^2&\xi_3X^1X^3-2S X^1+2N^2\xi_1\\ 0&S&N^2\xi_1-SX^1&N^2\xi_2-SX^2&N^2\xi_3-SX^3~&0&0&0 \end{bmatrix}$},$$ We now carry out the following row and column operation to simplify $\hat B$. First, multiply the first row of $\hat B$ by $-X^1$ and then add it to the second row. Multiply the first row by $2N^2$ and then add it to the fifth row. The matrix becomes: $$\hat B_1=\resizebox{.97\textwidth}{!}{$ \begin{bmatrix} 0&0&0&0&0~&0&-\xi_2&-\xi_3\\ 0&0&0&\xi_2&\xi_3~&0&0&0\\ -2N^2\xi_2&0&\xi_2&\xi_1&0~&-\xi_2X^1&\xi_3X^3&-\xi_2X^3\\ -2N^2\xi_3&0&\xi_3&0&\xi_1~&-\xi_3X^1&-\xi_3X^2&\xi_2X^2\\ 0&\xi_1&2S-2\xi_1X^1&-2\xi_1X^2&-2\xi_1X^3~&\xi_1X^1X^1+N^2\xi_1-2S X^1&\xi_1X^1X^2-2S X^2&\xi_1X^1X^3-2S X^3\\ 0&\xi_2&-2\xi_2X^1&2S-2\xi_2X^2&-2\xi_2X^3~&\xi_2X^1X^1-N^2\xi_2&\xi_2X^1X^2-2S X^1+2N^2\xi_1&\xi_2X^1X^3\\ 0&\xi_3&-2\xi_3X^1&-2\xi_3X^2&2S-2\xi_3X^3~&\xi_3X^1X^1-N^2\xi_3&\xi_3X^1X^2&\xi_3X^1X^3-2S X^1+2N^2\xi_1\\ 0&S&N^2\xi_1-SX^1&N^2\xi_2-SX^2&N^2\xi_3-SX^3~&0&0&0 \end{bmatrix}$}.$$ In $\hat B_1$, multiply the second row by $(-N^2)$ and add it to the last row: $$\hat B_2=\resizebox{.97\textwidth}{!}{$ \begin{bmatrix} 0&0&0&0&0~&0&-\xi_2&-\xi_3\\ 0&0&0&\xi_2&\xi_3~&0&0&0\\ -2N^2\xi_2&0&\xi_2&\xi_1&0~&-\xi_2X^1&\xi_3X^3&-\xi_2X^3\\ -2N^2\xi_3&0&\xi_3&0&\xi_1~&-\xi_3X^1&-\xi_3X^2&\xi_2X^2\\ 0&\xi_1&2S-2\xi_1X^1&-2\xi_1X^2&-2\xi_1X^3~&\xi_1X^1X^1+N^2\xi_1-2S X^1&\xi_1X^1X^2-2S X^2&\xi_1X^1X^3-2S X^3\\ 0&\xi_2&-2\xi_2X^1&2S-2\xi_2X^2&-2\xi_2X^3~&\xi_2X^1X^1-N^2\xi_2&\xi_2X^1X^2-2S X^1+2N^2\xi_1&\xi_2X^1X^3\\ 0&\xi_3&-2\xi_3X^1&-2\xi_3X^2&2S-2\xi_3X^3~&\xi_3X^1X^1-N^2\xi_3&\xi_3X^1X^2&\xi_3X^1X^3-2S X^1+2N^2\xi_1\\ 0&S&N^2\xi_1-SX^1&-SX^2&-SX^3~&0&0&0 \end{bmatrix}$}.$$ In $\hat B_2$, multiply the second column by $N^2$ and add it to the sixth column. Then multiply the second column by $X^i$ and add it to the $(2+i)$th column ($i=1,2,3$): $$\hat B_3=\resizebox{.97\textwidth}{!}{$ \begin{bmatrix} 0&0&0&0&0~&0&-\xi_2&-\xi_3\\ 0&0&0&\xi_2&\xi_3~&0&0&0\\ -2N^2\xi_2&0&\xi_2&\xi_1&0~&-\xi_2X^1&\xi_3X^3&-\xi_2X^3\\ -2N^2\xi_3&0&\xi_3&0&\xi_1~&-\xi_3X^1&-\xi_3X^2&\xi_2X^2\\ 0&\xi_1&2S-\xi_1X^1&-\xi_1X^2&-\xi_1X^3~&\xi_1X^1X^1+2N^2\xi_1-2S X^1&\xi_1X^1X^2-2S X^2&\xi_1X^1X^3-2S X^3\\ 0&\xi_2&-\xi_2X^1&2S-\xi_2X^2&-\xi_2X^3~&\xi_2X^1X^1&\xi_2X^1X^2-2S X^1+2N^2\xi_1&\xi_2X^1X^3\\ 0&\xi_3&-\xi_3X^1&-\xi_3X^2&2S-\xi_3X^3~&\xi_3X^1X^1&\xi_3X^1X^2&\xi_3X^1X^3-2S X^1+2N^2\xi_1\\ 0&S&N^2\xi_1&0&0~&N^2S&0&0 \end{bmatrix}$}.$$ In $\hat B_3$, multiply the $i$th column by $X^1$ and add it to the $(i+3)$th column ($i=3,4,5$). Then multiply the second column by $X^i$ and add it to column $(i+1),~(i=1,2,3)$. $$\hat B_4=\resizebox{.97\textwidth}{!}{$ \begin{bmatrix} 0&0&0&0&0~&0&-\xi_2&-\xi_3\\ 0&0&0&\xi_2&\xi_3~&0&\xi_2X^1&\xi_3X^1\\ -2N^2\xi_2&0&\xi_2&\xi_1&0~&0&\xi_3X^3+\xi_1X^1&-\xi_2X^3\\ -2N^2\xi_3&0&\xi_3&0&\xi_1~&0&-\xi_3X^2&\xi_2X^2+\xi_1X^1\\ 0&\xi_1&2S-\xi_1X^1&-\xi_1X^2&-\xi_1X^3~&2N^2\xi_1&-2S X^2&-2S X^3\\ 0&\xi_2&-\xi_2X^1&2S-\xi_2X^2&-\xi_2X^3~&0&2N^2\xi_1&0\\ 0&\xi_3&-\xi_3X^1&-\xi_3X^2&2S-\xi_3X^3~&0&0&2N^2\xi_1\\ 0&S&N^2\xi_1&0&0~&N^2S+N^2\xi_1X^1&0&0 \end{bmatrix}$}.$$ In $\hat B_4$, multiply column $2$ by $X^i$ and add it to column $(i+2)$, $(i=1,2,3)$. Multiply column 1 by $(2N^2)^{-1}$ and add it to column 3. Then multiply the first row by $X^1$ and add it to row 2: $$\hat B_5=\resizebox{.97\textwidth}{!}{$ \begin{bmatrix} 0&0&0&0&0~&0&-\xi_2&-\xi_3\\ 0&0&0&\xi_2&\xi_3~&0&0&0\\ -2N^2\xi_2&0&0&\xi_1&0~&0&\xi_3X^3+\xi_1X^1&-\xi_2X^3\\ -2N^2\xi_3&0&0&0&\xi_1~&0&-\xi_3X^2&\xi_2X^2+\xi_1X^1\\ 0&\xi_1&2S&0&0~&2N^2\xi_1&-2S X^2&-2S X^3\\ 0&\xi_2&0&2S&0~&0&2N^2\xi_1&0\\ 0&\xi_3&0&0&2S~&0&0&2N^2\xi_1\\ 0&S&N^2\xi_1+SX^1&SX^2&SX^3~&N^2S+N^2\xi_1X^1&0&0 \end{bmatrix}$}.$$ This is the matrix given in (4.5). Calculation in the projection formalism --------------------------------------- $~~$ Take a general stationary metric in $V^{(4)}$ expressed in the projection formalism as, $$g^{(4)}=-u^2(dt+\theta)^2+g_S.$$ In this section, we use $\nabla$ to denote the Levi-Civita connection of $g^{(4)}$, and $\nabla_{g_S}$ to denote that of $g_S$. We first state two simple facts. 1\. Since $\partial_t$ is a Killing vector field, it follows that for any vector field $Y\in TV^{(4)}$, we have $\langle\nabla_{\partial t}\partial_t, Y\rangle=-\langle\nabla_Y\partial_t,\partial_t\rangle=uY(u)$. Thus, $$\nabla_{\partial_t}\partial_t=u\nabla u.$$ Notice that $\nabla u$ is a vector field on $S$ because $u$ is independent of $t$. 2\. For any horizontal vector fields $v,w\in TS$, one has $\langle v,\partial_t\rangle=0$, $L_{\partial t}v=0$, and hence $$\langle\nabla_vw,\partial_t\rangle=-\langle w,\nabla_v\partial_t\rangle=\langle v,\nabla_w\partial_t\rangle=-\langle \nabla_wv,\partial_t\rangle.$$ Let $\xi=-u^2(dt+\theta)$ be the dual of $\partial_t$. Then by the definition of exterior derivative, we have $d\xi(v,w)=\langle\nabla_v\partial_t,w\rangle-\langle\nabla_w\partial_t,v\rangle$. Combining the quality above, we obtain $d\xi(v,w)=2\langle\nabla_wv,\partial_t\rangle$. On the other hand, $d\xi=d[-u^2(dt+\theta)]=-u^2d\theta-2udu\wedge(dt+d\theta)=-u^2d\theta+2u^{-1}du\wedge\xi$. Thus we can derive that, $$\begin{split} &2\langle\nabla_wv,\partial_t\rangle=d\xi(v,w)=-u^2d\theta(v,w)\\ &\xi([v,w])=\langle \nabla_vw-\nabla_wv,\partial_t\rangle=2\langle\nabla_vw,\partial_t\rangle=u^2d\theta(v,w). \end{split}$$ Next we give a proof for the formula (5.12): Let $\alpha=dt+\theta=-u^{-2}\xi$, so $\alpha(\partial_t)=1,~\alpha(v)=0~\forall v\in TS$. Then according to the the following Lie-derivative formula for time-independent vector feilds $A,B,Y$ in the spacetime: $$\begin{split} L_{Y}\alpha^2(A,B)=Y[\alpha^2(A,B)]-\alpha^2([Y,A],B)-\alpha^2(A,[Y,B]), \end{split}$$ it is easy to see that $$\begin{cases} L_{ Y}\alpha^2(\partial t,\partial t)=0\\ [L_{Y}\alpha^2]^T=0. \end{cases}$$ As for the mixed component of $L_Y\alpha^2$, we can carry out the following computation for $v\in TS$, $$\begin{split} L_Y\alpha^2(\partial_t,v)&=-\alpha^2([Y,v],\partial_t)\\ &=-\alpha([Y,v])\\ &=u^{-2}\xi([Y,v]). \end{split}$$ As discussed in §5, any vector field $Y\in T^{m,\alpha}_{\delta}(V^{(4)})$ can be decomposed as, $$Y=Y^T-\frac{Y^{\perp}}{u}\partial t,~\text{with}~Y^T\in TS \text{ and }Y^{\perp}=\frac{1}{u}\langle Y,\partial_t\rangle.$$ Thus, for $v\in TS$, one has, $$\begin{split} \xi[Y,v]&=\xi([Y^T,v])-\xi([\frac{Y^{\perp}}{u}\partial_t,v])=\xi([Y^T,v])+\xi[v(\frac{Y^{\perp}}{u})\partial_t]\\ &=u^2d\theta(Y^T,v)-u^2v(\frac{Y^{\perp}}{u}). \end{split}$$ In the last equality above, we use the formula in (7.6) to compute $\xi([Y^T,v])$. Plugging this to equation (7.7) we obtain $$[L_Y\alpha^2(\partial_t)]^T=d\theta(Y^T)-d(\frac{Y^{\perp}}{u})$$ This completes the proof of (5.12). Using the same notation as above, we give a proof of the formula (5.13) as follows. Based on the decomposition (7.8), we have $$\begin{split} 2\delta^*_{g^{(4)}}Y=L_{Y^T}g^{(4)}-L_{\frac{Y^{\perp}}{u}\partial_t}g^{(4)}. \end{split}$$ In the following, we assume $v,w\in TS$. For the first term in (7.9), we have $$\begin{split} &L_{Y^T}g^{(4)}(\partial_t,\partial_t)=2\langle\nabla_{\partial_t}Y^T,\partial_t\rangle=2\langle\nabla_{Y^T}\partial_t,\partial_t\rangle=-2uY^T(u),\\ &L_{Y^T}g^{(4)}(v,w)=\langle\nabla_vY^T,w\rangle+\langle\nabla_wY^T,v\rangle=L_{Y^T}g_S(v,w),\\ &L_{Y^T}g^{(4)}(\partial_t,v)=\langle\nabla_{\partial_t}Y^T,v\rangle+\langle\nabla_vY^T,\partial_t\rangle\\ &\quad\quad=\langle\nabla_{Y^T}\partial_t,v\rangle+\langle\nabla_vY^T,\partial_t\rangle=-\langle\nabla_{Y^T}v,\partial_t\rangle+\langle\nabla_vY^T,\partial_t\rangle\\ &\quad\quad=2\langle\nabla_vY^T,\partial_t\rangle=-u^2d\theta(Y^T,v). \end{split}$$ In the last equality, we apply (7.6) to the term $2\langle\nabla_vY^T,\partial_t\rangle$. Summing up the equations, $$\begin{cases} L_{Y^T}g^{(4)}(\partial_t,\partial_t)=-2uY^T(u)\\ [L_{Y^T}g^{(4)}(\partial_t)]^T=-u^2d\theta(Y^T)\\ [L_{Y^T}g^{(4)}]^T=L_{Y^T}g_S. \end{cases}$$ As for the second term on the right side of (7.9), basic calculation yields, $$\begin{split} L_{\frac{Y^{\perp}}{u}\partial_t}g^{(4)}=\frac{Y^{\perp}}{u}L_{\partial_t}g^{(4)}+d(\frac{Y^{\perp}}{u})\odot\xi=d(\frac{Y^{\perp}}{u})\odot\xi. \end{split}$$ Thus, $$\begin{cases} L_{\frac{Y^{\perp}}{u}\partial_t}g^{(4)}(\partial_t,\partial_t)=0\\ [L_{\frac{Y^{\perp}}{u}\partial_t}g^{(4)}(\partial_t)]^T=-u^2d(\frac{Y^{\perp}}{u})\\ [L_{\frac{Y^{\perp}}{u}\partial_t}g^{(4)}]^T=0. \end{cases}$$ Equations (7.10) and (7.11) together give (5.13). At last we derive the decomposition (5.9) of the Bianchi gauge operator. We assume $g^{(4)}$ is in addition vacuum, which is equivalent to the following system in the projection formalism, (cf.\[G\],\[H1\],\[H2\]), $$\begin{cases} Ric_{g_S}=\frac{1}{u}D^2_{g_S}u+2u^{-4}(\omega^2-|\omega|^2_{g_S}\cdot g_S)\\ \Delta_{g_S} u=2u^{-3}|\omega|^2_{g_S}\\ \delta_{g_S}\omega+3u^{-1}\langle du,\omega\rangle_{g_S}=0\\ d\omega=0 \end{cases},$$ where $\omega$ is the twist tensor defined as, $$\omega=-\frac{1}{2}u^{3}\star_{g_S}d\theta.$$ Here we use subscript $'$$'$$_{g_S}$$'$$'$ to denote geometric operators (connection and Laplacian) of the Riemannian metric $g_S$ on the quotient manifold $S$. First observe that, from the last equation in (7.12), it follows that $$\begin{split} 0=d\omega=d(u^3\star_{g_S}d\theta)=d\star_{g_S} (u^3d\theta)=\delta_{g_S}(u^3d\theta)=u^3\delta_{g_S} d\theta-3u^2d\theta(\nabla u). \end{split}$$ Thus, we obtain $$\begin{split} u\delta_{g_S} d\theta=3d\theta(\nabla u). \end{split}$$ Moreover, based on the second equation in (7.12), one easily obtains, $$\begin{split} \Delta_{g_S} u=\frac{1}{2}u^3|d\theta|_{g_S}^2. \end{split}$$ Now we analyze the operator $\beta_{g^{(4)}}\delta^*_{g^{(4)}}$ acting on a time-independent vector field $Y$, which is decomposed as in (7.8). To begin with, because the metric $g^{(4)}$ is vacuum, a standard Bochner-Weitzenbock formula gives, $$\begin{split} 2\beta_{g^{(4)}}\delta^*_{g^{(4)}}Y=\nabla^*\nabla Y-Ric_{g^{(4)}}(Y)=\nabla^*\nabla Y. \end{split}$$ Based on the formula of the Laplace operator, we have, $$\begin{split} \nabla^*\nabla Y=\frac{1}{u^2}[\nabla_{\partial_t}\nabla_{\partial_t}Y-\nabla_{\nabla_{\partial_t}\partial_t}Y]-\Sigma_i[\nabla_{e_i}\nabla_{e_i}Y-\nabla_{\nabla_{e_i}e_i}Y], \end{split}$$ where $e_i~(i=1,2,3)$ are taken to be geodesic normal basis on $S$. In the following, we compute the tensors on the right side of (7.15) term by term.\ \ 1.We start with the first two terms in (7.15). Since $[Y,\partial_t]=0$, we have $\nabla_{\partial_t}Y=\nabla_Y\partial_t$. Thus, the first term in (7.15) gives $$\begin{split} \nabla_{\partial_t}\nabla_{\partial_t}Y&=\nabla_{\partial_t}\nabla_Y\partial_t=\nabla_{\nabla_Y\partial_t}\partial_t=\nabla_{\nabla_{Y^T}\partial_t}\partial_t-\frac{Y^{\perp}}{u}\nabla_{\nabla_{\partial_t}\partial_t}\partial_t\\ &=\nabla_{\nabla_{Y^T}\partial_t}\partial_t-\frac{Y^{\perp}}{u}\nabla_{u\nabla u}\partial_t. \end{split}$$ In the above, we use the decomposition (7.8) and the fact that $\nabla_{\partial_t}\partial_t=u\nabla u$. In the same way, the second term in (7.15) gives, $$\begin{split} &\nabla_{\nabla_{\partial_t}\partial_t}Y=\nabla_{u\nabla u}Y=\nabla_{u\nabla u}Y^T-\nabla_{u\nabla u}(\frac{Y^{\perp}}{u}\partial_t)\\ &=\nabla_{u\nabla u}Y^T-\langle u\nabla u,\nabla \frac{Y^{\perp}}{u}\rangle\cdot\partial_t-\frac{Y^{\perp}}{u}\nabla_{u\nabla u}\partial_t \end{split}$$ Subtract (7.17) from (7.16). We get $$\begin{split} \nabla_{\partial_t}\nabla_{\partial_t}Y-\nabla_{\nabla_{\partial_t}\partial_t}Y &=\nabla_{\nabla_{Y^T}\partial_t}\partial_t-u\nabla_{\nabla u}Y^T+\langle u\nabla u,\nabla \frac{Y^{\perp}}{u}\rangle\cdot\partial_t \end{split}$$ Based on (7.6) and (7.8), $$\nabla_v\partial_t=(\nabla_v\partial_t)^T-u^{-2}\langle\nabla_v\partial_t,\partial_t\rangle\cdot\partial_t\\ =-\frac{1}{2}u^2d\theta(v)+u^{-1}v(u)\cdot\partial_t\quad\forall v\in TS.$$ Thus the first term on the right side of (7.18) can be written as, $$\begin{split} &\nabla_{\nabla_{Y^T}\partial_t}\partial_t=\nabla_{-\frac{1}{2}u^2d\theta(Y^T)+u^{-1}Y^T(u)\cdot\partial_t}\partial_t\\ &=\nabla_{-\frac{1}{2}u^2d\theta(Y^T)}\partial_t+u^{-1}Y^T(u)\nabla_{\partial_t}\partial_t\\ &=\frac{1}{4}u^4d\theta(d\theta(Y^T))-\frac{1}{2}ud\theta(Y^T,\nabla u)\cdot\partial_t+Y^T(u)\nabla u. \end{split}$$ For two vector fields $v,w\in TS$, we have $$\nabla_vw=[\nabla_vw]^T+\langle\nabla_vw,\partial_t\rangle\cdot\frac{\partial_t}{-u^2}=(\nabla_{g_S})_vw+\frac{1}{2}d\theta(w,v)\cdot\partial_t.$$ Applying the decomposition above to the second term on the right side of (7.18) yields, $$\begin{split} u\nabla_{\nabla u}Y^T=u(\nabla_{g_S})_{\nabla u}Y^T-\frac{1}{2}ud\theta(\nabla u,Y^T)\cdot\partial_t \end{split}$$ Plug (7.20),(7.22) into (7.18), $$\begin{split} &\nabla_{\partial_t}\nabla_{\partial_t}Y-\nabla_{\nabla_{\partial_t}\partial_t}Y\\ &= \frac{1}{4}u^4d\theta(d\theta(Y^T))-\frac{1}{2}ud\theta(Y^T,\nabla u)\cdot\partial_t+Y^T(u)\nabla u\\ &\quad\quad-u(\nabla_{g_S})_{\nabla u}Y^T+\frac{1}{2}ud\theta(\nabla u,Y^T)\cdot\partial_t +\langle u\nabla u,\nabla \frac{Y^{\perp}}{u}\rangle\cdot\partial_t\\ &=\frac{1}{4}u^4d\theta(d\theta(Y^T))+Y^T(u)\nabla u-u(\nabla_{g_S})_{\nabla u}Y^T +\langle u\nabla u,\nabla \frac{Y^{\perp}}{u}\rangle\cdot\partial_t \end{split}$$ $~~~$\ 2.As for the third term in (7.15), we first use the decomposition (7.8) to get $$\begin{split} \nabla_{e_i}\nabla_{e_i}Y=\nabla_{e_i}\nabla_{e_i}Y^T-\nabla_{e_i}\nabla_{e_i}(\frac{Y^{\perp}}{u}\partial_t). \end{split}$$ Apply formula (7.21) to the first term on the right side above, $$\begin{split} &\nabla_{e_i}\nabla_{e_i}Y^T\\ &=\nabla_{e_i}[(\nabla_{g_S})_{e_i}Y^T+\frac{1}{2}d\theta(Y^T,e_i)\cdot\partial_t]\\ &=(\nabla_{g_S})_{e_i}(\nabla_{g_S})_{e_i}Y^T+\frac{1}{2}d\theta((\nabla_{g_S})_{e_i}Y^T,e_i)\cdot\partial_t+[\nabla_{e_i}\frac{1}{2}d\theta(Y^T,e_i)]\cdot\partial_t\\ &\quad\quad+\frac{1}{2}d\theta(Y^T,e_i)\cdot\nabla_{e_i}\partial_t\\ &=(\nabla_{g_S})_{e_i}(\nabla_{g_S})_{e_i}Y^T+\frac{1}{2}d\theta((\nabla_{g_S})_{e_i}Y^T,e_i)\cdot\partial_t+[\nabla_{e_i}\frac{1}{2}d\theta(Y^T,e_i)]\cdot\partial_t\\ &\quad\quad+\frac{1}{2}d\theta(Y^T,e_i)\cdot(-\frac{1}{2}u^2d\theta(e_i)+u^{-1}e_i(u)\cdot\partial_t)\\ &=(\nabla_{g_S})_{e_i}(\nabla_{g_S})_{e_i}Y^T-\frac{1}{4}u^2d\theta(Y^T,e_i)\cdot d\theta(e_i)\\ &\quad\quad+[\frac{1}{2}d\theta((\nabla_{g_S})_{e_i}Y^T,e_i)+\frac{1}{2}u^{-1}d\theta(Y^T, e_i)e_i(u)+\frac{1}{2}\nabla_{e_i}d\theta(Y^T,e_i)]\cdot\partial_t. \end{split}$$ Here in the third equality, we use the formula (7.19). As for the second term in (7.24), we first write it as, $$\begin{split} \nabla_{e_i}\nabla_{e_i}(\frac{Y^{\perp}}{u}\partial_t)&=\nabla_{e_i}[e_i(\frac{Y^{\perp}}{u})\partial_t+\frac{Y^{\perp}}{u}\nabla_{e_i}\partial_t]\\ &=e_i(e_i(\frac{Y^{\perp}}{u}))\partial_t+2e_i(\frac{Y^{\perp}}{u})\nabla_{e_i}\partial_t+\frac{Y^{\perp}}{u}\nabla_{e_i}\nabla_{e_i}\partial_t. \end{split}$$ Apply formula (7.19) to the second term above, $$\begin{split} 2e_i(\frac{Y^{\perp}}{u})\nabla_{e_i}\partial_t=2e_i(\frac{Y^{\perp}}{u})[-\frac{1}{2}u^2d\theta(e_i)+u^{-1}e_i(u)\cdot\partial_t]. \end{split}$$ Apply formula (7.19) twice to the third term in (7.26) gives $$\begin{split} &\frac{Y^{\perp}}{u}\nabla_{e_i}\nabla_{e_i}\partial_t\\ &=\frac{Y^{\perp}}{u}\nabla_{e_i}[-\frac{1}{2}u^2d\theta(e_i)+u^{-1}e_i(u)\cdot\partial_t]\\ &=\frac{Y^{\perp}}{u}(\nabla_{g_S})_{e_i}[-\frac{1}{2}u^2d\theta(e_i)]-\frac{1}{4}uY^{\perp}d\theta(d\theta(e_i),e_i)\cdot\partial_t.\\ &\quad\quad+\frac{Y^{\perp}}{u}{e_i}(u^{-1}e_i(u))\cdot\partial_t+\frac{Y^{\perp}}{u^2}e_i(u)(-\frac{1}{2}u^2d\theta(e_i)+u^{-1}e_i(u)\cdot\partial_t)\\ &=\frac{Y^{\perp}}{u}(\nabla_{g_S})_{e_i}[-\frac{1}{2}u^2d\theta(e_i)]-\frac{1}{2}Y^{\perp}e_i(u)d\theta(e_i)\\ &\quad\quad+[\frac{1}{4}uY^{\perp}d\theta(e_i,d\theta(e_i))+\frac{Y^{\perp}}{u^2}{e_i}(e_i(u))]\cdot\partial_t. \end{split}$$ Summarizing equations (7.24-28) gives, $$\begin{split} &\nabla_{e_i}\nabla_{e_i}Y\\ &=(\nabla_{g_S})_{e_i}(\nabla_{g_S})_{e_i}Y^T-\frac{1}{4}u^2d\theta(Y^T,e_i)\cdot d\theta(e_i)\\ &\quad\quad+[\frac{1}{2}d\theta((\nabla_{g_S})_{e_i}Y^T,e_i)+\frac{1}{2}u^{-1}d\theta(Y^T, e_i)e_i(u)+\frac{1}{2}\nabla_{e_i}d\theta(Y^T,e_i)]\cdot\partial_t\\ &\quad\quad-e_i(e_i(\frac{Y^{\perp}}{u}))\partial_t-2e_i(\frac{Y^{\perp}}{u})[-\frac{1}{2}u^2d\theta(e_i)+u^{-1}e_i(u)\cdot\partial_t]\\ &\quad\quad-\frac{Y^{\perp}}{u}(\nabla_{g_S})_{e_i}[-\frac{1}{2}u^2d\theta(e_i)]+\frac{1}{2}Y^{\perp}e_i(u)d\theta(e_i)\\ &\quad\quad-[\frac{1}{4}uY^{\perp}d\theta(e_i,d\theta(e_i))+\frac{Y^{\perp}}{u^2}{e_i}(e_i(u))]\cdot\partial_t\\ &=(\nabla_{g_S})_{e_i}(\nabla_{g_S})_{e_i}Y^T-\frac{1}{4}u^2d\theta(Y^T,e_i)\cdot d\theta(e_i)+e_i(\frac{Y^{\perp}}{u})u^2d\theta(e_i)\\ &\quad\quad-\frac{Y^{\perp}}{u}(\nabla_{g_S})_{e_i}[-\frac{1}{2}u^2d\theta(e_i)]+\frac{1}{2}Y^{\perp}e_i(u)d\theta(e_i)\\ &\quad\quad+[\frac{1}{2}d\theta((\nabla_{g_S})_{e_i}Y^T,e_i)+\frac{1}{2}u^{-1}d\theta(Y^T, e_i)e_i(u)+\frac{1}{2}\nabla_{e_i}d\theta(Y^T,e_i)]\cdot\partial_t\\ &\quad\quad-[e_i(e_i(\frac{Y^{\perp}}{u}))+2u^{-1}e_i(\frac{Y^{\perp}}{u})e_i(u)+\frac{1}{4}uY^{\perp}d\theta(e_i,d\theta(e_i))+\frac{Y^{\perp}}{u^2}{e_i}(e_i(u))]\cdot\partial_t \end{split}$$ Take negative trace of the expression above, $$\begin{split} &-\Sigma_i\nabla_{e_i}\nabla_{e_i}Y\\ &=(\nabla_{g_S})^*\nabla_{g_S}Y^T+\frac{1}{4}u^2d\theta(d\theta(Y^T))-u^2d\theta(\nabla\frac{Y^{\perp}}{u})\\ &\quad\quad+\frac{Y^{\perp}}{2u}\delta_{g_S}[u^2d\theta]-\frac{1}{2}Y^{\perp}d\theta(\nabla u)\\ &\quad\quad+[\langle \frac{1}{2}d\theta,\nabla_{g_S}Y^T\rangle+\frac{1}{2}\delta_{g_S}(d\theta(Y^T))-\Delta_{g_S}(\frac{Y^{\perp}}{u})+\frac{1}{4}uY^{\perp}|d\theta|^2]\cdot\partial_t\\ &\quad\quad+[-\frac{1}{2}u^{-1}d\theta(Y^T,\nabla u)+2u^{-1}\langle\nabla\frac{Y^{\perp}}{u},\nabla u\rangle-\frac{Y^{\perp}}{u^2}\Delta_{g_S} u]\cdot\partial_t. \end{split}$$ Notice that in the third line of (7.29), $\delta_{g_S}[u^2d\theta]=u^2\delta_{g_S}d\theta-2ud\theta(\nabla u)$. In the fourth line of (7.29), $\delta_{g_S}(d\theta(Y^T))=-\delta_{g_S}d\theta(Y^T)+\langle d\theta,\nabla_{g_S}Y^T\rangle$. Thus (7.29) can be rewritten as, $$\begin{split} &-\Sigma_i\nabla_{e_i}\nabla_{e_i}Y\\ &=(\nabla_{g_S})^*\nabla_{g_S}Y^T+\frac{1}{4}u^2d\theta(d\theta(Y^T))-u^2d\theta(\nabla\frac{Y^{\perp}}{u})\\ &\quad\quad+\frac{1}{2}Y^{\perp}u\delta_{g_S}[d\theta]-\frac{3}{2}Y^{\perp}d\theta(\nabla u)\\ &\quad\quad+[\langle d\theta,\nabla_{g_S}Y^T\rangle-\frac{1}{2}\delta_{g_S}d\theta(Y^T)-\Delta_{g_S}(\frac{Y^{\perp}}{u})+\frac{1}{4}uY^{\perp}|d\theta|^2]\cdot\partial_t\\ &\quad\quad+[-\frac{1}{2}u^{-1}d\theta(Y^T,\nabla u)+2u^{-1}\langle\nabla\frac{Y^{\perp}}{u},\nabla u\rangle-\frac{Y^{\perp}}{u^2}\Delta_{g_S} u]\cdot\partial_t. \end{split}$$\ 4.The last term in (7.15) is zero because $\nabla_{e_i}e_i=0$ based on formula (7.21).\ $~~$ Adding up the equations (7.23) and (7.30), we have $$\begin{cases} [\nabla^*\nabla Y]^T=(\nabla_{g_S})^*\nabla_{g_S}Y^T+u^{-2}Y^T(u)\nabla u -u^{-1}(\nabla_{g_S})_{\nabla u}Y^T\\ \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad+\frac{1}{2}u^2d\theta(d\theta(Y^T))-u^2d\theta(\nabla\frac{Y^{\perp}}{u})\\ \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad+\frac{1}{2}Y^{\perp}u\delta_{g_S}d\theta-\frac{3}{2}Y^{\perp}d\theta(\nabla u)\\ ~~~\\ \langle \nabla^*\nabla Y,u^{-2}\partial_t\rangle=\Delta_{g_S}(\frac{Y^{\perp}}{u})-3u^{-1}\langle\nabla\frac{Y^{\perp}}{u},\nabla u\rangle-\langle d\theta,\nabla_{g_S}Y^T\rangle\\ \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad-\frac{1}{4}uY^{\perp}|d\theta|^2+\frac{Y^{\perp}}{u^2}\Delta_{g_S} u\\ \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad-\frac{1}{2}u^{-1}d\theta(\nabla u,Y^T)+\frac{1}{2}\delta_{g_S}d\theta(Y^T). \end{cases}$$ According to equations (7.13) and (7.14), the equations above can be simplified as, $$\begin{cases} [\nabla^*\nabla Y]^T=(\nabla_{g_S})^*\nabla_{g_S}Y^T+u^{-2}Y^T(u)\nabla u -u^{-1}(\nabla_{g_S})_{\nabla u}Y^T\\ \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad+\frac{1}{2}u^2d\theta(d\theta(Y^T))-u^2d\theta(\nabla\frac{Y^{\perp}}{u})\\ ~~~\\ \langle \nabla^*\nabla Y,u^{-2}\partial_t\rangle=\Delta_{g_S}(\frac{Y^{\perp}}{u})-3u^{-1}\langle\nabla\frac{Y^{\perp}}{u},\nabla u\rangle+\frac{1}{4}uY^{\perp}|d\theta|^2\\ \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad-\langle d\theta,\nabla_{g_S}Y^T\rangle+u^{-1}d\theta(\nabla u,Y^T), \end{cases}$$ which is the formula (5.9). We note that in the case where $\tilde g^{(4)}=\tilde g^{(4)}_0$, the standard flat (Minkowski) metric on $\mathbb R\times (\mathbb R^3\setminus B)$. Because $\theta=0,~u=1$ for $\tilde g^{(4)}_0$, equations in (7.31) can be simplified as $$\begin{cases} [\nabla^*\nabla Y]^T=(\nabla_{g_0})^*\nabla_{g_0}Y^T\\ [\nabla^*\nabla Y]^{\perp}=\Delta_{g_0}Y^{\perp}. \end{cases}$$ Here $g_0$ denotes the flat metric in $\mathbb R^3\setminus B$. Based on the decomposition above, it is easy to see that the solution to $\nabla^*\nabla Y=0$ with trivial Dirichlet boundary condition must be $Y=0$. Therefore, the operator $\beta_{\tilde g_0^{(4)}}\delta^*_{\tilde g_0^{(4)}}$ is invertible, i.e. the Assumption 3.1 holds for $\tilde g_0^{(4)}$. Perturbation of the operator $\beta_{\tilde g^{(4)}}\delta^*_{\tilde g^{(4)}}$ ------------------------------------------------------------------------------ $~~$ Here we show that in the beginning of the proof of Proposition 5.3, if it is assumed that the system (5.6) admits a nontrivial solution for all $\epsilon\in I$, then there exists a smooth curve $Y(\epsilon)$ solving it. In the following discussion, we work with the weighted Sobolev spaces. Since a vector $Y$ solving BVP (5.6) must be $C^{\infty}$ smooth by elliptic regularity, the Banach space we choose does not affect the final conclusion. Let $M$ and $V^{(4)}$ be the same as in §7.1. For fixed $p,\delta$, the weighted Sobolev spaces are defined as, $$\begin{split} &L_{\delta}^p(M)=\{\text{functions u on }M:~||u||_{p,\delta}=(\int_{M}|u|^pr^{\delta p-n}dx)^{1/p}<\infty \},\\ &W_{\delta}^{k,p}(M)=\{\text{functions $u$ on }M:~\Sigma_{i=0}^k||D^iu||_{p,\delta+i}<\infty \},\\ &W^{k,p}(TV^{(4)})=\{\text{vector fields }Y\text{ in }V^{(4)}:~L_{\partial_t}Y=0, Y^{\gamma}\in W^{k,p}_{\delta}(M), \gamma=0,1,2,3\}. \end{split}$$ Let $W$ be the space of vector fields that vanish on the boundary: $$\begin{split} W=\{Y\in W^{2,2}_{\delta}(TV^{(4)}): Y=0 \text{ on }\partial M\}. \end{split}$$ The operator $\beta_{\tilde g^{(4)}}\delta^*_{g_{\epsilon}^{(4)}}$ give rise to a family of map $T_{\epsilon}$ defined as, $$\begin{split} &T_{\epsilon}:W\to L^2_{\delta}(TV^{4})\\ &T_{\epsilon}(Y)=r^{2}\beta_{\tilde g^{(4)}}\delta^*_{g_{\epsilon}^{(4)}}(Y) \end{split}$$ It is obvious that $T_{\epsilon}$ is an analytic curve of linear operators parametrized by $\epsilon$. It has been proved in §5 that $\beta_{\tilde g^{(4)}}\delta^*_{g_{\epsilon}^{(4)}}$ is formally self-adjoint, thus so is $T_{\epsilon}$. Moreover, by standard theory of elliptic operators on non-compact manifold (cf.\[MV\],\[Le\]), $T_{\epsilon}$ has compact resolvent. According to \[K\] (Chapter 7, Theorem 3.9), for an analytic curve $T_{\epsilon}$ of self-adjoint operators that have compact resolvent, all (repeated) eigenvalues can be represented by analytic functions $u_n(\epsilon)$ and there is a sequence of analytic vector-valued functions $Y_n(\epsilon)$ representing the eigenvectors to $u_n(\epsilon)$. If, as assumed in the proof of Proposition 5.3, there is an interval $I$ such that for all $\epsilon\in I$ the system (5.6) admits a nonzero solution, then 0 is an eigenvalue of $T_{\epsilon}$ for all $\epsilon\in I$. Based on the analysis above, for each $\epsilon_0\in I$ there must be a function $u_n$ such that $u_n(\epsilon_0)=0$. However, there are only countably many eigenvalues $u_n$. Thus, among the eigenfunctions $u_n(\epsilon)$, there must be some $u_{n_0}$ such that $u_{n_0}(\epsilon)= 0$ for uncountably $\epsilon$. Since $u_{n_0}(\epsilon)$ is analytic in $\epsilon$, $u_{n_0}\equiv 0$ for $\epsilon\in I$. Correspondingly, $Y_{n_0}(\epsilon)$ is a smooth curve of 0-eigenvectors for $T_{\epsilon}$ ($\epsilon\in I$). This directly implies that there is a smooth curve $Y(\epsilon)$ solving the system (5.6). Bianchi operator in the Minkowski spacetime ------------------------------------------- $~~$ We give the proof of equation (6.6). Recall that the spacetime is $(V^{(4)}=\mathbb R\times(\mathbb R^3\setminus B),\tilde g_0^{(4)})$, where $\tilde g_0^{(4)}$ is the standard Minkowski metric. The metric is varied along the infinitesimal deformation $h^{(4)}$ such that $$\beta_{\tilde g_0^{(4)}}h^{(4)}=0.$$ Under the standard coordinate $\{t,x^i\}$ of the flat spacetime $(V^{(4)},\tilde g_0^{(4)})$, the Killing vector field $\partial_t$ is of unit norm and it is perpendicular to the hypersurface $M=\{t=0\}$. On the hypersurface, ${\partial_{x^i}}~ (i=1,2,3)$ is a orthonormal basis of the tangent bundle. Let $\tilde{\nabla}$ denote the Levi-Civita connection of the flat metric. Then $\tilde{\nabla}_{\partial_t}\partial_t=0,~\tilde{\nabla}_{\partial_t}\partial_{x^i}=0$ and $\tilde{\nabla}_{\partial_{x^i}}\partial_{x^j}=0$. As in §6, we use $g_0$ to denote the induced (flat) metric on $M$. While the infinitesimal variation of the spacetime metric is $h^{(4)}$, the shift vector is deformed by the vector field $Y\in TM$ such that $\langle Y,\partial_{x^i}\rangle_{g_0}=h^{(4)}(\partial_t,\partial_{x^i})$. Pairing (7.32) with $\partial_t$, we obtain, $$\begin{split} 0=\beta_{\tilde g_0^{(4)}}h^{(4)}(\partial_t)&=[\delta_{\tilde g_0^{(4)}}h^{(4)}+\frac{1}{2}dtrh^{(4)}](\partial_t)=\delta_{\tilde g_0^{(4)}}h^{(4)}(\partial_t)\\ &=\tilde{\nabla}_{\partial_t}h^{(4)}(\partial_t,\partial_t)-\Sigma_i\tilde{\nabla}_{\partial_{x^i}}h^{(4)}(\partial_{x^i},\partial_t)\\ &={\partial_t}\big(h^{(4)}(\partial_t,\partial_t)\big)-\Sigma_i{\partial_{x^i}}\big(h^{(4)}(\partial_{x^i},\partial_t)\big)\\ &=-\Sigma_i{\partial_{x^i}}\big(h^{(4)}(\partial_{x^i},\partial_t)\big)\\ &=\delta_{g_0} Y. \end{split}$$ This gives the first equation in (6.6). In the calculation above, the second equality uses the fact that $h^{(4)}$ is time-independent. The third and last equality are based on that the metric $\tilde g_0^{(4)}$ and $g_0$ are flat. Under the deformation $h^{(4)}$, the induced metric on $M$ is deformed by $h$ which is the restriction of $h^{(4)}$ on $M$. The lapse function is deformed by $v$ so that $h^{(4)}(\partial_t,\partial_t)=-2v$. Pair (7.32) with $\partial_{x^i}$. Similar calculation as above gives, $$\begin{split} 0=\beta_{\tilde g_0^{(4)}}h^{(4)}(\partial_{x^i})&=[\delta_{\tilde g_0^{(4)}}+\frac{1}{2}dtrh^{(4)}](\partial_{x^i})\\ &=\delta_{\tilde g_0^{(4)}}h^{(4)}(\partial_{x^i})+\frac{1}{2}\partial_{x^i}(trh^{(4)})\\ &=\tilde{\nabla}_{\partial_t}h^{(4)}(\partial_t,\partial_{x^i})-\Sigma_k\tilde{\nabla}_{\partial_{x^k}}h^{(4)}(\partial_{x^k},\partial_{xi})+\frac{1}{2}\partial_{x^i}(trh^{(4)})\\ &={\partial_t}\big(h^{(4)}(\partial_t,\partial_{x^i})\big)-\Sigma_k{\partial_{x^k}}\big(h^{(4)}(\partial_{x^k},\partial_{x^i})\big)+\frac{1}{2}\partial_{x^i}(trh^{(4)})\\ &=-\Sigma_k{\partial_{x^k}}\big(h^{(4)}(\partial_{x^k},\partial_{x^i})\big)+\frac{1}{2}\partial_{x^i}(tr_{g_0}h^{(4)}-h^{(4)}(\partial_t,\partial_t))\\ &=\delta_{g_0} h(\partial_{x^i})+\frac{1}{2}d(tr_{g_0}h+2v) \end{split}$$ This gives the second equation in (6.6). $$\begin{split} \end{split}$$ [10]{} $[ADN]$ S. Agmon, A. Douglis, and L.Nirenberg. 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--- abstract: 'We study a multi-armed bandit problem with covariates in a setting where there is a possible delay in observing the rewards. Under some reasonable assumptions on the probability distributions for the delays and using an appropriate randomization to select the arms, the proposed strategy is shown to be strongly consistent.' author: - 'Sakshi Arya and Yuhong Yang\' bibliography: - 'mybibfile\_rev.bib' title: 'Randomized Allocation with Nonparametric Estimation for Contextual Multi-Armed Bandits with Delayed Rewards' --- Introduction ============ Multi-armed bandits were first introduced in the landmark paper by [@robbins1952]. The development of multi-armed bandit methodology has been partly motivated by clinical trials with the aim of balancing two competing goals, 1) to effectively identify the best treatment (exploration) and 2) to treat patients as effectively as possible during the trial (exploitation). The classic formulation of the multi-armed bandit problem in the context of clinical practice is as follows: there are $\ell \geq 2$ treatments (arms) to treat a disease. The doctor (decision maker) has to choose for each patient, one of the $\ell$ available treatments, which result in a reward (response) of improvement in the condition of the patient. The goal is to maximize the cumulated rewards as much as possible. In the classic multi-armed bandit terminology, this is achieved by devising a policy for sequentially pulling arms out of the $\ell$ available arms, with the goal of maximizing the total cumulative reward, or minimizing the regret. Substantial amount of work has been done both on standard context-free bandit problems ([@gittins1979bandit], [@berry1985bandit], [@lai1985asymptotically], [@auer2002finite]) and on contextual bandits or multi-armed bandits with covariates (MABC) ([@woodroofe1979one], [@sarkar1991one], [@yang2002randomized], [@langford2008epoch], [@li2010contextual] and [@slivkins2014contextual]). The MABC problems have been studied in both parametric and nonparametric frameworks. Our work follows nonparametric framework of MABC in [@yang2002randomized] where the randomized strategy is an annealed $\epsilon$-greedy strategy, which is a popular heuristic in bandits literature ([@sutton2018reinforcement], Chapter 2). Some of the other notable work in studying finite time analysis for MABC problems in a nonparametric framework are [@perchet2013multi; @qian2016kernel; @qian2016randomized]. Some insightful overviews and bibliographic remarks can be found in [@bubeck2012regret], [@cesa2006prediction], [@lattimore2018bandit]. In most multi-armed bandit settings it is assumed that the rewards related to each treatment allocation are achieved before the next patient arrives. This is not realistic since in most cases the treatment effect is seen at some delayed time after the treatment is provided. Most often, it would be the case that while waiting for treatment results of one patient, other patients would have to be treated. In such a situation, all past patient information and feedback is not yet available to make the best treatment choices for the patients being treated at present. While an overwhelming amount of work has been done assuming instantaneous observations in both contextual and non-contextual multi-armed bandit problems, not much work has been done for the case with delayed rewards. The importance of considering delays was highlighted by [@anderson1964sequential] and [@suzuki1966sequential]. They used Bayesian multi-armed bandits to devise optimal policies. Thompson sampling ([@agrawal2012analysis; @russo2018tutorial]) is another commonly used Bayesian heuristic. [@chapelle2011empirical] conducted an empirical study to illustrate robustness of Thompson sampling in the case of constant delayed feedback. Most of the work that has been done in the recent years is motivated by reward delays in online settings like advertisement and news article recommendations. [@Dudik:2011:EOL:3020548.3020569] considered a constant known delay which resulted in an additional additive penalty in the regret for the setting with covariates. [@joulani2013online] propose some black box multi-armed bandit algorithms that use the algorithms for the non-delayed case to handle the delayed case. Their finite time results show an additive increase in the regret for stochastic multi-armed bandit problems. More recently, [@pike2018bandits] proposed a variant of delayed bandits with aggregated anonymous feedback. They show that with their proposed algorithm and with the knowledge of the expected delay, an additive regret increase like in [@joulani2013online] can still be maintained. Some other work related to delayed bandits can be found in [@mandel2015queue], [@cesa2016delay] and [@vernade2017stochastic]. In our knowledge, there does not seem to be any work on delayed MABCs using a nonparametric framework. In this work, we propose an algorithm accounting for delayed rewards with optimal treatment decision making as the motivation. We use nonparametric estimation to estimate the functional relationship between the rewards and the covariates. We show that the proposed algorithm is strongly consistent in that the cumulated rewards almost surely converge to the optimal cumulated rewards. Problem setup ============= Assume that there are $\ell \geq 2$ arms available for allocation. Each arm allocation results in a reward which is obtained at some random time after the arm allocation. For each time $j \geq 1$, a treatment $I_j$ is alloted based on the data observed previously and the covariate $X_j$. We assume that the covariates are $d$-dimensional continuous random variables and take values in the hypercube $[0,1]^d$. Since the rewards can be obtained at some delayed time, we denote $\{t_j \in \mathbb{R}^+, j\geq 1\}$ to be the observation time for the rewards for arms $\{I_j, j\geq 1\}$ respectively. Let $Y_{i,j}$ be the reward obtained at time $t_j \geq j$ for arm $i = I_j$. The mean reward with covariate $X_j$ for the $i$[th]{} arm is denoted as $f_i(X_j), 1\leq i \leq \ell$. The observed reward with covariate $X_j$ by pulling the $i$th arm is modeled as, $ Y_{i,j} = f_{i}(X_j) + \epsilon_{i,j}$, where $\epsilon_{i,j}$ denotes random error with $\text{E}(\epsilon_{i,j}) = 0$ and $\text{Var}(\epsilon_{i,j}) < \infty$ for all $1\leq i \leq \ell$ and $j\in \mathbb{N}$. The functions $f_i$ are assumed to be unknown and not of any given parametric form. The rewards are observed at delayed times $t_j$; the delay in the reward for arm $I_j$ pulled at the $j$[th]{} time is given by a random variable $d_j:= t_j - j$. Assume that these delays are mutually independent, independent of the covariates, and could be drawn from different distributions. That is, let $\{d_j, j\geq 1\}$ be a sequence of independent random variables with probability density functions $\{g_j, j \geq 1\}$ and the cumulative distribution functions $\{G_j, j \geq 1\}$, respectively. Let $\{X_j,j\geq 1\}$ be a sequence of covariates independently generated according to an unknown underlying probability disribution $P_X$, from a population supported in $[0,1]^d$. Let $\delta$ be a sequential allocation rule, which for each time $j$ chooses an arm $I_j$ based on the previous observations and $X_j$. The total mean reward up to time $n$ is $\sum_{j=1}^n f_{I_j}(X_j)$. To evaluate the performance of the allocation strategy, let $i^*(x) = \operatorname*{\arg\!\max}_{1\leq i \leq \ell} f_i(x)$ and $f^*(x) = f_{i^*(x)}(x)$. Without the knowledge of the random errors, the ideal performance occurs when the choices of arms selected $I_1,\hdots, I_n$ match the optimal arms $i^*(X_1), \hdots, i^*(X_n)$, yielding the optimal total reward $\sum_{j=1}^n f^*(X_j)$. The ratio of these two quantities is the quantity of interest, $$R_n(\delta) = \dfrac{\sum_{j=1}^n f_{I_j}(X_j)}{\sum_{j=1}^n f^*(X_j)}.$$ It can be seen that $R_n$ is a random variable no bigger than 1. An allocation rule $\delta$ is said to be strongly consistent if $R_n(\delta) \rightarrow 1$ with probability 1, as $n \rightarrow \infty$. In Section \[algorithm\], we propose an allocation rule which takes into account reward delays. Then in Sections \[consistency\] and \[consistency\_histogram\], we discuss the consistency of the proposed allocation rule under some assumptions and then validate those assumptions when the histogram method is used to estimate the regression functions respectively. The proposed strategy {#algorithm} ===================== Let $Z^{n,i}$ denote the set of observations for arm $i$ whose rewards have been obtained up to time $n$, that is, $Z^{n,i}:= \{(X_j,Y_{i,j}): 1\leq t_j \leq n \ \text{and} \ I_j = i\}$. Let $\hat{f}_{i,n}$ denote the regression estimator of $f_i$ based on the data $Z^{n,i}$. Let $\{\pi_j, j \geq 1 \}$ be a sequence of positive numbers in $[0,1]$ decreasing to zero. 1. **Initialize.** Allocate each arm once, w.l.o.g., we can have $I_1 = 1, I_2 = 2, \hdots, I_\ell = \ell$. Since the rewards are not immediately obtained for each of these $\ell$ arms, we continue these forced allocations until we have at least one reward observed for each arm. Suppose, that happens at time $m_0$. 2. **Estimate the individual functions $f_i$.** For $n = m_0$, based on $Z^{n,i}$, estimate $f_i$ by $\hat{f}_{i,n}$ for $1 \leq i \leq \ell$ using the chosen regression procedure. 3. **Estimate the best arm.** For $X_{n+1}$, let $\hat{i}_{n+1}(X_{n+1}) = \arg\max_{1\leq i \leq \ell} \hat{f}_{i,n}(X_{n+1})$. 4. \[randomization\_step\] **Select and pull.** Randomly select an arm with probability $1-(\ell-1)\pi_{n+1}$ for $i = \hat{i}_{n+1}$ and with probability $\pi_{n+1}$, for all other arms, $i \neq \hat{i}_{n+1}$. Let $I_{n+1}$ denote this selected arm. 5. **Update the estimates.** 1. If a reward is obtained at the $(n+1)$[th]{} time (could be one or more rewards corresponding to one or more arms $I_j, 1\leq j \leq (n+1)$), update the function estimates of $f_i$ for the respective arm (or arms) for which the reward (or rewards) are obtained at $(n+1){\textsuperscript}{th}$ time. 2. If no reward is obtained at the $(n+1)$[th]{} time, use the previous function estimators, i.e. $\hat{f}_{i,n+1} = \hat{f}_{i,n} \ \forall \ i \in \{1,\hdots,\ell\}$. 6. **Repeat.** Repeat steps 3-5 when the next covariate $X_{n+2}$ surfaces and so on. The choice of $\pi_n$ in the randomization step \[randomization\_step\] is crucial in determining how much exploration and exploitation is done at any phase of the trial. To emphasize the role of $\pi_n$, we may use $\delta_{\pi}$ to denote the allocation rule. In order to select the best arm as time progresses, $\pi_n$ needs to decrease to zero but the rate of decrease will play a key role in determining how well the allocations work. For example, if in our set-up we have large delays for some arms then it might be beneficial to decrease $\pi_n$ at a slower rate so that there is enough exploration and the accuracy of our estimates is not affected in the long run. We use a user-determined choice of $\pi_n$ in this work, that is, the sequence $\pi_n$ does not adapt to the data. Consistency of the proposed strategy {#consistency} ------------------------------------ Let $A_n := \{j: t_j \leq n\}$, denote the time points for which rewards were obtained by time $n$. If $A_n$ is known, then the total number of observed rewards until time $n$, denoted by $N_{n}$, is also known. Recall that it is possible to observe multiple rewards at the same time point. Given $A_n$, let $\{s_k, k=1,\hdots, N_n\}$ be the reordered sequence of these observed reward timings, $\{t_k, k\in A_n\}$, arranged in a non-decreasing order. **Assumption 1.** The regression procedure is strongly consistent in $L_\infty$ norm for all individual mean functions $f_i$ under the proposed allocation scheme. That is, $||\hat{f}_{i,n} - f_i||_\infty \overset{\text{a.s.}}{\rightarrow} 0$ as $n \rightarrow \infty$ for each $1\leq i \leq \ell$. As described in the allocation strategy in Section \[algorithm\], $\hat{f}_{i,n}$ is the estimator based on all previously observed rewards. That is, after initialization, the mean reward function estimators are only updated at the time points $\{s_k, k=1,\hdots N_n\}$ where $N_n$ is the number of rewards observed by time $n$. Therefore, this condition is equivalent to saying $||\hat{f}_{i,s_n} - f_i||_\infty \overset{\text{a.s.}}{\rightarrow} 0$ as $n\rightarrow \infty$. **Assumption 2.** Mean functions satisfy $f_i(x) \geq 0$, $A = {\displaystyle}\sup_{1\leq i\leq \ell} \sup_{x \in [0,1]^d} (f^*(x) - f_i(x)) < \infty \ \text{and}\ {\text{E}}(f^*(X_1)) > 0.$ \[Theorem 1\] Under Assumptions 1 and 2, the allocation rule $\delta_\pi$ is strongly consistent as $n\rightarrow \infty$. Note that consistency holds only when the sequence $\{\pi_n, n\geq 1\}$ is chosen such that $\pi_n \rightarrow 0$ as $n\rightarrow \infty$. The proof is very similar to the proof in [@yang2002randomized]. The details can be found in the supplementary material (see *Appendix A.1* in Appendix). Note that Assumption 1, seemingly natural, is a strong assumption and it requires additional work to verify this assumption for a particular regression setting. We verify this assumption for the histogram method in Section \[consistency\_histogram\]. On the other hand, Assumption 2 does not involve the estimation procedure and does not require any verification. The Histogram method {#histogram} ==================== In this section, we explain the histogram method for the setting with delayed rewards. Partition $[0,1]^d$ into $M = (1/h)^d$ hyper-cubes with side width $h$, assuming $h$ is chosen such that $1/h$ is an integer. For some $x \in [0,1]^d$, let $J(x)$ denote the set of time points, for which the corresponding design points observed until time $n$ fall in the same cube as $x$, say $B(x)$, and for which the corresponding rewards are observed by time $n$. Let $N(x)$ denote the size of $J(x)$. That is, let $J(x) = \{j: X_j \in B(x), t_j \leq n\}$ and $N(x) = \sum_{j=1}^n I\{X_j \in B(x), t_j \leq n\}$. Furthermore, let $\bar{J}_i(x)$ be the subset of $J(x)$ corresponding to arm $i$ and $\bar{N}_i(x)$ is the number of such time points, that is, $\bar{J}_i(x) = \{j\in J(x): I_j = i \}$ and $\bar{N}_i(x) = \sum_{j=1}^n I\{I_j = i, X_j \in B(x), t_j \leq n\}$. Then the histogram estimate for $f_i(x)$ is defined as, $$\begin{aligned} \hat{f}_{i,n}(x) = \frac{1}{\bar{N}_i(x)} \sum_{j \in \bar{J}_i(x)} Y_j.\end{aligned}$$ For the estimator to behave well, a proper choice of the bandwidth, $h = h_n$ is necessary. Although one could choose different widths $h_{i,n}$ for estimating different $f_i$’s, for simplicity, the same bandwidth $h_n$ is used in the following sections. For notational convenience, when the analysis is focused on a single arm, $i$ is dropped from the subscript of $\hat{f}$, $\bar{N}$ and $\bar{J}$. Other nonparametric methods like nearest-neighbors, kernel method, spline fitting and wavelets can also be considered for estimation. Assumption 1 could be verified for these methods using the same broad approach as illustrated in the following sections for the Histogram method, along with some method specific mathematical tools and assumptions. Allocation with histogram estimates {#consistency_histogram} ----------------------------------- Here, we show that the histogram estimation method along with the allocation scheme described in Section \[algorithm\], leads to strong consistency under some reasonable conditions on random errors, design distribution, mean reward functions and delays. As already discussed in Section \[consistency\], we only need to verify that Assumption 1 holds for histogram method estimators. Along with Assumption 2, we make the following assumptions. **Assumption 3.** \[ass\_design\_distribution\] The design distribution $P_X$ is dominated by the Lebesgue measure with a density $p(x)$ uniformly bounded above and away from 0 on $[0,1]^d$; that is, $p(x)$ satisfies $\underbar{c} \leq p(x) \leq \bar{c}$ for some positive constants $\underbar{c} < \bar{c}$. **Assumption 4.** \[ass\_bernstein\_errors\] The errors satisfy a moment condition that there exists positive constants $v$ and $c$ such that, for all $m \geq 2$, the Bernstein condition is satisfied, that is, ${\text{E}}|\epsilon_{ij}|^m \leq \frac{m!}{2} v^2 c^{m-2}.$ **Assumption 5.**\[ass\_delay\_independence\] The delays, $\{d_j, j\geq 1\}$, are independent of each other, the choice of arms and also of the covariates. **Assumption 6.** \[assump\_delay\] Let the partial sums of delay distributions satisfy, $\sum_{j=1}^n G_j(n-j) = \Omega(n^\alpha \log^\beta{n})$ [^1] for some $\alpha > 0$, $\beta \in \mathbb{R}$ or for $\alpha = 0$ and $\beta > 1$. Note that, the choice $n^\alpha \log^\beta{n}$ could be generalized to a sub-linear function $q(n)$ with a growth rate faster than $\log{n}$. \[mod\_of\_continuity\] Let $x_1, x_2 \in [0,1]^d$. Then $w(h;f)$ denotes a modulus of continuity defined by, $ w(h;f) = \sup\{|f(x_1) - f(x_2)|: |x_{1k} - x_{2k}| \leq h \ \text{for all}\ 1 \leq k \leq d\}.$ Number of observations in a small cube for histogram estimation. {#A.3} ---------------------------------------------------------------- From Assumption 3 and Assumption 5, we have that for a fixed cube $B$ with side width $h_n$ at time $n$, $ P(X_j \in B, t_j \leq n) = P(X_j \in B)P(t_j \leq n) \geq \underbar{c}h_n^dG_j(n-j)$. Let $N$ be the number of observations that fall in $B$ and are observed by time $n$, that is $N = \sum_{j=1}^n I_{\{X_j \in B, t_j\leq n\}}$. It is easily seen that $N$ is a random variable with expectation $\beta \geq \sum_{j=1}^{n}\underbar{c}h_n^dG_j(n-j)$. From the extended Bernstein inequality (see *Appendix A.3* in \[Appendix\]), we have $$\begin{aligned} \text{P}\left(N \leq \dfrac{\underbar{c}h_n^d \sum_{j=1}^n G_j(n-j)}{2}\right) \leq \exp\left(-\dfrac{3\underbar{c}h_n^d\sum_{j=1}^n G_j(n-j)}{28} \right). \label{no_of_obs}\end{aligned}$$ \[lemma\_theorem\] Let $\epsilon > 0$ be given. Suppose that $h$ is small enough such that $w(h;f) < \epsilon$. Then the histogram estimator $\hat{f}_n$ satisfies, $$\begin{aligned} \text{P}_{A_n,X^n}(||\hat{f}_n - f||_\infty \geq \epsilon) &\leq M \exp\left(-\dfrac{3\pi_n \min_{1\leq b \leq M} N_b}{28} \right)\\ &\quad \quad +2M \exp\left(- \dfrac{\min_{1\leq b \leq M} N_b \pi_n^2 (\epsilon - w(h;f))^2}{8(v^2 + c(\pi_n/2)(\epsilon-w(h;f)))} \right),\end{aligned}$$ where the probability $P_{A_n,X^n}$ denotes conditional probability given design points $X^n = (X_1,X_2,\hdots,X_n)$ and $A_n = \{j: t_j\leq n\}$. Here, $N_b$ is the number of design points for which the rewards have been observed by time $n$ such that they fall in the $b$th small cube of the partition of the unit cube at time $n$. The proof of Lemma \[lemma\_theorem\] is included in the supplementary materials (*Appendix A.2* in Appendix). \[Theorem2\] Suppose Assumptions 2-6 are satisfied. If for some $\alpha > 0$ and $\beta \in \mathbb{R}$ or $\alpha =0$ and $\beta > 1$, $h_n$ and $\pi_n$ are chosen to satisfy, $$n^\alpha (\log{n})^{\beta -1} h_n^d \pi_n^2 \rightarrow \infty, \label{condition_for_assumptionA}$$ then the allocation rule $\delta_\pi$ is strongly consistent. The histogram technique partitions the unit cube into $M = (1/h)^d$ small cubes. For each small cube $B_b, \ 1\leq b\leq M$, in the partition of the unit cube, let $N_b$ denote the number of time points, for which the corresponding design points fall in the cube $B_b$ and corresponding arm rewards are observed by time $n$. In other words, $N_b = \sum_{j=1}^n I_{\{X_j \in B_b, t_j \leq n\}}$. Using inequality we have, $$\begin{aligned} P&\left(N_b \leq \dfrac{ \underbar{c}h_n^d\sum_{j=1}^n G_j(n-j)}{2}\right) \leq \exp\left(-\dfrac{3 \underbar{c}h_n^d \sum_{j=1}^n G_j(n-j)}{28} \right) \nonumber \\ \Rightarrow P&\left(\min_{1\leq b \leq M}N_b \leq \dfrac{ \underbar{c}h_n^d\sum_{j=1}^n G_j(n-j)}{2}\right) \leq M \exp\left(-\dfrac{3 \underbar{c}h_n^d \sum_{j=1}^n G_j(n-j)}{28} \right). \label{min_no_of_points} \end{aligned}$$ Let $W_1,\hdots,W_n$ be Bernoulli random variables indicating whether the $i$th arm is selected $(W_j =1)$ for time point $j$, or not $(W_j =0)$. Note that, conditional on the previous observations and $X_j$, the probability of $W_j =1$ is almost surely bounded below by $\pi_j \geq \pi_n$ for $1\leq j \leq n$. Let $w(h_n;f_i)$ be the modulus of continuity as in Definition \[mod\_of\_continuity\]. Note that, under the continuity assumption of $f_i$, we have $w(h_n;f_i) \rightarrow 0$ as $h_n\rightarrow 0$. Thus, for any $\epsilon >0$, when $h_n$ is small enough, $\epsilon - w(h_n;f_i) \geq \epsilon/2$. Consider, $$\begin{aligned} P(||\hat{f}_{i,n} - f_i||_\infty > \epsilon) &= P\left(||\hat{f}_{i,n} - f_i||_\infty > \epsilon,\min_{1\leq b \leq M}N_b \geq \dfrac{\underbar{c}h_n^d\sum_{j=1}^n G_j(n-j)}{2} \right) \\ &\quad \quad + P\left(||\hat{f}_{i,n} - f_i||_\infty > \epsilon,\min_{1\leq b \leq M}N_b < \dfrac{ \underbar{c}h_n^d\sum_{j=1}^n G_j(n-j)}{2} \right)\\ &\leq {\text{E}}P_{A_n,X^n}\left(||\hat{f}_{i,n} - f_i||_\infty > \epsilon, \min_{1\leq b \leq M}N_b \geq \dfrac{ \underbar{c}h_n^d\sum_{j=1}^n G_j(n-j)}{2} \right)\\ &\quad \quad + P\left(\min_{1\leq b \leq M}N_b < \dfrac{ \underbar{c}h_n^d\sum_{j=1}^n G_j(n-j)}{2} \right), \end{aligned}$$ where $P_{A_n, X^n}$ denotes conditional probability given the design points until time $n$, $X^n = \{X_1,X_2,\hdots,X_n\}$ and the event, $A_n := \{j: t_j \leq n\}$. From Lemma \[lemma\_theorem\], we have that given the design points and the time points for which rewards were observed, for any $\epsilon > 0$, when $h$ is small enough, $$\begin{aligned} \text{P}_{A_n,X^n}(||\hat{f}_n - f||_\infty \geq \epsilon) &\leq M \exp\left(-\dfrac{3\pi_n \min_{1\leq b \leq M} N_b}{28} \right)\\ & \quad \quad +2M \exp\left(- \dfrac{\min_{1\leq b \leq M} N_b \pi_n^2 (\epsilon - w(h_n;f))^2}{8(v^2 + c(\pi_n/2)(\epsilon-w(h_n;f)))} \right). \end{aligned}$$ Using the above inequality and , we have, $$\begin{aligned} P(||\hat{f}_{i,n} - f_i||_\infty > \epsilon)&\leq 2M \exp\left(-\dfrac{\underbar{c}h_n^d(\sum_{j=1}^n G_j(n-j)) \pi_n^2 (\epsilon - w(h_n;f_i))^2}{16 (v^2 + c\pi_n/2(\epsilon - w(h_n;f_i)))} \right)\\ &\quad + M \exp\left(-\dfrac{3\underbar{c}h_n^d\pi_n \sum_{j=1}^n G_j(n-j)}{56} \right)+ \exp\left( -\dfrac{3 \underbar{c}h_n^d \sum_{j=1}^n G_j(n-j)}{28} \right). \end{aligned}$$ It can be shown that the above upper bound is summable in $n$ under the condition, $$\begin{aligned} \dfrac{h_n^d\pi_n^2 \sum_{j=1}^nG_j(n-j)}{\log{n}} \rightarrow \infty. \label{condition_on_delaydist} \end{aligned}$$ It is easy to see that this follows from Assumption 6 and . Since $\epsilon$ is arbitrary, by the Borel-Cantelli lemma, we have that $||\hat{f}_{i,n} - f_i||_\infty \rightarrow 0$. This is true for all arms $1\leq i \leq \ell$. Hence, this completes the proof of Theorem \[Theorem2\]. Effects of reward delay distributions ------------------------------------- As one would expect, the amount of delay in observing the rewards will have a considerable effect on the speed of sequential learning. In terms of treatment allocation, if there are substantial delays in observing patient responses for a particular treatment, the learning for that treatment will slow down and as a result the efficiency of the allocation strategy will decrease. Therefore, Assumption 6 imposes some restrictions on the delay distributions to ensure that at least a small proportion of rewards will be obtained in finite time. It is of interest to see how the delay distribution affects the rate at which $\pi_n$ and $h_n$ are allowed to decrease. This relationship can be understood by examining condition for Theorem \[Theorem2\]. Note that Assumption 6 and in Theorem \[Theorem2\] can be generalized to include any function $q(x)$ with at least a growth rate faster than logarithmic growth rate. We assume $\sum_{j=1}^n G_j(n-j) = \Omega\left(q(n)\right)$ where $q(n)$ satisfies, $q(n)/\log(n) \rightarrow \infty$  as  $n \rightarrow \infty$. Then it is easy to see that $h_n$ and $\pi_n$ can be chosen such that, $$\begin{aligned} \dfrac{h_n^d \pi_n^2 q(n)}{\log(n)} \rightarrow \infty \ \text{as} \ n \rightarrow \infty. \end{aligned}$$ which implies condition holds. A possible advantage of this is that we allow a wide range of possible delay distributions with mild restrictions on the delays. Below, we consider some cases of the delay distributions and see how they effect exploration $(\pi_n)$ and bandwidth $(h_n)$ of the histogram estimator as time progresses. 1. \[case1\] In condition , $q(n) = n^\alpha \log^\beta{n}$ for $\alpha > 0$ and $\beta \in \mathbb{R}$ or $\alpha = 0$ and $\beta > 1$. Let us first consider the case when $\alpha = 0$ and $\beta > 1$, we have $q(n) = \log^\beta{n}$ for $\beta > 1$ and we want $\sum_{j=1}^n G_j(n-j) = \Omega(\log^\beta{n})$. Consider, $\pi_n = (\log{n})^{-(\beta - 1)/(2+d)}$ for $n > m_0$ and $\beta > 1$, then for to hold we need the bandwith $h_n$ also to be of order $\Omega((\log{n})^{-(\beta - 1)/(2+d)})$. For example, $h_n = (\log{n})^{-(\beta - 1)/\beta(2+d)}$ would guarantee consistency. Notice that with these $\pi_n$ and $h_n$, one would spend a lot of time in exploration and the bandwidth would also decay very slowly which would effect the accuracy of the reward function estimates until $n$ is sufficiently large. Notice that the restriction of partial sum of probability distributions for the delays, being at least of the order $\log^\beta{n}$ gives the possibility of modeling cases with extremely large delays. For example, in clinical studies when the outcome of interest is survival time and we want to administer treatments for a disease such that the survival time is maximized. With the unprecedented advances in drug development, the life expectancy of patients is more likely to increase, hence the survival time for a patient given any treatment would be large. Therefore, the assumption that partial sums of probability distributions for the delays until time $n$ need only be at least $\log^\beta{n}$ seems to be quite reasonable when the expected waiting times (in this case survival times) are long. For example, diseases like diabetes and hypertension which have a long survival time, since they cannot be cured, but can be controlled with medications. These diseases also have fairly high prevalence, so a large sample size to be able to get close to optimality would not be a problem. For such diseases, assuming that one would only observe the responses (survival times) of a small fraction of patients in finite time seems reasonable. 2. For the case when $\alpha > 0$ and $\beta \in \mathbb{R}$, we have that $\sum_{j=1}^n G_j(n-j) = \Omega(n^\alpha\log^\beta{n})$. Consider, $\pi_n = n^{-{\alpha}/{(2+d)}}$ for $n > m_0$, then for the condition to hold we need $h_n$ to also be of order $\Omega(n^{-{\alpha}/{(2 + d)}})$. For example, $h_n = n^{-{\alpha}/{2(2+d)}}$ results in $ h_n^d \pi_n^2 n^\alpha \log^{\beta - 1}{(n)} = n^{{\alpha d}/{2(2 + d)}} \log^{\beta - 1}{(n)} \rightarrow \infty \ \text{as n $\rightarrow \infty$}$, irrespective of the value of $\beta$. Here the lower bound on the partial sums of probability distributions for the delays can grow faster than the previous case, depending on the values of $\alpha$ and $\beta$. This restriction of order $n^\alpha (\log^\beta{n})$ can model cases with moderately large delays. From a clinical point of view, one could model diseases in which treatments show their effect in a short to moderate duration of time, for examples diseases like diarrhea, common cold, headache, and nutritional deficiencies. Here the response of interest would be improvement in the condition of a patient as a result of a treatment. For such diseases, one can expect to see the treatment effects on patients in a short period of time. Hence, the delay in observing treatment results will not be too long. If the response considered was survival (survived or not), then stroke could also fall in this category because of high mortality. Note that, Assumption 6 only restricts on the proportion of rewards expected to be observed in the long run. Therefore, it is possible for strong consistency to be achieved even when there is infinite delay in observing the rewards of some arms (non-observance of some rewards). Simulation study ================ We conduct a simulation study to compare the effect of different delay scenarios on the per-round average regret of our proposed strategy. The per-round regret is given by, $ r_n(\delta) = \frac{1}{n} \sum_{j=1}^n (f^*(X_j) - f_{I_j}(X_j))$. Note that if $\frac{1}{n} \sum_{j=1}^n f^*(X_j)$ is eventually bounded above and away from 0 with probability 1, then $R_n(\delta) \rightarrow 1$ a.s. is equivalent to $r_n(\delta) \rightarrow 0$ a.s. ![Per-round regret for the proposed strategy for different delay scenarios. The grid of plots represent 4 different combination of choices for $\{\pi_n\}$ and $\{h_n\}$. For a given row, $\pi_n$ remains fixed and $h_n$ varies and vice versa for columns.[]{data-label="fig:simulations"}](a0_31stAug.pdf "fig:"){width=".40\textwidth"} ![Per-round regret for the proposed strategy for different delay scenarios. The grid of plots represent 4 different combination of choices for $\{\pi_n\}$ and $\{h_n\}$. For a given row, $\pi_n$ remains fixed and $h_n$ varies and vice versa for columns.[]{data-label="fig:simulations"}](a3_31stAug.pdf "fig:"){width=".40\textwidth"}\ ![Per-round regret for the proposed strategy for different delay scenarios. The grid of plots represent 4 different combination of choices for $\{\pi_n\}$ and $\{h_n\}$. For a given row, $\pi_n$ remains fixed and $h_n$ varies and vice versa for columns.[]{data-label="fig:simulations"}](b0_31stAug.pdf "fig:"){width=".40\textwidth"} ![Per-round regret for the proposed strategy for different delay scenarios. The grid of plots represent 4 different combination of choices for $\{\pi_n\}$ and $\{h_n\}$. For a given row, $\pi_n$ remains fixed and $h_n$ varies and vice versa for columns.[]{data-label="fig:simulations"}](b3_31stAug.pdf "fig:"){width=".40\textwidth"}\ Simulation setup ---------------- Consider number of arms, $\ell = 3$, and the covariate space to be two-dimensional, $d= 2$. Let $X_n = (X_{n1}, X_{n2})$ where $X_{ni} \overset{i.i.d.}{\sim} $ Unif$(0,1)$. We assume that the errors $\epsilon_n \sim 0.5 $N(0,1). The first 30 rounds were used for initialization. The following true mean reward functions are used, $$\begin{aligned} f_1(\mathbf{x}) = 0.7 (x_1 + x_2),\ f_2(\mathbf{x}) = 0.5 x_1^{0.75} + \sin(x_2), \ f_3(\mathbf{x})= \frac{2 x_1}{0.5 + (1.5 + x_2)^{1.5}}.\end{aligned}$$ We consider the following delay scenarios and run simulations until $N = 10000$. 1) *No delay*; 2) *Delay 1:* Geometric delay with probability of success (observing the reward) $p = 0.3$; 3) *Delay 2:* Every 5[th]{} reward is not observed by time $N$ and other rewards are obtained with a geometric ($p = 0.3$) delay; 4) *Delay 3:* Each case has probability 0.7 to delay and the delay is half-normal with scale parameter, $\sigma = 1500$; 5) *Delay 4:* In this case we increase the number of non-observed rewards. Divide the data into four equal consecutive parts (quarters), such that, in part 1, we only observe every 10[th]{} (with Geom(0.3) delay) observation by time $N$ and not observe the remaining; in part 2, we only observe every 15[th]{} observation; in part 3, only observe every 20[th]{} observation; in part 4, only observe every 25[th]{} observation. In Figure \[fig:simulations\], we plot the per-round regret vs time by delay type for four combinations of $\pi_n$ and $h_n$. As one would expect (see Figure \[fig:simulations\]), the severity of delay has a clear effect on the regret, and for delay scenarios where a large number of rewards are not observed in finite time, the regret is comparatively higher. Note that most delay scenarios for which a substantial number of rewards can be obtained in finite time, tend to converge in quite similar patterns. **Choice of $\{\pi_n\}$ and $\{h_n\}$:** According to Theorem 2, if $\pi_n$ and $h_n$ are chosen such that condition is met, consistency of the allocation rule follows. Therefore, for the case with $d=2$, which is the case of the simulation setting, we have to choose sequences slower than ($\pi_n = n^{-1/2}, h_n = n^{-1/2}$), even in the case of no delays. Keeping this in mind, we chose two different choices of sequences for $\pi_n$ ($n^{-1/4}, n^{-1/6}$) and two choices of $h_n ((\log{n})^{-1}, n^{-1/6})$. Note that, in Figure \[fig:simulations\], for a given row, $\pi_n$ remains fixed while $h_n$ varies and vice versa for columns. It can be seen that the regret gets worse when $h_n$ decays too fast (in our range of n as $N = 10000$), specially for the scenario (Delay 4) with increasing number of non-observed rewards, possibly because of violation of condition . Also notice that, slow decaying $\pi_n$ has higher regret (last row). This could be because of large randomization error that leads to high exploration price. In general, there are a large pool of choices for $h_n$ and $\pi_n$ that satisfy equation as can be seen from the Figure \[fig:simulations\]. However, a thorough understanding of the finite-time regret rates and further research would be needed to evaluate optimal choices of $\{\pi_n\}$ and $\{h_n\}$ for a given scenario. Conclusion ========== In this work we develop an allocation rule for multi-armed bandit problem with covariates when there is delay in observing rewards. We show that strong consistency can be established for the proposed allocation rule using the histogram method for estimation, under reasonable restrictions on the delay distributions and also illustrate that using a simulation study. Our approach on modeling reward delays is different from the previous work done in this field because, 1) we use nonparametric estimation technique to estimate the functional relationship between the rewards and covariates and 2) we allow for delays to be unbounded with some assumptions on the delay distributions. The assumptions impose mild restrictions on the delays in the sense that they allow for the possibility of non-observance of some rewards as long as a certain proportion of rewards are obtained in finite time. With this general setup, it is possible to model many different situations including the one with no delays. The conditions on the delay distributions easily allow for large delays as long as they grow at a certain minimal rate. This obviously will result in slower rate of convergence because of longer time spent in exploration. Ideally, we would like our allocation scheme to devise the optimal treatments sooner, for which we would need to impose stricter conditions on the delay distributions. Therefore, working on finite-time analysis for the setting with delayed rewards seems to be an immediate future direction. In addition, we assume some knowledge on the delay distributions, so for situations where there is little understanding of the delays, a different approach might be needed, such as a methodology which adaptively updates the delay distributions. Appendix {#Appendix} ======== **Proof of consistency of the proposed strategy** {#Proof_consistency_YangZhu} ------------------------------------------------- Since the ratio $R_n(\delta_\pi){}$ is always upper bounded by 1, we only need to work on the lower bound direction. Note that, $$\begin{aligned} R_n(\delta_\pi) &= \dfrac{\sum_{j=1}^n f_{\hat{i}_j}(X_j)}{\sum_{j=1}^n f^*(X_j)} + \dfrac{\sum_{j=1}^n (f_{I_j}(X_j) - f_{\hat{i}_j}(X_j))}{\sum_{j=1}^n f^*(X_j)}\\ &\geq \dfrac{\sum_{j=1}^n f_{\hat{i}_j}(X_j)}{\sum_{j=1}^n f^*(X_j)} - \dfrac{\frac{1}{n} \sum_{j=1}^n A I_{\{I_j \neq \hat{i}_j\}}}{\frac{1}{n} \sum_{j=1}^n f^*(X_j)}, \label{inequality_begin}\end{aligned}$$ where the inequality follows from Assumption 2. Let $U_j = I_{\{I_j \neq \hat{i}_j\}}$. Since $(1/n)\sum_{j=1}^n f^*(X_j)$ converges a.s. to ${\text{E}}f^*(X) > 0$, the second term on the right hand side in the above inequality converges to zero almost surely if $({1}/{n}) \sum_{j=1}^n U_j \overset{\text{a.s.}}{\rightarrow} 0$. Note that for $j \geq m_0 +1$, $U_j$’s are independent Bernoulli random variables with success probability $(\ell-1)\pi_j$. Since, $$\begin{aligned} \sum_{j=m_0+1}^\infty \text{Var} \left(\dfrac{U_j}{j}\right) = \sum_{j=m_0+1}^{\infty} \dfrac{(\ell - 1)\pi_j (1-(\ell-1)\pi_j)}{j^2} < \infty.\end{aligned}$$ we have that $\sum_{m_0+1}^\infty ((U_j - (\ell -1)\pi_j)/j)$ converges almost surely. It then follows by Kronecker’s lemma that, $$\begin{aligned} \dfrac{1}{n} \sum_{j=1}^n (U_j - (\ell-1)\pi_j) \overset{\text{a.s.}}{\rightarrow} 0.\end{aligned}$$ We know that $\pi_j \rightarrow 0$ as $j \rightarrow \infty$ (the speed depending on the delay times). Thus, we will have ${1}/{n} \sum_{j=1}^n (\ell-1)\pi_j \rightarrow 0$ since $\pi_j\rightarrow 0$ as $j\rightarrow \infty$. Hence, ${1}/{n} \sum_{j=1}^n U_j\rightarrow 0$ a.s. To show that $R_n(\delta_\pi) \overset{\text{a.s.}}{\rightarrow} 1$, it remains to show that $$\begin{aligned} \dfrac{\sum_{j=1}^n f_{\hat{i}_j}(X_j)}{\sum_{j=1}^n f^*(X_j)} \overset{\text{a.s.}}{\rightarrow} 1 \ \text{or equivalently,}\ \dfrac{\sum_{j=1}^n (f_{\hat{i}_j}(X_j) - f^*(X_j))}{\sum_{j=1}^n f^*(X_j)} \overset{\text{a.s.}}{\rightarrow} 0. \end{aligned}$$ Recall from Section 3.1, given the observed reward timings $\{t_j: t_j \leq n , 1\leq j \leq n\}$, let $\{s_k: k =1,\hdots, N_n\}$ be the reordered sequence of the observed reward timings, arranged in an increasing order. Then for any $j, m_0+1\leq j \leq n$, there exists an $s_{k_j}, k_j\in \{1,\hdots, N_n\}$ such that $s_{k_j} \leq j < s_{k_j+1}$. Also, note that as $j \rightarrow \infty$, we also have that $k_j \rightarrow \infty$. By the definition of $\hat{i}_j$, for $j \geq m_0 + 1$, $\hat{f}_{\hat{i}_j,s_{k_j}}(X_j) \geq \hat{f}_{i^*(X_j),s_{k_j}}(X_j)$ and thus, $$\begin{aligned} f_{\hat{i}_j}(X_j) - f^*(X_j) &= f_{\hat{i}_j}(X_j) - \hat{f}_{\hat{i}_j,s_{k_j}}(X_j) + \hat{f}_{\hat{i}_j, s_{k_j}}(X_j) - \hat{f}_{i^*(X_j),s_{k_j}}(X_j)\\ &\quad \quad \quad + \hat{f}_{i^*(X_j),s_{k_j}}(X_j) - f^*(X_j)\\ &\geq f_{\hat{i}_j}(X_j) - \hat{f}_{\hat{i}_j, s_{k_j}}(X_j) + \hat{f}_{i^*(X_j),s_{k_j}}(X_j) - f_{i^*(X_j)}(X_j)\\ &\geq -2 \sup_{1\leq i \leq \ell} ||\hat{f}_{i,s_{k_j}} - f_i||_\infty.\end{aligned}$$ For $1\leq j \leq m_0$, we have $f_{\hat{i}_j}(X_j) - f^*(X_j) \geq -A$. Based on Assumption A, $||\hat{f}_{i,s_{k_j}} - f_i||_\infty \overset{\text{a.s.}}{\rightarrow} 0$ as $j \rightarrow \infty$ for each $i$, and thus $\sup_{1\leq i \leq \ell} || \hat{f}_{i,s_{k_j}} - f_i||_\infty \overset{\text{a.s.}}{\rightarrow} 0$. Then it follows that, for $n > m_0$, $$\begin{aligned} &\dfrac{\sum_{j=1}^n (f_{\hat{i}_j}(X_j) - f^*(X_j))}{\sum_{j=1}^n f^*(X_j)} \\ &\quad \quad \geq \dfrac{-Am_0/n - (2/n)\sum_{j=m_0+1}^n \sup_{1\leq i \leq \ell} ||\hat{f}_{i,s_{k_j}} - f_i||_\infty}{(1/n)\sum_{j=1}^n f^*(X_j)}.\end{aligned}$$ The right hand side converges to 0 almost surely and hence the conclusion follows. **A probability bound on the performance of the histogram method** {#A.1} ------------------------------------------------------------------ Consider the regression model as in Section 2, with $i$ dropped for notational convenience. $$\begin{aligned} Y_{j} = f(x_j) + \epsilon_{j},\end{aligned}$$ where $\epsilon_j$’s are independent errors satisfying the moment condition in Assumption 4 of Section 4.1. Let $W_1,\hdots, W_n$ are Bernoulli random variables that decide if arm $i$ is observed or not, that is $W_j = I_{\{I_j = i\}}$. Assume, for each $1\leq j \leq n$, $W_j$ is independent of $\{\epsilon_k: k\geq j\}$. Let $\hat{f}_n$ be the histogram estimator of $f$. Let $w(h;f)$ denote a modulus of continuity defined by, $$w(h;f) = \sup\{|f(x_1) - f(x_2)|: |x_{1k} -x_{2k}| \leq h \ \text{for all} \ 1\leq k \leq d\}.$$ Let $A_n$ denote the event consisting of the indices (time points) for which the rewards were observed by time $n$, that is $A_n := \{j: t_j \leq n\}$ and $X^n = \{X_1,X_2,\hdots,X_n\}$, the design points until time $n$. \[lemma1\_appendix\] Let $\epsilon > 0$ be given. Suppose that $h$ is small enough that $w(h;f) < \epsilon$. Then the histogram estimator $\hat{f}_n$ satisfies, $$\begin{aligned} \text{P}_{A_n,X^n}(||\hat{f}_n - f||_\infty \geq \epsilon) &\leq M \exp\left(-\dfrac{3\pi_n \min_{1\leq b \leq M} N_b}{28} \right) \\ & \quad \quad +2M \exp\left(- \dfrac{\min_{1\leq b \leq M} N_b \pi_n^2 (\epsilon - w(h;f))^2}{8(v^2 + c(\pi_n/2)(\epsilon-w(h;f)))} \right).\end{aligned}$$ where the probability $P_{A_n,X^n}$ denotes conditional probability given design points $X^n = (X_1,X_2,\hdots,X_n)$ and $A_n = \{j: t_j\leq n\}$. Here, $N_b$ is the number of design points for which the rewards have been obtained by time $n$ such that they fall in the $b$th small cube of the partition of the unit cube at time $n$. Note that the above inequality trivially holds if $\min_{1\leq b \leq M} N_b = 0$. Therefore, let’s assume that $\min_{1\leq b\leq M} N_b > 0$. Let $N(x)$ denote the number of time points, for which the corresponding design points $x_j$’s fall in the same cube as $x$ and for which the corresponding rewards are observed by time $n$. Let $J(x)$ denote the set of indices $1\leq j \leq n$ of such design points. Let $\bar{J}(x)$ be the subset of $J(x)$ where arm $i$ is chosen (i.e. where $W_j =1$) and let $\bar{N}(x)$ be the number of such design points (note that $i$ is dropped for notational convenience). For arm $i$, we consider the histogram estimator $$\begin{aligned} \hat{f}_{n}(x) &= \dfrac{1}{\bar{N}(x)}\sum_{j \in \bar{J}(x)} Y_j \\ & = f(x) + \dfrac{1}{\bar{N}(x)}\sum_{j \in \bar{J}(x)} (f(x_j) - f(x)) + \dfrac{1}{\bar{N}(x)}\sum_{j \in\bar{J}(x)} \epsilon_j\\ \Rightarrow |\hat{f}_n(x)& - f(x)| \leq w(h;f) + \left|\dfrac{1}{\bar{N}(x)}\sum_{j \in \bar{J}(x)}\epsilon_j \right|, \end{aligned}$$ where $w(h;f)$ is the modulus of continuity. For any $\epsilon > w(h;f)$, with the given design points and the time points for which rewards have been observed by time $n$, $$\begin{aligned} P_{A_n,X^n}(||\hat{f}_n - f||_\infty \geq \epsilon) \leq P_{A_n,X^n}\left(\sup_x \left|\dfrac{1}{\bar{N}(x)}\sum_{j \in \bar{J}(x)} \epsilon_j \right| \geq \epsilon -w(h;f) \right). \end{aligned}$$ Note that, in the same small cube $B$, $N(x)\ \text{and}\ \bar{N}(x), J(x)\ \text{and}\ \bar{J}(x)$ are the same for any $x$, respectively. Let $x_0$ be a fixed point in $B$. Then consider, $$\begin{aligned} P_{A_n,X^n}&\left(\sup_{x \in B} \left|\dfrac{1}{\bar{N}(x)} \sum_{j\in \bar{J}(x)} \epsilon_j \right| \geq \epsilon- w(h;f)\right)\\ & = P_{A_n,X^n}\left(\left| \sum_{j\in \bar{J}(x_0)} \epsilon_j \right| \geq \bar{N}(x_0) (\epsilon- w(h;f)) \right)\\ & = P_{A_n,X^n}\left(\left| \sum_{j\in J(x_0)} W_j \epsilon_j \right| \geq N(x_0) \dfrac{\bar{N}(x_0)}{N(x_0)} (\epsilon- w(h;f)) \right)\\ & = P_{A_n,X^n}\left(\left| \sum_{j\in J(x_0)} W_j \epsilon_j \right| \geq N(x_0) \dfrac{\bar{N}(x_0)}{N(x_0)} (\epsilon- w(h;f)), \dfrac{\bar{N}(x_0)}{N(x_0)} > \dfrac{\pi_n}{2} \right) \\ &\quad \quad+ P_{A_n,X^n}\left(\left| \sum_{j\in J(x_0)} W_j \epsilon_j \right| \geq N(x_0) \dfrac{\bar{N}(x_0)}{N(x_0)} (\epsilon- w(h;f)), \dfrac{\bar{N}(x_0)}{N(x_0)} \leq \dfrac{\pi_n}{2} \right) \\ & \leq P_{A_n,X^n}\left(\left| \sum_{j\in J(x_0)} W_j \epsilon_j \right| \geq N(x_0) \dfrac{\pi_n}{2} (\epsilon- w(h;f)) \right) + P_{A_n,X^n}\left(\dfrac{\bar{N}(x_0)}{N(x_0)} \leq \dfrac{\pi_n}{2} \right)\\ & \leq 2 \exp\left(- \dfrac{N(x_0) \pi_n^2 (\epsilon - w(h;f))^2}{8(v^2 + c\pi_n/2 (\epsilon - w(h;f)))} \right) + \exp\left(- \dfrac{3 N(x_0) \pi_n}{28} \right),\end{aligned}$$ where the last inequality follows from inequality in \[A.4\] and in \[A.2\] respectively. For applying , we used the fact that $W_j$ is independent of the $\epsilon_{ik}$’s for all $k\geq j$ since $W_j$ depends only on the previous observations and $X_j$. Given that $N_b$ be the number of design points in the $b$th small cube whose rewards are observed by time $n$, we have $$\begin{aligned} P_{A_n,X^n}(||\hat{f}_n - f||_\infty) &\leq M \exp\left(- \dfrac{3 (\min_{1\leq b \leq M}N_b) \pi_n}{28} \right) \\ & \quad \quad + 2M \exp\left(- \dfrac{(\min_{1\leq b \leq M}N_b) \pi_n^2 (\epsilon - w(h;f))^2}{8(v^2 + c(\pi_n/2) (\epsilon - w(h;f)))} \right). \end{aligned}$$ This concludes the proof of Lemma 1. **An inequality for Bernoulli trials.** {#A.2} --------------------------------------- For $1\leq j \leq n$, let $\tilde{W}_j$ be Bernoulli random variables, which are not necessarily independent. Assume that the conditional probability of success for $\tilde{W}_j$ given the previous observations is lower bounded by $\beta_j$, that is, $$\begin{aligned} P(\tilde{W}_j = 1|\tilde{W}_i, 1 \leq i \leq j-1) \geq \beta_j \ \text{a.s.},\end{aligned}$$ for all $1\leq j\leq n$. Appylying the extended Bernstein’s inequality as described in [@qian2016kernel], we have $$\begin{aligned} P\left(\sum_{j=1}^n \tilde{W}_j \leq \left(\sum_{j=1}^n \beta_j\right)/2 \right) \leq \exp\left(- \dfrac{3\sum_{j=1}^n \beta_j}{28} \right). \label{binomial_inequality}\end{aligned}$$ **A probability inequality for sums of certain random variables.** {#A.4} ------------------------------------------------------------------ Let $\epsilon_1,\epsilon_2,\hdots$ be independent random variables satisfying the refined Bernstein condition in Assumption 3. Let $I_1,I_2,\hdots$ be Bernoulli random variables such that $I_j$ is independent of $\{\epsilon_l: l\geq j\}$ for all $j \geq 1$. For any $\epsilon >0$, $$\begin{aligned} P\left(\sum_{j=1}^n I_j \epsilon_j \geq n\epsilon\right) \leq \exp\left(-\dfrac{n\epsilon^2}{v^2 + c\epsilon}\right).\label{bernstein_inequality}\end{aligned}$$ The proof for this lemma can be found in [@yang2002randomized]. [^1]: $f(n) = \Omega{(g(n))}$ if for some positive constant $c$,$f(n) \geq cg(n)$ when $n$ is large enough

--- abstract: 'This paper reviews stability analysis techniques by using the Nyquist and Nichols charts. The relationship between the Nyquist and Nichols stability criteria is fully described by using the crossing concept. The results are demonstrated through several numerical examples. This tutorial provides useful insights into the loop-shaping based control systems design such as Quantitative Feedback Theory.' author: - 'S.M.Mahdi Alavi and Mehrdad Saif [^1][^2] [^3]' title: On stability analysis by using Nyquist and Nichols Charts --- [Shell : Bare Demo of IEEEtran.cls for Journals]{} Stability analysis, Nyquist diagram, Nichols chart, Quantitative feedback theroy (QFT), loop-shaping. Introduction ============ Stability is always a major concern in analysis and design of feedback control systems. Consider the Linear Time Invariant (LTI) feedback system as shown in Fig. \[Fig:1DOF\_standard\_Feedback\_structure\]. $P(s)$ and $G(s)$ represent plant and controller Transfer Functions (TFs). \[c\]\[c\][$P(s)$]{} \[c\]\[c\][$G(s)$]{} ![The standard feedback system.[]{data-label="Fig:1DOF_standard_Feedback_structure"}](standard_feedbakc_system.eps "fig:") For the stability of the feedback system Fig. \[Fig:1DOF\_standard\_Feedback\_structure\], the poles of the closed-loop system must have negative real parts and lie in the Left Half Plane, (LHP). The analogous is that the zeros of the characteristic function, $Q(s)$, given by $$\begin{aligned} \label{system_characteristics_function} Q(s)=1+L(s), \end{aligned}$$ must be located in LHP to ensure the stability of the feedback system Fig. \[Fig:1DOF\_standard\_Feedback\_structure\], where $L(s)=P(s)G(s)$ refers to the ‘loop function’ in the control literature. A variety of stability criteria are available in the literature, [@Houpis-LinearCont]. The Routh-Hurwitz and Jury tests determine the stability of the LTI systems without finding the roots of the characteristic equation in continuous and discrete time domains, respectively. Nyquist diagram is an alternative stability analysis method, providing some graphical information on how the poles of the closed-loop control system move by changing the gain of the controller in the complex plane. Nichols chart is an alternative coordinates for presentation of the system’s frequency response. In the Nichols chart, the loop-function’s magnitude (in dB) versus its phase (in Degrees) is plotted. Since the Nichols chart presents a complete simultaneous information in relation to magnitude and phase of the system frequency response, a control approach that employs it as the design environment is more efficient than other control approaches that are only based on the magnitude or phase information. The use of both magnitude and phase information into the design of the feedback system significantly eases stability analysis of non-minimum phase, unstable plants and the plants that suffer from time-delay. Despite the Nyquist stability has very well been established in the linear control text books, there is still significant demands for more discussions and examples to illustrate the Nichols chart stability criterion. This paper summarizes the results in [@Cohen-1994] and [@Chen-Ballance1997] in a tutorial fashion. The relationship between the Nyquist and Nichols stability criteria is exactly described through several illustrative examples. As an application, the Quantitative Feedback Theory (QFT) [@Houpis-QFT] employs the Nichols chart as a tool [@Borhgesani-QFTToolbox] and [@Garcia-Sanz-QFTToolbox] for the design of the feedback compensator. In this approach, desired control objectives are given in terms of some graphical design bounds on the system’s loop-function. The feedback controller is then obtained by loop-function shaping such that these design bounds are satisfied. Some applications of QFT include: fault diagnosis and control in chemical processes [@Alavi-2012; @Alavi-2007] and in power systems [@Alavi-2008], power control in wireless sensor networks [@Alavi-2009a], active queue management in communications systems [@Alavi-2009b] to name but a few. From quantitative feedback design perspective, this tutorial provides useful insights into the loop-shaping process. The rest of the paper is organized as follows. In section \[sec:Stability Analysis Using Nyquist Plot\], the Nyquist stability test is described. In section \[sec:Stability Analysis Using Nichols Chart\], the stability criterion by using the Nichols chart, and its relation to the Nyquist stability test are presented. Several examples are given in section \[sec:Examples\], demonstrating the results. Stability Analysis Using Nyquist Plot {#sec:Stability Analysis Using Nyquist Plot} ===================================== The Nyquist stability criterion is based on the Cauchy’s principle. For the feedback system Fig. \[Fig:1DOF\_standard\_Feedback\_structure\] with the loop-function $L(s)$, consider a contour in s-plane enclosing the entire Right-Half Plane (RHP) including the imaginary axis. This contour refers to the ‘standard Nyquist contour’ and is denoted by $\Gamma$ hereafter. The requirement for the Cauchy’s principle is that $\Gamma$ must not pass through any pole of $L(s)$. Its radius should be large enough to enclose all Right-Half Plane (RHF) poles of $L(s)$. To avoid passing through the poles located on the imaginary axis, $\Gamma$ is constructed such that it encircles these poles by a semicircular arc with a very small radius tending to zero. Fig. \[Fig:standard\_Nyquist\_contour\] shows how the standard Nyquist contour is chosen in the presence of different locations of poles. Zeros of $L(s)$ do not affect the selection of $\Gamma$. The Nyquist plot is then the map of $\Gamma$ trough $L(s)$ into the complex plane. For the sake of simplicity, it is assumed that there is no pole-zero cancellation between the numerator and denominator of $L(s)$. \[c\]\[c\][$Re\{s\}$]{} \[c\]\[c\][$\Gamma$]{} \[c\]\[c\][$s$-plane]{} ![The standard Nyquist contour $\Gamma$. ‘$\times$’s and ‘$\circ$’s represent the possible poles and zeros of $L(s)$, respectively. $\Gamma$ must be large enough to enclose all RHP poles of $L(s)$. The poles located on the imaginary axis must be detoured by $\Gamma$.[]{data-label="Fig:standard_Nyquist_contour"}](nyquist_contour.eps "fig:") By using Cauchy’s principle, the following equation holds for the obtained Nyquist plot of $L(s)$ in the complex plane, $$\begin{aligned} \label{Cauchy_principle_Eq} N_z=N+N_p,\end{aligned}$$ where, $N_z$ and $N_p$ are the number of zeros and poles of $L(s)$ inside the contour $\Gamma$. $N$ is the number of times that the Nyquist plot encircles the origin. According to (\[Cauchy\_principle\_Eq\]), the number of times that Nyquist plot of $L(s)$ encircles the critical point $(-1,0)$ plus the number of poles of $L(s)$ determine the number of zeros of $Q(s)$ which are located inside the contour $\Gamma$. From equation (\[system\_characteristics\_function\]), the zeros of $Q(s)$ which are located inside the contour $\Gamma$ are unstable poles of the feedback system Fig. \[Fig:1DOF\_standard\_Feedback\_structure\]. Therefore, the number of times that Nyquist plot of $L(s)$ encircles the critical point $(-1,0)$ plus the number of poles of $L(s)$ determine the number of unstable poles of the closed-loop system through the following theorem. \[Theorem:Nyquist\_stability\_Criterion\] Consider the feedback system as shown in Fig. \[Fig:1DOF\_standard\_Feedback\_structure\]. The followings are then equivalent: - The number of unstable poles of the feedback system Fig. \[Fig:1DOF\_standard\_Feedback\_structure\], $N_z$, is given by: $$\begin{aligned} \label{Nyquist_Criterion_Eq} N_z=N+N_p,\end{aligned}$$ $N_p$ is the number of poles of $L(s)$ inside the contour $\Gamma$ and $N$ is the number of times that the Nyquist plot encircles the critical point $(-1,0)$. - The feedback system is stable if and only if the Nyquist diagram of $L(s)$ does not intersect the critical point $(-1,0)$ and encircles it ‘$n$’ times in the counterclockwise direction. This is a direct consequence of the Cauchy’s principle. For a more comprehensive information interested readers are directed to [@Houpis-LinearCont]. It should be noted that an encirclement of the point $(-1,0)$ is counted as positive if it is in the same direction as the standard Nyquist contour and negative if it is in the opposite direction. In this context, since the standard Nyquist contour is clockwise, if $L(s)$ generates clockwise encirclement of the point $(-1,0)$, then $N$ is added with $+1$. Otherwise if $L(s)$ encircles the point $(-1,0)$ in counterclockwise, then $N$ is added with $-1$. In the following, an approach is presented that simplifies counting $N$ in (\[Nyquist\_Criterion\_Eq\]). It provides the basis for development of the stability criterion using Nichols chart. \[Def:Smooth\_curve\] A Nyquist diagram is a smooth curve if it is differentiable for all $\theta\in [0,2\pi]$ where, $\theta$ represents the phase of the loop-function at each frequency, and is given by: $$\label{phase_in_complex_plane_Eq} \theta=\tan^{-1}(Im\{L(j\omega)\}/Re\{L(j\omega)\}).$$ \[Def:Crossing\] Let $\mathcal{R}_{(-\infty,-1)}$ be the ray on the real axis from $-\infty$ to $-1$ in the complex plane. A crossing occurs when $L(s)$ intersects $\mathcal{R}_{(-\infty,-1)}$. The crossing is said to be positive if the tangent to the Nyquist plot has a positive imaginary part. Consequently, it is negative if the tangent to the Nyquist plot has a negative imaginary part, as shown in Fig. \[Fig:crossing\_Nyquist\_plot\]. \[c\]\[c\][$Re\{L(s)\}$]{} \[c\]\[c\][$-1$]{} \[c\]\[c\][$\mathcal{R}_{(-\infty,-1)}$]{} \[c\]\[c\][$L$-plane]{} ![The crossing notation of a Nyquist diagram $L(s)$.[]{data-label="Fig:crossing_Nyquist_plot"}](crossing_in_nyquist_plot.eps "fig:") If the Nyquist plot is a smooth curve, the sign of crossing is obviously $+1$ or $-1$. However, when the Nyquist plot approaching a point on the real axis, there might be a [*cusp*]{} due to local singularity of the Nyquist plot. The crossing is not [*well-defined*]{} as the Nyquist plot is not differentiable at such a point. This situation might occur at $\omega=0$ or $\omega =\pm \infty$. To address this issue when there exists a cusp at $\omega=0$, the sign of the first nonzero derivative of $\theta$, given by (\[phase\_in\_complex\_plane\_Eq\]), with respect to $\omega$ at $\omega=0$ is adopted as the sign of the crossing. If this derivative is positive, the crossing is supposed to be positive, otherwise, it is selected as negative. For the cusp at $\omega=\pm \infty$, because all derivatives of $\theta$ approach to zero as $\omega \rightarrow \pm \infty$, the above solution is not feasible. Instead, $\omega$ is replaced with $1/\omega$ and the derivation technique is again applied but at this time with respect to $1/\omega$. It is then a direct consequence of the above statements that: ‘A crossing is said to be positive if the direction of the Nyquist plot is upward, otherwise the crossing is negative.’ \[Theorem:Modified\_Nyquist\_stability\_Criterion\] Consider the feedback system as shown in Fig. \[Fig:1DOF\_standard\_Feedback\_structure\]. Assume that $L(s)$ has ‘$n \geq 0$’ unstable poles. By using the crossing concepts as discussed above, the followings are then equivalent. - The sum of crossings is equal to the number of encirclements of the critical point $(-1,0)$ by the Nyquist diagram of $L(s)$. - The feedback system is stable if and only if the Nyquist diagram of $L(s)$ does not intersect the critical point $(-1,0)$ and sum of its crossings is equal to ‘$-n$’. This is an immediate consequence of the crossing concept and Theorem \[Theorem:Nyquist\_stability\_Criterion\]. The number of RHP zeros of $Q(s)$ is obtained by substituting the sum of crossings as well as the number of poles of $L(s)$ into (\[Nyquist\_Criterion\_Eq\]). Stability Analysis Using Nichols Chart {#sec:Stability Analysis Using Nichols Chart} ====================================== In the following, the stability criterion of the feedback system Fig. \[Fig:1DOF\_standard\_Feedback\_structure\] is studied which is based on the Nichols chart [@Nichols-Theory-of-Servomechanisms]. Again, it is assumed that no unstable pole/zero cancellation takes place in $L(s)$. The stability criterion is obtained by mapping the critical point $(-1,0)$, the ray $\mathcal{R}_{(-\infty,-1)}$ and the Nyquist plot of $L(s)$ into the Nichols chart, and by using the crossing concept. $\Upsilon$ is the map that transforms any point $A(x,y)$ on the Nyquist diagram of $L(s)$ into its equivalent point, $A^\prime(\phi,r)$, in the Nichols chart, through: $$\label{Map_from_Nyquist_To_NC_Eq} \Upsilon: A(x,y) \rightarrow A^\prime(\phi,r)$$ $$\begin{aligned} \label{Phase_in_NC_inTerms_xy}& \phi=\tan^{-1}(y/x),\\ \label{Mag_in_NC_inTerms_xy} & r=20\log(\sqrt{x^2+y^2}).\end{aligned}$$ $x$ and $y$ denote the real and imaginary parts of the Nyquist plot of $L(s)$ in the complex plane. Stability Analysis Using Single-Sheeted Nichols Chart ----------------------------------------------------- There are two aspects in the presentation of the system’s frequency response in the Nichols chart. In the first aspect, the exact phase obtained through $\Upsilon$ is plotted inside the Nichols chart. The Nichols chart that covers phase information outside the range of $[-360^\circ,0)$, is so-called as the multiple-sheeted Nichols chart. Subsequently, the Nichols curve that is plotted in multiple-sheeted Nichols chart refers to multiple-sheeted Nichols plot. The second aspect relies on this fact that the system’s characteristics do not change if the phase response is shifted by $(\pm360k)^\circ$ for $k=0,1,\cdots$. Therefore, any segment of the multiple-sheeted Nichols plot outside the range of $[-360^\circ,0)$ can be horizontally shifted by integer multiples of $\pm360^\circ$ to be located inside $[-360^\circ,0)$. The Nichols chart with horizontal phase axis within the range of $[-360^\circ,0)$ is so-called as the single-sheeted Nichols chart. Subsequently, the Nichols curve that is plotted in the single-sheeted Nichols chart refers to single-sheeted Nichols plot. In this section, the stability criterion of the feedback system Fig. \[Fig:1DOF\_standard\_Feedback\_structure\] using single-sheeted Nichols chart, is derived as follows. Let $\mathcal{R}_{(-180^\circ,r>0dB)}$ be the ray of all points on the line $r>0dB$ and $\phi=-180^\circ$ in the single-sheeted Nichols chart. By simple calculations, the critical point $(-1,0)$ and the ray $\mathcal{R}_{(-\infty,-1)}$ in complex plane are respectively mapped through $\Upsilon$ to the point $(-180^\circ,0dB)$ and the ray $\mathcal{R}_{(-180^\circ,r>0dB)}$ in the single-sheeted Nichols chart as shown in Fig. \[Fig:crossing\_Nichols\_chart\]. \[c\]\[c\][$20\log r$dB]{} \[c\]\[c\][$0^\circ$]{} \[c\]\[c\][$-180^\circ$]{} \[c\]\[c\][$-360^\circ$]{} \[c\]\[c\][$0$dB]{} \[c\]\[c\][Nichols chart]{} ![Illustration of the ray $\mathcal{R}_{(-180^\circ,r>0dB)}$ and crossing signs in the single-sheeted Nichols chart.[]{data-label="Fig:crossing_Nichols_chart"}](map_of_ray_in_nc.eps "fig:") Consider the Nyquist plot of a given $L(s)$ which is a curve denoted by $\Psi_{ny}$ in the complex plane. As mentioned earlier, $\Psi_{ny}$ is a continues and closed curve, symmetric with respect to the real axis. When mapping $\Psi_{ny}$ through $\Upsilon$ into the single-sheeted Nichols chart, the resulting curve, which is denoted by $\Psi_{nc}$, intersects the ray $\mathcal{R}_{(-180^\circ,r>0dB)}$ at the points that are maps of the crossings of $\Psi_{ny}$ in the complex plane. For a positive crossing of $\Psi_{ny}$ in the complex plane, $\Psi_{nc}$ will pass $\mathcal{R}_{(-180^\circ,r>0dB)}$ to the left side in the single-sheeted Nichols chart. Similarly, For a negative crossing of $\Psi_{ny}$ in the complex plane, $\Psi_{nc}$ will pass $\mathcal{R}_{(-180^\circ,r>0dB)}$ to the right side in the single-sheeted Nichols chart. For a given Nyquist diagram of $L(s)$ and its types of crossings in complex plane as shown in Fig. \[Fig:crossing\_Nyquist\_plot\], Fig. \[Fig:crossing\_Nichols\_chart\] demonstrates the corresponding Nichols curve and types of the equivalent crossings in the single-sheeted Nichols chart. \[Def:Crossing\_in\_NC\]Let $\mathcal{R}_{(r>0dB,-180^\circ)}$ be the ray on the line $r>0dB$ and $\phi=-180^\circ$ in the single-sheeted Nichols chart. A crossing occurs when the single-sheeted Nichols plot of $L(s)$ intersects $\mathcal{R}_{(r>0dB,-180^\circ)}$. For consistency with the Definition \[Def:Crossing\], the crossing is said to be positive if the direction of the single-sheeted Nichols plot of $L(s)$ is toward left of $\mathcal{R}_{(r>0dB,-180^\circ)}$, otherwise it is negative. \[Theorem:Stability\_in\_Single\_Sheeted\_NC\] Consider the feedback system as shown in Fig. \[Fig:1DOF\_standard\_Feedback\_structure\]. Assume that $L(s)$ has ‘$n \geq 0$’ unstable poles. By using the crossing concepts as discussed above, the followings are equivalent. - The sum of crossings in the single-sheeted Nichols chart demonstrates the number of times that the Nyquist plot of $L(s)$ encircles the critical point $(-1,0)$ in the Nyquist diagram. - The feedback system is stable if and only if the single-sheeted Nichols plot of $L(s)$ does not intersect the critical point $(-180^\circ,0dB)$ and sum of its crossing with $\mathcal{R}_{(-180^\circ,r>0dB)}$ is equal to ‘$-n$’. This is a direct consequence of the above discussion. Stability Analysis Using Multiple-Sheeted Nichols Chart ------------------------------------------------------- In this section, the proposed stability criterion given by Theorem \[Theorem:Stability\_in\_Single\_Sheeted\_NC\] is extended to the multiple-sheeted Nichols chart, where the phase of the system frequency response exceeds the range of $[-360^\circ,0)$. In this case, the map of the critical point $(-1,0)$ through $\Upsilon$, from the complex plane into the multiple-sheeted Nichols chart, is repeated at the points $((-180\pm 360k)^\circ,0dB)$, $k=0,1, \cdots$. In addition, the map of the ray $\mathcal{R}_{(-\infty,-1)}$ through $\Upsilon$, from the complex plane into the multiple-sheeted Nichols chart will result in multiple rays on lines $((-180\pm 360k)^\circ,r>0dB)$, $k=0,1, \cdots$. By comparing the single and multiple sheeted Nichols plots of $L(s)$, it is seen that the sum of crossings will be constant for both plots. There is only one difference that the Nichols plot of one is horizontally shifted with respect to another. Therefore, the following stability criterion is derived when using the multiple-sheeted Nichols chart. \[Theorem:Stability\_in\_Multiple\_Sheeted\_NC\] Consider the feedback system as shown in Fig. \[Fig:1DOF\_standard\_Feedback\_structure\]. Assume that $L(s)$ has ‘$n \geq 0$’ unstable poles. The followings are equivalent when using the multiple-sheeted Nichols chart. - The sum of crossings illustrates the number of times that the Nyquist diagram of $L(s)$ encircles the critical point (-1,0) in the complex plane. - The feedback system is stable if and only if the multiple-sheeted Nichols plot of $L(s)$ does not intersect the critical points $((-180 \pm 360k)^\circ,0dB)$ and sum of its crossing with the rays located on the lines $((-180\pm 360k)^\circ,r>0dB)$ is equal to ‘$-n$’, $k=0,1, \cdots$. This is a direct consequence of the above discussion and extension of Theorem \[Theorem:Stability\_in\_Single\_Sheeted\_NC\] to the multiple-sheeted Nichols chart. ### Multiple-Sheeted Nichols Chart versus Single-Sheeted Nichols Chart By comparing the single and multiple Nichols charts, the following features are highlighted. - There is no priority between multiple-sheeted and single-sheeted Nichols charts or plots. Only in some cases, the plot of the loop-function’s frequency response into the multiple-sheeted Nichols will result in the closed curve. However, multiple crossing ray and multiple critical points appear that must be taken into account when analyzing and designing the control system using multiple-sheeted Nichols chart. - What is very important is that the system characteristics will be preserved by shifting the phase response by $\pm(360k)^\circ$ for $k=0,1,\cdots$. Thus, the use of multiple or single sheeted Nichols chart is a matter of convenience only and do not affect the analysis and design of the feedback systems. - In general, there exist a vertical line, $\phi=k$, in which both multiple-sheeted and single-sheeted Nichols plots are symmetric to that. Half of the Nichols chart, located in one side of $\phi=k$, is related to the system frequency response for positive $\omega$, and the other half part, located in the other side of $\phi=k$, is related to the system frequency response for negative $\omega$. Since both full and half multiple-sheeted or single-sheeted Nichols charts result in same number of crossings, half of the multiple-sheeted or single-sheeted Nichols is enough for the stability analysis. Examples {#sec:Examples} ======== In the following several examples are provided that cover a large variety of the loop-function models, stable and unstable zeros and poles. \[c\]\[c\][$Im\{L(s)\}$]{} \[r\]\[r\][$\omega=0^{+}$]{}\ \[MP\_Stable\_Ex\] Consider the feedback system Fig. \[Fig:1DOF\_standard\_Feedback\_structure\] with a stable, minimum phase loop-function as given by: $$\label{Eq:Stable_MP_NC_Ex} L(s)=\frac{K}{(\frac{s}{1}+1)(\frac{s}{2}+1)(\frac{s}{3}+1)}.$$ Determine the system stability for $K=5$ and $K=15$ by using the crossing concept, Nyquist and single-sheeted Nichols plots. The given loop-function does not have unstable mode, then $N_p=0$. Fig. \[SubFig:Stable\_MP\_Ny\_Ex1\] shows the Nyquist diagram of $L(s)$ for $K=5$. The number of crossings on the ray $R_{(-\infty,-1)}$ is $N=0$, meaning that the Nyquist diagram does not encircle the critical point $(-1,0)$. Using (\[Nyquist\_Criterion\_Eq\]), $N_z=N+N_p=0+0=0$ and therefore, the characteristic function $Q(s)$ does not have any zero in the RHP. This implies that the feedback system is stable for $K=5$. Fig. \[SubFig:Stable\_MP\_NC\_Ex1\] shows the single-sheeted Nichols plot of $L(s)$ for $K=5$. The number of crossings on the ray $R_{(-180^\circ,r>0dB)}$ is $N=0$, meaning that the equivalent Nyquist diagram in the complex plane does not encircle the critical point $(-1,0)$ and the feedback system is stable for $K=5$. Similar method is employed for when $K=15$. Fig. \[SubFig:Stable\_MP\_Ny\_Ex2\] shows the Nyquist diagram of $L(s)$ for $K=15$. The number of crossings of $R_{(-\infty,-1)}$ is $N=2$, meaning that the Nyquist diagram encircles the critical point $(-1,0)$ two times in clockwise direction. By using (\[Nyquist\_Criterion\_Eq\]), $N_z=N+N_p=2+0=2$ implying that the characteristics function $Q(s)$ has two zeros in the RHP. Then, the feedback system has two unstable poles with $K=15$. Fig. \[SubFig:Stable\_MP\_NC\_Ex2\] shows single-sheeted Nichols plot of $L(s)$ for $K=15$. The number of crossings of $R_{(-180^\circ,r>0dB)}$ is $N=2$, meaning that the corresponding Nyquist diagram of $L(s)$ encircles the critical point $(-1,0)$ two times in clockwise direction in the complex plane. $N \neq N_p$ thus, the feedback system is unstable for $K=15$. \[MP\_Unstable\_Ex\] Consider the feedback system Fig. \[Fig:1DOF\_standard\_Feedback\_structure\] with an unstable, minimum phase loop-function as given by: $$\label{Eq:Unstable_MP_NC_Ex} L(s)=\frac{K(\frac{s}{3}+1)(\frac{s}{5}+1)}{(\frac{s}{2}-1)(\frac{s}{4}-1)}.$$ Determine the system stability for $K=5$ and $K=1$ by using the crossing concept, Nyquist and single-sheeted Nichols plots. At this example, the loop-function has two unstable poles, then $N_p=2$. Fig. \[SubFig:Unstable\_MP\_Ny\_Ex1\] shows the Nyquist diagram of $L(s)$ for $K=5$. The number of crossings of $R_{(-\infty,-1)}$ is $N=-2$, meaning that the Nyquist diagram encircles the critical point $(-1,0)$ two times in counterclockwise direction. Using (\[Nyquist\_Criterion\_Eq\]), $N_z=N+N_p=-2+2=0$, implying that and the system characteristic function $Q(s)$ does not have any pole in the RHP. Then, the feedback system is stable for $K=5$. Fig. \[SubFig:Unstable\_MP\_NC\_Ex1\] shows the single-sheeted Nichosl chart of $L(s)$ for $K=5$. The number of crossings of $R_{(-180^\circ,r>0dB)}$ is $N=-2$, meaning that the Nyquist diagram encircles the critical point $(-1,0)$ two times in counterclockwise direction. Since $N_z=-N_p$ then, the feedback system is stable for $K=5$. Fig. \[SubFig:Unstable\_MP\_Ny\_Ex2\] shows the Nyquist diagram of $L(s)$ for $K=1$. The number of crossings of $R_{(-\infty,-1)}$ is $N=0$. Therefore, the Nyquist diagram does not encircle the critical point $(-1,0)$. By using (\[Nyquist\_Criterion\_Eq\]), $N_z=N+N_p=0+2=2$, implying that the feedback system is unstable for $K=1$ with two RHP poles. Fig. \[SubFig:Unstable\_MP\_NC\_Ex2\] shows the Nyquist diagram of $L(s)$ for $K=1$. The number of crossings of $R_{(-\infty,-1)}$ is $N=0$, meaning that the Nyquist diagram does not encircle the critical point $(-1,0)$. Thus, $N\neq N_p$ and the feedback system is unstable for $K=1$. \[c\]\[c\][$Im\{L(s)\}$]{} \[r\]\[r\][$\omega=0^{+}$]{}\ \[NMP\_Stable\_Ny\_Ex\]Consider the feedback system Fig. \[Fig:1DOF\_standard\_Feedback\_structure\] with a stable but non-minimum phase loop-function as given by: $$\label{Eq:Stable_NMP_Ny_Ex} L(s)=\frac{K(\frac{s}{0.5}-1)}{(\frac{s}{2}+1)(\frac{s}{3}+1)}$$ Determine the system stability for $K=0.5$ and $K=1.5$ by using the crossing concept, Nyquist and Nichols plots. The given loop-function is stable, then $N_p=0$. According to the Nyquist diagram of $L(s)$ as shown Fig. \[SubFig:Stable\_NMP\_Ny\_Ex1\], the number of crossings of $R_{(-\infty,-1)}$ is $0$ for $K=0.5$. It means that the Nyquist diagram does not encircle the critical point $(-1,0)$. By using (\[Nyquist\_Criterion\_Eq\]), $N_z=N+N_p=0+0=0$, which implies the feedback system is stable for $K=0.5$. As Fig. \[SubFig:Stable\_NMP\_NC\_Ex1\] shows, the number of crossings of $R_{(-180^\circ,r>0dB)}$ is $0$ for $K=0.5$ and then $N=0$. Since $N=-N_p=0$, the feedback system is stable for $K=0.5$. Subsequently, Fig. \[SubFig:Stable\_NMP\_Ny\_Ex2\] shows the Nyquist diagram of $L(s)$ for $K=1.5$. The number of crossings of $R_{(-\infty,-1)}$ is $N=+1$, meaning that the Nyquist diagram encircles the critical point $(-1,0)$ once in the clockwise direction. By using (\[Nyquist\_Criterion\_Eq\]), $N_z=N+N_p=1+0=1$, showing that the feedback system is unstable for $K=1.5$. As shown in Fig. \[SubFig:Stable\_NMP\_NC\_Ex2\], the number of crossing of the ray $R_{(-180^\circ,r>0dB)}$ by the single-sheeted Nichols plot of $L(s)$ will be $N=1$. Thus, $N \neq N_p$ and the feedback system is unstable for $K=1.5$. \[c\]\[c\][$Im\{L(s)\}$]{} \[r\]\[r\][$\omega=0^{+}$]{}\ \[c\]\[c\][$Im\{L(s)\}$]{} \[r\]\[r\][$\omega=0^{+}$]{}\ \[NMP\_UnStable\_Ny\_Ex\]Consider the feedback system Fig. \[Fig:1DOF\_standard\_Feedback\_structure\] with an unstable and nonminimum phase loop-function as given by: $$\label{Eq:Unstable_NMP_Ny_Ex} L(s)=\frac{K(\frac{s}{0.5}-1)}{(\frac{s}{2}-1)(\frac{s}{3}-1)}$$ Determine the system stability for $K=0.5$ and $K=1.5$ by using the crossing concept, Nyquist and Nichols plots. The given loop-function has two unstable poles, then $N_p=2$. The Nyquist diagram of $L(s)$ for $K=0.5$ is shown in Fig. \[SubFig:Unstable\_NMP\_Ny\_Ex1\]. The number of crossings of $R_{(-\infty,-1)}$ is $-2$. It means that the Nyquist diagram encircles the critical point $(-1,0)$ two times in counterclockwise direction. Using (\[Nyquist\_Criterion\_Eq\]), $N_z=N+N_p=-2+2=0$ which implies the feedback system does not have any pole at the RHP for $K=0.5$ and is therefore stable. The single-sheeted Nichols plot of $L(s)$ shows two positive crossings that occur on $R_{(-180^\circ,r>0dB)}$, for $K=0.5$, that is $N=-2$. Thus, $N=-N_p$ and the feedback system is stable for $K=0.5$. Fig. \[SubFig:Unstable\_NMP\_Ny\_Ex2\] shows the Nyquist diagram of $L(s)$ for $K=1.5$. At this example, there are three crossings of $R_{(-\infty,-1)}$, one positive and two negative, so $N=-1$. It means that the Nyquist diagram encircles the critical point $(-1,0)$ once in the counterclockwise direction. Again, using (\[Nyquist\_Criterion\_Eq\]), $N_z=N+N_p=-1+2=1$, saying that the feedback system has one RHP pole and them is unstable for $K=1.5$. Fig. \[SubFig:Unstable\_NMP\_NC\_Ex2\] shows the single-sheeted Nichols plot of $L(s)$ for $K=1.5$. At this example, there are three crossings of $R_{(-\infty,-1)}$, two negative and one positive which lead to $N=+1$. Since $N \neq N_p$, then the feedback system is unstable for $K=1.5$. \[c\]\[c\][$Im\{L(s)\}$]{} \[r\]\[r\][$\omega=0^{+}$]{}\ \[Integral\_MP\_Stable\_Ny\_Ex\] The use of integrator in the control structure is necessary if tracking is one of the desired objectives. It makes the tracking error to be zero. In this example, a feedback system with integrator is considered. Let $L(s)$ in feedback system Fig. \[Fig:1DOF\_standard\_Feedback\_structure\] be as follows: $$\label{Eq:Integral_Stable_MP_Ny_Ex} L(s)=\frac{K}{s(\frac{s}{0.5}+1)(\frac{s}{2}+1)}$$ Using the crossing concept, determine the system stability for $K=1$ and $K=5$. The loop-function has a pole at the origin. As mentioned earlier, to plot the Nyquist diagram which is the map of the standard Nyquist contour through $L(s)$ into the complex plan, $\Gamma$ must detour around the poles on the imaginary axis. Thus, $N_p=0$ as $L(s)$ does not have any RHP pole with the real part greater than zero. According to the Nyquist diagram of $L(s)$ as shown Fig. \[SubFig:Integral\_Stable\_MP\_Ny\_Ex1\], the number of crossings of $R_{(-\infty,-1)}$ is $0$ for $K=1$. It means that the Nyquist diagram does not encircle the critical point $(-1,0)$. Using (\[Nyquist\_Criterion\_Eq\]), $N_z=N+N_p=0+0=0$ which implies the feedback system is stable for $K=1$. The single-sheeted Nichols plot of $L(s)$, as shown Fig. \[SubFig:Integral\_Stable\_MP\_NC\_Ex1\], demonstrates that there is no crossing and $N=0$. Then, $N \neq -N_p=0$ and the feedback system is stable for $K=1$. Subsequently, Fig. \[SubFig:Integral\_Stable\_MP\_Ny\_Ex2\] shows the Nyquist diagram of $L(s)$ for $K=5$. The number of crossings of $R_{(-\infty,-1)}$ is $N=+2$, meaning that the Nyquist diagram encircles the critical point $(-1,0)$ once in the clockwise direction. By using (\[Nyquist\_Criterion\_Eq\]), $N_z=N+N_p=2+0=2$, saying that the feedback system has two unstable poles and is unstable for $K=5$. As it is shown by Fig. \[SubFig:Intehral\_Stable\_MP\_NC\_Ex2\], the sum of crossing is $N=2$ for $K=5$. Since $N \neq -N_p$, then the feedback system is unstable for $K=5$. \[c\]\[c\][$Im\{L(s)\}$]{} \[r\]\[r\][$\omega=0^{+}$]{}\ \[Integral\_NMP\_Stable\_Ny\_Ex\] In this example, the stability conditions is investigated for a system with an integrator and NMP zero. Consider the feedback system \[Fig:1DOF\_standard\_Feedback\_structure\] with an unstable and non-minimum phase loop-function as given by: $$\label{Eq:Integral_Stable_NMP_Ny_Ex} L(s)=\frac{K(\frac{s}{2}-1)}{s(\frac{s}{1}+1)}$$ Using the crossing concept, determine the system stability for $K=-1$ and $K=-5$. The system does not have nay pole inside the standard Nyquist contour, then $N_p=0$. According to the Nyquist diagram of the $L(s)$ as shown Fig. \[SubFig:Integral\_Stable\_NMP\_Ny\_Ex1\], the number of crossings of $R_{(-\infty,-1)}$ is $0$ for $K=-1$. It means that the Nyquist diagram does not encircle the critical point $(-1,0)$. Using (\[Nyquist\_Criterion\_Eq\]), $N_z=N+N_p=0+0=0$ which implies the feedback system is stable for $K=1$. There is also no crossing as shown by Fig. \[SubFig:Integral\_Stable\_NMP\_NC\_Ex1\]. Then, $N \neq -N_p=0$ and the feedback system is stable for $K=-1$. Subsequently, Fig. \[SubFig:Integral\_Stable\_NMP\_Ny\_Ex2\] shows the Nyquist diagram of $L(s)$ for $K=-5$. The number of crossings of $R_{(-\infty,-1)}$ is $N=+2$, implying that the Nyquist diagram encircles the critical point $(-1,0)$ once in the clockwise direction. By using (\[Nyquist\_Criterion\_Eq\]), $N_z=N+N_p=2+0=2$, saying that the feedback system has two unstable poles and is unstable for $K=5$. For $K=-5$, the sum of crossing is $N=2$ as shown by Fig. \[SubFig:Intehral\_Stable\_NMP\_NC\_Ex2\]. Thus, $N \neq -N_p$ and the feedback system is unstable. Conclusions {#sec:Conclusions} =========== In this paper stability analysis techniques based on Nyquist and Nichols charts have been reviewed. The relationship between theses two methodologies has fully been described through several numerical examples. This tutorial provides useful insights into the loop-shaping based control systems design such as Quantitative Feedback Theory. Appendix ======== Harry Nyquist (1889-1976) ------------------------- Harry Nyquist was born in the Stora Kil parish of Nilsby, Värmland, Sweden in 1889. He emigrated to the USA in 1907. He entered the University of North Dakota in 1912 and received B.S. and M.S. degrees in electrical engineering in 1914 and 1915, respectively. He received a Ph.D. in physics at Yale University in 1917. He worked at AT&T’s Department of Development and Research from 1917 to 1934, and continued when it became Bell Telephone Laboratories that year, until his retirement in 1954. As an engineer at Bell Laboratories, Nyquist contributed to many communications problems. In 1932, he published a classic paper on stability of feedback amplifiers, and his Nyquist stability criterion has been the reference in all textbooks of feedback control theory. Nyquist lived in Pharr, Texas after his retirement, and passed away on April 4, 1976. ![Harry Nyquist (1889–1976)[]{data-label="Pic-Harry Nyquist"}](Harry_Nyquist.eps) Nathaniel B. Nichols (1914-1997)[@Obituary-NickNichols] ------------------------------------------------------- Nathaniel B. Nichols was born in Michigan in 1914 and earned a BS degree from Central Michigan University in 1936. He went from there to the University of Michigan, earning a MS degree in physics in 1937. He received two Doctor of Philosophy honorary degrees from Case Western Reserve University and from Central Michigan University. In 1942, John Ziegler and Nichols published a paper in the ASME Transactions (Vol. 64, Pg. 759) describing a set of parameters (later known as the Ziegler-Nichols tuning parameters) which were rough approximations to optimal settings for open loop transfer functions to ensure prescribed behavior in closed loop PID realizations. Nichols and his colleagues W.P. Manger and E.H. Krohn published their idea of reading closed loop gain and phase directly from a plot of open loop logrithmic gain and phase which is known as the Nichols chart. The idea was published entitled “General Design Principles for Servomechanisms" in Chapter 4 of the now famous Volume 25, “Theory of Servomechanisms" [@Nichols-Theory-of-Servomechanisms]. He was President of the IEEE Control Systems Society in 1968 and the American Automatic Control Council in 1974 and in 1975 when the IFAC Congress was first held in the US. After so many technical contributions and services to control engineering community, Nichols was retired from the Control Analysis Department of Aerospace Corporation in El Segundo, California in 1974, and passed away on 17 April 1997. ![Nathaniel B. Nichols (1914-1997)[]{data-label="Pic-Nichols"}](Nichols.eps) [1]{} C.H. Houpis, S.N. Sheldon, Linear Control System Analysis and Design with MATLAB, Sixth Edition, CRC Press, 2013, ISBN 9781466504264. N. Cohen, Y. Chait, O. Yaniv, C. Borghesani, Stability analysis using Nichols charts, International Journal of Robust and Nonlinear Control Volume 4, Issue 1, p.p. 3-20, 1994. W. Chen, and D.J. Ballance, Stability Analysis On the Nichols Chart and Its Application in QFT, Technical Report No. CSC-98013, Center for Systems and Control, Department of Mechanical Engineering, University of Glasgow. C.H. Houpis, S.J. Rasmussen, M. Garcia-Sanz, *Quantitative Feedback Theory: Fundamentals and Applications*, 2nd Edition. CRC Press, Taylor and Francis, USA, 2006, ISBN: 0-8493-3370-9. C. Borhgesani, Y. Chait, O. Yaniv, Quantitative feedback theory toolbox, The MATH WORKS Inc. 1994. M. Garcia-Sanz, A. Mauch and C. Philippe, “The QFT Control Toolbox (QFTCT) for Matlab”, CWRU, UPNA and ESA-ESTEC, Version 5.02, August 2014, http://cesc.case.edu S.M.M. Alavi, M. Saif, A QFT-Based Decentralized Design Approach for Integrated Fault Detection and Control, [*IEEE Transactions on Control Systems Technology*]{}, Vol. 20, Issue 5, pp. 1366 - 1375, 2012. S.M.M. Alavi, A. Khaki-Sedigh, B. Labibi and M.J. Hayes, Improved Multivariable Quantitative Feedback Design for Tracking Error Specifications, [*IET Control Theory & Applications*]{}, Vol. 1, No. 4, pp. 1046-1053, 2007. S.M.M. Alavi, R. Izadi-Zamanabadi, M.J. Hayes, Robust Fault Detection and Isolation Technique for SISO Closed-loop Control systems that Exhibit Actuator and Sensor Faults, [*IET Control Theory & Applications*]{}, Vol. 2, No. 11, pp. 951-965, 2008. S.M.M. Alavi, M.J. Walsh, M.J. Hayes, Robust distributed active power control technique for IEEE 802.15.4 wireless sensor networks - A quantitative feedback theory approach, [*Control Engineering Practice*]{}, Vol. 17, No. 7, pp. 805-814, 2009. S.M.M. Alavi, M.J. Hayes, Robust active queue management design: a loop-shaping approach, [*Computer Communications*]{}, 32(2), pp. 324-331, 2009. H.M. James, N.B. Nichols, R.S. Phillips, *Theory of Servomechanisms*, Dover Publications; New Ed edition, 1965. Stephen Kahne, Obituary : Nathaniel B. Nichols (1914-1997), [*Automatica*]{}, Volume 33, No. 12, December 1997. [^1]: S.M.M. Alavi is with the Brain Stimulation Engineering Laboratory, Duke University, Durham, NC 27710, USA. Email to: [mahdi.alavi@duke.edu]{}. [^2]: M. Saif is with the Department of Electrical and Computer Engineering, University of Windsor, Windsor, ON N9B 3P4, Canada. Email to: [msaif@uwindsor.ca]{} [^3]: This work was accomplished during the appointment of the first author at the University of Windsor between 2012 and 2013. The authors would like to acknowledge the partial support from Natural Sciences and Engineering Research Council of Canada (NSERC).

--- author: - bibliography: - 'main.bib' title: 'AI-Mediated Exchange Theory' --- &lt;ccs2012&gt; &lt;concept&gt; &lt;concept\_id&gt;10003120.10003130.10003131&lt;/concept\_id&gt; &lt;concept\_desc&gt;Human-centered computing Collaborative and social computing theory, concepts and paradigms&lt;/concept\_desc&gt; &lt;concept\_significance&gt;500&lt;/concept\_significance&gt; &lt;/concept&gt; &lt;/ccs2012&gt;

--- abstract: | We consider dissipative systems resulting from the Gaussian and $alpha$-stable noise perturbations of measure-preserving maps on the $d$ dimensional torus. We study the dissipation time scale and its physical implications as the noise level $\vep$ vanishes. We show that nonergodic maps give rise to an $O(1/\vep)$ dissipation time whereas ergodic toral automorphisms, including cat maps and their $d$-dimensional generalizations, have an $O(\ln{(1/\vep)})$ dissipation time with a constant related to the minimal, [*dimensionally averaged entropy*]{} among the automorphism’s irreducible blocks. Our approach reduces the calculation of the dissipation time to a nonlinear, arithmetic optimization problem which is solved asymptotically by means of some fundamental theorems in theories of convexity, Diophantine approximation and arithmetic progression. We show that the same asymptotic can be reproduced by degenerate noises as well as mere coarse-graining. We also discuss the implication of the dissipation time in kinematic dynamo. [**Keyword**]{}. dissipation, noise, toral automorphisms, dynamo author: - Albert Fannjiang - 'Lech Wo[ł]{}owski' title: 'Noise induced dissipation in Lebesgue-measure preserving maps on $d-$dimensional torus' --- [^1] Introduction {#sI} ============ Irreversibility and approach to equilibrium are fundamental problems in statistical mechanics and dynamical systems and its complete solution is still elusive (see, e.g., [@Kr]). There are possibly many routes to irreversibility. One view is that macroscopic systems are exceedingly difficult to isolate from their environments for a time comparable to their dynamical time scales. The noise as a result of interaction with environment may further trigger irreversibility, such as approach to equilibrium, in the systems. The initial uncertainty involved in preparing a physical system and the random perturbation due to measurements as well as Gibbs’ coarse-graining procedure can all be viewed as certain noises. The point is that noises, intrinsic as a result of internal stochasticity or extrinsic as a result of random influence from surrounding environment, can induce effects that would be weak or absent without noise. In this paper we investigate one such effect, called dissipation, for discrete time, conservative dynamical systems under the influence of noise. In particular we study the time scale, called the dissipation time, on which the dissipation as measured in $L^p-$norm, $1<p<\infty$, has an [*order one*]{} effect even as the magnitude of noise vanishes. Clearly the dissipation time depends on the ergodic properties of the noiseless dynamics as well as the noise level. The noisy dynamical system considered in this paper can be viewed as a discrete generalization of the dynamics of a passive scalar in a periodic, incompressible velocity field $\mathbf{v}$ \[SDE\] &&d\^(t) =(\^(t))dt + d(t)\ &&()=0 , where the standard Brownian motion $\mathbf{w}$ and the molecular diffusivity $\vep$ represent the stochastic perturbations as a result of random molecular collisions (see, e.g., [@Ki], [@F]). The discrete-time dynamical system will be defined on the $d$-dimensional torus $\IT^d=\IR^d/\IZ^d$. The velocity field $\mathbf{v}$ will be replaced by arbitrary Lebesgue-measure preserving map $F$ defined on $\IT^{d}$ (periodicity condition). In order to study the dynamics generated by $F$ it is useful to consider its Koopman operator $U_{F}$ defined by a composition $U_{F}f:=f \circ F$, with $f$ belonging to some Banach space of functions on $\IT^d$. We will be mainly concerned with the standard Banach spaces $L^p(\IT^d), 1\leq p\leq \infty,$ and their subspaces $L^p_0(\IT^d)$ of functions with zero mean $\la f\ra=0$, where $\la f\ra$ denotes the average of $f$ w.r.t. the Lebesgue measure. In case of $L^{1}(\IT^{d})$ one can consider $U_{F}$ as the Frobenius-Perron operator associated with $F^{-1}$. In the time-discrete version we consider general $\alpha-$stable noise operator $G_{\vep,\alpha}:L_{0}^{2}(\IT^{d}) \mapsto L_{0}^{2}(\IT^{d})$, with $\alpha\in (0,1]$, defined by means of the Fourier transform of corresponding $\alpha$-stable noise kernel $g_{\vep,\alpha}$ G\_[,]{}f() = \_[\^[d]{}]{}g\_[,]{}(-)f()d = \_[\^[d]{}]{}e\^[-||\^[2]{}]{}()\_(), where \[kernel\] g\_[,]{}():=\_[\^[d]{}]{}e\^[-||\^[2]{}]{}\_(), with $\bfe_{\bk}(\bx):=e^{2 \pi i \bk \cdot \bx}$, $\bk \in \IZ^{d}$. Here, just like in (\[SDE\]), $\vep>0$ represents the level of the noise. Putting $\alpha=1$ one recovers standard heat kernel. The operator $T_{\vep,\alpha}$ on $L_{0}^2(\IT^{d})$ generating the noise-perturbed dynamical system considered in this paper is thus given by \[T\] T\_[,]{}f:=G\_[,]{}U\_[F]{}f=g\_[,]{}\*(f F). Simple computations yield \[contr\] T\_[,]{}\^[n]{}=G\_[,]{}U\_[F]{}...G\_[,]{}U\_[F]{} G\_[,]{}\^[n]{}=e\^[-n]{}. Here and throughout the paper $\|\cdot\|$ denotes the standard $L_0^{2}$-norm or the corresponding operator norm (any other norm will be equipped with suitable index). We define the dissipation time as the time on which the contraction (\[contr\]) becomes of order one: \[tdiss1\] n\_[diss]{}:={n \_[+]{}:T\_[,]{}\^[n]{} &lt; 1/e}. Hence the dissipation time is a function of $\ep,\alpha$ as well as the underlying dynamics. The choice of the threshold $e^{-1}$ in the definition is a convenient one and for the purpose of the paper can be any positive number less than one (see Proposition \[prop1\]). The fact that for all $\vep>0$, $\|T_{\vep,\alpha}^{n}\|$ is monotonically decreasing ensures that $n_{diss}$ is well defined. By contrast, when $\vep=0$, the fine-grained Boltzmann-Gibbs entropy as well as $L^p$-norm of the initial state remains constant in the course of evolution. In other words this is a “dissipation” effect and hence the term “dissipation time”. On the dissipation time scale the system is, in a sense, “half way” through its irreversible route to the equilibrium state. The dissipation time provides a measure of the instability of the dynamics w.r.t. the stochastic perturbations which result in the “aging” of the system toward the final state. The main purpose of this paper is to investigate the asymptotics of the dissipation time as $\ep$ tends to zero. Due to the non-normality of the operator $T_{\vep,\alpha}$, the dissipation time can not be determined from its spectral radius. Indeed, as we will see below, the operator $T_{\vep,\alpha}$ corresponding to any ergodic toral automorphism $F$ is quasi-nilpotent for any $\vep>0$ (thus the time scale estimated from the spectral radius is infinite) whereas the dissipation time is of the order $\ln{(1/\ep)}$. To briefly describe the main results we pause to note the following asymptotic notation. Given two sequences $a_{\vep},b_{\vep}$ indexed by the parameter $\vep>0$ we write a\_ && b\_ \_[0]{} &lt;\ a\_ && b\_ \_[0 ]{}=1 Moreover we write $a_{\vep} \sim b_{\vep}$, if both $a_{\vep} \lesssim b_{\vep}$ and $b_{\vep} \lesssim a_{\vep}$ hold simultaneously. The first result is that the dissipation time $n_{diss}\sim 1/\vep$ for nonergodic or, more general, non-weakly-mixing maps (cf. Theorem \[nonergodic\] and its corollaries, Section \[sDTA\]), which is also the longest possible time scale for dissipation in view of (\[contr\]) (see also Lemma \[tdlub\], Section \[sDCal\]). In other word, such systems are most stable w.r.t. stochastic perturbations. The main aim, however, is to investigate the cases in which the dissipation is much faster due to rapid mixing in $F$. We show (Theorem 2, Section \[sDTA\]) that, for a toral automorphism $F$, $n_{diss} \sim \log(1/\vep)$ if and only if the map $F$ is ergodic (which in this case is also an Anosov diffeomorphism). In particular our results hold for all classical cat maps (hyperbolic automorphisms of $2$-torus) and their $d$-dimensional generalizations. In addition, we provide a general lower bound for the constant of the logarithmic asymptotics (Theorem 2). We further show that the lower bound is achieved for [*diagonalizable*]{} automorphisms, namely \[open\] n\_[diss]{}(2(F))\^[-1]{} (1/) where $\hat{h}(F)$ denotes the minimal, dimensionally averaged entropy among $F$’s irreducible blocks (Theorem 3, Section \[sDTA\]). Dimensionally averaged entropy for each irreducible sub-block of the toral automorphism is the Kolmogorov-Sinai (KS) entropy per dimension of an irreducible factor of the whole map. Our method involves solving asymptotically a quadratic arithmetic optimization (i.e. quadratic integer programming) problem by obtaining sharp upper and lower bounds using number theoretical tools including multidimensional Diophantine approximation theorems (Schmidt’s subspace theorem), Minkowski’s theorem on linear forms and Van der Waerden’s theorem on arithmetic progressions. This is done in Section \[sA\]. In Section \[coarse\] we show that the same result (\[open\]) holds when the noise is replaced by coarse-graining the initial and terminal states. This is reminiscent of the well-know results of statistical stability in the literature, namely, the Bernoulli systems are stable under the sufficiently small intrinsic random perturbation in the rough sense that the perturbed system is close to the direct product of the unperturbed one and some auxiliary viewer system (see [@Ki], [@OW]). In other words, for those systems, the process that results from small intrinsic random perturbation can be reproduced exactly by looking at the unperturbed system through a viewer that distorts randomly but slightly. In spite of the above the asymptotic (\[open\]) indicates the perturbed system is irreversibly far from the unperturbed one even on the relatively short dissipation time scale. From this perspective, such a system is statistically unstable. In Section \[degnoise\] we consider a class of highly degenerate noises and show that the same conclusions about the dissipation hold if the degenerate noises satisfy an additional generic condition. In Section \[dynamo\] we consider the relation between the dissipation time and some characteristic time scales relevant to kinematic dynamo. We show that fast dissipation generally inhibits dynamo action. When there is no fast dynamo action the noisy push-forward map dissipates the magnetic field energy on the dissipation time scale. However, the magnetic field energy can still grow to relatively large magnitude as inverse power-law of the small noise with the exponent proportional to the ratio of the logarithmic spectral radius of the toral automorphism to the minimal, dimensionally averaged entropy among the automorphism’s irreducible blocks (cf. (\[anti\])). The notion of dissipation time has a natural bearing on the problems of quantum chaos with noise. The family of symplectic toral automorphisms constitute important examples of quantizable chaotic dynamics on compact manifolds for which various quantization procedures have been intensively studied (see, for example, [@RO] and [@Mez]). We will address the issue of decoherence time for quantized symplectic toral automorphisms with noise in a forthcoming paper. The organization of the rest of the paper is as follows. In Section \[sD\] we develop the general theory of dissipation time and its relation to the Boltzmann-Gibbs entropy. We also formulate the dissipation time calculation for total automorphisms as an arithmetic minimization problem and state the main results. In Appendix \[aAT\] we generalize the dissipation time asymptotic result to the affine transformations. The proofs of some elementary facts are presented in Appendix \[aP\] for the sake of completeness. Dissipation time {#sD} ================ In its general form the dissipation time $n_{diss}(p)$ can be defined in terms of the norm $\|\cdot\|_{p,0}$ on the space $L^p_0(\IT^d)$ w.r.t. a threshold $\eta\in(0,1)$ \[tdiss2\] n\_[diss]{}(p,):={n \_[+]{}:T\_[,]{}\^[n]{}\_[p,0]{}&lt; }, 1p. First we show that the value of the threshold $\eta$ in (\[tdiss2\]) does not affect the order of divergence of $n_{diss}(p,\eta)$, as $\vep$ tends to zero. \[prop1\] For any $0<\tilde{\eta},\eta<1$, $n_{diss}(p,\tilde{\eta}) \sim n_{diss}(p,\eta)$. **Proof.** Assume $0<\tilde{\eta}<\eta<1$. Obviously $n_{diss}(p,\tilde{\eta}) \geq n_{diss}(p,\eta)$. On the other hand let $k$ be a positive integer such that $\eta^{k}<\tilde{\eta}$. Then T\_[,]{}\^[n\_[diss]{}(p,)]{}\_[p,0]{}&lt;T\_[,]{}\^[kn\_[diss]{}(p)]{}\_[p,0]{} &lt;\^[k]{}&lt;. Hence $kn_{diss}(p,\eta) \geq n_{diss}(p,\tilde{\eta})$, which implies $n_{diss}(p,\eta) \sim n_{diss}(p,\tilde{\eta})$.$\qquad \blacksquare$ Following the argument of [@Ros] one can use the Riesz convexity theorem to establish also the asymptotic equivalence of the $n_{diss}(p)$, for all $1<p<\infty$. \[pdiss\] i\) For any $1<q,p<\infty$, $n_{diss}(q) \sim n_{diss}(p)$. ii\) For any $1<p<\infty$, $n_{diss}(p) \lesssim n_{diss}(1)$ and $n_{diss}(p) \lesssim n_{diss}(\infty)$. The details of the proof can be found in Appendix \[aP\]. Our particular choice of the exponent $p=2$ and threshold $\eta=e^{-1}$ in (\[tdiss1\]) is computationally convenient and will be used throughout the paper. We will use the convention that $n_{diss}(p)=n_{diss}(p,e^{-1})$. We say that operator $T_{\vep,\alpha}$ or associated with it measure preserving map $F$ has a [*simple (slow)*]{} dissipation time when $n_{diss} \sim 1/\vep$ and that it has a [*logarithmic (fast)*]{} dissipation time when $n_{diss} \sim \ln(1/\vep)$. In the particular case of fast dissipation, with a logarithmic dissipation time, in order to estimate precisely the rate of dissipation, one needs to determine the value of the [*dissipation rate constant*]{} $R_{diss}$, defined as \[Rdiss\] R\_[diss]{}:=\_[0]{} . Similarly in case of simple dissipation time the dissipation rate constant can be defined as \[Rdissnon\] R\_[diss]{}:=\_[0]{} n\_[diss]{}. Dissipation time and Boltzmann-Gibbs entropy {#sDB-G} -------------------------------------------- In this section we briefly discuss the connection between dissipation time and Boltzmann-Gibbs entropy. First we note that on the scale of $n_{diss}$ the Boltzmann-Gibbs entropy approaches the maximal equilibrium value (i.e. $0$) as can be seen from the following simple estimate [@LM]. Let us first restrict considerations to bounded initial states, i.e., $f\geq 0, f\in L^\infty$ and $\|f\|_{1}=1$. Let $$\eta(u)=\begin{cases} -u\ln{u},& u>0\\ 0,& u=0 \end{cases}$$ and let $D_n=\{\bx\in \IT^d: 1\leq T^n_{\vep,\alpha}f\}$. On one hand, we have &&|\_[D\_n]{}(T\^n\_[,]{}f())d| \[fi\]\ &&\_[D\_n]{}|\_[1]{}\^[T\^n\_[,]{}f()]{}du| d\ &&\_[1u[T\_[,]{}]{}\^n f\_]{}(1+)\_[D\_n]{} |[T\_[,]{}]{}\^n f()-1|d\ && (1+) [T\_[,]{}]{}\^nf-1\_1\ &&(1+)[T\_[,]{}]{}\^nf-1\_1 \[li\]. On the other hand, we have $$0 \geq \int_{\IT^d}\eta({T_{\vep,\alpha}}^n f(\bx))d\bx \geq \int_{D_n}\eta({T_{\vep,\alpha}}^n f(\bx))d\bx.$$ In view of the inclusion relation: $L^\infty(\IT^d)\subset L^2(\IT^d) \subset L^1(\IT^d)$, we then obtain that for $n\gg n_{diss}$ $$\sup_{f\geq 0,\|f\|_\infty\leq c}\left|\int_{\IT^d}\eta({T_{\vep,\alpha}}^n f(\bx))d\bx \right| \stackrel{\vep \downarrow 0}{\longrightarrow} 0, \quad \forall c>0.$$ For unbounded initial states, we note that, by Young’s inequality, $$\|{T_{\vep,\alpha}}^n f\|_\infty \leq \|{T_{\vep,\alpha}}f\|_\infty \leq \|g_{\vep,\alpha}\|_\infty \|f\|_{1}=\|g_{\vep,\alpha}\|_\infty$$ from which we have, instead of (\[li\]), the following estimate $$\left|\int_{D_n}\eta(T^n_{\vep,\alpha}f(\bx))d\bx\right| \leq (1+\ln{\|g_{\vep,\alpha}\|_\infty})\|{T_{\vep,\alpha}}^nf-1\|_1.$$ where $$\ln\|g_{\vep,\alpha}\|_\infty \sim \ln(1/\vep).$$ Therefore for sufficiently fast diverging $n\gg n_{diss}(1)$ such that \[entropy2\] (1/)[T\_[,]{}]{}\^n(f-1)\_[1,0]{} 0 one obtains $$\sup_{f\geq 0,\|f\|_1=1}\left|\int_{\IT^d}\eta({T_{\vep,\alpha}}^n f(\bx))d\bx \right| \stackrel{\vep \downarrow 0}{\longrightarrow} 0.$$ The condition (\[entropy2\]) typically results in a slightly longer time scale than $n_{diss}(1)$. On the other hand, we can bound the $L_1$ distance between the probability density function $f$ and the Lebesgue measure by their relative entropy via Csiszár’s inequality [@Cs] $$\int_{\IT^d} |f(\bx)-g(\bx)|d\bx \leq \sqrt{2\int_{\IT^d} f(\bx)\ln{(f(\bx)/g(\bx))}d\bx}$$ with $g(\bx)=1$. We see immediately that the decay rate of $$\sup_{f\geq 0,\|f\|_1=1}\left|\int_{\IT^d}\eta({T_{\vep,\alpha}}^n f(\bx))d\bx \right|$$ provides an estimate for $n_{diss}(1)$ and, consequently, for $n_{diss}(p), p\in (1,\infty)$. Calculating the dissipation time {#sDCal} -------------------------------- For greater generality and transparency of arguments we consider, in this section, a slightly more general family of operators $T_{\vep,\alpha}$ defined, as previously, by the first equality in (\[T\]), but with arbitrary unitary or isometric (not necessary Koopman) operator $U$ (and hence in these cases we drop the subscript $F$). \[tdlub\] For any isometric operator $U$, the dissipation time of $T_{\vep,\alpha}$ satisfies following constraints \[tdisslub\] R(1;T\_[,]{}) n\_[diss]{} 1/, where $R(1;T_{\vep,\alpha})$ denotes the resolvent of $T_{\vep,\alpha}$ at 1. **Proof.** In view of (\[tdiss1\]) and (\[contr\]), for $n=n_{diss}$ one has e\^[-1]{} T\_[,]{}\^[(n\_[diss]{}-1)]{} e\^[-(n\_[diss]{}-1)]{}, which clearly implies the second estimate of (\[tdisslub\]). In order to prove the other inequality we proceed as follows. &&R(1;T\_[,]{})= \_[n=0]{}\^T\_[,]{}\^[n]{}= \_[n=0]{}\^[n\_[0]{}-1]{}T\_[,]{}\^[n]{} + T\_[,]{}\^[n\_[0]{}]{}\_[n=0]{}\^T\_[,]{}\^[n]{}\ &&\_[n=0]{}\^[n\_[0]{}-1]{}T\_[,]{}\^[n]{} + T\_[,]{}\^[n\_[0]{}]{}\_[n=0]{}\^T\_[,]{}\^[n]{} n\_[0]{} + T\_[,]{}\^[n\_[0]{}]{}R(1;T\_[,]{}). Hence taking in the above inequality $n_{0}=n_{diss}$ one gets R(1;T\_[,]{})(1-e\^[-1]{}) R(1;T\_[,]{})(1-T\_[,]{}\^[n\_[diss]{}]{}) n\_[diss]{}, which gives the first estimate of (\[tdisslub\]). $\qquad \blacksquare$ The above lemma provides an absolute upper bound for dissipation time. Taking $F=I$ one easily finds that this bound is best possible in general. The lower bound is useful in the case when one can estimate from below the norm of the resolvent (see proof of Theorem \[nonergodic\]). \[nonergodic\] If $U$ acting on $L^{2}_{0}(\IT^{d})$ possesses nonempty pure point spectrum and at least one of its eigenfunctions belongs to $H^{2\alpha}(\IT^{d})$, then $T_{\vep,\alpha}$ has simple dissipation time. **Proof**. In view of Lemma \[tdlub\] it is enough to find a lower bound for the norm of the resolvent $R(1;T_{\vep,\alpha})$. Let $h \in H^{2\alpha}$ be one of the eigenfunctions of $U$. Since $U$ is isometric we have Uh=e\^[i]{}h. We first assume that $\phi=0$. Since $1 \not \in \sigma (T_{\vep,\alpha})$, $I-T_{\vep,\alpha}$ is a homeomorphism and hence R(1;T\_[,]{})= \_[fL\^[2]{}\_[0]{}]{}= \_[fL\^[2]{}\_[0]{}]{} . Now expressing $h$ in the Fourier series we get (I-T\_[,]{})h\^[2]{}= \_[0 =\^[d]{}]{}|()|\^[2]{}|1-e\^[-||\^[2]{}]{}|\^[2]{} \_[0 =\^[d]{}]{}(|()|||\^[2]{})\^[2]{}= \^[2]{} h\_[H\^[2]{}]{}\^[2]{}. Hence R(1;T\_[,]{}) =: , Thus in view of (\[tdisslub\]) and above calculations 1/R(1;T\_[,]{}) n\_[diss]{} 1/, which ends the proof in the case $\phi=0$. If $\phi \not=0$, we put $\hat{U}= e^{-i\phi}U$, which implies $\hat{U}h=h.$ The proof is completed by applying the above reasoning to operator $\hat{T}_{\vep,\alpha}=G_{\vep,\alpha}\hat{U}$ and observing that the dissipation times for $T_{\vep,\alpha}$ and $\hat{T}_{\vep,\alpha}$ are identical. $\qquad \blacksquare$ When $U$ is a Koopman operator associated with a map $F$, then the property that $U$ considered on $L^{2}_{0}(\IT^{d})$ possesses nonempty pure point spectrum is equivalent to the fact that $F$ is not weakly mixing (see [@CoFoSi]). Thus we have If $F$ is not weakly mixing and its Koopman operator possesses $H^{2\alpha}$ eigenfunction in $L^2_0(\IT^{d})$, then $T_{\vep,\alpha}$ has simple dissipation time. Another immediate consequence is \[ne\] If $F$ is not ergodic and its nontrivial invariant measure possesses $H^{2\alpha}$ density function, then $T_{\vep,\alpha}$ has simple dissipation time. A typical example of ergodic but not weakly mixing transformations for which the above corollary applies is the family of ’irrational’ shifts on $\IT^{d}$ i.e. maps $F\bx=\bx+\mf{c}$ on $\IT^{d}$, where $\mf{c}=(c_{1},..,c_{d})$ is a constant vector such that the numbers $1,c_{1},..,c_{d}$ are linearly independent over rationals. More general and less trivial examples of ergodic maps giving rise to a simple dissipation time will be discussed in Appendix A (cf. Remark \[remerg\]). In general the problem of computing the dissipation time is rather complicated. In some cases it can be reformulated as an asymptotic optimization problem. To see it, one can represent the action of a given unitary operator $U$ in the Fourier basis \[uk\] U\_=\_[0 = ’ \^[d]{}]{}u\_[,’]{}\_[’]{}, where for each $\bk$ \[ul2\] \_[0 = ’ \^[d]{}]{}|u\_[,’]{}|\^[2]{}=1. Next we introduce the notation \_[n]{}(\_0,\_[n]{})&=&\_[0 = \_[1]{},...,\_[n-1]{} \^[d]{}]{} u\_[\_[0]{},\_[1]{}]{}...u\_[\_[n-1]{},\_[n]{}]{} e\^[-\_[l=1]{}\^[n]{}|\_[l]{}|\^[2]{}]{}\ \_n(\_[n]{})&=&{\_[0]{} \^[d]{}\\{0} : \_[n]{}(\_0,\_[n]{})= 0}. Then for any $f\in L_{0}^{2}(\IT^{d})$ we have T\_[,]{}\^[n]{}f\^[2]{} &=& \_[0 = \_[0]{} \^[d]{}]{}(\_[0]{})T\_[,]{}\^[n]{} \_[\_[0]{}]{}\^[2]{}= \_[0 = \_[0]{} \^[d]{}]{}(\_[0]{})\_[0 = \_[n]{} \^[d]{}]{} \_[n]{}(\_0,\_[n]{})\_[\_[n]{}]{}\^[2]{}\ \[Tnorm1\] &=&\_[0 = \_[n]{} \^[d]{}]{}| \_[0 = \_[0]{} \^[d]{}]{}(\_[0]{}) \_[n]{}(\_0,\_[n]{})|\^[2]{}= \_[0 = \_[n]{} \^[d]{}]{}| \_[\_[0]{} \_[n]{}(\_[n]{})]{}(\_[0]{}) \_[n]{}(\_0, \_[n]{})|\^[2]{}. The following general upper bound for $\|T_{\vep,\alpha}^{n}f\|$ holds. \[gub\] For any $f\in L_{0}^{2}(\IT^{d})$, \[uppernorm\] T\_[,]{}\^[n]{}f\^[2]{} && \_[0 = \_[n]{} \^[d]{}]{} \_[\_[0]{} \_[n]{}(\_[n]{})]{} |(\_[0]{})|\^[2]{} \_[\_[0]{} \_[n]{}(\_[n]{})]{} |\_n(\_[0]{},\_[n]{})|\^[2]{}. For the proof we refer the reader to Appendix \[aP\]. When $u_{\bk,\bk'}$ is a Kronecker’s delta function \[linear\] u\_[,’]{}=\_[A,’]{}, where $A:\IZ^{d}\mapsto \IZ^{d}$ is a linear surjective map, the upper bound (\[uppernorm\]) can be used to obtain an identity for $\|T_{\vep,\alpha}^{n}\|$. First observe that \[linear1\] \_n(\_[0]{},\_[n]{})=e\^[-\_[l=1]{}\^[n]{}|A\^[l]{}\_[0]{}|\^[2]{}]{}\_[A\^[n]{}\_[0]{},\_[n]{}]{} and hence (\[uppernorm\]) becomes T\_[,]{}\^[n]{}f\^[2]{} && \_[0 = \_[0]{} \^[d]{}]{} |(\_[0]{})|\^[2]{}e\^[-2\_[l=1]{}\^[n]{}|A\^[l]{}\_[0]{}|\^[2]{}]{} f\^[2]{}\_[0 =\^[d]{}]{} e\^[-2\_[l=1]{}\^[n]{}|A\^[l]{}|\^[2]{}]{}. On the other hand for any nonzero $\bk\in \IZ^{d}$, one can take in (\[Tnorm1\]) $f=\bfe_{\bk}$ and get T\_[,]{}\^[n]{}f\^[2]{}= e\^[-2\_[l=1]{}\^[n]{}|A\^[l]{}|\^[2]{}]{} and therefore \[Tnorm3\] T\_[,]{}\^[n]{} &=& \_[0 =\^[d]{}]{} e\^[-\_[l=1]{}\^[n]{}|A\^[l]{}|\^[2]{}]{} = e\^[-\_[0 =\^[d]{}]{} \_[l=1]{}\^[n]{}|A\^[l]{}|\^[2]{}]{}. Let us now determine the class of maps $F$ such that the corresponding Koopman operator $U_{F}$ satisfies (\[linear\]). The relation (\[linear\]) implies U\_[F]{}\_=\_[A]{}=e\^[2i A,]{}. On the other hand U\_[F]{}\_()=\_(F)=e\^[2i ,F]{}. Thus $$\la \bk,F\bx\ra=\la A\bk,\bx\ra \,\,\hbox{mod}\,\, 1,\quad\forall \bx \in \IR^d,\,\, \bk\in \IZ^d,$$ that is, $A$ is linear and $A^\dagger$ equals the lifting of $F$ from $\IT^{d}$ onto $\IR^{d}$. Moreover, the matrix $A$ has integer entries and determinant equal to $\pm 1$, i.e., $A$ (and $F$) is a toral automorphism. Hence, for toral automorphisms, the calculation of the dissipation time reduces to the following nonlinear, asymptotic (large $n$) arithmetic minimization problem \[artminA\] \_[0 =\^[d]{}]{} \_[l=1]{}\^[n]{}|A\^[l]{}|\^[2]{}. We will show in Section \[sA\] that for any ergodic toral automorphism this minimum value grows geometrically in $n$ with the base related to the dimensionally-averaged KS-entropy of the total automorphism. Dissipation time of toral automorphisms {#sDTA} --------------------------------------- It is well known (see [@AP]) that (the lifting map corresponding to) any toral homeomorphism $H:\IT^{d} \mapsto \IT^{d}$ can be decomposed into three parts $H=L+P+c$, where $L$, the linear part, is an element of $SL(d,\IZ)$ - the set of all matrices with integer entries and determinant equal to $\pm 1$, $P$ is periodic i.e. $P(\bx+{{\bf v}})=P(\bx)$ for any ${{\bf v}}\in \IZ^{d}$, and $c$ is a constant shift vector. Every algebraic and measurable automorphism of the torus is continuous. Each continuous toral automorphism is a homeomorphism with zero periodic and constant parts and hence can be identified with an element of $SL(d,\IZ)$. And vice versa, each element of $SL(d,\IZ)$ uniquely determines a measurable, algebraic toral automorphism. Thus from now on the term [*toral automorphism*]{} will simply be reserved for elements of $SL(d,\IZ)$. We recall here that all Anosov diffeomorphisms on $\IT^d$ are topologically conjugate to the toral automorphisms ([@Fr], [@Man]). Below we summarize some ergodic properties of toral automorphisms (cf. [@Katok] p. 160, [@Katznelson] and [@Arnold]). \[ergodicTA\] Let $F$ be a toral automorphism. The following statements are equivalent a\) no root of unity is an eigenvalue of $F$. b\) $F$ is ergodic. c\) $F$ is mixing. d\) $F$ is a K-system. e\) $F$ is a Bernoulli system. In the sequel we will use the following result (cf. [@Yuz]). The entropy $h(F)$ of any toral endomorphism $F$ is computed by the formula \[entropy\] h(F)=\_[|\_[j]{}| 1]{}, where $\lam_{j}$ denote the eigenvalues of $A$. &gt;From the formula (\[entropy\]) one immediately sees that a toral automorphism has zero entropy if and only if all its eigenvalues are of modulus 1. In fact much stronger result holds. \[zeroentropy\] A toral automorphism has zero entropy if and only if all its eigenvalues are roots of unity. In particular all ergodic toral automorphisms have positive entropy. Given any toral automorphism $F$ we denote by $P$ its characteristic polynomial and by $\{P_{1},...,P_{s}\}$ the complete set of its distinct irreducible (over $\IQ$) factors. Let $d_{j}$ denote the degree of polynomial $P_{j}$ and $h_{j}$ the KS-entropy of any toral automorphism with the characteristic polynomial $P_{j}$. For each $P_{j}$ we define its dimensionally averaged KS-entropy as \_[j]{}=. For the whole matrix $F$ we define its minimal dimensionally averaged entropy (denoted $\hat{h}(F)$) as (F)=\_[j=1,...,s]{}\_[j]{} Now we state two main theorems of the present paper. \[thm2\] Let $F$ be any toral automorphism, $U_{F}$ the Koopman operator associated with $F$, $G_{\vep,\alpha}$ $\alpha$-stable noise operator and $T_{\vep,\alpha}=G_{\vep,\alpha}U_{F}$. Then i\) $T_{\vep,\alpha}$ has simple dissipation time if and only if $F$ is not ergodic. ii\) $T_{\vep,\alpha}$ has logarithmic dissipation time if and only if $F$ is ergodic. iii\) If $T_{\vep,\alpha}$ has logarithmic dissipation time then the dissipation rate constant satisfies the following constraint R\_[diss]{} , where $\tilde{h}(F)$ is a positive constant satisfying $\tilde{h}(F)\leq\hat{h}(F)$. Part i) of the above theorem follows immediately from Theorem \[nonergodic\]. For details of a simple proof we refer to appendix \[aP\]. The natural question arises, whether the lower bound for the dissipation rate constant given in the above theorem is best possible. The next theorem and its corollary provides a strong argument in favor of this conjecture. \[thm3\] If $F$ is ergodic and diagonalizable then n\_[diss]{}(1/). That is, the dissipation rate constant of $T_{\vep,\alpha}$ is given by R\_[diss]{}=. The proof of parts ii) and iii) of Theorem \[thm2\] and of Theorem \[thm3\] constitute the most important part of this work and will be presented in Section \[sAT\] after necessary tools are developed. We end this section with the the results for two and three dimensional tori. Ergodicity of two dimensional toral automorphisms is equivalent to hyperbolicity. Two dimensional hyperbolic toral automorphisms are often referred to as the *cat maps*. Using Corollary \[23dim\] and applying Theorem \[thm3\] to two and three dimensions one gets the following \[cat\] Let $F$ be any ergodic, two or three dimensional toral automorphism. Then n\_[diss]{}(1/), Asymptotic arithmetic minimization problem {#sA} ========================================== In this section we find the asymptotics, as n goes to infinity, of the following quadratic arithmetic minimization problem \[artmin0\] \_[0 =\^[d]{}]{} \_[l=1]{}\^[n]{}|A\^[l]{}|\^[2]{}, where $A\in SL(d,\IZ)$. When $A$ is not ergodic the asymptotics of (\[artmin0\]) is of the order $O(n)$. For the rest of the paper we will consider only the ergodic case. For $d=2$ the problem (\[artmin0\]) can be solved easily as follows. Consider first the case that $A$ is symmetric and $\alpha=1$. &gt;From $det(A)=1$ we see that eigenvalues are $\lam, \lam^{-1}$ with $|\lam|>1$. We have \_[0 =\^[d]{}]{} \_[l=1]{}\^[2n+1]{}|A\^[l]{}|\^[2]{}&=& \_[0 =\^[d]{}]{} \_[l=-n]{}\^[n]{}|A\^[l]{}|\^[2]{}\ &=&\_[0 ==\_[1]{}+\_[2]{} \^[d]{}]{} ( ||\^[2]{} + \_[l=1]{}\^[n]{}||\^[2l]{}|\_[1]{}|\^[2]{}+||\^[-2l]{}|\_[2]{}|\^[2]{} +\_[l=1]{}\^[n]{}||\^[-2l]{}|\_[1]{}|\^[2]{}+||\^[2l]{}|\_[2]{}|\^[2]{})\ &=& \_[0 =\^[d]{}]{}\_[l=-n]{}\^[n]{}||\^[2l]{}||\^[2]{} =\_[l=-n]{}\^[n]{}||\^[2l]{}. Hence there exist constants $C_{1}$ and $C_{2}$ such that C\_[1]{}e\^[h(A)n]{} \_[0 =\^[d]{}]{} \_[l=1]{}\^[n]{}|A\^[l]{}|\^[2]{} C\_[2]{}e\^[h(A)n]{}. where $h(A)$ denotes the KS-entropy of $A$. The estimates for the general case of non-symmetric $A$ and $\alpha \neq 1$ are similar. In higher dimensions, the solution to (\[artmin0\]) is much more involved because of the presence of different eigenvalues with absolute values bigger than one. We have the following general estimate \[thmart\] Let $A\in SL(d,\IZ)$ be ergodic. There exist constants $C_{1}$ and $C_{2}$ such that for any $0<\del<1$ and sufficiently large $n$ \[ulestimateA\] C\_[1]{} e\^[(1-)2(A) n]{} \_[0 =\^[d]{}]{} \_[l=1]{}\^[n]{}|A\^[l]{}|\^[2]{} C\_[2]{}ne\^[2(A)n]{} where as before $\hat{h}(A)$ denotes minimal dimensionally averaged entropy of $A$ and $\tilde{h}(A)$ denotes a constant satisfying $0<\tilde{h}(A)\leq \hat{h}(A)$, with equality achieved for all diagonalizable matrices $A$. The question whether the equality $\tilde{h}(A)=\hat{h}(A)$ holds for all ergodic matrices remains open. The proof of the theorem relies on nontrivial use of three number-theoretical results stated below. **I. Minkowski’s Theorem on linear forms** [*Let $L_{1},...,L_{d}$ be linearly independent linear forms on $\IR^{d}$ which are real or occur in conjugate complex pairs. Suppose $a_{1},a_{2},...,a_{d}$ are real positive numbers satisfying $a_{1}a_{2}...a_{d}=1$ and $a_{i}=a_{j}$, whenever $L_{i}=\bar{L}_{j}$. Then there exists a nonzero integer vector $\bk \in \IZ^{d}$ such that for every $j=1,...,d$, \[minko\] |L\_[j]{}| Da\_[j]{}, where $D=|\det[L_{1},...,L_{d}]|^{1/d}$.* ]{} Minkowski’s Theorem on linear forms will be used to obtain a sharp upper bound on the asymptotic solution of the arithmetic minimization problem. The proof of the above theorem and its generalization to arbitrary lattices can be found in [@Newman] (Chap. VI). **II. Schmidt’s Subspace Theorem** [*Let $L_{1},...,L_{d}$ be linearly independent linear forms on $\IR^{d}$ with real or complex algebraic coefficients. Given $\del>0$, there are finitely many proper rational subspaces of $\IR^{d}$ such that every nonzero integer vector $\bk$ with \[Schmi\] \_[j=1]{}\^[d]{}|L\_[j]{}|&lt;||\^[-]{} lies in one of these subspaces.* ]{} Schmidt’s Subspace Theorem will be used in conjunction with Van der Waerden’s Theorem on arithmetic progressions (see below) to obtain a sharp lower bound for the asymptotic solution of the arithmetic minimization problem. The proof of Schmidt’s Subspace Theorem can be found in [@Schmidt] (Theorem 1F, p. 153). \[exept\] For a given set of linear forms and for fixed $\del>0$, the smallest collection of proper rational subspaces of $\IR^{d}$ which contain all nonzero integer vectors satisfying (\[Schmi\]), is called the exceptional set and denoted by $E_{\del}$. A main difficulty to be resolved in using Schmidt’s Subspace Theorem is to show that the minimizer of either the original problem (\[artmin0\]) or an equivalent problem does not lie in the respective exceptional set which is in general unknown. We will pursue the latter route by using Van der Waerden’s Theorem on arithmetic progressions to show that one can always construct an equivalent minimization problem whose minimizer is guaranteed to lie outside the corresponding exceptional set. To this end we note that Schmidt’s Subspace Theorem is true when the standard lattice $\IZ^{d}$ is replaced by any other rational lattice, that is any lattice of the form $\Lambda=Q(\IZ^{d})$ where $Q\in GL(d,\IQ)$. Schmidt’s subspace theorem can be generalized to this situation by considering the set of new forms $\tilde{L}_{j}=L_{j}Q$. The fact that $Q \in GL(d,\IQ)$ implies immediately that $\tilde{L}_{j}$ are still linearly independent forms on $\IR^{d}$ with real or complex algebraic coefficients. **III. Van der Waerden’s Theorem on arithmetic progressions** [*Let $k$ and $d$ be two arbitrary natural numbers. Then there exists a natural number $n_*(k,d)$ such that, if an arbitrary segment of length $n\geq n_*$ of the sequence of natural numbers is divided in any manner into $k$ (finite) subsequences, then an arithmetic progression of length $d$ appears in at least one of these subsequences.* ]{} The original proof was published in [@vdW]; Lukomskaya’s simplification can be found in [@Lukomskaya]. Before presenting the proof of our main results we state a number of technical facts concerning the structure of toral automorphisms. Algebraic structure of toral automorphisms {#sAA} ------------------------------------------ In this section we denote by $GL(d,\IQ)$ the group of nonsingular $d\times d $ matrices with rational entries or the group of linear operators on Euclidean space $\IR^{d}$, which are represented in standard basis by such matrices. We generally use the same symbol to denote both operator and its matrix. In the sequel a vector $x \in \IR^{d}$ will be called an integer (or integral) vector if all its components are integers, and similarly a rational, an algebraic vector if all its components are rational or respectively algebraic numbers. The term [*rational subspace of $\IR^{d}$*]{} will then refer to a linear subspace of $\IR^{d}$ spanned by rational vectors (cf. [@Schmidt] p. 113). $A \in GL(d,\IQ)$ is called irreducible (over $\IQ$) if its characteristic polynomial is irreducible in $\IQ[x]$. \[irred\] The following statements about a matrix $A \in GL(d,\IQ)$ are equivalent. a\) $A$ is irreducible. b\) $A$ does not possess any proper rational $A$-invariant subspaces of $\IR^{d}$. c\) No rational proper subspace of $\IR^{d}$ is contained in any proper $A$-invariant subspace of $\IR^{d}$. d\) For any nonzero $\bq \in \IQ^{d}$ and any arithmetic progression of integer numbers $n_{1},...,n_{d}$, the set $\{A^{n_{1}}\bq,A^{n_{2}}\bq,...,A^{n_{d}}\bq\}$ forms a basis of $\IR^{d}$. e\) $A^\dagger$ is irreducible. f\) No nonzero $\bq \in \IQ^{d}$ is orthogonal to any proper $A$-invariant subspace of $\IR^{d}$. g\) No proper $A$-invariant subspace of $\IR^{d}$ is contained in any proper rational subspace of $\IR^{d}$. We say that operator $A \in GL(d,\IQ)$ is completely decomposable over $\IQ$ if there exists a rational basis of $\IR^{d}$ in which $A$ admits the following block diagonal form \[blockdiag\] A\_[1]{} & 0 & ...& 0\ 0 & A\_[2]{} & ...& 0\ ... & ... & ...& ...\ 0 & 0 & ...& A\_[r]{} , where for each $j=1,...,r \leq d$, $A_{j} \in GL(d_{j},\IQ)$ is irreducible and $\sum_{j=1}^r d_{j}=d$. In general, any matrix $A \in GL(d,\IQ)$ admits a rational block diagonal representation $[A_{j}]_{j=1,...,r}$. The smallest rational blocks to which $A$ can be decomposed are called elementary divisor blocks. The characteristic polynomial corresponding to any elementary divisor block is of the form $p^{m}$, where $p$ is an irreducible (over $\IQ$) polynomial (see, e.g., [@Dummit]). Although elementary divisor blocks cannot be decomposed over $\IQ$ into smaller invariant blocks, some elementary divisor blocks may not be irreducible. This happens iff $m>1$ iff $A$ is not completely decomposable over $\IQ$. One has the following elementary fact (see Appendix \[aP\] for a proof). \[red\] $A \in GL(d,\IQ)$ is completely decomposable over $\IQ$ iff $A$ is diagonalizable. However, even if $A\in GL(d,\IQ)$ is not completely decomposable, each elementary divisor block of $A$ can be uniquely represented (in a rational basis) in the following block upper triangular form \[but\] B & C\ 0 & D , where $B$ is the unique rational irreducible sub-block associated with $A$-invariant rational subspace of that elementary divisor and $C$, $D$ denote some rational matrices. \[distinct\] All the eigenvalues of an irreducible matrix $A\in GL(d,\IQ)$ are distinct (complex) algebraic numbers. In particular all irreducible matrices are diagonalizable. The proofs of the above propositions can be found in Appendix B. Finally we note that since the leading coefficient and constant term of a characteristic polynomial of any toral automorphism are equal to 1, the only possible rational eigenvalues of such map are $\pm 1$ or $\pm i$. The latter fact implies that ergodic toral automorphisms do not possesses rational eigenvalues. Thus we have the following \[23dim\] Let $F$ be an ergodic, two or three dimensional toral automorphism. Then $F$ is irreducible (and hence diagonalizable). Proof of Theorem \[thmart\] {#sAP} --------------------------- This section is entirely devoted to the proof of Theorem \[thmart\]. Let $[A_{j}]_{j=1,...,r}$ be a rational block-diagonal decomposition of $A$ into elementary divisor blocks. Since $A\in SL(d,\IZ)$, there exist a transition matrix $Q\in SL(d,\IQ)$ such that for every $l\in\IZ$, A\^[l]{}=Q\^[-1]{}(\[A\_[j]{}\])\^[l]{}Q and moreover each elementary divisor block $[A]_{j}$ is represented in its block upper triangular form (\[but\]). The matrix $Q$ defines a new lattice $\Lambda=Q(\IZ^{d})$ and acts bijectively between this lattice and the standard lattice $\IZ^{d}$. Hence \_[0 =\^[d]{}]{} \_[l=1]{}\^[n]{}|A\^[l]{}|\^[2]{}= \_[0 =\^[d]{}]{} \_[l=1]{}\^[n]{}|Q\^[-1]{}(\[A\_[j]{}\])\^[l]{}Q|\^[2]{}= \_[0 =]{} \_[l=1]{}\^[n]{}|Q\^[-1]{}(\[A\_[j]{}\])\^[l]{}|\^[2]{}. Moreover Q\^[-2]{}|(\[A\_[j]{}\])\^[l]{}|\^[2]{}|Q\^[-1]{}(\[A\_[j]{}\])\^[l]{}|\^[2]{}Q\^[-1]{}\^[2]{}|(\[A\_[j]{}\])\^[l]{}|\^[2]{}, l, j, . Now we decompose $\Lambda$ into the direct sum of lower dimensional sublattices $\Lambda_{j}$ corresponding to invariant blocks $[A_{j}]$. So that \[minimal\] \_[0 =]{} \_[l=1]{}\^[n]{}|(\[A\_[j]{}\])\^[l]{}|\^[2]{}= \_[j{1,...,r}]{}\_[0 =\_[j]{}]{} \_[l=1]{}\^[n]{}|(A\_[j]{})\^[l]{}|\^[2]{}. Thus, without loss of generality, we may specialize to the case that $A$ is already indecomposable over $\IQ$ i.e. $A$ does not possesses any proper elementary divisor blocks. To simplify the notation we will work with the standard lattice $\Lambda=\IZ^{d}$. According to the remarks following the statements of Minkowski’s and Schmidt’s Theorems the proof can be easily adapted for any rational lattice $\Lambda=Q(\IZ^{d})$. Since the technique of the proof differs depending on diagonalizability of $A$ we consider two cases: ### Diagonalizable case Here we concentrate on the case when $A$ is diagonalizable and hence due to its in-decomposability irreducible (cf. Proposition \[red\]). We denote by $\lam_{j}$ ($j=1,...,d$) the eigenvalues of $A$. Following Proposition \[distinct\] we note that $\lam_{j}$ are distinct (possibly complex) algebraic numbers and hence there exists a basis (of $\IC^{d}$) $\{{{\bf v}}_{j}\}_{j=1,...,d}$ composed of normalized algebraic eigenvectors corresponding to eigenvalues $\lam_{j}$. We denote by $[P_{j}]_{j=1}^d$ the projections on $[{{\bf v}}_{j}]$, and by $[L_{j}]$ the corresponding linear forms. It is easy to check that $[L_{j}]$ are given, in the Riesz identification, by the eigenvectors $[{{\bf u}}_j]$ of the matrix $A^\dagger$ which are co-orthogonal to $[{{\bf v}}_j]$, i.e., $\la {{\bf u}}_i,{{\bf v}}_j\ra=0$ for $i\not=j$. $[{{\bf u}}_j]$ and $[{{\bf v}}_j]$ are real or occur in complex conjugate pairs. We have =\_[j=1]{}\^[d]{}P\_[j]{}=\_[j=1]{}\^[d]{}(L\_[j]{})[[**v**]{}]{}\_[j]{} =\_[j=1]{}\^d,[[**u**]{}]{}\_j[[**v**]{}]{}\_j, \^d. The equivalence between any two norms in a finite dimensional vector space, implies the existence of absolute constants $C_{1},C_{2}$ such that C\_[1]{} \_[j=1]{}\^[d]{}|P\_[j]{}|\^[2]{} ||\^[2]{} C\_[2]{} \_[j=1]{}\^[d]{}|P\_[j]{}|\^[2]{}. Using the above inequalities, the monotonicity of a map $\bx \mapsto \bx^{\alpha}$ and an obvious inequality $(a+b)^{\alpha} \leq a^{\alpha}+ b^{\alpha}$, which holds for all positive $a,b$ and $\alpha \in (0,1]$ one obtains the following estimates \_[l=1]{}\^[n]{}|A\^[l]{}|\^[2]{} && \_[l=1]{}\^[n]{}(C\_[2]{}\_[j=1]{}\^[d]{}|P\_[j]{}A\^[l]{}|\^[2]{})\^= C\_[2]{}\^\_[l=1]{}\^[n]{}(\_[j=1]{}\^[d]{}|\_[j]{}|\^[2l]{}|P\_[j]{}|\^[2]{})\^\ && C\_[2]{}\^\_[l=1]{}\^[n]{}\_[j=1]{}\^[d]{}|\_[j]{}|\^[2l]{}|P\_[j]{}|\^[2]{}= C\_[2]{}\^\_[j=1]{}\^[d]{}(\_[l=1]{}\^[n]{}|\_[j]{}|\^[2l]{})|P\_[j]{}|\^[2]{} and on the other hand \_[l=1]{}\^[n]{}|A\^[l]{}|\^[2]{} && (\_[l=1]{}\^[n]{}|A\^[l]{}|\^[2]{})\^ (\_[l=1]{}\^[n]{} C\_[1]{}\_[j=1]{}\^[d]{}|P\_[j]{}A\^[l]{}|\^[2]{})\^\ &=&C\_[1]{}\^(\_[l=1]{}\^[n]{}\_[j=1]{}\^[d]{}|\_[j]{}|\^[2l]{}|P\_[j]{}|\^[2]{})\^= C\_[1]{}\^(\_[j=1]{}\^[d]{}(\_[l=1]{}\^[n]{}|\_[j]{}|\^[2l]{})|P\_[j]{}|\^[2]{})\^. Now we introduce some notation \[lamhat\] \_[j]{}&:=&max{1,|\_[j]{}|},\ \[hatlamgeo\] \_[geo]{}&:=&(\_[j=1]{}\^[d]{}\_[j]{} )\^[1/d]{}. One can easily observe that there exists a constant $C$ such that C\_[j]{}\^[2n]{}\_[l=1]{}\^[n]{}|\_[j]{}|\^[2l]{} n\_[j]{}\^[2n]{}. In the sequel we do not distinguish between particular values of constants appearing in computations. The symbols $C_{1},C_{2},..$ are used to denote any generic constants independent of $n$. The normalization condition $|{{\bf v}}_j|=1$ implies the following relation \[normalization\] |P\_[j]{}|=|L\_[j]{}|. Combining the above estimates one gets the following general bounds \[ulestimate\] C\_[1]{}(\_[j=1]{}\^[d]{} \_[j]{}\^[2 n]{}|L\_[j]{}|\^[2]{})\^ \_[l=1]{}\^[n]{}|A\^[l]{}|\^[2]{} C\_[2]{}n\_[j=1]{}\^[d]{} \_[j]{}\^[2n]{}|L\_[j]{}|\^[2]{}. Therefore in order to estimate (\[artmin0\]) it suffices, essentially, to estimate \[artmin2\] \_[0 =\^[d]{}]{} \_[j=1]{}\^[d]{}\_[j]{}\^[2n]{}|L\_[j]{}|\^[2]{}. We denote by ${\bz}_{n}$ the sequence of minimizers i.e. nonzero integral vectors solving (\[artmin2\]). **Upper bound.** For the upper bound we assign to the set of linear forms $L_{j}$ the set $\mathcal{A}$ composed of all real vectors $\ba=(a_{1},...,a_{d})$ satisfying the conditions $a_{j}>0$, for $j=1,...,d$ and $a_{i}=a_{j}$ whenever $L_{i}=\bar{L}_{j}$ and \[constr\] \_[j=1]{}\^[d]{}a\_[j]{}=1. &gt;From Minkowski’s theorem on linear forms, we know that for any $\ba\in\mathcal{A}$, there exists nonzero integral vector $\bk_{\ba}$ satisfying $|L_{j}\bk_{\ba}| \leq Da_{j}$, $j=1,...,d$, where $D=|\det[L_{1},...,L_{d}]|^{1/d}$. Thus \[up1\] \_[j=1]{}\^[d]{}\_[j]{}\^[2n]{}|L\_[j]{}\_|\^[2]{} D\_[j=1]{}\^[d]{}\_[j]{}\^[2n]{}a\_[j]{}\^[2]{}. The minimizing property of $\bz_{n}$ implies that for any $\ba \in \mathcal{A}$, \[up3\] \_[j=1]{}\^[d]{}\_[j]{}\^[2n]{}|L\_[j]{}\_[n]{}|\^[2]{} \_[j=1]{}\^[d]{}\_[j]{}\^[2n]{}|L\_[j]{}\_|\^[2]{}. Thus combining (\[up1\]) and (\[up3\]), and applying the Lagrange multipliers minimization with the constraint (\[constr\]) (and using the fact that $\hat{\lam}_{i}=\hat{\lam}_{j}$ whenever $L_{i}=\bar{L}_{j}$), we get \[ubound1\] \_[j=1]{}\^[d]{}\_[j]{}\^[2n]{}|L\_[j]{}\_[n]{}|\^[2]{} D\_\_[j=1]{}\^[d]{}\_[j]{}\^[2n]{}a\_[j]{}\^[2]{}= dD(\_[j=1]{}\^[d]{}\_[j]{}\^[2n]{})\^[1/d]{}= dD\_[geo]{}\^[2 n]{}. Thus the following upper bound holds \[ubound\] \_[0 =\^[d]{}]{} \_[l=1]{}\^[n]{}|A\^[l]{}|\^[2]{} C\_[2]{} n\_[geo]{}\^[2 n]{}. **Lower bound.** Let $m$ denote an arbitrary natural number. Using the fact that $A$ acts bijectively on $\IZ^{d}$ we can restate the minimization problem (\[artmin2\]) in the following form \[artmin3\] \_[0 =\^[d]{}]{} \_[j=1]{}\^[d]{}\_[j]{}\^[2n]{}|L\_[j]{}|\^[2]{} &=& \_[0 =\^[d]{}]{} \_[j=1]{}\^[d]{}\_[j]{}\^[2n]{}|L\_[j]{}A\^[-m]{}A\^[m]{}|\^[2]{}\ &=&\_[0 =\^[d]{}]{} \_[j=1]{}\^[d]{}\_[j]{}\^[2n]{}|\_[j]{}|\^[-2m]{}|L\_[j]{}A\^[m]{}|\^[2]{} That is \[eqref3\] \_[j=1]{}\^[d]{}\_[j]{}\^[2n]{}|L\_[j]{}\_[n]{}|\^[2]{}= \_[j=1]{}\^[d]{}\_[j]{}\^[2n]{}|\_[j]{}|\^[-2m]{}|L\_[j]{}A\^[m]{}\_[n]{}|\^[2]{}. We choose arbitrary $\del>0$ and consider the exceptional set $E_{\del}$ (see Definition \[exept\]) associated with the system of linear forms $[L_{j}]$. Since $[L_{j}]$ correspond to the eigen-pairs $[\bar{\lam}_j, {{\bf u}}_j]$ of $A^\dagger$ they are linearly independent linear forms with (real or complex) algebraic coefficients. Thus the subspace theorem asserts that $E_{\del}$ is a finite collection of proper rational subspaces of $\IR^{d}$. We denote by $k_{\del}$ the number of subspaces forming $E_{\del}$. Now we want to show that for all sufficiently large $n$ there exist an integer $m\leq n$ such that $A^{m}\bz_{n}$ does not lie in any element of $E_{\del}$. To this end we assume to the contrary that all $A^{m}\bz_{n}$ lie in the subspaces forming $E_{\del}$ and we divide the sequence of natural numbers $1,...,n$ into $k_{\del}$ classes in such a way that two numbers $m_{1}$ and $m_{2}$ are in the same class if $A^{m_{1}}\bz_{n}$ and $A^{m_{2}}\bz_{n}$ lie in the same element of $E_{\del}$. Now let $n_*(k_{\del},d)$ be the number given in the van der Waerden theorem and let $n\geq n_*$. Then there exists an arithmetic progression $m_{1},...,m_{d}$ in one of these subsequences. By Lemma \[irred\] d) the set of vectors $\{A^{m_{1}}\bz_{n},A^{m_{2}}\bz_{n},...,A^{m_{d}}\bz_{n}\}$ forms a basis of the whole space $\IR^{d}$, which contradicts the fact that they lie in one fixed rational proper subspace. Hence for any $\del>0$ and $n\geq n_*$ there exists $m_*\leq n$ such that $A^{m_*}{\bz}_{n}$ does not lie in any element of $E_{\del}$. Now, introducing the notation \[minrel1\] \_[n]{}=A\^[m\_\*]{}\_[n]{} one concludes from (\[eqref3\]) that for any $\del >0$ and all $n \geq n_*$ the following equality and estimate hold \[minrel2\] \_[j=1]{}\^[d]{}\_[j]{}\^[2n]{}|L\_[j]{}\_[n]{}|\^[2]{}&=& \_[j=1]{}\^[d]{}\_[j]{}\^[2n]{}|\_j|\^[-2m\_[\*]{}]{}|L\_[j]{}\_[n]{}|\^[2]{}\ \[constr1\] \_[j=1]{}\^[d]{}|L\_[j]{}\_[n]{}| && . Inequality (\[constr1\]) may be rewritten as \[f\] \_[j=1]{}\^[d]{} |L\_[j]{}\_[n]{}|= with some $f:\IR^+\to\IR^+$ such that $f(r)\leq r, \forall r>0$. Using (\[minrel1\]) and (\[ubound1\]) we obtain the existence of a constant $\lam>1$ such that \[minbound\] f(|\_[n]{}|) |\_[n]{}|= |A\^[m\_\*]{}\_[n]{}| \_[max]{}\^[m\_\*]{}|\_[n]{}| \_[max]{}\^[n]{}\_[j=1]{}\^[d]{}\_[j]{}\^[n]{}|L\_[j]{}\_[n]{}| dD(\_[max]{}\_[geo]{})\^[n]{} \^[n]{}. Note that $\prod_{j}\lam_j=1$. So, by (\[f\]) the quantities $B_{j,n}=\left(|\lam_{j}|^{-m_*}f(|\hat{\bz}_{n}|)^{\del/d}|L_{j}\hat{\bz}_{n}|\right)^{2\alpha}, j=1,...,d$ satisfy the constraint \[newc\] \_[j=1]{}\^[d]{} B\_[j,n]{}=1,n&gt;n\_\*. Thus applying (\[minbound\]) and the Lagrange multipliers minimization with the constraint (\[newc\]) one gets \_[j=1]{}\^[d]{}\_[j]{}\^[2n]{}|\_[j]{}|\^[-2m\_[\*]{}]{}|L\_[j]{}\_[n]{}|\^[2]{} = f(|\_[n]{}|)\^[-2/d]{}\_[j=1]{}\^[d]{}\_[j]{}\^[2n]{}B\_[j,n]{} \^[-2 n/d]{}\_[geo]{}\^[2n]{}=: \_[geo]{}\^[2n(1-)]{}. This and equality (\[minrel2\]) yields the following lower bound for (\[artmin2\]) \[dlbound\] \_[0 =\^[d]{}]{} \_[j=1]{}\^[d]{}\_[j]{}\^[2n]{}|L\_[j]{}|\^[2]{} \_[geo]{}\^[2 n(1-)]{}. ### Non-diagonalizable case We move on to the general case where $A$ is not irreducible (but, as assumed at the beginning of the proof, indecomposable over $\IQ$). We denote by $B$ the invariant irreducible sub-block of $A$ given by its block upper triangular decomposition (\[but\]) and by $S$ the rational invariant subspace associated with this block. We note that $B$ as an irreducible matrix is diagonalizable. **Upper bound.** Note that \_[0 =\^[d]{}]{} \_[l=1]{}\^[n]{}|A\^[l]{}|\^[2]{} \_[0 =S \^[d]{}]{} \_[l=1]{}\^[n]{}|B\^[l]{}|\^[2]{}. The corresponding upper bound (\[ubound\]) for $B$ is then also an upper bound for the whole matrix $A$. We note that geometric average of $\hat{\lam}_j$ over $S$ is equal to the geometric average of all $\hat{\lam}_j$ associated with matrix $A$ (i.e. over the whole space $\IR^{d}$). **Lower bound.** According to our assumption $A$ is indecomposable and thus the characteristic polynomial of $A$ is of the form $p^m$ for some irreducible $p$. All Jordan blocks of $A$ have the same size $m$ and different Jordan blocks correspond to distinct eigenvalues. Denote by $b$ the number of the Jordan blocks in $A$ and by $\lam_{j}$, where $j=1,..,l$ all these distinct eigenvalues . Since each $\lam_{j}$ has algebraic multiplicity $m$, we get $d=mb.$ Let $\{{{\bf v}}_{j,h}\}_{j=1,...,b;h=0,...,m-1}$ be a basis (of $\IC^{d}$) in which $A$ admits the Jordan canonical form. As usually $L_{j,h}$ will denote the corresponding linear forms. Each ${{\bf v}}_{j,h}$ can be regarded as a generalized eigenvector of $A$ associated with an eigenvalue $\lam_{j}$. We assume that these generalized eigenvectors are ordered according to their degree i.e. ${{\bf v}}_{j,h}$ satisfies the equation $(A-\lam_{j}I)^{1+h}{{\bf v}}_{j,h}=0$. Reordering the eigenvalues, if necessary, we can also assume that $\lam_{1}$ has the largest modulus among all eigenvalues of $A$ and hence $\hat{\lam}_{1}=|\lam_{1}|$. Let ${\bz}_{n}$ be the sequence of minimizers solving (\[artmin0\]). We first note that for each $n$ there exists $0 \leq h \leq m-1$ such that $L_{1,h}{\bz}_{n} \not =0$. Indeed, otherwise for all $h=0,...,m-1$, $L_{1,h}{\bz}_{n} =0$ and consequently for any $n$ and $h$ $L_{1,h}A^{n}{\bz}_{n}=0$. The latter implies that the set of consecutive iterations $\{{\bz}_{n},A^{1}{\bz}_{n},A^{2}{\bz}_{n},...\}$ spans a proper rational $A$-invariant subspace of $\IR^{d}$ which does not have any intersection with the subspace spanned by the generalized eigenvectors of $A$ associated with eigenvalue $\lam_{1}$. This clearly contradicts the irreducibility of $p$. Now, for given $n$ we denote by $h(n)$ the biggest index $h$ for which the condition $L_{1,h}{\bz}_{n} \not =0$ holds. We have the following estimate \[est1\] &&\_[1]{}\^[2n]{}|L\_[1,h(n)]{}\_[n]{}|\^[2]{} (\_[j=1]{}\^[b]{}\_[h=0]{}\^[m-1]{} |\_[i=0]{}\^[m-1-h]{}\_[j]{}\^[n-i]{}L\_[j,h+i]{}\_[n]{}|\^[2]{})\^\ &&C\_[1]{}|A\^[n]{}\_[n]{}|\^[2]{} C\_[1]{}\_[l=1]{}\^[n]{}|A\^[l]{}\_[n]{}|\^[2]{} C\_[2]{}n\_[geo]{}\^[2n]{}, where the last inequality follows from previously established upper bound. &gt;From the Diophantine approximation and the assumption that $|L_{1,h(n)}{\bz}_{n}|\not=0$, there exists $\beta>0$ such that (see [@Schmidt] p. 164) \[est2\] |L\_[1,h(n)]{}\_[n]{}| . Thus combining (\[est1\]) with (\[est2\]) one gets \_[1]{}\^[2n]{}|\_[n]{}|\^[-2]{} \_[1]{}\^[2n]{}|L\_[1,h(n)]{}\_[n]{}|\^[2]{} C\_[2]{}n\_[geo]{}\^[2n]{}. After rearrangements one obtains the following lower bound estimate for (\[artmin0\]) \[nlbound\] \^[2n]{} C|\_[n]{}|\^[2]{} \_[l=1]{}\^[n]{}|A\^[l]{}\_[n]{}|\^[2]{}= \_[0 =\^[d]{}]{} \_[l=1]{}\^[n]{}|A\^[l]{}|\^[2]{}, where =()\^[1/]{}. We note that ergodicity of $A$ implies $\hat{\lam}_{1}>\hat{\lam}_{geo}>1$ (see (\[hatlamgeo\]), (\[entropy\]) and Proposition \[zeroentropy\]) which ensures non-triviality of this lower bound. Now in order to finish the proof is suffices to combine the estimates (\[ubound\]), (\[dlbound\]) and (\[nlbound\]), and note that \_[geo]{}\^[2n]{}=e\^[2n]{}=e\^[2(A)n]{} which yields (\[ulestimateA\]). $\qquad \blacksquare$ Proofs of Theorem \[thm2\] ii), iii) and Theorem \[thm3\] {#sAT} --------------------------------------------------------- In this section we apply Theorem \[thmart\] to prove main theorems of Section \[sD\]. In order to determine the dissipation time of $T_{\vep,\alpha}$ one has to determine the asymptotics of $\|T_{\vep,\alpha}^{n}\|$ when $n$ goes to infinity. According to formulas (\[Tnorm3\]) and (\[artminA\]) this problem reduces to problem (\[artmin0\]) solved in previous sections. Thus in view of Theorem \[thmart\]) there exist constants $C_{1}$ and $C_{2}$ such that for any $\del,\del'>0$ and sufficiently large $n$ \[ulestimate1\] C\_[1]{} e\^[(1-)2(A) n]{} \_[0 =\^[d]{}]{} \_[l=1]{}\^[n]{}|A\^[l]{}|\^[2]{} C\_[2]{}ne\^[2(A)n]{}C\_[2]{}e\^[(1+’)2(A)n]{} Using formula (\[Tnorm3\]) e\^[-C\_[2]{}e\^[(1+’)2(A)n]{}]{} T\_[,]{}\^[n]{} e\^[-C\_[1]{}e\^[(1-)2(A)n]{}]{}. Now when $n=n_{diss}$, we have C\_[1]{} e\^[(1-)2(A)n\_[diss]{}]{} C\_[2]{} e\^[(1+’)2(A)n\_[diss]{}]{} and ((1/) -C\_[2]{}) n\_[diss]{} ((1/)-C\_[1]{}), which proves part ii) of Theorem \[thm2\] i.e. the logarithmic growth of dissipation time as a function of $1/\vep$. Moreover, using the definition of dissipation rate constant R\_[diss]{}=\_[0]{} we obtain R\_[diss]{} . Finally letting $\del \rightarrow 0$ and $\del' \rightarrow 0$ we arrive at the following results: - The general case - Theorem \[thm2\] iii) R\_[diss]{} - The diagonalizable case - Theorem \[thm3\] R\_[diss]{}=. This completes the proof. $\qquad \blacksquare$ Degenerate Noise {#degnoise} ================ In this section we compute the dissipation time for non-strictly contracting generalizations of $\alpha$-stable transition operators. Instead of considering standard $\alpha$-stable kernels of the form (\[kernel\]) one can allow for some degree of degeneracy of noise in chosen directions by introducing the following family of noise kernels \[noker\] g\_[,,B]{}():=\_[\^[d]{}]{}e\^[-|B|\^[2]{}]{}\_(), Where $B$ denotes any $d\times d$ matrix with $\det B=0$. We denote by $G_{\vep,\alpha,B}$ the noise operator associated with $g_{\vep,\alpha,B}$. The degeneracy of $B$ immediately implies that $\|G_{\vep,\alpha,B}\|=1$ and hence the general considerations of sections 1 and 2 do not apply here. The answer to the question whether or not the dissipation time is finite depends on the choice of matrix $B$. For simplicity we concentrate on the case when $B$ is diagonalizable. We call the eigenvector of $B$ nondegenerate if it corresponds to nonzero eigenvalue. Let $F$ be any toral automorphism and $T_{\vep,\alpha,B}=G_{\vep,\alpha,B}U_{F}$. Assume that $B$ is diagonalizable. Then i\) If all nondegenerate eigenvectors of $B^{*}$ lie in one proper invariant subspace of $F$ then dissipation does not take place i.e. $n_{diss}=\infty$. ii\) Otherwise the following statements hold. a\) $T_{\vep,\alpha}$ has simple dissipation time iff $F$ is not ergodic. b\) $T_{\vep,\alpha}$ has logarithmic dissipation time iff $F$ is ergodic. c\) If $T_{\vep,\alpha,B}$ has logarithmic dissipation time then the dissipation rate constant satisfies the following bounds R\_[diss]{} , with some constant $\tilde{h}(F)\leq\hat{h}(F)$. The equality is achieved for all diagonalizable automorphisms $F$. **Proof.** We continue to use the convention $A=F^{\dagger}$. The general formula derived previously for $\|T_{\vep,\alpha}^{n}\|$ (see (\[Tnorm3\])), will now take the form \[Tnorm3B\] T\_[,,B]{}\^[n]{} &=& \_[0 =\^[d]{}]{} e\^[-\_[l=1]{}\^[n]{}|BA\^[l]{}|\^[2]{}]{} = e\^[- \_[l=1]{}\^[n]{}|BA\^[l]{}|\^[2]{}]{}. Thus we need to estimate \[Binf\] \_[0 =\^[d]{}]{} \_[l=1]{}\^[n]{}|BA\^[l]{}|\^[2]{}. To this end we denote by $\mu_{j}$ ($j=1,...,d$) the eigenvalues of $B$ and we construct a basis (of $\IC^{d}$) $\{{{\bf v}}_{j}\}_{j=1,...,d}$ composed of normalized eigenvectors corresponding to eigenvalues $\mu_{j}$. We denote by ${P_{j}}_{j=1,...,d}$ the set of eigen-projections on ${{{\bf v}}_{j}}$, and by ${L_{j}}$ the set of corresponding linear forms, given by the eigenvectors ${{\bf u}}_j$ of $B^{\dagger}$, which are of course co-orthogonal to ${{{\bf v}}_j}$, i.e. $\la {{\bf u}}_i,{{\bf v}}_j\ra=0$ for $i\not=j$. We have =\_[j=1]{}\^[d]{}P\_[j]{}=\_[j=1]{}\^[d]{}(L\_[j]{})[[**v**]{}]{}\_[j]{} =\_[j=1]{}\^d,[[**u**]{}]{}\_j[[**v**]{}]{}\_j, \^d. In subsequent computations the symbols $C_{1}$, $C_{2}$ denote some absolute constants values of which are subject to change during calculations. We consider two cases. i\) All nondegenerate eigenvectors of $B^{\dagger}$ lie in one proper subspace of $F$. We have the following estimates |BA\^[l]{}|\^[2]{} C\_[1]{} \_[j=1]{}\^[d]{}|P\_[j]{}BA\^[l]{}|\^[2]{} = C\_[1]{} \_[j=1]{}\^[d]{}|\_[j]{}|\^[2]{}|P\_[j]{}A\^[l]{}|\^[2]{} = C\_[1]{} \_[j=1]{}\^[d]{}|\_[j]{}|\^[2]{}|A\^[l]{},[[**u**]{}]{}\_j|\^[2]{} = C\_[1]{} \_[j=1]{}\^[d]{}|\_[j]{}|\^[2]{}|,F\^[l]{}[[**u**]{}]{}\_j|\^[2]{} and |BA\^[l]{}|\^[2]{} C\_[2]{} \_[j=1]{}\^[d]{}|P\_[j]{}BA\^[l]{}|\^[2]{}= C\_[2]{} \_[j=1]{}\^[d]{}|\_[j]{}|\^[2]{}|P\_[j]{}A\^[l]{}|\^[2]{}= C\_[2]{} \_[j=1]{}\^[d]{}|\_[j]{}|\^[2]{}|A\^[l]{},[[**u**]{}]{}\_j|\^[2]{}= C\_[2]{} \_[j=1]{}\^[d]{}|\_[j]{}|\^[2]{}|,F\^[l]{}[[**u**]{}]{}\_j|\^[2]{} Since at least one of $\mu_{j}$ is zero and all nondegenerate vectors ${{\bf u}}_j$ lie in a proper invariant subspace of $F$, one easily sees that for each fixed $n$ \_[0 =\^[d]{}]{}\_[l=1]{}\^[n]{}|BA\^[l]{}|\^[2]{}= \_[0 =\^[d]{}]{}\_[l=1]{}\^[n]{}\_[j=1]{}\^[d]{}|\_[j]{}|\^[2]{}|,F\^[l]{}[[**u**]{}]{}\_j|\^[2]{}=0. ii) In this case we have the following upper bound \[ubinf\] \_[0 =\^[d]{}]{} \_[l=1]{}\^[n]{}|BA\^[l]{}|\^[2]{} B\^[2]{}\_[0 =\^[d]{}]{} \_[l=1]{}\^[n]{}|A\^[l]{}|\^[2]{} = B\^[2]{}\_[0 =\^[d]{}]{} \_[l=1]{}\^[n]{}|A\^[l]{}|\^[2]{}. In order to provide an appropriate lower bound we note that the set of vectors $\{F^{h}{{\bf u}}_{j}\}$, where $1\leq h \leq d$ and $j$ runs through the indices of all nondegenerate eigenvectors of $B$, spans the whole space (otherwise all nondegenerate ${{\bf u}}_{j}$ would lie in one proper invariant subspace of $F$). We denote by $\{F^{h_{i}}{{\bf u}}_{j_{i}}\}$ ($1\leq i \leq d$) a basis extracted from the above set. We can define now a new norm $|\cdot|_{{{\bf u}}}$ on $\IR^{d}$ by ||\^[2]{}\_[[[**u**]{}]{}]{}=\_[i=1]{}\^[d]{}|,F\^[h\_[i]{}]{}[[**u**]{}]{}\_[j\_[i]{}]{} |\^[2]{} and compute \_[l=1]{}\^[dn]{}|BA\^[l]{}|\^[2]{} &=&\_[l=0]{}\^[n-1]{}\_[h=1]{}\^[d]{}|BA\^[dl+h]{}|\^[2]{} \_[l=0]{}\^[n-1]{}\_[h=1]{}\^[d]{}C\_[1]{}\_[j=1]{}\^[d]{}|P\_[j]{}BA\^[dl+h]{}|\^[2]{}\ &=&C\_[1]{}\_[l=0]{}\^[n-1]{}\_[h=1]{}\^[d]{}\_[j=1]{}\^[d]{}|\_[j]{}|\^[2]{}|P\_[j]{}A\^[dl+h]{}|\^[2]{} C\_[1]{} \_[l=0]{}\^[n-1]{}\_[i=1]{}\^[d]{}|A\^[dl+h\_[i]{}]{},[[**u**]{}]{}\_[j\_[i]{}]{} |\^[2]{}\ &=&C\_[1]{}\_[l=0]{}\^[n-1]{}\_[i=1]{}\^[d]{}|A\^[dl]{},F\^[h\_[i]{}]{}[[**u**]{}]{}\_[j\_[i]{}]{} |\^[2]{} =C\_[1]{}\_[l=0]{}\^[n-1]{}|A\^[dl]{}|\^[2]{}\_[[[**u**]{}]{}]{} Using the equivalence between norms $|\cdot|$ and $|\cdot|_{{{\bf u}}}$ and combining (\[ubinf\]) with the above estimate we get C\_[1]{}\_[0 =\^[d]{}]{} \_[l=0]{}\^[n-1]{}|A\^[dl]{}|\^[2]{}\_[0 =\^[d]{}]{}\_[l=1]{}\^[dn]{}|BA\^[l]{}|\^[2]{}B\^[2]{}\_[0 =\^[d]{}]{} \_[l=1]{}\^[dn]{}|A\^[l]{}|\^[2]{}. This together with the obvious fact that $\hat{h}(A^{d})=d\hat{h}(A)$ and the general estimate (\[ulestimateA\]) reduces the proof back to the nondegenerate case considered in the previous section. $\qquad \blacksquare$ Time of decay of $\vep$-coarse-grained states {#coarse} ============================================= The uncertainties in the initial preparation and the final measurement of the noiseless system give rise to non-cumulative random perturbations to the system. Alternatively, one can coarse-grain the initial and final states of the noiseless system by convoluting with the $\veps$-noise kernel. That is, instead of the original operator $T_{\vep,\alpha}$, we consider the operator $\hat{T}_{\vep,\alpha}$ defined as \_[,]{}\^[n]{}=G\_[,]{}U\_[F]{}\^[n]{}G\_[,]{}=G\_[,]{}U\_[F\^[n]{}]{}G\_[,]{}. and compute the number of iterations required to have the $L^2$-norm of the final state being $e^{-1}$ times that of the initial state. We will show that for ergodic toral automorphisms the required number of iterations is essentially the same asymptotically as the dissipation time computed in the previous sections. One can represent the action of $U_{F}$ or more generally $U_{F}^{n}$ in the Fourier series U\_[F]{}\^[n]{}\_=\_[0 = ’ \^[d]{}]{}u\_[,’]{}\^[(n)]{}\_[’]{}, where $u_{\bk,\bk'}^{(1)}$ coincides with $u_{\bk,\bk'}$ defined previously (cf. (\[uk\])) and $$u^{(n)}_{\bk,\bk'}=\sum_{0\neq \bk_1,...,\bk_{n-1}\in \IZ^d} u_{\bk,\bk_1}u_{\bk_1,\bk_2}...u_{\bk_{n-1},\bk'}$$ which satisfies \[ul3\] \_[0 = ’ \^[d]{}]{}|u\^[(n)]{}\_[,’]{}|\^[2]{}=1, n,. Then \_[,]{}\^[n]{}\_[\_[0]{}]{}&=& G\_[,]{}U\_F\^[n]{}G\_[,]{}\_[\_[0]{}]{}= G\_[,]{}U\_F\^[n]{}e\^[-|\_[0]{}|\^[2]{}]{} \_[\_[0]{}]{}= e\^[-|\_[0]{}|\^[2]{}]{}G\_[,]{}\_[0 = \_[n]{} \^[d]{}]{} u\^[(n)]{}\_[\_[0]{},\_[n]{}]{}\_[\_[n]{}]{}\ &=&e\^[-(|\_[0]{}|\^[2]{}+|\_[n]{}|\^[2]{})]{} \_[0 = \_[n]{} \^[d]{}]{}u\^[(n)]{}\_[\_[0]{},\_[n]{}]{}\_[\_[n]{}]{}. Now we define S\_n(\_[n]{})={\_[0]{} \^[d]{}\\{0} : u\^[(n)]{}\_[\_[0]{},\_[n]{}]{}= 0}. Similar computations to these performed in Section \[sD\] give the following general upper bound for $\|\hat{T}_{\vep,\alpha}^{n}\|$ \[uppernorm1\] \_[,]{}\^[n]{}f\^[2]{} \_[0 = \_[n]{} \^[d]{}]{} \_[\_[0]{} S\_n(\_[n]{})]{} |(\_[0]{})|\^[2]{} \_[\_[0]{} S\_n(\_[n]{})]{} |u\_[\_[0]{},\_[n]{}]{}\^[(n)]{}|\^[2]{}. For a toral automorphism one easily sees that u\_[\_[0]{},\_[n]{}]{}\^[(n)]{}= e\^[-(|\_[0]{}|\^[2]{}+|A\^[n]{}\_[0]{}|\^[2]{})]{}\_[\_[n]{},A\^[n]{}\_[0]{}]{} and hence \_[,]{}\^[n]{} = e\^[-\_[0 =\^[d]{}]{}(|\_[0]{}|\^[2]{}+|A\^[l]{}\_[0]{}|\^[2]{})]{}. The arithmetic minimization problem (\[artminA\]) corresponding to the dissipation time of $\hat{T}_{\vep,\alpha}$ now becomes \[artminA1\] \_[0 =\^[d]{}]{}(||\^[2]{}+|A\^[n]{}|\^[2]{}). The key observation is that, by the same arguments as before, similar estimates to these given in (\[ulestimate\]) hold \[ulestimate2\] C\_[1]{}(\_[j=1]{}\^[d]{} \_[j]{}\^[2 n]{}|L\_[j]{}|\^[2]{})\^ ||\^[2]{}+| A\^[n]{}|\^[2]{} C\_[2]{}\_[j=1]{}\^[d]{} \_[j]{}\^[2n]{}|L\_[j]{}|\^[2]{}. The remaining computations are the same verbatim so the dissipation time of $T_{\vep,\alpha}$ and $\hat{T}_{\vep,\alpha}$ are equal asymptotically. Time Scales in Kinematic Dynamo {#dynamo} =============================== In this section we briefly discuss the connection between the dissipation time and some characteristic time scales associated with kinematic dynamo, which concerns the generation of electromagnetic fields by mechanical motion. For a general setup and discussion we refer the reader to [@CG] and [@KY] and references therein. Here we restrict ourselves only to necessary definitions. Let $\bB\in L^{2}_{0}(\IT^{d},\IR^{d})$ denote periodic, zero mean and divergence free magnetic field and let $F$ be the time-1 map associated with the fluid velocity. We define the push-forward map F\_[\*]{}()=dF(F\^[-1]{}())(F\^[-1]{}()). The noisy push-forward map $P_{\vep,\alpha}$ on $L_{0}^2(\IT^{d},\IR^{d})$ is then given by \[Tdyn\] P\_[,]{}:=G\_[,]{}F\_[\*]{}, where the convolution (the action of $G_{\vep,\alpha}$) is applied component-wise. It is said that the kinematic dynamo action (positive dynamo effect) occurs if the dynamo growth rate is positive i.e. R\_[dyn]{}=\_[n]{}P\_[,]{}\^[n]{}&gt;0. Moreover if \_[0]{}\_[n]{}P\_[,]{}\^[n]{}&gt;0, then the dynamo action is said to be fast; otherwise it is slow. The anti-dynamo action takes place if \_[n]{}P\_[,]{}\^[n]{}e P\_[,]{}\^[n-1]{} P\_[,]{}\^[n+1]{} e}. The threshold time $n_{th}(\vep)$ is of order $O(1)$ as $\vep\rightarrow 0$ for all fast kinematic dynamo systems. In the case of anti-dynamo action, $n_{th}(\vep)$ captures the longest time scale on which the generation of the magnetic field still takes place. Finally $n_{th}(\vep)$ is not defined for systems with do not exhibit any growth of magnetic field throughout the evolution. In the case of anti-dynamo we consider the time scale on which the generation of the magnetic field achieves its maximal value n\_[p]{}={n: P\_[,]{}\^[n]{}=\_[m]{}P\_[,]{}\^[m]{}}. which is called the [*peak time*]{} of the anti-dynamo action. Our next theorem establishes the relation between $n_{p}$, $n_{th}$ and $n_{diss}$ for toral automorphisms. Thus $dF=F$ and P\_[,]{}=g\_[,]{}\*F(F\^[-1]{}). Let $F$ be any toral automorphism. Then i\) If $F$ is nonergodic and has positive entropy then for all $0<\vep<R_{diss}\ln\rho_{F}$ the fast dynamo action takes place with dynamo growth rate satisfying R\_[dyn]{}= \_[F]{}-R\_[diss]{}\^[-1]{}\_[F]{}&gt;0, where $\rho_{F}$ denotes the spectral radius of $F$. The threshold time $n_{th}$ is of order $O(1)$ and if $F$ is diagonalizable then $n_{th}\approx [R_{dyn}^{-1}]$. ii\) If $F$ is nonergodic and has zero entropy then anti-dynamo action occurs and for nondiagonalizable $F$, n\_[p]{}&\~&\ &\~& n\_[diss]{}\ && R\_[diss]{}. Moreover there exists a constant $0< \gamma \leq d$ such that $\|P_{\vep,\alpha}^{n_{p}}\|\sim (1/\vep)^{\gamma}$. If $F$ is diagonalizable then $\|P_{\vep,\alpha}^{n}\|$ is strictly decreasing (in $n$) and, hence, $n_{p}=0$ and $n_{th}$ is not defined. iii\) If $F$ is ergodic then anti-dynamo action occurs and n\_[p]{}n\_[diss]{}. In particular if $F$ is diagonalizable then n\_[p]{} && n\_[th]{}-R\_[diss]{}(n\_[th]{})\ && R\_[diss]{}(1/)\ &=& (1/) and \[anti\] P\_[,]{}\^[n\_[p]{}]{}\~(1/)\^. We see that even in the case of anti-dynamo action the magnetic field can still grow to relatively large magnitude when the noise is small (power-law in $1/\ep$). **Proof.** Representing the initial magnetic field $\bB=(\bb_{1},...,\bb_{d})$ in Fourier basis =\_[0 = \^[d]{}]{}()e\_ one obtains P\_[,]{}=\_[0 = \^[d]{}]{}F() e\^[-|A|\^[2]{}]{}e\_[A]{}, where we set $A=(F^{-1})^{\dagger}$. After $n$ iterations P\_[,]{}\^[n]{}=\_[0 = \^[d]{}]{}F\^[n]{}() e\^[-\_[l=1]{}\^[n]{}|A\^[l]{}|\^[2]{}]{} e\_[A\^[n]{}]{}. Thus &&P\_[,]{}\^[n]{}\^[2]{} \_[0 = \^[d]{}]{}|F\^[n]{}()|\^[2]{} e\^[-2\_[l=1]{}\^[n]{}|A\^[l]{}|\^[2]{}]{} \_[0 =\^[d]{}]{}e\^[-2\_[l=1]{}\^[n]{}|A\^[l]{}|\^[2]{}]{} \_[0 = \^[d]{}]{}|F\^[n]{}()|\^[2]{}\ &=&e\^[-2\_[0 =\^[d]{}]{}\_[l=1]{}\^[n]{}|A\^[l]{}|\^[2]{}]{} |F\^[n]{}|\^[2]{}=e\^[-2\_[l=1]{}\^[n]{}|A\^[l]{}\_[n]{}|\^[2]{}]{} |F\^[n]{}|\^[2]{} e\^[-2\_[l=1]{}\^[n]{}|A\^[l]{}\_[n]{}|\^[2]{}]{} F\^[n]{}\^[2]{}||\^[2]{}, where $\bk_{n}$ denotes a solution of the minimization problem \_[0 =\^[d]{}]{}\_[l=1]{}\^[n]{}|A\^[l]{}|\^[2]{}. The above calculation provides the following upper bound P\_[,]{}\^[n]{} && e\^[-\_[l=1]{}\^[n]{}|A\^[l]{}\_[n]{}|\^[2]{}]{} F\^[n]{}. On the other hand let ${{\bf v}}_{n}$ denote a unit vector satisfying $\|F^{n}\|=|F^{n}{{\bf v}}|$. One immediately sees that the above upper bound for $\|P_{\vep,\alpha}^{n}\|$ is achieved for magnetic field of the form $\bB={{\bf v}}_{n}e_{\bk_{n}}$. Thus P\_[,]{}\^[n]{}= e\^[-\_[l=1]{}\^[n]{}|A\^[l]{}\_[n]{}|\^[2]{}]{} F\^[n]{}. Now we consider the cases mentioned in the statement of the theorem i\) Nonergodic, nonzero entropy case. For any nonergodic map we have \_[l=1]{}\^[n]{}|A\^[l]{}\_[n]{}|\^[2]{} R\_[diss]{}\^[-1]{} n. This implies the following asymptotics \[na\] P\_[,]{}\^[n]{}e\^[-R\_[diss]{}\^[-1]{} n]{} F\^[n]{} e\^[(-R\_[diss]{}\^[-1]{} + \_[F]{})n + c\_[1]{}n +c\_[2]{}]{}, where $c_{1},c_{2}\geq 0$ are constants (both equal $0$ iff $F$ is diagonalizable). Thus for $\vep<R_{diss}{\ln\rho_{F}}$ we have R\_[dyn]{}= \_[F]{}-R\_[diss]{}\^[-1]{} \_[F]{}&gt;0. The threshold time is clearly of order $O(1)$ and in diagonalizable case can be written as n\_[th]{} . ii\) Nonergodic, zero entropy case. In this case $\ln\rho_{F}=0$. Thus if $F$ is nondiagonalizable then (\[na\]) reads P\_[,]{}\^[n]{}e\^[-R\_[diss]{}\^[-1]{} n]{} F\^[n]{} e\^[-R\_[diss]{}\^[-1]{}n + c\_[1]{}n +c\_[2]{}]{}, with $0<c_{1}\leq d$. This immediately yields n\_[p]{}R\_[diss]{}, \~. And moreover $\|P_{\vep,\alpha}^{n_{p}}\|\sim (1/\vep)^{c_{1}}$. If $F$ is diagonalizable then $\|F^{n}\|=1$ and in this case $\|P_{\vep,\alpha}^{n}\|\approx e^{-\vep R_{diss}^{-1} n}$ which implies $n_{p}=0$. iii\) If $F$ is diagonalizable, then form (\[ulestimateA\]) we know that for any $0<\del<1$ and sufficiently large $n$ \[est\] \_[-]{}\^[n]{}=e\^[(1-)2(A) n]{} \_[0 =\^[d]{}]{} \_[l=1]{}\^[n]{}|A\^[l]{}|\^[2]{} e\^[(1+)2(A)n]{}=\_[+]{}\^[n]{}. Thus for large $n$ we have \_[n]{} e\^[-\_[+]{}\^[n]{}]{}\_[F]{}\^[n]{} \_[n]{}P\_[,]{}\^[n]{} \_[n]{} e\^[-\_[-]{}\^[n]{}]{}\_[F]{}\^[n]{}. We obtain the following constraints for $n_{p}$ () +() n\_[p]{}() +(). This gives \_[0]{} . Now since $\lam_{\pm \del} \rightarrow e^{2\alpha\hat{h}(F)}$ for $\del\rightarrow 0$ the above estimation yields the following asymptotics n\_[p]{} (1/)n\_[diss]{} ,n\_[th]{}-R\_[diss]{}(n\_[th]{})n\_[diss]{}. Similar asymptotic estimates (except for the constant) hold for nondiagonalizable $F$. $\qquad \blacksquare$ **Appendices** Affine transformations {#aAT} ====================== In this appendix we present a slight generalization of the results obtained in the paper. We consider here general affine transformations of the torus. The term [*affine transformations*]{} will be used here to refer to homeomorphisms of the torus with zero periodic but not necessary zero constant part (cf. Section \[sDTA\]) i.e. transformations of the form $\tilde{F}=F+{{\bf c}}$, where $F$ is a toral automorphism and ${{\bf c}}$ is a constant shift vector. We begin with a short discussion of the ergodicity of affine transforms. The relation between ergodicity of a given affine transform $\tilde{F}$ and associated with it toral automorphism $F$ is summarized in the following proposition (for the proof we refer to appendix B) \[affinerg\] Let $F$ be any toral automorphism. Then i\) If $F$ is ergodic then $\tilde{F}$ is also ergodic. ii\) If $F$ is not ergodic then $\tilde{F}$ is ergodic iff $1$ is the only root of unity in the spectrum of $F$ and ${{\bf c}}\cdot\bk\not\in\IZ^{d}$ for any integer eigenvector $\bk$ of $F^{\dagger}$. [**Proof.**]{} i) Assume $F$ is ergodic and for some ${{\bf c}}$, $\tilde{F}=F+{{\bf c}}$ is not ergodic. Then there exists non-constant $f\in L_{0}^{2}(\IT^{d})$ satisfying $f=f\circ \tilde{F}$ or in the Fourier representation \[inv\] \_[\^[d]{}]{}()\_=\_[\^[d]{}]{} e\^[2 i A\^[-1]{}[[**c**]{}]{}]{}(A\^[-1]{})\_ where $A=F^{\dagger}$. Comparing the absolute values of the coefficients we get \[inv1\] |()|=|(A\^[-n]{})| for any integer $n$ and any $\bk$. However, ergodicity of $F$ implies that $A^{-n}\bk\not=\bk$ for all $\bk\not=0$, which contradicts our assumption that $f\in L_{0}^{2}(\IT^{d})$. ii\) We will use the following fact, which can be proved by simple application of rational canonical decomposition. For any $A\in SL(d,\IZ)$ the following conditions are equivalent a\) $A$ possesses in its spectrum a root of unity not equal to one. b\) There exists nonzero $\bk \in \IZ^{d}$ and a positive integer $n$ such that $\bk+A\bk+...+A^{n-1}\bk=0$. Now assume that $1$ is the only root of unity in spectrum of $F$ (and hence of $A$) and ${{\bf c}}\cdot\bk\not\in\IZ^{d}$ for any integer eigenvector $\bk$ of $A$, and that both $F$ and $\tilde{F}$ are not ergodic. The latter assumption implies the existence of a non-constant $f\in L_{0}^{2}(\IT^{d})$ satisfying equations (\[inv\]) and (\[inv1\]). Relation (\[inv1\]) clearly implies that if $\hat{f}(\bk)\not= 0$ then $A^{n}\bk=\bk$ for some $n$. Moreover, since $1$ is the only root of unity in spectrum of $A$, we have, in view of b) that $A\bk=\bk$. Thus the only possible non-constant invariant functions of $\tilde{F}$ are single Fourier modes $\bfe_{\bk}$ corresponding to integer eigenvectors of $A$. But if such a Fourier mode is invariant under $\tilde{F}$ then directly form (\[inv\]) one concludes that $e^{2 \pi i \bk\cdot {{\bf c}}}=1$ or equivalently $\bk\cdot {{\bf c}}\in\IZ^{d}$, for some integer eigenvector of $A$. To prove the converse we assume that $F$ is not ergodic and consider two cases: Case 1. $A$ possesses in its spectrum a root of unity not equal to one. In this case according to condition b) there exists nonzero $\bk \in \IZ^{d}$ and a positive integer $n$ such that $\bk+A\bk+...+A^{n-1}\bk=0$, which implies in particular that $A^{n}\bk=\bk$ and $A\bk\not=\bk$. Now we define the function f=\_+e\^[2i [[**c**]{}]{}]{}\_[A]{}+...+ e\^[2i (\_[l=0]{}\^[n-2]{}A\^[l]{})[[**c**]{}]{}]{}\_[A\^[n-1]{}]{} which clearly satisfies the condition $f=f\circ \tilde{F}$. This proves that $\tilde{F}$ is not ergodic. Case 2. There exists integer eigenvector of $A$ such that $\bk\cdot {{\bf c}}\in\IZ^{d}$. Then clearly for such $\bk$, $f=e_{\bk}$ is $\tilde{F}$-invariant and hence again $\tilde{F}$ is not ergodic. $\qquad \blacksquare$ We recall that $\mf{c}=(c_{1},..,c_{d})$ generates ergodic shift on the torus iff $1,c_{1},..,c_{d}$ are linearly independent over rationals. Thus as a direct consequence of the above proposition we get \[corerg\] If $F$ is not ergodic and $1$ is the only root of unity in the spectrum of $F$ then $\tilde{F}$ is ergodic for all vectors ${{\bf c}}$ generating ergodic shifts on the torus. Now we are in a position to state and prove the generalization of Theorem \[thm2\] from Section \[sDTA\] to the case of affine transforms (the corresponding generalizations of Theorem \[thm3\] and Corollary \[cat\] are straightforward) \[thm2’\] Let $\tilde{F}$ be any affine transformation on the torus $\IT^{d}$, $F$ associated with $\tilde{F}$ toral automorphism and $T_{\vep,\alpha}=G_{\vep,\alpha}U_{\tilde{F}}$. Then i\) $T_{\vep,\alpha}$ has simple dissipation time iff $F$ is not ergodic. ii\) $T_{\vep,\alpha}$ has logarithmic dissipation time iff $F$ is ergodic. iii\) If $T_{\vep,\alpha}$ has logarithmic dissipation time then the dissipation rate constant satisfies the following constraint R\_[diss]{} , where $\tilde{h}(F)\leq\hat{h}(F)$ is certain positive constant. \[remerg\] The dissipation time of an affine transformation $\tilde{F}$ is determined by ergodic properties of its linear part $F$ and hence not by ergodic properties of $\tilde{F}$ itself. In particular all ergodic affine transformations associated with nonergodic toral automorphisms (cf. Proposition \[affinerg\]) have simple dissipation time. **Proof of Theorem \[thm2’\]** Specializing the general calculations of dissipation time presented in Section \[sDCal\] to the case of affine transformations $\tilde{F}=F+{{\bf c}}$, with nonzero ${{\bf c}}$, one easily finds the following counterparts of formulas (\[linear\]) and (\[linear1\]) u\_[,’]{}&=&e\^[2i [[**c**]{}]{}]{}\_[A,’]{},\ \_[n]{}(\_[0]{},\_[n]{}) &=&e\^[2i (\_[l=0]{}\^[n-1]{}A\^[l]{}) [[**c**]{}]{}]{} e\^[-\_[l=1]{}\^[n]{}|A\^[l]{}|\^[2]{}]{}\_[A\^[n]{}\_[0]{},\_[n]{}]{}. Now, in order to determine the dissipation time of $T_{\vep,\alpha}=G_{\vep,\alpha}U_{\tilde{F}}$ one has to determine the asymptotics of $\|T_{\vep,\alpha}^{n}\|$ as $n$ goes to infinity. According to the above formulas and formulas (\[uppernorm\]) and (\[Tnorm3\]) from Section \[sDCal\] the value of $\|T_{\vep,\alpha}^{n}\|$ does not depend on ${{\bf c}}$, which reduces the proof to the case ${{\bf c}}=0$ considered in the main body of the paper. $\qquad \blacksquare$ Proofs of some elementary facts {#aP} =============================== **Proof of Proposition \[pdiss\]** The proof will be based on the Riesz convexity theorem (see [@Zyg], pp. 93-100) which states that for any operator $T$ defined on $L^p(\IT^d), 1\leq p\leq \infty,$ $\ln{\|T\|_p}$ is a convex function of $p^{-1}$. On the space $L^p(\IT^d)$ we consider the operator $\widetilde{T}:=T_{\vep,\alpha}-\la\cdot\ra$ and we have the relation $\widetilde{T}^n f= T^n_{\vep,\alpha} (f-\la f\ra), \forall f\in L^p(\IT^d), n\geq 1$ because $T_{\vep,\alpha}$ is conservative. Now since $\|f -\la f\ra\|_p\leq 2\|f\|_p$, it follows that \[14\] \^n\_p &&2 T\^n\_[,]{}\_[p,0]{}2\ \[15\] T\^n\_[,]{}\_[p,0]{} &&\^n\_p for $1\leq p\leq \infty, n\geq 1$. The Riesz convexity theorem implies that if $p<q<\infty$ \[16\] +(1- ) while if $1<q<p$ \[17\] () +(1- ). &gt;From (\[16\])-(\[17\]) we have the interpolation relations \[16’\] \^n\_q && \^n\_p\^[p/q]{}\^n\_\^[1-p/q]{}, p&lt;q&lt;\ \[17’\] \^n\_q && \^n\_p\^[(1-q\^[-1]{})/(1-p\^[-1]{})]{}\^n\_1\^[1-(1-q\^[-1]{})/(1-p\^[-1]{})]{},1&lt;q&lt;p which, along with (\[14\])-(\[15\]), imply T\^n\_[,]{}\_[q,0]{}&& 2T\^n\_[,]{}\_[p,0]{}\^[p/q]{},p&lt;q&lt;\ T\^n\_[,]{}\_[q,0]{}&& 2T\^n\_[,]{}\_[p,0]{}\^[(1-q\^[-1]{})/(1-p\^[-1]{})]{}, 1&lt;q&lt;p This proves that the order of divergence of $n_{diss}(p)$ are the same for $1<p<\infty$. Estimates (\[16’\])-(\[17’\]) also show that the order of divergence of $n_{diss}(1)$ and $n_{diss}(\infty)$ is at least as high as $n_{diss}(p), 1<p<\infty$. $\qquad \blacksquare$ **Proof of Lemma \[gub\]** Using the notation introduced in Section \[sDCal\] one has &&T\_[,]{}\^[n]{}\_[\_[0]{}]{}= (G\_[,]{}U)\^[n]{}\_[\_[0]{}]{} =(G\_[,]{}U)\^[n-1]{}\_[0 = \_[1]{} \^[d]{}]{}u\_[\_[0]{},\_[1]{}]{} e\^[-|\_[1]{}|\^[2]{}]{}\_[\_[1]{}]{}\ &=& \_[0 = \_[1]{},...,\_[n]{} \^[d]{}]{}u\_[\_[0]{},\_[1]{}]{} u\_[\_[1]{},\_[2]{}]{}...u\_[\_[n-1]{},\_[n]{}]{} e\^[-\_[l=1]{}\^[n]{}|\_[l]{}|\^[2]{}]{} \_[\_[n]{}]{}=\_[0 = \_[n]{} \^[d]{}]{} \_[n]{}(\_0,\_[n]{})\_[\_[n]{}]{}. We note that for any $n$ and $\bk_{n} \in \IZ^{d}$, the sequence $\ml{U}_n(\bk_{0},\bk_{n})$ (indexed by $\bk_{0}\in \IZ^{d}$) belongs to $l^{2}(\IZ^{d})$. Indeed, using the Cauchy-Schwarz inequality and identity (\[ul2\]) one gets for $n=2$, &&\_[0 = \_[0]{} \^[d]{}]{}|\_2(\_[0]{},\_[2]{})|\^[2]{}= \_[0 = \_[0]{} \^[d]{}]{}|\_[0 = \_[1]{} \^[d]{}]{} u\_[\_[0]{},\_[1]{}]{}u\_[\_[1]{},\_[2]{}]{} e\^[-(|\_[1]{}|\^[2]{}+|\_[2]{}|\^[2]{})]{}|\^[2]{}\ &&\_[0 = \_[0]{} \^[d]{}]{}\_[0 = \_[1]{} \^[d]{}]{} |u\_[\_[0]{},\_[1]{}]{}|\^[2]{}e\^[-|\_[1]{}|\^[2]{}]{} \_[0 = \_[1]{} \^[d]{}]{}|u\_[\_[1]{},\_[2]{}]{}|\^[2]{} e\^[-|\_[1]{}|\^[2]{}]{}e\^[-2|\_[2]{}|\^[2]{}]{}\ &&\_[0 = \_[1]{} \^[d]{}]{}e\^[-|\_[1]{}|\^[2]{}]{} \_[0 = \_[1]{} \^[d]{}]{}e\^[-|\_[1]{}|\^[2]{}]{} e\^[-2|\_[2]{}|\^[2]{}]{} = Ke\^[-2|\_[2]{}|\^[2]{}]{}, where K denotes a constant. Similar estimates hold for $n>2$. Now applying the Cauchy-Schwarz inequality in (\[Tnorm1\]) we get T\_[,]{}\^[n]{}f\^[2]{} && \_[0 = \_[n]{} \^[d]{}]{} \_[\_[0]{} \_[n]{}(\_[n]{})]{} |(\_[0]{})|\^[2]{} \_[\_[0]{} \_[n]{}(\_[n]{})]{} |\_n(\_[0]{},\_[n]{})|\^[2]{}. $\qquad \blacksquare$ **Proof of part i) of Theorem \[thm2\].** In view of theorem \[nonergodic\] it suffices to construct an eigenfunction of $U_{F}$ which belongs to $L^{2}_{0}(\IT^{d}) \cap H^{2\alpha}(\IT^{d})$. Directly from Proposition \[ergodicTA\] one concludes that $F$, and hence also $A$, possesses a root of unity in its spectrum. This means that $A^{m}\bk_{0}=\bk_{0}$, for some $m$ and certain nonzero vector $\bk_{0}$, which can be chosen to be an integer. Now we define f=\_[\_[0]{}]{}+\_[A\_[0]{}]{}+...+\_[A\^[m-1]{}\_[0]{}]{} Obviously $f\in L^{2}_{0}(\IT^{d}) \cap H^{2\alpha}(\IT^{d})$, for any $\alpha$. To complete the proof it suffices to notice that U\_[F]{}f=\_[A\_[0]{}]{}+\_[A\^[2]{}\_[0]{}]{}+...+\_[A\^[m]{}\_[0]{}]{} =\_[\_[0]{}]{}+\_[A\_[0]{}]{}+...+\_[A\^[m-1]{}\_[0]{}]{}=f. $\qquad \blacksquare$ **Proof of Proposition \[irred\].** For the purposes of the proof we use the following abbreviation - $PRS(\IR^{d})$ - proper rational subspace of $\IR^{d}$. - $PIS(A,\IR^{d})$ - proper $A$-invariant subspace of $\IR^{d}$. - $PRIS(A,\IR^{d})$ - proper rational $A$-invariant subspace of $\IR^{d}$. a) $\Rightarrow $ b) : . Suppose there exists $PRIS(A,\IR^{d})$ $S_{1}$. Let $A_{1}$ be a matrix representing invariant rational block associated with $S_{1}$. Then $A_{1}$ is rational matrix and its characteristic polynomial $P_{1}$ belongs to $\IQ[x]$. Let $P$ denote the characteristic polynomial of $A$. Then $P=P_{1}P_{2}$ and since both $P,P_{1}\in \IQ[x]$ then also $P_{2}\in \IQ[x]$, which means $P$ and hence $A$ is not irreducible. b) $\Rightarrow $ c) : . Assume there exists $PRS(\IR^{d})$ $S$ contained in $PIS(A,\IR^{d})$ $V$. Take any rational vector $\bq\in S$ and let $d_{0}=dimV$ then the set $\{\bq,A\bq,..,A^{d_{0}-1}\bq\}$ spans $PRIS(A,\IR^{d})$. c) $\Rightarrow$ d) : . Assume that for given $\bq$ and an arithmetic sequence $n_{1},...,n_{d}$, the set $S=\{A^{n_{1}}\bq,A^{n_{2}}\bq,...,A^{n_{d}}\bq\}$ does not form a basis. Since for some fixed integer $r$, $n_{l}=n_{1}+(l-1)r$, we have $A^{n_{l}}\bq=(A^{r})^{l-1}A^{n_{1}}\bq=(A^{r})^{l-1}\hat{\bq}$, where $\hat{\bq}=A^{n_{1}}\bq$. Now consider the biggest subset $S_{0}=\{\hat{\bq},A^{r}\hat{\bq},(A^{r})^{2}\hat{\bq},...,(A^{r})^{d_{0}-1}\hat{\bq}\}$ such that $d_{0}<d$ and $S_{0}$ is linearly independent. Obviously $S_{0}$ spans a $PRIS(A^{r})$ which is also a $PRIS(A)$. d) $\Rightarrow $ a) : . Suppose that characteristic polynomial $P$ of $A$ is not irreducible in $\IQ[x]$. Then $P=P_{1}P_{2}$, with $P_{1},P_{2}\in \IQ[x]$. From the Cayley-Hamilton theorem we get that $0=P(A)=P_{1}(A)P_{2}(A)$. Hence for any nonzero rational vector $\bq$, either 1) $P_{2}(A)\bq=0$ or 2) $\hat{\bq}:=P_{2}(A)\bq \not = 0$ and $P_{1}(A)\hat{\bq}=0$. Since $\max\{deg(P_{1},P_{2})\}<d$, there exists a nonzero rational vector $\tilde{\bq}$ (namely $\bq$ or $\hat{\bq}$) such that the set of iterations $\{\tilde{\bq},A\tilde{\bq},A^{2}\tilde{\bq},...,A^{d-1}\tilde{\bq}\}$ does not form a basis of $\IR^{d}$. e) $\Rightarrow $ f) : . Assume there exist nonzero $\bq \in \IQ^{d}$ orthogonal to certain $PIS(A,\IR^{d})$ $V$. Then for any $n$ and any $f\in V$, $\la (A^{\dagger})^{n}\bq, f \ra= \la \bq, A^{n}f \ra = 0$ and hence $S=\{\bq,A^{\dagger}\bq,(A^{\dagger})^{2}\bq,...,(A^{\dagger})^{d-1}\bq\}$, cannot form a basis, which in view of equivalence a) $\Leftrightarrow$ d) implies reducibility of $A^{\dagger}$. f) $\Rightarrow$ g) : . Suppose there exists $PIS(A,\IR^{d})$ $V$ contained in certain $PRS(\IR^{d})$ $S$. Since $S$ is rational, $S^{\perp}$ is also rational. Consider any rational vector $\bq \in S^{\perp}$, then $\la \bq,f \ra = 0$ for any $f \in V$. g) $\Rightarrow$ b) : . If there exists $PRIS(\IR^{d})$, then this subspace is $A$-invariant and contained in $PRS(\IR^{d})$ i.e in itself. Now since b) is equivalent to a) it is enough to establish the equivalence between a) and e) to complete the proof. But the latter equivalence is obvious in view of the fact that $A$ and $A^{\dagger}$ have the same characteristic polynomial. $\qquad \blacksquare$ **Proof of Proposition \[zeroentropy\].** Suppose $A$ is a toral automorphism of zero entropy. The latter property is equivalent to the fact that all the eigenvalues of $A$ are of modulus $1$. Let $P_{A}$ be a characteristic polynomial of $A$. Consider any irreducible over $\IZ$ factor $P$ of polynomial $P_{A}$ and construct a toral automorphism $B$ such that its characteristic polynomial is equal to $P$ . Obviously all the eigenvalues of $B$ are also the eigenvalues of $A$, and each eigenvalue of $A$ can be found among eigenvalues of some matrix $B$ of this type. Irreducibility of $P$ implies irreducibility and hence diagonalizability of $B$. Thus for any nonzero vector $\bk \in \IZ^{d}$ and any positive integer $n$ the following estimate holds $|B^{n}\bf{k}| \leq |\bf{k}|$, which implies the existence (for each $\bf{k}$) of some integer $r$ such that $ B^{r}\bf{k}=\bf{k}$. The latter shows that all the eigenvalues of $B$ (and hence also of $A$) are roots of unity. $\qquad \blacksquare$ **Proof of Proposition \[red\].** We first show that irreducible polynomials $P\in\IQ[x]$ do not have repeated roots. Indeed suppose $\lam$ is a root of $P$ of multiplicity greater that 1, then $\lam$ is also a root of a derivative polynomial $P'\in\IQ[x]$. Since the minimal polynomial of $\lam$ must divide both $P$ and $P'$ and $deg(P')<deg(P)$ one immediately concludes that $P$ is not irreducible. Now, suppose $A \in GL(d,\IQ)$ is completely decomposable over $\IQ$ and let (\[blockdiag\]) be its block diagonal decomposition into irreducible blocks. Each $P_{A_{j}}$, as a characteristic polynomial of $A_{j}$, is irreducible over $\IQ$ and hence does not possesses repeated roots, which implies diagonalizability of each $A_{j}$ and hence of $A$. On the other hand if $A$ is diagonalizable then its minimal polynomial does not possesses repeated roots, which implies that all characteristic polynomials associated with elementary divisors are (first powers of) irreducible polynomials. This implies irreducibility of each block in representation (\[blockdiag\]). $\qquad \blacksquare$ **Proof of Proposition \[distinct\].** Let $P_{A}$ be the characteristic polynomial of an irreducible matrix $A \in GL(d,\IQ)$. Since $P_{A}$ is an irreducible element of $\IQ[x]$ it does not possesses repeated roots (see the proof of Proposition \[red\]). $\qquad \blacksquare$ [99]{} Adler, R.L. and P. 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Young: [*Rigorous Bounds on the Fast Dynamo Growth Rate Involving Topological Entropy*]{}. [*Commun. Math. Phys.*]{} [173]{} 623-646 (1995) Lasota, A. and M. Mackey: [*Chaos, Fractals, and Noise : Stochastic Aspects of Dynamics*]{}, 2nd ed. Spring-Verlag, New York, 1994. Lukomskaya, M. A.:[*A new proof of the theorem of van der Waerden on arithmetic progressions and some generalizations of this theorem*]{}. (Russian) [*Uspehi Matem. Nauk (N.S.)*]{} [3:6(28)]{}(1948), 201–204. Manning, A.: [*There are no new Anosov diffeomorphisms on tori*]{}. [*Amer. J. Math.*]{} [96:3]{} (1974), pp. 422-429. Mezzadri, F.: [*On the multiplicativity of quantum cat maps*]{}. [*Nonlinearity*]{} [15]{} (2002), pp. 905-922. Newman, M.: [*Integral Matrices*]{}. Pure and Applied Mathematics Vol. 45, Academic Press, 1972. Ornstein, D.S. and B. Weiss: [*Statistical properties of chaotic systems*]{}. Bull. Amer. Math. Soc. [24:1]{}, 11-116 (1991). Rivas, A. and Ozorio de Almeida, A.M.: [*The Weyl representation on the torus*]{} [*Ann. Phys.*]{}[276]{} 123 (1999). Rosenblatt, M.: [*Markov Processes. Structure and Asymptotic Behavior*]{}. Springer-Verlag, 1971. Schmidt, W. M.: [*Diophantine Approximation*]{}. Lecture Notes in Mathematics Vol. 785, Springer-Verlag, 1980. van der Waerden, B. L.: [*Beweis einer Baudetschen Vermutung*]{}, [*Nieuw Arch. Wiskunde*]{} [(2)15]{} (1927), 212-216. Yuzvinskii, S. A.: [*Computing the entropy of a group of endomorphisms*]{}. [*Siberian Math. Jour.*]{} [8]{} (1967), 172-178. Zygmund, A.: [*Trigonometric Series*]{} [**2**]{}. Cambridge, 1959. [^1]: Department of Mathematics, University of California at Davis, Davis, CA 95616, Internet: fannjian@math.ucdavis.edu, wolowski@math.ucdavis.edu. The research of AF is supported in part by the grant from U.S. National Science Foundation, DMS-9971322 and UC Davis Chancellor’s Fellowship

--- abstract: 'We discuss two questions related to the concept of weak values as seen from the standard quantum-mechanics point of view. In the first part of the paper, we describe a scenario where unphysical results similar to those encountered in the study of weak values are obtained using a simple experimental setup that does not involve weak measurements. In the second part of the paper, we discuss the correct physical description, according to quantum mechanics, of what is being measured in a weak-value-type experiment.' author: - 'S. Ashhab' - Franco Nori title: 'How the result of a measurement of a component of the spin of a spin-1/2 particle can turn out to be 100 without using weak measurements' --- Introduction ============ The first part of the title of this paper (all but the last four words) is taken from the title of a paper written by Aharonov, Albert and Vaidman (AAV) over twenty years ago [@Aharonov]. In that paper AAV introduced the concept of weak values. This concept immediately caused controversy [@Comments], but over the years it has proved to be a useful paradigm for considering questions related to quantum measurement and the foundations of quantum mechanics. For example, the observation of paradoxical values in a weak-value-type measurement has been linked to the violation of the Leggett-Garg inequality, which can be used to test realism [@LeggettGarg; @Williams; @Romito]. In the setup considered by AAV, a beam of spin-1/2 particles propagates through a non-uniform magnetic field in a Stern-Gerlach-type experiment, where the trajectory of a given particle is affected by the spin state of the particle. The modification from the original Stern-Gerlach experiment is that, in the path of its propagation, the beam encounters two regions in space with magnetic fields. The magnetic field gradient in the first region is designed such that it creates a tendency for particles whose $x$-component of the spin (which we denote by $S_x$) is positive to develop a finite component of the momentum in the positive $x$ direction and for particles whose $S_x$ is negative to develop a finite component of the momentum in the negative $x$ direction. After exiting this region in space, the beam enters a second region where a $z$-component in the momentum develops based on the $z$-component of the spin ($S_z$). Either one of these stages would constitute a measurement of the spin along some direction: by setting up a screen that the beam hits sufficiently far from the field-gradient region, the position where a given particle hits the screen serves as an indicator of the particle’s spin state. When combined, they create a situation where two non-commuting variables are being measured in succession. If (1) the first measurement stage is designed to be a weak measurement, (2) the particles in the beam are created in a certain initial state \[e.g. close to being completely polarized along the positive $z$-axis\] and (3) only those particles for which the second measurement produces a certain outcome \[in this example, a negative $z$-component of the spin\], then the average value of the spin’s $x$-component indicator can suggest values of this component of spin being much larger than 1/2, a situation that seems paradoxical. A number of studies have already pointed out that since in the AAV setup two non-commuting variables are being measured in succession, quantum mechanics forbids treating them as independent measurements whose outcomes do not affect one another [@Comments]. In this paper we start by presenting an example that demonstrates the role of interpretation in obtaining unphysical results in a weak-measurement-related setup. The setup is chosen to be very simple in order to remove any complications in the analysis related to the successive measurement of non-commuting variables. In the second part of the paper, we present the proper analysis (from the point of view of quantum mechanics) of the measurement results obtained in an AAV setup. Question 1: Unphysical results of the AAV type in an alternative setup ====================================================================== Let us consider the following situation: An experimenter purchases a device for measuring the $z$-component of a spin-1/2 particle. The device produces one of two readings, 0 or 1. The experimenter goes to the lab and calibrates the device. The calibration is done by preparing $10^6$ particles in the spin up state, measuring them one by one, and then doing the same for the spin down state. Let us say that the result of the calibration procedure is that for the spin up state the device shows the reading “1” in 50.25% of the experimental runs and the reading “0” in 49.75% of the runs. For the spin down state, the probabilities are reversed. Clearly, the reading of the measurement device is only weakly correlated with the spin state of the measured particle. The experimenter takes this fact into account and reaches the following conclusion: If I have a large number of identically prepared spin-1/2 particles and measure them using this device, I will obtain a probability for the reading “1”. Using the results of the calibration procedure, the expectation value of the spin $z$-component for the prepared state will be given by the formula: $${\left\langle S_z \right\rangle} = \left({\rm Prob_1}-0.5\right) \times 200.$$ If the probability of obtaining the outcome “1” is 0.5025, the above formula gives 1/2. If the probability of obtaining the outcome “1” is 0.4975, the above formula gives -1/2. It looks like the device is ready to be used. The experimenter now performs an experiment that involves, as its final step, a measurement of $S_z$. Surprisingly, the measurement device shows the reading “1” every time the experiment is repeated, leading the experimenter to conclude that the value of the spin is in fact 100. Thus one has a paradox. The resolution of the paradox in the above story lies in the fact that the device was not a weak-measurement device as the experimenter assumed, but a strong-measurement device whose reading is perfectly correlated with the spin state of the measured particle. The only problem is that at some point before the measurement device was calibrated, its spin-sensing part was rotated from being parallel to the $z$-axis to an axis that makes an angle 89.7135 with the $z$-axis (note here that $\cos^2 (89.7135/2) \approx 0.5025$). Not surprisingly, the calibration procedure produced the probabilities 0.5025 and 0.4975. In the “real” experiment, the spins were all aligned with the measurement axis of the device, and the reading “1” was observed in all the runs. The paradox is therefore resolved. An unquestioning believer in quantum mechanics might say that the situation discussed in Ref. [@Aharonov] has a large amount of overlap with the story presented above. In both cases a perfectly acceptable measurement is performed. The reason for obtaining a paradoxical measurement result is simply the wrong interpretation of what the measurement device is measuring and the resulting erroneous mapping from measurement outcomes to values of the measured quantity. Question 2: Correct explanation of results in an AAV setup ========================================================== {width="6.0cm"} We now turn to the question of the correct interpretation of the AAV experiment according to quantum mechanics. Instead of the original, Stern-Gerlach-type experiment analyzed by AAV, we formulate the problem slightly differently. We consider a spin-1/2 particle that is subjected to two separate measurements. As a first step, a weak measurement is performed in the basis $\left\{ {\left| + \right\rangle},{\left| - \right\rangle} \right\}$, where ${\left| \pm \right\rangle} = ({\left| \uparrow \right\rangle} \pm {\left| \downarrow \right\rangle})/\sqrt{2}$ and the states ${\left| \uparrow \right\rangle}$ and ${\left| \downarrow \right\rangle}$ are the eigenstates of $\hat{S}_z$. This measurement can produce any one of a large number of possible outcomes, with probability distributions as shown in Fig. 1. This measurement constitutes a weak measurement of $\hat{S}_x$. As discussed in [@Ashhab], each possible outcome is associated with a measurement matrix $\hat{U}_{x,k}$, where the index $k$ represents the outcome that is observed in a given run of the experiment. If the outcome $k$ occurs with probability $P_{x,k}$ for the system’s maximally mixed state, i.e. when averaged over all possible initial states, and it provides measurement fidelity $F_{x,k}$ (in favor of the state ${\left| + \right\rangle}$), the measurement matrix $\hat{U}_{x,k}$ is given by $$\begin{aligned} \hat{U}_{x,k} & = & \sqrt{P_{x,k}} \left\{ \sqrt{1+F_{x,k}} {\left| + \right\rangle}{\left\langle + \right|} + \sqrt{1-F_{x,k}} {\left| - \right\rangle}{\left\langle - \right|} \right\} \nonumber \\ & = & \frac{\sqrt{P_{x,k}}}{2} \left( \begin{array}{cc} \sqrt{1+F_{x,k}} + \sqrt{1-F_{x,k}} & \sqrt{1+F_{x,k}} - \sqrt{1-F_{x,k}} \\ \sqrt{1+F_{x,k}} - \sqrt{1-F_{x,k}} & \sqrt{1+F_{x,k}} + \sqrt{1-F_{x,k}} \\ \end{array} \right).\end{aligned}$$ We shall use the convention where a measurement that favors the state ${\left| - \right\rangle}$ has a negative value of $F_{x,k}$ and $\hat{U}_{x,k}$ is given by the same expression as above. It is worth mentioning here that the overall, or average, fidelity of this measurement can be obtained by averaging over all possible initial states and all possible outcomes: $${\left\langle F_x \right\rangle} = \sum_k P_{x,k} \left| F_{x,k} \right|.$$ After the weak $x$-basis measurement, a strong measurement in the $\left\{ {\left| \uparrow \right\rangle},{\left| \downarrow \right\rangle} \right\}$ basis is performed. This strong measurement step can be described by two outcomes with corresponding measurement matrices $$\begin{aligned} \hat{U}_{z,1} & = & {\left| \uparrow \right\rangle}{\left\langle \uparrow \right|} \nonumber \\ & = & \left( \begin{array}{cc} 1 & 0 \\ 0 & 0 \\ \end{array} \right) \nonumber \\ \hat{U}_{z,2} & = & {\left| \downarrow \right\rangle}{\left\langle \downarrow \right|} \nonumber \\ & = & \left( \begin{array}{cc} 0 & 0 \\ 0 & 1 \\ \end{array} \right).\end{aligned}$$ As mentioned above, paradoxes arise if one treats the $x$-basis and $z$-basis measurements as two separate measurements that provide complementary information. Instead, one should treat each pair of outcomes as a single *combined-measurement* outcome. The maximum amount of information in a given run of the experiment can be extracted as follows [@Ashhab]: given that the outcome pair $\{k,l\}$ was observed, one can construct the combined-measurement matrix $$\hat{U}_{{\rm Total},k,l} = \hat{U}_{z,l} \hat{U}_{x,k}.$$ From the matrices $\hat{U}_{{\rm Total},k,l}$ one can construct a so-called positive operator-valued measure (POVM) defined by the matrices $\hat{M}_{k,l}$: $$\hat{M}_{k,l} = \hat{U}_{{\rm Total},k,l}^{\dagger} \hat{U}_{{\rm Total},k,l},$$ where the superscript $\dagger$ represents the transpose conjugate of a matrix. In particular, $$\begin{aligned} \hat{M}_{k,1} & = & \frac{P_{x,k}}{2} \left( \begin{array}{cc} \left( \sqrt{1+F_{x,k}} + \sqrt{1-F_{x,k}} \right)^2 & 2 F_{x,k} \\ 2 F_{x,k} & \left( \sqrt{1+F_{x,k}} - \sqrt{1-F_{x,k}} \right)^2 \\ \end{array} \right) \nonumber \\ & = & P_{x,k} \left( 1 + {\left| \psi_{k,1} \right\rangle}{\left\langle \psi_{k,1} \right|} - {\left| \overline{\psi}_{k,1} \right\rangle}{\left\langle \overline{\psi}_{k,1} \right|} \right) \nonumber \\ & = & 2 P_{x,k} {\left| \psi_{k,1} \right\rangle}{\left\langle \psi_{k,1} \right|}\end{aligned}$$ where $$\begin{aligned} {\left| \psi_{k,1} \right\rangle} & = & \cos\frac{\theta_{k}}{2} {\left| \uparrow \right\rangle} + \sin\frac{\theta_{k}}{2} {\left| \downarrow \right\rangle} \nonumber \\ {\left| \overline{\psi}_{k,1} \right\rangle} & = & \sin\frac{\theta_{k}}{2} {\left| \uparrow \right\rangle} - \cos\frac{\theta_{k}}{2} {\left| \downarrow \right\rangle} \nonumber \\ \sin\theta_{k} & = & F_{x,k}.\end{aligned}$$ Similarly one can find that $$\begin{aligned} \hat{M}_{k,2} & = & 2 P_{x,k} {\left| \psi_{k,2} \right\rangle}{\left\langle \psi_{k,2} \right|} \nonumber \\ {\left| \psi_{k,2} \right\rangle} & = & \sin\frac{\theta_{k}}{2} {\left| \uparrow \right\rangle} + \cos\frac{\theta_{k}}{2} {\left| \downarrow \right\rangle},\end{aligned}$$ with $\theta_k$ given by the same expression as above. As discussed in Ref. [@Ashhab], one can obtain the measurement basis and fidelity that correspond to the outcome defined by $\{k,l\}$ by diagonalizing the matrix $\hat{M}_{k,l}$. Since $\hat{M}_{k,l}$ is a hermitian matrix, its two eigenvalues ($m_{k,l,1}$ and $m_{k,l,2}$, with $m_{k,l,1} \geq m_{k,l,2}$) will be real and its two eigenstates (${\left| \psi_{k,l} \right\rangle}$ and ${\left| \overline{\psi}_{k,l} \right\rangle}$) will be orthogonal quantum states that define a basis (the measurement basis). Note that because the second measurement in the problem considered here is a strong measurement, we always have $m_{k,l,2}=0$. The different outcomes produce different measurement bases, thus this measurement cannot be thought of in the usual sense of measuring $S_{\bf n}$ with ${\bf n}$ being some fixed direction. Therefore, the measurement basis is determined stochastically for each (combined) measurement (note that after the strong $z$-basis measurement, the system always ends up in one of the states $\{{\left| \uparrow \right\rangle},{\left| \downarrow \right\rangle}\}$, even though the combined-measurement basis can be different from the basis $\{{\left| \uparrow \right\rangle},{\left| \downarrow \right\rangle}\}$). By analyzing all the measurement data, one can perform partial quantum state tomography and determine the $x$ and $z$-components in the initial state of the system (assuming of course that all copies are prepared in the same state, which can be pure or mixed). Note that in this setup no information about $S_y$ can be obtained from the measurement outcome. We now ask whether information can be extracted from the $x$-basis and $z$-basis measurements separately, i.e. by disregarding the outcome of one of the two measurement steps. The answer is yes, provided care is taken in interpreting the results. Extracting an $x$-basis measurement from a given measurement outcome is straightforward. All one has to do is disregard the outcome of the $z$-basis measurement, since this measurement is performed after the $x$-basis measurement and cannot affect the outcome of the $x$-basis measurement. Therefore, by disregarding the outcome of the $z$-basis measurement, one obtains an $x$-basis measurement with overall fidelity ${\left\langle F_x \right\rangle}$. The situation is somewhat trickier if one wants to extract a $z$-basis measurement from the measurement outcome. One can disregard the outcome of the $x$-basis measurement, but one must take into account the fact that this measurement generally changes the state of the system before the $z$-basis measurement is performed. The effect of the $x$-basis measurement is to reduce the fidelity of the $z$-basis measurement. One can calculate this reduced fidelity as follows: Let us assume that the system starts in the initial state ${\left| \uparrow \right\rangle}$. After the $x$-basis measurement is performed and the outcome $k$ (with fidelity $F_{x,k}$) is observed, the state of the system is transformed into a new pure state ${\left| \psi_{\rm int} \right\rangle}$ with $\left| {\left\langle \psi_{\rm int} \right|} \hat{\sigma}_x {\left| \psi_{\rm int} \right\rangle} \right| = F_{x,k}$. Since $$\left| {\left\langle \psi_{\rm int} \right|} \hat{\sigma}_x {\left| \psi_{\rm int} \right\rangle} \right|^2 + \left| {\left\langle \psi_{\rm int} \right|} \hat{\sigma}_y {\left| \psi_{\rm int} \right\rangle} \right|^2 + \left| {\left\langle \psi_{\rm int} \right|} \hat{\sigma}_z {\left| \psi_{\rm int} \right\rangle} \right|^2 = \frac{1}{4}$$ for any pure state and here we have $\left| {\left\langle \psi_{\rm int} \right|} \hat{\sigma}_y {\left| \psi_{\rm int} \right\rangle} \right| = 0$, we find that after the $x$-basis measurement $4 \left| {\left\langle \psi_{\rm int} \right|} \hat{\sigma}_z {\left| \psi_{\rm int} \right\rangle} \right|$ is reduced from 1 to $\sqrt{1-F_{x,k}^2}$. If $F_{x,k}$ is independent of $k$, one obtains the relation (in this context, see e.g. Ref. [@Kurotani]) $${\left\langle F_x \right\rangle}^2 + {\left\langle F_z \right\rangle}^2 = 1.$$ We now take one final look at the AAV gedankenexperiment. We choose a specific form for the $x$-basis measurement, which is essentially the same one used by AAV $$\begin{aligned} P_{x,k} & = & \frac{1}{\sqrt{2\pi k_{\rm rms}^2}} \exp \left\{ -\frac{k^2}{2k_{\rm rms}^2} \right\} \nonumber \\ F_{x,k} & = & \sqrt{\frac{\pi}{2}} \frac{{\left\langle F_x \right\rangle}}{k_{\rm rms}} k,\end{aligned}$$ with $k$ running over all integers from $-\infty$ to $+\infty$ and $k_{\rm rms}$ assumed to be a large number. Note that the above expression violates the constraint that $F_{x,k}<1$. However, provided that ${\left\langle F_x \right\rangle} \ll 1$, the above expression can be treated as a good approximation of the realistic situation for all practical purposes. A simple calculation shows that in this case $$\begin{aligned} {\left\langle F_z \right\rangle} & = & \sum_{k=-\infty}^{\infty} \sqrt{1-F_{x,k}^2} P_{x,k} \nonumber \\ & \approx & 1 - \frac{\pi {\left\langle F_x \right\rangle}^2}{4},\end{aligned}$$ such that $${\left\langle F_x \right\rangle}^2 + {\left\langle F_z \right\rangle}^2 \approx 1 - \frac{\pi - 2}{2} {\left\langle F_x \right\rangle}^2.$$ If the measured system is prepared in one of the states ${\left| \pm \right\rangle}$, the average value of $k$ that is obtained in an ensemble of measurements (all with the same initial state) is $${\left\langle k \right\rangle}_{{\left| \pm \right\rangle}} = \pm \frac{{\left\langle F_x \right\rangle} k_{\rm rms}}{2}.$$ The small difference between ${\left\langle k \right\rangle}_{{\left| + \right\rangle}}$ and ${\left\langle k \right\rangle}_{{\left| - \right\rangle}}$ is the reason why the $x$-basis measurement qualifies as a weak measurement of $S_x$. We now consider the full measurement procedure. If one prepares the measured system in a state that is very close to ${\left| \uparrow \right\rangle}$, most $z$ basis measurements will produce the outcome $l=1$. Only a small fraction of the experimental runs will produce the outcome $l=2$. If the initial state deviates slightly from ${\left| \uparrow \right\rangle}$, i.e. $${\left| \psi_i \right\rangle} = \cos\frac{\alpha}{2} {\left| \uparrow \right\rangle} + \sin\frac{\alpha}{2} {\left| \downarrow \right\rangle},$$ then outcomes with negative values of $k$ and $l=2$ will be suppressed the most (assuming $\alpha$ is positive), because these outcomes correspond to states that are orthogonal or almost orthogonal to the initial state (making their occurrence probabilities particularly small). One therefore finds that among the measurements that produced $l=2$, the average value of $k$ can be much larger than ${\left\langle k \right\rangle}_{{\left| + \right\rangle}}$ for properly chosen parameters. This situation leads to the AAV paradox. Conclusion ========== In conclusion, we have presented explanations according to quantum mechanics of two questions that are relevant to discussions of weak values. First we presented an example that emphasizes the role of interpretation in obtaining unphysical results in an AAV setup. We have also presented the correct interpretation (according to quantum mechanics) of the measurement results obtained in an AAV setup. We believe that our discussion is useful for understanding the origin of the possible observation of unphysical values in a weak-value experimental setup. This work was supported in part by the National Security Agency (NSA), the Laboratory for Physical Sciences (LPS), the Army Research Office (ARO) and the National Science Foundation (NSF) grant No. EIA-0130383. [99]{} Y. Aharonov, D. Z. Albert, and L. Vaidman, Phys. Rev. Lett. [**60**]{}, 1351 (1988). A. J. Leggett, Phys. Rev. Lett. [**62**]{}, 2325 (1989); A. Peres, [*ibid*]{} [**62**]{}, 2326 (1989); Y. Aharonov and L. Vaidman, [*ibid*]{} [**62**]{}, 2327 (1989). A. J. Leggett and A. Garg, Phys. Rev. Lett. [**54**]{}, 857 (1985); A. J. Leggett, J. Phys. Condens. Matter [**14**]{}, R415 (2002). N. S. Williams and A. N. Jordan, Phys. Rev. Lett. [**100**]{}, 026804 (2008). For other studies on the subject, see e.g. A. Romito, Y. Gefen, and Y. M. Blanter, Phys. Rev. Lett. [**100**]{}, 056801 (2008); V. Shpitalnik, Y. Gefen, and A. Romito, Phys. Rev. Lett. [**101**]{}, 226802 (2008). S. Ashhab, J. Q. You, and F. Nori, Phys. Rev. A [**79**]{}, 032317 (2009); arXiv:0903.2319. Y. 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--- abstract: 'Digital mathematical libraries (DMLs) such as arXiv, Numdam, and [EuDML](https://eudml.org) contain mainly documents from STEM fields, where mathematical formulae are often more important than text for understanding. Conventional information retrieval (IR) systems are unable to represent formulae and they are therefore ill-suited for math information retrieval (MIR). To fill the gap, we have developed, and open-sourced the MIR system. MIaS is based on the full-text search engine Apache Lucene. On top of text retrieval, MIaS also incorporates a set of tools for preprocessing mathematical formulae. We describe the design of the system and present speed, and quality evaluation results. We show that MIaS is both efficient, and effective, as evidenced by our victory in the NTCIR-11 Math-2 task.=-1' author: - Petr Sojka - Michal Ružička - Vít Novotný bibliography: - 'main.bib' - 'sojka.bib' title: '*MIaS*: Math-Aware Retrieval in Digital Mathematical Libraries' --- =1 renderers= headingOne= \#1 {#section .unnumbered} === , headingTwo= \#1 {#section-1 .unnumbered} --- , strongEmphasis=, \# CIKM 2018 \#\# TODOs - For the review: - \*\*Convert to the \[ACM LaTeX template\](https://www.acm.org/publications/proceedings-template), use ‘sample-sigconf.tex‘ as example, aim for 4 pages.\*\* - \*\*Make sure the article is properly anonymized. (not necessary for Demo paper\*\* - \*\*Make sure the bibliography is properly formatted.\*\* - \*\*Extend the evaluation section with results from NTCIR\*\* - \*\*Describe query expansion, and striping\*\* - For the camera-ready: \#\# Links - \[Topics of Interest\](http://www.cikmconference.org/\#topics): - Performance evaluation - Information storage and retrieval and interface technology - Digital libraries - \[Call for Demonstrations\](http://www.cikm2018.units.it/callfordemo.html) \# MIaS Figures - WebMIaS web interface: - \[@dml:Liskaetal2011, Figure 2\] - \[@dl:liska2010eng, Figure 4.1\] - \[@MIR:MIRMU, Figure 1\] \*\[shows two extra webpage screenshots\]\* - \[@dml:cicm2014liskaetal, Figure 1\] - \[@Ruzicka17Math, Figure 4\] - Scheme of system workflow: - \[@dml:Liskaetal2011, Figure 1\] - \[@dl:liska2010eng, Figure 3.1\] \*\[in Slovak\]\* - \[@mir:LiskaMasters2013, Figures 4.1, and 4.2\] - \[@dml:sojkaliska2011, Figures 1, and 2\] - \[@dml:doceng2011SojkaLiska, Figure 1\] - \[@Ruzicka17Math, Figures 1, and 2\] \*\[shows extra detail\]\* - Formula preprocessing: - \[@dl:liska2010eng, Appendix B\] \*\[in Slovak\]\* - \[@dml:sojkaliska2011, Figure 3\] - \[@dml:doceng2011SojkaLiska, Figure 2\] - Unpublished CICM 2017 article \*„Towards Math-Aware Automated Classification and Similarity Search of Scientific Publications“\* (Figure 1) - \[@Ruzicka17Math, Figure 3\] - Relative number of results found using different subqueries in NTCIR-11 CMath run of MIR MU: - \[@LiskaSojkaRuzicka15Combining, Figure 1\] - \[@mir:MIaSNTCIR-11, Figure 1\] - \[@Ruzicka17Math, Figure 5\] - NTCIR-11 BPREF evaluation results: - \[@RuzickaSojkaLiska16Math, Figure 2\] - MathML structural unification: - \[@RuzickaSojkaLiska16Math, Figure 1\] - \[@Ruzicka17Math, Figure 7\] - Scalability diagrams: - \[@dml:sojkaliska2011, Figure 4\] \*\[MREC dataset\]\* - \[@mir:LiskaMasters2013, Figures 5.1–5.3\] \*\[MIR workshop dataset\]\* - MIaS UML diagrams: - \[@mir:LiskaMasters2013, Figures B.1–B.3\] - MSC topic modelling: - Unpublished CICM 2017 article \*“Towards Math-Aware Automated Classification and Similarity Search of Scientific Publications”\* (Figures 2, and 3) - \[@Ruzicka17Math, Figures 26, and 27\] - Relevance density estimates: - Unpublished COLING 2018 article \*“Weighted Averaging in Disjoint Passage Retrieval for Question Answering”\* (Figure 4). \# Top-level outline - Abstract 1. Background 2. Aim 3. Methods 4. Results 5. Conclusion 1. Introduction (following the organizational pattern set forth by \*“Academic Writing for Graduate Students: Essential Tasks and Skills”\* by Swales, and Feak, 1994) - Move 1a: Establishing a research territory - Move 1b: Introducing previous research - Move 2: Establishing a niche - Move 3: Occupying a niche 2. System Description - MIaS - MathML Canonicalizer - MathML Unificator - MIaSMath - WebMIaS - Figure 1: WebMIaS web interface 3. Evaluation - MREC evaluation results \[@dml:Liskaetal2011, Section 5\] - MIR workshop evaluation results \[@mir:LiskaMasters2013, Chapter 5\] - NTCIR-10 evaluation results: - \[@MIR:MIRMU, Sections 4, and 5\] - \[@mir:NTCIR-10-Overview, Table 7\] - NTCIR-11 evaluation results: - \[@mir:MIaSNTCIR-11, Sections 4, and 5\] - \[@NTCIR11Math2overview, Table 8\] - NTCIR-12 evaluation results: - \[@RuzickaSojkaLiska16Math, Section 5\] - \[@ZanibbiEtAl16NTCIR, Tables 8, and 9\] 4. Conclusion and Future Work - Extending the techniques to general semi-structured text - Incorporating static relevance estimates based on the position of resulting documents - Figure 2: Relevance density estimates - Migrating MIaS to ElasticSearch - Extending ordering to a rewriting system in a computer algebra system (CAS) \[@cohl2017semantic\] - References Introduction ============ In mathematical discourse, formulae are often more important than text for understanding. As a result, digital mathematical libraries (DMLs) require math information retrieval (MIR) systems that recognize both text and math in documents and queries. Conventional IR systems represent both text, and formulae using the bag-of-words vector-space model (VSM). However, the VSM captures neither the structural, nor the semantic similarity between mathematical formulae, which makes it ill-suited for MIR. To fill the gap, new math-aware IR systems started to appear after the pioneering workshop on DMLs [@dml:dml2008proceedings]. Springer’s [[^1]]{} system takes formulae from papers with available LaTeX sources, and hashes the formulae to obtain a text representation. Zentralblatt Math uses the system[[^2]]{} [@kohlhase2008mathwebsearch], which represents formulae with substitution trees. We have developed and open-sourced the MIaS (Math Indexer and Searcher) system[[^3]]{} [@mir:MIaSNTCIR-11; @Ruzicka17Math] using the robust highly-scalable full-text search engine [Apache Lucene](https://lucene.apache.org/) [@bialecki12] and our own set of tools for the preprocessing of mathematical formulae. Since 2012, MIaS has been deployed in [the European Digital Mathematical Library (EuDML)[[^4]]{}]{}, making it historically the first system to be deployed in a DML. {width="\textwidth"} System Description ================== MIaS processes text and math separately. The text is tokenized and stemmed to unify inflected word forms. Math is expected to be in [the MathML format[[^5]]{}]{}. Open tools such as [Tralics[[^6]]{}]{}, [LaTeXML[[^7]]{}]{} convert documents in the popular math authoring language of LaTeX to MathML. Other tools such as [InftyReader](http://www.inftyreader.org/) [@dml:suzukietal2003] , and [MaxTract](https://github.com/zorkow/MaxTract) [@DBLP:conf/aisc/BakerSS12] convert raster, and vector PDF documents, respectively, to MathML. The math is then canonicalized, ordered, tokenized, and unified (see Figure \[fig:system\]). We will describe each of these processing steps in detail in the following paragraphs. #### Canonicalization As explained above, MathML can originate from multiple sources and each can encode equivalent mathematical formulae a little differently. To obtain a single *canonical* representation, we initially used the third-party MathML canonicalizer from the UMCL library that converts math to a subset of MathML called the Canonical MathML [@acc:archambaultmoco06cmathml]. However, since the conversion speed and accuracy did not match our expectations, we have developed and open-sourced our own MathML canonicalizer[^8] [@FormanekEtAl:OpenMathUIWiP2012]. #### Ordering MathML canonicalization only affects the encoding of mathematical formulae and does not result in any syntactic manipulation. We go a step further and reorder the operands of commutative operators alphabetically. For example, we convert the formulae $a+b$, and $b+a$ to a single canonical form $a+b$.=-1 #### Tokenization A user of our system may not know the precise form of a formula they are searching for. To enable partial matches, we index not only the original formula, but also all its *subformulae*, which correspond to all the XML subtrees of the original formula XML tree. To penalize partial matches, the weight of subformulae is inversely proportional to their depth in the XML tree. [@dml:sojkaliska2011] A user is likely interested in documents that contain either the query formula itself, or larger formulae with the query formula as a subformula. On the other hand, a user is unlikely to be interested in documents that contain only small parts of the query formula, such as isolated numbers, and symbols. For that reason, we only tokenize formulae in indexed documents, not in user queries. #### Unification In theory, the naming of variables does not affect the meaning of formulae. To match formulae in different notations, we replace each variable with a numbered identifier. For example, we convert the formulae $a+b^a,$ and $x+y^x$ to a single *unified* form $\text{id}_1+\text{id}_2^{\text{id}_1}$. In practice, many fields have an established notation and variable names are meaningful. To encourage precise matches, we keep the original formulae in addition to the unified formulae. Two formulae that only differ in numeric constants are often related. For example, both $3x^2-2x+2,$ and $8x^2-3x+6$ are quadratic polynomials. We replace every numeric constant with a constant identifier. For example, we convert the above formulae to a single unified form $\text{const}x^2-\text{const}x+\text{const}$. To encourage precise matches, we keep the original formulae in addition to the unified formulae. In predicate logic, a variable can represent an arbitrary formula. For example, the formulae $a^2+{}$, and $a^2+{}$ are equivalent if $x$ equals $\sqrt b$. Starting with the deepest subformulae, we replace all subformulae at a given depth with a unifying identifier. [@RuzickaSojkaLiska16Math] For example, we convert the formula $a^2+{}$ to a sequence of *structurally unified* formulae $a^2+{}$$,\Unif^{\unif}+{}$, and $\Unif+\Unif$ and the formula $a^2+{}$ to a sequence of structurally unified formulae $\Unif^{\unif}+{}$, and $\Unif+\Unif$. To penalize partial matches, the weight of the formulae is proportional to the depth of replacement. To encourage precise matches, we keep the original formulae in addition to the unified formulae. We have open-sourced [the MathML structural unificator[[^9]]{}]{}. ------------- ------- ------- ------- ------- ------- Subquery 1: $f_1$ $f_2$ $t_1$ $t_2$ $t_3$ Subquery 2: $f_1$ $f_2$ $t_1$ $t_2$ Subquery 3: $f_1$ $f_2$ $t_1$ Subquery 4: $f_1$ $f_2$ Subquery 5: $f_1$ $t_1$ $t_2$ $t_3$ Subquery 6: $t_1$ $t_2$ $t_3$ ------------- ------- ------- ------- ------- ------- adddotafter #### After preprocessing, a query consists of a weighted set of terms, and formulae. Since we are now going to search for documents that match at least one term, and at least one formula from the query, ill-posed terms, and formulae will negatively impact the recall of our system. To overcome this problem, we remove selected terms and formulae to produce a set of *subqueries*. Figure \[fig:query-expansion\] shows an example strategy for producing subqueries. @LiskaSojkaRuzicka15Combining describe other strategies that we use. We then submit the subqueries to Apache Lucene and receive ranked lists of resulting documents. Since the scores of the resulting documents are incomparable between subqueries, we cannot merge and rerank the individual result lists. Instead, we interleave them to obtain the final search results that we present to the user. {width="\textwidth"} --------- ------------- --------------- ---------- ---------- Docs Input Indexed Real CPU 10,000 3,406,068 64,008,762 35.75 35.05 50,000 18,037,842 333,716,261 189.71 181.19 100,000 36,328,126 670,335,243 384.44 366.54 200,000 72,030,095 1,326,514,082 769.06 733.44 300,000 108,786,856 2,005,488,153 1,197.75 1,116.64 350,000 125,974,221 2,318,482,748 1,386.66 1,298.10 439,423 158,106,118 2,910,314,146 1,747.16 1,623.22 --------- ------------- --------------- ---------- ---------- : Speed evaluation results on the NTCIR-11 Math-2 dataset using the same computer as above.[]{data-label="tab:speed-eval-ntcir11"} ----------- ------------ --------------- --------- ---------- Docs Input Indexed Real CPU 8,301,545 59,647,566 3,021,865,236 1940.07 3,413.55 ----------- ------------ --------------- --------- ---------- : Speed evaluation results on the NTCIR-11 Math-2 dataset using the same computer as above.[]{data-label="tab:speed-eval-ntcir11"} adddotafter #### To provide a web user interface to MIaS, we have developed and open-sourced [WebMIaS[[^10]]{}]{}$^,$[[^11]]{} [@mir:MIaSNTCIR-11; @dml:cicm2014liskaetal]. Users can input their query in a combination of text, and math with a native support for LaTeX provided by Tralics, and [MathJax](https://www.mathjax.org/) [@cervone2012mathjax]. Matches are conveniently highlighted in the search results. The user interface of WebMIaS is shown in Figure \[fig:webmias\]. We have deployed a demo of [the latest development version of WebMIaS[[^12]]{}]{} using the [Apache Tomcat[[^13]]{}]{} implementation of the Java Servlet. The demo uses an index built from a subset of the arXMLiv dataset [@dml:arXMLiv2010] made available to the NTCIR-12 conference participants and will serve as the basis for our live demonstration at the conference. Evaluation ========== We performed a speed evaluation of MIaS on the MREC dataset of 439,423 documents [@dml:liska2011] (see Table \[tab:speed-eval-mrec\]), a quality and speed evaluation on the NTCIR-10 Math [@mir:NTCIR-10-Overview; @MIR:MIRMU] dataset of 100,000 documents, and a quality and speed evaluation on the NTCIR-11  [@NTCIR11Math2overview; @mir:MIaSNTCIR-11] (see Tables \[tab:speed-eval-ntcir11\], and \[tab:quality-eval-ntcir11\]), and NTCIR-12 MathIR [@ZanibbiEtAl16NTCIR; @RuzickaSojkaLiska16Math] dataset of 105,120 documents that were split into 8,301,578 paragraphs. Speed evaluation shows that the indexing time of our system is linear in the number of indexed documents and that the average query time is 469ms. With respect to quality evaluation, MIaS has notably won the NTCIR-11 Math-2 task. Measure Level PMath CMath PCMath LaTeX --------- ------- -------- ---------------- ---------------- -------- MAP 3 0.3073 **0.3630 /1/** 0.3594 0.3357 P@10 3 0.3040 **0.3520 /1/** 0.3480 0.3380 P@5 3 0.5120 **0.5680 /1/** 0.5560 0.5400 MAP 1 0.2557 **0.2807 /2/** 0.2799 0.2747 P@10 1 0.5020 0.5440 **0.5520 /1/** 0.5400 P@5 1 0.8440 **0.8720 /2/** 0.8640 0.8480 : Quality evaluation results on the NTCIR-11 Math-2 dataset. The mean average precision (MAP), and precisions at ten (P@10), and five (P@5) are reported for queries formulated using Presentation (PMath), and Content MathML (CMath), a combination of both (PCMath), and LaTeX. Two different relevance judgement levels of $\geq1$ (partially relevant), and $\geq3$ (relevant) were used to compute the measures. Number between slashes (/$\cdot$/) is our rank among all teams.[]{data-label="tab:quality-eval-ntcir11"} Conclusion and Future Work ========================== With the growing importance of DMLs, there is a growing demand for effective MIR systems. The evaluation shows that our open-source system is both efficient, and effective while building on industrial-strength full-text search engine Apache Lucene. The system allows low-latency responses even on the big math corpora as proved by its deployment in EuDML. The speed of indexing and response latency of MIR will be further increased by the migration of from Apache Lucene to the distributed full-text search engine [ElasticSearch[[^14]]{}]{}. The idea of indexing structures rather than terms can be generalized from mathematical formulae to semi-structured text. Reordering the operands of associative operators is only a simple transformation. For example, to convert $\sqrt[n]{a}$, and $a^{1/n}$ to a single canonical representation, a general computer algebra system (CAS) can be used. We experiment [@rygletal16] with improving the vector space representations of document passages, aiming to add support for mathematics in the future. Embeddings can also be computed for equations [@mir:krstowski2018arXiv] now, which presents new possibilities of using language modeling for the semantic segmentation of STEM articles, and weighting the segments [@rygletal16]. Grasping the meaning of mathematical formulae is crucial: content is king. =-1 ### Acknowledgements {#acknowledgements .unnumbered} We gratefully acknowledge the support by the European Union under the FP7-CIP program, project 250,503 (EuDML), and by the ASCR under the Information Society R&D program, project 1ET200190513 (DML-CZ). We also sincerely thank three anonymous reviewers for their insightful comments. [^1]: <https://www.ams.org/notices/201004/rnoti-apr10-cov4.pdf> [^2]: <https://zbmath.org/formulae/> [^3]: <https://github.com/MIR-MU/MIaS> [^4]: <https://eudml.org/search> [^5]: <https://www.w3.org/TR/MathML3/> [^6]: <https://www-sop.inria.fr/marelle/tralics/> [^7]: <https://dlmf.nist.gov/LaTeXML/> [^8]: <https://github.com/MIR-MU/MathMLCan> [^9]: <https://github.com/MIR-MU/MathMLUnificator> [^10]: <https://mir.fi.muni.cz/webmias/> [^11]: <https://github.com/MIR-MU/WebMIaS> [^12]: <https://mir.fi.muni.cz/webmias-demo/> [^13]: <https://tomcat.apache.org/> [^14]: <https://elastic.co>

--- abstract: 'Fermat’s principle and variational analysis is used to analyze the trajectories of light propagating in a radially inhomogeneous medium with a singularity in the center. It is found that the light trajectories are similar to those around a black hole, in the sense that there exists a critical radius within which the light cannot escape, but spirals into the singularity.' author: - | M. Marklund, D. Anderson, F. Cattani, M. Lisak and L. Lundgren\ *Department of Electromagnetics, Chalmers University of Technology,*\ *SE–412  96 Göteborg, Sweden* date: '()' title: 'Fermat’s principle and variational analysis of an optical model for light propagation exhibiting a critical radius' --- Introduction ============ There has recently, within the scientific community, been much interest focused on the extra-ordinary properties associated with Bose-Einstein condensates - clouds of atoms cooled down to nano-Kelvin temperatures where all atoms are in the same quantum state and macroscopic quantum conditions prevail [@Leonhardt-Piwnicki-99; @Leonhardt-Piwnicki-00; @Hau-etal-99]. One of the most dramatic experiments which has been performed in Bose-Einstein condensates is the demonstration of optical light pulses traveling at extremely small group velocities, e.g. velocities as small as 17 m/s have been reported [@Hau-etal-99]. In a physically suggestive application of the optical properties of Bose-Einstein condensates, it was recently suggested that they could be used to create, in the laboratory, dielectric analogues of relativistic astronomical phenomena like those associated with a black hole [@Leonhardt-Piwnicki-00]. The astrophysical concept of a black hole is one of the most fascinating and intriguing phenomena related to the interaction between light and matter through the curvature of space-time within the framework of general relativity. Black holes are believed to form when compact stars undergo complete gravitational collapse due to, e.g., accretion of matter from its surroundings (in general relativity, the pressure contributes to the gravitational mass of a fluid, and an increased pressure will, after a certain point, therefore only help to accelerate the collapse phase). In ordinary dielectric media, unrealistic physical conditions would be required in order to demonstrate any of the spectacular effects of general relativity. However, in dielectric media characterized by very small light velocities, new possibilities appear. In particular, it was recently suggested that by creating a vortex structure in such a dielectric medium it is possible to mimic the properties of an optical black hole [@Leonhardt-Piwnicki-00]. What this study in fact suggested was that one could construct, by using the aforementioned vortex, an unstable photon orbit, much like the orbit at a radius $r = 3M$ in the Schwarschild geometry [@Misner-Thorne-Wheeler]. In order to construct an event horizon, one would need to supplement the vortex flow by a radial motion of the fluid [@Visser-00; @Leonhardt-Piwnicki-00b]. The vortex, which involves a rotating cylindrical velocity field, tends to attract light propagating perpendicular to its axis of rotation and to make it deviate from its straight path. In the model considered in Refs.[@Leonhardt-Piwnicki-99; @Leonhardt-Piwnicki-00], the increasing, in fact diverging, rotational velocity field of the vortex core attracts light by the optical Aharanov–Bohm effect and causes a bending of the light ray similar to that of the gravitational field in general relativity. It is also shown that there exist a critical radius, $r_{crit}$, from the vortex core with the properties that if light rays come closer to the core than $r_{crit}$,the light will fall towards the singularity of the vortex core. In this sense, the critical radius, $r_{crit}$, plays the role of an optical unstable photon orbit, analogous to the unstable orbit in the Schwarzschild geometry. The basic physical effect involved in the analysis of the ray propagation in Refs.[@Leonhardt-Piwnicki-99; @Leonhardt-Piwnicki-00] is the fact that the refractive index, $ n$, of a medium changes when the medium is moving. In Refs.[@Leonhardt-Piwnicki-99; @Leonhardt-Piwnicki-00], the medium is assumed to have a cylindrical vortex velocity field $ \bf V= V(\bf r)$ given by $${\bf V(r)} = \frac{W}{r}{\bf \widehat{\varphi}}$$ where $r$ is the radius from the vortex core centre, $\widehat{\varphi }$ is the azimuthal unit vector and $2\pi W$ is the vorticity. Advanced physical concepts like Bose-Einstein condensates, vortex velocity fields and black holes are fascinating physical effects, which however may be difficult to present in a simplified manner for undergraduate students. Nevertheless, it is important to try to convey inspiration for and arouse curiosity in such phenomena. An interesting example of such an effort was recently made in Ref. [@McDonald-00], where the basic physical mechanism, in the form of a classical two level system was used to demonstrate the possibility of the very low light velocities observed in Bose- Einstein condensates. In a later work by the same author, Ref.  [@McDonald-01], an inspiring investigation is given of a simple mechanical model that exhibits a gravitational critical radius. The purpose of the present work is similar to that of Refs. [@McDonald-00; @McDonald-01]. We will try to discuss a simple classical example of a medium where the refractive index has a divergence at $r = 0$ and which exhibits some of the dramatic properties of the light behaviour around a black hole as discussed in Refs.. In addition we want to illustrate the power and beauty of the classical principle of Fermat and of variational methods in connection with this new fascinating concepts. Fermat’s principle {#sec:Fermat} ================== It is well known that light tends to be deflected towards regions with higher refractive index. Consider light propagating in a cylindrically symmetric medium where the refractive index increases towards the centre. In such a situation we expect the light path to look qualitatively as in Fig. 1. The actual light path is determined by Fermat’s principle i.e.$$\delta\int n({\bf r})ds=0$$ where$ ds$ is an infinitesimal element along the light ray. We will consider light propagation in a medium where the refractive index is of the qualitative form, cf Eq. (1) $$n(r)\simeq \left\{ \begin{array} {r@{\quad as\quad}l}1 & r\gg r_{0}\\ r_0/r & r\ll r_{0}\end{array} \right.$$ One possible realization of $n(r)$ with the desired properties is given by $$n^{2}(r) = 1 + \left(\frac{r_{0}}{r}\right)^2$$ Using Fermat’s principle and expressing the light path as the relation $\theta=\theta(r)$ where $\theta$ is the polar angle, Eq. (2) implies $$\delta\int{n(r)\sqrt{1+r^{2}\left(\frac{d\theta}{dr}\right)^2}}dr = 0 \ .$$ The Euler-Lagrange variational equation corresponding to Eq. (5) reads $$\frac{d}{dr}\left[ n(r)\frac{r^2 \, d\theta/dr}% {\sqrt{1 + r^2(d\theta/dr)^2}}\right]=0$$ which determines the trajectory of the light. Solution of the light trajectory ================================ Equation (6) directly implies that $$\frac{r^2n(r) \, d\theta/dr }% {\sqrt{1 + r^2 (d\theta/dr)^2}} = r_{i}$$ where the constant $r_{i}$ is determined by initial conditions, i.e. the properties of the incident light. Equation (7) is easily inverted to read $$\frac{d\theta}{dr}=\pm\frac{r_{i}}{\sqrt{r^4n^2(r) - r_{i}^2r^2}}$$ For a light ray incident as shown in Fig. 1, we clearly have $d\theta/dr <0$ (at least up to some minimum radius $ r=r_{min}$. It is illustrative to first consider the trivial case of a homogeneous medium with $n(r)\equiv1$. In this case Eq. (9) becomes $$\frac{d\theta}{dr}=-\frac{r_{i}}{\sqrt{r^4 - r_{i}^2r^2}}$$ which can easily be interpreted to yield the light path in the form $$\theta=\mathrm{arccot}\sqrt{\frac{r^2}{r_{i}^2} - 1}$$ or simpler $$r=\frac{r_{i}}{\sin \theta}$$ Clearly this is the straight line solution $$y=r_{i}$$ where the parameter $r_{i}$ plays the role of “impact parameter” or minimum distance from the centre. Let us now consider the model variation for $n(r)$ as given by Eq. (5). Within this model Eq. (10) becomes $$\frac{d\theta}{dr} = - \frac{r_i}{\sqrt{r^4 - (r_i^2 - r_0^2)r^2}}$$ Clearly the solution of Eq. (13) will depend crucially on the relative magnitude of $r_i$ and $r_0$, i.e. the impact parameter relative to the characteristic radial extension of the inhomogeneity. Consider first the case when $r_0 < r_i$. Equation (13) can then be rewritten as $$\begin{aligned} \frac{b_1}{r_i}\frac{d\theta}{dr} &=& -\frac{b_1}{\sqrt{r^4 -b_1^2r^2}} \ , \\ b_1 &\equiv& \sqrt{r_{i}^{2}-r_{0}^{2}} \ .\end{aligned}$$ Equation (15) is of the same form as Eq. (9) and we directly infer the following solution $$r = \frac{\sqrt{r_{i}^{2} - r_{0}^{2}}}% {\sin\left[\theta\sqrt{1-r_{0}^{2}/r_{i}^{2}}\,\right]}$$ As $\theta\rightarrow0$, we still have asymptotically $$r\simeq\frac{r_{i}}{\sin \theta}$$ However, the trajectory is now bending towards the origin and the minimum distance occurs at the polar angle $\theta=\theta_{m}$ given by $$\theta_m = \frac{\pi}{2}\frac{1}{\sqrt{1-r_0^2/r_i^2}}$$ The corresponding minimum distance, $r_m$ is $$r_m \equiv r{\theta_m} = \sqrt{r_i^2 - r_0^2}$$ We also note that the trajectory is symmetrical around the angle $\theta_{m}$ and that the asymptotic angle of the outgoing light ray is $$\theta_{\infty}\equiv \lim_{r\rightarrow\infty} \theta(r) =\frac{\pi}{\sqrt{1 - r_0^2/r_i^2}} = 2 \theta_m$$ The solution given by eq.(10) describes a trajectory which is bent towards the centre of attraction at $ r=0$. Depending on the ratio $r_{0}/r_{i}$, the trajectory is either more or less bent or may even perform a number of spirals towards the centre before again turning outwards and escaping, cf. Fig. 2. The number of turns, $N$,which the light ray does around the origin before escaping is simply $$N = \left\lfloor \frac{2\theta_{m}}{2\pi} \right\rfloor = \left\lfloor \frac{1}{2\sqrt{1-r_{0}^{2}/r_{i}^{2}}} \right\rfloor$$ where $\left\lfloor x \right\rfloor$ denotes the largest integer less than $x$. Let us now consider the special case when the impact parameter equals the characteristic width of the refractive index core i.e.$r_{i}=r_{0}$. The equation for the trajectory now simplifies to $$\frac{d\theta}{dr} = -\frac{r_{i}}{r^{2}}$$ with the simple solution $$\theta=\frac{r_{i}}{r}$$ i.e. the trajectory describes a path in the form of Arkimede’s spiral as the light falls towards the origin, cf. Fig. 3. The form of the light trajectory in the situation when $ r_{i}< r_{0}$ is now obvious, it will spiral into the singularity more or less directly depending on the magnitude of the ratio $r_{0}/r_{i}>1$. The actual trajectory in this case is given by a slight generalization of Eq. (16) viz $$r=\frac{\sqrt{r_{0}^{2}-r_{i}^{2}}}{\sinh [\theta\sqrt{1-r_{i}^{2}/r_{0}^{2}}]}$$ This solution does indeed convey the expected behaviour, the trajectory spirals monotonously into the singularity of the refractive index. Final comments ============== The present analysis is inspired by recent discoveries and discussions about light propagation in Bose-Einstein condensates where extremely low light velocities can be obtained. This has triggered speculations about possible laboratory demonstrations of effects, which normally are associated with general relativistic conditions. In particular, it has been suggested that it may be possible to create the analogue of a black hole using a divergent in-spiral of a Bose-Einstein condensate. In the present work we have analyzed a simple classical example of light propagation as determined by Fermat’s principle in a medium characterized by a radially symmetric refractive index. In analogy with the variation of the vortex velocity field suggested in [@Leonhardt-Piwnicki-00], the refractive index is here assumed to diverge towards the centre. This classical example exhibit some of the characteristic properties of light propagating around a black hole where the gravitational attraction deflects the light and where, under certain conditions, the light may be “swallowed” by the black hole. In the example analyzed here, the unstable photon orbit of the Schwarzschild black hole is similar to the characteristic radius, $r_0$, of the refractive index variation, which together with the impact parameter, $r_i$, of the incident light completely determines the light trajectory. If $r_0 < r_i$, the light is more or less deflected, but ultimately escapes. However, if $r_0 \geq r_i$, the light spirals into the singularity.It should be cautioned that this result depends crucially on the presence of the singularity in the refractive index (4). If this is removed the in-spiraling photon orbit will eventually turn and start spiraling outwards. In the recent discussion about the possibility of generating analogues of optical black holes in Bose-Einstein condensates, it has been suggested that the vortex motion must also have a velocity component in the radial direction. In the classical model considered here, a number of additional physical effects will obviously affect the light path when it comes close to the axis and will in fact remove the mathematical singularity. Nevertheless, the model provides a simple example of light dynamics, which resembles some of the properties of light propagation around a black hole. Furthermore,the investigation is based on Fermat’s principle and variational analysis, in this way illustrating the use of classical methods in connection with very new and fascinating concepts at the front line of modern research. Figure Captions. {#figure-captions. .unnumbered} ================ Fig.1 Qualitative plot of a light ray trajectory in a cylindrically symmetric medium with a refractive index, which increases towards the centre.\ Fig.2 Light trajectories for the refractive index model of eq.() for different impact radii $r_{i}>r_{0}$\ Fig.3 Light trajectory in the case of $r_{i}=r_{0}$, the spiral of Arkimede.\ [99]{} U. Leonhardt and P. Piwnicki, Phys. Rev. A **60**, 4301 (1999) U. Leonhardt and P. Piwnicki, Phys. Rev. Letters **84**, 822 (2000) L. V. Hau, S. E. Harris, Z. Dutton and C. H. Behroozi, Nature (London) **397**, 594 (1999) C. Misner, K. S. Thorne and J. A. Wheeler *Gravitation* (Freeman, 1973) M. Visser, Phys. Rev. Letters **85**, 5252 (2000) U. Leonhardt and P. Piwnicki, Phys. Rev. Letters **85**, 5253 (2000) K. T. Mc Donald, Am. J. Phys. **68**, 293 (2000) K. T. Mc Donald, Am. J. Phys. **69**, 617 (2001)

--- abstract: 'The Lithium Depletion Boundary (LDB) is a robust method for accurately determining the ages of young clusters, but most pre-main-sequence models used to derive LDB ages do not include the effects of magnetic activity on stellar properties. In light of this, we present results from our spectroscopic study of the very low-mass members of the southern open cluster Blanco 1 using the Gemini-North telescope, program IDs: GN-2009B-Q-53 and GN-2010B-Q-96. We obtained GMOS spectra at intermediate resolution for cluster candidate members with $I$$\approx$13–20 mag. From our sample of 43 spectra, we find 14 probable cluster members by considering proximity to the cluster sequence in an $I/I-K\rm_s$ color-magnitude diagram, agreement with the cluster’s systemic radial velocity, and magnetic activity as a youth indicator. We systematically analyze the [$\rm H\alpha$]{} and Li features and update the LDB age of Blanco 1 to be $126^{+13}_{-14}$ Myr. Our new LDB age for Blanco 1 shows remarkable coevality with the benchmark Pleiades open cluster. Using available empirical activity corrections, we investigate the effects of magnetic activity on the LDB age of Blanco 1. Accounting for activity, we infer a corrected LDB age of $114^{+9}_{-10}$ Myr. This work demonstrates the importance of accounting for magnetic activity on LDB inferred stellar ages, suggesting the need to re-investigate previous LDB age determinations.' author: - 'Aaron J. Juarez, Phillip A. Cargile, David J. James, Keivan G. Stassun' bibliography: - 'LDB-B1.bib' title: An Improved Determination of the Lithium Depletion Boundary Age of Blanco 1 and a First Look on the Effects of Magnetic Activity --- Introduction ============ As low-mass stars ($\lesssim 1 M_\odot$) contract along the pre-main-sequence (PMS), their internal temperature rises. When the temperature of the stellar interior reaches $\sim$$2.5 \times 10^6$ K, lithium is destroyed by $^{7}$Li($p,\alpha)^{4}$He and $^{6}$Li($p,\alpha)^{3}$He proton capture reactions [[[*e.g.*]{},]{} @bodenheimer:1965]. The elapsed time to reach Li-burning temperatures is a sensitive function of mass and thus, depends very sensitively on the luminosity [@bildsten:1997; @ushomirsky:1998]. PMS low-mass stars are fully convective, so the mixing timescale is short, and since the temperature dependence of the nuclear reactions is steep, these stars rapidly deplete their Li content. For coeval stellar groups, like open clusters or moving groups, the determination of the luminosity at which stars transition from exhibiting Li in their atmospheres to being fully depleted provides a very precise age estimate. Moreover, the LDB technique is relatively model-insensitive, rendering similar ages to within $\pm$10% [@burke:2004vf], making it a highly robust method which can lead to the identification of missing input physics when compared with other age-dating methods. The LDB method originated with @basri:1996 who first applied it to the Pleiades cluster, leading to the discovery of the first brown dwarf using the lithium test. Ages of open clusters are traditionally determined by matching their Hertzsprung Russell diagrams (HRDs) to distance-dependent, theoretical stellar isochrones. In particular, stars in and near the region close to the main-sequence turn-off (MSTO) are the most sensitive determinants of the age because they are evolving quickly. MSTO fitting can be improved in precision through statistical techniques that take account of the usually small number of stars at the turn-off, but the method remains limited by several factors. For very young clusters ($< 100$ Myr), the MSTO corresponds to the minimum number of objects for their initial mass function, and un-resolved, undetected binary/multiple systems can significantly affect MSTO ages. At the same time, derived ages are highly dependent on the input models used and the physical constraints bounding them. One major influence, for example, has been the inclusion of core mixing in intermediate-mass stars that led to ages being systematically increased by $\sim$50% for clusters with ages less than 1–2 Gyr [@maeder:1974; @naylor:2009]. The LDB method offers a means to critically test the MSTO technique and it has several advantages. First, the physical processes involved in MSTO and LDB stars are completely different and so are independent. Second, the fundamental physics underpinning the LDB method is much more simple and straight-forward to compute and calibrate than for hot, high-mass stars because stars lying close to the LDB point in young clusters are fully convective. While the exteriors of very-low-mass stars ($< 0.2 M_\odot$) may host complex magnetic activity phenomena that are challenging to understand, their interiors are fairly straightforward. Third, the nature of open cluster mass functions implies that there are many more stars that can be exploited to establish the LDB than there are at the MSTO [@soderblom:2010; @soderblom:2013]. However, one may obtain a relatively sparse data set due to efforts of removing field star contaminants. The LDB method has limitations in its applicability. It can only be applied to very young clusters because of the rapidity of Li depletion. Furthermore, very-low-mass stars at the LDB are extremely faint, so only very nearby clusters are amenable to observation, usually by 8–10m class telescopes. Such stars are typically mid-M dwarfs in the cluster, and for physical and practical reasons, the LDB method is limited in its utility for ages in the range $20 <\tau< 200$ Myr. Currently, seven other open clusters and two moving group associations have LDB age determinations, but only the Pleiades is similar in age [125$\pm$8 Myr, @stauffer:1998aa] [126$\pm$11 Myr, @burke:2004vf] [gyrochronology age of $134^{+9}_{-10}$ Myr, @cargile:2014] to Blanco 1. As yet, no open cluster with an isochrone age $>$130 Myr has been investigated using the LDB method. Blanco 1 is an open cluster whose near-solar composition, \[Fe/H\] $=+0.04\pm0.04$ [@ford:2005], and age similarity to the Pleiades make it ideal for direct comparison and systematic characterization of age diagnostics. Previous age estimates for Blanco 1 suggest that it is a relatively young open cluster [100–150 Myr, @panagi:1997; @moraux:2007]. Blanco 1 is also considered nearby at a modest distance of 207 pc [@van-leeuwen:2009], and lies at high Galactic latitude ($b=-79^\circ$). Initially, a subset of low-mass Blanco 1 candidates were analyzed, and the LDB age was determined to be 132$\pm$24 Myr [@cargile:2010]. In this manuscript, we present additional Gemini-N spectra of Blanco 1 LDB candidates, and describe a consistent analysis for the full sample of Blanco 1 spectroscopic observations, which allows us to further resolve the LDB location and derive a more precise LDB age for the cluster. Despite the similarity of derived LDB ages among different PMS models, most models do not account for physical processes in an inclusive and realistic stellar environment, such as rotation and magnetism prevalent in low-mass star PMS evolution. Such omissions have the potential to affect the rate of Li depletion and thus, the age inferred from the LDB. It is well-established that magnetic activity can influence stellar parameters, particularly the radius ($R$) and effective temperature ($T_{\rm eff}$) of a star [@morales:2008]. One additional goal of this project is to quantify the extent that activity influences these stellar parameters, allowing us to correct for the age determinations based on the LDB technique. Using empirical relationships presented in @stassun:2012, we will account for the magnetic activity by effectively determining the properties of inactive Blanco 1 stars. In doing so, we enable a consistent LDB age determination from standard PMS models. In Section \[data\], we describe the data arising from our new medium-resolution spectroscopic campaign of additional Blanco 1 low-mass candidate members. In Section \[analysis\], we present our analysis, emphasizing the [$\rm H\alpha$]{} and [Li[i]{}]{} features, which are important to the astrophysical interpretations for inferring the LDB age of Blanco 1. In Section \[results\], we present the details of a clear methodology for establishing LDB boundaries as well as the derivation of the LDB age for our sample; we then present the empirical corrections for magnetic activity and derive a new LDB age based on the changes found in $T_{\rm eff}$ and $R$ for the stars which define the LDB boundaries. We conclude the manuscript with a summary of our work in Section \[summ\]. Targets, Observations, and Data Reduction {#data} ========================================= A photometric catalog of the very-low-mass members of Blanco 1 was compiled by @moraux:2007, where they selected cluster candidates on the basis of their location in CMDs compared to theoretical isochrones (100 and 150 Myr). @moraux:2007 furthermore took low-resolution optical spectra for 17 of the brightest brown dwarf candidates and found [$\rm H\alpha$]{} in emission for 5 of them, which is an initial indicator of youth. Their list of 15 probable members straddling the substellar boundary provides us with an ideal sample for investigating the LDB of Blanco 1. These probable low-mass cluster members have $I$$\approx$18–20, corresponding to the expected luminosity of Blanco 1 LDB, which in a cluster of age $\sim$100 Myr at a distance of $\sim$200 pc should be $I$$\approx$19 [@burke:2004vf; @cargile:2010]. For a subset of these objects, @cargile:2010 have previously presented medium-resolution spectra of the [Li[i]{}]{}(6707.8Å) region. In this work, we have obtained additional spectra using the same instrument and setup as were employed for the Cargile et al. study. Both the previous and new spectra were obtained with the Gemini Multi-Object Spectrograph (GMOS) in queue schedule mode on the Gemini-North telescope [@hook:2004], under program IDs GN-2009B-Q-53 and GN-2010B-Q-96. We used $1''$ slitlets to yield a two-pixel resolving power of $\simeq$4400 over a spectral wavelength range of 5700–8000Å and dispersion of 0.67Å per pixel. Moreover, a recent optical survey performed using the SMARTS 1.0m telescope at CTIO provided additional candidates with $I$$\approx$13.0–17.5; these targets were identified as photometric candidate members from their location near to the cluster sequence in an optical color magnitude diagram (CMD, James et al. in prep). Altogether, our sample contains 43 spectra (13 targets selected from Moraux et al., 30 from the SMARTS survey), from which, we find 14 high confidence members of the Blanco 1 cluster. In addition, we retain spectra of the radial velocity (RV) standard star GJ 905 (M6) from our initial GN-2009B-Q-53 program, which was observed and analysed in an identical manner to the Blanco 1 candidates. All of our GMOS spectra are reduced using standard reduction routines in the IRAF[^1] Gemini-GMOS package, including bias removal, flat-fielding, aperture extraction, and wavelength calibration. Our spectral signal-to-noise ratios (SNR) per pixel ranged from approximately 10 to 500 for the faintest and brightest targets, respectively. RVs for each Blanco 1 target were measured using the `fxcor` procedure in IRAF by cross-correlating target GMOS spectra with the RV standard star, GJ 905. We performed cross-correlation in the wavelength region $\sim$6600–7000Å, masking out regions rich in telluric features. Uncertainties on these RVs can be relatively large (up to $\sim$15 km s$^{-1}$), which is primarily due to the low SNR of the target spectra and the medium resolution of our observations. Analysis ======== We developed a spectral analysis code in Python to completely automate the analysis method in order to consistently derive equivalent widths and spectral indices. The spectral type of the object is determined via TiO and CaH spectral indices, whose methodology we describe in § \[spec\]. Equivalent widths (EWs) of the [$\rm H\alpha$]{}(6562.8Å) and [Li[i]{}]{}(6707.8Å) spectral features are measured systematically, using established wavelength regions flanking both features to carry out the linear normalization procedure to the pseudo-continuum. Systematic EW measurement and error estimation for the [$\rm H\alpha$]{} line is discussed in § \[ha\]. Cluster membership criteria are laid out in § \[mem\], where we identify high confidence cluster members. In § \[lithium\], we describe the measurement of the [Li[i]{}]{} feature, which we then place in the context of the curve of growth to derive lithium abundance, A(Li), in section \[liab\], allowing us to provide insight as to the stage of Li depletion for our targets. In section \[activity\], we describe how we obtain $\log L_{\rm H\alpha}/L_{\rm bol}$ values, which are used to account for magnetic activity and determine their concomitant changes in $R$ and $T_{\rm eff}$. Observational and empirical data for the cluster members are summarized in Table \[b1\], which include positions, photometric properties, spectral types, RVs, EW([$\rm H\alpha$]{}), and EW([Li[i]{}]{}) for each object. We include 1$\sigma$ errors for both the [$\rm H\alpha$]{} and [Li[i]{}]{} EW measurements for completeness, with 3$\sigma$ upper limits reported for Li non-detections. [lccccccccccc]{} B1opt-[**6335**]{} & 00:00:28.868 & -30:08:30.01 &$13.156\pm0.011$&$1.724\pm0.024$& K5.5 & $-10 \pm 8$ & $-2.39 \pm 0.06$ & $-3.6759$ &$0.087\,^{+0.087}_{-0.077}$ &$ 0.945\,^{+0.461}_{-0.592}$\ B1opt-[**18229**]{} & 00:01:39.768 & -30:04:38.24 &$13.315\pm0.016$&$1.703\pm0.029$& K5.3 & $ 15 \pm 12$ & $-1.19 \pm 0.12$ & $-3.9698$ & $<0.249$ &$<-1.348$\ B1opt-[**2156**]{} & 00:07:40.790 & -30:05:56.58 &$14.45\pm0.030$&$2.09\pm0.042$& M0.4 & $ 7 \pm 6$ & $-3.88 \pm 0.06$ & $-3.6883$ & $<0.177$ &$<1.593$\ B1opt-[**13328**]{} & 00:04:22.733 & -30:23:06.00 &$15.86\pm0.001$&$2.27\pm0.051$& M3.8 & $ 1 \pm 5$ & $-7.89 \pm 0.04$ & $-3.5062$ & $<0.054$ &$<0.698$\ CFHT-BL-[**16**]{} & 00:01:28.438 & -30:06:06.95 & 18.30 & 2.85 & M5.1 & $ 4 \pm 6$ & $-4.50 \pm 0.14$ & $-4.1908$ & $<0.357$ &$<1.727$\ CFHT-BL-[**22**]{} & 00:00:02.661 & -30:20:15.90 & 18.47 & 2.90 & M5.6 & $ 24 \pm 5$ & $-6.22 \pm 0.07$ & $-4.0863$ &$0.320\,^{+0.137}_{-0.148}$ &$1.387\,^{+0.587}_{-0.566}$\ CFHT-BL-[**24**]{} & 00:07:50.539 & -30:05:09.46 & 18.51 & 2.95 & M6.0 & $ 3 \pm 8$ & $-6.47 \pm 0.13$ & $-4.1053$ & $<0.291$ &$<1.374$\ CFHT-BL-[**25**]{} & 00:00:02.844 & -30:17:43.98 & 18.62 & 3.06 & M5.6 & $ 28 \pm 6$ & $-6.26 \pm 0.16$ & $-4.1991$ & $<0.294$ &$<1.313$\ CFHT-BL-[**29**]{} & 00:00:17.351 & -30:46:20.32 & 18.77 & 3.03 & M6.2 & $ 29 \pm 7$ & $-5.58 \pm 0.15$ & $-4.2268$ & $<0.333$ &$<1.490$\ CFHT-BL-[**38**]{} & 00:05:13.037 & -30:27:35.78 & 18.98 & 3.10 & M6.4 & $ 7 \pm 11$ & $-4.31 \pm 0.22$ & $-4.3900$ &$0.957\,^{+0.170}_{-0.145}$ &$2.790\,^{+0.241}_{-0.285}$\ CFHT-BL-[**43**]{} & 00:04:32.849 & -30:18:41.40 & 19.02 & 3.13 & M6.3 & $ 7 \pm 10$ & $-6.39 \pm 0.18$ & $-4.2404$ &$1.183\,^{+0.182}_{-0.122}$ &$2.977\,^{+0.245}_{-0.243}$\ CFHT-BL-[**36**]{} & 00:00:28.585 & -30:06:41.94 & 19.06 & 3.37 & M6.0 & $ 24 \pm 7$ & $-5.53 \pm 0.09$ & $-4.4539$ & $<0.213$ &$<0.827$\ CFHT-BL-[**45**]{} & 00:01:35.611 & -30:03:09.90 & 19.23 & 3.27 & M6.2 & $ 25 \pm 12$ & $-5.26 \pm 0.37$ & $-4.4150$ &$1.521\,^{+0.183}_{-0.117}$ &$3.163\,^{+0.207}_{-0.215}$\ CFHT-BL-[**49**]{} & 00:04:28.858 & -30:20:37.00 & 19.46 & 3.56 & M6.3 & $ 3 \pm 18$ & $-2.42 \pm 0.66$ & $-4.9226$ &$1.930\,^{+0.120}_{-0.166}$ &$3.197\,^{+0.190}_{-0.192}$\ \[b1\] Spectral Indices {#spec} ---------------- Spectral types are estimated from the value of the TiO (7140Å) and CaH (6975Å) narrow-band spectral indices. They are defined as $$\rm TiO(7140\AA) = \frac{C(7020\!-\!7050\AA)}{TiO(7125\!-\!7155\AA)},\;\; CaH(6975\AA) = \frac{C(7020\!-\!7050\AA)}{CaH(6960\!-\!6990\AA)},$$ where C(7020–7050Å) represents the pseudo-continuum, and TiO(7125–7155Å) and CaH(6960–6990Å) represent the molecular absorption bands, integrated in the indicated wavelength intervals [@briceno:1998; @oliveira:2003]. The CaH index is sensitive to gravity and helps us verify that the objects that we observed are, in fact, dwarfs. However, these narrow-band indices are not by themselves good indicators of cluster membership since the sample is sure to be contaminated with other foreground field M-dwarfs with similar index values [@jeffries:2004]. The spectral type of each target is estimated from the relationship between TiO (7140Å) index and spectral type calibrated from standards in @montes:1997fk and @barrado-y-navascues:1999sj [see Table 6 in @jeffries:2013]. The resulting spectral types are reported in Table \[b1\]. We adopt a typical uncertainty of half a spectral subclass [@oliveira:2003; @jeffries:2005]. The CaH versus TiO spectral indices for the Blanco 1 sample are plotted in Figure \[f1\], with reference to an H$\alpha$ feature annotated (see § \[ha\]). Stars with [$\rm H\alpha$]{} in absorption are likely cluster non-members because at the age of Blanco 1, we expect such low-mass [*bona fide*]{} cluster members to be chromospherically active. In addition, zero-[$\rm H\alpha$]{} stars can be very active stars with strong chromospheres as the [$\rm H\alpha$]{} core may be filled-in by active region emission [@panagi:1997]; hence, such objects may be young cluster members as well. We return to [$\rm H\alpha$]{} as a membership criterion in section \[mem\]. ![Spectral indices CaH versus TiO for our sample of low-mass Blanco 1 candidates. The spectral type for a given object is determined with the TiO index, while the CaH index can be used to eliminate background giant stars from out sample. Note the transition of the [$\rm H\alpha$]{} feature from absorption to zero to emission as one tends to higher CaH-TiO index (or later M spectral types). The solid line represents the locus of approximate positions for giant stars in CaH-vs-TiO space [@allen:1995].[]{data-label="f1"}](spec.eps) The H$\boldsymbol\alpha$ feature {#ha} -------------------------------- As well as establishing cluster membership, [$\rm H\alpha$]{} EW can be employed in empirical corrections for magnetic activity (discussed in section \[results\]). Our method of obtaining [$\rm H\alpha$]{} EW consistently is achieved by performing continuum normalization with a 10Å-span of wavelength neighboring the [$\rm H\alpha$]{} feature. We use the intervals 6545–6555Å and 6570–6580Å and find no significant issues with other spectral features within these intervals. The mean is calculated from each 10Å-segment, and the line connecting both mean values establishes the continuum level. The [$\rm H\alpha$]{} EW is then determined by measuring excess from the normalized continuum using a Gaussian line-profile, which is obtained from a minimized least-squares fit. A simple interpolation is performed at the boundaries of the [$\rm H\alpha$]{} feature so that the baseline will exactly measure flux above unity (for emission) or below unity (for absorption). Figure \[22ha\] demonstates such an EW measurement process for star 22. EW uncertainties are estimated from $$\label{dew} \rm \sigma_{\rm EW} \simeq 1.5 \times \sqrt{FWHM \times p}/SNR,$$ where FWHM, p, and SNR are the full-width half-maximum of the Gaussian fit, the pixel dispersion scale in Å, and the signal-to-noise ratio, respectively [@cayrel:1988]. ![[$\rm H\alpha$]{} emission from low-mass Blanco 1 member, target 22. The [$\rm H\alpha$]{} EW is measured over the wavelength range where the normalized continuum exceeds unity.[]{data-label="22ha"}](22-4ha.eps) Membership Selection {#mem} -------------------- The stars from our GMOS sample can be classified as Blanco 1 cluster members upon consideration of three different membership criteria: (1) that the photometry of a candidate member is consistent with the cluster sequence in an $I/I-K\rm_s$ CMD; (2) its 3$\sigma$ RV must be within the range of +2 to +10 km s$^{-1}$; (3) the [$\rm H\alpha$]{} line EW must be in emission or zero and comparable to similar-mass stars in the Pleiades cluster. We note that Blanco 1 has a measured systemic velocity around +6 km s$^{-1}$ [@mermilliod:2008; @gonzalez:2009], but given the difficulty of determining RVs from low SNR spectra, we consider all stars within +2 to +10 km s$^{-1}$ as candidate cluster members. Each of these membership criterion has its own level of field star contamination, so each individual property is considered necessary but not sufficient for cluster membership. We combine the three criteria so that stars exhibiting these properties are considered high confidence single-star Blanco 1 members. In Figure \[cmd0\], we plot an intrinsic $I/I-K \rm_s$ cluster sequence for cluster members. We correct for reddening and extinction by applying $E(I-K_{\rm s})=0.02$ and $A_I=0.03$ to obtain the intrinsic photometry. The RV distribution for Blanco 1 members is shown in Figure \[rv\]. Most of these low-mass stars fall safely within the range of our RV criterion (shown by the shaded band), but a few of them appear to be marginal RV members. Figure \[has\] shows the distribution of EW([$\rm H\alpha$]{}) for our sample of Blanco 1 members and some low-mass members in the Pleiades [@stauffer:1998aa]. These clusters share a similar age, and a given EW([$\rm H\alpha$]{}) is expected to be comparable to similar-mass stars amongst these populations. Our sample of low-mass members of Blanco 1 exhibits very active chromospheres at mid-M spectral types. This appears to be comparable to the low-mass activity found in the Pleiades. Recorded in Table \[b1\] are our measurements of EW([$\rm H\alpha$]{}) for high confidence Blanco 1 members. ![Intrinsic photometry for Blanco 1 low-mass members. The Pleiades single-star locus is plotted as the dashed line [@stauffer:2007] and is shifted appropriately for the distance to Blanco 1 [207 pc, @van-leeuwen:2009]. The stand-alone error bar represents the estimated uncertainty in photometry.[]{data-label="cmd0"}](cmd_0.eps) ![RV distribution for Blanco 1 low-mass members (brighter to fainter going upward). The shaded band represents our velocity range for determining RV membership. Bold error bars convey 1$\sigma$ errors, while thin error bars show the 3$\sigma$ range.[]{data-label="rv"}](Blanco1_RV.eps) ![[$\rm H\alpha$]{} EW versus spectral type for high confidence low-mass members of Blanco 1. All of the stars exhibit [$\rm H\alpha$]{} emission as expected from chromospheric activity in young stars. For comparison, we plot several active low-mass Pleiades members from @stauffer:1998aa.[]{data-label="has"}](ewha.eps) Lithium ------- ### EW Measurement via Spectral Subtraction {#liew} A spectral subtraction technique is carried out in our study by using a catalog of M-dwarf templates from the Sloan Digital Sky Survey [SDSS, @bochanski:2007]. These templates were produced by averaging over 4000 SDSS stellar spectra for spectral types M0–L0. In particular, we use the catalog of [*inactive*]{} spectra for spectral types M0–M7, where the measured EW of the [$\rm H\alpha$]{} feature was $<$1Å in emission. Moreover, since the majority of the combined spectra used for these templates are field M-dwarfs, they are expected to be old enough ($\sim$Gyrs) to have destroyed their initial lithium. Due to the lack of K-type templates, we must compare the K-type stars with the M0 SDSS template. Otherwise, we paired a given GMOS spectrum with a template by rounding to the nearest spectral type determined from the TiO spectral index described in section \[spec\]. In the determination of the [Li[i]{}]{} EW, the spectrum of the target is shifted to the rest frame, normalized, smoothed, and compared with a SDSS template spectrum. Both the target and template are normalized by using small wavelength spans bounding [Li[i]{}]{} (6707.8Å), specifically 6703–6706Å and 6710–6712Å. Data were smoothed with a Gaussian kernel, and the template is convolved to match the resolution of our GMOS spectrum. EWs are measured in the residual spectrum over an interval of $\sim$4Å centered on [Li[i]{}]{}. Figure \[rli\] shows an example of measuring the Li EW for star 22 (M5.6) and 6335 (K5.5), confirming detectable lithium in these objects for the first time. Present in some of our spectra with low SNR are telluric sky absorption lines near S[ii]{} that could not be removed because of poor sky-subtraction. The error quoted for EW(Li) in Figure \[rli\] is estimated using equation \[dew\]. ![Analysis of the [Li[i]{}]{} feature. Spectral subtraction of SDSS inactive template spectrum (teal line) from our GMOS target spectrum (blue line) enables the measurement of EW(Li). The TiO index determines which template is most appropriate for subtraction. The measured EW(Li) is indicated by the shaded region in the residual. The absorption features denoted by Earth symbols are telluric lines near S[ii]{} that could not be subtracted.[]{data-label="rli"}](22-4rli.eps "fig:") ![Analysis of the [Li[i]{}]{} feature. Spectral subtraction of SDSS inactive template spectrum (teal line) from our GMOS target spectrum (blue line) enables the measurement of EW(Li). The TiO index determines which template is most appropriate for subtraction. The measured EW(Li) is indicated by the shaded region in the residual. The absorption features denoted by Earth symbols are telluric lines near S[ii]{} that could not be subtracted.[]{data-label="rli"}](36-5rli.eps "fig:") In figure \[lipanel\], we show the six low-mass Blanco 1 members that contain detectable Li. Telluric S[ii]{} features are indicated. It is evident that as one goes fainter, the signal in Li becomes more significant. On the other hand, the SNR diminishes, increasing the difficulty to match the continuum. Targets 38, 43, 45, and 49 were reported to have Li detected $>$$3\sigma$ in @cargile:2010, although EW measurements at the time were not possible. Now, we have identified two additional members (targets 22 and 6335). In section \[results\], we explain how target 22 in particular influences the location of the LDB. ![Blanco 1 low-mass members exhibiting [Li[i]{}]{} absorption (grey dashed line). Each Blanco 1 GMOS spectrum is indicated along with its template (solid black and grey lines, respectively) and intrinsic $I$ magnitude.[]{data-label="lipanel"}](lipanel.eps) ### EW Measurement via MCMC {#liMCMC} Due to the low signal-to-noise of the spectrum around the [Li[i]{}]{} line (typically $\sim$10 for the faintest Blanco 1 stars), we sought to provide a robust characterization of our Li EW measurements. Here, we incorporate an affine-invariant MCMC to sample the posterior probability distribution functions for our [Li[i]{}]{} equivalent widths using the `emcee`[^2] package developed by @foreman-Mackey:2013. After we subtract the appropriate template for each target spectrum, we model the resulting residual with a Gaussian as a likelihood function, and calculate 80,000 samples (400 MCMC “walkers” $\times$ 200 iteration steps) of the posterior probability distribution. We place an uninformative prior on the amplitude of our Gaussian model, constraining it to only consider Li in absorption, as well as normal priors on the Gaussian $\sigma$ and centroid based on [*a priori*]{} knowledge of the GMOS instrument resolution and predicted 6707.8Å Li line center, respectively. For each star, we set a conservative estimate of the variance in our flux measurement based on a SNR$=$10. We show in Figure \[limcmc\] examples of the marginalized posterior distributions for a Gaussian model of the Li absorption line, as well as an inferred equivalent width distribution based on the predicted cumulative function. The Li equivalent widths we report in Table \[b1\] are determined from the mode of the marginalized distribution with uncertainties based on the inter-68$\rm^{th}$ percentile range (1$\sigma$ errors). ![ Posterior probability distributions for the Gaussian parameters for our modeling of Targets 22 and 6335. Best-fit values for the parameters are based on the mode of the distributions (red line) and formal 68$\rm^{th}$ percentile uncertainty ranges are indicated with shaded regions. \[limcmc\] ](22.eps "fig:") ![ Posterior probability distributions for the Gaussian parameters for our modeling of Targets 22 and 6335. Best-fit values for the parameters are based on the mode of the distributions (red line) and formal 68$\rm^{th}$ percentile uncertainty ranges are indicated with shaded regions. \[limcmc\] ](6335.eps "fig:") In Figure \[lis\], we display the distribution of EW(Li) for our modeling of Blanco 1 cluster members. A clear pattern is apparent is these data; namely, we detect little or no Li in earlier spectral types ($\lesssim$M6), but measure significant Li absorption in the latest spectral type stars in Blanco 1. In Section \[results\], we further investigate the quantitative nature of this distribution in the context of predictions from PMS Li models. However, the overall spectral type dependent transition in the EW(Li) of Blanco 1 stars is qualitatively consistent with the identification of the lithium depletion boundary in the cluster. ![ Measured [Li[i]{}]{} equivalent width versus spectral type for low-mass members of Blanco 1. Stars with detected Li are shown as blue points with 1$\sigma$ errors, and downward arrows indicate 3$\sigma$ upper limits for stars with no significant Li absorption based on their GMOS spectra. \[lis\]](ewli.eps) ### Lithium Abundance {#liab} In order to calculate Li abundances, it is necessary to convert intrinsic color to $T_{\rm eff}$. For the stars in our spectroscopic survey, we used BT-Settl models [@allard:2011] to obtain $T_{\rm eff}$. @jeffries:1999 performed a lithium study on G- and K-dwarf members of Blanco 1, but we use the empirical relationship given in @casagrande:2010 to derive $T_{\rm eff}$ for these stars. Abundances were calculated from EW(Li) using an appropriate curve of growth for the [Li[i]{}]{} (6707.8Å) feature; for hotter stars ($T_{\rm eff}$$>$4000K), we used the calculations given in @soderblom:1993, and for cooler objects, we used the models presented in @pavlenko:1995 and @pavlenko:1996. We note that our procedure of measuring EW(Li) after subtracting a template spectrum has the effect of mitigating the contribution of the nearby contaminating Fe line at 6707Å, as well as the large molecular TiO absorption that is present in the Li region. Non-LTE corrections for Li abundances presented in @carlsson:1994oa were applied to the hotter Blanco 1 stars. For the cooler stars, we did not correct the abundances for non-LTE effects as these are negligible at cool temperatures [@pavlenko:1995; @zapatero-osorio:2002]. We adopt an initial Li abundance of $\log N_{0}({\rm Li}) = 3.1$ for the cluster [@zapatero-osorio:2002].\ Figure \[abundance\] shows the distribution of Li abundance for Blanco 1 versus absolute $I$ magnitude, $M_{I_C}$. Here, the three regimes of Li depletion are present: (1) stars more massive than 0.6 $M_\odot$ gain radiative interiors and only lose a small amount of Li depletion [green squares, @jeffries:1999]; (2) stars in the ‘Li chasm’ [@basri:1997] that have fully depleted their initial Li supply ($7\lesssim M_{I_C}\lesssim11$); (3) low-mass stars that still exhibit Li content ($M_{I_C}\gtrsim12$). It is evident that for hotter stars ($M_{I_C}<6$), with the exception of a few points, the models fail to reproduce the overall observed abundance distribution in the cluster. The Li detections near the substellar boundary ($M_{I_C}\approx12$) suggest that target 22 is currently depleting Li, while the fainter Li detections lie near full natal Li abundance. ![Li abundances for low-mass stars in Blanco 1 shown versus $M_{I_C}$ magnitudes for the HIPPARCOS distance modulus to the cluster [6.58 mag, @van-leeuwen:2009]. Blue circles represent stars in our new sample with Li detections (downward arrows are 3$\sigma$ upper limits), and data from @jeffries:1999 [green squares] show Li abundances for G- and K-dwarf cluster members. The Li depletion boundary is located near $M_{I_C}$$\approx$12 with fainter stars retaining their full natal Li abundance.[]{data-label="abundance"}](liab.eps) Results ======= Locating the LDB {#locate} ---------------- In the identification of the LDB for this study, we establish a set of rules to demarcate the boundaries of the LDB. First, having identified the cluster members, we concentrate on the targets that contain lithium. Based on the insight we have gained from the lithium abundance of Blanco 1 members, the LDB boundaries are set in the following way: the target currently depleting lithium (star [**22**]{}) establishes the bright, blue (upper left) corner; the nearest target in the CMD with full lithium content (star [**38**]{}) establishes the faint, red (lower right) corner. The edges of the LDB box incorporate the photometric uncertainties in the stars defining these corners (stars 22 and 38): $\sigma_{K\rm_s}$=0.03, $\sigma_{I}$=0.04, $\sigma_{I-K\rm_s}$=0.05. We define the center of this box to be the location of the LDB in Blanco 1, the brightest luminosity at which Li content still remains unburned in the atmospheres of low-mass stars. Figure \[cmd\] shows the CMD for intrinsic $I$-band magnitude versus $I-K\rm_s$ for the stars in our sample that have Li detections among the Blanco 1 low-mass members. The Li detection at $I_0\approx13$ is from a K-dwarf in the cluster. This star formed a radiative core early enough in its PMS evolution to stop convection down to the stellar depth necessary to burn Li, and thus still retains some of its Li content. We also illustrate in Figure \[cmd\] our definition of the LDB region as the shaded box. ![The intrinsic $I/I-K\rm_s$ CMD for the low-mass members of Blanco 1 which exhibit Li in their spectra. The shaded rectangle represents the uncertainty on the position of the LDB (red star) as established by targets 22 and 38. The stand-alone error bar represents the photometric uncertainty. The Pleiades single-star locus from @stauffer:2007 is plotted (dashed line), as well as the @baraffe:1998aa predicted luminosity loci for the LDB (solid lines) corresponding to the given ages in Myr.[]{data-label="cmd"}](cmd-li.eps) LDB Ages for Blanco 1 --------------------- ### Standard LDB Age Previously in @cargile:2010, with a limited data set, the authors provided a preliminary identification of the LDB in Blanco 1 and found it to be located at $I=18.78\pm0.24$ and $I-K\rm_s=3.05\pm0.10$. They calculated the absolute $I$ magnitude, $M_{I_C}$, of the LDB using the distance modulus from HIPPARCOS [6.58$\pm$0.12, @van-leeuwen:2009] and corrected for reddening and extinction by adopting $E(I-K\rm_s)=0.02$ and $A_I=0.03$. Using predicted Li-depletion rates from PMS models, specifically @chabrier:1997 and @baraffe:1998aa [hereafter BCAH], @cargile:2010 used the luminosity at which 99% of the star’s natal Li is destroyed to measure the LDB age for Blanco 1 to be 132$\pm$24 Myr. We designate their method as the ‘standard’ LDB age determination technique. Here, we determine the standard LDB age using a similar approach to @cargile:2010 Using our stars 22 and 38 to establish the LDB boundaries, the updated Blanco 1 LDB is located at $I_0 = 18.69\pm0.26$ and $(I-K\rm_s)_0 = 2.95\pm0.10$. We determine $M_{I_C}$ using the same distance modulus from HIPPARCOS, as well as extinction and reddening corrections used in @cargile:2010 We first calculate the LDB age of Blanco 1 using the BCAH models and synthetic photometry from the DUSTY model atmospheres [@baraffe:2002]. Alternatively, we also calculate the LDB age using the empirical bolometric corrections from @pecaut:2013 [hereafter P&M] to derive luminosity directly from our absolute $I$ magnitudes. In Table \[ldbp\], we list our measured LDB parameters for Blanco 1 using the BCAH PMS models with both synthetic photometry and using empirical corrections from P&M. For clarity, Figure \[zoomI\] shows the region of the $I/I-K\rm_s$ CMD near the LDB of Blanco 1. As in Figure \[cmd\], the LDB is established by the Li detections in targets 22 and 38. Using the ‘standard’ LDB technique, the BCAH PMS models with synthetic photometry, and our new Li detections, the LDB in Blanco 1 is found at a $\log(L) = -2.910 L_\odot$, resulting in an updated LDB age of $126^{+13}_{-14}$ Myr. We have included a 126 Myr BCAH LDB luminosity locus in Figure \[zoomI\] to illustrate this age measurement, which is strikingly similar to the age of the Pleiades [126$\pm$11 Myr, @burke:2004vf]. One might instead consider that the position of the LDB could be defined entirely by Target 22, given that this object evidently lies within the Li depletion zone (see Figure \[abundance\]). For stars at full natal Li abundance, A(Li) = 3.1, so the LDB (defined at 99% depletion) is found when A(Li) = 1.1. Conceivably, the absolute $I$ magnitude range for the LDB would be bounded by the abundance errors for star 22. We interpolate over the abundance isochrones to calculate $M_{I_C}$ for A(Li) = 0.821, 1.1, and 1.974 dex, which correspond with the error bounds of star 22 and the 99% Li depletion level. The 110 Myr model isochrone matches well with the data, and we find at A(Li) = 1.1 that $dM_{I_C}/d{\rm A(Li)} = 0.214$ mag/dex. Thus, the LDB using this interpretation is $I_0 = 18.45^{+ 0.19}_{-0.09}$ mag, and the corresponding LDB age is $114\pm7$ Myr. While this abundance-derived age is in statistical agreement with our standard LDB age, the reported error (6.1%) is smaller than the 10% systematic error found for the uncertainties associated with the stellar evolution models and bolometric corrections [@burke:2004vf]. Therefore, we prefer the more conservative approach described above since our method results in the observed precision of the LDB age that is no better than the predicted accuracy of the LDB technique at $\sim$120 Myr. ### Activity-Corrected LDB Age {#activity} Investigations have found that the fundamental properties of low-mass stars can be altered in the presence of strong magnetic activity [@lopez-morales:2007; @ribas:2006]. @morales:2008 have provided observational evidence that active stars are cooler than inactive stars of similar luminosity, therefore, implying that active stars have a larger radius. Their results generalize for all active low-mass stars – single or binary. In the context of the LDB, we thus expect that active stars would be more massive than initially thought, and their associated ages would be younger. @stassun:2012 provide empirical relations to determine the amount by which the effective temperatures ($T_{\rm eff}$) and radii ($R$) of low-mass stars and brown dwarfs are altered due to chromospheric activity. Their results presented a strong correlation between the strength of [$\rm H\alpha$]{} emission in active M-dwarfs, and the degree to which their temperatures are [*suppressed*]{} and radii [*inflated*]{} compared to inactive stars. In order to determine the change in $T_{\rm eff}$ and $R$ as a result of stellar activity, the following empirical relations were implemented: $$\label{delt} \Delta T_{\rm eff}/T_{\rm eff} = m_T \times (\log L_{\rm H\alpha}/L_{\rm bol} + 4) + b_T$$ $$\label{delr} \Delta R/R = m_R \times (\log L_{\rm H\alpha}/L_{\rm bol} + 4) + b_R,$$ where $m$ and $b$ are linear coefficients. The averaged values, as defined in @stassun:2012, are in percent units: $m_T = -4.71 \pm 2.33$, $b_T = -4.4 \pm 0.6$, $m_R = 15.37 \pm 2.91$, and $b_R = 7.1 \pm 0.6$. We translate our measured [$\rm H\alpha$]{} EW to $\log L_{\rm H\alpha}/L_{\rm bol}$ using a grid of BT-Settl model atmospheres from @allard:2011 for $T_{\rm eff}$ in the range 2200–5000K, assuming Solar composition and $\log(g)=5.0$ (appropriate for very-low-mass stars in Blanco 1). First, we compute the bolometric flux ($F_{\rm bol}$) for these model atmospheres. Then, for a given GMOS target, we use its color to estimate $T_{\rm eff}$ from a BCAH 135 Myr isochrone. This $T_{\rm eff}$ is overestimated since the activity would suppress it, but this is a small effect ($\sim$0.1 dex in $\log L_{\rm H\alpha}/L_{\rm bol}$ for a $\sim$200K shift). We use this $T_{\rm eff}$ to interpolate over the model atmospheres to estimate the atmospheric continuum flux at the [$\rm H\alpha$]{} feature ($F_{\rm \lambda, H\alpha}$). The [$\rm H\alpha$]{} flux ($F_{\rm H\alpha}$) is computed by convolving $F_{\rm \lambda, H\alpha}$ with the [$\rm H\alpha$]{} EW of our target. Finally, by computing $\log F_{\rm H\alpha}/F_{\rm bol}$, we also obtain the equivalent $\log L_{\rm H\alpha}/L_{\rm bol}$. Propagating the photometric uncertainty in color, we find an error of $\sim$0.03 dex in $\log L_{\rm H\alpha}/L_{\rm bol}$, but this is much smaller than the systematic contribution of $\sim$0.4 dex in the transformation of color to temperature. Hence, the total systematic error for $\log L_{\rm H\alpha}/L_{\rm bol}$ is about 0.5 dex. From the empirical relationships, we find the percent change in $T_{\rm eff}$ (suppression) and $R$ (inflation) as a result of magnetic activity along with the percent change in luminosity. Due to the nature of how equations \[delt\] and \[delr\] were derived and calibrated, the activity corrections should only be applied to stars with [$\rm H\alpha$]{} in emission. We then use this information to determine the 135 Myr BCAH magnitudes and colors of our Blanco 1 sample as if these stars were inactive; we remove the effects of activity. Using the same logic as before, we set the ‘corrected’ LDB boundaries using the corrected, inactive photometry for targets 22 and 38. We determine the LDB parameters at this new LDB location and record these values in Table \[ldbp\]. We infer the activity-corrected LDB is located at $I_0 = 18.45 \pm 0.16$ and $(I-K\rm_s)_0 = 2.77 \pm 0.12$, which corresponds to $\log(L)=-2.818 L_\odot$ and the BCAH LDB age of $114^{+9}_{-10}$ Myr. Figure \[zoomI\] shows a closer view on the regions of the LDB for the intrinsic $I/I-K\rm_s$ CMD. This plot shows the Li detections for both the original (black points) and activity-corrected (yellow points) photometric positions. For simplicity, vectors showing the direction of the activity corrections are drawn only for targets 22 and 38, which establish the LDB boundaries in the CMD, and this is shown by the shaded boxes. The LDB positions (red stars) are marked within these boxes, and BCAH isochrones are drawn to show the age we infer from their predicted luminosities. Accounting for the effects of chromospheric activity mainly shifts the data upward along the cluster sequence. Additionally, the boundaries of the LDB are compacted when the effects of magnetic activity are removed and renders a corrected, more precise age. We carried out this same process of characterizing the LDB for $K\rm_s$ versus $I-K\rm_s$. LDB parameters derived from PMS models using $K\rm_s$ are recorded in Table \[ldbp\]. ![Zoomed in on the regions of the LDB for the Blanco 1 stars with Li detections. Stars are shown at their observed CMD position (black points), and our ‘Standard LDB’ location is indicated. Also shown are the shifted locations of the Blanco 1 stars after removing the effects of magnetic activity (yellow points). For simplicity, arrows showing the effect of activity are drawn only for targets 22 and 38. Accounting for stellar activity generally shifts the star’s colors/magnitudes brighter and blueward along the cluster sequence. We also show the LDB position after removing the effects of activity on Blanco 1 stars (‘Corrected LDB’). The error bar represents the photometric uncertainty, and the slanted error bars on the corrected data show the error due to the uncertainty in the @stassun:2012 empirical relationships.[]{data-label="zoomI"}](zoom-I.eps) Despite consistent treatments, use of the BCAH models along with DUSTY synthetic photometry renders statistically different LDB ages and parameters depending on whether we use $I_0$ or $K\rm_{s,0}$ magnitudes versus the $(I-K\rm_s)_0$ color as presented in Table \[ldbp\]. For the standard BCAH results, the $I_0$ and $K\rm_{s,0}$ LDB parameter values are statistically compatible to within 1$\sigma$ of their errors. Conversely, for the corrected BCAH results, the $I_0$ and $K\rm_{s,0}$ results differ greater than 1$\sigma$. Further work on understanding the reason for different age and parameter determinations at the LDB depending on the choice of photometry is required. [lcccc]{} & $I_0 = 18.685 \pm 0.255$ & $I_0 = 18.450 \pm 0.159$ & $K\rm_{s,0} = 15.685 \pm 0.155$ & $K\rm_{s,0} = 15.635 \pm 0.039$\ [**Location**]{}&$(I-K\rm_s)_0 = 2.950 \pm 0.100$ & $(I-K\rm_s)_0 = 2.765 \pm 0.120$ & $(I-K\rm_s)_0 = 2.950 \pm 0.100$ & $(I-K\rm_s)_0 = 2.765 \pm 0.120$\ & $M_{I_C}$ $= 12.114^{+0.280}_{-0.281}$, $^{+0.479}_{-0.575}$ & $M_{I_C} = 11.867\pm0.200$, $^{+0.285}_{-0.347}$ & $M_{K\rm_s} = 12.491^{+0.282}_{-0.290}$ & $M_{K\rm_s} = 12.408^{+0.182}_{-0.184}$\ [**Age**]{} \[Myr\]\ BCAH & $126^{+13}_{-14}$ & $114^{+9}_{-10}$ & $145^{+14}_{-15}$ & $141^{+9}_{-10}$\ P&M & $152^{+24}_{-32}$ & $124^{+14}_{-17}$ & &\ $\bm{\log(L)}$ \[$L_\odot$\]\ BCAH & $-2.910^{+0.062}_{-0.105}$ & $-2.818^{+0.077}_{-0.075}$ & $-2.993^{+0.066}_{-0.063}$ & $-2.975^{+0.042}_{-0.040}$\ P&M & $-3.026^{+0.120}_{-0.108}$ & $-2.898^{+0.080}_{-0.109}$ & &\ \[ldbp\] Summary and Conclusion {#summ} ====================== In this paper, we have expanded upon the initial identification of the Blanco 1 LDB [@cargile:2010]. We obtain the full sample of Blanco 1 candidates for our GMOS survey and analyzed both previous and new data consistently to update the inferred LDB age. This was done by developing spectral analysis software to systematically analyze the [$\rm H\alpha$]{} and [Li[i]{}]{} features. Moreover, we analyze the [Li[i]{}]{} feature using `emcee` to perform MCMC sampling on the gaussian parameters that measure EW(Li). We find that for Li detections with $I$$>$17, the error is reduced by up to a factor of 2 using MCMC as opposed to relying on the SNR estimate from equation \[dew\]. Since the [$\rm H\alpha$]{} region is simpler and has a higher SNR, a gaussian fit via least-squares to the [$\rm H\alpha$]{} line is sufficient.\ Out of the 43 spectra from our GMOS survey, we find 14 high confidence low-mass members belonging to Blanco 1, and 6 of these stars exhibit detectable Li features. Based on our systematic analysis of the [Li[i]{}]{} feature, we verify the findings of @cargile:2010 that targets 38, 43, 45, and 49 exhibit Li absorption with confidence $>$3$\sigma$. We have also obtained two new Li detections for low-mass Blanco 1 members; namely, the K-dwarf target 6335 and M-dwarf target 22. Importantly, target 22 influences how we determine the LDB age as it appears to currently be in the process of depleting its initial Li content.\ Using targets 22 and 38 to establish the LDB boundaries, we derive parameters at the LDB using the ‘standard’ technique. We first determine the LDB age of Blanco 1 using the BCAH models and synthetic photometry from the DUSTY model atmospheres, and also obtain the LDB age using the empirical bolometric corrections from P&M. Using the BCAH models and the synthetic photometry, we measure an updated standard LDB age for Blanco 1 of $126^{+13}_{-14}$ Myr. Compared with the Pleiades [126$\pm$11 Myr, @burke:2004vf], these open clusters share remarkable coevality.\ For the low-mass Blanco 1 members in our sample, empirical corrections from @stassun:2012 were used to determine the amount of suppression in $T_{\rm eff}$ and inflation in $R$ due to chromospheric activity as indicated by [$\rm H\alpha$]{} emission. We remove these effects and determine the photometric properties of our targets as if they were not active. Using the inactive properties of targets 22 and 38, we identify a ‘corrected’ LDB and infer a new age of $114^{+9}_{-10}$ Myr from BCAH models.\ This corrected age for Blanco 1 brings the LDB age and MSTO isochrone age [$\tau^2$ isochrone fitting with moderate convective-core overshoot; @naylor:2006; @naylor:2009] into close agreement ($\sim$110 Myr, James et al. in prep). On the other hand, the gyrochronology method from @cargile:2014 determined the age of Blanco 1 to be 146$^{+13}_{-14}$ Myr. Understanding the reasons for this disagreement is beyond the scope of this paper.\ We find that applying empirical relationships to account for magnetic activity slightly increases the LDB luminosity, and subsequently results in a $\sim$10% decrease in the predicted age. This systematic is comparable to the typical measurement error quoted by other LDB age determinations [[[*e.g.*]{},]{} @burke:2004vf] but has not been included in any previous LDB study. Our work prompts the need to re-investigate previous LDB determinations in an effort to produce more accurate ages, and recalibrate the stellar age scale relying on LDB ages. We thank David Soderblom for helpful discussions. A.J.J. and P.A.C. acknowledge support from the National Science Foundation Grant AAG Grant AST-1109612. We gratefully acknowledge the staff at the Cerro Tololo Observatories and those of the SMARTS Consortium. Our research is based on observations obtained at the Gemini Observatory, which is operated by the Association of Universities for Research in Astronomy, Inc., under a cooperative agreement with the NSF on behalf of the Gemini partnership: the National Science Foundation (United States), the Science and Technology Facilities Council (United Kingdom), the National Research Council (Canada), CONICYT (Chile), the Australian Research Council (Australia), Ministério da Ciência e Tecnologia (Brazil) and Ministerio de Ciencia, Tecnología e Innovación Productiva (Argentina). [lccccccccc]{} 18237 & 00:01:30.043 & -30:02:17.56 &$16.193\pm0.021$&$1.519\pm0.113$&K4.0&$-105 \pm 13$ &$ 1.81 \pm 0.08$ & $ 0.021 \pm 0.044$\ 222 & 00:08:07.963 & -30:16:55.78 &$11.964\pm0.005$&$1.299\pm0.020$&K4.1&$-156 \pm 5$ &$ 1.29 \pm 0.05$ & $-0.023 \pm 0.038$\ 3005a & 00:00:16.708 & -30:44:36.56 & & & K4.1 & $ -78 \pm 10$ & $ 2.36 \pm 0.21$ & $ 0.110 \pm 0.127$\ 3001a & 00:00:16.656 & -30:47:08.81 & & & K4.2 & $-132 \pm 17$ & $ 1.43 \pm 0.07$ & $-0.002 \pm 0.053$\ 3004a & 00:00:17.533 & -30:45:11.62 & & & K4.2 & $-21 \pm 5$ & $ 1.18 \pm 3.24$ & $-0.002 \pm 2.506$\ 3006 & 00:00:17.947 & -30:45:20.74 & & & K4.2 & $-10 \pm 13$ & $ 1.76 \pm 0.07$ & $-0.014 \pm 0.048$\ 3007 & 00:00:17.856 & -30:48:56.34 & & & K4.3 & $ 1 \pm 9$ & $ 2.13 \pm 0.14$ & $-0.012 \pm 0.090$\ 2250 & 00:08:11.556 & -30:15:30.27 & & & K4.4 & $ 26 \pm 10$ & $ 1.37 \pm 0.08$ & $ 0.008 \pm 0.071$\ 300a & 00:08:10.980 & -30:17:54.67 &$16.979\pm0.042$&$1.528\pm0.198$&K4.5&$6 \pm 18$ & $ 2.81 \pm 0.10$ & $ 0.004 \pm 0.062$\ 3001b & 00:00:29.608 & -30:07:35.47 & & & K5.0 & $-228 \pm 12$ & $ 2.31 \pm 0.10$ & $ 0.080 \pm 0.088$\ 2019 & 00:00:03.377 & -30:18:19.04 & & & K5.0 & $-138 \pm 8$ & $ 1.59 \pm 0.04$ & $ 0.001 \pm 0.023$\ 2001 & 00:00:02.299 & -30:20:25.55 & & & K5.1 & $-120 \pm 10$ & $ 2.47 \pm 0.05$ & $ 0.040 \pm 0.032$\ 3004b & 00:00:29.008 & -30:08:07.26 & & & K5.1 & $-129 \pm 99$ & $ 1.96 \pm 0.13$ & $ 0.006 \pm 0.227$\ 2002 & 00:00:02.713 & -30:21:41.51 &$16.23$&$1.635$&K7.3&$-7 \pm 6$&$ 0.09 \pm 0.70$&$ 0.056 \pm 0.076$\ 3005b & 00:00:28.639 & -30:07:27.73 &$16.807\pm0.04$&$2.189\pm0.05$ &M0.2&$ 20 \pm 5$& $ 0.03 \pm 0.94$ & $ 0.062 \pm 0.732$\ 3002a & 00:00:16.786 & -30:48:20.92 & & & M0.9 & $-100 \pm 10$ & $ 0.17 \pm 0.97$ & $ 0.000 \pm 0.752$\ 18184 & 00:01:33.739 & -30:06:20.05 &$15.79\pm0.015$&$1.86\pm0.064$ &M2.0&$ 28 \pm 6$& $ 0.13 \pm 1.14$ & $-0.062 \pm 0.885$\ 300b & 00:00:28.379 & -30:09:34.52 & & & M2.2 & $ 8 \pm 12$ & $ 0.68 \pm 1.74$ & $ 0.062 \pm 1.344$\ 2003 & 00:00:03.452 & -30:19:04.01 & & & M2.3 & $-44 \pm 7$ & $ 0.06 \pm 0.87$ & $-0.077 \pm 0.672$\ 3002b & 00:00:28.299 & -30:08:47.11 & & & M2.5 & $-68 \pm 31$ & $ 0.82 \pm 1.60$ & $-0.175 \pm 1.239$\ 250 & 00:07:56.902 & -30:04:16.57 &$14.473\pm0.036$&$2.081\pm0.048$&M2.9&$-73 \pm 4$& $-0.14 \pm 0.82$ & $-0.124 \pm 0.638$\ 993 &00:05:22.171 & -30:27:59.51&$16.82\pm0.031$&$2.26\pm0.093$&M3.5&$-2 \pm 6$& $-0.19 \pm 0.86$ & $ 0.000 \pm 0.161$\ 9424&00:05:13.306 & -30:26:28.72&$16.37\pm0.022$&$2.26\pm0.068$&M4.0&$ 2 \pm 6$& $-5.06 \pm 0.06$ & $-0.019 \pm 0.094$\ 1868&00:01:36.322 & -30:05:55.39&$17.34\pm0.053$&$2.50\pm0.149$&M4.0&$ 15 \pm 4$& $-0.21 \pm 0.75$ & $ 0.029 \pm 0.186$\ 28 & 23:59:55.379 & -30:02:32.28 &$18.75\pm0.04$&$2.80\pm0.05$& M5.4 & $-39 \pm 8$ & $-0.57 \pm 0.69$ & $ 0.311 \pm 0.535$\ 3 & 23:59:40.898 & -30:01:56.67 &$17.80\pm0.04$&$2.56\pm0.05$& — & — & — & —\ 50 & 23:59:50.002 & -30:01:58.52 &$19.66\pm0.04$&$3.45\pm0.05$& — & — & — & —\ 3003 & 00:00:17.026 & -30:47:43.55 & & & — & — & — & —\ 9152 & 00:05:26.484 & -30:26:03.77 &$17.357\pm0.051$&$1.864\pm0.228$& — & — & — & —\ \[nonmem\] [^1]: [iraf]{} is distributed by the National Optical Astronomy Observatories, which are operated by the Association of Universities for Research in Astronomy, Inc., under cooperative agreement with the National Science Foundation. [^2]: http://dan.iel.fm/emcee

--- abstract: | We present a near-infrared cooled grating spectrometer that has been developed at the Arcetri Astrophysical Observatory for the 1.5 m Infrared Telescope at Gornergrat (TIRGO). The spectrometer is equipped with cooled reflective optics and a grating in Littrow configuration. The detector is an engineering grade Rockwell NICMOS3 array (256$\times$256 pixels of $40 \mu$m). The scale on the focal plane is 1.73 arcsec/pixel and the field of view along the slit is 70 arcsec. The accessible spectral range is $0.95-2.5\mu$m with a dispersion, at first order, of about 11.5 [Å]{}/pixel. This paper presents a complete description of the instrument, including its optics and cryo-mechanical system, along with astronomical results from test observations, started in 1994. Since January 1996, LonGSp is offered to TIRGO users and employed in several Galactic and extragalactic programs. author: - 'L. Vanzi, M. Sozzi, G. Marcucci, A. Marconi, F. Mannucci, F. Lisi, L. Hunt, E. Giani, S. Gennari, V. Biliotti, and C. Baffa' date: 'Received ...' title: 'LonGSp: the Gornergrat Longslit Infrared Spectrometer ' --- 2.0cm Introduction ============ The development of the spectrometer LonGSp (Longslit Gornergrat Spectrometer) was part of a project aimed at providing the 1.5-m Infrared Telescope at Gornergrat (TIRGO) with a new series of instruments based on NICMOS3 detectors. The infrared (IR) camera, ARNICA, developed in the context of this project is described in Lisi et al. (1995). LonGSp is an upgrade of GoSpec (Lisi et al. 1990), the IR spectrometer operating at TIRGO since October 1988. Thanks to the NICMOS detector, the new spectrometer enables longslit spectroscopy with background limited performance (BLP). The GoSpec characteristics of compactness and simplicity are maintained in the new instrument. Only a subsection of the engineering grade array (40$\times$256 pixels) is used. A description of the optics, cryogenics, and mechanics is presented in Section \[opt\] and \[cry\]; the electronics, software and the performance of the detector are presented in Section \[elect\] and \[detec\]. Finally, in Section \[obs\] we present details regarding the observations and data reduction, and in Section \[astro\] the results of the first tests at the telescope. Optical Design {#opt} ============== The optical scheme of the instrument is sketched in Fig. 1; it is designed to match the f/20 focal ratio of the TIRGO telescope. Following the optical path from the telescope, the beam encounters the window of the dewar, the order sorting filter, a field lens, and the slit; the latter resides at the focal plane of telescope. The window and field lens are composed of calcium fluoride. Filters and slits are respectively mounted on two wheels and can be quickly changed during the observations. The field lens images the pupil on the secondary mirror of an inverted cassegrain (with focal length of 1400 mm) that produces a parallel beam 70 mm in diameter. This beam is reflected onto the grating by a flat mirror tilted by $10^{\circ}$. The grating, arranged in Littrow configuration, has 150 grooves/mm and a blaze wavelength of $2 \mu$m at first order; rotation around the $10^{\circ}$ tilted axis allows the selection of wavelengths and orders. A modified Pfund camera (with focal length of 225 mm) following the grating, collects the dispersed beams on the detector. The sky-projected pixel size is 1.73 arcsec, and the total field covered along the slit direction is 70 arcsec. The back face of the grating is a flat mirror so that, when the grating is rotated by 180 degrees, the instrument functions as a camera, in the band defined by the filters, with a field of view of about 1.5 arcmin square. This facility can be useful for tests, maintenance, and for centering weak sources on the slit. All the mirrors are gold coated to provide good efficiency over a wide spectral range, and the optics are acromatic at least up to $5\mu$m. The optical components are cooled to about 80 K by means of thermal contact with a cryogenic vessel filled with liquid nitrogen at atmospheric pressure as described below. The mounting of optical elements is designed to take into account the dimensional changes between mirrors (in pyrex) and supports (in aluminium) generated by the cooling and the differences in thermal expansion coefficents. The resolving power is (for first order) about 600 in the center of J band, and 950 in the center of the K band, using a slit of two pixels (3.46 arcsec). Cryogenics and Mechanics {#cry} ======================== As can be seen in Fig. 2, where we show some parts of the mechanical structure of the instrument, the core of the instrument is the liquid nitrogen reservoir, which has a toroidal shape with rectangular cross section and a capacity of 3 liters. It provides support and cooling for two optical benches, which are located on opposite sides of the vessel. The central hole of the toroid allows the beam to pass from one optical bench to the other. The grating motion is assured by an external stepper motor via a ferrofluidic feedthrough, and the position is controlled by an encoder connected to the motor axis outside the dewar. Two springs acting on the worm gear guarantee good stability of the grating position. Two internal stepper motors, modified to operate at cryogenic temperatures (Gennari et al. 1993), drive the filter and slit wheels. Mechanics and optics are enclosed in a radiation shield. The internal cold structure is supported by nine low-thermal-conductivity rods, which are fixed between the internal liquid nitrogen reservoir and the external vacuum shield, and are rigidly linked to the focal plane adaptor of the telescope. Externally, the instrument has the form of a cylinder with a base of about 40 cm in diameter and length of about 60 cm. A small amount of active charcoal is present to maintain the value of the pressure required (less than 10$^{-4}$ mb) for a sufficiently long time (more than 20 days). The charcoal is cooled by an independent cryogenic system; in the rear optical bench there is a smaller nitrogen vessel ($\sim$ 0.5 l) thermally insulated from the surrounding environment. The regeneration of the charcoal must be carried out once a month in order to maintain a sufficiently high absorption rate. This operation consists of heating the charcoal to 300 K, while the pressure inside the dewar is maintained below 10$^{-1}$ mb by means of a rotary vacuum pump. Because the charcoal is cooled by an independent cryogenic system, the heating of the optics and the main part of the mechanical structure is not necessary and the operation can be completed in about four hours. To cool and warm the entire instrument reasonably quickly, the dewar is filled with gaseous nitrogen at a pressure of about 200 mb during the cooling and heating phases: in this way the thermal transients prove to be shorter than seven hours. The rate of evaporation of the nitrogen from the main reservoir allows about 16 hours of operation in working conditions, more than a winter night of observation. Electronics and Software {#elect} ======================== The electronics of LonGSp comprise two main parts: “upper” electronics, that are situated near the instrument, and the “lower” electronics in the control room. The connection between the two parts is assured via a fiber-optics link. Two boards, close to the cryostat, house part of the interface electronics, that is a set of four preamplifiers and level shifters and an array of drivers and filters that feed the clock signals and the bias to the array multiplexer. The “upper” electronics are composed of an intelligent multi–part sequence generator and a data acquisition section. The first is controlled by a microprocessor (a Rockwell 65C02), and the sequencer is capable of generating many different waveforms template (at 8 bits depth) stored in an array of 128 Kbytes of memory. The final waveform is generated by selecting, via software, the templates needed together with their repetitions. The data acquisition segment consists of a bank of four analog-to-digital converters at 16 bits, and the logic for converting them to serial format. A transceiver sends data to the telescope control room through a fiber optics link. Data are sent as groups of four, one for each quadrant, and are presented together with the quadrant identifier (two bits) to the frame grabber. The fiber optics link is bidirectional, so that it is possible to send instructions to the control microprocessor in the “upper” electronics, and to communicate with the motor control through a serial connection (RS-232) encoded on the same fiber-optics link. The “upper” electronics are completed by the box which contains the power supply, the stepper motors controllers, and the temperature controller of the array. The “lower” electronics implement the logic to decode the serial data protocol, in order to correctly reconstruct the frame coming from the array detector. Data are collected by the custom frame grabber (known as the “PingPong”) which is capable of acquiring up to four images in each of its two banks. When a bank is written, the other can be read, enabling continuous fast acquisitions. Also of note is the ability to re-synchronize data acquisition to the quadrant address, virtually eliminating mis–aligned frames. The instrument is controlled by an MS-DOS PC equipped with a 80486 CPU (33 MHz clock), high-resolution monitor, and 600 Mbytes of hard disk space. At the end of a data acquisition sequence, each single frame or the stack average of a group of frames is stored on the PC hard disk, and are later transferred to optical disk (WORM) storage. In the near future, the WORM cartridges will be superseded by more standard writable CD-ROM cartridges. A local Ethernet network connects the control computer to the TIRGO Sun workstation, so that it is possible to transfer the data for preliminary reduction using standard packages. The software developed for this instrument is “layer organized”, that is to say organized as a stack of many layers of subroutines of similar levels of complexity. To accomplish its task, each routine need rely only on the immediately adjacent level and on global utility packages. Such a structure greatly simplifies the development and maintenance of the software. Our efforts were directed towards several different requirements. Our first priority was to have a flexible laboratory and telescope engine, capable of acquiring easily the large quantity of data a panoramic IR array can produce. The human interface is realized through a fast character-based menu interface. The operator is presented only with the options which are currently selectable, and the menu is rearranged on the basis of user choices or operations. We have also stressed the auto–documentation of data. After the decision to store data in standard FITS format, it was deemed useful to fully exploit the header facility to label each frame with all relevant information, such as telescope status, instrument status, and user acquisition choices. Data are also labelled with the observer name in order to facilitate data retrieval from our permanent archive. In particular, the form of the FITS file is completely compatible with the context IRSPEC of the ESO package MIDAS. Finally, one of our main goals was to produce an easy-to-use software and with the smallest “learning curve”. Our idea was that data acquisition must [*disappear*]{} from observer attention, giving him/her the possibility to concentrate on the details of the observations; in this way, observing efficiency is much higher. As a result, we have implemented automatic procedures such as multi–position (“mosaic”), and multi–exposure (stack of many frames). Detector Performance {#detec} ==================== Although the spectrometer was initially designed to use a subsection 40x100 of a NICMOS3 detector, we found later that very good performance can be obtained on an even larger area. Using 256 pixels in the wavelength direction, we have a spectral coverage of almost 0.3 $\mu m$. This means that with a single grating setting we can measure a complete J spectrum and have good coverage in H and K. The best 40x256 subsection was selected on the basis of good cosmetics (low percentage of bad pixels) and low dark current and readout noise. We measured the percentage of bad pixels, the dark current, and the readout noise via laboratory tests based on sets of images taken at a series of exposure times of a spatially uniformly illuminated scene, and without any illumination (by substituting the filter with a cold stop). The readout noise is determined as the mean standard deviation of each pixel in the stack of short integration times where the dark current is negligible. The dark current and gain measurement are based on two linear regressions: values of dark frames as a function of exposure time in the first case, and spatial medians of the stack variance relative to the stack median in the second one. Details of these tests are presented in Vanzi et al. (1995). In Table 1, we present the results of further tests carried out in April 1995. \[riv\] ---------------- --------------- Bad pixels 2.9% Dark current 0.9 $e^-$/sec Read out noise 45 $e^-$ ---------------- --------------- : Measured parameters of the detector Observations and Data Reduction {#obs} =============================== The procedures for LonGSp observations are those commonly used in NIR spectroscopy, optimized for the characteristics of the instrument. For compact sources, observations consist of several groups of frames with the object placed at different positions along the slit. In the case of extended sources, observations consist of several pairs of object and sky frames. On-chip integration time is 60 sec or less for a background level of roughly 6000 counts/pixel because of ensuing problems with sky line subtraction (see below). At a given position along the slit, several frames can be coadded. The main steps in the reduction of NIR spectroscopic data are flat–field correction, subtraction of sky emission, wavelength calibration, correction for telluric absorption, and correction for optical system $+$ detector efficency. Data reduction can be performed using the IRSPEC context in MIDAS, the ESO data analysis package, properly modified to take into account the LonGSp instrumental setup. We have found it useful to acquire dark and flat frames at the beginning and the end of the night; we obtain flat-field frames by illuminating the dome with a halogen lamp. Observations of a reference star are taken for a fixed grating position. An early type, featureless star (preferably an O star) is needed to correct for telluric absorption and differential efficency of the system, and a photometric standard star is needed if one wants to flux calibrate the final spectrum (only one grating position in each band is required). An alternative technique, proposed by Maiolino et al (1996), consists of using a G star corrected through data of solar spectrum. Both methods have been succesfully tested. Flat field frames are first corrected for bad pixels, then dark-current subtracted, and normalized. Dark current is subtracted from all raw frames, and then divided by the normalized flat field. For compact sources (the frames taken at the different positions along the slit are denoted by A, B and C), the sky is subtracted by considering $A-B$, $B-(A+C)/2$, and $C-B$, and taking a median of the three differences. In case of extended objects, if A and B denote object and sky frames, the sky is subtracted by considering $(A1+A2)/2-(B1+B2)/2$ (the order of observations is $A1\,B1\,B2\,A2$). However, a simple sky subtraction is almost never sufficient to properly eliminate the bright OH lines whose intensity varies on time scales comparable with object and sky observations. Moreover, mechanical instabilities can produce movements of spectra (usually a few hundreds of a pixel) which are nevertheless enough to produce residuals which exceed the detector noise. To correct for these two effects, the sky frames are multiplied by a correcting factor and shifted along the dispersion direction by a given amount. These factors and shifts are chosen automatically by minimizing the standard deviation in selected detector areas where only sky emission is present. Because this effect increases with the integration time, it is advisable not to exceed 60 seconds for each single integration. Slit images at various wavelengths are tilted as a consequence of the off-axis mount of the grating. Sky subtracted frames are corrected by computing analytically the tilt angle from the instrumental calibration parameters, or by directly measuring it from the data. Wavelength calibration in LonGSp data is performed using OH sky emission lines. The wavelength dispersion on the array is linear to within a small fraction of the pixel size and is computed analytically once the central wavelength of the frame is known. At the beginning of the data reduction, the nominal central wavelength used in the observations is assigned to a properly chosen sky frame. Then the calibration is refined using the bright OH sky lines (precise wavelengths of OH lines as well as a discussion of their use as calibrators are given in Oliva & Origlia 1992). The same procedures are applied to the reference stars frames to obtain the calibration spectra, and the spectrum of a photometric standard star can be used to flux calibrate the final frames. Astronomical Results {#astro} ==================== The first tests at the telescope took place successfully in early 1994. From these observations we measured the efficiency of the instrument (through the observation of photometric standard stars) and its sensitivity (1$\sigma$ in 60 sec of integration time); these are reported in table 2. \[efi\] [cccc]{} Band(order) & Efficency & Line$^{1}$ & Continuum$^{2}$\ J (I) & 0.045 & 4$\times 10^{-14}$ & 2$\times 10^{-15}$\ H (I) & 0.10 & 2$\times 10^{-14}$ & 8$\times 10^{-16}$\ K (I) & 0.08 & 2$\times 10^{-14}$ & 8$\times 10^{-16}$\ \ Since January 1996, LonGSp is offered to TIRGO users and employed in several galactic and extragalactic programs. To give an impression of the capabilities of the instrument, we show (in Figs. 3,4 and 5) some acquired spectra of various type of sources: extended, compact and extragalactic, without comment as to their astrophysical significance. Gennari S., Mannucci, F., Vanzi L., 1993, Cryogenic stepper motors for infrared astronomical instrumentation. In: A. M. Fowler (ed.) Proc. SPIE 1946, International Symposium on Optical Engigneering and Photonics in Aerospace and Remote Sensing, p. 610 Gennari S., Vanzi L, 1993, LonGSp: the new infrared spectrometer of TIRGO. In: R. Bandiera (ed.) Proc. XXXVII Annual Meeting of the S.A.It., p. 752 Gennari S., Vanzi L, 1994, LonGSp: a near infrared spectrometer. In: I. McLean (ed.) Infrared astronomy with arrays, Kluwer Academic Publishers, p. 351 Lisi F, Baffa C., Biliotti V., et al., 1995, PASP 108, 364 Maiolino R., Rieke G.H., Rieke M.J., 1996, AJ 111, 537 Oliva E., Origlia L. 1992, A&A 254, 466 Vanzi L., Gennari S., Marconi A., 1994, NICMOS3 detector for astronomical spectroscopy, IAU Symp. 167, Com. N. 167.Or.031

--- author: - | [Yoonbai Kim$^{\ast}$]{}\ \[2mm\] [*Department of Physics, Nagoya University, Nagoya 464-01, Japan*]{} title: | \   --- =16.5cm -1.1cm -1cm =.285in ‘=11 maketitle to 2em **title** 1.5em .5em -------- author -------- 1.5em \#1[preprint[\#1]{}]{} ‘=12 [**Abstract**]{}\ The first-order phase transition of $O(3)$ symmetric models is considered in the limit of high temperature. It is shown that this model supports a new bubble solution where the global monopole is formed at the center of the bubble in addition to the ordinary $O(3)$ bubble. Though the free energy of it is larger than that of normal bubble, the production rate can considerably be large at high temperatures. Since a general theory of the decay of the metastable phase was developed [@Lan; @Col1], the study of first-order phase transitions have attracted the attention due to their possible relevance to the physics of the early universe. The semiclassical expression of the tunneling rate from false vacuum to true vacuum is given by the bubble solution of lowest (Euclidean) action both for the first-order phase transitions at zero temperature field theories [@Col1] and those at finite temperature [@Lin; @KT]. However, the above case does not include the possibility that there exist various decay modes between two given classical vacua. In this note, we explore such problem by examining a scalar model of internal $O(3)$ symmetry at finite temperature and examine a new bubble solution with a global monopole. Suppose that the model of our interest contains a series of stationary points which support bubble solutions, a decay probability per unit time per unit volume is given by \[decay\] /V=\_[n]{}A\_[n]{}e\^[-B\_[n]{}/]{}, where $n$ represents the $n$-th local minimum. For each $n$, $B_{n}$ is given by a value of Euclidean action for $n$-th bubble solution and $A_{n}$ is estimated by integrating out the fluctuations around a given $n$-th configuration. We choose the $O(3)$-symmetric scalar models described in terms of an isovector field $\phi^{a}$ $(a=1,2,3)$ as the first possibility for sample calculations. At finite temperature the Euclidean action is \[action\] S\_[E]{}=\^\_[0]{} dt\_[E]{} { ()\^[2]{} +(\^[a]{})\^[2]{}+V(\^[a]{}) }, where $\beta=\hbar /k_{B}T$. Although our argument followed is general and does scarcely depend on the detailed form of scalar potential if it has the vacuum structure with both a true and a false vacuum, we will consider a specific model, [*i.e.*]{} a $\phi^{6}$-potential V()=(\^[2]{}+v\^[2]{})(\^[2]{}-v\^[2]{})\^[2]{} +V\_[0]{} where $\phi$ denotes the amplitude of scalar fields $\phi^{a}$ defined by $\phi=\sqrt{\phi^{a}\phi^{a}}$. Here we consider the transition from the symmetric vacuum to the broken vacuum, [*i.e.*]{} $0<\alpha<1/2$, and choose $V_{0}$ as $-\lambda\alpha v^{4}$ in order to make the value of $S_{E}$ coinside with $B_{n}$ in Eq.(\[decay\]). This does not lose generality since the case of the transition from the broken vacuum to the symmetric one ($-1<\alpha<0$) also possesses the same type of bubble solutions if we replace $\phi$ to $v-\phi$. It has been known that the rate of vacuum transitions at finite temperature has relevant amount of contributions from $O(3)$-symmetric sphaleron-type bubbles [@Lin; @KT]. In high temperature limit this contribution dominates and then we can neglect the dependence along $t_{E}$-axis. Here we are interested in the regime where thermal fluctuations are much larger than quantum fluctuations and address the problem for different kind of finite temperature bubbles which connect global $O(3)$ internal symmetry to that of spatial rotation. We now ask for a solution of the field equations that is time-independent and spherically symmetric, apart from the angle dependence due to the mapping between the $(\theta,\varphi)$ angles in space and those of isovector space $\hat{\phi}^{a}_{n} (\equiv \phi^{a}/\phi)$ such as \[angle\] \^[a]{}\_[n]{}=(nn,nn, n), where the allowed $n$ is 0 or 1 in order to render the scalar amplitude $\phi$ a function of $r$ only. Under this assumption, the Euler-Lagrange equation becomes \[equation\] + -\_[n1]{}=, where $\delta_{n1}$ in the third term of Eq.(\[equation\]) denotes Kronecker delta for $n=0$ or 1. The condition that the theory should be in false vacuum at spatial infinity fixes the boundary value of field, [*i.e.*]{} $\lim_{r\rightarrow\infty}\phi\rightarrow 0$. To be nonsingular solution at the origin of coordinates, the boundary condition is { [ll]{} .|\_[r=0]{}=0 &\ (0)=0 & . When $n=0$, it is well-known bubble solution at finite temperature [@Lin]. We will analyze $n=1$ case and show that there always be $n=1$ solution if the equation contains $n=0$ solution. A brief argument of the existence of $n=1$ solution is as follows. If we regard the radius $r$ as time and the scalar amplitude $\phi (r)$ as the coordinate of a particle, Eq.(\[equation\]) describes a one-dimensional motion of a unit-mass hypothetical particle under the conserved force due to the potential $-V(\phi)$ and two nonconservative forces, [*i.e.*]{} one is the friction of time-dependent coefficient $-\frac{2}{r}\frac{d\phi}{dr}$ and the other is time-dependent repulsion $\frac{2}{r^{2}}\phi$. Hence, in the terminology of Newton equation, $n=1$ solution in Fig. 1 is interpreted as the motion of a particle that starts at time zero at the origin $(\phi(0)=0)$, turns at an appropriate nonzero position at a certain time $(\phi(t_{turn})=\phi_{turn})$, and stops at the origin at infinite time $(\phi(\infty)=0)$. At first, if one considers a hypothetical particle at the origin at time zero, it is accelerated by the time-dependent repulsion of which coefficient is divergently large for small $r$. Since one initial condition is fixed by the starting point $(\phi(0)=0)$, such motions near the origin are characterized by another initial condition $C$ \[phi0\] (r)C{ r+[O]{}(r\^[3]{})+}, where $C$ is the initial velocity of a particle which should be tuned by the proper boundary condition at infinite time $(\phi(\infty)=0)$. From now on let us consider a set of solutions specified by a real parameter $C$ and show that there always exists the unique motion of $\phi(\infty)=0$ for an appropriate $C$. When $C$ is sufficiently large, the acceleration near the origin due to time-dependent repulsion and conservative force is too strong and then the particle overshoots the top of potential $-V(\phi=v)$ and goes to infinity $(\phi(\infty)=\infty)$, despite the deceleration due to the friction. Since the solution of our interest is the motion which includes a return, we will look at the motions of which $C$ is smaller than a critical value $C_{top}$. Here $C_{top}$ gives the motion that the particle stops at the hilltop of the potential $-V$ at infinite time $\phi(\infty)=v$. Suppose that $C$ be too small, [*i.e.*]{} $C$ is smaller than another critical value $C_{0}\; (C_{0} <C_{top})$, particle turns at a point smaller than $\phi_{0}$ ($0<\phi_{0}<v$) where $-V(\phi=\phi_{0})=0$ and thereby it can not return to the origin. Moreover, the particle which turns at a point too close to $v$ arrives at the origin at a finite time, since the time-dependent friction and repulsion during returning do not play a role due to the vanishing of their coefficients $1/r$ and $1/r^{2}$ while the particle stays near $v$ for sufficiently long time. By continuity for an appropriate $C\;(C_{0}<C<C_{top})$ there is a solution which describes the motion that starts at time zero at the origin turns at a position $\phi_{turn}$ between $\phi_{0}$ and $v$, and then stops at the origin at infinite time. Here for the existence proof of $n=1$ solutions, we have used the properties of scalar potential $V(\phi)$ that it has a local minimum at $\phi=0$, a local maximum at $\phi_{bottom}$ ($0<\phi_{bottom}<\phi_{0}$) and an absolute minimum at $v$. It is exactly the same condition as that for the existence of $n=0$ solution. Hence we completes our argument that the equation always contains $n=1$ solutions when it has $n=0$ solution. A rigorous proof for the existence of $n=1$ solution is demonstrated in ref.[@KKK]. We have no analytic form of solutions, so numerical solutions for $n=0$ and $n=1$ bubbles are given in Fig. 1. Suppose that each classical solution describes a critical bubble corresponding to a different decay channel, from the profile of energy density $T^{0}_{\;0}$ (see Fig. 2) we can read the following characteristics of $n=1$ bubbles. First, the boundary value of scalar field at the origin always has that of false vacuum, [*i.e.*]{} $\phi(0)=0$, so this implies that there remains a false-vacuum core inside the true-vacuum region of the bubble due to the winding between $O(3)$ internal symmetry and spatial rotation. Second, the value of energy density has a local minimum at the origin, [*i.e.*]{} $3C^{2}/2+V(0)$ in which $C$ is a constant appeared in Eq.(\[phi0\]), increases to the maximum at $R_{m}$ in Fig. 2 and decays below $V(0)$. This implies that a matter droplet is created inside the bubble and is surrounded by inner bubble wall with size of order $R_{m}\sim 1/m_{Higgs} =\sqrt{4(3+2\alpha)\lambda}v$. Third, if we read the expression of energy density when the scalar amplitude has maximum value $\phi=\phi_{turn}$, its derivative term vanishes and then it becomes \[turn\] T\^[0]{}\_[0]{}=+V(\_[turn]{}). The potential term $V(\phi_{turn})$ can be neglected in thin-wall limit, so the object inside the bubble has a long-range hair which penetrates the inner bubble wall. Since the central region of this matter is in false symmetric vacuum due to the hedgehog ansatz of scalar field in Eq.(\[angle\]) and for large $r$ the scalar field goes to the true broken vacuum but the long-range tail of energy proportional to $r$ has a cutoff at the outer bubble wall at $R_{n=1}$ in Fig. 2, we can interpret the matter aggregate inside the $n=1$ bubble as a global monopole of size $R_{m}$ [@Vil]. Even though there is a no-go theorem for static scalar objects in spacetime dimensions more than two [@Der], the global monopole of this case can be supported inside the Euclidean bubble configuration as a smooth finite-energy configuration due to a natural cutoff introduced by the outer bubble wall. Fig. 2 shows that the radius of $n=1$ bubble is larger than that of $n=0$ bubble. It can be easily understood by the conservation of energy, [*i.e.*]{} the additional energy used to make a matter aggregate is equal to the loss of energy due to the increase of the radius of bubble. Though we do not have any analytic solution, we can estimate $B_{n}$ in Eq.(\[decay\]) by use of the obtained bubble configurations through numerical analysis for given parameters of theory. It has already been proved that $n=0$ solution describes the nontrivial solution of lowest action [@CGM] and from Fig. 3 we read that the value of action for $n=1$ solution, $vB^{'}_{1}/T$, is larger than that of $n=0$, $vB^{'}_{0}/T$, irrespective of the shape of scalar potential. Fig. 3 also shows that the ratio $B^{'}_{1}/B^{'}_{0}$ becomes small while the difference $B^{'}_{1}- B^{'}_{0}$ increases as the size of bubble becomes large in comparison with the mass scale of theory, [*i.e.*]{} thin-wall limit. This can be understood as follows; the amount of energy consumed to support a global monopole at the center of bubble is proportional to the radius of bubble due to its long-range tail (see Eq.(\[turn\])), however the free energy to support the bubble itself, $vB^{'}_{0}$ or $vB^{'}_{1}$, is proportional to the cubic of bubble radius in thin-wall limit. The next task is to estimate the pre-exponential factor $A_{n}$ in Eq.(\[decay\]). The scheme of computing $A_{0}$ for $n=0$ bubble solution which is the lowest action solution with one negative mode was given in the second paper of Ref.[@Col1]. Here, let us attempt to calculate the pre-exponential factor $A_{1}$ for $n=1$ bubble. Considering the small fluctuation around $n=1$ bubble $\delta\phi^{a}_{n=1}=\sum_{k}c^{a}_{k} \psi^{a}_{k}$, we obtain a Schrödinger-type equation for three particles in three dimensions \[sch\] (-\^[2]{}\^[ab]{}+\^[a]{}\^[b]{} |\_[\^[a]{}\_[n=1]{}]{}+(\^[ab]{}-\^[a]{}\^[b]{}) |\_[\^[a]{}\_[n=1]{}]{})\^[b]{}\_[k]{}=\_[k]{} \^[a]{}\_[k]{}. It looks too difficult to solve this equation since it includes $\theta$ and $\varphi$ dependent off-diagonal terms in its Hamiltonian and the potential form is only determined numerically, however the eigenfunctions of six zero modes due to three translations and three rotations are explicitly given in a form \[zero\] \^[a]{}\_[k, i]{}&=&N\_[t]{}\_[i]{}\^[a]{}\_[n=1]{}\ &=&N\_[t]{}{\_[i]{}\^[a]{} +(\_[i]{}\^[a]{}-\_[i]{}\^[a]{}) }, and \[azero\] \^[a]{}\_[k, i]{}=N\_[r]{}\_[ijk]{}x\^[j]{}\^[k]{}\^[a]{}\_[n=1]{} =N\_[r]{}\_[ija]{}\^[j]{}\_[n=1]{}, where $N_{t}$ and $N_{r}$ are normalization constants. Here we give few comments on the number of negative modes. First, every component of translational zero-mode eigenfunction in Eq.(\[zero\]) has $(\theta,\varphi)$-dependence where its radial part, $\frac{d\phi_{n=1}}{dr}-\frac{\phi_{n=1}}{r}$, has single node at the origin, and each $a=i$ component includes an additional $(\theta,\varphi)$-independent part, $\frac{\phi_{n=1}}{r}$, which has no node. Every $a\neq i$ component of rotational ones, $\phi_{n=1}$, has one node at the origin. For the sake of simplicity, let us examine the problem for the perturbation with specific direction; $(i)$ one for amplitude ($\delta\phi^{a}_{n=1}= \hat{r}^{a}\delta\phi_{n=1})$ and $(ii)$ two for transverse ($\delta\phi^{a}_{n=1}=(\delta^{ab}-\hat{r}^{a}\hat{r}^{b}) \delta\phi^{b}_{n=1}$). Under these fluctuations Eq.(\[sch\]) reduces to \[nonz\] (-\^[2]{}+U(r))\^[a]{}\_[k]{}=\_[k]{}\^[a]{}\_[k]{}, where $U(r)=\left.\frac{d^{2}V}{d\phi^{2}}\right|_{\phi_{n=1}}$ for the first case $(i)$ and $U(r)=\left.\frac{1}{\phi}\frac{dV}{d\phi} \right|_{\phi_{n=1}}$ for the second case $(ii)$. When $ U(r)=\left.\frac{d^{2}V}{d\phi^{2}}\right|_{\phi_{n=0}}$, each single ‘$a$’ component of Eq.(\[nonz\]) is nothing but the fluctuation equation for $n=0$ bubble which contains a nodless $s$-wave mode as the unique negative mode [@Col1; @SCol]. For the fluctuation of scalar amplitude for $n=1$ bubble, the lowest mode can not be nodless $s$-wave mode but $l=1$ mode with single node at $r=0$ since $\psi^{a}_{k}$ should have $\theta$ and $\varphi$ dependence proportional to $\hat{r}^{a}$ even though the operator in Eq.(\[nonz\]) is a scalar operator. It is analogous for the perturbation to the transversal directions. It implies that the eigenfunction of negative mode may take a somewhat complicated form which depends on angels. Second, the $n=1$ solution which is a parity-odd bounce solution, $\phi(r=0)=\phi(r=\infty)$, is supported by assuming the winding between three spatial rotations and those in internal space. It seems that the argument for the system of quantum mechanics with one time variable in Ref.[@SCol] does not forbid directly the existence of $n=1$ solution as a bubble configuration. We know very little on the counting of negative modes, and then there should be further work on this issue. Since the operator in Eq.(\[sch\]) is even under parity transformation and is covariant under the rotation, [*i.e.*]{} $x^{i}\rightarrow O^{ij}x^{j}$ and $\psi^{a}_{k}(x^{'})=O^{ab}\psi^{b}_{k}(x)$ where both $O^{ij}$ and $O^{ab}$ are the elements of $O(3)$ group, vector spherical harmonics is a method to investigate the modes of which the eigenfunctions can be chosen to be either even or odd in parity [@Tam]. In order to compute the decay rate accurately, we should calculate the nonzero modes $\lambda_{k}$, which is extremely difficult even for $n=0$ bubbles. However, the dimensional estimate may reach a rough result such as [@Lin] \~( )\^. At high temperature limit, $T>>v$, both decay rates for $n=0$ and 1 bubbles, of course, increase and, moreover, the relative decay rate of $n=1$ bubble to that of $n=0$ is also enhanced exponentially. When $v/T\sim 10^{-1}$, the order of $\Gamma^{(1)}/\Gamma^{(0)}$ is around 0.17 in a thin-wall case ($\lambda=1$, $\alpha=0.12$) and it is around 28 in a thick-wall case ($\lambda=1$, $\alpha=0.47$). Therefore, at high temperatures and in thick-wall case, this $n=1$ bubbles can be preferred to those of $n=0$, and obviously the existence of another decay channel can enhance considerably the total nucleation rate of bubbles, $\Gamma=\Gamma^{(0)}+\Gamma^{(1)}$, except for the bubbles with extremely thin wall. The main consideration of the paper was to gain an understanding as to how bubbles with global monopoles nucleated in first-order phase transitions at high temperatures and the next question will be how the high-temperature bubbles grow [@Ste], particularly at the site of global monopole. Once we assume these bubbles in early universe where the gravity effect should be included, the bubbles with solitons may result in interesting phenomena [@KMS] in relation with inflationary models, [*i.e.*]{} the inflation in the cores of topological defects [@Vil2]. The author would like to express deep gratitude to Jooyoo Hong, K. Ishikawa, Chanju Kim, Kimyeong Lee, K. Maeda, V. P. Nair, Q. Park, N. Sakai, A. I. Sanda, S. Tanimura, E. J. Weinberg, Y. S. Wu and K. Yamawaki for valuable discussions and also thanks Kyung Hee University and Hanyang University for their hospitality. This work is supported by JSPS under $\#$93033. \#1 References {#references .unnumbered} ----------- 40004000 ‘=1000 = [100]{} E-mail address: yoonbai$@$eken.phys.nagoya-u.ac.jp J. S. Langer, Ann. Phys. [**41**]{}, 108 (1967). S. Coleman, Phys. Rev [**D15**]{}, 2929 (1977); C. Callan and S. Coleman, [*ibid*]{} [**D16**]{}, 1762 (1977). A. D. Linde, Phys. Lett. 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[**B327**]{}, 208 (1994). **Figure Captions** {#figure-captions .unnumbered} =================== FIG. 1. Plot of bubble solutions. $n=0$ and $n=1$ configurations are shown as dotted and solid lines, respectively. The parameters chosen in the figures are: $\lambda=1$, $\alpha=0.12$, and $V_{0}=-0.12v^{4}$. FIG. 2. Plot of energy density $T^{0}_{0}$. The parameters chosen in the figures are: $\lambda=1$, $\alpha=0.12$, and $V_{0}=-0.12v^{4}$ which is the minimum of energy density. FIG. 3. Plot of action $S_{E}$ (or equivalently $B_{n}$) as a function of $\alpha$. Another parameters are chosen as $\lambda=1$ and $V_{0}=-\lambda\alpha v^{4}$. 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------------------------------------------------------------------------ (268.58,361.00)(0.492,1.864)[19]{} ------------------------------------------------------------------------ (267.17,361.00)(11.000,36.773)[2]{} ------------------------------------------------------------------------ (279.58,401.00)(0.492,1.746)[21]{} ------------------------------------------------------------------------ (278.17,401.00)(12.000,37.956)[2]{} ------------------------------------------------------------------------ (291.58,442.00)(0.492,1.817)[19]{} ------------------------------------------------------------------------ (290.17,442.00)(11.000,35.849)[2]{} ------------------------------------------------------------------------ (302.58,481.00)(0.492,1.817)[19]{} ------------------------------------------------------------------------ (301.17,481.00)(11.000,35.849)[2]{} ------------------------------------------------------------------------ (313.58,520.00)(0.492,1.530)[21]{} ------------------------------------------------------------------------ (312.17,520.00)(12.000,33.302)[2]{} ------------------------------------------------------------------------ (325.58,556.00)(0.492,1.581)[19]{} ------------------------------------------------------------------------ (324.17,556.00)(11.000,31.226)[2]{} ------------------------------------------------------------------------ (336.58,590.00)(0.492,1.272)[21]{} ------------------------------------------------------------------------ (335.17,590.00)(12.000,27.717)[2]{} ------------------------------------------------------------------------ (348.58,620.00)(0.492,1.251)[19]{} ------------------------------------------------------------------------ (347.17,620.00)(11.000,24.755)[2]{} ------------------------------------------------------------------------ (359.58,647.00)(0.492,1.013)[21]{} ------------------------------------------------------------------------ (358.17,647.00)(12.000,22.132)[2]{} ------------------------------------------------------------------------ (371.58,671.00)(0.492,0.920)[19]{} ------------------------------------------------------------------------ (370.17,671.00)(11.000,18.283)[2]{} ------------------------------------------------------------------------ (382.58,691.00)(0.492,0.712)[21]{} ------------------------------------------------------------------------ (381.17,691.00)(12.000,15.616)[2]{} ------------------------------------------------------------------------ (394.58,708.00)(0.492,0.637)[19]{} ------------------------------------------------------------------------ (393.17,708.00)(11.000,12.736)[2]{} ------------------------------------------------------------------------ (405.00,722.58)(0.496,0.492)[21]{} ------------------------------------------------------------------------ (405.00,721.17)(10.962,12.000)[2]{} ------------------------------------------------------------------------ (417.00,734.58)(0.547,0.491)[17]{} ------------------------------------------------------------------------ (417.00,733.17)(9.879,10.000)[2]{} ------------------------------------------------------------------------ (428.00,744.59)(0.692,0.488)[13]{} ------------------------------------------------------------------------ (428.00,743.17)(9.651,8.000)[2]{} ------------------------------------------------------------------------ (439.00,752.59)(0.874,0.485)[11]{} ------------------------------------------------------------------------ (439.00,751.17)(10.369,7.000)[2]{} ------------------------------------------------------------------------ (451.00,759.59)(1.155,0.477)[7]{} ------------------------------------------------------------------------ (451.00,758.17)(8.966,5.000)[2]{} ------------------------------------------------------------------------ (462.00,764.59)(1.267,0.477)[7]{} ------------------------------------------------------------------------ (462.00,763.17)(9.800,5.000)[2]{} ------------------------------------------------------------------------ (474.00,769.60)(1.505,0.468)[5]{} ------------------------------------------------------------------------ (474.00,768.17)(8.509,4.000)[2]{} ------------------------------------------------------------------------ (485.00,773.61)(2.472,0.447)[3]{} ------------------------------------------------------------------------ (485.00,772.17)(8.472,3.000)[2]{} ------------------------------------------------------------------------ (497.00,776.61)(2.248,0.447)[3]{} ------------------------------------------------------------------------ (497.00,775.17)(7.748,3.000)[2]{} ------------------------------------------------------------------------ (508,779.17) ------------------------------------------------------------------------ (508.00,778.17)(6.811,2.000)[2]{} ------------------------------------------------------------------------ (520,781.17) ------------------------------------------------------------------------ (520.00,780.17)(6.226,2.000)[2]{} ------------------------------------------------------------------------ (531,782.67) ------------------------------------------------------------------------ (531.00,782.17)(6.000,1.000)[2]{} ------------------------------------------------------------------------ (543,784.17) ------------------------------------------------------------------------ (543.00,783.17)(6.226,2.000)[2]{} ------------------------------------------------------------------------ (554,785.67) ------------------------------------------------------------------------ (554.00,785.17)(5.500,1.000)[2]{} ------------------------------------------------------------------------ (565,786.67) ------------------------------------------------------------------------ (565.00,786.17)(6.000,1.000)[2]{} ------------------------------------------------------------------------ (577,787.67) ------------------------------------------------------------------------ (577.00,787.17)(5.500,1.000)[2]{} ------------------------------------------------------------------------ (588,788.67) ------------------------------------------------------------------------ (588.00,788.17)(6.000,1.000)[2]{} ------------------------------------------------------------------------ (600,789.67) ------------------------------------------------------------------------ (600.00,789.17)(5.500,1.000)[2]{} ------------------------------------------------------------------------ (611,790.67) ------------------------------------------------------------------------ (611.00,790.17)(6.000,1.000)[2]{} ------------------------------------------------------------------------ (634,791.67) ------------------------------------------------------------------------ (634.00,791.17)(6.000,1.000)[2]{} ------------------------------------------------------------------------ (623.0,792.0) ------------------------------------------------------------------------ (669,792.67) ------------------------------------------------------------------------ (669.00,792.17)(5.500,1.000)[2]{} ------------------------------------------------------------------------ (646.0,793.0) ------------------------------------------------------------------------ (726,792.67) ------------------------------------------------------------------------ (726.00,793.17)(5.500,-1.000)[2]{} ------------------------------------------------------------------------ (680.0,794.0) ------------------------------------------------------------------------ (749,791.67) ------------------------------------------------------------------------ (749.00,792.17)(5.500,-1.000)[2]{} ------------------------------------------------------------------------ (760,790.67) ------------------------------------------------------------------------ (760.00,791.17)(6.000,-1.000)[2]{} ------------------------------------------------------------------------ (772,789.17) ------------------------------------------------------------------------ (772.00,790.17)(6.226,-2.000)[2]{} ------------------------------------------------------------------------ (783,787.17) ------------------------------------------------------------------------ (783.00,788.17)(6.811,-2.000)[2]{} ------------------------------------------------------------------------ (795.00,785.95)(2.248,-0.447)[3]{} ------------------------------------------------------------------------ (795.00,786.17)(7.748,-3.000)[2]{} ------------------------------------------------------------------------ (806.00,782.94)(1.505,-0.468)[5]{} ------------------------------------------------------------------------ (806.00,783.17)(8.509,-4.000)[2]{} ------------------------------------------------------------------------ (817.00,778.93)(1.033,-0.482)[9]{} ------------------------------------------------------------------------ (817.00,779.17)(10.132,-6.000)[2]{} ------------------------------------------------------------------------ (829.00,772.93)(0.943,-0.482)[9]{} ------------------------------------------------------------------------ (829.00,773.17)(9.270,-6.000)[2]{} ------------------------------------------------------------------------ (840.00,766.93)(0.669,-0.489)[15]{} ------------------------------------------------------------------------ (840.00,767.17)(10.685,-9.000)[2]{} ------------------------------------------------------------------------ (852.00,757.92)(0.496,-0.492)[19]{} ------------------------------------------------------------------------ (852.00,758.17)(9.962,-11.000)[2]{} ------------------------------------------------------------------------ (863.58,745.65)(0.492,-0.582)[21]{} ------------------------------------------------------------------------ (862.17,746.82)(12.000,-12.824)[2]{} ------------------------------------------------------------------------ (875.58,731.02)(0.492,-0.779)[19]{} ------------------------------------------------------------------------ (874.17,732.51)(11.000,-15.509)[2]{} ------------------------------------------------------------------------ (886.58,713.68)(0.492,-0.884)[21]{} ------------------------------------------------------------------------ (885.17,715.34)(12.000,-19.340)[2]{} ------------------------------------------------------------------------ (898.58,691.96)(0.492,-1.109)[19]{} ------------------------------------------------------------------------ (897.17,693.98)(11.000,-21.981)[2]{} ------------------------------------------------------------------------ (909.58,667.57)(0.492,-1.229)[21]{} ------------------------------------------------------------------------ (908.17,669.79)(12.000,-26.786)[2]{} ------------------------------------------------------------------------ (921.58,637.75)(0.492,-1.486)[19]{} ------------------------------------------------------------------------ (920.17,640.38)(11.000,-29.377)[2]{} ------------------------------------------------------------------------ (932.58,605.30)(0.492,-1.628)[19]{} ------------------------------------------------------------------------ (931.17,608.15)(11.000,-32.151)[2]{} ------------------------------------------------------------------------ (943.58,570.47)(0.492,-1.573)[21]{} ------------------------------------------------------------------------ (942.17,573.23)(12.000,-34.233)[2]{} ------------------------------------------------------------------------ (955.58,532.85)(0.492,-1.769)[19]{} ------------------------------------------------------------------------ (954.17,535.92)(11.000,-34.924)[2]{} ------------------------------------------------------------------------ (966.58,495.19)(0.492,-1.659)[21]{} ------------------------------------------------------------------------ (965.17,498.09)(12.000,-36.094)[2]{} ------------------------------------------------------------------------ (978.58,455.85)(0.492,-1.769)[19]{} ------------------------------------------------------------------------ (977.17,458.92)(11.000,-34.924)[2]{} ------------------------------------------------------------------------ (989.58,418.47)(0.492,-1.573)[21]{} ------------------------------------------------------------------------ (988.17,421.23)(12.000,-34.233)[2]{} ------------------------------------------------------------------------ (1001.58,381.45)(0.492,-1.581)[19]{} ------------------------------------------------------------------------ (1000.17,384.23)(11.000,-31.226)[2]{} ------------------------------------------------------------------------ (1012.58,348.16)(0.492,-1.358)[21]{} ------------------------------------------------------------------------ (1011.17,350.58)(12.000,-29.579)[2]{} ------------------------------------------------------------------------ (1024.58,316.21)(0.492,-1.345)[19]{} ------------------------------------------------------------------------ (1023.17,318.60)(11.000,-26.604)[2]{} ------------------------------------------------------------------------ (1035.58,287.85)(0.492,-1.142)[21]{} ------------------------------------------------------------------------ (1034.17,289.92)(12.000,-24.924)[2]{} ------------------------------------------------------------------------ (1047.58,260.96)(0.492,-1.109)[19]{} ------------------------------------------------------------------------ (1046.17,262.98)(11.000,-21.981)[2]{} ------------------------------------------------------------------------ (1058.58,237.41)(0.492,-0.967)[19]{} ------------------------------------------------------------------------ (1057.17,239.21)(11.000,-19.207)[2]{} ------------------------------------------------------------------------ (1069.58,216.96)(0.492,-0.798)[21]{} ------------------------------------------------------------------------ (1068.17,218.48)(12.000,-17.478)[2]{} ------------------------------------------------------------------------ (1081.58,198.17)(0.492,-0.732)[19]{} ------------------------------------------------------------------------ (1080.17,199.58)(11.000,-14.585)[2]{} ------------------------------------------------------------------------ (1092.58,182.51)(0.492,-0.625)[21]{} ------------------------------------------------------------------------ (1091.17,183.75)(12.000,-13.755)[2]{} ------------------------------------------------------------------------ (1104.58,167.62)(0.492,-0.590)[19]{} ------------------------------------------------------------------------ (1103.17,168.81)(11.000,-11.811)[2]{} ------------------------------------------------------------------------ (1115.00,155.92)(0.543,-0.492)[19]{} ------------------------------------------------------------------------ (1115.00,156.17)(10.887,-11.000)[2]{} ------------------------------------------------------------------------ (1127.00,144.92)(0.547,-0.491)[17]{} ------------------------------------------------------------------------ (1127.00,145.17)(9.879,-10.000)[2]{} ------------------------------------------------------------------------ (1138.00,134.93)(0.758,-0.488)[13]{} ------------------------------------------------------------------------ (1138.00,135.17)(10.547,-8.000)[2]{} ------------------------------------------------------------------------ (1150.00,126.93)(0.692,-0.488)[13]{} ------------------------------------------------------------------------ (1150.00,127.17)(9.651,-8.000)[2]{} ------------------------------------------------------------------------ (1161.00,118.93)(0.874,-0.485)[11]{} ------------------------------------------------------------------------ (1161.00,119.17)(10.369,-7.000)[2]{} ------------------------------------------------------------------------ (1173.00,111.93)(1.155,-0.477)[7]{} ------------------------------------------------------------------------ (1173.00,112.17)(8.966,-5.000)[2]{} ------------------------------------------------------------------------ (1184.00,106.93)(1.155,-0.477)[7]{} ------------------------------------------------------------------------ (1184.00,107.17)(8.966,-5.000)[2]{} ------------------------------------------------------------------------ (1195.00,101.93)(1.267,-0.477)[7]{} ------------------------------------------------------------------------ (1195.00,102.17)(9.800,-5.000)[2]{} ------------------------------------------------------------------------ (1207.00,96.94)(1.505,-0.468)[5]{} ------------------------------------------------------------------------ (1207.00,97.17)(8.509,-4.000)[2]{} ------------------------------------------------------------------------ (1218.00,92.95)(2.472,-0.447)[3]{} ------------------------------------------------------------------------ (1218.00,93.17)(8.472,-3.000)[2]{} ------------------------------------------------------------------------ (1230.00,89.95)(2.248,-0.447)[3]{} ------------------------------------------------------------------------ (1230.00,90.17)(7.748,-3.000)[2]{} ------------------------------------------------------------------------ (1241,86.17) ------------------------------------------------------------------------ (1241.00,87.17)(6.811,-2.000)[2]{} ------------------------------------------------------------------------ (1253.00,84.95)(2.248,-0.447)[3]{} ------------------------------------------------------------------------ (1253.00,85.17)(7.748,-3.000)[2]{} ------------------------------------------------------------------------ (1264,81.67) ------------------------------------------------------------------------ (1264.00,82.17)(6.000,-1.000)[2]{} ------------------------------------------------------------------------ (1276,80.17) ------------------------------------------------------------------------ (1276.00,81.17)(6.226,-2.000)[2]{} ------------------------------------------------------------------------ (1287,78.17) ------------------------------------------------------------------------ (1287.00,79.17)(6.811,-2.000)[2]{} ------------------------------------------------------------------------ (1299,76.67) ------------------------------------------------------------------------ (1299.00,77.17)(5.500,-1.000)[2]{} ------------------------------------------------------------------------ (1310,75.67) ------------------------------------------------------------------------ (1310.00,76.17)(5.500,-1.000)[2]{} ------------------------------------------------------------------------ (1321,74.67) ------------------------------------------------------------------------ (1321.00,75.17)(6.000,-1.000)[2]{} ------------------------------------------------------------------------ (1333,73.67) ------------------------------------------------------------------------ (1333.00,74.17)(5.500,-1.000)[2]{} ------------------------------------------------------------------------ (1344,72.67) ------------------------------------------------------------------------ (1344.00,73.17)(6.000,-1.000)[2]{} ------------------------------------------------------------------------ (737.0,793.0) ------------------------------------------------------------------------ (1367,71.67) ------------------------------------------------------------------------ (1367.00,72.17)(6.000,-1.000)[2]{} ------------------------------------------------------------------------ (1356.0,73.0) ------------------------------------------------------------------------ (1390,70.67) ------------------------------------------------------------------------ (1390.00,71.17)(6.000,-1.000)[2]{} ------------------------------------------------------------------------ (1379.0,72.0) ------------------------------------------------------------------------ (1413,69.67) ------------------------------------------------------------------------ (1413.00,70.17)(6.000,-1.000)[2]{} ------------------------------------------------------------------------ (1402.0,71.0) ------------------------------------------------------------------------ (1425.0,70.0) ------------------------------------------------------------------------ (1306,767)[(0,0)\[r\][$n=0$]{}]{} (1328,767)(20.756,0.000)[4]{} (1394,767) (176,803) (176.00,803.00) (196.76,803.00) (199,803)(20.756,0.000)[0]{} (217.51,803.00) (222,803)(20.756,0.000)[0]{} (238.27,803.00) (245,803)(20.756,0.000)[0]{} (259.02,803.00) (268,803)(20.756,0.000)[0]{} (279.78,803.00) (300.53,803.00) (302,803)(20.756,0.000)[0]{} (321.29,803.00) (325,803)(20.756,0.000)[0]{} (342.04,803.00) (348,803)(20.756,0.000)[0]{} (362.80,803.00) (371,803)(20.756,0.000)[0]{} (383.55,803.00) (404.31,803.00) (405,803)(20.756,0.000)[0]{} (425.07,803.00) (428,803)(20.756,0.000)[0]{} (445.82,803.00) (451,803)(20.756,0.000)[0]{} (466.58,803.00) (474,803)(20.756,0.000)[0]{} (487.33,803.00) (497,803)(20.756,0.000)[0]{} (508.09,803.00) (528.84,803.00) (531,803)(20.756,0.000)[0]{} (549.60,803.00) (554,803)(20.670,-1.879)[0]{} (570.31,802.00) (577,802)(20.670,-1.879)[0]{} (591.02,801.00) (600,801)(20.670,-1.879)[0]{} (611.73,799.94) (632.40,798.15) (634,798)(20.473,-3.412)[0]{} (652.74,794.16) (657,793)(20.136,-5.034)[0]{} (672.73,788.64) (680,786)(18.895,-8.589)[0]{} (691.81,780.53) (709.04,769.06) (724.58,755.31) (726,754)(12.823,-16.320)[0]{} (737.59,739.17) (749,723)(9.631,-18.386)[2]{} (768.39,685.21) (776.50,666.13) (783,649)(7.288,-19.434)[2]{} (795,617)(6.223,-19.801)[2]{} (806,582)(5.771,-19.937)[2]{} (817,544)(6.104,-19.838)[2]{} (829,505)(5.634,-19.976)[2]{} (840,466)(6.104,-19.838)[2]{} (857.40,408.84) (863,390)(6.732,-19.634)[2]{} (875,355)(6.563,-19.690)[2]{} (890.72,310.60) (898,293)(7.831,-19.222)[2]{} (915.29,253.42) (924.57,234.86) (934.08,216.41) (944.77,198.64) (957.19,182.02) (969.97,165.70) (984.31,150.69) (999.84,136.96) (1001,136)(16.064,-13.143)[0]{} (1016.35,124.46) (1034.04,113.61) (1035,113)(18.564,-9.282)[0]{} (1052.65,104.43) (1058,102)(19.506,-7.093)[0]{} (1072.01,97.00) (1091.88,91.03) (1092,91)(20.136,-5.034)[0]{} (1111.97,85.83) (1115,85)(20.473,-3.412)[0]{} (1132.36,82.03) (1138,81)(20.684,-1.724)[0]{} (1152.94,79.47) (1161,78)(20.684,-1.724)[0]{} (1173.51,76.95) (1194.19,75.07) (1195,75)(20.684,-1.724)[0]{} (1214.86,73.29) (1218,73)(20.756,0.000)[0]{} (1235.58,72.49) (1241,72)(20.684,-1.724)[0]{} (1256.27,71.00) (1264,71)(20.756,0.000)[0]{} (1277.03,70.91) (1297.74,70.00) (1299,70)(20.756,0.000)[0]{} (1318.50,70.00) (1321,70)(20.684,-1.724)[0]{} (1339.21,69.00) (1344,69)(20.756,0.000)[0]{} (1359.96,69.00) (1367,69)(20.756,0.000)[0]{} (1380.72,69.00) (1401.48,69.00) (1402,69)(20.756,0.000)[0]{} (1422.20,68.23) (1425,68)(20.756,0.000)[0]{} (1436,68) (1500,900)(0,0) =cmr10 at 10pt (-40,480)[(0,0)\[1\]]{} (800,-100)[(0,0)[$rv$]{}]{} (290,80)[(0,0)[$\cdot$]{}]{} (290,100)[(0,0)[$\cdot$]{}]{} (290,120)[(0,0)[$\cdot$]{}]{} (290,140)[(0,0)[$\cdot$]{}]{} (290,160)[(0,0)[$\cdot$]{}]{} (290,180)[(0,0)[$\cdot$]{}]{} (290,200)[(0,0)[$\cdot$]{}]{} (290,220)[(0,0)[$\cdot$]{}]{} (290,240)[(0,0)[$\cdot$]{}]{} (290,260)[(0,0)[$\cdot$]{}]{} (290,280)[(0,0)[$\cdot$]{}]{} (290,300)[(0,0)[$\cdot$]{}]{} (290,320)[(0,0)[$\cdot$]{}]{} (290,340)[(0,0)[$\cdot$]{}]{} (290,360)[(0,0)[$\cdot$]{}]{} (290,380)[(0,0)[$\cdot$]{}]{} (290,400)[(0,0)[$\cdot$]{}]{} (290,420)[(0,0)[$\cdot$]{}]{} (290,440)[(0,0)[$\cdot$]{}]{} (290,460)[(0,0)[$\cdot$]{}]{} (290,480)[(0,0)[$\cdot$]{}]{} (290,500)[(0,0)[$\cdot$]{}]{} (290,520)[(0,0)[$\cdot$]{}]{} (290,540)[(0,0)[$\cdot$]{}]{} (290,560)[(0,0)[$\cdot$]{}]{} (290,580)[(0,0)[$\cdot$]{}]{} (290,600)[(0,0)[$\cdot$]{}]{} (290,620)[(0,0)[$\cdot$]{}]{} (290,640)[(0,0)[$\cdot$]{}]{} (290,660)[(0,0)[$\cdot$]{}]{} (290,680)[(0,0)[$\cdot$]{}]{} (290,700)[(0,0)[$\cdot$]{}]{} (290,720)[(0,0)[$\cdot$]{}]{} (290,740)[(0,0)[$\cdot$]{}]{} (290,760)[(0,0)[$\cdot$]{}]{} (290,780)[(0,0)[$\cdot$]{}]{} (290,800)[(0,0)[$\cdot$]{}]{} (290,820)[(0,0)[$\cdot$]{}]{} (290,-30)[(0,0)[$R_{m}$]{}]{} (845,80)[(0,0)[$\cdot$]{}]{} (845,100)[(0,0)[$\cdot$]{}]{} (845,120)[(0,0)[$\cdot$]{}]{} (845,140)[(0,0)[$\cdot$]{}]{} (845,160)[(0,0)[$\cdot$]{}]{} (845,180)[(0,0)[$\cdot$]{}]{} (845,200)[(0,0)[$\cdot$]{}]{} (845,220)[(0,0)[$\cdot$]{}]{} (845,240)[(0,0)[$\cdot$]{}]{} (845,260)[(0,0)[$\cdot$]{}]{} (845,280)[(0,0)[$\cdot$]{}]{} (845,300)[(0,0)[$\cdot$]{}]{} (845,320)[(0,0)[$\cdot$]{}]{} (845,340)[(0,0)[$\cdot$]{}]{} (845,360)[(0,0)[$\cdot$]{}]{} (845,380)[(0,0)[$\cdot$]{}]{} (845,400)[(0,0)[$\cdot$]{}]{} (845,420)[(0,0)[$\cdot$]{}]{} (845,440)[(0,0)[$\cdot$]{}]{} (845,460)[(0,0)[$\cdot$]{}]{} (845,480)[(0,0)[$\cdot$]{}]{} (845,500)[(0,0)[$\cdot$]{}]{} (845,520)[(0,0)[$\cdot$]{}]{} (845,540)[(0,0)[$\cdot$]{}]{} (845,560)[(0,0)[$\cdot$]{}]{} (845,580)[(0,0)[$\cdot$]{}]{} (845,600)[(0,0)[$\cdot$]{}]{} (830,-30)[(0,0)[$R_{n=0}$]{}]{} (980,80)[(0,0)[$\cdot$]{}]{} (980,100)[(0,0)[$\cdot$]{}]{} (980,120)[(0,0)[$\cdot$]{}]{} (980,140)[(0,0)[$\cdot$]{}]{} (980,160)[(0,0)[$\cdot$]{}]{} (980,180)[(0,0)[$\cdot$]{}]{} (980,200)[(0,0)[$\cdot$]{}]{} (980,220)[(0,0)[$\cdot$]{}]{} (980,240)[(0,0)[$\cdot$]{}]{} (980,260)[(0,0)[$\cdot$]{}]{} (980,280)[(0,0)[$\cdot$]{}]{} (980,300)[(0,0)[$\cdot$]{}]{} (980,320)[(0,0)[$\cdot$]{}]{} (980,340)[(0,0)[$\cdot$]{}]{} (980,360)[(0,0)[$\cdot$]{}]{} (980,380)[(0,0)[$\cdot$]{}]{} (980,400)[(0,0)[$\cdot$]{}]{} (980,420)[(0,0)[$\cdot$]{}]{} (980,440)[(0,0)[$\cdot$]{}]{} (980,460)[(0,0)[$\cdot$]{}]{} (980,480)[(0,0)[$\cdot$]{}]{} (980,500)[(0,0)[$\cdot$]{}]{} (980,520)[(0,0)[$\cdot$]{}]{} (980,540)[(0,0)[$\cdot$]{}]{} (980,560)[(0,0)[$\cdot$]{}]{} (980,580)[(0,0)[$\cdot$]{}]{} (980,600)[(0,0)[$\cdot$]{}]{} (995,-30)[(0,0)[$R_{n=1}$]{}]{} (800,-400)[(0,0)[Figure 2]{}]{} (176.0,289.0) ------------------------------------------------------------------------ (176.0,68.0) ------------------------------------------------------------------------ (176.0,68.0) ------------------------------------------------------------------------ (154,68)[(0,0)\[r\][-0.15]{}]{} (1416.0,68.0) ------------------------------------------------------------------------ (176.0,142.0) ------------------------------------------------------------------------ (154,142)[(0,0)\[r\][-0.1]{}]{} (1416.0,142.0) ------------------------------------------------------------------------ (176.0,215.0) ------------------------------------------------------------------------ (154,215)[(0,0)\[r\][-0.05]{}]{} (1416.0,215.0) ------------------------------------------------------------------------ (176.0,289.0) ------------------------------------------------------------------------ (154,289)[(0,0)\[r\][0]{}]{} (1416.0,289.0) ------------------------------------------------------------------------ (176.0,362.0) ------------------------------------------------------------------------ (154,362)[(0,0)\[r\][0.05]{}]{} (1416.0,362.0) ------------------------------------------------------------------------ (176.0,436.0) ------------------------------------------------------------------------ (154,436)[(0,0)\[r\][0.1]{}]{} (1416.0,436.0) ------------------------------------------------------------------------ (176.0,509.0) ------------------------------------------------------------------------ (154,509)[(0,0)\[r\][0.15]{}]{} (1416.0,509.0) ------------------------------------------------------------------------ (176.0,583.0) ------------------------------------------------------------------------ (154,583)[(0,0)\[r\][0.2]{}]{} (1416.0,583.0) ------------------------------------------------------------------------ (176.0,656.0) ------------------------------------------------------------------------ (154,656)[(0,0)\[r\][0.25]{}]{} (1416.0,656.0) ------------------------------------------------------------------------ (176.0,730.0) ------------------------------------------------------------------------ (154,730)[(0,0)\[r\][0.3]{}]{} (1416.0,730.0) ------------------------------------------------------------------------ (176.0,803.0) ------------------------------------------------------------------------ (154,803)[(0,0)\[r\][0.35]{}]{} (1416.0,803.0) ------------------------------------------------------------------------ (176.0,877.0) ------------------------------------------------------------------------ (154,877)[(0,0)\[r\][0.4]{}]{} (1416.0,877.0) ------------------------------------------------------------------------ (176.0,68.0) ------------------------------------------------------------------------ (176,23)[(0,0)[0]{}]{} (176.0,857.0) ------------------------------------------------------------------------ (405.0,68.0) ------------------------------------------------------------------------ (405,23)[(0,0)[2]{}]{} (405.0,857.0) ------------------------------------------------------------------------ (634.0,68.0) ------------------------------------------------------------------------ (634,23)[(0,0)[4]{}]{} (634.0,857.0) ------------------------------------------------------------------------ (863.0,68.0) ------------------------------------------------------------------------ (863,23)[(0,0)[6]{}]{} (863.0,857.0) ------------------------------------------------------------------------ (1092.0,68.0) ------------------------------------------------------------------------ (1092,23)[(0,0)[8]{}]{} (1092.0,857.0) ------------------------------------------------------------------------ (1321.0,68.0) ------------------------------------------------------------------------ (1321,23)[(0,0)[10]{}]{} (1321.0,857.0) ------------------------------------------------------------------------ (176.0,68.0) ------------------------------------------------------------------------ (1436.0,68.0) ------------------------------------------------------------------------ (176.0,877.0) ------------------------------------------------------------------------ (176.0,68.0) ------------------------------------------------------------------------ (1306,812)[(0,0)\[r\][$n=1$]{}]{} (1328.0,812.0) ------------------------------------------------------------------------ (176,604) (176.00,604.60)(1.505,0.468)[5]{} ------------------------------------------------------------------------ (176.00,603.17)(8.509,4.000)[2]{} ------------------------------------------------------------------------ (187.58,608.00)(0.492,0.539)[21]{} ------------------------------------------------------------------------ (186.17,608.00)(12.000,11.893)[2]{} ------------------------------------------------------------------------ (199.58,621.00)(0.492,0.967)[19]{} ------------------------------------------------------------------------ (198.17,621.00)(11.000,19.207)[2]{} ------------------------------------------------------------------------ (210.58,642.00)(0.492,1.186)[21]{} ------------------------------------------------------------------------ (209.17,642.00)(12.000,25.855)[2]{} ------------------------------------------------------------------------ (222.58,670.00)(0.492,1.534)[19]{} ------------------------------------------------------------------------ (221.17,670.00)(11.000,30.302)[2]{} ------------------------------------------------------------------------ (233.58,703.00)(0.492,1.530)[21]{} ------------------------------------------------------------------------ (232.17,703.00)(12.000,33.302)[2]{} ------------------------------------------------------------------------ (245.58,739.00)(0.492,1.628)[19]{} ------------------------------------------------------------------------ (244.17,739.00)(11.000,32.151)[2]{} ------------------------------------------------------------------------ (256.58,774.00)(0.492,1.272)[21]{} ------------------------------------------------------------------------ (255.17,774.00)(12.000,27.717)[2]{} ------------------------------------------------------------------------ (268.58,804.00)(0.492,0.873)[19]{} ------------------------------------------------------------------------ (267.17,804.00)(11.000,17.358)[2]{} ------------------------------------------------------------------------ (279.00,823.60)(1.651,0.468)[5]{} ------------------------------------------------------------------------ (279.00,822.17)(9.302,4.000)[2]{} ------------------------------------------------------------------------ (291.58,824.32)(0.492,-0.684)[19]{} ------------------------------------------------------------------------ (290.17,825.66)(11.000,-13.660)[2]{} ------------------------------------------------------------------------ (302.58,806.60)(0.492,-1.534)[19]{} ------------------------------------------------------------------------ (301.17,809.30)(11.000,-30.302)[2]{} ------------------------------------------------------------------------ (313.58,771.67)(0.492,-2.133)[21]{} ------------------------------------------------------------------------ (312.17,775.33)(12.000,-46.333)[2]{} ------------------------------------------------------------------------ (325.58,719.38)(0.492,-2.854)[19]{} ------------------------------------------------------------------------ (324.17,724.19)(11.000,-56.188)[2]{} ------------------------------------------------------------------------ (336.58,658.31)(0.492,-2.866)[21]{} ------------------------------------------------------------------------ (335.17,663.16)(12.000,-62.157)[2]{} ------------------------------------------------------------------------ (348.58,590.62)(0.492,-3.090)[19]{} ------------------------------------------------------------------------ (347.17,595.81)(11.000,-60.811)[2]{} ------------------------------------------------------------------------ (359.58,526.14)(0.492,-2.607)[21]{} ------------------------------------------------------------------------ (358.17,530.57)(12.000,-56.572)[2]{} ------------------------------------------------------------------------ (371.58,465.58)(0.492,-2.477)[19]{} ------------------------------------------------------------------------ (370.17,469.79)(11.000,-48.792)[2]{} ------------------------------------------------------------------------ (382.58,414.36)(0.492,-1.918)[21]{} ------------------------------------------------------------------------ (381.17,417.68)(12.000,-41.679)[2]{} ------------------------------------------------------------------------ (394.58,370.00)(0.492,-1.722)[19]{} ------------------------------------------------------------------------ (393.17,373.00)(11.000,-34.000)[2]{} ------------------------------------------------------------------------ (405.58,334.43)(0.492,-1.272)[21]{} ------------------------------------------------------------------------ (404.17,336.72)(12.000,-27.717)[2]{} ------------------------------------------------------------------------ (417.58,304.96)(0.492,-1.109)[19]{} ------------------------------------------------------------------------ (416.17,306.98)(11.000,-21.981)[2]{} ------------------------------------------------------------------------ (428.58,281.72)(0.492,-0.873)[19]{} ------------------------------------------------------------------------ (427.17,283.36)(11.000,-17.358)[2]{} ------------------------------------------------------------------------ (439.58,263.37)(0.492,-0.669)[21]{} ------------------------------------------------------------------------ (438.17,264.69)(12.000,-14.685)[2]{} ------------------------------------------------------------------------ (451.58,247.62)(0.492,-0.590)[19]{} ------------------------------------------------------------------------ (450.17,248.81)(11.000,-11.811)[2]{} ------------------------------------------------------------------------ (462.00,235.92)(0.543,-0.492)[19]{} ------------------------------------------------------------------------ (462.00,236.17)(10.887,-11.000)[2]{} ------------------------------------------------------------------------ (474.00,224.93)(0.611,-0.489)[15]{} ------------------------------------------------------------------------ (474.00,225.17)(9.778,-9.000)[2]{} ------------------------------------------------------------------------ (485.00,215.93)(0.669,-0.489)[15]{} ------------------------------------------------------------------------ (485.00,216.17)(10.685,-9.000)[2]{} ------------------------------------------------------------------------ (497.00,206.93)(0.798,-0.485)[11]{} ------------------------------------------------------------------------ (497.00,207.17)(9.488,-7.000)[2]{} ------------------------------------------------------------------------ (508.00,199.93)(1.033,-0.482)[9]{} ------------------------------------------------------------------------ (508.00,200.17)(10.132,-6.000)[2]{} ------------------------------------------------------------------------ (520.00,193.93)(1.155,-0.477)[7]{} ------------------------------------------------------------------------ (520.00,194.17)(8.966,-5.000)[2]{} ------------------------------------------------------------------------ (531.00,188.93)(1.267,-0.477)[7]{} ------------------------------------------------------------------------ (531.00,189.17)(9.800,-5.000)[2]{} ------------------------------------------------------------------------ (543.00,183.93)(1.155,-0.477)[7]{} ------------------------------------------------------------------------ (543.00,184.17)(8.966,-5.000)[2]{} ------------------------------------------------------------------------ (554.00,178.94)(1.505,-0.468)[5]{} ------------------------------------------------------------------------ (554.00,179.17)(8.509,-4.000)[2]{} ------------------------------------------------------------------------ (565.00,174.95)(2.472,-0.447)[3]{} ------------------------------------------------------------------------ (565.00,175.17)(8.472,-3.000)[2]{} ------------------------------------------------------------------------ (577.00,171.94)(1.505,-0.468)[5]{} ------------------------------------------------------------------------ (577.00,172.17)(8.509,-4.000)[2]{} ------------------------------------------------------------------------ (588.00,167.95)(2.472,-0.447)[3]{} ------------------------------------------------------------------------ (588.00,168.17)(8.472,-3.000)[2]{} ------------------------------------------------------------------------ (600.00,164.95)(2.248,-0.447)[3]{} ------------------------------------------------------------------------ (600.00,165.17)(7.748,-3.000)[2]{} ------------------------------------------------------------------------ (611,161.17) ------------------------------------------------------------------------ (611.00,162.17)(6.811,-2.000)[2]{} ------------------------------------------------------------------------ (623.00,159.95)(2.248,-0.447)[3]{} ------------------------------------------------------------------------ (623.00,160.17)(7.748,-3.000)[2]{} ------------------------------------------------------------------------ (634,156.17) ------------------------------------------------------------------------ (634.00,157.17)(6.811,-2.000)[2]{} ------------------------------------------------------------------------ (646,154.17) ------------------------------------------------------------------------ (646.00,155.17)(6.226,-2.000)[2]{} ------------------------------------------------------------------------ (657,152.17) ------------------------------------------------------------------------ (657.00,153.17)(6.811,-2.000)[2]{} ------------------------------------------------------------------------ (669,150.17) ------------------------------------------------------------------------ (669.00,151.17)(6.226,-2.000)[2]{} ------------------------------------------------------------------------ (680,148.67) ------------------------------------------------------------------------ (680.00,149.17)(5.500,-1.000)[2]{} ------------------------------------------------------------------------ (691,147.17) ------------------------------------------------------------------------ (691.00,148.17)(6.811,-2.000)[2]{} ------------------------------------------------------------------------ (703,145.67) ------------------------------------------------------------------------ (703.00,146.17)(5.500,-1.000)[2]{} ------------------------------------------------------------------------ (714,144.17) ------------------------------------------------------------------------ (714.00,145.17)(6.811,-2.000)[2]{} ------------------------------------------------------------------------ (726,142.67) ------------------------------------------------------------------------ (726.00,143.17)(5.500,-1.000)[2]{} ------------------------------------------------------------------------ (737,141.67) ------------------------------------------------------------------------ (737.00,142.17)(6.000,-1.000)[2]{} ------------------------------------------------------------------------ (749,140.67) ------------------------------------------------------------------------ (749.00,141.17)(5.500,-1.000)[2]{} ------------------------------------------------------------------------ (772,139.67) ------------------------------------------------------------------------ (772.00,140.17)(5.500,-1.000)[2]{} ------------------------------------------------------------------------ (760.0,141.0) ------------------------------------------------------------------------ (795,139.67) ------------------------------------------------------------------------ (795.00,139.17)(5.500,1.000)[2]{} ------------------------------------------------------------------------ (806,141.17) ------------------------------------------------------------------------ (806.00,140.17)(6.226,2.000)[2]{} ------------------------------------------------------------------------ (817.00,143.60)(1.651,0.468)[5]{} ------------------------------------------------------------------------ (817.00,142.17)(9.302,4.000)[2]{} ------------------------------------------------------------------------ (829.00,147.59)(0.798,0.485)[11]{} ------------------------------------------------------------------------ (829.00,146.17)(9.488,7.000)[2]{} ------------------------------------------------------------------------ (840.00,154.58)(0.600,0.491)[17]{} ------------------------------------------------------------------------ (840.00,153.17)(10.796,10.000)[2]{} ------------------------------------------------------------------------ (852.58,164.00)(0.492,0.779)[19]{} ------------------------------------------------------------------------ (851.17,164.00)(11.000,15.509)[2]{} ------------------------------------------------------------------------ (863.58,181.00)(0.492,1.056)[21]{} ------------------------------------------------------------------------ (862.17,181.00)(12.000,23.063)[2]{} ------------------------------------------------------------------------ (875.58,206.00)(0.492,1.581)[19]{} ------------------------------------------------------------------------ (874.17,206.00)(11.000,31.226)[2]{} ------------------------------------------------------------------------ (886.58,240.00)(0.492,1.961)[21]{} ------------------------------------------------------------------------ (885.17,240.00)(12.000,42.610)[2]{} ------------------------------------------------------------------------ (898.58,286.00)(0.492,2.713)[19]{} ------------------------------------------------------------------------ (897.17,286.00)(11.000,53.415)[2]{} ------------------------------------------------------------------------ (909.58,344.00)(0.492,2.780)[21]{} ------------------------------------------------------------------------ (908.17,344.00)(12.000,60.295)[2]{} ------------------------------------------------------------------------ (921.58,409.00)(0.492,3.137)[19]{} ------------------------------------------------------------------------ (920.17,409.00)(11.000,61.736)[2]{} ------------------------------------------------------------------------ (932.58,476.00)(0.492,2.901)[19]{} ------------------------------------------------------------------------ (931.17,476.00)(11.000,57.113)[2]{} ------------------------------------------------------------------------ (943.58,538.00)(0.492,2.047)[21]{} ------------------------------------------------------------------------ (942.17,538.00)(12.000,44.472)[2]{} ------------------------------------------------------------------------ (955.58,586.00)(0.492,1.345)[19]{} ------------------------------------------------------------------------ (954.17,586.00)(11.000,26.604)[2]{} ------------------------------------------------------------------------ (966.00,615.59)(0.758,0.488)[13]{} ------------------------------------------------------------------------ (966.00,614.17)(10.547,8.000)[2]{} ------------------------------------------------------------------------ (978.00,621.92)(0.496,-0.492)[19]{} ------------------------------------------------------------------------ (978.00,622.17)(9.962,-11.000)[2]{} ------------------------------------------------------------------------ (989.58,607.99)(0.492,-1.099)[21]{} ------------------------------------------------------------------------ (988.17,609.99)(12.000,-23.994)[2]{} ------------------------------------------------------------------------ (1001.58,580.45)(0.492,-1.581)[19]{} ------------------------------------------------------------------------ (1000.17,583.23)(11.000,-31.226)[2]{} ------------------------------------------------------------------------ (1012.58,546.33)(0.492,-1.616)[21]{} ------------------------------------------------------------------------ (1011.17,549.16)(12.000,-35.163)[2]{} ------------------------------------------------------------------------ (1024.58,507.85)(0.492,-1.769)[19]{} ------------------------------------------------------------------------ (1023.17,510.92)(11.000,-34.924)[2]{} ------------------------------------------------------------------------ (1035.58,470.88)(0.492,-1.444)[21]{} ------------------------------------------------------------------------ (1034.17,473.44)(12.000,-31.440)[2]{} ------------------------------------------------------------------------ (1047.58,437.06)(0.492,-1.392)[19]{} ------------------------------------------------------------------------ (1046.17,439.53)(11.000,-27.528)[2]{} ------------------------------------------------------------------------ (1058.58,407.66)(0.492,-1.203)[19]{} ------------------------------------------------------------------------ (1057.17,409.83)(11.000,-23.830)[2]{} ------------------------------------------------------------------------ (1069.58,382.68)(0.492,-0.884)[21]{} ------------------------------------------------------------------------ (1068.17,384.34)(12.000,-19.340)[2]{} ------------------------------------------------------------------------ (1081.58,362.02)(0.492,-0.779)[19]{} ------------------------------------------------------------------------ (1080.17,363.51)(11.000,-15.509)[2]{} ------------------------------------------------------------------------ (1092.58,345.79)(0.492,-0.539)[21]{} ------------------------------------------------------------------------ (1091.17,346.89)(12.000,-11.893)[2]{} ------------------------------------------------------------------------ (1104.00,333.92)(0.496,-0.492)[19]{} ------------------------------------------------------------------------ (1104.00,334.17)(9.962,-11.000)[2]{} ------------------------------------------------------------------------ (1115.00,322.93)(0.758,-0.488)[13]{} ------------------------------------------------------------------------ (1115.00,323.17)(10.547,-8.000)[2]{} ------------------------------------------------------------------------ (1127.00,314.93)(0.943,-0.482)[9]{} ------------------------------------------------------------------------ (1127.00,315.17)(9.270,-6.000)[2]{} ------------------------------------------------------------------------ (1138.00,308.93)(1.267,-0.477)[7]{} ------------------------------------------------------------------------ (1138.00,309.17)(9.800,-5.000)[2]{} ------------------------------------------------------------------------ (1150.00,303.94)(1.505,-0.468)[5]{} ------------------------------------------------------------------------ (1150.00,304.17)(8.509,-4.000)[2]{} ------------------------------------------------------------------------ (1161.00,299.95)(2.472,-0.447)[3]{} ------------------------------------------------------------------------ (1161.00,300.17)(8.472,-3.000)[2]{} ------------------------------------------------------------------------ (1173,296.17) ------------------------------------------------------------------------ (1173.00,297.17)(6.226,-2.000)[2]{} ------------------------------------------------------------------------ (1184,294.17) ------------------------------------------------------------------------ (1184.00,295.17)(6.226,-2.000)[2]{} ------------------------------------------------------------------------ (1195,292.67) ------------------------------------------------------------------------ (1195.00,293.17)(6.000,-1.000)[2]{} ------------------------------------------------------------------------ (1207,291.67) ------------------------------------------------------------------------ (1207.00,292.17)(5.500,-1.000)[2]{} ------------------------------------------------------------------------ (1218,290.67) ------------------------------------------------------------------------ (1218.00,291.17)(6.000,-1.000)[2]{} ------------------------------------------------------------------------ (1230,289.67) ------------------------------------------------------------------------ (1230.00,290.17)(5.500,-1.000)[2]{} ------------------------------------------------------------------------ (783.0,140.0) ------------------------------------------------------------------------ (1264,288.67) ------------------------------------------------------------------------ (1264.00,289.17)(6.000,-1.000)[2]{} ------------------------------------------------------------------------ (1241.0,290.0) ------------------------------------------------------------------------ (1276.0,289.0) ------------------------------------------------------------------------ (1306,767)[(0,0)\[r\][$n=0$]{}]{} (1328,767)(20.756,0.000)[4]{} (1394,767) (176,112) (176.00,112.00) (196.76,112.00) (199,112)(20.756,0.000)[0]{} (217.51,112.00) (222,112)(20.756,0.000)[0]{} (238.27,112.00) (245,112)(20.756,0.000)[0]{} (259.02,112.00) (268,112)(20.756,0.000)[0]{} (279.78,112.00) (300.53,112.00) (302,112)(20.756,0.000)[0]{} (321.29,112.00) (325,112)(20.756,0.000)[0]{} (342.04,112.00) (348,112)(20.756,0.000)[0]{} (362.80,112.00) (371,112)(20.756,0.000)[0]{} (383.55,112.00) (404.31,112.00) (405,112)(20.756,0.000)[0]{} (425.07,112.00) (428,112)(20.756,0.000)[0]{} (445.82,112.00) (451,112)(20.756,0.000)[0]{} (466.58,112.00) (474,112)(20.756,0.000)[0]{} (487.33,112.00) (497,112)(20.756,0.000)[0]{} (508.09,112.00) (528.84,112.00) (531,112)(20.756,0.000)[0]{} (549.60,112.00) (554,112)(20.756,0.000)[0]{} (570.35,112.00) (577,112)(20.756,0.000)[0]{} (591.11,112.00) (600,112)(20.756,0.000)[0]{} (611.86,112.07) (632.58,113.00) (634,113)(20.756,0.000)[0]{} (653.30,113.66) (657,114)(20.473,3.412)[0]{} (673.71,117.28) (680,119)(19.506,7.093)[0]{} (693.18,124.27) (709.64,136.64) (722.41,152.92) (732.06,171.22) (737,182)(6.563,19.690)[2]{} (749,218)(4.730,20.209)[2]{} (760,265)(4.137,20.339)[3]{} (772,324)(3.314,20.489)[3]{} (783,392)(3.459,20.465)[4]{} (795,463)(3.412,20.473)[3]{} (806,529)(4.218,20.322)[3]{} (822.49,596.65) (830.57,615.57) (845.70,620.25) (857.06,603.50) (863,590)(6.732,-19.634)[2]{} (875,555)(5.771,-19.937)[2]{} (886,517)(6.250,-19.792)[2]{} (898,479)(6.065,-19.850)[2]{} (915.68,426.29) (921,413)(8.087,-19.115)[2]{} (941.15,369.53) (952.30,352.05) (965.26,335.87) (966,335)(15.300,-14.025)[0]{} (980.70,322.04) (998.38,311.31) (1001,310)(18.895,-8.589)[0]{} (1017.45,303.18) (1024,301)(20.024,-5.461)[0]{} (1037.42,297.60) (1057.86,294.02) (1058,294)(20.670,-1.879)[0]{} (1078.54,292.21) (1081,292)(20.670,-1.879)[0]{} (1099.21,290.40) (1104,290)(20.756,0.000)[0]{} (1119.95,290.00) (1127,290)(20.670,-1.879)[0]{} (1140.66,289.00) (1150,289)(20.756,0.000)[0]{} (1161.42,289.00) (1182.17,289.00) (1184,289)(20.756,0.000)[0]{} (1202.93,289.00) (1207,289)(20.756,0.000)[0]{} (1223.68,289.00) (1230,289)(20.756,0.000)[0]{} (1244.44,289.00) (1253,289)(20.756,0.000)[0]{} (1265.20,289.00) (1285.95,289.00) (1287,289)(20.756,0.000)[0]{} (1306.71,289.00) (1310,289)(20.756,0.000)[0]{} (1327.46,289.00) (1333,289)(20.756,0.000)[0]{} (1348.22,289.00) (1356,289)(20.756,0.000)[0]{} (1368.97,289.00) (1389.73,289.00) (1390,289)(20.756,0.000)[0]{} (1410.48,289.00) (1413,289)(20.756,0.000)[0]{} (1431.24,289.00) (1436,289) (1500,900)(0,0) =cmr10 at 10pt (0,500)[(0,0)\[1\]]{} (0,420)[(0,0)\[1\]]{} (800,-40)[(0,0)[$\alpha$]{}]{} (800,-400)[(0,0)[Figure 3]{}]{} (176.0,68.0) ------------------------------------------------------------------------ (176.0,68.0) ------------------------------------------------------------------------ (154,68)[(0,0)\[r\][0]{}]{} (1416.0,68.0) ------------------------------------------------------------------------ (176.0,158.0) ------------------------------------------------------------------------ (154,158)[(0,0)\[r\][10]{}]{} (1416.0,158.0) ------------------------------------------------------------------------ (176.0,248.0) ------------------------------------------------------------------------ (154,248)[(0,0)\[r\][20]{}]{} (1416.0,248.0) ------------------------------------------------------------------------ (176.0,338.0) ------------------------------------------------------------------------ (154,338)[(0,0)\[r\][30]{}]{} (1416.0,338.0) ------------------------------------------------------------------------ (176.0,428.0) ------------------------------------------------------------------------ (154,428)[(0,0)\[r\][40]{}]{} (1416.0,428.0) ------------------------------------------------------------------------ (176.0,517.0) ------------------------------------------------------------------------ (154,517)[(0,0)\[r\][50]{}]{} (1416.0,517.0) ------------------------------------------------------------------------ (176.0,607.0) ------------------------------------------------------------------------ (154,607)[(0,0)\[r\][60]{}]{} (1416.0,607.0) ------------------------------------------------------------------------ (176.0,697.0) ------------------------------------------------------------------------ (154,697)[(0,0)\[r\][70]{}]{} (1416.0,697.0) ------------------------------------------------------------------------ (176.0,787.0) ------------------------------------------------------------------------ (154,787)[(0,0)\[r\][80]{}]{} (1416.0,787.0) ------------------------------------------------------------------------ (176.0,877.0) ------------------------------------------------------------------------ (154,877)[(0,0)\[r\][90]{}]{} (1416.0,877.0) ------------------------------------------------------------------------ (176.0,68.0) ------------------------------------------------------------------------ (176,23)[(0,0)[0.1]{}]{} (176.0,857.0) ------------------------------------------------------------------------ (334.0,68.0) ------------------------------------------------------------------------ (334,23)[(0,0)[0.15]{}]{} (334.0,857.0) ------------------------------------------------------------------------ (491.0,68.0) ------------------------------------------------------------------------ (491,23)[(0,0)[0.2]{}]{} (491.0,857.0) ------------------------------------------------------------------------ (648.0,68.0) ------------------------------------------------------------------------ (648,23)[(0,0)[0.25]{}]{} (648.0,857.0) ------------------------------------------------------------------------ (806.0,68.0) ------------------------------------------------------------------------ (806,23)[(0,0)[0.3]{}]{} (806.0,857.0) ------------------------------------------------------------------------ (963.0,68.0) ------------------------------------------------------------------------ (963,23)[(0,0)[0.35]{}]{} (963.0,857.0) ------------------------------------------------------------------------ (1121.0,68.0) ------------------------------------------------------------------------ (1121,23)[(0,0)[0.4]{}]{} (1121.0,857.0) ------------------------------------------------------------------------ (1278.0,68.0) ------------------------------------------------------------------------ (1278,23)[(0,0)[0.45]{}]{} (1278.0,857.0) ------------------------------------------------------------------------ (1436.0,68.0) ------------------------------------------------------------------------ (1436,23)[(0,0)[0.5]{}]{} (1436.0,857.0) ------------------------------------------------------------------------ (176.0,68.0) ------------------------------------------------------------------------ (1436.0,68.0) ------------------------------------------------------------------------ (176.0,877.0) ------------------------------------------------------------------------ (176.0,68.0) ------------------------------------------------------------------------ (1306,812)[(0,0)\[r\][$n=0$]{}]{} (1328,812)(20.756,0.000)[4]{} (1394,812) (239,518) (239,518)(8.827,-18.785)[5]{} (278,435)(11.125,-17.522)[4]{} (318,372)(12.925,-16.240)[3]{} (357,323)(15.237,-14.094)[2]{} (397,286)(16.156,-13.029)[2]{} (428,261)(17.184,-11.641)[2]{} (459,240)(18.090,-10.176)[2]{} (506.71,213.89) (522,206)(19.229,-7.812)[2]{} (554,193)(19.434,-7.288)[2]{} (602.99,175.52) (617,171)(19.932,-5.787)[2]{} (662.70,157.87) (680,153)(20.276,-4.435)[2]{} (723.33,143.81) (743,140)(20.377,-3.944)[2]{} (774,134)(20.400,-3.825)[2]{} (825.46,125.57) (838,124)(20.491,-3.305)[2]{} (887.08,116.74) (901,115)(20.585,-2.656)[2]{} (948.91,109.36) (963,108)(20.665,-1.937)[2]{} (1010.90,103.51) (1027,102)(20.659,-1.999)[2]{} (1072.88,97.56) (1089,96)(20.715,-1.295)[2]{} (1134.98,93.10) (1152,92)(20.692,-1.617)[3]{} (1216,87)(20.712,-1.336)[2]{} (1259.19,84.21) (1278,83)(20.715,-1.295)[2]{} (1321.33,80.27) (1341,79)(20.715,-1.295)[2]{} (1373,77) (1306,767)[(0,0)\[r\][$n=1$]{}]{} (1328.0,767.0) ------------------------------------------------------------------------ (239,818) (239.58,812.55)(0.497,-1.527)[59]{} ------------------------------------------------------------------------ (238.17,815.28)(31.000,-91.275)[2]{} ------------------------------------------------------------------------ (270.58,719.59)(0.497,-1.210)[61]{} ------------------------------------------------------------------------ (269.17,721.79)(32.000,-74.795)[2]{} ------------------------------------------------------------------------ (302.58,643.32)(0.497,-0.989)[61]{} ------------------------------------------------------------------------ (301.17,645.16)(32.000,-61.158)[2]{} ------------------------------------------------------------------------ (334.58,580.80)(0.497,-0.841)[59]{} ------------------------------------------------------------------------ (333.17,582.40)(31.000,-50.400)[2]{} ------------------------------------------------------------------------ (365.58,529.25)(0.497,-0.704)[61]{} ------------------------------------------------------------------------ (364.17,530.62)(32.000,-43.625)[2]{} ------------------------------------------------------------------------ (397.58,484.55)(0.497,-0.613)[59]{} ------------------------------------------------------------------------ (396.17,485.77)(31.000,-36.775)[2]{} ------------------------------------------------------------------------ (428.58,446.82)(0.497,-0.531)[59]{} ------------------------------------------------------------------------ (427.17,447.91)(31.000,-31.909)[2]{} ------------------------------------------------------------------------ (459.00,414.92)(0.551,-0.497)[55]{} ------------------------------------------------------------------------ (459.00,415.17)(30.876,-29.000)[2]{} ------------------------------------------------------------------------ (491.00,385.92)(0.620,-0.497)[47]{} ------------------------------------------------------------------------ (491.00,386.17)(29.763,-25.000)[2]{} ------------------------------------------------------------------------ (522.00,360.92)(0.697,-0.496)[43]{} ------------------------------------------------------------------------ (522.00,361.17)(30.637,-23.000)[2]{} ------------------------------------------------------------------------ (554.00,337.92)(0.804,-0.496)[37]{} ------------------------------------------------------------------------ (554.00,338.17)(30.464,-20.000)[2]{} ------------------------------------------------------------------------ (586.00,317.92)(0.866,-0.495)[33]{} ------------------------------------------------------------------------ (586.00,318.17)(29.363,-18.000)[2]{} ------------------------------------------------------------------------ (617.00,299.92)(0.977,-0.494)[29]{} ------------------------------------------------------------------------ (617.00,300.17)(29.184,-16.000)[2]{} ------------------------------------------------------------------------ (648.00,283.92)(1.079,-0.494)[27]{} ------------------------------------------------------------------------ (648.00,284.17)(30.021,-15.000)[2]{} ------------------------------------------------------------------------ (680.00,268.92)(1.250,-0.493)[23]{} ------------------------------------------------------------------------ (680.00,269.17)(29.749,-13.000)[2]{} ------------------------------------------------------------------------ (712.00,255.92)(1.315,-0.492)[21]{} ------------------------------------------------------------------------ (712.00,256.17)(28.648,-12.000)[2]{} ------------------------------------------------------------------------ (743.00,243.92)(1.315,-0.492)[21]{} ------------------------------------------------------------------------ (743.00,244.17)(28.648,-12.000)[2]{} ------------------------------------------------------------------------ (774.00,231.92)(1.642,-0.491)[17]{} ------------------------------------------------------------------------ (774.00,232.17)(29.136,-10.000)[2]{} ------------------------------------------------------------------------ (806.00,221.92)(1.642,-0.491)[17]{} ------------------------------------------------------------------------ (806.00,222.17)(29.136,-10.000)[2]{} ------------------------------------------------------------------------ (838.00,211.93)(1.776,-0.489)[15]{} ------------------------------------------------------------------------ (838.00,212.17)(27.933,-9.000)[2]{} ------------------------------------------------------------------------ (869.00,202.93)(2.079,-0.488)[13]{} ------------------------------------------------------------------------ (869.00,203.17)(28.472,-8.000)[2]{} ------------------------------------------------------------------------ (901.00,194.93)(2.013,-0.488)[13]{} ------------------------------------------------------------------------ (901.00,195.17)(27.575,-8.000)[2]{} ------------------------------------------------------------------------ (932.00,186.93)(2.013,-0.488)[13]{} ------------------------------------------------------------------------ (932.00,187.17)(27.575,-8.000)[2]{} ------------------------------------------------------------------------ (963.00,178.93)(2.399,-0.485)[11]{} ------------------------------------------------------------------------ (963.00,179.17)(27.997,-7.000)[2]{} ------------------------------------------------------------------------ (995.00,171.93)(2.399,-0.485)[11]{} ------------------------------------------------------------------------ (995.00,172.17)(27.997,-7.000)[2]{} ------------------------------------------------------------------------ (1027.00,164.93)(2.323,-0.485)[11]{} ------------------------------------------------------------------------ (1027.00,165.17)(27.116,-7.000)[2]{} ------------------------------------------------------------------------ (1058.00,157.93)(2.751,-0.482)[9]{} ------------------------------------------------------------------------ (1058.00,158.17)(26.503,-6.000)[2]{} ------------------------------------------------------------------------ (1089.00,151.93)(2.841,-0.482)[9]{} ------------------------------------------------------------------------ (1089.00,152.17)(27.365,-6.000)[2]{} ------------------------------------------------------------------------ (1121.00,145.93)(2.751,-0.482)[9]{} ------------------------------------------------------------------------ (1121.00,146.17)(26.503,-6.000)[2]{} ------------------------------------------------------------------------ (1152.00,139.93)(2.841,-0.482)[9]{} ------------------------------------------------------------------------ (1152.00,140.17)(27.365,-6.000)[2]{} ------------------------------------------------------------------------ (1184.00,133.93)(2.841,-0.482)[9]{} ------------------------------------------------------------------------ (1184.00,134.17)(27.365,-6.000)[2]{} ------------------------------------------------------------------------ (1216.00,127.93)(2.751,-0.482)[9]{} ------------------------------------------------------------------------ (1216.00,128.17)(26.503,-6.000)[2]{} ------------------------------------------------------------------------ (1247.00,121.93)(2.751,-0.482)[9]{} ------------------------------------------------------------------------ (1247.00,122.17)(26.503,-6.000)[2]{} ------------------------------------------------------------------------ (1278.00,115.93)(2.841,-0.482)[9]{} ------------------------------------------------------------------------ (1278.00,116.17)(27.365,-6.000)[2]{} ------------------------------------------------------------------------ (1310.00,109.93)(2.323,-0.485)[11]{} ------------------------------------------------------------------------ (1310.00,110.17)(27.116,-7.000)[2]{} ------------------------------------------------------------------------

--- abstract: 'By exploiting the fermionic qubit parity measurement, we present a scheme to realize quantum non-demolition (QND) measurement of Bell-states and generate n-party GHZ state in quantum dot. Compared with the original protocol, the required electron transfer before and after parity measurement can be nonadiabatic, which may speed up the operation speed and make the omitting of spin-orbit interaction more reasonable. This may help us to construct CNOT gate without highly precise control of coupling as the way of D. Gottesman and I. L. Chuang.' address: | Key Laboratory of Quantum Information, University of Science and\ Technology\ of China, CAS, Hefei 230026, People’s Republic of China author: - 'Guo-Ping Guo[^1], Hui Zhang, and Guang-Can Guo' title: 'Quantum Non-Demolition Bell State Measurement and N-party GHZ State Preparation in Quantum Dot' --- \#1\#2 Introduction ============ Since D. Loss and D. P. DiVicenzo proposed a quantum computation protocol based on trapping spin electrons in semiconductor quantum dot (QD) in 1998, the potential of QDs for implementing tasks in quantum information processing (QIP) has been intensely studied both theoretically and experimentally[@D.; @Loss; @and; @D.; @P.; @DiVincenzo; @B.; @E; @Kane]. The spin of an excess electron in the dot represents a promising qubit realization in such systems and it can be more accessible and scalable compared with other microscopic systems such as atoms or ions[@Jeroen; @Martijn; @Elzerman; @1]. In work[@D.; @P.; @DiVincenzo], DiVincenzo put forward five criteria, which must be all satisfied for any physical implementation of a quantum computer. There is a detailed review about the experimental progress on the spin qubit proposal using these five criteria[@J.; @M.; @Elzerman; @and; @etc.; @9]. It has been claimed that three criteria (well defined qubits, initialization and readout) have been already achieved and future experiments may focus on measuring the coherence time via the coherent manipulation of single spin and the coherent coupling via the manipulation of spins in neighboring dots. Although it has been shown[@E.; @Knill; @et; @al.; @Nature; @409; @M.; @A.; @Nielsen; @cnot] that partial measurement is sufficient for quantum computation with photons, the extension of this paradigm to other systems is highly desirable and challenging. In the so-called measurement-based quantum computation, gates coupling qubits are no longer required. This eliminates the need of highly precise control of the strength and pulsing between qubits. Recently, H.-A. Engel and D. Loss presented a novel protocol of electrons’ spin parity measurement[@Hans-Andres; @Engel; @and; @Daniel; @Loss], which explores the fact that resonant tunneling between the dots with different Zeeman splitting is only possible when the spins are antiparallel. By measuring the charge distributions between the two dots with a charge detector, such as quantum point contact (QPC)[@J.; @M.; @Elzerman; @M.; @Kroutvar; @et; @al.; @M.; @Field; @and; @et; @al.], one can figure out that two electrons initially loaded in one QD have either parallel or antiparallel spin configuration without demolition. In addition, they suggested a construction of CNOT gates in the way of C. W. J. Beenakker [*et. al*]{}[@cnot], which requires two parity measurements, an ancillary qubit, a single-qubit measurement, and the application of single-qubit operations depending on the measurement outcomes. However, this CNOT construction requires that the electron transferring between two dots should be adiabatic to keep the electrons’ spin states in each dot unchanged after a processing of loading the two dots electrons together and then separating them[@guof]. Consider two identical quantum dots 1 and 2 each with one electron initially in state $(a\left| \uparrow \right\rangle _1+b\left| \downarrow \right\rangle _1)$ and $(c\left| \uparrow \right\rangle _2+d\left| \downarrow \right\rangle $$_2)$ . The subscripts 1 and 2 depict the electrons in dot 1 and 2 respectively. The electron in dot 1 is transferred into dot 2 by operation, which acts on electron charge degree of freedom and keeps the electron spin state unchanged. Later, separate out one of the two electrons in dot 2 back into dot 1. If these processes of transferring electron into and out of dot 2 are adiabatic, the two electrons in dot 1 and 2 can respectively remain in spin states $(a\left| \uparrow \right\rangle _1+b\left| \downarrow \right\rangle _1)$ and $(c\left| \uparrow \right\rangle _2+d\left| \downarrow \right\rangle $$_2$ $)$ after these processes[@Hans]. It is noted that the electron transfer between dots is made by acting on electron charge degree of freedom, which preserves electrons’ spin state. If these electron transfer processes are nonadiabatic, the electron in dot 1 can also be transferred into dot 2 while its spin state is preserved. And the two electrons in dot 2 respectively in spin states $(a\left| \uparrow \right\rangle _2+b\left| \downarrow \right\rangle _2)$ and $(c\left| \uparrow \right\rangle _2+d\left| \downarrow \right\rangle $$_2)$ will have equal probability to be separated into dot 1 in these nonadiabatic transfer processes. Then there is 50% probability that the two electrons in dot 1 and 2 will remain in spin states $$(a\left| \uparrow \right\rangle _1+b\left| \downarrow \right\rangle _1)(c\left| \uparrow \right\rangle _2+d\left| \downarrow \right\rangle _2)$$ when one electron is transferred back into dot 1 with nonadiabatic processes. There is also 50% probability that the separated out electron into dot 1 initially stays at dot 2 and in spin state $(c\left| \uparrow \right\rangle _2+d\left| \downarrow \right\rangle $$_2)$. Then the two electrons in dot 1 and 2 will changed into spin states $$(c\left| \uparrow \right\rangle _1+d\left| \downarrow \right\rangle _1)(a\left| \uparrow \right\rangle _2+b\left| \downarrow \right\rangle _2).$$ In this case, it seems as if the two electrons in dot 1 and 2 have exchanged their spin states after all these processes. Although the nonadiabatic electron transfer process can be performed faster than the adiabatic one, we can not employ the C. W. J Beenakker [*et. al* ]{} way[@cnot] to construct CNOT gate if the two electrons after separating can be either state (1) or state (2)[@guof; @Hans]. Here we propose a scheme to realize Bell-state QND measurement and prepare n-party GHZ state with H.-A. Engel and D. Loss electron spin parity measurement. Compared with the original protocol, the present scheme emplores nonadiabatic electron transfer processes are emplored and there is uncertainty in separating the two electrons with different spins from one dot after the parity measurement. As single qubit rotations of electron in individual quantum dot can be straightly achieved with radio-frequency field, we can thus construct CNOT gate without highly precise controlled coupling by exploring the electron parity measurement in the way of D. Gottesman and I. L. Chuang [@Hans-Andres; @Engel; @and; @Daniel; @Loss; @D.; @Gottesman; @and; @I.; @L.; @Chuang]. As the electron transferring between dots, which is realized by acting on electron charge degree of freedom and preserves the electron spin, does not need to be adiabatic, the operations speeds may be enhanced and the neglecting of spin-orbital interaction may be more reasonable. The Bell-state QND Measurement ============================== The idea of Bell-state QND measurement is firstly proposed by G.-P. Guo and C.-F.Li[@G.-P.; @Guo; @and; @C.-F.; @Li].If two qubits are initially in a Bell state, the measurement can check which Bell state they are in without destruction. And if the two qubits are not in Bell states, they can be prepared in any Bell state. In this sense, the QND measurement can be used as both a complete Bell-state analyzer and a Bell states generator. In the original QND protocol, CNOT gates are employed, which challenges its realization under the present experimental conditions. Now through the electron spin parity measurement of H.-A. Engel and D. Loss, the Bell states QND measurement can be implemented straightforwardly as shown in Fig 1, even although nonadiabatic electron transferring is emplored. In the first step, two electrons 1 and 2 are both loaded into dot A. This electron transfer process can be made by acting on electron charge degree of freedom and then accordingly can preserve electron spin state. In this stage, the gate between the dots A and B is closed and the two electrons stay in dot A. The two quantum point contact(QND) charge detectors $% D_1(0)=1\ $and $D_2(0)=0$. In the second step, we open the gate for some time $t$ (about 20ns[@Hans-Andres; @Engel; @and; @Daniel; @Loss]). If their spins are antiparallel, the two electrons can tunnel to dot B. Then two charge detectors will click as $D_1(t)=0\ $and $D_2(t)=1$. On the other hand, if their spin are parallel, the two electrons will stay in dot A and the detectors remains as $D_1(t)=1\ $and $D_2(t)=0$. In this way, we know whether the two-electron spin state$\left| AA\right\rangle $ belongs to the Hilbert space of states $\{\Phi ^{\pm }=\left( \left| \uparrow \uparrow \right\rangle \pm \left| \downarrow \downarrow \right\rangle \right) /\sqrt{2% }\}$ or $\{\Psi ^{\pm }=\left( \left| \uparrow \downarrow \right\rangle \pm \left| \downarrow \uparrow \right\rangle \right) /\sqrt{2}$$\}$. This is just the H.-A. Engel and D. Loss’ original partial Bell-state measurement. To realize Bell-state QND measurement, we proceed to separate the two electrons into two dots with some static field acting on electron charge degree of freedom. As the two electrons are separated in two dots, radio-frequency fields can address individual electron spin. We can preform single qubit operations and respectively rotate the two electrons spins as $% \left| \uparrow \right\rangle \Rightarrow (\left| \uparrow \right\rangle +\left| \downarrow \right\rangle )/\sqrt{2}$ , $\left| \downarrow \right\rangle \Rightarrow (\left| \uparrow \right\rangle -\left| \downarrow \right\rangle )/\sqrt{2}$. Accordingly, these Hadamard operations will rotate the two-electron state as[@G.-P.; @Guo; @and; @C.-F.; @Li]: $$\hat{H}_1\hat{H}_2\left( \begin{array}{c} \Phi ^{+} \\ \Phi ^{-} \\ \Psi ^{+} \\ \Psi ^{-} \end{array} \right) =\left( \begin{array}{c} \Phi ^{+} \\ \Psi ^{+} \\ \Phi ^{-} \\ -\Psi ^{-} \end{array} \right) .$$ Then we reload the two electrons into one dot (for example dot A ), and re-open the gate for some time $t.$ We observe the two detectors $D_1(2t)$ and $D_2(2t)$. Combined with $D_1(t)$ and $D_2(t)$, we can determine exactly which Bell state the two electrons initially belong to just as shown in Table 1. $$\text{Table 1: The states of the two detectors corresponding to each Bell state}$$ $$\begin{tabular}{|lllll|} \hline \multicolumn{1}{|l|}{} & \multicolumn{1}{l|}{$\Psi ^{+}$} & \multicolumn{1}{l|}{$\Psi ^{-}$} & \multicolumn{1}{l|}{$\Phi ^{+}$} & $\Phi ^{-}$ \\ \hline \multicolumn{1}{|l|}{$\left| D_1(t)D_2(t)\right\rangle $} & \multicolumn{1}{l|}{$\left| 01\right\rangle $} & \multicolumn{1}{l|}{$\left| 01\right\rangle $} & \multicolumn{1}{l|}{$\left| 10\right\rangle $} & $% \left| 10\right\rangle $ \\ \hline \multicolumn{1}{|l|}{$\left| D_1(2t)D_2(2t)\right\rangle $} & \multicolumn{1}{l|}{$\left| 01\right\rangle $} & \multicolumn{1}{l|}{$\left| 10\right\rangle $} & \multicolumn{1}{l|}{$\left| 10\right\rangle $} & $% \left| 01\right\rangle $ \\ \hline \end{tabular}$$ Lastly, we separate the two electrons and irradiate them to perform Hadamard operations as the above steps. In this way, the final output state will recover to the initial state if the two electrons is originally in a Bell state and a complete Bell-state measurement is implemented without destruction. As all operations are identical to the two electrons, we need not know which electron (with different spin state) is separated out from dot B in this Bell-state QND measurement. Distinguished with the original protocol, all the electron transfer processes between dots can thus be non-adiabatic and directly realized by acting on electron charge degree of freedom. Obviously, if the two electrons are initially in an arbitrary state $\Phi _{12}=a\Phi ^{+}+b\Phi ^{-}+c\Psi ^{+}+d\Psi ^{-},$where $a,b,c,d\in C$, the above measurement process will project them into one of the Bell states: $$\begin{aligned} &&(a\Phi ^{+}+b\Phi ^{-}+c\Psi ^{+}+d\Psi ^{-})\otimes \left| D_1(t)D_2(t)\right\rangle \otimes \left| D_1(2t)D_2(2t)\right\rangle \\ &\rightarrow &a\Phi ^{+}|10\rangle \otimes |10\rangle +b\Phi ^{-}|10\rangle \otimes |01\rangle +c\Psi ^{+}|01\rangle \otimes |01\rangle +d\Psi ^{-}|01\rangle \otimes |10\rangle . \nonumber\end{aligned}$$ For example, we have a probability of $\left| d\right| ^2$ to project the two electrons into state $\Psi ^{-}$. With some single qubit rotations, we can thus get any other Bell-state. In this sense, the above measurement can be also regarded as spin-electron Bell states generation protocol, which theoretically has unit efficiency. Actually, there is no need for two charge detectors, as the signal from D1 is always anti-correlated with the signal from D2. Therefore a single charge detector is sufficient. This would simplify the experimental setup. Preparation of the n-particle GHZ state ======================================= After the demonstration of two electrons full Bell states measurements, we now discuss the preparation of the prior required GHZ entanglement states for the CNOT gate construction[@D.; @Gottesman; @and; @I.; @L.; @Chuang]. We assume that we have gotten two electrons $i,$ $j$ in the state $\Phi _{ij}^{+}=\left( \left| \uparrow \uparrow \right\rangle +\left| \downarrow \downarrow \right\rangle \right) /\sqrt{2}$ in dot B from the above Bell-state QND measurement. Then we can separate one electron from dot B to dot A by static fields, which preserve their total spin-state $\Phi ^{+}$[@W.; @G.; @van; @der; @Wiel]. Adjusting the bias voltage and the gate voltage between dot B and dot C, we can load another electron $k$ from dot C to dot B as shown in Fig.2. The initial state of the electron in dot C is $\Psi _k=% \frac 1{\sqrt{2}}\left( \left| \uparrow \right\rangle _k+\left| \downarrow \right\rangle _k\right) $. Thus the three-electron state can be written as (only the electron spin wavefunctions are shown): $$\begin{aligned} \Psi _{ijk} &=&\Phi _{ij}^{+}\otimes \Psi _k=\frac 1{\sqrt{2}}\left( \left| \uparrow \uparrow \right\rangle _{ij}+\left| \downarrow \downarrow \right\rangle _{ij}\right) \otimes \frac 1{\sqrt{2}}\left( \left| \uparrow \right\rangle _k+\left| \downarrow \right\rangle _k\right) \nonumber \\ &=&\frac 12(\left| \uparrow \right\rangle _i\left| \uparrow \uparrow \right\rangle _{jk}+\left| \downarrow \right\rangle _i\left| \downarrow \downarrow \right\rangle _{jk}+\left| \uparrow \right\rangle _i\left| \uparrow \downarrow \right\rangle _{jk}+\left| \downarrow \right\rangle _i\left| \downarrow \uparrow \right\rangle _{jk}).\end{aligned}$$ The two electrons (electron 2 and 3) in dot B have $1/2$ probabilities to be measured in parallel spins to get state $(\left| \uparrow \right\rangle _i\left| \uparrow \uparrow \right\rangle _{jk}+\left| \downarrow \right\rangle _i\left| \downarrow \downarrow \right\rangle _{jk})/\sqrt{2}$. In this case, when we separate one electron from dot B to dot C, the three electrons ( $i,j,k$ named the three electrons respectively in dot A, B and C) are in the state $(\left| \uparrow \right\rangle _i\left| \uparrow \right\rangle _j\left| \uparrow \right\rangle _k+\left| \downarrow \right\rangle _i\left| \downarrow \right\rangle _j\left| \downarrow \right\rangle _k)/\sqrt{2},$ which can be rotated into any other GHZ state with single qubit operation. Without doubt, the two electrons $j,k$ in dot B have $1/2$ probabilities to be measured in anti-parallel spins to get state $% (\left| \uparrow \right\rangle _i\left| \uparrow \downarrow \right\rangle _{jk}+\left| \downarrow \right\rangle _i\left| \downarrow \uparrow \right\rangle _{jk})/\sqrt{2}$. In view of the indistinguishability of two electrons $j$ and $k$ in dot B and the uncertainty in separating them, we then have equal probability to get the state $(\left| \uparrow \right\rangle _i\left| \uparrow \right\rangle _j\left| \downarrow \right\rangle _k+\left| \downarrow \right\rangle _i\left| \downarrow \right\rangle _j\left| \uparrow \right\rangle _k)/\sqrt{2}$ and $(\left| \uparrow \right\rangle _i\left| \downarrow \right\rangle _j\left| \uparrow \right\rangle _k+\left| \downarrow \right\rangle _i\left| \uparrow \right\rangle _j\left| \downarrow \right\rangle _k)/\sqrt{2}$, when we separate one electron from dot B to dot C. In this case, we can then reload the two electrons $i$ and $j$ into one dot and measure their parity again. After separating them, we then again have $1/2$ probabilities to get the three electrons GHZ state $(\left| \uparrow \right\rangle _i\left| \uparrow \right\rangle _j\left| \downarrow \right\rangle _k+\left| \downarrow \right\rangle _i\left| \downarrow \right\rangle _j\left| \uparrow \right\rangle _k)/\sqrt{2}$. Two electrons $% i $ and $j$ also have $1/2$ probabilities to be measured in antiparallel spins to get state $(\left| \uparrow \downarrow \right\rangle _{ij}\left| \uparrow \right\rangle _k+\left| \downarrow \uparrow \right\rangle _{ij}\left| \downarrow \right\rangle _k)/\sqrt{2}$. In this case, when we separate electrons $i,j$, we can have again equal probability to get the state $(\left| \uparrow \right\rangle _i\left| \downarrow \right\rangle _j\left| \uparrow \right\rangle _k+\left| \downarrow \right\rangle _i\left| \uparrow \right\rangle _j\left| \downarrow \right\rangle _k)/\sqrt{2}$ and $% (\left| \downarrow \right\rangle _i\left| \uparrow \right\rangle _j\left| \uparrow \right\rangle _k+\left| \uparrow \right\rangle _i\left| \downarrow \right\rangle _j\left| \downarrow \right\rangle _k)/\sqrt{2}$. If necessary, we can again measure the spin parity of electron $j$ and $k$. In the process of repeating comparing the three electrons’ spin, we can have a success probability $p$ of $1-\frac 1{2^m}$ to get the electron GHZ state, where $m$ represents the times of comparing the electrons spins. When $m$ is large enough, $p\rightarrow 1$ and we can almost prepare three electrons GHZ state. Of course, if we only compare the spins of the two electrons $j,k$ initially in dot B one time, we have $1/2$ probabilities to get the GHZ state $(\left| \uparrow \right\rangle _i\left| \uparrow \right\rangle _j\left| \uparrow \right\rangle _k+\left| \downarrow \right\rangle _i\left| \downarrow \right\rangle _j\left| \downarrow \right\rangle _k)/\sqrt{2}.$ Obviously, the above process can be directly performed on the case of n-electron GHZ states preparation ($n>3$) by comparing the spins of the electrons between two dots in turn. In this way, the generation efficiency is $p^{n-2},$where $p$ is the success probability of get three electron GHZ states. In fact, the above GHZ state generation procedure is very similar to the idea of S. Bose[* et. al.[@swapping],* ]{}which shows that[* *]{}the entangled states involving higher number of particles can be generated from entangled states involving lower number of particles by employing the same procedure as entanglement swaps. The basic ingredient of the original paper[@Guo; @swapping] is a Bell state measuring device and some L-particle (particle of lower number ($n=3$) ) entanglement states. However, it can be proved that the required lower numbers of particles entanglement states can be just 2-particle entangled states with the present non-demolition partial Bell state measurement. Firstly, we prepare two copies of entangled states of $n/2$ ( for example, assume $n$ is even) electrons in the form $\Psi _{n/2}=(\left| \uparrow \uparrow \uparrow ...\uparrow \right\rangle _{123...\frac n2}+\left| \downarrow \downarrow \downarrow ...\downarrow \right\rangle _{123...\frac n2}$$)/\sqrt{2}$. Secondly, we coherently draw one electron from each of the two copies into a dot. By checking these two electrons spins (if it is parallel), we can get the entangled state of $n$ electrons $\Psi _n=(\left| \uparrow \uparrow \uparrow ...\uparrow \right\rangle _{123...n}+\left| \downarrow \downarrow \downarrow ...\downarrow \right\rangle _{123...n}$$)/\sqrt{2}$ with 1/2 probability. As the success probability of each time parity checking is 1/2, this $n$ ($n\geq 2$) electrons preparation protocol has an efficiency of $% p^{n/2-1}$ ($n$ is even) or $p^{^{(n+1)/2-1}}$ (n is odd). As the $n$ electrons in $\Psi _n$ are respectively in $n$ quantum dots, we can rotate it into any other $n$-particle GHZ entangled states with single-qubit operations. Conclusion ========== Here we propose a protocol to realize the QND measurement of Bell-state, prepare $n$-electron GHZ states and then construct CNOT gate by employing the novel partial Bell-state measurement of Fermionic qubits in the article[@Hans-Andres; @Engel; @and; @Daniel; @Loss]. Distinguished with the previous protocol[@Hans-Andres; @Engel; @and; @Daniel; @Loss], the electron transfer processes before and after the spin parity measurement can nonadiabatic, which may make the omitting of the spin-orbit interaction effects in electron transport more sensible and speed up the computation operations. Furthermore, the present modified protocol has similar robustness to the experimental noises, such as the effect of extra phases from the inhomogeneous Zeeman splitting, finite J and different tunnel couplings of singlet and triplet. It is noted that the precision of the charge detectors or the fidelity of QPC measurements, which greatly influence the success of the electrons spin parity checking and the present protocols, have been recently analyzed in detail[@J.; @M.; @Elzerman]. [**Acknowledgments**]{} This work is funded by the National Fundamental Research Program (2001CB309300), National Nature Science Foundation of China (10304017), the Innovation Funds from Chinese Academy of Sciences. D. Loss and D. P. DiVincenzo, Phys. Rev. A 57, 120 (1998). B. E. Kane Nature 393, 119-120 (1998). Jeroen Martijn Elzerman’s PHD thesis, [*Electron spin and charge in semiconductor quantum dots*]{}, chapter 1. Delft University of Technology, The Netherlands (2004). D. P. DiVincenzo, Fortschr. Phys. 48, 771 (2000). Jeroen Martijn Elzerman’s PHD thesis, [*Electron spin and charge in semiconductor quantum dots*]{} , chapter 9. Delft University of Technology, The Netherlands (2004). E. Knill, R. Laflamme, G. Milburn, Nature 409,46 (2001). M. A. Nielsen, I. L. Chuang, [*Quantum Computation and Quantum Information*]{} (Cambridge Univ. Press, New York, 2000). C. W. J. Beenakker, [*et.al.* ]{}, Phys. Rev. Lett. 93, 020501 (2004). Hans-Andres Engel and Daniel Loss, Science 309, 22 July 2005, VOL 309. J. M. Elzerman, et al. Nature 430, 431-435 (2004). Petta et al. Science 309, 2180, 2005. M. Kroutvar et al., Nature 432, 81 (2004). M. Field et al., Phys. Rev. Lett. 70, 1311 (1993). In C. W. J. Beenakker [*et. al*]{}[@cnot] original propose of constructing CNOT gate, the photon with state $(a\left| 0\right\rangle _u+b\left| 1\right\rangle _u)$ input from up-path determinately comes out from up-path and the photon with state $(c\left| 0\right\rangle _d+d\left| 1\right\rangle $$_d$ $)$ input from down-path determinately comes out from down-path after parity measurement. These two output photons will then undergo different operations. If the output photons from parity measurement circuit up-path has equal probability to be in either state $(a\left| 0\right\rangle _u+b\left| 1\right\rangle _u)$ or $% (c\left| 0\right\rangle _u+d\left| 1\right\rangle $$_u$ $)$, (the output photon from down-path correspondingly will be in either state $(c\left| 0\right\rangle _d+d\left| 1\right\rangle $$_d$ $)$ or $(a\left| 0\right\rangle _d+b\left| 1\right\rangle _d)$ ), we then cannot determinate the following operations. Hans-Andres Engel and Daniel Loss, Science 309, 22 July 2005, VOL 309. D. Gottesman and I. L. Chuang, Nature 402, 390 (1999). G.-P. Guo, C.-F. Li ,Phys. Lett. A 286, 401 (2001). H. -A. Engel et al., Phys. Rev. Lett. 93, 106804 (2004). W. Lu , Z. Ji , L. Pfeiffer , K. W. West , A. J. Rimberg , Nature 423 , 422 (2003). R. Schleser et al., Appl. Phys. Lett. 85 , 2005 (2004). W. G. van der Wiel, et al. Reviews of Modern Physics, VOL. 75, JANUARY 2003. G. P. Guo, et. al Phys. Rev. A 65, 042102 (2002). S. Bose, V. Vedral, and P. L. Knight, Phys. Rev. A [**57**]{}, 822 (1998). [**Fig. 1.**]{} Schematic picture of the Bell-state analyzer, which includes two coupled quantum dots (circle A and B), two quantum point contact charge detectors (triangle D$_1$ and D$_2$) and a gate (solid vertical line). Dot A and B are assumed to have different Zeeman splitting, and the individual tunneling events can be efficiently identified with a time resolved measurement[@J.; @M.; @Elzerman; @H.; @-A.; @Engel; @et; @al.; @W.; @Lu; @and; @Z.; @Ji; @and; @L.; @Pfeiffer; @and; @K.; @W.; @West; @A.; @J.; @Rimberg; @R.; @Schleser; @et; @al.]. The gate switches on and off the coupling between dot A and B. Here we consider only the case that the two electrons are both in dot A or in dot B $\left| AA\right\rangle $ and $\left| BB\right\rangle $, as the coupling to the state $\left| AB\right\rangle $ is small[@Hans-Andres; @Engel; @and; @Daniel; @Loss]. The two electrons tunneling between states $\left| AA\right\rangle $ and $% \left| BB\right\rangle $ is only resonant, when they have antiparallel spins and in the Hilbert space $\{\Psi ^{\pm }=(\left| \uparrow \downarrow \right\rangle \pm \left| \downarrow \uparrow \right\rangle $$)/\sqrt{2}\}$. If they are in space $\{$ $\Phi ^{\pm }=(\left| \uparrow \uparrow \right\rangle $ $\pm \left| \downarrow \downarrow \right\rangle $$)/% \sqrt{2}\}$, these two electrons will remain on the initial dot. This requirement is the key principle of electrons spin parity measurements. If the two electrons are both in the dot A (or in the dot B), $D_1=1\ $and $% D_2=0\ $(or $D_2=1\ $and $D_1=0\ $). [**Fig. 2.**]{} Schematic picture for preparation of the $3$-electron GHZ states. Initially, dot A, B and C each has one electron. The electron transfer between quantum dots is required to preserve electron spin state, which can be achieved by static electrical fields acting on the electron charge degree of freedom. Spin parity measurement of two electrons is made in dot B. [^1]: Electronic address: gpguo@ustc.edu.cn

--- abstract: 'We study the dynamical behaviour of the quantum cellular automaton of Refs. [@darianopla; @BDTqcaI], which reproduces the Dirac dynamics in the limit of small wavevectors and masses. We present analytical evaluations along with computer simulations, showing how the automaton exhibits typical Dirac dynamical features, as the Zitterbewegung and the scattering behaviour from potential that gives rise to the so-called Klein paradox. The motivation is to show concretely how pure processing of quantum information can lead to particle mechanics as an emergent feature, an issue that has been the focus of solid-state, optical and atomic-physics quantum simulator.' author: - Alessandro - Giacomo Mauro - Alessandro bibliography: - 'bibliography.bib' title: 'The Dirac Quantum Cellular Automaton in one dimension: Zitterbewegung and scattering from potential' --- Introduction ============ The idea of reproducing the evolution of a macroscopic system starting from a simple rule of local interaction among its elementary constituents was first formalized in the pioneering von Neumann’s paper [@neumann1966theory] with the notion of *Cellular Automaton*. The automaton is a regular lattice of cells with a finite number of states, equipped with a rule that updates the cell states from time $t$ to time $t+1$. Such rule must be *local*, namely the state of the $x$ cell at $t+1$ depends only on the states of a finite number of neighboring cells at $t$. Cellular automata have been a popular topic for many years, as a new paradigm for complex systems, and many books have been devoted to the subject (see eg. Refs. [@wolfram2002new; @toffoli1987cellular]). One of the reasons of its first success, which eventually has become its own weakness, is the chaotic behaviour of the automaton dynamics [@sep-cellular-automata]. Differently from classical cellular automata, *Quantum Cellular Automata* (QCA) exhibit a less chaotic behaviour, which makes them predictable for large number of steps [@ambainis2001one]. Here the cells are finite-dimensional quantum systems interacting locally and unitarily. Being locality of interactions an essential ingredient of any physical evolution, QCA have been considered already by Feynman as candidates for simulating physics [@feynman1965quantum; @feynman1982simulating]. More recently QCA earned interest in the quantum information community leading to many results on its mathematical theory [@schumacher2004reversible; @arrighi2011unitarity; @gross2012index], and on their general dynamical features [@ambainis2001one; @knight2004propagating; @valcarcel2010tailoring; @ahlbrecht2011asymptotic; @reitzner2011quantum]. In quantum field theory, after the first appearance of a prototype of QCA in the Feynman chessboard [@feynman1965quantum] for solving the path-integral for the Dirac field, a similar framework has appeared in the work of Nakamura [@nakamura1991nonstandard] motivated by a rigorous formulation of the Feynman path integral, and later in the seminal work of Bialynicki-Birula [@bialynicki1994weyl], as a lattice theory for Weyl, Dirac, and Maxwell fields. Then the possibility of using automata for describing the evolution of relativistic fields emerged in the context of lattice-gas simulations, especially in the work of Meyer [@meyer1996quantum], where a notion of “field automaton” first appeared, and in the papers of Yepez [@Yepez:2006p4406]. More recently QCA have been considered for extending quantum field theory [@darianopla] to the Plank scale. Similar to lattice-gas theories, here the quantum cell corresponds to the evaluation $\psi(x)$ of a quantum field on the site $x$ of a lattice, with the dynamics updated in discrete time steps by a local unitary evolution. However, differently from lattice-gas theory, here the continuum limit is not taken, whereas, instead, the asymptotic large-scale (Fermi) evolution is considered. The main difference is then that Lorentz covariance holds exactly in the relativistic limit of small momentum, whereas generally it is distorted, in a fashion analogous to Refs. [@magueijo2003generalized; @amelino2001planck1; @amelino2001planck]. In this context the one dimensional Dirac automaton has been derived from symmetry principles for the QCA [@BDTqcaI] showing how the usual Dirac dynamics emerges at the Fermi scale, though relativistic covariance and other symmetries are violated at the Planck/ultrarelativistic scale. In the present paper we analyze in detail the one-particle sector of the automaton of Refs. [@darianopla; @BDTqcaI]. Here, particle states are “smooth” states peaked around a momentum eigenstate of the QCA. We will consider dynamical quantities as the particle position, momentum and velocity, along with their evolution both in the free case and in the presence of a potential, recovering typical features of Dirac quantum field evolution–as [*Zitterbewegung*]{} and [ *Klein paradox*]{}–from the pure quantum information processing of the QCA. Recently there has been a renewed interest in Dirac features in solid-state and atomic physics, which provide a physical hardware to simulate the dynamics. Zitterbewegung can be seen in the response of electrons to external fields [@huang1952zitterbewegung] and can appear for nonrelativistic particles in a crystal [@cannata1991effects; @ferrari1990nonrelativistic; @cannata1990dirac], quasiparticles in superconductors [@lurie1970zitterbewegung] and systems with spin-orbit coupling [@PhysRevLett.95.187203; @PhysRevLett.99.076603]. Proving that the oscillation behavior is not unique to Dirac electrons, but rather is a generic feature of spinor systems with linear dispersion relations, these works opened the way for possible simulation of Zitterbewegung using for example trapped ions [@lamata2007dirac; @gerritsma2010quantum], two-band crystalline structure such as graphene [@cserti2006unified; @rusin2007transient] or semiconductors [@schliemann2005zitterbewegung; @zawadzki2005zitterbewegung; @zawadzki2010nature; @geim2007rise; @zawadzki2011zitterbewegung], ultra cold atoms [@vaishnav2008obserVing], and finally photonic crystals [@PhysRevLett.100.113903]. On the other hand, the Klein paradox (tunneling of relativistic particles) provides insight in the mechanics of relativistic particles propagating through potential barriers, along with vacuum polarization effects, and has been a focus in the hot topic of graphene as a simulator for Dirac equation, as in Ref. [@katsnelson2007graphene], and [@gerritsma2010quantum] for trapped ions. Recently also microfabricated optical waveguide circuits have become an alternative physical simulator for particle dynamics [@sansoni2012two]. After reviewing the Dirac QCA in 1d in Section \[s:DiracQCA\], in Section \[sec:zitterbewegung\] we present the evolution of position and momentum operators for the automaton, showing the Zitterbewegung behaviour produced by the interference between positive and negative frequencies. In Section \[s:barrier\] we modify the QCA in order to insert a potential in the free evolution, and show the automaton dynamics in the presence of a barrier for one particle states. We end the paper with a summary and some concluding remarks in Section\[s:concl\]. The Dirac Automaton {#s:DiracQCA} =================== The quantum automaton corresponding to the Dirac equation in 1d, first introduced in [@darianopla], has been derived from the discrete automaton symmetries of parity and time-reversal in Ref. [@BDTqcaI], where also the Dirac equation has been recovered as the large-scale relativistic limit of the automaton. The cell of the quantum automaton is given by the evaluation $\psi(x)$ of the two-component field operator $\psi$, and the unitary evolution of one step of the automaton is given by $$\begin{aligned} \psi(x)\to U\psi(x),\quad \psi(x) := \left(\begin{array}{c} \psi_r(x) \\ \psi_l(x) \end{array} \right)\quad \end{aligned}$$ where $\psi_l$ and $\psi_r$ denote the [*left*]{} and [*right*]{} mode of the field, whereas the unitary matrix $U$ is given by $$\label{U} U=\begin{pmatrix} n S &-im\\ -im & n S^\dag \end{pmatrix},\\ \quad n^2+m^2=1,$$ with $S$ denoting the shift operator $S f(x) = f(x+1)$. The constants $n$ and $m$ in the last equation can be chosen positive. As shown in Refs. [@darianopla; @BDTqcaI], the parameter $m$ plays the role of an a-dimensional inertial mass, and is bounded by unit. We remark that the automaton description is completely a-dimensional, and a conversion to the usual physical dimensions needs a length, a time and a mass, which one can take as the Planck length $\ell_P$, the Planck time $\tau_P$, and the Planck mass $m_P$, the latter playing the role of the bound for the inertial mass. The maximal speed of propagation of information is one cell per step ($c=\ell_P/\tau_P$ in dimensional units, corresponding to the speed of light). The quantum field can be taken generally as Fermionic, Bosonic, or even Anyonic. However, in the present case it will not be relevant, since we will consider only single-particle states, which span the Hilbert space $\mathbb{C}^2\otimes l_2(\Z)$, and for which we will use the factorized orthonormal basis $\ket{s}\ket{x}$, where for $\ket{s}$ we consider the canonical basis corresponding to $s=l,r$. These states can be also obtained as $\psi^\dag_s(x)\ket{\Omega}$ upon introducing a vacuum $|\Omega\>$ which is annihilated by the field operator, and invariant under the automaton evolution. Similarly also $N$-particle states with $N>1$ can be obtained by acting with products of $N$ evaluations of the field operator, building up the Fock space in the usual way. Notice that the evolution of the field is restricted to be linear, and there exists a unitary operator $U$ such that the field evolution is given by $V\psi_s(x)V^\dag=U\psi_s(x)$, with $V|\Omega\>=|\Omega\>$, whereas for product of field evaluations the evolution is given by tensor powers of $U$ as $V\psi_{s_1}(x_1) \ldots \psi_{s_N}(x_N) V^\dag=U^{\otimes N}\psi_{s_1}(x_1)\otimes \ldots\otimes\psi_{s_N}(x_N)$. In the $\ket{s}\ket{x}$ representation the unitary matrix $U$ can be written as follows $$\begin{aligned} U := \sum_x \begin{pmatrix} n \ketbra{x-1}{x} & -im \ketbra{x}{x} \\ -im \ketbra{x}{x} & n \ketbra{x+1}{x} \end{pmatrix},\end{aligned}$$ describing a [*Quantum Walk*]{} on the Hilbert space $\mathbb{C}^2\otimes l_2(\Z)$ [@ambainis2001one]. Tanks to the translational invariance of $U$, it is convenient to move to the momentum representation $$\begin{aligned} \label{vk} \ket{\psi_s}\ket{k} :=\frac{1}{\sqrt{2\pi}} \sum_{x} e^{-ikx} \ket{\psi_s}\ket{x},\quad k\in[-\pi,\pi],\end{aligned}$$ and $U$ becomes $$\begin{aligned} \label{eq:automaton-Uk} U= \int_{\minus \pi}^{\pi} \d k U(k)\otimes\ketbra{k}{k},\:\, {U}(k)= \begin{pmatrix} n e^{ik} & -i m\\ -i m & n e^{-ik} \end{pmatrix}.\end{aligned}$$ Notice that discreteness bound momenta to the Brillouin zone, as in solid-state theory. By diagonalizing the unitary matrix ${U}(k)$ $$\begin{aligned} \label{eq:eigenstates} &U(k) \ket{s}_k = e^{-is \omega(k)}\ket{s}_k,\qquad \omega(k) = \arccos(n \cos k)\\\nonumber &\ket{s}_k:=\tfrac{1}{\sqrt{2}} \begin{bmatrix} \sqrt{1-sv(k)}\\s\sqrt{1+sv(k)} \end{bmatrix}, \quad s=\pm,\quad v(k) := \partial_k \omega(k)\end{aligned}$$ it is easy to evaluate the logarithm of $U$ ($e^{-i H}:= U$) as follows $$\begin{aligned} \label{eq:hamiltonian} H = \int_{-\pi}^{\pi} \d k H(k)&\otimes\ketbra{k}{k},\\\nonumber H(k) &=\omega(k)\left( \ket{+}_k\bra{+}_k - \ket{-}_k\bra{-}_k\right) \\\nonumber &=\mathrm{sinc^{-1}}{\omega(k)} (-n\sin k\, \sigma_3+m\sigma_1),\end{aligned}$$ where $\sigma_i$ $i=1,2,3$ denote the usual Pauli matrices. The function $\omega(k)$ is the dispersion relation of the automaton, which recovers the usual Dirac one $\omega(k)=\sqrt{k^2+m^2}$ in the limit $k,m\ll 1$ and $k/m\gg 1$ as shown in [@BDTqcaI]. This is also clear in Fig. \[fig:disp\] where the dispersion relation as a function of $k$ is reported for four different values of the mass. The derivative $v(k)$ in Eq. (\[eq:eigenstates\]) is then the group velocity of the wavepacket. The $s=+1$ eigenvalues correspond to positive-energy particle states, whereas the negative $s=-1$ eigenvalues correspond to negative-energy anti-particle states. Notice that the operator $H$ regarded as an Hamiltonian would interpolate the evolution to continuous time as $U(t)\equiv U^t$, which, however, in this context should be considered unphysical [^1] ![The Dirac automaton dispersion relation in Eq. for four different values of the mass: $m=0.1,\,0.2,\,0.4,\,0.8$.[]{data-label="fig:disp"}](DispRel-0_1.pdf "fig:"){width=".21\textwidth"}![The Dirac automaton dispersion relation in Eq. for four different values of the mass: $m=0.1,\,0.2,\,0.4,\,0.8$.[]{data-label="fig:disp"}](DispRel-0_2.pdf "fig:"){width=".21\textwidth"} ![The Dirac automaton dispersion relation in Eq. for four different values of the mass: $m=0.1,\,0.2,\,0.4,\,0.8$.[]{data-label="fig:disp"}](DispRel-0_4.pdf "fig:"){width=".21\textwidth"}![The Dirac automaton dispersion relation in Eq. for four different values of the mass: $m=0.1,\,0.2,\,0.4,\,0.8$.[]{data-label="fig:disp"}](DispRel-0_8.pdf "fig:"){width=".21\textwidth"} In the following sections we will analyze two typical aspects of the Dirac-field dynamics, namely the Zitterbewegung and the Klein paradox. Position and Momentum Operators and Zitterbewegung {#sec:zitterbewegung} ================================================== The QCA (\[U\]) describes very precisely the Dirac field dynamics for customary relativistic wavevectors and energies (consider that e.g. a ultra-high-energy cosmic ray has $k\simeq 10^{-8}$) [@BDTqcaI]. In this section we will show how efficiently it reproduces a typical feature of the one-particle Dirac dynamics, namely the Zitterbewegung. The Zitterbewegung was first recognized by Schrödinger in 1930 [@schrodinger1930kraftefreie] who noticed that in the Dirac equation describing the free relativistic electron the velocity operator does not commute with the Dirac Hamiltonian: the evolution of the position operator, in addition to the classical motion shows a very fast periodic oscillation with frequency $2mc^2$ and amplitude equal to the Compton wavelength $\hbar/mc$ with $m$ the rest mass of the relativistic particle. This jittering motion first encountered in the Dirac theory of the electron was then shown [@huang1952zitterbewegung] to arise from the interference of states corresponding to the positive and negative energies resulting from the Dirac equation with the trembling disappearing with time [@lock1979zitterbewegung] for a wavepacked particle state. Zitterbewegung oscillations cannot be directly observed by current experimental techniques for a Dirac electron since the amplitude should by very small $\approx 10^{-12}$ m. However, it can be seen in a number of solid-state, atomic-physics, photonic-cristal and optical waveguide simulators, as quoted in the introduction. The “position” operator $X$ providing the representation $|x\>$ (i.e. such that $X|s\>|x\>=x|s\>|x\>$) is defined as follows $$\begin{aligned} X=\sum_{x\in\mathbb{Z}}x(I\otimes\ketbra{x}{x}).\end{aligned}$$ Generally $X$ provides the average location of a wavepacket in terms of $\<\psi|X|\psi\>$. The conjugated “momentum” operator is given by $$\begin{aligned} P=\int_{-\pi}^{\pi}\;\df{k}{2\pi} k(I\otimes\ketbra{k}{k}).\end{aligned}$$ One can verify that $X$ and $P$ obey the usual canonical commutation rule $[X,P]=i$. In the following it will be convenient to work with the continuous time $t$ interpolating exactly the discrete automaton evolution, namely $U^t$. However, all numerical results will be given only for discrete $t$, namely for repeated applications of the automaton unitary $U$ in Eq. (\[U\]). The time evolution of the position operator ${X}(t)={U}^{ - t} {X}{U}^{t}$ can be more easily computed by integrating the differential equation $A(t) = [H,[H,X(t)]]$ where $H$ was defined in Eq. . We have $$\begin{aligned} \nonumber &A(t) = \int_{-\pi}^{\pi} \d k A(k,t) \otimes\ketbra{k}{k} \quad A(k,t) = e^{2i{H(k)}t}{A}(k)\\ &A(k) = -\frac{2\omega^2}{\sin^2\omega}nm\cos{k}\, \sigma_2\end{aligned}$$ which leads to $$\begin{aligned} \label{e:Zx} {X}(t)={X}(0)+{V} t+{Z}_{{X}}(t)-{Z}_{{X}}(0)\\ {V}(k)=-v(k)^2\sigma_3+v(k)\sqrt{1-v(k)^2}\sigma_1\\ {Z}_{{X}}(k,t)=-\frac{1}{4}{H}^{-2}(k){A}(k,t)\end{aligned}$$ where $V$ is the classical component of the velocity operator which, in the base diagonalizing the Hamiltonian , is $V(k)=v(k)\sigma_3$ and is proportional to the group velocity $v(k)$. Since a generic one-particle state $\ket{\psi}$ is a superposition of a positive and a negative energy state, i.e. $\ket{\psi_+} + \ket{\psi_-}$, the evolution of the mean value of the position operator $X(t)$, can be written as $$\begin{aligned} \nonumber x_\psi(t) := \bra{\psi} X(t) \ket{\psi} = x_{\psi}^+(t) + x_{\psi}^-(t) + x_{\psi}^{\rm{int}}(t)\\\nonumber x_{\psi}^{\pm}(t) : = \bra{\psi _\pm} X(0) + Vt \ket{\psi_\pm}\\\label{e:zitterbewegung} x_{\psi}^{\rm{int}}(t) := 2 \Re [\bra{\psi_+} X(0) - {Z}_{{X}}(0) + {Z}_{{X}}(t) \ket{\psi_-}]\end{aligned}$$ where $\Re$ denotes the real part. The interference between positive and negative frequency is responsible of the $x_{\psi}^{\rm{int}}(t)$. The magnitude of $x_{\psi}^{\rm{int}}(t)$ is bounded by $1/m$ (see appendix \[a:zitterbewegung\]) which in the usual dimensional units correspond to the Compton wavelength $\hbar/ m c$. Moreover the stationary phase approximation shows that for $t \to \infty$ the term $2 \Re [\bra{\psi_+} {Z}_{{X}}(t) \ket{\psi_-}]$, which is responsible of the oscillation, goes to $0$ as $1/\sqrt{t}$ (see appendix \[a:zitterbewegung\]) and only the shift contribution coming from $2 \Re [\bra{\psi_+} X(0) - {Z}_{{X}}(0) \ket{\psi_-}]$ survives. These results show that $x_{\psi}^{\rm{int}}(t)$ is the automaton analogue of the so-called Zitterbewegung. As already noticed in the introduction this phenomenon was never observed for a free relativistic electron because of the small value of $x_{\psi}^{\rm{int}}(t)$ which is bounded by the electron Compton wave length $10^{-12}$ m in natural units. The results of this section are in agreement with the one for the Hadamard walk [@kurzynski2008relativistic]. In Fig. \[fig:Zitt11-12-21-22\] we have considered the evolution of states with particle and antiparticle components smoothly peaked around some momentum eigenstate, namely $$\begin{aligned} \label{e:states} c_+\ket{\psi_+}+c_-\ket{\psi_-},\quad \ket{\psi_\pm}= \int \df{k}{\sqrt{2\pi}} g_{k_0}(k) \ket{\pm}_k\ket{k}\end{aligned}$$ where $c_+^2 \mkern-6mu+\mkern-6mu c_-^2 \mkern-14mu = \mkern-14mu 1$ and $g_{k_0}$ is a Gaussian peaked around the momentum $k_0$ with width $\sigma$. An easy computation shows that for these states the shift contribution reduces to $2\Re[\bra{\psi}X(0)+ Z_{X}(0)\ket{\psi}]=\Im{(c^{*}_+c_-)}/(2\pi)\int_{-\pi}^{\pi}\,\d{k} |g_{k_0}(k)|^2 z(k)$ with the function $z(k)=m\cos\omega(k)/\sin^2\omega(k)$ bounded again by the Compton wavelength $1/m$ and the oscillation frequency given by $\omega(0)/\pi$ (see also Fig. \[fig:z-frequency\]). ![(Colors online) Plots of $z(k)$ (left) and $\omega(k)/\pi$ (right) related to the oscillation amplitude and frequency of the position expectation value in Eq. . In both cases the plots are reported for different values of the mass ($m=0.1,\,0.2,\,0.4,\,0.8$ from the top in the figure on the left and from the bottom in the figure on the right).[]{data-label="fig:z-frequency"}](z.pdf "fig:"){width=".23\textwidth"}![(Colors online) Plots of $z(k)$ (left) and $\omega(k)/\pi$ (right) related to the oscillation amplitude and frequency of the position expectation value in Eq. . In both cases the plots are reported for different values of the mass ($m=0.1,\,0.2,\,0.4,\,0.8$ from the top in the figure on the left and from the bottom in the figure on the right).[]{data-label="fig:z-frequency"}](frequency.pdf "fig:"){width=".23\textwidth"} ![ Automaton evolution of a state as in Eq. showing the Zitterbewegung of the position expectation value. [ **Top:**]{} $m=0.15$, $c_+=1/ \sqrt{2}$, $c_-=i/\sqrt{2}$, $k_0=0$, and $\sigma=40^{-1}$. The calculated shift and oscillation frequency are respectively $\bra{\psi} X(0)+ Z_{X}(0)\ket{\psi}=3.2$ and $\omega(0)/\pi=0.05$, accordingly to the simulation. [**Middle:**]{} $m=0.15$, $c_+=1/\sqrt{2}$, $c_-=1/\sqrt{2}$, $k_0=0$, $\sigma=40^{-1}$. The calculated shift and oscillation frequency are $0$ and $0.13$, respectively. [**Bottom:**]{} $m=0.13$, $c_+=\sqrt{2/3}$, $c_-=1/\sqrt{3}$, $k_0=10^{-2}\pi$, $\sigma=40^{-1}$. In this case the particle and antiparticle contribution are not balanced and the average position drift velocity is thus $\bra{\psi _+} V \ket{\psi_+}+\bra{\psi _-} V \ket{\psi_-}=(|c_+|^2-|c_-|^2)v(k_0)=0.08$, corresponding to an average position $x_{\psi}^+(800) + x_{\psi}^-(800)=464$ (see Eq. ). Notice that for $t \to \infty$ the term $2 \Re [\bra{\psi_+} {Z}_{{X}}(t) \ket{\psi_-}$, which is responsible of the oscillation, goes to $0$.[]{data-label="fig:Zitt11-12-21-22"}](Zitt11.pdf "fig:"){width=".23\textwidth"}![ Automaton evolution of a state as in Eq. showing the Zitterbewegung of the position expectation value. [ **Top:**]{} $m=0.15$, $c_+=1/ \sqrt{2}$, $c_-=i/\sqrt{2}$, $k_0=0$, and $\sigma=40^{-1}$. The calculated shift and oscillation frequency are respectively $\bra{\psi} X(0)+ Z_{X}(0)\ket{\psi}=3.2$ and $\omega(0)/\pi=0.05$, accordingly to the simulation. [**Middle:**]{} $m=0.15$, $c_+=1/\sqrt{2}$, $c_-=1/\sqrt{2}$, $k_0=0$, $\sigma=40^{-1}$. The calculated shift and oscillation frequency are $0$ and $0.13$, respectively. [**Bottom:**]{} $m=0.13$, $c_+=\sqrt{2/3}$, $c_-=1/\sqrt{3}$, $k_0=10^{-2}\pi$, $\sigma=40^{-1}$. In this case the particle and antiparticle contribution are not balanced and the average position drift velocity is thus $\bra{\psi _+} V \ket{\psi_+}+\bra{\psi _-} V \ket{\psi_-}=(|c_+|^2-|c_-|^2)v(k_0)=0.08$, corresponding to an average position $x_{\psi}^+(800) + x_{\psi}^-(800)=464$ (see Eq. ). Notice that for $t \to \infty$ the term $2 \Re [\bra{\psi_+} {Z}_{{X}}(t) \ket{\psi_-}$, which is responsible of the oscillation, goes to $0$.[]{data-label="fig:Zitt11-12-21-22"}](Zitt12.pdf "fig:"){width=".23\textwidth"} ![ Automaton evolution of a state as in Eq. showing the Zitterbewegung of the position expectation value. [ **Top:**]{} $m=0.15$, $c_+=1/ \sqrt{2}$, $c_-=i/\sqrt{2}$, $k_0=0$, and $\sigma=40^{-1}$. The calculated shift and oscillation frequency are respectively $\bra{\psi} X(0)+ Z_{X}(0)\ket{\psi}=3.2$ and $\omega(0)/\pi=0.05$, accordingly to the simulation. [**Middle:**]{} $m=0.15$, $c_+=1/\sqrt{2}$, $c_-=1/\sqrt{2}$, $k_0=0$, $\sigma=40^{-1}$. The calculated shift and oscillation frequency are $0$ and $0.13$, respectively. [**Bottom:**]{} $m=0.13$, $c_+=\sqrt{2/3}$, $c_-=1/\sqrt{3}$, $k_0=10^{-2}\pi$, $\sigma=40^{-1}$. In this case the particle and antiparticle contribution are not balanced and the average position drift velocity is thus $\bra{\psi _+} V \ket{\psi_+}+\bra{\psi _-} V \ket{\psi_-}=(|c_+|^2-|c_-|^2)v(k_0)=0.08$, corresponding to an average position $x_{\psi}^+(800) + x_{\psi}^-(800)=464$ (see Eq. ). Notice that for $t \to \infty$ the term $2 \Re [\bra{\psi_+} {Z}_{{X}}(t) \ket{\psi_-}$, which is responsible of the oscillation, goes to $0$.[]{data-label="fig:Zitt11-12-21-22"}](Zitt21.pdf "fig:"){width=".23\textwidth"}![ Automaton evolution of a state as in Eq. showing the Zitterbewegung of the position expectation value. [ **Top:**]{} $m=0.15$, $c_+=1/ \sqrt{2}$, $c_-=i/\sqrt{2}$, $k_0=0$, and $\sigma=40^{-1}$. The calculated shift and oscillation frequency are respectively $\bra{\psi} X(0)+ Z_{X}(0)\ket{\psi}=3.2$ and $\omega(0)/\pi=0.05$, accordingly to the simulation. [**Middle:**]{} $m=0.15$, $c_+=1/\sqrt{2}$, $c_-=1/\sqrt{2}$, $k_0=0$, $\sigma=40^{-1}$. The calculated shift and oscillation frequency are $0$ and $0.13$, respectively. [**Bottom:**]{} $m=0.13$, $c_+=\sqrt{2/3}$, $c_-=1/\sqrt{3}$, $k_0=10^{-2}\pi$, $\sigma=40^{-1}$. In this case the particle and antiparticle contribution are not balanced and the average position drift velocity is thus $\bra{\psi _+} V \ket{\psi_+}+\bra{\psi _-} V \ket{\psi_-}=(|c_+|^2-|c_-|^2)v(k_0)=0.08$, corresponding to an average position $x_{\psi}^+(800) + x_{\psi}^-(800)=464$ (see Eq. ). Notice that for $t \to \infty$ the term $2 \Re [\bra{\psi_+} {Z}_{{X}}(t) \ket{\psi_-}$, which is responsible of the oscillation, goes to $0$.[]{data-label="fig:Zitt11-12-21-22"}](Zitt22.pdf "fig:"){width=".23\textwidth"} ![ Automaton evolution of a state as in Eq. showing the Zitterbewegung of the position expectation value. [ **Top:**]{} $m=0.15$, $c_+=1/ \sqrt{2}$, $c_-=i/\sqrt{2}$, $k_0=0$, and $\sigma=40^{-1}$. The calculated shift and oscillation frequency are respectively $\bra{\psi} X(0)+ Z_{X}(0)\ket{\psi}=3.2$ and $\omega(0)/\pi=0.05$, accordingly to the simulation. [**Middle:**]{} $m=0.15$, $c_+=1/\sqrt{2}$, $c_-=1/\sqrt{2}$, $k_0=0$, $\sigma=40^{-1}$. The calculated shift and oscillation frequency are $0$ and $0.13$, respectively. [**Bottom:**]{} $m=0.13$, $c_+=\sqrt{2/3}$, $c_-=1/\sqrt{3}$, $k_0=10^{-2}\pi$, $\sigma=40^{-1}$. In this case the particle and antiparticle contribution are not balanced and the average position drift velocity is thus $\bra{\psi _+} V \ket{\psi_+}+\bra{\psi _-} V \ket{\psi_-}=(|c_+|^2-|c_-|^2)v(k_0)=0.08$, corresponding to an average position $x_{\psi}^+(800) + x_{\psi}^-(800)=464$ (see Eq. ). Notice that for $t \to \infty$ the term $2 \Re [\bra{\psi_+} {Z}_{{X}}(t) \ket{\psi_-}$, which is responsible of the oscillation, goes to $0$.[]{data-label="fig:Zitt11-12-21-22"}](Zitt31.pdf "fig:"){width=".23\textwidth"}![ Automaton evolution of a state as in Eq. showing the Zitterbewegung of the position expectation value. [ **Top:**]{} $m=0.15$, $c_+=1/ \sqrt{2}$, $c_-=i/\sqrt{2}$, $k_0=0$, and $\sigma=40^{-1}$. The calculated shift and oscillation frequency are respectively $\bra{\psi} X(0)+ Z_{X}(0)\ket{\psi}=3.2$ and $\omega(0)/\pi=0.05$, accordingly to the simulation. [**Middle:**]{} $m=0.15$, $c_+=1/\sqrt{2}$, $c_-=1/\sqrt{2}$, $k_0=0$, $\sigma=40^{-1}$. The calculated shift and oscillation frequency are $0$ and $0.13$, respectively. [**Bottom:**]{} $m=0.13$, $c_+=\sqrt{2/3}$, $c_-=1/\sqrt{3}$, $k_0=10^{-2}\pi$, $\sigma=40^{-1}$. In this case the particle and antiparticle contribution are not balanced and the average position drift velocity is thus $\bra{\psi _+} V \ket{\psi_+}+\bra{\psi _-} V \ket{\psi_-}=(|c_+|^2-|c_-|^2)v(k_0)=0.08$, corresponding to an average position $x_{\psi}^+(800) + x_{\psi}^-(800)=464$ (see Eq. ). Notice that for $t \to \infty$ the term $2 \Re [\bra{\psi_+} {Z}_{{X}}(t) \ket{\psi_-}$, which is responsible of the oscillation, goes to $0$.[]{data-label="fig:Zitt11-12-21-22"}](Zitt32.pdf "fig:"){width=".23\textwidth"} Evolution with a square potential barrier {#s:barrier} ========================================= In order to study the scattering with a potential, we modify the automaton adding a position dependent phase representing a square potential barrier, as in Refs. [@kurzynski2008relativistic; @meyer1997quantum]. We will provide explicitly the transmission $T$ and reflection $R$ coefficients as functions of the energy and mass of the incident wavepacket and of the potential barrier’s height. We will find a general behavior independently on the regime, namely on the energy and mass of the incident particle. Increasing the value of the potential barrier beyond a certain threshold a transmitted wave reappears and the reflection coefficient starts decreasing. The width of the $R=1$ region is an increasing function of the mass which is proportional to the gap between positive and negative frequency eigenvalues of the unitary evolution. For a generic potential $\phi(x)$, the unitary evolution becomes $$\begin{aligned} U_\phi := \sum_x e^{-i \phi(x)} \left( \begin{array}{ll} n \ketbra{x-1}{x} & -im \ketbra{x}{x} \\ -im \ketbra{x}{x} & n \ketbra{x+1}{x} \end{array} \right).\end{aligned}$$ We will analyze the simple case $\phi(x) := \phi \, \theta(x)$ ($\theta(x)$ is the Heaviside step function) that is a potential step which is $0$ for $x <0 $ (region $\mathrm{I}$) and has a constant value $\phi \in [0, 2\pi]$ for $x \geq 0$ (region $\mathrm{II}$) as illustrated in Fig. \[fig:potential\]. ![Schematic of the potential[]{data-label="fig:potential"}](potential6.pdf){width=".30\textwidth"} ![Reflection coefficient as a function of the potential barrier height $\phi$ and of the momentum $k$ of the incident particle state. From the top-left to the bottom-right the reflection coefficient is depicted for different values of the mass: $m=0.1,\,0.2,\,0.4,\,0.8$.[]{data-label="fig:Reflection"}](R-0_1.pdf "fig:"){width=".23\textwidth"}![Reflection coefficient as a function of the potential barrier height $\phi$ and of the momentum $k$ of the incident particle state. From the top-left to the bottom-right the reflection coefficient is depicted for different values of the mass: $m=0.1,\,0.2,\,0.4,\,0.8$.[]{data-label="fig:Reflection"}](R-0_2.pdf "fig:"){width=".23\textwidth"} ![Reflection coefficient as a function of the potential barrier height $\phi$ and of the momentum $k$ of the incident particle state. From the top-left to the bottom-right the reflection coefficient is depicted for different values of the mass: $m=0.1,\,0.2,\,0.4,\,0.8$.[]{data-label="fig:Reflection"}](R-0_4.pdf "fig:"){width=".23\textwidth"}![Reflection coefficient as a function of the potential barrier height $\phi$ and of the momentum $k$ of the incident particle state. From the top-left to the bottom-right the reflection coefficient is depicted for different values of the mass: $m=0.1,\,0.2,\,0.4,\,0.8$.[]{data-label="fig:Reflection"}](R-0_8.pdf "fig:"){width=".23\textwidth"} ![Group velocity of the transmitted wave packet as a function of the potential barrier height $\phi$ and of the momentum $k$ of the incident particle state. From the top-left to the bottom-right the transmitted group velocity for different values of the mass: $m=0.1,\,0.2,\,0.4,\,0.8$.[]{data-label="fig:Momentum"}](Vt-0_1.pdf "fig:"){width=".23\textwidth"}![Group velocity of the transmitted wave packet as a function of the potential barrier height $\phi$ and of the momentum $k$ of the incident particle state. From the top-left to the bottom-right the transmitted group velocity for different values of the mass: $m=0.1,\,0.2,\,0.4,\,0.8$.[]{data-label="fig:Momentum"}](Vt-0_2.pdf "fig:"){width=".23\textwidth"} ![Group velocity of the transmitted wave packet as a function of the potential barrier height $\phi$ and of the momentum $k$ of the incident particle state. From the top-left to the bottom-right the transmitted group velocity for different values of the mass: $m=0.1,\,0.2,\,0.4,\,0.8$.[]{data-label="fig:Momentum"}](Vt-0_4.pdf "fig:"){width=".23\textwidth"}![Group velocity of the transmitted wave packet as a function of the potential barrier height $\phi$ and of the momentum $k$ of the incident particle state. From the top-left to the bottom-right the transmitted group velocity for different values of the mass: $m=0.1,\,0.2,\,0.4,\,0.8$.[]{data-label="fig:Momentum"}](Vt-0_8.pdf "fig:"){width=".23\textwidth"} Let us now study the eigenvector of $U_\phi$ of the form $$\begin{aligned} &\ket{\Phi_k} = \Pi_\mathrm{I} \ket{+}_k\ket{k} +\Pi_\mathrm{I} \beta_k \ket{+}_{-k}\ket{k} + \gamma_k \Pi_\mathrm{II}\ket{+}_{k'}\ket{k'} \\ &\Pi_\mathrm{I} := \sum _{x<0} I \otimes \ketbra{x}{x} \qquad \Pi_\mathrm{II} := \sum _{x \geq 0} I \otimes \ketbra{x}{x} \end{aligned}$$ where $\beta_k$, $\gamma_k$ and $k'$ are functions of $k$. The condition that $\ket{\Phi_k}$ is genuinely an eigenstate of $U_\phi$, i.e. $ U_\phi \ket{\Phi_k} = e^{-i \omega(k)} \ket{\Phi_k}$, implies that $$\begin{aligned} &\omega(k') = \omega(k) -\phi \label{eq:kprime}\\ &\beta_k=\frac{e^{-ik}\sqrt{(1+ v)(1-v^\prime)}-e^{-ik^\prime}\sqrt{(1- v)(1+v^\prime)}}{-e^{ik}\sqrt{(1- v)(1- v^\prime)}+e^{-ik^\prime}\sqrt{(1+v)(1+v^\prime)}} \nonumber \\ &\nonumber \gamma_k =\frac{2e^{i\xi} (v\cos{k}-i\sin{k})}{-e^{ik}\sqrt{(1- v)(1- v^\prime))}+e^{-ik^\prime}\sqrt{(1+v)(1+v^\prime)}}\end{aligned}$$ with $v:=v(k)$ and $v':=v(k')$ the group velocities of the incident and transmitted wave. Let us now consider the superposition $$\begin{aligned} \ket{\Psi(0)}:= \int \df{k}{\sqrt{2\pi}} g_{k_0}(k) \ket{\Phi_k}\end{aligned}$$ where $g_{k_0}(k)$ is a function in $C_0^{\infty}[-\pi, \pi]$ which we assumed to be smoothly peaked around $k_0$. The state at time $t$ is then $$\begin{aligned} \ket{\Psi(t)}:= \int \df{k}{\sqrt{2\pi}} g_{k_0}(k) e^{-i \omega(k)t} \ket{\Phi_k}\end{aligned}$$ and one can verify that for $t \ll 0$ the state is negligible in region $\mathrm{II}$ while the only appreciable contribution in region $\mathrm{II}$ comes from the term $e^{i k_0 x}$ which describes a wavepacket that moves at group velocity $v(k_0)$ and hits the barrier form the left. When $t \gg 0$ the state can be approximated by a superposition of a reflected and a transmitted wavepacket as follows $$\begin{aligned} \begin{split} \ket{\Psi(t)} \xrightarrow{ t \gg 0} \beta(k_0) \int \df{k}{\sqrt{2\pi}} g_{k_0}(k) e^{-i \omega(k)t} \ket{+}_{-k}\ket{k} +\\ {}+ \tilde{\gamma}(k_0) e^{-i \phi t} \int \df{k}{\sqrt{2\pi}} \tilde{g}_{k'_0}(k') e^{-i \omega(k')t} \ket{+}_{k'}\ket{k'} \end{split}\end{aligned}$$ where we defined $$\begin{aligned} &k'_0 \mbox{ s.t. } \omega(k'_0) = \omega(k_0) -\phi, \\ &\tilde{\gamma}(k_0) := {\gamma}(k_0) \sqrt{\frac{v(k'_0)}{v(k_0)}}, \qquad \tilde{g}_{k'_0}(k') = \sqrt{\frac{v(k'_0)}{v(k_0)}}{g}_{k'_0}(k') \end{aligned}$$ (one can check $ \int \df{k}{\sqrt{2\pi}} |\tilde{g}_{k'_0}(k')|^2 = 1$), whose group velocities are $-v(k_0)$ for the reflected wave packet and $v(k^\prime_0)$ for the transmitted wave packet (see Fig. \[fig:Momentum\]). The probability of finding the particle in the reflected wavepacket is $R = |\beta(k_0)|^2$ (reflection coefficient) while the probability of finding the particle in the transmitted wavepacket is $ T= |\tilde{\gamma}(k_0)|^2$ (trasmission coefficient). The consistency of the result can be verified by checking that $R+T = 1$. For $k \ll m \ll 1 $ (Schröedinger regime) we recover the usual reflection and transmission coefficient for the Schröedinger equation with a potential step. In Fig. \[fig:Reflection\] we plot the reflection coefficient $R$ as a function of $\phi$ and $k$ for different values of the mass $m$. Clearly when $\phi=0$ we have $R = 0$ and increasing $\phi$ while fixing $k$ the value increases up to $R=1$. One notice that when $ \omega(k) - \arccos(n) < \phi < \omega(k) + \arccos(n)$ Eq.  has solution for imaginary $k'$ which implies an exponential damping of the transmitted wave and pure reflection. By further increasing the value of $\phi$ beyond the threshold $\omega(k) + \arccos(n)$, Eq.  have solution for real $k'$ and negative $\omega(k')$, and then a transmitted wave reappears and the reflection coefficient decreases. This is the so called “Klein paradox” which is originated by the presence of positive and negative frequency eigenvalues of the unitary evolution. The width of the $R=1$ region is an increasing function of the mass equal to $2\arccos(n)$ which is the gap between positive and negative frequency solution solutions (see Fig. \[fig:disp\] ). ![Reflection coefficient for $m=0.4$ and momentum of the incident particle $k_0=2$ as a function of the potential barrier height $\phi$ (section of plots in Figs. \[fig:Reflection\]-\[fig:Momentum\] for $m=0.4$, $k_0=2$).[]{data-label="fig:Section"}](R-m0_4-K2.pdf "fig:"){width=".23\textwidth"}![Reflection coefficient for $m=0.4$ and momentum of the incident particle $k_0=2$ as a function of the potential barrier height $\phi$ (section of plots in Figs. \[fig:Reflection\]-\[fig:Momentum\] for $m=0.4$, $k_0=2$).[]{data-label="fig:Section"}](Vt-m0_4-K2.pdf "fig:"){width=".23\textwidth"} ![Simulations of the Dirac automaton evolution with a square potential barrier. Here the automaton mass is $m=0.2$ while the barrier turns on at $x=140$. In the simulation the incident state is a smooth state of the form $\ket{\psi(0)}= \int \df{k}{\sqrt{2\pi}} g_{k_0}(k) \ket{+}_k$ peaked around the positive energy eigenstate $\ket{+}_{k_0}$ with $k_0=2$ and with $g_{k_0}$ a Gaussian having width $\sigma=15^{-1}$. The incident group velocity is $v(k_0)=0.90$. The simulation is run for four increasing values of the potential $\phi$. [**Top-Left:**]{} Potential barrier height $\phi=1.42$, reflection coefficient $R=0.25$, velocity of the transmitted particle $v(k^\prime_0)=0.63$. [**Top-Right:**]{} $\phi=1.55$, $R=0.75$, $v(k^\prime_0)=0.1$. [**Bottom-Left:**]{} $\phi=2$, $R=0.1$, $v(k^\prime_0)=0$. [**Bottom-Right:**]{} $\phi=2.4$, $R=0.50$, $v(k^\prime_0)=0.33$.[]{data-label="fig:Simulation"}](R0_25-m0_4-V1_42-k2.pdf "fig:"){width=".23\textwidth"}![Simulations of the Dirac automaton evolution with a square potential barrier. Here the automaton mass is $m=0.2$ while the barrier turns on at $x=140$. In the simulation the incident state is a smooth state of the form $\ket{\psi(0)}= \int \df{k}{\sqrt{2\pi}} g_{k_0}(k) \ket{+}_k$ peaked around the positive energy eigenstate $\ket{+}_{k_0}$ with $k_0=2$ and with $g_{k_0}$ a Gaussian having width $\sigma=15^{-1}$. The incident group velocity is $v(k_0)=0.90$. The simulation is run for four increasing values of the potential $\phi$. [**Top-Left:**]{} Potential barrier height $\phi=1.42$, reflection coefficient $R=0.25$, velocity of the transmitted particle $v(k^\prime_0)=0.63$. [**Top-Right:**]{} $\phi=1.55$, $R=0.75$, $v(k^\prime_0)=0.1$. [**Bottom-Left:**]{} $\phi=2$, $R=0.1$, $v(k^\prime_0)=0$. [**Bottom-Right:**]{} $\phi=2.4$, $R=0.50$, $v(k^\prime_0)=0.33$.[]{data-label="fig:Simulation"}](R0_75-m0_4-V1_55-k2.pdf "fig:"){width=".23\textwidth"} ![Simulations of the Dirac automaton evolution with a square potential barrier. Here the automaton mass is $m=0.2$ while the barrier turns on at $x=140$. In the simulation the incident state is a smooth state of the form $\ket{\psi(0)}= \int \df{k}{\sqrt{2\pi}} g_{k_0}(k) \ket{+}_k$ peaked around the positive energy eigenstate $\ket{+}_{k_0}$ with $k_0=2$ and with $g_{k_0}$ a Gaussian having width $\sigma=15^{-1}$. The incident group velocity is $v(k_0)=0.90$. The simulation is run for four increasing values of the potential $\phi$. [**Top-Left:**]{} Potential barrier height $\phi=1.42$, reflection coefficient $R=0.25$, velocity of the transmitted particle $v(k^\prime_0)=0.63$. [**Top-Right:**]{} $\phi=1.55$, $R=0.75$, $v(k^\prime_0)=0.1$. [**Bottom-Left:**]{} $\phi=2$, $R=0.1$, $v(k^\prime_0)=0$. [**Bottom-Right:**]{} $\phi=2.4$, $R=0.50$, $v(k^\prime_0)=0.33$.[]{data-label="fig:Simulation"}](R1-m0_4-V2-k2.pdf "fig:"){width=".23\textwidth"}![Simulations of the Dirac automaton evolution with a square potential barrier. Here the automaton mass is $m=0.2$ while the barrier turns on at $x=140$. In the simulation the incident state is a smooth state of the form $\ket{\psi(0)}= \int \df{k}{\sqrt{2\pi}} g_{k_0}(k) \ket{+}_k$ peaked around the positive energy eigenstate $\ket{+}_{k_0}$ with $k_0=2$ and with $g_{k_0}$ a Gaussian having width $\sigma=15^{-1}$. The incident group velocity is $v(k_0)=0.90$. The simulation is run for four increasing values of the potential $\phi$. [**Top-Left:**]{} Potential barrier height $\phi=1.42$, reflection coefficient $R=0.25$, velocity of the transmitted particle $v(k^\prime_0)=0.63$. [**Top-Right:**]{} $\phi=1.55$, $R=0.75$, $v(k^\prime_0)=0.1$. [**Bottom-Left:**]{} $\phi=2$, $R=0.1$, $v(k^\prime_0)=0$. [**Bottom-Right:**]{} $\phi=2.4$, $R=0.50$, $v(k^\prime_0)=0.33$.[]{data-label="fig:Simulation"}](R0_5-m0_4-V2_4-k2.pdf "fig:"){width=".23\textwidth"} In Fig. \[fig:Section\] we plot the reflection $R$ coefficient and the transmitted wave velocity group $v(k_0')$ as a function of the potential barrier height $\phi$ with the incident wave packet having $k_0=2$ and $m=0.4$. From the figure it is clear that after a plateau with $R=1$ the reflection coefficient starts decreasing for higher potentials. In Fig. \[fig:Simulation\] we show the scattering simulation for four increasing values of the potential, say $\phi=1.42,\,1.55,\,2,\,2.4$ (see the caption to figure for the details). Conclusions {#s:concl} =========== In this paper we studied the dynamics of the quantum cellular automaton of Refs. [@darianopla; @BDTqcaI], which gives the Dirac dynamics as emergent in the limit of small wavevectors. We presented computer simulations and analytical evaluations, focusing on typical features of the Dirac dynamics, in particular the Zitterbewegung and the scattering from potential. Our automaton covers all regimes of masses and energy-momenta, beyond the same validity range of the Dirac equation, with the possibility of considering arbitrary input states, enabling to investigate and visualize a wide range of fundamental processes. This facts, in addition to the discreteness of the automaton, makes of it the ideal theoretical counterpart for the experimental simulators in the literature. A similar quantum cellular automaton can be also developed in two dimensions [@DP], corresponding to the graphene as quantum simulator. Bound of the oscillating term and its asymptotic behavior {#a:zitterbewegung} ========================================================= Here we provide an upper bound for the oscillating term $x_{\psi}^{\rm{int}}(t)$ in Eq. (\[e:zitterbewegung\]) in the position operator evolution derived in Section \[sec:zitterbewegung\] and we derive its behaviour for very long time steps. The jittering of the position expectation value is caused by the operator ${Z}_{{X}}(t)$ which in the base diagonalizing the automaton Hamiltonian $H$ can be written as $$\begin{aligned} \nonumber Z_X(t)=\int_{-\pi}^{\pi}\d k e^{2i\omega(k) \sigma_z t}Z_X(k)\otimes\ketbra{k}{k},\\\nonumber Z_X(k)=z(k)\sigma_2,\qquad z(k)=\frac{m\cos{\omega(k)}}{2\sin^2{\omega}(k)}\end{aligned}$$ with $z(k)\in L^1(-\pi,\pi)$ for any $m\neq 0$. By defining $$\begin{aligned} \nonumber \ket{\psi_\pm}= \int_{-\pi}^{\pi}\; \df{k}{\sqrt{2\pi}} g_{\pm}(k) \ket{\pm}_k\ket{k},\qquad g_{\pm}(k)\in C_0^{\infty}[-\pi, \pi]\end{aligned}$$ we have $$\begin{aligned} \nonumber 2\Re[\bra{\psi_+}Z_{X}{(t)}\ket{\psi_-}]=\int_{-\pi}^{\pi} \df{k}{\pi} z(k) \Re{\left[i g^*_+(k) g_-(k) e^{2i\omega(k) t}\right]}\end{aligned}$$ Since, for any $m\neq 0$, $\omega(k)$ has three stationary points in $k=0,\,\pm\pi$ ($\omega^{(1)}(0)=\omega^{(1)}(\pm\pi)=0$ and $\omega^{(1)}(k)\neq 0$ elsewhere in the closed interval $k\in[-\pi,\pi]$, with $\omega^{(2)}(0),\omega^{(2)}(\pm\pi)\neq 0$), the stationary phase approximation gives $$\begin{gathered} \label{e:asymptotic}\nonumber 2\Re[\bra{\psi_+}Z_{X}{(t)}\ket{\psi_-}]\xrightarrow{ t \gg 0}\\\nonumber \qquad\qquad\sum_{k=0,\pm\pi} z(k)\Re{\left[ig^*_+(k) g_-(k) e^{2i\omega(k) t}\sqrt{\frac{i}{\pi\omega^{(2)}(k)t}}\;\right]}\end{gathered}$$ showing that the term $2 \Re [\bra{\psi_+} {Z}_{{X}}(t) \ket{\psi_-}]$, goes to $0$ as $1/\sqrt{t}$. In order to find an upper bound for $x_{\psi}^{\rm{int}}(t)$ notice that $$\begin{aligned} \nonumber | x_{\psi}^{\rm{int}}(t)|&\leq 2 | \bra{\psi_+} X(0) - {Z}_{{X}}(0) + {Z}_{{X}}(t) \ket{\psi_-}|\\\nonumber &\leq 2(|\bra{\psi_+} X(0) \ket{\psi_-}| + | {Z}_{{X}}(0)| + |{Z}_{{X}}(t)|) \end{aligned}$$ and, according to the expression of $Z_X(k)$, we get $$\begin{aligned} \nonumber | {Z}_{{X}}(0)| + |{Z}_{{X}}(t)| \leq 2| {Z}_{{X}}(0)|\\\nonumber | {Z}_{{X}}(0)|\leq \max_{k\in[-\pi,\pi]} |z(k)|=z(0)=\frac{\sqrt{1-m^2}}{2m}.\end{aligned}$$ Now defining the $C_0^{\infty}[-\pi, \pi]$ test function $\varphi(k,k')=g^*_+(k) g_-(k') \bra{+}{}_k \ket{-}_{k'}$, we have $$\begin{aligned} \nonumber &|\bra{\psi_+} X(0) \ket{\psi_-}| =\\\nonumber &\left|\BraKet{ \frac {\;{\mathop{\!\! \mathrm{d}}}\delta(k-k')}{\;{\mathop{\!\! \mathrm{d}}}(k-k')} }{\varphi(k,k')}\right|= \left|\BraKet{\delta(k-k')} { \frac {\;{\mathop{\!\! \mathrm{d}}}\varphi(k,k')}{\;{\mathop{\!\! \mathrm{d}}}(k-k')} }\right|=\\\nonumber &\left|\int_{-\pi}^{\pi} \;\df{k}{2\pi}\, \d k' \delta(k-k') g^*_+(k) g_-(k') \frac{\;{\mathop{\!\! \mathrm{d}}}}{\;{\mathop{\!\! \mathrm{d}}}(k-k')}\bra{+}{}_k \ket{-}_{k'}\right|=\\\nonumber &\left|\int_{-\pi}^{\pi}\; \df{k}{2\pi} g^*_+(k) g_-(k) f(k) \right| \leq \max_{k\in[-\pi,\pi]} |f(k)|=f(0)\\\nonumber &f(k):=\frac{n}{\sin^2{\omega}},\qquad f(0)= \frac{\sqrt{1-m^2}}{m^2}.\end{aligned}$$ which finally gives $$\begin{aligned} | x_{\psi}^{\rm{int}}(t)| \leq \frac{2}{m}+\frac{2}{m^2}.\end{aligned}$$ [^1]: The interactions between cells in the interpolation interval would be nonlocal, and, in addition, the Hamiltonian would involve distant cells.

--- abstract: 'A general analysis of the $n$-vertex loop amplitude in a strong magnetic field is performed, based on the asymptotic form of the electron propagator in the field. As an example, the photon-neutrino processes are considered, where one vertex in the amplitude is of a general type, and the other vertices are of the vector type. It is shown, that for odd numbers of vertices only the $SV_1 \ldots V_{n-1}$ amplitude grows linearly with the magnetic field strength, while for even numbers of vertices the linear growth takes place only in the amplitudes $PV_{1} \ldots V_{n-1}$, $VV_{1} \ldots V_{n-1}$ and $AV_{1} \ldots V_{n-1}$. The general expressions for the amplitudes of the processes $\gamma \gamma \to \nu \bar\nu$ (in the framework of the model with the effective $\nu \nu e e$ – coupling of a scalar type) and $\gamma \gamma \to \nu \bar\nu \gamma$ (in the framework of the Standard Model) for arbitrary energies of particles are obtained. A comparison with existing results is performed.' author: - | A. V. Kuznetsov, N. V. Mikheev, D. A. Rumyantsev\ [*Division of Theoretical Physics, Yaroslavl State (P.G. Demidov) University,*]{}\ [*Sovietskaya 14, 150000 Yaroslavl, Russian Federation*]{}\ [E-mail: avkuzn@uniyar.ac.ru, mikheev@uniyar.ac.ru, rda@uniyar.ac.ru]{} title: | [Yaroslavl State University\ Preprint YARU-HE-02/09\ hep-ph/0210029]{}\ General amplitude of the $n$ – vertex\ one-loop process in a strong magnetic field --- [*Talk presented at the 12th International Seminar\ “Quarks-2002”,\ Valday and Novgorod, Russia, June 1-7, 2002*]{} Introduction ============ Nowadays, there exists a growing interest to astrophysical objects, where the strong magnetic fields with the strength $B>B_e$ can be generated $(B_e = m^2/e \simeq 4.41 \cdot 10^{13}$ Gs [^1] is the so-called critical field value). The influence of a strong external field on quantum processes is interesting because it catalyses the processes, it changes the kinematics and it induces new interactions. It is especially important to investigate the influence of external field on the loop quantum processes where only electrically neutral particles in the initial and the final states are presented, such as neutrinos, photons and hypotetical axions, familons and so on. The external field influence on these loop processes is provided by the sensitivity of the charged virtual fermion to the field and by the change of the photon dispersion properties and, therefore, the photon kinematics. The research of the loop processes of this type has a rather long history. The two-vertex loop processes (the photon polarization operator in an external field, the decays $\gamma\to\nu\bar{\nu}$, $\nu\to\nu\gamma$ and so on) were studed in the papers [@Tsai:1974; @Shabad:88; @Skobelev:1995; @Gvozdev:1996; @Ioannisian:1997]. The general expression for the two-vertex loop amplitude $j \to f \bar f \to j'$ in the homogeneous magnetic and in the crossed field was obtaned in the paper [@Borovkov:1999], where all combinations of scalar, pseudoscalar, vector and axial-vector interactions of the generalized currents $j,\,j'$ with fermions were considered. A loop process with three vertices is also intresting for theoreticians. For example, the photon splitting in a magnetic field $\gamma \to \gamma \gamma$ is forbidden in vacuum. The review [@Papanian:1986] and the recent papers [@Adler:1996; @Baier:1996; @Baier:1997; @Chistyakov:1998; @Kuznetsov:1999] were devoted to this process. One more three-vertex loop process is the conversion of the photon pair into the neutrino pair, $\gamma\gamma\to\nu\bar{\nu}$. This process is interesting as a possible channel of stellar cooling. A detailed list of references on this process can be found in our paper [@Kuznetsov:2002a]. It is well-known (the so-called Gell-Mann theorem [@Gell-Mann:1961]), that for massless neutrinos, for both photons on-shell and in the local limit of the standard-model weak interaction, the process $\gamma\gamma\to\nu\bar{\nu}$ is forbidden. Because of this, the four-vertex loop process with an additional photon $\gamma\gamma\to\nu\bar{\nu}\gamma$ was considered by some authors. In spite of the extra factor $\alpha$, this process has the probability larger than the two-photon process. The process $\gamma\gamma\to\nu\bar{\nu}\gamma$ was studied both in vacuum (from the first paper [@VanHieu:1963] to the recent Refs. [@Dicus:1997; @Dicus:1999; @Abada:1999a; @Abada:1999b; @Abada:1999c]), and under the stimulating influence of a strong magnetic field [@Loskutov:1987; @Skobelev:2001; @Kuznetsov:2002b]. So, the calculation of the amplitude of the $n$-vertex loop quantum processes ($\gamma \gamma \to \nu \bar\nu$, $\gamma \gamma \to \gamma \nu \bar\nu$, the axion and familon processes $\gamma \gamma \to \gamma a$, $\gamma \gamma \to \gamma \varPhi$ and so on) in a strong magnetic field is important, because these results can be useful for astrophysical applications. The paper is organized as follows. A general analysis of the $n$-vertex one-loop process amplitude in a strong magnetic field is performed in Section 2. The amplitude, in which the one vertex is of a general type (scalar $S$, pseudoscalar $P$, vector $V$ or axial-vector $A$), and the other vertices are of the vector type (contracted with photons), is calculated in Section 3. This amplitude is the main result of the paper. The analitical expressions for the amplitudes of the processes $\gamma \gamma \to \nu \bar\nu$ and $\gamma \gamma \to \nu \bar\nu \gamma$ are presented in Sections 4 and 5. General analysis of the $n$-vertex one-loop processes in a strong magnetic field ================================================================================ We use the effective Lagrangian for the interaction of the generalized currents $j$ with electrons in the form: $$\begin{aligned} {\cal L}(x) \, = \, \sum \limits_{i} g_i [\bar {\psi_e}(x) \Gamma_i \psi_e(x)] j_i, \label{eq:L}\end{aligned}$$ where the generic index $i = S, P, V, A $ numbers the matrices $\Gamma_i$, e.g. $\Gamma_S = 1, \, \Gamma_P = \gamma_5, \, \Gamma_V = \gamma_{\alpha}, \, \Gamma_A = \gamma_{\alpha} \gamma_5 $, $j$ is the corresponding quantum object (the current or the photon polarisation vector), $g_i$ are the coupling constants. In particular, for the electron - photon interaction we have $g_V = e,\,\Gamma_V = \gamma_{\alpha},\,j_{V\alpha}(x) = A_\alpha(x)$. A general amplitude of the process, corresponding to the effective Lagrangian (\[eq:L\]), is described by fig. \[fig:loop1\]. In the strong field limit, after integration over the coordinates, the amplitude takes the form $$\begin{aligned} {\cal M}_n \simeq \frac{i \,(-1)^n \,e B}{(2 \pi)^3} \exp \! \left (-\frac{R_{\mprp n}}{2eB}\right ) \int d^2 p_{\mprl} \, \mbox{Tr} \, \big \{\prod \limits^{n}_{k=1} g_k \Gamma_k j_k S_{\mprl}(p-Q_k) \big \}, \label{eq:M2}\end{aligned}$$ where $d^2 p_{\mprl} = dp_0 dp_z$, $S_{\mprl}(p) = \Pi_{-} ((p\gamma)_{\mprl} + m)/(p_{\mprl}^2 - m^2)$ is the asymptotic form of the electron propagator in the limit $eB /\vert m^2 - p_{\mprl}^2 \vert \gg 1$, $$R_{\mprp 2} = q_{\mprp 1}^2, \quad R_{\mprp 3} = q_{\mprp 1}^2 + q_{\mprp 2}^2 + (q_1 \varphi \varphi q_2) + i(q_1 \varphi q_2),$$ $$R_{\mprp n}(n\ge 3) = \sum \limits^{n-1}_{k=1} Q_{\mprp k}^2 - \sum \limits^{n-1}_{k=2} \sum \limits^{k-1}_{j=1} \left [(Q_k \varphi \varphi Q_j) + i(Q_k \varphi Q_j)\right ],$$ $$\quad Q_k = \sum \limits^{k}_{i=1} q_i, \quad Q_n = 0,$$ $q_{\mprl}^2 = (q \tilde \varphi \tilde \varphi q)$, $q_{\mprp}^2 = (q \varphi \varphi q)$, $\varphi_{\alpha \beta} = F_{\alpha \beta} /B$ is the dimensionless field tensor, ${\tilde \varphi}_{\alpha \beta} = \frac{1}{2} \varepsilon_{\alpha \beta \mu \nu} \varphi_{\mu \nu}$ is the dual tensor, and the indices of the four-vectors and tensors standing inside the parentheses are contracted consecutively, e.g. $(a \varphi b) = a_\alpha \varphi_{\alpha \beta} b_\beta$. As is seen from Eq. (\[eq:M2\]), the amplitude depends only on the longitudinal components of the momenta, if the magnetic field strength is the maximal physical parameter $e B \gg q_{\mprp}^2, q_{\mprl}^2$. The photons processes ===================== Let the vertices $\Gamma_1 \ldots \Gamma_{n-1}$ are of the vector type, and the vertex $\Gamma_n$ is arbitrary. It can be shown that in the limit $q_{\mprp}^2 \ll e B $, for odd numbers of vertices, only the $S V_1 \ldots V_{n-1}$ amplitude grows linearly with the magnetic field strength, while for even numbers of vertices the linear growth takes place only in the amplitudes $PV_{1} \ldots V_{n-1}$, $VV_{1} \ldots V_{n-1}$ and $AV_{1} \ldots V_{n-1}$. It should be noted, that in the amplitude (\[eq:M2\]) the projecting operators $\Pi_-$ separate out the photons of only one polarization $(\perp)$ of the two possible (in Adler’s notation [@Ad71]) $$\begin{aligned} \varepsilon^{(\mprl)}_{\alpha} = \frac{\varphi_{\alpha\beta} q_{\beta}}{\sqrt{(q\varphi \varphi q)}}, \qquad \varepsilon^{(\mprp)}_{\alpha} = \frac{\widetilde \varphi_{\alpha\beta} q_{\beta}} {\sqrt{(q \widetilde \varphi \widetilde \varphi q)}}. \label{eq:vect}\end{aligned}$$ As can be deduced from the corresponding analysis, the calculation of any type of the amplitude can be reduced to the evaluation of the scalar integrals $$\begin{aligned} S_{n}(Q_{1 \mprl},\ldots ,Q_{n \mprl}) = \int \, \frac{d^2 p_{\mprl}}{(2\pi)^2} \, \prod \limits^{n}_{i=1} \frac{1}{(p - Q_i)^2_{\mprl} - m^2 + i\varepsilon}. \label {eq:s1}\end{aligned}$$ Notice that the use of the standart method of Feynman parametrization in calculation of the integrals (\[eq:s1\]) can be non-optimal, because the number of integrations grows. For example, if $n=3$, the double integral (\[eq:s1\]) is transformed into the integral over the two Feynman variables. If $n=4$, the double integral (\[eq:s1\]) is transformed into the integral over the three Feynman variables and so on. Here we suggest another way. By integrating (\[eq:s1\]) over $dp_0dp_z$, we obtain $$\begin{aligned} S_{n}(Q_{1 \mprl},\ldots ,Q_{n \mprl}) = \, \frac{i}{8m^2\pi} \sum \limits^{n}_{i=1} \underset{l \ne i}{\sum \limits^{n}_{l=1}} \, \left [ H \left (\frac{d_{il}^2}{4m^2} \right) + 1 \right ] Re \, \bigg \{ \underset{k \ne i,l} {\prod \limits^{n}_{k=1}} \frac{1}{Y_{ilk}} \bigg \} , \label {eq:s4}\end{aligned}$$ where $$Y_{ilk} = (d_{lk} d_{ik}) \, + \, i (d_{lk} \tilde \varphi d_{ik}) \sqrt{\frac{4m^2}{d_{il}^2} - 1}, \quad d_{il}^{\alpha} = Q_{\mprl \, i}^{\alpha} - Q_{\mprl \, l}^{\alpha}.$$ The function $H(z)$ is defined by the expressions $$\begin{aligned} &&H(z) = \frac{1}{2\sqrt{-z(1 - z)}} \ln{\frac{\sqrt{1 - z} + \sqrt{-z}}{\sqrt{1 - z} - \sqrt{-z}}} - 1,\quad z<0, \nonumber \nonumber \\ &&H(z) = \frac{1}{\sqrt{z(1 - z)}} \arctan{\sqrt{\frac{z}{1 - z}}} - 1,\quad 0<z<1, \nonumber \nonumber \\ &&H(z) = - \frac{1}{2\sqrt{z(z - 1)}} \ln{\frac{\sqrt{z} + \sqrt{z - 1}}{\sqrt{z} - \sqrt{z - 1}}} - 1 + \frac{i\pi}{2\sqrt{z(z - 1)}} ,\quad z>1, \nonumber \end{aligned}$$ and it has the asymptotics $$\begin{aligned} &&H(z) \simeq \frac{2}{3}z + \frac{8}{15}z^2 + \frac{16}{35}z^3, \quad |z| \ll 1, \label {eq:s5}\end{aligned}$$ $$\begin{aligned} &&H(z) \simeq -1 - \frac{1}{2z}\ln{4|z|}, \quad |z| \gg 1. \label {eq:s6}\end{aligned}$$ The process $\gamma \gamma \to \nu \bar\nu$ =========================================== Let us apply the results obtained to the calculation of some quantum processes. For the amplitude of the process $\gamma \gamma \to \nu \bar\nu$ in the framework of the model with the effective $\nu \nu e e$ – coupling of a scalar type we obtain from Eqs. (\[eq:M2\]), (\[eq:s1\]), (\[eq:s4\]) $$\begin{aligned} {\cal M}_{3}^s \,& = &\, \frac{2\alpha}{\pi} \, \frac{B}{B_e} \, g_{s} \, j_s \, m \, \frac{(q_1 \widetilde \varphi \varepsilon^{(1)}) (q_2 \widetilde \varphi \varepsilon^{(2)})} {4m^2 [(q_1 q_3)^2_{\mprl} - q^2_{1\mprl} q^2_{3\mprl}] + q^2_{1\mprl} q^2_{2\mprl} q^2_{3\mprl}} \times \nonumber \nonumber \\ &\times &\, \left \{ \left [q^2_{1\mprl} q^2_{3\mprl} - 2m^2 (q^2_{3\mprl} + q^2_{1\mprl} - q^2_{2\mprl})\right ] H \left (\frac{q_{1\mprl}^2}{4m^2} \right ) \right. \, + \nonumber \nonumber \\ & + & \, \left [q^2_{2\mprl} q^2_{3\mprl} - 2m^2 (q^2_{3\mprl} + q^2_{2\mprl} - q^2_{1\mprl})\right ] H \left (\frac{q_{2\mprl}^2}{4m^2} \right ) \, + \nonumber \nonumber \\ & + & \, \left. q^2_{3\mprl}(4m^2 -q^2_{3\mprl}) H \left (\frac{q_{3\mprl}^2}{4m^2} \right ) - 2q^2_{3\mprl}(q_1q_2)_{\mprl} \, \right \}, \label{eq:MS3}\end{aligned}$$ where $g_s = - 4 \; \zeta \; G_{\mbox{\normalsize{F}}}/\sqrt{2}$ is the effective $\nu \nu e e$ – coupling constant in the left-right-symmetric extension of the Standard Model, $\zeta$ is the small mixing angle of the charged $W$ bosons, $j_s = [\bar \nu_e(p_1) \nu_e(-p_2)]$ is Fourier transform of the scalar neutrino current, $q_3 = p_1 + p_2$ is the neutrino pair momentum. Substituting the photon polarization vector $\varepsilon^{(\mprp)}_{\alpha}$ from Eq. (\[eq:vect\]) into (\[eq:MS3\]) and using (\[eq:s5\]) and (\[eq:s6\]), we obtain the asymptotics: - at low photon energies, $\omega_{1,2} \lesssim m$ $$\begin{aligned} {\cal M}_{3}^{s} \simeq \frac{8 \alpha}{3 \pi} \, \frac{G_{\mbox{\normalsize{F}}}}{\sqrt{2}}\, \frac{\zeta}{m}\;\frac{B}{B_e}\, \left [\bar\nu_e (p_1) \, \nu_e (- p_2) \right ]\, \sqrt{q_{1\mprl}^2 q_{2\mprl}^2} ; \label{eq:M<}\end{aligned}$$ - at high photon energies, $\omega_{1,2} \gg m$, in the leading log approximation: $$\begin{aligned} {\cal M}_{3}^{s} \simeq \frac{16 \alpha}{\pi} \, \frac{G_{\mbox{\normalsize{F}}}}{\sqrt{2}}\, \zeta\;\frac{B}{B_e}\,m^3 \, \left [\bar\nu_e (p_1) \, \nu_e (- p_2) \right ]\, \frac{1}{\sqrt{q_{1\mprl}^2 q_{2\mprl}^2}}\, \ln \frac{\sqrt{q_{1\mprl}^2 q_{2\mprl}^2}}{m^2}. \label{eq:M>}\end{aligned}$$ These expressions coincide with the results obtained in the paper [@Kuznetsov:2002a]. The process $\gamma \gamma \to \nu \bar\nu \gamma $ =================================================== The process of this type, where one initial photon is virtual, namely, the photon conversion into neutrino pair on nucleus was considered, in the framework of the Standard Model, in the papers [@Skobelev:2001; @Kuznetsov:2002b]. This process can be studied by using the amplitude of the transition $\gamma \gamma \to \nu \bar\nu \gamma $, which can be obtained from Eq. (\[eq:M2\]) in the form: $$\begin{aligned} {\cal M}_{4}^{VA} \,& = &\, - \, \frac{8 i e^3}{\pi^2} \, \frac{B}{B_e} \, \frac{G_{\mbox{\normalsize{F}}}}{\sqrt{2}} \, m^2 \times \nonumber \nonumber \\ &\times & (q_1 \widetilde \varphi \varepsilon^{(1)}) (q_2 \widetilde \varphi \varepsilon^{(2)}) (q_3 \widetilde \varphi \varepsilon^{(3)}) [g_V (j\widetilde \varphi q_4) + g_A (j\widetilde \varphi \widetilde \varphi q_4)] \times \nonumber \nonumber \\ &\times & \frac{1}{D} \left \{ I_4(q_{1\mprl},q_{2\mprl},q_{3\mprl}) + I_4(q_{2\mprl},q_{1\mprl},q_{3\mprl}) + I_4(q_{1\mprl},q_{3\mprl},q_{2\mprl}) \right \}, \label{eq:MVA4}\end{aligned}$$ where $g_V,\;g_A$ are the vector and axial-vector constants of the effective $\nu \nu e e$ Lagrangian of the Standard Model, $g_V = \pm 1/2 + 2 \sin^2 \theta_W, \; g_A = \pm 1/2$, here the upper signs correspond to the electron neutrino, and lower signs correspond to the muon and tau neutrinos; $j_{\alpha} = [\bar \nu_e(p_1) \gamma_{\alpha} (1 + \gamma_5) \nu_e(-p_2)]$ is the Fourier transform of the neutrino current; $q_4 = p_1 + p_2$ is the neutrino pair momentum; $$D = (q_1q_2)_{\mprl}(q_3q_4)_{\mprl} + (q_1q_3)_{\mprl}(q_2q_4)_{\mprl} + (q_1q_4)_{\mprl}(q_2q_3)_{\mprl}.$$ The formfactor $I_4 (q_{1 \mprl} ,q_{2 \mprl} ,q_{3 \mprl})$ is expressed in terms of the integrals (\[eq:s1\]), (\[eq:s4\]) $$\begin{aligned} &&I_4 (q_{1 \mprl}, q_{2 \mprl}, q_{3 \mprl}) \, = \, S_3 (q_{1 \mprl} + q_{2 \mprl}, q_{4 \mprl},0) + S_3 (q_{1 \mprl}, q_{4 \mprl},0) + \nonumber \nonumber \\ && + S_3 (q_{1 \mprl} + q_{2 \mprl}, q_{1 \mprl},0) + S_3 (q_{2 \mprl} - q_{3 \mprl}, q_{2 \mprl},0) + \nonumber \nonumber \\ && + [6 m^2 - (q_1 + q_2)_{\mprl}^2 - (q_2 - q_3)_{\mprl}^2] S_4 (q_{1 \mprl}, q_{1 \mprl}+q_{2 \mprl}, q_{4 \mprl},0). \end{aligned}$$ Using the asymptotics of the functions $H(z)$, we obtain - at low photon energies, $\omega_{1,2,3} \ll m$ $$\begin{aligned} &&{\cal M}_{4}^{VA} \, \simeq \, - \frac{2 e^3}{15\pi^2} \, \frac{B}{B_e} \, \frac{G_{\mbox{\normalsize{F}}}}{\sqrt{2}} \, \frac{1}{m^4} \times \nonumber \nonumber \\ &&\times (q_1 \widetilde \varphi \varepsilon^{(1)}) (q_2 \widetilde \varphi \varepsilon^{(2)}) (q_3 \widetilde \varphi \varepsilon^{(3)}) [g_V (j\widetilde \varphi q_4) + g_A (j\widetilde \varphi \widetilde \varphi q_4)], \label{eq:I_4_low}\end{aligned}$$ which is in agreement with the result of the paper [@Kuznetsov:2002b]; - at high photon energies, $\omega_{1,2,3} \gg m$, in the leading log approximation we obtain: $$\begin{aligned} &&{\cal M}_{4}^{V,A} \, \simeq \, - \frac{8 e^3}{3\pi^2} \, \frac{G_{\mbox{\normalsize{F}}}}{\sqrt{2}} \, \frac{B}{B_e} \, m^4 \times \nonumber \nonumber \\ &&\times (q_1 \widetilde \varphi \varepsilon^{(1)}) (q_2 \widetilde \varphi \varepsilon^{(2)}) (q_3 \widetilde \varphi \varepsilon^{(3)}) [g_V (j\widetilde \varphi q_4) + g_A (j\widetilde \varphi \widetilde \varphi q_4)] \times \nonumber \nonumber \\ &&\times \frac{1}{q^2_{1 \mprl}q^2_{2 \mprl}q^2_{3 \mprl}q^2_{4 \mprl}} \ln \frac{\sqrt{q_{1 \mprl}^2 q_{2 \mprl}^2 q_{3 \mprl}^2}}{m^3}. \label{eq:I_4_hig}\end{aligned}$$ To the best of our knowledge, this result is obtained for the first time. Conclusions =========== We have obtained the general expressions (\[eq:MS3\]) and (\[eq:MVA4\]) for the amplitudes of the processes $\gamma \gamma \to \nu \bar\nu$ (in the framework of the model with the effective $\nu \nu e e$ coupling of a scalar type) and $\gamma \gamma \to \nu \bar\nu \gamma$ (in the framework of the Standard Model) for arbitrary energies of particles. A comparison with the existing results has been performed. [**Acknowledgements**]{} We express our deep gratitude to the organizers of the Seminar “Quarks-2002” for warm hospitality. This work was supported in part by the Russian Foundation for Basic Research under the Grant No. 01-02-17334 and by the Ministry of Education of Russian Federation under the Grant No. E00-11.0-5. [99]{} W.-Y. Tsai, Phys. Rev. [**D 10**]{}, 2699 (1974). A. E. Shabad, Tr. Fiz. Inst. Akad. Nauk SSSR [**192**]{}, 5 (1988). V. V. Skobelev, Zh. Eksp. Teor. Fiz. [**108**]{}, 3 (1995) \[JETP [**81**]{}, 1 (1995)\]. A. A. Gvozdev, N. V. 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--- author: - | Diego Marin[^1]\ with contributions by\ Pangea Association[^2] $\qquad\qquad$ Fabrizio Coppola[^3],\ Marcello Colozzo[^4] $\qquad\qquad$ Istituto Scientia[^5] title: | Arrangement Field Theory\ beyond Strings and Loop Gravity --- Abstracts ========= This paper regroups all contributions to the arrangement field theory (AFT), together with a philosophical introduction by Dr. Fabrizio Coppola. AFT is an unifying theory which describes gravitational, gauge and fermionic fields as elements in the super-symmetric extension of Lie algebra $Sp(12,\mathbf{C})$. $\pt$ **Paper number 1** We introduce the concept of non-ordered space-time” and formulate a quaternionic field theory over such generalized non-ordered space. The imposition of an order over a non-ordered space appears to spontaneously generate gravity, which is revealed as a fictitious force. The same process gives rise to gauge fields that are compatible with those of Standard Model. We suggest a common origin for gravity and gauge fields from a unique entity called arrangement matrix” ($M$) and propose to quantize all fields by quantizing $M$. Finally we give a proposal for the explanation of black hole entropy and area law inside this paradigm. $\pt$ **Paper number 2** In this work we apply the formalism developed in the previous paper (The arrangement field theory”) to describe the content of standard model plus gravity. The resulting scheme finds an analogue in supersymmetric theories but now all quarks and leptons take the role of gauginos for $Sp(12,\mathbf{C})$ gauge fields. Moreover we discover a triality between *Arrangement Field Theory*, *String Theory* and *Loop Quantum Gravity*, which appear as different manifestations of the same theory. Finally we show as three families of fields arise naturally and we discover a new road toward unification of gravity with gauge and matter fields. $\pt$ **Paper number 3** We show how antigravity effects emerge from arrangement field theory. AFT is a proposal for an unifying theory which joins gravity with gauge fields by using the Lie group $Sp(12,\mathbf{C})$. Details of theory have been exposed in the papers number 1 and number 2. The philosophy of arrangement field theory {#intro} ========================================== Classical Physics \[classicalphy\] ---------------------------------- In classical physics, space and time are fundamental entities, providing a preordained structure in which interactions between physical objects can occur. In short, space and time are absolute". Moreover, the physical properties of a body or system are supposed to be objective and independent from a possible observation. In this paradigm, reality exists independently of classical measurements and is not significantly influenced by measurements, unless these are particularly invasive". But even in such cases, it is assumed that the observed systems had their own pre-existing characteristics. These were obvious and implicit tenets in classical physics, which influenced whole science, aimed to be purely objective. Space and time according to philosophers \[philosophers\] --------------------------------------------------------- Despite the rapid and successful development of classical physics and science in general, firmly based on the fixed concepts of space and time, between late 17th century and early 19th century respectable philosophers such as Locke, Hume, Leibniz, Kant and Schopenhauer, conceptualized space and time not as objective and universal entities, but as concepts defined by our own intellect, aimed to interpret the external reality perceived by our senses. This idea was radically different from the founding conception of classical physics, based on full objectivity, and appeared quite extravagant to several scientists at that time. Nevertheless Kant, who had a scientific background, exposed his conception in a profound and rational way. In 1781 Kant distinguished two main activities of conscious mind [@kant]: analytic propositions“ and synthetic propositions”. In an oversimplied interpretation, analytic propositions" are the elements of rational, logical reasoning, in which thoughts proceed by deduction, starting from known facts and finding consequences which, anyway, were implicit in the premises and only had to made explicit by reasoning. Synthetic propositions", instead, are new, non-deductible informations, coming from perceptions and sensations. For instance we can not deduce whether an apple is sweet, or a radiator is hot, but we must check that through our senses. Kant also proposed a distinction between a priori“ propositions, meaning in advance”, ie before“ an experience is performed; and a posteriori” propositions, meaning after" an experience. According to Kant, all analytic propositions are a priori“. A trivial example is given by any sum, such as 4 + 7 = 11. This analytic proposition is true a priori”: the result is already 11 before we make the calculation. Kant states that no analytic proposition can be a posteriori“. Synthetic propositions, on the other side, are generally a posteriori”, since perceptions come from experience. Now, an interesting question remains: may a priori“ synthetic propositions exist? Kant answers that they do actually exist. Certain categories” that human mind applies to events, such as the principle of cause and effect“, are a priori”. In fact we perceive events and relate to each other according to a category, causality", which, according to Kant, already exists in our intellect. Kant states that space“ and time” are also a priori“ synthetic forms. Even if space, time and causality are related to experience, Kant does not consider them as inherent to the objective phenomena, but as subjective tools (even if they manifest themselves as universal) that our intellect uses to order” the experiences. After Kant’s definitions, anyway, classical, mechanistic science continued to achieve extraordinary results. However, in the early twentieth century, physics started to face unexpected problems and contradictions, that forced scientists to formulate new principles and accept radical changes. Relativistic physics \[relativistic\] ------------------------------------- In 1632 Galileo had intuited and enunciated the principle of relativity", stating that the laws of physics are the same in every inertial frame of reference [@galileo1]. Later developments of physics, including several discoveries in optics and electromagnetism, suggested instead that a privileged, steady, fundamental frame of reference should exist. This issue especially afflicted electromagnetism, that was an excellent theory but included certain unsolved inconsistencies. In 1905 Einstein solved the whole problem, restarting from the Galileo’s principle of relativity and applying it to the new knowledge of electromagnetism and optics, thus developing an original, consistent theory, special relativity"[@einstein1]. His theory also accounted for the results of the Michelson-Morley experiment [@mmorley], conducted in 1887, which had demonstrated that speed of light does not follow the classical laws of velocity addition. Einstein solved all the inconsistencies by proposing that the speed of light $c$ is independent from the motion of the emitting body. The universal constant $c$ became an insurmountable speed limit in physics. Einstein’s theory also implied new, counter-intuitive ideas: for example, time flows differently in different inertial reference systems, and perception of space also depends on the frame of reference of the observer. In light of such new discoveries, Kant’s ideas do not seem to be so extravagant anymore. Space and time lose their absolute characteristics if considered independently from each other, but, adequately considered as components (coordinates) of four-dimensional points, remain absolute“ (invariant”) in a single entity, space-time“ or chronotope”, ruled by a generalized geometrical entity including time as the fourth coordinate. In 1908 such a four-dimensional structure was perfected and named Minkowski space" [@minko]. In 1916 Einstein expanded the principle of relativity to non-inertial reference frames, thus defining the new theory of general relativity“, in which the four-dimensional geometry is curved by the presence of the masses [@einstein3]. Hence, even the (linear) Minkowski space had to be considered as an approximation, valid only in small regions of the (curved) universe. In this perspective, gravitational forces” find their natural explanation in geometrical terms, based on a specific concept of metric. This approach also affected the interpretation of the principle of cause-effect, to the point that Einstein, in paragraph $a2$, wrote: The law of causality has not the signicance of a statement as to the world of experience, except when observable facts ultimately appear as causes and effects“ [@einstein3]. Kant had exposed this extravagant” idea a long time before [@kant]. In this paper we suggest a new step in the direction of relativization" (so to say), by questioning the absolute ordering of the space-time points, that we believe is an imposition made by our intellect, rather than a proper quality of Nature. Such conjecture might open new unexpected perspectives for understanding the fundamental fields of physics, as we are going to see. Quantum limitation of objectivity \[limitation\] ------------------------------------------------ In 1900 Planck had proposed quantization“ of energy to explain the electromagnetic emission of a black body” [@planck]. In 1905 quantization of energy was also applied by Einstein to explain the photoelectric effect" [@einstein2]. The several discoveries that clarified the structure of the atom from 1905 to the 1930’s included the Rutherford’s experiment [@rutherford] in 1911, and the consequent Bohr model [@bohr1] in 1913. Bohr started from the results of the Rutherford’s experiment, and imposed quantization to the angular momentum of electrons, instead of quantizing energy directly. As a consequence, energy also turned out to be quantized, and the calculated levels were in excellent agreement with the experimental values. The agreement was nearly perfect in the case of hydrogen, the simplest atom in Nature. In the case of more complex and heavier chemical elements, the mathematical frame was more difficult and the results were less precise. To solve these problems, the complete theory of Quantum Mechanics (QM) was gradually developed (mainly by the Copenhagen school" directed by Bohr himself during the 1920’s), which came out to be intuitively abstruse, offering no image of the motion of the electrons around the atomic nucleus. While developing QM, it began to emerge that the experiments inevitably influenced the observed systems. Bohr, Heisenberg and other physicists of the Copenhagen school" suspected that physical properties of quantum systems could no longer be assumed to be completely predefined and ontologically independent from observation. In the first version of the Copenhagen interpretation" they assumed that free will of the conscious observers played a decisive role in the collapse of a quantum state into an eigenstate [@heisenberg1]. This appeared as an unacceptable extravagance to many physicists, including Einstein, because of the unexpected restrictions that the supposed objectivity of the universe had to suffer, as a consequence of the new theory. Quantum states evolve deterministically according to the Schrödinger equation [@schr], formulated in 1926, but remain devoided of certain characteristics, which can be revealed (objectivated“) only when the quantum state collapses into an eigenstate” of the measured physical quantity. This is the main reason why physical quantities in QM are called observables". QM works fine“ only if it is accepted that such hidden properties are not objectively defined before the measurement and are partly created by observation itself, when the state is reduced to an eigenstate. The eigenvalues calculated according to QM are in excellent agreement with the possible outcomes given by experiments, even though the theory can not predict which eigenvalue will come out: only the respective probabilities can be calculated, as pointed out by Born [@born] in 1926. This led in 1927 to the Heisenberg’s uncertainty principle” [@heisenberg2], which put an end to the absolute determinism that was implicit in classical physics. QM thus introduced a margin of uncertainty“, in which Nature may reserve a small room for Her non-predictable caprice” or willingness“, according to Jordan [@jordan], and secondarily [@heisenberg1] accepted by Pauli, Wigner, Eddington, and von Neumann [@von], and years later by Wheeler [@wheeler], Stapp [@stapp1], and other physicists. For example, Stapp in 1982 defined human mental activity as creative”, because it only partially undergoes the course of causal mechanisms, having a margin for free choices [@stapp1]. Another important consequence concerns the act of measurement, after which, the subsequent course of the physical system under observation is unavoidably modified by the measurement itself, so that observations inevitably imprint different directions to events. In 1932 von Neumann, after reordering and formalizing QM into a consistent theory, stated that a distinctive element was necessary to trigger the quantum collapse“or reduction”, and declared that the consciousness of an observer could be such an element, distinctive enough from the usual physical quantities [@von]. In 2001 Stapp consistently explained this concept in detailed and clear terms [@stapp2]. In 1935 the discussion about the interpretation of QM faced the problem introduced by the Einstein, Podolski and Rosen (EPR) paradox [@epr], [@bridge], that later, in 1951, was better defined by Bohm [@bohm]. In this well-known thought experiment, two particles in quantum entanglement" but far away from each other, produce instant, non-local influences, in contradiction with the upper limit set by relativity at the speed of light: E., P. and R. considered that as absurd and impossible. Nevertheless, the experimental version that was defined by the Bell’s theorem [@bell] in 1964, and implemented in 1982 by Aspect et al. [@aspect], confirmed the existence of non-local influences due to the entanglement. Thus, a conflict seems to exist between special relativity (that does not allow non-local influences) and QM (which includes and reveals such influences). The subsequent theories have not been able to solve in a convincing way such a dissonance. The conjecture exposed in this paper, however, may offer a new framework where such conflict can be finally overcome. Fabrizio Coppola, Istituto Scientia The arrangement field theory (AFT) ================================== Introduction to formalism {#sec:1} ------------------------- The arrangement field paradigm describes the universe be means of a graph (ie an ensemble of vertices and edges). However there is a considerable difference between this framework and the usual modeling with spin-foams or spin-networks. The existence of an edge which connects two vertices is in fact probabilistic. In this way we consider the vertices as fundamental physical quantities, while the edges become dynamic fields. In section \[reciprocal\] we introduce the concept of non-ordered space-time, ie an ensemble of vertices without any information on their mutual positions. In section \[arrmatrix\] we define the arrangement matrix” ($M$), which is a matricial field whose entries define the probability amplitudes for the existence of edges. The arrangement matrix regulates the order of vertices in the space-time, determining the topology of space-time itself. In the same section we extend the concept of derivative on such non-ordered space-time. In section \[ord\] we define a simple toy-action” for a quaternionic field in a non-ordered space-time. We show how the imposition of an arrangement in such space-time generates automatically a metric $h$ which is strictly determined by $M$. In section \[local\] we discover a low energy limit under which the toy-action” becomes a local action after the arrangement imposition. In section \[spin\] we show that a new interpretation of spin nature arises spontaneously from our framework. In the same section, the role of arrangement matrix” is compared to the role of an external observer. In section \[symmetry\] we anticipate some unpublished results regarding the availment of our framework to describe all standard model interactions. In section \[entropy\] we apply a second quantization to the arrangement matrix”, turning it in an operator which creates or annihilates edges. We show how this process can give a new interpretation to black hole entropy and area law. We infer that quantization of $M$ automatically quantizes $h$, apparently without renormalization problems. A non-ordered universe \[unordered space\] ------------------------------------------ ### Reciprocal relationship between space-time points \[reciprocal\] Every euclidean $4$-dimensional space can be approximated by a graph $\L$, that is a collection of vertices connected by edges of length $\Delta$. We recover the continuous space in the limit $\Delta \ra 0$. Moreover we can pass from the euclidean space to the lorenzian space-time by extending holomorphically any function in the fourth coordinate $x_4 \ra ix_4$ [@minko]. In non commutative geometry, one can assume that a first vertex is connected to a second, without the second is connected to the first. This means that connections between vertices are made by two oppositely oriented edges, which we can represent by a couple of arrows. We assume the vertices as fundamental quantities. Then we can select what couples of vertices are connected by edges; different choices of couple generated different graphs, which in the limit $\Delta \ra 0$ correspond to different spaces. Our fundamental assumption is that the existence of an edge follows a probabilistic law, like any other quantity in QM. We draw any pair of vertices, denoted by $v_{1}$ and $v_{2}$, and we connect each other by a couple of arrows oriented in opposite directions. Before proceeding, we extend the common definition of amplitude probability. Usually this is a complex number, whose square module represents a probability and so is minor or equal to one. We define instead the amplitude probability as an element in the division ring of quaternionic numbers, commonly indicated with $\mathbf{H}$. Its square module represents yet a probability and so is minor or equal to one. A quaternion $q$ have the form $q = a+ib+jc+kd$ with $a,b,c,d \in \mathbf{R}$, $i^2 = j^2 = k^2 = -1$ and $ij = -ji = k$, $jk = -kj =i$, $ki = -ik =j$. We write a quaternionic number near the arrow which moves from $v_1$ to $v_2$. It corresponds to the probability amplitude for the existence of an edge which connects $v_1$ with $v_2$. We do the same thing for the other arrow, writing the probability amplitude for the existence of an edge which connects $v_2$ with $v_1$ \[geometry\]. {width="40.00000%"} A non-drawn arrow corresponds to an arrow with number $0$. In principle, for every pair of vertices exists a couple of arrows which connect each other, eventually with label $0$. We can describe our universe by means of vertices connected by couple of arrows, with a quaternionic number next to each arrow, as shown in figure \[eq: network\], below. ![We can describe our universe by means of vertices connected by couple of arrows, with a quaternionic number next to each arrow.[]{data-label="eq: network"}](fig2.jpg){width="70.00000%"} What we are building is another variation of the Penrose’s spin-network model [@spinnet] or the Spin-Foam models [@spinfoam1], [@spinfoam2] in Loop Quantum Gravity [@loopgravity], which generalize Feynman diagrams. ### The Matrix relating couples of points \[arrmatrix\] Given a spin-network, like the one in figure \[eq: network\], we can move from picture to the Arrangement Matrix” $M$, which is a simple table constructed as follows. We enumerate all the vertices in the graph at our will, provided we enumerate all of them. Typically we think of indexing the vertices by the usual sequence of integers $1,2,3,4,5,\ldots.$ Thus we create such matrix, whose rows and columns are enumerated in the same way as the vertices in the graph. Then we look at the vertices $v_{i}$ and $v_{j}$: in the entry $(i,j)$ we report the number situated near the arrow which moves from $v_i$ to $v_j$. Similarly, in the entry $(j,i)$ we report the number written near the opposite arrow. Remember that an absent arrow is an arrow with number $0$ and consider for the moment $|M^{ij}|\leq 1$ for every $ij$. {width="30.00000%"} \[figuranuova\] In principle, we can image an entry $M_{ij} \neq M_{ji}$, even with $|M_{ij}|^2 \neq |M_{ji}|^2$. This means that $v_i$ may be connected to $v_j$ even if $v_j$ is not connected to $v_i$. In that case, a non-commutative geometry is involved. The probability amplitude that $v_i$ and $v_j$ are mutually connected (we could talk about classical” connection), is: $$Cl.ampl. \propto M_{ij} M_{ji}$$ The probability amplitude for the vertex $v_i$ to be classically connected with any other vertex (hence it will be not isolated) is: $$Cl.ampl. \propto \sum_j M_{ij} M_{ji} = (M \cdot M)_{ii}$$ We can imagine our table with elements $M_{ij}$ as a machine which creates” jointures between vertices, by connecting each other or closing a single vertex onto itself through a loop. The loops are obviously represented by diagonal elements of matrix, with the form $(i,i)$. Now let’s ask ourselves: is it necessary to know where the vertices are located? Let’s look at the Standard Model action: it is given by a sum (or more properly, an integral), over $all$ the points of the universe, of locally defined terms. Any term is defined on a single point. Since the terms are separated - a term for each point - and we integrate all of them, we do not need to know where the points physically are. However, there are terms which are not strictly local, ie those containing the derivative operator $\partial$. The operator $\partial$, acting on a field $\varphi$ in the point $v_{j}$, calculates the difference between the value of $\varphi$ in a point immediately after” $v_{j}$, and the value of $\varphi$ immediately  before” $v_{j}$. In the discretized theory, the integral over points becomes a sum over vertices of the graph. Similarly, the derivative becomes a finite difference. Hence, for terms containing $\partial$, we need a clear definition of before” and after”, that is an arrangement of the vertices, as defined by the matrix $M$. We consider a scalar field but don’t represent it with the usual function (or distribution) $\varphi\left( x\right) $. Instead we denote it with a column of elements (an array) where each element is the value of the field in a specific vertex of the graph. For example (with only 7 vertices): [1.1]{} $$\varphi=\left( \begin{array} [c]{c}\varphi\left( p_{0}\right) \\ \varphi\left( p_{1}\right) \\ \varphi\left( p_{2}\right) \\ \varphi\left( p_{3}\right) \\ \varphi\left( p_{4}\right) \\ \varphi\left( p_{5}\right) \\ \varphi\left( p_{6}\right) \end{array} \right)$$ [1.5]{} For simplicity, we start with a one-dimensional graph: it’s easy to see how the derivative operator is proportional to an antisymmetric matrix $\tilde{M}$ whose elements are different from zero only immediately above the diagonal (where they count +1), and immediately below (where they count -1). We can see this, for example, in a toy-graph” formed by only $12$ separated vertices (figure \[cerchio\]). The argument remains true while increasing the number of vertices. [1.1]{} $$\! \partial\varphi \! = \!\fr 1 {2\Delta} \!\left( \begin{array} [c]{cccccccccccc}0 & +1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1\\ -1 & 0 & +1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & -1 & 0 & +1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & -1 & 0 & +1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -1 & 0 & +1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -1 & 0 & +1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & -1 & 0 & +1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & +1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & +1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & +1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & +1\\ +1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 \end{array} \right) \!\!\! \left( \begin{array} [c]{c}\varphi\left( 0\right) \\ \varphi\left( 1\right) \\ \varphi\left( 2\right) \\ \varphi\left( 3\right) \\ \varphi\left( 4\right) \\ \varphi\left( 5\right) \\ \varphi\left( 6\right) \\ \varphi\left( 7\right) \\ \varphi\left( 8\right) \\ \varphi\left( 9\right) \\ \varphi\left( 10\right) \\ \varphi\left( 11\right) \end{array} \right)$$ = ( \[c\][c]{}( 1) -( 11)\ ( 2) -( 0)\ ( 3) -( 1)\ ( 4) -( 2)\ ( 5) -( 3)\ ( 6) -( 6)\ ( 7) -( 5)\ ( 8) -( 6)\ ( 9) -( 7)\ ( 10) -( 8)\ ( 11) -( 9)\ ( 0) -( 10) )\[eq: matrix\_dev\] $$\pt$$ ![[A simple graph with $12$ vertices which approximates a circular one-dimensional space.]{}[]{data-label="cerchio"}](fig4.jpg){width="40.00000%"} $\Delta$ is the length of graph edges. In the continuous limit, $\Delta \ra 0$ (that occurs in Hausdorff spaces, where matricial product turns into a convolution), we obtain \(x) &=& \_[0]{}1 [2]{} (x,y) (y) dy\ (x) &=& \_[0]{}1 [2]{} (y) dy\ (x) &=& \_[0]{} = (x) In this way our definition is consistent with the usual definition of derivative. While increasing the number of points, a $(-1)$ still remains in the up right corner of the matrix, and a $(+1)$ in the down left corner as well. To remove those two non-null terms, it is sufficient to make them unnecessary, by imposing boundary conditions that make the field null in the first and in the last point. In fact we can describe an open universe (a straight line in one dimension), starting from a closed universe (a circle) and making the radius to tend to infinity. Hence we see that the conditions of null field in the first and in the last point become the traditional boundary conditions for the Standard Model fields. Note that in spaces with more than one dimension, a derivative matrix $\tilde{M}_\mu$ assumes the form (\[eq: matrix\_dev\]) only if we number the vertices progressively along the coordinate $\mu$. However, two different numberings can be always related by a vertices permutation. A quaternionic field action in a non-ordered space-time \[ord\] --------------------------------------------------------------- For any graph $\L$ we define its *associated non-ordered space* $\mathbf{S}_\Lambda$ as the ensemble of all its vertices. The graph includes vertices plus edges (ordered connections between vertices), while the *associated non-ordered space* contains only vertices. In some sense, $\S$ doesn’t know where any vertex is. Consider a *numbering function* $\pi$, that is whatever bijection from $X \subset\mathbf{N}$ to the non-ordered space. &:& X §\ && i v\_i = (i) In this way, every vertex $v_i$ in $\S$ is one to one with an integer $i \in X\subset\mathbf{N}$. This means that the ensemble of vertices has to be at most numerable. We consider a generic invertible matrix $M$ and interpret any entry $M^{ij}$ of $M$ as the probability amplitude for the existence in $\L$ of an edge which connects $\pi(i)$ with $\pi(j)$. Remember that a couple of vertices can be connected by at most two oriented edges with different orientations. $M^{ij}$ defines the probability amplitude for the edge which moves from $\pi(i)$ to $\pi(j)$, while $M^{ji}$ defines the probability amplitude for the edge which moves from $\pi(j)$ to $\pi(i)$. Take care that in four dimensions we have to number the vertices by elements $(i,j,k,l)$ in $\mathbf{N}^4$ before taking the limit $\Delta \ra 0$. In this way $\sum_{(i,j,k,l)}\Delta^4$ becomes $\int dx^0 dx^1 dx^2 dx^3$. If, as we have suggested, the vertices have been already numbered with elements of $\mathbf{N}$, we can change the numbering by using the natural bijection $\vartheta$ between $\mathbf{N}$ and $\mathbf{N}^4$, with $(i,j,k,l) = \vartheta (a)$, $(i,j,k,l) \in \mathbf{N}^4$ and $a \in \mathbf{N}$. Given any skew hermitian matrix $A_\mu$, with entries in $\mathbf{H}$, and a skew hermitian matrix $\tilde{M}_\mu$, which assumes the form (\[eq: matrix\_dev\]) when the vertices are numbered along the coordinate $\mu$, their associated covariant derivative is $$\nabla_{\mu}=\tilde{M}_{\mu}+ A_{\mu}.$$ We indicate with $n$ the number of elements inside $X \subset \mathbf{N}$. Given a normal matrix $\hat{M}$ and four covariant derivatives $\nabla_\mu$ ($\mu = 0,1,2,3$) with dimensions $n \times n$, an *arrangement* for $\hat{M}$ is a quadruplet of couples $(\hat{D}^\mu,\hat{U})$, with $\hat{D}^\mu$ diagonal and $\hat{U}$ hyperunitary, such that $$\hat{M} = \sum_\mu \hat{U}\hat{D}^\mu \nabla_\mu \hat{U}^{\dag}.$$ We require that covariant derivative will be form-invariant under the action of a transformation $V \in U(n,\mathbf{H})$ which acts both on $\tilde{M}_\mu$ and $A_\mu$. We explicit $V\nabla_{\mu}V^{\dag}$: $$\begin{aligned} V\nabla_{\mu}V^{\dag} & = V\left( \tilde{M}_{\mu}+A_{\mu}\right) V^{\dag}\\ & = \underset{=1}{\underbrace{VV^{\dag}}}\tilde{M}_{\mu}+V\left[ \tilde {M}_{\mu},V^{\dag}\right] +VA_{\mu}V^{\dag}\nonumber .\end{aligned}$$ Setting $$A_{\mu}^{\prime}= V\left[ \tilde{M}_{\mu},V^{\dag}\right] +VA_{\mu}V^{\dag},$$ we obtain $$V\nabla_{\mu}V^{\dag}=\tilde{M}_{\mu}+A_{\mu}^{\prime}\overset{def}{=}\nabla_{\mu}^{\prime} \label{eq: der_A}$$ that means $$V\nabla_{\mu}[A]V^{\dag} = \nabla_\mu [A'].$$ Hence the transformation law for the matrix $A_{\mu}$ is like we expect: $$A_{\mu}\rightarrow {A'}_\mu = V\left[ \tilde{M}_{\mu},V^{\dag}\right] +VA_{\mu}V^{\dag} . \label{eq: trasforma_A}$$ We observe that (\[eq: trasforma\_A\]) preserves the hermiticity of $A_{\mu}$. In fact \^\_&=& (V\[\_,V\^\] + VA\_V\^)\^\ &=& (V\_V\^- \_+ VA\_V\^)\^\ &=& V\_\^V\^- \_\^+ V A\_\^V\^\ &=& - V \_V\^+ \_- V A\_V\^\ &=& -(V\[\_,V\^\] + VA\_V\^) = -[A’]{}\_It’s easy to see that (\[eq: trasforma\_A\]) reduces to the usual transformation for a gauge field ${A'}_{\mu} = V\partial_{\mu}V^{\dag}+VA_{\mu}V^{\dag}$ in the limit $\Delta \ra 0$. For every invertible normal matrix $\hat{M}$ and every covariant derivative $\nabla[A]_\mu$ which is invertible (in the matricial sense), there exist 1. A new quadruplet of covariant derivatives ${\nabla'}_\mu = \nabla[A']_\mu$ such that $D^\mu {\nabla'}_\mu = 1$ for some diagonal matrix $D^\mu$, where $A^\prime_\mu$ is the gauge transformed of $A_\mu$ for some unitary transformation $U$; 2. An arrangement $(\hat{D}^\mu,\hat{U})$ between $\hat{M}$ and $\nabla^\prime_\mu$. \[existence\] According to spectral theorem, $\forall \hat{M} \in\mathbb{M}^{(N)}$ $\exists \hat{U}$ hyperunitary such that $\hat{U}\hat{M}\hat{U}^{\dag}= K$ with $K$ diagonal. $\hat{M}$ is invertible, so the same is true for $K$. Setting $\hat{D}= K^{-1}$: $$\begin{aligned} \hat{U}\hat{M}\hat{U}^{\dag}\hat{D} = K\hat{D} = KK^{-1} = 1\label{ordinamento}\\ \hat{D}\hat{U}\hat{M}\hat{U}^{\dag}= \hat{D}K = K^{-1}K = 1 .\nonumber\end{aligned}$$ At this point we choice a covariant derivative $\nabla_\mu$ (which is also a normal matrix) and we reason as we did above for $\hat{M}$, putting $$1 =D^{\mu}U\nabla_{\mu}U^{\dag}=U\nabla_{\mu}U^{\dag}D^{\mu} \label{eq: riduzioneB}$$ for some $D^\mu$ diagonal and $U$ unitary. No sum over repeated indices is implied. A well known theorem states that $U$ can be chosen in such a way that $D^\mu$ takes values in $\mathbf{C}$. Moreover we can always find a quaternion $s$ with $|s|=1$ such that, if $D^\mu$ takes values in $\mathbf{C} = \mathbf{R} \oplus i\mathbf{R}$, then $s^* D^\mu s$ will take values in $\mathbf{C} = \mathbf{R} \oplus (ri+tj+pk)\mathbf{R}$, with fixed $r,t,p \in \mathbf{R}$ and $r^2+t^2+p^2 =1$. Every $s$ with $|s|=1$ describes in fact a rotation in the $3$ dimensional space with base elements $i,j,k$. Introducing such $s$, the equation (\[eq: riduzioneB\]) becomes s1s = s\^\*D\^ss\^[\*]{}U\_ U\^s. Now we note that $s^*U$ is another hyperunitary transformation. Redefining $s^* D_\mu s \ra D_\mu$, $s^* U \ra U$ we obtain newly 1 = D\^U\_ U\^. In this way we can always choose in what complex plane is $D_\mu$. In the following we call this propriety $s$-invariance”. Using (\[eq: der\_A\]) into (\[eq: riduzioneB\]): $$1=D^\mu \nabla_{\mu}^{\prime}=\nabla_{\mu}^{\prime}D^\mu \Longrightarrow\left[ \nabla_{\mu}^{\prime},D^\mu \right] = 0 .$$ Taking into account (\[ordinamento\]): $$\begin{aligned} \hat{D}\hat{U}\hat{M} \hat{U}^{\dag} & =D^\mu \nabla_{\mu}^{\prime}\\ \hat{U}\hat{M} \hat{U}^{\dag}\hat{D} & =\nabla_{\mu}^{\prime}D^\mu \nonumber .\end{aligned}$$ Summing on $\mu$ we obtain: $$\begin{aligned} 4\hat{D}\hat{U}\hat{M} \hat{U}^{\dag} & = \sum_\mu D^\mu \nabla_{\mu}^{\prime}\\ 4\hat{U}\hat{M} \hat{U}^{\dag}\hat{D} & = \sum_\mu \nabla_{\mu}^{\prime}D^\mu . \nonumber\end{aligned}$$ Solving for $\hat{M}$: = 14 \_\^ \^[-1]{} D\^ \_\^ = 14 \_\^ \_\^ D\^\^[-1]{} . Defining $\hat{D}^\mu$ as $\fr 14 \hat{D}^{-1} D^\mu $ = \_\^ \^ \_\^ \[finale-dim\] QED Note that in general $\hat{M} \neq \sum_\mu \hat{U}^{\dag} \nabla_{\mu}^{\prime} \hat{D}^{\mu}\hat{U}$ because $\hat{D}^{-1} D^\mu \neq D^\mu \hat{D}^{-1}$ for the non commutativity of quaternions. For every invertible matrix $M$ with entries in $\mathbf{H}$, a normal matrix $\hat{M} = U_M M$ exists, where $U_M$ is unitary and $\hat{M}$ is neither hermitian nor skew hermitian. \[normal\] Given an invertible matrix $M$, a unique choice of matrices $U$ and $P$ always exists, with $U$ unitary and $P$ hermitian positive, such that $UM = P$. Moreover, a well known theorem states that, for every hermitian matrix $P$ with entries in $\mathbf{H}$, there exist $I,J,K$ skew hermitian unitary matrices which commute with $P$. Moreover $I,J,K$ achieve the same algebra of quaternionic imaginary unities $i,j,k$. [1.5]{} Consider then the unitary matrix $p = exp((bI + cJ + dK)P)$, with $b,c,d \in \mathbf{R}$. It’s easy to see that $[p,P]=0$. Moreover the matrix $\hat{M} = pP$ is normal and it is neither hermitian or skew hermitian. In fact $$(pP)^\dag = p^{\dag} P = p^{-1} P = \neq \pm pP$$ $$(pP)(pP)^\dag = (Pp)(Pp)^\dag = Ppp^\dag P^\dag = PP = Pp^\dag pP = P^\dag p^\dag pP = (pP)^\dag (pP)$$ Moreover $$\hat{M} = pUM = U_M M\qquad U_M = pU\,\,\, unitary.$$ $\pt\!\!\!\!$ For every invertible matrix $M$, we define an *associated normal matrix* as a normal matrix obtained trough the construction above. We indicate it with $\hat{M}$ and use the notation $U_M$ for the unitary transformation which transforms $M$ in $\hat{M} = U_M M$. For every $n \times n$ invertible matrix $M$ with entries in $\mathbf{H}$ and every quadruplet of covariant derivatives $\nabla[A]_\mu$ which are invertible (in the matricial sense), there exist 1. An associated normal matrix $\hat{M} = U_M M$ with $U_M$ unitary; 2. A new quadruplet of covariant derivatives ${\nabla'}_\mu = \nabla[A']_\mu$ such that $D^\mu {\nabla'}_\mu = 1$ for some diagonal matrix $D^\mu$, where $A^\prime_\mu$ is the gauge transformed of $A_\mu$ for some unitary transformation $U$; 3. An arrangement $(\hat{D}_\mu,\hat{U})$ between $\hat{M}$ and $\nabla^\prime_\mu$ such that && S = (M)\^(M) =\_[i=1]{}\^n \_[,]{} h\^(x\_i) ([’]{}\_\^(x\_i))\^\* ([’]{}\_\^(x\_i)) .\ && \[azione-sca\] Here $\phi$ is a one-component quaternionic field, while && x\_i (i)\ && \^(x\_[i]{})= [\^]{}\^[i]{}(x)= \_[j]{} \^[ij]{}\^[j]{}(x) = \_[j]{} \^[ij]{}(x\_[j]{})\ && h h\^(x\_i) = 12 d\^d\^[\*]{}(x\_i) +c.c.\_\^[ij]{} = d\^(x\_i)\_[ij]{}.\ && \[transf-scalar\] The existence of $\nabla^\prime_\mu = \nabla[A']_\mu$ follows from the proof of theorem \[existence\], while the existence of an associated normal matrix $\hat M = U_M M$ descends from theorem \[normal\]. Hence we see that the first action in (\[azione-sca\]) is invariant for transformations $(U_1,U_2)$ in $U(n,\mathbf{H}) \otimes U(n,\mathbf{H})$ which send $M$ in $U_2 M U_1^\dag$ and $\phi$ in $U_1\phi$. In fact [1.3]{} S\[\] &=& \^M\^M\ && \^U\_1\^(U\_2 M U\_1\^)\^(U\_2 M U\_1\^) U\_1\ &=& \^U\_1\^U\_1 M\^U\_2\^U\_2 M U\_1\^U\_1\ &=& \^ M\^M = S\[\]If we set $U_1 = 1$ and $U_2 = U_M$ we have S\[\] \^M\^U\_M\^U\_M M = { [ll]{} = \^M\^M = S\[\]\ = \^\^ . . \[sca-nuova\] We substitute (\[finale-dim\]) in (\[sca-nuova\]) with $\hat{M}$ in place of $M$. S\[\] &=& [\_[,]{}]{} ( \^\^\_\^)\^( \^\^\_\^)\ &=& [\_[,]{}]{} ( \^\^\_\^\^ \^\^\_\^)\ &=& [\_[,]{}]{} ( \^\^\_\^\^ \^\_\^ )\ &=& [\_[,]{}]{} ( \^\^\_\^\^ \^\_\^)\ &=& [\_[,]{}]{} ( \^\_\^ \^\^ \_\^\^) . In the last step we have taken in account the definition (\[transf-scalar\]). Finally S = 12 \_[,]{} \^[\^]{} \_\^[\^]{} ( \^\^ +c.c. ) \_\^\^. It is remarkable that $\hat{D}_{\mu}$ is diagonal: $$\hat{D}^{\mu}_{ij}=d^{\mu}\left( x_{i}\right) \delta_{ij} .$$ We can set $$\sqrt{\left\vert h\right\vert }h^{\mu\nu}\left( x_{i}\right) = \fr 12 d^{\mu *}d^{\nu}\left( x_{i}\right) +c.c.$$ and then $$S= {\displaystyle\sum\limits_{i,\mu,\nu}} \sqrt{\left\vert h\right\vert }h^{\mu\nu}(x_{i})\left( \nabla_{\mu}^{\prime }\phi^{\prime}\right)^{*i} \left( \nabla_{\nu}^{\prime} \phi^{\prime}\right)^{i} . \label{ultima2}$$ QED. The action of a transformation $(U_1, U_2)$ on $\nabla'$ follows from its action on $M$. We can always use the invariance under $U(n,\mathbf{H}) \otimes U(n,\mathbf{H})$ to put $M$ in the form $M =\sum_\mu \hat{D}^\mu \nabla'_\mu$. Starting from this we have U\_2 M U\_1\^= \_U\_2 \^’\_U\_1\^= \_U\_2 \^U\_1\^U\_1’\_U\_1\^.\[transf\]We define ${\nabla''}_\mu = U_1\nabla'_\mu U_1^\dag$ the transformed of $\nabla'$ under $(U_1, U_2)$ and $\hat{D}^{\prime\mu} = U_2 \hat{D}^\mu U_1^\dag$ the transformed of $\hat{D}^\mu$. We assume that $A^\prime_\mu$ inside $\nabla^\prime_\mu$ transforms correctly as a gauge field, so that $$\nabla^\prime [A^\prime]_\mu \phi' = \nabla^\prime [A^\prime]_\mu U_1^\dag \phi'' = U_1^\dag \nabla'' [A^\prime]_\mu \phi'' = U_1^\dag \nabla^\prime [A^\prime_{U1}]_\mu \phi''$$ $$\phi'' = U_1 \phi' .$$ We want $\hat{D}^{\prime\mu}$ remain diagonal and $h' = h[\hat{D}'] = h[\hat{D}]$. In this case there are two relevant possibilities: 1. $\hat{D}$ is a matrix made by blocks $m \times m$ with $m$ integer divisor of $n$ and every block proportional to identity. In this case the residual symmetry is $U(1,\mathbf{H})^n \times U(m,\mathbf{H})^{n/m}$ with elements $(sV, V)$, $s$ both diagonal and unitary, $V \in U(m,\mathbf{H})^{n/m}$; 2. $h$ is any diagonal matrix. The symmetry reduces to $U(1,\mathbf{H})^n \otimes U(1,\mathbf{H})^n$ which is local $U(1,\mathbf{H}) \otimes U(1,\mathbf{H}) \sim SU(2) \otimes SU(2) \sim SO(4)$. In this way, if we keep fixed the metric $h$ and keep diagonal $\hat{D}$, the new action will be invariant at least under $U(1,\mathbf{H})^n \otimes U(1,\mathbf{H})^n$ which doesn’t modify $h$. Note however that the action (\[ultima2\]) is highly non local, because the fields $A_\mu(x^a, x^b)$ with $a \neq b$ can relate couples of vertices very far each other. In fact the transformations in $U(n,\mathbf{H})$ mix all the vertices in the universe independently from their position. In the next section we’ll discover in what limit (besides $\Delta \ra 0$) the (\[ultima2\]) becomes a local action. Let us now pause on the metric $h^{\mu\nu}$. We observe how the metric $h$ has appeared from nowhere. We get the impression” that the metric does not exist a priori”, but is generated by the matrices $\hat{D}$. In other words: the metric is simply the result of our desire to see an ordered universe at any cost. Note that we have chosen the matrix $\nabla$ between skew hermitian matrices, so that the gauge fields $AR_i$ have real eigenvalues, corresponding to effectively measurable quantities[^6]. Conversely, $\hat{M}$ must remain generically normal. In fact, if $\hat{M}$ was (skew) hermitian, the fields $d$ would become (imaginary) real, and there would not be enough degrees of freedom to construct the metric $h$. We focus on the relationship: $$\sqrt{\left\vert h\right\vert }h^{\mu\nu}\left( x_{i}\right) = \fr 12 d^{\mu *}d^{\nu}\left( x_{i}\right) +c.c. \label{eq: metrica_ord}$$ We set: d =( \[c\][c]{}a\_[0]{}+ib\_[0]{}+jc\_0+kd\_0\ a\_[1]{}+ib\_[1]{}+jc\_1+kd\_1\ a\_[2]{}+ib\_[2]{}+jc\_2+kd\_2\ a\_[3]{}+ib\_[3]{}+jc\_3+kd\_3 ) It’s easy to see how $s$-invariance permits us to choose the $D_\mu$ in such a way that the real vectors [1.3]{} ( \[c\][c]{}a\_[0]{}\ a\_[1]{}\ a\_[2]{}\ a\_[3]{} ), ( \[c\][c]{}b\_[0]{}\ b\_[1]{}\ b\_[2]{}\ b\_[3]{} ), ( \[c\][c]{}c\_0\ c\_1\ c\_2\ c\_3 ), ( \[c\][c]{}d\_0\ d\_1\ d\_2\ d\_3 ) will be linearly independent. h\^[-1]{} &=& ( \[c\][cc]{} a\_[0]{}\^[2]{}+b\_[0]{}\^[2]{}+c\_0\^2+d\_0\^2 & a\_[0]{}a\_[1]{}+b\_[0]{}b\_[1]{}+c\_0 c\_1+d\_0 d\_1\ a\_[1]{}a\_[0]{}+b\_[1]{}b\_[0]{}+c\_1 c\_0+d\_1 d\_0 & a\_1\^2+b\_1\^2+c\_1\^2+d\_1\^2\ a\_2 a\_0+b\_2 b\_0+c\_2 c\_0 +d\_2 d\_0 & a\_2 a\_1+b\_2 b\_1 +c\_2 c\_1+d\_2 d\_1\ a\_3 a\_0+b\_3 b\_0+c\_3 c\_0+d\_3 d\_0 & a\_3 a\_1+b\_3 b\_1+c\_3 c\_1+d\_3 d\_1 .\ && . \[c\][cc]{} a\_[0]{}a\_[2]{}+b\_[0]{}b\_[2]{}+c\_0 c\_2+d\_0 d\_2 & a\_[0]{}a\_[3]{}+b\_[0]{}b\_[3]{}+c\_0 c\_3+d\_0 d\_3\ a\_1 a\_2 + b\_1 b\_2+c\_1 c\_2+d\_1 d\_2 & a\_1 a\_3 +b\_1 b\_3+c\_1 c\_3+d\_1 d\_3\ a\_2\^2+b\_2\^2+c\_2\^2+d\_2\^2 & a\_2 a\_3 + b\_2 b\_3 + c\_[2]{}c\_[3]{}+d\_[2]{}d\_[3]{}\ a\_3 a\_2+b\_3 b\_2 +c\_3 c\_2+d\_3 d\_2 & a\_3\^2+b\_3\^2+c\_3\^2+d\_3\^2 ) Note that we have $10$ independent metric components as it should be. What would have happened if the entries of $M$ were been simply complex numbers? In that case we could always take a one-form $X_\nu$ such that $X_\nu (Im$ $d^\nu) = X_\nu (Re\,d^\nu) = 0$. The contraction of $X_\nu$ with the metric would be $$\sqrt h h^{\mu\nu} X_\nu = d^{*\mu} (d^\nu X_\nu) + d^\mu (d^{*\nu} X_\nu) = 0 .$$ Hence the metric would be degenerate. For $d^\mu \in \mathbf{H}$ this can’t happen, because no one-form can be orthogonal to $4$ vectors linearly independent in a $4$-dimensional space. Moreover a such one-form exists in spaces with dimension $>4$. For this reason our theory hasn’t meaning in presence of extra dimensions. A local action from the quaternionic field action {#local} ------------------------------------------------- Here we expose how to get a local action from the quaternionic field action in the limit of low energy. We can add to action quadratic $\sim M^2$ and quartic $\sim M^4$ terms, provided they are gauge invariant. In general we obtain a non-trivial potential of form $\alpha M^4 - \beta M^2$. We suppose that a minimum for such potential breaks the symmetry $U(n,\mathbf{H}) \otimes U(n,\mathbf{H})$ and provides a mass to gauge fields $A_\mu$. To view it is sufficient to rewrite $M$ as a function of $A_\mu$ and consider a quartic term: $$h^{\mu\alpha}A_{\mu}A_{\alpha}h^{\nu\beta}A_{\nu}A_{\beta} . \label{eq: quartico}$$ For a minimum of $M$ there is a minimum of $A$ which gives sense to the expansion: $$A_{\mu}=A_{\mu}^{\min}+\delta A_{\mu} .$$ Therefore the (\[eq: quartico\]) generates a factor: $$m\left( x\right) ^{2}h^{\nu\beta}A_{\nu}A_{\beta}$$ $$m\left( x\right) ^{2} = h^{\mu\alpha}A_{\mu}^{\min}A_{\alpha}^{\min}$$ Hence the gauge fields acquire a mass, varying from point to point in the universe and essentially dependent on the metric. Given a potential for $M$, which is both hermitian and invariant for $U(n,\mathbf{H})\otimes U(n,\mathbf{H})$, his minimum configurations are always invariant at least for $U(1,\mathbf{H})^n \otimes U(1,\mathbf{H})^n$, that is a local $U(1,\mathbf{H}) \otimes U(1,\mathbf{H})$. A such potential contains only terms of type $tr((MM^\dag)^j)$, $j \in \mathbf{N}$. All we can measure are eigenvalues of hermitian operators, and a hermitian operator has only real eigenvalues $q$ which are invariant under $U(1,\mathbf{H})^n$, ie $sqs^* = qss^* =q$ for $|s| =1$. The simpler hermitian operators made by $M$ are $MM^\dag$ and $M^\dag M$, whose eigenvalues are invariant under $$M \ra s_1 M s_2^*\qquad (s_1, s_2) \in U(1,\mathbf{H}) \otimes U(1,\mathbf{H})$$ $$MM^\dag \ra s_2 M s_1^* s_1 M^\dag s_2^* = s_2 MM^\dag s_2^*$$ $$M^\dag M \ra s_1 M^\dag s_2^* s_2 M s_1^* = s_1 M^\dag M s_1^*$$ In this manner we have always $m =0$ for diagonal fields $A_\mu (x^a, x^a) \overset{!}{=} A_\mu (x^a)$. A transformation $(s_1,s_2) \in U(1, \mathbf{H})\otimes U(1, \mathbf{H})$ acts inside action in the expected way (see formula (\[transf\])) $$\phi' \ra s_1 \phi' \overset{!}{=} \phi''$$ $${\nabla'}[A']_\mu \ra s_1 {\nabla'}[A']_\mu s_1^* = {\nabla'}[A^\prime_{s1}]_\mu$$ $$d^\mu \ra s_2 d^\mu s_1^*$$ S\[’, A’\] = S’\[”, A\^\_[s1]{}\] = \_(s\_2 d\^s\_1\^\* \^\_”)\^(s\_2 d\^s\_1\^\* \^\_”) \[locale1\] We use the natural correspondence (1, i, j, k) i(\^0, \^1, \^2, \^3), \^0 = -i, \[correspondence\]and define the complex field $\hat{\phi}$ as a complex $2 \times 2$ matrix, $$\hat{\phi}^a = \left( \begin{array}[c]{cc} \phi_1^a +i\phi_2^a & \phi_3^a+i\phi_4^a \\ -\phi_3^a +i\phi_4^a & \phi_1^a -i\phi_2^a \\ \end{array} \right)$$ with $\phi'' = \phi_1 + i \phi_2 +j\phi_3 +k \phi_4$ and $\phi_1, \phi_2, \phi_3, \phi_4 \in \mathbf{R}$. Every term between parenthesis becomes W\_2 (i\^k) d\_k\^W\_1\^ \[A\^\_[s1]{}\]\_ \[SU2\]where $\s$ are Pauli matrices and $(W_1,W_2) \in SU(2)\otimes SU(2)$. For every $SO(4)$ transformation $\Lambda$, a transformation $(W_1,W_2) \in SU(2) \otimes SU(2)$ exists, such that for every vector $d_j \in R^4$ we find $$\Lambda_i^{\pt j} d_j \sigma^i = d_i W_2 \sigma^i W_1^\dag .$$ We write $W_1 = U^{\prime\dag}_1 U_1$ and $W_2 = U^\prime_1 U_1$. In this manner we decompose $SU(2)\otimes SU(2)$ in $SU(2)_{rot} \otimes SU(2)_{boosts}$. $SU(2)_{rot}$ is generated by the couples $(U_1,U_1)$, while $SU(2)_{boosts}$ by the couples $(U^{\prime\dag}_1, U^{\prime}_1)$. After a wick rotation, the first one describes rotation in $R^3$, while the second one describes boosts. A generic vector $d = \left( \begin{array}[c]{cccc} d_0 & d_1 & d_2 & d_3 \end{array} \right)$ gives $$d_i (i\sigma^i) = \left( \begin{array}[c]{cc} d_0 + id_3 & id_1+d_2 \\ id_1-d_2 & d_0-id_3 \end{array} \right)$$ with $|d|^2 = det\,d_i (i\sigma^i)$. A transformation in $SO(4)$ doesn’t change the norm $|d|$. Moreover, for every $d$ exists a transformation in $SO(4)$ which put it in the normal form $$d = \left( \begin{array}[c]{cccc} |d| & 0 & 0 & 0 \end{array} \right) .$$ The same properties have to be true for $SU(2) \otimes SU(2)$. The first one is banally verified because $det\,W_1 = det\,W_2 = 1$ and then $det\, d_i(i\sigma^i) = det\,d_i(W_2 i\sigma^i W_1^\dag)$. Being $d_i(i\sigma^i)$ normal, we can use a transformation in $SU(2)_{rot}$ to put it in a diagonal form $$U_1 d_i (i\sigma^i) U_1^\dag = \left( \begin{array}[c]{cc} d_0 + id_3 & 0 \\ 0 & d_0-id_3 \end{array} \right) .$$ Define now the matrix $U^\prime_1$ as $$U^\prime_1 = \fr 1 {\sqrt{|d|}}\left( \begin{array}[c]{cc} \sqrt{d_0 + id_3} & 0 \\ 0 & \sqrt{d_0-id_3} \end{array} \right) .$$ It’s easy to verify that $U^\prime_1 U^{\prime\dag}_1 = 1$ and $det\,U^\prime_1 = 1$. Applying to $U_1 d_i (i\sigma^i) U_1^\dag$ this transformation in $SU(2)_{boosts}$ we obtain $$U_1^\prime U_1 d_i (i\sigma^i) U_1^\dag U_1^\prime = \left( \begin{array}[c]{cc} |d| & 0 \\ 0 & |d| \end{array} \right) .$$ So, for every $d$, a transformation in $SU(2) \otimes SU(2)$ exists, which puts it in the normal form. In this way, $d$ transforms exactly as a vielbein field in the Palatini formulation of General Relativity, giving then the correspondence \_i\^[j]{} d\_j (i\^i) &=& U\^\_1 U\_1 (i\^j) U\_1\^U\^\_1 d\_j\ \_i\^[j]{} d\_j \^i &=& W\_2 \^j W\_1\^d\_j\ 12 tr (\_i\^[j]{} d\_j \^i \^k) &=& 12 tr (W\_2 \^j W\_1\^\^k) d\_j\ \_k\^[j]{} d\_j &=& 12 tr (W\_2 \^j W\_1\^\^k) d\_j\ \_k\^[j]{} &=& 12 tr (W\_2 \^j W\_1\^\^k) .So, at every $\Lambda \in SO(4)$ corresponds a couple $(U_1, U_2) \in SU(2) \otimes SU(2)$. Applying this to (\[SU2\]), it becomes W\_2 (i\^k) d\_k\^W\_1\^ \[A\^\_[s1]{}\]\_ = \_k\^[i]{} d\_i\^\^k [’]{}\[A\^\_[s1]{}\]\_ \[SU2-second\] . Note that if we write $\hat{\phi} = (\hat{\phi}_1\,\,\,\hat{\phi}_2)$, with $\hat{\phi}_1, \hat{\phi}_2$ complex column arrays $1 \times 2$, then $\hat{\phi}_2 = i\s_2 \hat{\phi}_1^*$. This implies that the column array $1 \times 4$ $\left( \begin{array}[c]{c} \hat{\phi}_1 \\ \hat{\phi}_2 \\ \end{array} \right)$ transforms under $SO(4)$ as a Majorana spinor. Applying newly the correspondence (\[correspondence\]) to (\[SU2-second\]), we obtain $$s_2 d^\mu s_1^* \nabla^\prime [A^\prime_{s1}]_\mu \phi'' = \Lambda d^\mu {\nabla'} [A^\prime_{s1}]_\mu \phi''.$$ Inserting it in the action (\[locale1\]) S’\[A\^\_[s1]{},”\] &=& \_([’]{} \[A\^\_[s1]{}\]\_”)\^d\^[\*]{} \^ d\^([’]{} \[A\^\_[s1]{}\]\_”)\ &=& \_([’]{} \[A\^\_[s1]{}\]\_”)\^d\^[\*]{} d\^([’]{} \[A\^\_[s1]{}\]\_”)\ &=& \_(d\^[’]{} \[A\^\_[s1]{}\]\_”)\^(d\^ \[A\^\_[s1]{}\]\_”)\ &=& S\[A\^\_[s1]{},”\] . The diagonal gauge field $A(x^a)$ compensates the action of $SU(2)\otimes SU(2)$ inside $\nabla'$. Moreover we have just demonstrated that the field $d^\mu$ transforms under this group as a vielbein field in the Palatini formulation of General Relativity. This implies $A(x^a)$ is a gravitational spin-connection. Consequently, every purely imaginary quaternion defines a spin operator $\vec{S}$ via the correspondence $(i,j,k) \leftrightarrow 2i(S_1, S_2,$ $S_3)$. In fact, each element in $U(1,\mathbf{H})$ is the exponential of a purely imaginary quaternion, in the same way as an element in $SU(2)$ is the exponential of $i\vec{\a} \cdot \vec{S}$ for some real vector $\vec{\a}$. Note that a majorana spinor in an euclidean space can’t distinguish if $s_2$ belongs to $SU(2)_{rot}$, $SU(2)_{boosts}$ or if it is a mixed combination. Only after the wick rotation it feels a difference, because the generator of $SU(2)_{boosts}$ moves from $i\sigma^i$ to $\sigma^i$, while $SU(2)_{rot}$ remains unchanged. Someone can infer that, if $\phi$ transforms as a majorana spinor, our action has not the standard form. We don’t care this now: what exposed is only a toy model. In another work (under review) we show explicitly how to get the correct Dirac action for these and all the others fields (both fermions and bosons). To finish, we suppose that masses of other fields ($A(x^a, x^b)$ with $a \neq b$) are sufficiently large, so that the experimental physics of nowadays is unable to locate them. For the same reason, in the low energy approximation, they can be omitted from the action. Neglecting the ultra-massive” fields, the scalar field action becomes a local action $$S=\sum_{i=1}^n \sum_{\mu,\nu} \sqrt{\left\vert h\right\vert }h^{\mu\nu}\left( x_i\right) \left( \overset{G}{\nabla}_{\mu}\phi(x_i)\right)^* \left( \overset{G}{\nabla}_{\nu}\phi(x_i)\right)$$ where $\overset{G}{\nabla}$ are standard gravitational covariant derivatives. The origin of spin {#spin} ------------------ Consider the spin operator $S_{3}$ $$\hat{S}_{3} = \frac{\hslash}2\left( \begin{array} [c]{cc}1 & 0\\ 0 & -1 \end{array} \right)$$ and calculate the normalized eigenvectors and eigenstates. $$\begin{aligned} \left\vert \uparrow\right\rangle & =e^{i\phi}\left( \begin{array} [c]{c}1\\ 0 \end{array} \right) \text{, \ with eigenvalue }\lambda_{1}=+\frac{1}{2}\text{ \ (in unit }\hslash=1\text{)}\label{eq: autketS3}\\ \left\vert \downarrow\right\rangle & =e^{i\phi}\left( \begin{array} [c]{c}0\\ 1 \end{array} \right) \text{, \ with eigenvalue }\lambda_{2}=-\frac{1}{2}\text{ \ }\end{aligned}$$ where $\phi$ is an arbitrary phase. The eigenvectors completeness guarantees that the field $\hat{\phi}_1$, which appears in the precedent section, can be always decomposed in a sum of such eigenstates. The projectors on a single eigenstate of $S_{3}$ are $$\hat{\pi}^{+} =\frac{1}{2}\left( \begin{array} [c]{cc} 1 & 0 \\ 0 & 0 \end{array} \right) ,$$ $$\hat{\pi}^{-} =\frac{1}{2}\left( \begin{array} [c]{cc} 0 & 0 \\ 0 & 1 \end{array} \right) .$$ We see that $\hat{\pi}^{\pm}$ are idempotent, while $\hat{\pi}^{+}\hat{\pi }^{-}=0$, as it should be. A rotation by an angle $\theta$ around the axe $1$ is represented by the unitary matrix: $$U_{1}\left( \theta\right) =\left( \begin{array} [c]{cc} \cos(\theta/2) & -i\sin(\theta/2) \\ -i\sin(\theta/2) & \cos(\theta/2) \end{array} \right)$$ where $$U_1(\theta)\hat{\phi}_1 = \widehat{(s(U_1)\phi)}_1\qquad \hat{\phi}_1=\widehat{(\phi)}_1$$ for some quaternion $s(U)$ with $|s|=1$. In the special case of a rotation by $\pi$: $$U_{1}\left( \pi\right) =\left( \begin{array} [c]{cc} 0 & -i \\ -i & 0 \end{array} \right) \label{eq: rotazione}.$$ We suppose now that the system is in the eigenstate $\left\vert \uparrow\right\rangle $; following a rotation around the axis $1$ the state will be: $$\left\vert \uparrow\right\rangle _{R}= U_{1}(\theta)\left\vert \uparrow \right\rangle .$$ For $\theta=\pi$: $$\left\vert \uparrow\right\rangle _{R}= U_{1}\left( \pi\right) \left\vert \uparrow\right\rangle = -i\left\vert \downarrow\right\rangle =e^{-i\pi /2}\left\vert \downarrow\right\rangle \rightarrow\left\vert \downarrow \right\rangle \text{,} \label{eq: scambio_spin}$$ $$\left\vert \downarrow\right\rangle _{R}= U_{1}\left( \pi\right) \left\vert \downarrow\right\rangle = -i\left\vert \uparrow\right\rangle =e^{-i\pi /2}\left\vert \uparrow\right\rangle \rightarrow\left\vert \uparrow \right\rangle \text{,} \label{eq: scambio_spin2}$$ since the state is defined up to an inessential phase factor. We observe that a rotation by $\pi$ around the axe $1$ is equivalent to exchange $\left\vert \uparrow\right\rangle $ with $\left\vert \downarrow\right\rangle $, as we have just verified by (\[eq: scambio\_spin\]) and (\[eq: scambio\_spin2\]). Surely we can expand the matrix $M$ as follows $$M\left( x^{a},x^{b}\right) = M'\left( x^{a},x^{b}\right) + |s(x^a)|e^{r(x^a)} \d^{ab}$$ with $M'\left( x^{a},x^{b}\right) = 0$ for $a=b$. The element $r(x^a) = arg[s(x^a)]$ is a purely imaginary quaternion: when it acts on $\phi$, it determines uniquely the result of a spin measure, exchanging the states $\left\vert \uparrow\right\rangle $ - $\left \vert\downarrow\right\rangle $. This seems to suggest an identification between the arrangement field $M$ and the observer who performs the measurement. Indeed the operator $M$ can simulate a measurement operation when it presents the form $M^{ab} = u^a w^b$: $$\begin{aligned} M^{ab} &=& u^a w^b \overset{continuous}{\longrightarrow} M(x,y) = \psi(x) \psi^{\ast}(y) \nonumber \\ M^{ab}\varphi_b &=& u^a (w^b \varphi_b) \overset{continuous}{\longrightarrow} \int dy \, M(x,y) \varphi(y) \nonumber \\ &=& \psi(x)\int dy \, \psi^{\ast}(y)\varphi(y) = \psi(x)(\psi,\varphi) \nonumber\end{aligned}$$ $\psi\left( x\right)$ is any eigenstate, while $\left( \psi ,\varphi\right)$ denotes the scalar product between $\psi$ and $\varphi$. We see that $M$ projects $\varphi$ along the eigenstate $\psi$, and in quantum mechanics a measurement is just a projection. The latter argument gives also an indication about the spin nature. Consider the entries of $M$ closest to the diagonal: they are the $M^{ij+1}$ and $M^{ij-1}$ which compose $\tilde{M}$. Moreover, they represent the probability amplitudes for the existence of connections between (numerically) consecutive vertices. In the limit $\Delta \ra 0$, $\tilde{M}$ becomes $\pa$, which is proportional to $i\pa$, an operator which acts on a wave function $\psi(x)$ and returns the momentum $p$ of the corresponding particle: $$i\pa \psi(x) = p\psi(x) .$$ In this way, the entries of $\tilde{M}$ represent both a momentum and a probability amplitude for connections between (numerically) consecutive vertices. In a certain sense, $\tilde{M}$ draws continuous paths and measures the momentums along these paths (figure \[percorso\_continuo\]). If we describe a particle with a wave function $\phi$, its spin is determined by diagonal components of $M$: in fact, $exp(r)$ acts on $\phi$ as a rotation in the tangent space. Consequently, if $r$ is applied to $\phi$, it returns the spin of the associated particle. The diagonal components of $M$ represent also the probability amplitudes for a connection between a vertex and itself. Reasoning in analogy with the components of $\tilde{M}$, we associate at every such pointwise” loop a circumference $S^1$: we interpret the spin as the rotational momentum due to the motion along these circumferences (figure \[Loop\_1\]). ![$\tilde{M}$ behaves as a derivative, that is proportional to a momentum operator. The non-empty entries of $\tilde{M}$ represent both a momentum and a probability amplitude for connections between (numerically) consecutive vertices. In a certain sense, $\tilde{M}$ draws continuous paths and measures the momentums along them.[]{data-label="percorso_continuo"}](fig5.jpg){width="60.00000%"} ![Each diagonal component of $M$ represents the probability amplitude for a connection between a vertex and itself. The spin is a momentum along such pointwise loops.[]{data-label="Loop_1"}](fig6.jpg){width="60.00000%"} It is remarkable that there exist two types of pointwise loops: the one in figure \[Loop\_1\], where a particle assumes the same aspect after a complete rotation, and the one in figure \[Loop\_2\], where a particle assumes the same aspect after two complete rotations. The first case suggests a relationship with gauge fields of spin $1$, the second with fermionic fields of spin $1/2$. ![Pointwise loop associable with fermionic field.[]{data-label="Loop_2"}](fig7.jpg){width="60.00000%"} Symmetry breaking {#symmetry} ----------------- We imagine that the symmetry breaking of $U(n,\mathbf{H})\otimes U(n,\mathbf{H})$ is not complete, but a residual symmetry remains for transformations in $U(1,\mathbf{H})^n \times U(m,\mathbf{H})^{n/m}$. Here $m$ is an integer divisor of $n$. In this case, it is possible to regroup the $n$ points into $n/m$ ensembles $\mathcal{U}^a$, with $a = 1, 2, \ldots, n/m$. $$\mathcal{U}^a = \mathcal{U}^a (x^a_1, x^a_2, \ldots, x^a_m)$$ $$\varphi = (\varphi (x^a_i)) = \left( \begin{array}[c]{ccccc} \varphi(x^1_1) & \varphi(x^1_2) & \varphi(x^1_3) & \ldots & \varphi(x^1_m) \\ \varphi(x^2_1) & \varphi(x^2_2) & \varphi(x^2_3) & \ldots & \varphi(x^2_m) \\ \varphi(x^3_1) & \varphi(x^3_2) & \varphi(x^3_3) & \ldots & \varphi(x^3_m) \\ \vdots & \vdots & \vdots & \vdots & \vdots\\ \varphi(x^{n/m}_1) & \varphi(x^{n/m}_2) & \varphi(x^{n/m}_3) & \ldots & \varphi(x^{n/m}_4) \end{array} \right)$$ $$A = (A^{ab}_{ij}) = (A(x^a_i, x^b_j)) .$$ Now the indices $a,b$ of $A$ act on the columns of $\varphi$, while the indices $i,j$ act on the rows. The fields $A^{ab}_{ij}$ with $a=b$ maintain null masses and so they continue to behave as gauge fields for $U(m,\mathbf{H})^{n/m}$. Every $U(m,\mathbf{H})$ term in $U(m,\mathbf{H})^{n/m}$ acts independently inside a single $\mathcal{U}^a$. So, if we consider the ensembles $\mathcal{U}^a$ as the real physical points, we can interpret $U(m,\mathbf{H})^{n/m}$ as a local $U(m,\mathbf{H})$. It’s simple to verify: h\^(x\^a\_i) &=& h\^(x\^a\_j) x\_i, x\_j \^a\ h(x\^a) && h(\^a) = h\^(x\^a\_i) x\^a\_i \^a\ A(x\^a\_i, x\^a\_j) &=& Tr, where\ A(x\^a) &=& \_[ij]{} A(x\^a\_i, x\^a\_j) T\^[(ij)]{}, with $T^{(ij)}$ generator of $U(m,\mathbf{H})$. Using these relations, in the next work we’ll show how the terms $tr\,(MM^\dag)$ and $tr\,(MM^\dag MM^\dag)$ generate respectively the Ricci scalar and the kinetic term for gauge fields. Extending $M$ to grassmanian elements we have (up to a generalized $U(n, \mathbf{H})$ transformation) $$M = \theta(\pa + \psi) + d^\mu (\pa_\mu + A_\mu)$$ $$M^\dag = (\pa^\dag +\psi^\dag)\theta^\dag + (\pa_\mu^\dag + A^\dag_\mu)d^{*\mu} .$$ $\theta, \theta^\dag$ are at the same time grassmanian coordinates and grassmanian equivalents of $d, d^*$. $\pa, \pa^\dag$ are grassmanian derivatives and $\psi, \psi^\dag$ grassmanian fields (ie fermions). Our final action will be $$S = tr\,\left(\fr {MM^\dag}{16\pi G}-\fr 1 4 MM^\dag MM^\dag \right)$$ This action resembles the action of a $\lambda \phi^4$ theory. Some preliminary results suggest that we can treat it by means of Feynman graphs, apparently without renormalization problems. We will see how the quartic term includes automatically the kinetic terms for gauge fields of $SO(4) \otimes SU(3)\otimes SU(2) \otimes U(1)$ and the dirac action for exactly three fermionic families. Second quantization and black hole entropy {#entropy} ------------------------------------------ It is remarkable that in our model the gauge fields and the gravitational fields have different origins, although they are both born from $M$. The gravitational field in fact appears as a multiplicative factor for moving from $M$ to the covariant derivative $\nabla'$. The gauge fields are instead some additive elements in $\nabla'$. This could be the reason for which the gravitational field seems non quantizable in the standard way. On the other side, quantizing the gauge fields is equivalent to quantize a partial piece of $M$ in a flat space. But a similar equivalence does not exist for the gravitational field. In our framework this doesn’t create problems, since we will quantize $M$ directly, rather than gravitational and gauge fields. What does it mean to quantize” $M$? It’s true that a matrice $M$ is a quantum object from its birth, as they are quantum objects the wave functions which describe particles. However, we will impose commutation relations on $M$, in the same way we impose commutation relations on the wave functions. This is the so called second quantization”. The wave functions, which first had described the probability amplitude to find a particle, then have become operators which create or annihilate particle. Similarly, $M$ describes first the probability amplitude for the existence of connections between vertices. After the second quantization it will become an operator which creates or annihilate connections. In particular, the operator $M(x^a,x^b)$ creates a connection between the vertices $x^a$ and $x^b$. $M$ corresponds to $D^\mu\nabla^\prime_\mu$ (by invariance respect $U(n,\mathbf{H})$): so it contains the various fields $A_\mu$ and $h^{\mu\nu}$. If we second quantize $M$, then, indirectly, we quantize the other fields, including the gravitational field. To quantize $M$ we put $[M^{ij}, M^{kl\dag}]=\d^{ik}\d^{jl}$. Here the symbol $\dag$ indicates the adjoint operator respect only the scalar product between states in the Fock space. The condition $[M^{ij}, M^{kl\dag}]=\d^{ik}\d^{jl}$ means that every entry $M^{ij}$ expands in a sum of $4$ operators $$M^{ij} = a + i(b_1 +b_2+ b_3)\qquad\quad b_1^\dag = b_1,\,\,\,b_2^\dag = b_2,\,\,\,b_3^\dag = b_3$$ The $b$’s realize the $SU(2)$ algebra implicit in the imaginary part of quaternions. $$[b_1,b_2]=b_3 ;\qquad [b_2,b_3] = b_1;\qquad [b_3,b_1]=b_2$$ The operators $a^\dag$ and $b^\dag = b_1 + ib_2$ create an edge which connects the vertex $i$ with the vertex $j$. The number operator is $$N^{ij} = M^{ij\dag} M^{ij} = a^\dag a + |\vec{b}|^2\qquad \text{no sum on} \pt ij$$ $a^\dag a$ has eigenvalues $q \in \mathbf{N}$ with multiplicity $1$. Moreover the eigenvalues of $|\vec{b}|^2$ are in the form $j(j+1)$ for $j \in \mathbf{N}/2$, with multiplicity $(2j+1)$. How about $N > 1$? We can consider a surface immersed into the graph. Its area is $\Delta^2$ times the number of edges which pass through it. If we admit the possibility for the creation of many superimposed edges, we can interpret this superimposition as a super-edge” which carries an area equal to $N \Delta^2$. Regarding diagonal components, we suggest a slightly different interpretation: $a^\dag$ could create loops, while $b^\dag$ could create perturbations which travel through the loops (ie particles with spin $j$). This suggest a duality between a loop on vertex $v_i$ and a closed string (as intended in *String Theory*) situated approximately on the same vertex. Note that the two interpretations can be accommodated if we consider quanta of area as non-local perturbations. The only Black Hole information detectable from the exterior, is the information coded in the Horizon. So, the only distinguishable states of a Black Hole are distinguishable states of its horizon. For the Black Hole horizon we consider all the edges which pass through it, oriented only from the interior to the exterior. If the horizon is crossed only by edges with $N = q+j(j+1)$ and $a^\dag a =q$, the number of its distinguishable states is $$num_S = \left( 2j+1 \right)^{A /(q+j(j+1))} .$$ We suppose now a generic partition with $A = \sum_{j,q} A_{j,q}$, where an area $A_{j,q}$ is crossed only by edges with $N=q+j(j+1)$ and $a^\dag a =q$. The number of distinguishable states becomes $$num_S = \sum_{\{A_{j,q}\}}\prod_{j,q} \left( 2j+1 \right)^{A_{j,q} /((q+j(j+1))\Delta^2)}$$ where the sum is over all the possible partitions of $A$. The classical” contribution comes from $j=0$ and gives $num_S =1$ (We call it classical” because it is the only one with $N =1$). This implies no entropy and is related to the fact that $tr \,M_H^\dag M_H \sim \int_H \sqrt {h_H} R(h_H) = 2\pi \chi_H$, where $M_H$ is the restriction of $M$ to the edges which cross the horizon, $h_H$ is the induced metric on the horizon and $\chi$ is the Euler characteristic. The dominant contribution comes from $q=0$ and $j = 1/2$, which gives $$num_S = 2^{4A_{1/2,0} /(3\Delta^2)}$$ So we can define entropy as $$S = k_B\,log\,2^{4A/{3\Delta^2}} = \fr {4 \,log\,2 \, k_B A }{3\Delta^2}.$$ Our approach gives thus a proposal for the explanation of area law. Indeed our entropy formula corresponds to the one given by Bekenstein and Hawking if $3\Delta^2 = 16 G\,log\,2$. What is our interpretation of black hole radiation? The proximity between vertices is probabilistic: we can have a high probability of receiving two vertices as neighbors”, but never a certainty. We look at a large number of vertices for a long time: some vertex, which first seems to be adjacent to some other, suddenly can appears far away. For this reason, some internal vertices in a Black Hole may happen to be found outside, so that the Black Hole slowly evaporates. We can consider also the contribution from ($q = j = 0$). If it exists, clearly it is the dominant one. Indeed, an horizon means absence of connections between the exterior and the interior. For an external observer, the universe finishes with the horizon. In fact, respect the coordinate system of a statical observer infinitely distant from the horizon, every object, falling in the black hole, sits on the horizon for an infinite time. In relation to the proper time of the statical far away observer, the object never surpasses the horizon. If nothing surpasses the horizon, this means that the Hawking radiation comes from the deposit of all the objects fallen in the black hole, ie from the horizon. This resolves the information paradox proposed by Hawking. Someone can infer that absence of connections is only illusory, because the horizon singularity is of the type called apparent”: it doesn’t exist in several coordinate systems, as the system comoving with a free falling object. We reply that it’s true, because also the absence of connections depends strictly from the state on which the number operator acts. Every state can be associated to a particular coordinates system and, if we change coordinate system, we have to change the state. In this way, the connections can exist for an observer and not exist for some others. It’s the same which happens for the particles. The same particle can exist in a coordinate system and not exists in an another system (see Unruh effect). This is because the same number operator acts on different states. Calculate now $num_S$ for $q = 0, j \ra 0$. It is num\_S &=& \_[j0]{} ( 2j+1 )\^[A /(j(j+1)\^2)]{}\ &=& \_[j0]{} ( 1+2j )\^[A /(j\^2)]{}\ &=& \_[j0]{} ( 1+2j )\^[2A /(2j\^2)]{}\ &=& \_[x]{} ( 1+1x )\^[2Ax/\^2]{}\ &=& e\^[2A/\^2]{} The entropy becomes $$S =k_B \,log\,e^{2A/\Delta^2} = \fr {2k_B A}{\Delta^2}$$ This corresponds to the Bekenstein-Hawking result for $\Delta^2 = 8G$. Conclusion ---------- In this paper we have abandoned the preconceived existence of an order in the space-time structure, taking a probabilistic approach also to its topology and its homology. This framework gives new suggestions about the origin of space-time metric and particles spin. At the same time it hints a possible emersion of all fields from an unique entity, ie the arrangement matrix, after the imposition of an order. Unfortunately, there isn’t space here to post an explicit calculation of terms $tr\,(M^\dag M)$ and $tr\,(M^\dag M M^\dag M)$. We have already said that they generate the Ricci scalar, the kinetic terms for gauge fields and the Dirac actions for exactly three fermionic families. In a next future we’ll show how several phenomena can find a possible explanation inside this paradigm, as we have seen earlier for black hole entropy. These deal with the galaxy rotation curves, the inflation, the quantum entanglement, the values of matrices CKM and PMNS and the value of Newton constant $G$. Here we have given a simple example by using a one-component field. Nevertheless, a potential for $M$ causes a symmetry breaking which gives mass to gauge fields without need of Higgs mechanism. In the end, the one-component field action results unnecessary. Acknowledgements {#acknowledgements .unnumbered} ---------------- I thank professor Valter Moretti, Dr. Fabrizio Coppola and Dr. Marcello Colozzo for the useful discussions and suggestions. The arrangement field theory (AFT). Part 2 ========================================== Introduction ------------ The arrangement field paradigm describes universe by means of a graph, ie an ensemble of vertices and edges. However there is a considerable difference between this framework and the usual modeling with spin-foams or spin-networks. The existence of an edge which connects two vertices is in fact probabilistic. In this framework the fundamental quantity is an invertible matrix $M$ with dimension $n \times n$, where $n$ is the number of vertices. In the entry $ij$ of such matrix we have a quaternionic number which gives the probability amplitude for the existence of an edge connecting vertex $i$ to vertex $j$. In the introductory work [@Arrangement] we have developed a simple scalar field theory in this probabilistic graph (we call it non-ordered space”). We have seen that a space-time metric emerges spontaneously when we fix an ensemble of edges. Moreover, the quantization of metric descends naturally from quantization of $M$ in the non-ordered space. In section \[formalism\] we summarize these results. In section \[ricciscalar\] we express Ricci scalar as a simple quadratic function of $M$. We discover how the gravitational field emerges from diagonal components of $M$, in contrast to gauge fields which come out from non-diagonal components. In section \[kinetic\] we define a quartic function of $M$ which develops a Gauss Bonnet term for gravity and the usual kinetic term for gauge fields. In section \[string\] we discover a triality between *Arrangement Field Theory*, *String Theory* and *Loop Quantum Gravity* which appear as different manifestations of the same theory. In section \[electroweak\] we show that a grassmanian extension of $M$ generates automatically all known fermionic fields, divided exactly in three families. We see how gravitational field exchanges homologous particles in different families. The resulting scheme finds an analogue in supersymmetric theories, with known fermionic fields which take the role of gauginos for known bosons. In the subsequent sections we explore some practical implications of arrangement field theory, in connection to inflation, dark matter and quantum entanglement. Moreover we explain how deal with theory perturbatively by means of Feynman diagrams. We warmly invite the reader to see introductory work [@Arrangement] before proceeding. Formalism --------- In paper [@Arrangement] we have considered an euclidean $4$-dimensional space represented by a graph with $n$ vertices. In this section we retrace the fundamental results of that work, moving to Lorentzian spaces in the next section. Since now we assume the Einstein convention, summing over repeated indices. In proof of **theorem 8** in [@Arrangement] we have demonstrated the equivalence between the following actions: S\_1 = (M )\^(M ) \[iniziale\]S\_2 = \_[i=1]{}\^n h\^ (x\^i) (\_\^i)\^\*(\_\^i). \[action-1\] $M$ is any invertible matrix $n \times n$ while the field $\varphi$ is represented by a column array $1\times n$, with an entry for every vertex in the graph: = ( [c]{} (x\^1)\ (x\^2)\ (x\^3)\ \ (x\^N) ) . The entries of both $M$ and $\phi$ take values in the division ring of quaternions, usually indicated with $\mathbf{H}$. The first action considers the universe as an abstract ensemble of vertices, numbered from $1$ to $n$, where $n$ is the total number of space vertices. The entry $(ij)$ in the matrix $M$ represents the probability amplitude for the existence of an edge which connects the vertex number $i$ to the vertex number $j$. We admit non-commutative geometries, which in this framework implies a possible inequivalence $|M^{ij}| \neq |M^{ji}|$. More, the first action is invariant under transformations $(U_1,U_2) \in U(n,\mathbf{H}) \otimes U(n,\mathbf{H})$ which send $M$ in $U_2 M U_1^\dag$. In action (\[action-1\]) a covariant derivative for $U(n,\mathbf{H}) \otimes U(n,\mathbf{H})$ appears, represented by a skew hermitian matrix $\nabla$ which expands according to $\nabla_\mu = \tilde{M}_\mu + A_\mu$. Here $\tilde{M}_\mu$ is a linear operator such that $lim_{\Delta \ra 0} \tilde{M}_\mu = \pa_\mu$, where $\Delta$ is the graph step. If we number the space vertices along direction $\mu$, $\tilde{M}_\mu$ becomes \^[ij]{}\_= 1 [2]{} \[dderiv\]$$\sum_j \tilde{M}^{ij} \varphi^j = \fr 1 {2\Delta} \sum_j \delta^{(i+1)j} \varphi^j - \delta^{(i-1)j} \varphi^j = \fr {\varphi(i+1) - \varphi(i-1)} {2\Delta} .$$ The gauge fields $A$ act as skew hermitian matrices too: $$A = (A^{ij}) = (A(x^i, x^j))$$ $$(A\phi)^i = A^{ij}\phi^j .$$ In proof of **theorem 5** we have discovered that for every normal matrix $\hat{M}$, which is neither hermitian nor skew hermitian, four couples $(U_1,D^\mu)$ exist, with $U_1$ unitary and $D^\mu$ diagonal, such that U\_1\^D\^\_U\_1 = \[fondamentale\] h\^(x\^i) = 1 2 d\^[\*]{}\_i d\_i\^+c.c. D\^[ij]{}\_= d\^\_i \^[ij]{} .Here $h$ is a non degenerate metric while the first relation determines uniquely the values of gauge fields. The matrices $\nabla_\mu, U_1, D^\mu$ act on field arrays via matricial product and the ensemble of four couples $(U_1, D^\mu)$ is called space arrangement”. Further, in proof of **theorem 6**, we have seen that for every invertible matrix $M$ we can always find an unitary transformation $U_M$ and a normal matrix $\hat{M}$, which is neither hermitian nor skew hermitian, such that $M = U_M \hat{M}$. If we define $U_2 = U_1 U_M^\dag$, we have M\^M = \^ \[matemat\] U\_2\^D\^\_U\_1 = M .\[fond\] It’s sufficient to substitute (\[fond\]) in (\[iniziale\]) to verify its equivalence with (\[action-1\]). We have called $\hat{M}$ the associated normal matrix” of $M$. The action of a transformation $(U_1, U_2)$ on $\nabla$ follows from its action on $M$. We can always use the invariance under $U(n,\mathbf{H}) \otimes U(n,\mathbf{H})$ to put $M$ in the form $M = D^\mu \na_\mu$. Starting from this we have $$U_2 M U_1^\dag = U_2 D^\mu \na_\mu U_1^\dag = U_2 D^\mu U_1^\dag U_1\na_\mu U_1^\dag.$$ We define $\nabla' = U_1\na_\mu U_1^\dag$ the transformed of $\nabla$ under $(U_1, U_2)$ and $D^{\prime\mu} = U_2 D^\mu U_1^\dag$ the transformed of $D^\mu$. We assume that $A_\mu$ inside $\nabla_\mu$ transforms correctly as a gauge field, so that $$\na [A]_\mu \phi = \na [A] U_1^\dag \phi' = U_1^\dag \na [A_{U1}]_\mu \phi'$$ $$\phi' = U_1 \phi .$$ We want $D^{\prime\mu}$ remain diagonal and $h' = h[D'] = h[D]$. In this case there are two relevant possibilities: 1. $D$ is a matrix made by blocks $m \times m$ with $m$ integer divisor of $n$ and every block proportional to identity. In this case the residual symmetry is $U(1,\mathbf{H})^n \times U(m,\mathbf{H})^{n/m}$ with elements $(sV, V)$, $s$ both diagonal and unitary, $V \in U(m,\mathbf{H})^{n/m}$; 2. $h$ is any diagonal matrix. The symmetry reduces to $U(1,\mathbf{H})^n \otimes U(1,\mathbf{H})^n$ which is local $U(1,\mathbf{H}) \otimes U(1,\mathbf{H}) \sim SU(2) \otimes SU(2) \sim SO(4)$. In this way, if we keep fixed the metric $h$ and keep diagonal $D$, the action (\[action-1\]) will be invariant at least under $U(1,\mathbf{H})^n \otimes U(1,\mathbf{H})^n$ which doesn’t modify $h$. We have supposed that a potential for $M$ breaks the $U(n,\mathbf{H})\otimes U(n,\mathbf{H})$ symmetry in $U(1,\mathbf{H})^n \otimes U(m,\mathbf{H})^{n/m}$ where $m$ is an integer divisor of $n$. We’ll see in fact that the more natural potential has the form $tr\,(\a M^\dag M - \b M^\dag MM^\dag M)$, known as mexican hat potential”. This potential is a very typical potential for a spontaneous symmetry breaking. In this way all the vertices are grouped in $n/m$ ensembles $\mathcal{U}^a$: $$\mathcal{U}^a = \{x^a_1, x^a_2, x^a_3, \ldots, x^a_m\}$$ $$\varphi = (\varphi (x^a_i)) = \left( \begin{array}[c]{ccccc} \varphi(x^1_1) & \varphi(x^1_2) & \varphi(x^1_3) & \ldots & \varphi(x^1_m) \\ \varphi(x^2_1) & \varphi(x^2_2) & \varphi(x^2_3) & \ldots & \varphi(x^2_m) \\ \varphi(x^3_1) & \varphi(x^3_2) & \varphi(x^3_3) & \ldots & \varphi(x^3_m) \\ \vdots & \vdots & \vdots & \vdots & \vdots\\ \varphi(x^{n/m}_1) & \varphi(x^{n/m}_2) & \varphi(x^{n/m}_3) & \ldots & \varphi(x^{n/m}_4) \end{array} \right)$$ $$A = (A^{ab}_{ij}) = (A(x^a_i, x^b_j)) .$$ Now the indices $a,b$ of $A$ act on the columns of $\varphi$, while the indices $i,j$ act on the rows. The fields $A^{ab}_{ij}$ with $a=b$ maintain null masses and then they continue to behave as gauge fields for $U(m,\mathbf{H})^{n/m}$. Every $U(m,\mathbf{H})$ term in $U(m,\mathbf{H})^{n/m}$ acts independently inside a single $\mathcal{U}^a$. So, if we consider the ensembles $\mathcal{U}^a$ as the real” physical points, we can interpret $U(m,\mathbf{H})^{n/m}$ as a local $U(m,\mathbf{H})$. It’s simple to verify: $$h^{\mu\nu}(x^a_i) = h^{\mu\nu}(x^a_j) \qquad \forall x^a_i, x^a_j \in \mathcal{U}^a$$ $$h^{\mu\nu}(x^a) \overset{!}{=} h^{\mu\nu}(\mathcal{U}^a) = h^{\mu\nu}(x^a_i) \qquad \forall x^a_i \in \mathcal{U}^a$$ $$A_{ij} (x^a) \overset{!}{=} Tr\,\left[ A(x^a)T^{ij} \right],\quad where$$ A(x\^a) &=& \_[ij]{} A(x\^a\_i, x\^a\_j) T\^[ij]{},\ && \[definitions\] Ricci scalar in the arrangement field paradigm {#ricciscalar} ---------------------------------------------- ### Hyperions In this subsection we define an extension of $\mathbf{H}$ by inserting a new imaginary unit $I$. It satisfies: $$I^2 = -1 \qquad I^\dag = -I$$ $$[I,i] = [I,j] = [I,k] = 0$$ In this way a generic number assumes the form $$v = a + Ib+ ic + jd + ke + iIf+ jIg + kIh, \qquad a,b,c,d,e,f,g,h \in \mathbf{R}$$ $$v = p + Iq, \qquad p,q \in \mathbf{R}$$ We call this numbers Hyperions” and indicate their ensemble with $Y$. It’s straightforward that such numbers are in one to one correspondence with even products of Gamma matrices. Explicitly: $$1 \Leftrightarrow \g_0 \g_0 = 1 \qquad I \Leftrightarrow \g_5 = \g_0 \g_1 \g_2 \g_3$$ $$i \Leftrightarrow \g_2 \g_1 \qquad iI \Leftrightarrow \g_0 \g_3$$ $$j \Leftrightarrow \g_1 \g_3 \qquad jI \Leftrightarrow \g_0 \g_2$$ $$k \Leftrightarrow \g_3 \g_2 \qquad kI \Leftrightarrow \g_0 \g_1$$ Note that imaginary units $i,j,k,iI,jI,kI$ satisfy the Lorentz algebra, with $i,j,k$ which describe rotations and $iI, jI, kI$ which describe boosts. The bar-conjugation is an operation which exchanges $I$ with $-I$ (or $\g_0$ with $-\g_0$ in the $\g$-representation). Explicitly, if $v = a + Ib+ ic + jd + ke + iIf+ jIg + kIh$ with $a,b,c,d,e,f,g,h \in \mathbf{R}$, then $\bar{v} = a - Ib+ ic + jd + ke - iIf - jIg - kIh$. The pre-norm is a complex number with $I$ as imaginary unit (we say I-complex number”). Given an hyperion $v$, its pre-norm is $|v| = (\bar{v}^\dag v)^{1/2}$. If $v \in \mathbf{H}$, its pre-norm coincides with usual norm $(v^\dag v)^{1/2}$. Note that every hyperion $v$ can be written in the polar form $$v = |v|e^{ia+jb+kc+iId+jIe+kIf} \qquad a,b,c,d,e,f$$ $$|v|^2 = \bar{v}^\dag v = |v|e^{-(ia+jb+kc+iId+jIe+kIf)} |v|e^{ia+jb+kc+iId+jIe+kIf}= |v|^2.$$ If $M$ takes values in $\mathbf{Y}$, the probability for the existence of an edge $(ij)$ can be defined as $||M^{ij}||$, which is the norm of pre-norm. $\pt$ The fundamental relation (\[fondamentale\]) descends uniquely from spectral theorem in $\mathbf{H}$. You can see from work of Yongge Tian [@Tian] that spectral theorem is still valid in $Y$ in the following form: Every normal matrix $M$ with entries in $\mathbf{Y}$ is diagonalizable by a transformation $U \in U(n,\mathbf{Y})$ which sends $M$ in $UM\bar{U}^\dag$”. Here $U(n,\mathbf{Y})$ is the exponentiation of $u(n, \mathbf{Y}) = u(n, \mathbf{H})\cup Iu(n, \mathbf{H})$ and $M$ satisfies a generalized normality condition. Explicitly, $\bar{U}^\dag = U^{-1}$ and $\bar{M}^\dag M = M \bar{M}^\dag$. This implies that (\[fondamentale\]) is valid too in the form $$\bar{U}^\dag D^\mu \na_\mu U = M$$ Matrix $\na$ is now in $u(n, \mathbf{Y})$ and then it satisfies $\bar{\na}^\dag = -\na$. Accordingly, its diagonal entries belong to Lorentz algebra (they don’t comprise real and $I$-imaginary components). To conclude, we don’t know if an associate normal matrix exists for any invertible matrix with entries in $\mathbf{Y}$. Fortunately, in lorentzian spaces there is no reason for using such machinery and we can start from the beginning with a normal arrangement matrix. \[sinv\]$\pt$ It follows from spectral theorem that eigenvalues $\lambda$ of $M$ are equivalence classes $$\lambda \sim s\lambda \bar{s}^\dag \qquad s \in \mathbf{Y}, \bar{s}^\dag s = 1.$$ As a consequence, we can choose freely the diagonal matrix $D$ inside the equivalence class $SD\bar{S}^\dag$, where $S$ is both diagonal and unitary ($\bar{S}^\dag = S^{-1}$). This choice does’t affect the metric $\sqrt h h^{\mu\nu} = Re\,(\bar{D}^{\dag\mu} D^\nu)$, granting for the persistence of a symmetry $U(1,Y)^n = SO(1,3)^n$, ie local $SO(1,3)$. Clearly this is a reworking of the usual gauge symmetry which acts on the tetrads, sending $e^\mu_a$ in $\Lambda_a^{\pt b} e^\mu_b$ via the lorentz transformation $\Lambda$. In what follows we exploit $SO(1,3)$-symmetry to satisfy two conditions: && tr({|\^\_, \_} |[D]{}\^ D\^) = 0 \[scond\]\ && tr(D\^{\_, |\^\_} |[D]{}\^ D\^\_|\^\_|[D]{}\^) = 0 Note that these are global conditions because operator *tr* is analogous to a space-time integration. ### Ricci scalar with hyperions In this subsection we simplify the form of Ricci scalar by means of hyperions, in order to make it suitable for the arrangement field formalism. Given a gauge field $\w_\mu$ in $so(1,3)$ and a complex tetrad $e^\mu$, we define A\_= \^[ab]{}\_\_[a]{}\_[b]{} h\^ = Re(e\^\_a e\^\_b \^[ab]{})\[gaugey\] $$d^\mu = \sqrt e e^{\mu a} \g_0 \g_a \qquad e = \left[ det(- e^{\dag \mu}_a e^\nu_b \eta^{ab}) \right]^{-1/2} \in \mathbf{R^+}$$ $$\bar{d}^\mu = d^\mu (\g_0 \rightarrow-\g_0)$$ $$\Rightarrow \bar{d}^{\dag \mu} d^{\nu} = ee^{\dag \mu a}e^{\nu b} \g_{a} \g_{b} \quad \Rightarrow \sqrt h h^{\mu\nu} = \fr 14 Re\,\left[ tr(\bar{d}^{\dag \mu} d^{\nu})\right]$$ Note that our definitions are the same to require $\bar{A}^\dag =-A$ in the hyperions framework. The Ricci scalar can be written as $$\sqrt h R(x) = -\fr 18 tr\left(\left(\pa_\mu A_\nu - \pa_\nu A_\mu + [A_\mu, A_\nu]\right) \bar{d}^{\dag\mu} d^\nu\right)$$ To verify its correctness we expand first the commutator &=& \^[ab]{}\_\^[cd]{}\_( \_a\_b\_c\_d - \_c \_d \_a \_b )\ &=& 12 \^[ab]{}\_\^[cd]{}\_( \_a{\_b,\_c}\_d - \_c {\_d, \_a} \_b ) +\ && +1 [2]{}\^[ab]{}\_\^[cd]{}\_( \_a \[\_b,\_c \]\_d - \_c \[\_d, \_a\] \_b ) &=& (\^[ab]{}\_\_[b]{}\^[d]{} - \^[ab]{}\_\_[b]{}\^[d]{} )( \_a \_d ) +\ && +1 [4!]{}\^[ab]{}\_\^[cd]{}\_( \_[abcd]{} \^[efgh]{} \_e \_f \_g \_h )\ &=& \[\_, \_\]\^[ab]{}\_a\_b + \^[ab]{}\_\^[(D)]{}\_[ab]{} \_5 In the last line we have defined $\w^{(D)}_{ab \nu} = \e_{abcd} \w_\nu^{cd}$. Hence R(x) &=& -18 tr(\_a \_b \_c \_d)( \_\^[ab]{}\_- \_\^[ab]{}\_+ \[\_,\_\]\^[ab]{})e\^[c]{} e\^[d]{} -\ && -18 tr(\_5 \_b \_c) \^[ab]{}\_\^[(D)]{}\_[ab]{} e\^[c]{} e\^[d]{} Consider now the relations $$\fr 14 tr(\g_a \g_b \g_c \g_d) = \eta_{ab}\eta_{cd} - \eta_{ac}\eta_{bd} + \eta_{ad}\eta_{bc}$$ $$tr(\g_5 \g_b \g_c) = 0$$ We obtain $$R(x) = \left( \pa_\mu \w^{ab}_\nu - \pa_\nu \w^{ab}_\mu + [\w_\mu,\w_\nu]^{ab}\right)e^{\dag \mu}_a e^{\nu}_b$$ which is the usual definition. A complex tetrad implies that tangent space is the complexification of Minkowsky space (usually indicated with $\mathcal{CM}$). This fact gives a strict connection with theory of **twistors**[@twistor], where massless particles move on trajectories which have an imaginary component proportional to helicity. We can move freely from matrices $\g$ to hyperions, substituting $tr$ with $4$. In this way h R(x) &=& -12 (\_A\_- \_A\_+ \[A\_, A\_\]) |[d]{}\^ d\^\ &=& -12 \[\_,\_\]|[d]{}\^ d\^$$\na_\mu =\pa_\mu + A_\mu \qquad\qquad A_\mu, d^\mu \in \mathbf{Y}$$ $$e^\mu_a = Re\,e^\mu_a + I\,Im\,e^\mu_a$$ d\^&=& Ree\^[0]{} + i I Ree\^[3]{} + j I Ree\^[2]{} + k I Ree\^[1]{} +\ && + IIme\^[0]{} - i Ime\^[3]{} - j Ime\^[2]{} - k Ime\^[1]{} ### Ricci scalar in the new paradigm We try to define Hilbert-Einstein action as $$S_{HE} = tr\,(\bar M^\dag M).$$ We insert in $S_{HE}$ the usual expansion $M = UD^\mu \na_\mu \bar{U}^\dag$, obtaining S\_[HE]{} &=& tr\[(|U |[D]{}\^ |\_U\^)\^(U D\^\_|[U]{}\^)\]\ &=& tr\[ U |\^\_|[D]{}\^ |[U]{}\^U D\^\_|[U]{}\^\]\ &=& tr\[\_|\^\_|[D]{}\^ D\^\].\[HEfirst\] Now we can impose the first condition in (\[scond\]) which gives S\_[HE]{} &=& 12 tr{\[\_, |\^\_\] |[D]{}\^ D\^} \[espansione\] .Expanding the covariant derivatives we obtain S\_[HE]{} &=& 12 \_[a,b,c]{} { \^\_A\_(x\^a, x\^b)-\_|[A]{}\_\^(x\^a, x\^b) +\ && + \[|[A]{}\^\_, A\_\](x\^a, x\^b)}|[d]{}\^(x\^b)\^[bc]{} d\^(x\^c)\^[ca]{}\ &=& 12 \_a { \^\_A\_(x\^a)-\_|[A]{}\^\_(x\^a) +\ && +\[|[A]{}\^\_, A\_\](x\^a, x\^a)}|[d]{}\^(x\^a)d\^(x\^a)\ &=& 12\_[a,ba]{} { \^\_A\_(x\^a)-\_|[A]{}\_\^(x\^a) + \[|[A]{}\_\^(x\^a), A\_(x\^a)\] +\ && +\[|[A]{}\_\^(x\^a,x\^b),A\_(x\^b,x\^a)\]} |[d]{}\^(x\^a)d\^(x\^a)\ && Consider now a symmetry breaking with residual group $U(m,\mathbf{Y})^{n/m}$ which regroups vertices in ensembles $\mathcal{U}^a = \{x^a_1, x^a_2,\ldots,x^a_m\}$. We assume that fields $A(x^a_i,x^b_j)$ with $a \neq b$ acquire big masses and thus we can neglect them. The symbol $\sum_a$ becomes $\sum_{a,i}$, while $\sum_{a,b\neq a}$ becomes $\sum_{a,i,b,j|(a,i)\neq (b,j)}$. After neglecting heavy fields, the last one is simply $\sum_{a,i,j\neq i}$. S\_[HE]{} &=& 12\_[a]{} { \^\_trA\_(x\^a)-\_tr|[A]{}\^\_(x\^a) + \[tr|[A]{}\^\_(x\^a), trA\_(x\^a)\] +\ && +\_[i,ji]{}\[|[A]{}\_\^[ij]{}(x\^a) A\^[ji]{}\_(x\^a) - A\_\^[ij]{}(x\^a) |[A]{}\^[ji]{}\_\](x\^a)} |[d]{}\^(x\^a)d\^(x\^a)\ && For what follows we write $S_{HE} = \fr 12 \sum_a R^{ik}_{\mu\nu} \d^{ik}\bar{d}^{\dag\mu}d^\nu $ with R\^[ik]{}\_ &=& \^\_trA\_(x\^a)-\_tr|[A]{}\^\_(x\^a) + \[tr|[A]{}\^\_(x\^a), trA\_(x\^a)\] +\ && +\_[i,ji,kj]{}\[|[A]{}\_\^[ij]{}(x\^a) A\^[jk]{}\_(x\^a) - A\_\^[ij]{}(x\^a) |[A]{}\^[jk]{}\_\](x\^a) .\ && \[curvature\]$R^{ik}_{\mu\nu}$ is a generalization of curvature tensor. We have indicated with $tr\,A$ the track on $ij$, ie $\d^{ij}A^{ij}(x^a) = \d^{ij}A(x^a_i,x^a_j)$. Note that $[\bar{A}^{\dag ii},A^{jj}]$ is equal to zero when $i \neq j$ and then $$\sum_{i} [\tilde{A}^{^\dag ii}_\mu, A^{ii}_\nu] = \sum_{ij}[\bar{A}^{^\dag ii}_\mu, A^{jj}_\nu]= [tr\,\bar{A}^\dag_\mu, tr\,A_\nu].$$ Consider now any skew hermitian matrix $W_\mu$ with elements $W_\mu^{ij} = A_\mu^{ij}$ for $i \neq j$ and $W_\mu^{ij} = 0$ for $i = j$. It belongs to the subalgebra of $u(m,\mathbf{Y})$ made by all null track generators. This means that commutators between null track generators are null track generators too. In this way $$\sum_{i,i\neq j} [\bar{A}^\dag_\mu (x^i,x^j),A_\nu (x^j,x^i)] = tr [\bar{W}^\dag_\mu, W_\nu] = 0 .$$ Hence we can delete the mixed term in $S_{EH}$. S\_[HE]{} &=& 12 \_[a]{} { \^\_trA\_(x\^a)-\_tr|[A]{}\^\_(x\^a) + \[tr|[A]{}\^\_(x\^a), trA\_(x\^a)\]}\ &&|[d]{}\^(x\^a)d\^(x\^a) In the arrangement field paradigm, the operator $\dag$ transposes also rows with columns in matrices which represent $\pa$ and $A$. As we have seen, the fields $A$ which intervene in $R$ are only the diagonal ones, so the transposition of rows with columns is trivial. Note that $\na$ satisfies a generalized condition of skew-hermiticity ($\bar{\na}^\dag = -\na$) and then its diagonal components belong to lorentz algebra. This implies $tr\,\bar{A}^\dag = -tr\,A$, matching exactly with our request in (\[gaugey\]). Finally, if we consider the matrix which represents $\pa$ (we have called it $\tilde M$), we note that $\bar{\pa}^\dag = \pa^T = -\pa$. Explicitly $$\na_\nu^\dag = (\pa_\nu + tr\,\bar{A}_\nu)^\dag = \pa_\nu^\dag + tr\,\bar{A}_\nu^\dag = -\pa_\nu - tr\,A_\nu = -\na_\nu .$$ Applying this to $S_{HE}$, S\_[HE]{} &=& -12 \_[a]{} { \_trA\_(x\^a)-\_trA\_(x\^a) + \[trA\_(x\^a), trA\_(x\^a)\]}\ &&|[d]{}\^(x\^a)d\^(x\^a)\ &=& -12 \[\_,\_\]|[d]{}\^(x\^a)d\^(x\^a)\ &=& \_a R(x\^a) d\^4 x R(x). Here $\overset{G}{\na}$ is the gravitational covariant derivative $\overset{G}{\na} = \pa + tr\,A$. It’s very remarkable that gauge fields in $R$ are only the diagonal ones. First, this is the unique possibility to obtain ${\overset{G}{\na^\dag}_\nu} = -\overset{G}{\na}_\nu$. Moreover, while gauge fields in $R$ are tracks of matrices $(A_{ij})(x^a)$, we’ll see as the other gauge fields in Standard Model correspond to non diagonal components. The kinetic term {#kinetic} ---------------- Until now we have obtained no terms which describe gauge interactions. In this section we find a such term, with the condition that it hasn’t to change Einstein equations. One option is as follows: S\_[GB]{} &=& -tr(|[M]{}\^M |[M]{}\^M) \[eq: opz\]\ &=& -tr\ &=& -tr We assume a residual symmetry under $U(m,\mathbf{Y})^{n/m}$. This means that $D^\mu$ are matrices made of blocks $m \times m$ where every block is a hyperionic multiple of identity. We use newly the correspondence between $(1,I,i,j,k,iI,jI,kI)$ and gamma matrices: S\_[GB]{} &=& -14 tr(\_a\_b\_c\_d\_e\_f\_g\_h) We use letters $a,b,c,d$ for indices which run on Gamma matrices, $\a,\b,\mu,\nu$ for spatial coordinates indices and $ijk$ for gauge indices (ie indices which run inside a single $\mathcal{U}^a$). Pay attention to not confuse the index $a$ in the first group with the index $a$ which runs over the vertices like in $x^a_i$. We will see that physical fields arise in three families, determined by the choice of a subspace inside $Y$. This is true both for fermionic and bosonic fields. Thus the indices with letters $a,b,c,d$ run over the three families. We proceed by imposing the second condition in (\[scond\]), in such a way to ignore terms proportional to $\{\na_\b, \bar{\na}^\dag_\mu\}$ inside $S_{GB}$. We take $$S_{GB} = \sum_a L_{GB} (x^a)$$ Then L\_[GB]{} &=& R\^[ij]{}\_[ab]{} R\^[ab ji]{}\_ |[d]{}\^\_c d\^[c]{} |[d]{}\^\_d d\^[d]{} - 4R\^[ij]{}\_[ac]{} |[d]{}\^[a]{} R\^[cb ji]{}\_ d\^\_b d\^[d]{} |[d]{}\^\_d +\ && + R\^[ij]{}\_[ac]{} |[d]{}\^[a]{} d\^[c]{} R\^[cb ji]{}\_ |[d]{}\^\_c d\^\_b\ &=& h R\^[ij]{}\_[ab]{} R\^[ab ji ]{} - 4 h R\^[ij]{}\_[c ]{} R\^[c ji ]{} + h R\^[ij]{}R\^[ji]{} $R^{ij}_{\b\mu}$ was defined in (\[curvature\]), while $\sqrt[4] h R^{ij}_\mu = R^{ij}_{\b\mu} d^{\b}$ and $\sqrt h R^{ij} = R^{ij}_{\b\mu} d^{\b} d^{*\mu}$. You understand in a moment that for $i \neq j$ we have $R^{ij}_{ac\b\mu} R^{ji ac}_{\nu\a} h^{\mu\a}h^{\nu\b} = tr\,\sum_{(ac)} F^{(ac)}_{\mu\nu} F^{(ac) \mu\nu}$. The index $(ac)$ runs over three fields families and $F_{(ac)\mu\nu}$ is a strength field tensor. In this way the terms $R^{ij \nu}_\b R^{ji \b}_\nu$ and $R^{ij}R_{ji}$ are terms which mix families. The trouble with $S_{GB}$ is that it generates a factor $h$ instead of $\sqrt{h}$. However, we can solve the problem imposing the gauge condition $h=1$. Note that for $i=j$ we have $$L_{GB} = R_{ac\b\mu} R^{ac\b\mu} + R^2 - 4 R^{\a}_\mu R^{\mu}_\a$$ which is a topological term and it doesn’t change the Einstein equations. The combination of $S_{HE}$ and $S_{GB}$ gives to gravitational gauge field $\overset{G}{A}$ a potential with form $$\overset{G}{A^2} - \overset{G}{A^4}.$$ This potential has non trivial minimums which imply a non-trivial expectation value for $\overset{G}{A}$. Moreover, inside $S_{GB}$ we find the following kind of terms for other fields $A$: $$\langle \overset{G}{A^2} \rangle A^2 - A^4.$$ In this way we have a mass for gauge fields $A$ and another potential with non-trivial minimums. Therefore, also gauge fields $A$ have non-trivial expectation values. Finally, such expectation values give mass to fermionic fields via terms $$\psi^\dag \langle A \rangle \psi.$$ There is no need for a scalar Higgs boson. Connections with Strings and Loop Gravity {#string} ----------------------------------------- We have seen in [@Arrangement], at **Remark 13**, that some similarities exist between diagonal components of $M$ (loops) and closed strings in string theory. Now we have discovered that such diagonal components describe a gravitational field. Is then a case that the lower energy state for closed string is the graviton? We think no. Moreover, we have seen that gauge fields correspond to non-diagonal components of $M$, ie open edge in the graph. This finds also a connection with open strings, whose lower energy states are gauge fields. We have shown that a symmetry $U(m,\mathbf{Y})$ arises when vertices are grouped in ensembles $\mathcal{U}^a$ containing $m$ vertices. This seems to represent a superimposition of $m$ universes or branes. Gauge fields for such symmetry correspond to open edge which connect vertices in the same $\mathcal{U}^a$. Is then a case that the same symmetry arises in open strings with endpoints in $m$ superimposed branes? We still think no. Until now we have supposed that open edges between vertices in the same $\mathcal{U}^a$ have length zero, so that we haven’t to introduce extra dimensions. However, by $T-duality$ such edges correspond to open strings with $U(m,\mathbf{Y})$ Chan-Paton which moves in an infinite extended extra dimension. This happens because an absente extra dimension is a compactified dimension with $R = 0$ and $T-duality$ sends $R$ in $1/R$. Regarding edges between vertices in different $\mathcal{U}^a$, we see that they have a mass proportional to separation between endpoints. This is true both in our model and string theory. $\pt$ The following two theorems emphasize a triality between *Arrangement Field Theory*, *String Theory* and *Loop Quantum Gravity*. We can see as they are different manifestations of the same theory. \[Loop\] Every element $M^{ij}$ in the arrangement matrix can be written as a state in the Hilbert space of *Loop Quantum Gravity*, ie an holonomy for a $SO(1,3)$ gauge field[^7]. In this way, every field (gauge or gravitational) becomes a manifestation of only gravitational field. An element $M^{ij}$ can always be written in the following form: M\^[ij]{} = |M\^[ij]{}| exp (\_[x\_i]{}\^[x\_j]{} A\_dx\^) \[defM\] with $\mu = 1,2,3$ and $$|M^{ij}|= exp \left(\int_{x_i}^{x_j} A_0 dx^0\right).$$ Here $A_\mu$ is a $SO(1,3)$ connection and $A_0$ is an $I$-complex field. Obviously, we take $A_\mu$ hyperionic by using the usual correspondence with Gamma matrices. In this way $A_\mu$ is purely imaginary. The integration is intended over the edge which goes from vertex $i$ to vertex $j$, parametrized by any $\t \in [0,1]$. If you look (\[defM\]), you see on the left a discrete space (the graph) with discrete derivatives and fields which are defined only on the vertices. On the right you find instead a Hausdorff space with continuous paths, continuous derivatives and fields which are defined everywhere. Applying eventually a transformation in $U(n,\mathbf{Y})$, we have $$M^{ij} = D^{ik\mu}\na^{kj}_\mu = D^{ii\mu} \na^{ij}_\mu = d^\mu(x_i) \na^{ij}_\mu.$$ In the following we introduce a real constant $\lambda$, with length dimensions, in order to make $M$ dimensionless: M\^[ij]{} = D\^[ik]{}\^[kj]{}\_= D\^[ii]{} \^[ij]{}\_= d\^(x\_i) \^[ij]{}\_. \[scompose\]In *Loop Quantum Gravity* we consider any space-time foliation defined by some temporary parameter and then we quantize the theory on a tridimensional slice. The simpler choice is a foliation along $x_0$: in this case the metric on the slice is simply the spatial block $3\times 3$ inside the four dimensional metric when it’s taken in temporary gauge. In such framework we have $d^0 = \mathbf{1}$ and $[d^\mu(x), A_\nu(x')] = G\d^\mu_\nu \d^3(x-x')$ with $\mu,\nu = 1,2,3$. We deduce the relation $d^\mu(x) = G\d / {\d A_\mu(x)}$ and apply it to (\[scompose\]) when vertices $i$ and $j$ sit on the same slice. We obtain d\^(x\_i) \^[ij]{}\_= G \^[ij]{}\_= 1 |M\^[ij]{}| exp ([\_[x\_i]{}\^[x\_j]{} A\_dx\^]{})\[appl\]with $\mu = 1,2,3$. Note that $x_0 (x_i) = x_0 (x_j)$ when $i$ and $j$ sit on the same slice. Hence $$|M^{ij}|= exp \left(\int_{x_i}^{x_j} A_0 dx^0\right) = exp \left(\oint A_0 dx^0\right).$$ Consider now the following relation: exp ([\_[x\_i]{}\^[x\_j]{} A\_dx\^]{}) = \_d\^2s n\_exp ([\_[x\_i]{}\^[x\_j]{} A\_dx\^]{})\[relaz\]with $$n_\nu = \fr 12 \e_{\nu\mu\a} \fr {\pa x^\mu}{\pa s^a} \fr {\pa x^\a}{\pa s^b}\e^{ab}.$$ $\Omega$ is a two dimensional surface parametrized by coordinates $s^a$ with $a=1,2$ and $\int_\Omega d^2 s = G$. We assume that $\Omega$ contains the vertex $x_i$ and no other point which is a vertex or sits along an edge. Substituting (\[relaz\]) in (\[appl\]) we obtain $$\fr \d {\d A_\nu^i} \na^{ij}_{\nu} = \fr 1 {\lambda G}\fr \d {\d A_\nu} \int_\Omega d^2s \, n_\nu |M^{ij}| exp \left({\int_{x_i}^{x_j} A_\mu dx^\mu}\right)$$ and then \^[ij]{}\_ &=& 1 [G]{}\_d\^2s n\_|M\^[ij]{}| exp ([\_[x\_i]{}\^[x\_j]{} A\_dx\^]{}) +K\_(x\_i, x\_j)\ &=& 1 [G]{} \_d\^2s n\_exp ([\_[x\_i]{}\^[x\_j]{} A\_dx\^]{}) + K\_(x\_i, x\_j).$K_\nu$ is any function of $x_i$ and $x_j$ independent from $A_\mu$. In the second line we have taken $\mu =0,1,2,3$. For diagonal components this becomes A\_\^[ii]{} = 1 [G]{}\_d\^2s n\_exp ([A\_dx\^]{})+K\_(x\_i).\[una\]We have used $\pa^{ii} = 0$ because the matrix which represents the discrete derivative is null along diagonal. We choose loops and surfaces $\Omega$ in such a way to have $$n_\nu \oint A_\mu dx^\mu = \lambda A_\nu (x_i) + O(\lambda^2).$$ Applying this into (\[una\]), it becomes A\^[ii]{}\_&=& 1 [G]{} \_d\^2s n\_( 1 + A\_dx\^+ O(\^2) ) +K\_(x\_i)\ &=& 1 [G]{} (G n\_+ G A\_(x\_i)+GO(\^2)) + K\_(x\_i)\ &=& 1 (n\_+ A\_(x\_i)+ O(\^2))+ K\_(x\_i).If we set $K_\nu (x_i) = -n_\nu (x_i) / \lambda$, we obtain $$A^{ii}_\nu = A_\nu (x_i) + O(\lambda).$$ This verifies the consistence of our definition and proves the theorem. Note that $\l$ could be taken equal to $\Delta$ because $M$ contains a factor $\Delta^{-1}$ from definition (\[dderiv\]) of $\tilde{M}$. In such case we obtain $$A^{ii}_\nu = A_\nu (x_i)$$ in the continuous limit. Note that canonical quantization of gauge fields implies $$\left[ \pa_0 A^{ij}_\a (x_a), A^{ij}_\nu (x_b) \right] = \left[\left(\int d^4 x \pa_0 A_\mu (x) \fr {\d \na^{ij}_\a} {\d A_\mu (x)}\right)(x_a), \na^{ij}_\nu(x_b)\right] = \d_{\a\nu}\d^3(x_a -x_b).$$ Integration in the first factor is over continuous coordinates of Hausdorff space. Conversely, the argument $x_a$ indicates simply to what ensemble $\mathcal{U}^a$ the edge $(ij)$ belongs. Here we have used $\pa^{ij} = 0$, which holds not only for $i = j$ but also for $x_i$ and $x_j$ in the same ensemble $\mathcal{U}^a$. This implies $\na^{ij} = A^{ij}$. Moreover $\na^{ij}$ is a state in the Hilbert space of Loop Quantum Gravity and hence we have a sort of third quantization which applies on gravitational states and creates gauge fields: $$\left[\left( \int d^4 x \dot{A}_\mu(x) \fr {\d \Psi[\Lambda,A]}{\d A_\mu(x)}\right), \Psi^\dag[\Lambda',A]\right] = \d (\Lambda - \Lambda').$$ $$\left[\left( \int d^4 x \dot{A}_\mu(x) \fr {\d \Psi[\Lambda, A]}{\d A_\mu(x)}\right), \Psi^\dag[\Lambda, A']\right] = \d (A - A').$$ This implies $$\Psi[A] = \int D[d^\mu]\, a(d)\, exp\left(\fr 1 G{\int d^4 x \,d^\mu A_\mu}\right) + b^\dag(d) \, exp\left(\fr 1 G{\int d^4 x \,d^{\dag\mu} A^\dag_\mu}\right)$$ $$\left[a(d), a^\dag (d')\right] = \fr 1 {\int d^4 x \dot A_\nu d^\nu} \d(d-d^{\dag\prime})$$ $$\left[b(d), b^\dag (d')\right] = \fr 1 {\int d^4 x \dot A_\nu d^\nu} \d(d-d^{\dag\prime})$$ ![A spin network with symmetry $U(6,\mathbf{Y}$). The six vertices are assumed superimposed.[]{data-label="Spin-network"}](Spin-network.jpg){width="60.00000%"} In figure \[Spin-network\] we see a spin network which defines a $U(6,\mathbf{Y})$ gauge field $A^{ij}$ with $i,j =1,2,3,4,5,6$. The vertices are assumed superimposed. The symmetry group is bigger than $U(1,\mathbf{Y})^6 \sim SO(1,3)^6$ which acts separately on the single vertices. The group grows in fact to $U(6,\mathbf{Y})$ because we can exchange the vertices without change the graph. We have the same situation with open strings: six strings with endpoints on six separated branes define a state with symmetry $U(1)^6$ but, if the branes are superimposed, the symmetry becomes $U(6)$. Generators in $u(6,\mathbf{Y})$ are generators in $u(6,\mathbf{H})$ multiplied by $1$ or $I$. In turn, generators in $u(6,\mathbf{H})$ can be divided in three families of generators in $u(6)$, one for every choice of imaginary unit ($i,j$ or $k$). Note that commutation relations for $U(6)$ are satisfied if and only if $$U^{ij}U^{jk} = U^{ik},$$ where $U^{ij}$ is the holonomy from $x_i$ to $x_j$. Hence $$A_\mu = \pa_\mu \Gamma \qquad \,\,\,\text{with}\,\,\,\Gamma\,\,\,\text{scalar.}$$ This means that gauge fields in $U(6)$ could exist without gravity, ie when $A$ is a pure gauge. Otherwise, an holonomy with $A \neq \pa \Gamma$ exchanges gauge fields between different families. The actions $tr\,(M^\dag M)$ and $tr\,(M^\dag M M^\dag M)$ are sums of exponentiated string actions. We obtain from theorem \[Loop\]: M\^[ij]{} M\^[\*jk]{} M\^[kl]{} M\^[\*li]{} &=& exp(\_ A\_dx\^)\ &=& exp(\_ F\_ dx\^dx\^)\ &=& exp(\_ \^[ab]{} F\_ X\^\_[,a]{} X\^\_[,b]{} d\^2 s) This is the exponential of an action for open strings whose worldsheet is a square made by edges $(ij)$, $(jk)$, $(kl)$, $(li)$. The strings move in a curved background with antisymmetric metric $F_{\mu\nu} = (d \wedge A)_{\mu\nu}$. In a similar manner M\^[ij]{} M\^[\*jk]{} M\^[ki]{} &=& exp(\_ \^[ab]{} F\_ X\^\_[,a]{} X\^\_[,b]{} d\^2 s) This is the exponential of an action for open strings whose worldsheet is a triangle. M\^[ij]{} M\^[\*ji]{} &=& exp(\_[O]{} \^[ab]{} F\_ X\^\_[,a]{} X\^\_[,b]{} d\^2 s) This is the exponential of an action for open strings whose worldsheet is a circle. M\^[ii]{} &=& exp(\_[O]{} \^[ab]{} F\_ X\^\_[,a]{} X\^\_[,b]{} d\^2 s) The same of above. M\^[ii]{}M\^[jj]{} &=& exp(\_[Cil]{} \^[ab]{} F\_ X\^\_[,a]{} X\^\_[,b]{} d\^2 s) This is the exponential of an action for closed strings whose worldsheet is a cilinder. This concludes the proof. Standard model interactions \[electroweak\] ------------------------------------------- We suppose that a residual symmetry for $U(6,\mathbf{Y})^{n/6}$ survives. If we consider the ensembles $\mathcal{U}^a = (x^a_1, x^a_2, x^a_3, x^a_4, x^a_5, x^a_6)$ as the real physical points, $U(6,\mathbf{Y})^{n/6}$ can be considered as a local $U(6,\mathbf{Y})$. We have defined $u(6,\mathbf{Y})$ as the complexified Lie algebra of $U(6,\mathbf{H})$, generated by all matrices in $u(6,\mathbf{H})$ and $Iu(6,\mathbf{H})$. By exponentiating $u(6,\mathbf{Y})$ we obtain a simple Lie group with complex dimension $78$. This group is the symplectic group $Sp(12,\mathbf{C})$ and $U(6,\mathbf{H})$ is its real compact form, sometimes called $Sp(6)$. We consider the fields $A(x^a_i, x^b_j)$ with $a = b$ (we call them $A(x^a)$). They are $6 \times 6$ skew adjoint hyperionic matrices $\bar{A}^\dag = -A$. These matrices form the $Sp(12,\mathbf{C})$ algebra which has $156$ generators $\w$ with $\bar{\w}^\dag = -\w$. $$\w = \left( \begin{array}[c]{cccccc} \vec{y} & b+\vec{b} & c+\vec{c} & d+\vec{d} & e+\vec{e} & m+\vec{m} \\ -b+\vec{b}& \vec{a}_1 & f+\vec{f} & g+\vec{g} & h+\vec{h} & p+\vec{p} \\ -c+\vec{c}& -f+\vec{f} & \vec{a}_2 & s+\vec{s} & q+\vec{q} & r+\vec{r} \\ -d+\vec{d}& -g+\vec{g} & -s+\vec{s} & \vec{a}_3 & k+\vec{k} & t+\vec{t} \\ -e+\vec{e}& -h+\vec{h} & -q+\vec{q} & -k+\vec{k} & \vec{a}_4 & v+\vec{v} \\ -m+\vec{m}& -p+\vec{p} & -r+\vec{r} & -t+\vec{t} & -v+\vec{v} & \vec{a}_5 \\ \end{array} \right)$$ Consider now the subalgebra of the following form with complex (not hyperionic) components except for $y$ which remains hyperionic: $$\w = \left( \begin{array}[c]{cccccc} \vec{y} & 0 & 0 & 0 & 0 & 0 \\ 0 & \vec{a}_1 & f+\vec{f} & g+\vec{g} & h+\vec{h} & p+\vec{p} \\ 0 & -f+\vec{f} & \vec{a}_2 & s+\vec{s} & q+\vec{q} & r+\vec{r} \\ 0 & -g+\vec{g} & -s+\vec{s} & \vec{a}_3 & k+\vec{k} & t+\vec{t} \\ 0 & -h+\vec{h} & -q+\vec{q} & -k+\vec{k} & \vec{a}_4 & v+\vec{v} \\ 0 & -p+\vec{p} & -r+\vec{r} & -t+\vec{t} & -v+\vec{v} & \vec{a}_5 \\ \end{array} \right)$$ Moreover we put the additional condition $\vec{a} = \sum_l \vec{a}_l = 0$. The field $y = tr\,\w$ is the only one which contributes to Ricci scalar. Conversely, all other fields belong to a $SU(5)$ subgroup, which defines the Georgi - Glashow grand unification theory. The symmetry breaking in Georgi - Glashow model is induced by Higgs bosons in representations which contain triplets of color. These color triplet Higgs can mediate a proton decay that is suppressed by only two powers of GUT scale. However, our mechanism of symmetry breaking doesn’t use such Higgs bosons, but descends from the expectation values of quadratic terms $AA$, which derive from non trivial minimums of a potential $AA - AAAA$. So we circumvent the problem. Restrict now the attention to the $SO(1,3) \otimes SU(2) \otimes U(1) \otimes SU(3)$ generators, that are the generators of standard model plus gravity. $$\w = \left( \begin{array}[c]{cccccc} \vec{y} & 0 & 0 & 0 & 0 & 0 \\ 0 & \vec{a}_1 & f+\vec{f} & 0 & 0 & 0 \\ 0 & -f+\vec{f} & \vec{a}_2 & 0 & 0 & 0 \\ 0 & 0 & 0 & \vec{a}_3 & k+\vec{k} & t+\vec{t} \\ 0 & 0 & 0 & -k+\vec{k} & \vec{a}_4 & v+\vec{v} \\ 0 & 0 & 0 & -t+\vec{t} & -v+\vec{v} & \vec{a}_5 \\ \end{array} \right)$$ We’ll show in a moment that all standard model fields transform under this subgroup in the adjoint representation. In this way themselves are elements of $Sp(12,\mathbf{C})$ algebra, explicitly: $$\psi = \psi^1 + I\psi^2 = \left( \begin{array}[c]{cccccc} 0 & e & -\nu & d^c_{R} & d^c_{G} & d^c_{B} \\ -e^* & 0 & e^c & -u_{R} & -u_{G} & -u_{B} \\ \nu^* & -e^{c*} & 0 & -d_{R} & -d_{G} & -d_{B} \\ -d^{c*}_R & u^*_R & d^*_R & 0 & u^c_{B} & -u^c_{G} \\ -d^{c*}_G & u^*_G & d^*_G & -u^{c*}_{B} & 0 & u^c_{R} \\ -d^{c*}_B & u^*_B & d^*_B & u^{c*}_{G} & -u^{c*}_{R} & 0 \\ \end{array} \right)$$ We have used the convention of Georgi - Glashow model, where the basic fields of $\psi^1$ are all left and the basic fields of $I\psi^2$ are all right. We have indicated with $^c$ the charge conjugation. The subscripts $R,G,B$ indicates the color charge for the strong interacting particles (R=red, G=green, B=blue). In Georgi - Glashow model the fermionic fields are divided in two families. The first one transforms in the representation $\bar{5}$ of $SU(5)$ (the fundamental representation). It is exactly the array $(\w^{1j})$ in the matrix above, with $j = 2,3,4,5,6$. This array transforms in fact in the fundamental representation for transformations in every $SU(5) \subset Sp(12,\mathbf{C})$ which acts on indices values $2 \div 6$. The second family transforms in the representation $10$ of $SU(5)$ (the skew symmetric representation). Unfortunately it isn’t the sub matrix $(\w^{ij})$ with $i,j = 2,3,4,5,6$. This is in fact the skew adjoint representation of $U(5,\mathbf{Y})$, which is skew hermitian and not skew symmetric. Do not lose heart. We’ll see in a moment that such adjoint representation is a quaternionic combination of three skew symmetric representations, one for every fermionic family. This concept could appears cumbersome, but it will be clear along the following calculations. The skew adjoint representation of $U(m,\mathbf{H})$ is a quaternionic combination of three skew symmetric representations of $U(m) = U(m,\mathbf{C})$ plus a real skew symmetric representation (which is also skew hermitian). Consider a fermionic matrix $\psi$ which transforms in the adjoint representation of $U(m,\mathbf{H})$: $$\psi \ra U\psi U^\dag$$ Take then a matrix $\psi'$ with $\psi' k =\psi$. Its transformation law under $U(m) = U(m,\mathbf{C})$ is easily derived when this group is constructed by using imaginary unit $i$ or $j$: $$\psi' k \ra U \psi' k U^\dag = U \psi' U^T k .$$ Here we have used the relation $k \lambda = \lambda^* k$ for $\lambda \in \mathbf{H}$ without $k$ component. We see that $\psi'$ transforms in the skew symmetric representation: $$\psi' \ra U \psi' U^T$$ We obtain a complex matrix $\psi'$ (with $i$ as imaginary unit) when $\psi$ has the form $Ak+Bj$ with $A,B$ real matrices. Indeed: $$\psi' = - \psi k = -Akk-Bjk = A - Bi$$ Sending $\psi$ in $\psi^*$ we bring $\psi'$ to $-\psi'$ and so we satisfy the skew symmetry. Finally we can always write $$\psi = \psi_0 + \psi_1 k + \psi_2 i + \psi_3 j$$ In this decomposition $\psi_1, \psi_2, \psi_3$ are complex matrices with complex unit respectively $i, j, k$. Explicitly: \_1 &=& \_1 - i\_1 = \_1\^1 - i\_1\^1 + I(\_1\^2 - i\_1\^2)\ \_2 &=& \_2 - j\_2 = \_2\^1 - j\_2\^1 + I(\_2\^2 - j\_2\^2)\ \_3 &=& \_3 - k\_3 = \_3\^1 - k\_3\^1 + I(\_3\^2 - k\_3\^2). Here all $\phi^1$, $\phi^2$ and $\xi^1$, $\xi^2$ are real fields. In this way $\psi_{1,2,3}$ transform in the skew symmetric representation of $U(m)$ when we construct this group by using the correspondent imaginary unit ($i$ for $\psi_1$, $j$ for $\psi_2$ and $k$ for $\psi_3$). Hence they define the famous three fermionic families, relate each other by $U(1,\mathbf{H})$ transformations. Moreover $\psi_0$ is a real skew symmetric field. $\pt$ Consider the following lagrangian tr(\^ ) &=& tr(k\^\* \^\_1 \_1 k) + tr(i\^\* \^\_2 \_2 i) +tr(j\^\* \^\_3 \_3 j)\ && - tr(i\^\* \_2\^\_3 i) - tr(j\^\* \_3\^\_1 j) - tr(k\^\* \_1\^\_2 k)\ &&- tr(\_0\^\_0) &=& tr(\^\_1 \_1 kk\^\*) + tr(\^\_2 \_2 ii\^\*) +tr(\^\_3 \_3 jj\^\*)\ && - tr(\_2\^\_3 ii\^\*) - tr(\_3\^\_1 jj\^\*) - tr(\_1\^\_2 kk\^\*)\ && - tr(\_0\^\_0)\ &=& tr(\^\_1 \_1) + tr(\^\_2 \_2) +tr(\^\_3 \_3)\ && - tr(\_2\^\_3) - tr(\_3\^\_1) - tr(\_1\^\_2)\ &&- tr(\_0\^\_0) In the third last line we have the fermionic terms in Georgi-Glashow model for three families of fields in representation $10$. In this way we can use the lagrangian $tr(\psi^{\dag} \na \psi)$, with $\psi$ in the adjoint representation, in place of Georgi-Glashow terms with $\psi_{1,2,3}$ in the skew symmetric representation. Mixed terms in the second last line give a reason to CKM and PMNS matrices which appear in standard model. Consider now the equivalence $$tr(\psi^{\dag} \psi \na ) = tr( \psi \na \psi^{\dag}) = tr( (-\psi^{\dag}) \na (-\psi) )= tr(\psi^{\dag} \na \psi).$$ Hence tr(\^ ) = 12 tr(\^ {, }).\[anticom\] In this formalism, given $\w \in su(3)\otimes su(2) \otimes u(1)$, the transformation $\d\psi = [\w,\psi]$ corresponds to the usual transformation $\d\psi = \w\psi$ in the standard model formalism. We see that the only fields which transform correctly under $SO(1,3)$ are $e$, $\nu$ and $d^c$. For now we do not care. We note rather that, when we restrict the elements of $\w$ from the hyperions to the complex numbers, we have $3$ possibilities to do it. A complex number is not only in the form $a+ib$, with $a,b \in R$, but also $a+jb$ and $a+kb$. The same is true for a fixed linear combination $a+(ci+dj+fk)b$, where $c,d,f \in R$ and $c^2 + d^2 + f^2 =1$. The choice of $j$ in place of $i$ determines another set of ($\w,\psi)$ isomorphic to the first one. In the same way we obtain a third set choosing $k$. Note that for a $i$-complex left field we have an $Ii$-complex right field and so on for $j$ and $k$. The three sets are related by the group $SU(2)$ which rotates an unitary vector in $R^3$ with coordinates $(c,d,f)$. Its generators are $$\w = \fr {\vec{y}}{6} \left( \begin{array}[c]{cccccc} 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \\ \end{array} \right) .$$ Their diagonal form suggests an identification between this group and the gravitational group $SU(2)^{\subset SO(1,3)}$. If the two groups coincided, all fields would transform correctly under $SU(2)^{\subset SO(1,3)}$. By extending this group to the entire $SO(1,3)$, we see that boosts exchange left fields with right fields. Note that three families have to exist also for bosonic particles (photon, $W^\pm$, $Z$, gluons) although they are probably indistinguishable. Other interesting thing is that we have no warranty for the persistence of $Sp(12,\mathbf{C})$ in the entire universe. However we have surely at least the symmetry $U(1,\mathbf{Y}) = SO(1,3)$, which implies the secure existence of gravity. ### Fermions from an extended arrangement matrix We introduce the following entities: I-complex grassmannian coordinates $\theta = \theta^1 + I\theta^2$ and $\bar{\theta} = \theta^1 - I\theta^2$; Grassmannian derivatives $\pa_g$ and $\bar{\pa}_g$, with $\pa_g \theta = \bar{\pa}_g \bar{\theta} =1$ and $\pa_g \bar{\theta} = \bar{\pa}_g \theta =0$; Grassmannian covariant derivatives $\na_g = \pa_g +\psi$ and $\bar{\na}_g^\dag = \bar{\pa}_g+ \bar{\psi}^\dag$. The fundamental products return $$\theta\theta = \theta^1 \theta^1 + \theta^1 I \theta^2 + I \theta^2 \theta^1 - \theta^2\theta^2 = 0 + I \theta^1 \theta^2 - I \theta^1 \theta^2 - 0 = 0 \nonumber$$ || &=& \^1 \^1 - \^1 I \^2 - I \^2 \^1 - \^2\^2 = 0 - I \^1 \^2 + I \^1 \^2 - 0 = 0\ | &=& \^1 \^1 - \^1 I \^2 + I \^2 \^1 + \^2\^2 = - I \^1 \^2 - I \^1 \^2 = -2 I \^1 \^2 In the arrangement field formalism, covariant derivatives descend from a grassmanian matrix $M_g$ or $\bar{M}_g^\dag$. We can consider a unique generalized matrix $M_T = M_g + M$ that, up to a generalized $U(n,\mathbf{Y})$, becomes M\_T &=& \_g + d\^\_= \_g + + d\^\_\ |[M]{}\^\_T &=& |\^\_g | + |\^\_|[d]{}\^ = |\_g | + |\^| + |\^\_|[d]{}\^.\[deriv\] Expanding $tr\,(\bar{M}^\dag_T M_T)$ we obtain tr(|[M]{}\^\_T M\_T) &=& tr(d\^|[d]{}\^ |\^\_\_) = \_a h R(x\^a) \[ricci2\].To calculate $tr\,(\bar{M}^\dag_T M_T\bar{M}^\dag_T M_T)$ we write first $\bar{M}_T^{\dag 2}$ and $M_T^{2}$. M\^2\_T &=& \_g + + d\^{ \_, } + d\^\_d\^\_\ |[M]{}\^[2]{}\_T &=& |\_g | + |\^| + {|\^, |\^\_} |[d]{}\^ | + |\^\_|[d]{}\^|\^\_|[d]{}\^ If $M$ has the form (\[deriv\]), then $[M_T,\bar{M}^\dag_T] = 0$. This implies $$tr\,(\bar{M}^\dag_T M_T\bar{M}^\dag_T M_T) = tr\,(M_T^2 \bar{M}_T^{\dag 2}).$$ We calculate its value starting from the following product tr(d\^{\_, }{|\^, |\^\_}|[d]{}\^| ) &=& tr(| d\^{\_, }{|\^, |\^\_}|[d]{}\^).\ && Remember that operator $tr$ acts as a sum over vertices. Now every vertex is labeled by a couple $(\theta, x_i)$ and then $$tr\, (\theta \bar{\theta} (***)) = \left( \int d\bar{\theta} d\theta\,\theta \bar{\theta} \right)tr\,(***) = tr\,(***)$$ Hence tr(d\^{\_, }{|\^, |\^\_}|[d]{}\^|) &=& tr(d\^{\_, }{|\^, |\^\_}|[d]{}\^)\ &=& tr(|[d]{}\^ d\^|\^)\ &=& \_a h R(x\^a) |\^ In this way tr(|[M]{}\^\_T M\_T|[M]{}\^\_T M\_T) &=& tr (|\^d\^ {\_, } + {|\^, |\^\_} |[d]{}\^) +\ && + \_a h R(x\^a) |\^+ S\_[GB]{} \[ora\] We have seen that every family distinguishes itself by the choice of complex unity. Inserting in $\psi$ the definitions of $\psi_{1,2,3}$ we can write &=& \_0\^1 + i(\_2\^1 + \_3\^1) + j(\_3\^1 + \_1\^1) +k(\_1\^1 + \_2\^1) +\ && + I\_0\^2 + iI(\_2\^2 + \_3\^2) + jI(\_3\^2 + \_1\^2) + kI(\_1\^2 + \_2\^2) Using the correspondences $(1,I,i,j,k,iI,jI,kI) \leftrightarrow \g\g$ and $4 \leftrightarrow tr$, the first term in (\[ora\]) becomes $$2 \times \fr 14 \times tr\, \left( {\psi}^{lm} \overline{(\g_l\g_m)^\dag} \left( \g_0 \g_s e^{\mu s} \overset{G}{\na}_\mu \psi^{np} (\g_n \g_p) + A_\mu \psi \right)\right)$$ === $$\fr 12 \, tr\, \left( \psi^{lm} (\g_m\g_l) \left( \g_0 \g_s e^{\mu s} \overset{G}{\na}_\mu, \psi^{np} (\g_n \g_p) + A_\mu \psi_0 + \sum_{q,q'=1}^3 A^q_\mu \psi_{q'} i_{q'} \right)\right)$$ Here we have deleted the anticommutator by means of (\[anticom\]). In the covariant derivative we have included only the gravitational (track) contribution, while $A_\mu$ is intended to have null track. Moreover $i_1 =k$, $i_2 =i$ and $i_3 =j$. In the second line we have divided the $75$ generators $A_\mu$ in three families of $35$ generators. Obviously, only two families are linearly independent. When they act on spinorial fields which belong to their own family, they behave exactly as the $35$ generators of $SU(6)$ (which comprise the $24$ generators of $SU(5)$). Conversely, when a generator $A^q$ acts on a $q'$-field (with $q \neq q'$), it mimics the application of some generator $A^{q'}$ followed by a rotation in $SU(2)_{GRAVITY}$ which sends the family $q'$ in the remaining family $q''$. We explicit now one entry of $\psi = \psi^1 + I\psi^2$ by exploiting the correspondence with $\g\g$. We have $$\psi = \left( \begin{array}[c]{cc} \psi_0^1 + i(\phi^1_2+\xi_3^1)&(\phi_3^1+\xi_1^1) +i(\phi^1_1 +\xi^1_2) \\ -(\phi_3^1+\xi_1^1)+i(\phi^1_1 +\xi^1_2)&\psi_0^1-i(\phi^1_2+\xi_3^1) \\ i\psi_0^2-(\phi^2_2+\xi_3^2)&i(\phi_3^2+\xi_1^2)+(\phi^2_1 +\xi^2_2) \\ -i(\phi_3^2+\xi_1^2)+(\phi^2_1 +\xi^2_2)& i\psi_0^2+(\phi^2_2+\xi_3^2) \\ \end{array}\right.$$ $$\left. \begin{array}[c]{cc} i\psi_0^2-(\phi^2_2+\xi_3^2) & i(\phi_3^2+\xi_1^2)+(\phi^2_1 +\xi^2_2) \\ -i(\phi_3^2+\xi_1^2)+(\phi^2_1 +\xi^2_2) &i\psi_0^2+(\phi^2_2+\xi_3^2) \\ \psi_0^1+i(\phi^1_2+\xi_3^1) &(\phi_3^1+\xi_1^1)+i(\phi^1_1 +\xi^1_2) \\ -(\phi_3^1+\xi_1^1)+i(\phi^1_1 +\xi^1_2) &\psi_0^1-i(\phi^1_2+\xi_3^1) \\ \end{array}\right)$$ If we define the four components spinor $$\hat{\psi} = \left(\begin{array}[c]{c} \psi_0^1 + i(\phi^1_2+\xi_3^1) \\ -(\phi_3^1+\xi_1^1)+i(\phi^1_1 +\xi^1_2) \\ i\psi_0^2-(\phi^2_2+\xi_3^2) \\ -i(\phi_3^2+\xi_1^2)+(\phi^2_1 +\xi^2_2) \end{array} \right)$$ the derivative term can be rewritten as 2 \^ \_0 \_s e\^[s]{} \_ This is the Dirac action, although with a new interpretation of spinorial components. Moreover $$\psi^{AB} = W^{ABC} \hat{\psi}^C \qquad ; \qquad W^{ABC} W^{DBC} = \mathbf{1}_{AD}$$ $$\left(W^{ABC}\right) =$$ $$\pt$$ $$\left( \left( \begin{array}[c]{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ \end{array}\right), \left( \begin{array}[c]{cccc} 0 & -\ast & 0 & 0 \\ \ast & 0 & 0 & 0 \\ 0 & 0 & 0 & \ast \\ 0 & 0 & -\ast & 0 \\ \end{array}\right), \left( \begin{array}[c]{cccc} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ \end{array}\right), \left( \begin{array}[c]{cccc} 0 & 0 & 0 & \ast \\ 0 & 0 & -\ast & 0 \\ 0 & -\ast & 0 & 0 \\ \ast & 0 & 0 & 0 \\ \end{array}\right) \right)$$ $$\pt$$ with $\ast \hat{\psi} = \hat{\psi}^\ast$. Adding the other terms $$tr\,(\hat{M}^\dag \hat{M} \hat{M}^\dag \hat{M}) = S_{GB} +$$ $$+ 2 \int \left( \hat{\psi}^\dag\, \g_0 \g_s e^{\mu s} \overset{G}{\na}_\mu \hat{\psi} + \hat{\psi} \sum_{q,q'} A^q_\mu \hat{\psi}_{q'} i_{q'} + \sqrt h R(x)\sum_q \hat{\psi}_q^\dag \hat{\psi}_q \right) dx$$ In this way we include all the contents of standard model as elements in the generalized $Sp(12,\mathbf{C})$ algebra. Terms which mix families can be used to calculate values in CKM and PMNS matrices. Masses for fermionic fields arise, as usual, from non null expectation values of $A_\mu(x^a_i, x^b_j)$ with $a\neq b$ in $\na_\mu$. We obtain a contribute to Hilbert-Einstein action also from term $\int d^4x \sqrt h R \bar{\psi} \psi$, due to a non null expectation value of $\sum_q \bar{\psi}_q \psi_q $. It contains in fact the chiral condensate, whose non null vacuum value breaks the chiral flavour symmetry of QCD Lagrangian. Note that known fermionic fields fill up a matrix $\psi$ with null track. However, only if $tr\,\psi \neq 0$ our action has an extra invariance under A\_&& d\_\^[-1]{}\ && \_g d\^A\_.\[super\]Here $\overleftarrow{\pa}_g$ is a $\pa_g$ which acts backwards. This means we have the same number of fermions and bosons, so that the vacuum energies erase each other. Invariance (\[super\]) predicts the existence of a new colored fermionic sextuplet which sits on diagonal in $\psi$. Inside it we can include a conjugate neutrino ($\nu^c$), a sterile neutrino ($N$) and a conjugate sterile neutrino ($N^c$). Explicitly $$\psi = \left( \begin{array}[c]{cccccc} N & 0 & 0 & 0 & 0 & 0 \\ 0 & \nu^c & 0 & 0 & 0 & 0 \\ 0 & 0 & \nu^c & 0 & 0 & 0 \\ 0 & 0 & 0 & N^c & 0 & 0 \\ 0 & 0 & 0 & 0 & N^c & 0 \\ 0 & 0 & 0 & 0 & 0 & N^c \end{array} \right).$$ This field commutes with any gauge field in $U(1) \otimes SU(2) \otimes SU(3)$ and so it hasn’t electromagnetic, weak or strong interactions. Moreover it gives a Dirac mass to neutrinos via the term $$tr\,(\bar{\psi}^\dag d^\mu A_\mu \psi) = \bar{\psi}^{\dag ij} d^\mu A_\mu^{kl} \psi^{mn} f^{(ij)(kl)(mn)}.$$ Here $f^{(ij)(kl)(mn)}$ are structure constants for $SU(6)$ and masses for neutrinos are eigenvalues of $<d^\mu A_\mu >$. ### The vector superfield The invariance (\[super\]) suggests a connection with super-symmetric theories. We redefine the supersymmetry algebra as follows: Q = \_g - d\^| \_ &;& \[Q, \_\] = - d\^| \_\ |[Q]{} = |\_g - |[d]{}\^ \_ &;& {Q, |Q } = -2d\_H\^\_- \^ \_\ 2 d\_H\^&=& d\^+ |[d]{}\^ Here $\widetilde{\na}$ is a compatible covariant derivative which acts as a skew-adjoint operator. It is a functional of $d^\mu$ with $[\widetilde{\na}_\nu, d^\mu] = [\widetilde{\na}_\nu, \bar{d}^{\dag\mu}] = 0$. Note that off-shell we have $\widetilde{\na}_\nu \neq \overset{G}{\na}_\nu$. Moreover $${\Sigma}^{\mu\nu} = d^\mu \bar{d}^{\dag\nu} \theta \bar{\theta}$$ $\widetilde{R}_{\mu\nu}$ is the curvature tensor made with $\widetilde{\na}$, ie $\widetilde{R}_{\mu\nu} = [\widetilde{\na}_\mu, \widetilde{\na}_\nu]$. Consider that locally we can find a coordinate system where $\widetilde{R}_{\mu\nu} = 0$, recovering the usual SUSY algebra with $-i\widetilde{\na}_\mu$ in place of $P_\mu$. The vector superfield assumes the simple form[^8] V &=& + d\^A\_\ V\_[AB]{} &=& \_[AD]{} W\^[D]{}\_[BC]{} \^C + (d\^A\_)\_[AB]{} with $\theta_{AD} = \mathbf{1}_{AD}\,\theta^1 + \g_{5\,AD} \,\theta^2$. Note that $M_T = V+ \theta \pa_g + d^\mu \pa_\mu$ and then the usual kinetic term for $V$ includes the same terms we have found in $tr\,(\bar{M}_T^\dag M_T) - tr\,(\bar{M}^\dag_T M_T\bar{M}^\dag_T M_T)$. It’s remarkable that all the known fermionic fields take the role of gauginos for all the known bosonic fields. In this way the right up quarks are gauginos for gluons, while right electrons are gauginos for $W$ bosons. This is permitted because both fermions and bosons in AFT transform in the adjoint representation of $Sp(12,\mathbf{C})$. In this way our theory includes SUSY $N = 1$ with no need for new unknown particles. Inflation --------- Our final action is \[inflation\] $$S = tr\,\left(\fr {\bar{M}^\dag M }{16\pi G}- \bar{M}^\dag M \bar{M}^\dag M\right)$$ This is also an action for an $U(n,\mathbf{Y})$ gauge theory with coupling constant $1/G$ in a mono-vertex space-time. In these theories the scaling of coupling constant can be calculated exactly in the limit of large $n$. In several cases the coupling constant changes its sign for big values of scale: this has considerable consequences for the first times after Big Bang, when a measurement of $G$ has sense only at very high energies (very small distances). What said suggests that such measurement can return a negative value of $G$, which implies a repulsive force of gravity. In turn, repulsive gravity implies an accelerate expansion for the universe. Because the entries of $M$ are probability amplitudes, we would be it was dimensionless. However, when we pass from $M$ to $\na$, we need a scale $\Delta$ to define the matrix $\pa$. This justify the inclusion of $\Delta^{-1}$ inside $M$. If we extract this factor, the Hilbert Einstein action becomes $$\fr {\Delta^4}{16\pi G \Delta^2}tr\,(\bar{M}^\dag M) = \fr {\Delta^2}{16\pi G}tr\,(\bar{M}^\dag M)$$ where we have also added the correct volume form $\Delta^4$. This seems a more natural formulation when $M$ represents probability amplitudes. In this way we can take $\Delta$ very small but not zero. The most natural choice is $\Delta^2 \sim G$. In this case, what does it mean that $G$ is negative? Negative $G$ implies negative $\Delta^2 = ds^2$. In lorentzian spaces $\Delta^2 = dt^2 -ds^2 <0$. For purely temporal intervals we’ll have $dt^2 < 0$, so the time becomes imaginary. An imaginary time is indistinguishable from space. This hypothesis of a spatial” time had already been espoused by Hawking as a solution for eliminate the singularity in the Big Bang [@Hawking]. Classical solutions {#classical} ------------------- We rewrite our action in the form $$S = \fr 12 tr\, (\bar{M}^\dag M) - \fr 1{4g} tr\,(\bar{M}^\dag M \bar{M}^\dag M)$$ where we have defined $g = \fr {\Delta^2} {32\pi G}$. We diagonalize $M$ with a transformation in $U(n,\mathbf{Y})$ and define $M^{ii} \equiv \f(x_i)$, $\f(x) = a(x)+\vec{b}(x)$. The lagrangian becomes: $$L = \fr 12 \left[ a(x_i)^2 + |\vec{b}(x_i)|^2\right] -\fr 1{4g} \left[ a(x_i)^4 + |\vec{b}(x_i)|^2 + 2a(x_i)^2 |\vec{b}(x_i)|^2 \right]$$ The motion equations are $$g a(x) - a(x)^3 - a(x)|\vec{b}(x)|^2 = 0$$ $$g \vec{b}(x) - \vec{b}(x)|\vec{b}(x)|^2 - a(x)^2\vec{b}(x) = 0$$ There are two solutions: $$(1) \qquad a(x) = \vec{b}(x) = 0$$ $$(2) \qquad a(x)^2 + |\vec{b}(x)|^2 = \bar{M}^\dag M = g$$ The first one corresponds to the vacuum (all non-gravitational fields equal to zero) plus a solution of Einstein equations in the vacuum: $$\psi = A_\mu = 0 \qquad R(x) = 0$$ The solution $\bar{M}^\dag M = g$ corresponds to a vacuum expectation value for $\bar{M}^\dag M$ equal to $g$. $M$ contains a factor $A$, so that an expectation value for $\bar{M}^\dag M$ corresponds to an expectation value for $AA$. This means that $$AAAA = <AA>AA + \text{quantum perturbations}$$ $<AA>$ gives a mass for $A$. More precisely, for $A \in U(n,\mathbf{Y})/U(m,\mathbf{Y})^{n/m}$, $$m_A^2 \sim \fr {<\bar{M}^\dag M>}{\Delta^2} = \fr g {\Delta^2} = \fr 1 {32\pi G}$$ So the fields $A \in U(n,\mathbf{Y})/U(m,\mathbf{Y})^{n/m}$ have a mass in the order of Planck mass $m_P$. Moreover, in the primordial universe, when $k_B T \approx m_p$, all the fields behave like null mass fields. In that time the symmetry was then $U(n,\mathbf{Y})$ and no arrangement exists. Our conclusion is that Quantum Gravity cannot be treated as a quantum field theory in an ordinary space. In what follows we explain how overcome this trouble. Quantum theory {#quantize} -------------- [1]{} Quantum theory is defined via the following path integral: && D\[M(x,y)\]D\[|M\^\* (x,y)\]\ && Oe\^[M(x,y)|[M]{}\^\*(x,y) dx dy - M(x,y)|M\^\*(x,y’)M(x’,y’)|M\^\*(x’,y) dx dy dx’ dy’]{} with && Oe\^[F(x,y)dx dy]{} =\ && = 1+ F(x,y)dx dy + 12 F(x,y)F(x\^1,y\^1)dx dy dx\^1 dy\^1 +\ &&+ …+ 1 [n!]{} F(x,y)F(x\^1,y\^1)…F(x\^[n-1]{},y\^[n-1]{}) dx dy dx\^1 dy\^1 …\ && …dx\^[n-1]{} dy\^[n-1]{} Oe\^[F(x,x’,y,y’)dx dx’ dy dy’]{} &=& 1+ F(x,x’,y,y’)dx dy dx’ dy’+\ && + 12 F(x,x’,y,y’)F(x\^1,[x’]{}\^1,[y’]{}\^1,y\^1)dx dy dx’ dy’ dx\^1 dy\^1 d[x’]{}\^1 d[y’]{}\^1 +\ && + …+ 1 [n!]{} F(x,x’,y,y’)F(x\^1,[x’]{}\^1,y\^1,[y’]{}\^1)…\ && …F(x\^[n-1]{},[x’]{}\^[n-1]{},y\^[n-1]{},[y’]{}\^[n-1]{}) dx dy dx’ dy’ dx\^1 dy\^1 d[x’]{}\^1 d[y’]{}\^1…\ && …dx\^[n-1]{} dy\^[n-1]{} d[x’]{}\^[n-1]{} d[y’]{}\^[n-1]{}\ && 1 [n!]{} F\^n [2]{} The integration of $F^n$ is very simple and gives $$\fr 1 {n!} \int D^2[M] e^{\int M^2 dx dy} \,\,F^n = \fr {(4n)!}{n! 2^{2n}(2n)!} = \fr 1 {n!} P(4n)$$ Here $P(4n)$ gives the number of different ways to connect in couples $4n$ points. It’s clear that any universe configuration corresponds to an $F^k$ inside which some connections have been fixed and the corresponding integrations have been removed. For example: {width="100.00000%"} If the fixed connections are $m$, then $$<\hat F^k > = \fr{\sum_n \fr 1 {n!} P(4(n+k)- 2m)}{\sum_n \fr 1 {n!} P(4n)}$$ At relatively low energies we can tract $\overset{G}{A}$ as an ordinary gauge field. The arrangement field theory is then approximated with a common quantum theory on a curved background, determined by $e^{\mu a}$. Quantum Entanglement and Dark Matter\[entanglement\] ---------------------------------------------------- The elements of $M$ which do not reside in or near the diagonal, describe connections between points that are not necessarily adjacent to each other, in the common sense. These connections construct discontinuous paths as in figure \[cammini-entanglement\] and can be considered as quantum perturbations of the ordered space-time. Such components permit us to describe the quantum entanglement effect, as it could be shown in detail in a complete coverage that goes beyond the purpose of the present paper. It is remarkable that in this framework the discontinuity of paths is only apparent, and it is a consequence of imposing an arrangement to the space-time points. These discontinuous paths can be considered as continuous paths which cross wormholes. The trait of path inside a wormhole is described by a component of $M$ far away from diagonal. The information seems to travel faster than light, but in reality it only takes a byway. ![Discontinuous paths. The connection between $x_3$ and $x_4$ is done by a component of $M$ far away from diagonal.[]{data-label="cammini-entanglement"}](cammini-entanglement.jpg){width="60.00000%"} Imagine now a gravitational source with mass $M_S$ which emits some gravitons with energy $\sim E_{PLANCK}$, directed to an orbiting body with mass $M_B$ at distance $r$. In this case (respect such gravitons) the fields $M(x^a, x^b)$ with $a \neq b$ would behave as they had null mass. This implies the probable existence of connections (practicable by such gravitons) between every couple of vertices in the path from the source to the orbiting body. This means that if $r = \Delta j$, $j \in \mathbf{N}$, the graviton could reach the orbiting body by traveling a shorter path $\Delta j'$, $j>j' \in \mathbf{N}$. The question is: what is the average gravitational force perceived by the orbiting body? The probability for a graviton to reach a distance $r$ passing through $m$ vertices is $$P_m = (1-a)^{m-1}a \qquad with\,\,\,\sum_{m=1}^\infty P_m = 1$$ where $a = 1/j$. These are the probabilities for extracting one determined object in a box with $j$ objects at the $m$-th attempt. In this way the effective length traveled by the graviton will be $\Delta m$. We use these probabilities to compute the average gravitational force in a semiclassical approximation. &=& G a [1-a]{} \_1\^ dm\ &=& G a [1-a]{} \[log(1-a)\] \_[log(1-a)]{}\^[-]{} dx\ && \[dark\]The last integral gives $$\int_{log(1-a)}^{-\infty} \fr{e^x}{x^2}dx = -Ei(log(1-a)) + \fr {1-a}{log(1-a)}$$ We expand $\langle F \rangle$ near $a=0$ (which implies $j >> 1$), obtaining $$\fr a {(1-a)} [log(1-a)]\int_{log(1-a)}^{-\infty} \fr{e^x}{x^2}dx \approx a + a^2(log(a) + \g) + O(a^3).$$ Here $\g$ is the Eulero-Mascheroni constant. The dominant contribution is then && G a (1+alog(a) + a)\ && ( 1 - r ( log( r ) - ) ) If the massive object orbits at a fix distance $r$, its centrifugal force has to be equal to the gravitational force. This gives $$<F> \approx \fr{G}\Delta\fr{M_{B}M_{S}}{r}\left( 1 - \fr \Delta r \left( log\left( \fr r \Delta \right) - \g \right) \right) = \fr {M_B v^2} r$$ $$v^2 = \fr{G M_{B}M_{S}}{\Delta}\left( 1 - \fr \Delta r \left( log\left( \fr r \Delta \right) - \g \right) \right)$$ We see that, varying the radius, the velocity remains more or less constant (It increases slightly with $r$). Can this explain the rotation curves of galaxies without introducing dark matter? Surely not all gravitons have energy $> E_{PLANCK}$; at the same time we have to consider that $G$ scales for small distances (hence for small $m$ in (\[dark\])). It’s possible that these factors reduces the extremely high value of $r/\Delta$. Conclusion ---------- In this work we have applied the framework developed in [@Arrangement] to describe the contents of our universe, ie forces and matter. Doing this, we have discovered an unexpected road toward unification, which shows similarities with Loop Gravity, String Theory and Georgi - Glashow model. For the first time a natural symmetry justifies the existence of three particles families, not one more, not one less. Moreover a new version of supersymmetry seems to couple gauge fields with all known fermions, without necessity of imagining new particles never seen by experiments. Clearly this fact closes the door to dark matter. To compensate this big absence, AFT proposes an explanation to galaxy rotation curves which doesn’t make use of dark matter. Another considerable implication of AFT regards tangent space, which has symmetry $SO(1,3)$ only when gravity decouples from other forces. At that point also the real space-time can obtain the same symmetry. This fact is coherent with *no-go theorem* of Coleman-Mandula[@nogo], under which $S$-matrix is Lorentz invariant if and only if the action symmetry is $SO(1,3) \otimes internal\,\,symmetries$”. We don’t say that this theory is exact. However there are several good signals which must be taken into account. We hope that a future teamwork can verify this theory in detail, deepening all its implications. Antigravity in AFT ================== Introduction ------------ Arrangement field theory is a quantum theory defined by means of probabilistic spin-networks. These are spin-networks where the existence of an edges is regulated by a quantum amplitude. AFT is a proposal for an unifying theory which joins gravity with gauge fields. See [@Arrangement] and [@Arrangement2] for details. The unifying group is $Sp(12,\mathbf{C})$ for the lorentzian theory and its compact real form $Sp(6)$ for the euclidean theory. The unifying group contains three indistinguishable copies of gauge fields, mixed by gravitational field. Moreover, commutators between gravitational and gauge fields are non null and give new terms for the Einstein equations. In what follows we focus on the term which mixes gravity with electromagnetism, showing that its contribution to Einstein equations could generate antigravity. In the end we verify that new interactions don’t affect the making of nucleus and nucleons. Antigravity {#formalism2} ----------- [1.5]{} The term which mixes gravity with electromagnetism is given by space-time integration of the following expression: $$-\fr 14 f^{(G)(EM1)(EM2)} A^{(G)}_\mu A^{(EM1)}_\nu \bigg( F^{(EM2)\mu\nu} + \a f^{(EM3)(EM1)(EM2)} A^{(EM3)\mu} A^{(EM1)\nu} \bigg)$$ \[prima\]Remember that AFT includes three indistinguishable electro-magnetic fields, with non-trivial commutators. In this way $A^{(G)}$ is the gravitational gauge field, $A^{(EMn)}$ is the n-th electromagnetic field and $\a$ is the fine structure constant. In the realistic case of null torsion, the gravitational gauge field can be rewritten in function of the tetrad field: $$A_\mu^{(G)bc} = \fr 12 e^{\nu [b} \pa_{[\mu} e^{c]}_{\nu]} + \fr 14 e_{\mu d} e^{\nu b} e^{\sigma c} \pa_{[\sigma} e^d_{\nu]}$$ From now we take a low energy limit so defined: $e_{ii} = 1$ with $i=1,2,3$, $e_{00} = \theta(x)$ and $\pa_0 \theta(x) =0$. Varying with respect to $e$ we obtain: $$\fr {\d A_\mu^{(G)bc}}{\d e^s_\tau} = \fr 12 e^{\nu [b} \d^{c]}_s \d^\tau_{[\nu} \pa_{\mu]} + \fr 14 e_{\mu s} e^{\nu b} e^{\sigma c} \d^\tau_{[\nu} \pa_{\sigma]}$$ $$\fr {\d A_\mu^{(G)bc}}{\d g_{\w\tau}} = 2e^{\w s} \fr {\d A_\mu^{(G)bc}}{\d e^s_\tau} = e^{\w [c} e^{b] \nu} \d^\tau_{[\nu} \pa_{\mu]} + \fr 12 \d^\w_{\mu} e^{\nu b} e^{\sigma c} \d^\tau_{[\nu} \pa_{\sigma]}$$ The component with $c = \w =\tau = 0$ and $b \neq 0$ results: $$\fr {\d A_\mu^{(G)b0}}{\d g_{00}} = -\theta^{-1} \d^0_\mu \pa_b - \fr 12 \theta^{-1} \d^0_\mu \pa_b = -\fr 3{2\theta} \d^0_\mu \pa_b$$ $$A^{(EM)\rho}A^{(EM)}_\rho A^{(EM)\mu}\fr {\d A_\mu^{(G)b0}}{\d g_{00}} = \fr 3{2\theta} \pa_b A^{(EM)0} A^{(EM)\rho}A^{(EM)}_\rho$$ The minus sign has disappeared because we have reversed the derivative. The variation of quartic term in (\[prima\]) with respect to $\d g_{00}$ is then given by $$-\fr \a 4 \cdot \fr 3{2\theta} \pa_b f^b A^{(EM)0} A^{(EM)\rho}A^{(EM)}_\rho = -\pa_b f^b \fr {3\a} {8\theta} V(\theta^2 V^2 - A^2)$$ $$f^b = \sum_{cade} f^{(bo)ca} f^{dea} \approx 4\fr {x^b}{r}.$$ Here we have indicated with $V$ the electric potential and with $A$ the magnetic vector potential. The sum inside $f$ is over the three electromagnetic fields. It’s so clear that varying the complete action with respect to $g_{\mu\nu}$ we obtain a new term for Einstein equations. In the Newtonian limit we can substitute $g_{00} = -(1-2\phi)$ and $R_{00} - (1/2)Rg_{00} = \na^2 \phi$ where $\phi$ is the newtonian potential. Hence: 2 \^2 && 8T\^[00]{} = 8\ && \_b 24V(V\^2 - \^[-1]{} A\^2) For radial potential we have $$\pa_b \phi = \fr {x^b}{r} \pa_r \phi .$$ In such case $$C_G = \pa_r \phi \approx 12\pi\a V(\theta V^2 -\theta^{-1} A^2)$$ Now we insert the appropriate universal constants and approximate $\theta$ with $1$: C\_G 12 V(V\^2 - c\^2 A\^2) = k V(V\^2 - c\^2 A\^2) \[ultima\] Here $L_p$ is the Planck length, equal to $\sqrt {\hbar G/c^3}$. The multiplicative constant is $$k = \fr{12\pi}{137}\cdot \fr{(6,67\cdot 10^{-11}\cdot 8,85\cdot 10^{-12})^{3/2}}{(3\cdot 10^8)^4 \cdot(1,62\cdot 10^{-35})} = 30,27\cdot 10^{-33} \,\left(\fr{C^3 s^4}{Kg^3 m^5}\right).$$ This means that for having a weight variation (on Earth) of about $10\%$ ($\Delta C_G =1$) we need an electrical potential of $10^{11}$ Volts. These are $100$ billions of Volts. For $V = Q/r$ and $A=0$ we have: $$C_G = \fr {k}{(4\pi\e_0)^3}\cdot \fr {Q^3}{r^3} = 2,198 \cdot 10^{-2} \left(\fr {m^4}{s^2 C^3}\right)\fr {Q^3}{r^3}$$ Note that the sign of $C_G$ is the sign of $Q$ and then we obtain antigravity for negative $Q$. We associate to this interaction an equivalent mass $m$, substituting $C_G = Gm/{r^2}$. We have $$m = \fr k G V^3 r^2 = \fr {k}{G(4\pi \e_0)^3}\fr {Q^3} r = 3,293\cdot 10^8 \left(\fr {Kg\, m}{C^3}\right) \fr {Q^3}{r}$$ which is a negative mass for negative $Q$. Negative mass implies negative energy via the relation $E =mc^2$. Intuitively, if we search a similar relation for gravi-magnetic field (which is $\na \times (g^{0i})$, $i=1,2,3$), we should find the same formula (\[ultima\]) with an exchange between $V$ and $cA$. We calculate now at what distance the gravitational attraction between two protons is equal to their electromagnetic repulsion. $$G\fr {m^2}{r^2} = \fr {k^2}{G^2 (4\pi\e_0)^6} \fr {Q_p^6}{r^4} = \fr 1 {4\pi \e_0} \fr {Q_p^2}{r^2}$$ $$\fr {k^2 Q_p^4}{G^2 (4\pi\e_0)^5} = r^2$$ $$\Longrightarrow r^2 = 79,49 \cdot 10^{-70} m^2 \Longrightarrow r = 8,916 \cdot 10^{-35} m = 5,516\, L_p$$ Note that we are $20$ orders of magnitude under the range of strong force and $23$ orders of magnitude under the range of weak force. In this way the gravitational force doesn’t affect the making of nucleus and nucleons. Conclusion ---------- We have seen that a potential of $10^{11}$ Volts can induce relevant gravitational effects. They are too many for notice variations in the experiments with particles accelerators. However they sit at the border of our technological capabilities. 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--- abstract: 'In this letter we analyze two local extensions of a model introduced some time ago to obtain a path integral formalism for Classical Mechanics. In particular, we show that these extensions exhibit a nonrelativistic local symmetry which is very similar to the well known $\kappa$-symmetry introduced in the literature almost 20 years ago. Differently from the latter, this nonrelativistic local symmetry gives no problem in separating $1^{\text{st}}$ from $2^{\text{nd}}$-class constraints.' author: - | E. Deotto\ Dipartimento di Fisica Teorica, Università di Trieste,\ Strada Costiera 11, P.O.Box 586, Trieste, Italy\ and INFN, Sezione di Trieste title: 'Classical Mechanics and $\kappa$-Symmetry' --- =15.5pt Introduction ============ The dynamics of relativistic superparticles [@Brink] has been deeply analyzed in the last 20 years because of the profound relation between these simple systems and the more realistic models of supersymmetric field theories and strings. Almost 20 years ago an important symmetry of the massless supersymmetric particle was discovered by Siegel [@Siegel]. This symmetry, which was also found in superstrings and D-branes, allows to gauge away half of the fermionic degrees of freedom involved in the formalism and has been analyzed in detail in many following papers [@DeAz]-[@Moshe]. In particular, a lot of work has been done to understand the geometry of the constraints and to solve the problem of quantizing the system. In fact it is not trivial to quantize the massless superparticle (as well as superstrings and D-branes) because, due to the presence of the $\kappa$-symmetry, $1^{\text{st}}$-class and $2^{\text{nd}}$-class constraints cannot be separated covariantly; many attempts have been performed to solve this problem [@Sorokin][@Kallosh]. In this letter we continue the analysis (see Ref.[@DG]) of the symmetries of a model introduced some time ago to describe Classical Mechanics in terms of path integrals. This model possesses a universal [*global*]{} supersymmetry generated by two charges $\QH$ and $\QBH$. Here we focus on two other fermionic charges which we call $D_{\s H}$ and $\o{D}_{\s H}$, which are strictly related to $\QH$ and $\QBH$. In fact in superspace $D_{\s H}$ and $\o{D}_{\s H}$ are represented by the covariant derivatives associated to the Susy charges mentioned above. Following the lines of Ref.[@DG] we make these two symmetries ($D_{\s H}$ and $\o{D}_{\s H}$) local and we note that the new nonrelativistic local Susy we get is very similar to the famous $\kappa$-symmetry introduced by Siegel. The main difference with respect to the latter becomes manifest after imposing the invariance under local time reparametrization, as one does in Siegel’s model. In fact, in our nonrelativistic framework, there is no difficulty in separating $1^{\text{st}}$-class from $2^{\text{nd}}$-class constraints, simply because no $2^{\text{nd}}$-class constraint survives after imposing the invariance under local reparametrizations of time. There are two simple ways to make local the symmetries $D_{\s H}$ and $\o{D}_{\s H}$ above. The two models we obtain are two gauge theories which differ in the physical Hilbert space. We show that one model selects, as physical states, only the distributions built up with the constants of motion only, while the other is more restrictive and selects only the Gibbs distributions of the canonical ensemble. The $\kappa$-symmetry ===================== The model studied by Siegel [@Siegel] for the massless relativistic superparticle is characterized by the following ($1^{\text st}$ order) action: $$\label{1-1} S=\int d\tau\left\{p_{\mu}\left[\dot{x}^{\mu}-\frac{i}{2}\left(\zb \gamma^{\mu}\dot{\z}-\dot{\zb}\gamma^{\mu}\z\right)\right]-\frac{1}{2}\l p^2\right\},$$ where $x^{\mu}$ are $n$-dimensional space-time coordinates, $\z^{\,\s a}$ and $\zb_{\:\s a}$ are Dirac spinors and $\l$ is a Lagrange multiplier introduced to implement the $p^2=0$ constraint. This action is invariant under the following transformations: $$\begin{aligned} & \text{\bf{$\tau$-reparametrization} (local)} \nonumber \\ & \begin{array}{lll} \delta x^{\mu}=\epsilon\dot{x}^{\mu}\,;\hspace{2.8cm} & \delta p_{\mu}=\epsilon\dot{p}_{\mu}\,; \hspace{.45cm}& \delta \l=\dot{(\epsilon\l)}\,; \\ \delta \z=\epsilon\dot{\z}\,; & \delta \zb=\epsilon\dot{\zb}\,; & \end{array} \label{1-2}\vspace{.1cm} \\ & \text{\bf{Supersymmetry} (global)}\nonumber \\ & \begin{array}{lll} \delta x^{\mu}=\displaystyle\frac{i}{2} \left(\o{\ve}\gamma^{\mu}\z-\zb\gamma^{\mu}\ve\right)\,;\hspace{.6cm} & \delta p_{\mu}=0\,; \hspace{.7cm}& \delta \l=0\,; \\ \delta \z=\varepsilon\,; & \delta \zb=\o{\varepsilon}\,; & \\ \end{array}\label{1-3}\vspace{.1cm} \\ & \text{\bf{$\kappa$-symmetry} (local)}\nonumber\\ & \begin{array}{lll} \delta x^{\mu}=\displaystyle\frac{i}{2} \left(\zb\gamma^{\mu}\pslsh\k-\kb\pslsh\gamma^{\mu}\z\right)\,; & \delta p_{\mu}=0\,; & \hspace{.7cm}\delta \l=2i(\dot{\zb}\k-\kb\dot{\z})\,; \\ \delta \z=\pslsh\k \,;& \delta \zb=\kb\pslsh\; . &\\ \end{array}\label{1-4} \end{aligned}$$ In (\[1-2\]) the dot means derivation with respect to $\tau$ and $\pslsh$ is obviously $p_{\mu}\gamma^{\mu}$. As specified above, $\epsilon$ and $\kappa,\o{\kappa}$ are local parameters (the first is a commuting scalar, the others are anticommuting spinors) while $\ve$ and $\o{\ve}$ are two global (i.e. they do not depend on the base space $\tau$) spinorial parameters. We are particularly interested in the structure of the third symmetry, which has been deeply analyzed in the literature. Here we want to give a pedagogical description of the structure of the transformation in phase space, and we want to highlight the role of the various operators and various commutation structures (Dirac Brackets) involved. This will turn out to be useful when we will analyze the analog of the $\kappa$-symmetry in Classical Mechanics. First of all we notice that the first and third symmetries above are strictly related. In fact, if we introduce a mass $m$ in (\[1-1\]) turning the $p^2=0$ constraint into $p^2-m^2=0$, we get $$\label{1-5} S_{m}=\int d\tau\left\{p_{\mu}\left[\dot{x}^{\mu}-\frac{i}{2}\left(\zb \gamma^{\mu}\dot{\z}-\dot{\zb}\gamma^{\mu}\z\right)\right]-\frac{1}{2}\l (p^2-m^2)\right\}.$$ $S_m$ is still invariant under (\[1-3\]) but the other two symmetries are lost. This is easy to see in phase space if we apply the Dirac procedure to the actions (\[1-1\]) and (\[1-5\]). Consider first the massive model. The constraints are the following: $$\text{$1^{\text{st}}$-Class} \begin{cases} \Pi_{\s \l} = 0 & (a) \\ p^2-m^2=0 & (b) \end{cases} \vspace{.5cm} \hspace{1cm} \text{$2^{\text{nd}}$-Class} \begin{cases} \Pi_{\s p}^{\mu} = 0 & (c)\\ (\Pi_{\s x})_{\mu}-p_{\mu} = 0 & (d) \vspace{.1cm}\\ D^{\s a}\equiv(\Pi_{\s\zb})^{\s a}+\displaystyle\frac{i}{2}(\pslsh\,\z)^{\s a} = 0 &(e) \vspace{.15cm}\\ \o{D}_{\s a}\equiv(\Pi_{\s\z})_{\s a} +\displaystyle\frac{i}{2}(\zb\pslsh)_{\s a} = 0 & (f), \end{cases}\label{1-7}$$ where $\Pi_{(\ldots)}$ are the momenta conjugated[^1] to the variables indicated as $({\s\ldots})$, which satisfy the following (graded) Poisson Brackets[^2]: $$\begin{split} & \big[\l,\Pi_{\l}\big]_{-}=1; \hspace{2cm} \big[x^{\mu},p_{\nu}\big]_{-}=\delta^{\mu}_{\nu}; \\ & \big[\z^{\,\s a},(\Pi_{\z})_{\s b}\big]_{+}=\delta^{\s a}_{\s b}; \hspace{1.2cm} \big[\zb_{\:\s a},(\Pi_{\zb})^{\s b}\big]_{+}=\delta_{\s a}^{\s b}. \end{split}$$ The first thing to do is to construct the Dirac Brackets associated to the $2^{\text{nd}}$-class constraints. If we define the matrix $$\label{delta} \Delta_{ij}=[\phi_i,\phi_j]_{\s PB}$$ where $\phi_k$ are the second class constraints, then the Dirac Brackets between two generic variables $A,B$ of phase space are defined as: $$\label{1-8} [A,B]_{\s DB}=[A,B]_{\s PB}-[A,\phi_i]_{\s PB} (\Delta^{-1})^{ij}[\phi_j,B]_{\s PB}.$$ Once we have built the correct structure in phase space, it is not difficult to realize that the generators of the [*global*]{} supersymmetry are the following operators: $$\begin{aligned} \label{1-18} & Q=\pslsh\,\z; &\o{Q}=\zb\pslsh~\:; \end{aligned}$$ which reproduce precisely the transformations (\[1-3\]) if we define: $$\label{1-18c} \delta({\s\ldots})\equiv\big[({\s\ldots}),i\o{\varepsilon}Q-i\o{Q}\varepsilon\big]_{\s DB}.$$ Note that the minus sign in the RHS of the previous equation is chosen because of the anticommuting character of the parameter $\varepsilon$. Moreover we have: $$\label{1-19} \big[ Q,\o{Q}\big]_{DB}=i\pslsh,$$ which confirms that $Q$ and $\o{Q}$ are two supersymmetry charges. Notice that we can induce the same SUSY-transformations through the following operators: $$\begin{aligned} \label{1-20} & Q^{\prime}=i\Pi_{\zb}+\frac{1}{2}\pslsh\,\z; & \o{Q}^{\prime}=i\Pi_{\z}+\frac{1}{2}\zb\pslsh; \end{aligned}$$ which is obvious because $Q\approx Q^{\prime}$ and $\o{Q}\approx \o{Q}^{\prime}$ in the Dirac sense. Let us now switch to the massless case (\[1-1\]). The main difference is that we cannot repeat all the steps of the previous analysis. In fact the new constraint $p^2=0$ implies that the matrix $\Delta$ of Eq.(\[delta\]) is no longer invertible. This is due to the fact that $\det\Delta\propto\det(\pslsh)=p^{\mu}p_{\mu}=0$. Thus the construction of the Dirac Brackets is not as simple as in the massive case. In fact half of the constraints in Eqs.(\[1-7\]-c) and (\[1-7\]-d) are now $1^{\text{st}}$-class while the other half remains $2^{\text{nd}}$-class and the separation of the two sets is not quite easy (see for example Refs.[@Kallosh]). Nevertheless we can list the generators of the $\kappa$-transformations of Eq.(\[1-4\]): $$\begin{aligned} & K=i\pslsh D=i\pslsh\Pi_{\zb}-\frac{1}{2}\pslsh^{\;2}\z; & \o{K}=i\o{D}\pslsh=i\Pi_{\z}\pslsh-\frac{1}{2}\zb\pslsh^{\;2}. \label{1-22} \end{aligned}$$ ($K$ and $\o{K}$ generate the transformation (\[1-4\]) through commutators like those in (\[1-18c\]).) Obviously we should remember that $(K,\o{K})$ are not a set of independent constraints, as we explained before, because $\pslsh$ is not invertible on the shell of the contraints. Note that we can write down the form of the generators $K$, $\o{K}$ even if we do not know exactly the form of the Dirac Brackets in this particular case. We can do that because the $K,\o{K}$ constraints commute (weakly) with all the constraints in (\[1-7\][*c*]{}) and (\[1-7\][*d*]{}) and therefore we have $[K,({\s\ldots})]_{DB}\approx [K,({\s\ldots})]_{PB}$ (and the same holds for $\o{K}$) whatever are the surviving $2^{\text{nd}}$-class constraints determining the Dirac Brackets at hand. The Functional Approach To Classical Mechanics. =============================================== In this section we shall briefly review the path integral approach to Classical Mechanics which was originally developed in Ref.[@Ennio]. The idea originated from the fact that whenever a theory has an operatorial formulation, it also possesses a corresponding path integral. Now Classical Mechanics (CM) does have an operatorial formulation [@Koop] and therefore it is reasonable to look for the corresponding path integral formalism. The strategy to build this Classical Path Integral (CPI) is simple. In CM we have a $2n$-dimensional phase space ${\cal M}$ whose coordinates we denote by $\varphi^{a}$$(a=1,\ldots, 2n)$, i.e.: $\varphi^{a}=(q^1,\ldots, q^n; p^1,\ldots, p^n)$, and we indicate with $H(\varphi)$ the Hamiltonian of the system. Then, the equations of motion have the form: $$\label{uno} {\dot\varphi }^{a}=\omega^{ab}{\partial H\over\partial\varphi ^{b}}\equiv\w^{ab}\p_b H(\v) ~~~~~ \text{$\w^{ab}=$ symplectic matrix}.$$ The classical kernel (i.e. the probability for the system to be in the configuration $\v_f$ at time $t_f$ if it was in the configuration $\v_i$ at time $t_i$) has the following expression: $$\label{due} K_{cl}(f|i)=\delta(\varphi^{a}_{f}-\phi^{a}_{cl}(t_{f}|\varphi_{i},t_{i}))$$ where $\phi^{a}_{cl}(t|\varphi_{i},t_{i})$ is the classical trajectory at time $t$ (that is the solution of the Hamilton equations) having $\varphi_{i}$ as initial condition at time $t_i$. Since $K_{cl}(f|i)$ is a classical probability we can rewrite it as follows: $$\label{tre} \begin{array}{rl} K_{cl}(f|i) & =\displaystyle\sum_{k_{i}} K_{cl}(f|k_{N-1})K_{cl}(k_{N-1}|k_{N-2})\cdot...\cdot K_{cl}(k_{1}|i) \vspace{.2cm} \\ &=\displaystyle\prod^N_{j=1}\int d^{2n}\varphi_{j}~\delta^{(2n)}[\varphi^{a}_{j}-\phi^{a}_{cl}(t_{j} |\varphi_{j-1},t_{j-1})] \vspace{.2cm}\\ \xrightarrow{N\rightarrow\infty} &=\displaystyle\int{\mathscr D}\varphi~\tilde{\delta}[\varphi^{a}(t)-\phi^{a}_{cl}(t)] \end{array}$$ where in the first equality $k_i$ denotes formally an intermediate configuration $\v_{k_i}$ between $\v_{i}$ and $\v_{f}$ and in the last equality the symbol $\tilde{\delta}$ represents a [*functional*]{} Dirac delta. The last formula in (\[tre\]) is already a path integral but we can give it a more familiar form if we rewrite the Dirac delta as: $$\label{quattro} {\tilde\delta}[\varphi^a -\phi^a_{cl}]={\tilde\delta}[{\dot\varphi ^{a}-\omega^{ab} \partial_{b}H]~\det [\delta^{a}_{b}\partial_{t}-\omega^{ac}\partial_{c}\partial _{b}H}]$$ where we have used the functional analog of the relation $\delta[f(x)]=\frac{\delta[x-x_i]}{\Bigm|\frac{\partial f}{\partial x}\Bigm|_{x_i}}$. Next (see Ref.[@Ennio] for details) we can exponentiate both terms of the RHS of Eq.(\[quattro\]) via a Lagrange multiplier $\lambda$ (the first term) and a couple of Grassmannian variables $(c,\o{c})$ (the second term). What we finally get is the following expression: $$\label{cinque} K_{cl}(f|i)=\int{\mathscr D}\varphi ^{a}{\mathscr D}\lambda_{a}{\mathscr D} c^{a}{\mathscr D}{\o c}_{a}~\exp\biggl[i\int dt~{\widetilde{\cal L}}\biggr]$$ where $\widetilde{\cal L}$ is the Lagrangian characterizing the CPI: $$\label{sei} {\widetilde{\cal L}}=\lambda_{a}[{\dot\varphi }^{a}-\omega^{ab}\partial_{b}H]+ i{\o c}_{a}[\delta^{a}_{b}\partial_{t}-\omega^{ac}\partial_{c}\partial_{b}H] c^{b},$$ and the $8n$ variables $(\v^a,\l_a,c^a,\bc_a)$ form the new [*enlarged*]{} phase space which we denote by $\widetilde{\cal M}$. It is easy to Legendre transform the Lagrangian $\LT$ and obtain the corresponding Hamiltonian: $$\label{sette} \widetilde{\cal H}=\lambda_a\omega^{ab}\partial_bH+i\o{c}_a\omega^{ac} (\partial_c\partial_bH)c^{b}.$$ From the path integral (\[cinque\]) we can easily derive [@Ennio] the following commutator structure: $$\label{otto} \big[\v^{a},\l_{b}\big]=i\delta^{a}_{b}~~;~~\big[ c^{a}, {\o c}_{b}\big]=\delta^{a}_{b}. ~~~ \text{(all others are zero)}.$$ Via these commutators we can realize the $\l$ and ${\o c}$ variables as differential operators: $$\label{dieci} \lambda_{a}=-i{\partial\over\partial\varphi ^{a}};~~~~{\o c}_{a}={\partial\over\partial c^{a}}$$ and these in turn can be used to construct the operatorial version of the Hamiltonian (\[sette\]): $$\label{undici} {\widehat{\widetilde{\cal H}}}\equiv - i\omega^{ab}\partial_{b}H\frac{\p}{\p\v^a} -i\w^{ab}\p_b\p_d H c^{d}\frac{\p}{\p c^a}$$ and the corresponding “Schrödinger-type" equation for the probability density $\rho(\v,c;t)$: $$\label{dodici} {\widehat{\widetilde{\cal H}}}\rho(\v,c;t)=i\frac{\p}{\p t}\rho(\v,c;t).$$ For a nice interpretation of the geometry of the formalism we refer the reader to Ref.[@Geom]. For our purposes here it is sufficient to say that $\HT$ has a very precise geometrical meaning, being the [*Lie derivative*]{} along the Hamiltonian vector field $h\equiv \w^{ab}\p_bH\p_a$. We end this brief review with some remarks about the symmetries of the Lagrangian (\[sei\]) and the Hamiltonian (\[sette\]). It is easy to check that they are both invariant under the supersymmetry transformations generated by the following operators[^3]: $$\begin{aligned} \QH & =\Qb-\beta\NH = ic^a\l_a-\beta c^a\p_aH \label{sedici}\\ \QBH & =\QBb+\beta\NHB= i\o{c}_a\w^{ab}\l_b + \beta\o{c}_a\w^{ab}\p_bH. \label{diciassette} \end{aligned}$$ ($\beta$ is a dimensional parameter). It is also not difficult to represent all the formalism developed so far on a suitable superspace composed by the time $t$ and two Grassmannian partners $\theta$ and $\o{\theta}$. We refer the reader to Ref.[@Ennio] for all the details. For our purposes it is sufficient to say that we can introduce a classical superfield $$\label{diciannove} \Phi^a(t,\t,\tb)=\v^a + \t c^a + \tb\w^{ab}\bc_b +i\tb\t\w^{ab}\l_b.$$ on which the susy charges (\[sedici\])(\[diciassette\]) and the Hamiltonian (\[sette\]) act as[^4]: $$\begin{aligned} &\mathscr{Q}_{\s H} =-\frac{\p}{\p\t}-\beta\tb\frac{\p}{\p t}; &&\o{\mathscr{Q}}_{\s H} =\frac{\p}{\p\tb}+\beta\t\frac{\p}{\p t}. &&\widetilde{\mathscr{H}} =i\frac{\p}{\p t}; \end{aligned}$$ It is also easy to work out the covariant derivatives associated to $\mathscr{Q}_{\s H}$ and $\o{\mathscr{Q}}_{\s H}$: $$\begin{aligned} &\mathscr{D}_{\s H} =-i\frac{\p}{\p\t}+i\beta\tb\frac{\p}{\p t}; &\o{\mathscr{D}}_{\s H} =i\frac{\p}{\p\tb}-i\beta\t\frac{\p}{\p t}\label{ventitre}; \end{aligned}$$ which correspond (in $\widetilde{\cal M}$) to the following operators: $$\begin{aligned} \label{DD} & D_{\s H} =i\Qb+i\beta\NH & \o{D}_{\s H} =i\QBb-i\beta\NHB, \end{aligned}$$ where $\Qb$, $\QBb$, $\NH$ and $\NHB$ are defined in Eqs.(\[sedici\]) and (\[diciassette\]). $\kappa$-symmetry and CPI ========================= In the previous Section we have shown that the formalism of the Classical Path Integral exhibits a universal [*global*]{} Supersymmetry. However, differently from the model of Siegel, it does not possess any local invariance. If we want to build up a nonrelativistic analog of the model introduced in Section 1, we first must inject the local $t$-reparametrization invariance into the Lagrangian (\[sei\]) by adding the corresponding constraint via a Lagrange multiplier $g$: $$\label{3-1} \LT_1\equiv\LT + g\HT.$$ In fact it is easy to see that the previous Lagrangian is [*locally*]{} invariant under $$\label{3-2} \begin{cases} \delta({\s\ldots})=\big[({\s\ldots}),\e(t)\HT\big] \\ \delta g=-i\dot{\e}(t). \end{cases}$$ Here and in the sequel $({\s\ldots})$ denotes any one of the variables $(\v^a,\l_b,c^a,\bc_b)$. Moreover it is easy to check that it remains [*globally*]{} invariant under the $N=2$ classical Susy of Eqs.(\[sedici\])(\[diciassette\]). Nevertheless, in this simple model no other local symmetry is present. If we want to complete the analogy, we must add (following the lines of Ref.[@DG]) two further constraints to the Lagrangian (\[3-1\]) and we get: $$\label{3-16} \LT_2\equiv\LT + \xi D_{\s H} + \o{\xi}\o{D}_{\s H} + g\HT.$$ In the previous equation $D_{\s H}$ and $\o{D}_{\s H} $ are the operators introduced in Eq.(\[DD\]). We want to analyze this model following the same steps we used in Section 1 for the Lagrangian (\[1-1\]). First of all we remember again that, in our non-relativistic case, the analog of the “$p^2=0$" constraint is represented by the term $g\HT$ in (\[3-16\]) which produces the constraint $\HT=0$. Thus, as we did in Eq.(\[1-5\]), we start our analysis by releasing this constraint in the following way: $$\label{3-17} \LT^{\prime}_2\equiv\LT + \xi D_{\s H} + \o{\xi}\o{D}_{\s H} + g(\HT-\widetilde{E}),$$ which is the analog of Eq.(\[1-5\]). It should be remembered that $\widetilde{E}$ is not the energy of the system, but just a parameter related to the invariance under local time reparametrization: if $\widetilde{E}=0$ this symmetry is present, while if $\widetilde{E}\neq 0$ this symmetry is lost. One can immediately work out the constraints: $$\begin{aligned} &\text{$1^{\text{st}}$-Class\hspace{.25cm}} \begin{cases} \Pi_{\s \xi} =\Pi_{\s \o{\xi}}=\Pi_{\s g}=0; \\ \HT-\widetilde{E}=0; \end{cases} &\text{$2^{\text{nd}}$-Class\hspace{.25cm}} \begin{cases} D_{\s H}= 0; \\ \o{D}_{\s H} = 0. \end{cases}\label{3-18} \end{aligned}$$ Now we can compare the previous constraints with those in Eq.(\[1-7\]). Concerning the $1^{\text{st}}$-class constraints, we notice that $\HT-\widetilde{E}=0$ is the classical analog of the relativistic mass-shell constraint $p^{\mu}p_{\mu}-m^2=0$. This implies that $\Pi_{\s g}=0$ plays the same role as $\Pi_{\s\l}=0$ in the relativistic case, while the remaining two constraints ($\Pi_{\s\xi}=0$ and $\Pi_{\s\o{\xi}}=0$) have no analog in the relativistic case. Consider now the $2^{\text{nd}}$-class constraints. The first thing to point out is that $D_{\s H}= 0$ and $\o{D}_{\s H} = 0$ are precisely the classical analogs of $D^{\s a}=0$ and $\o{D}_{\s b}=0$ in the relativistic case. We can say that because $D_{\s H}$ and $\o{D}_{\s H}$ are related to the classical Susy charges $\QH$ and $\QBH$ in the same way in which $D^{\s a}$ and $\o{D}_{\s b}$ are related to the relativistic Susy charges $Q^{\s a}$ and $\o{Q}_{\s b}$. In fact it is easy to see that in the relativistic framework $D^{\s a}$ and $\o{D}_{\s b}$ commute with $Q^{\s a}$ and $\o{Q}_{\s b}$ and $[D^{\s a},\o{D}_{\s b}]=[Q^{\s a},\o{Q}_{\s b}]=i\pslsh~^{\s a}_{\s b}$ in the same way in which, in the nonrelativistic context, $D_{\s H}$ and $\o{D}_{\s H}$ commute with $Q_{\s H}$ and $\o{Q}_{\s H}$ and $[D_{\s H},\o{D}_{\s H}]=[Q_{\s H},\o{Q}_{\s H}]=2i\beta\HT$. This is actually the heart of the analogy. We start from a model which possesses a universal SUSY generated by $\QH$ and $\QBH$ and we want to check whether it is possible to implement a classical analog of the relativistic $\kappa$-symmetry of Siegel. Since in the relativistic case the $2^{\text{nd}}$-class constraints are $D^{\s a}=0$ and $\o{D}_{\s b}=0$, we have modified the CPI-Lagrangian (\[sei\]) in such a way that the resulting extension provides as $2^{\text{nd}}$-class constraints the classical analogs of $D^{\s a}$ and $\o{D}_{\s b}$, that is $D_{\s H}$ and $\o{D}_{\s H}$. This is precisely the model (\[3-17\]). If we go on with the same steps as in Section 1 we find that the matrix $\Delta_{ij}=[\phi_i,\phi_j]$ has the form: $$\label{3-20a} \Delta = \begin{pmatrix} 0 & 2i\beta\HT \\ 2i\beta\HT & 0 \end{pmatrix} \Longrightarrow \Delta^{-1}= \begin{pmatrix} 0 & (2i\beta\HT)^{-1} \\ (2i\beta\HT)^{-1} & 0 \end{pmatrix}$$ and consequently the Dirac Brackets deriving from (\[3-18\]) are: $$\label{3-20b} \big[A,B\big]_{\s DB}=\big[A,B\big]- \big[A,\o{D}_{\s H}\big](2i\beta\HT)^{-1} \big[D_{\s H},B\big]-\big[A,D_{\s H}\big](2i\beta\HT)^{-1} \big[\o{D}_{\s H},B\big].$$ Now that we have the correct structure of our phase space we can proceed with the analogy with the relativistic case. First of all we can prove that the two supersymmetry charges $\QH$ and $\QBH$ introduced in Eqs.(\[sedici\])(\[diciassette\]) become weakly equal to the $\Qb$ and $\QBb$ charges: $$\begin{aligned} & \QH\approx 2\Qb=2ic^a\l_a; \label{3-21} \\ & \QBH\approx 2\QBb=2i\bc_a\w^{ab}\l_b; \label{3-22} \end{aligned}$$ and consequently: $$\begin{aligned} & \big[\Qb,\QBb\big]_{\s DB}=\displaystyle\frac{1}{4}\big[\QH,\QBH\big]_{\s DB} =\frac{i\beta}{2}\HT. \label{3-23} \end{aligned}$$ This shows that $\QH$ and $\QBH$ are, more precisely, the analogs[^5] of the charges $Q^{\prime}$ $\o{Q}^{\prime}$ of Eq.(\[1-20\]) while the $\Qb$ and $\QBb$ charges are analogous to the $Q$ and $\o{Q}$ charges of Eq.(\[1-18\]). Consider now the case in which $\widetilde{E}=0$. We get down to the Lagrangian (\[3-16\]) and we see that something happens which is similar to the mechanism of $\kappa$-symmetry discussed in Section 1. In fact in that case we saw that half of the $2^{\text{nd}}$-class constraints became $1^{\text{st}}$-class. Here, on the other hand, we notice that both the $2^{\text{nd}}$-class constraints $D_{\s H}=\o{D}_{\s H}=0$ become $1^{\text{st}}$-class. This can be easily seen if one remembers that $\big[ D_{\s H},\o{D}_{\s H}\big]=2i\beta\HT\approx 0$ because now the constraint $\HT-\widetilde{E}=0$ has turned into $\HT=0$. In other words all the constraints in the model (\[3-16\]) are gauge constraints and contribute to restrict the space of the physical states. Therefore we see that in our nonrelativistic framework there is no difficult in separating $1^{\text{st}}$-class from $2^{\text{nd}}$-class constraints (like in the relativistic case). This is simply due to the fact that no $2^{\text{nd}}$-class constraint remains after imposing the constraint $\HT=0$ (which is the classical analog of $p_{\mu}p^{\mu}=0$)[^6]. Proceeding with the analogy it is very easy to construct the CPI-analogs of $K$ and $\o{K}$ of Eq.(\[1-22\]), that is the generators of the nonrelativistic $\kappa$-symmetry. They are simply: $$\begin{aligned} & K_{\s NR}=\HT D_{\s H}; & \o{K}_{\s NR}=\HT\o{D}_{\s H}; \end{aligned}$$ (“$NR$" stands for “Non Relativistic") and the local transformations (under which the Lagrangian (\[3-16\]) is invariant) generated by $K_{\s NR}$ and $\o{K}_{\s NR}$ are: $$\begin{cases} \delta({\s\ldots})=\big[({\s\ldots}),\varkappa(t)K_{\s NR}+\o{\varkappa}(t)\o{K}_{\s NR}\big] \\ \delta\xi=-i\dot{\varkappa}\HT \\ \delta\o{\xi}=-i\dot{\o{\varkappa}}\HT \\ \delta g=2i\beta(\o{\xi}\varkappa+ \xi\o{\varkappa})\HT. \end{cases}$$ It is interesting to determine the physical states selected by the theory defined by Eq.(\[3-16\]). Since all the constraints are now $1^{\text{st}}$-class, we must impose them strongly on the states as follows: $$\begin{aligned} &\Pi_{\s \xi} \,\rho(\v,c,\xi,\o{\xi},g) =\Pi_{\s \o{\xi}}\,\rho(\v,c,\xi,\o{\xi},g) =\Pi_{\s g}\,\rho(\v,c,\xi,\o{\xi},g)=0; \\ &D_{\s H}\,\rho(\v,c,\xi,\o{\xi},g)= 0; \\ &\o{D}_{\s H} \,\rho(\v,c,\xi,\o{\xi},g)= 0; \\ &\HT\,\rho(\v,c,\xi,\o{\xi},g)=0; \end{aligned}$$ and it is not difficult to prove that the resulting (normalizable[^7]) physical states have the following form: $$\label{3-28} \rho(\v,c,\xi,\o{\xi},g)\propto \exp[-\beta H(\v)].$$ This is precisely the [*Gibbs*]{} distribution characterizing the [*canonical*]{} ensemble, provided we interpret the $\beta$ constant of Eqs.(\[sedici\])(\[diciassette\]) as $(k_{\s B}T)^{-1}$, where $T$ plays the role of the temperature at which the system is in equilibrium. In fact we should remember that up to now the dimensional parameter $\beta$ introduced in Eqs.(\[sedici\]) and (\[diciassette\]) has not been restricted by any constraint. It is a completely free parameter with a dimension of $(\text{\it Energy})^{-1}$ which characterizes the particular $N=2$ classical supersymmetry. The canonical Gibbs state made its appearance earlier in the context of the CPI and precisely in Ref.[@Ergo]. There it was shown that, in the pure CPI model (\[sei\]), the zero eigenstates of $\HT$ which are also Susy-invariant are precisely the canonical Gibbs states. In our model instead we have obtained the Gibbs states as the entire set of physical states associated to the gauge theory described by the Lagrangian (\[3-16\]). However the model (\[3-16\]), though interesting for the peculiar physical subspace it determines, is not the nonrelativistic Lagrangian which is closest to the Siegel model. We mean that one should remember that the Lagrangian (\[3-16\]) gives rise to a canonical Hamiltonian of the form: $$\label{3-29} \HT_{2}\equiv\HT -\xi D_{\s H} -\o{\xi}\o{D}_{\s H} - g\HT,$$ but on the other hand we have already checked that the two couples of operators $(\QH,\QBH)$ and $(D_{\s H},\o{D}_{\s H})$ close on $\HT$ and not on $\HT_{2}$. Therefore, if we want to construct a more precise nonrelativistic analog of the model of Siegel, we should consider a slightly modified version of the Lagrangian (\[3-16\]) which is: $$\label{3-30} \LT_3\equiv\LT + \dot{\xi} D_{\s H} + \dot{\o{\xi}}\o{D}_{\s H} + g\HT.$$ One can easily check that the Lagrangian (\[3-30\]) yields, a part from a factor $(1-g)$, the same Hamiltonian as the CPI. Therefore we can proceed following the same steps as before: we turn the $\HT=0$ constraint into $\HT-\widetilde{E}=0$ $$\label{3-31} \LT^{\prime}_3\equiv\LT + \dot{\xi} D_{\s H} + \dot{\o{\xi}}\o{D}_{\s H} + g(\HT-\widetilde{E})$$ and we find out that the new constraints are: $$\begin{aligned} &\text{$1^{\text{st}}$-Class\hspace{.25cm}} \begin{cases} \Pi_{\s g}=0; \\ \HT-\widetilde{E}=0; \end{cases} &\text{$2^{\text{nd}}$-Class\hspace{.25cm}} \begin{cases} \Pi_{\s\xi}+D_{\s H}\equiv D^{\prime}_{\s H}= 0; \\ \Pi_{\s\o{\xi}}+\o{D}_{\s H}\equiv D^{\prime}_{\s H}= 0. \end{cases}\label{3-32} \end{aligned}$$ Then, it is easy to check that we can repeat all the considerations we did below Eq.(\[3-18\]), if we replace $D_{\s H}$ and $\o{D}_{\s H}$ with $D^{\prime}_{\s H}$ and $\o{D}^{\prime}_{\s H}$. As a second remark, we notice that the two constraints $\Pi_{\s\xi}=\Pi_{\s\o{\xi}}=0$, which had no analog in the relativistic context, have now disappeared. Moreover, because $\big[D^{\prime}_{\s H},\o{D}^{\prime}_{\s H}\big]= \big[D_{\s H},\o{D}_{\s H}\big]=2i\beta\HT$, we have also that the Dirac Brackets remain the same as those in Eq.(\[3-20b\]), which lead to Eqs.(\[3-21\])-(\[3-23\]). Again, when we put $\widetilde{E}=0$, we obtain that the two $2^{\text{nd}}$-class constraints $D^{\prime}_{\s H}=\o{D}^{\prime}_{\s H}=0$ become both $1^{\text{st}}$-class, differently from the relativistic case. However, the two models described by the two Lagrangians (\[3-16\]) and (\[3-30\]) are not equivalent. There are basically two differences. The first is the new form of the nonrelativistic $\kappa$-symmetry which now reads: $$\begin{aligned} &\begin{cases} \delta({\s\ldots})=\HT\big[({\s\ldots}),\varkappa(t)D_{\s H}+\o{\varkappa}(t)\o{D}_{\s H}\big] \approx \big[({\s\ldots}),\varkappa(t)K^{\prime}_{\s NR}+\o{\varkappa}(t)\o{K}^{\prime}_{\s NR}\big] \\ \delta\xi=\big[\xi,\varkappa(t)K^{\prime}_{\s NR}+\o{\varkappa}(t)\o{K}^{\prime}_{\s NR}\big] =-i\varkappa\HT \\ \delta\o{\xi}=\big[\o{\xi},\varkappa(t)K^{\prime}_{\s NR}+\o{\varkappa}(t)\o{K}^{\prime}_{\s NR}\big] =-i\o{\varkappa}\HT \\ \delta g=2i\beta\HT(\dot{\o{\xi}}\varkappa+ \dot{\xi}\o{\varkappa}). \end{cases} \end{aligned}$$ where “$\approx$" is understood in the Dirac sense and $$\begin{aligned} & K^{\prime}_{\s NR}\equiv\HT D^{\prime}_{\s H}; \hspace{2cm} \o{K}^{\prime}_{\s NR}\equiv\HT \o{D}^{\prime}_{\s H}. \end{aligned}$$ The second difference, which is the most important, is represented by the two physical spaces associated to the two models (\[3-16\]) and (\[3-30\]). In fact we have already seen that the physical states associated to the first model are the Gibbs distributions $\rho(\v)\propto\exp(-\beta H(\v))$; on the other hand the physical states determined by the Lagrangian (\[3-30\]) must obey the following conditions: $$\begin{aligned} &\Pi_{\s g}\,\rho(\v,c,\xi,\o{\xi},g)=0~; &D^{\prime}_{\s H}\,\rho(\v,c,\xi,\o{\xi},g) =\big(-i\p_{\xi}+D_{\s H}\big) \rho(\v,c,\xi,\o{\xi},g)=0~; \label{3-35}\\ &\HT\,\rho(\v,c,\xi,\o{\xi},g)=0~; &\o{D}^{\prime}_{\s H}\,\rho(\v,c,\xi,\o{\xi},g) =\big(-i\p_{\o{\xi}}+ \o{D}_{\s H}\big)\rho(\v,c,\xi,\o{\xi},g)=0~. \label{3-38} \end{aligned}$$ It is not difficult to realize that the solution of Eqs.(\[3-35\])-(\[3-38\]) has the form: $$\rho(\v,c,\xi,\o{\xi},g)\propto\exp\big(-i\xi D_{\s H}-i\o{\xi}\o{D}_{\s H}\big)\tilde{\rho}(\v,c)~,\label{3-39}$$ where $$\HT\,\tilde{\rho}(\v,c)=0~, \label{3-40}$$ which implies that $\tilde{\rho}(\v,c)$ is a function of constants of motion only. Therefore we can say that the physical states associated to the Lagrangian (\[3-30\]) are isomorphic to the functions $\tilde{\rho}(\v,c)$ which are annihilated by the Hamiltonian $\HT$ and are consequently constants of motion. Obviously the Gibbs distributions are a subset of them. This allows us to claim that the model (\[3-30\]) is actually more general than that characterized by the Lagrangian (\[3-16\]). More precisely the theory described by (\[3-30\]) is equivalent to that characterized by the Lagrangian (\[3-1\]). In fact it is easy to see that the physical Hilbert space associated to the latter is characterized by the distributions $\tilde{\rho}(\v,c,g)$ obeying to the constraints: $$\begin{aligned} \label{3-41} &\displaystyle\frac{\p}{\p g}\tilde{\rho}(\v,c,g)=0; &\HT\,\tilde{\rho}(\v,c,g)=0; \end{aligned}$$ and the physical space is precisely the same as that in (\[3-40\]), which is isomorphic to that determined by Eqs.(\[3-35\])-(\[3-38\]). Conclusions =========== In this paper we have analyzed two local versions of a model introduced some years ago to describe Classical Mechanics in terms of path integrals. In particular, we have built two nonrelativistic models which exhibit a universal local supersymmetry which is very similar to the famous $\kappa$-symmetry introduced almost 20 years ago by Siegel. Differently from the relativistic case, in our non relativistic framework the constraint $\HT=0$, which is analogous to the relativistic $p^2=0$, promotes to $1^{\text{st}}$-class all the $2^{\text{nd}}$-class constraints present in the case in which $\HT=\widetilde{E}\neq 0$. Consequently there is no difficulty in treating the constraints, differently from what happened in the relativistic case. In our first model the physical states of the theory turn out to be the Gibbs distributions characterizing the canonical ensemble, while in the second one the physical Hilbert space is formed by all the generic functions of the constants of motion of the theory. Acknowledgments {#acknowledgments .unnumbered} =============== We wish to thank Ennio Gozzi for many fruitful discussions. This work has been supported by grants from MURST and INFN of Italy. [99]{} L. Brink, S. Deser, P. Di Vecchia, P. Howe and B. Zumino, Phys. Lett. B [**64**]{} (1976) 369; W. Siegel, Phys. Lett. B [**128**]{} (1983) 397; J. A. de Azcarraga and J. Lukierski, Phys. Lett. B [**113**]{} (1982) 170; D. P. Sorokin, V. I. Tkach, D. V. Volkov and A. A. Zheltukhin, Phys. Lett. B [**216**]{} (1989) 302;\ D. P. Sorokin, V. I. Tkach, D. V. Volkov, Mod. Phys. Lett. A [**4**]{} (1989) 901; R. Kallosh, Phys. Rev. D[**56**]{} 3515 (1997);\ M. Hatsuda and K. Kamimura, Nucl Phys. [**B490**]{} 145 (1998); M. Moshe and N. Sakai, Phys. Rev. D[**62**]{}:086004 (2000); E. Deotto and E. Gozzi, Int. J. Mod. Phys. A [**16**]{} (2001) 2709; E. Gozzi, M. Reuter and W.D. Thacker, Phys. Rev. D [**40**]{} 3363 (1989);\ E. Gozzi, M. Reuter and W.D. Thacker, Phys.Rev.D [**46**]{} 757 (1992); B.O. Koopman, Proc. Nat. Acad. Sci. USA [**17**]{}, 315 (1931);\ J. von Neumann, Ann.Math. [**33**]{},587 (1932); E. Gozzi, M. Reuter, Phys. Lett. B [**240**]{} (1,2) 137 (1990);\ E. Gozzi, M. Regini, Phys.Rev.D [**62**]{} 067702 (2000) (hep-th/9903136); E. Deotto, E. Gozzi and D. Mauro, in preparation. A.A.Abrikosov (jr.), E.Gozzi, Nucl.Phys. B (Proc.Supp.) [**vol.88**]{} 369 (2000); K. Sundermeyer, “[*Constrained Dynamics*]{}", Springer-Verlag Heidelberg, 1982;\ M. Hanneaux, C. Teitelboim, “[*Quantization of Gauge Systems*]{}", Princeton Univ. Press, Princeton, N.J. 1992; E. Gozzi, M. Reuter, Phys.Lett.[**233B**]{} 383 (1989);\ E. Gozzi, Prog. Theor. Phys. (Suppl.) [**111**]{}, 115, (1993);\ E. Gozzi, M. Reuter, W.D. Thacker, Chaos, Solitons and Fractals [**2**]{} 441 (1992); ibid. [**4**]{} 1117 (1994). [^1]: Here and in the sequel we choose right derivatives for Grassmannian variables: $\Pi_{\z}:=\frac{\overleftarrow{\p}L}{\p\z}$. [^2]: In the sequel we shall omit the subcripts $+$ and $-$. [^3]: Here we use the same notation as in Ref.[@Ennio]. [^4]: According to the formula: $\mathscr{Q}\Phi^a(t,\t,\tb)\equiv[\Phi^a(t,\t,\tb),\e Q]$. [^5]: This is not in contradiction with what we said few lines above, that is that $\QH$ and $\QBH$ are the nonrelativistic analogs of $Q^{\s a}$ and $\o{Q}_{\s b}$. In fact it should be remembered that on the shell of the contraints we have $Q\approx Q^{\prime}$, $\o{Q}\approx \o{Q}^{\prime}$ (in the relativistic case) and $\QH\approx 2\Qb$, $\QBH\approx 2\QBb$ (in the nonrelativistic case). [^6]: This could be expected somehow, because here we have only two $2^{\text{nd}}$-class constraints and consequently it cannot happen that only half of these become $1^{\text{st}}$-class, like it happens in the Siegel model. In fact, if this were the case, we would remain with an odd number (that is 1) of $2^{\text{nd}}$-class constraints which is absurd because this number must always be even. [^7]: Also a state of the form $\rho(\v,c)\propto \exp[\beta H(\v)]c^1c^2\ldots c^{2n}$ would be admissible, but it is not normalizable in $\v$.

--- abstract: 'In the framework of minimal flavor violation (MFV), we discuss the decay properties of a supersymmetric scalar top (stop) in the presence of a light gravitino. Given a small mass difference between the lighter stop and lightest neutralino and an otherwise sufficiently decoupled spectrum, the stop may be long–lived and thus can provide support to MFV at hadron colliders. For a bino–like lightest neutralino, we apply bounds from searches in the $\gamma\gamma{\ensuremath{\slashed{E}_T}}$ channel (ATLAS with [$\unit[1]{fb^{-1}}$]{} and [D$\slashed{\text{0}}$]{} with [$\unit[6.3]{fb^{-1}}$]{}) and give a [$\unit[5]{fb^{-1}}$]{} projection for the ATLAS search.' author: - 'J. S. Kim' - 'H. Sedello' title: 'Probing Minimal Flavor Violation with Long–Lived Stops and Light Gravitinos at Hadron Colliders' --- =1 Introduction {#sec:intro} ============ Supersymmetry (SUSY) [@Ferrara:1974ac] is an attractive extension of the standard model (SM). However the simplest version of a supersymmetric SM, the minimal supersymmetric standard model (MSSM) [@Haber:1997if], does not predict a specific flavor structure; all superrenormalizable soft supersymmetry breaking terms allowed by gauge and Lorentz symmetry as well as $R$ parity are present in its Lagrange density [@Haber:1997if]. However, it is clear that a supersymmetric extension of the standard model must have a non–generic flavor structure to be compatible with experimental results [@Gabbiani:1996hi; @Amsler:2008zzb]. The way the standard model flavor structure is extended to the MSSM is not unique, yet a widely discussed flavor scheme is minimal flavor violation [@D'Ambrosio:2002ex] (MFV). In MFV the standard model Yukawa couplings are promoted to spurion fields transforming under the SM flavor group to restore the SM’s flavor symmetry. If all additional flavor structure of a new physics model can be understood as higher dimensional flavor invariant operators including these spurions and the model’s fields, the model is called MFV. As the LHC is running and eventually will find supersymmetry, it will be challenging to investigate the flavor structure at a hadron collider due to the detectors’ limited flavor identification abilities and the complexity of the recorded events. In [@Hiller:2008wp] it was pointed out that the third generation’s squarks decouple from the first two generations in MFV. As a result, a light stop can be long–lived decaying through the flavor changing neutral current channel [@Hikasa:1987db; @Muhlleitner:2011ww] $${\ensuremath{\tilde{t}_1}}\rightarrow c{\ensuremath{\tilde{\chi}_1^0}}, \label{eq:mfv-decay}$$ if all flavor diagonal channels are kinematically closed. (${\ensuremath{\tilde{t}_1}}$ denotes the light stop, ${\ensuremath{\tilde{\chi}_1^0}}$ the lightest neutralino and $c$ a charm quark.) An observation of long–living light stops thus would hint in the direction of MFV. In MFV, the coupling $Y$ between ${\ensuremath{\tilde{t}_1}}$, $c$ and ${\ensuremath{\tilde{\chi}_1^0}}$ is $$Y\propto\lambda_b^2 V_{cb}V_{tb}^*, \label{eq:defY}$$ where $\lambda_b$ and $V_{ij}$ denotes the bottom Yukawa coupling and elements of the Cabibbo Kobayashi Maskawa (CKM) matrix respectively. The precise value of $Y$ depends on the stop left–right composition, the neutralino decomposition, and on a numeric factor stemming from the MFV expansion; see Ref. [@Hiller:2008wp] for details. In [@Hiller:2009ii] it is shown that the average transverse impact parameters for the stop decay products can be expected to be ${\mathcal{O}(\unit[1800]{\mu m})}$ for stop lifetimes of the order of ten ps in the production channel $pp\rightarrow\bar t\bar t {\ensuremath{\tilde{t}_1}}{\ensuremath{\tilde{t}_1}}$ [@Kraml:2005kb] . When both top quarks in this channel decay leptonically, the pair of same signed leptons in the final state allows to separate the signal process from its SM background; however, the small leptonic branching ratio of top quarks suppresses this process so that, after applying all kinematic cuts, only few events are left in this channel. Ref. [@Carena:2008mj] proposes an alternative collider signature assuming stop pair production in association with one hard jet. Demanding a minimum transverse momentum of 1 TeV for the additional jet, the whole parameter region consistent with electroweak baryogenesis can be probed. Ref. [@Bornhauser:2010mw] considers an analogous process, stop pair production in association with two b–jets. However, these two studies do not consider stops in the MFV framework. If we consider local SUSY instead of a global implementation of SUSY, we can have distinct collider signal signatures with little SM background: In local SUSY, a massive gravitino emerges in the supersymmetric mass spectrum [@Nilles:1983ge]. Its interactions with other particles are severely suppressed by the reduced Planck mass $${\ensuremath{m_{\rm{Pl}}}}=(8\pi G_N)^{-\frac{1}{2}}=2.4\times{\unit[10^{18}]{GeV}},$$ where $G_N$ is Newton’s constant. Depending on the exact breaking mechanism in the hidden sector, the gravitino can be very light. A light gravitino interacts through its goldstino components with couplings proportional to $$({\ensuremath{m_{3/2}}}{\ensuremath{m_{\rm{Pl}}}})^{-1},$$ where ${\ensuremath{m_{3/2}}}$ is the gravitino mass. In models of gauge mediation [@Dine:1981za; @Dine:1981gu; @Dine:1993yw; @Meade:2008wd; @Buican:2008ws], ${\ensuremath{m_{3/2}}}$ is generally much smaller than the sparticle mass scale; thus, the gravitino is the lightest supersymmetric particle (LSP). Its goldstino interactions are enhanced and can be of the electroweak order. Consequently, the lightest neutralino decays via $${\ensuremath{\tilde{\chi}_1^0}}\rightarrow X{\ensuremath{\tilde{G}}}, \label{eq:neutr-decay}$$ where $X$ denotes a photon, $Z$, or a Higgs boson; ${\ensuremath{\tilde{G}}}$ denotes a gravitino [@Ambrosanio:1996jn]. If $X$ is a photon ($\gamma$), this decay leads to very clear collider signatures with high $p_T$ isolated photons plus missing transverse energy (${\ensuremath{\slashed{E}_T}}$) stemming from gravitinos leaving the detector unseen. Several studies with light gravitinos at hadron colliders were performed in the past. Ref. [@Shirai:2009kn; @Baer:1998ve; @Ambrosanio:1996jn] consider the diphoton plus ${\ensuremath{\slashed{E}_T}}$ channel at hadron colliders. In Ref. [@Hamaguchi:2006vu; @Ellis:2006vu] the authors examine a stau NLSP and a gravitino LSP. A sneutrino NLSP and a gravitino LSP scenario is investigated in Ref. [@Katz:2009qx; @Santoso:2009qa]. Ref. [@Meade:2009qv] considers the discovery potential of a neutralino NLSP and a gravitino LSP at the Tevatron, where they consider a general decomposition of the lightest neutralino. In Ref. [@Ambrosanio:1996jn] the authors consider a light stop NNLSP and a light neutralino NLSP and a gravitino LSP. Ref. [@Kats:2011it] investigate a stop NLSP and a gravitino LSP scenario for the Tevatron as well as the LHC. A chargino NLSP and a gravitino LSP is considered in Ref. [@Kribs:2008hq]. Depending on the size of the gravitino mass, non–pointing photons can be measured. The discovery potential of sparticle decays with a finite decay length are investigated in [@Hamaguchi:2007ji; @Feng:2010ij; @Meade:2010ji]. A recent experimental search for sparticles with finite decay lengths is published in [@Aad:2011zb]. In this paper we investigate the parameter region where the decay in Eq.  is dominant in the context of a light gravitino, and how the stop masses are constrained in this framework by recent collider searches, assuming that ${\ensuremath{\tilde{t}_1}}$, ${\ensuremath{\tilde{\chi}_1^0}}$, and ${\ensuremath{\tilde{G}}}$ are the only light supersymmetric particles. We discuss the stop and ${\ensuremath{\tilde{\chi}_1^0}}$ decay patterns in section \[sec:patterns\]. In section \[sec:collider\] we apply collider bounds in the $\gamma\gamma{\ensuremath{\slashed{E}_T}}$ channel [@Abazov:2010us; @arXiv:1111.4116]. The cuts adapted from the experimental studies are discussed in the two appendices. Decay patterns {#sec:patterns} ============== Light stop decays {#sec:stop} ----------------- Here we discuss possible decay patterns of light stops and their implications on the parameter space. We start with decays of the light stop via Yukawa and gauge couplings and then discuss direct stop decays into a gravitino. The lighter stop mass eigenstate ${\ensuremath{\tilde{t}_1}}$ is the lightest squark state in many supersymmetry breaking scenarios. On one hand, the large top Yukawa coupling can induce a sizable left-right mixing in the stop sector leading to light ${\ensuremath{\tilde{t}_1}}$ masses. On the other hand, if the soft squark mass terms are unified at a high scale, the stop mass terms are prominently reduced by the top Yukawa coupling in the running of the renormalization group equations to the electroweak scale [@Ibanez:1984vq]. In this case, ${\ensuremath{\tilde{t}_1}}$ is mostly a $SU(2)$ singlet state since its mass term does not receive any contributions from $SU(2)$ gaugino loops. In addition, right–handed stop loop contributions to the rho parameter are sufficiently suppressed [@Drees:1990dx]. Surprisingly light masses of ${\mathcal{O}({\unit[100]{GeV}})}$ are still consistent with experimental searches for stops decaying to $c{\ensuremath{\tilde{\chi}_1^0}}$ at the Tevatron if the mass splitting $$\Delta m = m_{{\ensuremath{\tilde{t}_1}}}-m_{{\ensuremath{\tilde{\chi}_1^0}}} \label{eq:massSplitting}$$ is smaller than $\approx{\unit[30]{GeV}}$ [@CDFexotic]. Since we want the flavor changing decay in Eq.  to be the dominant decay and the stop to be long–lived, we must ensure that the potentially dominant decays ${\ensuremath{\tilde{t}_1}}\rightarrow t {\ensuremath{\tilde{\chi}_1^0}}$, ${\ensuremath{\tilde{t}_1}}\rightarrow b{\ensuremath{\tilde{\chi}_1}}^\pm$, and ${\ensuremath{\tilde{t}_1}}\rightarrow b{\ensuremath{\tilde{\chi}_1^0}}W$ are kinematically closed or sufficiently suppressed, $$m_{{\ensuremath{\tilde{t}_1}}}<m_b+m_{{\ensuremath{\tilde{\chi}_1^0}}}+ m_{W},\quad m_{{\ensuremath{\tilde{t}_1}}}<m_b+m_{{\ensuremath{\tilde{\chi}_1}}^\pm}.$$ Depending on the chargino and slepton masses , the four–body decay ${\ensuremath{\tilde{t}_1}}\rightarrow b \ell \nu{\ensuremath{\tilde{\chi}_1^0}}$ may be dominant if $\Delta m$ exceeds a few . For small stop neutralino mass splittings, the tree–level four–body decay is strongly phase space suppressed and Eq.  remains the dominant decay mode [@Hiller:2008wp]. The MFV decay width can be written as $$\label{eq:sigdecay} \Gamma({\ensuremath{\tilde{t}_1}}\rightarrow c{\ensuremath{\tilde{\chi}_1^0}}) = \frac{m_{{\ensuremath{\tilde{t}_1}}} Y^2}{4\pi}\left(\frac{\Delta m}{m_{{\ensuremath{\tilde{t}_1}}}}\right)^2$$ in the limit of $\Delta m \ll m_{{\ensuremath{\tilde{t}_1}}}$ [@Hiller:2008wp]. Thus, if this decay is dominant, the stop lifetime is governed by $\Delta m Y$. The kinetic energy of the hadronic stop decay remnants depends on the size of the mass splitting $\Delta m$. In a study with same-sign leptons [@Hiller:2009ii], a major reduction of event numbers due to a minimal $p_T$ cut on these low energy decay products has been found. In the presence of a light gravitino, restrictions on $\Delta m$ and $Y$ are not sufficient to guarantee the dominance of the FCNC (flavor changing neutral current) decay mode Eq. . If $\Delta m Y$ is too small, the two–body decays of the stop into the gravitino, $${\ensuremath{\tilde{t}_1}}\rightarrow{\ensuremath{\tilde{G}}}c,\quad{\ensuremath{\tilde{t}_1}}\rightarrow {\ensuremath{\tilde{G}}}t, \label{eq:2bGrav}$$ and—if the resonant decay to tops is kinematically closed—the three–body decay mode of the stop, $${\ensuremath{\tilde{t}_1}}\rightarrow {\ensuremath{\tilde{G}}}b W, \label{eq:3bGrav}$$ can have a significant contribution to the full stop decay width. These decays can invalidate our assumption, that the stop dominantly decays via a FCNC process with a finite impact parameter. For the flavor diagonal two and three–body decay channels, the decay rates are [@Ambrosanio:1996jn; @Sarid:1999zx] \[eq:gravdecay\] $$\begin{aligned} \label{eq:grav2bdecay} \Gamma({\ensuremath{\tilde{t}_1}}\rightarrow t{\ensuremath{\tilde{G}}}) & = \frac{1}{48\pi}\frac{m_{{\ensuremath{\tilde{t}_1}}}^5}{{\ensuremath{m_{\rm{Pl}}}}^2{\ensuremath{m_{3/2}}}^2} \left(1-x_t^2\right)^4\\ \label{eq:grav3bdecay} \Gamma({\ensuremath{\tilde{t}_1}}\rightarrow W^+b{\ensuremath{\tilde{G}}}) & = \frac{V_{tb}^2\alpha_{em}}{384\pi^2\sin^2{\theta_W}} \frac{m_{{\ensuremath{\tilde{t}_1}}}^5}{{\ensuremath{m_{\rm{Pl}}}}^2{\ensuremath{m_{3/2}}}^2}\nonumber\\ \cdot\left[|c_L|^2\right.I\left(\right.&\!\!\!\left.\left.x_W^2,x_t^2\right) +|c_R|^2J\left(x_W^2,x_t^2\right)\right], \end{aligned}$$ where the gravitino mass is neglected in the phase space integrals. $\alpha_{em}$, and $\theta_W$ denote the fine-structure constant and the Weinberg angle, respectively. Further $x_W=m_W/m_{{\ensuremath{\tilde{t}_1}}}$ and $x_t=m_t/m_{{\ensuremath{\tilde{t}_1}}}$ with the top mass $m_t$. $c_L$ and $c_R$ parametrize the $\tilde{t}_L$ and $\tilde{t}_R$ contribution to ${\ensuremath{\tilde{t}_1}}$; due to the 3rd generation’s decoupling in MFV the other squarks’ admixture is small, *i.e.* $|c_L|^2+|c_R|^2\approx 1$. The functions $I(x_W^2,x_t^2)$ and $J(x_W^2,x_t^2)$ are phase space integrals and can be found in [@Sarid:1999zx]. Note that Eq.  does not comprise the finite top width and diverges at the top mass threshold. We use this formula for $m_{{\ensuremath{\tilde{t}_1}}}<m_t$ only. As the three–body decay proceeds trough a virtual top quark, its rate is largest for a right–handed stop, because the chirality flipping top mass dominates the propagator. For a bino–like ${\ensuremath{\tilde{\chi}_1^0}}$, the flavor structure of the ${\ensuremath{\tilde{t}_1}}-{\ensuremath{\tilde{G}}}-c$ coupling stemming from the MFV expansion is the same as in the ${\ensuremath{\tilde{t}_1}}-{\ensuremath{\tilde{\chi}_1^0}}-c$ coupling, thus the decay rate for ${\ensuremath{\tilde{t}_1}}\rightarrow {\ensuremath{\tilde{G}}}c$ can be written as [@Hiller:2009ii] $$\label{eq:cgravdecay} {\ensuremath{\Gamma({\ensuremath{\tilde{t}_1}}\rightarrow{\ensuremath{\tilde{G}}}c)}}= \frac{Y^{\prime^2}}{48\pi}\frac{m_{{\ensuremath{\tilde{t}_1}}}^5}{m_{\rm{Pl}}^2{\ensuremath{m_{3/2}}}^2},$$ where $Y'$ and $Y$ are related by a factor dependent on the stop composition as $Y$ comprises the hypercharges of the left- and right–handed stop fields. The factor is $$\left|\frac{Y'}{Y}\right|=\frac{1}{\sqrt{2}g'Y_Q}\approx \begin{cases} 3&\text{(right--handed ${\ensuremath{\tilde{t}_1}}$)}\\ 12&\text{(left--handed ${\ensuremath{\tilde{t}_1}}$)} \end{cases}$$ with the SM ${U}(1)$ coupling $g'$ and the left–handed (right–handed) stop hypercharge $Y_Q=\frac{1}{6}\left(\frac{2}{3}\right)$. We show the branching ratio ${\mbox{\ensuremath{\mathcal{B}({\ensuremath{\tilde{t}_1}}\rightarrowc{\ensuremath{\tilde{\chi}_1^0}})}}}$ and the stop lifetime in the $m_{{\ensuremath{\tilde{G}}}}$–$Y$ plane for three different masses of a right–handed stop in Fig. \[fig:brs\] using the decay rates , , and . To generate the plots, we keep $\Delta m$ fixed at ${\unit[10]{GeV}}$ and use $\sin^2\theta_W=0.23$, $\alpha_{em}=1/128$, and $m_t={\unit[173]{GeV}}$. The plot for left–handed stops does not differ significantly from the one shown. Due to the $m_{{\ensuremath{\tilde{t}_1}}}^5$ dependence of the decay widths in Eqs ,  and the weaker $m_{{\ensuremath{\tilde{t}_1}}}^{-1}$ dependence of $\Gamma({\ensuremath{\tilde{t}_1}}\rightarrow c{\ensuremath{\tilde{\chi}_1^0}})$, the minimal gravitino mass necessary to account for a sizable increases with larger stop masses. The $m_{{\ensuremath{\tilde{t}_1}}}^{-1}$ dependence of $\Gamma({\ensuremath{\tilde{t}_1}}\rightarrow c{\ensuremath{\tilde{\chi}_1^0}})$ also causes the smallness of the shifts of the lifetime regions in Fig. \[fig:brs\] to larger $Y$ values when the stop mass is increased. As it is clearly visible from Fig. \[fig:brs\], very small values of $Y\lesssim{\mathcal{O}(10^{-5})}$ and at least gravitino masses of ${\mathcal{O}({\unit[0.1-1]{keV}})}$ are required in addition to the mass hierarchy $${\ensuremath{m_{3/2}}}\ll m_{{\ensuremath{\tilde{\chi}_1^0}}}\leq m_{{\ensuremath{\tilde{t}_1}}}\leq m_{{\ensuremath{\tilde{\chi}_1}}^\pm} \label{eq:massOrder2}$$ for the stop to be long–lived and to decay dominantly to $c{\ensuremath{\tilde{\chi}_1^0}}$. As the charmed hadron produced in the decay has a macroscopic lifetime of ${\mathcal{O}({\unit[1]{ps}})}$ itself, however, a macroscopic stop lifetime might turn out to be accessible experimentally only if it exceeds this timescale. NLSP neutralino composition and decays {#sec:neutralino} -------------------------------------- Neutralinos are mass eigenstates of the $U(1)$ gauge fermion (bino), the neutral $SU(2)$ gauge fermion (wino), and the neutral up– and down–type Higgs fermion (higgsino). The fields’ individual contributions to ${\ensuremath{\tilde{\chi}_1^0}}$ as well as the neutralino mass spectrum depends on the bino mass $M_1$, the wino mass $M_2$, the Higgs mixing parameter $\mu$ and the ratio between the up–type and down–type Higgs vacuum expectation value (VEV) $\tan\beta$. If ${\ensuremath{\tilde{\chi}_1^0}}$ is the NLSP and ${\ensuremath{\tilde{G}}}$ the LSP, ${\ensuremath{\tilde{\chi}_1^0}}$ decays via ${\ensuremath{\tilde{\chi}_1^0}}\rightarrow X {\ensuremath{\tilde{G}}}$, where $X$ is either the photon, the Z boson, or a neutral Higgs boson. Branching ratios into the various decay channels are fixed by phase space suppression factors and the decomposition of the lightest neutralino. General formulae for the decay widths are given in Refs [@Ambrosanio:1996jn; @Meade:2009qv]. In the previous subsection, we argued that the mass of the light chargino ${\ensuremath{\tilde{\chi}_1}}^\pm$, a mass eigenstate of charged winos and higgsinos, has to be larger than the light stop mass in order to suppress the flavor diagonal stop decay to ${\ensuremath{\tilde{\chi}_1}}^\pm b$. This requirement cannot be satisfied if the ${\ensuremath{\tilde{\chi}_1^0}}$ is wino–like, *i.e.* if $M_2\ll M_1,|\mu|$, as in this case both the ${\ensuremath{\tilde{\chi}_1^0}}$ and the ${\ensuremath{\tilde{\chi}_1}}^\pm$ mass are $\approx M_2$. The mass splitting between wino–like ${\ensuremath{\tilde{\chi}_1}}^\pm$ and ${\ensuremath{\tilde{\chi}_1^0}}$ is of the order of $\frac{m_Z^5}{\mu^4}$ [@Martin:1993ft], given $M_1 \ll |\mu|$, and thus is extremely suppressed for $|\mu|\gtrsim\text{few }{\unit[100]{GeV}}$. Similarly, if ${\ensuremath{\tilde{\chi}_1^0}}$ is higgsino–like, ${\ensuremath{\tilde{\chi}_1^0}}$ and ${\ensuremath{\tilde{\chi}_1}}^\pm$ have masses of the same order of magnitude given by $\mu$. The mass splitting $\Delta m_{{\ensuremath{\tilde{\chi}_1}}^\pm{\ensuremath{\tilde{\chi}_1^0}}}$ is of the order of $\frac{m_Z^2}{M_2}$ [@Martin:1993ft] for $|\mu|\ll M_1, M_2$. If ${\ensuremath{\tilde{\chi}_1^0}}$ is bino–like, its mass is $M_1$ approximately, while the ${\ensuremath{\tilde{\chi}_1}}^\pm$ mass is given by $|\mu|$ or $M_2$. As the mass gap depends on two different supersymmetric mass parameters, it can be sizable depending on the details of high scale physics. While in a mass region close to the $Z$ mass, also a higgsino–like ${\ensuremath{\tilde{\chi}_1^0}}$ may respect the anticipated mass hierarchy in Eq. , we focus on a bino–like ${\ensuremath{\tilde{\chi}_1^0}}$ in discussing experimental bounds as 1) in the bino case the mass hierarchy can exist over a large stop mass scale and 2) binos have a large branching fraction to photons. For $m_{{\ensuremath{\tilde{\chi}_1^0}}}>m_{Z}$ and negligible phase space suppression, the branching ratio is ${\mbox{\ensuremath{\mathcal{B}({\ensuremath{\tilde{\chi}_1^0}}\rightarrow\gamma{\ensuremath{\tilde{G}}})}}}\approx\cos^2\theta_W$. This value is obvious as ${\ensuremath{\tilde{G}}}$ is a gauge singlet and $\gamma$ a mixed state of the hypercharge gauge boson and the neutral $SU(2)$ gauge boson where the mixture is parametrized by the Weinberg angle. Including the phase space suppression from $m_{Z}$ and assuming that the higgsino sector is decoupled, the bino decay rates are [@Ambrosanio:1996jn; @hep-ph/9609434] \[eq:neutrdecay\] $$\begin{aligned} \Gamma({\ensuremath{\tilde{\chi}_1^0}}\rightarrow \gamma{\ensuremath{\tilde{G}}}) &= \cos^2{\theta_W}\frac{m_{{\ensuremath{\tilde{\chi}_1^0}}}^5}{48\pi{\ensuremath{m_{3/2}}}^2{\ensuremath{m_{\rm{Pl}}}}^2}\\ \Gamma({\ensuremath{\tilde{\chi}_1^0}}\rightarrow Z{\ensuremath{\tilde{G}}}) &=\nonumber\\ \sin^2&\theta_W\frac{m_{{\ensuremath{\tilde{\chi}_1^0}}}^5}{48\pi{\ensuremath{m_{3/2}}}^2{\ensuremath{m_{\rm{Pl}}}}^2}\left(1-\frac{m_{Z}^2}{m_{{\ensuremath{\tilde{\chi}_1^0}}}^2}\right)^4. \end{aligned}$$ For reference we show the bino–neutralino lifetime as a function of the lightest neutralino mass for gravitino masses 1, 10, 100, 1000 eV in Fig. \[fig:neutr-ltime\]. Collider bounds {#sec:collider} =============== When ${\ensuremath{\tilde{t}_1}}$, ${\ensuremath{\tilde{\chi}_1^0}}$, and ${\ensuremath{\tilde{G}}}$ are the only light supersymmetric particles, stops are dominantly produced in pairs, both at $\bar pp$ and $pp$ colliders, $$\bar pp\rightarrow{\ensuremath{\tilde{t}_1}}{\ensuremath{\tilde{t}_1}}^*,\qquad pp\rightarrow{\ensuremath{\tilde{t}_1}}{\ensuremath{\tilde{t}_1}}^*.$$ The production cross sections are given in Fig. \[fig:xsec\] as a function of the stop mass for the LHC at as well as for Tevatron and are calculated with [`Prospino`]{} [@Beenakker:1997ut] using the built-in CTEQ6.6M [@Nadolsky:2008zw] parton distribution functions (PDFs). We also show the next–to–leading order uncertainty by varying the factorization scale ($\mu_F$) and the renormalization scale ($\mu_R$) between $\frac{1}{2}m_{{\ensuremath{\tilde{t}_1}}}$ and $2m_{{\ensuremath{\tilde{t}_1}}}$ while keeping $\mu_R$ equal to $\mu_F$. Given a bino–like ${\ensuremath{\tilde{\chi}_1^0}}$, the final state signatures of a decay chain via Eq.  and Eq.  are $$\gamma\gamma c\bar c{\ensuremath{\tilde{G}}}{\ensuremath{\tilde{G}}},\quad \gamma Z c\bar c{\ensuremath{\tilde{G}}}{\ensuremath{\tilde{G}}},\quad Z Z c\bar c{\ensuremath{\tilde{G}}}{\ensuremath{\tilde{G}}}.$$ In general, with a small mass gap $\Delta m$, the charm jets are too soft to be useful for event selection. Thus constrains on our parameter space can stem from searches for an excess in the $\gamma\gamma{\ensuremath{\slashed{E}_T}}$, $\gamma Z{\ensuremath{\slashed{E}_T}}$ and $ZZ{\ensuremath{\slashed{E}_T}}$ channels. As binos dominantly decay to $\gamma{\ensuremath{\tilde{G}}}$, the SM background is negligible for energetic photons, and large ${\ensuremath{\slashed{E}_T}}$ and high $p_T$ photons are efficiently identified in multipurpose detectors, we concentrate on the $\gamma\gamma{\ensuremath{\slashed{E}_T}}$ channel in this work. Several experimental searches for the diphoton and ${\ensuremath{\slashed{E}_T}}$ channel have been published [@Aad:2010qr; @Aaltonen:2009tp; @Chatrchyan:2011wc; @Abazov:2010us; @arXiv:1111.4116]. So far, no excess above the SM expectation has been found. In the following, we present exclusion limits in the stop–gravitino mass plane derived from the latest search in the $\gamma\gamma{\ensuremath{\slashed{E}_T}}$ channel for a luminosity ($\mathcal L$) of ${\ensuremath{\unit[1.07]{fb^{-1}}}}$ [@arXiv:1111.4116]. We derive also bounds from the [D$\slashed{\text{0}}$]{} search with $\mathcal L={\ensuremath{\unit[6.3]{fb^{-1}}}}$ [@Abazov:2010us], and give a $\mathcal L={\ensuremath{\unit[5]{fb^{-1}}}}$ projection for the bound. The dominant SM background with ${\ensuremath{\slashed{E}_T}}$ originating from the hard process stems from $W+\gamma$, $W+{\rm jets}$, and $W/Z\gamma(e) \gamma(e)$ production. Here, electrons/jets are misidentified as photons. SM background with ${\ensuremath{\slashed{E}_T}}$ from mismeasurements emerges from multijet production and direct photon production. A supersymmetric background can only arise from ${\ensuremath{\tilde{\chi}_1^0}}{\ensuremath{\tilde{\chi}_1^0}}$ pair production as all other colored sparticles, sleptons, heavier neutralinos and charginos are assumed to be heavy and thus will have a negligible contribution. However, also the ${\ensuremath{\tilde{\chi}_1^0}}{\ensuremath{\tilde{\chi}_1^0}}$ production cross section is severely suppressed. In the limiting case of vanishing higgsino admixture to ${\ensuremath{\tilde{\chi}_1^0}}$, the cross section vanishes even at ${\mathcal{O}(\alpha_{\text{EW}}^2\alpha_s)}$. Consequently we do not take ${\ensuremath{\tilde{\chi}_1^0}}{\ensuremath{\tilde{\chi}_1^0}}$ pair production into account in the following. Calculation of exclusion limits ------------------------------- To constrain the stop mass, the gravitino mass, and the MFV coupling $Y$, we calculate $\sigma_{{\ensuremath{\tilde{t}_1}}{\ensuremath{\tilde{t}_1}}^*}$, the total cross section for ${\ensuremath{\tilde{t}_1}}{\ensuremath{\tilde{t}_1}}^*$ production, in a grid of the light stop mass $m_{{\ensuremath{\tilde{t}_1}}}$ using [`Prospino`]{}. Note here that the light stop mass is the dominant SUSY parameter in the cross section [@Beenakker:1997ut], both for $p\bar p$ and $pp$ initial states. For each stop mass in the grid, we generate 100$\,$000 ${\ensuremath{\tilde{t}_1}}{\ensuremath{\tilde{t}_1}}^*$ pair events with [`pythia 6.4.25`]{} [@Sjostrand:2006za] using the CTEQ6.6M [@Nadolsky:2008zw] parton distribution functions. With the hadron level events we simulate the efficiency/acceptance for the $\gamma\gamma c\bar c{\ensuremath{\tilde{G}}}{\ensuremath{\tilde{G}}}$ final state in the and [D$\slashed{\text{0}}$]{} analyses employing a slightly modified version of [`Delphes 1.9`]{} [@Ovyn:2009tx][^1]. The photon energy, and therefore our signal’s detection efficiency, depends on the mass splitting $\Delta m$. As in the previous sections, we fix $\Delta m ={\unit[10]{GeV}}$. In appendices \[sec:dzero\] and \[sec:atlas\], we give details on the cuts adopted from the experimental studies and on further simulation parameters. As a result of this simulation step, we obtain efficiencies $\epsilon_n$ in bins of ${\ensuremath{\slashed{E}_T}}$ and can calculate a signal cross section in bin $n$ from $$\sigma^{sig}_{n}=\epsilon_n{\mbox{\ensuremath{\mathcal{B}({\ensuremath{\tilde{t}_1}}\rightarrow{\ensuremath{\tilde{\chi}_1^0}}c)}}}^2{\mbox{\ensuremath{\mathcal{B}({\ensuremath{\tilde{\chi}_1^0}}\rightarrow{\ensuremath{\tilde{G}}}\gamma)}}}^2\sigma_{{\ensuremath{\tilde{t}_1}}{\ensuremath{\tilde{t}_1}}^*}.$$ Using Eqs  to calculate ${\mbox{\ensuremath{\mathcal{B}({\ensuremath{\tilde{\chi}_1^0}}\rightarrow{\ensuremath{\tilde{G}}}\gamma)}}}$, we finally employ the $\mathrm {CL}_s$ method [@Junk:1999kv; @599622][^2] to calculate the 95% exclusion limits for ${\mbox{\ensuremath{\mathcal{B}({\ensuremath{\tilde{t}_1}}\rightarrow{\ensuremath{\tilde{\chi}_1^0}}c)}}}$. Those are depicted in Fig. \[fig:br-excl\], where we use the measurements plus background predictions of the experimental studies as enlisted in Tab. \[tab:data\]. When calculating the exclusion limit, we treat the errors on the luminosity and the background as Gaussian nuisance parameters, see Tab. \[tab:data\], but do not take into account theory uncertainties stemming from scale variations and the choice of PDF sets. The projection for the ATLAS study with a luminosity of ${\ensuremath{\unit[5]{fb^{-1}}}}$ is calculated using the prescription of [@Conway:2000ju]. ${\ensuremath{\slashed{E}_T}}$ Bin \[\] ------------------------------------------------ ----------------------------------------- ------ -------------- [D$\slashed{\text{0}}$]{} [@Abazov:2010us] $35-50$ $18$ $11.9\pm2.0$ ${\ensuremath{\unit[(6.3\pm0.4)]{fb^{-1}}}}$ $50-75$ $3$ $5.0\pm0.9$ $>75$ $1$ $1.9\pm0.4$ ATLAS [@arXiv:1111.4116] $>125$ $5$ $4.1\pm0.6$ ${\ensuremath{\unit[(1.07\pm0.04)]{fb^{-1}}}}$ : ATLAS and [D$\slashed{\text{0}}$]{}measurements and background (bgd) predictions.[]{data-label="tab:data"} Numerical analysis and discussion --------------------------------- As can be seen in Fig. \[fig:br-excl\], the ATLAS search (solid pink curve) gives a bound on ${\mbox{\ensuremath{\mathcal{B}({\ensuremath{\tilde{t}_1}}\rightarrow{\ensuremath{\tilde{\chi}_1^0}}c)}}}$ for stop masses up to ${\unit[560]{GeV}}$. For a luminosity of ${\ensuremath{\unit[5]{fb^{-1}}}}$, this mass is projected to ${\unit[660]{GeV}}$. For larger masses, a dominant tree level FCNC decay ${\ensuremath{\tilde{t}_1}}\rightarrow{\ensuremath{\tilde{\chi}_1^0}}c$ is in agreement with the measurement. As can be seen from Fig. \[fig:br-excl\], the [$\unit[1.07]{fb^{-1}}$]{} ATLAS data already excludes a larger parameter region than the [$\unit[6.3]{fb^{-1}}$]{} [D$\slashed{\text{0}}$]{} data [^3]. For each stop mass, the bound in Fig. \[fig:br-excl\] can be mapped to a bound on the maximal gravitino mass for given values of $Y$ using Eqs , , and . We plot these bounds for $Y=10^{-7}$, $10^{-6}$, $10^{-5}$ in Fig. \[fig:excl\]. As Eqs  do not provide the correct stop width for masses in the threshold region close to the top mass, we exclude this region from the mapping and interpolate our result in the region $m_t\pm{\unit[30]{GeV}}$ ($m_t={\unit[173]{GeV}}$). The mass difference $\Delta m$ enters the bounds in Fig. \[fig:excl\] through Eq.  and through the hardness of the photon $p_T$ spectrum. For small changes in $\Delta m$, the latter dependency can be neglected, and the bounds depend on the product $(Y\Delta m)$ only. Therefore the bounds for other viable values of $\Delta m$ can be derived from those shown for $\Delta m ={\unit[10]{GeV}}$ by rescaling $Y$. Obviously, the smaller we choose $Y$ the larger ${\ensuremath{m_{3/2}}}$ can be to generate a branching ratio below a certain value. The bound on ${\ensuremath{m_{3/2}}}$ resulting from the mapping in Fig. \[fig:excl\] varies over a wide mass range and thus potentially implies different regimes of ${\ensuremath{\tilde{t}_1}}$ and ${\ensuremath{\tilde{\chi}_1^0}}$ lifetimes; therefore, we show slopes of fixed values: For the ${\ensuremath{\tilde{\chi}_1^0}}$ lifetime, we show slopes for , PS. [1]{}, and (black dotted) following approximately ${\ensuremath{m_{3/2}}}\propto m_{{\ensuremath{\tilde{t}_1}}}^{5/2}$. For ${\ensuremath{\tilde{t}_1}}$, we show slopes for ${\unit[1/5]{ps}}$, ${\unit[1]{ps}}$, and ${\unit[5]{ps}}$ (black solid). At $Y=10^{-5}$, the stop lifetime is below PS. [1]{} already induced by ${\ensuremath{\tilde{t}_1}}\rightarrow {\ensuremath{\tilde{\chi}_1^0}}c$ irrespective of the gravitational decay channels; therefore, only the ${\unit[1/5]{ps}}$ slope can be drawn here. Similarly, the stop lifetime drops below ${\unit[1/5]{ps}}$ for stop masses smaller than $\approx{\unit[230]{GeV}}$ for $Y=10^{-5}$, as $\Gamma({\ensuremath{\tilde{t}_1}}\rightarrow c{\ensuremath{\tilde{\chi}_1^0}})\propto m_{{\ensuremath{\tilde{t}_1}}}^{-1}$ for fixed $\Delta m$. For both smaller values of $Y$, the stop lifetime slopes shown only depend weakly on $Y$ because the corresponding total stop widths are dominated by the gravitational decay ${\ensuremath{\tilde{t}_1}}\rightarrow t{\ensuremath{\tilde{G}}}$ resp. ${\ensuremath{\tilde{t}_1}}\rightarrow W^+b{\ensuremath{\tilde{G}}}$. Fig. \[fig:excl\] shows that the gravitino mass region promoted in [@Hiller:2009ii] as the region where the decay ${\ensuremath{\tilde{t}_1}}\rightarrow{\ensuremath{\tilde{\chi}_1^0}}c$ dominates for stop masses between $100$ and ${\unit[150]{GeV}}$ is now disfavored. More generally, Fig. \[fig:excl\] allows to discuss two regimes of different orders of ${\ensuremath{\tilde{t}_1}}$ and ${\ensuremath{\tilde{\chi}_1^0}}$ lifetimes: - In the stop mass region where ${\mbox{\ensuremath{\mathcal{B}({\ensuremath{\tilde{t}_1}}\rightarrow{\ensuremath{\tilde{\chi}_1^0}}c)}}}$ is bounded, *i.e.* for $m_{{\ensuremath{\tilde{t}_1}}}\lesssim{\unit[500]{GeV}}$, the gravitino channels have a significant contribution to the stop decay. If this contribution is dominant, both—the ${\ensuremath{\tilde{\chi}_1^0}}$ and the ${\ensuremath{\tilde{t}_1}}$ decay—are governed by the same coupling $\sim1/{{\ensuremath{m_{3/2}}}^2}$. As the stop decay width is suppressed by phase space ($t{\ensuremath{\tilde{G}}}$ channel) or top propagator (${\ensuremath{\tilde{G}}}W^+b$ channel), the stop lifetime is expected to be larger than, or at the same order of magnitude as, the neutralino lifetime in this region. - In the mass region where ${\mbox{\ensuremath{\mathcal{B}({\ensuremath{\tilde{t}_1}}\rightarrow{\ensuremath{\tilde{\chi}_1^0}}c)}}}$ is allowed to dominate, *i.e.* for $m_{{\ensuremath{\tilde{t}_1}}}\gtrsim{\unit[500]{GeV}}$, the phase space suppression of the stop decay width Eq.  is less pronounced; thus, the stop and neutralino gravitational partial decay widths are of the same order of magnitude. Consequently, if ${\ensuremath{\tilde{t}_1}}\rightarrow c{\ensuremath{\tilde{\chi}_1^0}}$ is the dominant stop decay in this mass region, the stop lifetime is significantly smaller than the neutralino lifetime. The relation between the stop and neutralino lifetimes ($\tau_{{\ensuremath{\tilde{t}_1}}}$ and $\tau_{{\ensuremath{\tilde{\chi}_1^0}}}$) described above is summarized in Fig. \[fig:lt-ratios\] where we plot the allowed ratio of both lifetimes within the bounds. Along the solid black line bounding the grey area, the contribution of Eq.  to the stop decay width vanishes; thus, this line represents the smallest value the ratio can have. The plot is generated for $Y=10^{-6}$; however, only the leftmost parts of the exclusion areas depend weakly on the specific value of $Y$. For smaller masses, the bounds are driven by the phase space dependence of Eqs  and . Note that for large ${\ensuremath{\tilde{t}_1}}$ or ${\ensuremath{\tilde{\chi}_1^0}}$ lifetimes, care must be taken in the interpretation of the bounds in Figs \[fig:excl\] and \[fig:lt-ratios\]: The selection criteria for photon candidates in the ATLAS publication are chosen for prompt photons [@arXiv:1012.4389; @atlas-conf-2010-005]. Also, more explicitly, the [D$\slashed{\text{0}}$]{} study requires that photon candidates point to a reconstructed primary vertex. We assumed in our calculation that all signal photons fulfill these criteria as if they were prompt. In a more realistic simulation, for increasing neutralino lifetimes, the signal photons’ selection efficiency should decrease. For longitudinal neutralino decay lengths of ${\mathcal{O}(\unit[1000]{mm})}$ ATLAS simulations show that for several photon selection criteria [^4] used in [@arXiv:1111.4116] the efficiency drops from ${\mathcal{O}({)}85\%}$ to ${\mathcal{O}({)}55\%}$ [@Aad:2009wy]. Consequently, for lifetimes $\gtrsim{\mathcal{O}({\unit[]{ns}})}$, we overestimate the number of photons accepted, and the bounds on ${\ensuremath{m_{3/2}}}$ presented should be regarded with care in this lifetime region. While for a bino–like neutralino, the $\gamma\gamma{\ensuremath{\slashed{E}_T}}$ channel offers the highest sensitivity for setting mass limits, it may be difficult to measure the neutralino lifetime in this channel due to lack of photon tracks. To construct the photons’ impact parameters, CMS focuses on converted photons in a [$\unit[2.1]{fb^{-1}}$]{} search in the $\gamma\gamma{\ensuremath{\slashed{E}_T}}+\text{jets}$ channel in [@CMS-PAS-EXO-11-067]. As pointed out in [@Meade:2010ji], the ${\ensuremath{\tilde{\chi}_1^0}}\rightarrow Z{\ensuremath{\tilde{G}}}$ channel may be used to investigate the neutralinos’ lifetimes, as the $Z$’s decay products allow to reconstruct the ${\ensuremath{\tilde{\chi}_1^0}}$’s trajectory. Summary {#sec:summary} ======= In SUSY models with MFV, the third generation of squarks decouples from the other two generations. This decoupling opens the opportunity to support the MFV hypotheses with the measurement of a macroscopic stop decay length if the stop decay can only proceed through a generation–changing channel due to kinematic constraints [@Hiller:2008wp]. In this work, we investigate the decay of a stop into a charm and a bino–like lightest neutralino with a subsequent bino decay to a photon and a light gravitino, $${\ensuremath{\tilde{t}_1}}\rightarrow{\ensuremath{\tilde{\chi}_1^0}}(\rightarrow\gamma{\ensuremath{\tilde{G}}}) c, \label{eq:complete_decay}$$ given a sufficiently small mass difference between ${\ensuremath{\tilde{t}_1}}$ and ${\ensuremath{\tilde{\chi}_1^0}}$. While a macroscopic stop decay length serves as a hint for MFV, the neutralino’s decay to a photon leaves a distinct signature in LHC and Tevatron detectors offering a good signal isolation. Assuming that the remainder of the SUSY spectrum is decoupled, we find that the ATLAS search in the $\gamma\gamma{\ensuremath{\slashed{E}_T}}$ channel based on a luminosity of ${\ensuremath{\unit[1]{fb^{-1}}}}$ [@arXiv:1111.4116] implies a bound on ${\mbox{\ensuremath{\mathcal{B}({\ensuremath{\tilde{t}_1}}\rightarrowc{\ensuremath{\tilde{\chi}_1^0}})}}}$ for stop masses up to ${\unit[560]{GeV}}$, see Fig. \[fig:br-excl\]. In a ${\ensuremath{\unit[5]{fb^{-1}}}}$ projection, the bound is raised to ${\unit[660]{GeV}}$. We find that stops lighter than $\sim{\unit[400]{GeV}}$ are still compatible with the $\gamma\gamma{\ensuremath{\slashed{E}_T}}$ searches; however, in this region, a significant fraction of the stops decays into gravitinos and quarks of the third generation. Here the stops are expected to have larger lifetimes than the lightest neutralinos; though, a macroscopic stop decay length is governed by the gravitino mass scale ${\ensuremath{m_{3/2}}}$ and is no hint for a decoupled stop flavor mixing structure. Stops heavier than $\sim{\unit[500]{GeV}}$ can dominantly decay through the decay chain in Eq.  and eventually support MFV if they are a long–lived. As the lifetime depends on both the stops’s flavor mixing and the gravitino mass, it can serve as a hint to MFV only if the contribution of the gravitino modes to the total decay width is negligible. This is assured if the neutralino lifetime is much larger than the stop lifetime. We have discussed a split spectrum where the only light particles are a light gravitino, a light bino, and a light right–handed stop. It may be possible, however, to separate supersymmetric background processes in a scenario with additional light sparticles by vetoing on additional jets and/or isolated leptons. We thank Manuel Drees, Sebastian Grab, and Gudrun Hiller for discussions in the initial phase of this project and for the comments on the manuscript. JSK also thanks the Bethe Center of Theoretical Physics and the Physikalisches Institut at the University of Bonn for their hospitality. This work has been supported in part by the Initiative and Networking Fund of the Helmholtz Association, Contract No. HA-101 (“Physics at the Terascale”) and by the ARC Centre of Excellence for Particle Physics at the Terascale. D0 cuts {#sec:dzero} ======= Irrespective of the very clear signal event structure, we employ the fast detector simulation [`Delphes`]{} as a framework to calculate the signal efficiency. For [D$\slashed{\text{0}}$]{}we use a simplified calorimeter layout composed from cells of dimension $0.1\times\frac{2\pi}{64}$ in $\eta\times\phi$ space covering $|\eta|\le 4.2$ and $\phi\in[0,2\pi[$ where $\eta$ denotes pseudorapidity and $\phi$ the azimuthal angle. Jets are constructed employing the iterative midpoint algorithm with $R=0.5$. We adopt the cuts of [@Abazov:2010us] by requiring - at least two isolated photons with $p_T>{\unit[25]{GeV}}$ and $|\eta| < 1.1$, - the azimuthal angle between $\vec{\ensuremath{\slashed{E}_T}}$ and the hardest jet, if existent, is $< 2.5$, - the azimuthal angles between $\vec{\ensuremath{\slashed{E}_T}}$ and both photons are $> 0.2$, - ${\ensuremath{\slashed{E}_T}}>{\unit[35]{GeV}}$. Photons must have 95% of their energy deposited in the electromagnetic calorimeter. For a photon to be isolated, the calorimetric isolation variable $I$ defined in [@Abazov:2010us] must fulfill $I<0.1$ and the scalar sum of transverse momenta of the tracks in a distance of $0.05<R<0.4$ from the photon must be smaller than . Here is $R=\sqrt{\Delta \phi^2+\Delta\eta^2}$, where $\Delta \phi$ is a track’s azimuthal distance from the photon and $\Delta \eta$ is the corresponding distance in pseudorapidity. Note that nearly all signal events fulfill the isolation criteria hinting that the hadronic stop decay products are well separated from the photon stemming from the subsequent neutralino decay. This is expected as the neutralino decay products $\gamma$ and ${\ensuremath{\tilde{G}}}$ are massless and thus the photon can have a large $p_T$ relative to the stop flight direction. Also note that we assume that a primary vertex can be reconstructed, and that the photon trajectories point to this vertex. The latter assumption should be regarded with care when the neutralino lifetime is large. ATLAS cuts {#sec:atlas} ========== We use the default detector layout implemented in [`Delphes`]{} and apply the following simplified cuts: - At least two isolated photons exist with $p_T>{\unit[25]{GeV}}$ and $|\eta| < 1.81$, but outside the transition region $1.37 < |\eta| < 1.52$, - ${\ensuremath{\slashed{E}_T}}>{\unit[125]{GeV}}$ The study employs a tight photon selection criterion on photon candidates, where the efficiency to identify a true (prompt) photon is approximately 85% in the kinematic region considered [@arXiv:1012.4389; @atlas-conf-2010-005]. We mimic this selection criterion by removing photons from our Monte Carlo sample with a probability of 15%. We consider a photon to be isolated if in a cone of width $R=0.2$, the scalar $E_T$ sum is less than . Here we sum the $E_T$ of the [`Delphes`]{} calorimeter cells and exclude the cell the photon is mapped to. As for the [D$\slashed{\text{0}}$]{}case, nearly all signal photons fulfill the isolation requirement. 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--- abstract: 'We realize a phenomenological study to examine the sensitivity on the magnetic moment and electric dipole moment of the top quark through the processes $\gamma\gamma\rightarrow t \bar{t}$, $e \gamma \rightarrow e \gamma^{*} \gamma \rightarrow e t \bar{t}$ and $e^{-} e^{+}\rightarrow e^{-} \gamma^{*} \gamma^{*} e^{+} \rightarrow e^{-} t \bar{t}e^{+}$ at the CLIC. We find that with a center-of-mass energy of the CLIC-1.4$\hspace{0.8mm}TeV$, integrated luminosity of ${\cal L}=1500\hspace{0.8mm}fb^{-1}$ and CLIC-3$\hspace{1mm}TeV$, integrated luminosity of ${\cal L}=2000\hspace{0.8mm}fb^{-1}$ with systematic uncertainties of $\delta_{sys}=0, 5, 10\hspace{1mm}\%$ at the $95\%\hspace{1mm}C. L.$, it is possible the CLIC may put limits on the electromagnetic dipole moments of the top quark $\hat a_V$ and $\hat a_A$ with a sensitivity of ${\cal O}(10^{-3}-10^{-2})$. Therefore, we show that the sensitivity with the CLIC data is much greater than that for the LHC data.' author: - 'A. A. Billur[^1]' - 'M. Köksal[^2]' - 'A. Gutiérrez-Rodríguez[^3]' title: 'Improved sensitivity on the electromagnetic dipole moments of the top quark in $\gamma\gamma$, $\gamma\gamma^*$ and $\gamma^*\gamma^*$ collisions at the CLIC' --- Introduction ============ The Standard Model (SM) has been tested in many important experiments and has been quite successful, particularly after the discovery of a particle consistent with the Higgs boson with a mass of about $125\pm 0.4 \hspace{0.8mm}GeV$. On the other hand, some of the most fundamental questions still remain unanswered. For instance, the CP problem, neutrino oscillations and matter-antimatter asymmetry have not been adequately clarified by the SM. For this reason, it is often thought that the SM is embedded in a more fundamental theory with which its effects can be observed at higher energy scales. The top quark is the most massive of all observed elementary particles in the SM. Because of the top quark’s large mass, its couplings are expected to be more sensitive to new physics beyond the SM with respect to other particles. New physics can manifest itself in different forms. One possibility is that the new physics may lead to the appearance or a huge increase of new types of interactions like $tH^+b$ or anomalous Flavor Changing Neutral Current $tqg$, $tq\gamma$ and $tqZ$ ($q=u, c$) interactions. Another possibility is the modification of the SM couplings that involve $t\bar t g$, $t\bar t \gamma$, $t\bar t Z$ and $tW b$ vertices. CP violation was first discovered in a small fraction of the kaon decays. This phenomenology can be easily introduced by the Cabibbo-Kobayashi-Maskawa matrix (CKM) in the quark sector. CP violation in this sector is not enough to clarify the baryon asymmetry in the universe. This asymmetry is one of the basic problems in the SM that has not been resolved even in the heavy quarks decay processes. Therefore, the measurement of large amounts of CP violation in the top quark processes in colliders can demonstrate new physics. The existence of new physics can be analyzed by investigating the electromagnetic properties of the top quark that are determined with its dipole moments such as the Magnetic Dipole Moment (MDM) and Electric Dipole Moment (EDM) defined as a source of CP violation. The projection in the SM for the MDM of the top quark is $a^{SM}_t=0.02$ [@Benreuther], and can be tested in the current and future colliders such as the Large Hadron Collider (LHC) and the Compact Linear Collider (CLIC). In contrast, the EDM of the top quark is strongly suppressed with a value of less than $10^{-30}\hspace{0.8mm}e \hspace{1mm}{\mbox{cm}}$ [@Hoogeveen; @Pospelov; @Soni], and is much too small to be observed. However, it is very attractive for probing new physics. If there is a new physics beyond the SM, the top quark may have a higher EDM value than $10^{-30}\hspace{0.8mm}e \hspace{1mm}{\mbox{cm}}$. It is worth mentioning that the sensitivity to the EDM has been studied in models with vector like multiplets which predicted the top quark EDM close to $1.75\times 10^{-3}$ [@Ibrahim]. The studies performed through the $t\bar t\gamma$ production for the LHC at $\sqrt{s}=14\hspace{0.8mm}TeV$, ${\cal L}=300\hspace{0.8mm}fb^{-1}$ and $3000\hspace{0.8mm}fb^{-1}$ reported the limits of $\pm 0.2$ and $\pm 0.1$, respectively [@Baur]. The limits $-2.0\leq \hat a_V\leq 0.3$ and $-0.5\leq \hat a_A\leq 1.5$ are obtained from the branching ratio and the CP asymmetry from radiative $b \to s\gamma$ transitions [@Bouzas]. However, the authors of Ref. [@Bouzas1] obtained the bounds on $|\hat a_V|< 0.05\hspace{0.8mm}(0.09)$ and $|\hat a_A|< 0.20\hspace{0.8mm}(0.28)$ from measurements of the $\gamma^{*} p\to t\bar t$ cross section with $10\%$ $(18\%)$ uncertainty at the Large Hadron electron Collider (LHeC), respectively. Bounds on the dipole moments of the top quark were recently reported in literature through the process $pp \to p\gamma^* \gamma^* p \to p t\bar tp$ for the energy and luminosity of the LHC of $\sqrt{s}=14\hspace{0.8mm}TeV$, ${\cal L}=3000\hspace{0.8mm}fb^{-1}$ and $68\%$ C.L.: $-0.6389\leq \hat a_V\leq 0.0233$ and $|\hat a_A|\leq 0.1158$ [@Sh]. Moreover, in the case of the $e^+e^-$ collider as the International Linear Collider (ILC), the sensitivity bounds at $1\sigma$ for the anomalous couplings of the top quark through top quark pair production $e^+e^- \to t\bar t$ at $\sqrt{s}=500\hspace{0.8mm}GeV$, ${\cal L}=200\hspace{0.8mm}fb^{-1}$, ${\cal L}=300\hspace{0.8mm}fb^{-1}$ and ${\cal L}=500\hspace{0.8mm}fb^{-1}$ are predicted to be of the order ${\cal O}(10^{-3})$, indicating that measurements at an electron positron collider lead to a significant improvement in comparison with the LHC. Thorough and detailed discussions on the dipole moments of the top quark in top quark pair production at the ILC are reported in the literature [@Atwood; @Polouse; @Choi; @Polouse1; @Aguilar0; @Amjad; @Juste; @Asner; @Abe; @Aarons; @Brau; @Baer]. On the other hand, Ref. [@Grzadkowski:2005ye] have found that the process $e^{-}e^{+} \rightarrow t\bar{t}$ will do slightly better than $\gamma \gamma\rightarrow t\bar{t}$ for the determination of the anomalous $tt \gamma$ couplings. In Ref. [@murat], bounds are estimated on the electromagnetic dipole moments of the top quark through the processes $\gamma e^- \to \bar t b\nu_e$ and $e^+e^- \to e^-\gamma^* e^+ \to \bar t b\nu_e e^+$ with unpolarized and polarized electron beams at the CLIC. For the systematic uncertainties of $\delta_{sys}=0\%,\hspace{1mm}5\%$, $b-\mbox{tagging efficiency}=0.8$, center-of-mass energy of $\sqrt{s}=3\hspace{0.8mm}TeV$, integrated luminosity of ${\cal L}=2000\hspace{0.8mm}fb^{-1}$ and $2\sigma\hspace{1mm}(3\sigma)$, the bounds obtained on the electromagnetic dipole moments $\hat a_V$ and $\hat a_A$ of the top quark are of the order ${\cal O}(10^{-2}-10^{-1})$ and are highly competitive with those reported in previous studies. The advantage of the linear $e^{-}e^{+}$ colliders with respect to the hadron colliders is in the general cleanliness of the events where two elementary particles, electrons and positrons beams, collide at high energy, and the high resolutions of the detector made possible by the relatively low absolute rate of background events. In addition, these colliders will complement the physics program of the LHC, especially for precision physics. Therefore, precise measurements of the top quark properties, such as the mass, charge, spin and dipole moments will become possible. The CLIC is a proposed future $e^{-}e^{+}$ collider, designed to fulfill $e^{-}e^{+}$ collisions at center-of-mass energies of 0.35, 1.4 and 3 TeV planned to be constructed with a three main stage research region [@Abramowicz]. This enables the investigation of the $\gamma\gamma$ and $e\gamma$ interactions by converting the original $e^{-}$ or $e^{+}$ beam into a photon beam through the Compton backscattering mechanism. The other well-known applications of the linear colliders are the processes $e\gamma^{*}$, $\gamma \gamma^{*}$ $\gamma^{*} \gamma^{*}$ where the emitted quasireal photon $\gamma^{*}$ is scattered with small angles from the beam pipe of $e^{-}$ or $e^{+}$ [@Ginzburg; @Ginzburg1; @Brodsky; @Budnev; @Terazawa; @Yang]. Since these photons have a low virtuality, they are almost on the mass shell. These processes can be described by the Weizsacker-Williams Approximation (WWA). The WWA has a lot of advantages such as providing the skill to reach crude numerical predictions via simple formulae. In addition, it may principally ease the experimental analysis because it enables one to directly achieve a rough cross section for $\gamma^{*} \gamma^{*}\rightarrow X$ process via the examination of the main process $e^{-}e^{+}\rightarrow e^{-} X e^{+}$ where X represents objects produced in the final state. The production of high mass objects is particularly interesting at the linear colliders and the production rate of massive objects is limited by the photon luminosity at high invariant mass while $\gamma^{*}\gamma^{*}$ and $e\gamma^{*}$ processes at the linear colliders arise from quasireal photon emitted from the incoming beams. Hence, $\gamma^{*} \gamma^{*}$ and $e\gamma^{*}$ are more realistic than $\gamma\gamma$ and $e\gamma$. These processes have been observed experimentally at LEP, Tevatron and LHC [@Abulencia; @Aaltonen1; @Aaltonen2; @Chatrchyan1; @Chatrchyan2; @Abazov; @Chatrchyan3; @Inan; @Inan1; @Inan2; @Sahin1; @Atag; @Sahin2; @Sahin4; @Senol; @Senol1; @Fichet; @Sun; @Sun1; @Sun2; @Senol2; @Atag1; @Koksal1; @koksal2; @koksal3; @koksal4; @billur1; @koksal5; @koksal6; @arı1; @koksal7; @billur2; @billur3; @Fichet1; @Sun3]. In this paper, we perform a phenomenological study for determining the sensitivity on the magnetic moment and electric dipole moment of the top quark through the $t\bar t$ pair production in $e^+e^-$ colliders, specifically for center-of-mass energy and luminosity of CLIC-1.4$\hspace{0.8mm}TeV$, ${\cal L}=1500\hspace{0.8mm}bf^{-1}$ and CLIC-3$\hspace{0.8mm}TeV$, ${\cal L}=2000\hspace{0.8mm}bf^{-1}$ with systematic uncertainty of $\delta_{sys}=0, 5, 10\hspace{0.8mm}\%$ and $95\%\hspace{0.8mm}C.L.$. Here, we consider that the top quark pair production in $e^+e^-$ interactions are given through three different processes $\gamma\gamma\rightarrow t \bar{t}$, $e \gamma \rightarrow e \gamma^{*} \gamma \rightarrow e t \bar{t}$, $e^{-} e^{+}\rightarrow e^{-} \gamma^{*} \gamma^{*} e^{+} \rightarrow e^{-} t \bar{t}e^{+}$ where $\gamma$ and $\gamma^{*}$ are Compton backscattered and Weizsacker-Williams photons, respectively. These processes are one of the most important sources of $t\bar t$ pair production and may represent new physics effects at a high energy and high luminosity linear electron positron collider such as the CLIC and also isolate anomalous $t\bar t\gamma$ coupling from $t\bar t Z$. This work is structured as follows. In Section II, we introduce the top quark effective electromagnetic interactions. In Section III, we study the dipole moments of the top quark through the processes $\gamma\gamma\rightarrow t \bar{t}$, $e \gamma \rightarrow e \gamma^{*} \gamma \rightarrow e t \bar{t}$ and $e^{-} e^{+}\rightarrow e^{-} \gamma^{*} \gamma^{*} e^{+} \rightarrow e^{-} t \bar{t}e^{+}$. Finally, we present our conclusions in Section IV. Top quark pair production processes in photon-photon collisions =============================================================== General Effective Coupling $t\bar t \gamma$ ------------------------------------------- The most general effective coupling $t\bar t\gamma$ which includes the SM coupling and contributions from dimension-six effective operators can be written as [@Sh; @Kamenik; @Baur; @Aguilar; @Aguilar1]: $${\cal L}_{t\bar t\gamma}=-g_eQ_t\bar t \Gamma^\mu_{ t\bar t \gamma} t A_\mu,$$ where $g_e$ is the electromagnetic coupling constant, $Q_t$ is the top quark electric charge and the Lorentz-invariant vertex function $\Gamma^\mu_{t\bar t \gamma}$, which describes the interaction of a $\gamma$ photon with two top quarks, can be parameterized by $$\Gamma^\mu_{t\bar t\gamma}= \gamma^\mu + \frac{i}{2m_t}(\hat a_V + i\hat a_A\gamma_5)\sigma^{\mu\nu}q_\nu,$$ where $m_t$ is the mass of the top quark, $q$ is the momentum transfer to the photon and the couplings $\hat a_V$ and $\hat a_A$ are real and related to the anomalous magnetic moment and the electric dipole moment of the top quark, respectively. Theoretical Calculations ------------------------ Schematic diagrams for the processes $e \gamma \rightarrow e \gamma^{*} \gamma \rightarrow e t \bar{t}$ and $e^{-} e^{+}\rightarrow e^{-} \gamma^{*} \gamma^{*} e^{+} \rightarrow e^{-} t \bar{t}e^{+}$ are given in Fig. \[Fig.1\]. With these processes, $ \gamma (\gamma^*) \gamma(\gamma^*) \rightarrow t \bar{t} $ have two Feynman diagrams which are shown in detail in Fig. \[Fig.2\]. For $\gamma\gamma$, $\gamma\gamma^{*}$ and $\gamma^{*}\gamma^{*}$ collisions including the effective Lagrangian in Eq. 1, the polarization summed amplitude square is given in function of the Mandelstam invariants $\hat{s}$, $\hat{t}$ and $\hat{u}$ as follows, $$\begin{aligned} |M_{1}|^{2}&=&\frac{16\pi^{2}Q_{t}^2\alpha^{2}_e}{2m_{t}^{4}(\hat{t}-m_{t}^{2})^{2}}\biggl[48\hat{a}_{V}(m_{t}^{2}-\hat{t}) (m_{t}^{2}+\hat{s}-\hat{t})m_{t}^{4}-16(3m_{t}^{4}-m_{t}^{2}\hat{s}+\hat{t}(\hat{s}+\hat{t})) m_{t}^{4}\nonumber\\ &+&2(m_{t}^{2}-\hat{t})(\hat{a}_{V}^{2}(17m_{t}^{4}+(22\hat{s}-26\hat{t})m_{t}^{2} +\hat{t}(9\hat{t}-4\hat{s})) \nonumber\\ &+&\hat{a}_{A}^{2}(17m_{t}^{2}+4\hat{s}-9\hat{t})(m_{t}^{2}-\hat{t}))m_{t}^{2}+12\hat{a}_{V}(\hat{a}_{V}^{2}+\hat{a}_{A}^{2})\hat{s}(m_{t}^{3}-m_{t}\hat{t})^{2}\nonumber\\ &-&(\hat{a}_{V}^{2}+\hat{a}_{A}^{2})^{2}(m_{t}^{2}-\hat{t})^{3}(m_{t}^{2}-\hat{s}-\hat{t})\biggr],\end{aligned}$$ $$\begin{aligned} |M_{2}|^{2}&=&\frac{-16\pi^{2}Q_{t}^2\alpha^{2}_e}{2m_{t}^{4}(\hat{u}-m_{t}^{2})^{2}}\biggl[48\hat{a}_{V}(m_{t}^{4}+(\hat{s}-2\hat{t})m_{t}^{2}+\hat{t}(\hat{s}+\hat{t}))m_{t}^{4}\nonumber\\ &+&16(7m_{t}^{4}-(3\hat{s}+4\hat{t})m_{t}^{2}+\hat{t}(\hat{s}+\hat{t})) m_{t}^{4}\nonumber\\ &+&2(m_{t}^{2}-\hat{t})(\hat{a}_{V}^{2}(m_{t}^{4}+(17\hat{s}-10\hat{t})m_{t}^{2}+9\hat{t}(\hat{s}+\hat{t})) \nonumber\\ &+&\hat{a}_{A}^{2}(m_{t}^{2}-9\hat{t})(m_{t}^{2}-\hat{t}-\hat{s}))m_{t}^{2}\nonumber\\ &+&(\hat{a}_{V}^{2}+\hat{a}_{A}^{2})^{2}(m_{t}^{2}-\hat{t})^{3}(m_{t}^{2}-\hat{s}-\hat{t})\biggr],\end{aligned}$$ $$\begin{aligned} M_{1}^{\dag}M_{2}+M_{2}^{\dag}M_{1}&=&\frac{16\pi^{2}Q_{t}^2\alpha^{2}_e}{m_{t}^{2}(\hat{t}-m_{t}^{2})(\hat{u}-m_{t}^{2})} \nonumber \\ &\times &\biggl[-16(4m_{t}^{6}-m_{t}^{4}\hat{s})+8\hat{a}_{V}m_{t}^{2}(6m_{t}^{4}-6m_{t}^{2}(\hat{s}+2\hat{t})-\hat{s})^{2} \nonumber \\ &+&6\hat{t})^{2}+6\hat{s}\hat{t})+(\hat{a}_{V}^{2}(16m_{t}^{6}-m_{t}^{4}(15\hat{s}+32\hat{t})+m_{t}^{2}(15\hat{s})^{2} \nonumber \\ &+&14\hat{t}\hat{s}+16\hat{t})^{2})+\hat{s}\hat{t}(\hat{s}+\hat{t}))+\hat{a}_{A}^{2}(16m_{t}^{6}-m_{t}^{4}(15\hat{s}+32\hat{t}) \nonumber\\ &+&m_{t}^{2}(5\hat{s})^{2}+14\hat{t}\hat{s}+16\hat{t})^{2})+\hat{s}\hat{t}(\hat{s}+\hat{t})))-4\hat{a}_{V}\hat{s}(\hat{a}_{V}^{2}+\hat{a}_{A}^{2})\nonumber\\ &\times& (m_{t}^{4}+m_{t}^{2}(\hat{s}-2\hat{t})+\hat{t}(\hat{s}+\hat{t}))-4\hat{a}_{A}(\hat{a}_{V}^{2}+\hat{a}_{A}^{2})(2m_{t}^{2} -\hat{s}-2\hat{t}) \nonumber\\ &\times& \epsilon_{\alpha \beta \gamma \delta} p_{1}^{\alpha}p_{2}^{\beta}p_{3}^{\gamma}p_{4}^{\delta}-2\hat{s}(\hat{a}_{V}^{2}+\hat{a}_{A}^{2})^{2} (m_{t}^{4}-2\hat{t}m_{t}^{2}+\hat{t}(\hat{s}+\hat{t}))\biggr],\end{aligned}$$ where $\hat s=(p_1 + p_2)^2=(p_3 + p_4)^2$, $\hat t=(p_1 - p_3)^2=(p_4 - p_2)^2$, $\hat u=(p_3 - p_2)^2=(p_1 - p_4)^2$ and $p_{1}$ and $p_{2}$ are the four-momenta of the incoming photons, $p_{3}$ and $p_{4}$ are the momenta of the outgoing top quarks, $Q_{t}$ is the top quark charge, $\alpha_e=g^2_e/4\pi$ is the fine-structure constant, $m_t$ is the mass of top and $\hat a_V$ ($\hat a_A$) are their dipole moments. The most promising mechanism to generate energetic photon beams in a linear collider is Compton backscattering. Compton backscattered photons interact with each other and generate the process $\gamma \gamma \rightarrow t \bar{t}$. The spectrum of Compton backscattered photons is given by $$\begin{aligned} f_{\gamma}(y)=\frac{1}{g(\zeta)}[1-y+\frac{1}{1-y}- \frac{4y}{\zeta(1-y)}+\frac{4y^{2}}{\zeta^{2}(1-y)^{2}}] , \end{aligned}$$ where $$\begin{aligned} g(\zeta)=(1-\frac{4}{\zeta}-\frac{8}{\zeta^2})\log{(\zeta+1)}+ \frac{1}{2}+\frac{8}{\zeta}-\frac{1}{2(\zeta+1)^2} , \end{aligned}$$ with $$\begin{aligned} y=\frac{E_{\gamma}}{E_{e}} , \;\;\;\; \zeta=\frac{4E_{0}E_{e}}{M_{e}^2} ,\;\;\;\; y_{max}=\frac{\zeta}{1+\zeta}. \end{aligned}$$ Here, $E_{0}$ and $E_{e}$ are energy of the incoming laser photon and initial energy of the electron beam before Compton backscattering and $E_{\gamma}$ is the energy of the backscattered photon. The maximum value of $y$ reaches 0.83 when $\zeta=4.8$. WWA is another possibility for top pair production. The quasireal photons emitted from both lepton beams collide with each other and produce the process $\gamma^{*} \gamma^{*} \rightarrow t \bar{t}$. In WWA, the photon spectrum is given by $$\begin{aligned} f_{\gamma^{*}}(x)=\dfrac{\alpha}{\pi E_{e}}\lbrace [\dfrac{1-x+x^2/2}{x}]log(\dfrac{Q^2_{max}}{Q^2_{min}})-\dfrac{m_{e}^2x}{Q^2_{min}}(1-\dfrac{Q^2_{min}}{Q^2_{max}})-\dfrac{1}{x}[1-\dfrac{x}{2}]^2log(\dfrac{x^2 E_{e}^2+Q^2_{max}}{x^2 E_{e}^2+Q^2_{min}})\rbrace,\end{aligned}$$ where $x=E_{\gamma}/E_{e}$ and $Q^2_{max}$ is maximum virtuality of the photon. In this work, we have taken into account the maximum virtuality of the photon is $Q^2_{max}=2\hspace{0.8mm}GeV^2$. The larger values of $Q^2_{max}$ do not make a significant contribution to the sensitivity limits which is similar to results in previous works [@alp; @Gutierrez; @Gutierrez1; @atagx]. The minimum value of the $Q^2_{min}$ is given by $$\begin{aligned} Q^2_{min}=\dfrac{m_{e}^2x^2}{1-x}.\end{aligned}$$ The $Q^2_{min}$ value is very small due to the electron mass. However, the scattering angles of the electrons are so small that the transverse momentum is close to zero. Due to the momentum conservation, the transverse momentum of the emitted photons also have small values. In light of all these arguments, virtuality of the photons in WWA is very small and the photons are almost on mass shell. The process $\gamma^{*} \gamma^{*} \rightarrow t \bar{t}$ participates as a subprocess in the main process $e^{-} e^{+}\rightarrow e^{-} \gamma^{*} \gamma^{*} e^{+} \rightarrow e^{-} t \bar{t}e^{+}$. However, an $\gamma^{*}$ photon emitted from either of the incoming leptons can interact with the Compton backscattered photon and the subprocess $\gamma \gamma^{*} \rightarrow t \bar{t}$ can take place. Hence, we calculate the process $e \gamma \rightarrow e \gamma^{*} \gamma \rightarrow e t \bar{t}$ by integrating the cross section for the subprocess $\gamma \gamma^{*} \rightarrow t \bar{t}$. The total cross sections are, $$\begin{aligned} \sigma=\int f_{\gamma(\gamma^{*})}(x)f_{\gamma(\gamma^{*})}(x)d\hat{\sigma}dE_{1}dE_{2}.\end{aligned}$$ The total cross sections of these processes as functions of anomalous $\hat a_V$ and $\hat a_A$ are shown in Figs. \[Fig.3\]-\[Fig.5\]. In these figures, the cross sections depending on the anomalous couplings were obtained by varying only one of the anomalous couplings at a time while the other was fixed to zero. We understand from Figs. \[Fig.3\]-\[Fig.5\] that the total cross sections show a clear dependence on the dipole moments of the top quark. Anomalous $\hat a_V$ and $\hat a_A$ parameters have different CP properties which can be seen in Eqs. 3-5. The cross section with respect to the $\hat a_A$ parameter is even power and a nonzero value of this parameter allows a constructive effect on the total cross section. In addition, the contribution of $\hat a_V$ coupling to the cross sections is proportional to odd power. In Fig. \[Fig.3\], there are small intervals around $\hat a_V$ in which the cross section that includes new physics is smaller than the SM cross section. For this reason, the $\hat a_V$ parameter has a partially destructive effect on the total cross section. The scattering amplitudes can be given in Eqs. (3)-(5) as a polynomial in powers of $\hat a_V ( \hat a_A )$. Therefore, the cross section as a polynomial in powers of $\hat a_V$$ ( \hat a_A )$ for the three modes $\gamma\gamma \to t\bar t$, $e^+\gamma \to e^+\gamma^* \gamma \to e^+ t \bar t$ and $e^+e^- \to e^+\gamma^* \gamma^* e^- \to e^+ t \bar t e^-$ are given by $$\begin{aligned} \sigma_{Tot}(\hat a_V)&=&\sigma_4 \hat a^4_V + \sigma_3 \hat a^3_V + \sigma_2 \hat a^2_V + \sigma_1 \hat a^1_V + \sigma_0, \\ \sigma_{Tot}(\hat a_A)&=&\sigma'_4 \hat a^4_A + \sigma'_2 \hat a^2_A + \sigma_0,\end{aligned}$$ where $\sigma^i (\sigma^{'i})$ $i=1,..,4$ is the anomalous contribution, while $\sigma_0$ is the contribution of the SM at $\hat a_V= \hat a_A =0$, respectively. This provides more precise and convenient information for each process. The numerical computations of the coefficients of $\hat a_V$ and $\hat a_A $ of the Eqs. (12) and (13) are presented in Table I. Mode $\sigma_4$ $\sigma_3$ $\sigma_2$ $\sigma_1$ $\sigma_0$ $\sigma'_4$ $\sigma'_2$ ------------------------------------------------------------ ------------ ------------ ------------ ------------ ------------ ------------- ------------- -- $\gamma\gamma \to t\bar t$ $4.52$ $4.51$ $5.24$ $0.97$ $0.38$ $4.52$ $4.75$ $e^+\gamma \to e^+\gamma^* \gamma \to e^+ t \bar t$ $0.24$ $0.42$ $0.56$ $0.19$ $0.07$ $0.24$ $0.46$ $e^+e^- \to e^+\gamma^* \gamma^* e^- \to e^+ t \bar t e^-$ 0.012 0.027 0.039 0.016 0.006 0.012 0.031 : Numerical computations of the total cross sections versus $\hat a_V$ and $\hat a_A $ at $\sqrt{s}=$3 TeV. When comparing the three processes in Figs. \[Fig.3\]-\[Fig.5\], the largest deviation from the SM of the anomalous cross sections, including anomalous $\hat a_V$ and $\hat a_A$ couplings, is the process $\gamma\gamma \to t \bar t$. The best sensitivities on anomalous $\hat a_V$ and $\hat a_A$ couplings are obtained from the process $\gamma\gamma \to t \bar t$. Similarly, the sensitivities obtained on anomalous couplings through the process $e \gamma \rightarrow e \gamma^{*} \gamma \rightarrow e t \bar{t}$ are expected to be more restrictive than the sensitivities on the process $e^{-} e^{+}\rightarrow e^{-} \gamma^{*} \gamma^{*} e^{+} \rightarrow e^{-} t \bar{t}e^{+}$. When making a direct comparison of our results for the total cross section as a function of the dipole moments $\hat a_V$ and $\hat a_A$ reported in Figs. \[Fig.3\]-\[Fig.5\] with those reported in Ref. [@Sh] (see Figs. \[Fig.3\]-\[Fig.4\]), we find that our results, using processes $\gamma\gamma \to t \bar t$, $\gamma\gamma^* \to t \bar t$ and $\gamma^*\gamma^* \to t \bar t$ at CLIC energies, with respect to process $pp\to p\gamma^* \gamma^* p \to pt\bar t p$ at LHC energies, show a significant improvement. In addition, with our processes the total cross sections are of 3-4 orders of magnitude better than those reported in Ref. [@Sh]. This shows that the bounds on the anomalous couplings $\hat a_V$ and $\hat a_A$ can be improved at a linear collider such as the CLIC by a few orders of magnitude when compared to what is possible at the LHC. Dipole moments of the top quark in $\gamma\gamma$, $\gamma\gamma^*$ and $\gamma^*\gamma^*$ collisions ===================================================================================================== To investigate the sensitivity to the anomalous $\hat a_V$ and $\hat a_A$ couplings we use the chi-squared distribution: $$\chi^2=\biggl(\frac{\sigma_{SM}-\sigma_{NP}(\hat a_V, \hat a_A)}{\sigma_{SM}\delta}\biggr)^2,$$ where $\sigma_{NP}(\hat a_V, \hat a_A)$ is the total cross section including contributions from the SM and New physics, $\delta=\sqrt{(\delta_{st})^2+(\delta_{sys})^2}$, $\delta_{st}=\frac{1}{\sqrt{N_{SM}}}$ is the statistical error and $\delta_{sys}$ is the systematic error. The number of events for each of the three processes is given by $N_{SM}={\cal L}_{int}\times BR \times \sigma_{SM}\times \epsilon_{b-tag}\times \epsilon_{b-tag}$, where ${\cal L}_{int}$ is the integrated luminosity and b-jet tagging efficiency is 0.8 [@atlas]. The top quarks decay almost 100$\%$ to $W$ boson and b quark. For top quark pair production we can categorize decay products according to the decomposition of $W$. In this work, we assume that one of the $W$ bosons decays leptonically and the other hadronically for the signal. This phenomenon has already been studied by ATLAS and CMS Collaborations [@cmstop1; @cmstop2; @atlastop]. Thus, we assume that the branching ratio of the top quark pair in the final state to be BR = 0.286. For our numerical computation, we take a set of independent input parameters which are known from current experiments. The input parameters are $\alpha=\frac{1}{137.4}$, $m_b=4.18\hspace{0.8mm}GeV$, $m_t=173.21\hspace{0.8mm}GeV$ [@Data2014] and for our analysis, we consider a $95\%$ C.L. sensitivity on the dipole moments $\hat a_V$ and $\hat a_A$ of the top quark via the processes $\gamma\gamma\rightarrow t \bar{t}$, $e \gamma \rightarrow e \gamma^{*} \gamma \rightarrow e t \bar{t}$ and $e^{-} e^{+}\rightarrow e^{-} \gamma^{*} \gamma^{*} e^{+} \rightarrow e^{-} t \bar{t}e^{+}$ at the CLIC-1.4 $TeV$ with ${\cal L}_{int}=1500\hspace{0.8mm}fb^{-1}$ and CLIC-3 $TeV$ with ${\cal L}_{int}=2000\hspace{0.8mm}fb^{-1}$. Tables I-VI illustrate the sensitivity obtained at $95\%$ $C.L.$ on the dipole moments $\hat a_V$ and $\hat a_A$ of the top quark through the processes $\gamma\gamma\rightarrow t \bar{t}$, $e \gamma \rightarrow e \gamma^{*} \gamma \rightarrow e t \bar{t}$ and $e^{-} e^{+}\rightarrow e^{-} \gamma^{*} \gamma^{*} e^{+} \rightarrow e^{-} t \bar{t}e^{+}$. The bounds are obtained assuming that the center-of-mass energy of CLIC-1.4$\hspace{0.8mm}TeV$ and luminosities of ${\cal L}=50, 300, 500, 1000, 1500\hspace{0.8mm}fb^{-1}$ for the second stage of operation of the collider. For the third stage, we consider the center-of-mass energy of CLIC-3$\hspace{0.8mm}TeV$ and luminosities of ${\cal L}=50, 300, 500, 1000, 1500, 2000\hspace{0.8mm}fb^{-1}$. An important part of our analysis is the inclusion of theoretical uncertainties as there may be several experimental and systematic uncertainty sources in top quark identification. This situation has not been studied experimentally in linear colliders. For hadron colliders, especially LHC, the process of determining the cross section of top pair production has been experimentally studied [@topuncertanintyatlas; @topuncertaintycms]. As seen from these studies, the total systematic uncertainty value is about 10$\%$ and is increasingly improved when it is compared with previous experimental studies [@cmstop2]. We use three scenarios for the systematic uncertainties in our entire set of Tables: (1) we assume a systematic uncertainty of $\delta_{sys} = 0\%$, (2) we estimate future results for $\hat a_V$ and $\hat a_A$ with $5\%$ systematic uncertainty and (3) we consider a systematic uncertainty of as much as $\delta_{sys} = 10\%$. We find in Tables I-VI that the most prominent mode of top quark pair production that imposes stronger limits on the dipole moments is the production process $\gamma\gamma \to t \bar t$, followed in order of importance by the processes $e \gamma \rightarrow e \gamma^{*} \gamma \rightarrow e t \bar{t}$ and $e^{-} e^{+}\rightarrow e^{-} \gamma^{*} \gamma^{*} e^{+} \rightarrow e^{-} t \bar{t}e^{+}$, respectively. In conclusion, it is possible that the CLIC may put limits on the electromagnetic dipole moments of the top quark with a sensitivity of the order ${\cal O}(10^{-3}-10^{-2})$ at the $95\%\hspace{0.8mm}C.L.$. We can see from the Figs 3-5, the cross section for the negative values of the $\hat{a}_v$ are smaller than their positive values. This can easily be seen on sensitivity tables: the bounds for the negative values of the $\hat{a}_v$ for increasing luminosity values do not change much. It is worthwhile to compare the results obtained here with those of Ref. [@Sh] which consider the process $pp\to p\gamma^* \gamma^* p \to pt\bar t p$ with the LHC running at $\sqrt{s}=14, 33\hspace{0.8mm}TeV$ and with integrated luminosities of ${\cal L}=100, 300,3000\hspace{0.8mm}fb^{-1}$. Varying one coupling at a time, they find constraints at $68\%\hspace{0.8mm}C.L.$ of the order ${\cal O}(10^{-2}-10^{-1})$. We also note that, while we do consider three systematic errors in our study, the quoted sensitivities in Ref. [@Sh] do not include theoretical uncertainty. Also, the CLIC sensitivity is even better for our processes than for those reported in Ref. [@Sh]. Finally, in Figs. \[Fig.6\]-\[Fig.8\] we show the $95\%$ C.L. contours for anomalous $\hat a_V-\hat a_A$ couplings for the processes $\gamma\gamma\rightarrow t \bar{t}$, $e \gamma \rightarrow e \gamma^{*} \gamma \rightarrow e t \bar{t}$ and $e^{-} e^{+}\rightarrow e^{-} \gamma^{*} \gamma^{*} e^{+} \rightarrow e^{-} t \bar{t}e^{+}$ at the CLIC for various integrated luminosities and center-of-mass energies. Among the three combinations shown in these figures, we find that the strongest simultaneous limits come from the reaction $\gamma\gamma \to t\bar t$ at the CLIC-$3\hspace{0.8mm}TeV$ and ${\cal L}_{int}=2000\hspace{0.8mm}fb^{-1}$ with the $3\sigma$ level. Conclusions =========== The LHC is expected to provide answers to some fundamental questions of the SM. However, high precision measurements may not be available due to remnants from the strong interactions of proton-proton collisions. For this reason, the linear collider with high luminosity and energy is a good choice to complement and extend the LHC physics program. This collider with high luminosity and energy is extremely important to examine new physics beyond the SM. The anomalous $t\bar{t}\gamma$ coupling have very strong energy dependencies since they are characterized by effective Lagrangians that contrains dimensional-high operators. Thus, the cross section including the anomalous $t\bar{t}\gamma$ coupling has a higher energy dependence than the SM cross section. The anomalous $t\bar{t}\gamma$ coupling can be analyzed through the process $e^{-}e^{+}\rightarrow t \bar{t}$ at the linear colliders. This process receives contributions from both anomalous $t\bar{t}\gamma$ and $t\bar{t}Z$ couplings. However, the processes $\gamma\gamma\rightarrow t \bar{t}$, $e \gamma \rightarrow e \gamma^{*} \gamma \rightarrow e t \bar{t}$ and $e^{-} e^{+}\rightarrow e^{-} \gamma^{*} \gamma^{*} e^{+} \rightarrow e^{-} t \bar{t}e^{+}$ isolate $t\bar{t}\gamma$ coupling which provides the possibility to analyze the $t\bar{t}\gamma$ coupling separately from the $t\bar{t}Z$ coupling. Any signal which conflicts with the SM predictions would be convincing evidence for new physics effects in $t\bar{t}\gamma$. In this paper, we carry out a phenomenological study to investigate the sensitivity of the CLIC to the anomalous $t\bar t \gamma$ coupling through the $\gamma\gamma$, $\gamma\gamma^*$ and $\gamma^*\gamma^*$ collision modes followed by the semileptonic decay of the top pair production. We find that with a center-of-mass energy of CLIC-$1.4\hspace{0.8mm}TeV$, integrated luminosity of ${\cal L}=1500\hspace{0.8mm}fb^{-1}$ and CLIC-$3\hspace{0.8mm}TeV$ and integrated luminosity of ${\cal L}=2000\hspace{0.8mm}fb^{-1}$ with systematic uncertainties of $\delta_{sys}=0, 5, 10\hspace{1mm}\%$ at the $95\%\hspace{1mm}C. L.$, it is possible that the CLIC may put limits on the electromagnetic dipole moments of the top quark $\hat a_V$ and $\hat a_A$ with a sensitivity of the order ${\cal O}(10^{-3}-10^{-2})$. In addition, it is noteworthy that our bounds on the dipole moments of the top quark $\hat a_V$ and $\hat a_A$ at $1\hspace{0.8mm}\sigma$ are predicted to be of the order ${\cal O}(10^{-4}-10^{-3})$, which is an order of magnitude better than those reported in Refs. [@Atwood; @Polouse; @Choi; @Polouse1; @Aguilar0; @Amjad; @Juste; @Asner; @Abe; @Aarons; @Brau; @Baer]. Finally, we highlight that the sensitivity with the CLIC data is much stronger than that reported in the literature for the LHC [@Juste] and the ILC [@Abe; @Aguilar0; @Amjad] data. In conclusion, despite the fact that the LHC prospects are already strong due to its excellent statistic, the sensitivity of ILC and the CLIC is even stronger. [cccc]{}\ ${\cal L}\hspace{0.8mm}(fb^{-1})$ & $\delta_{sys}$ & $\hat a_V$ & $\hspace{1.7cm} |\hat a_A|$\ 500 & $0\%$ & \[-0.5170, 0.0034\] & 0.0385\ 500 & $5\%$ & \[-0.5650, 0.0347\] & 0.1286\ 500 & $10\%$ & \[-0.6122, 0.0641\] & 0.1811\ 1000 & $0\%$ & \[-0.5155, 0.0024\] & 0.0324\ 1000 & $5\%$ & \[-0.5649, 0.0346\] & 0.1285\ 1000 & $10\%$ & \[-0.6122, 0.0641\] & 0.1811\ 1500 & $0\%$ & \[-0.5149, 0.0020\] & 0.0293\ 1500 & $5\%$ & \[-0.5648, 0.0346\] & 0.1284\ 1500 & $10\%$ & \[-0.6121, 0.0640\] & 0.1811\ [cccc]{}\ ${\cal L}\hspace{0.8mm}(fb^{-1})$ & $\delta_{sys}$ & $\hat a_V$ & $\hspace{1.7cm} |\hat a_A|$\ 500 & $0\%$ & \[-0.2225, 0.0040\] & 0.0291\ 500 & $5\%$ & \[-0.2564, 0.0331\] & 0.0892\ 500 & $10\%$ & \[-0.2870, 0.0585\] & 0.1254\ 1000 & $0\%$ & \[-0.2212, 0.0029\] & 0.0245\ 1000 & $5\%$ & \[-0.2563, 0.0330\] & 0.0891\ 1000 & $10\%$ & \[-0.2869, 0.0585\] & 0.1254\ 1500 & $0\%$ & \[-0.2206, 0.0024\] & 0.0221\ 1500 & $5\%$ & \[-0.2563, 0.0330\] & 0.0890\ 1500 & $10\%$ & \[-0.2869, 0.0585\] & 0.1254\ 2000 & $0\%$ & \[-0.2203, 0.0020\] & 0.0206\ 2000 & $5\%$ & \[-0.2563, 0.0330\] & 0.0879\ 2000 & $10\%$ & \[-0.2869, 0.0585\] & 0.1254\ [cccc]{}\ ${\cal L}\hspace{0.8mm}(fb^{-1})$ & $\delta_{sys}$ & $\hat a_V$ & $\hspace{1.7cm} |\hat a_A|$\ 500 & $0\%$ & \[-0.7300, 0.0121\] & 0.0848\ 500 & $5\%$ & \[-0.7717, 0.0358\] & 0.1496\ 500 & $10\%$ & \[-0.8244, 0.0647\] & 0.2068\ 1000 & $0\%$ & \[-0.7241, 0.0086\] & 0.0713\ 1000 & $5\%$ & \[-0.7701, 0.0349\] & 0.1477\ 1000 & $10\%$ & \[-0.8236, 0.0643\] & 0.2061\ 1500 & $0\%$ & \[-0.7215, 0.0070\] & 0.0644\ 1500 & $5\%$ & \[-0.7697, 0.0346\] & 0.1470\ 1500 & $10\%$ & \[-0.8234, 0.0641\] & 0.2059\ [cccc]{}\ ${\cal L}\hspace{0.8mm}(fb^{-1})$ & $\delta_{sys}$ & $\hat a_V$ & $\hspace{1.7cm} |\hat a_A|$\ 500 & $0\%$ & \[-0.4626, 0.0088\] & 0.0610\ 500 & $5\%$ & \[-0.4961, 0.0034\] & 0.1249\ 500 & $10\%$ & \[-0.5333, 0.0630\] & 0.1740\ 1000 & $0\%$ & \[-0.4593, 0.0063\] & 0.0512\ 1000 & $5\%$ & \[-0.4955, 0.0343\] & 0.1240\ 1000 & $10\%$ & \[-0.5332, 0.0629\] & 0.1737\ 1500 & $0\%$ & \[-0.4579, 0.0052\] & 0.0463\ 1500 & $5\%$ & \[-0.4953, 0.0342\] & 0.1237\ 1500 & $10\%$ & \[-0.5331, 0.0628\] & 0.1736\ 2000 & $0\%$ & \[-0.4570, 0.0045\] & 0.0431\ 2000 & $5\%$ & \[-0.4952, 0.0341\] & 0.1236\ 2000 & $10\%$ & \[-0.5330, 0.0627\] & 0.1735\ [cccc]{}\ ${\cal L}\hspace{0.8mm}(fb^{-1})$ & $\delta_{sys}$ & $\hat a_V$ & $\hspace{1.7cm} |\hat a_A|$\ 500 & $0\%$ & \[-0.9123, 0.0490\] & 0.1878\ 500 & $5\%$ & \[-0.9298, 0.0580\] & 0.2057\ 500 & $10\%$ & \[-0.9690, 0.0774\] & 0.2415\ 1000 & $0\%$ & \[-0.8859, 0.0357\] & 0.1581\ 1000 & $5\%$ & \[-0.9090, 0.0478\] & 0.1850\ 1000 & $10\%$ & \[-0.9558, 0.0709\] & 0.2299\ 1500 & $0\%$ & \[-0.8739, 0.0295\] & 0.1429\ 1500 & $5\%$ & \[-0.9014, 0.0437\] & 0.1762\ 1500 & $10\%$ & \[-0.9510, 0.0686\] & 0.2256\ [cccc]{}\ ${\cal L}\hspace{0.8mm}(fb^{-1})$ & $\delta_{sys}$ & $\hat a_V$ & $\hspace{1.7cm} |\hat a_A|$\ 500 & $0\%$ & \[-0.6212, 0.0291\] & 0.1259\ 500 & $5\%$ & \[-0.6417, 0.0434\] & 0.1561\ 500 & $10\%$ & \[-0.6773, 0.0679\] & 0.2000\ 1000 & $0\%$ & \[-0.6097, 0.0210\] & 0.1059\ 1000 & $5\%$ & \[-0.6353, 0.0390\] & 0.1472\ 1000 & $10\%$ & \[-0.6739, 0.0656\] & 0.1962\ 1500 & $0\%$ & \[-0.6044, 0.0173\] & 0.0957\ 1500 & $5\%$ & \[-0.6330, 0.0374\] & 0.1439\ 1500 & $10\%$ & \[-0.6728, 0.0648\] & 0.1948\ 2000 & $0\%$ & \[-0.6013, 0.0151\] & 0.0890\ 2000 & $5\%$ & \[-0.6317, 0.0365\] & 0.1420\ 2000 & $10\%$ & \[-0.6721, 0.0644\] & 0.1942\ [**Acknowledgments**]{} A. 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[^1]: abillur@cumhuriyet.edu.tr [^2]: mkoksal@cumhuriyet.edu.tr [^3]: alexgu@fisica.uaz.edu.mx

--- abstract: 'Online socio-technical systems can be studied as proxy of the real world to investigate human behavior and social interactions at scale. Here we focus on Instagram, a media-sharing online platform whose popularity has been rising up to gathering hundred millions users. Instagram exhibits a mixture of features including social structure, social tagging and media sharing. The network of social interactions among users models various dynamics including follower/followee relations and users’ communication by means of posts/comments. Users can upload and tag media such as photos and pictures, and they can “like” and comment each piece of information on the platform. In this work we investigate three major aspects on our Instagram dataset: *(i)* the structural characteristics of its network of heterogeneous interactions, to unveil the emergence of self organization and topically-induced community structure; *(ii)* the dynamics of content production and consumption, to understand how global trends and popular users emerge; *(iii)* the behavior of users labeling media with tags, to determine how they devote their attention and to explore the variety of their topical interests. Our analysis provides clues to understand human behavior dynamics on socio-technical systems, specifically users and content popularity, the mechanisms of users’ interactions in online environments and how collective trends emerge from individuals’ topical interests.' author: - | Emilio Ferrara\ \ \ Roberto Interdonato, Andrea Tagarelli\ \ \ title: | Online Popularity and Topical Interests\ through the Lens of Instagram --- Introduction ============ The study of society through the lens of social media allows us to uncover questions about human behavior at scale [@lazer2009life]. Recent results unveiled complex dynamics in human behavior [@vespignani2009predicting; @centola2010spread], interactions [@aral2009distinguishing; @crandall2010inferring] and influence [@aral2012identifying; @bond201261]. Still, many open questions remain: for example, how do social interactions affect individual and collective behavior? Or, how does connectivity affect individual and collective topical interests? Yet, how do trends and popular content emerge from individuals’ interactions? In this paper we address these questions by studying an emerging socio-technical system, namely Instagram. The popularity of this platform has been growing during recent years: as of the beginning of 2014 Instagram gathers over one hundred million users. Instagram users generate an unparalleled amount of media content. Hence, it should not be surprising that Instagram has recently attracted the attention of the research community, fostering results in different areas including cultural analytics [@hochman2012visualizing; @hochman2013zooming] and urban social behavior [@silva2013comparison]. Instagram represents an unprecedented environment of study, in that it mixes features of various social media and online social networks (including the ability of creating user-generated content in the form of visual media), the option of social tagging, and the possibility of establishing social relations (*e.g.*, followee/follower relationships), and social interactions (*e.g.*, commenting or liking media of other users.) A natural comparison arises between Instagram and other photo sharing systems, particularly Flickr. The two systems appear rather different in terms of features and target of users. Flickr offers more professional-oriented features (*e.g.*, high-quality photos, thematic groups and communities, advanced media organization features.) Instagram, being designed for mobile users, resembles an amateur photo-blog, as it incorporates features to quickly take photos and apply visual effects, and it offers a minimal interface. In other words, Flickr can be seen as a more complete photo sharing platform with social network features, while Instagram resembles a Twitter-like online social network based on photo sharing. Following the lead of studies based on similar platforms such as Flickr [@rattenbury2007towards; @crandall2009mapping; @mislove08wosn; @cha2009www], in this paper we address five different research questions, discussed in the following, spanning different areas of network-, semantic- and topical-based data analysis using signals from user activities and interactions. Contribution and outline ------------------------ We provide a framework to analyze the Instagram ecosystem, incorporating in our model the unique mixture of social interactions, social tagging and media sharing features provided by the platform. By using this framework, we conduct a rigorous analysis focusing on the following main aspects: *(i)* the structural characteristics of the Instagram network, *(ii)* the dynamics of content production and consumption, and *(iii)* the users’ interests modeled via the social tagging mechanisms available to label media with topical tags. We elaborate on each and all these aspects to answer the following research questions: **Q1** : *Network and community structure*:  What are the salient structural features in the network built on the users’ interactions? **Q2** : *Content production and consumption*:  How do users produce and consume content? That is, how do users get engaged on the platform and how do they interact with content produced by others? **Q3** : *Social tagging*: How diverse is the set of tags exploited by each user? In other words, what is the user tagging behavior? **Q4** : *Topical clusters of interest*:   How can users be grouped based on the tags they use to annotate media? **Q5** : *Popularity and topicality*:   How does the topical interests of users affect their popularity? And, how large is the variety of topics covered by each user or by each media? Scope of this work ------------------ To the best of our knowledge, this work is the first to study the Instagram network of users’ interactions, social tagging activities, and topical interests. Therefore, our major goal is to fill a lack of knowledge concerning a number of research issues in Instagram. Within this view, we aimed at providing a first understanding of the above listed aspects of the Instagram network, being aware that all such aspects are interrelated and hence they should be preferably addressed together. It should also be noted that our experimental findings depend on the particular sampling mechanism used to build our dataset; as we shall discuss in the next section, this introduces a bias that does not allow us to provide an analysis of the full Instagram ecosystem, but only of users (and associated media) that are engaged in a public Instagram initiative. Methodology =========== In this section we describe the challenges that we faced in gathering data from the Instagram network, and the technical choices that we adopted to build our dataset. Analogously to other studies, we had to cope with the impossibility of obtaining data directly from the network administrators; therefore, we collected an Instagram sample by querying the Instagram API.[^1] Various features are made publicly available, including: *(i)* the *users API*, which allows sampling from the Instagram user space by querying for specific user account details; *(ii)* the *relationships API*, which retrieves information about specific users, their followers and followees; *(iii)* the *media API*, which queries for specific or popular media; *(iv,v)* the *comments* and the *likes* APIs, respectively, to extract comments and likes from specific media; and *(vi)* the *tags API*, which extracts the keywords associated with specific media, as attributed by the social tagging process of Instagram users. -------------------- --------------- No. Media 1,686,349 No. Distinct users 2,081 No. Tags 8,919,630 No. Distinct tags 269,359 No. Likes 1,242,923,022 No. Comments 41,341,783 -------------------- --------------- : Statistics on the Instagram media dataset. \[tab:statistics\] Crawling strategy ----------------- Our primary objective in crawling the Instagram network was to ensure adequate levels of consistency in user relationships as well as topical variety in media properties, over a timespan possibly larger than the actual crawling period. We expected to detect a user interaction graph having topological properties (*e.g.*, clustering coefficient, average path length) as close as possible to those typically exhibited by other (directed) social media networks [@WilsonBSPZ09; @MisloveMGDB07]; at the same time, we aimed at collecting media whose thematic subjects could span over a predetermined, relatively large classification, while capturing time information about media and user relationships that would allow for trend evolution analysis. Our initial crawling attempt consisted in retrieving media geolocalized w.r.t. a list of touristic/popular locations, which were selected based on their presumed potential to attract users with very different (photographic) tastes, concerning, *e.g.*, art and culture, entertainment and night life, wild life (sea/mountain), etc. Then, the user relations underlying the authors of the retrieved media were taken into account to build a user network. Our hypothesis here was that two users who take pictures within a limited area are more likely to be connected via a follower/followee relation (they may know each other in real life.) Unfortunately, despite the spatial proximity between the authors of the collected media, a poor number of followships were identified, resulting in a network overly disconnected (*e.g.*, clustering coefficient of 2.0E-6). Note that, by trying different sets of touristic locations, we obtained similar results in terms of connectivity. We changed our crawling strategy based on retrieving users that belong to a relatively large “community” in Instagram. Here, our usage of term community corresponds to that of thematic channel, which is typical in many other social media networks (*e.g.*, YouTube); Instagram does not offer an explicit group/community feature, therefore we exploited the existence of public initiatives officially organized by Instagram. We focused our crawling on the Instagram *weekend hashtag project* (WHP) promoted by the Instagram’s official blog.[^2] The characteristics of the WHP initiative and their implications on our data crawling are described next. Dataset construction -------------------- Every Friday, the Instagram team runs a photographic contest, through the Instagram’s official blog. Each contest is assigned a specific topic, which is expressed by a unique (hash)tag prefixed with *\#whp*. According to the project rules, submitted photos need to be marked with no more than one contest-specific tag. We selected $72$ popular contests and randomly picked up about $2,100$ users that participated in at least one of those contests. All media uploaded by these users (including media that were not tagged with *\#whp*-hashtags) were gathered and their information retrieved and stored into the *media dataset*. For each media, we retrieved its unique ID, the ID of the user who posted it, the timestamp of media creation, the set of tags assigned to the media, the number of likes and comments it received. We constructed the *Relational Instagram Network* (RIN) as a directed weighted graph. Edges were drawn to model asymmetric relationships of the form follower-followee, and edge weights were calculated proportionally to the number of likes and comments generated by a user (follower) towards media created by her/his followee. The users selected to build the media dataset were used as seed nodes for the construction of the RIN. Note that we conceived the RIN so to model (asymmetric) relationships that hold strictly among the participants in the contests. The reason for this choice is that including the whole topological neighborhood of the candidate nodes (*e.g.*, the individual egonets also including non-participants) would have resulted again in highly disconnected networks (with clustering coefficient of the order of 1.0E-6). Therefore, we started a breadth-first search process from the set of seed nodes, filtering out any user who did not participate in at least a *\#whp* contest. Our data were crawled over about one-month period (from Jan 20 to Feb 17, 2014). The obtained media dataset contains full information about over 2 thousand users and almost 1.7 million media, with about 9 million tags, 1.2 billion likes, and 41 million comments (see Table \[tab:statistics\].) Details on our RIN are reported in Table \[tab:network\_statistics\]. Here it can be noted that the network of user relations shows a negative, close-to-zero assortativity, which would indicate no tendency of users with similar degree to connect each other. Moreover, the characteristic path length and clustering coefficient are both low, while the modularity is rather high, which would indicate that the RIN has small modules, with moderately dense connections between the nodes within modules and sparse connections between nodes in different modules.[^3] ------------------------ --------- No. Nodes 44,766 No. Links 677,686 Avg. In-degree 15.14 Avg. Path length 3.16 Clustering coefficient 4.1E-2 Diameter 11 Assortativity index -0.097 No. Communities 151 Network modularity 0.578 ------------------------ --------- : Relational Instagram network statistics. \[tab:network\_statistics\] #### Limitations As previously discussed, our dataset is intentionally built around the set of users and media that belong to a competition-driven, large, community in Instagram. Unlike previous work on the Flickr network (a major competitor of Instagram) [@mislove08wosn], we were not able to perform a number of analyses such as, *e.g.*, preferential creation/reception and proximity bias in link creation, which rely on fellowship creation timestamps. This information is missing in our dataset, as the Instagram APIs do not make it available. Flickr APIs do not make it available either, but those authors inferred such temporal information by crawling the Flickr network daily, and monitoring the creation of new links [@mislove08wosn]. Another limitation concerns the analysis of latent interactions (*e.g.*, profile browsing), which has been shown to be a prominent activity in OSNs [@BenevenutoRCA09; @SchneiderFKW09; @JiangWWHSDZ10]: unfortunately, this information is not publicly available for Instagram, while obtaining significant clickstream data (like that used other studies [@BenevenutoRCA09; @SchneiderFKW09]) is challenging. Analysis and Results ==================== ![Distribution of node degree and community size of the Relational Instagram Network.[]{data-label="fig:node_cs_distribution"}](./node_cs_distribution_rev){width="\columnwidth"} We begin with explaining the five research questions that we will address to unveil the characteristics of Instagram. ### Q1: Network and community structure Our first question aims at understanding what are the structural features of the Relational Instagram Network and the characteristic of its community structure. We want to determine the dynamics of social relations and interactions on the system and how they shape (if they do) the structure of the network. In addition, we want to determine whether or not the community structure reflects the self-organization principle [@luhmann1995social] by which individuals in social networks tend to aggregate in communities oriented to topical discussions, and if this, in turn, yields the emergence of a *topically-induced community structure*. ### Q2: Content production and consumption We want also to understand how the cycle of production and consumption of information (*e.g.*, media) is characterized on Instagram. We first aim at understanding what is the driving mechanism of content production; then, we aim to unveil whether content consumption, measured in some way (*e.g.*, via social interactions), follows similar patterns or if any striking difference emerges. ### Q3: Social tagging dynamics In the third research question our goal is to study the dynamics of social tagging on Instagram. We want to study both the patterns of tag adoption at the user level, and at the global level, to characterize how popular tags emerge from the adoption of independent users. We also want to describe the variety of tagging usage by the users, to determine whether users focus their attention on few rather than many contexts. ### Q4: Topical clusters of interests A fourth research questions aims at determining whether it is possible to cluster users exploiting their tagging behavior, and, in turn, if topical clusters emerge by means of such procedure. ### Q5: Popularity and topicality Our final research questions aims at unveiling the dynamics of user popularity and how this relates to topical interests. We hypothesize that popular users might exhibit different patterns of attention and therefore different topical interests. We want to determine whether we can characterize user popularity as function of the variety of their interests, and, in turn, learn how topicality relates to social interactions. ![Visualization of the community structure of the Relational Instagram Network.[]{data-label="fig:cvis_community"}](./cvis_community){width="\columnwidth"} Structural features of the Instagram Network -------------------------------------------- ![Distribution of user content production.[]{data-label="fig:media_distribution"}](./media_distribution_rev){width="\columnwidth"} We discuss the analysis of the Relational Instagram Network (RIN) we carried out to answer our first research question (**Q1**). Our goal here is to study its topological features and determine whether they reflect any particular social process. In particular, we aim at unveiling whether this particular environment, at the boundary between a social network and a sharing media platform, exhibits any characteristic feature: for example, we will drive our attention on the effect of topical interests of users and how these reflect on the network structure. Figure \[fig:node\_cs\_distribution\] shows the distribution of node degree (in blue) and community size (in green) for the RIN. The community detection task has been carried out using two algorithms: the *Louvain method* [@blondel2008fast], and OSLOM [@lancichinetti2011finding]. Results obtained with both methods are consistent (the plot shows the results from the former algorithm.) Both the node degree and the community size distributions are broad and exhibit a fat-tail. A broad degree distribution suggests that the Relational Instagram Network growth may follow a preferential-attachment mechanism [@barabasi1999emergence]: new social relations and social interactions are disproportionately more likely to occur between individuals who previously grew their social network and invested in interacting with others, rather than between users less prone to connect [@simon1955class]. The formation of communities of heterogeneous size suggests the emergence of self organization [@luhmann1995social], a principle explaining that individuals tend to aggregate in units (the communities) optimized for efficiency of communication (*e.g.*, around specific topics of conversation.) A self-organized network structure enjoys crucial properties, including that of enhancing the topicality of interests, or their scope, to smaller sets of individuals rather than to the entire system. By addressing research questions **Q2** and **Q3** in the following sections, we will determine whether these communities emerge from user relations and interactions around certain topics of interest; in other words, we will investigate whether the network exhibits a *topically-induced community structure*. To visualize the community structure of the RIN we produced a graphical representation in Figure \[fig:cvis\_community\], by means of a circular hierarchical algorithm.[^4] Here nodes (*i.e.*, users) belonging to the same community have the same colors, and the hue of the edges transitions from the color of the community of the source node to that of the target one. The RIN community structure clearly separates close clusters of individuals (*e.g.*, bottom-right ones) from clusters of isolated individuals (*e.g.*, top-right ones.) Note that the RIN has (multi)edges weighted by means of social relations and interactions (*i.e.*, follower/followee, likes and comments), being these weights accounted in the community detection and visualization tasks. Differently from other social networks [@ferrara2012large; @grabowicz2012social], Instagram does not exhibit a tight core-periphery structure, whereas communities of large size exist in peripheral areas of the network and they are interconnected with other communities of comparable size. Other basic statistics of the Relational Instagram Network are reported in Table \[tab:network\_statistics\]. ![Distribution of social interactions.[]{data-label="fig:actions_distribution"}](./actions_distribution){width="\columnwidth"} Content production and consumption ---------------------------------- We continue our analysis of the Instagram ecosystem by investigating how users produce and consume content (**Q2**.) Our goal is to determine whether any particular pattern emerges to describe how individuals’ get engaged on the platform and how they interact with content produced by others. To this aim, we study *content production* from the user perspective. Figure \[fig:media\_distribution\] shows the probability density function (pdf($x$)) of the amount $x$ of media posted by each Instagram user in our dataset. This plot suggests peculiar content production dynamics on Instagram: users who already uploaded a large number of media are more likely to do so, causing the presence of a fat tail showing users with a disproportionate amount of media posted on the platform. Individuals exhibit higher tendency to posting new content if they already did that in the past. The lack of a scale-invariant content production dynamics differentiates Instagram from other platforms [@mislove08wosn] (even if some caution is required given how the sample was constructed.) If our observation holds in general, this has an interesting impact from the perspective of system design, in that it suggests a neat separation between active and inactive users: those who are already engaged in using the platform are more likely to keep staying active users. Strategies to engage inactive users could be designed and implemented based on these findings to lower the heterogeneity (*i.e.*, the imbalance) in users involved in content production on the platform. We now investigate *content consumption* on Instagram. Here with content consumption we intend that a given user on the platform has performed some specific action toward a media produced by another user (*e.g.*, liking or commenting it.) This draws an interesting parallel between content production and social interactions, and provides a slightly different perspective from usual studies on platform like social media such as Twitter, where content consumption is intended as users rebroadcast others’ content (*e.g.*, via retweets) aiming at information diffusion rather than interactions. Figure \[fig:actions\_distribution\] shows the distribution of two consumption dynamics, namely “like” and comment, of Instagram users. The plot includes the best fit of a power law to the likes distribution,[^5] with an exponent $\gamma=1.391$ ($x_{min} = 3$, $\sigma = 0.001$), whereas no significant power law fit has been found for the comment distribution that clearly shows two different regimes, $x\lessapprox 250$ and $x\gtrapprox 250$. The “likes” distribution shows a cut-off in the tail due to the finite system size, and suggests that the behavior of likes and comments on Instagram might follow two different dynamics. Popularity of media measured by the number of likes grows by preferential attachment similarly to how, for example, scientific papers acquire citations [@jeong2003measuring]: resources with large number of likes (resp., citations) are more likely to acquire even more. Differently, the ecosystem is less prone to trigger large conversations (based on comments); this is consistent with the theory of user communication efficiency: the different costs (*e.g.*, in terms of time required to perform the action) between “liking” some content and writing a comment affect the nature of interactions among individuals on the platform. ![Tag adoption and global popularity.[]{data-label="fig:tags_distribution"}](./tags_distribution_rev){width="\columnwidth"} Social tagging dynamics {#sec:social_tagging} ----------------------- To answer our question about the dynamics of social tagging on Instagram (**Q3**) we investigated three aspects: *(i)* the tag popularity at the global level and the distribution of tags per media; *(ii)* the distribution of total tags used by the users and their vocabulary size; and, *(iii)* the diversity in tag usage by each individual. Our first goal is to understand how tags emerge in the system at the global level from the tagging patterns of individual media. To this end, we derived the distribution of tag popularity, as represented by the probability density function of observing a given total number of tag occurrences across all media. Then, we obtained the distribution of the number of tags assigned to each media. The results are shown in Figure \[fig:tags\_distribution\]. The plot reports the best fitting of a power law to the distribution of tag popularity with an exponent equal to $\gamma=1.865$ ($x_{min} = 2$, $\sigma = 0.002$), whereas the tags-per-media distribution best fits an exponential-decay function. Two main observations stand out. First, the tag usage mechanism seems to follow an information economy principle of least effort, that is that the majority of media are labeled with just a few tags, and larger sets of tags assigned to the same media are increasingly more unlikely to be observed. Second, although the mechanism describing the assignment of tags is not quite by preferential attachment, the outcome of the process, that is the overall tag popularity, follows a power law behavior. Similar findings have been observed in other popular systems, like Twitter, where popular (hash)tags emerge from individuals’ adoption [@weng2012competition]. Limited attention of users and competition among (hash)tags have been hypothesized as explanation of the emergence of such broad distributions. Moreover, we seek to understand what is the emerging behavior at the user level. We want to determine what patterns of tag adoption users follow, in terms of how many total tags they use, and how many of these tags are distinct. In other words, we establish their vocabulary size (*i.e.*, the number of “words” they are aware of) and we compare it against the total number of tags they produce. Figure \[fig:user\_tags\_distribution\] shows the distribution of, respectively, total and distinct tags used by each user. Both distributions are fat-tailed and show similar slopes. Vocabulary size reflects the information economy principle: the distribution of distinct tags per user spans above one order of magnitude less if compared with that of the total tags usage. This suggests that the actual user vocabulary size is limited, with a large majority of users adopting only few tags. This can be explained by considering that users cannot keep track of all tags emerging on the platform. ![Tag usage and tagset size distributions.[]{data-label="fig:user_tags_distribution"}](./user_tags_distribution_rev){width="\columnwidth"} Finally, to the aim of studying how diverse is the set of tags used by each individual we proceeded as follows. First, we described each given user $u$ in our dataset by means of a vector $T_u$ where each entry represents the frequency $f(t)$ of adoption of tag $t$ (*i.e.*, the total number of times user $u$ adopted tag $t$ to label one of the media she/he uploaded to Instagram), for all tags used by $u$. We define the entropy value $H(\cdot)$, to describe each user’s entropy in the adoption of tags, in the classic Shannon way $$\small\label{eq:user_tag_entropy} H(u) = -\sum_{t\in T_u}{p(t)\cdot \log p(t)}, \quad \mbox{with } p(t)=\frac{f(t)}{\sum_{t\in T_u}{f(t)}}.$$ Afterwards, we determined the probability density function of the distribution of users’ tag adoption entropy, as shown in Figure \[fig:user\_tags\_entropy\]. Note that the entropy ranges between $0$ and the logarithm of the total number of tags of each user. The lower the entropy, the more focused a user’s tagging pattern is (that is, she/he tends to adopt less tags in a more concentrated ways), the more diverse is her/his tagging behavior. Figure \[fig:user\_tags\_entropy\] shows that the entropy is roughly normally distributed with a peak between $5$ and $6$, and a skewness towards lower values of entropy. This suggests that, while a fraction of about 50% of the users tend to exhibit an average tagging variety (corresponding to entropy values $4\lessapprox x\lessapprox 7$), the remainder are either focused ($x\lessapprox 4$) or extremely heterogeneous ($x\gtrapprox 7$) in their tagging adoption. The analysis of tag adoption entropy reveals crucial features from the perspective of modeling user attention: tagging entropy is a proxy to measure how spread or focused users’ attention is towards few or several contexts. A more refined analysis, that will take into account not only tags but the topics that emerge from their co-occurrences is presented later to address **Q5**. ![User-Tag entropy distribution.[]{data-label="fig:user_tags_entropy"}](./user_tags_entropy_rev){width="\columnwidth"} Topical clusters of interest ---------------------------- To answer **Q4**, we conducted a number of experiments aimed at evaluating how users in the media dataset can be grouped together. Users were represented as term-frequency vectors in the space of media tags. We performed the clustering of these users based on Bisecting $k$-Means [@steinbach2000comparison], which is well-suited to produce high-quality (hard) clustering solutions in high-dimensional, large datasets [@zhao2004empirical]. We used the CLUTO clustering toolkit[^6] which provides a globally-optimized version of Bisecting $k$-Means. Feature selection was carried out to retain only the features (*i.e.*, tags) that accounted for 80% of the overall similarity of clusters. We experimented by varying the number $k$ of clusters from 2 to 50, with unitary increment of $k$ at each run. Our evaluation was both quantitative, based on standard within-cluster and across-cluster similarity criteria, and qualitative, based on the cluster characterization in terms of descriptive and discriminating features. The best-quality clustering solution corresponded to $k=5$. ![5-way clustering of the users in the media dataset.[]{data-label="fig:topics_5_users"}](./user-clust_k5){width="45.00000%"} Figure \[fig:topics\_5\_users\] shows a color-intensity plot of the relations between the different clusters of users and features (*i.e.*, tags), corresponding to a 5-way clustering solution. Only a subset of the features is displayed, which corresponds to the union of the most descriptive and discriminating features of each cluster. Moreover, features are re-ordered according to a hierarchical clustering solution, which is visualized on the left-hand side of the figure. A brighter red cell corresponding to a pair feature-cluster indicates higher power of that feature to be, for that cluster, descriptive (*i.e.*, the fraction of within-cluster similarity that this feature can explain) and discriminating (*i.e.*, the fraction of dissimilarity between the cluster and the rest of the objects this feature can explain.) The width of each cluster-column is proportional to the logarithm of the corresponding cluster’s size. It can be noted that the five clusters are quite well-balanced. The first two clusters (*i.e.*, the two left-most columns) are strongly characterized by hashtags denoting the use of popular *applications*, namely VSCO Cam and Latergram. The former is commonly used to modify pictures by adding filters, while the latter is used to schedule the upload of a picture at different (later) time than that of its shot. The \#latergram cluster is also characterized by another popular hashtag, \#tbt, which is an acronym of Throwback Thursday (a “throwback” theme can pertain to some event that happened in the past), and at higher levels in the induced feature-cluster hierarchy, by geographical hashtags (*e.g.*, \#nyc, \#california.) While the fifth cluster is labeled by *subject-based* tags that are evocative of feelings (\#love) or nature (\#sky, \#nature), the third and fourth clusters are instead characterized by either *attention-seeking* tags or *microcommunity-focused* tags: \#photooftheday, \#igmaster are representative of the former category, as users are seeking approval from their peers, whereas \#amselcom, and \#justgoshoot fall into the latter category along with \#iphonesia (originally used by East-Asia users who share photos taken with their iPhones) and \#instagramhub (which aims at helping users understand best practices and sharing tips.) Yet, \#jj, which is run by prominent Instagram user Josh Johnson, denotes a community which asks their users to abide by the rule “for every one photo posted, comment on two others and like three more”. Note that, in general, members of such microcommunities are often asked to share photos on a specific theme, and motivated to create more effective images. These challenges posed by the community continuously prompt their members to play active roles in Instagram. User popularity and topicality ------------------------------ Our final research question (**Q5**) aims at exploring the topical interests space of users and how this affects their popularity. To learn the topics of interest exhibited by the users we employed a topic model which adopts the tags assigned by users to their media as the topical characterization feature. We filtered out tags occurring only once in our corpus, that account for roughly 20% of the total. ![User and media topical entropy.[]{data-label="fig:topic_entropy"}](./ent__lsi_tfidf_t10){width="\columnwidth"} After experimenting with various topic models available in the *gensim* *python* library,[^7] (including LDA and HDP), we adopted Latent Semantic Indexing (LSI) that provided the most interpretable model for a suitable number of topics set to $10$. Note that, differently from topic modeling applications where the impact of the choice of the number of topics might affect the results, in our case we obtained consistent results by using larger number of topics as well (we tested with 5, 10, 20 and 30 topics obtaining consistent results.) We set up our topic model inferring the posterior probability distribution over the topics for each media in our dataset. To determine the topical interests of each user $u$, we simply averaged the probabilities of each topic being exhibited by the media produced by $u$. As concerns the variety of topics covered by each user (as well as that exhibited by a given media), we adopted the Shannon entropy. Similarly to the formula used in Section \[eq:user\_tag\_entropy\] for users and tags, we calculated the probability of observing the topics (rather than the tags.) Afterwards, we estimated the probability distribution of user (respectively, media) topical entropy, as illustrated in Figure \[fig:topic\_entropy\]. Here we observe that the topical entropy (both for users and media) is very concentrated and spans values between $2.5$ and $3.5$ as opposed to the broader entropy interval of user tags, which ranges between $0$ and $9$ (see Figure \[fig:user\_tags\_entropy\].) This suggests that, although users are equally likely to adopt either a narrow or broad vocabulary of tags, their topical interests tend to be in general more concentrated. At the end of this section we will discuss if there are deviations from this pattern, and how they relate to users’ popularity. In other words, we will seek to understand whether popularity can be described by variety of topical interests. Note that user topical entropy and media topical entropy are similarly distributed, as it should be, suggesting the goodness of our approach to build users topical interest profiles. In order to investigate the popularity of users, we measured the total number of likes and comments received by a user’s media. We also account for the total number of times a user likes or comments someone else media, namely the number of *social actions* that this user performs. Such measures are clearly correlated since one is complementary to the other. In Figure \[fig:populary\_actions\_distribution\] we show the distribution of user popularity and user social actions. From the two distributions some interesting facts emerge. First, they are both broadly distributed. The slope of the user popularity distribution is small. This implies the presence of many users with approximately the same (small) popularity. Around $x\gtrapprox 1000$, the slope of the user popularity distribution drastically changes, becoming steeper as of identifying a cut-off due to the finite size of the sample. Values larger than this point coincide with the few extremely popular users who receive a lot many likes and comments to their media. The social actions distribution is still broad but with a steeper slope. This implies that there exist relatively less users (with respect to the popularity distribution) who produce many likes or comments to others’ media. ![User popularity and social actions.[]{data-label="fig:populary_actions_distribution"}](./populary_actions_distribution_rev2){width="\columnwidth"} Our final experiment aims to understand whether user popularity can be explained by means of variety of users’ topical interests. Our goal is to determine whether different classes of popular users emerge, according to their topical interests. To this aim, we correlate user popularity with their topical entropy values discussed above. Figure \[fig:gmm\_entropy\_popularity\] shows a boxplot that separates users in five logarithmic bins. For each bin, the corresponding box extends from the lower to upper quartile values of the data, whereas the whiskers extend from the box to show the range of the popularity values for that bin. A red line corresponds to the median value for each bin. Popularity once again is measured as the sum of likes and comments received from the media produced by each user. Results do not vary when considering the count instead of the sum of social actions, or when varying the number of topics in the topic model. The values of topical entropy span between $2.7$ and $3.3$ bits, in a spectrum of $0.6$ bit overall. From Figure \[fig:gmm\_entropy\_popularity\] two interesting findings emerge. First, user popularity is somewhat affected by topical entropy. As popularity grows, the topical entropy increases accordingly. For example, the median topical entropy for very popular users ($768 < x \leq 7039$) is around $0.1$ bits larger than that of unpopular users ($x \leq 9$). By comparing these two distributions we observe a statistically very significant difference: a two-sided t-test of the two independent samples yields a t-statistic of $3.674$ corresponding to a p-value of $0.0005$. The second observation is that various outliers are present among the popular users; this causes the presence of popular users with topical entropy much lower or much higher than average. Our findings suggest that unpopular users tend to be more focused in their interests with respect to more popular users. However, there exist popular users who are either extremely specialized (very low values of topical entropy) or have extremely broad topical interests. These results complement the intriguing hypothesis, recently advanced by other studies [@weng2013virality; @weng2014predicting], that popularity might be affected by structural features and information diffusion patterns in addition to content production and topical interests. ![Boxplot on popularity and topical entropy.[]{data-label="fig:gmm_entropy_popularity"}](./ent_popularity__lsi_tfidf_t10){width="\columnwidth"} Discussion ========== In this section we summarize the results obtained addressing the five research questions we posed at the beginning of the paper, providing a final memorandum to the reader with the main findings of this work. **A1** : *Network and community structure*:  The network structure of the Relational Instagram Network exhibits two relevant characteristics: a scale-free distribution of node degree and a broad distribution of community size. This suggests that the network growth might happen by preferential attachment, whereas the emergence of the community structure might be driven by self organization of users around topics of interest. **A2** : *Content production and consumption*:  Regarding content production, the life-cycle of information generation on the platform might be explained in Simon’s terms with a heightened likelihood that already engaged users produce more content. Content consumption, on the other hand, might be driven by the information economy principle of least effort: users tend to adopt the “like” behavior strikingly more than producing comments, in line with the intuition that a greater effort (in terms of time and communication economy) must be done to drive the social interaction towards conversation. **A3** : *Social tagging*: Tag usage too is in line with the principle of least effort: the majority of media are labeled with just a few tags, yielding a power law distribution of tagging activity. A similar effect was recently observed in other platforms, like Twitter [@weng2012competition] due to limited attention in combination with competition among tags. **A4** : *Topical clusters of interest*:   Clusters of Instagram users can be detected by means of the tags they adopt to label the contents they produce (and how contents are produced), to indicate their intention to seek approval from other users, or to denote the microcommunity the users belong to. **A5** : *Popularity and topicality*:   User popularity is mildly affected by the breath of topical interests. Increasing values of topical entropy are positively correlated to higher user popularity; however, popular users exhibit more extreme topical entropy values, which means that some popular users are highly specialized, whereas other have very broad interests. This translates in the principle that users with general interests have the same chance to become popular than the specialized ones. Related Work ============ In recent literature, social media and online communities have been used as proxy to study human communication and behavior at scale in different scenarios, including social protests or mobilizations, social influence and political interests and much more [@bond201261; @conover2013geospatial; @conover2013digital]. Other research highlighted how trends emerge and diffuse in socio-technical systems, and how individuals’ interacting in such environments devote their attention [@leskovec2009meme; @budak2011structural]. In this work, we addressed popularity and trends emerging in Instagram. Trends are used to represent popular topics of interest as they are considered indicators of collective attention [@lehmann2012dynamical], and have been studied to detect exogenous real-world events [@sayyadi2009event; @becker2011beyond; @ferrara2013traveling]. Our work explores network features and diffusion patterns of social media content. Information diffusion and the network structure in social media have been extensively studied [@lerman2010information; @grabowicz2012social; @weng2013role; @de2013analyzing]. A lot of attention has been devoted to explore the community structure of such socio-technical systems [@ferrara2012large] and to study the formation and evolution of social groups therein [@backstrom2006group]. The interplay between the dimensions of social interactions and those of topical interests of users have been also investigated showing a mutual reshaping based on mutual feedback mechanisms [@romero2013interplay]. Moreover, our study touches on a mixture of ingredients commonly exhibited by socio-technical systems that digitally mediate communications among individuals: content, topics of discussion, language and tags. Content in social media well reflects socio-economic indicators of users, in that languages highlight patterns of linguistic homogeneity [@quercia2012social], individual and collective satisfaction, demographic characteristics [@mislove2011understanding]. Social network data also exhibit cues of users’ evolution, as discussed in this work. Existing literature witnesses that online content reflects the intuition that users are susceptible to changing their behavior along with experience, and common patterns of evolution emerge over time [@danescu2013no; @mcauley2013amateurs]. Narrowing our focus on research investigating social media features similar to those of Instagram, an extensive analysis of the Flickr social network is reported in [@mislove08wosn; @cha2009www]. Particular attention here is devoted to the understanding of the temporal evolution of network topology, picture popularity, and relating processes of information propagation. However, unlike in our work, no content information (*e.g.*, tags, comments) is taken into account. A study concerning user interactions in the 22 largest regional networks on Facebook is conducted in [@WilsonBSPZ09]. Results show that interaction activity on Facebook is significantly skewed towards a small portion of each user’s social links, and consequently the interaction graph reveals different characteristics than the corresponding social graph. Note that we also leverage the importance of user interactions, as our RIN takes into account like and comment actions. In [@BenevenutoRCA09], clickstream data obtained by an aggregator of social networks (Orkut, Myspace, Hi5 and LinkedIn) are exploited to analyze various aspects of online lifetime of users, such as frequency and length of online sessions, activity sequences and types, dynamics of social interactions. The analysis of workloads has showed that the user browsing is the dominant behavior (accounting for $92\%$ of all requests.) A similar study is performed in [@SchneiderFKW09], where clickstream data from Facebook, LinkedIn, Hi5, and StudiVZ are used to characterize actual user interactions within the sites, in terms of session length, feature popularity and active/inactive time. In our work, we do not consider clickstream data as such information is not made available through the Instagram APIs. User latent interactions in Renren are investigated in [@JiangWWHSDZ10]. Results show that latent interactions (*e.g.*, browsing of users profiles) are more numerous, non-reciprocal and they often connect non-friend strangers if compared to visible ones. In contrast to previously discussed works, the authors in [@JiangWWHSDZ10] did not use clickstream data, but they exploited public data about visits to the crawled user profiles. Conclusions =========== In this paper we presented a broad analysis of the Instagram ecosystem, exploiting its heterogeneous structure, part social network, part tagging environment, and part media sharing platform. We exploited users signals in the form of relationships and interactions to investigate a number of research questions spanning network-, semantic- and topical-based analysis on users, media and how these two dimensions are interrelated. We first focused on the network and community structure, observing that the topical interests exhibited by the users might affect their inter-connectivity and interactions, shaping the network structure around topical communities that can possibly be explained by users’ self organization. We then studied the patterns of content production and consumption on the platform, putting into evidence a strong heterogeneity in the mechanism of production of new information, and the emergence of an information economy principle in the case of content consumption. Our analysis shifted toward the study of social tagging behavior, and we highlighted that users exhibit vocabularies of limited size reflecting their limited attention capabilities, but, nonetheless, popular trends emerge. This can be explained by limited attention of individuals and competition among “memes” (*i.e.*, popular tags.) The study concludes by investigating topics and topicality in the network, and relating it to user popularity. We showed that clusters of users can be found around the tags. 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--- abstract: 'We study the Ginzburg-Landau model of type-I superconductors in the regime of small external magnetic fields. We show that, in an appropriate asymptotic regime, flux patterns are described by a simplified branched transportation functional. We derive the simplified functional from the full Ginzburg-Landau model rigorously via $\Gamma$-convergence. The detailed analysis of the limiting procedure and the study of the limiting functional lead to a precise understanding of the multiple scales contained in the model.' author: - 'Sergio Conti[^1]' - 'Michael Goldman[^2]' - 'Felix Otto[^3]' - 'Sylvia Serfaty[^4]' bibliography: - 'CGOS.bib' title: '[A branched transport limit of the Ginzburg-Landau functional]{}' --- Introduction ============ In 1911, K. Onnes discovered the phenomenon of superconductivity, manifested in the complete loss of resistivity of certain metals and alloys at very low temperature. W. Meissner discovered in 1933 that this was coupled with the expulsion of the magnetic field from the superconductor at the critical temperature. This is now called the Meissner effect. After some preliminary works of the brothers F. and H. London, V. Ginzburg and L. Landau proposed in 1950 a phenomenological model describing the state of a superconductor. In their model (see below), which belongs to Landau’s general theory of second-order phase transitions, the state of the material is represented by the order parameter $u: \Omega\to {\mathbb{C}}$, where $\Omega$ is the material sample. The density of superconducting electrons is then given by $\rho:= |u|^2$. A microscopic theory of superconductivity was first proposed by Bardeen-Cooper-Schrieffer (BCS) in 1957, and the Ginzburg-Landau model was derived from BCS by Gorkov in 1959 (see also [@frankseiringer] for a rigorous derivation). One of the main achievements of the Ginzburg-Landau theory is the prediction and the understanding of the mixed (or intermediate) state below the critical temperature. This is a state in which, for moderate external magnetic fields, normal and superconducting regions coexist. The behavior of the material in the Ginzburg-Landau theory is characterized by two physical parameters. The first is the coherence length $\xi$ which measures the typical length on which $u$ varies, the second is the penetration length $\lambda$ which gives the typical length on which the magnetic field penetrates the superconducting regions. The Ginzburg-Landau parameter is then defined as $\kappa:= \frac{\lambda}{\xi}$. The Ginzburg-Landau functional is given by $$\label{GLorigin} \int_{\Omega} |\nabla_A u|^2 +\frac{\kappa^2}{2} (1-|u|^2)^2 dx +\int_{\R^3} |\nabla\times A- {B_{\mathrm{ext}}}|^2 dx$$ where $A: \R^3\to \R^3$ is the magnetic potential (so that $B:=\nabla\times A$ is the magnetic field), $ \nabla_A u:= \nabla u -i Au$ is the covariant derivative of $u$ and ${B_{\mathrm{ext}}}$ is the external magnetic field. In these units, the penetration length $\lambda$ is normalized to $1$. As first observed by A. Abrikosov this theory predicts two types of superconductors. On the one hand, when $\kappa < 1/\sqrt{2}$, there is a positive surface tension which leads to the formation of normal and superconducting regions corresponding to $\rho\simeq 0$ and $\rho\simeq 1$ respectively, separated by interfaces. These are the so-called type-I superconductors. On the other hand, when $\kappa>1/\sqrt{2}$, this surface tension is negative and one expects to see the magnetic field penetrating the domain through lines of vortices. These are the so-called type-II superconductors. In this paper we are interested in better understanding the former type but we refer the interested reader to [@tinkham; @SanSerf; @Serfrev] for more information about the latter type. In particular, in that regime, there has been an intensive work on understanding the formation of regular patterns of vortices known as Abrikosov lattices. In type-I superconductors, it is observed experimentally [@ProzHo; @Proz; @Proal] that complex patterns appear at the surface of the sample. It is believed that these patterns are a manifestation of branching patterns inside the sample. Although the observed states are highly history-dependent, it is argued in [@ChokKoOt; @Proal] that the hysteresis is governed by low-energy configurations at vanishing external magnetic field. The scaling law of the ground-state energy was determined in [@ChokConKoOt; @ChokKoOt] for a simplified sharp interface version of the Ginzburg-Landau functional and in [@CoOtSer] for the full energy, these results indicate the presence of a regime with branched patterns at low fields. This paper aims at a better understanding of these branched patterns by going beyond the scaling law. Starting from the full Ginzburg-Landau functional, we prove that in the regime of vanishing external magnetic field, low energy configurations are made of nearly one-dimensional superconducting threads branching towards the boundary of the sample. In a more mathematical language, we prove $\Gamma-$convergence [@braides; @dalmaso] of the Ginzburg-Landau functional to a kind of branched transportation functional in this regime. We focus on the simplest geometric setting by considering the sample $\Om$ to be a box $Q_{\Lz,T}:=(-\frac{\Lz}{2}, \frac{\Lz}{2})^2\times(-T,T)$ for some $T,\Lz>0$ and consider periodic lateral boundary conditions. The external magnetic field is taken to be perpendicular to the sample, that is ${B_{\mathrm{ext}}}:= {b_{\mathrm{ext}}}e_3$ for some ${b_{\mathrm{ext}}}>0$ and where $e_3$ is the third vector of the canonical basis of $\R^3$. After making an isotropic rescaling, subtracting the bulk part of the energy and dropping lower order terms (see and the discussion after it), minimizing can be seen as equivalent to minimizing $$\begin{gathered} \label{Etilde} E_T(u,A):=\frac{1}{L^2} \int_{Q_{L,1}} |\nabla_{TA} u |^2 + \left(B_3 - \a (1-\rho)\right)^2 + |B'|^2 dx + \| B_3-\a \b\|_{H^{-1/2}(\{x_3=\pm 1\})}^2,\,\end{gathered}$$ where we have let $B:=(B',B_3):=\nabla \times A$, $$\kappa T:=\sqrt{2}\alpha, \qquad {b_{\mathrm{ext}}}:=\frac{\beta \kappa}{\sqrt{2}} \qquad \textrm{ and } \qquad L:=\Lz/T.$$ If $u=\rho^{1/2} \exp(i\theta)$, since $|\nabla_{TA} u |^2 =|\nabla \rho^{1/2}|^2 +\rho|\nabla \theta-T A|^2$, in the limit $T\to +\infty$ we obtain, at least formally, that $A$ is a gradient field in the region where $\rho>0$ and therefore the Meissner condition $\rho B=0$ holds. Moreover, in the regime $\alpha\gg 1$, from we see that $B_3 \simeq \a (1-\rho)$ and $\rho$ takes almost only values in $\{0,1\}$. Hence $\Div B=0$ can be rewritten as $$ \partial_3 \chi + \frac1\alpha\Divp \chi B'=0,$$ where $\chi:= (1-\rho)$ and $\Divp$ denotes the divergence with respect to the first two variables. Therefore, from the Benamou-Brenier formulation of optimal transportation [@AGS; @villani] and since from the Meissner condition, $B'\simeq \frac{1}{\chi} B'$, the term $$\int_{Q_{L,1}} |B'|^2 dx\simeq \int_{Q_{L,1}} \frac{1}{\chi}|B'|^2 dx$$ in the energy can be seen as a transportation cost. We thus expect that inside the sample (this is, in $Q_{L,1}$), superconducting domains where $\rho\simeq1$ and $B\simeq 0$ alternate with normal ones where $\rho\simeq 0$ and $B_3\simeq \alpha$. Because of the last term $\| B_3-\a \b\|_{H^{-1/2}(\{x_3=\pm 1\})}^2$ in the energy , one expects $B\simeq \a \b e_3$ outside the sample. This implies that close to the boundary the normal domains have to refine. The interaction between the surface energy, the transportation cost and the penalization of an $H^{-1/2}$ norm leads to the formation of complex patterns (see Figure \[figscales\]). It has been proven in [@CoOtSer] that in the regime $T\gg 1$, $\a \gg 1$ and $\b \ll 1$, $$\label{scallawintro} \min E_T(u,A)\sim \min\{ \a^{4/3} \b^{2/3}, \a^{10/7} \b\}\,.$$ The scaling $\min E_T(u,A)\sim \a^{4/3} \b^{2/3}$ (relevant for $\a^{-2/7}\ll \b$) corresponds to uniform branching patterns whereas the scaling $\min E_T(u,A)\sim \a^{10/7} \b$ corresponds to non-uniform branching ones. We focus here for definiteness on the regime $\min E_T(u,A)\sim \a^{4/3} \b^{2/3}$, although we believe that our proof can be extended to the other one. Based on the construction giving the upper bounds in , we expect that in the first regime there are multiple scales appearing (see Figure \[figscales\]): $$\label{scalesepar}\stackrel{\displaystyle\textrm{penetration}}{\textrm{ length}}\,\stackrel{\displaystyle\ll}{\, } \, \stackrel{\displaystyle\textrm{ coherence}}{\textrm{ length}}\, \stackrel{\displaystyle\ll}{\,} \, \stackrel{\displaystyle\textrm{ diameter of the}}{\textrm{ threads in the bulk}} \, \, \stackrel{\displaystyle\ll}{\,} \, \, \stackrel{\displaystyle\textrm{distance between the}}{\textrm{ threads in the bulk}},$$ which amounts in our parameters to $$T^{-1} \ll \alpha^{-1}\ll \alpha^{-1/3}\beta^{1/3}\ll \alpha^{-1/3}\beta^{-1/6}.$$ \[figscales\] In order to better describe the minimizers we focus on the extreme region of the phase diagram $T,T \a^{-1}, \beta^{-1}, \a \b^{7/2}\to +\infty$, with $L={\widetilde}L \alpha^{-1/3}\beta^{-1/6}$ for some fixed ${\widetilde}L>0$. In this regime, we have in particular $\a^{-1}\ll \a^{-1/3}\b^{1/3}$ so that the separation of scales holds. We introduce an anisotropic rescaling (see Section \[Sec:GL\]) which leads to the functional $$\begin{aligned} \label{eqdefwe} \wE(u,A)&:=\frac{1}{\widetilde{L}^2}\Big[\int_{Q_{{\widetilde}L,1}} \a^{-2/3}\b^{-1/3}\lt|\nabla_{\a^{1/3}\b^{-1/3}TA}'u\rt|^2+ \a^{-4/3}\b^{-2/3}\lt|(\nabla_{\a^{1/3}\b^{-1/3}TA} u)_3\rt|^2\\ \nonumber + & \lt.\a^{2/3}\b^{-2/3}\left({B_3} -(1-|{u}|^2)\right)^2 + \b^{-1}|{B'}|^2 dx +\a^{1/3}\b^{7/6}\|\b^{-1}{B_3}-1\|_{H^{-1/2}(x_3=\pm1)}^2\rt].\end{aligned}$$ Our main result is a $\Gamma-$convergence result of the functional $\wE$ towards a functional defined on measures $\mu$ living on one-dimensional trees. These trees correspond to the normal regions in which $\rho\simeq 0$ and where the magnetic field $B$ penetrates the sample. Roughly speaking, if for a.e. $x_3\in (-1,1)$ the slice of $\mu=\mu_{x_3}\otimes dx_3$ has the form $\mu_{x_3}=\sum_i {\varphi}_i \delta_{X_i(x_3)}$ where the sum is at most countable, then we let (see Section \[Sec:limiting\] for a precise definition) $$\label{eqdefIintro} I(\mu):=\frac1{\widetilde{L}^2}\int_{-1}^1 K_* \sum_i \sqrt{{\varphi}_i}+ {\varphi}_i |{\dot{X}}_i|^2 dx_3,$$ where $K_*=8\sqrt\pi/3$ and ${\dot{X}}_i$ denotes the derivative (with respect to $x_3$) of $X_i(x_3)$. The $X_i$’s represent the graphs of each branch of the tree (parametrized by height) and the ${\varphi}_i$’s represent the flux carried by the branch. We can now state our main result \[maintheo\] Let $T_n, \a_n, \b_n^{-1}\to+\infty$ with $T_n \a_n^{-1}, \a_n \b_n^{7/2}\to +\infty$, ${\widetilde}L>0$, then: 1. \[maintheolb\] For every sequence $(u_n, A_n)$ with $\sup_n \wE(u_n,A_n)<+\infty$, up to subsequence , $\beta_n^{-1}(1-|u_n|^2)$ weakly converges to a measure $\mu$ of the form $\mu=\mu_{x_3}\otimes dx_3$ with $\mu_{x_3}=\sum_i {\varphi}_i \delta_{X_i}$ for a.e. $x_3\in (-1,1)$, $\mu_{x_3} \weaklim dx'$ (where $dx'$ denotes the two dimensional Lebesgue measure on $Q_{{\widetilde}L}$) when $x_3\to \pm 1$ and such that $$\liminf_{n\to +\infty} \wE(u_n,A_n)\ge I(\mu).$$ 2. \[maintheoub\] If in addition $L_n^{2} \a_n \b_n T_n\in 2\pi \Z$, where $L_n:= {\widetilde}L \a_n^{-1/3}\b_n^{-1/6}$, then for every measure $\mu$ such that $I(\mu)<+\infty$ and $\mu_{x_3} \weaklim dx'$ as $x_3\to \pm1$ , there exists $(u_n,A_n)$ such that $\b_n^{-1}(1-|u_n|^2)\weaklim \mu$ and $$\limsup_{n\to +\infty} \, \wE(u_n,A_n)\le I(\mu).$$ By scaling, it suffices to consider the case ${\widetilde}L=1$. The first assertion follows from Proposition \[gammaliminf\], the second one from Proposition \[gammalimsup-v2\]. Let us stress once again that our result could have been equivalently stated for the full Ginzburg-Landau energy instead of $\wE$ (see Section \[Sec:GL\]).\ Within our periodic setting, the quantization condition $L_n^2 \a_n \b_n T_n\in 2\pi \Z$ for the flux is a consequence of the fact that the phase circulation of the complex-valued function in the original problem is naturally quantized. It is necessary in order to make our construction but we believe that it is also a necessary condition for having sequences of bounded energy (see the discussion in Section \[Sec:GL\] and the construction in Section \[recoveryGL\]). We remark that scaling back to the original variables this condition is the physically natural one $L_0^2{b_{\mathrm{ext}}}\in 2\pi\Z$.\ Before going into the proof of Theorem \[maintheo\] we address the limiting functional $I(\mu)$, which has many similarities with irrigation (or branched transportation) models that have recently attracted a lot of attention (see [@BeCaMo] and more detailed comments in Section \[Sec:reliri\] or the recent papers [@BraWirth; @BraWir16] where the connection is also made to some urban planning models). In Section \[Sec:limiting\], we first prove that the variational problem for this limiting functional is well-posed (Proposition \[existmu\]) and show a scaling law for it (Proposition \[branchingmu\] and Proposition \[branchinglowerbound\]). In Proposition \[def:subsystem\], we define the notion of subsystems which allows us to remove part of the mass carried by the branching measure. This notion is at the basis of Lemma \[noloop\] and Proposition \[finitebranch\] which show that minimizers contain no loops and that far from the boundary, they are made of a finite number of branches. From the no-loop property, we easily deduce Proposition \[reg\] which is a regularity result for minimizers of $I$. The main result of Section \[Sec:limiting\] is Theorem \[theodens\] which proves the density of “regular” measures in the topology given by the energy $I(\mu)$. As in nearly every $\Gamma-$convergence result, such a property is crucial in order to implement the construction for the upper bound \[maintheoub\]. We now comment on the proof of Theorem \[maintheo\]. Let us first point out that if the Meissner condition $\rho B =0$ were to hold, and $A$ could be written as a gradient field in the set $\{\rho>0\}$, then $|\nabla_{TA} u|^2= |\nabla \rho^{1/2}|^2$ and we would have $$\label{introlowerbound}\int_{Q_{L,1}} |\nabla_{TA} u |^2 + \left(B_3 - \a (1-\rho)\right)^2dx=\int_{Q_{L,1}} |\nabla \rho^{1/2} |^2 + \a^2 \chi_{\rho>0} (1-\rho)^2 +\chi_{\rho=0} (B_3-\a)^2 dx.$$ This is a Modica-Mortola [@ModMort] type of functional with a degenerate double-well potential given by $W(\rho):= \chi_{\rho>0} (1-\rho)^2$. Thanks to Lemma \[lemmameissner\], one can control how far we are from satisfying the Meissner condition. From this, we deduce that almost holds (see Lemma \[lemmafirstlowerbound\]). This implies that the Ginzburg-Landau energy gives a control over the perimeter of the superconducting region $\{\rho>0\}$. In addition, $\b\ll 1$ imposes a small cross-area fraction for $\{\rho>0\}$. Using then isoperimetric effects to get convergence to one-dimensional objects (see Lemma \[lowerbound2d\]), we may use Proposition \[gammaliminf\] to conclude the proof of \[maintheolb\]. In order to prove \[maintheoub\], thanks to the density result in Theorem \[theodens\], it is enough to consider regular measures. Given such a measure $\mu$, we first approximate it with quantized measures (Lemma \[lemmaquantize\]). Far from branching points the construction is easy (see Lemma \[lemmacurve3\]). At a branching point, we need to pass from one disk to two (or vice-versa); this is done passing through rectangles (see Lemma \[Lembranch3\] and Figure \[fig1\]). Close to the boundary we use instead the construction from [@CoOtSer], which explicitly generates a specific branching pattern with the optimal energy scaling; since the height over which this is done is small the prefactor is not relevant here (Proposition \[propboundraylayer\]). The last step is to define a phase and a magnetic potential to get back to the full Ginzburg-Landau functional. This is possible since we made the construction with the Meissner condition and quantized fluxes enforced, see Proposition \[thirdupperbound\]. From and the discussion around , for type-I superconductors, the Ginzburg-Landau functional can be seen as a non-convex, non-local (in $u$) functional favoring oscillations, regularized by a surface term which selects the lengthscales of the microstructures. The appearance of branched structures for this type of problem is shared by many other functionals appearing in material sciences such as shape memory alloys [@KohnMuller92; @KohnMuller94; @Conti00; @KnuepferKohnOtto2013; @BelGol; @Zwicknagl2014; @ChanConti2015; @ContiZwicknagl], uniaxial ferromagnets [@ChoKoOtmicro; @ViehOtt; @KnMu] and blistered thin films [@BCDM00; @JinSternberg; @BCDM02]. Most of the previously cited results on branching patterns (including [@ChokConKoOt; @ChokKoOt; @CoOtSer] for type-I superconductors) focus on scaling laws. Here, as in [@ViehOtt; @CDZ], we go one step further and prove that, after a suitable anisotropic rescaling, configurations of low energy converge to branched patterns. The two main difficulties in our model with respect to the one studied in [@ViehOtt] are the presence of an additional lengthscale (the penetration length) and its sharp limit counterpart, the Meissner condition $\rho B=0$ which gives a nonlinear coupling between $u$ and $B$. Let us point out that for the Kohn-Müller model [@KohnMuller92; @KohnMuller94], a much stronger result is known, namely that minimizers are asymptotically self-similar [@Conti00] (see also [@Viehmanndiss; @AlChokOt] for related results). In [@Gol], the optimal microstructures for a two-dimensional analogue of $I(\mu)$ are exactly computed. The paper is organized as follows. In section \[sec:not\], we set some notation and recall some notions from optimal transport theory. In Section \[Sec:GL\], we recall the definition of the Ginzburg-Landau functional together with various important quantities such as the superconducting current. We also introduce there the anisotropic rescaling leading to the functional $\widetilde{E}_T$. In Section \[Sec:intermediate\], we introduce for the sake of clarity intermediate functionals corresponding to the different scales of the problem. Let us stress that we will not use them in the rest of the paper but strongly believe that they help understanding the structure of the problem. In Section \[Sec:limiting\], we carefuly define the limiting functional $I(\mu)$ and study its properties. In particular we recover a scaling law for the minimization problem and prove regularity of the minimizers. We then prove the density in energy of ’regular’ measures. This is a crucial result for the main $\Gamma-$ convergence result which is proven in the last two sections. As customary, we first prove the lower bound in Section \[Sec:gammaliminf\] and then make the upper bound construction in Section \[Sec:upperbound\]. Notation and preliminary results {#sec:not} ================================= In the paper we will use the following notation. The symbols $\sim$, $\ges$, $\les$ indicate estimates that hold up to a global constant. For instance, $f\les g$ denotes the existence of a constant $C>0$ such that $f\le Cg$, $f\sim g$ means $f\les g$ and $g\les f$. In heuristic arguments we use $a\simeq b$ to indicate that $a$ is close (in a not precisely specified sense) to $b$. We use a prime to indicate the first two components of a vector in $\R^3$, and identify $\R^2$ with $\R^2\times\{0\}{\subseteq}\R^3$. Precisely, for $a\in \R^3$ we write $a'=(a_1,a_2,0)\in\R^2{\subseteq}\R^3$; given two vectors $a,b\in\R^3$ we write $a'\times b'=(a\times b)_3 = (a'\times b')_3$. We denote by $(e_1,e_2,e_3)$ the canonical basis of $\R^3$. For $L>0$ and $T>0$, $Q_L:=(-\frac{L}{2},\frac{L}{2})^2$ and $Q_{L,T}:=Q_L\times(-T,T)$. For a function $f$ defined on $Q_{L,T}$, we denote $f_{x_3}$ the function $f_{x_3}(x'):=f(x',x_3)$ and we analogously define for $\Omega {\subseteq}Q_{L,T}$, the set $\Omega_{x_3}$. For $x=(x',x_3)$ and $r>0$ we let $\B_r(x)=\B(x,r)$ be the ball of radius $r$ centered at $x$ (in $\R^3$) and $\B'_r(x')=\B'(x',r)$ be the analogue two-dimensional ball centered at $x'$. Unless specified otherwise, all the functions and measures we will consider are periodic in the $x'$ variable, i.e., we identify $Q_L$ with the torus $\R^2/L\Z^2$. In particular, for $x',y'\in Q_L$, $|x'-y'|$ denotes the distance for the metric of the torus, i.e., $|x'-y'|:=\min_{k\in \Z^2} |x'-y'+Lk|$. We denote by $\H^k$ the $k-$dimensional Hausdorff measure. We let $\cP(Q_L)$ be the space of probability measures on $Q_L$ and $\M(Q_{L,T})$ be the space of finite Radon measures on $Q_{L,T}$, and similarly $\M(Q_L)$. Analogously, we define $\M^+(Q_{L,T})$ and $\M^+(Q_L)$ as the spaces of finite Radon measures which are also positive. For a measure $\mu$ and a function $f$, we denote by $f\sharp \mu$ the push-forward of $\mu$ by $f$. We recall the definition of the (homogeneous) $H^{-1/2}$ norm of a function $f\in L^2(Q_{L})$ with $\int_{Q_{L}} f dx'=0$, $$\label{defHdem} \|f\|_{H^{-1/2}}^2:=\inf\lt\{ \int_{Q_{L}\times [0,+\infty)} |B|^2 dx \ : \ \Div B=0, \ B_3(\cdot,0)=f \rt\},$$ which can be alternatively given in term of the two-dimensional Fourier series as (see, e.g., [@ChokKoOt]) $$ \|f\|_{H^{-1/2}}^2=\frac{1}{2\pi} \sum_{ k'\in \lt(\frac{2\pi}{L}\Z\rt)^2\backslash\{0\}} \frac{ |\hat f (k')|^2}{|k'|}.$$ We shall write $\|f\|^2_{H^{-1/2}(Q_L\times\{\pm T\})}$ for $\|f\|^2_{H^{-1/2}(Q_L\times\{T\})}+ \|f\|^2_{H^{-1/2}(Q_L\times\{-T\})}$. The $2$-Wasserstein distance between two measures $\mu$ and $\nu\in {\mathcal{M}}^+(Q_L)$ with $\mu(Q_L)=\nu(Q_L)$ is given by $$W_2^2(\mu,\nu):=\min\lt\{ \mu(Q_L)\int_{Q_L\times Q_L} |x-y|^2 \, d\Pi(x,y) \, : \, \Pi_1=\mu, \ \Pi_2=\nu\rt\},$$ where the minimum is taken over measures on $Q_L\times Q_L$ and $\Pi_1$ and $\Pi_2$ are the first and second marginal of $\Pi$[^5], respectively. For measures $\mu,\nu\in{\mathcal{M}}^+(Q_{L,T})$, the $2$-Wasserstein distance is correspondingly defined. We now introduce some notions from metric analysis, see [@AGS; @villani] for more detail. A curve $\mu:(a,b)\to \cP(Q_L)$, $z\mapsto \mu_z$ belongs to $AC^2(a,b)$ (where $AC$ stands for absolutely continuous) if there exists $m\in L^2(a,b)$ such that $$\label{metricderiv} W_2(\mu_{z},\mu_{{\widetilde}{z}})\le \int_{z}^{{\widetilde}{z}} m(t) dt \ \qquad \forall a<z\le {\widetilde}{z} <b.$$ For any such curve, [the [*speed*]{}]{} $$|\mu'|(z):=\lim_{{\widetilde}{z}\to z}\frac{W_2(\mu_{{\widetilde}{z}},{\mu}_{z})}{|z-{\widetilde}{z}|}$$ exists for $\H^1-$a.e. $z\in (a,b)$ and $|\mu'|(z)\le m(z)$ for $\H^1$-a.e. $z\in (a,b)$ for every admissible $m$ in . Further, there exists a Borel vector field $B$ such that $$\label{eqexistsB} B(\cdot,z)\in L^2(Q_L,\mu_{z}), \qquad \| B(\cdot,z)\|_{L^2(Q_L,\mu_{z})}\le |\mu'|(z) \quad \textrm{for } \H^1\text{-a.e. } \ z\in (a,b)$$ and the continuity equation $$\label{conteqini} \partial_3 \mu_{z}+\Divp (B \mu_{z})=0$$ holds in the sense of distributions [@AGS Th. 8.3.1]. Conversely, if a weakly continuous curve $\mu_{z} : (a,b)\to \cP(Q_L)$ satisfies the continuity equation for some Borel vector field $B$ with $\| B(\cdot,z)\|_{L^2( Q_L,\mu_{z})}\in L^2(a,b)$ then $\mu\in AC^2(a,b)$ and $|\mu'|(z)\le \| B(\cdot,z)\|_{L^2(Q_L,\mu_{z})}$ for $\H^1$-a.e. $z\in (a,b)$. In particular, we have $$\label{eqw22b} W_2^2(\nu,\hat\nu)= \min_{\mu,B} \lt\{ 2T\int_{Q_{L,T}} |B|^2 d\mu_{z} dz \ :\ \mu_{-T}=\nu, \ \mu_T= \hat\nu \textrm{ and \eqref{conteqini} holds}\rt\}\,,$$ where by scaling the right-hand side does not depend on $T$.\ For a (signed) measure $\mu\in \M(Q_L)$, we define the Bounded-Lipschitz norm of $\mu$ as $$\label{BLdefin} \|\mu\|_{BL}:=\sup_{\|\psi\|_{Lip}\le 1} \int_{Q_L} \psi d\mu,$$ where for a $Q_L-$periodic and Lipschitz continuous function $\psi$, $\|\psi\|_{Lip}:=\|\psi\|_{\infty}+\|\nabla \psi\|_{\infty}$. By the Kantorovich-Rubinstein Theorem [@villani Th. 1.14], the $1-$Wasserstein and the Bounded-Lipschitz norm are equivalent. The Ginzburg-Landau functional {#Sec:GL} ============================== In this section we recall some background material concerning the Ginzburg-Landau functional and introduce the anisotropic rescaling leading to $\wE$.\ For a (non necessarily periodic) function $u:Q_{\Lz,T}\to{\mathbb{C}}$, called the order parameter, and a vector potential $A: Q_{\Lz}\times\R\to \R^3$ (also not necessarily periodic), we define the covariant derivative $$\nabla_A u:= \nabla u-iAu,$$ the magnetic field $$B:=\nabla\times A,$$ and the [superconducting]{} current $$\label{defj} j_A:= \frac12\left( -i\bar u (\nabla_A u)+ i u (\overline{\nabla_A u})\right)=\textrm{Im}(iu \overline{\nabla_A u})\,.$$ Let us first notice that $|\nabla_A u|^2$ and the observable quantities $\rho$, $B$ and $j_A$ are invariant under change of gauge. That is, if we replace $u$ by $u e^{i\varphi}$ and $A$ by $A+\nabla \varphi$ for any function $\varphi$, they remain unchanged. We also point out that if $u$ is written in polar coordinates as $u=\rho^{1/2} e^{i\theta}$, then $$ |\nabla_A u|^2=|\nabla \rho^{1/2}|^2+\rho|\nabla \theta-A|^2.$$ For any admissible pair $(u,A)$, that is such that $\rho$, $B$ and $j_A$ are $Q_{\Lz}$-periodic, we define the Ginzburg-Landau functional as $$ {E_\mathrm{GL}}(u,A):=\int_{Q_{\Lz,T}} |\nabla_A u|^2 + \frac{\kappa^2}{2} (1-|u|^2)^2 dx + \int_{Q_{\Lz}\times \R} |\nabla\times A-{B_{\mathrm{ext}}}|^2 dx\,.$$ We remark that $u$ and $A$ need not be (and, if ${B_{\mathrm{ext}}}\ne 0$, cannot be) periodic. See [@CoOtSer] for more details on the functional spaces we are using. Here ${B_{\mathrm{ext}}}:={b_{\mathrm{ext}}}e_3$ is the external magnetic field and $\kappa\in(0,1/\sqrt2)$ is a [material constant, called the Ginzburg-Landau parameter]{}. From periodicity and $\Div B=0$ it follows that $ \int_{Q_{\Lz}\times\{x_3\}} B_3 \, dx'$ does not depend on $x_3$ and therefore, if the energy is finite, necessarily $$\label{B2bex} \int_{Q_{\Lz}\times\{x_3\}} B_3 \, dx'=\Lz^2 {b_{\mathrm{ext}}}\hskip1cm\text{ for all }x_3\in\R\,.$$ We first remove the bulk part from the energy ${E_\mathrm{GL}}$. In order to do so, we introduce the quantity $${\mathcal{D}}_A^3u := (\partial_2 u - i A_2 u) - i (\partial_1 u - i A_1 u) = (\nabla_A u)_2 - i (\nabla_A u)_1$$ and, more generally, $${\mathcal{D}}_A^ku := (\partial_{k+2} u - i A_{k+2} u) - i (\partial_{k+1} u - i A_{k+1} u) = (\nabla_A u)_{k+2} - i (\nabla_A u)_{k+1} \, ,$$ where components are understood cyclically (i.e., $a_k=a_{k+3}$). The operator ${\mathcal{D}}_A$ (which corresponds to a creation operator for a magnetic Laplacian) was used by Bogomol’nyi in the proof of the self-duality of the Ginzburg-Landau functional at ${\kappa}=\frac{1}{\sqrt{2}} $ (cf. e.g. [@jaffetaubes]). His proof relied on identities similar to the next ones, which will be crucial in enabling us to separate the leading order part of the energy. Expanding the squares, one sees (for details see [@CoOtSer Lem. 2.1]) that (recall that $\rho:=|u|^2$) $$\label{magic} |\nabla'_A u|^2 = |{\mathcal{D}}_A^3u|^2 + \rho B_3 + \nabla'\times j'_A$$ and, for any $k=1,2,3$, $$\label{magic2} |(\nabla_A u)^{k+1}|^2 + |(\nabla_A u)^{k+2}|^2= |{\mathcal{D}}_A^ku|^2 + \rho B_k + (\nabla\times j_A)_k \,.$$ This implies $$|\nabla_A' u |^2 = (1-{\kappa}\sqrt2) |\nabla_A' u |^2 + {\kappa}\sqrt2 |{\mathcal{D}}_A^3u|^2 +{\kappa}\sqrt2\rho B_3 + {\kappa}\sqrt2\nabla'\times j_A'\,.$$ The last term integrates to zero by the periodicity of $j_A$. Therefore, for each fixed $x_3$, using , we have $$\int_{Q_{\Lz}} |\nabla_A' u |^2\, dx' = \int_{Q_{\Lz}} (1-{\kappa}\sqrt2) |\nabla_A' u |^2 + {\kappa}\sqrt2 |{\mathcal{D}}_A^3u|^2 +{\kappa}\sqrt2(\rho-1) B_3 \, dx'+ \Lz^2 {\kappa}\sqrt2 {b_{\mathrm{ext}}}\,.$$ We substitute and obtain, using $\int_{Q_{\Lz}} (B_3-{b_{\mathrm{ext}}})^2 dx'=\int_{Q_{\Lz}} B_3^2-{b_{\mathrm{ext}}}^2 dx'$ [and completing squares]{}, $$\label{eqEGLE} {E_\mathrm{GL}}(u,A)=2T\Lz^2\left({\kappa}\sqrt2 {b_{\mathrm{ext}}}-{b_{\mathrm{ext}}}^2\right) + E(u,A) + {\kappa}\sqrt2 \int_{Q_{\Lz,T}}|{\mathcal{D}}_A^3u|^2-|\nabla_A'u|^2 dx,$$ where $$\begin{aligned} E(u,A)&:=& \int_{Q_{\Lz,T}} |\nabla_A u |^2 + \left(B_3 - \frac{{\kappa}}{\sqrt2} (1-\rho)\right)^2 dx \\&& + \int_{Q_{\Lz}\times \R} |B'|^2 dx+ \int_{Q_{\Lz}\times(\R\setminus(-T,T))} (B_3 -{b_{\mathrm{ext}}})^2 dx\,.\end{aligned}$$ In particular, the bulk energy is $2{\Lz}^2T({\kappa}\sqrt2 {b_{\mathrm{ext}}}-{b_{\mathrm{ext}}}^2) $. Since we are interested in the regime $\kappa\ll 1$ and since $ |{\mathcal{D}}_A^3u|^2\le 2 |\nabla_A'u|^2$, the contribution of the last term in (\[eqEGLE\]) to the energy is (asymptotically) negligible with respect to the first term in $ E$, and therefore it can be ignored in the following. Applying to $B-{b_{\mathrm{ext}}}e_3 $ and minimizing outside $Q_{\Lz} \times [-T,T]$ if necessary, the last two terms in $E(u,A)$ can be replaced by $$\int_{Q_{{\Lz},T}} |B'|^2 dx+ \| B_3-{b_{\mathrm{ext}}}\|_{H^{-1/2}(\{x_3= T\})}^2+\| B_3-{b_{\mathrm{ext}}}\|_{H^{-1/2}(\{x_3=-T\})}^2\,,$$ so that $ E(u,A)$ becomes $$\label{E} E(u,A)= \int_{Q_{\Lz,T}} |\nabla_A u |^2 + \left(B_3 - \frac{{\kappa}}{\sqrt2} (1-\rho)\right)^2 + |B'|^2 dx+\| B_3-{b_{\mathrm{ext}}}\|_{H^{-1/2}(\{x_3=\pm T\})}^2\,.$$ Let us notice that the normal solution $\rho=0$, $B={b_{\mathrm{ext}}}e_3$ (for which we can take $A(x_1,x_2,x_3)={b_{\mathrm{ext}}}x_1 e_2$) is always admissible but has energy equal to $${E_\mathrm{GL}}(u,A)= \Lz^2 T \kappa^2\gg 2\Lz^2T({\kappa}\sqrt2 {b_{\mathrm{ext}}}-{b_{\mathrm{ext}}}^2),$$ in the regime $\kappa\gg {b_{\mathrm{ext}}}$ that we consider here. The following scaling law is established in [@CoOtSer]. \[theoCoOtSer\] For ${b_{\mathrm{ext}}}<\kappa/8$, $\kappa\le 1/2$, $\kappa T \ge 1$, $\Lz$ sufficiently large, if the quantization condition $$\label{quantificationini} {b_{\mathrm{ext}}}\Lz^2 \in 2\pi \Z,$$ holds then $$\label{minE} \min E(u,A) \sim \min \left\{{b_{\mathrm{ext}}}^{2/3} \kappa^{2/3} T^{1/3} , {b_{\mathrm{ext}}}\kappa^{3/7} T^{3/7} \right\}\Lz^2.$$ We believe that is also a necessary condition for to hold. Indeed, we expect that if $(u,A)$ is such that $$E(u,A) \les \left\{{b_{\mathrm{ext}}}^{2/3} \kappa^{2/3} T^{1/3} \Lz^2, {b_{\mathrm{ext}}}\kappa^{3/7} T^{3/7} \Lz^2 \right\},$$ then the normal phase $\rho\simeq 0$ is the minority phase (typically disconnected on every slice) and there exist $x_3\in(-T,T)$ and (periodic) curves $\gamma_1$ and $\gamma_2$ such that $\Gamma_1:=\{(\gamma_1(s), s, x_3) : s\in[ 0, 1]\} {\subseteq}\{\rho\simeq 1\}$ and $\Gamma_2:=\{(s,\gamma_2(s), x_3) : s\in[ 0, 1]\} {\subseteq}\{\rho\simeq 1\}$, with $\gamma_i(1)=\gamma_i(0)+L_0 e_i$. If this holds then using Stokes Theorem on large domains the boundary of which is made of concatenations of the curves $\Gamma_i$, it is possible to prove that must hold. As in [@CoOtSer], we will need to assume in order to build the recovery sequence in Section \[recoveryGL\].\ The first regime in corresponds to uniform branching patterns while the second corresponds to well separated branching trees (see [@ChokConKoOt; @ChokKoOt; @CoOtSer]). We focus here on the first regime, that is ${\kappa}^{5/7}\ll {b_{\mathrm{ext}}}T^{2/7}$, and replace $\kappa$ and ${b_{\mathrm{ext}}}$ by the variables $\alpha$, $\beta$, defined according to $$\kappa T = \sqrt{2}\a \qquad {b_{\mathrm{ext}}}= \frac{\beta \kappa }{\sqrt{2}}=\frac{\alpha\beta}{T}\,,$$ and then rescale ------------------------------------------------------- --------------------------- $\hat{x}:= x/T$, $ L:=\Lz/T$, \[6pt\] $\hat{u}(\hat{x}):= u(x)$, $\hat{A}(\hat{x}):= A(x)$ \[6pt\] $E_T(\hat u,\hat A):=\frac{1}{T L^2} E(u,A)$, ------------------------------------------------------- --------------------------- so that in particular $ \hat{B}(\hat{x})= \widehat{{\nabla}}\times \hat{A}(\hat{x})= TB(x)$ and ${\nabla}_A u(x)= T^{-1} \widehat{{\nabla}} \hat{u}( x/T)-i \hat{A}(x/T)\hat u (x/T)$. Changing variables and removing the hats yields $$E_T(u,A)= \frac{1}{L^2}\lt(\int_{Q_{L,1}} |{\nabla}_{T{A}} {u}|^2+ \left({B_3} - \a(1-\rho)\right)^2 + |{B'}|^2 dx+\|{B_3}- \a\b\|_{H^{-1/2}(\{x_3=\pm1\})}^2\rt)\,.$$ as was anticipated in (\[Etilde\]). In these new variables, the scaling law becomes $E_T\sim \min\{\alpha^{4/3}\beta^{2/3},\a^{10/7}\beta \}$ and the uniform branching regime corresponds to $E_T\sim \alpha^{4/3}\beta^{2/3}$ which amounts to $\a^{-2/7}\ll \b\ll 1$, see also (\[scallawintro\]). Constructions (leading to the upper bounds in [@CoOtSer; @ChokKoOt; @ChokConKoOt]), suggest that in this regime, typically, the penetration length of the magnetic field inside the superconducting regions is of the order of $T^{-1}$, the coherence length (or domain walls) is of the order of $\a^{-1}$, the width of the normal domains in the bulk is of the order of $\a^{-1/3} \b^{1/3}$ and their separation of order $\a^{-1/3}\b^{-1/6}$. These various lengthscales motivate the anisotropic rescalings that we will introduce in Section \[Sec:intermediate\]. In closing this section we present the anisotropic rescaling that will lead to the functional defined in (\[eqdefwe\]), postponing to the next section a detailed explanation of its motivation. We set for $x \in Q_{L,1}$, --------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -------------------------------------------- $\begin{pmatrix}{\widetilde}{x}'\\ {\widetilde}{x}_3\end{pmatrix}:= \begin{pmatrix}\a^{1/3} \b^{1/6} x'\\x_3\end{pmatrix}$, $ {\widetilde}L:= \a^{1/3}\b^{1/6}L$, \[8pt\] $\begin{pmatrix}{\widetilde}{A}'\\ {\widetilde}{A}_3\end{pmatrix}({\widetilde}{x}):=\begin{pmatrix}\a^{-2/3}\b^{1/6} A'\\ \a^{-1/3} \b^{1/3}A_3\end{pmatrix}(x)$, $ {\widetilde}{u}({\widetilde}{x}):=u(x),$ --------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -------------------------------------------- to get ${\widetilde}B_3({\widetilde}x)=\alpha^{-1}B_3(x)$, ${\widetilde}B'({\widetilde}x)=\alpha^{-2/3}\beta^{1/6}B'(x)$ inside the sample. Outside the sample, i.e. for $|x_3|\ge 1$, we make the isotropic rescaling ${\widetilde}{x}:= \pm e_3+\a^{1/3}\b^{1/6}( x \mp e_3)$ to get ${\widetilde}B({\widetilde}x)=\a^{-1} B(x)$. A straightforward computation leads to $\wE({\widetilde}u, {\widetilde}A)=\a^{-4/3}\b^{-2/3} E_T(u,A)$, where $$\begin{gathered} \wE(u,A):=\frac{1}{\widetilde{L}^2}\Big[\int_{Q_{{\widetilde}L,1}} \a^{-2/3}\b^{-1/3}\lt|\nabla_{\a^{1/3}\b^{-1/3}TA}'u\rt|^2+ \a^{-4/3}\b^{-2/3}\lt|(\nabla_{\a^{1/3}\b^{-1/3}TA} u)_3\rt|^2\\ + \a^{2/3}\b^{-2/3}\left({B_3} -(1-|{u}|^2)\right)^2 + \b^{-1}|{B'}|^2 dx +\a^{1/3}\b^{7/6}\|\b^{-1}{B_3}-1\|_{H^{-1/2}(x_3=\pm1)}^2\Big],\end{gathered}$$ with ${\nabla}\times A=B$ (and in particular $\Div B=0$). We assume that ${\widetilde}L$ is a fixed quantity of order 1. For simplicity of notation, the detailed analysis is done only for the case ${\widetilde}L=1$.\ Let us point out that in these units, the penetration length is of order $T^{-1}\a^{1/3}\b^{1/6}$, the coherence length of order $\a^{-2/3}\b^{1/6}$, the width of the normal domains in the bulk of order $\b^{1/2}$ and the distance between the threads of order one. That is, the scale separation reads now $$\label{scalesepar2} T^{-1}\a^{1/3}\b^{1/6}\ll \a^{-2/3}\b^{1/6}\ll \b^{1/2}\ll1.$$ The intermediate functionals {#Sec:intermediate} ============================ In this section we explain the origin of the rescaling leading from $E_T$ to ${\widetilde}E_T$, and the different functionals which appear at different scales. This material is not needed for the proofs but we think it is important to illustrate the meaning of our results. We carry out the scalings in detail but the relations between the functionals are here discussed only at a heuristic level. We want to successively send $T\to+\infty$, $\alpha\to +\infty$ and $\beta\to 0$. For this we are going to introduce a hierarchy of models starting from $E_T(u,A)$ and finishing at $I(\mu)$. When sending first $T\to +\infty$ with fixed $\alpha$ and $\beta$, the functional $E_T$ approximates $$ F_{\alpha,\beta}(\rho,B):=\frac{1}{L^2} \lt(\int_{Q_{L,1}} |\nabla \rho^{1/2}|^2 +(B_3-\alpha (1-\rho))^2 +|B'|^2 dx +\|{B_3}- \a\b\|_{H^{-1/2}(x_3=\pm1)}^2\rt),$$ with the constraints $$\label{meissner4} \Div B=0 \qquad\textrm{and } \qquad \rho B=0.$$ The main difference between $E_T$ and $F_{\alpha,\beta}$ is that for the latter, since the penetration length (which corresponds to $T^{-1}$) was sent to zero, the Meissner condition is enforced. We now want to send the coherence length (of order $\a^{-1}$) to zero at fixed $\beta$, while keeping superconducting domains of finite size. Since the typical domain diameter is of order $\a^{-1/3}\b^{1/3}$ and their distance is of order $\a^{-1/3}\b^{-1/6}$, we are led to the anisotropic rescaling: -------------------------------------------------------------------------------------------------------------------------------- --------------------------------- $\begin{pmatrix}\hat{x}'\\ \hat{x}_3\end{pmatrix}:= \begin{pmatrix}\a^{1/3} x'\\x_3\end{pmatrix}$, $\hat{L}:= \a^{1/3}L$, \[6pt\] $\begin{pmatrix}\hat{B}'\\ \hat{B}_3\end{pmatrix}(\hat{x}):=\begin{pmatrix}\a^{-2/3} B'\\ \a^{-1}B_3\end{pmatrix}(x)$, $\hat{\rho}(\hat{x}):=\rho(x)$, \[6pt\] $\wF=\a^{-4/3} F_{\a,\b}$. -------------------------------------------------------------------------------------------------------------------------------- --------------------------------- In these variables, the coherence length is of order $\a^{-2/3}\ll 1$ (at least horizontally) while the diameter of the normal domains is of order $\beta^{1/3}$ and their separation of order $\beta^{-1/6}$. Dropping the hats (we just keep them on the functional and on $L$ to avoid confusion) we obtain $$\begin{gathered} \wF(\rho,B):=\frac{1}{\hL^2} \lt(\int_{Q_{\hL,1}} \a^{-2/3}\left|\begin{pmatrix} \nabla' \rho^{1/2}\\ \a^{-1/3} \partial_3 \rho^{1/2} \end{pmatrix}\right|^2 +\a^{2/3}\lt(B_3- (1-\rho)\rt)^2 +|B'|^2 dx\rt.\\ \lt. +\a^{1/3}\|{B_3}- \b\|_{H^{-1/2}(x_3=\pm1)}^2\rt),\end{gathered}$$ with the constraints . The scaling of Theorem \[theoCoOtSer\] indicates that $\wF$ behaves as $ \min\{\beta^{2/3},\a^{2/21}\beta \}$ which is of order $\beta^{2/3}$ if $\a\gg 1$ and $\beta$ is fixed. We remark that, letting $\eta:= \alpha^{-1/3}$ and $\delta:=\eta^2=\alpha^{-2/3}$, one has $$\begin{gathered} \wF(\rho,B)= \frac{1}{\hL^2} \left(\int_{Q_{\hL,1}} \delta \lt|\begin{pmatrix}\nabla'\rho^{1/2}\\ \eta \partial_3 \rho^{1/2}\end{pmatrix}\rt|^2+\frac{1}{\delta} \lt(B_3-(1-\rho)\rt)^2 +|B'|^2 dx \rt.\\ \lt.+ \eta^{-1} \|B_3-\beta\|_{H^{-1/2}(x_3=\pm1)}^2\rt).\end{gathered}$$ In this form, $ \wF(\rho,B)$ is very reminiscent of the functional studied in [@ViehOtt]. Notice however that besides the Meissner condition which makes our functional more rigid, the scaling $\delta=\eta^2$ is borderline for the analysis in [@ViehOtt]. Recalling that $\lt(B_3-(1-\rho)\rt)^2 = \chi_{\rho>0} (1-\rho)^2$, the corresponding term in $\wF$ has the form of a double well-potential, and so in the limit $\alpha\to+\infty$ the functional $\wF$ approximates $$ G_\beta(\chi,B'):=\frac{1}{\hL^2} \lt( \int_{Q_{\hL,1}} \frac{4}{3}|D'\chi| +|B'|^2 dx \rt),$$ with the constraints $\chi\in \{0,1\}$, $\chi(\cdot,x_3) \weaklim \beta dx'$ when $x_3\to \pm1$ and $$ \partial_3 \chi +\Divp B'=0 \qquad \textrm{and } \qquad \chi B'=B'.$$ This is similar to the simplified sharp-interface functional that was studied in [@ChokKoOt; @ChokConKoOt]. In the definition of $G_\beta$, we used the notation $$ \int_{Q_{\hL,1}} |D' u|:= \sup_{\stackrel{\xi \in C^{\infty}(Q_{\hL,1}),}{ |\xi|_\infty \le 1}} \, \int_{Q_{\hL,1}} u \Divp \xi \, dx,$$ for the horizontal $BV$ norm of a function $u\in L^1(Q_{\hL,1})$. By definition it is lower semicontinuous for the $L^1$ convergence and it is not hard to check that if we let $u_{x_3}(x'):= u(x',x_3)$, then $$\int_{Q_{\hL,1}} |D'u|=\int_{-1}^1 \left(\int_{Q_{\hL}} |D' u_{x_3}|\right) dx_3,$$ where $\int_{Q_{\hL}} |D' u_{x_3}|$ is the usual $BV$ norm of $u_{x_3}$ in $Q_{\hL}$ [@AFP]. From this and the usual co-area formula [@AFP Th. 3.40], we infer that $$\label{coarea} \int_{Q_{\hL,1}} |D'u|=\int_{\R} \int_{-1}^1 \H^1( \partial \{u_{x_3}>s\}) dx_3 ds.$$ In , $\partial \{u_{x_3}>s\}$ represents the measure-theoretic boundary of $\{u_{x_3}>s\}$ in $Q_{\hL}$. We finally want to send the volume fraction of the normal phase to zero and introduce the last rescaling in $\beta$ for which we let -------------------------------------------------------------------------------------------------------------------- ---------------------------------------------- $\begin{pmatrix}{\widetilde}{x}'\\ {\widetilde}{x}_3\end{pmatrix}:= \begin{pmatrix}\b^{1/6} x'\\x_3\end{pmatrix}$, $ {\widetilde}L:= \beta^{1/6}\hL$, \[8pt\] $\tB'({\widetilde}{x}):=\beta^{1/6} B'(x)$ $\tchi({\widetilde}{x}):=\beta^{-1}\chi(x)$, \[6pt\] $\wG:=\b^{-2/3} G_{\b}$. -------------------------------------------------------------------------------------------------------------------- ---------------------------------------------- After this last rescaling, the domain width is of order $\b^{1/2}\ll1$, the separation between domains of order $1$. We obtain (dropping the tildas again) the order-one functional $$ \wG(\chi,B'):= \frac{1}{\widetilde{L}^2}\int_{Q_{{\widetilde}L,1}} \frac{4}{3} \beta^{1/2}|D'\chi| +\chi|B'|^2 dx$$ under the constraints $\chi\in \{0,\beta^{-1}\}$, $\chi(\cdot,x_3) \weaklim dx'$ when $x_3\to \pm1$, and $$ \partial_3 \chi +\Divp (\chi B')=0 \qquad \textrm{and } \qquad \chi B'=\beta^{-1}B'.$$ This functional converges to $I(\mu)$ as $\beta\to 0$. Let us point out that since we are actually passing directly from the functional $\wE$ to $I$ in Theorem \[maintheo\], we are covering the whole parameter regime of interest. In particular, our result looks at first sight stronger than passing first from $ E_T$ to $F_{\a,\b}$, then from $\wF$ to $G_\beta$ and finally from $\wG$ to $I$. However, because of the Meissner condition, we do not have a proof of density of smooth objects for $F_{\a,\b}$ and $G_\b$. Because of this, we do not obtain the $\Gamma-$convergence of the intermediate functionals (the upper bound is missing). The limiting energy {#Sec:limiting} =================== Before proving the $\Gamma$-limit we study the limiting functional $I$ that was mentioned in (\[eqdefIintro\]) and motivated in the previous section. We give here a self-contained treatment of the functional $I$, which is motivated by the analysis discussed above, and will be crucial in the proofs that follow. However, in this discussion we do not make use of the relation to the Ginzburg-Landau functional. For $L,T>0$ we denote by ${\mathcal{A}}_{L,T}$ the set of pairs of measures ${\mu}\in{\mathcal{M}}^+(Q_{L,T})$, $m\in {\mathcal{M}}(Q_{L,T};\R^2)$ with $m\ll\mu$, satisfying the continuity equation $$\label{conteq0} \pa_3 {\mu}+\Divp m=0 \qquad \textrm{in } Q_{L,T},$$ and such that ${\mu}={\mu}_{x_3}\otimes d{x_3}$ where, for a.e. $x_3\in(-T,T)$, ${\mu}_{x_3}=\sum_i {\varphi}_i \delta_{X_i}$ for some ${\varphi}_i> 0$ and $X_i\in Q_L$. We denote by ${\mathcal{A}}_{L,T}^*:=\{\mu: \exists m, (\mu,m)\in{\mathcal{A}}_{L,T}\}$ the set of admissible $\mu$. We define the functional $I:{\mathcal{A}}_{L,T}\to[0,+\infty]$ by $$\label{Imum} I(\mu,m):= \frac{K_*}{L^2} \int_{-T}^T \sum_{x'\in Q_L} \left(\mu_{x_3}(x')\right)^{1/2} \, dx_3 + \frac{1}{L^2}\int_{Q_{L,T}} \left(\frac{dm}{d{\mu}} \right)^2 d{\mu},$$ where $K_*:= \frac{8 \sqrt{\pi}}{3}$ and (with abuse of notation) $I:{\mathcal{A}}_{L,T}^*\to[0,\infty]$ by $$\label{Imu} I({\mu}):=\min \{ I(\mu,m)\ : \ m\ll{\mu}, \ \pa_3 {\mu}+ {\Divp} m=0\}.$$ Condition (\[conteq0\]) is understood in a $Q_L$-periodic sense, i.e., for any $\psi\in C^1(\R^3)$ which is $Q_L$-periodic and vanishes outside $\R^2\times (0,T)$ one has $\int_{Q_{L,T}} \partial_3\psi d\mu + \nabla'\psi \cdot dm=0$. If $\mu_{x_3}=\sum_i {\varphi}_i \delta_{X_i}$ then $\sum_{x'\in Q_L} \left(\mu_{x_3}(x')\right)^{1/2}=\sum_i {\varphi}_i^{1/2}$. Because of , $\mu_{x_3}(Q_L)$ does not depend on $x_3$. Let us point out that the minimum in (\[Imu\]) is attained thanks to (\[eqexistsB\]). Moreover, the minimizer is unique by strict convexity of $m\to\int_{Q_{L,T}} \left(\frac{dm}{d{\mu}} \right)^2 d{\mu}$. As proven in Lemma \[lemmacurves\] below, if $\mu$ is made of a finite union of curves then there is actually only one admissible measure $m$ for . More generally, Since every measure $\mu$ with finite energy is rectifiable (see Corollary \[coroXia\]), we believe that it is actually always the case. For $\mu$ an admissible measure and $z,{\widetilde}{z}\in[-T,T]$, we let $$\label{Iztz} I^{(z,{\widetilde}{z})}(\mu):= \frac{K_*}{L^2} \int_{z}^{{\widetilde}{z}} \sum_{x'\in Q_L} \left(\mu_{x_3}(x')\right)^{1/2} \, dx_3 + \frac{1}{L^2}\int_{Q_{L}\times[z,{\widetilde}{z}]} \left(\frac{dm}{d{\mu}} \right)^2 d{\mu},$$ where $m$ is the optimal measure for ${\mu}$ on $[z,{\widetilde}{z}]$ (which coincides with the restriction to $[z,{\widetilde}{z}]$ of the optimal measure on $[-T,T]$). From (\[eqw22b\]) one immediately deduces for every measure $\mu$, and every $x_3, {\widetilde}x_3\in[-T,T]$, the following estimate on the Wasserstein distance $$\label{HolderW2} W_2^2(\mu_{x_3}, {\mu}_{{\widetilde}x_3})\le L^2 I(\mu) |x_3-{\widetilde}x_3|.$$ In particular for every measure $\mu$ with $I(\mu)<+\infty$, the curve $x_3\mapsto \mu_{x_3}$ is Hölder continuous with exponent $1/2$ in ${\mathcal{M}}^+( Q_L)$ (endowed with the metric $W_2$) and the traces $\mu_{\pm T}$ are well defined. Existence of minimizers ----------------------- Given two measures $\bar {\mu}_{\pm}$ in $\M^+(Q_L)$ with $\bar \mu_+(Q_L)=\bar \mu_-(Q_L)$, we are interested in the variational problem $$\label{limitProb} \inf\lt\{ I(\mu) \ : \ \mu_{\pm T}= \bar {\mu}_{\pm} \rt\}.$$ We first prove that any pair of measures with equal flux can be connected with finite cost and that there always exists a minimizer. The construction is a branching construction which gives the expected scaling (see [@ChokConKoOt; @CoOtSer]) if the boundary data is such that $\bar {\mu}_+=\bar{\mu}_-$. \[branchingmu\] For every pair of measures $ \bar {\mu}_{\pm}\in {\mathcal{M}}^+(Q_L)$ with $\bar\mu_+(Q_L)=\bar\mu_-(Q_L)=\Phi$, there is $\mu\in {\mathcal{A}}^*_{L,T}$ such that $\mu_{\pm T}=\bar\mu_\pm$ and $$I(\mu)\les \frac{T\Phi^{1/2}}{L^2}+\frac{\Phi}{T}.$$ If $\bar\mu_+=\bar\mu_-$, then there is a construction with $$I(\mu)\les \frac{T \Phi^{1/2}}{L^2} +\frac{ T^{1/3} \Phi^{2/3} }{L^{4/3}},$$ and such that the slice at $x_3=0$ is given by $\mu_0=\Phi N^{-2}\sum_j \delta_{X_j}$, with $X_j$ the $N^2$ points in $[-L/2,L/2)^2\cap ((L/N)\Z)^2$, and $N:=\lfloor1+ \Phi^{1/6}L^{2/3}/T^{2/3}\rfloor$. The measure $\mu$ is supported on countably many segments, which only meet at triple points. By periodicity we can work on $[0,L)^2$ instead of $[-L/2,L/2)^2$. We first perform the construction for $x_3\ge 0$. The idea is to approximate $\bar\mu_+$ by linear combinations of Dirac masses, which become finer and finer as $x_3$ approaches $T$. Fix $N\in\N$, chosen below. For $n\in\N$, fix $x_{3,n}:=T(1-3^{-n})$, and let $T_n:= x_{3,n}-x_{3,n-1}=\frac{2}{3^{n}}T$ be the distance between two consecutive planes. At level $x_{3,n}$ we partition $Q_L$ into squares of side length $L_n:= \frac{L}{2^nN}$. More precisely, for $i,j=0,...,2^nN-1 $, we let $x'_{ij,n}:=\lt(L_n\, i, L_n \, j\rt)$ be a corner of the square $Q_{ij,n}:= x'_{ij,n}+ [0,L_n)^2$, and we let $\Phi_{ij,n}:=\bar\mu_+(Q_{ij,n})$ be the flux associated to this square. We define the measures $\mu^\mathrm{br}$ and $m^\mathrm{br}$ (here the suffix $\mathrm{br}$ stands for branching) by $$\mu^\mathrm{br}_{x_3}:= \sum_{ij} \Phi_{ij,n} \delta_{X_{ij,n}(x_3)} \text{ and } m^\mathrm{br}_{x_3}:= \sum_{ij} \frac{d X_{ij,n}}{dx_3}(x_3) \Phi_{ij,n} \delta_{X_{ij,n}(x_3)} \quad \text{for } x_3\in [x_{3,n-1}, x_{3,n}),$$ where $X_{ij,n}:[x_{3,n-1}, x_{3,n}]\to Q_L$ is a piecewise affine function such that $X_{ij,n}(x_{3,n})=x'_{ij,n}$, $X_{ij,n}(x_{3,n}-\frac12 T_n)=x'_{i_*j,n}$, and $X_{ij,n}(x_{3,n-1})=x'_{i_*j_*,n}$, where $i_*=2\lfloor i/2\rfloor$, $j_*=2\lfloor j/2\rfloor$. Four such curves end in every $i_*$, $j_*$ (which corresponds to the pair $i_*/2$, $j_*/2$ at level $n-1$), but they are pairwise superimposed for $x_3\in [x_{3,n-1},x_{3,n}-\frac12 T_n]$, therefore all junctions are triple points (one curve goes in, two go out). Using that $\sum_{i j} \Phi_{ij,n}=\Phi$ and $\sum_{ij}\sqrt{\Phi_{ij,n}} \le (\sum_{ij} \Phi_{ij,n})^{1/2}(\sum_{ij} 1)^{1/2} = \Phi^{1/2} 2^nN$, we get that the energy of $\mu^\mathrm{br}$ is given by $$\begin{aligned} I(\mu^\mathrm{br})&=\frac{1}{L^2} \sum_{n=1}^{+\infty} \sum_{ij} \left( K_* T_n \sqrt{\Phi_{ij,n}}+ \Phi_{ij,n} T_n \frac{2 L_n^2}{ T_n^2}\rt)\\ & \les L^{-2} TN\Phi^{1/2} \sum_{n=0}^{+\infty} \lt(\frac{2}{3}\rt)^n + \frac{\Phi}{TN^2}\sum_{n=0}^{+\infty} \lt(\frac{3}{4}\rt)^n\,.\end{aligned}$$ If we choose $N=1$, then there is only one point in the central plane, $\mu_0=\Phi \delta_0$. Therefore the top and bottom constructions can be carried out independently, since by assumption the total flux is conserved, and we obtain the first assertion. If $\bar\mu_+=\bar\mu_-$, we can choose the value of $N$ which makes the energy minimal. Up to constants this is the value given in the statement. Inserting in the estimate above gives the second assertion. If the boundary densities are maximally spread, in the sense that they are given by the Lebesgue measure, the scaling is optimal, as the following lower bound shows. \[branchinglowerbound\] For every measure $\mu\in {\mathcal{A}}^*_{L,T}$ such that $\mu_{\pm T}=\Phi L^{-2} dx'$ one has $$\label{lower} I(\mu)\ges \frac{T \Phi^{1/2}}{L^2}+ \frac{ T^{1/3}\Phi^{2/3}}{ L^{4/3}}\, .$$ The bound $I(\mu)\ge L^{-2} T\Phi^{1/2}$ follows at once from the subadditivity of the square root. Hence we only need to prove the other one. We give two proofs of this bound. The first uses only elementary tools while the second is based on an interpolation inequality. [*First proof:*]{} Let $I:=I(\mu,v\mu)$, where $v:=dm/d\mu$. Fix $\lambda>0$, chosen below. Choose $x_3\in (-T,T)$ such that $\mu_{x_3}=\sum_i\varphi_i \delta_{X_i}$ obeys $$\label{eqlbchoicex3} \sum_i \varphi_i^{1/2} \le \frac{L^2 I}{T}\,.$$ For some set ${\mathcal{I}}{\subseteq}\N$ to be chosen below, let $\psi:Q_L\to \R$ be a mollification of the function $\max\{ (\lambda-\dist(x',X_i))_+: i\in {\mathcal{I}}\}$, where as usual the distance is interpreted periodically. By the divergence condition, $$\int_{Q_L} \psi d\mu_{x_3} = \frac{\Phi}{L^{2}} \int_{Q_L} \psi dx'+\int_{-T}^{x_3} \int_{Q_L} \nabla'\psi\cdot v d\mu$$ (to prove this, pick $\xi_\eps\in C^1_c((-T,x_3))$ which converge pointwise to $1$ and use $\xi_\eps\psi$ as a test function in (\[conteq0\]) and then pass to the limit). Since $|\nabla'\psi|\le 1$, $$\sum_{i\in {\mathcal{I}}} \lambda \varphi_i \le \frac{\Phi}{L^{2}} \sum_{i\in {\mathcal{I}}}\frac{ \pi}{3} \lambda^3+ \int_{-T}^{x_3} \int_{Q_L} |v| d\mu \le \frac{\Phi}{L^2}\sum_{i\in {\mathcal{I}}} \pi \lambda^3+ L (T\Phi)^{1/2} I^{1/2},$$ where in the second step we used Hölder’s inequality and flux conservation. We choose ${\mathcal{I}}=\{i\in \N: \varphi_i\ge 4 \Phi \lambda^2/L^2\}$. From the definition of ${\mathcal{I}}$, we have $$\frac{\Phi}{L^2}\sum_{i\in {\mathcal{I}}} \pi \lambda^3\le \frac{\pi}{4} \sum_{i\in {\mathcal{I}}} \lambda {\varphi}_i.$$ Therefore, since $\pi<4$, we obtain $$\label{E1} \sum_{i\in {\mathcal{I}}} \lambda \varphi_i {\stackrel{<}{\sim}}L(T\Phi I)^{1/2}\,.$$ At the same time, again by the definition of ${\mathcal{I}}$ and (\[eqlbchoicex3\]), $$\label{E2} \sum_{i\not\in {\mathcal{I}}} \varphi_i\le \frac{2\lambda\Phi^{1/2}}{L} \sum_{i\not\in {\mathcal{I}}} \varphi_i^{1/2} \le 2\lambda L \Phi^{1/2}\, \frac{I}{T}\,.$$Adding and , we obtain $$\sum_{i\in {\mathcal{I}}} \varphi_i {\stackrel{<}{\sim}}\frac{1}{\lambda} L(T\Phi I)^{1/2}+ \lambda L \Phi^{1/2}\, \frac{I}{T}\, ,$$ hence $$\Phi^{1/2}{\stackrel{<}{\sim}}\frac{1}{\lambda} L(T I)^{1/2}+ \lambda L \, \frac{I}{T}\, ,$$ and optimizing over $\lambda$ by choosing $\lambda = \frac{T^{3/4}}{I^{1/4}}$ yields $I\ges \Phi^{2/3} T^{1/3}L^{-4/3}$. [*Second Proof:*]{} As before, let $x_3\in (-T,T)$ be such that $\mu_{x_3}=\sum_i\varphi_i \delta_{X_i}$ obeys . By Young’s inequality and , we have $$I\ges \frac{T}{L^2} \left( \sum_i \varphi_i^{1/2} \right) +\frac{W_2^2(\mu_{x_3}, \Phi L^{-2} dx')}{L^2 T}\ges L^{-2} T^{1/3}\left( \sum_i \varphi_i^{1/2} \right)^{2/3} \left(W_2^2(\mu_{x_3}, \Phi L^{-2} dx')\right)^{1/3}.$$ The desired lower bound would then follow if we can show that for every measure $\mu\in\M^+(Q_L)$ with $\mu=\sum_i {\varphi}_i \delta_{X_i}$ and $\sum_i {\varphi}_i = \Phi$, $$\label{interpolBVW2} \left( \sum_i \varphi_i^{1/2} \right)^{2/3} \left(W_2^2(\mu, \Phi L^{-2} dx')\right)^{1/3}\ges \Phi^{2/3} L^{-4/3}.$$ By rescaling it is enough considering $\Phi=L=1$. The optimal transport map is necessarily of the form $\psi(x')=X_i$ if $x'\in E_i$, where $E_i$ is a partition of $Q_1$ with $|E_i|={\varphi}_i$ (the corresponding transport plan is $(Id\times \psi)\sharp dx'$). By definition, it holds $$W_2^2(\mu, dx')\ge \sum_{i} \int_{E_i} |x'-X_i|^2 dx'.$$ But since $|E_i|={\varphi}_i=|\B'(X_i,({\varphi}_i/\pi)^{1/2})|$, $$\sum_{i} \int_{E_i} |x'-X_i|^2 dx'\ge \sum_{i} \int_{\B(X_i,({\varphi}_i/\pi)^{1/2})} |x'|^2 dx'\ges \sum_{i} {\varphi}_i^2 .$$ By Hölder’s inequality, we conclude that $$1=\sum_i {\varphi}_i \le \left(\sum_i {\varphi}_i^{1/2}\right)^{2/3} \left(\sum_i {\varphi}_i^2\right)^{1/3}\les \left( \sum_i \varphi_i^{1/2} \right)^{2/3} \left(W_2^2(\mu, dx')\right)^{1/3},$$ as desired. The lower bound can also be obtained as a consequence of the scaling law proven in [@CoOtSer] for the Ginzburg-Landau model combined with our lower bound in Section \[Sec:gammaliminf\] (which does not use this lower bound). However, since the proof here is much simpler and contains some of the main ideas behind the proofs of [@ChokConKoOt; @CoOtSer], we decided to include it. Similarly, the interpolation inequality can be obtained by approximation from a similar inequality proven in [@CintiOt] (where it is used in the same spirit as here to re-derive the lower bounds of [@ChokConKoOt]). We end this section by proving existence of minimizers. \[existmu\] For every pair of measures $ \bar {\mu}_{\pm }$ with $\bar {\mu}_+(Q_L)=\bar {\mu}_-(Q_L)$, the infimum in is finite and attained. In this proof we assume $L=T=1$ and $\bar\mu_+(Q_{1})=1$. By Proposition \[branchingmu\] the infimum is finite. Let now $\mu^n$ be a minimizing sequence for $I$. Since $\sup_n I(\mu^n)<+\infty$, thanks to , the functions $x_3\mapsto \mu^n_{x_3}$ are equi-continuous in $\cP(Q_1)$ (recall that $W_2$ metrizes the weak convergence in $\mathcal{P}(Q_1)$) hence by the Arzelà-Ascoli theorem there exists a subsequence, still denoted $\mu^n$, uniformly converging (in $x_3$) to some measure $\mu$ which also satisfies the given boundary conditions. Moreover, if $m^n$ is an optimal measure in for $\mu^n$, since by the Cauchy-Schwarz inequality we have $$\int_{Q_{1,1}} d|m^n|\le \lt(\int_{Q_{1,1}} \lt(\frac{dm^n}{d\mu^n}\rt)^2 d\mu^n\rt)^\hal \lt(\int_{Q_{1,1}} d\mu^n\rt)^\hal\les 1,$$ there also exists a subsequence $m^n$ converging to some measure $m$ satisfying . By [@AFP Th. 2.34 and Ex. 2.36] we deduce that $m\ll \mu$ and $$\liminf_{n\to +\infty} \int_{Q_{1,1}} \lt(\frac{dm^n}{d\mu^n}\rt)^2 d\mu^n\ge \int_{Q_{1,1}} \lt(\frac{dm}{d\mu}\rt)^2 d\mu.$$ It remains to prove that $\mu_{x_3}=\sum_i {\varphi}_i(x_3) \delta_{X_i(x_3)}$ for a.e. $x_3$ and that $$\label{semicontperi} \liminf_{n\to +\infty} \int_{-1}^1 \sum_{x'\in Q_{1}} \left(\mu^n_{x_3}(x')\right)^{1/2} \, dx_3 \ge \int_{-1}^1 \sum_{x'\in Q_{1}} \left(\mu_{x_3}(x')\right)^{1/2} \, dx_3.$$ If $\mu^n_{x_3}=\sum {\varphi}_i^n(x_3) \delta_{X^n_i}(x_3)$, with ${\varphi}_i^n$ ordered in a decreasing order, we let $f_n(x_3):= \sum_i \sqrt{{\varphi}_i^n(x_3)}$ and observe that $\int_{-1}^1 f_n dx_3 \le I(\mu^n)\les 1 $. Hence, by Fatou’s lemma, $$\label{fatouf} 1\ges \liminf_{n\to +\infty} \int_{-1}^1 f_n(x_3) dx_3\ge \int_{-1}^1 \liminf_{n\to +\infty} f_n(x_3) dx_3,$$ from which we infer that $g(x_3):=\liminf_{n\to+\infty} f_n(x_3)$ is finite for a.e. $x_3$. Consider such an $x_3$ and let $\psi(n)$ be a subsequence (which depends on $x_3$) such that $g(x_3)=\lim_{n\to+\infty} f_{\psi(n)}(x_3)$. Up to another subsequence, still denoted $\psi(n)$, we may assume that for every $i\in \N$, ${\varphi}_i^{\psi(n)}(x_3)$ converges to some ${\varphi}_i(x_3)$ and $X_i^{\psi(n)}(x_3)$ converges to some $X_i(x_3)$. By Lemma \[lemsqrt\] (see below), for every $N\in \N$, $$\sum_{i\le N} {\varphi}_i^{\psi(n)}(x_3)\ge 1- \frac{f_{\psi(n)}(x_3)}{\sqrt{N}}.$$ This implies, by tightness, $\mu_{x_3}^{\psi(n)}\weaklim \sum_i {\varphi}_i(x_3) \delta_{X_i(x_3)}$ and $\sum_i ({\varphi}_i(x_3))^{1/2}\le g(x_3)$. But since $\mu_{x_3}^{\psi(n)}\weaklim {\mu}_{x_3}$, we have ${\mu}_{x_3}=\sum_i {\varphi}_i(x_3) \delta_{X_i(x_3)}$. Finally, by the subadditivity of the square root, and the definition of $g$ we obtain . \[lemsqrt\] If a nonincreasing sequence of positive numbers $\gamma_i$ is such that $$\sum_i \gamma_i = c_0 \qquad \text{ and }\qquad \sum_i \sqrt{\gamma_i} \le C_0,$$ then for all $N\in \N$ one has $$\sum_{i\le N} \gamma_i \ge c_0- C_0 \sqrt{\frac{c_0}{N}}.$$ Indeed $\sum_{i >N} \gamma_i \le \sqrt{\gamma_N} \sum_{i >N} \sqrt{\gamma_i} \le C_0 \sqrt{\gamma_N},$ while $c_0 \ge \sum_{i\le N} \gamma_i \ge N \gamma_N$. Regularity of minimizers ------------------------ We now want to prove regularity of the minimizing measures $\mu$. In order to prove that we can restrict our attention to measures containing no loops, we first define the notion of subsystem. \[Existence of subsystems\]\[def:subsystem\] Given a point $x:=(X, x_3) \in Q_{L,T}$ and $\mu\in{\mathcal{A}}^*_{L,T}$ with $I(\mu)<+\infty$, there exists a subsystem ${\widetilde}{{\mu}}$ of ${\mu}$ emanating from $x$, meaning that there exists ${\widetilde}{{\mu}} \in {\mathcal{A}}^*_{L,T}$ such that 1. \[def:subsystempos\] $ {\widetilde}{{\mu}}\le {\mu}$ in the sense that ${\mu}-{\widetilde}{{\mu}}$ is a positive measure, 2. \[def:subsystemdelta\] ${\widetilde}{{\mu}}_{x_3}=a\delta_{X}$, where $a={\mu}_{x_3}(X)$, 3. \[def:subsystemconti\] $$\pa_3 {\widetilde}{{\mu}}+ \div' \lt(\frac{dm}{d{\mu}} {\widetilde}{{\mu}}\rt)=0.$$ In particular, \[def:subsystemdelta\] implies that $({\mu}_{x_3} - {\widetilde}{{\mu}}_{x_3}) \perp \delta_{X}$ in the sense of the Radon-Nikodym decomposition. Let us for notational simplicity assume that $x_3=0$, $L=T=1$, $\mu(Q_{1,1})=2$. Let us denote $v= \frac{d m}{d{\mu}}$. According to [@AGS Th. 8.2.1 and (8.2.8)], since $v\in L^2(Q_{1,1},\mu)$, there exists a positive measure $\sigma$ on $ C^0([-1, 1];Q_1)$ (endowed with the sup norm), whose disintegration [@AGS Th. 5.3.1] with respect to $\mu_0$, i.e. $\sigma=\int_{Q_1} \sigma_{x'} d\mu_0(x')$, is made of probability measures $\sigma_{x'}$ concentrated on the set of curves $ \gamma$ solving $$\left\{ \begin{array}{l} \dot{\gamma}(x_3)= v(\gamma(x_3))\\ \gamma(0)= x',\end{array}\right.$$ and such that for every $x_3\in [-1,1]$, ${\mu}_{x_3}= (e_{x_3})_\# \sigma$, where $e_{x_3}$ denotes the evaluation at $x_3$, in the sense that $$\int_{Q_1}\varphi d\mu_{x_3} = \int_{ C^0([-1, 1];Q_1)} \varphi(\gamma(x_3)) d\sigma (\gamma) \hskip1cm \text{ for all }\varphi\in C^0(Q_1)\,.$$ Then, the measure ${\widetilde}{{\mu}}= {\widetilde}{{\mu}}_{x_3}\otimes dx_3$ with ${\widetilde}{\mu}_{x_3}= (e_{x_3})_\# (a \sigma_{X})$, where $a=\mu_{0}(X)$, satisfies all the required properties. \[noloop\] Let ${\mu}$ be a minimizer for the Dirichlet problem (\[limitProb\]), $\bar x_3\in (-T,T)$. Let $x_1=(X_1,\bar{x}_3)$, $x_2=(X_2,\bar{x}_3)$ be two points in the plane $\{x:x_3=\bar{x}_3\}$. Let ${\mu}_1$ and ${\mu}_2$ be subsystems of ${\mu}$ emanating from $x_1$ and $x_2$. Let $x_+:=(X_+,z_+)$ be a point with $ z_+ > \bar{x}_3$ and $x_-:=(X_-,z_-)$ a point with $ z_-<\bar{x}_3$, and such that ${\mu}_{1,z_+}, {\mu}_{2,z_+}$ both have Diracs at $X_+$, and ${\mu}_{1,z_-}, {\mu}_{2,z_-}$ both have Diracs at $X_-$ (with nonzero mass). Then $X_1=X_2$. Let ${\varphi}_1:=\mu_{\bar x_3}(X_1)$ be the mass of ${\mu}_1$ and ${\varphi}_2$ be the mass of $\mu_2$. Let ${\varphi}_{1, +}:={\mu}_{1,\bar x_3}(X_+)$ be the mass of ${\mu}_1$ at $x_+$, ${\varphi}_{2, +}$ the mass of ${\mu}_2$ at $x_+$, ${\varphi}_{1, - } $ the mass of ${\mu}_1$ at $x_-$, ${\varphi}_{2,-}$ the mass of ${\mu}_2$ at $x_-$. Let ${\varphi}:= \min \{{\varphi}_{1, +},{\varphi}_{1, -}, {\varphi}_{2, +}, {\varphi}_{2,-}\}$ which by assumption is positive. We define ${\mu}_{1, +}$ as the subsystem of ${\mu}_1$ coming from $x_+$, it is thus of mass ${\varphi}_{1, +}$, and at level $\bar{x}_3$ all its mass is at $X_1$ (since it is a subsystem of ${\mu}_1$ for which this is the case). Similarly with ${\mu}_{1, -}$, ${\mu}_{2, +}, {\mu}_{2,-}$. We can now define ${\widetilde}{{\mu}}_1:= \frac{{\varphi}}{{\varphi}_{1, +} } {\mu}_{1, +}$ for $x_3\ge\bar{x}_3$ and ${\widetilde}{{\mu}}_1:=\frac{{\varphi}}{{\varphi}_{1, -} }{\mu}_{1, -}$ for $x_3 <\bar{x}_3$, and the same with ${\widetilde}{{\mu}}_2$. The measures ${\widetilde}{{\mu}}_1$ and ${\widetilde}{{\mu}}_2$ are “systems” of mass ${\varphi}$ that join $X_-$ and $X_+$. By construction, we have $$\partial_{x_3} ( {\widetilde}{{\mu}}_1- {\widetilde}{{\mu}}_2 ) \restr[z_-,z_+]+\div' \lt( ( {\widetilde}{{\mu}}_1- {\widetilde}{{\mu}}_2 ) \restr[z_-,z_+]\frac{dm}{d\mu}\rt)=0$$ and $$-\frac{\varphi}{\min(\varphi_{2,+}, \varphi_{2,-})}\mu \le ( {\widetilde}{{\mu}}_1- {\widetilde}{{\mu}}_2 ) \restr[z_-,z_+]\le \frac{\varphi}{\min(\varphi_{1,+}, \varphi_{1,-})}\mu.$$ We now define $\hat{{\mu}}_\eta:= {\mu}+ \eta ( {\widetilde}{{\mu}}_1- {\widetilde}{{\mu}}_2 ) \restr[z_-,z_+]$, which is admissible for $\eta$ small enough [(and different from $\mu$ unless $X_1=X_2$)]{}, and evaluate $$\begin{gathered} I(\hat{{\mu}}_\eta)- I({\mu}) = K_*\int_{z_-}^{z_+} \sum_{x'\in Q_1}\left( \mu_{x_3} (x') +\eta ({\widetilde}{\mu}_1)_{x_3}(x')-\eta ({\widetilde}{\mu}_2)_{x_3}(x') \right)^\hal - \sum_{x'\in Q_1}\left( \mu_{x_3} (x') \right)^\hal \, dx_3 \\ +\eta \int_{Q_1 \times [z_-, z_+] } \left(\frac{dm}{d{\mu}} \right)^2 ( d{\widetilde}{{\mu}}_1- \, d{\widetilde}{{\mu}}_2)\,. \end{gathered}$$ But the function $\eta \mapsto \sqrt{a+\eta b}$ is strictly concave for $a>0$ and $b\ne0$, therefore $I(\hat{{\mu}}_\eta)+I(\hat{{\mu}}_{-\eta})<2I({\mu})$ for any $\eta\ne 0$, a contradiction with the minimality of ${\mu}$. A consequence of this lemma is the following. Consider a minimizing measure $\mu$ of $I$. Let $z_-$ and $z_+$ be any two slices and let $X_-$ be one of the Diracs at slice $z_-$. Let ${\widetilde}{{\mu}}$ be a subsystem emanating from $(X_-,z_-)$. Let $X_+$ be any point in the slice $z_+$ where ${\widetilde}{{\mu}}$ carries mass. Then, there is a unique “path" connecting $X_-$ to $X_+$ (otherwise there would be a loop). Since this is true for any couple of “sources" in two different planes, this means that there are at most a countable number of absolutely continuous curves (absolutely continuous because of the transport term) on which ${\mu}\restr[z_-, z_+]$ is concentrated. So we have a representation of the form $$\label{representationmu} {\mu}= \sum_i \frac{\varphi_i}{\sqrt{1+|{\dot{X}}_i|^2}} \, \mathcal{H}^1\restr\Gamma_i \, ,$$ where the sum is countable and $\Gamma_i=\{ (X_i(x_3),x_3) : x_3\in [a_i,b_i]\} $ with $X_i$ absolutely continuous and almost everywhere non overlapping. Another consequence is that if there are two levels at which ${\mu}$ is a finite sum of Diracs, then it is the case for all the levels in between. So, if there is a slice with an infinite number of points, then either it is also the case for all the slices below or for all the slices above. For measures which are concentrated on finitely many curves we obtain a simple representation formula for $I(\mu)$. \[lemmacurves\] Let $\mu=\sum_{i=1}^N \frac{\varphi_i}{\sqrt{1+|{\dot{X}}_i|^2}} \, \mathcal{H}^1\restr \Gamma_i \in{\mathcal{A}}^*_{L,T}$ with $\Gamma_i=\{ (X_i(x_3),x_3) : x_3\in [a_i,b_i]\}$ for some absolutely continuous curves $X_i$, almost everywhere non overlapping. Every ${\varphi}_i$ is then constant on $[a_i,b_i]$ and we have conservation of mass. That is, for $x:=(x',x_3)$, letting $$\begin{aligned} {\mathcal{I}}^+(x)&:=\{ i\in[1,N] \ : \ x_3=b_i, \ X_i(b_i)=x'\}, \\ {\mathcal{I}}^-(x)&:=\{ i\in[1,N] \ : \ x_3=a_i, \ X_i(a_i)=x'\}, \end{aligned}$$ it holds $$\sum_{i\in{\mathcal{I}}^-(x)} {\varphi}_i=\sum_{i\in{\mathcal{I}}^+(x)} {\varphi}_i.$$ Moreover, $m=\sum_i \frac{\varphi_i}{\sqrt{1+|{\dot{X}}_i|^2}} {\dot{X}}_i \, \mathcal{H}^1\restr\Gamma_i$ and $$\label{Iparticul} I(\mu)=\frac{1}{L^2}\sum_i \int_{a_i}^{b_i} K_* \sqrt{{\varphi}_i}+ {\varphi}_i |{\dot{X}}_i|^2 dx_3.$$ Let $\bar x=(\bar x',\bar x_3)$ with $\bar x_3 \in (-T,T)$ be such that $\mu_{\bar x_3}(\bar x')\neq 0$. Then, by continuity of the $X_i$’s, there exist $\delta>0, \eps>0$ such that every curve $\Gamma_i$ with $\Gamma_i\cap\lt( \B_\eps'(\bar x')\times [\bar x_3-\delta,\bar x_3+\delta]\rt) \neq \emptyset$ satisfies $X_i(\bar x_3)=\bar x',$ and such that $\mu\restr (\B'_{2\eps}(\bar x')\backslash \B_\eps'(\bar x'))\times [\bar x_3-\delta,\bar x_3+\delta]=0$ (and thus also $m\restr (\B'_{2\eps}(\bar x')\backslash \B_\eps'(\bar x'))\times [\bar x_3-\delta,\bar x_3+\delta]=0$ since $m\ll {\mu}$). Consider then $\psi_1\in C^\infty_c(\B'_{2\eps}(\bar x'))$ with $\psi_1=1$ in $\B_\eps(\bar x')$ and $\psi_2\in C^\infty_c( \bar x_3-\delta,\bar x_3+\delta)$ and test with $\psi:=\psi_1 \psi_2$ to obtain $$\begin{aligned} \int_{\bar x_3-\delta}^{\bar x_3+\delta} \frac{d\psi_2}{dx_3} (x_3)\lt(\sum_{X_i(x_3)\in \B_\eps'(\bar x')} {\varphi}_i\rt) dx_3&=\int_{\bar x_3-\delta}^{\bar x_3+\delta}\int_{\B'_\eps(\bar x')} \frac{d\psi_2}{dx_3}(x_3)\psi_1(x') d\mu\\ &=\int_{Q_{1,1}} \frac{\partial \psi}{\partial x_3} d\mu=-\int_{Q_{1,1}} \nabla'\psi \cdot dm\\ &=-\int_{\bar x_3-\delta}^{\bar x_3+\delta}\int_{\B_{2\eps}(\bar x')\backslash \B_\eps(x')} \psi_2 \nabla' \psi_1 \cdot dm =0,\end{aligned}$$ from which the first two assertions follow. It can be easily checked that this implies that $\bar m:= \sum_i \frac{\varphi_i}{\sqrt{1+|{\dot{X}}_i|^2}} {\dot{X}}_i \, \mathcal{H}^1\restr\Gamma_i$ satisfies . Let $m$ be any other measure satisfying and let us prove that $\nu:=\bar m -m=0$. Since $\Divp \nu_{x_3}=0$, we have for every $\psi\in C^{\infty}(Q_1)$ $$\sum_i \nabla' \psi(X_i(x_3))\cdot \nu_i(x_3)=0,$$ where $\nu_{x_3}=\sum_i \nu_i (x_3) \delta_{X_i(x_3)}$, from which the claim follows. We remark that Corollary \[coroXia\] below will imply that representation holds for every measure $\mu$ with $I(\mu)<+\infty$. The previous results lead to the following. \[reg\] A minimizer of the Dirichlet problem (\[limitProb\]) with boundary conditions $\bar{\mu}_+= \sum_{i =1}^N {\varphi}_i^+\delta_{X_i^+}$ and $\bar{\mu}_{-} = \sum_{i=1}^N {\varphi}_i^- \delta_{X_i^-} $ (some ${\varphi}_i$ may be zero) satisfies 1. $\mu= \sum_{i=1}^M \frac{\varphi_i}{\sqrt{1+|{\dot{X}}_i|^2}} \, \mathcal{H}^1\restr\Gamma_i $ for some $M\in\N$, where $\Gamma_i=\{ (X_i(x_3),x_3) : x_3\in [a_i,b_i]\}$ are disjoint up to the endpoints, and the $X_i$ are absolutely continuous. 2. Each $X_i$ is affine. 3. If $\bar{\mu}_{-}= \bar{\mu}_{+} $ then there exists a symmetric minimizer with respect to the $x_3=0$ plane. Let $\mu^{ij}$ be the subsystem emanating from $(X_i^-,-T)$ of the subsystem emanating from $(X_j^+,T)$ of $\mu$, so that $\mu=\sum_{ij}\mu^{ij}$ by $\mu^{ij}_{x_3}\le \mu_{x_3}$ and conservation of mass. By Lemma \[noloop\] we have $\mu^{ij}_{x_3}={\varphi}_{ij}(x_3)\delta_{X^{ij}}(x_3)$ for all $x_3$, otherwise there would be loops. By Lemma \[lemmacurves\], ${\varphi}_{ij}(x_3)$ does not depend on $x_3$. By (\[HolderW2\]), if ${\varphi}_{ij}>0$ then $X^{ij}$ is absolutely continuous. After a relabeling, (i) is proven. Assertion (ii) follows from minimizing $I({\mu})$ as given by with respect to ${\dot{X}}_i$. Let now $\bar{\mu}_{-}= \bar{\mu}_{+}$. If $I(\mu,(-T,0))\le I(\mu,(0,T))$ we obtain a symmetric minimizer $\hat\mu$ by reflection of $\mu\LL(-T,0)$ across $\{x_3=0\}$, and analogously in the other case. This proves (iii). We now show that for symmetric minimizers, at arbitrarily small distance from the boundary we have a finite number of Diracs. We already know that at arbitrarily small distance we have a countable number, and then that we have a representation of ${\mu}$ of the form . Let us point out that we will not use this proposition but rather include it for its own interest. \[finitebranch\] Fix $\bar\mu\in {\mathcal{M}}^+(Q_L)$. Let $\mu$ be a symmetric minimizer of $I$ subject to $\mu_{\pm T}=\bar\mu$. Then for any $\delta>0$ sufficiently small, the number of Diracs in each slice $x_3 \in [-T + \delta T , T- \delta T]$ is ${\stackrel{<}{\sim}}\delta^{-4}$. We may assume $L=T=1$, $\mu(Q_{1,1})=2$. By symmetry, we need only to consider the interval $[0,1-\delta]$. If $\mu_{1-\delta}=\sum_{i} \p_i \delta_{X_i}$, it suffices to prove that ${\varphi}_i{\stackrel{>}{\sim}}\delta^{4}$ for every $i$. For the rest of the proof we fix a point $X_i$ and in order to ease notation we write ${\varphi}:={\varphi}_i$ and $X:= X_i$. Let ${\widetilde}{{\mu}}$ be the subsystem emanating from $(X,1-\delta)$. Thanks to the symmetry of ${\mu}$ and to the no-loop condition, $\mu$ and $\mu-{\widetilde}{{\mu}}$ are disjoint for $x_3> 1-\delta$. Indeed, if this was not the case, by symmetry they would meet also for $x_3<- 1+\delta$, and there would be a loop, which is excluded by Lemma \[noloop\]. Therefore $$\label{secondestimI} I({\mu})- I({\mu}- {\widetilde}{{\mu}})\ge \int_{1-\delta}^1 \sum_{x'\in Q_1} \left( {\widetilde}{{\mu}}_{x_3}(x')\right)^\hal \, dx_3\ge \delta \sqrt{\p},$$ where in the second step we used subadditivity of the square root. Let now ${\widetilde}{\mu}_{1}$ be the trace of ${\widetilde}{{\mu}}$ on $x_3={1}$ and for $z:={\varphi}^{1/4}$, let $\hat\mu$ be the symmetric comparison measure constructed as follows: - in $[0, 1-z]$, $\hat\mu_{x_3}:= (1+\frac{\p}{1-\p})(\mu_{x_3}-{\widetilde}{\mu}_{x_3})$ and - in $[1-z,1]$, $\hat\mu_{x_3}:=\mu_{x_3}-{\widetilde}{\mu}_{x_3}+\nu_{x_3}$ where $\nu$ is a measure connecting $\frac{\p}{1-\p} (\mu-{\widetilde}{{\mu}})_{1-z}$ to ${\widetilde}{{\mu}}_{1}$ constructed in Proposition \[branchingmu\], so that (recall ) $$ I^{(1-z,1)}(\nu)\les z\sqrt{\p}+\frac{\p}{z}\sim \p^{3/4}.$$ Since ${\mu}$ is a minimizer it follows by subadditivity of the energy that, for some universal (but generic) constant $C$, $$\begin{aligned} I(\mu)\le I(\hat\mu)&\le \lt(1+\frac{{\varphi}}{1-\p}\rt) I(\mu-{\widetilde}{{\mu}})+ C\p^{3/4}\\ &\le I(\mu-{\widetilde}{{\mu}})+C\p^{3/4}. \end{aligned}$$ Indeed, $I(\mu-{\widetilde}{{\mu}}) \le I(\mu) {\stackrel{<}{\sim}}1$ while $\varphi \ll 1$ without loss of generality. Recalling , we deduce that $C\p^{3/4}\ge \delta \p^{1/2}$, which yields the result. \[defpolygonal\] We say that a measure $\mu$ is polygonal if $$\mu= \sum_i \frac{\varphi_i}{\sqrt{1+|{\dot{X}}_i|^2}} \, \mathcal{H}^1\restr\Gamma_i,$$ where the sum is countable, $\Gamma_i$ are segments of the form $\Gamma_i=\{ (X_i(x_3),x_3) : x_3\in [a_i,b_i]\}$ disjoint up to the endpoints, and for any $z\in (0,T)$ only finitely many segments intersect $Q_L\times (-z,z)$. We say it is finite polygonal if the total number of segments is finite. For any polygonal measure, the representation formula holds. Let $\bar {\mu}\in {\mathcal{M}}^+(Q_L)$. Then, every symmetric minimizer of $I$ with boundary data $\mu_{\pm T}=\bar {\mu}$ is polygonal. It suffices to show that for any $z\in (0,T)$ the measure $\mu$ is polygonal in $Q_L\times (-z,z)$. By Proposition \[finitebranch\] the measures $\mu_{z}$ and $\mu_{-z}$ are finite sums of Diracs. Since $\mu$ is a minimizer, it minimizes $I$ restricted to $(-z,z)$ with boundary data $\mu_{\pm z}$. By Proposition \[reg\] we conclude. Density of regular and quantized measures ----------------------------------------- In this section we want to prove that when $\bar \mu_{\pm T }= \Phi L^{-2} dx'$, the set of “regular" measures is dense in energy. \[defregular\] We denote by $\Mreg(Q_{L,T}){\subseteq}{\mathcal{A}}^*_{L,T}$ the set of regular measures, i.e., of measures $\mu$ such that: - The measure $\mu$ is finite polygonal, according to Def. \[defpolygonal\]. - All branching points are triple points. This means that any $x\in Q_{L,T}$ belongs to the closures of no more than three segments. For $N\in \N$, we say that $\mu$ is $N$-regular, $\mu\in \MNreg(Q_{L,T}){\subseteq}\Mreg(Q_{L,T})$, if in addition - The traces obey $\mu_{T}=\mu_{-T}=\Phi N^{-2}\sum_{j} \delta_{X_j}$, where the $X_j$ are $N^2$ points on a square grid, spaced by $L/N$, and $\Phi\ge0$. We can now state the main theorem of this section: \[theodens\] For every measure $\mu\in {\mathcal{A}}^*_{L,T}$ with $I(\mu)<+\infty$ and $ \mu_{\pm T}= \Phi L^{-2} dx'$, there exists a sequence of measures $\mu_N \in\MNreg$, with $\mu_N \weaklim \mu$ and such that $\limsup_{N \to +\infty} I(\mu_N)\le I(\mu)$, $\mu_N(Q_{L,T})=\mu(Q_{L,T})$. The proof will be based on the following intermediate result. \[lemmamudensityinside\] For every measure $\mu\in {\mathcal{A}}^*_{L,T}$ with $I(\mu)<+\infty$, and such that the traces $\mu_T$ and $\mu_{-T}$ are finite sums of Diracs, there exists a sequence of regular measures $\mu^{(N)}\in \Mreg(Q_{L,T})$ with $\mu^{(N)} \weaklim \mu$, $\mu^{(N)}_{\pm T}=\mu_{\pm T}$, and such that $\limsup_{N \to +\infty} I(\mu^{(N)})\le I(\mu)$. We shall modify $\mu$ in two steps to make it polygonal: first on finitely many layers, to have finitely many Diracs on each of them, and then in the rest of the volume, using local minimization. Fix $N\in\N$ and $\delta>0$, both chosen later. We choose levels $z_j\in (Tj/N, T(j+1)/N)$, for $j=-N+1,\dots, N-1$, with the property that $\mu_{z_j}=\sum_{k\in\N} \varphi_{j,k}\delta_{x'_{j,k}}$, with $\sum_{k\in \N} \varphi_{j,k}^{1/2}<+\infty$ for every $j$. We shall iteratively truncate the measure at these levels so that it is supported on finitely many points; for notational simplicity we also define $z_{\pm N}:=\pm T$. The measures $\mu^j$, $j=-N,\dots, N$ will all satisfy $\mu^j\ll\mu$ and $I(\mu^j)\le I(\mu)$. We start with $\mu^{-N}:=\mu$. In order to construct $\mu^{j+1}$ from $\mu^j$, we first choose $K_j$ such that $\sum_{k\ge K_j} \varphi_{j,k}^{1/2}\le \delta/N$. Then we define $\mu^{j+1}$ as the sum of the subsystems of $\mu^j$ originating from the points $(x'_{j,k},z_j)$ with $k<K_j$. Clearly $I(\mu^{j+1})\le I(\mu^j)\le I(\mu)$. At the same time, since $\sum_{k=K_j}^{+\infty} {\varphi}_{j,k} \le \Phi^{1/2} \sum_{k=K_j}^{+\infty} {\varphi}_{j,k}^{1/2}$, $$\mu^{j+1}_{z_j}(Q_L) = \mu^{j}_{z_j}(Q_L)- \sum_{k=K_j}^{+\infty} {\varphi}_{j,k}\ge \mu^{j}_{z_j}(Q_L)-\frac{ \Phi^{1/2} \delta}{N},$$ so that $|\mu^{j+1}-\mu^j|(Q_{L,T})\le 2\Phi^{1/2}\delta T/N$. Therefore $|\mu^N-\mu|(Q_{L,T})\les \delta T \Phi^{1/2}$. We define $\hat\mu^N$ as the minimizer with boundary data $\mu^N_{z_j}$ and $\mu^N_{z_{j+1}}$ in each stripe $Q_L\times (z_j, z_{j+1})$. Then $I(\hat\mu^N)\le I(\mu^N)\le I(\mu)$, and $\hat\mu^N_{\pm T}=\mu^N_{\pm T}$. At this point we fix the boundary data. For this, we let ${\widetilde}{\mu}$ be the minimizer on $Q_{L,T}$ with boundary data ${\widetilde}{\mu}_\pm:=\mu_{\pm T}-\mu^N_{\pm T}$. These boundary data are finite sums of Diracs, and their flux is $|{\widetilde}{\mu}_\pm|(Q_L)=|\mu_T-\mu^N_T|(Q_L)\les \delta \Phi^{1/2}$. By Proposition \[reg\] the minimizer is finite polygonal, by Proposition \[branchingmu\] it has energy no larger than a constant times $\delta \Phi^{1/2}T^{-1} + \delta^{1/2} T \Phi^{1/4} L^{-2}$. Finally we set $\mu^{(N)}:=\hat\mu^N+{\widetilde}{\mu}$. Then $\mu^{(N)}_{\pm T}=\mu_{\pm T}$, and $$I(\mu^{(N)})\le I(\hat\mu^N)+I({\widetilde}{\mu}) \le I(\mu) + C\left(\delta \Phi^{1/2}T^{-1} + \delta^{1/2} T \Phi^{1/4} L^{-2}\rt) .$$ Up to a small perturbation, we may further assume that all junctions are triple. We can now choose for instance $\delta=1/N$. It only remains to show that $\mu^{(N)}\weaklim\mu$ as $N\to+\infty$. Recalling that $|\mu^N-\mu|(Q_{L,T})+|{\widetilde}{\mu}|(Q_{L,T})\les \delta \Phi^{1/2}T$, we only need to show that $\hat\mu^N-\mu^N\weaklim 0$. For $x_3\in (z_j,z_{j+1})$ we have by definition of $\hat \mu^N$ $$W_2(\mu_{x_3}^N, \hat \mu_{x_3}^N) \le W_2(\mu_{x_3}^N, \mu_{z_j}^N) +W_2(\hat \mu_{x_3}^N, \hat\mu_{z_j}^N),$$ and by $$W_2^2(\mu_{x_3}^N, \mu_{z_j}^N)+W_2^2(\hat \mu_{x_3}^N, \hat \mu_{z_j}^N) \le L^2 (z_{j+1}-z_j) \lt( I(\mu^N)+I(\hat \mu^N)\rt),$$ so that $$\max_{x_3} W_2^2(\mu_{x_3}^N, \hat \mu_{x_3}^N) {\stackrel{<}{\sim}}\frac{L^2T}{N}I(\mu).$$ For $x_3\in (-T,T)$, let $\Pi_{x_3}$ be an optimal transport plan from $\mu^N_{x_3}$ to $\hat{\mu}^N_{x_3}$. Considering then the transport plan $\Pi:=\Pi_{x_3}\otimes dx_3$ between $\mu^N$ and $\hat{\mu}^N$, we get $$W_2^2(\mu^N,\hat{\mu}^N)\le \int_{-T}^T W^2_2(\mu_{x_3}^N,\hat\mu_{x_3}^N) dx_3\les \frac{L^2T^2}{N}I(\mu) \,,$$ which yields that indeed $\hat\mu^N-\mu^N\weaklim 0$. Let $\eps\in (0,1/4)$, chosen such that it tends to zero as $N\to\infty$. We define $\hat\mu$ in $Q_{L,(1-2\eps)T}$ as a rescaling by $(1-2\eps)$ in the vertical direction, $\hat\mu_{(1-2\eps)x_3}=\mu_{x_3}$. An easy computation shows that $I(\hat\mu, -(1-2\eps)T, (1-2\eps)T)\le \frac{1}{1-2\eps} I(\mu)$. In particular, $\hat\mu_{(1-2\eps)T}=\Phi L^{-2} dx'$. For $x_3\in ((1-2\eps)T, T)$ we define $\hat\mu$ as the result of Proposition \[branchingmu\]. Then we set ${\widetilde}{\mu}=\hat\mu$ on $(-(1-\eps)T,(1-\eps)T)$ and extend it constant outside, in the sense that ${\widetilde}{\mu}_{x_3}=\hat\mu_{(1-\eps)T}$ for $x_3\in ((1-\eps)T,T)$, and the same on the other side. Since $\hat \mu_{(1-\eps)T}$ is the midplane configuration of the branching measure constructed in Proposition \[branchingmu\], ${\widetilde}{\mu}_{x_3}$ is a finite sum of Diracs for $|x_3|\ge (1-\eps) T$. We obtain $$I({\widetilde}{\mu}) \le \frac{1}{1-2\eps} I(\mu) + C \eps^{1/3} \left(\Phi^{2/3}T^{1/3} L^{-4/3}+\eps^{2/3} \Phi^{1/2} T L^{-2} \right)\,.$$ By Lemma \[lemmamudensityinside\] applied to the inner domain $(-(1-\eps)T,(1-\eps)T)$ there is a finite polygonal measure $\check\mu$ which is close to ${\widetilde}{\mu}$ and has the same boundary data at $x_3=\pm T(1-\eps)$. The measure given by $\check{\mu}$ inside, and ${\widetilde}{\mu}$ outside, has the required properties. We now turn to the quantization of the measures. We say that a regular measure $\mu\in\Mreg(Q_{L,T}){\subseteq}{\mathcal{A}}_{L,T}^*$ is $k$-quantized, for $k>0$, if for all $(x',z)\in Q_{L,T}$ one has $k\mu_z(\{x'\})\in 2\pi\N$. \[lemmaquantize\] Let $\mu\in \MNreg(Q_{L,T})$ and $\Phi=\mu_T(Q_L)$. For any $k>0$ such that $k\Phi\in 2\pi\N$ there is a $k$-quantized regular measure $\mu^{k}\in \Mreg$ such that $\mu^{k}(Q_{L,T})=\mu(Q_{L,T})$ and $$\left(1-\frac{ C(\mu)}k\right) \mu \le \mu^k \le \left(1+\frac{ C(\mu)}k\right)\mu\,.$$ This implies in particular $\mu^k\ll\mu$, $\mu^k\to\mu$ strongly, $W_2^2(\mu,\mu_k)\les C(\mu)k^{-1}$ and $I(\mu^k)\to I(\mu)$ as $k\to\infty$. The measure $\mu$ consists of finitely many segments, each with a flux. To prove the assertion it suffices to round up or down the fluxes to integer multiples of $2\pi/k$ without breaking the divergence condition, and without changing the total flux. Since $\mu\in\MNreg(Q_{L,T})$, we have $\mu_{T}=\Phi N^{-2}\sum_i \delta_{X_i}$. We select $\varphi_i^k$ as $2\pi \lfloor k \Phi /(2\pi N^2)\rfloor /k$ or $2\pi \lfloor k \Phi /(2\pi N^2)+1\rfloor /k$ , depending on $i$. Precisely, we choose the first value for $i=0$ and then, at each $i$, we choose the lower one if $\sum_{j<i} ({\varphi}_j^k- \Phi N^{-2})>0$, and the upper one otherwise. This concludes the definition of $\mu^k_T$. The fluxes in the interior of the sample are defined by propagating the rounding. At each point where a bifurcation occurs, if there is more then one outgoing branch we distribute the rounding as discussed for $\mu_T^k$. This increases the maximal error by at most $2\pi/k$, at each branching point. Since $\mu$ is finite polygonal, there is a finite number of branching points, hence the total error is bounded by a constant times $1/k$. Precisely $|{\varphi}_i^k-{\varphi}_i|\le C(\mu)/k$ for any segment $i$. Since ${\varphi}_i$ only takes finitely many values, $|{\varphi}_i^k-{\varphi}_i|\le {\varphi}_i C(\mu)/k$ for any segment $i$, which concludes the proof. Relation with irrigation problems {#Sec:reliri} --------------------------------- The functional $I(\mu)$ bears similarities with the so-called irrigation problems which have attracted a lot of interest (see for instance [@Xia; @BeCaMo]). Besides their applications to the modeling of communication networks and other branched patterns (see again [@BeCaMo] and the references therein), they have also been recently used in the study of Sobolev spaces between manifolds [@bethuel]. Let us recall their definition and for this, follow the notation of [@BeCaMo]. For $E(G)$ a set of oriented straight edges and $\p: E(G)\to (0,+\infty)$ we define the irrigation graph $G$ as the vector measure $$G:=\sum_{e\in E(G)} {\varphi}(e) \mathbf{e} \, \H^1\restr e \,$$ where $\mathbf{e}$ is the unit tangent vector to $e$. For $\a\in [0,1]$, we then define the Gilbert energy of $G$ by $$M^{\a}(G):=\sum_{e\in E(G)} \p(e)^\a \H^1(e).$$ Given two atomic probability measures $\mu^+=\sum_{i=1}^k a_i \delta_{X_i}$ and $\mu^-=\sum_{j=1}^l b_j \delta_{Y_j}$, we say that $G$ irrigates $(\mu^+,\mu^-)$ if $\div G= \mu^+-\mu^-$ in the sense of distributions (this implies in particular that $G$ satisfies Kirchoff’s law). If we are now given any two probability measures $(\mu^+,\mu^-)$ and a vector measure $G$, with $\div G=\mu^+-\mu^-$ (sometimes called an irrigation path between $\mu^+$ and $\mu^-$), we define $$M^{\a}(G):=\inf \{ \liminf_{i\to +\infty} M^{\a}(G_i)\},$$ where the infimum is taken among all the sequences of irrigation graphs $G_i$ with $G_i\weaklim G$ in the sense of measures and such that $\div G= \mu^+_i-\mu^-_i$ for some atomic measures $\mu^{\pm}_i$ tending to $\mu^{\pm}$. If no such sequence exists then we set $M^\a(G)=+\infty$. The irrigation problem then consists in minimizing $M^{\a}(G)$ among all the transport paths $G$ between $\mu^+$ and $\mu^-$. For $\a=0$ this is a generalization of the famous Steiner problem while for $\a=1$ it is just the Monge-Kantorovich problem. Using some powerful rectifiability criterion of B. White, the following theorem was proven by Q. Xia [@Xia]. \[theoXia\] Given $0<\a<1$, any transport path $G$ with $M^{\a}(G)+M^1(G)<+\infty$ is rectifiable in the sense that $$G={\varphi}\tau \, \H^1\restr\Gamma$$ for some density function $\p$ and some $1-$rectifiable set $\Gamma$ having $\tau$ as tangent vector. For minimal irrigation paths, much more is known about their interior and boundary regularity [@BeCaMo]. For instance, as for our functional $I(\mu)$ (see Proposition \[reg\]), it can also be proven that minimal irrigation paths contain no loops and that for $\a> 1-\frac{1}{n}$ (where $n$ is the dimension of the ambient space i.e. $n=3$ for us), any two probability measures $\mu^{\pm}$ can be irrigated at a finite cost (compare with Proposition \[existmu\]). Using Theorem \[theoXia\] and Lemma \[lemmamudensityinside\], we can obtain the following rectifiability result. \[coroXia\] Every measure $\mu$ for which $I(\mu)<+\infty$ is rectifiable. Using the construction of Lemma \[lemmamudensityinside\], we can find a sequence $\mu^n$ such that $\mu^n\weaklim \mu$, $\limsup_{n\to +\infty} I(\mu^n)\le I(\mu)$ and $\mu^n=\sum_{i=1}^N \frac{{\varphi}_i}{\sqrt{1+|{\dot{X}}_i|^2}} \H^1\restr \Gamma_i$ for some straight edges $\Gamma_i=\{(x_3,X_i(x_3)) \, :\, x_3\in(a_i,b_i)\}$. Letting $\widetilde{\mu}^n:=\sum_{i=1}^N \frac{{\varphi}_i}{\sqrt{1+|{\dot{X}}_i|^2}} \begin{pmatrix} {\dot{X}}_i\\ 1\end{pmatrix} \H^1\restr \Gamma_i$, we have for $\a\ge\frac{3}{4}$, $$\begin{aligned} M^\a({\widetilde}\mu^n)&=\sum_i \int_{a_i}^{b_i} {\varphi}_i^\alpha \sqrt{1+ |{\dot{X}}_i|^2} dx_3\\ &\les \sum_i \int_{a_i}^{b_i} {\varphi}_i (1+|{\dot{X}}_i|^2) + {\varphi}_i^{2\alpha-1} dx_3\\ &\les \sum_i \int_{a_i}^{b_i} {\varphi}_i (1+|{\dot{X}}_i|^2) + \sqrt{{\varphi}_i} dx_3\les I(\mu^n) +1\end{aligned}$$ so that $\liminf_{n\to +\infty} M^{\a}(\widetilde{\mu}^n)\les I(\mu)+1<+\infty$ and by Theorem \[theoXia\], the claim follows. In [@OudSant], an approximation of the functional $M^\a$ in the spirit of the Modica-Mortola [@ModMort] approximation of the perimeter was proposed. Even though their proofs and constructions are completely different from ours, this approach bears some similarities with our derivation of the functional $I({\mu})$ from the Ginzburg-Landau functional $E_T(u,A)$. Lower bound {#Sec:gammaliminf} =========== In the rest of the paper we consider sequences with $$\label{eqlbassumptcoeff} T_n\to +\infty, \hskip5mm \alpha_n\to +\infty,\hskip5mm \beta_n\to 0, \hskip5mm \frac{T_n}{\alpha_n} \to +\infty,\hskip5mm \alpha_n \beta_n^{7/2}\to +\infty\,.$$ No constant appearing in the sequel will depend on the specific choice of the sequence. We observe that (\[eqlbassumptcoeff\]) immediately implies $\alpha_n\beta_n^2\to+\infty$ and $\alpha_n^2\beta_n\to+\infty$. Let us recall that in this proof we set ${\widetilde}L=1$ and that (see (\[eqdefwe\])) $$\begin{gathered} \wE(u,A)=\int_{Q_{1,1}} \a^{-2/3}\b^{-1/3}\lt|\nabla_{\a^{1/3}\b^{-1/3}TA}'u\rt|^2+ \a^{-4/3}\b^{-2/3}\lt|(\nabla_{\a^{1/3}\b^{-1/3}TA} u)_3\rt|^2\\ + \a^{2/3}\b^{-2/3}\left({B_3} -(1-|{u}|^2)\right)^2 + \b^{-1}|{B'}|^2 dx +\a^{1/3}\b^{7/6}\|\b^{-1}{B_3}-1\|_{H^{-1/2}(x_3=\pm1)}^2.\end{gathered}$$ In this section, we prove the following compactness and lower bound result. \[gammaliminf\] Fix sequences of positive numbers $\alpha_n$, $\beta_n$, $T_n$ such that (\[eqlbassumptcoeff\]) holds, and let $(u_n, A_n)$ be such that $\sup_{n} \wE(u_n,A_n)<+\infty$. Then up to a subsequence, the following holds : 1. $\b_n^{-1}(1-\rho_n)\weaklim \mu$ for some measure $\mu$, $\b_n^{-1} B'_n\weaklim m$ for some vector-valued measure $m\ll {\mu}$ satisfying the continuity equation . 2. For almost every $x_3\in (-1,1)$, there exists some probability measure $\mu_{x_3}$ on $Q_1$ with $\mu=\mu_{x_3}\otimes dx_3$ and such that $\mu_{x_3}\weaklim dx'$ as $x_3\to \pm 1$. 3. For almost every $x_3\in (-1,1)$, ${\mu}_{x_3}=\sum_{i\in {\mathcal{I}}} \p_i \delta_{X_i}$ with ${\mathcal{I}}$ at most countable and $\p_i>0$. 4. One has $(\mu,m)\in {\mathcal{A}}_{1,1}$ with $$\liminf_{n\to +\infty} \wE(u_n,A_n)\ge I({\mu},m).$$ Let us first show that the energy gives a quantitative control on the failure of the Meissner condition $\rho B=0$ in a weak sense. \[lemmameissner\] For every $Q_1$-periodic test function $\psi\in H^1_\mathrm{per}(Q_{1,1})$, if $\|\rho\|_\infty\le 1$ then $$\label{estimmeissner} \lt|\int_{Q_{1,1}} \rho B_3 \psi dx\rt|\les \frac{\a^{1/3}\b^{2/3}}{T}\wE(u,A)\|\psi\|_{L^\infty}+ \frac{\beta^{1/2}}{T}\wE(u,A)^\hal \|{\nabla}' \psi\|_{L^2},$$ and, if additionally $\psi(x',\pm1)=0$, for $k=1,2$ and $\alpha^2\beta\ge 1$, $$\label{estimmeissnerprime} \lt|\int_{Q_{1,1}} \rho B_k \psi dx\rt|\les \frac{\a^{2/3}\b^{5/6}}{T}\wE(u,A)\|\psi\|_{L^\infty}+ \frac{\alpha^{1/3}\beta^{2/3}}{T}\wE(u,A)^\hal \|{\nabla}\psi\|_{L^2}.$$ Moreover, if $\xi\in H^1_0(-1,1)$ and $\psi$ is a periodic Lipschitz continuous function on $Q_1$ then $$\label{estimmeissnerter} \lt|\int_{Q_{1,1}} \rho B'\cdot \nabla' \psi \, \xi dx\rt|\les \frac{\a^{2/3}\b^{5/6}}{T}\wE(u,A)\|\xi \nabla'\psi\|_{L^\infty}+ \frac{\b^{1/2}}{T}\wE(u,A)^\hal \| \partial_3 \xi \nabla'\psi \|_{L^2}.$$ Let $\lambda:=\a^{1/3}\b^{-1/3}T$. For (\[estimmeissner\]) we use formula with $A$ substituted by $\lambda A$, that is $|{\nabla}'_{\lambda A} u|^2 = |{\mathcal{D}}_{\lambda A}^3 u|^2 +\rho \lambda B_3+ {\nabla}'\times j'_{\lambda A}$. We integrate against a test function $\psi$, $$\begin{aligned} \lt|\int_{Q_{1,1}} \rho B_3 \psi dx\rt| & = & \frac{\a^{1/3}\b^{2/3}}{T}\lt|\int_{Q_{1,1}}\a^{-2/3}\b^{-1/3} \left(|{\nabla}_{\lambda A}'u|^2 - |{\mathcal{D}}_{\lambda A}^3 u|^2 - {\nabla}'\times j'_{\lambda A}\right) \psi dx\rt|\\ & \les & \frac{\a^{1/3}\b^{2/3}}{T}\lt(\wE(u,A)\|\psi\|_{L^\infty} + \a^{-1/3}\b^{-1/6}\int_{Q_{1,1}} \a^{-1/3}\b^{-1/6}|j'_{\lambda A}| |{\nabla}' \psi|dx\rt)\\ & \les & \frac{\a^{1/3}\b^{2/3}}{T}\lt(\wE(u,A)\|\psi\|_{L^\infty}+ \a^{-1/3}\b^{-1/6}\wE(u,A)^\hal \|{\nabla}' \psi\|_{L^2} \rt),\end{aligned}$$ [where we have used that $|j'_{\lambda A}|\le |\nabla'_{\lambda A} u| $ in view of the definition and the upper bound $\rho\le 1$.]{} We obtain similarly : One first checks from the definition of ${\mathcal{D}}_{\lambda A}u$ that $$\left| |{\mathcal{D}}^1_{\lambda A} u|^2- |(\nabla_{\lambda A} u)_3|^2 - |(\nabla_{\lambda A} u)_2|^2\right| \le 2 |(\nabla_{\lambda A} u)_2| \, |(\nabla_{\lambda A} u)_3|.$$ Testing (\[magic2\]) with $\psi$ and integrating by parts the term with $j_{\lambda A}$ as above gives $$\lambda \lt|\int_{Q_{1,1}} \rho B_k \psi dx\rt|\les \int_{Q_{1,1}} 2 |\psi| \, |(\nabla_{\lambda A} u)_2| \, |(\nabla_{\lambda A} u)_3| + |\nabla \psi| \, |\nabla_{\lambda A} u| dx\,.$$ Estimating $ 2|(\nabla_{\lambda A} u)_2| \, |(\nabla_{\lambda A} u)_3| \le \alpha^{1/3}\beta^{1/6}|(\nabla_{\lambda A} u)_2|^2 + \alpha^{-1/3}\beta^{-1/6}|(\nabla_{\lambda A} u)_3|^2 $ and $\|\nabla_{\lambda A} u\|_2^2\le \alpha^{4/3}\beta^{2/3} \wE(u,A)$ concludes the proof of (\[estimmeissnerprime\]).\ The proof of is very similar to the proof of . Arguing as above with $\xi\partial_{k}\psi$ playing the role of $\psi$, we get $$\lambda \lt|\int_{Q_{1,1}} \rho B'\cdot \nabla'\psi \xi dx\rt|\les \int_{Q_{1,1}} 2 |\xi \nabla' \psi | \, |(\nabla_{\lambda A} u)_2| \, |(\nabla_{\lambda A} u)_3| dx + \lt|\int_{Q_{1,1}} (\nabla \times j_{\lambda A})\cdot \nabla \psi \xi dx\rt|\,.$$ The first term is estimated exactly as before, while the second one gives after integration by parts of $\nabla \times $ and using $\nabla \times \nabla=0$, $$\lt|\int_{Q_{1,1}} (\nabla \times j_{\lambda A})\cdot \nabla \psi \xi dx\rt|= \lt|\int_{Q_{1,1}} \nabla \xi\cdot (j_{\lambda A}\times \nabla \psi) dx\rt|\le \|\partial_3 \xi \nabla' \psi\|_{L^2} \lt(\int_{Q_{1,1}} |j'_{\lambda A}|^2\rt)^{1/2},$$ from which we conclude the proof. We now prove that for admissible pairs $(u,A)$ of bounded energy, the corresponding curves $x_3\mapsto \beta^{-1}B_3(\cdot,x_3)$ satisfy a sort of uniform Hölder continuity. This is the analog of for the limiting energy. \[HolderE\] For every admissible pair $(u,A)$ with $\|\rho\|_\infty\le 1$, and every $x_3,{\widetilde}{x}_3\in(-1,1)$, letting $E:=\wE(u,A)$ it holds (recall ) $$\begin{gathered} \label{HolderE2} \|\b^{-1}B_3(\cdot,x_3) - \b^{-1}B_3(\cdot,{\widetilde}{x}_3) \|_{BL}\les E^{1/2} |x_3-{\widetilde}{x}_3|^{1/2} + \sigma(\alpha,\beta,T)(E^{1/2}+E), \end{gathered}$$ where $\sigma(\alpha,\beta,T):= \lt(\frac{\a}{T}\rt)^{1/2} \lt(\a^2\b^{5/2}\rt)^{-1/6}+ (\a^{1/2}\b)^{-1/3} + \lt(\frac{\alpha^{1/3}}{T\b^{5/6}} \rt)^{1/2}$, which goes to zero in the regime . In particular, in that regime, if $E\les 1$, for every $x_3,{\widetilde}{x}_3\in(-1,1)$ with $|x_3-{\widetilde}{x}_3|\ge \sigma^{1/2}(\alpha,\b,T)$, there holds $$\label{equiconB} \|\beta^{-1}B_3(\cdot,x_3)-\beta^{-1}B_3(\cdot,{\widetilde}{x}_3)\|_{BL}\les |x_3-{\widetilde}{x}_3|^{1/2}.$$ The proof resembles that of [@CoOtSer Lem. 3.13]. First, we show that for every $Q_1-$periodic and Lipschitz continuous function $\psi$ with $\|\psi\|_{Lip}\le 1$, $$\label{firstestimHolder} \lt|\int_{Q_1\times\{x_3\} } \b^{-1}B_3 \psi dx' -\int_{Q_1\times\{{\widetilde}{x}_3\} } \b^{-1}B_3 \psi dx'\rt|\le|x_3-{\widetilde}{x}_3|^{1/2} \b^{-1/2} E^{1/2}.$$ This follows from $\Div B=0$ and integration by parts, which yields $$\begin{aligned} \lt|\int_{Q_1\times\{x_3\} } \b^{-1}B_3 \psi dx' -\int_{Q_1\times\{{\widetilde}{x}_3\} } \b^{-1}B_3 \psi dx'\rt|&=\lt|\int_{Q_1\times(x_3,{\widetilde}{x}_3)} \b^{-1}\partial_3 B_3 \psi dx\rt|\\ &=\lt|\int_{Q_1\times(x_3,{\widetilde}{x}_3)}\beta^{-1} B' \cdot \nabla' \psi dx\rt|\\ &\le |x_3-{\widetilde}{x}_3|^{1/2} \beta^{-1/2} \lt(\int_{Q_1\times(x_3,{\widetilde}{x}_3)} \beta^{-1}|B'|^2 dx\rt)^{1/2}\\ &\le |x_3-{\widetilde}{x}_3|^{1/2} \beta^{-1/2}E^{1/2}.\end{aligned}$$ For $|x_3-{\widetilde}{x}_3|\le T^{-1} \a^{1/3}\b^{1/6}$, this implies that $$\|\b^{-1}B_3(\cdot,x_3) - \b^{-1}B_3(\cdot,{\widetilde}{x}_3) \|_{BL}\le \lt(\a^{1/3} T^{-1} \b^{-5/6}\rt)^{1/2} E^{1/2}\le \sigma(\a,\b,T) E^{1/2},$$ and is proven. Letting $\hat{\sigma}(\a,\b,T):= \lt(\frac{\a}{T}\rt)^{1/2} \lt(\a^2\b^{5/2}\rt)^{-1/6}+ (\a^{1/2}\b)^{-1/3}$, we are left to prove that for $Q_1-$periodic and Lipschitz continuous $\psi$ with $\|\psi\|_{Lip}\le 1$ and $|x_3-{\widetilde}{x}_3|\ge T^{-1}\a^{1/3}\b^{1/6}$, $$\label{toproveHolder} \lt|\int_{Q_{1}\times\{x_3\}} \beta^{-1}B_3 \psi dx' -\int_{Q_{1}\times\{{\widetilde}{x}_3\}} \beta^{-1}B_3 \psi dx' \rt|\les |x_3-{\widetilde}{x}_3|^{1/2}E^{1/2}+ \hat{\sigma}(\alpha,\beta,T)(E^{1/2}+E).$$ Up to translation we may assume ${\widetilde}{x}_3=0$ and $x_3>0$. Let $\delta\le x_3/2$ and define $\xi: \R\to \R$ by $$\xi(z):=\begin{cases} \frac{z}{\delta} & \textrm{if } 0<z<\delta\\ 1 & \textrm{if } \delta\le z \le x_3-\delta\\ \frac{x_3-z}{\delta} & \textrm{if } x_3-\delta\le z\le x_3\\ 0 & \textrm{otherwise}. \end{cases}$$ We then have using again $\Div B=0$ and integration by parts $$\label{integpart} \int_{Q_{1,1}} \beta^{-1} B_3 \psi \partial_3\xi dx=-\int_{Q_{1,1}} \b^{-1}\rho B'\cdot \nabla'\psi \, \xi dx -\int_{Q_{1,1}} \b^{-1}(1-\rho) B'\cdot\nabla'\psi\, \xi dx.$$ The first term on the right-hand side of is estimated by . For the second term, we now estimate $$\label{firsttermright} \int_{Q_{1,1}} \b^{-1}(1-\rho)|B'|\xi dx\le \lt(\int_{Q_1\times(0,x_3)} \b^{-1}(1-\rho) dx\rt)^{1/2} \lt(\int_{Q_{1,1}}\b^{-1}(1-\rho)|B'|^2 dx \rt)^{1/2}.$$ We rewrite the first factor as $$\int_{Q_1\times(0,x_3)} \b^{-1}(1-\rho) dx=\int_{Q_1\times(0,x_3)} \b^{-1}B_3 dx +\int_{Q_1\times(0,x_3)} \b^{-1}(B_3-(1-\rho)) dx,$$ from which we obtain $$\int_{Q_1\times(0,x_3)} \b^{-1}(1-\rho) dx\le \lt|\int_{Q_1\times(0,x_3)} \b^{-1}B_3 dx \rt| +|x_3|^{1/2} \lt(\int_{Q_{1,1}} \b^{-2} (B_3-(1-\rho))^2 dx\rt)^{1/2}.$$ This allows to make use of $$\int_{Q_{1,1}} \b^{-2} (B_3-(1-\rho))^2 dx\le \a^{-2/3}\b^{-4/3} E,$$ and $\int_{Q_1\times\{z\}} \beta^{-1} B_3 dx' =1$, yielding $$\label{firstestimrightbis} \int_{Q_1\times(0,x_3)} \b^{-1}(1-\rho) dx\le |x_3|+|x_3|^{1/2}\a^{-1/3}\b^{-2/3} E^{1/2}.$$ The second factor in is directly estimated by $$\int_{Q_{1,1}}\b^{-1}(1-\rho)|B'|^2 dx\le E,$$ so that inserting this and into gives $$\label{firstestimrightter} \lt| \int_{Q_{1,1}} \b^{-1}(1-\rho) B'\cdot\nabla'\psi \xi dx\rt|\les |x_3|^{1/2} E^{1/2} +|x_3|^{1/4} \a^{-1/6}\b^{-1/3}E^{3/4}.$$ Letting $f(z):=\int_{Q_1\times\{z\}}\beta^{-1} B_3 \psi dx'$, we thus obtain from , and , $$\label{firstestimf} \lt|\int_0^{x_3} f \partial_3 \xi dz\rt|\les \frac{\a^{2/3}\b^{-1/6}}{T} E +\frac{\a^{1/3}\b^{-1/3}}{T} E^{1/2} \|\partial_3\xi\|_{L^2}+ |x_3|^{1/2} E^{1/2} +|x_3|^{1/4} E^{3/4} \a^{-1/6}\b^{-1/3}.$$ Since by definition of $\xi$, $\int_0^{x_3} f \partial_3\xi dz=\frac{1}{\delta} \int_0^\delta f dz-\frac{1}{\delta}\int_{x_3-\delta}^{x_3} f dz$, we have $$f(x_3)-f(0)=\int_0^{x_3} f \partial_3 \xi dz +\frac{1}{\delta}\int_{0}^\delta (f-f(0)) dz+\frac{1}{\delta}\int_{x_3-\delta}^{x_3} (f(x_3)-f) dz,$$ so that $$|f(x_3)-f(0)|\le \lt|\int_0^{x_3} f \partial_3 \xi dz\rt|+\sup_{(0,\delta)}|f-f(0)| +\sup_{(x_3-\delta,x_3)}|f-f(x_3)|.$$ In view of this elementary inequality, the estimates and combine to $$\begin{gathered} |f(x_3)-f(0)|\les |x_3|^{1/2} E^{1/2} +\frac{\a^{1/3}\b^{-1/3}}{T} E^{1/2} \delta^{-1/2} + \delta^{1/2}\b^{-1/2}E^{1/2} +\frac{\a^{2/3}\b^{-1/6}}{T} E\\ + |x_3|^{1/4} E^{3/4} \a^{-1/6}\b^{-1/3}. \end{gathered}$$ We now optimize in $\delta $ by choosing $\delta=T^{-1}\a^{1/3}\b^{1/6}$, which combined with $\frac{\a^{2/3}\b^{-1/6}}{T}\ll \a^{-1/6}\b^{-1/3}$, yields in the form of $$|f(x_3)-f(0)|\les |x_3|^{1/2} E^{1/2} + (E^{1/2}+E)\lt( \lt(\frac{\a}{T}\rt)^{1/2} \lt(\a^2\b^{5/2}\rt)^{-1/6}+ \a^{-1/6}\b^{-1/3}\rt).$$ \[remarkequi\] We notice that thanks to the Kantorovich-Rubinstein Theorem [@villani Th. 1.14], if $B_3$ is non-negative then we can substitute the Bounded-Lipschitz norm in by a $1-$Wasserstein distance. In particular, it would imply that if $(\alpha_n,\beta_n,T_n)$ satisfy and if $(u_n,A_n)$ are admissible with $\|\rho_n\|_{\infty}\le 1$, $\wE(u_n,A_n)\les 1$, and $B^n_3\ge 0$, then the corresponding curves $x_3\mapsto \beta_n^{-1} B^n_3(\cdot,x_3)$ would be in some sense equi-continuous in the space of probability measures endowed with the Wasserstein metric. For $\eps>0$ fixed, we define the following regularization of the singular double well potential $\chi_{\rho>0}(1-\rho)^2$ : $$\label{doublewell} W_\eps(\rho):=\eta_\eps(\rho)(1-\rho)^2 \ \text{with } \ \eta_\eps(\rho):=\min\{\rho/\eps,1\} \,,$$ see and Figure \[figweps\]. We next show that the energy controls $W_\eps(\rho)$. Similar ideas have been used in the context of Bose-Einstein condensates [@GolMer]. ![The cutoff function $\eta_\eps$ and the two-well potential $W_\eps$ used in Lemma \[lemmafirstlowerbound\].[]{data-label="figweps"}](fig-weps-crop){height="4cm"} \[lemmafirstlowerbound\] For every $\eps>0$ there exists $C_\eps>0$ such that for every $(u,A)$ with $\|\rho\|_\infty\le 1$ it holds, $$\label{estimlower} \int_{Q_{1,1}} \a^{2/3}\b^{-2/3}W_\eps(\rho) dx \le \int_{Q_{1,1}} \a^{2/3}\b^{-2/3}\left(B_3 -(1-\rho)\right)^2 dx +C_\eps \wE(u,A) \frac{\a}{T}.$$ As above, to lighten notation, we let $E:= \wE(u,A)$. Writing $(1-\rho)= B_3 -(B_3-(1-\rho))$, we obtain by Young’s inequality $$\begin{aligned} (1-\rho)^2&= B_3 (1-\rho)-(B_3-(1-\rho)) (1-\rho)\\ &\le B_3 (1-\rho) +\frac{1}{2} (1-\rho)^2+\frac{1 }{2} (B_3-(1-\rho))^2.\end{aligned}$$ Multiplying by $2\eta_\eps(\rho)$ and using that $0\le\eta_\eps\le 1$ we obtain for $W_\eps(\rho)$ the estimate $$\label{wer} W_\eps(\rho) =\eta_\eps{(\rho)} (1-\rho)^2\le (B_3-(1-\rho))^2+2\eta_\eps{(\rho)} (1-\rho)B_3.$$ Let $\psi_\eps(s):=2\frac{\eta_\eps(s)}{s}(1-s)= 2\min \{ \frac{1}{\eps}, \frac{1}{s}\} (1-s) $ then $\psi_\eps$ is bounded by $1/\eps$ and is Lipschitz continuous in $s^{1/2} $ with constant of order $\eps^{-3/2}$ i.e. $\sup_t |(\psi_\eps(t^2))'|\les \eps^{-3/2}$. Since $2\eta_\eps {(\rho)} (1-\rho)B_3= \rho B_3 \psi_\eps(\rho)$, using with $\psi=\psi_\eps{(\rho)} $, we get $$\begin{aligned} \left| \int_{Q_{1,1}} 2\eta_\eps {(\rho)} (1-\rho)B_3 dx\right|&\les \frac{\a^{1/3}\b^{2/3}}{T}\lt(\eps^{-1} E+ \a^{-1/3}\b^{-1/6} E^{1/2} \|{\nabla}' (\psi_\eps(\rho))\|_{L^2} \rt) \\ &\les \frac{\a^{1/3}\b^{2/3}}{T}\lt(\eps^{-1}E+ \eps^{-3/2} \a^{-1/3}\b^{-1/6} E^{1/2} \|{\nabla}'\rho^{1/2}\|_{L^2} \rt) \\ &\les C_\eps \frac{\a^{1/3}\b^{2/3}}{T}\lt(E+ \int_{Q_{1,1}} \a^{-2/3}\b^{-1/3} |\nabla' \rho^{1/2}|^2 dx\rt) \\ &\les C_\eps \frac{\a^{1/3}\b^{2/3}}{T} E,\end{aligned}$$ where we used that $|\nabla' \rho^{1/2}|\le |\nabla'_{\a^{1/3}\b^{1/3}TA} u|$ and thus $\int_{Q_{1,1}} \a^{-2/3}\b^{-1/3} |\nabla' \rho^{1/2}|^2 dx\le E$. Estimate follows from inserting this estimate into . To prove the lower bound, we will need the following two dimensional result. \[lowerbound2d\] Let $\chi_n\in BV(Q_1, \{0,\b_n^{-1}\})$ be such that $\lim_{n\to +\infty}\int_{Q_1}\chi_n dx'=1$ and $$\sup_{n} \int_{Q_1} \b_n^{1/2} |D' \chi_n| <+\infty.$$ Then, up to a subsequence, $\chi_n\weaklim \sum_i {\varphi}_i \delta_{X_i}$ for some at most countable family of ${\varphi}_i> 0$ and $X_i \in Q_1$, and $$\label{liminfperibeta} \liminf_{n\to +\infty} \int_{Q_1} \b_n^{1/2} |D' \chi_n|\ge 2\sqrt{\pi}\sum_i \sqrt{{\varphi}_i}.$$ [[*Step 1 (Compactness):*]{}]{} For each $n$ we split the cube $Q_1$ into small cubes of side length $3\b_n^{1/2}$. Let $Q_i^{n}$ be an enumeration of theses cubes such that $$\varphi_i^{n} :=\int_{Q_i^{n}} \chi_n dx'$$ is nonincreasing in $i$. Since $|\{\chi_n=\b_n^{-1}\}|=\b_n \int_{Q_1} \chi_n dx'=\b_n+o({\beta_n})$, we have $|{Q_i^{n}}\cap\{\chi_n=\b_n^{-1}\}|\le\b_n+o(\b_n)\le \frac{1}{2}|{Q_i^{n}}|$ and thus by the relative isoperimetric inequality [@AFP Th. 3.46], we have on each ${Q_i^{n}}$ $$\varphi_i^{n}=\int_{{Q_i^{n}}} \chi_n dx'\les\left( \int_{{Q_i^{n}}} \b_n^{1/2} |D' \chi_n|\right)^2\,.$$ It follows that $$\sum_i \sqrt{\varphi_i^{n}} \les \int_{Q_1} \b_n^{1/2}|D' \chi_n| \le C,$$ by the energy bound. Arguing as in the proof of Proposition \[existmu\], we deduce from Lemma \[lemsqrt\] that up to extracting a subsequence, $\chi_n \weaklim \sum_{i\in {\mathcal{I}}}{\varphi}_i \delta_{X_i}$ for some ${\varphi}_i> 0$ and $X_i\in Q_1$, pairwise distinct. [[*Step 2 (Lower bound):*]{}]{} Assume now that the ${\varphi}_i$ are labeled in a decreasing order and fix $N\in \N$. Choose $r\in(0,1/4)$ sufficiently small so that $$\label{bxr} \B'(X_i,r)\cap \B'(X_j,r)=\emptyset \qquad \qquad \forall i,j \le N \text{ with } i\neq j,$$ and let $\psi\in C^\infty_c(\B'(X_i, r);[0,1])$ be a smooth function such that $\psi=1$ in $\B'(X_i, r/2)$ and $|{\nabla}\psi|\le C/r$. Let $C_\mathrm{iso}=(2\sqrt{\pi})^{-1}$ be the isoperimetric constant in dimension 2, then we may write $$\begin{aligned} \nonumber \left(\int_{\B'(X_i, r/2) } \b_n^{-1} \chi_n dx'\right)^\hal & \le &\left( \int_{Q_1} ( \psi \chi_n)^2 dx'\right)^{1/2} \le C_\mathrm{iso} \int_{Q_1} |D' (\psi \chi_n)|\\ \nonumber & \le & C_\mathrm{iso} \left( \int_{\B'(X_i, r) } |D' \chi_n| + \frac{C}{r} \int_{\B'(X_i,r)\backslash \B'(X_i, r/2)} \chi_n dx'\right)\\ \nonumber & \le & C_\mathrm{iso} \int_{\B'(X_i, r) } |D' \chi_n| + \frac{C}{r} \end{aligned}$$ since $\int_{Q_1} \chi_n dx'\to 1$. Multiplying by $\beta_n^{1/2}$ and summing over $N$, we get $$\label{estimisop} \sum_{i=1}^N \left(\int_{\B'(X_i, r/2) } \chi_n dx'\right)^{1/2} \le C_\mathrm{iso} \int_{Q_1} \beta_n^{1/2}|D' \chi_n| + \beta_n^{1/2}\frac{CN}{r}.$$ Next observe that since $\chi_n\rightharpoonup \sum_{i} {\varphi}_i \delta_{X_i}$, we have for every $i=1,\dots,N$, $$\liminf_{n\to +\infty}\lt(\int_{\B'(X_i, r/2)} \chi_n dx'\right)^{1/2}= {\varphi}_i^{1/2} \,.$$ Therefore, passing to the limit in , we obtain $$\sum_{i=1}^N {\varphi}^{1/2}_i\le C_\mathrm{iso} \liminf_{n \to +\infty} \int_{Q_1} \beta_n^{1/2}|D' \chi_n|.$$ Since $N$ was arbitrary this implies . With this lemma at hand, we can prove the compactness and lower bound result. We fix for the proof a sequence $(u_n,A_n)$ with $\wE(u_n,A_n)\les 1$. We then let $B_n:=(B'_n,B_3^n):=\nabla \times A_n$ and $\rho_n:=|u_n|^2$.\ [[*Step 1 (Compactness):*]{}]{} Notice first that $$\label{bounddoublewell} \int_{Q_{1,1}} \lt( \frac{B^n_3-(1-\rho_n)} {\beta_n} \rt)^2 dx\le \a_n^{-2/3} \b_n^{-4/3} \wE(u_n,A_n) \to0\,.$$ By [@CoOtSer Lem. 3.7] there is $\hat u_n$ with $\hat\rho_n:=|\hat u_n|^2=\min\{\rho_n,1\}$ such that $\wE(\hat u_n,A_n)\le (1+2\alpha_n/T_n)\wE(u_n,A_n)$ (the error comes from the last two terms in (\[eqEGLE\])). In particular, also $ (B^n_3-(1-\hat\rho_n))/\beta_n\to0$ in $L^2(Q_{1,1})$. Using this and $|B^n_3|\le |B^n_3-(1-\hat\rho_n)|$ on $\{ B^n_3\le 0\}$ we obtain $$\label{convneg}\lt|\int_{\{B^n_3<0\}} \b_n^{-1} B^n_3 dx\rt|\les \lt( \b_n^{-2} \int_{Q_{1,1}} (B^n_3-(1-\hat\rho_n))^2 dx\rt)^\hal \le \frac{1}{(\alpha_n\beta_n^2)^{2/3} } \wE(u_n,A_n) \to 0,$$ and since $\int_{Q_{1,1}} \b_n^{-1}B^n_3 dx=2$ the sequence $\b_n^{-1}B^n_3$ is bounded in $L^1$ and, after extracting a subsequence, $\b_n^{-1} B^n_3 \weaklim \mu$ for some measure $\mu$. From we also get $\b_n^{-1}(1-\rho_n)\weaklim \mu$, and the same for $\hat\rho_n$. It also follows from that $$\int_{Q_{1,1}} \b_n^{-1} (1- \hat\rho_n) dx= 2+\left|\beta_n^{-1} \int_{Q_{1,1}} \left(B_3^n-(1-\hat \rho_n) \right) dx\right|\to 2.$$ Moreover, since $\int_{Q_{1,1}} \b_n^{-1}(1-\hat\rho_n) |B_n'|^2 dx\le\int_{Q_{1,1}} \b_n^{-1} |B'_n|^2 dx\le \wE(u_n,A_n)$ it holds $$\int_{Q_{1,1}} \b_n^{-1}(1-\hat\rho_n) | B_n'| dx\le \lt(\int_{Q_{1,1}} \b_n^{-1}(1-\hat\rho_n) |B_n'|^2 dx\rt)^\hal \lt(\int_{Q_{1,1}} \b_n^{-1}(1-\hat\rho_n) dx\rt)^{\hal}\les 1,$$ thus (up to a subsequence) $\b_n^{-1}(1-\hat\rho_n) B_n'\weaklim m$ for some vector-valued measure $m$. By [@AFP Th. 2.34], $$\label{liminftrans} \liminf_{n\to +\infty} \int_{Q_{1,1}} \b_n^{-1}|B_n'|^2 dx\ge\liminf_{n\to +\infty} \int_{Q_{1,1}} \b_n^{-1}(1-\hat\rho_n)|B_n'|^2 dx\ge \int_{Q_{1,1}} \lt(\frac{dm}{d{\mu}}\rt)^2 d\mu,$$ and $m\ll \mu$. Moreover, from Lemma \[lemmameissner\] we have that $\b_n^{-1}\hat\rho_n B'_n\weaklim 0$ in a distributional sense and therefore $\b^{-1}_n B_n'$ itself converges to $m$. Letting $n\to +\infty$ in $$\b_n^{-1} \Div B_n=\partial_3 \left[\b_n^{-1} B^n_3\right]+\Divp \lt[\b_n^{-1}B_n'\rt]=0,$$ we obtain $\partial_3 \mu +\Divp m=0$. This proves (i).\ We now prove that $\mu=\mu_{x_3}\otimes dx_3$, that $\beta^{-1}_nB^n_3(\cdot,x_3)\weaklim \mu_{x_3}$ for a.e. $x_3\in(-1,1)$ and that $\mu_{x_3}\weaklim dx'$ as $x_3\to \pm 1$. By we have that, up to a subsequence in $n$, for a.e. $x_3\in (-1,1)$, $$\lim_{n\to +\infty} \int_{Q_1\cap \{B^n_3<0\}} \beta^{-1}_n B^n_3 dx=0.$$ Let $\mathcal{G}{\subseteq}(-1,1)$ be the set of $x_3$ for which this hold. For every $x_3\in \mathcal{G}$, the $L^1$ norm of $\beta^{-1}_nB^n_3(\cdot,x_3)$ is bounded thus if we fix a countable dense set $\mathcal{G}_d{\subseteq}\mathcal{G}$, we can assume up to extraction that for every $x_3\in \mathcal{G}_d$, $\beta^{-1}_n B^n_3(\cdot,x_3)\weaklim \nu_{x_3}$ for some probability measure $\nu_{x_3}$. For $x_3,{\widetilde}{x}_3\in \mathcal{G}_d$, thanks to the weak lower semi-continuity of $\|\cdot \|_{BL}$ and , $$\begin{aligned} \|\nu_{x_3}-\nu_{{\widetilde}{x}_3}\|_{BL}&\le \limsup_{n\to +\infty} \|\beta_n^{-1} B_3^n (\cdot, x_3) - \beta_n^{-1} B_3^n(\cdot, {\widetilde}{x}_3)\|_{BL} \\ &\les \lim_{n\to +\infty}\lt( |x_3-{\widetilde}{x}_3|^{1/2} +\sigma(\alpha_n,\beta_n,T_n)\rt) \\ &= |x_3-{\widetilde}{x}_3|^{1/2}. \end{aligned}$$ Therefore, there exists a unique Hölder-continuous extension of $\nu_{x_3}$ to $(-1,1) \ni x_3 \mapsto \nu_{x_3} \in \mathcal{P}(Q_1)$. We claim that $\mu=\nu_{x_3}\otimes dx_3$. For $K\to +\infty$, let $\{z^K_j\}_{j=1}^{K+1} \in \mathcal{G}_d$ be an increasing sequence such that $|z^K_j-z^K_{j+1}|\les K^{-1}$, $|z_1^K+1|\les K^{-1}$ and $|z_{K+1}^K-1|\les K^{-1}$. Notice that for $n$ large enough, we have for every $j$ that $|z^K_j-z^K_{j+1}|\ge \sigma^{1/2}(\a_n,\b_n,T_n)$ (where $\sigma$ is defined in Lemma \[HolderE\]) so that applies. Let $\psi$ be a $Q_1-$periodic and Lipschitz continuous function on $Q_{1,1}$ with $\|\psi\|_{Lip}\le 1$. By the continuity of $x_3 \mapsto \nu_{x_3}$, we have $$\begin{aligned} \int_{Q_{1,1}} \psi (d\mu- d\nu_{x_3}\otimes dx_3)&=\lim_{K\to +\infty} \int_{Q_{1,1}} \psi d\mu-\sum_{j=1}^K (z^K_{j+1}-z^K_j) \int_{Q_1} \psi(\cdot ,z_j^K) d\nu_{z_j^K}\\ &=\lim_{K\to +\infty} \lim_{n\to +\infty} \sum_{j=1}^K \int_{z_j^K}^{z_{j+1}^K} \int_{Q_1} \psi(x',x_3) \beta_n^{-1} B^n_3(x',x_3)\\ &\qquad \qquad \qquad \qquad \qquad -\psi(x',z_j^K)\beta_n^{-1} B^n_3(x',z_j^K) dx' dx_3.\end{aligned}$$ Using the finite difference version of Leibniz’ rule, $$\begin{gathered} \psi(x',x_3) B^n_3(x',x_3)-\psi(x',z_j^K) B^n_3(x',z_j^K)\\ = B^n_3(x',x_3)(\psi(x',x_3)-\psi(x', z_j^K))+ \psi(x',z_j^K)( B^n_3(x',x_3) - B^n_3(x',z_j^K))\end{gathered}$$ and using that $\|\psi\|_{Lip}\le 1$, we can estimate for fixed $K,j$ and $n$ large enough, $$\begin{aligned} \lt| \int_{z_j^K}^{z_{j+1}^K} \rt.& \lt.\int_{Q_1} \psi(x',x_3) \beta_n^{-1} B^n_3(x',x_3)-\psi(x',z_j^K)\beta_n^{-1} B^n_3(x',z_j^K) dx' dx_3\rt|\\ &\le \int_{z_j^K}^{z_{j+1}^K} \int_{Q_1} \beta_n^{-1} |B^n_3| |x_3-z^K_j| dx + \int_{z_j^K}^{z_{j+1}^K} \|\beta^{-1}_n( B^n_3(x',x_3) - B^n_3(x',z_j^K))\|_{BL}\\ &\les K^{-1} \int_{z_j^K}^{z_{j+1}^K} \int_{Q_1} \beta_n^{-1} |B^n_3|dx +K^{-1} K^{-1/2},\end{aligned}$$ where in the last line we have used that $|x_3-z^K_j|\les K^{-1}$ and . Summing this estimate over $j$, we obtain $$\left|\int_{Q_{1,1}} \psi (d\mu- d\nu_{x_3}\otimes dx_3)\right| \les \lim_{K\to +\infty} K^{-1}\lt[\lim_{n\to +\infty} \int_{Q_{1,1}}\beta_n^{-1} |B^n_3| dx + K^{-1/2}\rt]=0.$$ This establishes that $\mu= \nu_{x_3}\otimes dx_3$. Moreover, this proves that for every $x_3\in \mathcal{G}_d$, the whole sequence $\beta_n^{-1} B^n_3(\cdot,x_3)$ weakly converges to $\mu_{x_3}$. Since the set $\mathcal{G}_d$ was arbitrary, this proves the above convergence for all $x_3\in \mathcal{G}$.\ We finally show that the boundary conditions hold. For this we focus on $x_3=1$. For $x_3\in \mathcal{G}$, it holds by the weak lower semi-continuity of $\|\cdot \|_{BL}$, $$\begin{aligned} \|1-\mu_{x_3}\|_{BL}&\le \liminf_{n\to +\infty} \|1-\beta_n^{-1}B_3^n(\cdot, x_3)\|_{BL}\\ & \le \liminf_{n\to +\infty} \|1-\beta_n^{-1}B_3^n(\cdot, 1)\|_{BL}+\limsup_{n\to +\infty} \|\beta_n^{-1}B_3^n(\cdot, 1)-\beta_n^{-1}B_3^n(\cdot, x_3)\|_{BL}.\end{aligned}$$ By , the second right-hand side term is controlled by $|1-x_3|^{1/2}$. For the first right-hand side term we note that because of $\|\psi\|_{H^{1/2}}{\stackrel{<}{\sim}}\|\psi\|_{Lip}$, we have $$\|\beta_n^{-1}B_3^n (\cdot, 1)- 1\|_{BL}^2 {\stackrel{<}{\sim}}\|\beta_n^{-1} B_3^n (\cdot, 1)-1\|_{H^{1/2}}^2 \le \alpha_n^{-1/3}\beta_n^{-7/6} \wE(\hat u_n, A_n).$$ Hence, it is the last assumption in that ensures that this term vanishes in the limit $n\to +\infty$. We thus obtain the desired estimate $$\| 1- \mu_{x_3}\|_{BL}{\stackrel{<}{\sim}}|1-x_3|^{1/2}.$$ [[*Step 2 (Lower bound and structure of $\mu$):*]{}]{} The starting point is an application of the usual Modica-Mortola trick. In this step we only deal with $\hat u_n$ and $\hat\rho_n$, and drop the hats for brevity. By and $|\nabla' \rho^{1/2}|\le |\nabla'_{\lambda A} u|$ we obtain from the Cauchy-Schwarz inequality : $$\label{estimMM} \left(1 + C_\eps \frac{\alpha_n}{T_n}\right)\wE(u_n,A_n)\ge \int_{Q_{1,1}} \b_n^{-1/2}2\sqrt{W_\eps(\rho_n)} |\nabla' \rho_n^{1/2}| +\b_n^{-1}|B_n'|^2 dx,$$ for any $\eps>0$. We momentarily fix a small $\delta>0$ and estimate by the co-area formula , $$\begin{aligned} \int_{Q_{1,1}} 2\sqrt{W_\eps(\rho_n)}|\nabla' \rho_n^{1/2}| dx&\ge \int_{\delta}^{1-\delta} \int_{-1}^1 2\sqrt{W_\eps(s^2)} \H^1(\partial \{\rho_n(\cdot,x_3)>s^2\}) dx_3 ds.\end{aligned}$$ In particular there exists $s_n\in[\delta,1-\delta]$ depending on $n$ such that $$\begin{gathered} \int_{\delta}^{1-\delta} 2\sqrt{W_\eps(s^2)} \int_{-1}^1 \H^1(\partial \{\rho_n(\cdot,x_3)>s^2\}) dx_3 ds\\ \ge \lt(\int_{\delta}^{1-\delta} 2\sqrt{W_\eps(s^2)} ds\rt)\int_{-1}^1 \H^1(\partial \{\rho_n(\cdot,x_3)>s^2_n\}) dx_3.\end{gathered}$$ Letting $\chi_n(x',x_3):=\b_n^{-1}(1-\chi_{\{\rho_n(\cdot,x_3)>s_n^2\}}(x'))$ this reads $$\label{boundBV} \int_{Q_{1,1}} \b_n^{-1/2}2\sqrt{W_\eps(\rho_n)}|\nabla' \rho_n^{1/2}| dx\ge C_{\delta,\eps}\int_{Q_{1,1}} \b_n^{1/2} |D' \chi_n| dx_3,$$ where $C_{\delta,\eps}:=\int_{\delta}^{1-\delta} 2\sqrt{W_\eps(s^2)} ds$. Let $\gamma_n\to 0$ to be chosen later. For $n$ large enough, if $\rho_n\le \gamma_n$, then $\chi_n=\beta_n^{-1}$ while if $\rho_n\ge 1-\gamma_n$, $\chi_n=0$ so that $$\begin{aligned} \int_{Q_{1,1}} |\chi_n-\beta_n^{-1}(1-\rho_n)| dx&\le\beta_n^{-1} \int_{\{\rho_n\le \gamma_n\}} \rho_n dx +\beta_n^{-1} \int_{\{\rho_n\ge 1-\gamma_n\}} (1-\rho_n) dx \\ & \qquad +\beta_n^{-1}|\{\gamma_n<\rho_n<1-\gamma_n\}| \\ &\le 2\beta_n^{-1}\gamma_n +\beta_n^{-1}|\{\gamma_n<\rho_n<1-\gamma_n\}|.\end{aligned}$$ By definition of $W_\eps$ (recall , $\min_{[\gamma_n,1-\gamma_n]} W_\eps=\min(\frac{\gamma_n}{\eps}, \gamma_n^2)=\gamma_n^2$ so that using that $\int_{Q_{1,1}} \alpha_n^{2/3}\beta_n^{-2/3} W_\eps(\rho_n) dx\les 1$, $$\beta_n^{-1}|\{\gamma_n<\rho_n<1-\gamma_n\}|\le \beta_n^{-1} \gamma_n^{-2} \int_{Q_{1,1}} W_\eps(\rho_n) dx \les \beta^{-1/3}_n\gamma_n^{-2}\alpha_n^{-2/3}.$$ Therefore, if we choose $\gamma_n$ such that $\beta_n\gg \gamma_n\gg \alpha_n^{-1/3}\beta_n^{-1/6}$, which is possible since by hypothesis $\alpha_n \beta_n^{7/2}\to +\infty$, we obtain that $$\lim_{n\to +\infty}\int_{Q_{1,1}} |\chi_n-\beta_n^{-1}(1-\rho_n)| dx=0.$$ Combining this with , we obtain that $$\lim_{n\to +\infty} \int_{Q_{1,1}}|\chi_n-\b_n^{-1}B_3^n| dx=0.$$ By Fubini, this implies that, after passing to a subsequence in $n$, for a.e. $x_3\in (-1,1)$, if $ \b_n^{-1}B_3^n(\cdot,x_3)\weaklim \mu_{x_3}$ then also $\chi_n(\cdot,x_3)\weaklim \mu_{x_3}$. Moreover, from $\int_{Q_1} \b_n^{-1} B^n_3(x',x_3) dx'=1$ for a.e. $x_3\in(-1,1)$, we obtain that $\lim_{n\to +\infty}\int_{Q_1} \chi_n(x',x_3) dx'=1$ for a.e. $x_3\in(-1,1)$. We thus can use Lemma \[lowerbound2d\] to prove that $\mu_{x_3}=\sum_i \p_i \delta_{X_i}$ for some $\p_i>0$ and $$\label{lowerboundperim} \liminf_{n\to +\infty} \int_{Q_{1,1}} \b_n^{1/2} |D'\chi_n| dx_3 \ge 2\sqrt{\pi} \int_{-1}^1 \sum_i \sqrt{\p_i} dx_3\, ,$$ where we used Fatou lemma. This shows (iii). Putting , , and together we find $$\liminf_{n\to +\infty} \wE(u_n,A_n)\ge \int_{-1}^1 2\sqrt{\pi} C_{\delta,\eps} \sum_i \sqrt{{\varphi}_i} dx_3 +\int_{Q_{1,1}} \lt(\frac{dm}{d\mu}\rt)^2 d\mu,$$ for any $\eps$ and $\delta$. Since $$\lim_{\eps\to0}\lim_{\delta\to0}C_{\delta,\eps}= 2\int_0^1 (1-t^2) dt=\frac{4}{3},$$ and $K_*=\frac{8\sqrt{\pi}}{3}$, this concludes the proof of (iv). Upper bound {#Sec:upperbound} =========== In this section we construct a recovery sequence for any sequences $T_n, \alpha_n, \beta_n$ which obey (\[eqlbassumptcoeff\]) and additionally the condition of quantization of the total flux $$\label{defKtotal} L_n^2 T_n\a_n \b_n\in 2\pi \N\,,$$ where $L_n:={\widetilde}L \a^{-1/3}_n \b_n^{-1/6}$. Recalling the form of the gradient term in the functional $\wE$ defined in (\[eqdefwe\]), to discuss quantization of the flux of individual domains it is convenient to introduce $$\label{defK} k_n:=\alpha_n^{1/3}\beta_n^{2/3}T_n\,.$$ The global flux quantization (\[defKtotal\]) then reads $k_n {\widetilde}L^ 2\in 2\pi\N$ and, in the ${\widetilde}L=1$ case we are considering here, simplifies to $k_n\in 2\pi \N$. Condition (\[eqlbassumptcoeff\]) implies $k_n\to +\infty$ and in particular $k_n/\beta_n\to+\infty$, so that the quantization condition becomes less and less stringent with increasing $n$. Aim of this section is to prove the following: \[gammalimsup-v2\] Assume ${\widetilde}L=1$, (\[eqlbassumptcoeff\]) and (\[defKtotal\]). Then, for every $\mu$ with $I(\mu)<+\infty$ and $\mu_1=\mu_{-1}=dx'$ there exist sequences $u_n: Q_{1,1}\to {\mathbb{C}}$ and $A_n: Q_{1,1}\to \R^3$ such that $$\limsup_{n\to +\infty} \, \wE(u_n,A_n)\le I(\mu).$$ The fields $\rho_n:=|u_n|^2$ and $B_n:= \nabla \times A_n$ are $Q_1$-periodic and it holds $$\b_n^{-1}(1-\rho_n)\weaklim \mu \qquad \textrm{and } \qquad \b_n^{-1} B_n'\weaklim m,$$ where $m$ is the measure such that $I(\mu)=I({\mu},m)$. The idea of the construction is to use the density result Theorem \[theodens\] to separate the construction in two regions. In the bulk, the measure will be approximated by a finite polygonal measure for which the construction is made in Section \[secconstrbulk\]. In the boundary layer, we plug in the construction of [@CoOtSer], see Section \[secconstrboundary\], which is optimal up to a factor. Since the energy in the boundary layer is small, its suboptimal effect disappears in the limit. We shall first construct the density $\rho$ and the magnetic field $B$. The appropriate energy is ${\widetilde}F_{\alpha,\beta}:={\widetilde}F_{\alpha,\beta}^{(-T,T)}+{\widetilde}F_{\alpha,\beta}^{\Ext}$, where $$\begin{gathered} \label{eqdeftildaF} {\widetilde}F_{\alpha,\beta}^{(a,b)}(\rho,B):=\int_{Q_1\times(a,b)} \Bigl( \a^{-2/3}\b^{-1/3}\lt|\nabla'\rho^{1/2}\rt|^2+ \a^{-4/3}\b^{-2/3}\lt|\partial_3\rho^{1/2}\rt|^2\\ + \a^{2/3}\b^{-2/3}\left({B_3} -(1-\rho)\right)^2 + \b^{-1}|{B'}|^2 \Bigr)dx\,,\end{gathered}$$ and $$\label{eqdeftildaFext} {\widetilde}F_{\alpha,\beta}^{\Ext}(B):= \alpha^{1/3}\beta^{7/6} \| \beta^{-1}B_3-1\|^2_{H^{-1/2}(Q_1\times \{\pm1\})}\,.$$ Their sum corresponds to the energy $\wE$, up to a reconstruction of $u$ and $A$ that will be discussed in Section \[recoveryGL\]. A pair $(\rho,B)$ is admissible for ${\widetilde}F_{\a,\b}^{(a,b)}$ if $\rho$ and $B$ are $Q_1$-periodic, $\div B=0$ and $\int_{Q_1} B_3(x',x_3)dx'=\beta$ for all $x_3$. We say that a pair $(\rho,B)$ is $k$-quantized if there is a closed set $\omega{\subseteq}Q_{1,1}$ such that $B=0$ outside $\omega$, $\rho=0$ in $\omega$, and the flux of $ \b^{-1}B_3$ over every connected component of $\omega\cap\{x_3=z\}$ is an integer multiple of $2\pi /k$, for all $z\in(-1,1)$. Construction in the bulk {#secconstrbulk} ------------------------ This section is concerned with the local construction of flux tubes. For notational simplicity we present this construction in $\R^2\times(a,b)$, without the periodicity assumption; since $\rho=1$ and $B=0$ outside a small region, its periodic extension is immediate (see proof of Proposition \[propubregular\] below). We start from the optimal profile at the boundary of the individual tubes, with the lengths measured in units of the coherence length (recall ). For the purpose of the upcoming constructions, we cut the profile at a lengthscale $R\gg1$ towards the normal region. \[lemmaVR\] Consider the functional $\displaystyle G(v):=2\sqrt\pi\int_0^{+\infty} |\dot{v}|^2+(1-v^2)^2 dt$. Then, $$\inf\left\{ G(v): v(0)=0, \lim_{t\to+\infty} v(t)=1\right\}=K_*=\frac83\sqrt\pi\,.$$ Furthermore, for all $R\ge3$ there is $v_R\in C^\infty(\R;[0,1])$ with $v_R(t)=0$ for $t\le 0$, $v_R(t)=1$ for $t\ge R$, $|\dot{v}_R|\le 2$, and, setting $K_R:=G(v_R)$, one has $$\lim_{R\to\infty} K_R=\frac83\sqrt\pi,$$ and $\int_0^{+\infty} t \left(|\dot{v}_R|^2+(1-v_R^2)^2\right) dt{\stackrel{<}{\sim}}1$. The lower bound follows from the usual Modica-Mortola type computation $$\int_0^{+\infty} |\dot{v}|^2+(1-v^2)^2dt\ge 2\int_0^{+\infty} (1-v^2)\dot{v}dt=2\int_0^1 (1-s^2) ds = \frac43\,.$$ To prove the upper bound, we recall that $v(t):= \tanh t$ is the minimizer of $G$ under the constraint $v(0)=0$. A direct computation shows that $G(v)=K_*$. We then define for $R>0$, $$\hat v_R(t):= \begin{cases} 0 & \text{ if } t<1/R,\\ \displaystyle\frac{\tanh (t-1/R)}{\tanh (R-2/R)}&\text{ if } t\in[1/R,R-1/R],\\ 1 & \text{ if } t>R-1/R, \end{cases}$$ and $v_R:=\psi_{1/R}\ast \hat v_R$, with $\psi_{1/R}\in C^\infty_c(-1/R,1/R)$ a mollifier. By construction, this has the desired properties and verifies $G(v_R)\to G(v)$ as $R\to\infty$. We start with the simple case in which the limiting measure comes from a Lipschitz curve; in practice this will be used only for affine or piecewise affine curves. \[lemmacurve3\] Let $X: (a,b)\to \R^2$ be a Lipschitz curve, ${\varphi}>0$, $R>0$. We define $$\rho(x):= v_R^2\left(\frac{|x'-X(x_3)|-\sqrt{\beta{\varphi}/\pi}}{\eta}\right),$$ where $\eta:=\alpha^{-2/3}\beta^{1/6}\ll 1$ is the coherence length (see ) and $v_R$ was introduced in Lemma \[lemmaVR\], and define $B$ through $$B_3(x):=\chi_{\B'(X(x_3),\sqrt{\beta {\varphi}/\pi} )}(x')\,, \hskip1cm B'(x):=B_3(x) \dot{X}(x_3) \,.$$ Then, $\rho B=0$ almost everywhere, $\div B=0$, and $${\widetilde}F^{(a,b)}_{\alpha,\beta}(\rho,B)\le \left(1+ \frac{C}{({\varphi}\alpha^{4/3}\beta^{2/3})^{1/2}} + \frac{C}{{\varphi}\alpha^{4/3}\beta^{2/3}}\right) \int_a^b \bigl( K_R\sqrt{{\varphi}}+{\varphi}|\dot{X}|^2\bigr) dx_3.$$ The constant $C$ is universal. Moreover, if $\mu:={\varphi}\delta_{X(x_3)}\otimes dx_3$, $$\label{distestim} W_2^2(\b^{-1} B_3, \mu)\le |b-a|\frac{\beta {\varphi}^2}{2\pi}.$$ The condition $\rho B=0$ follows from $v_R(t)=0$ for $t\le 0$. To check the divergence condition, pick $\psi\in C^1_c(\R^2\times(a,b))$ and compute $$\begin{aligned} \int_{\R^2\times(a,b)} (B_3\partial_3 \psi + B'\cdot \nabla'\psi) dx &=\int_a^b \int_{\B'(X(x_3), \sqrt{\beta{\varphi}/\pi})} ( \partial_3 \psi + \nabla'\psi \cdot \dot{X}(x_3)) dx'dx_3\\ & =\int_a^b \int_{\B'(0, \sqrt{\beta{\varphi}/\pi})} \frac{d}{dx_3} \psi(X(x_3)+y', x_3) dy' dx_3=0\,.\end{aligned}$$ We now estimate the energy. We start from the interfacial energy at fixed $x_3$ (see ), $$\begin{aligned} E_1(x_3):=\int_{\R^2} \alpha^{-2/3}\beta^{-1/3} |\nabla'\rho^{1/2}|^2 + \alpha^{2/3}\beta^{-2/3} (B_3-(1-\rho))^2 dx'\,.\end{aligned}$$ If $|x'-X(x_3)|< \sqrt{\beta{\varphi}/\pi}$ then $\rho=0$ and $B_3=1$, whereas if $|x'-X(x_3)|>\eta R +\sqrt{\beta{\varphi}/\pi}$ then $\rho=1$, $B_3=0$. In the intermediate region, we use $|\nabla'\rho^{1/2}|=|\dot{v}_R|/\eta$. Passing to polar coordinates and using $r=|x'-X(x_3)|$ as an integration variable, $$\begin{aligned} E_1(x_3)= \int_{\sqrt{\beta{\varphi}/\pi}}^{\sqrt{\beta{\varphi}/\pi}+\eta R} \Bigl[ \frac{\alpha^{-2/3}\beta^{-1/3}}{\eta^2} |\dot{v}_R|^2 + \alpha^{2/3}\beta^{-2/3} (1-v_R^2)^2\Bigr] \lt(\frac{r-\sqrt{\beta{\varphi}/\pi}}\eta\rt)2\pi r dr \,.\end{aligned}$$ We change variables according to $r=\sqrt{\beta{\varphi}/\pi}+s\eta$, insert the definition of $\eta$, and by Lemma \[lemmaVR\] obtain $$\begin{aligned} E_1(x_3)&= \int_0^R \beta^{-1/2} \left[|\dot{v}_R|^2+ (1-v_R^2)^2\right]2\pi (\sqrt{\beta{\varphi}/\pi}+s\eta) ds \\ & \le K_R \sqrt{\varphi}+ C \alpha^{-2/3}\beta^{-1/3} \le \sqrt{\varphi}\left(K_R+\frac{C}{{\varphi}^{1/2}\alpha^{2/3}\beta^{1/3}}\right)\,.\end{aligned}$$ The other contributions to the energy are the cost of transport and the vertical part of the gradient, $$\begin{aligned} E_2(x_3):=\int_{\R^2} \alpha^{-4/3}\beta^{-2/3} |\partial_3\rho^{1/2}|^2 + \beta^{-1}|B'|^2 dx'\,.\end{aligned}$$ By definition of $B'$, we have for the second term, $$\begin{aligned} \int_{\R^2} \beta^{-1}|B'|^2 dx'= |\dot{X}(x_3)|^2 {\varphi}\,.\end{aligned}$$ For the first one we use $|\partial_3\rho^{1/2}|\le |\dot{v}_R||\dot{X}|/\eta$ and change variables as above to obtain $$\begin{aligned} \int_{\R^2} \alpha^{-4/3}\beta^{-2/3} |\partial_3\rho^{1/2}|^2 dx' &{\stackrel{<}{\sim}}\frac{ \alpha^{-4/3}\beta^{-2/3}}{\eta} |\dot{X}(x_3)|^2 \int_0^R ( \sqrt{\beta{\varphi}/\pi}+s\eta)|v_R'|^2 ds\\ &{\stackrel{<}{\sim}}\alpha^{-4/3}\beta^{-2/3} |\dot{X}(x_3)|^2 (1 + \sqrt{\beta{\varphi}}/\eta)\\ &= |\dot{X}(x_3)|^2 {\varphi}\left( \frac{1}{{\varphi}\alpha^{4/3}\beta^{2/3}}+ \frac{1}{{\varphi}^{1/2}\alpha^{2/3}\beta^{1/3}} \right) \,.\end{aligned}$$ To prove we consider the transport map $T(x',x_3)=(X(x_3),x_3)$ which gives $$W_2^2(\b^{-1}B_3,\mu)\le \int_a^b \b^{-1} \int_{\B'(X(x_3),\sqrt{\b {\varphi}/\pi})} |x'-X(x_3)|^2 dx'dx_3=|b-a| \frac{\b {\varphi}^2}{2\pi}.$$ We now turn to the construction around branching points. Since the total length around branching points is small, the construction here does not need to achieve the optimal constant but only the optimal scaling. The idea of the construction is the following. We first transform disks into squares, then split the square into two rectangles and then retransform each rectangle into a disk. The construction is sketched in Figure \[fig1\]. We start with the transformation from a rectangle to a disk. ![Construction around a branching point[]{data-label="fig1"}](figsep2-crop){width="6cm"} \[lemmacurveendpoints\] Let $a\le b\in \R$, $\gamma>0$, $X\in \R^2$, ${\varphi}>0$, and $R\ge 3$. Let then $Q {\subseteq}\R^2$ be a rectangle with side lengths $w$ and $h$ centered in $X$, such that $wh=\beta{\varphi}$, $w/h+h/w\le \gamma$ and $\a^{-4/3}\b^{-2/3}\le {\varphi}$. Let as before $\eta:=\alpha^{-2/3}\beta^{1/6}$ be the coherence length. Then there are $\rho\in L^\infty(\R^2\times [a,b];[0,1])$ and $B\in L^2(\R^2\times [a,b];\R^3)$ such that $\div B=0$, $\rho B=0$, with $B=0$ and $\rho=1$ on $(\R^2\setminus \B'(X,r))\times[a,b]$ for some $r\sim \eta R + \sqrt{\beta{\varphi}}$, and $${\widetilde}F^{(a,b)}_{\alpha,\beta}(\rho, B) {\stackrel{<}{\sim}}\sqrt{\varphi} |a-b| R^2+ \frac{r^2}{|a-b|}\varphi R^2,$$ where the implicit constants only depend on $\gamma$. Further, $\rho$ and $B$ satisfy the boundary conditions $$\rho(x',a)= v_R^2\left(\frac{|x'-X|-\sqrt{\beta{\varphi}/\pi}}{\eta}\right), \hskip1cm \rho(x',b)= \min\left\{1,\frac{\dist^2(x', Q)}{\eta^2}\right\}\,,$$ and $$B_3(x',a)= \chi_{\B'(X, \sqrt{\beta{\varphi}/\pi})}(x')\,, \hskip3mm\text{ }\hskip3mm B_3(x',b)= \chi_{Q}(x')\,.$$ We note that our assumptions on the parameters $w, h, {\varphi},$ just mean $w\sim h\sim \sqrt{\beta {\varphi}}\ge \eta$. That is, the thread diameter is large compared to the coherence length. Note that $r\sim \eta R+\sqrt{\beta {\varphi}}$ behaves like the maximum between the thread diameter $w\sim h$ and the cut-off scale $\eta R$. The proof of Lemma \[lemmacurveendpoints\] is based on an explicit construction for a bilipschitz bijection with unit determinant that transforms a rectangle into a circle, which we first present. ![The construction in Lemma \[lemmasquarecircle\] transforms a rectangle into a circle, keeping $\det \nabla'u=1$. Each curve corresponds to a different value of $x_3$, and plots the solution of $\hat r(r,\theta)=t$, which corresponds to $r(\theta)=t/[\lambda(x_3)\cos(\theta x_3)]$ for $\theta\in (-\pi/4,\pi/4)$.[]{data-label="fig-square"}](fig-square-crop){width="5cm"} \[lemmasquarecircle\] Assume that $z_-,z_+,h,w>0$, $X\in\R^2$ are given, $z_-< z_+$. Then there is $u:\R^2\times[z_-,z_+]\to\R^2$ such that $$u(x',z_-)=x'\,,\hskip5mm u(X+(-\frac12w,\frac12w)\times(-\frac12h,\frac12h),z_+)=\B'(X,\sqrt{hw/\pi})\,,\hskip5mm \det \nabla' u =1 \text{ a.e.}\,.$$ The function $x\mapsto (u(x),x_3)$ is bilipschitz, its inverse is of the form $y\mapsto (U(y),y_3)$, and $u(X,x_3)=X$ for all $x_3$. If additionally $h/w+w/h\le \gamma$, then the bounds $|\nabla'u|+|\nabla'U|{\stackrel{<}{\sim}}1$, $|\partial_3 u|+|\partial_3 U|{\stackrel{<}{\sim}}h/|z_+-z_-|$ hold, with constants which only depend on $\gamma$. By scaling and translation we may assume that $hw=\pi$, $z_+=1$, $z_-=0$, $X=0$. We can further assume $h=w=\sqrt\pi$, as the general case is obtained by taking the composition at each $x_3$ with the linear map ${\mathop{\mathrm {diag}}}(g(x_3), 1/g(x_3))$, where $g(x_3):=\sqrt{h/w}(1-x_3)+x_3$. We work in polar coordinates, and construct functions $\hat r$, $\hat\theta$ of $r$, $\theta$ and $x_3$ such that $$u(r\cos\theta,r\sin\theta,x_3)=(\hat r\cos\hat\theta, \hat r\sin\hat\theta),$$ with $r\ge 0$ and $0\le\theta\le \pi/4$, and then extend by symmetry. We set $$\hat \theta = f(\theta, x_3) \,,\hskip5mm \hat r = r \lambda(x_3) \cos(x_3\theta),$$ where $\lambda$ and $f$ are two functions still to be determined (see Figure \[fig-square\]). The extension of the $\frac{1}{8}-$sectors by reflection is feasible provided that $f(0,x_3)=0$ and $f(\pi/4,x_3)=\pi/4$ for all $x_3$, the boundary data are attained provided that $f(\theta,0)=\theta$, $\lambda(0)=1$, $\lambda(1)=2/\sqrt\pi$. The latter ensures that indeed the straight segment $r \cos \theta=\frac{w}{2}=\frac{\sqrt{\pi}}{2}$ is mapped into the unit circle $\hat{r}=1$. The determinant condition is equivalent to $$1=\frac{\hat r}{r}\partial_r \hat{r} \partial_\theta f= \lambda^2(x_3)\cos^2(x_3\theta) \partial_\theta f(\theta, x_3),$$ which can be solved (using $f(0,x_3)=0$) to give $$f(\theta,x_3):=\frac{1}{x_3\lambda^2(x_3)}\tan (\theta x_3)\,,$$ smoothly extended to $f(\theta,0)=\theta/\lambda^2(0)$. The condition $f(\pi/4,x_3)=\pi/4$ determines $\lambda$, $$\lambda(x_3):=\left(\frac{\tan (\pi x_3/4)}{\pi x_3/4}\right)^{1/2},$$ which obeys $\lambda(1)=2/\sqrt\pi$ and smoothly extends to $\lambda(0)=1$. Clearly $\lambda\sim 1$ so that $\partial_\theta f\sim 1$. This implies that the change of variables defines a smooth deformation of the $\frac{1}{8}-$sector which smoothly depends on $x_3\in[0,1]$. After a rotation and a translation, we may assume that $X=0$, $a=0$ and $b>0$; so that $Q=(-w/2,w/2)\times (-h/2,h/2)$. We treat two regions separately. In the lower region $\R^2\times [0,b/2]$, we interpolate between $\B'_0:=\B'(0, \sqrt{\beta{\varphi}/\pi})$ and $Q$. To do this, let $u$ and $U$ be the functions from Lemma \[lemmasquarecircle\], using it for the rectangle $Q$ and $z_-=0$, $z_+=b/2$. The magnetic field is defined by $$B_3(x):=\chi_{\B'_0}(u(x))\,,\hskip1cm B'(x):=\chi_{\B'_0}(u(x))\partial_3 U(u(x),x_3)\hskip3mm \text{ for } x_3\in[0,b/2]\,.$$ The density is defined by $$\rho(x):=v_R^2\left(\frac{|u(x)|-\sqrt{\beta{\varphi}/\pi}}{\eta}\right) \hskip3mm \text{ for } x_3\in[0,b/2]\,.$$ The condition $\rho B=0$ follows immediately, as well as the boundary data at $x_3=0$. Since $B=0$ and $\rho=1$ whenever $|u(x)|\ge \eta R + \sqrt{\beta{\varphi}/\pi}$ and since $|u(x)|\ge |x'|/\|\nabla'U\|_\infty$, we have $B=0$ and $\rho=1$ whenever $|x'| \ge \|\nabla'U\|_\infty (\eta R + \sqrt{\beta{\varphi}})$. In order to check $\div B=0$, we fix $\psi\in C^1_c(\B'_r\times(0,b/2))$. Performing then a change of variables at each $x_3$ gives, since $\det \nabla'U=1$, $$\begin{aligned} 1 \int_{\R^2\times(0,b/2)} \left[\partial_3 \psi B_3 + \nabla'\psi\cdot B' \right]dx &= \int_{\R^2\times(0,b/2)} (\partial_3\psi B_3 + \nabla' \psi \cdot B')(U(y),y_3) dy \\ &= \int_{\R^2\times(0,b/2)} \chi_{\B'_0} \left[\partial_3 \psi( U(y),y_3) + \partial_3 U(y)\cdot \nabla'\psi( U(y),y_3)) \right] dy \\ &= \int_{\B'_0} \int_{(0,b/2)} \frac{d}{dx_3} (\psi(U(y),y_3)) dy_3 dy'=0\,.\end{aligned}$$ By the properties of $u$ we obtain $B_3(x',b/2)=\chi_Q(x')$. In the upper region $\R^2\times[b/2,b]$, we keep $B(x',x_3):=e_3 \chi_Q(x')$ concentrated on $Q$ and linearly interpolate the profile between $\rho^{1/2}(x',b/2)$ and the profile $\rho^{1/2}(x',b)$ given in the statement: $$\rho^{1/2}(x):=\frac{2}{b}\lt[ \rho^{1/2}(x',b/2) (b-x_3)+ \rho^{1/2}(x',b)(x_3-b/2)\rt] \hskip3mm \text{ for } x_3\in(b/2,b)\,.$$ It is immediate to check that $B_3$ and $\rho$ match continuously at all interfaces, $\rho B=0$ and $\div B=0$. This concludes the construction. We estimate the energy similarly to the proof of Lemma \[lemmacurve3\]. Lemma \[lemmasquarecircle\] gives $|\nabla'u|{\stackrel{<}{\sim}}1$, $|\partial_3 u|{\stackrel{<}{\sim}}h/b$, $|\partial_3 U|{\stackrel{<}{\sim}}h/b$, with constants depending only on $\gamma$. This yields $|B'|\les h/b$. Furthermore, by Lemma \[lemmaVR\] $|v_R|\le1$ and $|\dot{v}_R|\le 2$, so that $\rho\le 1$, $|\nabla'\rho^{1/2}|{\stackrel{<}{\sim}}1/\eta $ and $|\partial_3\rho^{1/2}|{\stackrel{<}{\sim}}h/(\eta b)$. We start with the region $x_3\in [0,b/2]$. All integrals in $x'$ can be restricted to the set $$\Omega_{x_3}:=\{x': \sqrt{ \beta{\varphi}/\pi}\le |u(x',x_3)|\le \eta R+\sqrt{\beta{\varphi}/\pi}\}.$$ Since $\det \nabla u'=1$, we have $$|\Omega_{x_3}|=|\B'(0,\eta R+\sqrt{\beta{\varphi}/\pi})\setminus \B'_0| = \pi \eta^2R^2+2\sqrt{\beta{\varphi}\pi}\eta R{\stackrel{<}{\sim}}\eta \sqrt{\beta{\varphi}} R^2,$$ (in the last step we used the assumption on ${\varphi}$ and $R\ge 3$). Therefore $$\begin{aligned} {\widetilde}F_{\alpha,\beta}^{(0,b/2)}(\rho,B) &{\stackrel{<}{\sim}}\int_{0}^{b/2}\int_{\Omega_{x_3}} \left( \frac{\alpha^{-2/3}\beta^{-1/3}}{\eta^2} + \frac{h^2\alpha^{-4/3}\beta^{-2/3}}{\eta^2 b^2} + \alpha^{2/3}\beta^{-2/3} + \frac{h^2}{\beta b^2}\right) dx' dx_3\\ &{\stackrel{<}{\sim}}\int_{0}^{b/2}|\Omega_{x_3}| dx_3 \left( \alpha^{2/3}\beta^{-2/3} + \frac{h^2}{\beta b^2}\right) \\ &{\stackrel{<}{\sim}}\left( b + \frac{h^2}b \alpha^{-2/3}\beta^{-1/3}\right) R^2 \sqrt{{\varphi}}\,\\ &{\stackrel{<}{\sim}}\sqrt{{\varphi}} b R^2 +\frac{r^2}{b} {\varphi}R^2, \end{aligned}$$ where in the last line we have used that $h^2\les r^2$ and $\a^{-2/3}\b^{-1/3}\les \sqrt{{\varphi}}$. The region $(b/2,b)$ is simpler, as the $|B'|^2$ term does not appear, the others are the same with the exception of $|\partial_3 \rho^{1/2}|\les 1/b$, which is smaller by the factor $\eta/h\les 1$. Using this building block we can finally produce the construction that will be used at branching points. \[Lembranch3\] Let $a<b\in \R$, $\gamma>0$, $\ell>0$, $R\ge 3$, $X_i\in\R^2$ and $\varphi_i>0$ for $i=0,1,2$ with ${\varphi}_0={\varphi}_1+{\varphi}_2$, ${\varphi}_0/{\varphi}_1+{\varphi}_0/{\varphi}_2\le \gamma$ and $\sqrt{\beta {\varphi}_0}\ge \eta$, where as above $\eta:=\alpha^{-2/3}\beta^{1/6}$. Then, if $|X_0-X_1|,|X_0-X_2|\le \ell/4$, $\sqrt{\beta \varphi_0}\ll |X_1-X_2|$ and $ \eta R\ll \ell$, then there are $\rho\in L^\infty(\R^2\times [a,b];[0,1])$ and $B\in L^2(\R^2\times [a,b];\R^3)$ such that $\div B=0$, $\rho B=0$, $B=0$ and $\rho=1$ on $(\R^2\setminus \B'(X_0,3\ell/4))\times[a,b]$, $${\widetilde}F^{(a,b)}_{\alpha,\beta}(\rho, B) {\stackrel{<}{\sim}}\sqrt{\varphi_0} |a-b| R^2+ \frac{\ell^2}{|a-b|}\varphi_0R^2,$$ where the implicit constants only depend on $\gamma$, and which satisfy the boundary conditions$$\rho(x',a)=v_R^2\left(\frac{|x'-X_0|-\sqrt{\beta{\varphi}_0/\pi}}{\eta}\right) \hskip1mm\text{ }\hskip1mm \rho(x',b)=\min_{i=1,2}v_R^2\left(\frac{|x'-X_i|-\sqrt{\beta{\varphi}_i/\pi}}{\eta}\right)\,,$$ and $$B_3(x',a)= \chi_{\B'(X_0, \sqrt{\beta{\varphi}_0/\pi})}(x')\,, \hskip3mm\text{ }\hskip3mm B_3(x',b)=\sum_{i=1,2} \chi_{\B'(X_i, \sqrt{\beta{\varphi}_i/\pi})}(x')\,.$$ Note that our assumptions on the parameters ${\varphi}_0, {\varphi}_1, {\varphi}_2$ and $h$ just mean that $\eta\les \sqrt{\beta {\varphi}_i}\ll |X_1-X_2|\les \ell$ and $\eta R\ll \ell$. That is, the thread diameter is at least as large as the coherence length $\eta$ but small compared to the distance between the threads. Likewise, $\ell$ is large compared to the cut-off scale $\eta R$. After a translation and a rotation we may assume that $a=0$, $b>0$, $X_0=0$, $X_2-X_1=\zeta e_1$, with $\zeta>0$. Let $$w_0:=h:=(\b {\varphi}_0)^{1/2}, \qquad w_1:= \frac{{\varphi}_1}{{\varphi}_0} w_0 \qquad \textrm{ and } \qquad w_2:= \frac{{\varphi}_2}{{\varphi}_0}w_0.$$ Let then $Q^i:=X_i+[-w_i/2,w_i/2]\times[-h/2,h/2]$, for $i=0,1,2$. Notice that since $|X_i|\le \ell/4$ and $w_i\le w_0=h\ll \ell$, $Q^i{\subseteq}\B'(3\ell/8)$.\ We divide the interval $(0,b)$ in three parts (see again Figure \[fig1\]). In $(0,b/3)$, we apply Lemma \[lemmacurveendpoints\] to transform $\B'(\sqrt{\beta{\varphi}_0/\pi})$ into $Q^0$ (in particular we have $\rho=1$ on $\lt(\R^{2}\backslash \B'(\frac{3\ell}{4})\rt)\times (0,b/3)$). In $(b/3,2b/3)$, we connect $Q^0$ to $Q^1$ and $Q^2$ by an explicit construction (see below). Finally, in $(2b/3,b)$ we apply again Lemma \[lemmacurveendpoints\] to transform $Q^1$ and $Q^2$ back into $\B'(X_1, \sqrt{\beta {\varphi}_1/\pi})$ and $\B'(X_2, \sqrt{\beta {\varphi}_2/\pi})$. Notice that in $(2b/3,b)$, if $\rho(x',x_3)\ne 1$ then by Lemma \[lemmacurveendpoints\] and our hypothesis on the parameters, necessarily$$|x'|\le \max_i |X_i|+O(\eta R+\sqrt{\beta{\varphi}_0})\le \frac{\ell}{4} +o(\ell).$$ Thus, $\rho=1$ on $\lt(\R^{2}\backslash \B'(\frac{3\ell}{4})\rt)\times (2b/3,b)$.\ It only remains to discuss the construction in the central region. Let $y_1:=X_1+w_2e_1/2$, $y_2:=X_2-w_1e_1/2$ and, for $i=1,2$, ${\widetilde}Q^i:=Q^i-y_i$, so that up to a null set, $Q^0$ is the disjoint union of ${\widetilde}Q^1$ and ${\widetilde}Q^2$ (see Figure \[figseparsquares\]). Since $y_2-y_1=X_2-X_1-w_0e_1/2$, and by assumption $w_0=(\beta{\varphi}_0)^{1/2} \ll|X_1-X_2|$, we have $(y_2-y_1)\cdot e_1>0$.\ For $i=1,2$ and $x_3\in(b/3,2b/3)$ we set ${\widetilde}{Q}^i(x_3):= {\widetilde}{Q}_i+\frac{x_3-b/3}{b/3} y_i$. Since $(y_2-y_1)\cdot e_1>0$, we have ${\widetilde}{Q}^1(x_3)\cap{\widetilde}{Q}^2(x_3)=\emptyset$ for all $x_3$. Furthermore, since $Q^i{\subseteq}\B'(3\ell/8)$, also ${\widetilde}{Q}^i(x_3){\subseteq}\B'(3\ell/8)$ for $x_3\in(b/3,2b/3)$. We finally let $$\rho^{1/2}(x):=\min \lt\{ 1,\eta^{-1}\dist(x',{\widetilde}{Q}^1(x_3)\cup {\widetilde}{Q}^2(x_3))\rt\}\hskip3mm \text{ for } x_3\in[b/3,2b/3],$$ and correspondingly $$(B',B_3)(x):= \sum_{i=1,2} \lt( \frac{y_i}{b/3},1\rt)\chi_{{\widetilde}{Q}^i(x_3)}(x')\,.$$ All admissibility conditions are easily checked. In particular, $\rho=1$ if $\dist(x',{\widetilde}{Q}^1\cup{\widetilde}{Q}^2(x_3))\ge \eta $ which holds if $|x'|\ge 3\ell/4$. The energy estimate is immediate. Boundary layer {#secconstrboundary} -------------- \[propboundraylayer\] Let $N\in\N$, $\alpha,\beta,T,t>0$ be given. Let $k:=\a^{1/3}\b^{2/3}T$ and let ${\varphi}_1, \dots, {\varphi}_{N^2}$ be positive numbers such that $$k {\varphi}_i \in 2\pi \N \, \qquad \textrm{and } \qquad\sum_{i=1}^{N^2} {\varphi}_i=1.$$ Assume that in the regime $\beta\le \alpha$, $\alpha \sqrt{2}/T\le 1$, we have $t\gg \alpha^{-1}$, $1/N\ges \alpha^{-2/3}\beta^{-1/3}$ and ${\varphi}_i\sim 1/N^2$ for every $i$. Then there are $\rho$ and $B$, admissible for ${\widetilde}F_{\alpha,\beta}^{(0,t)}$ and $k$-quantized, such that $$\label{eqboundarylint} {\widetilde}F_{\alpha,\beta}^{(0,t)}(\rho,B) \les tN + \frac{1}{N^2t},$$ $$\label{eqboundarylext} \alpha^{1/3}\beta^{7/6} \|\beta^{-1}{B_3}- 1\|_{H^{-1/2}(x_3=0)}^2\les \frac{\beta^{1/3}}{\alpha^{1/3}} + \frac{\beta^{1/2}}{N^2t} +\alpha^{1/3}\beta^{7/6},$$ and, denoting by $Q^i$ the squares centered in the $N^2$ points of the square grid of spacing $N^{-1}$ with $|Q^i|=\beta {\varphi}_i$, $$\label{eqrandwb3bdly} B_3(x',t)=\sum_i \chi_{Q^i}(x') \text{ and } \rho^{1/2}(x',t)=\min\{ 1,\eta^{-1} \dist(x',\cup_i Q^i) \}\,.$$ Note that the assumptions on the parameters $N$ and ${\varphi}_i$ mean that the sidelength $\sqrt{\beta {\varphi}_i}$ of the squares is larger than the coherence length $\a^{-2/3}\b^{1/6}$ (see ) and that both of them are small compared to the distance (equal to $1/N$) between the squares. The assumption on the parameter $t$ means that the thickness $t$ of the boundary layer is large with respect to the coherence length, where we recall that vertical and horizontal lengths have different units. The key construction is described in [@CoOtSer Lem. 4.7]. However, the notation and the scalings are different in that paper. Indeed, since there was no need to rescale $x_3$ by the thickness and $x'$ by the distance between the threads $T \alpha^{-1/3} \beta^{-1/6}$, in that paper length was measured in terms of the penetration length. We first let $r_i$ be a square of side $1/N$ centered on the square grid of spacing $N^{-1}$, $Q^i{\subseteq}r_i$ be a square with the same center and area given by $ |Q^i|=\beta{\varphi}_i$, and $b_i:= \beta{\varphi}_iN^2$. Denoting with a star the quantities from [@CoOtSer], we set $T_*:=Tt/2$, $\kappa_*:=\alpha \sqrt2/T$, $L_*:=T\alpha^{-1/3}\beta^{-1/6}$, $r_i^*:=L_*r_i$, $\hat r_i^*:=L_*Q^i$, $d_0^*:=L_*/N$, $\rho_0^*:=\beta^{1/2}L_*/N$, $N_*=N$, and $ b_i^*:=(|\hat r_i^*|/|r_i^*|)\kappa_*/\sqrt2 =b_i \kappa_*/\sqrt2$. [@CoOtSer Lem. 4.7] gives $Q_{L_*}$-periodic fields $\chi_*\in BV_\loc(\R^2\times (0,T_*);\{0,1\})$ and $B_*=(B'_*,B_3^*)\in L^2_\loc(\R^3;\R^3)$ such that $\div B_*=0$, $B_*(1-\chi_*)=0$, and, $$\chi_*(x',T_*)=\sum_i \chi_{\hat r_i^*}(x') \qquad \textrm{and} \qquad B^*_3(x',T_*)=\frac{\kappa_*}{\sqrt2} \sum_i \chi_{\hat r_i^*}(x') \hskip5mm\text{ for } x'\in Q_{L_*}\,,$$ with energy $$\label{estimstar} \frac{1}{L_*^2} \int_{Q_{L_*}\times(0,T_*)} \kappa_* |D\chi_*|+|B_*'|^2+\chi_* \lt(B_3^*-\frac{\kappa_*}{\sqrt2}\rt)^2 dx \les E_*^{int}:= \lt( \frac{\kappa_* \rho_0^* T_*}{\sqrt2 } + \frac{\kappa_*^2(\rho_0^*d_0^*)^2}{2T_*}\rt)\lt(\frac{N}{L_*}\rt)^2$$ and $$\frac{1}{L_*^2} \|B_3^*-\sum_i b_i^*\chi_{q_j^*}\|^2_{H^{-1/2}(Q_{L_*})}\les E_*^{ext}:=\lt( \kappa_*(\rho_0^*)^2+ \frac{\kappa_*^2(\rho_0^*)^3 d_0}{T_*}\rt)\lt(\frac{N}{L_*}\rt)^2\,.$$ We extend $\chi_*$ and $B_*$ to $Q_{L_*}\times(T_*,2T_*)$ by $$\chi_*(x):=\chi_*(x',T_*) \qquad \textrm{and} \qquad B_*=(0,B_3^*(x',T_*)) \hskip5mm\text{ for } x'\in Q_{L_*}\times(T_*,2T_*),$$ so that holds in $Q_{L_*}\times(0,2T_*)$. We set $$\rho_*(x):=\min\{1,\kappa_*^2d(x,\omega_*)^2\},$$ where $\omega_*=\{x: \chi_*(x)=1\}$ (here $d(\cdot,\omega_*)$ is the 3D distance, periodic in the tangential directions). Since $\omega_*$ is invariant in the $x_3-$direction inside $(T_*,2T_*)$ and since $t\gg \a^{-1}$, giving $T_*\gg \kappa_*^{-1}$, for $x_3=2T_*$, $$\min\{1,\kappa_*^2d(x,\omega_*)^2\}=\min\{1,\kappa_*^2\dist(x',\omega_*\cap\{x_3=2T_*\})^2\} \qquad \textrm{for } x=(x',2T_*).$$ Hence, $$\rho_*(x',2T_*)=\min_{i} \lt\{\min(1,\kappa_*^2\dist(x',\hat{r}^i_*)\rt\}.$$ The same computation as in the proof of [@CoOtSer Th. 4.9] leads to $$\begin{aligned} 1 \frac{1}{L_*^2} \int_{Q_{L_*}\times(0,2T_*)} |\nabla\rho_*^{1/2}|^2+|B_*'|^2+ (B_3^*-\frac{\kappa_*}{\sqrt2}(1-\rho_*))^2 dx \les E_*^{int}\,. \end{aligned}$$ We scale back in the tangential direction to obtain $$\rho(x',x_3):=\rho^*(L_*x',x_3)\quad \text{ and } \quad (B',B_3)(x',x_3):=\frac{\sqrt2}{\kappa_*}\lt(\frac{B'_*}{L_*},B_3^*\rt)(L_*x',x_3)\,.$$ We have $\div B=0$ and (\[eqrandwb3bdly\]) holds. Changing variables gives $\nabla'\rho^{1/2}(x)=L_*\nabla'\rho_*^{1/2}(L_*x',x_3)$ and $\partial_3\rho^{1/2}(x)=\partial_3 \rho^{1/2}_*(L_*x',x_3)$, so that $$\begin{gathered} {\widetilde}F^{(0,t)}_{\alpha,\beta}(\rho,B)\le \alpha^{-4/3}\beta^{-2/3} \frac{1}{L_*^2} \int_{Q_{L_*}\times(0,2T_*)} |\nabla \rho_*^{1/2}|^2 + |B'_*|^2 + \lt(B_3^*-\frac{\kappa_*}{\sqrt2}(1-\rho_*)\rt)^2 dx\\ \les\alpha^{-4/3}\beta^{-2/3} E_*^{int}.\end{gathered}$$ Since $E_*^{int}=\alpha^{4/3}\beta^{2/3} (Nt/2+2/(N^2t))$, this concludes the estimate for ${\widetilde}F^{(0,t)}_{\alpha,\beta}$. We now estimate the boundary term. By the embedding of $L^\infty$ into $H^{-1/2}$ (recall that since $\sum_i {\varphi}_i =1$, $\int_{Q_1} \sum_i(b_i-\beta) \chi_{r_i} dx'=0$) we have $$\|\sum_i (b_i-\beta)\chi_{r_i}\|^2_{H^{-1/2}(Q_{1})}\les \|b_i-\beta\|_\infty^2 \,.$$ Rescaling the boundary estimate for $B_3^*$ leads to $$\|B_3-\sum_i b_i\chi_{r_i}\|^2_{H^{-1/2}(Q_1)}\les \frac{1}{L_*^3} \frac2{\kappa_*^2}\|B_3^*-\sum_i b_i^*\chi_{r_j^*}\|^2_{H^{-1/2}(Q_{L_*})}\les \frac{\beta^{7/6}}{\alpha^{2/3}} + \frac{\beta^{4/3}}{\alpha^{1/3} N^2t}$$ Adding terms and using that $\sum_i \chi_{r_i}\equiv 1$, we conclude that $$\alpha^{1/3}\beta^{7/6} \|\beta^{-1}{B_3}- 1\|_{H^{-1/2}(Q_1)}^2\les \frac{\beta^{1/3}}{\alpha^{1/3}} + \frac{\beta^{1/2}}{N^2t} +\frac{\alpha^{1/3}}{\beta^{5/6}} \|b_i-\beta\|_\infty^2 \,.$$ Back to the full GL functional {#recoveryGL} ------------------------------ In this section, we relate the functional ${\widetilde}F_{\alpha,\beta}$, as defined in (\[eqdeftildaF\]–\[eqdeftildaFext\]), with the functional ${\widetilde}E_T$, as defined in (\[eqdefwe\]). \[thirdupperbound\] Let $k=\alpha^{1/3}\beta^{2/3}T$ and let $\rho,B$ be $k$-quantized admissible functions for ${\widetilde}F_{\alpha,\beta}$ with $\rho B=0$. Let $\omega:=\{\rho=0\}$ and then for $z\in(-1,1)$, $\omega_{z}:=\omega\cap \{x_3=z\}$. Assume that $\omega$ is closed, that $\omega^c$ is connected and that for every $x_3\in(-1,1)$, $\omega_{x_3}^c$ is also connected. Then, there is a pair $(u,A)$, admissible for ${\widetilde}E_T$, such that $\rho= |u|^2$, $B=\nabla\times A$, and $$\label{limsupT} {\widetilde}E_T(u,A)={\widetilde}F_{\a,\b}(\rho,B).$$ We shall construct a function $A$ on $\R^2\times (-1,1)$ so that $B=\nabla\times A$ everywhere, and then a multivalued function $\theta$ on the set $\Omega^c:=\Z^2\times\{0\}+\omega^c$ such that $\nabla\theta=\b^{-1}kA$. Here $B$ and $\rho$ are $Q_1$-periodic, but $A$ and $\theta$ not necessarily. Setting $u:=\rho^{1/2} e^{i\theta}$ we then have $\nabla_{\b^{-1}kA}u= e^{i\theta}\nabla\rho^{1/2}$, which directly implies (\[limsupT\]). Notice that thanks to the hypothesis on $\omega$, the set $\Omega^c$ is a connected open set. We start from the construction of $A$. By [@CoOtSer Lem. 4.8] applied to $B-\beta e_3$ there is a $Q_1$-periodic potential $A_\mathrm{per}$ such that $\nabla \times A_\mathrm{per}= B- \b e_3$ and $\Div A=0$. We define $ A(x):= A_\mathrm{per}(x)+\b x_1 e_2$ so that $\nabla \times A=B$. We remark that on any open set where $B=0$ the vector field $A$ is curl-free and divergence-free, therefore harmonic, and in particular smooth. In particular, since $\Omega^c$ is open and since by the Meissner condition, $B=0$ in $\Omega^c$, $A$ is smooth in $\Omega^c$. We now turn to the existence of $\theta$. For a fixed level $x_3$, let $h+\omega^i_{x_3}$, $h\in\Z^2$, $i=1,..., I(x_3)$ be the connected components of $(\R^2\times\{x_3\}) \cap (\Z^2+\omega_{x_3})$. Denote the flux going through $\omega^i_{x_3}$ by $$\Phi^i(x_3):=\int_{\omega^i_{x_3}} B_3\, dx'.$$ By assumption we have $$\label{condfluxi} \b^{-1}k \Phi^i(x_3)\in 2\pi \Z \qquad \textrm{ for every $i$. }$$ Fix a smooth curve $\Gamma_0:=\{(\gamma_0(x_3),x_3) \ : \ x_3\in(-1,1)\}{\subseteq}\Omega^c$. For $x_3,y_3\in(-1,1)$, let $\Gamma_0^{x_3,y_3}:=\{(\gamma_0(t),t) \ : \ t\in(x_3,y_3)\}{\subseteq}\Gamma_0$. For $x=(\gamma_0(x_3),x_3)$, let $\theta(x):= \b^{-1}k \int_{\Gamma_0^{0,x_3}} A\cdot \tau d\H^1$. Now, for a generic $x=(x',x_3)\in \Omega^c$, let $$\theta(x):= \theta(\gamma_0(x_3),x_3)+\b^{-1}k\int_{\Gamma^x} A\cdot \tau d\H^1,$$ where $\Gamma^x$ is any (horizontal) curve in $\Omega^c\cap(\R^2\times\{x_3\})$ connecting $x$ to $(\gamma_0(x_3),x_3)$. This gives a well defined $\theta:\Omega^c\to\R/2\pi\Z$ since for every closed (horizontal) curve $\Gamma$ in $\Omega^c\cap(\R^2\times\{x_3\})$, $$\label{condfluxcurvebis} \b^{-1}k\int_{\Gamma} A\cdot \tau d\H^1\in 2\pi \Z.$$ Indeed, this follows by Stokes’ Theorem and . Let us show that $\nabla \theta=\b^{-1}k A $ in $\Omega^c$. Let $x=(x',x_3)$ be fixed and let $\Gamma^x$ be a fixed simple smooth curve joining $x$ to $(\gamma_0(x_3),x_3)$ inside $\Omega^c\cap(\R^2\times\{x_3\})$. Since $\Omega^c$ is open, there is a simply connected neighborhood $V$ of $\Gamma^x$ such that $V{\subseteq}\Omega^c$. Let then $y=(y',y_3)\in V$. Upon shrinking $V$, we may assume that $(\gamma_0(y_3),y_3)\in V$. Let $\Gamma^y{\subseteq}V$ be a smooth curve joining $y$ to $ (\gamma_0(y_3),y_3)$ and let $\Gamma^{x,y}{\subseteq}V$ be a smooth curve joining $x$ to $y$. By definition, we have $$\theta(x)=\theta(\gamma_0(x_3),x_3)+ \b^{-1}k \int_{\Gamma^x}A\cdot \tau d\H^1, \quad \theta(y)=\theta(\gamma_0(y_3),y_3)+ \b^{-1}k\int_{\Gamma^y} A\cdot \tau d\H^1$$ and $$\theta(\gamma_0(y_3),y_3)=\theta(\gamma_0(x_3),x_3)+ \b^{-1}k\int_{\Gamma_0^{x_3,y_3}} A\cdot \tau d\H^1,$$ so that $$\theta(y)-\theta(x)= \b^{-1}k\int_{\Gamma^y} A\cdot \tau d\H^1+ \b^{-1}k\int_{\Gamma_0^{x_3,y_3}} A\cdot \tau d\H^1-\b^{-1}k\int_{\Gamma^x} A\cdot \tau d\H^1.$$ However, by Stokes Theorem (and $B=0$ in $V$), $$\int_{\Gamma^y}A\cdot \tau d\H^1+\int_{\Gamma_0^{x_3,y_3}} A\cdot \tau d\H^1-\int_{\Gamma^x} A\cdot \tau d\H^1=\int_{\Gamma^{x,y}} A\cdot \tau d\H^1,$$ so that $$\theta(y)-\theta(x)=\b^{-1}k\int_{\Gamma^{x,y}} A\cdot \tau d\H^1,$$ proving that indeed, $\nabla \theta=\b^{-1}k A$ in $\Omega^c$. Proof of the upper bound ------------------------ We start from a construction for an $N$-regular measure and finite $R$. \[propubregular\] Let $\mu\in \MNreg(Q_{1,1})$ for some $N\in\N$, $R\ge 3$, and assume (\[eqlbassumptcoeff\]) and (\[defKtotal\]) hold. Then, there exist sequences $u_n: Q_{1,1}\to {\mathbb{C}}$ and $A_n: Q_{1,1}\to \R^3$ such that $$\label{estimenerregular} \limsup_{n\to +\infty} \, \wE(u_n,A_n)\le \frac{K_R}{K_*} I(\mu) + \frac{C}{N^{1/2}}(1+I(\mu)),$$ with $|u_n|$ and $\nabla\times A_n$ $Q_1-$periodic. Here $K_R$ and $K_*$ are as in Lemma \[lemmaVR\], $C$ is universal. Moreover, $$\label{estimW2regular} \limsup_{n\to +\infty} W_2^2(\b^{-1}_n B^n_3,\mu)\les N^{-3/2}.$$ We first modify slightly $\mu$ in order to be able to use Proposition \[propboundraylayer\] for the boundary layer. Set $\varepsilon_N:=N^{-3/2}$. For $x_3\in [-1+\eps_N,1-\varepsilon_N]$ let $\hat{\mu}_{x_3}:=\mu_{x_3/(1-\varepsilon_N)}$ and for $x_3\in [-1,-1+\eps_N]\cup[1-\varepsilon_N,1]$, let $\hat{\mu}_{x_3}:=\mu_1=\mu_{-1}=N^{-2}\sum_l \delta_{X_l}$, where the $\{X_l\}_{l=1}^{N^2}$ form a regular grid with spacing $1/N$ (recall Def. \[defregular\]). We then have $$I(\hat{\mu})\le \frac{1}{1-\varepsilon_N} I(\mu)+2K_*\varepsilon_N N.$$ Moreover, since $|\hat \mu|(Q_1\times (1-\varepsilon_N,1))= \varepsilon_N$, the same on the other side, and $$W_2^2((1-\varepsilon_N)\mu,\hat{\mu}\LL(Q_1\times (-1+\varepsilon_N,1-\varepsilon_N))\les \eps_N^2,$$ so that $$W_2^2(\mu, \hat{\mu})\les \eps_N+\eps^2_N\les N^{-3/2},$$ it is enough to prove the estimates for $\hat{\mu}$ instead of $\mu$. We start by characterizing the geometry of the construction, which will not depend on $n$. The measure $\hat{\mu}$ is supported on finitely many polygonal curves, parametrized by $X_i:[a_i,b_i]\to Q_1$, which are disjoint up to the endpoints and carry a flux ${\varphi}_i>0$. The endpoints are either on the boundary of $Q_{1,1}$, or they are triple points. Those on the boundary constitute a regular grid. We let $\phimin:=\min_i{\varphi}_i$, $\phimax:=\max_i {\varphi}_i$, and $\gamma:=8\phimax/\phimin$. Since there are finitely many curves, these quantities are finite and positive. We define as before $\eta_n:=\alpha_n^{-2/3}\beta_n^{1/6}$ to be the coherence length; by (\[eqlbassumptcoeff\]) we have $\eta_n\to0$. Let $y_j:=(Y_j,z_j)\in Q_{1,1}$ denote the internal endpoints of the curves. For $\ell>0$ sufficiently small one has that, for any $j$, the only curves which intersect $\B'_\ell(Y_j)\times(z_j-\ell,z_j+\ell)$ are those with an endpoint in $y_j$. Since $y_j$ is a triple point, there are three such curves $\Gamma_0$, $\Gamma_1$ and $\Gamma_2$, intersecting only at $y_j$. If we let $M$ be the maximal slope of all curves and $t_\ell:=\ell/(8M)$, then no curve intersects $\partial\B'_{\ell/8}(Y_j)\times (z_j-t_\ell,z_j+t_\ell)$ (this means, they all “exit” from the top and bottom faces). Without loss of generality, we can assume that $(X_0,z_j-t_\ell)=\Gamma_0\cap\lt( \B'(Y_j,\ell)\times\{z_j-t_\ell\}\rt)$, $(X_1,z_j+t_\ell)=\Gamma_1\cap \lt(\B'(Y_j,\ell)\times\{z_j+t_\ell\}\rt)$ and $(X_2,z_j+t_\ell)=\Gamma_2\cap \lt(\B'(Y_j,\ell)\times\{z_j+t_\ell\}\rt)$ (see Figure \[figtriplepoint\]). By definition of $t_\ell$, it holds $|X_0-Y_j|\le \frac{\ell}{8}$ and $|X_i-X_0|\le \frac{\ell}{4}$ for $i=1,2$. We set $\omega_j:=\B'_{\ell}(Y_j)\times (z_j-t_\ell,z_j+t_\ell)$ and let $\delta_\ell$ be the minimum distance between any two curves outside $\cup_j\omega_j$. To explain the strategy, we first carry out a construction which ignores quantization. For sufficiently large $n$ we have $ \eta_nR+\sqrt{\phimax \beta_n/\pi}<\delta_\ell/2$ and can thus use the construction of Lemma \[lemmacurve3\] in a $\delta_\ell/2$-neighborhood of each curve (outside of $\cup_j \omega_j$), extending by $\rho_n=1$ and $B_n=0$ to the complement. In the cylinders $\omega_j$, since the geometry is fixed, for $n$ sufficiently large the conditions $\sqrt{\beta_n \phimin}\ge \eta_n$, $\sqrt{\beta_n\phimax}\ll |X_1-X_2|$ and $ \eta_n R\ll \ell$ are satisfied and we can use Lemma \[Lembranch3\]. We then have that $\rho_n=1$ outside $\B'(X_0, 3\ell/4)\times(z_j-t_\ell,z_j+t_\ell){\subseteq}\omega_j$. In order to obtain a quantized field, we define $k_n$ as in (\[defK\]), which obeys $ k_n\in 2\pi\N$ and $k_n\to+\infty$. Let $\hat{\mu}^n$ be the $k_n$-quantized approximation of $\hat{\mu}$, as given by Lemma \[lemmaquantize\]. For sufficiently large $n$ we have $\frac12 \hat{\mu}\le \hat{\mu}^n\le 2\hat{\mu}$ and for $|x_3|\ge 1-\varepsilon_N$, $$\hat{\mu}_{x_3}^n=\sum_l{\varphi}_l^n\delta_{X_l},$$ where $|{\varphi}_l^n-1/N^2|\le C(\hat{\mu})/k_n$, ${\varphi}_l^n\le 2/N^2$. The fluxes ${\varphi}_i^n$ obey $\frac12{\varphi}_i\le {\varphi}_i^n\le 2{\varphi}_i$. We then construct $B_n$ and $\rho_n$ using Lemma \[lemmacurve3\] and Lemma \[Lembranch3\] as discussed above. The geometry is the one determined by $\hat{\mu}$. In particular, the points $y_j$, and the constants $M$, $\phimin$, $\phimax$, $\delta_\ell$ and $t_\ell$ do not depend on $n$, and only $\delta_\ell$ and $t_\ell$ depend on $\ell$. Adding terms gives for sufficiently large $n$ (as a geometry dependent function of $\ell$ and $R$) $$\begin{aligned} {\widetilde}F_{\alpha_n,\beta_n}^{(-1,1)}(\rho_n,B_n)\le & \sum_i \left(1+ \frac{C}{({\varphi}_i^n \alpha_n^{4/3}\beta_n^{2/3})^{1/2}} + \frac{C}{{\varphi}_i^n \alpha_n^{4/3}\beta_n^{2/3}}\right) \int_{a_i}^{b_i} \bigl( K_R\sqrt{{\varphi}_i^n}+{\varphi}_i^n |\dot{X}_i|^2\bigr) dx_3 \\ &+C(\gamma) \sum_j \sqrt{\phimax} t_\ell R^2+ \frac{\ell^2}{t_\ell}\phimax R^2,\end{aligned}$$ where the first sum runs over all curves and the second over the cylinders and where $C(\gamma)>0$ is a constant depending on $\gamma$. In the limit $n\to+\infty$ we have ${\varphi}_i^n\to{\varphi}_i$, $\alpha_n^2\beta_n\to+\infty$ and therefore, inserting the definition of $t_\ell$, $$\begin{aligned} \limsup_{n\to+\infty} {\widetilde}F_{\alpha_n,\beta_n}^{(-1,1)}(\rho_n,B_n)\le & \sum_i \int_{a_i}^{b_i} \bigl( K_R\sqrt{{\varphi}_i}+{\varphi}_i |\dot{X}_i|^2\bigr) dx_3 +C(\gamma) R^ 2 \ell \sum_j \left(\frac{\sqrt{\phimax}}{M} +M\phimax\right)\,.\end{aligned}$$ The sum over $j$ depends only on $\hat{\mu}$. Therefore if $\ell$ is chosen sufficiently small we have $$\limsup_{n\to +\infty} \, {\widetilde}F_{\alpha,\beta}^{(-1,1)}(\rho_n,B_n)\le \frac{K_R}{K_*} I(\mu)+\frac1{N^{1/2}}\,.$$ Moreover, thanks to and Lemma \[lemmaquantize\], we have $$\begin{aligned} W_2^2(\beta^{-1}_n B^n_3,\hat{\mu})&\les W_2^2(\beta^{-1}_n B^n_3,\hat{\mu}^n)+W_2^2(\hat{\mu},\hat{\mu}^n)\\ &\les C(\hat{\mu})\b_n + t_\ell \, \sharp\{Y_j\}+ C(\hat{\mu})k_n^{-1}\\ &\les C(\hat{\mu})\b_n + \frac{\ell}{M} \, \sharp\{Y_j\}+ C(\hat{\mu})k_n^{-1}.\end{aligned}$$ Again, since $ \sharp\{Y_j\}/M$ only depends on $\hat\mu$, if we choose $\ell$ sufficiently small then $$\label{estimW2inside} \limsup_{n\to+\infty} W_2^2(\beta^{-1}_n B^n_3,\hat{\mu})\le N^{-3/2}.$$ We finally address the boundary layer, focusing for definiteness on the side $x_3>0$. We first apply once more Lemma \[lemmacurveendpoints\] to each curve for $x_3\in (1-\eps_N, 1-\eps_N/2)$, so that the resulting fields $B_n$ and $\rho_n$ obey for all $x_3\in (1-\eps_N/2,1)$, $$B_n(x)=\sum_l e_3 \chi_{Q^l}(x') \text{ and } \rho_n(x)= \min\{1,\eta_n^{-1} \dist(x',\cup_l Q^l)\},$$ where $Q^l$ are squares centered in the $N^2$ points $X_l$ with $|Q^l|= \b{\varphi}_l^n $ , and $|{\varphi}_l^n-1 /N^2|\le C(\hat{\mu})/k_n$. Then we modify $\rho_n$ and $B_n$ in the set $x_3\in (1-\eps_N/2,1)$ using Proposition \[propboundraylayer\]. This results in new fields $\hat \rho_n$, $\hat B_n$ which obey $${\widetilde}F_{\alpha_n,\beta_n}^{(1-\eps_N/2,1)}(\hat\rho_n,\hat B_n)+ \alpha_n^{1/3}\beta_n^{7/6} \|\beta_n^{-1}{(\hat{B}_n)_3}- 1\|_{H^{-1/2}(x_3=1)}^2\les \frac{1}{N^{1/2}}+ \frac{\beta_n^{1/3}}{\alpha_n^{1/3}} + \frac{\beta_n^{1/2}}{N^{1/2}} +\frac{\alpha_n^{1/3}}{\beta_n^{5/6}} \frac{C(\hat\mu)N^4}{k_n^2} \,.$$ By (\[eqlbassumptcoeff\]) and (\[defK\]) all terms up to the first one tend to zero as $n\to+\infty$. Therefore $$\limsup_{n\to +\infty} \, {\widetilde}F_{\alpha_n,\beta_n}(\hat\rho_n,\hat B_n)\le \frac{K_R}{K_*} I(\hat{\mu}) +\frac{C}{N^{1/2}}\,.$$ Finally, we pass from $(\hat\rho_n,\hat B_n)$ to $(u_n,A_n)$ via Proposition \[thirdupperbound\] and conclude the proof of . We also obtain since $$W^2_2(\beta^{-1}_n (\hat{B}_n)_3, \hat{\mu})\les W^2_2(\beta^{-1}_n B^n_3, \hat{\mu})+N^{-3/2}.$$ It only remains to combine the different steps. By density (Theorem \[theodens\]) there is a sequence $\mu^N\in \MNreg(Q_{1,1})$ of $N$-regular measures converging weakly to $\mu$, with $\limsup_{N\to+\infty} I(\mu^N)\le I(\mu)$. Fix $R\ge 3$ and let $(u_n^N,A_n^N)$ be as in Proposition \[propubregular\]. Taking a diagonal subsequence (first with $N$, then with $R$) we obtain $$\limsup_{n\to +\infty} \, \wE(u_n^{N(n)},A_n^{N(n)})\le I(\mu),$$ and $\b_n^{-1}(B_n^{N(n)})_3\weaklim \mu $. From the compactness statement of Proposition \[gammaliminf\] and uniqueness of $m$ one obtains $\b_n^{-1} (B_n^{N(n)})'\weaklim m$ and $\b_n^{-1}(1-|u_n^{N(n)}|^2)\weaklim \mu $. Acknowledgment {#acknowledgment .unnumbered} ============== M. G. thanks E. Esselborn, F. Barret and P. Bella for stimulating discussions about optimal transportation, and the FMJH PGMO fundation for partial support through the project COCA. The work of S.C. was partially supported by the Deutsche Forschungsgemeinschaft through the Sonderforschungsbereich 1060 [*“The mathematics of emergent effects”*]{}. [^1]: Institut für Angewandte Mathematik, Universität Bonn, Germany, email: sergio.conti@uni-bonn.de [^2]: LJLL, Université Paris Diderot, CNRS, UMR 7598, France email: goldman@math.univ-paris-diderot.fr, part of his research was funded by a Von Humboldt PostDoc fellowship [^3]: Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany, email: otto@mis.mpg.de [^4]: Courant Institute, NYU & Institut Universitaire de France & UPMC, Paris, France, email: serfaty@cims.nyu.edu [^5]: for $p\ge 1$, we analogously define $W_p^p(\mu,\nu):=\min\lt\{ \mu(Q_L)\int_{Q_L\times Q_L} |x-y|^p \, d\Pi(x,y) \, : \, \Pi_1=\mu, \ \Pi_2=\nu\rt\}$.

--- abstract: 'We discuss in detail the option to access the transversity distribution function $h_1(x)$ by utilizing the analyzing power of interference fragmentation functions in two-pion production inside the same current jet. The transverse polarization of the fragmenting quark is related to the transverse component of the relative momentum of the hadron pair via a new azimuthal angle. As a specific example, we spell out thoroughly the way to extract $h_1(x)$ from a measured single spin asymmetry in two-pion inclusive lepton-nucleon scattering. To estimate the sizes of observable effects we employ a spectator model for the fragmentation functions. The resulting asymmetry of our example is discussed as arising in different scenarios for the transversity.' address: - | Dipartimento di Fisica Nucleare e Teorica, Università di Pavia, and\ Istituto Nazionale di Fisica Nucleare, Sezione di Pavia, I-27100 Pavia, Italy - 'Fachbereich Physik, Universität Wuppertal, D-42097 Wuppertal, Germany' - | Dipartimento di Chimica e Fisica per i Materiali e per l’Ingegneria,\ Università di Brescia, I-25133 Brescia, Italy author: - Marco Radici - Rainer Jakob - Andrea Bianconi title: | [Preprint\ WU B 01-09]{} Accessing transversity with\ interference fragmentation functions --- =-1cm Introduction {#sec:intro} ============ At leading power in the hard scale $Q$, the quark content of a nucleon state is completely characterized by three distribution functions (DF). They describe the quark momentum and spin with respect to a preferred longitudinal direction induced by a hard scattering process. Two of them, the momentum distribution $f_1$ and the longitudinal spin distribution $g_1$, have been reliably extracted from experiments and accurately parametrized. Their knowledge has deeply contributed to the studies of the quark-gluon substructure of the nucleon. The third one, the transversity distribution $h_1$, measures the probability difference to find the quark polarization parallel versus antiparallel to the transverse polarization of a nucleon target. Therefore, it correlates quarks with opposite chiralities and is usually referred to as a “chiral-odd” function. Since hard scattering processes in QCD preserve chirality at leading twist, the $h_1$ is difficult to measure and is systematically suppressed like ${\cal O}(1/Q)$, for example, in inclusive deep inelastic scattering (DIS). A chiral-odd partner is needed to filter the transversity out of the cross section. Historically, the so-called double spin asymmetry (DSA) in Drell-Yan processes with two transversely polarized protons ($p^\uparrow$) was suggested first [@Ralston:1979ys]. However, the transversity distribution $h_1$ for antiquarks in the proton is presumably small [@Jaffe:1997yz]. Moreover, an upper limit for the DSA derived in a next-to-leading order analysis by using the Soffer bounds on transversity was found to be discouraging low [@Martin:1998rz]. As for DIS, semi-inclusive reactions need to be considered in order to provide the chiral-odd partner to $h_1$. In fact, in this case new functions enter the game, the fragmentation functions (FF), which give information on the hadronic structure complementary to the one delivered by the DF. At leading twist, the FF describe the hadron content of quarks and, more generally, they contain information on the hadronization process leading to the detected hadrons; as such, they give information also on the quark content of hadrons that are not (or even do not exist as) stable targets. The FF are also universal, but are presently less known than the DF because a very high resolution and good particle identification are required in the detection of the final state. Since pions are the most abundant particles detected in the calorimeter, it would be natural to consider semi-inclusive processes where a single collinear pion is detected together with the final lepton inelastically scattered from a transversely polarized nucleon target. However, the $h_1$ would appear convoluted with a chiral-odd fragmentation function only at twist three and, therefore, suppressed like ${\cal O}(1/Q)$ [@Jaffe:1992ra]. It seems more convenient to select the more rare final state where a polarized $\Lambda$ decays into protons and pions [@Jaffe:1993xb]. The analysis of the decay products reveals the $\Lambda$ polarization and a DSA isolates at leading twist a contribution proportional to the product of $h_1$ and a chiral-odd FF, $H_1$, which describes how a transversely polarized quark $q^\uparrow$ fragments into a transversely polarized $\Lambda^\uparrow$. But again, as in the case of the Drell-Yan DSA, low rates are expected here too because of the few $\Lambda$ particles produced in a hard reaction. Moreover, the theoretical knowledge of the mechanisms that determine the polarization transfer $q^\uparrow \rightarrow \Lambda^\uparrow$ (i.e. of $H_1$) is not yet firmly established. For all these reasons, building single spin asymmetries (SSA) seems to be a better strategy, i.e. considering DIS or $p-p$ processes where only one particle (the target) is transversely polarized, but selecting more complicated final states. The most famous example is the Collins effect: the analyzing power of the transverse polarization of the fragmenting quark is represented by the transverse component of the momentum of the detected hadron with respect to the current jet axis. The typical reactions would be, therefore, a semi-inclusive DIS, $ep^\uparrow \rightarrow e'\pi X$, or $p^\uparrow p \rightarrow \pi X$, where the pion is detected not collinear with the jet axis. At leading twist, a specific SSA allows for the deconvolution of $h_1$ from the so-called Collins function $H_1^\perp$, the prototype of a new class of FF, the interference FF, which are not only chiral-odd, but also [*naive*]{} T-odd: in absence of two or more reaction channels with a significant relative phase, they are forbidden by time-reversal invariance [@Collins:1993kk]. From the experimental point of view, extraction of $h_1$ via the Collins effect is quite a demanding task, because it requires the complete determination of the transverse momentum of the detected hadron (though first observations of a non-zero SSA have been reported [@Airapetian:2000tv]). On the other side, it is not sufficient to limit the theoretical analysis at leading order. Because of the explicit dependence on an intrinsic transverse momentum, some soft gluon divergencies (introduced by loop corrections to the tree level result) do not cancel and must be summed up in Sudakov form factors. The net result is a dilution of the transverse momentum distribution of the fragmenting quark and a final suppression of the SSA, particularly when in the fragmentation process there is another scale very different from the hard one $Q$, as for instance the transverse momentum of the produced hadron in the Collins effect. The same phenomenon happens “squared” in $e^+e^-$ processes, that, consequently, do not help in determining the Collins function [@Boer:2001he]. Moreover, modelling this interference FF by definition requires the ability of giving a microscopic description of the relevant phase produced by the quantum interference of different channels leading to the same detected hadron: a very difficult task that implies a description of the structure of the residual jet (as discussed in [@Bianconi:2000cd]), or the introduction of dressed quark propagators [@Collins:1993kk] which may be effectively modelled, for instance, by pion loop corrections [@Bacchetta:2001di]. As a better alternative, the SSA with detection of two unpolarized leading hadrons inside the same jet was suggested [@Collins:1994ax; @Collins:1994kq; @Jaffe:1998hf]. In a previous work, we have discussed the general framework for the interference FF arising in this case [@Bianconi:2000cd]. Assuming that the residual interactions between each leading hadron and the undetected jet is of higher order than the one between the two hadrons themselves, the main result was that $h_1$ gets factorized at leading twist through a novel interference FF, $H_1^{{{<\kern -0.3 em{\scriptscriptstyle )}}}}$, that relates the transverse polarization of the fragmenting $q^\uparrow$ to the relative motion of the two detected hadrons. This new analyzing power, $H_1^{{{<\kern -0.3 em{\scriptscriptstyle )}}}}$, filters out the $h_1$ in a very advantageous way, because collinear factorization holds, which leads to an exact cancellation of all collinear divergences, and makes the evolution equations much simpler. Moreover, it is also easier to model the residual interaction between the two hadrons only. In another previous work, we have presented a model calculation for the case of the two hadrons being a $\pi$ and a $p$ with invariant mass close to the Roper resonance [@Bianconi:2000uc]. In the present paper we carry the calculation on to the experimentally more relevant case of $\pi^+ \pi^-$ production with invariant mass close to the $\rho$ resonance, and we discuss some of the practical details for the extraction of $h_1$ from a spin asymmetry in semi-inclusive lepton-nucleon DIS. This observable should be accessible, for instance, at HERMES (when the transversely polarized target will be operative) or even better at COMPASS (because of higher counting rates); it will also be a very interesting quantity for the future options in hadronic physics like ELFE, TESLA-N and EIC. However, as we like to emphasize, the calculation of the $\pi^+\pi^-$ fragmentation is process independent and could be most useful also for the spin physics program at RHIC, where the extraction of transversity is planned via a SSA in $p-p$ reactions. The rest of the paper is organized as follows. In Sec. \[subsec:iff\] we briefly recall the kinematics and the properties of the FF arising when a transversely polarized quark fragments into two unpolarized leading hadrons in the same current jet. Then, in Sec. \[subsec:h1fromssa\] we specialize the formulae to the case of semi-inclusive lepton-nucleon DIS and detail the strategy for building a SSA that allows for the extraction of $h_1$ at leading twist. In Sec. \[sec:spectator\] we consider the two hadrons to be two charged pions with invariant mass around the $\rho$ resonance and we explicitly calculate both the (process independent) interference FF and the SSA (for semi-inclusive lepton-nucleon DIS) in the spectator model approximation. In Sec. \[sec:results\] results are presented and commented. Conclusions and outlooks are given in Sec. \[sec:end\]. Single spin asymmetry for two hadron-inclusive lepton-nucleon DIS {#sec:ssa} ================================================================= In this Section, we discuss the general properties of two-hadron interference FF when the kinematics is specialized to semi-inclusive DIS, and for this process we work out the formula for a SSA that isolates the transversity at leading twist. However, we emphasize that under the assumption of factorization the soft parts of the process, i.e. the DF and the interference FF, are universal objects and, therefore, the results can be generalized to other hard processes, such as proton-proton scattering. Interference Fragmentation Functions in semi-inclusive DIS {#subsec:iff} ---------------------------------------------------------- At leading order, the hadron tensor for two unpolarized hadron-inclusive lepton-nucleon DIS reads [@Bianconi:2000cd] $$\begin{aligned} 2M\, {\cal W}^{\mu\nu} &=& \int \d p^-\,\d k^+\,\d^2{\vec p}_{{\scriptscriptstyle T}}^{}\,\d^2{\vec k}_{{\scriptscriptstyle T}}^{}\; \delta^2\!\left({\vec p}_{{\scriptscriptstyle T}}^{}+{\vec q}_{{\scriptscriptstyle T}}^{}-{\vec k}_{{\scriptscriptstyle T}}^{}\right) \mbox{Tr}\big[ \; \Phi(p;P,S) \; \gamma^\mu \; \Delta(k;P_1,P_2) \; \gamma^\nu \; \big] \Big|_{\tiny\begin{array}{c} p^+ = x P^+ \\ k^- = P_h^-/z \end{array}} {\nonumber}\\ & & {}+ \left(\begin{array}{c} q\leftrightarrow -q \\ \mu \leftrightarrow \nu \end{array} \right) \; , \label{eq:tensor}\end{aligned}$$ where $M$ is the target mass. The kinematics, also depicted in Fig. \[fig:handbag\], represents a nucleon with momentum $P (P^2=M^2)$ and a virtual hard photon with momentum $q$ that hits a quark carrying a fraction $p^+ = x P^+$ of the parent hadron momentum. We describe a 4-vector $a$ as ${\left[\;a^-\;,\;a^+\;,\;{\vec a}_{{\scriptscriptstyle T}}\;\right]}$ in terms of its light-cone components $a^\pm = (a^0\pm a^3)/\sqrt{2}$ and a transverse bidimensional vector ${\vec a}_{{\scriptscriptstyle T}}$. Because of momentum conservation in the hard vertex, the scattered quark has momentum $k=p+q$, and it fragments into two unpolarized hadrons, which carry a fraction $(P_1+P_2)^-\equiv P_h^- = z k^-$ of the “parent quark” momentum, and the rest of the jet. The quark-quark correlator $\Phi$ describes the nonperturbative processes that make the parton $p$ emerge from the spin-1/2 target, and it is symbolized by the lower shaded blob in Fig. \[fig:handbag\]. Using Lorentz invariance, hermiticity and parity invariance, the partly-integrated $\Phi$ can be parametrized at leading twist in terms of DF as $$\begin{aligned} \Phi(x,{\vec p}_{{\scriptscriptstyle T}}) &\equiv & \left. \int \d p^-\;\Phi(p;P,S) \right|_{p^+ = x P^+} \!\!\!\!=\frac{1}{2}\,\Biggl\{ f_1\, {{\kern 0.2 em n\kern -0.45em /}}_+ + f_{1T}^\perp\, \epsilon_{\mu \nu \rho \sigma}\gamma^\mu \frac{n_+^\nu p_{{\scriptscriptstyle T}}^\rho S_{{{\scriptscriptstyle T}}}^\sigma}{M} - \left(\lambda\,g_{1L} +\frac{({\vec p}_{{\scriptscriptstyle T}}\cdot{\vec S}_{{{\scriptscriptstyle T}}})}{M}\,g_{1T}\right) {{\kern 0.2 em n\kern -0.45em /}}_+ \gamma_5 {\nonumber}\\[2 mm] && {}- h_{1T}\,i\sigma_{\mu\nu}\gamma_5 S_{{{\scriptscriptstyle T}}}^\mu n_+^\nu - \left(\lambda\,h_{1L}^\perp +\frac{({\vec p}_{{\scriptscriptstyle T}}\cdot{\vec S}_{{{\scriptscriptstyle T}}})}{M}\,h_{1T}^\perp\right)\, \frac{i\sigma_{\mu\nu}\gamma_5 p_{{\scriptscriptstyle T}}^\mu n_+^\nu}{M} + h_1^\perp \, \frac{\sigma_{\mu\nu} p_{{\scriptscriptstyle T}}^\mu n_+^\nu}{M}\Biggl\} \; , \label{eq:phi}\end{aligned}$$ where the DF depend on $x, {\vec p}_{{\scriptscriptstyle T}}$ and the polarization state of the target is fully specified by the light-cone helicity $\lambda = M S^+ / P^+$ and the transverse component ${\vec S}_{{\scriptscriptstyle T}}$ of the target spin. Similarly, the correlator $\Delta$, symbolized by the upper shaded blob in Fig. \[fig:handbag\], represents the fragmentation of the quark into the two detected hadrons and the rest of the current jet and can be parametrized as [@Bianconi:2000cd] $$\begin{aligned} \Delta &\equiv & \left.\frac{1}{4z}\int \d k^+ \; \Delta(k;P_1,P_2) \right|_{k^-=P_h^-/z} {\nonumber}\\[2mm] &= & \frac{1}{4}\left\{ D_1\,{\kern 0.2 em n\kern -0.45em /}_- - G_1^\perp\, \frac{{\epsilon}_{\mu\nu\rho{\sigma}}\,{\gamma}^\mu\,n_-^\nu\,k_{{\scriptscriptstyle T}}^\rho\,R_{{\scriptscriptstyle T}}^{\sigma}} {M_1 M_2}\,{\gamma}_5 + H_1^{{{<\kern -0.3 em{\scriptscriptstyle )}}}}\, \frac{{\sigma}_{\mu\nu}\, R_{{\scriptscriptstyle T}}^\mu\, n_-^\nu}{M_1 M_2} + H_1^{\perp}\, \frac{{\sigma}_{\mu\nu}\, k_{{\scriptscriptstyle T}}^\mu\, n_-^\nu}{M_1 M_2} \right\} \; , \label{eq:delta}\end{aligned}$$ where $n_\pm={\left[\;1\mp 1\;,\;1\pm 1\;,\;{\vec 0}_{{\scriptscriptstyle T}}\;\right]}/2$ are light-cone versors and $R\equiv (P_1-P_2)/2$ is the relative momentum of the hadron pair. For convenience, we will choose a frame where, besides $\vec P_{{\scriptscriptstyle T}}= 0$, we have also $\vec P_{h{{\scriptscriptstyle T}}} = 0$. By defining the light-cone momentum fraction $\xi = P_1^-/P_h^-$, we can parametrize the final-state momenta as $$\begin{aligned} k&=& {\left[\;\frac{P_h^-}{z}\;,\;z\frac{k^2+{\vec k}_{{\scriptscriptstyle T}}^2}{2P_h^-}\;,\;{\vec k}_{{\scriptscriptstyle T}}\;\right]} \;,{\nonumber}\\ P_1&=& {\left[\;\xi\,P_h^-\;,\;\frac{M_1^2+{\vec R}_{{\scriptscriptstyle T}}^2}{2\,\xi\,P_h^-}\;,\;{\vec R}_{{\scriptscriptstyle T}}\;\right]} \;,{\nonumber}\\ P_2&=& {\left[\;(1-\xi)\,P_h^-\;,\;\frac{M_2^2+{\vec R}_{{\scriptscriptstyle T}}^2}{2\,(1-\xi)\,P_h^-}\;,\;-{\vec R}_{{\scriptscriptstyle T}}\;\right]} \;. \label{eq:vectors}\end{aligned}$$ From the definition of the invariant mass of the hadron pair, i.e. $M_h^2 \equiv P_h^2 = 2 P_h^+ P_h^-$, and the on-shell condition for the two hadrons themselves, $P_1^2=M_1^2 , P_2^2=M_2^2$, we deduce the relation $${\vec R}_{{\scriptscriptstyle T}}^2=\xi\,(1-\xi)\,M_h^2-(1-\xi)\,M_1^2-\xi\,M_2^2 \label{eq:rt2}$$ which in turn puts a constraint on the invariant mass from the positivity requirement ${\vec R}_{{\scriptscriptstyle T}}^2 \geq 0$: $$M_h^2 \geq \frac{M_1^2}{\xi}+\frac{M_2^2}{1-\xi} \; . \label{eq:mh2}$$ After having given all the details of the kinematics, we can specify the actual dependence of the quark-quark correlator $\Delta$ and of the FF. From the frame choice $\vec P_{h{{\scriptscriptstyle T}}} = 0$, the on-shell condition for both hadrons, Eq. (\[eq:rt2\]), the constraint on $k^-$ and the integration over $k^+$ implied by the definition of $\Delta$ in Eq. (\[eq:delta\]), we deduce that the actual number of independent components of the three 4-vectors $k,P_1,P_2$ is five (cf. [@Bianconi:2000cd]). They can conveniently be chosen as the fraction of quark momentum carried by the hadron pair, $z$, the subfraction in which this momentum is further shared inside the pair, $\xi$, and the “geometry” of the pair in the momentum space. Namely, the “opening” of the pair momenta, ${\vec R}_{{\scriptscriptstyle T}}^2$, the relative position of the jet axis and the hadron pair axis, ${\vec k}_{{\scriptscriptstyle T}}^2$, and the relative position of hadron pair plane and the plane formed by the jet axis and the hadron pair axis, ${\vec k}_{{\scriptscriptstyle T}}\cdot {\vec R}_{{\scriptscriptstyle T}}$ (see Fig. \[fig:kin\]). Both DF and FF can be deduced from suitable projections of the corresponding quark-quark correlators. In particular, by defining $$\Delta^{[\Gamma]} (z,\xi,{\vec k}_{{\scriptscriptstyle T}}^2,{\vec R}_{{\scriptscriptstyle T}}^2,{\vec k}_{{\scriptscriptstyle T}}\cdot {\vec R}_{{\scriptscriptstyle T}}) \equiv \frac{1}{4z}\left.\int \d k^+\;\mbox{Tr}[\Gamma \, \Delta(k,P_1,P_2)] \right|_{k^-=P_h^-/z} \; , \label{eq:proj}$$ we can deduce \[eq:ff\] $$\begin{aligned} \Delta^{[{\gamma}^-]} &=& D_1(z_h,\xi,{\vec k}_{{\scriptscriptstyle T}}^{\,2},{\vec R}_{{\scriptscriptstyle T}}^{\,2}, {\vec k}_{{\scriptscriptstyle T}}\cdot {\vec R}_{{\scriptscriptstyle T}}) \label{eq:d1} \\[2mm] \Delta^{[{\gamma}^- {\gamma}_5]}&=& \frac{{\epsilon}_{{\scriptscriptstyle T}}^{ij} \,R_{{{\scriptscriptstyle T}}i}\,k_{{{\scriptscriptstyle T}}j}}{M_1\,M_2}\; G_1^\perp (z_h,\xi,{\vec k}_{{\scriptscriptstyle T}}^{\,2},{\vec R}_{{\scriptscriptstyle T}}^{\,2}, {\vec k}_{{\scriptscriptstyle T}}\cdot {\vec R}_{{\scriptscriptstyle T}}) \label{eq:g1} \\[2mm] \Delta^{[i{\sigma}^{i-} {\gamma}_5]} &=& {\epsilon_{{\scriptscriptstyle T}}^{ij}R_{{{\scriptscriptstyle T}}j}\over M_1+M_2}\, H_1^{{{<\kern -0.3 em{\scriptscriptstyle )}}}}(z_h,\xi,{\vec k}_{{\scriptscriptstyle T}}^{\,2},{\vec R}_{{\scriptscriptstyle T}}^{\,2}, {\vec k}_{{\scriptscriptstyle T}}\cdot {\vec R}_{{\scriptscriptstyle T}}) + {\epsilon_{{\scriptscriptstyle T}}^{ij}k_{{{\scriptscriptstyle T}}j}\over M_1+M_2}\, H_1^\perp(z_h,\xi,{\vec k}_{{\scriptscriptstyle T}}^{\,2},{\vec R}_{{\scriptscriptstyle T}}^{\,2}, {\vec k}_{{\scriptscriptstyle T}}\cdot {\vec R}_{{\scriptscriptstyle T}}) \; . \label{eq:h1} \end{aligned}$$ The leading-twist projections give a nice probabilistic interpretation of FF related to the Dirac operator $\Gamma$ used. Hence, $D_1$ is the probability for a unpolarized quark to fragment into the unpolarized hadron pair, $G_1^\perp$ is the probability difference for a longitudinally polarized quark with opposite chiralities to fragment into the pair, both $H_1^\perp$ and $H_1^{{{<\kern -0.3 em{\scriptscriptstyle )}}}}$ give the same probability difference but for a transversely polarized fragmenting quark. A different interpretation for $H_1^\perp$ and $H_1^{{{<\kern -0.3 em{\scriptscriptstyle )}}}}$ comes only from the possible origin for a non-vanishing probability difference, which is induced by the direction of $k_{{\scriptscriptstyle T}}$ and $R_{{\scriptscriptstyle T}}$, respectively. $G_1^\perp , H_1^\perp, H_1^{{{<\kern -0.3 em{\scriptscriptstyle )}}}}$ are all [*naive*]{} T-odd and $H_1^\perp, H_1^{{{<\kern -0.3 em{\scriptscriptstyle )}}}}$ are further chiral-odd. $H_1^{{{<\kern -0.3 em{\scriptscriptstyle )}}}}$ represents a genuine new effect with respect to the Collins one, because it relates the transverse polarization of the fragmenting quark to the orbital angular motion of the transverse component of the pair relative momentum ${\vec R}_{{\scriptscriptstyle T}}$ via the new angle $\phi$ defined by $$\sin \phi = \frac{{\vec S}'_{{\scriptscriptstyle T}}\cdot {\vec P}_2 \times {\vec P}_1} {|{\vec S}'_{{\scriptscriptstyle T}}| |{\vec P}_2 \times {\vec P}_1|} = \frac{{\vec S}'_{{\scriptscriptstyle T}}\cdot {\vec P}_h \times {\vec R}} {|{\vec S}'_{{\scriptscriptstyle T}}| |{\vec P}_h \times {\vec R}|} \equiv \frac{{\vec S}'_{{\scriptscriptstyle T}}\cdot {\vec P}_h \times {\vec R}_{{\scriptscriptstyle T}}} {|{\vec S}'_{{\scriptscriptstyle T}}| |{\vec P}_h \times {\vec R}_{{\scriptscriptstyle T}}|} = \cos \left( \phi_{S'_{{\scriptscriptstyle T}}} - \frac{\pi}{2} - \phi_{R_{{\scriptscriptstyle T}}} \right) = \sin (\phi_{S_{{\scriptscriptstyle T}}} + \phi_{R_{{\scriptscriptstyle T}}}) \; , \label{eq:angle}$$ where we have used the condition ${\vec P}_{h{{\scriptscriptstyle T}}} = 0$ and $\phi_{S_{{\scriptscriptstyle T}}}$ ($\phi_{S^\prime_{{\scriptscriptstyle T}}}$), $\phi_{R_{{\scriptscriptstyle T}}}$ are the azimuthal angles of the initial (final) quark transverse polarization and of ${\vec R}_{{\scriptscriptstyle T}}$ with respect to the scattering plane, respectively (see also Fig. \[fig:kin\]). Isolating transversity from the SSA {#subsec:h1fromssa} ----------------------------------- Usually, the analysis of experimental observables is better accomplished in the frame where the target momentum $P$ and the momentum transfer $q$ are collinear and with no transverse components. Using a different notation, we have ${\vec P}_\perp = {\vec q}_\perp = 0$ and ${\vec P}_{h\perp} \neq 0$. An appropriate transverse Lorentz boost transforms this frame to the previous one where ${\vec P}_{{\scriptscriptstyle T}}= {\vec P}_{h{{\scriptscriptstyle T}}} = 0$ and ${\vec q}_{{\scriptscriptstyle T}}= -{\vec P}_{h\perp}/z$ [@Bianconi:2000cd]. However, the difference between the components of vectors in each frame is suppressed like ${\cal O}(1/Q)$. Since we are here considering expressions for the observables at leading twist only, this difference can be safely neglected. By using Eq. (\[eq:rt2\]), the complete cross section at leading twist for the two-hadron inclusive DIS of a unpolarized beam on a transversely polarized target, where two unpolarized hadrons are detected in the same quark current jet, is given by $$\begin{aligned} \lefteqn{ \frac{\d\sigma} {\d\Omega\,\d x\,\d z\,\d\xi\,\d^2{\vec P}_{h\perp}\, \d M_h^2\,\d\phi_{R_\perp}} \, =\frac{\xi (1-\xi)}{2} \, \frac{\d\sigma} {\d\Omega\,\d x\,\d z\,\d\xi\,\d^2{\vec P}_{h\perp}\, \d^2{\vec R}_\perp}} {\nonumber}\\[2mm] \lefteqn{ \phantom{ \frac{\d\sigma} {\d\Omega\,\d x\,\d z\,\d\xi\,\d^2{\vec P}_{h\perp}\, \d M_h^2\,\d\phi_{R_\perp}} \,} = \frac{\d\sigma_{OO}} {\d\Omega\,\d x\,\d z\,\d\xi\,\d^2{\vec P}_{h\perp}\, \d M_h^2\,\d\phi_{R_\perp}} \, + \, |{\vec S}_\perp| \, \frac{\d\sigma_{OT}} {\d\Omega\,\d x\,\d z\,\d\xi\,\d^2{\vec P}_{h\perp}\, \d M_h^2\,\d\phi_{R_\perp}}} {\nonumber}\\[2mm] & = & \frac{\alpha_{em} sx}{(2\pi )^3 2 Q^4} \, \Bigg\{ {}A(y)\;{\cal F}\left[f_1 \, D_1\right] {\nonumber}\\[2mm] && {}+|{\vec R}_\perp|\;B(y)\;\sin(\phi_h+\phi_{R_\perp})\; {\cal F}\left[\,{\hat g}\!\cdot \!\vec p_{{\scriptscriptstyle T}}^{}\, \frac{h_1^{\perp} \, H_1^{{{<\kern -0.3 em{\scriptscriptstyle )}}}}}{M(M_1+M_2)}\right] {}-|{\vec R}_\perp|\;B(y)\;\cos(\phi_h+\phi_{R_\perp})\; {\cal F}\left[\,{\hat h}\!\cdot \!\vec p_{{\scriptscriptstyle T}}^{}\, \frac{h_1^{\perp} \, H_1^{{{<\kern -0.3 em{\scriptscriptstyle )}}}}}{M(M_1+M_2)}\right] {\nonumber}\\[2mm] && {}-B(y)\;\cos(2\phi_h)\; {\cal F}\left[\left(2\,{\hat h}\!\cdot\!\vec p_{{\scriptscriptstyle T}}^{}\, \,{\hat h}\!\cdot \! \vec k_{{\scriptscriptstyle T}}^{}\, -\,\vec p_{{\scriptscriptstyle T}}^{}\!\cdot \! \vec k_{{\scriptscriptstyle T}}^{}\,\right) \frac{h_1^{\perp} \, H_1^{\perp}}{M(M_1+M_2)}\right] {\nonumber}\\[2mm] && {}-B(y)\;\sin(2\phi_h)\; {\cal F}\left[\left( \,{\hat h}\!\cdot\!\vec p_{{\scriptscriptstyle T}}^{}\, \,{\hat g}\!\cdot \! \vec k_{{\scriptscriptstyle T}}^{}\, +\,{\hat h}\!\cdot\!\vec k_{{\scriptscriptstyle T}}^{}\, \,{\hat g}\!\cdot \! \vec p_{{\scriptscriptstyle T}}^{}\,\right) \frac{h_1^{\perp} \, H_1^{\perp}}{M(M_1+M_2)}\right] \Bigg\} {\nonumber}\\[2mm] & + &\frac{\alpha_{em} sx}{(2\pi )^3 2 Q^4} \, |{\vec S}_\perp| \, \Bigg\{ A(y)\;\sin(\phi_h-\phi_{S_\perp})\; {\cal F}\left[\,{\hat h}\!\cdot \!\vec p_{{\scriptscriptstyle T}}^{}\, \frac{f_{1T}^{\perp} \, D_1}{M}\right] \, + \, A(y)\;\cos(\phi_h-\phi_{S_\perp})\; {\cal F}\left[\,{\hat g}\!\cdot \!\vec p_{{\scriptscriptstyle T}}^{}\, \frac{f_{1T}^{\perp} \, D_1}{M}\right]{\nonumber}\\ && \quad {}+ B(y)\;\sin(\phi_h+\phi_{S_\perp}) {\cal F}\left[\,{\hat h}\!\cdot \!\vec k_{{\scriptscriptstyle T}}^{}\, \frac{h_1 \, H_1^{\perp}}{M_1+M_2}\right] \, + \, B(y)\;\cos(\phi_h+\phi_{S_\perp}) {\cal F}\left[\,{\hat g}\!\cdot \!\vec k_{{\scriptscriptstyle T}}^{}\, \frac{h_1 \, H_1^{\perp}}{M_1+M_2}\right]{\nonumber}\\ && \quad {}+ |{\vec R}_\perp|\;B(y)\;\sin(\phi_{R_\perp}+\phi_{S_\perp})\; {\cal F}\left[\frac{h_1 \, H_1^{{{<\kern -0.3 em{\scriptscriptstyle )}}}}}{M_1+M_2}\right]{\nonumber}\\ && \quad {}- |{\vec R}_\perp|\;A(y)\; \cos(\phi_h-\phi_{S_\perp})\;\sin(\phi_h-\phi_{R_\perp})\; {\cal F}\left[\,{\hat h}\!\cdot \!\vec k_{{\scriptscriptstyle T}}^{}\, \,{\hat h}\!\cdot \!\vec p_{{\scriptscriptstyle T}}^{}\, \frac{g_{1T} \, G_1^{\perp}}{MM_1M_2}\right]{\nonumber}\\ && \quad {}+ |{\vec R}_\perp|\;A(y)\; \sin(\phi_h-\phi_{S_\perp})\;\sin(\phi_h-\phi_{R_\perp})\; {\cal F}\left[\,{\hat h}\!\cdot \!\vec k_{{\scriptscriptstyle T}}^{}\, \,{\hat g}\!\cdot \!\vec p_{{\scriptscriptstyle T}}^{}\, \frac{g_{1T} \, G_1^{\perp}}{MM_1M_2}\right]{\nonumber}\\ && \quad {}- |{\vec R}_\perp|\;A(y)\; \cos(\phi_h-\phi_{S_\perp})\;\cos (\phi_h-\phi_{R_\perp})\; {\cal F}\left[\,{\hat g}\!\cdot \!\vec k_{{\scriptscriptstyle T}}^{}\, \,{\hat h}\!\cdot \!\vec p_{{\scriptscriptstyle T}}^{}\, \frac{g_{1T} \, G_1^{\perp}}{MM_1M_2}\right]{\nonumber}\\ && \quad {}+ |{\vec R}_\perp|\;A(y)\; \sin(\phi_h-\phi_{S_\perp})\;\cos(\phi_h-\phi_{R_\perp})\; {\cal F}\left[\,{\hat g}\!\cdot \!\vec k_{{\scriptscriptstyle T}}^{}\, \,{\hat g}\!\cdot \!\vec p_{{\scriptscriptstyle T}}^{}\, \frac{g_{1T} \, G_1^{\perp}}{MM_1M_2}\right]{\nonumber}\\ && \quad {}+ B(y)\;\cos(3\phi_h-\phi_{S_\perp})\; {\cal F}\left[\,{\hat h}\!\cdot \!\vec k_{{\scriptscriptstyle T}}^{}\, \,{\hat h}\!\cdot \!\vec p_{{\scriptscriptstyle T}}^{}\, \,{\hat g}\!\cdot \!\vec p_{{\scriptscriptstyle T}}^{}\, \frac{h_{1T}^{\perp} \, H_1^{\perp}}{M^2(M_1+M_2)}\right]{\nonumber}\\ && \quad {}+ B(y)\;\sin(2\phi_h)\,\cos(\phi_h-\phi_{S_\perp})\; {\cal F}\left[\,{\hat h}\!\cdot \!\vec k_{{\scriptscriptstyle T}}^{}\, \left(\,{\hat h}\!\cdot \!\vec p_{{\scriptscriptstyle T}}^{}\,\right)^2 \frac{h_{1T}^{\perp} \, H_1^{\perp}}{M^2(M_1+M_2)}\right]{\nonumber}\\ && \quad {}- B(y)\;\cos(2\phi_h)\,\sin(\phi_h-\phi_{S_\perp})\; {\cal F}\left[\,{\hat h}\!\cdot \!\vec k_{{\scriptscriptstyle T}}^{}\, \left(\,{\hat g}\!\cdot \!\vec p_{{\scriptscriptstyle T}}^{}\,\right)^2 \frac{h_{1T}^{\perp} \, H_1^{\perp}}{M^2(M_1+M_2)}\right]{\nonumber}\\ && \quad {}- B(y)\;\sin(3\phi_h-\phi_{S_\perp})\; {\cal F}\left[\,{\hat g}\!\cdot \!\vec k_{{\scriptscriptstyle T}}^{}\, \,{\hat h}\!\cdot \!\vec p_{{\scriptscriptstyle T}}^{}\, \,{\hat g}\!\cdot \!\vec p_{{\scriptscriptstyle T}}^{}\, \frac{h_{1T}^{\perp} \, H_1^{\perp}}{M^2(M_1+M_2)}\right]{\nonumber}\\ && \quad {}+ B(y)\;\cos(2\phi_h)\,\cos(\phi_h-\phi_{S_\perp})\; {\cal F}\left[\,{\hat g}\!\cdot \!\vec k_{{\scriptscriptstyle T}}^{}\, \left(\,{\hat h}\!\cdot \!\vec p_{{\scriptscriptstyle T}}^{}\,\right)^2 \frac{h_{1T}^{\perp} \, H_1^{\perp}}{M^2(M_1+M_2)}\right]{\nonumber}\\ && \quad {}+ B(y)\;\sin(2\phi_h)\,\sin(\phi_h-\phi_{S_\perp})\; {\cal F}\left[\,{\hat g}\!\cdot \!\vec k_{{\scriptscriptstyle T}}^{}\, \left(\,{\hat g}\!\cdot \!\vec p_{{\scriptscriptstyle T}}^{}\,\right)^2 \frac{h_{1T}^{\perp} \, H_1^{\perp}}{M^2(M_1+M_2)}\right]{\nonumber}\\ && \quad {}+ |{\vec R}_\perp|\;B(y)\; \sin(2\phi_h+\phi_{R_\perp}-\phi_{S_\perp}) {\cal F}\left[\left(({\hat h}\!\cdot\!\vec p^{}_{{\scriptscriptstyle T}})^2 -({\hat g}\!\cdot\!\vec p^{}_{{\scriptscriptstyle T}})^2 +2\,{\hat h}\!\cdot \!\vec p_{{\scriptscriptstyle T}}^{}\, \,{\hat g}\!\cdot \!\vec p_{{\scriptscriptstyle T}}^{}\,\right) \frac{h_{1T}^{\perp} \, H_1^{{{<\kern -0.3 em{\scriptscriptstyle )}}}}}{2M^2(M_1+M_2)}\right] \Bigg\} \, , \label{eq:cross}\end{aligned}$$ where $\alpha_{em}$ is the fine structure constant, $s=Q^2/xy=-q^2/xy$ is the total energy in the center-of-mass system and $$A(y) = \left( 1-y+\frac{1}{2} y^2 \right) \quad , \quad B(y) = (1-y) \quad , \quad C(y) = y (2-y) \label{eq:abc}$$ with the lepton invariant $y=(P \cdot q)/ (P \cdot l) \approx q^-/l^-$. The convolution of distribution and fragmentation functions is defined as $${\cal F}\left[w({\vec p}_{{\scriptscriptstyle T}}^{},{\vec k}_{{\scriptscriptstyle T}}^{})\; f\, D\right] \equiv \; \sum_a e_a^2\; \int\d^{2}{\vec p}_{{\scriptscriptstyle T}}^{}\; \d^{2}{\vec k}_{{\scriptscriptstyle T}}^{}\; \delta^2 ({\vec k}_{{\scriptscriptstyle T}}^{}-{\vec p}_{{\scriptscriptstyle T}}^{}+\frac{{\vec P}_{h\perp}}{z}) \; w({\vec p}_{{\scriptscriptstyle T}}^{},{\vec k}_{{\scriptscriptstyle T}}^{})\; f^a(x,{\vec p}_{{\scriptscriptstyle T}}^{\;2})\,D^a(z_h,\xi,{\vec k}_{{\scriptscriptstyle T}}^{\,2},{\vec R}_{{\scriptscriptstyle T}}^{\,2}, {\vec k}_{{\scriptscriptstyle T}}\cdot {\vec R}_{{\scriptscriptstyle T}}) \;,$$ where $w({\vec p}_{{\scriptscriptstyle T}}^{},{\vec k}_{{\scriptscriptstyle T}}^{})$ is a weight function and the sum runs over all quark (and anti-quark) flavors, with $e_a$ the electric charges of the quarks. The versors appearing in the weight function $w$ are defined as ${\hat h} = {\vec P}_{h\perp} / |{\vec P}_{h\perp}|$ and ${\hat g}^i = \epsilon_{{\scriptscriptstyle T}}^{ij} \, {\hat h}^j$ (with $\epsilon_{{\scriptscriptstyle T}}^{ij} \equiv \epsilon^{-+ij}$), respectively, and they represent the two independent directions in the $\perp$ plane perpendicular to ${\hat z} \parallel {\vec q}/|{\vec q}|$. All azimuthal angles $\phi_{S_\perp}, \phi_{R_\perp}$ and $\phi_h$ (relative to ${\vec P}_{h\perp}$) lie in the $\perp$ plane and are measured with respect to the scattering plane (see Fig. \[fig:reaction\]). Eq. (\[eq:cross\]) corresponds to the sum of Eqs. (B1) and (B4) in Ref. [@Bianconi:2000cd], where, however, the expressions are simpler because they rely on the assumption of a symmetrical cylindrical distribution of hadron pairs around the jet axis, in order to have fragmentation functions depending on even powers of ${\vec k}_{{\scriptscriptstyle T}}$ only (this assumption would make all terms including the ${\hat g}$ versor disappear from Eq. (\[eq:cross\]); see also Ref. [@Barone:2001sp] for a comparison). During experiments the scattering plane changes (different scales $Q$ imply different positions of the scattered beam). Therefore, it is better to define the laboratory frame as the plane formed by the beam and the direction of the target polarization. All azimuthal angles are conveniently reexpressed with respect to the laboratory frame as $$\begin{aligned} \phi_{R_\perp} &= &\phi_{R_\perp}^L-\phi^L {\nonumber}\\ \phi_{S_\perp} &= &-\phi^L {\nonumber}\\ \phi_h &= &\phi_h^L - \phi^L \, , \label{eq:azangles}\end{aligned}$$ where the superscript $^L$ indicates the new reference frame. The oriented angle between the scattering plane and the laboratory frame is $\phi^L$ (see Fig. \[fig:reaction\]). At leading order, the azimuthal angle of Eq. (\[eq:angle\]) becomes $\phi = \phi_{R_\perp}^L- 2\phi^L$ in the new frame. The new expression for the cross section is obtained by simply replacing Eq. (\[eq:azangles\]) inside the angular dependence of Eq. (\[eq:cross\]). After replacement and apart from phase space coefficients, each term of the cross section will look like $$d\sigma^{tw} \, \propto \, t (\phi_{R_\perp}^L,\phi^L,\phi_h^L) \; {\cal F} \left[ w \; \mbox{DF} \; \mbox{FF} \right] \, = \, t(\phi_{R_\perp}^L,\phi^L,\phi_h^L) \; I(z,\xi,{\vec R}_{{\scriptscriptstyle T}}^{\, 2}) \, , \label{eq:term}$$ where $t$ is a trigonometric function, $w$ is the specific weight function for each combination of distribution and fragmentation functions (DF and FF, respectively), and $I$ is the result of the convolution integral. It is easy to verify that folding the cross section by $$\frac{1}{2\pi} \int_0^{2\pi} \d\phi^L \d\phi_{R_\perp}^L \; \sin(\phi_{R_\perp}^L -2\phi^L) \; \frac{\d\sigma} {\d\Omega\,\d x\,\d z\,\d\xi\,\d^2{\vec P}_{h\perp}\, \d M_h^2\,\d\phi_{R_\perp}} \label{eq:fold}$$ makes only those $d\sigma^{tw}$ terms survive where $H_1^{{{<\kern -0.3 em{\scriptscriptstyle )}}}}$ shows up in the convolution, i.e. for the following combinations \[eq:tws\] $$\begin{aligned} t = \cos (\phi_h^L + \phi_{R_\perp}^L -2 \phi^L) &\quad , &\quad w = {\hat h} \cdot {\vec p}_{{\scriptscriptstyle T}}\quad ; \label{eq:OOh} \\ t = \sin (\phi_h^L + \phi_{R_\perp}^L -2 \phi^L) &\quad , &\quad w = {\hat g} \cdot {\vec p}_{{\scriptscriptstyle T}}\quad ; \label{eq:OOg} \\ t = \sin (\phi_{R_\perp}^L - 2\phi^L) &\quad , &\quad w = 1 \quad ; \label{eq:OT1} \\ t = \sin (2\phi_h^L + \phi_{R_\perp}^L - 2\phi^L) &\quad , &\quad w = \left(({\hat h}\cdot {\vec p}_{{\scriptscriptstyle T}})^2 - ({\hat g}\cdot {\vec p}_{{\scriptscriptstyle T}})^2 + 2 \, {\hat h}\cdot {\vec p}_{{\scriptscriptstyle T}}\, {\hat g}\cdot {\vec p}_{{\scriptscriptstyle T}}\right) \label{eq:OThg} \, .\end{aligned}$$ Similarly, it is straightforward to proof that integrating these surviving terms upon $\d^2{\vec P}_{h\perp}$, and performing the integrals in the convolution ${\cal F}[w\;\mbox{DF}\; \mbox{FF}]$, makes only the combination (\[eq:OT1\]) to survive presenting the transversity in a factorized form. In fact, by integrating also upon $\d\xi$ we finally have $$\begin{aligned} \frac{<\d\sigma_{OT} >}{\d y\,\d x\,\d z\,\d M_h^2} &\equiv & \frac{1}{2\pi} \, \int_0^{2\pi} \d\phi^L\,\d\phi_{R_\perp}^L \int \d^2{\vec P}_{h\perp} \; \int \d\xi \; \sin(\phi_{R_\perp}^L -2\phi^L) \; \frac{\d\sigma}{\d\Omega\,\d x\,\d z\,\d\xi\,\d M_h^2\, \d\phi_{R_\perp}^L\,\d^2{\vec P}_{h\perp}} {\nonumber}\\ &= & \frac{\pi \alpha_{em}^2 s x}{(2 \pi)^3 Q^4} \; \frac{B(y) \, |{\vec S}_\perp |}{2(M_1+M_2)} \; \sum_a e_a^2\; \int \d^2{\vec p}_{{\scriptscriptstyle T}}\; h_1^a(x,{\vec p}_{{\scriptscriptstyle T}}^{\; 2}) {\nonumber}\\ & &\times \int \d\xi \; |{\vec R}_\perp | \int_0^{2 \pi} \d\phi_{R_\perp}^L \int \d^2{\vec k}_{{\scriptscriptstyle T}}\; H_1^{{{<\kern -0.3 em{\scriptscriptstyle )}}}\, a} (z,\xi,M_h^2,{\vec k}_{{\scriptscriptstyle T}}^{\; 2}, {\vec k}_{{\scriptscriptstyle T}}\cdot{\vec R}_{{\scriptscriptstyle T}}) {\nonumber}\\ &= & \frac{\pi \alpha_{em}^2 s}{(2 \pi)^3 Q^4} \; \frac{B(y) \, |{\vec S}_\perp |}{2(M_1+M_2)} \; \sum_a e_a^2\; x \; h_1^a(x) \; H_{1\, (R)}^{{{<\kern -0.3 em{\scriptscriptstyle )}}}\, a} (z,M_h^2) \, , \label{eq:crossfact}\end{aligned}$$ where, for sake of simplicity, the same notations are kept for DF and FF before and after integration, distinguished by the explicit arguments only; the subscript $_{(R)}$ reminds of the additional weighting factor $|\vec R_\perp|$. Analogously, $$\begin{aligned} \frac{<\d\sigma_{OO} >}{\d y\,\d x\,\d z\,\d M_h^2} &\equiv & \frac{1}{2\pi} \, \int_0^{2\pi} \d\phi^L\, \d\phi_{R_\perp}^L \int \d^2{\vec P}_{h\perp} \; \int \d\xi \; \frac{\d\sigma}{\d\Omega\,\d x\,\d z\,\d\xi\,\d M_h^2\, \d\phi_{R_\perp}^L\,\d^2{\vec P}_{h\perp}}{\nonumber}\\ &= & \frac{\pi \alpha_{em}^2 s x}{(2 \pi)^3 Q^4} \, A(y) \, \sum_a e_a^2\; \int \d^2{\vec p}_{{\scriptscriptstyle T}}\; f_1^a (x,{\vec p}_{{\scriptscriptstyle T}}^{\; 2}) {\nonumber}\\ & &\times \int \d\xi \int_0^{2 \pi} \d\phi_{R_\perp}^L \int \d^2{\vec k}_{{\scriptscriptstyle T}}\; D_1^a (z,\xi,M_h^2,{\vec k}_{{\scriptscriptstyle T}}^{\; 2}, {\vec k}_{{\scriptscriptstyle T}}\cdot {\vec R}_{{\scriptscriptstyle T}}) {\nonumber}\\ &= & \frac{\pi \alpha_{em}^2 s}{(2 \pi)^3 Q^4} \, A(y) \, \sum_a e_a^2\; x \; f_1^a(x) \; D_1^a (z,M_h^2) \, , \label{eq:crossfact2}\end{aligned}$$ from which we can build the single spin asymmetry $$\begin{aligned} A^{\sin \phi} (y,x,z,M_h^2) &\equiv & \frac{<\d\sigma_{OT} >}{\d y\,\d x\,\d z\,\d M_h^2} \, \left[ \frac{<\d\sigma_{OO} >}{\d y\,\d x\,\d z\,\d M_h^2} \right]^{-1} {\nonumber}\\ &= & \frac{B(y)}{A(y)} \; \frac{|{\vec S}_\perp |}{2(M_1+M_2)} \; \frac{\sum_a e_a^2 \; x \, h_1^a (x) \, H_{1 \, (R)}^{{{<\kern -0.3 em{\scriptscriptstyle )}}}\, a} (z,M_h^2)} {\sum_a e_a^2 \; x \, f_1^a (x) \, D_1^a (z,M_h^2)} \, . \label{eq:ssa}\end{aligned}$$ Spectator model for $\pi^+ \pi^-$ fragmentation {#sec:spectator} =============================================== In the field theoretical description of hard processes, the FF represent the soft processes that connect the hard quark to the detected hadrons via fragmentation, i.e. they are hadronic matrix elements of nonlocal operators built from quark (and gluon) fields [@Soper:1977jc]. For a quark fragmenting into two hadrons inside the same current jet, the appropriate quark-quark correlator (in the light-cone gauge) reads [@Collins:1994kq; @Collins:1994ax] $$\Delta_{ij}(k,P_1,P_2)= {\kern 0.2 em {\textstyle\sum} \kern -1.1 em \int_X}\; \int \frac{\d^{4\!}\zeta}{(2\pi)^4} \; e^{ik\cdot\zeta}\; \langle 0|\psi_i(\zeta)|P_1,P_2,X\rangle \langle X,P_2,P_1|{\overline}{\psi}_j(0)|0\rangle \, , \label{eq:defDelta}$$ where the sum runs over all the possible intermediate states containing the hadron pair. \ The basic idea of the spectator model is to make a specific ansatz for this spectral decomposition by replacing the sum with an effective spectator state with a definite mass and quantum numbers [@Meyer:1991fr; @Jakob:1997wg; @Bianconi:2000uc]. By specializing the model to the case of $\pi^+ \pi^-$ fragmentation with $P_1=P_{\pi^+}$ and $P_2=P_{\pi^-}$, the spectator has the quantum numbers of an on-shell valence quark with a constituent mass $m_q=340$ MeV. Consequently, the quark-quark correlator (\[eq:defDelta\]) simplifies to $$\begin{aligned} \Delta_{ij}(k,P_{\pi^+},P_{\pi^-}) &\approx& \frac{\theta\!\left((k-P_h)^+\right)}{(2\pi)^3} \; \delta\left((k-P_h)^2-m_q^2\right)\; \langle 0|\psi_i(0)|P_{\pi^+},P_{\pi^-},q\rangle \langle q,P_{\pi^-},P_{\pi^+}| {\overline}{\psi}_j(0)|0\rangle {\nonumber}\\ &\equiv& \widetilde\Delta_{ij}(k,P_{\pi^+},P_{\pi^-})\; \delta(\tau_h-{\sigma}_h+M_h^2-m_q^2) \;, \label{eq:specDelta}\end{aligned}$$ where $\tau_h = k^2$ and $\sigma_h = 2k\cdot P_h$. When inserting Eq. (\[eq:specDelta\]) into Eq. (\[eq:proj\]), the projections drastically simplify to $$\Delta^{[\Gamma]}(z_h,\xi,{\vec k}_{{\scriptscriptstyle T}}^{\,2},\vec R_{{\scriptscriptstyle T}}^{\, 2}, {\vec k}_{{\scriptscriptstyle T}}\cdot {\vec R}_{{\scriptscriptstyle T}})= \left.\frac{\mbox{Tr}[\Gamma \, \widetilde\Delta]}{8(1-z)P_h^-} \right|_{\tau_h=\tau_h(z,{\vec k}_{{\scriptscriptstyle T}}^{\,2})} \; , \label{eq:projspect}$$ with $$\tau_h(z,{\vec k}_{{\scriptscriptstyle T}}^{\,2})=\frac{z}{1-z}{\vec k}_{{\scriptscriptstyle T}}^{\,2} +\frac{m_q^2}{1-z}+\frac{M_h^2}{z} \;. \label{eq:tauspect}$$ We will consider the $\pi^+ \pi^-$ system with an invariant mass $M_h$ close to the $\rho$ resonance, specifically $m_\rho - \Gamma_\rho/2 \leq M_h \leq m_\rho + \Gamma_\rho/2$, where $\Gamma_\rho$ is the width of the $\rho$ resonance. Hence, the most appropriate and simplest diagrams that can replace the quark decay of Fig. \[fig:handbag\] at leading twist, and leading order in $\alpha_s$, are represented in Fig. \[fig:diagspec\]: the $\pi^+ \pi^-$ can be produced from the $\rho$ decay or directly via a quark exchange in the $t$-channel (the background diagram); the quantum interference of the two processes generates the [*naive*]{} T-odd FF described in Sec. \[subsec:iff\]. A suitable selection of “Feynman” rules for the vertices and propagators of the diagrams in Fig. \[fig:diagspec\] allows for the analytic calculation of the matrix elements defining $\widetilde \Delta$ in Eq. (\[eq:specDelta\]) and, consequently, of the projections $\Delta^{[\Gamma]}$ defining the FF. Propagators {#sec:propagators} ----------- The propagators involved in the diagrams of Fig. \[fig:diagspec\] are: - quark with momentum $\kappa$ $$\begin{aligned} \left(\frac{i}{{\kern 0.2 em \kappa\kern -0.45em /}-m_q}\right)_{ij} \end{aligned}$$ The propagator occurs with $\kappa^2 = \tau_h \equiv k^2$ or $\kappa^2 = (k-P_{\pi^+})^2$. In both cases, the off-shell condition $k^2 \neq m_q^2$ is guaranteed by Eq. (\[eq:tauspect\]). - $\rho$ with momentum $P_h$ $$\begin{aligned} & &\frac{i}{P_h^2-m_\rho^2+im_\rho\Gamma_\rho} \left(-g^{\mu\nu}+\frac{P_h^\mu\,P_h^\nu}{P_h^2}\right) \end{aligned}$$ where $\Gamma_{\rho} = \displaystyle{\frac{f^2_{\rho \pi \pi}}{4\pi} \frac{m_{\rho}}{12} \left( 1 - \frac{4 m_{\pi}^2}{m_{\rho}^2} \right)^{\frac{3}{2}}}$ [@Ioffe:1984ep]. Vertices {#sec:vertices} -------- In analogy with previous works on spectator models [@Jakob:1997wg; @Bianconi:2000uc], we choose the vertex form factors to depend on one invariant only, generally denoted $\kappa^2$, that represents the virtuality of the external entering quark line. Therefore, we can have $\kappa^2 = \tau_h \equiv k^2$ or $\kappa^2 = (k-P_{\pi^+})^2$. The power laws are such that the asymptotic behaviour is in agreement with the expectations based on dimensional counting rules. Finally, the normalization coefficients have dimensions such that $\int \d^2{\vec k}_{{\scriptscriptstyle T}}\int \d^2{\vec R}_{{\scriptscriptstyle T}}\, D_1(z,\xi,{\vec k}_{{\scriptscriptstyle T}}^2, {\vec R}_{{\scriptscriptstyle T}}^2,{\vec k}_{{\scriptscriptstyle T}}\cdot {\vec R}_{{\scriptscriptstyle T}})$ is a pure number to be interpreted as the probability for the hadron pair to carry a $z$ fraction of the valence quark momentum and to share it in $\xi$ and $1-\xi$ parts. - $\rho \pi \pi$ vertex $$\begin{aligned} & &\Upsilon^{\rho\pi\pi, \mu} = f_{\rho\pi\pi} R^\mu \end{aligned}$$ where $\displaystyle{\frac{f_{\rho\pi\pi}^2}{4\pi}}=2.84\pm 0.50$ [@Ioffe:1984ep]. - $q \rho q$ vertex $$\begin{aligned} \Upsilon^{q\rho q,\mu}_{ij}&=&\frac{f_{q\rho q} (\kappa^2)}{\sqrt{2}} \; [{\gamma}^\mu]_{ij} {\nonumber}\\ &= &\frac{N_{q\rho}}{\sqrt{2}} \; \frac{1}{|\kappa^2-\Lambda_{\rho}^2|^{\alpha}} \; [{\gamma}^\mu]_{ij} \end{aligned}$$ where $\Lambda_\rho$ excludes large virtualities of the quark. The power $\alpha$ is determined consistently with the quark counting rule that determines the asymptotic behaviour of the FF at large $z$ [@Ioffe:1984ep], i.e. $$(1-z)^{2\alpha -1} = (1-z)^{-3+2r+2|\lambda|} \, , \label{eq:alpha}$$ where $r$ is the number of constituent quarks in the considered hadron, and $\lambda$ is the difference between the quark and the hadron helicities. Thus, here we have $\alpha = 3/2$. The normalization $N_{q\rho}$ is such that the sum rule $$\int_0^1 \d z \; z\,D_1(z) \leq 1 \label{eq:sumrule}$$ is satisfied. In fact, in the infinite momentum frame the integral in Eq. (\[eq:sumrule\]) represents the total fraction $z$ of the quark energy taken by all hadron pairs of the type under consideration. Since in this frame low-energy mass effects can be neglected, we estimate that charged pion pairs with an invariant mass inside the $\rho$ resonance width represent $\sim 50\%$ of the total pions detected in the calorimeter, which in turn can be considered $\sim 80\%$ of all particles detected. Neglecting mass effects, we may assume that the fraction of quark energy taken by charged pions, relative to the energy taken by other hadrons, follows their relative numbers. Therefore, we chose two values, $N_{q\rho} = 0.9$ GeV$^3$ and $1.6$ GeV$^3$, which correspond to rather extreme scenarios where the integral Eq. (\[eq:sumrule\]) amounts to 0.14 and 0.48, respectively. - $q \pi q$ vertex $$\begin{aligned} \Upsilon^{q\pi q}_{ij}&=&\frac{f_{q\pi q} (\kappa^2)}{\sqrt{2}} \; [{\gamma}_5]_{ij} \nonumber \\ &= &\frac{N_{q\pi}}{\sqrt{2}} \, \frac{1}{|\kappa^2-\Lambda_{\pi}^2|^\alpha} \;[{\gamma}_5]_{ij} \end{aligned}$$ where $\Lambda_\pi$ excludes large virtualities of the quark, as well. From quark counting rules, still $\alpha = 3/2$. The normalization $N_{q\pi}$ can be deduced from $N_{q\rho}$ by generalizing the Goldberger-Treiman relation to the $\rho$-quark coupling [@Glozman:1998fs]: $$\begin{aligned} \frac{g_{\pi qq}^2}{4\pi} &= &\left( \frac{g_q^A}{g_N^A} \right)^2 \left( \frac{m_q}{m_N} \right)^2 \frac{g_{\pi NN}^2}{4\pi} = \left( \frac{3}{5} \right)^2 \left( \frac{340}{939} \right)^2 14.2 = 0.67 {\nonumber}\\ \frac{(g_{\rho qq}^V+g_{\rho qq}^T)^2}{4\pi} &= & \left( \frac{g_q^A}{g_N^A} \right)^2 \left( \frac{m_q}{m_N} \right)^2 \frac{(g_{\rho NN}^V+g_{\rho NN}^T)^2}{4\pi} = \left( \frac{3}{5} \right)^2 \left( \frac{340}{939} \right)^2 27.755 = 1.31 \, , \label{eq:g-t} \end{aligned}$$ where $g^A_N, m_N$ are the nucleon axial coupling constants and mass, respectively, as well as $g^A_q, m_q$ the quark ones. The $\pi NN$ coupling is $g_{\pi NN}^2 /4\pi = 14.2$; the vector $\rho NN$ coupling is $(g_{\rho NN}^V)^2 /4\pi = 0.55$ and its ratio to the tensor coupling is $g_{\rho NN}^T / g_{\rho NN}^V=6.105$ [@Ericson:1988gk]. From the above relations, we deduce $$\frac{g_{\pi qq}}{(g_{\rho qq}^V+g_{\rho qq}^T)} \equiv \frac{N_{q\pi}}{N_{q\rho}} = 0.715 \, . \label{eq:qpi-coup}$$ As a final comment, we have explicitly checked that with the above rules the background diagram leads to a cross section that qualitatively shows the same $s$ dependence of experimental data for $\pi \pi$ production in the relative $L=0$ channel when $s$ is inside the $\rho$ resonance width, in any case below the first dip corresponding to the resonance $f_0 (980)$ [@Pennington:1999fa]. If we reasonably assume that the resonant diagram exhausts almost all of the $\pi \pi$ production in the relative $L=1$ channel and we also assume that in the given energy interval the $L=0,1$ channels approximate the whole strength for $\pi \pi$ production, we can safely state that the diagrams of Fig. \[fig:diagspec\] give a satisfactory reproduction of the $\pi \pi$ cross section, with invariant mass in the given interval, without invoking any scalar $\sigma$ resonance (cf. [@Collins:1994ax; @Jaffe:1998hf]). Interference FF {#sec:speciff} --------------- With the above rules applied to the diagrams of Fig. \[fig:diagspec\], we can calculate all the matrix elements of Eq. (\[eq:specDelta\]) and, consequently, all the projections (\[eq:projspect\]) leading to the FF. The [*naive*]{} T-odd $G_1^\perp, H_1^\perp, H_1^{{{<\kern -0.3 em{\scriptscriptstyle )}}}}$ receive contributions from the interference diagrams only. In particular, they result proportional to the imaginary part of the $\rho$ propagator ($\sim m_\rho \Gamma_\rho$), while the real part ($\sim M_h^2 - m_\rho^2$) contributes to $D_1$. Therefore, contrary to the findings of Ref. [@Jaffe:1998hf], a complex amplitude with a resonant behaviour is needed here to produce nonvanishing interference FF. For a $u$ quark fragmenting into $\pi^+ \pi^-$, we have at leading twist $$\begin{aligned} \lefteqn{ D_1^{u\rightarrow \pi^+ \pi^-} (z,\xi,M_h^2,k_{{\scriptscriptstyle T}}^2,{\vec k}_{{\scriptscriptstyle T}}\cdot {\vec R}_{{\scriptscriptstyle T}}) \;=\; \frac{N_{q\rho}^2 \, f_{\rho\pi\pi}^2 \, z^2 \, (1-z)^2} {4(2\pi)^3 \, [(M_h^2-m_\rho^2)^2+m_\rho^2\,\Gamma_\rho^2] \, a^2 \, \vert a+b\vert^3} {\nonumber}}\\ & & \qquad\qquad\times \Bigg\{ \frac{c}{4} [ \, c-za(2\xi -1) ] + z^2 (1-z) \left( \frac{M_h^2}{4}-m_\pi^2 \right) [ \, a-(1-z)M_h^2 \, ] \Bigg\} {\nonumber}\\[3mm] & & {}+ \frac{N_{q\pi}^4 \, z^7 \, (1-z)^7} {8 (2\pi )^3 \, a^2 \, d^2 \, \vert d+{\tilde b} \vert^3 \, \vert a+{\tilde b}\vert^3} {\nonumber}\\[2mm] & & \qquad\times \Bigg\{ -az \, [ z\xi (1-z) + (1-\xi) d ] -z(1-z) m_\pi^2 \, \frac{a-c-z(1-z)M_h^2}{2} \, + \, d \, \frac{a-c+z(1-z)M_h^2}{2} \Bigg\}{\nonumber}\\[3mm] & & {} + \frac{\sqrt{2} \, (M_h^2-m_\rho^2) \, z^{\frac{9}{2}} \, (1-z)^{\frac{9}{2}} \, N_{q\pi}^2 \, N_{q\rho} \, f_{\rho \pi \pi}} {8 (2\pi)^3 [(M_h^2-m_\rho^2)^2+m_\rho^2\,\Gamma_\rho^2] \, a^2 \, d \, \vert a+b \vert^{\frac{3}{2}} \, \vert a+{\tilde b} \vert^{\frac{3}{2}} \, \vert d+{\tilde b} \vert^{\frac{3}{2}}} {\nonumber}\\[2mm] & & \qquad\times \Bigg\{ az(1-z) \left( 2m_\pi^2 - \frac{M_h^2}{2}\right) + \frac{a(1-2z\xi) +c +z(1-z)M_h^2}{4} \, [ \, d+z(1-z) (M_h^2-5m_\pi^2) ] {\nonumber}\\[2mm] & & \qquad\qquad {}+ \, \frac{a \, [2z(1-\xi)-1]+c-z(1-z)M_h^2}{4} \, [\, 3d -a +z(1-z)m_\pi^2 \, ] \Bigg\} \label{eq:d1spec}\\[5mm] \lefteqn{ H_1^{{{<\kern -0.3 em{\scriptscriptstyle )}}}\, u\rightarrow \pi^+ \pi^-} (z,\xi,M_h^2,k_{{\scriptscriptstyle T}}^2,{\vec k}_{{\scriptscriptstyle T}}\cdot {\vec R}_{{\scriptscriptstyle T}}) =} {\nonumber}\\[2mm] & & {}-\, \frac{m_\rho \, \Gamma_\rho \, m_\pi \, m_q \, z^{\frac{13}{2}} \, (1-z)^{\frac{11}{2}} \, N_{q\pi}^2 \, N_{q\rho} \, f_{\rho \pi \pi}} {2\sqrt{2} (2\pi)^3 [(M_h^2-m_\rho^2)^2+m_\rho^2\Gamma_\rho^2] \, a \, d \, \vert a+b\vert^{\frac{3}{2}} \, \vert a+{\tilde b}\vert^{\frac{3}{2}} \, \vert d + z(1-z)(m_q^2-\Lambda_\pi^2)\vert^{\frac{3}{2}}} \label{eq:h1spec} \\[5mm] \lefteqn{ H_1^{\perp \, u\rightarrow \pi^+ \pi^-} (z,\xi,M_h^2,k_{{\scriptscriptstyle T}}^2,{\vec k}_{{\scriptscriptstyle T}}\cdot {\vec R}_{{\scriptscriptstyle T}}) = 0 } \label{eq:h10} \\[5mm] & & G_1^{\perp \, u\rightarrow \pi^+ \pi^-} (z,\xi,M_h^2,k_{{\scriptscriptstyle T}}^2,{\vec k}_{{\scriptscriptstyle T}}\cdot {\vec R}_{{\scriptscriptstyle T}}) = - \frac{m_\pi}{2m_q} \, H_1^{{{<\kern -0.3 em{\scriptscriptstyle )}}}\, u\rightarrow \pi^+ \pi^-} (z,\xi,M_h^2,k_{{\scriptscriptstyle T}}^2,{\vec k}_{{\scriptscriptstyle T}}\cdot {\vec R}_{{\scriptscriptstyle T}}) \, , \label{eq:g1perp}\end{aligned}$$ where $$\begin{aligned} a & = & z^2(k_{{\scriptscriptstyle T}}^2+m_q^2)+(1-z) M_h^2 \quad , \quad b= z(1-z)(m_q^2-\Lambda_\rho^2) \quad , \quad {\tilde b} = z(1-z)(m_q^2 - \Lambda_\pi^2) {\nonumber}\\ c & = &(2\xi -1) [z^2(k_{{\scriptscriptstyle T}}^2+m_q^2) -(1-z)^2M_h^2] -4z(1-z) {\vec k}_{{\scriptscriptstyle T}}\cdot {\vec R}_{{\scriptscriptstyle T}}{\nonumber}\\ d & = & z^2(1-\xi) (k_{{\scriptscriptstyle T}}^2+m_q^2) +\xi (1-z)^2 M_h^2 +z(1-z)(m_\pi^2 + 2{\vec k}_{{\scriptscriptstyle T}}\cdot {\vec R}_{{\scriptscriptstyle T}}) \, . \label{eq:coeffs}\end{aligned}$$ The simplifications induced by the spectator model reduce the number of independent FF, Eq. (\[eq:g1perp\]), and make $H_1^\perp$ vanish, i.e. the analogue of the Collins effect in this context turns out to be a higher-order effect. The structure induced by the model is simply not rich enough to produce a non-vanishing $H_1^\perp$. Moreover, the FF do not depend on the flavor of the fragmenting valence quark, provided that the charges of the final detected pions are selected according to the diagrams of Fig. \[fig:diagspec\]. Hence, the FF are the same for $u\rightarrow \pi^+ \pi^-$ and for $d\rightarrow \pi^- \pi^+$, where the final state differs only by the interchange of the two pions, i.e. by leaving everything unaltered but ${\vec R}_{{\scriptscriptstyle T}}\rightarrow -{\vec R}_{{\scriptscriptstyle T}}$ and $\xi \rightarrow (1-\xi)$: $$\begin{aligned} D_1^{\,u\rightarrow \pi^+\pi^-} (z,\xi,M_h^2,k_{{\scriptscriptstyle T}}^2,{\vec k}_{{\scriptscriptstyle T}}\cdot {\vec R}_{{\scriptscriptstyle T}}) & = & D_1^{\,d\rightarrow \pi^- \pi^+} (z,\xi,M_h^2,k_{{\scriptscriptstyle T}}^2,{\vec k}_{{\scriptscriptstyle T}}\cdot {\vec R}_{{\scriptscriptstyle T}}) {\nonumber}\\ & = & D_1^{\,d\rightarrow \pi^+ \pi^-} (z,(1-\xi),M_h^2,k_{{\scriptscriptstyle T}}^2,{\vec k}_{{\scriptscriptstyle T}}\cdot(-{\vec R}_{{\scriptscriptstyle T}})) {\nonumber}\\[2mm] H_1^{{{<\kern -0.3 em{\scriptscriptstyle )}}}\,u\rightarrow \pi^+ \pi^-} (z,\xi,M_h^2,k_{{\scriptscriptstyle T}}^2,{\vec k}_{{\scriptscriptstyle T}}\cdot {\vec R}_{{\scriptscriptstyle T}}) & = & H_1^{{{<\kern -0.3 em{\scriptscriptstyle )}}}\,d\rightarrow \pi^- \pi^+} (z,\xi,M_h^2,k_{{\scriptscriptstyle T}}^2,{\vec k}_{{\scriptscriptstyle T}}\cdot {\vec R}_{{\scriptscriptstyle T}}) {\nonumber}\\ & = & H_1^{{{<\kern -0.3 em{\scriptscriptstyle )}}}\, d\rightarrow \pi^+ \pi^-} (z,(1-\xi),M_h^2,k_{{\scriptscriptstyle T}}^2,{\vec k}_{{\scriptscriptstyle T}}\cdot (-{\vec R}_{{\scriptscriptstyle T}})) \, . \label{eq:flavorsym}\end{aligned}$$ When integrating the FF over $\d^2{\vec k}_{{\scriptscriptstyle T}}$ and $\d \xi$, the dependence on the direction of ${\vec R}_{{\scriptscriptstyle T}}$ is lost $$\begin{aligned} D_1^{\,u\rightarrow \pi^+ \pi^-}(z,M_h^2) & \equiv & \int_0^1\d\xi \int \d^2{\vec k}_{{\scriptscriptstyle T}}\; D_1^{\,u\rightarrow \pi^+ \pi^-} (z,\xi,M_h^2,k_{{\scriptscriptstyle T}}^2,{\vec k}_{{\scriptscriptstyle T}}\cdot {\vec R}_{{\scriptscriptstyle T}}) {\nonumber}\\ & = & \int_0^1\d\xi \int \d^2{\vec k}_{{\scriptscriptstyle T}}\; D_1^{\,d\rightarrow \pi^+ \pi^-} (z,(1-\xi),M_h^2,k_{{\scriptscriptstyle T}}^2,{\vec k}_{{\scriptscriptstyle T}}\cdot (- {\vec R}_{{\scriptscriptstyle T}})) {\nonumber}\\ & = & \int_0^1\d\xi \int \d^2{\vec k}_{{\scriptscriptstyle T}}\; D_1^{\,d\rightarrow \pi^+ \pi^-} (z,\xi,M_h^2,k_{{\scriptscriptstyle T}}^2,{\vec k}_{{\scriptscriptstyle T}}\cdot (- {\vec R}_{{\scriptscriptstyle T}})) {\nonumber}\\ & \equiv & D_1^{\,d\rightarrow \pi^+ \pi^-}(z,M_h^2) \, ,\end{aligned}$$ and similarly for $H_1^{{{<\kern -0.3 em{\scriptscriptstyle )}}}}$. Therefore, we can conclude that the integrated FF do not depend in general on the flavor of the fragmenting quark. Consequently, the SSA of Eq. (\[eq:ssa\]) simplifies to $$A^{\sin \phi} (y,x,z,M_h^2) = \frac{B(y)}{A(y)} \; \frac{|{\vec S}_\perp |}{4m_\pi} \; \frac{\left[ \displaystyle{ \frac{8}{9} x \, h_1^u (x) + \frac{1}{9} x \, h_1^d (x) }\right] \; H_{1 \, (R)}^{{{<\kern -0.3 em{\scriptscriptstyle )}}}\, u} (z,M_h^2)} {\left[ \displaystyle{ \frac{8}{9} x \, f_1^u (x) + \frac{1}{9} x \, f_1^d (x) }\right] \; D_1^u (z,M_h^2)} \, . \label{eq:ssaspec}$$ In the following, we will discuss the SSA without the inessential $|{\vec S}_\perp| B(y)/A(y)$ factor and after integrating away the $z$ dependence and, in turn, the $x$ or $M_h$ dependence according to $$\begin{aligned} A^{\sin \phi}_x (x) &\equiv & \frac{1}{4m_\pi} \; \frac{\displaystyle{ \left[ \frac{8}{9} x \, h_1^u (x) + \frac{1}{9} x \, h_1^d (x) \right] \; \int \d z \,\d M_h^2 }\; H_{1 \, (R)}^{{{<\kern -0.3 em{\scriptscriptstyle )}}}\, u} (z,M_h^2)} {\displaystyle{\left[ \frac{8}{9} x \, f_1^u (x) + \frac{1}{9} x \, f_1^d (x) \right] \; \int \d z \,\d M_h^2 }\; D_1^u (z,M_h^2)} \label{eq:ssa-x} \\[2mm] A^{\sin \phi}_{M_h} (M_h) &\equiv & \frac{1}{4m_\pi} \; \frac{\displaystyle{\int \d x \, \left[ \frac{8}{9} x \, h_1^u (x) + \frac{1}{9} x \, h_1^d (x) \right]\; \int \d z }\; H_{1 \, (R)}^{{{<\kern -0.3 em{\scriptscriptstyle )}}}\, u} (z,M_h^2)} {\displaystyle{\int \d x \, \left[ \frac{8}{9} x \, f_1^u (x) + \frac{1}{9} x \, f_1^d (x) \right]\; \int \d z }\; D_1^u (z,M_h^2)} \label{eq:ssa-mh} \, .\end{aligned}$$ Numerical Results {#sec:results} ================= In the remainder of the paper, we present numerical results in the context of the spectator model for both the process-independent FF and the SSA of Eqs. (\[eq:ssa-x\]) and (\[eq:ssa-mh\]) for semi-inclusive lepton-nucleon DIS. Considering different possible scenarios for $h_1$, we discuss the implications for an experimental search for transversity. The input parameters of the calculation can basically be grouped in three classes: - values of masses and coupling constants taken from phenomenology, as $m_\pi = 0.139$ GeV, $m_\rho = 0.785$ GeV, with $f_{\rho \pi \pi}$ and $\Gamma_\rho$ as described in Sec. \[sec:vertices\] and Sec. \[sec:propagators\], respectively; - values consistent with other works on the spectator model and the constituent quark model, as $\Lambda_\pi = 0.4$ GeV, $\Lambda_\rho = 0.5$ GeV and $m_q = 0.34$ GeV [@Jakob:1997wg; @Bianconi:2000uc]; - parameters, such as $N_{q\pi}$ and $N_{q\rho}$, without constraints that are firmly established, or at least usually adopted, in the literature. As previously anticipated in Sec. \[sec:vertices\], the last ones are constrained using the integral (\[eq:sumrule\]) and the proportionality (\[eq:qpi-coup\]) derived from the Goldberger-Treiman relation. All results will be plotted according to two extreme scenarios, where the integral (\[eq:sumrule\]) amounts to 0.14 ($N_{q\rho} = 0.9$ GeV$^3$, corresponding to solid lines in the figures) and 0.48 ($N_{q\rho} = 1.6$ GeV$^3$, corresponding to dashed lines in the figures). Because of the high degree of arbitrariness due to the lack of any data, the results should be interpreted as the indication not only of the sensitivity of the considered observables to the input parameters, but also of the degree of uncertainty that can be reached within the spectator model. In the same spirit, when dealing with the SSA of Eqs. (\[eq:ssa-x\],\[eq:ssa-mh\]), $f_1$ and $h_1$ are calculated consistently within the spectator model [@Jakob:1997wg] or, alternatively, $f_1$ and $g_1$ are taken from consistent parametrizations and $h_1$ is calculated again according to two extreme scenarios: the nonrelativistic prediction $h_1 = g_1$ or the saturation of the Soffer inequality, $h_1 = (f_1+g_1)/2$. The parametrizations for $f_1, g_1,$ are extracted at the same lowest possible scale ($Q^2 = 0.8$ GeV$^2$), consistently with the valence quark approximation assumed for the calculation of the FF. In Fig. \[fig:ffplot\] the integrated $D_1^u (z)$ and $H_{1\, (R)}^{{{<\kern -0.3 em{\scriptscriptstyle )}}}\, u} (z)$ are shown. Again, we recall that the solid line corresponds to a weaker $q\rho q$ coupling than the dashed line. The choice of the form factors at the vertices also guarantees the regular behaviour at the end points $z=0,1$. The strongest asymmetry in the fragmentation (recall that $H_1^{{{<\kern -0.3 em{\scriptscriptstyle )}}}}$ is defined as the probability difference for the fragmentation to proceed from a quark with opposite transverse polarizations) is reasonably reached at $z\sim 0.4$. Once again, we stress that this result, particularly its persistent negative sign, does not depend on a specific hard process and can influence the corresponding azimuthal asymmetry. In fact, the SSA (\[eq:ssa-x\]) and (\[eq:ssa-mh\]) for two-pion inclusive lepton-nucleon DIS as shown in Figs. \[fig:ssa-xplot\] and \[fig:ssa-mhplot\], respectively, turn out to be negative due to the sign of $H_1^{{{<\kern -0.3 em{\scriptscriptstyle )}}}\,u}$. The solid and dashed lines again refer to the weaker or stronger $q\rho q$ couplings in the FF, respectively. For each parametrization, three different choices of DF are shown. The label SP refers to the DF calculated in the spectator model [@Jakob:1997wg]. The label NR indicates that $f_1$ and $g_1$ are taken consistently from the leading-order parametrizations of Ref. [@Gluck:1998xa] and Ref. [@Gluck:1996yr], respectively, with $h_1 = g_1$. The label SO indicates the same parametrizations but with the Soffer inequality saturated, i.e. $h_1 = (f_1 + g_1)/2$. In the lower plot of each figure the “uncertainty band” is shown as a guiding line. It is built by taking, for each $z$ or $M_h$, the maximum and the minimum among the six curves displayed in the corresponding upper plot. The first obvious comment is that even the simple mechanism described in Fig. \[fig:diagspec\] produces a measurable asymmetry. For the HERMES experiment the size of the asymmetry may be at the lower edge of possible measurements, given the observed rather small average multiplicity which does not favor the detection of two pions in the final state. On the other hand, the planned transversely polarized target clearly will improve the situation of azimuthal spin-asymmetry measurements compared to the present one. COMPASS or possible future experiments at the ELFE, TESLA-N, or EIC facilities will have less problems because of higher counting rates. The second important result is that the sensitivity of the SSA to the parameters of the model calculation for the FF and to the different parametrizations for the DF is weak enough that the unambigous message of a negative asymmetry emerges through all the range of both $x$ and $m_\rho -\Gamma_\rho /2 = 0.69$ GeV $\leq M_h \leq m_\rho +\Gamma_\rho /2 = 0.84$ GeV. In particular, we do not find any change in sign for $A^{\sin \phi}_{M_h}$, contrary to what is predicted in Ref. [@Jaffe:1998hf]. Outlooks {#sec:end} ======== In this paper we have discussed a way for addressing the transversity distribution $h_1$ that we consider most advantageous compared to other strategies discussed in the literature. At present, the SSA seem anyway preferable to the DSA. But the fragmentation of a transversely polarized quark into two unpolarized leading hadrons in the same current jet looks less complicated than the Collins effect, both experimentally and theoretically. Collinear factorization implies an exact cancellation of the soft divergencies, avoiding any dilution of the asymmetry because of Sudakov form factors, and in principle makes the QCD evolution simpler, though we have not addressed this subject in the present paper. The new effect, that allows for the extraction of $h_1$ at leading twist through the new interference FF $H_1^{{{<\kern -0.3 em{\scriptscriptstyle )}}}}$, relates the transverse polarization of the quark to the transverse component of the relative momentum of the hadron pair via a new azimuthal angle. This is the only key quantity to be determined experimentally, while the Collins effect requires the determination of the complete transverse momentum vector of the detected hadron. We have shown also quantitative results for $H_1^{{{<\kern -0.3 em{\scriptscriptstyle )}}}}$ in the case of $\pi^+ \pi^-$ detection, and the related SSA for the example of lepton-nucleon scattering, because modelling the interference between different channels leading to the same final state is simpler than describing the Collins effect, where a microscopic knowledge of the structure of the residual jet is required. We have adopted a spectator model approximation for $\pi^+ \pi^-$ with an invariant mass inside the $\rho$ resonance width, limiting the process to leading-twist mechanisms. The interference between the decay of the $\rho$ and the direct production of $\pi^+ \pi^-$ is enough to produce sizeable and measurable asymmetries. Despite the theoretical uncertainty due to the arbitrariness in fixing the input parameters of the calculation of FF and in choosing the parametrizations for the DF, the unambigous result emerges that in the explored ranges in $x$ and invariant mass $M_h$ the SSA are always negative and almost flat. Anyway, it should be stressed again that, even if there are good arguments for considering the mechanisms depicted in Fig. \[fig:diagspec\] a good representation of $\pi^+ \pi^-$ production in the considered energy range, still the calculation has been performed at leading twist and in a valence-quark scenario. 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--- abstract: 'Using the momentum-dependent MDI effective interaction for nucleons, we have studied the transition density and pressure at the boundary between the inner crust and liquid core of hot neutron stars. We find that their values are larger in neutrino-trapped neutron stars than in neutrino-free neutron stars. Furthermore, both are found to decrease with increasing temperature of a neutron star as well as increasing slope parameter of the nuclear symmetry energy, except that the transition pressure in neutrino-trapped neutron stars for the case of small symmetry energy slope parameter first increases and then decreases with increasing temperature. We have also studied the effect of the nuclear symmetry energy on the critical temperature above which the inner crust in a hot neutron star disappears and found that with increasing value of the symmetry energy slope parameter, the critical temperature decreases slightly in neutrino-trapped neutron stars but first decreases and then increases in neutrino-free neutron stars.' author: - Jun Xu - 'Lie-Wen Chen' - Che Ming Ko - 'Bao-An Li' title: Transition density and pressure in hot neutron stars --- Introduction ============ Studying the properties of neutron stars allows us to test our knowledge on the properties of nuclear matter under extreme conditions. Theoretical studies have shown that a neutron star is expected to have a liquid core surrounded by an inner crust [@Cha08], which extends outward to the neutron drip-out region. While the neutron drip-out density $\rho _{\rm out}$ has been relatively well determined [@Rus06], the transition density $\rho _{t}$ at the inner edge of the crust is still quite uncertain because of our limited knowledge on the nuclear equation of state (EOS), especially the density dependence of the symmetry energy ($E_{\rm sym}(\rho)$) of neutron-rich nuclear matter [@Lat00; @Lat07]. Recently, significant progress has been made in constraining the EOS of neutron-rich nuclear matter using terrestrial laboratory experiments (See Ref. [@LCK08] for a recent review). In particular, from analyses of experimental data on neutron skin thickness, isobaric analogue states, Pygmy dipole resonances, and giant dipole resonances in nuclei as well as on isospin diffusion, isoscaling, and neutron-proton to triton-$^3$He ratio in intermediate-energy nuclear reactions, significant constraints on $E_{\rm sym}(\rho)$ have been obtained for the same sub-saturation density region as expected in the inner edge of neutron star crusts. The extracted slope parameter $L=3\rho_0 (\partial E_{\rm sym}(\rho)/\partial\rho)_{\rho=\rho_0}$ of the nuclear symmetry energy from these studies has values in the range $30$ MeV $< L < 80$ MeV [@She10]. With the MDI interaction together with the value $L=86\pm25$ MeV constrained from an analysis of the isospin-diffusion data [@Tsa04; @Che05a; @LiBA05; @Tsa09] in heavy-ion collisions using the isospin-dependent Boltzmann-Uehling-Uhlenbeck (IBUU) transport model with the momentum-dependent MDI interaction [@Das03], the density and pressure at the inner edge of the crust of cold neutron stars were studied by considering the boundary of the instability region or the spinodal boundary between the liquid core and inner crust of a cold neutron star in both the thermodynamical approach [@Kub07; @Lat07] and the dynamical approach [@BPS71; @BBP71; @Pet95a; @Pet95b; @Oya07]. This leads to the constraints $0.040$ fm$^{-3}<\rho_t<0.065$ fm$^{-3}$ and $0.01$ MeV/fm$^{3} < P_t < 0.26$ MeV/fm$^{3}$, respectively, for the transition density and pressure. Together with the crustal fraction of the total moment inertia of the Vela pulsar extracted from its glitches [@Lin99], a tighter constraint on the mass-radius relation of cold neutron stars was obtained [@XCLM09]. Because of the initial high temperature and appreciable proton fraction in a newly-formed neutron star immediately after gravitational collapse of a massive star [@Bur86; @Bet90; @Bur03], neutrinos are abundantly produced from the Urca process in its inner core. Although high energy neutrinos can be trapped at densities as low as 10$^{12}$ g/cm$^3$ [@BBLA79], the stars cools by neutrino emissions. As neutrinos are emitted from this so-called proto-neutron star, which has an initial temperature of $\sim 10^{11}K$ (about $10$ MeV) [@Bur88; @Hor04], its temperature drops to $\sim 10^{10} K$ (about $1$ MeV) and even lower. Afterwards, the neutron star becomes transparent to neutrinos as their mean free path increases with decreasing energy, and the cooling of the neutron star continues to be dominated by neutrino emission for a long time. It is thus of interest to study the transition density and pressure in newly-born hot neutron stars, as this would help to understand the cooling mechanism and structural evolution of neutron stars. In this paper, we extend the study of Ref. [@XCLM09] to finite temperature and study the dependence of the transition density and pressure of hot neutron stars on the nuclear symmetry energy, particularly its slope at nuclear saturation density. This paper is organized as follows. We first review in Sec. \[model\] the momentum-dependent MDI interaction for nucleons, in Sec. \[nsmatter\] the properties of hot neutron star matter, and in Sec. \[approaches\] the dynamical approach for locating the inner edge of the crust of a hot neutron star. We then show in Sec. \[results\] the results and conclude with a summary in Sec. \[summary\]. The MDI interaction {#model} =================== The MDI interaction is an effective nuclear interaction with its density and momentum dependence constrained from the phenomenological finite-range Gogny interaction [@Das03]. In the mean-field approximation, the potential energy density of a nuclear matter of density $\rho$ and isospin asymmetry $\delta=(\rho_n-\rho_p)/\rho$ , with $\rho_n$ and $\rho_p$ being, respectively, the neutron and proton densities, can be expressed as [@Das03; @Che05a] $$\begin{aligned} V(\rho,\delta ) &=&\frac{A_{u}(x)\rho _{n}\rho _{p}}{\rho _{0}} +\frac{A_{l}(x)}{2\rho _{0}}(\rho _{n}^{2}+\rho_{p}^{2})\notag\\ &+&\frac{B}{\sigma +1}\frac{\rho ^{\sigma +1}}{\rho _{0}^{\sigma }}(1-x\delta ^{2})\notag\\ &+&\frac{1}{\rho _{0}}\sum_{\tau ,\tau^{\prime}}C_{\tau ,\tau ^{\prime }} \int \int d^{3}pd^{3}p^{\prime }\frac{f_{\tau }(\vec{r},\vec{p})f_{\tau ^{\prime }}(\vec{r}^{\prime },\vec{p}^{\prime })}{1+(\vec{p}-\vec{p}^{\prime })^{2}/\Lambda ^{2}}.\notag\\ \label{MDIVB}\end{aligned}$$In the above equation, $\tau(\tau^\prime)$ is the nucleon isospin taken to be $1/2$ for neutron and $-1/2$ for proton; $f_{\tau}(\vec{r},\vec{p})=(2/h^{3})\{\exp [(p^2/2m+U_{\tau }-\mu _{\tau })/T]+1\}^{-1}$ is the nucleon phase-space distribution function in a thermally equilibrated nuclear matter with $m=939$ MeV being the nucleon mass, $\mu_\tau$ being the chemical potential of nucleon of isospin $\tau$, $T$ being the temperature and $U_\tau$ being the nucleon mean-field potential to be introduced below; and $\rho_0=0.16$ fm$^{-3}$ is the saturation density of normal nuclear matter. Values of the parameters $A_u(x)$, $A_l(x)$, $B$, $\sigma$, $\Lambda$, $C_l=C_{\tau,\tau}$ and $C_u=C_{\tau,-\tau}$ can be found in Refs. [@Das03; @Che05a]. For symmetric nuclear matter, this interaction gives a binding energy of $-16$ MeV per nucleon and an incompressibility $K_0$ of $212$ MeV at saturation density. Taking the derivative of Eq. (\[MDIVB\]) with respect to the proton or neutron density leads to the following single-particle potential for a nucleon of isospin $\tau $: $$\begin{aligned} U(\rho,\delta ,\vec{p},\tau ) &=&A_{u}(x)\frac{\rho _{-\tau }}{\rho _{0}}+A_{l}(x)\frac{\rho _{\tau }}{\rho _{0}}\notag\\ &+&B\left(\frac{\rho }{\rho _{0}}\right)^{\sigma }(1-x\delta ^{2})\notag\\ &-&8\tau x\frac{B}{\sigma +1}\frac{\rho ^{\sigma -1}}{\rho _{0}^{\sigma }} \delta \rho_{-\tau }\notag\\ &+&\frac{2C_{\tau ,\tau }}{\rho _{0}} \int d^{3}p^{\prime }\frac{f_{\tau }(\vec{r},\vec{p}^{\prime })}{1+(\vec{p}-\vec{p}^{\prime })^{2}/\Lambda ^{2}}\notag\\ &+&\frac{2C_{\tau ,-\tau }}{\rho _{0}} \int d^{3}p^{\prime }\frac{f_{-\tau }(\vec{r},\vec{p}^{\prime })}{1+(\vec{p}-\vec{p}^{\prime })^{2}/\Lambda ^{2}}, \label{MDIU}\end{aligned}$$which is seen to depend on the momentum $\vec{p}$ of the nucleon. Although the properties of cold nuclear matter can be essentially determined analytically using Eqs. (\[MDIVB\]) and (\[MDIU\]), these equations needs to be solved numerically and self-consistently by the iteration method [@Xu07] to obtain the thermodynamical quantities of hot nuclear matter. In cold nuclear matter, the symmetry energy from the MDI interaction is given by $$\begin{aligned} \label{esymmdi} E_{\rm sym}(\rho) &=& \frac{1}{2} \left(\frac{\partial^2 E} {\partial \delta^2}\right)_{\delta=0} \notag\\ &=& \frac{8 \pi}{9 m h^3 \rho} p^5_f + \frac{\rho}{4 \rho_0} (-24.59+4Bx/(\sigma +1)) \notag\\ &-& \frac{B x}{\sigma + 1} \left(\frac{\rho}{\rho_0}\right)^\sigma + \frac{C_l}{9 \rho_0 \rho} \left(\frac{4 \pi}{h^3}\right)^2 \Lambda^2 \notag\\ &\times& \left[4 p^4_f - \Lambda^2 p^2_f \ln \left(\frac{4 p^2_f + \Lambda^2}{\Lambda^2}\right)\right] \notag\\ &+& \frac{C_u}{9 \rho_0 \rho} \left(\frac{4 \pi}{h^3}\right)^2 \Lambda^2 \notag\\ &\times& \left[4 p^4_f - p^2_f (4 p^2_f + \Lambda^2) \ln \left(\frac{4 p^2_f + \Lambda^2}{\Lambda^2}\right)\right],\notag\\\end{aligned}$$ where $p_f=\hbar(3\pi^2\rho/2)^{1/3}$ is the nucleon Fermi momentum in symmetric nuclear matter. The first term in Eq. (\[esymmdi\]) is the contribution from the kinetic part while other terms are from the potential part. The symmetry energy is fixed to be $30.5$ MeV at normal nuclear density, and the parameter $x$ is used to model the density dependence of the symmetry energy away from the saturation density without changing the properties of symmetric nuclear matter. The resulting slope parameter of the symmetry energy has values of about $15$ MeV, $60$ MeV, and $106$ MeV for $x=1$, $0$, and $-1$, respectively. ![(Color online) (a) Density dependence of symmetry energies, (b) single-particle potentials in symmetric nuclear matter and (c) symmetry potentials in cold nuclear matter at $\rho_0/2$ for the MDI interaction.[]{data-label="EsymU"}](fig1.EPS) The density dependence of the symmetry energy from the MDI interaction is shown in panel (a) of Fig. \[EsymU\] for $x=1$, $x=0$ and $x=-1$. With $x=1$ ($x=-1$) the symmetry energy is larger (smaller) at subsaturation densities but smaller (larger) at suprasaturation densities, and it is called a ’soft’ (’stiff’) symmetry energy. For $x=0$, the value of the symmetry energy lies between those of $x=1$ and $x=-1$. In panels (b) and (c), the momentum dependence of the nucleon single-particle potential $U_0$ in symmetric nuclear matter at half saturation density is displayed. Also shown in the figure is the symmetry potential, defined as $U_{\rm sym}=(U_n-U_p)/2\delta$ and calculated at $\delta=0.2$ and the same density. It is seen that $U_0$ has negative values at low momenta but increases with increasing momentum and saturates at high momenta. The symmetry potential $U_{\rm sym}$, which measures the difference between the neutron and proton single-particle potentials in an asymmetric nuclear matter, decreases with increasing momentum and is larger for $x=1$ than for $x=0$ and $x=-1$, which is consistent with the behavior of the symmetry energy at subsaturation densities. Hot neutron star matter {#nsmatter} ======================= For a newly-born neutron star, its cooling is dominated by the emission of neutrinos, which are produced through the Urca process, leading to the following charge neutrality and $\beta$-stable conditions: $$\begin{aligned} \rho_p &=& \rho_e,\\ \mu_p + \mu_e &=& \mu_n + \mu_{\nu_e}.\end{aligned}$$ The production of muons is negligible for the density and temperature present in the hot neutron star crust. Neutrino trapping in the early stage of a supernova has been extensively studied in the literature [@BBLA79], and it was found that the fraction of leptons $$Y_l = \frac{\rho_e+\rho_{\nu_e}}{\rho}$$ is about $0.35 \sim 0.4$ at the onset of trapping [@Bur88; @Bet90] and decreases as neutrinos leave the star. In the present study, we consider the two extreme cases of neutrino-trapped and neutrino-free hot neutron star matters. In the former case, we choose $Y_l$ to be $0.4$ as an example and consider the typical temperature of $5$ or $10$ MeV. In the case of neutrino-free hot neutron star matter, we set the neutrino chemical potential to be $0$ and the temperature to be $0$ or $1$ MeV, corresponding to the later stage of a neutron star’s evolution. The abundance of each species in both cases can be calculated from above equations together with the baryon number conservation condition $\rho=\rho_n+\rho_p$. ![(Color online) The proton fraction $x_p$ as a function of baryon density $\rho$ from the MDI interaction for both neutrino-trapped matter (left panel) and neutrino-free matter (right panel) with different $x$ parameters and at different temperatures. Note that different scales for $x_p$ are used for the neutrino-free and the neutrino-trapped matter.[]{data-label="rhodelta"}](fig2.EPS) Assuming that both electrons and neutrinos are massless, we determine the proton fraction $x_p=(1-\delta)/2$ in both the neutrino-trapped matter and the neutrino-free matter as a function of baryon density $\rho$, and the results are shown in Fig. \[rhodelta\]. It is seen that the neutrino-free matter is much more neutron-rich than the neutrino-trapped matter. The critical proton fraction $11 \sim 15$ % for the direct Urca process [@Bog79; @Lat91] is smaller than the proton fraction in the neutrino-trapped matter but larger than the proton fraction in the neutrino-free matter. The proton fraction increases with increasing density in all cases and slightly increases with increasing temperature for a fixed density, especially at low densities. These can be understood from the increase of the symmetry free energy with increasing temperature and density [@Xu07], which makes the neutron star matter more symmetric at higher temperatures and densities. Furthermore, the stiff symmetry energy ($x=-1$) makes the system more neutron-rich at subsaturation densities as expected. The total pressure $P$ in a hot neutron star matter can be written as $$P = P_b + P_l,$$ where the baryon contribution $P_b(\rho ,T,\delta )$ is calculated from the thermodynamic relation $$\begin{aligned} P_b(\rho ,T,\delta ) &=&\left[ T{\sum_{\tau }}s_{\tau }(\rho ,T,\delta)-V(\rho ,T,\delta )\right. \notag\\ &&\left.- V_{\rm kin}(\rho,T,\delta)\right]+\sum_{\tau }\mu _{\tau }\rho _{\tau }. \label{Pb}\end{aligned}$$ In the above equation, $V(\rho ,T,\delta )$ and $V_{\rm kin}(\rho,T,\delta)$ are, respectively, the potential and kinetic contributions to the total energy density with the latter given by $$V_{\rm kin}(\rho,T,\delta) = {\sum_{\tau }}\int d^{3}p\frac{p^{2}}{2m}f_{\tau }(\vec{r},\vec{p}),$$ and $s_{\tau }(\rho ,T,\delta )$ is the entropy density, which is given by $$s_{\tau }(\rho ,T,\delta )=-\frac{8\pi }{{ }h^{3}}\int_{0}^{\infty }p^{2}[n_{\tau }\ln n_{\tau }+(1-n_{\tau })\ln (1-n_{\tau })]dp, \label{S}$$with the particle occupation number$$n_{\tau }=\frac{1}{\exp [(p^{2}/2m+U_{\tau }-\mu _{\tau })/T]+1}.$$Above formula can also be used for the calculation of $P_l$ by using $l$ ($l=e, \nu_e$) instead of $\tau$, and leptons are treated as non-interacting ultra-relativistic particles. Locating the inner edge of the neutron star crust {#approaches} ================================================= We briefly review in this section the dynamical approach that will be used to locate the inner edge of the neutron star crust [@Cha08], and discuss its application to the case of finite temperature. We neglect muons in the neutron star crust as their number is small compared to that of electrons at low densities and temperatures. In the dynamical approach, the stability condition for a homogeneous neutron star matter against small periodic density perturbations can be well approximated by [BPS71,BBP71,Pet95a,Pet95b,Oya07]{} $$V_{\rm dyn}(k)=V_{0}+\beta k^{2}+\frac{4\pi e^{2}}{k^{2}+k_{TF}^{2}}>0, \label{Vdyn}$$where $k$ is the wavevector of the spatially periodic density perturbations and $$\begin{aligned} V_{0} &=&\frac{\partial \mu _{p}}{\partial \rho _{p}}-\frac{(\partial \mu _{n}/\partial \rho _{p})^{2}}{\partial \mu _{n}/\partial \rho _{n}},\text{ }k_{TF}^{2}=\frac{4\pi e^{2}}{\partial \mu_e /\partial \rho _{e}}, \notag \\ \beta &=&D_{pp}+2D_{np}\zeta +D_{nn}\zeta ^{2},~~\zeta =-\frac{\partial \mu _{n}/\partial \rho _{p}}{\partial \mu _{n}/\partial \rho _{n}}.\notag\\\end{aligned}$$The three terms in Eq. (\[Vdyn\]) represent, respectively, contributions from the bulk nuclear matter, the density-gradient terms, and the Coulomb interaction. The empirical values for the coefficients of density-gradient terms are $D_{pp}=D_{nn}=D_{np}=132$ MeV$\cdot $fm$^{5}$ [@Oya07; @XCLM09]. At $k_{\min }=[(4\pi e^2/\beta )^{1/2}-k_{TF}^{2}]^{1/2}$, $V_{\rm dyn}(k)$ has the minimal value of $V_{\rm dyn}(k_{\min })=V_{0}+2(4\pi e^{2}\beta )^{1/2}-\beta k_{TF}^{2}$ [@BPS71; @BBP71; @Pet95a; @Pet95b; @Oya07], and $\rho _{t}$ is then determined from $V_{\rm dyn}(k_{\min })=0$. We note that the first term in Eq. (\[Vdyn\]) gives the dominant contribution in the determination of the transition density, and including other terms lowers the transition density. Also, although at low temperatures the electron density $\rho_e$ can be written as an expansion of the electron chemical potential $$\label{rhoelT} \rho_e \approx \frac{8\pi}{3h^3} \mu_e^3 [1+\pi^2(T/\mu_e)^2],$$ which gives an analytical expression for $\partial \mu_e/\partial \rho_e$, at high temperatures numerical calculations are needed. The dynamical approach reduces to the so-called thermodynamical approach [@Kub07; @Lat07] in the long wave length limit when the density gradient terms and the Coulomb interaction are neglected [@Mar03; @XCLM09], which leads to the stability condition $$\label{ther} V_{\rm ther}=\frac{\partial \mu _{p}}{\partial \rho _{p}}-\frac{(\partial \mu _{n}/\partial \rho _{p})^{2}}{\partial \mu _{n}/\partial \rho _{n}}>0.$$ ![(Color online) The instability region and the relative neutron-proton abundance of hot neutron star matter for different temperatures and nuclear symmetry energy parameters. Regions where they cross each other are shown in the insets in enlarged scales.[]{data-label="rhonrhop"}](fig3.EPS) To illustrate the relation between the transition density and the area of the spinodal region, we show in Fig. \[rhonrhop\] the instability region of nuclear matter with the boundary determined by $V_{\rm ther}=0$ and the relative neutron-proton abundance of a hot neutron star in the $(\rho_n,\rho_p)$ plane. The cross point, also shown with enlarged scales in the insets, is the transition density from the thermodynamical approach. It is seen that although the neutron star matter becomes less neutron-rich with increasing temperature, the area of the instability region shrinks more quickly with increasing temperature. Furthermore, the stiffness of the symmetry energy also affects the shape and area of the spinodal region. As temperature increases, the spinodal boundaries from different values of $x$ cross with the relative neutron-proton abundance curves at decreasingly small proton densities. For a more detailed discussion on the symmetry energy and temperature effects on the spinodal region, we refer readers to Refs. [@LiBA01; @LiBA97b; @Bar98; @Duc07; @Duc08; @Ava04]. The following analysis of the transition density and pressure in hot neutron stars is, however, carried out by using the more realistic dynamical approach. Results and discussions {#results} ======================= In this section, we show the temperature dependence of the transition density and pressure in newly-born hot neutron stars by using the MDI interaction with different values for the symmetry energy parameter $x$ or the slope parameter $L$ of the symmetry energy. ![(Color online) Transition densities $\rho_t$ ((a) and (c)) and pressure $P_t$ ((b) and (d)) as functions of the slope parameter $L$ of the symmetry energy at different temperatures for both the neutrino-trapped matter ((a) and (b)) and the neutrino-free matter ((c) and (d)).[]{data-label="rhotL"}](fig4.EPS) The dependence of the transition density and pressure in hot neutron stars on the slope parameter $L$ of the symmetry energy at different temperatures is shown in Fig. \[rhotL\]. It is seen that the transition density $\rho_t$ generally decreases with increasing value of the slope parameter $L$ of the symmetry energy. As the transition density can be viewed approximately as the beginning of a first-order liquid-gas phase transition, a stiffer symmetry energy, which corresponds to a softer equation of state at subsaturation densities, leads thus to a smaller phase transition density and therefore a lower core-crust transition density. Furthermore, the transition density decreases with increasing temperature, and for the neutrino-free matter this is more pronounced for larger values of $L$. The temperature effect can again be understood by the decreasing phase transition density with increasing temperature. For the neutrino-trapped matter, the $L$-dependence of the transition density is relatively weak. The weak $L$-dependence is mainly due to the fact that the isospin asymmetry is not so large as shown in Fig. \[rhodelta\]. We note that a similar temperature dependence of the transition density has been obtained in studies based on Skyrme interactions and relativistic mean field models [@Duc08; @Par09; @Ava09]. For the $L$-dependence of the transition density, results from these models are, however, not so clear as different values for the incompressibility $K_0$ of the symmetric matter at saturation density and $E_{sym}(\rho_0)$, which also affect the value of the transition density, have been used. For the transition pressure $P_t$, its value in the neutrino-trapped matter decreases only very slightly with increasing $L$ as a result of the weak $L$-dependence of $\rho_t$. Its temperature dependence shows, however, a complicated behavior of slightly higher and smaller values at higher temperatures for smaller and larger values of $L$, respectively. This is due to the fact that although the transition density $\rho_t$ decreases with increasing temperature, the contribution from leptons increases with increasing temperature. For the neutrino-free matter, $P_t$ is seen to decrease rapidly with increasing $L$. As to its temperature dependence, $P_t$ in the neutrino-free matter decreases with increasing temperature at larger values of $L$ but shows a weaker temperature dependence for smaller values of $L$. Also, $P_t$ is larger for the neutrino-trapped matter than for the neutrino-free matter. Interestingly, $P_t$ becomes very small and even negative at $T=1$ with larger $L$ for the neutrino-free matter. This is due to the smaller contributions from the leptons and the asymmetric part of the nuclear interactions to the total pressure. Since the pressure at the inner edge of neutron star crust cannot be negative, our finding thus indicates that either the neutrino-free matter in hot neutron stars cannot reach a temperature above $T=1$ MeV or the symmetry energy cannot have a slope parameter larger than $L\sim 100$ MeV. ![(Color online) Transition densities $\rho_t$ ((a) and (c)) and pressure $P_t$ ((b) and (d)) as functions of temperature $T$ with $x=0$ and $x=-1$ for both the neutrino-trapped matter ((a) and (b)) and the neutrino-free matter ((c) and (d)). Note that different scales for temperatures are used for the neutrino-trapped matter and the neutrino-free matter.[]{data-label="rhotT"}](fig5.EPS) The temperature effect on the transition density and pressure for a fixed symmetry energy parameter is demonstrated in Fig. \[rhotT\] for the symmetry energy parameters $x=1$, $x=0$ and $x=-1$. For the neutrino-trapped matter, the temperature effect is similar for all three $x$ values, and the transition density $\rho_t$ decreases almost linearly with increasing temperature at lower $T$ and decreases quickly at higher $T$. For the neutrino-free matter, although the transition density $\rho_t$ decreases smoothly with increasing temperature for $x=1$, it decreases slowly (quickly) at lower temperatures with $x=0$ ($x=-1$) while quickly (slowly) at higher temperatures. The temperature effects on the transition density reflect those on the spinodal region and the abundance of particle species in the hot neutron star matter. For the neutrino-trapped matter, the transition pressure $P_t$ is seen to be insensitive to the temperature for $x=-1$ but increases slightly with increasing temperature for $x=0$ and $x=1$ at lower $T$, and it decreases with increasing temperature at higher $T$ for all values of $x$. For the neutrino-free matter, it is insensitive to temperature for $x=1$ but decreases with increasing temperature for $x=0$, while for $x=-1$ it drops to a negative value and becomes positive again as the temperature increases. Our results again show that the behavior of $P_t$ is dominated by that of $\rho_t$ and the contribution from the leptons, with the former decreasing and the latter increasing with increasing temperature. ![(Color online) Critical temperature $T_c$ as a function of the slope parameters $L$ of the symmetry energy for both the neutrino-trapped matter and the neutrino-free matter. []{data-label="rhotTc"}](fig6.EPS) We have seen in Fig. \[rhonrhop\] that as the temperature of the neutron star matter increases, there will be eventually no cross point between the curve of neutron-proton relative abundance and the boundary of the spinodal region, leading to the disappearance of the transition density in hot neutron stars. In such case, the inner crust (nuclear ’pasta’ phase) disappears and the liquid core expends directly to the outer crust. To determine the critical temperature $T_c$ at which the transition density $\rho_t$ disappears is thus useful for understanding the structural evolution of newly-born hot neutron stars. Figure \[rhotTc\] displays the $L$-dependence of the critical temperature for both the neutrino-trapped matter and the neutrino-free matter. One sees that the critical temperature $T_c$ decreases slightly with increasing value of $L$ in the neutrino-trapped matter, but it first decreases and then increases with increasing $L$ in the neutrino-free matter. This complicated behavior is due to the isospin and temperature effects on the spinodal region and the relative neutron-proton abundance as shown in Fig. \[rhonrhop\]. Our results thus indicate that for neutrino-trapped neutron stars of temperatures higher than $12$ MeV or for neutrino-free neutron stars of temperatures higher than $1.5$ MeV, there exists no inner crust if the value of $L$ is $70 \sim 80$ MeV. As newly-born hot neutron stars cool, the temperature at which the inner crust can form thus depends on the density dependence of the symmetry energy at subsaturation densities. Again, the magnitude of the critical temperature for both neutrino-trapped and neutrino-free matter is similar to those from Skyrme interactions and relativistic mean field models [@Duc08; @Par09]. The above results were obtained using the dynamical approach that includes the effects of both the density gradient terms and the Coulomb interaction. Neglecting these effects, the resulting thermodynamical approach given by Eq. (\[ther\]) gives higher values for both the transition density and pressure. This is especially the case for the neutrino-trapped matter and/or for smaller values of $L$ as more electrons are present in such hot neutron stars and the effect due to the Coulomb interaction becomes more important. Also, there are recently some studies on the transition density in neutron stars using various nucleon-nucleon interactions [@Sur09; @Vid09]. To see the effect of momentum dependence in the nucleon-nucleon interaction on the transition density and pressure, we have also repeated above calculations using the momentum-independent MID interaction [@Xu08], which gives the same equation of state for asymmetric nuclear matter but different single-particle mean-field potential in comparison with the momentum-dependent MDI interaction used in the present study, and we find that the momentum-dependent effect on the transition density and pressure in hot neutron stars is small. Summary ======= We have studied the transition density and pressure at the boundary that separates the liquid core from the inner crust of neutron stars using the momentum-dependent MDI interaction in both the neutrino-trapped matter and the neutrino-free matter at finite temperatures, which are expected to exist during the early evolution of neutron stars. In particular, we have investigated the effect of nuclear symmetry energy by varying the parameter $x$ in the MDI interaction from $1$ to $-1$, corresponding to the values $15<L<106$ MeV for the slope of nuclear symmetry energy at normal density that were constrained by both the isospin diffusion data [@Tsa04; @Che05a; @LiBA05] and other experimental observables [@She10]. We have found that the transition density and pressure are larger in the neutrino-trapped matter than in the neutrino-free matter. Furthermore, the transition density and pressure are found to roughly decrease with increasing temperature and $L$ for both the neutrino-trapped and the neutrino-free matter, except that the transition pressure shows a complicated relation to the temperature for the neutrino-trapped matter. Also, negative values of the pressure at the transition density have been obtained, which can be used to rule out a very stiff symmetry energy at subsaturation densities. We have also studied the critical temperature above which the inner crust (nuclear ’pasta’ phase) cannot be formed in newly-born neutron stars and found that it depends sensitively on the density dependence of the nuclear symmetry energy at subsaturation densities. This work was supported in part by U.S. National Science Foundation under Grant No. PHY-0758115, PHY-0652548 and PHY-0757839, the Welch Foundation under Grant No. A-1358, the Research Corporation under Award No. 7123, the Texas Coordinating Board of Higher Education Award No. 003565-0004-2007, the National Natural Science Foundation of China under Grant Nos. 10675082 and 10975097, MOE of China under project NCET-05-0392, Shanghai Rising-Star Program under Grant No. 06QA14024, the SRF for ROCS, SEM of China, the National Basic Research Program of China (973 Program) under Contract No. 2007CB815004. [99]{} N. Chamel and P. Haensel, Living Rev. Relativity, **11**, 10 (2008). S. B. Ruster, M. Hempel, and J. Schaffner-Bielich, Phys. Rev. C **73**, 035804 (2006). J. M. Lattimer and M. Prakash, Phys. Rep. **333**, 121 (2000). J. M. Lattimer and M. Prakash, Phys. Rep. **442**, 109 (2007). B. A. Li, L. W. Chen, and C. M. Ko, Phys. Rep. **464**, 113 (2008). D. V. Shetty and S. J. 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--- bibliography: - 'BiblioTP.bib' --- [**Pseudodifferential extensions and adiabatic deformation of smooth groupoid actions**]{} [^1] Laboratoire de Mathématiques, UMR 6620 - CNRS -4pt Université Blaise Pascal, Campus des Cézeaux, BP [**80026**]{} -4pt F-63171 Aubière cedex, France -4pt claire.debord@math.univ-bpclermont.fr Université Paris Diderot, Sorbonne Paris Cité -4pt Sorbonne Universités, UPMC Paris 06, CNRS, IMJ-PRG -4pt UFR de Mathématiques, [CP]{} [**7012**]{} - Bâtiment Sophie Germain -4pt 5 rue Thomas Mann, 75205 Paris CEDEX 13, France -4pt skandalis@math.univ-paris-diderot.fr **Abstract** The adiabatic groupoid ${\mathcal{G}}_{ad}$ of a smooth groupoid ${\mathcal{G}}$ is a deformation relating ${\mathcal{G}}$ with its algebroid. In a previous work, we constructed a natural action of ${\mathbb{R}}$ on the C\*-algebra of zero order pseudodifferential operators on ${\mathcal{G}}$ and identified the crossed product with a natural ideal $J({\mathcal{G}})$ of $C^*({\mathcal{G}}_{ad})$. In the present paper we show that $C^*({\mathcal{G}}_{ad})$ itself is a pseudodifferential extension of this crossed product in a sense introduced by Saad Baaj. Let us point out that we prove our results in a slightly more general situation: the smooth groupoid ${\mathcal{G}}$ is assumed to act on a C\*-algebra $A$. We construct in this generalized setting the extension of order $0$ pseudodifferential operators $\Psi(A,{\mathcal{G}})$ of the associated crossed product $A\rtimes {\mathcal{G}}$. We show that ${\mathbb{R}}$ acts naturally on $\Psi(A,{\mathcal{G}})$ and identify the crossed product of $A$ by the action of the adiabatic groupoid ${\mathcal{G}}_{ad}$ with an extension of the crossed product $\Psi(A,{\mathcal{G}})\rtimes {\mathbb{R}}$. Note that our construction of $\Psi(A,{\mathcal{G}})$ unifies the ones of Connes (case $A={\mathbb{C}}$) and of Baaj (${\mathcal{G}}$ is a Lie group). [**Keywords:**]{} Noncommutative geometry; groupoids; pseudodifferential calculus. Introduction ============ Alain Connes in [@ConnesLNM [Chap. VIII]{}] pointed out that smooth groupoids offer a perfect setting for index theory. Since then, this fact has been explored and exploited by Connes as well as many other authors, in many geometric situations (see [@DebordLescureIndex] for a review). In [@ConnesNCG [section II.5]{}], A. Connes constructed a beautiful groupoid, that he called the “tangent groupoid”, which interpolates between the pair groupoid $M\times M$ of a (smooth, compact) manifold $M$ and the tangent bundle $TM$ of $M$. He showed that this groupoid describes the analytic index on $M$ in a way not involving (pseudo)differential operators at all, and gave a proof of the Atiyah-Singer Index Theorem based on this groupoid. This idea of a deformation groupoid was then used in [@HilSkFeuill [section III]{}], and extended in [@MonthPie; @NWX] to the general case of a smooth groupoid, where the authors associated to every smooth groupoid ${\mathcal{G}}$ an *adiabatic groupoid* ${\mathcal{G}}_{ad}$, which is obtained by applying the “deformation to the normal cone” construction to the inclusion ${\mathcal{G}}^{(0)}\to {\mathcal{G}}$ of the unit space of ${\mathcal{G}}$ into ${\mathcal{G}}$. Moreover, it was shown in [@MonthPie [Théorème 2.1]{}] that this adiabatic groupoid still describes the analytic index of the groupoid ${\mathcal{G}}$ in this generalized situation. In [@DS1], we further explored the relationship between pseudodifferential calculus on ${\mathcal{G}}$ and its adiabatic deformation ${\mathcal{G}}_{ad}$. An ideal $J({\mathcal{G}})\subset C^*({\mathcal{G}}_{ad})$ which sits in an exact sequence $0\to J({\mathcal{G}})\to C^*({\mathcal{G}}_{ad})\to C({\mathcal{G}}^{(0)})\to 0$ plays a crucial role in our constructions. We construct a canonical Morita equivalence between the algebra $\Psi^*({\mathcal{G}})$ of order $0$ pseudodifferential operators on ${\mathcal{G}}$ and the crossed product $J({\mathcal{G}})\rtimes {\mathbb{R}}_+^*$ of $J({\mathcal{G}})$ by the natural action of ${\mathbb{R}}_+^*$.\ It appeared that $J({\mathcal{G}})$ is canonically isomorphic to the crossed product $\Psi^*({\mathcal{G}})\rtimes {\mathbb{R}}$ associated with a natural action of ${\mathbb{R}}$ on the algebra $\Psi^*({\mathcal{G}})$. A natural question is then: can one recognize the $C^*$-algebra $C^*({\mathcal{G}}_{ad})$ in these terms ? In the present paper, we answer this question, thanks to [@BaajOPD1; @BaajOPD2], where Baaj constructed [an extension of pseudodifferential operators of order $0$ of the crossed product]{} of a $C^*$-algebra $A$ by the action of a Lie group $H$ - with Lie algebra ${\mathfrak{H}}$. Denote by $S^*{\mathfrak{H}}$ the sphere in ${\mathfrak{H}}^*$. Baaj’s exact sequence reads $$0\to A\rtimes H{\longrightarrow}\Psi_0^*(A,H)\overset{\sigma}{{\longrightarrow}}C(S^*{\mathfrak{H}})\otimes A\to 0.$$ Let $\mu :C({\mathcal{G}}^{(0)})\to \Psi^*({\mathcal{G}})$ be the inclusion by multiplication operators. In the present paper, we construct a commutative diagram, whose first line is Baaj’s exact sequence: $$\xymatrix{ 0\ar[r]&\Psi^*({\mathcal{G}})\rtimes {\mathbb{R}}\ar[r] &\Psi_0 ^*(\Psi^*({\mathcal{G}}),{\mathbb{R}})\ar[r]^{\sigma\ \ \ }& \Psi^*({\mathcal{G}})\oplus \Psi^*({\mathcal{G}})\ar[r]& 0\\ 0\ar[r]& J({\mathcal{G}})\ar[r]\ar[u]_{\simeq}&C^*({\mathcal{G}}_{ad})\ar[r]\ar[u] &C({\mathcal{G}}^{(0)})\ar[u]_{\mu _0}\ar[r]& 0 }\eqno (1)$$ where $\mu_0(f)=(\mu(f),0)$. Moreover, we show that all the morphisms of the above diagram are equivariant with respect to the natural actions of ${\mathbb{R}}_+^*$: - We consider ${\mathbb{R}}_+^*$ as the dual group of ${\mathbb{R}}$ and thus it acts on the crossed product $\Psi^*({\mathcal{G}})\rtimes {\mathbb{R}}$ via the dual action. This dual action extends (uniquely) to Baaj’s pseudodifferential extension $\Psi ^*_0(\Psi^*({\mathcal{G}}),{\mathbb{R}})$ and is trivial at the quotient level. - The action of ${\mathbb{R}}_+^*$ on the second line is the canonical action on the adiabatic groupoid by the natural rescaling, and the crossed product $C^*({\mathcal{G}}_{ad})\rtimes {\mathbb{R}}_+^*$ is the $C^*$-algebra of the “gauge adiabatic groupoid” ${\mathcal{G}}_{ga}$ considered in [@DS1]. In particular, this allows us to give also a description of the algebra $C^*({\mathcal{G}}_{ga})$ as a pseudodifferential extension. As a side construction, we define the pseudodifferential extension of an action $\alpha$ of a smooth groupoid ${\mathcal{G}}$ - in the setting introduced by Le Gall in [@PYLG; @PYLG2]. [This is a short exact sequence $$0\to A\rtimes _\alpha {\mathcal{G}}\longrightarrow \Psi^*(A,\alpha,{\mathcal{G}})\overset{\sigma_\alpha}\longrightarrow A\otimes _{C_0(M)}C(S^* {\mathfrak{A}}{\mathcal{G}})\to 0.\eqno(2)$$]{} This construction generalizes both the pseudodifferential calculus on a smooth groupoid of [@ConnesLNM; @ConnesNCG; @MonthPie; @NWX] and the pseudodifferential calculus of a crossed product by a Lie group of [@BaajOPD1; @BaajOPD2]. Our main result, Theorem \[maintheorem\], is stated (and proved) in this general frame: in diagram (1) we allow the groupoid ${\mathcal{G}}$ to act on a $C^*$-algebra $A$ and replace groupoid $C^*$-algebras by crossed products. We should note that the connecting map of extension (2) is the analytic index in this context. In the same way as in [@MonthPie; @NWX], the crossed product by the adiabatic groupoid allows to define the analytic index too. Here are some examples of natural actions of smooth groupoids which are relevant to our constructions. 1. Already an interesting case appears when $A=C_0(X)$ where $X$ is a smooth manifold, endowed with a smooth submersion $p:X\to M={\mathcal{G}}^{(0)}$ and ${\mathcal{G}}$ acts on the fibers. The action of ${\mathcal{G}}$ is given by a diffeomorphism $\alpha:{\mathcal{G}}{}_s\times_pX\to X{}_p\times_r{\mathcal{G}}$ of the form $(\gamma,x)\mapsto (\alpha_\gamma(x),\gamma)$, which satisfies $\alpha_{\gamma_1\gamma_2} =\alpha_{\gamma_1}\alpha_{\gamma_2}$. Here, ${\mathcal{G}}{}_s\times_pX$ is a smooth groupoid ${\mathcal{G}}_X$ with objects $X$, source and range maps given by $s(\gamma,x)=x$, $r(\gamma,x)=\alpha_\gamma(x)$ composition $(\gamma',\alpha_\gamma(x))(\gamma,x)=(\gamma'\gamma,x)$ and inverse $(\gamma,x)^{-1}=(\gamma^{-1},\alpha_\gamma(x))$. In that case, the crossed product $A\rtimes_\alpha {\mathcal{G}}$, the extension $\Psi^*(A,\alpha,{\mathcal{G}})$, the crossed product $(A\otimes {\mathbb{R}}_+)\rtimes {\mathcal{G}}_{ad}$ identify respectively with the groupoid $C^*$-algebra $C^*({\mathcal{G}}_X)$, the pseudodifferential extension $\Psi^*({\mathcal{G}}_X)$ and the $C^*$-algebra $C^*(({\mathcal{G}}_X)_{ad})$ of the adiabatic deformation of the groupoid ${\mathcal{G}}_X$. 2. Let $G$ be a Lie group acting on a $C^*$-algebra $A$. The corresponding adiabatic and gauge adiabatic deformations of $G$ are groupoids with objects ${\mathbb{R}}_+$. They naturally act on the $C_0({\mathbb{R}}_+)$ algebra $A\otimes C_0({\mathbb{R}}_+)$ - and the associated action is an important piece in our constructions - see section \[adgpdaction\]. 3. An interesting family of examples of groupoid actions comes from $1$-cocycles (generalized morphisms in the sense of [@HilSkFeuill section I], [@PYLG [Section 2.2]{}]) of a groupoid ${\mathcal{G}}$ to a Lie group. For instance, an equivariant vector bundle is equivalent to a cocycle from ${\mathcal{G}}$ to $GL_n({\mathbb{R}})$. Then every algebra $A$ endowed with an action of $G$ gives rise to a ${\mathcal{G}}$-algebra. This construction is studied in [@PYLG] where several examples connected with $K$-theory and index theory are studied. The corresponding pseudodifferential extension and associated actions of the adiabatic groupoid appear very naturally in this context. The paper is organized as follows: In the second section, we briefly review the action of a locally compact groupoid and the corresponding full and reduced crossed products ([[*cf.*]{} ]{}[@PYLG; @PYLG2; @Tu1; @Tu2; @Pat]). In the third section, we review Baaj’s construction and discuss the dual action. In the fourth section we generalize Baaj’s construction to the case of actions of smooth groupoids. The fifth section establishes the above mentioned equivariant commutative diagram. Finally, we gathered a few rather well known facts on unbounded multipliers in an appendix. If $A$ is a $C^*$-algebra, we denote by ${\mathcal{M}}(A)$ its multiplier algebra. Recall that, if $A$ and $B$ are $C^*$-algebras, a morphism $f:A\to {\mathcal{M}}(B)$ is said to be non degenerate if $f(A).B=B$; a non degenerate morphism extends uniquely to a morphism $\tilde f:{\mathcal{M}}(A)\to {\mathcal{M}}(B)$ - this extension is strictly continuous ([[*i.e.*]{} ]{}continuous with respect to the natural topologies of the multipliers). Recall that an ideal $J$ of a $C^*$-algebra $A$ is said to be essential if the morphism $A\to {\mathcal{M}}(J)$ is injective, [[*i.e.*]{} ]{}if $a\in A$ is such that $aJ=\{0\}$ then $a=0$. \[essential\] Note that if $\pi: A\to B$ is a surjective morphism of $C^*$-algebras and $J$ an essential ideal in $B$ then $\pi^{-1} (J)$ is essential in $A$. Actions of locally compact groupoids and crossed products ========================================================= In this section we briefly recall a few facts about actions of locally compact groupoids and the corresponding crossed products as defined by Le Gall in [@PYLG; @PYLG2]. See also [@Tu1; @Tu2; @Pat]. Actions of locally compact groupoids {#essonfrerK1} ------------------------------------ ### ${\mathbf C_0(X)}$-algebras {#soeur} ${\mathbf C_0(X)}$-algebras. : Recall ([@DG], [@Kasparov [Def. 1.5]{}]) that if $X$ is a locally compact space, a $C_0(X)$-algebra is a pair $(A,\theta)$, where $A$ is a $C^*$-algebra and $\theta$ is a non degenerate $*$-homomorphism $\theta:C_0(X)\to Z{\mathcal{M}}(A)$ from $C_0(X)$ to the center of the multiplier algebra of $A$. Fibers. : If $A$ is a $C_0(X)$-algebra, we define its fiber $A_x$ for every point $x\in X$ by setting $A_x=A/C_xA$ where $C_x = \{h\in C_0 (X);\ h(x) = 0\}$. Let $a\in A$ and denote by $a_x\in A_x$ its class; we have $\|a\|=\sup_{x\in X} \|a_x\|$. In particular $a $ is completely determined by the family $(a_x)_{x\in X}$ and the bundle $A$ is semi-continuous in the sense that for all $a\in A$ the map $x\mapsto \|a_x\|$ is upper semi-continuous. ${\mathbf C_0(X)}$-morphisms. : A $C_0(X)$-linear homomorphism $\alpha:A\to B$ of $C_0(X)$-algebras determines for each $x\in X$ a $*$-homomorphism $\alpha_x:A_x\to B_x$. Since $\alpha(a)$ is determined by the family $(\alpha(a))_x=\alpha_x(a_x)$, the morphism $\alpha$ is determined by the family $(\alpha_x)_{x\in X}$. Restriction to locally closed sets; pull back. : More generally, if $U\subset X$ is an open subset, we define the $C_0(U)$-algebra $A_U$ by putting $A_U=C_0(U)A$; if $F\subset X$ is a closed subset, we define the $C_0(F)$-algebra $A_F=A/A_{X\setminus F}$; if $Y=U\cap F$ is a locally closed subset of $X$ we put $A_Y=(A_U)_Y$ (which is canonically isomorphic to $(A_F)_Y$). Recall that if $f:Y\to X$ is a continuous map between locally compact spaces and $A$ is a $C_0(X)$-algebra, we may define $f^*(A)$ in the following way: we restrict the $C_0(X\times Y)$-algebra $A\otimes C_0(Y)$ to the graph $\{(x,y)\in X\times Y;\ f(y)=x\}$ of $f$ which is a closed subset of $X\times Y$ canonically homeomorphic with $Y$. As $f^*(A)$ is a quotient of $A\otimes C_0(Y)$, we have a non degenerate morphism $a\mapsto a\circ f$ from $A$ to the multiplier algebra of $f^*(A)$, where $a\circ f$ is the image of $a\otimes 1$ in the quotient $f^*(A)$ of $A\otimes C_0(Y)$. ### Actions of groupoids ([@PYLG2 Definition 2.2]). Let ${\mathcal{G}}$ be a locally compact groupoid with basis $X$. A continuous action of ${\mathcal{G}}$ on a $C_0(X)$-algebra $A$ is an isomorphism of $C_0({\mathcal{G}})$-algebras $\alpha : s^*A \to r^*A$ such that, for all $(\gamma_1,\gamma_2)\in {\mathcal{G}}^{(2)}$ we have $\alpha_{\gamma_1\gamma_2}=\alpha_{\gamma_1}\circ\alpha_{\gamma_2}$. An action of a non Hausdorff groupoid ${\mathcal{G}}$ on a $C_0(X)$-algebra $A$ (with $X={\mathcal{G}}^{(0)}$) is given by isomorphisms $\alpha_U:s_U^*(A)\to r_U^*(A)$ for every Hausdorff open subset $U$ of $X$ - where $s_U,r_U$ are the restrictions of $r$ and $s$ to $U$. These isomorphisms must agree on the intersection $U\cap V$ of two such sets. It follows that the family $(r_U)$ gives rise to isomorphisms $\alpha_\gamma:A_{s(\gamma)}\to A_{r(\gamma)}$ for $\gamma\in {\mathcal{G}}$. We further impose that these isomorphisms satisfy $\alpha_{\gamma_1\gamma_2}=\alpha_{\gamma_1}\circ\alpha_{\gamma_2}$ for all $(\gamma_1,\gamma_2)\in {\mathcal{G}}^{(2)}$.\ In the sequel of the paper, we will consider Hausdorff groupoids for simplicity of the exposition. Nevertheless, all our constructions and results extend in the usual way to the non Hausdorff case [@ConnesSurvey [section 6]{}], see also [@KhoshSk [section I.B]{}]. Note that the non trivial part of any kind of pseudodifferential calculus concentrates in a Hausdorff neighborhood of the space of units. Crossed products ---------------- The (full and reduced) crossed product $A\rtimes_\alpha {\mathcal{G}}$ of an action $\alpha $ of a groupoid ${\mathcal{G}}$ with (right) Haar system $(\nu ^x)_{x\in X}$ on a $C^*$-algebra $A$ is defined in [@PYLG2; @Pat2]. Let us briefly recall these constructions. ### The full crossed product The vector space $C_c(r^*A)=C_c({\mathcal{G}}).r^*(A)$ of elements of $r^*A$ with compact support is naturally a convolution $*$-algebra. For $f,g\in C_c(r^*A)$ and $\gamma\in {\mathcal{G}}$, we have $$(f\ast g)_\gamma=\int_{{\mathcal{G}}^{r(\gamma)}}f_{\gamma_1}\alpha_{\gamma_1}(g_{\gamma_1^{-1}\gamma})\,d\nu ^{r(\gamma)}(\gamma_1)\quad \hbox{and} \quad (f^*)_\gamma=\alpha_{\gamma}^{-1}(f_{{\gamma}^{-1}})$$ There is a $\|\ \|_1$ norm given by $$\|f\|_1=\sup_{x\in X} \max\left(\int_{{\mathcal{G}}^x}\|f_\gamma\|d\nu ^x(\gamma),\int_{{\mathcal{G}}^x}\|f_{\gamma^{-1}}\|d\nu ^x(\gamma)\right)$$on this algebra and the corresponding completion is a Banach $*$-algebra $L^1(r^*A,\nu )$ (recall that $X$ is the basis ${\mathcal{G}}^{(0)}$ of ${\mathcal{G}}$). The full crossed product $A\rtimes_\alpha {\mathcal{G}}$ is the enveloping $C^*$-algebra of $L^1(r^*A,\nu )$. The algebras $A$ and $C^*({\mathcal{G}})$ sit in the multipliers of $A\rtimes_\alpha {\mathcal{G}}$ in a non degenerate way, and $A\rtimes_\alpha {\mathcal{G}}$ is the closed vector span of products $a.f$ with $a\in A$ and $f\in C^*({\mathcal{G}})$. Note that $C_0(X)$ sits both in the multipliers of $C^*({\mathcal{G}})$ and of $A$; its images in ${\mathcal{M}}(A\rtimes_\alpha {\mathcal{G}})$ agree. ### Covariant representations (see [@Pat2 p. 1466] - see also [@Ren [section II.1]{}]) The representations of $A\rtimes _\alpha {\mathcal{G}}$ can easily be described as in [@Ren [Theorem 1.21, p. 65]{}]. Such a representation gives rise to representations of $A$ and $C^*({\mathcal{G}})$. We thus obtain: - The representation of $C_0(X)$ corresponds to a measure $\mu$ on $X$ and a measurable field of Hilbert spaces $(H_x)_{x\in X}$. - The representation of the $C_0(X)$-algebra $A$ is given by a measurable family $\pi=(\pi_x)_{x\in X}$ where $\pi_x:A_x\to {\mathcal{L}}(H_x)$ is a $*$-representation. - The representation of $C^*({\mathcal{G}})$ gives rise to a representation of ${\mathcal{G}}$ in the sense of [@Ren [def. 1.6, p. 52]{}]. In other words, the measure $\mu$ is quasi-invariant ([[*i.e.*]{} ]{}$\mu\circ\nu $ is quasi-invariant by the map $\gamma\mapsto \gamma^{-1}$) and we have a measurable family $U=(U_\gamma)_{\gamma\in {\mathcal{G}}}$ where $U_\gamma:H_{s(\gamma)}\to H_{r(\gamma)}$ is (almost everywhere) unitary and satisfies (almost everywhere) $U_{\gamma_1\gamma_2}=U_{\gamma_1}U_{\gamma_2}$. - The covariance property then reads: $\pi_{r(\gamma)}\circ \alpha_\gamma=Ad_{U_\gamma}\circ \pi_{s(\gamma)}$ (almost everywhere). Conversely, such data $(\mu,H,\pi,U)$ can be integrated to a representation of $A\rtimes _\alpha {\mathcal{G}}$. ### The reduced crossed product (see [@Ren; @KhoshSk]) The reduced crossed product $A\rtimes_{\alpha,red} {\mathcal{G}}$ is the quotient of $A\rtimes_\alpha {\mathcal{G}}$ corresponding to the family of regular representations on the Hilbert modules $A_x\otimes L^2({\mathcal{G}}^x;\nu ^x)$ for $x\in X$. If ${\mathcal{G}}$ is amenable (see [@ADR] [for a discussion on amenability of groupoids]{}) then the morphism $A\rtimes_\alpha {\mathcal{G}}\to A\rtimes_{\alpha,red} {\mathcal{G}}$ is an isomorphism. The reduced crossed product has a faithful representation on the Hilbert $A$-module ${\mathcal{E}}=L^2({\mathcal{G}};\nu )\otimes _{C_0(X)}A$ where $L^2({\mathcal{G}};\nu )$ is the Hilbert $C_0(X)$ module described in [@KhoshSk [Theorem 2.3]{}] (if ${\mathcal{G}}$ is Hausdorff). The module ${\mathcal{E}}$ is the completion of $C_c({\mathcal{G}};s^*A)$ with respect to the $A$-valued inner product satisfying $(\langle \xi|\eta\rangle )_x=\int _{{\mathcal{G}}_x}\xi ^*_\gamma\eta_\gamma d\nu _x(\gamma)$ (where $(\nu_x)_{x\in X}$ is the corresponding left Haar system given by $\int f(\gamma) d\nu_x(\gamma)=\int f(\gamma^{-1}) d\nu ^x(\gamma)$) and, right action given by $(\xi a)_\gamma=\xi_\gamma a_{s(\gamma)}$). Denote by $\lambda $ the action of $C_{red}^*({\mathcal{G}})$ by (left) convolution on the Hilbert $C_0(X)$-module $L^2({\mathcal{G}};\nu )$; the left action of $C^*({\mathcal{G}})$ is given by $f\mapsto \lambda(f)\otimes _{C_0(X)}1$. The action of $A$ is given by $a.\xi=\Big(\alpha^{-1}(a\circ r)\Big)\xi$: in other terms $(a.\xi)_\gamma=\alpha_\gamma^{-1}(a_{ r(\gamma)})\xi_\gamma$. It follows, that if $\pi =\int _{X}^\oplus \pi_x\,d\mu(x)$ is a faithful representation of $A$, the corresponding representation of $A\rtimes _{\alpha, red}{\mathcal{G}}$ on $\int_X^\oplus L^2({\mathcal{G}}_x,\nu_x)\otimes H_x\,d\mu(x)$ is faithful. ### Invariant ideals and exact sequences (see [@Pat2 Theorem 3]) Let $J\subset A$ be an ideal in $A$. Note that both $J$ and $A/J$ are then $C_0(X)$ algebras - recall that $X={\mathcal{G}}^{(0)}$. Assume that $J$ is invariant under the action of ${\mathcal{G}}$ which means that $\alpha(s^*(J))=r^*(J)$. Then $\alpha$ yields actions of ${\mathcal{G}}$ on $J$ and $A/J$. [@Pat2 Theorem 3] We have an exact sequence of *full* crossed products: $$0\to J\rtimes_\alpha {\mathcal{G}}\to A\rtimes_\alpha {\mathcal{G}}\to (A/J)\rtimes_\alpha {\mathcal{G}}\to 0.$$ The only thing which is not completely obvious in this sequence is that the morphism $(A\rtimes_\alpha {\mathcal{G}})/(J\rtimes_\alpha {\mathcal{G}})\to A/J\rtimes_\alpha {\mathcal{G}}$ is injective. To see that, take a faithful representation of $(A\rtimes_\alpha {\mathcal{G}})/(J\rtimes_\alpha {\mathcal{G}})$; it is a covariant representation of $A$ and ${\mathcal{G}}$ which vanishes on $J$, and therefore a covariant representation of $A/J$ and ${\mathcal{G}}$. If $J$ is a ${\mathcal{G}}$-invariant essential ideal in $A$, then at the level of *reduced* crossed products, the ideal $J\rtimes_{\alpha,red}{\mathcal{G}}$ of $A\rtimes_{\alpha,red}{\mathcal{G}}$ is essential. ### Invariant open sets Let $U$ be an open subset of ${\mathcal{G}}$, which is saturated for ${\mathcal{G}}$ ([[*i.e.*]{} ]{}for all $\gamma\in {\mathcal{G}}$, we have $s(\gamma)\in U\iff r(\gamma)\in U$). Put $F=X\setminus U$. Define the subgroupoids ${\mathcal{G}}_U=s^{-1}(U)=r^{-1}(U)$ and ${\mathcal{G}}_F=s^{-1}(F)=r^{-1}(F)$. The action $\alpha $ of ${\mathcal{G}}$ on $A$ gives actions $\alpha _U$ of ${\mathcal{G}}_U$ on $A_U$ and $\alpha _F$ of ${\mathcal{G}}_F$ on $A_F$. We may note that $A_U\rtimes _{\alpha_U}{\mathcal{G}}_U=A_U\rtimes _{\alpha}{\mathcal{G}}$ and $A_F\rtimes _{\alpha_v}{\mathcal{G}}_F=A_F\rtimes _{\alpha}{\mathcal{G}}$. Let us quote some results that we will use: 1. We have an exact sequence of full crossed products: $$0\to A_U\rtimes _{\alpha_U}{\mathcal{G}}_U\to A\rtimes_\alpha {\mathcal{G}}\to A_F\rtimes_{\alpha_F} {\mathcal{G}}_F\to 0.$$ 2. If ${\mathcal{G}}_F$ is amenable, the same is true for the reduced crossed products - exactness at the middle terms follows from the diagram $$\xymatrix{ 0\ar[r]&A_U\rtimes _{\alpha_U}{\mathcal{G}}_U\ar[r]\ar[d] &A\rtimes_\alpha {\mathcal{G}}\ar[r]\ar[d]& A_F\rtimes_{\alpha_F} {\mathcal{G}}_F\to 0\ar[r]\ar[d]^{\simeq}& 0\\ 0\ar[r]& A_U\rtimes_{\alpha_U,red} {\mathcal{G}}_U\ar[r]&A\rtimes_{\alpha,red} {\mathcal{G}}\ar[r] &A_F\rtimes_{\alpha_F,red} {\mathcal{G}}_F\ar[r]& 0 }$$ where the first line is exact and the vertical arrows are onto, the last one being an isomorphism. 3. If $A_U$ is an essential ideal in $A$, then $A_U\rtimes_{\alpha_U,red} {\mathcal{G}}_U$ is an essential ideal in $A\rtimes_{\alpha,red} {\mathcal{G}}$. 4. It follows from Rem. \[essential\] that, if ${\mathcal{G}}_F$ is amenable and $A_U$ is an essential ideal in $A$, then $A_U\rtimes_{\alpha_U} {\mathcal{G}}_U$ is an essential ideal in $A\rtimes_{\alpha} {\mathcal{G}}$. Baaj’s pseudodifferential extension =================================== In this section, we briefly review Baaj’s construction of the pseudodifferential extension of a crossed product by a Lie group $G$. We note that the dual action extends to the pseudodifferential extension (and is trivial at the symbol level) and discuss the corresponding crossed product. Although this is not necessary in our framework, we will not assume $G$ to be abelian, so that this dual action is a coaction of $G$, since this doesn’t really add any difficulty. We then establish an isomorphism between the crossed product of the algebra of the pseudodifferential operators by the dual action and a natural pseudodifferential extension. Finally, we examine the case where the Lie group is ${\mathbb{R}}$ - which is the relevant case for our results of section 5. Baaj’s pseudodifferential calculus for an action of a Lie group --------------------------------------------------------------- Let us begin by recalling the extension of pseudodifferential operators associated with a continuous action $\alpha$ by automorphisms of a Lie group $G$ on a $C^*$-algebra $A$ ([@BaajOPD1; @BaajOPD2], the results of Baaj concern the case $G={\mathbb{R}}^n$ - but immediately generalize to the general case of a Lie group). Recall first that the order $0$ pseudodifferential operators on a Lie group $G$ give rise to an exact sequence $$0\to C^*(G) \longrightarrow \Psi^*(G)\overset{\sigma }\longrightarrow C(S^*{\mathfrak{g}})\to 0$$ where [$C^*(G)$ is the (full) group $C^*$-algebra of $G$ and]{} $S^*{\mathfrak{g}}$ denotes the (compact) space of half lines in the dual space ${\mathfrak{g}}^*$ of the Lie algebra ${\mathfrak{g}}$. Now, the algebras $A$ and $C^*(G)$ sit in the multiplier algebra of $A\rtimes _\alpha G$ in a non degenerate way, and the elements $ax$ with $a\in A$ and $x\in C^*(G)$ span a dense subspace of $A\rtimes _\alpha G$. This holds for the full group algebra and crossed product, as well as for the reduced group algebra and crossed product. Note however that, at the level of full $C^*$-algebras, the morphism $C^*(G)\to {\mathcal{M}}(A\rtimes _\alpha G)$ needs not be injective in general - it is easily seen to be injective at the level of reduced $C^*$-algebras. We will somewhat abusively identify $C^*(G)$ and $A$ with their images in the multiplier algebra ${\mathcal{M}}(A\rtimes _\alpha G)$. In what follows, since we will consider the crossed product by the dual action, we will mainly use the reduced crossed product. Note also that we will mainly use Baaj’s construction in the case where $G$ is ${\mathbb{R}}$ which is amenable and there is no distinction between the full and the reduced case. In particular the morphism $C^*(G)\to {\mathcal{M}}(A\rtimes _\alpha G)$ is injective in that case (if $A\ne \{0\}$). The nondegenerate morphism $C^*(G)\to {\mathcal{M}}(A\rtimes_{\alpha} G)$ extends to the multiplier algebra of $C^*(G)$ and in particular to the subalgebra $\Psi^*(G)$ of order $0$ pseudodifferential operators of $G$. We still identify (abusively) the elements of $\Psi^*(G)$ with their images in ${\mathcal{M}}(A\rtimes_{\alpha} G)$. Recall that we have: [@BaajOPD1 [section 4]{}] 1. For every $P\in \Psi^*(G)$ and $a\in A$, the commutator $[P,a]$ belongs to $A\rtimes _\alpha G$. 2. The closure of the linear span of products of the form $Pa$ with $P\in \Psi^*(G)$ and $a\in A$ is a $C^*$-subalgebra $\Psi^*(A,\alpha,G)\subset {\mathcal{M}}(A\rtimes_{\alpha} G)$ and we have an exact sequence: $$0\to A\rtimes _\alpha G \longrightarrow \Psi^*(A,\alpha,G)\overset{\sigma_\alpha}\longrightarrow C(S^*{\mathfrak{g}}) \otimes A\to 0.\eqno(1)$$ Let us briefly discuss some naturality properties of this construction: \[etoile\] Let $(A,G,\alpha)$ and $(B,G,\beta)$ be $C^*$-dynamical systems and $\gamma:A\to {\mathcal{M}}(B)$ a $G$-equivariant morphism 1. We obtain a morphism $\widehat \gamma:\Psi ^*(A,\alpha,G)\to {\mathcal{M}}(\Psi^*(B,\beta,G))$ and a commutative diagram $$\xymatrix{ \Psi ^*(A,\alpha,G)\ar[r]^{\sigma_\alpha}\ar[d]^{\widehat {\gamma}} &C(S^*{\mathfrak{g}}) \otimes A\ar[d]^{{{\hbox{id}}}\otimes \gamma}\\ {\mathcal{M}}(\Psi^*(B,\beta,G))\ar[r]^{\widetilde{\sigma_\beta}} &{\mathcal{M}}(C(S^*{\mathfrak{g}}) \otimes B) . }$$ both for the full and the reduced versions - where we denoted by $\widetilde{\sigma_\beta}$ the extension of $\sigma_\beta$ to the multipliers. 2. If $\gamma(A)\subset B$ then $\widehat \gamma(\Psi ^*(A,\alpha,G)) \subset \Psi^*(B,\beta,G)$. Moreover, if $\gamma:A\to B$ is an isomorphism, then $\widehat \gamma:\Psi ^*(A,\alpha,G)\to \Psi^*(B,\beta,G)$ is an isomorphism. 3. If $\gamma$ is injective then so is the reduced version of $\widehat \gamma$. <!-- --> 1. By construction the inclusion of $B$ in $\Psi^*(B,\beta,G)$ is a nondegenerate morphism ([[*i.e.*]{} ]{}$B\Psi^*(B,\beta,G)=\Psi^*(B,\beta,G)$). It therefore extends to a morphism ${\mathcal{M}}(B)\to {\mathcal{M}}(\Psi^*(B,\beta,G))$. In this way, we find a representation $\widehat \gamma:A\to {\mathcal{M}}(\Psi^*(B,\beta,G))$. Now the images of $A$ and $G$ in ${\mathcal{M}}(B\rtimes _{\beta}G)\supset {\mathcal{M}}(\Psi^*(B,\beta,G))$ form a covariant representation so that we get a morphism $A\rtimes _{\alpha}G\to {\mathcal{M}}(B\rtimes _{\beta}G)$ (both for the reduced and full versions of the crossed products). The image of this morphism is spanned by elements $a.h$ with $a\in A$ and $h\in C^*(G)$; it therefore sits in ${\mathcal{M}}(\Psi^*(B,\beta,G))$. Finally, upon replacing $A$ by the algebra obtained by adjoining a unit, we may assume that $\gamma$ is non degenerate. It follows that $\widehat \gamma :A\rtimes _\alpha G\to {\mathcal{M}}(B\rtimes_\beta G)$ is non degenerate and therefore uniquely extends to the multiplier algebra. We thus get a morphism $\widehat \gamma :\Psi^*(A,\alpha,G)\to {\mathcal{M}}(B\rtimes _{\beta}G)$. The image of $a.P$ is $\widehat\gamma(a).P$ (for $a\in A$ and $P\in \Psi^*(G)$) and therefore $\widehat \gamma (\Psi^*(A,\alpha,G))\subset {\mathcal{M}}(\Psi^*(B,\beta,G))$. 2. This is obvious. 3. If $\gamma$ is one to one, then the reduced version $\gamma_{red}:A\rtimes _{\alpha,red}G\to {\mathcal{M}}(B\rtimes_{\beta,red} G)$ is injective. Therefore $\ker \widehat \gamma_{red}\cap A\rtimes _{\alpha, red}G=\{0\}$ whence $\ker \widehat \gamma_{red}=\{0\}$ since $A\rtimes _{\alpha,red}G$ is an essential ideal in $\Psi_{red}^*(A,\alpha,G)$ - see prop. \[orme\]. The dual action --------------- We now restrict to the reduced group algebras and crossed products. The coproduct of $C_{red}^*(G)$ is a non degenerate morphism $\delta:C_{red}^*(G)\to {\mathcal{M}}(C_{red}^*(G)\otimes C_{red}^*(G))$. It therefore extends to a morphism $\tilde \delta:{\mathcal{M}}(C_{red}^*(G))\to {\mathcal{M}}(C_{red}^*(G)\otimes C_{red}^*(G))$. The restriction of $\tilde \delta $ to $\Psi^*(G)$, is a coaction: for $P\in \Psi_{red}^*(G)$ and $f\in C_{red}^*(G)$, we have $\tilde\delta(P)(1\otimes f)\in \Psi_{red}^*(G)\otimes C_{red}^*(G)$ and the span of such products is dense in $\Psi_{red}^*(G)\otimes C_{red}^*(G)$. Moreover, for $P\in \Psi_{red}^*(G)$ and $f\in C_{red}^*(G)$, we have $(\tilde\delta(P)-P\otimes 1)(1\otimes f)\in C^*(G\times G)$. Let $(X_i)_{1\le i\le d}$ be an (orthonormal) basis of ${\mathfrak{g}}$ and let $\Delta =-\sum_i X_i^2$ be the associated (positive) laplacian, seen as an unbounded (elliptic, positive) multiplier of $C_{red}^*(G)$. The non degenerate morphism $\delta $ has an extension $\check\delta$ to unbounded multipliers: for $1\le i\le d$, set $p_i=X_i(1+\Delta)^{-1/2}\in \Psi_{red}^*(G)$.\ We let now $C^*_{red}(G\times G)$ act faithfully on $L^2(G\times G)$. The following equalities hold on the infinite domain of the laplacian of the group $G\times G$, which is a dense subspace of $L^2(G\times G)$. We have $\check\delta (X_i)=X_i\otimes 1+1\otimes X_i$. It follows that $\check\delta (\Delta)=\Delta\otimes 1+1\otimes \Delta-2\sum_iX_i\otimes X_i $. For $f\in C_c^\infty (G)$ [(acting as a convolution operator)]{}, we may then write: $$(1\otimes f)(\tilde \delta (p_i)-p_i\otimes 1)=(1\otimes fX_i)\delta((1+\Delta)^{-1/2})+(X_i\otimes f)(\delta((1+\Delta)^{-1/2})-(1+\Delta)^{-1/2}\otimes 1).$$ Now $fX_i$ and $(1+\Delta)^{-1/2}$ extend to elements of $C_{red}^*(G)$ therefore $C_i=(1\otimes fX_i)\delta((1+\Delta)^{-1/2})$ extends as well to an element of $C_{red}^*(G\times G)$. We write $(1+\Delta)^{-1/2}$ as an integral ([[*cf.*]{} ]{}[@BJcras]): $$(1+\Delta)^{-1/2} =\frac{2}{\pi}\int_0^{+\infty}(1+\Delta+\lambda^2)^{-1} d\lambda .$$ Write also $$(1+\Delta+\lambda^2)^{-1}\otimes 1-\delta(1+\Delta+\lambda^2)^{-1}=((1+\Delta+\lambda^2)^{-1}\otimes 1)(1\otimes \Delta+2\sum_jX_j\otimes X_j) \delta(1+\Delta+\lambda^2)^{-1}$$ Putting $D_i=(X_i\otimes f)\Big((1+\Delta)^{-1/2}\otimes 1-\delta((1+\Delta)^{-1/2})\Big)$, we find $$\begin{array}{ccl} D_i&=&\frac{2}{\pi}(X_i\otimes f)\int_0^{+\infty}((1+\Delta+\lambda^2)^{-1}\otimes 1)-\delta(1+\Delta+\lambda^2)^{-1}d\lambda \\ &=&\frac{2}{\pi} \int_0^{+\infty}(X_i(1+\Delta+\lambda^2)^{-1}\otimes f\Delta)\delta(1+\Delta+\lambda^2)^{-1}d\lambda\\&&-\frac{4}{\pi}\sum_j\int_0^{+\infty}(X_i(1+\Delta+\lambda^2)^{-1}X_j \otimes fX_j)\delta(1+\Delta+\lambda^2)^{-1}d\lambda \end{array}$$ Now all the terms appearing are bounded operators: - $X_i(1+\Delta+\lambda^2)^{-1}$ is pseudodifferential of order $-1$ and therefore $X_i(1+\Delta+\lambda^2)^{-1}\in C_{red}^*(G)$; - $f\Delta$ and $fX_j$ are smoothing therefore in $C_{red}^*(G)$; - $(1\otimes fX_j)\delta(1+\Delta+\lambda^2)^{-1}\in C_{red}^*(G)\otimes C_{red}^*(G)$. It follows that the integrand extends to an element of $C_{red}^*(G)\otimes C_{red}^*(G)$. Furthermore, $X_k(1+\Delta+\lambda^2)^{-1/2}=X_k(1+\Delta)^{-1/2}h_\lambda(\Delta)$ where $\|h_\lambda\|_\infty\le 1$, whence $\|X_i(1+\Delta+\lambda^2)^{-1}\|$ and $\|X_i(1+\Delta+\lambda^2)^{-1}X_j\|$ are bounded independently of $\lambda$. Hence, this integral is norm convergent and $D_i$ extends to an element $\bar D_i$ of $C_{red}^*(G)\otimes C_{red}^*(G)$.\ Thus, we have proved that $(1\otimes f)(\tilde \delta (p_i)-p_i\otimes 1)=C_i+\bar D_i$ belongs to $C_{red}^*(G)\otimes C_{red}^*(G)$. The set ${\mathcal{A}}$ of $P\in \Psi_{red}^*(G)$ such that $(1\otimes C_{red}^*(G))(\tilde \delta(P)-P\otimes 1)\subset C_{red}^*(G)\otimes C_{red}^*(G)$ and $(1\otimes C_{red}^*(G))(\tilde \delta(P^*)-P^*\otimes 1)\subset C_{red}^*(G)\otimes C_{red}^*(G)$ is a closed $*$-subalgebra of $\Psi_{red}^*(G)$; it contains $C_{red}^*(G)$. As $p_i+p_i^*\in C_{red}^*(G)$, it follows by the above calculation that $p_i\in {\mathcal{A}}$. Since the symbols of the $p_i$’s generate a dense subalgebra of the symbol algebra $C(S^*{\mathfrak{g}})$ we conclude that ${\mathcal{A}}= \Psi_{red}^*(G)$. Finally, the closed vector span of $(1\otimes f)\tilde \delta(P)$ contains the closed vector span of $(1\otimes f)\delta (h)$ (with $f,\ h\in C^*(G)$) hence, $C_{red}^*(G)\otimes C_{red}^*(G)$. Therefore $(1\otimes f)\tilde \delta(P)-P\otimes f$ is in this span: the same holds for $P\otimes f$. Isomorphisms {#sect3.4} ------------ Let $\alpha$ be an action of a Lie group $G$ on a $C^*$-algebra $A$. Denote by $\hat \alpha$ the dual action on the reduced crossed product $A\rtimes_{\alpha,red}G$ as well as its extension to $\Psi_{red}^*(A,\alpha,G)$ discussed above. [Recall that in the context on non abelian groups, $B \rtimes {\widehat G}$ is just a notation for the crossed product by a dual action, - it is a $C^*$-algebra generated by products $bf$ with $b\in B$ and $f\in C_0(G)$ subject to the equivariance condition.]{} The Takesaki-Takai duality ([@Takai]) for non abelian groups, (see [@Landstad1; @Landstad2]), is an isomorphism $(A\rtimes_{\alpha,red}G)\rtimes _{\hat \alpha}\widehat G\simeq A\otimes {\mathcal{K}}$ which is based on the following facts: 1. There are natural morphisms of the $C^*$-algebras $A$ and $C_0(G)$ to the multiplier algebra ${\mathcal{M}}((A\rtimes_{\alpha,red}G)\rtimes _{\hat \alpha}\widehat G)$, as well as a (strictly continuous) morphism of the group $G$ to the unitary group of this multiplier algebra, yielding a morphism of $C^*_r(G)$ to ${\mathcal{M}}((A\rtimes_{\alpha,red}G)\rtimes _{\hat \alpha}\widehat G)$. The double crossed product $(A\rtimes_{\alpha,red}G)\rtimes _{\hat \alpha}\widehat G$ is generated by the products $f.a.h$ with $a\in A$, $h\in C^*_r(G)$ and $f\in C_0(G)$ (sitting in the multiplier algebra of $(A\rtimes_{\alpha,red}G)\rtimes _{\hat \alpha}\widehat G$). Now, since the dual action is trivial on $A$, the images of $A$ and $C_0(G)$ commute so that we find in the multiplier algebra of $(A\rtimes_{\alpha,red}G)\rtimes _{\hat \alpha}\widehat G$ a copy of the $C^*$-tensor product $A\otimes C_0(G)$. The group $G$ acts on $A\otimes C_0(G)$ through the action $\alpha \otimes \lambda$ (where $\lambda$ denotes the action of $G$ on $C_0(G)$ by left translation). The morphisms of the $C^*$-algebra $A$ and the group $G$ ([[*resp.*]{} ]{}of $C_0(G)$ and $G$) to ${\mathcal{M}}((A\rtimes_{\alpha,red}G)\rtimes _{\hat \alpha}\widehat G)$ form a covariant representation of the $C^*$-dynamical system $(A,G,\alpha)$ ([[*resp.*]{} ]{}$(C_0(G),G,\lambda)$). It follows that the morphisms of $A\otimes C_0(G)$ and $G$ in the multiplier algebra ${\mathcal{M}}((A\rtimes_{\alpha,red}G)\rtimes _{\hat \alpha}\widehat G)$ form a covariant representation of the $C^*$-dynamical system $(A\otimes C_0(G),G,\alpha \otimes \lambda)$. In this way, we get an isomorphism $(A\rtimes_{\alpha,red}G)\rtimes _{\hat \alpha}\widehat G\simeq (A\otimes C_0(G))\rtimes_{\alpha\otimes \lambda,red}G$. 2. Now, on $A\otimes C_0(G)$, the actions $\alpha\otimes\lambda$ and ${{\hbox{id}}}\otimes \lambda$ are conjugate through the automorphism $\gamma $ of $C_0(G;A)=A\otimes C_0(G)$ given by the formula $(\gamma f)(x)=\alpha_x(f(x))$ for $f\in C_0(G;A)$ and $x\in G$. We find an isomorphism $(A\otimes C_0(G))\rtimes_{\alpha\otimes \lambda,red}G\simeq (A\otimes C_0(G))\rtimes_{{{\hbox{id}}}\otimes \lambda}G$. 3. Finally $(A\otimes C_0(G))\rtimes_{{{\hbox{id}}}\otimes \lambda}G\simeq A\otimes (C_0(G)\rtimes_{ \lambda}G)\simeq A\otimes {\mathcal{K}}$. \[prop2.4\] The isomorphism $f:(A\rtimes_{\alpha,red} G)\rtimes _{\hat \alpha}\widehat G\overset{\sim}{{\longrightarrow}} (A\otimes C_0(G))\rtimes _{\alpha\otimes \lambda,red}G$ extends to an isomorphism $\Psi_{red}^*(A,\alpha,G)\rtimes _{\hat \alpha}\widehat G\simeq \Psi_{red}^*(A\otimes C_0(G),\alpha\otimes \lambda ,G)$. Since $A\rtimes _{\alpha,red} G$ is an essential ideal in $\Psi_{red}^*(A,\alpha,G)$ (see \[orme\]), the algebra $\Psi_{red}^*(A,\alpha,G)\rtimes _{\hat \alpha}\widehat G$ sits in the multiplier algebra ${\mathcal{M}}((A\rtimes_{\alpha,red} G)\rtimes _{\hat \alpha}\widehat G).$ In the same way, the algebra $\Psi_{red}^*(A\otimes C_0(G),\alpha\otimes \lambda ,G)$ sits also in ${\mathcal{M}}((A\otimes C_0(G))\rtimes _{\alpha\otimes \lambda,red}G)$. Both algebras are generated by products $aPh$ where $a\in A$, $P\in \Psi_{red}^*(G)$ and $h\in C^0(G)$. Now the inclusions of $A$ and of $C_0(G)$ in ${\mathcal{M}}$ correspond to each other under the extension $\tilde f$ of $f$ to the multipliers. As the inclusions of $C_{red}^*(G)$ to ${\mathcal{M}}((A\rtimes_{\alpha,red} G)\rtimes _{\hat \alpha}\widehat G)$ and ${\mathcal{M}}((A\otimes C_0(G))\rtimes _{\alpha\otimes \lambda,red}G)$ correspond to each other under $\tilde f$, the same holds for the extension to the multipliers, and in particular for the inclusions of $\Psi_{red}^*(G)$. The actions $\alpha\otimes\lambda$ and ${{\hbox{id}}}\otimes \lambda$ of $G$ on $A\otimes C_0(G)$ are conjugate. Using prop. \[etoile\], we deduce isomorphisms $\Psi_{red}^*(A,\alpha,G)\rtimes _{\hat \alpha}\widehat G\simeq A\otimes \Psi_{red}^*(C_0(G),\lambda,G)\simeq A\otimes (\Psi_{red}^*(G)\rtimes _{\hat\lambda}\widehat G).$ Let $B$ be a subalgebra of $C(S^*{\mathfrak{g}})\otimes A $. We denote by $\Psi_{red}^*(A,\alpha,G;B)$ the *$B$-valued pseudodifferential extension of $\alpha$* [[*i.e.*]{} ]{}the subalgebra $$\Psi_{red}^*(A,\alpha,G;B)=\{P\in \Psi_{red}^*(A,\alpha,G);\ \sigma(P)\in B\}$$ of $\Psi_{red}^*(A,\alpha,G)$. In the case of the trivial action, $\Psi_{red}^*(A,{{\hbox{id}}},G;B)=\{P\in A\otimes \Psi_{red}^*(G);\ ( \sigma\otimes {{\hbox{id}}})(P)\in B\}$. The case of ${\mathbb{R}}$ {#caseofR} -------------------------- When $G={\mathbb{R}}$, then ${\mathfrak{g}}^*= {\mathbb{R}}$ which has two half lines, [[*i.e.*]{} ]{}$C(S^*{\mathfrak{g}})={\mathbb{C}}\oplus {\mathbb{C}}$. Extension (1) reads therefore $$0\to A\rtimes _\alpha{\mathbb{R}}\longrightarrow \Psi^*(A,\alpha,{\mathbb{R}})\overset{\sigma_\pm}\longrightarrow A\oplus A\to 0,$$ where $\sigma_+$ and $\sigma_-$ are morphisms from $\Psi^*(A,\alpha,{\mathbb{R}})\to A$. It is helpful for our discussion to identify the dual group of ${\mathbb{R}}$ with ${\mathbb{R}}^*_+$ through the pairing $\langle t|u\rangle=u^{it}$ for $u\in {\mathbb{R}}_+^*$ and $t\in {\mathbb{R}}$. Under this identification, $C^*({\mathbb{R}})\simeq C_0({\mathbb{R}}_+^*)$ and $\Psi_0^*({\mathbb{R}})\simeq C([0,+\infty])$. The maps $\sigma_-$ and $\sigma_+$ correspond to evaluation at $0$ and $+\infty$ in the sense that $\sigma_-(Pa)=P(0)a$ and $\sigma_+(Pa)=P(+\infty)a$, where $a\in A$ and $P\in C([0,+\infty])\simeq\Psi^*({\mathbb{R}})$. The algebra $A$ sits in ${\mathcal{M}}(A\rtimes_\alpha{\mathbb{R}})$ and we have a strictly continuous family $(u_t)_{t\in {\mathbb{R}}}$ in ${\mathcal{M}}(A\rtimes {\mathbb{R}})$. Then we can write $u_t=Q_\alpha^{it}$ where $Q_\alpha$ is a regular unbounded, selfadjoint, positive multiplier with dense range - [[*i.e.*]{} ]{}such that $Q_\alpha^{-1}$ is also densely defined, and therefore a regular unbounded, selfadjoint, positive multiplier. The algebra $A\rtimes {\mathbb{R}}$ is spanned by $af(Q_\alpha)$ with $f\in C_0({\mathbb{R}}_+^*)$ and $\Psi^*(A,\alpha,{\mathbb{R}})$ is spanned by $af(Q_\alpha)$ with $a\in A$ and $f\in C([0,+\infty])$. Let $A$ be a $C^*$-algebra and let $\alpha=(\alpha_t)_{t\in {\mathbb{R}}}$ be a continuous action of ${\mathbb{R}}$ on $A$ by $*$-automorphisms. Let $B$ be a $C^*$-subalgebra of $A$. We set $$\Psi^*(A,\alpha,{\mathbb{R}},0,B)=\{x\in \Psi^*(A,\alpha,{\mathbb{R}});\ \sigma_-(x)\in B,\ \sigma_+(x)=0\}.$$ The algebra $\Psi^*(A,\alpha,{\mathbb{R}},0,B)$ is spanned by elements $af(Q_\alpha)+b(1+Q_\alpha)^{-1}$ for $a\in A,\ b\in B,\ f\in C_0({\mathbb{R}}_+^*)=C^*({\mathbb{R}})$ all sitting naturally as multipliers of $A\rtimes _\alpha {\mathbb{R}}$. Pseudodifferential extension associated to an action of a smooth groupoid ========================================================================= In this section, we recall a few facts on smooth groupoids: the pseudodiffelential calculus, the adiabatic groupoid ${\mathcal{G}}$ of a smooth groupoid ${\mathcal{G}}$ [@MonthPie; @NWX], its ideal $J({\mathcal{G}})$ ([@DS1 [section 4.1]{}]), the action of ${\mathbb{R}}_+^*$. We then extend all these to the case of an action of ${\mathcal{G}}$ on a $C^*$-algebra $A$. [Recall that ${\mathfrak{A}}{\mathcal{G}}$ denotes the total space of the normal bundle of the inclusion of ${\mathcal{G}}^{(0)}\subset {\mathcal{G}}$, ${\mathfrak{A}}^*{\mathcal{G}}$ the total space of its dual bundle, and $S^*{\mathfrak{A}}{\mathcal{G}}$ the associated sphere bundle, [[*i.e.*]{} ]{}the set of half lines in ${\mathfrak{A}}^*{\mathcal{G}}$.]{} The extension of pseudodifferential operators {#grade} --------------------------------------------- On every Lie groupoid ${\mathcal{G}}$, there is a (longitudinal) pseudodifferential calculus. For every $m\in {\mathbb{R}}$ (and even for $m\in {\mathbb{C}}$ - [@chief [section 3]{}]) we have a space ${\mathcal{P}}_m({\mathcal{G}})$ of classical pseudodifferential operators of order $m$ (with polyhomogeneous symbol $\sigma\sim \sum_{k= 0}^{+\infty}a_{m-k}$ where $a_{m-k}$ is homogeneous of order $m-k$) and a symbol map which is a linear map $\sigma_m$ from ${\mathcal{P}}_m({\mathcal{G}})$ to homogeneous functions of order $m$ defined on ${\mathfrak{A}}^*{\mathcal{G}}$ (outside the zero section) - with kernel ${\mathcal{P}}_{m-1}({\mathcal{G}})$. The smooth functions of $M={\mathcal{G}}^{(0)}$ define elements of ${\mathcal{P}}_0({\mathcal{G}})$; the sections of the algebroid define elements of ${\mathcal{P}}_1({\mathcal{G}})$. The algebra generated by these is the algebra of differential operators. Given a positive definite quadratic form $q$ on the bundle ${\mathfrak{A}}^*{\mathcal{G}}$, we may find a (positive) laplacian $\Delta_{\mathcal{G}}\in {\mathcal{P}}_2({\mathcal{G}})$ which is a positive and whose principal symbol is $q$. At the level of $C^*$-algebras we obtain an extension $\Psi^*({\mathcal{G}})$ of $C^*({\mathcal{G}})$ and an exact sequence of order $0$ pseudodifferential operators $$0\to C^*({\mathcal{G}})\longrightarrow\Psi^*({\mathcal{G}})\overset{\sigma_0}\longrightarrow C(S^*{\mathfrak{A}}{\mathcal{G}})\to 0.$$ Recall ([[*cf.*]{} ]{}[@ConnesLNM; @MonthPie; @NWX]) that $\Psi^*({\mathcal{G}})$ is the closure of the algebra ${\mathcal{P}}_0({\mathcal{G}})$ of order zero pseudodifferential operators on ${\mathcal{G}}$ in the multiplier algebra of $C^*({\mathcal{G}})$ and $\sigma_0$ is the (extension by continuity of the) principal symbol map. The adiabatic groupoid and the ideal $J({\mathcal{G}})$ ------------------------------------------------------- Let ${\mathcal{G}}$ be a Lie groupoid. We denote by $M={\mathcal{G}}^{(0)}$ its set of objects. The associated *adiabatic groupoid* ${\mathcal{G}}_{ad}$ is obtained by applying the “deformation to the normal cone” construction to the inclusion $M\to {\mathcal{G}}$ of the unit space of ${\mathcal{G}}$ into ${\mathcal{G}}$. This construction was introduced by Connes in the case of a pair groupoid ${\mathcal{G}}=M\times M$ ([@ConnesNCG [section II.5]{}]), and generalized in [@MonthPie; @NWX]. As a set, and as a groupoid, ${\mathcal{G}}_{ad}={\mathfrak{A}}{\mathcal{G}}\times \{0\}\cup {\mathcal{G}}\times {\mathbb{R}}_+^*$ where ${\mathfrak{A}}{\mathcal{G}}$ is (the total space of) the Lie algebroid of ${\mathcal{G}}$, [[*i.e.*]{} ]{}the normal bundle of the inclusion in ${\mathcal{G}}$ of the space of objects $M$ of ${\mathcal{G}}$; its groupoid structure is given by addition of vectors - source and range coincide and are just the bundle map ${\mathfrak{A}}{\mathcal{G}}\to M$. These sets are glued using an exponential map ${\mathfrak{A}}{\mathcal{G}}\to {\mathcal{G}}$ (see [@MonthPie; @CR; @DS1] for further details). The $C^*$-algebra of the adiabatic groupoid of ${\mathcal{G}}$ sits in an exact sequence $$0\to C^*({\mathcal{G}})\otimes C_0({\mathbb{R}}_+^*)\longrightarrow C^*({\mathcal{G}}_{ad})\overset{ev_0}\longrightarrow C_0({\mathfrak{A}}^*{\mathcal{G}})\to0,$$ where ${\mathfrak{A}}^*{\mathcal{G}}$ denotes the total space of the dual bundle to the Lie algebroid ${\mathfrak{A}}{\mathcal{G}}$ of ${\mathcal{G}}$. Consider the morphism $\epsilon:C_0({\mathfrak{A}}^*{\mathcal{G}})\to C(M)$ which associates to a function on ${\mathfrak{A}}^*{\mathcal{G}}$ its value on the $0$-section $M$ of the bundle ${\mathfrak{A}}^*{\mathcal{G}}$ - [[*i.e.*]{} ]{}the trivial representation of the group ${\mathfrak{A}}_x {\mathcal{G}}$. We denote by $J({\mathcal{G}})$ the kernel of $\epsilon\circ ev_0$, which is an ideal of $C^*({\mathcal{G}}_{ad})$. We therefore have an exact sequence: $$0\to J({\mathcal{G}})\to C^*({\mathcal{G}}_{ad})\to C(M)\to 0.$$ \[Jessentiel\] It follows from [@KhoshSk Corollary 2.4], since $M\times {\mathbb{R}}_+^*$ is dense in $M\times {\mathbb{R}}_+$ that the ideal $C_0({\mathbb{R}}_+^*)\otimes C_{red}^*({\mathcal{G}})$ is essential in $C^*_{red}({\mathcal{G}}_{ad})$. Thanks to remark \[essential\] we deduce that $C_0({\mathbb{R}}_+^*)\otimes C^*({\mathcal{G}})$ is also an essential ideal in $C^*({\mathcal{G}}_{ad})$. As it contains $C_0({\mathbb{R}}_+^*)\otimes C^*({\mathcal{G}})$, the ideal $J({\mathcal{G}})$ is essential in $C^*({\mathcal{G}}_{ad})$ both for the reduced and the full $C^*$-norm. Note also that the subset ${\mathfrak{A}}^*{\mathcal{G}}\setminus M$ is dense in ${\mathfrak{A}}^*{\mathcal{G}}$ (unless the groupoid ${\mathcal{G}}$ is $r$-discrete in the sense of [@Ren [def. 2.6, p. 18]{}] - [[*i.e.*]{} ]{}the dimension of the algebroid is $0$), and therefore $\ker \epsilon$ is essential in $C_0({\mathfrak{A}}^*{\mathcal{G}})$. In this way we have another proof that $J({\mathcal{G}})$ is essential in $C^*({\mathcal{G}}_{ad})$. We denote by $\tau$ the action of the group ${\mathbb{R}}_+^*$ by groupoid automorphisms on ${\mathcal{G}}_{ad}$. This action is given by $\tau_t (\gamma,u) =(\gamma,tu)$ for $\gamma\in {\mathcal{G}}$ and $t,u\in {\mathbb{R}}_+^*$ $\tau_t(x,U,0)=(x,t^{-1} U,0)$ for $ (x,U)\in {\mathfrak{A}}{\mathcal{G}}\ \ (x\in M)$. We therefore get an action still denoted by $\tau$ of ${\mathbb{R}}_+^*$ on $C^*({\mathcal{G}}_{ad})$. Note that $J({\mathcal{G}})$ is invariant under this action and that the quotient action of ${\mathbb{R}}_+^*$ on $C^*({\mathcal{G}}_{ad})/J({\mathcal{G}})=C(M)$ is trivial. We will also use from [@DS1 section 3.1] the dense subspaces ${\mathcal{S}}({\mathcal{G}}_{ad})$ of $C^*({\mathcal{G}}_{ad}) $ and ${\mathcal{J}}({\mathcal{G}})$ of $J({\mathcal{G}})$ consisting of smooth functions with Schwartz decay properties. Recall ([@DS1 Theorem 3.7]) that for $f\in {\mathcal{J}}(G)$ and $m\in {\mathbb{R}}$, the operator $\displaystyle\int_0^{+\infty}\! f_t\, \displaystyle\frac {dt}{t^{m+1}}$ is an order $m$ pseudodifferential operator of the groupoid $G$ [[*i.e.*]{} ]{}an element of ${\mathcal{P}}_{m}(G)$; its principal symbol $\sigma$ is given by $\sigma(x,\xi)=\displaystyle\int_0^{+\infty} \hat f(x,t\xi,0)\displaystyle\frac {dt}{t^{m+1}}\cdot$ Pseudodifferential extension of smooth groupoid actions ------------------------------------------------------- We now extend Baaj’s construction of the pseudodifferential extension to the case of an action $\alpha$ of a smooth groupoid ${\mathcal{G}}$ on a $C^*$-algebra $A$ - in the sense of [@PYLG; @PYLG2] - see section \[essonfrerK1\]. ### Smooth elements Let ${\mathcal{G}}$ be a smooth groupoid with base $M$ acting on a $C_0(M)$ algebra $A$. We denote by $\alpha:s^*A\to r^*A$ this action. We may define elements of $A$ which are smooth along the action in the following way: - Let $W$ be an open subset in $ {\mathcal{G}}$ diffeomorphic to $U\times V$ where $U\subset M$ is open and $V$ is an open ball in ${\mathbb{R}}^k$, and such that $r(u,v)=u$. Then the $C_0(W)$ algebra $(r^*A)_W$ is isomorphic to $C_0(V;A_U)$; an element $a\in r^*A$ is said to be of class $C^{\infty,0}$ if for every such $W$ and $f\in C_c^\infty (W)$, we have $fa\in C_c^\infty(V;A_U)\subset C_0(V;A_U)\simeq A_W$. - An element $a\in A$ is said to be smooth for the action of ${\mathcal{G}}$ if for all $f\in C_c^\infty ({\mathcal{G}})$, the element $\alpha( f.(a\circ s))$ of $r^*A$ is of class $C^{\infty,0}$. Here $f.(a\circ s)$ is the class of $a\otimes f$ in $s^*A$ - [[*i.e.*]{} ]{}the restriction of $a\otimes f$ to the graph of $s$. In other words, we have $$\Big(\alpha(f.(a\circ s))\Big)_\gamma=f(\gamma)\alpha_\gamma(a_{s(\gamma)}).$$ The smooth elements form a dense sub-algebra $A^\infty$ of $A$. Indeed, if $a\in A$ and $f\in C_c^{\infty}({\mathcal{G}})$, the element $f\ast a$ given by $(f\ast a)_x=\int_{G_x}f(\gamma)\alpha_\gamma a_{s(\gamma)}d\nu ^x(\gamma)$ is easily seen to be smooth. Take then a sequence $f_n$ with $f_n\in C_c^\infty({\mathcal{G}})$ positive with support tending to $M$ and such that $\nu _x(f_n)=1$: we have $f_n\ast a\to a$. ### Crossed product by the adiabatic groupoid {#adgpdaction} Let ${\mathcal{G}}$ be a smooth groupoid with base $M$ acting on a $C_0(M)$ algebra $A$. Consider the morphism ${\mathcal{G}}_{ad}\to {\mathcal{G}}\times {\mathbb{R}}_+$ which is the identity on ${\mathcal{G}}\times {\mathbb{R}}_+^*$ and satisfies $(x,\xi,0)\mapsto (x,0)$ for $x\in M={\mathcal{G}}^{(0)} \subset {\mathcal{G}}$ and $\xi \in {\mathfrak{g}}_x$. Using this morphism, the adiabatic groupoid ${\mathcal{G}}_{ad}$ acts on the $C_0({\mathbb{R}}_+\times M)$-algebra $C_0({\mathbb{R}}_+)\otimes A$: we have $A_{x,t}=A_x$ (for $t\in {\mathbb{R}}_+ $ and $x\in M$) and, for $t\in {\mathbb{R}}_+^*$, $\gamma\in {\mathcal{G}}$ and $b\in A_{s(\gamma)}$, we have $\alpha_{\gamma,t}(b)=\alpha_\gamma(b)$; for $x\in M$, $\xi\in {\mathfrak{g}}_x$ and $b\in A_x$, we have $\alpha_{x,\xi,0}(b)=b$. We have an exact sequence $$0\to (A\rtimes _\alpha {\mathcal{G}})\otimes C_0({\mathbb{R}}_+^*)\to (A\otimes C_0({\mathbb{R}}_+))\rtimes_\alpha {\mathcal{G}}_{ad}\to A\otimes _{C_0(M)}C_0({\mathfrak{A}}^*{\mathcal{G}})\to 0.$$ As the groupoid ${\mathfrak{A}}{\mathcal{G}}$ is amenable, the same exact sequence holds with reduced crossed products. Note also that the action $\tau$ of ${\mathbb{R}}_+^*$ extends on $(A\otimes C_0({\mathbb{R}}_+))\rtimes_\alpha {\mathcal{G}}_{ad}$: it acts naturally on $(A\otimes C_0({\mathbb{R}}_+))=C_0({\mathbb{R}}_+;A)$ by $(\tau_t(a))(u)=a(t^{-1}u)$. We will also use the ideal $J({\mathcal{G}},A)\subset (A\otimes C_0({\mathbb{R}}_+))\rtimes_\alpha {\mathcal{G}}_{ad}$ which is the kernel of the morphism $(A\otimes C_0({\mathbb{R}}_+))\rtimes_\alpha {\mathcal{G}}_{ad}\to A$ obtained as the composition $$(A\otimes C_0({\mathbb{R}}_+))\rtimes_\alpha {\mathcal{G}}_{ad}\to A\otimes _{C_0(M)}C_0({\mathfrak{A}}^*{\mathcal{G}})\to A\otimes _{C_0(M)}C_0(M)=A.$$ It is the closed vector span of elements $f.a$ with $f\in J({\mathcal{G}})$ and $a\in A$. It is an essential ideal in $A\otimes C_0({\mathbb{R}}_+))\rtimes_\alpha {\mathcal{G}}_{ad}$ (see remark \[Jessentiel\]). \[ery\] If $a\in A$ is smooth for the ${\mathcal{G}}$ action and $f\in {\mathcal{S}}_c({\mathcal{G}}_{ad})$ ([[*cf.*]{} ]{}[@DS1 [section]{} 3.1]), then $\|[f_t,a]\|_{A\rtimes _\alpha {\mathcal{G}}}=O(t)$. Note that $f.a, a.f$ are in $(A\otimes C_0({\mathbb{R}}_+))\rtimes_\alpha {\mathcal{G}}_{ad}$ and since they are equal in $A\otimes _{C_0(M)}C_0({\mathfrak{A}}^*{\mathcal{G}})$, we find that $\|[f_t,a]\|_{A\rtimes _\alpha {\mathcal{G}}}\to 0$. Let $\theta :V'\to V$ be an “exponential map” which is a diffeomorphism of a (relatively compact) neighborhood $V'$ of the $0$ section $M$ in ${\mathfrak{A}}{\mathcal{G}}$ onto a tubular neighborhood $V$ of $M$ in ${\mathcal{G}}$. We assume that $r(\theta(x,U))=x$ for $x\in M$ and $U\in {\mathfrak{A}}_x {\mathcal{G}}$. Let $W'=\{(x,U,t)\in {\mathfrak{A}}{\mathcal{G}}\times{\mathbb{R}}_+;\ (x,tU)\in V'\}$ and $W$ be the open subset $W={\mathfrak{A}}{\mathcal{G}}\times\{0\}\cup V\times {\mathbb{R}}_+^*$ of ${\mathcal{G}}_{ad}$; finally let $\Theta:W'\times {\mathbb{R}}_+\to W$ be the diffeomorphism defined by $\Theta(x,U,0)=(x,U,0)$ and $\Theta(x,U,t)=(\theta(x,tU),t)$. If $f\in {\mathcal{S}}_c ({\mathbb{R}}_+^*\times {\mathcal{G}})$, then we have $\|[f_t,a]\|_{A\rtimes _\alpha {\mathcal{G}}}=O(t^n)$ for all $n$. We may therefore assume that $f$ is of the form $g\circ \Theta $ where $g\in {\mathcal{S}}_c(W')$; then $[f_t,a]$ is the image in $A\rtimes _{\alpha}{\mathcal{G}}$ of the function $b_t\in r^*A$, where $(b_t)_\gamma=f_t(\gamma)\Big(a_{r(\gamma)}-\alpha_\gamma\big(a_{s(\gamma)}\big)\Big)$. Note that there is a well defined element $c\in (r\circ\Theta)^*(A\otimes C_0({\mathbb{R}}_+))$ given by $c_{(x,U,t)}= g(x,U,t)\frac1t\Big(a_{x}-\alpha_{\theta(x,tU)}\big(a_{s(\theta(x,tU))}\big)\Big)$ for $t\ne 0$ and $-c_{(x,U,0)}$ is the derivative at $0$ of $t\mapsto \alpha_{\theta(x,tU)}\big(a_{s(\theta(x,tU))}\big)$, and $f.(c\circ \Theta^{-1})$ gives an element $d\in (A\otimes C_0({\mathbb{R}}_+))\rtimes_\alpha {\mathcal{G}}_{ad}$; we have $td_t=[f_t,a]$. ### Pseudodifferential extension \[orme\] 1. \[petita\] For $P\in \Psi^*({\mathcal{G}})$ and $a\in A$ sitting in ${\mathcal{M}}(A\rtimes _\alpha{\mathcal{G}})$, we have $[P,a]\in A\rtimes_\alpha {\mathcal{G}}$. 2. The closed vector span of products $aP$ where $a\in A$ and $P\in \Psi^*({\mathcal{G}})$ is a $C^*$-subalgebra $\Psi^*(A,\alpha,{\mathcal{G}})\subset {\mathcal{M}}(A\rtimes _\alpha {\mathcal{G}})$. 3. We have an exact sequence $$0\to A\rtimes _\alpha {\mathcal{G}}\longrightarrow \Psi^*(A,\alpha,{\mathcal{G}})\overset{\sigma_\alpha}\longrightarrow A\otimes _{C_0(M)}C(S^* {\mathfrak{A}}{\mathcal{G}})\to 0.$$ <!-- --> 1. We can assume $P$ is in a dense subalgebra of $\Psi^*({\mathcal{G}})$ and $a$ smooth. Whence, by [@DS1 Theorem 3.7], we may choose $P=\int_0^{+\infty} f_t\frac {dt}t$ where $f=(f_t)\in {\mathcal{J}}({\mathcal{G}})$. Then, by Lemma \[ery\], $[P,a]$ is a norm converging integral of elements in $A\rtimes _\alpha {\mathcal{G}}$. 2. This closed subspace contains $A\rtimes_\alpha {\mathcal{G}}$ and its image in ${\mathcal{M}}(A\rtimes_\alpha {\mathcal{G}})/(A\rtimes_\alpha {\mathcal{G}})$ is a $C^*$-algebra since $\Psi^*({\mathcal{G}})$ and $A$ commute in this quotient. 3. Using (\[petita\]) and the compatibility of the inclusions of $C_0(M)$ in $\Psi^*({\mathcal{G}})$ and in ${\mathcal{M}}(A)$, we find a morphism $\varpi:C(S^* {\mathfrak{A}}{\mathcal{G}})\otimes_{C_0(M)} A\to {\mathcal{M}}(A\rtimes_\alpha {\mathcal{G}})/(A\rtimes_\alpha {\mathcal{G}})$ such that $\varpi(\sigma (P)\otimes a)$ is the class of $Pa$. We just have to show that $\varpi$ is injective. Equivalently, we wish to show that $A\rtimes_\alpha {\mathcal{G}}$ is an essential ideal in the fibered product $\widetilde \Psi^*({\mathcal{G}};A)=\Psi^*({\mathcal{G}};A)\times_{\varpi(C(S^* {\mathfrak{A}}{\mathcal{G}}))}C(S^* {\mathfrak{A}}{\mathcal{G}})$. We have a representation of $\widetilde \Psi({\mathcal{G}},A)$ as multipliers of $J({\mathcal{G}},A)$ given, for $(T,\sigma)\in \widetilde \Psi^*({\mathcal{G}};A)$, by $((T,\sigma)f)_t=Tf_t$ for $t\ne 0$ and $\widehat{((T,\sigma)f)_0)}(x,\xi)=\sigma(x,\xi)\widehat{f_0}(x,\xi)$, where $T\in \Psi^*({\mathcal{G}},A)$ and $\sigma\in C(S^* {\mathfrak{A}}{\mathcal{G}})$. This representation is faithful: indeed, if $(T,\sigma)$ is in its kernel, taking its value at $0$ it follows that $\sigma =0$; therefore $T\in A\rtimes _\alpha G$; but the representation of $A\rtimes _\alpha {\mathcal{G}}$ in $J({\mathcal{G}};A)$ is faithful since $A\rtimes_\alpha {\mathcal{G}}\otimes C_0({\mathbb{R}}_+^*)\subset J({\mathcal{G}},A)$. Now as $C_0({\mathbb{R}}_+^*)\otimes A\rtimes_\alpha {\mathcal{G}}$ is an essential ideal in $J({\mathcal{G}};A)$, it follows that the representation $P\mapsto 1\otimes P$ of $\widetilde \Psi^*({\mathcal{G}};A)$ on $C_0({\mathbb{R}}_+^*)\otimes A\rtimes_\alpha {\mathcal{G}}$ is faithful, whence $A\rtimes_\alpha {\mathcal{G}}$ is essential in $\widetilde \Psi^*({\mathcal{G}};A)$. Action of the adiabatic groupoid and pseudodifferential extension ================================================================= Let ${\mathcal{G}}$ be a smooth groupoid acting on the $C^*$-algebra $A$. In this section we prove the main results of this paper: - We construct an action of ${\mathbb{R}}$ on the associated $C^*$-algebra $\Psi^*({\mathcal{G}},A)$ of pseudodifferential operators - extending a construction sketched in [@DS1 [Remark 4.10]{}]. - We establish the isomorphism $J({\mathcal{G}},A)\simeq \Psi^*({\mathcal{G}},A)\rtimes_\beta {\mathbb{R}}$ - which was sketched in [@DS1 [Remark 4.10]{}] in the case where $A=C_0(M)$ and the action is trivial. - Finally we identify $(A\otimes C_0({\mathbb{R}}_+))\rtimes _{\widetilde \alpha}G_{ad}$ as a pseudodifferential extension of the above crossed product. The unbounded multiplier $D$ of $C^*({\mathcal{G}}_{ad})$ {#vedere} --------------------------------------------------------- We first recall the construction of an unbounded multiplier $D$ of $C^*({\mathcal{G}}_{ad})$ which was given in [@DS1 [section]{} 4.4]. Let ${\mathcal{G}}$ be a longitudinally smooth groupoid with compact space of objects $M={\mathcal{G}}^{(0)}$. Fix a metric on ${\mathfrak{A}}{\mathcal{G}}$ (and therefore on ${\mathfrak{A}}^* {\mathcal{G}}$) and choose a positive invertible pseudodifferential operator $D_1$ on ${\mathcal{G}}$ with principal symbol $\sigma_{D_1}(x,\xi)=\|\xi\|$. It is shown in [@chief [Prop. 21]{}] that $D_1$ is a regular multiplier of $C^*({\mathcal{G}})$. ([[*cf.*]{} ]{}[@DS1 Prop. 4.8])\[afer\] Let ${\mathcal{G}}$ be a Lie groupoid with compact set of objects ${\mathcal{G}}^{(0)}=M$ and ${\mathcal{G}}_{ad}$ its adiabatic groupoid. Fix a metric on ${\mathfrak{A}}{\mathcal{G}}$ (and therefore on ${\mathfrak{A}}^* {\mathcal{G}}$) and choose a positive invertible pseudodifferential operator $D_1$ on ${\mathcal{G}}$ with principal symbol $\sigma_{D_1}(x,\xi)=\|\xi\|$. There is a unique regular unbounded multiplier $D$ of $C^*({\mathcal{G}}_{ad})$ satisfying: 1. the evaluation at $1$ of $D$ is $D_1$; 2. we have $\beta_u(D)=uD$ for $u\in {\mathbb{R}}^*_+$. Moreover, 1. The evaluation at $0$ of $D$, $D_0$, is the unbounded multiplier $q$ of $C_0({\mathfrak{A}}^*{\mathcal{G}})=C^*({\mathfrak{A}}{\mathcal{G}})$ where $q(x,\xi)=\|\xi\|$.\[propafera\] 2. The multiplier $(1+D)^{-1}$ is in fact a strictly positive element of $C^*({\mathcal{G}}_{ad})$.\[propaferb\] 3. For all $f\in C_0({\mathbb{R}}_+^*)$ we have $f(D)\in J({\mathcal{G}})$. Moreover, the representation $f\mapsto f(D)$ is non degenerate: if $h\in C_0({\mathbb{R}}_+^*)$ is strictly positive in ${\mathbb{R}}_+^*$, then $f(D)$ is a strictly positive element of $J({\mathcal{G}})$..\[propaferc\] If $D$ satisfies (i) and (ii), then $D_u=uD_1$ for all $u>0$, and this establishes uniqueness of $D$. Choose a finite family $(X_1,\ldots,X_m)$ of sections of ${\mathfrak{A}}{\mathcal{G}}$ in such a way that the embedding $\xi\mapsto \langle X_i|\xi\rangle$ is an isometry from ${\mathfrak{A}}^*{\mathcal{G}}$ to the trivial bundle. In [@DS1 prop. 4.8], we constructed an unbounded multiplier, call it $\widetilde D$ such that $\widetilde D_1=\Big(\sum X_i^*X_i+1\Big)^{1/2}$, $\widetilde D_0=q$ and $\widetilde D_u=u\widetilde D_1$ for $u\in {\mathbb{R}}_+^*$. Now, $D_1-\widetilde D_1$ is a $0$-order operator, whence bounded. We may then define an unbounded multiplier $D$ by putting $D_u=\widetilde D_u+u(D_1-\widetilde D_1)$ and $D_0=\widetilde D_0$. Let us prove property (\[propaferb\]).\ Let $c\in {\mathbb{R}}_+^*$. Since $M\times [0,c]$ is compact and $D$ is elliptic of order $1$ ([@chief [Th. 18 and Prop. 21]{}]), the restriction of $(1+D)^{-1}$ to $({\mathcal{G}}_{ad})_{|[0,c]}$ is in $C^*({\mathcal{G}}_{ad})_{|[0,c]}$. Let $m \in {\mathbb{R}}_+^*$ such that $D_1\geq m$, we have $1+D_u\geq 1+um$ and therefore $\|(1+D_u)^{-1}\| \leq (1+um)^{-1}$. It follows that $(1+D)^{-1}$ belongs to $C^*({\mathcal{G}}_{ad})$. Now, $(1+D)^{-1}C^*({\mathcal{G}}_{ad})$ is the domain of the multiplier $D$, whence it is dense, and $(1+D)^{-1}$ is strictly positive. Property (\[propaferc\]) follows from [@DS1 Prop. 4.8.b)]. Note that our $D_1$ here is slightly more general than the one used there, but the same proof applies. The Action of ${\mathbb{R}}$ on $\Psi^*({\mathcal{G}},A)$ --------------------------------------------------------- Let $S\in {\mathcal{P}}_{1/2}({\mathcal{G}})$ be a positive elliptic pseudodifferential operator of order $1/2$ (for instance $S$ such that $\sigma_{1/2}(S)=(\sigma_2(\Delta_{\mathcal{G}}))^{1/4}$ where $\Delta_{\mathcal{G}}$ is a laplacian as defined in the section \[grade\]. Denote by $\partial_{S}$ the associated derivation on ${\mathcal{M}}(A\rtimes_\alpha {\mathcal{G}})$ (see appendix - facts \[derivations\]). \[gueule\] Every smooth element $a\in A$ and every classical pseudodifferential $P$ on ${\mathcal{G}}$ of order $0$ are in the domain of the derivation $\partial_{S}$. We may write $S=R+S_1$ where $S_1=\int_0^{+\infty}f_t\,t^{-3/2}\,dt$, $(f_t)$ is a positive element in ${\mathcal{J}}({\mathcal{G}})$ and $R\in {\mathcal{P}}_{-1/2}({\mathcal{G}})$. This integral means that ${\mathrm{dom}}\,S_1$ is the set of $x\in A\rtimes_\alpha {\mathcal{G}}$ such that the integral $\int_{0}^{+\infty}f_tx\,t^{-3/2}\,dt $ converges in norm to some $y\in A\rtimes_\alpha {\mathcal{G}}$ and then $S_1x=y$. (Indeed, by [@chief [Prop. 21]{}], $S_1$ is selfadjoint regular and it is clear that $(x,y)$ is then in the graph of $S_1^*$). Since $a$ is assume to be smooth, the integral $\int_0^{+\infty}[f_t,a]\,t^{-3/2}\,dt$ converges in norm (by Lemma \[ery\]) to some element $b\in A\rtimes_\alpha {\mathcal{G}}$. Then, for $x\in {\mathrm{dom}}\,S={\mathrm{dom}}\,S_1$, the sequence $$\int_{1/n}^{+\infty}f_tax\,t^{-3/2}\,dt=\int_{1/n}^{+\infty}(af_t+[f_t,a])x\,t^{-3/2}\,dt$$ converges in norm to $aS_1x+bx$. Thus $ax\in {\mathrm{dom}}\,S_1={\mathrm{dom}}\, S$. It follows that $a\in {\mathrm{dom}}\, \partial_S$ and $\partial_S(a)=b+[R,a]$. If $P\in {\mathcal{P}}_0({\mathcal{G}})$, the operator $(S^2+1)^{1/2}P(S^2+1)^{-1/2}\in \Psi^*({\mathcal{G}})$; it follows that $P{\mathrm{dom}}\,S\subset {\mathrm{dom}}\,S$. Moreover, $[S,P]\in {\mathcal{P}}_{1/2}({\mathcal{G}})$ and since $\sigma_{1/2}[S,P]=[\sigma_{1/2}(S),\sigma_{1/2}(P)]=0$, we find $[S,P]\in {\mathcal{P}}_{-1/2}({\mathcal{G}})\subset C^*({\mathcal{G}})$. \[ephant\] Let $D_1\in {\mathcal{P}}_1({\mathcal{G}})$ be any positive invertible pseudodifferential operator elliptic of order $1$. Then we have an action $\beta$ of ${\mathbb{R}}$ on $\Psi^*({\mathcal{G}};A)$ given by $\beta_t(P)=D_1^{it}PD_1^{-it}$. This action is trivial at the symbol level. By [@chief [Theorem 41]{}] there exists $S\in {\mathcal{P}}_{1/2}$ positive elliptic of order $1/2$ and $T\in C^*({\mathcal{G}})$ such that $\sqrt{D_1}=S+T$. It follows by Lemma \[gueule\], that with $a,P$ as above $Pa\in {\mathrm{dom}}\,\partial_{\sqrt{D_1}}$. Since $D_1^{-1/2}\in A\rtimes_\alpha {\mathcal{G}}$, it follows from Lemma \[LemmeApp3\], that $Pa\in {\mathrm{dom}}\,\partial_{\ln D_1}$ and $[\ln D_1,aP]\in A\rtimes _\alpha {\mathcal{G}}$. The conclusion follows from Lemma \[LemmeApp1\]. Isomorphism $\Psi^*({\mathcal{G}},A)\rtimes {\mathbb{R}}\simeq J({\mathcal{G}},A)$ {#blonde} ---------------------------------------------------------------------------------- In [@DS1 prop. 4.2.b)], we constructed a morphism $\phi:\Psi^*({\mathcal{G}})\to {\mathcal{M}}(J({\mathcal{G}}))$ such that, for $P\in\Psi^*({\mathcal{G}})$ and $f=(f_u)$ in $J({\mathcal{G}})$ we have $(\phi(P)(f))_u=P\ast f_u$, for $u\ne 0$ and $(\phi(P)(f))_0=\sigma_0(P)f_0$ thanks to [@DS1 prop. 4.2.b)]. Now $J({\mathcal{G}})$ sits in a non degenerate way in ${\mathcal{M}}(J({\mathcal{G}},A))$. Also, by definition $A$ embeds in a compatible way in ${\mathcal{M}}(J({\mathcal{G}},A))$. In this way, we find a morphism $\phi:\Psi^*({\mathcal{G}},A)\to {\mathcal{M}}(J({\mathcal{G}},A))$ such that, for $P\in\Psi^*({\mathcal{G}},A)$ and $f=(f_t)$ in $J({\mathcal{G}},A)$ we have $(\phi(P)(f))_u=P\ast f_u$, for $u\ne 0$ and $(\phi(P)(f))_0=\sigma_0(P)f_0$. Furthermore, the operator $D$ recalled in section \[vedere\] yields a one parameter group $(D^{it})_{t\in {\mathbb{R}}}$ in ${\mathcal{M}}(J({\mathcal{G}}))$; we will still denote by $(D^{it})_{t\in {\mathbb{R}}}$ its image in ${\mathcal{M}}(J({\mathcal{G}},A))$. As $D_u$ and $D_1$ are scalar multiples of each other, we find in this way a covariant representation of the pair $(\Psi^*({\mathcal{G}}),\beta,{\mathbb{R}})$ (prop. \[ephant\]). Associated to this covariant representation is a morphism from $\Psi^*({\mathcal{G}},A)\rtimes_\beta {\mathbb{R}}$ into the multiplier algebra of $J({\mathcal{G}})$, but since the image of $C^*({\mathbb{R}})\subset \Psi^*({\mathcal{G}})\rtimes_\beta {\mathbb{R}}$ is contained in $J({\mathcal{G}})$, we get a homomorphism $\varphi :\Psi^*({\mathcal{G}},A)\rtimes_\beta {\mathbb{R}}\to J({\mathcal{G}},A)$. For $P\in \Psi^*({\mathcal{G}},A)\subset {\mathcal{M}}(\Psi^*({\mathcal{G}},A)\rtimes_\beta {\mathbb{R}})$ and $f\in C^*({\mathbb{R}})=C_0({\mathbb{R}}_+^*)\subset {\mathcal{M}}(\Psi^*({\mathcal{G}},A)\rtimes_\beta {\mathbb{R}})$, we have $(\varphi(Pf)) =\phi(P)f(D)$. The homomorphism $\varphi$ is an equivariant isomorphism from $(\Psi^*({\mathcal{G}},A)\rtimes_\beta {\mathbb{R}},\hat\beta)$ to $(J({\mathcal{G}},A),\tau)$. The images of the elements of $\Psi^*({\mathcal{G}},A)$ are translation invariant, [[*i.e.*]{} ]{}invariant by the extension $\overline\tau_u$ of $\tau_u$ to the multiplier algebra, and $\overline\tau_u(D^{it})=u^{it}D^{it}$. This shows that $\varphi$ is an equivariant morphism from $(\Psi^*({\mathcal{G}},A)\rtimes_\beta {\mathbb{R}},\hat\beta)$ to $(J({\mathcal{G}},A),\tau)$. Now $\beta_t$ restricts to an action of ${\mathbb{R}}$ on $C^*({\mathcal{G}})$, and according to [@DS1 prop. 4.2.a)] it follows that $\varphi$ extended to the multipliers defines a morphism from $C^*({\mathcal{G}})\rtimes_\beta {\mathbb{R}}$ into the ideal $C_0({\mathbb{R}}_+^*)\otimes C^*({\mathcal{G}})$ of $J({\mathcal{G}})$. It follows that $\varphi(A\rtimes_\alpha {\mathcal{G}})$ is contained in the ideal $A\rtimes_\alpha {\mathcal{G}}\otimes C_0({\mathbb{R}}_+^*)$ of $J({\mathcal{G}},A)$. We thus have the diagram: $$\xymatrix{ 0\ar[r]&(A\rtimes _{\alpha}{\mathcal{G}})\rtimes_\beta {\mathbb{R}}\ar[r]\ar[d]^{\varphi'} &\Psi^*({\mathcal{G}},A)\rtimes _\beta{\mathbb{R}}\ar[r]\ar[d]^{\varphi}& (A\otimes _{C(M)} C(S^* {\mathfrak{A}}{\mathcal{G}}))\rtimes _\beta{\mathbb{R}}\ar[r]\ar[d]^{\varphi''}& 0\\ 0\ar[r]& (A\rtimes _\alpha {\mathcal{G}})\otimes C_0({\mathbb{R}}_+^*)\ar[r]&J({\mathcal{G}},A)\ar[r] &A\otimes _{C(M)}C_0({\mathfrak{A}}^*{\mathcal{G}}\setminus M)\ar[r]& 0 }$$ As $D_1$ is an unbounded invertible multiplier of $C^*({\mathcal{G}})$ and therefore of $A\rtimes _\alpha {\mathcal{G}}$, the action $\beta$ of ${\mathbb{R}}$ on $A\rtimes _\alpha {\mathcal{G}}$ is inner. It follows that the crossed product $(A\rtimes _\alpha {\mathcal{G}})\rtimes _\beta {\mathbb{R}}$ identifies with $(A\rtimes _\alpha {\mathcal{G}})\otimes {\mathbb{R}}_+^*$. This isomorphism is defined in the following way: the canonical multipliers of the crossed product, [[*i.e.*]{} ]{}the generators $a\in A\rtimes _\alpha {\mathcal{G}}$ and $\lambda_t$ for $t\in {\mathbb{R}}$ map to the functions $u\mapsto a$ and $u\mapsto u^{it}D_1^{it}$ from ${\mathbb{R}}_+^*$ to ${\mathcal{M}}(A\rtimes _\alpha {\mathcal{G}})$. It follows, the image of $af$ with $a\in C^*({\mathcal{G}})$ and $f\in C^*({\mathbb{R}})=C_0({\mathbb{R}}_+^*)$ is $af(D)$. This isomorphism identifies thus with $\varphi'$. The action $\beta$ is trivial on symbols; thus $(A\otimes _{C(M)}C(S^* {\mathfrak{A}}{\mathcal{G}}))\rtimes _\beta{\mathbb{R}}$ is equal to $(A\otimes _{C(M)} C(S^* {\mathfrak{A}}{\mathcal{G}}))\otimes C_0({\mathbb{R}}_+^*)$, and $\varphi''(\sigma \otimes f)=\sigma f(q)$ is the isomorphism corresponding to the homeomorphism ${\mathfrak{A}}^*{\mathcal{G}}\setminus M\simeq S^* {\mathfrak{A}}{\mathcal{G}}\times {\mathbb{R}}_+^*$ given by $\xi\mapsto (\xi/q(\xi),q(\xi))$. The result follows. The crossed product by the adiabatic groupoid --------------------------------------------- The algebra $A$ sits in $\Psi^*({\mathcal{G}},A)$ as (the closure of) order $0$ differential operators. Denote by $\vartheta:A\to \Psi^*({\mathcal{G}};A)$ the corresponding morphism. The element $\vartheta(a)$ as a multiplier of $A\rtimes_\alpha {\mathcal{G}}$, is just the multiplication by $a$. \[mult=mult\] Using at the non degenerate morphism $\Psi^*({\mathcal{G}},A)\to {\mathcal{M}}(J({\mathcal{G}};A))$ we then obtain a morphism $\hat \vartheta:A\to {\mathcal{M}}(\Psi^*({\mathcal{G}},A)\rtimes {\mathbb{R}})$. Also the algebra $A$ is in the multiplier algebra of $A\otimes C_0({\mathbb{R}}_+)$ end thus we have an embedding $\tilde \vartheta :A\to {\mathcal{M}}((A\otimes C_0({\mathbb{R}}_+))\rtimes _{\tilde \alpha}{\mathcal{G}}_{ad})$ - which is a subalgebra of ${\mathcal{M}}(J({\mathcal{G}};A))$ since $J({\mathcal{G}};A)$ is an essential ideal in $(A\otimes C_0({\mathbb{R}}_+))\rtimes _{\tilde \alpha}{\mathcal{G}}_{ad}$. We now use the notation of paragraph \[caseofR\]. The main result of this paper is : \[maintheorem\] The isomorphism $\varphi:\Psi^*({\mathcal{G}},A)\rtimes_\beta {\mathbb{R}}\to J({\mathcal{G}},A)$ extends uniquely to an isomorphism of $\Psi^*(\Psi^*({\mathcal{G}},A),\beta,{\mathbb{R}},0,A)$ with $(A\otimes C_0({\mathbb{R}}_+))\rtimes_{\widetilde \alpha}{\mathcal{G}}_{ad}$. This isomorphism intertwines the actions $\beta $ and $\tau$ of ${\mathbb{R}}$. The isomorphism $\varphi:\Psi^*({\mathcal{G}},A)\rtimes _\beta{\mathbb{R}}\to J({\mathcal{G}},A)$ extends to an isomorphism $\Phi$ of the multiplier algebras. Since the ideals $\Psi^*({\mathcal{G}},A)\rtimes _\beta{\mathbb{R}}\subset \Psi^*(\Psi^*({\mathcal{G}},A),\beta,{\mathbb{R}})$ and $J({\mathcal{G}},A)\subset (A\otimes C_0({\mathbb{R}}_+))\rtimes_\alpha {\mathcal{G}}_{ad}$ are essential, we just need to show that $\Phi (\Psi^*(\Psi^*({\mathcal{G}},A),\beta,{\mathbb{R}},0,A))=(A\otimes C_0({\mathbb{R}}_+))\rtimes_{\widetilde \alpha}{\mathcal{G}}_{ad}$. It follows from proposition \[etoile\].a) that the morphism $\Phi$ coincides on $\Psi^*({\mathcal{G}},A)$ with the morphism $\phi : \Psi_*({\mathcal{G}},A)\rightarrow {\mathcal{M}}(J({\mathcal{G}},A))$ of [section]{} \[blonde\] and that the image of the unbounded multiplier $Q_\beta$ (see [section]{} \[caseofR\]) is $D$. With the notation introduced in remark \[mult=mult\], one easily checks that $\Phi\circ \hat\vartheta =\tilde\vartheta$. We deduce that $\Phi\Big(\Psi^*(\Psi^*({\mathcal{G}},A),\beta,{\mathbb{R}},0,A)\Big)$ is spanned by $\varphi(\Psi^*({\mathcal{G}},A)\rtimes _\beta{\mathbb{R}})= J({\mathcal{G}},A)$ and $(1+D)^{-1}\tilde\vartheta(a)$ where and $a$ over $A$. Since $(1+D)^{-1}\in C^*({\mathcal{G}}_{ad})$ (prop. \[afer\].\[propaferb\])), and for $a\in A$ we have $(1+D)^{-1}\tilde\vartheta(a))\in (A\otimes C_0({\mathbb{R}}_+))\rtimes _{\tilde \alpha}{\mathcal{G}}_{ad})$. Finally $\Phi$ induces a homomorphism $\widetilde \varphi:\Psi^*(\Psi^*({\mathcal{G}},A),\beta,{\mathbb{R}},0,A))\to (A\otimes C_0({\mathbb{R}}_+))\rtimes_{\widetilde \alpha}{\mathcal{G}}_{ad}$. Moreover, since ${{\rm ev}}_0(D)=q$ which vanishes at the $0$ section of ${\mathfrak{A}}^* {\mathcal{G}}$, we find that $\epsilon\circ {{\rm ev}}_0((1+D)^{-1})=1$, whence $\epsilon\circ {{\rm ev}}_0(\Phi((1+D)^{-1}\theta(a))=a$. We thus have a commutative diagram where the sequences are exact: $$\xymatrix{ 0\ar[r]&\Psi^*({\mathcal{G}},A)\rtimes_\beta {\mathbb{R}}\ar[r]\ar[d]^{\varphi} &\Psi^*(\Psi^*({\mathcal{G}},A),\beta,{\mathbb{R}},0,A)\ar[r]\ar[d]^{\Phi}& A\ar[r]\ar[d]^{{{\hbox{id}}}_A}& 0\\ 0\ar[r]& J({\mathcal{G}},A)\ar[r]&(A\otimes C_0({\mathbb{R}}_+))\rtimes _{\tilde \alpha}{\mathcal{G}}_{ad})\ar[r] &A\ar[r]& 0 }$$ Whence $\widetilde \varphi$ is an isomorphism. By uniqueness of the extension to multipliers, we deduce that $\hat \beta_t\circ \widetilde \varphi=\widetilde \varphi\circ \tau_t$ for all $t\in {\mathbb{R}}_+^*$. Recall that the gauge adiabatic groupoid ${\mathcal{G}}_{ga}$ is the semi-direct product ${\mathcal{G}}_{ga}={\mathcal{G}}_{ad}\rtimes_\tau {\mathbb{R}}_+^*$. If ${\mathcal{G}}$ acts on $A$, then ${\mathcal{G}}_{ga}$ acts on $A\otimes C_0({\mathbb{R}}_+)$. We have isomorphisms $$\begin{aligned} (A\otimes C_0({\mathbb{R}}_+))\rtimes_{\alpha} {\mathcal{G}}_{ga}&\simeq &\Psi^*(\Psi^*({\mathcal{G}},A),\beta,{\mathbb{R}},0,C(M))\rtimes_{\hat\beta}{\mathbb{R}}_+^*\\ &\simeq &\Psi^*(\Psi^*({\mathcal{G}},A)\otimes C_0({\mathbb{R}}),\beta\otimes \lambda,{\mathbb{R}},0,C(M)\otimes C_0({\mathbb{R}})).\end{aligned}$$ We have $ (A\otimes C_0({\mathbb{R}}_+))\rtimes_{\alpha} {\mathcal{G}}_{ga}= ((A\otimes C_0({\mathbb{R}}_+))\rtimes_{\alpha} {\mathcal{G}}_{ad})\rtimes_{\tau}{\mathbb{R}}_+^*$. The first isomorphism is a direct consequence of theorem \[maintheorem\]; the second one comes from prop. \[prop2.4\]. Let us drop the algebra $A$. The exact sequence $$0\to C^*({\mathcal{G}})\otimes {\mathcal{K}}\to C^*({\mathcal{G}}_{ga})\rtimes {\mathbb{R}}_+^* \to C_0({\mathfrak{A}}^*{\mathcal{G}})\rtimes {\mathbb{R}}_+^*\to 0$$ defines an “ext” element in $KK^1(C_0({\mathfrak{A}}^*{\mathcal{G}})\rtimes {\mathbb{R}}_+^*,C^*({\mathcal{G}})\otimes {\mathcal{K}})$. Using Connes’ Thom isomorphism ([[*cf.*]{} ]{}[@ConnesThom; @FaSkThom]), this group is isomorphic to $KK(C_0({\mathfrak{A}}^*{\mathcal{G}}),C^*({\mathcal{G}}))$. In fact, using again the Thom isomorphism, this element corresponds to the ext element in $KK^1(C_0({\mathfrak{A}}^*{\mathcal{G}}),C^*({\mathcal{G}})\otimes C_0({\mathbb{R}}_+^*))$ of the exact sequence $$0\to C^*({\mathcal{G}})\otimes C_0({\mathbb{R}}_+^*)\to C^*({\mathcal{G}}_{ad})\to C_0({\mathfrak{A}}^*{\mathcal{G}})\to 0.$$ [One easily sees (using e.g. [@MonthPie Theorem 2.1]) that]{} this element is the analytic index. Let $\mu:C(M)\to \Psi^*({\mathcal{G}})$ be the inclusion, and let $C_\mu$ be the corresponding mapping cone. We have an exact sequence $$0\to \Psi^*({\mathcal{G}})\otimes C_0({\mathbb{R}}_+^*)\to C_\mu\to C(M)\to 0.$$ The quotient of $C_\mu$ by the ideal $C^*({\mathcal{G}})\otimes C_0({\mathbb{R}}_+^*)$ is the cone of the inclusion $C(M)\to C(S^*{\mathfrak{g}})$, which is naturally isomorphic to $C_0({\mathfrak{A}}^*{\mathcal{G}})$. We thus find an exact sequence $$0\to C^*({\mathcal{G}})\otimes C_0({\mathbb{R}}_+^*)\to C_\mu\to C_0({\mathfrak{A}}^*{\mathcal{G}})\to 0.$$ The corresponding $KK$-element can be seen again to be the analytic index element in $KK(C_0({\mathfrak{A}}^*{\mathcal{G}}),C^*({\mathcal{G}}))$. Taking crossed product by the natural action of ${\mathbb{R}}_+^*$ on $C_\mu$ (just by rescaling), we find an exact sequence $$0\to {\mathcal{K}}\to C_\mu\rtimes {\mathbb{R}}_+^* \to C_0(T^*M)\rtimes {\mathbb{R}}_+^*\to 0.$$ In the case of the pair groupoid, we deduce an isomorphism $C_\mu\rtimes {\mathbb{R}}\simeq C^*({\mathcal{G}}_{ga})$ thanks to Voiculescu’s theorem ([@Voicu [Theorem 1.5]{}]). It is a natural question to decide whether this isomorphism extends to the general case. On the other hand, this isomorphism is not “natural”. Indeed, $C_\mu$ and $C^*({\mathcal{G}}_{ad})$ are not isomorphic in general, whence there is no isomorphism $C_\mu\rtimes {\mathbb{R}}\simeq C^*({\mathcal{G}}_{ga})(=C^*({\mathcal{G}}_{ad})\rtimes {\mathbb{R}}_+^*)$ equivariant with respect to the dual actions. Appendix: some facts on unbounded operators =========================================== In this appendix, we recall a few rather classical abstract facts about unbounded operators that we used in the text. These facts are presented here in a form suitable for our exposition and certainly not in their most general forms. They can be found in (or deduced directly from) [@BaajThese; @Woro] - see also [@chief]. Let $E$ be a $C^*$-module (over a $C^*$-algebra) and $L$ a regular (densely defined, unbounded) self-adjoint operator on $E$. \[ract\] Let us recall a few facts about unbounded functional calculus, $f\mapsto f(L)$ ([[*cf.*]{} ]{}[@BaajThese; @Woro]). 1. Put $h(t)=(i+t)^{-1}$; there exists a unique morphism $\pi_L:f\mapsto f(L)$ from $C_0({\mathbb{R}})$ to ${\mathcal{L}}(E)$ such that $\pi_L(h)=(L+i\,{{\hbox{id}}}_E)^{-1}$. 2. Since $h(L)$ has a dense range (${\mathrm{dom}}\, L$), this morphism is non degenerate, it extends to a morphism $f\mapsto f(L)$ from $C_b({\mathbb{R}})={\mathcal{M}}(C_0({\mathbb{R}}))$ to ${\mathcal{L}}(E)$. 3. If $f\in C({\mathbb{R}})$, define the operator $f(L)$ whose domain is the range of $g(L)$ where $g(t)=(|f(t)|+1)^{-1}$ and such that $f(L)g(L)=(fg)(L)$. 4. If $f,g\in C({\mathbb{R}})$ are such that $\frac{f}{|g|+1}$ is bounded, then ${\mathrm{dom}}\,g(L)\subset {\mathrm{dom}}f(L)$.\[ractaire\] 5. \[rigerateur\]If $(f_n)$ is an increasing sequence of positive elements of $C_b({\mathbb{R}})$ converging simply (and therefore uniformly on compact subsets of ${\mathbb{R}}$) to a continuous function $f$, then the domain of $f(L)$ is the set of $x\in E$ such that $(f_n(L)x)$ converges (in norm) and then $f(L)x$ is the limit of this sequence. Indeed, as $\frac{f_n+1}{f+1}=h_n$ converges to $1$ for the topology of $C_b({\mathbb{R}}): $ - if $x$ is in the domain of $f(L)$, it is of the form $x=(f(L)+1)^{-1}z$, and $x+f_n(L)x=h_n(L)z$ converges to $z$, therefore $f_n(L)x$ converges to $z-x$; - $(f(L)+{{\hbox{id}}}_E)^{-1}(f_n(L)x+x)=h_n(L)x$ converges to $x$; assume that $f_n(L)x$ converges to $y\in E$, then $(x,x+y)$ is the limit of the sequence $\Big(h_n(L)x,(f_n(L)x+x)\Big)$ of elements of the graph of $f(L)+{{\hbox{id}}}_E$; therefore $y=f(L)x$ since the graph of $f(L)$ is closed. \[mondo\] We have an equality $$L=\int_1^{+\infty}\Big(\frac{1}{s}-(e^L+s)^{-1}\Big)ds-\int_0^1(e^L+s)^{-1} ds$$ which means that ${\mathrm{dom}}\,L$ is the set of $x\in E$ such that the integrals $$\int_1^{+\infty}\Big(\frac{1}{s}-(e^L+s)^{-1}\Big)x \, ds\ \ \hbox{and}\ \ \int_0^1(e^L+s)^{-1} x\, ds$$ are norm convergent and $Lx$ is then the difference of these two integrals. Put $f_n(t)=\int_1^{n}\Big(\frac{1}{s}-(e^t+s)^{-1}\Big)ds$ and $f(t)=\lim f_n(t)=\ln (e^t+1)$; put also $g_n(t)=\int_{\frac1n}^1(e^t+s)^{-1} ds$ and $g(t)=\lim g_n(t)=\ln (e^t+1)- t$. Then as $\frac{\ln (e^t+1)}{|t |+1}$ is bounded, ${\mathrm{dom}}\, L={\mathrm{dom}}\, f(L)\cap {\mathrm{dom}}\, g(L)$ (by fact \[ract\].\[ractaire\]). The conclusion follows from fact \[ract\].\[rigerateur\]). Fact \[ract\].f). : Assume $L$ is positive with resolvent in ${\mathcal{K}}(E)$. Then $f\mapsto f(L)$ is a morphism $\pi_L:C_0({\mathbb{R}}_+^*)\to{\mathcal{K}}(E)$. Note that, for $t\in {\mathbb{R}}_+^*$, we have $\pi_{tL}=\pi_L\circ \lambda_t$ where $\lambda_t$ is the automorphism of $C_0({\mathbb{R}}_+^*)$ induced by the regular representation. Since $t\mapsto \frac{t}{t^2+1}$ is a strictly positive element of $C_0({\mathbb{R}}_+^*)$, it follows that $\pi_L(C_0({\mathbb{R}}_+^*))E$ is the closure of the image of $L(L^2+1)^{-1}$. \[derivations\]We will consider the (unbounded, skew adjoint) derivation $\partial_L$ associated with $L$: its domain is the $*$-subalgebra of the elements $a\in {\mathcal{L}}(E)$, such that there exists $\partial _L(a)\in {\mathcal{L}}(E)$ with $aL\subset La+\partial_L (a)$ (in other words $a\,{\mathrm{dom}}\, L\subset {\mathrm{dom}}\, L$ and $[a,L]$ defined on ${\mathrm{dom}}\, L$ extends to an operator $\partial _L(a)\in {\mathcal{L}}(E)$). Put $u_t=\exp(itL)$ and define for $a\in {\mathcal{L}}(E)$, $\beta_t(a)=u_tau_t^*$. 1. For $a\in {\mathcal{L}}(E)$, the map $t\mapsto \beta_t$ is of class $C^1$ (for the norm topology) if and only if $a\in {\mathrm{dom}}\, \partial_L$ and, in that case $d/dt(\beta_t(a))=i\partial_L (\beta_t(a))=i\beta_t(\partial_L (a))$. 2. The closure $\overline{{\mathrm{dom}}\, \partial_L}$ of ${\mathrm{dom}}\, \partial_L$ is a $C^*$-subalgebra of ${\mathcal{L}}(E)$ and $t\mapsto \beta_t(a)$ is a continuous action of ${\mathbb{R}}$ on it. \[LemmeApp1\] Let $Q$ be the norm closure of $\{a\in {\mathrm{dom}}\, \partial_L;\ \partial_L a\in {\mathcal{K}}(E)\}$. It is a $C^*$-subalgebra of $\overline{{\mathrm{dom}}\, \partial_L}$ invariant under the action $\beta$ of ${\mathbb{R}}$. The quotient action of ${\mathbb{R}}$ on $Q/{\mathcal{K}}(E)$ is trivial. In particular, every $C^*$-subalgebra of $Q$ containing ${\mathcal{K}}(E)$ is invariant by $\beta$. Denote by $q:{\mathcal{L}}(E)\to {\mathcal{L}}(E)/{\mathcal{K}}(E)$ the quotient map. If $a\in {\mathrm{dom}}\,\partial_L$ satisfies $\partial_L a\in {\mathcal{K}}(E)$, then $t\mapsto \beta_t(a)$ is $C^1$, and the derivative of $t\mapsto q(\beta_t(a))$ is zero. All other statements are clear. \[LemmeApp2\] Let $a\in {\mathrm{dom}}\,\partial_{e^L}\cap \partial_{e^{-L}}$. Then $a\in {\mathrm{dom}}\,\partial_L$. If the resolvent of $L$ is in ${\mathcal{K}}(E)$, then $\partial_L(a)\in {\mathcal{K}}(E)$. The integral $\int_1^{+\infty}\Big[\frac{1}{s}-(e^L+s)^{-1},a\Big]ds=\int_1^{+\infty}(e^L+s)^{-1}[e^L,a](e^L+s)^{-1}\,ds$ is norm convergent (since $\|(e^L+s)^{-1}\|\le s^{-1}$), as well as $$\begin{aligned} -\int_0^1\Big[(e^L+s)^{-1},a\Big] ds&=&\int_0^1(e^L+s)^{-1}\Big[e^L,a\Big](e^L+s)^{-1}\,ds\\ &=&-\int_0^1e^L(e^L+s)^{-1}\Big[e^{-L},a\Big]e^L(e^L+s)^{-1}\,ds.\end{aligned}$$ (since $\|e^L(e^L+s)^{-1}\|\le 1$). It follows, with the notation of Lemma \[mondo\] that $[(f_n-g_n)(L),a]$ converges to an element $b=\int_0^{+\infty}(e^L+s)^{-1}[e^L,a](e^L+s)^{-1}\,ds$. If $x\in {\mathrm{dom}}\,L$, then $(f_n-g_n)(L)ax$ converges to $aLx+bx$; therefore $ax\in {\mathrm{dom}}\,L$ and $\partial_L(a)=b$. Assume $L$ has compact resolvent [([[*i.e.*]{} ]{}in ${\mathcal{K}}(E)$)]{}. Put $q_s=(e^L+s)^{-1}[e^L,a](e^L+s)^{-1}$. Note that $e^Lq_s$ is bounded and, since $q_s=-(e^L+s)^{-1}e^L\Big[e^{-L},a\Big]e^L(e^L+s)^{-1}$, $e^{-L}q_s$ is also bounded. If $L$ has compact resolvant, then $(e^L+e^{-L})^{-1}\in {\mathcal{K}}(E)$, whence $q_s\in {\mathcal{K}}(E)$. \[LemmeApp3\] Assume $L$ is positive. Let $a\in {\mathcal{L}}(E)$ such that $a\,{\mathrm{dom}}\, e^L\subset {\mathrm{dom}}\, e^L$ and $e^{-L/2}[e^L,a]$ defined on ${\mathrm{dom}}\, e^L$ extends to an element of ${\mathcal{L}}(E)$. Then $a\in {\mathrm{dom}}\,\partial_L$. If moreover the resolvant of $L$ is in ${\mathcal{K}}(E)$, then $\partial_L(a)\in {\mathcal{K}}(E)$. The integral $\int_1^{+\infty}\Big[\frac{1}{s}-(e^L+s)^{-1},a\Big]ds=\int_1^{+\infty}(e^L+s)^{-1}[e^L,a](e^L+s)^{-1}\,ds$ is norm convergent, since $$\|(e^L+s)^{-1}[e^L,a](e^L+s)^{-1}\|\le \|e^{L/2}(e^L+s)^{-1}\|\|e^{-L/2}[e^L,a]\|\|(e^L+s)^{-1}\|\le s^{-1/2}Cs^{-1}$$ for $C=\|e^{-L/2}[e^L,a]\|$. Of course the integral $ -\int_0^1\Big[(e^L+s)^{-1},a\Big] ds$ is also norm convergent. It follows, with the notation of Lemma \[mondo\] that $[(f_n-g_n)(L),a]$ converges to an element $b=\int_0^{+\infty}(e^L+s)^{-1}[e^L,a](e^L+s)^{-1}\,ds$. If $x\in {\mathrm{dom}}\,L$, then $(f_n-g_n)(L)ax$ converges to $aLx+bx$; therefore $ax\in {\mathrm{dom}}\,L$ and $\partial_L(a)=b$. Assume $L$ has compact resolvant. Then, since $L$ is positive, $(e^L+s)^{-1}\in {\mathcal{K}}(E)$, whence $(e^L+s)^{-1}[e^L,a](e^L+s)^{-1}\in {\mathcal{K}}(E)$. [^1]: AMS subject classification: Primary 58H05. Secondary 46L89, 58J22.

--- abstract: 'We have observed the flare star YYGem simultaneously with [*XMM-Newton*]{} and [*Chandra*]{} as part of a multi-wavelength campaign aiming at a study of variability related to magnetic activity in this short-period eclipsing binary. Here we report on the first results from the analysis of the X-ray spectrum. The vicinity of the star provides high enough S/N in the CCD cameras onboard [*XMM-Newton*]{} to allow for time-resolved spectroscopy. Since the data are acquired simultaneously they allow for a cross-calibration check of the performance of the [*XMM-Newton*]{} RGS and the LETGS on [*Chandra*]{}.' author: - 'B. Stelzer, V. Burwitz, R. Neuhäuser' - 'M. Audard' - 'J. H. M. M. Schmitt' title: 'The joint [*XMM-Newton*]{} and [*Chandra*]{} view of YYGem' --- \#1[[*\#1*]{}]{} \#1[[*\#1*]{}]{} = \#1 1.25in .125in .25in Introduction ============ YYGem is the optically faintest of the three visual binaries in the Castor sextuplet. It is itself an eclipsing spectroscopic binary with period of $0.81$d. The two components of YYGem are both of spectral type dM1e, and belong to the class of BYDra variables. Indeed, [YYGem ]{}was the first stellar system on which periodic photometric variability was detected (Kron 1952). Since the discovery of X-ray emission from the Castor system by the [*Einstein*]{} satellite, the system was studied by virtually all X-ray observatories (Vaiana et al. 1981, Pallavicini et al. 1990, Gotthelf et al. 1994, Schmitt et al. 1994, Güdel et al. 2001). Flares on YYGem have been recorded from all parts of the electromagnetic spectrum. The extraordinary activity of this object may be related to its binarity (the frequency of photometric flares seems to be enhanced in the interbinary space suggesting interaction between the magnetospheres of the two stellar components; Doyle & Mathioudakis 1990), and makes it a prime target for simultaneous monitoring at different wavelengths. Observations and Data Analysis ============================== [YYGem ]{}was observed by both [*Chandra*]{} and [*XMM-Newton*]{} on Sep 29/30, 2000 for a total observing time of 59ksec and 55ksec, respectively. The [*XMM-Newton*]{} observations were obtained in the full-frame mode of EPIC-pn, with the thick filter inserted for both pn and MOS. We perform the data analysis with the standard [*XMM-Newton*]{} Science Analysis System (SAS). [*Chandra*]{} was used in the LETGS configuration, i.e. the Low Energy Transmission Grating (LETG) combined with the High Resolution Camera for Spectroscopy (HRC-S). We extracted the [*Chandra*]{} lightcurves and spectra using programs written in IDL version 5.4. The extraction areas for source and background spectrum are those defined in the [*Chandra*]{} User’s Guide. The time of observation for the individual X-ray instruments is given in Table 1. We display the corresponding X-ray lightcurves in Fig. 1. The orbital phase has been computed from the ephemeris of Torres & Ribas (2001). First inspection reveals strong variability throughout the whole observation, including two large flares, and two ‘high states’ (i.e. extended phases of enhanced emission) near the end of the observation. The secondary eclipse is clearly identified as a dip in the lightcurve close to orbital phase $0.5$. Note, that the minimum of the X-ray lightcurve is not exactly centered on $\Phi = 0.5$, but slightly offset towards earlier times. As we have observed simultaneously with two independent satellites a timing error is very unlikely. This shift may indicate an inhomogeneous distribution of emitting material in the coronae of the [YYGem ]{}binary. [lrrrrr]{}Instrument & & & Expo\ & Start & Stop & Start & Stop & \[ksec\]\ \ EPIC-pn & 18:56 & 08:57 & 0.2888 & 0.8730 & 50.47\ EPIC-MOS & 18:15 & 08:51 & 0.2604 & 0.8689 & 52.57\ RGS & 18:07 & 09:27 & 0.2542 & 0.8937 & 55.26\ \ LETGS & 21:30 & 13:54 & 0.3958 & 1.0792 & 59.00\ The combination of [*XMM-Newton*]{} and [*Chandra*]{} allows to examine the X-ray spectrum of YYGem with intermediate (EPIC) and high (LETGS, RGS) resolution, and to compare the performance of the grating instruments on both satellites. The CCD spectra obtained with the EPIC are analysed in the XSPEC environment (version 11.0.1). The [*XMM-Newton*]{} EPIC spectrum ================================== We start with the analysis of the quiescent spectrum observed prior to the first large flare in Fig. 1 (JD 2451817.256 $-$ JD 2451817.450). Following Güdel et al. (2001) we represent the quiescent EPIC spectrum of YYGem by a 3-temperature (3-T) model for thermal emission from an optically thin plasma (VMEKAL). In order to better constrain the spectral model we analyse the spectra from the pn and the two MOS detectors simultaneously. For the joint modeling of the spectrum from these three instruments we add a constant normalization factor to make up for uncertainties in the absolute calibration of the detectors. The EPIC spectrum for the pre-flare phase is shown in Fig. 2, and the best fit parameters from the 3-T model are summarized in Table 2. [rrrr]{} & & & \[${\rm keV}$\]\ $0.21^{+0.05}_{-0.07}$ & ------------------------------------------------------------------------ $0.64^{+0.01}_{-0.02}$ & $1.79^{+0.30}_{-0.24}$ &\ & & & \[$10^{51}\,{\rm cm^{-3}}$\]\ $2.24^{+1.87}_{-0.60}$ & ------------------------------------------------------------------------ $13.84^{+0.82}_{-2.88}$ & $2.88^{+1.05}_{-0.66}$ &\ & & &\ $0.64^{+0.19}_{-0.14}$ & ------------------------------------------------------------------------ $0.27^{+0.12}_{-0.07}$ &$0.47^{+0.16}_{-0.07}$ &\ & & & $\chi^2_{\rm red}$ (dof)\ $0.50^{+0.27}_{-0.20}$ & $0.23^{+0.04}_{-0.03}$ & $0.00^{+0.28}_{-0.00}$ & 1.24 (658)\ We use this spectrum as a baseline for time-resolved spectroscopy. The EPIC lightcurve of [YYGem ]{}is split in a total of 15 time intervals (listed in Table 3) ------- ------- ------------------------------ ---------- -- Start Stop Remarks Interval 0.290 0.450 pre-flare quiescence $t_1$ 0.450 0.475 hump before flare $t_2$ 0.475 0.484 rise flare 1 $t_3$ 0.484 0.493 decay (a) flare 1 $t_4$ 0.493 0.510 decay (b) flare 1 $t_5$ 0.510 0.525 mini-flare $t_6$ 0.525 0.630 post-flare quiescence $t_7$ 0.630 0.662 secondary eclipse (1st half) $t_8$ 0.662 0.688 secondary eclipse (2nd half) $t_9$ 0.688 0.710 post-eclipse feature (a) $t_{10}$ 0.710 0.735 post-eclipse feature (b) $t_{11}$ 0.735 0.775 post-eclipse feature (c) $t_{12}$ 0.775 0.790 rise flare 2 $t_{13}$ 0.790 0.805 decay flare 2 $t_{14}$ 0.805 0.869 ‘high state’ $t_{15}$ ------- ------- ------------------------------ ---------- -- : Time intervals selected for a systematic investigation of the evolution of spectral parameters throughout the [*XMM-Newton*]{} EPIC observation from 29/30 Sep 2000. representing different activity levels of the star, and the spectrum of each phase is modeled by a 3-T model. As the integrated light from the quiescent corona should be visible at all times we hold all temperatures and abundances fixed on the values given in Table 2, and vary only the emission measure. In some of the time segments, namely for the post-eclipse feature and during the large flares, the 3-T model does not provide an adequate description of the EPIC spectrum: A high energy excess stands out in the residuals suggesting the presence of higher temperature material in addition to the emission from the quiescent corona. Adding a fourth VMEKAL component does not lead to a significant improvement. Only a 5-T model represents the data well ($\chi^2_{\rm red} \sim 1$) during the phases of most intense emission. For the modeling of these time intervals we have fixed spectral components \#$\,1-3$ on their quiescent values (see Table 2). All abundances of components \#4 and \#5 have been held fixed on solar values because the statistics do not allow to constrain further parameters. The last time interval ($t_{\rm 15}$; the ‘high state’) is an exception: The signal at high energies is larger than for all other time segments, and broad Fe K-shell emission is clearly visible (see Fig. 2). We find an acceptable solution in this case for $\frac{\rm Fe}{\rm H} = 0.47^{+0.10}_{-0.09}$. Temperature - Emission Measure Diagrams --------------------------------------- The evolution of temperature and emission measure puts important constraints on the dynamics during flare decays. In a one-dimensional hydro-dynamic approach to model stellar flares developed by Reale et al. (1993) the duration of the heating determines the slope in the $\lg{T} - \lg{(\sqrt{EM})} - $diagram. We have derived $\lg{T} - \lg{(\sqrt{EM})} -$diagrams for the spectral components of the 5-T model that represent the heated plasma during the two large flares, i.e. VMEKAL components \#4 and \#5. Fig. 3 shows the evolution of the two large flares, both starting with the rise phase (time interval $t_3$ and $t_{13}$, respectively). Under the assumption that the flare emission is concentrated in a single loop the slope $\zeta$ observed during the decay phase can be used to obtain an estimate for the loop half-length $L$. This method has been calibrated for several instruments including EPIC-pn (F. Reale, priv. comm.). We apply the equivalent of Eq. 2 from Reale et al. (1997) to derive $L$ from the slope $\zeta$, the observed temperature ($T_{\rm max} = 39\,$MK), and the decay constant of the lightcurve ($\tau_{\rm lc} = 16 \pm 1$min). The resulting loop length is $L \sim 2 \cdot 10^9$cm. High-resolution Spectra: [*XMM-Newton*]{} RGS and [*Chandra*]{} LETGS ===================================================================== A comparison of the time-averaged first order X-ray spectra of [YYGem ]{}as observed with LETGS and RGS is given in Fig. 4. We only show the region between $10 - 26$Å, which contains the strongest lines. Line identifications are given on top of the diagram. The spectrum is given in units of cts/s/bin. Since the RGS and the LETGS observations overlap for about $75$% in time, the relative strength of the lines measured by both instruments should be similar, with some dependence of the line strength on the binsize, and the absolute numbers demonstrate directly the difference in sensitivity between RGS and LETGS. The Ly$\alpha$ line of H-like OVIII is by far the strongest line in the spectrum with the highest photon flux, i.e. taking account of the effective area. Next to a number of iron L-shell transitions we identify the He-like triplets of four elements: SiXIII, NeIX, OVII, and NVI. The OVII triplet is the strongest triplet and the only one which is clearly resolved and not blended with other lines. A detailed investigation of the properties of the coronal plasma making use of line ratios will be presented by Stelzer et al., in prep. Summary ======= The X-ray lightcurve of [YYGem ]{}shows that the object was subject to strong variability including two large outbursts during the time of observation. The parameters of a 3-T model for the quiescent emission are compatible with results from the analysis of an earlier [*XMM-Newton*]{} observation of YYGem presented by Güdel et al. (2001). Time-resolved modeling of the EPIC spectrum reveals the presence of a high temperature plasma ($kT_{\rm max} = 3.4\,$keV) in flares. According to a one-dimensional hydrodynamic model the flare emission arises in a semi-circular loop with $\sim 2 \cdot 10^9$cm length. This approach is certainly a simplification of the real situation which does involve a multi-temperature plasma and possibly complex loop systems. Nevertheless, the hydrodynamic approach is important: while simple quasi-static modeling tends to reproduce large loops the method applied here demonstrates that the coronal structures are likely to be much smaller than the radii of both stars in the [YYGem ]{}system. The simultaneous observation of [YYGem ]{}with [*Chandra*]{} and [*XMM-Newton*]{} demonstrates the different sensitivity of these instruments. Each of the two RGS provides roughly the same count rate as the LETGS first order spectrum. The LETGS is more sensitive at short wavelengths (see e.g. the region around the NeIX triplet), while the sensitivity of RGS is slightly higher towards longer wavelengths (e.g. near the OVIII Ly$\alpha$ line). Doyle J. G. & Mathioudakis M. 1990,  227, 130 Gotthelf E. V., Jalota L., Mukai K., et al. 1994,  436, L91 Güdel M., Audard M., Magee H., et al. 2001,  365, L344 Kron G. E. 1952,  115, 301 Ness J.-U., Mewe R., Schmitt J. H. M. M., et al. 2001,  367, 282 Pallavicini R., Tagliaferri G., Pollock A. M. T., et al. 1990,  227, 483 Reale F., Serio S., Peres G. 1993,  272, 486 Reale F., Betta R., Peres G., et al. 1997,  325, 782 Schmitt J. H. M. M., Güdel M., Predehl P. 1994,  287, 843 Torres G. & Ribas, 2001,  in press Vaiana G. S., Cassinelli J. P., Fabbiano G., et al. 1981,  244, 163

--- author: - 'Wu-Zhong Guo,' - 'Song He,' title: Rényi entropy of locally excited states with thermal and boundary effect in 2D CFTs --- Introduction ============ Many kinds of observables can be defined in Quantum field theories (QFTs). When we study global or non-local structures, entanglement entropy (EE) or the entanglement Rényi entropy (RE) are very helpful quantities. For a subsystem $A$, both of them are defined as a function of the reduced density matrix $\rho_A$. The reduced density matrix $\rho_A$ can be defined from the original density matrix $\rho$ by tracing out the subsystem $B$ which is the complementary of $A$. One may be curious about whether there is a kind of topological contribution in entanglement entropy even for gapless theories, particularly for conformal field theories (CFTs). For example, topological properties can be qualified by computing topological contributions in entanglement entropy called topological entanglement entropy [@wen]. In this paper, we focus on extracting such kind of topological quantity from both Rényi entropy and von-Neumann entropy of locally excited states in two dimensional rational CFTs with thermal effect and boundary effect. The $n$-th Rényi entanglement entropy is defined by $S^{(n)}_A=\log\mbox{Tr}[\rho_A^n]/(1-n)$ formally. The limit $n\to 1$ coincides with the von-Neumann entropy. This is standard replica trick method to calculate the entanglement entropy. The difference of $S^{(n)}_A$ between the locally excited states and the ground states with introducing thermal and boundary effects are main interest in this paper. The difference is denoted by $\Delta S^{(n)}_A$. Replica approach calculations of $\Delta S^{(n)}_A$ for states excited by operators have been given in [@UAM; @Nozaki:2014uaa; @Nozaki:2014hna]. We will closely follow the construction in [@Nozaki:2014hna][@He:2014mwa][@Nozaki:2014uaa]with introducing thermal and boundary effect. We would like to review the thermal effect and boundary effect respectively. There are many studies about thermal effect in 2D CFTs. For thermal states, EE is not a good measurable quantity, which is contaminated by the thermal entropy of the subregion. In high temperature limit, the EE will be dominated by thermal entropy. To reveal the quantum entanglement of system with thermal effect, one should identify the thermal contribution and other contributions of EE. Ref. [@Herzog:2012bw]conjectured universal form of correction of EE in any quantum system with mass gap. Ref.[@Cardy:2014jwa] provided the form of the coefficient of such correction in 2D CFT. To generalize studies in [@Cardy:2014jwa] to higher dimensions, Ref.[@Herzog:2014fra] and Ref.[@Herzog:2014tfa] considered thermal corrections to the entanglement entropy on spheres. On the other hand, the dynamics in 2D CFTs with a boundary have many new features comparing with 2D CFTs in full complex plane. The original works have been done by Cardy, who discussed surface critical behavior of correlation functions [@cardy1]. Ref.[@cardy2] studied the constraints on the operator content by imposing by boundary conditions and also the classification of boundary states in terms of the modular transformation. In [@cardy3][@cardylewellen], the concept of boundary operators have been introduced. Ref.[@3sp2] showed that the resulting set of boundary conditions to be complete. There are also nice correspondence called as AdS/BCFT proposed by [@Takayanagi:2011zk][@Fujita:2011fp]. The boundary effect can be also studied holographically, which is beyond the issue considered in this paper. In 2D rational CFTs, the authors in [@He:2014mwa] get an amazing result for the locally excited states, which relates the Rényi entropy to the quantum dimension of the primary operator which is kind of topological quantity. In this paper, we generalize the previous study [@He:2014mwa] on Rényi entropy with thermal and boundary effects. Firstly, there is a simple sum rule between the thermal correction and local excitation in low temperature limit. That is to say the total Rényi entropy are summing over Rényi entropy of local excitation and the one of thermal excitation in low temperature limit. Such kind of relation is similar to the sum rule related to the Rényi entropy in [@Nozaki:2014uaa]. One can generalize the result to local excitation in pure state in 2D CFT. We make use of a different approach [@Herzog:2012bw] to obtain the thermal correction to Rényi entropy which can be reduced to [@Caputa:2014eta]. Secondly, we investigate the the Rényi entropy for states excited by local primary operators in the rational CFTs with a boundary. These boundaries introduced here do not break the conformal symmetry. Such theories are called as BCFTs. We show the time evolution of the Rényi entropy in 2D free field theory and Ising model. Then we generalize to rational CFTs with a boundary. The boundary changes the time evolution of the Rényi entropy, but does not change the maximal value of the Rényi entropy. All these cases studied in this paper show that the Rényi entropy does not depend on the choice of boundary conditions. In 2D rational CFTs with a boundary, we also show that the maximal value of the Rényi entropy always coincides with the log of quantum dimension of the primary operator during some periods of the evolution. We give the physical understanding of boundary effect, which support the quasi-particles explanation of the local excitation. The boundary behaves as an infinite potential barrier which reflects the quasi-particle moving towards it. The layout of this paper is as follows. In section 2, we study the thermal effect on the Rényi entropy of the local excited state in low temperature limit. In section 3, we set up the local excitation in 2D CFT with a boundary and obtain the Rényi entropy of a subsystem with time evolution. We study the 2D free scalar, and Ising model as examples, then generalize the result to 2D rational CFT. In section 4, we devote to the conclusion and physical interpretations of such kinds of effects shown in this paper. Local excitation in non-vacuum states ===================================== In this section, we would like to study the local excitation of thermal state. We consider a system with temperature $T=1/\beta$ and assume the excitation is local at $x=-L$ by primary operator $O$ shown in fig.\[\[fig0\]\]. In this section, we just only consider the low temperature case with large $\beta$ [@Herzog:2014fra]. The subsystem $A$ is $-l<x<0$. The density matrix $\rho(t)$ is $$\begin{aligned} \label{T1} \rho(t)=N(t)\Big(O(\omega_2,\bar \omega_2)(\sum \Ket{n}\Bra{n} e^{-\beta E_n-2\beta \epsilon E_n})O^{\dag}(\omega_1,\bar \omega_1)\Big),\end{aligned}$$ where we have considered the real time evolution, $\epsilon$ is the ultraviolet regularization, $N(t)$ is fixed by normalization condition $tr\rho(t)=1$, $E_n$ are the energy of the excited states. The complex coordinates in $\omega$ plane are listed as follows. $$\begin{aligned} \omega_1=i({\epsilon-it})-L, \ \ \omega_2=-i(\epsilon+it)-L, \nonumber \\ \bar \omega_1=-i(\epsilon-it)-L,\ \ \bar \omega_2=i(\epsilon+it)-L.\end{aligned}$$ =8.0 cm =5.0 cm For later convenience, we define $$\begin{aligned} &&\rho_0(t)=tr_B( O(\omega_2,\bar \omega_2)\Ket{0}\Bra{0}O^{\dag}(\omega_1,\bar \omega_1)),\nonumber\\ &&\rho_1(t)=tr_B( e^{-\beta E_1-2\beta \epsilon E_1} O(\omega_2,\bar \omega_2)\Ket{1}\Bra{1}O^{\dag}(\omega_1,\bar \omega_1)),\end{aligned}$$ which can be taken as the reduced density matrix related to the vacuum and first excited state respectively, where we normalize the vacuum energy to be zero, $B$ is the complementary part of subsystem $A$. In the low temperature expansion with $\beta E_1 \ll 1$ $$\begin{aligned} \label{Expand} \rho_A(t)=tr_B \rho(t)=\frac{\rho_0(t)+\rho_1(t)+...}{tr_A (\rho_0(t)+\rho_1(t))+...}.\end{aligned}$$ In terms of the definition of the Rényi entropy $$\begin{aligned} \label{resultofthermal} S^{(n)}_A&=&\frac{\log tr \rho_A(t)^n }{1-n}\nonumber \\ &&\simeq \frac{1}{1-n}\log\Big[\frac{tr(\rho_0(t)^n)}{(tr\rho_0(t))^n}(1+\frac{n tr(\rho_0(t)^{n-1}\rho_1(t))}{tr(\rho_0(t)^n)}-\frac{n tr\rho_1(t)}{tr\rho_0(t)})\Big]\nonumber \\ &&\simeq \frac{1}{1-n}\Big[\log \frac{tr(\rho_0(t)^n)}{(tr\rho_0(t))^n}+\frac{n tr(\rho_0(t)^{n-1}\rho_1(t))}{tr(\rho_0(t)^n)}-\frac{n tr\rho_1(t)}{tr\rho_0(t)}\Big].\end{aligned}$$ When there is no local excitation, i.e., the operator $O=I$, the result is the same as [@Cardy:2014jwa]. The second and third terms of the last line in (\[resultofthermal\]) involve in the coupling between the local excitation and thermal environment. In terms of the state operator correspondence, one can denote $\Ket{1}=\lim_{t\to -\infty}\psi(x,t)\Ket{0}$ [@Cardy:2014jwa]for the excited state with energy $E_1$. Here we consider the whole system has an infrared cut-off $\Lambda$, and $l/\Lambda \ll 1$. Using the path integral language, (\[resultofthermal\]) is $$\begin{aligned} \label{ResultSncorrelation} S^{(n)}_A&=&\frac{1}{1-n} \log \frac{\langle O^{\dag}(\omega_1,\bar \omega_1)O(\omega_2,\bar \omega_2)...O(\omega_{2n},\bar \omega_{2n})\rangle_{C_n}}{(\langle O^{\dag}(\omega_1,\bar \omega_1)O(\omega_2,\bar \omega_2)\rangle_{C_1})^n}\nonumber \\ &+&\frac{ne^{-\beta E_1}}{1-n}\frac{\langle O^{\dag}(\omega_1,\bar \omega_1)O(\omega_2,\bar \omega_2)...O(\omega_{2n},\bar \omega_{2n})\psi(-\infty)\psi(+\infty)\rangle_{C_n}}{\langle O^{\dag}(\omega_1,\bar \omega_1)O(\omega_2,\bar \omega_2)...O(\omega_{2n},\bar \omega_{2n})\rangle_{C_n}}\nonumber \\ &-&\frac{n e^{-\beta E_1}}{1-n}\frac{\langle O^{\dag}(\omega_1,\bar \omega_1)O(\omega_2,\bar \omega_2) \psi(-\infty)\psi(+\infty)\rangle_{C_1}}{\langle O^{\dag}(\omega_1,\bar \omega_1)O(\omega_2,\bar \omega_2)\rangle_{C_1}},\end{aligned}$$ where $C_n$ is the $n$-sheet cylinder with circumference $\Lambda$ and $O(\omega_1,\bar \omega_1)...O(\omega_{2n},\bar \omega_{2n})$ are the operators that are inserted in the suitable place in the $n$-sheet cylinder. The first term (\[ResultSncorrelation\]) is given by [@He:2014mwa] for the local excitation in vacuum. We will study the second and third term of (\[ResultSncorrelation\]) in detail. The following transformation [@cag] can map the $n$-sheet cylinder to a cylinder with circumference $\Lambda$, $$\begin{aligned} \label{thermaltransformation} z=\Big(\frac{e^{2i\pi \omega/\Lambda}-1}{e^{2i\pi \omega/\Lambda}-e^{2i\pi l/\Lambda}}\Big)^{1/n}.\end{aligned}$$ The points $\omega=-\infty$ and $\omega=\infty$ are mapping to $z_{-\infty}=e^{-2i\pi l/\Lambda}$ and $z_{+\infty}=1$ respectively. For simplifying our analysis, we only consider the range $|\omega|\ll \Lambda$. Otherwise, the following calculation will be much more complicated. But the final statement does not change without this approximation. Thus (\[thermaltransformation\]) reduces to $$\begin{aligned} z\simeq \Big(\frac{w}{w-l}\Big)^{1/n},\end{aligned}$$ which is same as the one that is used in [@He:2014mwa]. The points $z_1,...,z_{2n}$ are given by $$\begin{aligned} z_{2k+1}&=&e^{2i\pi k/n}\Big(\frac{i\epsilon +t-L}{i\epsilon +t-L-l} \Big)^{1/n},\nonumber \\ z_{2k+2}&=&e^{2i\pi k/n}\Big(\frac{-i\epsilon +t-L}{-i\epsilon +t-L-l} \Big)^{1/n},\nonumber \\ \bar z_{2k+1}&=&e^{2i\pi k/n}\Big(\frac{-i\epsilon-t-L}{-i\epsilon -t-L-l} \Big)^{1/n},\nonumber \\ \bar z_{2k+2}&=&e^{2i\pi k/n}\Big(\frac{i\epsilon -t-L}{i\epsilon -t-L-l} \Big)^{1/n}.\label{transformationrule}\end{aligned}$$ In $t>l+L$ or $0<t<L$, $$\begin{aligned} \label{factor1thermal} z_{2k+1}-z_{2k+2}\simeq -\frac{2iL\epsilon}{n(t-L)(t-l-L)}z_{2k+1},\nonumber \\ \bar z_{2k+1}-\bar z_{2k+2}\simeq -\frac{2iL\epsilon}{n(t+L)(t-l+L)}\bar z_{2k+1}.\end{aligned}$$ In $L<t<L+l$, $$\begin{aligned} \label{factor1therma2} z_{2k}-z_{2k+1}\simeq -\frac{2iL\epsilon}{n(t-L)(t-l-L)}z_{2k},\nonumber \\ \bar z_{2k+1}-\bar z_{2k+2}\simeq -\frac{2iL\epsilon}{n(t+L)(t-l+L)}\bar z_{2k+1}.\end{aligned}$$ In $t>l+L$ or $0<t<L$ the second term of (\[ResultSncorrelation\]) is $$\begin{aligned} &&\frac{n e^{-\beta E_1}}{1-n}\frac{\langle O^{\dag}(\omega_1,\bar \omega_1)O(\omega_2,\bar \omega_2)...O(\omega_{2n},\bar \omega_{2n})\psi(-\infty)\psi(+\infty)\rangle_{C_n}}{\langle O^{\dag}(\omega_1,\bar \omega_1)O(\omega_2,\bar \omega_2)...O(\omega_{2n},\bar \omega_{2n})\rangle_{C_n}}\nonumber \\ &=&\frac{ne^{-\beta E_1}}{1-n}\frac{\langle O^{\dag}(z_1,\bar z_1)O(z_2,\bar z_2)...O(z_{2n},\bar z_{2n})\psi^{'}(z_{-\infty})\psi^{'}(z_{+\infty})\rangle_{C_1}}{\langle O^{\dag}(z_1,\bar z_1)O(z_2,\bar z_2)...O(z_{2n},\bar z_{2n})\rangle_{C_1}}\nonumber \\ &=&\frac{ne^{-\beta E_1}}{1-n}\langle \psi^{'}(z_{-\infty})\psi^{'}(z_{+\infty})\rangle_{C_1}.\end{aligned}$$ In the above formula, we label $\psi^{'}(z,\bar z)$ as the map of the operator $\psi(\omega,\bar \omega)$[^1]. In the second step due to (\[factor1thermal\]) we have used $$\begin{aligned} \label{Thermalcasefactorize} &&\langle O^{\dag}(z_1,\bar z_1)O(z_2,\bar z_2)...O(z_{2n},\bar z_{2n})\psi^{'}(z_{-\infty})\psi^{'}(z_{+\infty})\rangle_{C_1}\nonumber \\ &&\simeq \langle O^{\dag}(z_1,\bar z_1)O(z_2,\bar z_2)\rangle...\langle O^{\dag}(z_{2n-1},\bar z_{2n-1}) O(z_{2n},\bar z_{2n})\rangle \langle \psi^{'}(z_{-\infty})\psi^{'}(z_{+\infty})\rangle.\nonumber\\ \end{aligned}$$ In $L<t<L+l$, the correlation function can not be factorized as (\[Thermalcasefactorize\]) directly due to (\[factor1therma2\]). Following logic in [@He:2014mwa], one can make use of $n-1$ times fusion transformation $(z_1,z_2)(z_3,z_4)...(z_{2n-1},z_{2n}) \to (z_2,z_3)(z_4,z_5)...(z_{2n},z_1)$. The second terms of (\[ResultSncorrelation\]) is still given by in $\epsilon \rightarrow 0$ $$\begin{aligned} \frac{ne^{-\beta E_1}}{1-n}\langle \psi^{'}(z_{-\infty})\psi^{'}(z_{+\infty})\rangle_{C_1}.\end{aligned}$$ Here we assume that there are no nontrival correlation between $O$ and $\psi$ for simplifying analysis. The third term of (\[ResultSncorrelation\]) in the limit $\epsilon \to 0$ is $$\begin{aligned} &&-\frac{ne^{-\beta E_1}}{1-n}\frac{\langle O^{\dag}(\omega_1,\bar \omega_1)O(\omega_2,\bar \omega_2) \psi(-\infty)\psi(+\infty)\rangle_{C_1}}{\langle O^{\dag}(\omega_1,\bar \omega_1)O(\omega_2,\bar \omega_2)\rangle_{C_1}}\nonumber \\ &=&-\frac{n}{1-n}\langle \psi(-\infty)\psi(+\infty)\rangle_{C_1}=-\frac{n}{1-n}e^{-\beta E_1}.\end{aligned}$$ The sum of the second and third term is the same as the thermal correlation [@Cardy:2014jwa] for the short interval limit. (\[ResultSncorrelation\]) is the summation over the thermal correction of the vacuum state and local excitation in the vacuum state in low temperature limit. The Rényi entropy is summation of thermal effect and local excitation. This relationship can be considered as a sum rule for Rényi entropy of low temperature thermal state with local excitation, which is different with the sum rule proposed in [@Nozaki:2014hna]. For the pure state one could also get a similar sum rule. The local excitation in thermal states have been also studied in [@Caputa:2014eta] for free boson and Ising model. With a different method [@Cardy:2014jwa], we can reproduce the Rényi entropy [@Caputa:2014eta] for short interval in the low temperature with taking limit $\epsilon \to 0$ for general 2D rational CFT. Local excitation in 2D CFTs with a boundary =========================================== In this section, we would like to study the Rényi entropy of locally excited state in 2D CFT with a boundary which preserves conformal symmetry. The global property of the CFT with a boundary has been discussed in literature[@cag][@Cardy1][@Cardy2]. As we know, the Rényi entropy is sensitive to correlation function. The boundary will change the correlation functions. And the boundary conditions also affect the correlation functions. It is interesting to check what will happen to the Rényi entropy for the local excitation in 2D CFT with a boundary. Set-up of local exciation {#3.1} ------------------------- We begin with a CFT with a boundary at $x=0$ and the CFT is living in the range $x\le 0$. We divide the this region into two parts, one part is $-l<x<0$ denoted by A and the other is complement to the region A denoted by B. The time $t$ vary from $-\infty$ to $+\infty$, and the Hamiltonian $H$ is well defined as a operator to generate the time evolution. We assume that the local excitation of vacuum is at $x=-L$ and consider the Rényi entropy of the subsystem A. The time dependent density matrix can be written as $$\begin{aligned} \rho(t)=N O(\omega_2,\bar \omega_2)\Ket{0}\Bra{0}O^{\dag}(\omega_1,\bar \omega_1),\end{aligned}$$ where the coordinates are, $$\begin{aligned} \omega_1=i(\epsilon-it)-L, \ \ \omega_2=-i(\epsilon+it)-L, \nonumber \\ \bar \omega_1=-i(\epsilon-it)-L,\ \ \bar \omega_2=i(\epsilon+it)-L.\end{aligned}$$ =6.0 cm =5.0 cm We still make use of replica trick to study the variation of the Rényi entropy of the subsystem A in this section. By definition, the variance of $n$-th Rényi entropy can be calculated as $$\begin{aligned} \label{Mainformula} \Delta S^{(n)}_A&=&\frac{1}{1-n} \Big[\log \langle O^{\dag}(\omega_1,\bar \omega_1)O(\omega_2,\bar \omega_2)...O(\omega_{2n},\bar \omega_{2n})\rangle_{B_n}\nonumber \\ &-&n\log \langle O^{\dag}(\omega_1,\bar \omega_1)O(\omega_2,\bar \omega_2)\rangle_{B_1}\Big],\end{aligned}$$ where $B_n$ is the $n$-sheet Riemann surface that consists of n copies of original plane $x\le 0$ with gluing together along $-l \leqslant x \leqslant 0$, $t=0$. With the following conformal transformation, $$\begin{aligned} \label{Transformation1} z^n=\frac{\omega+l}{\omega-l}\end{aligned}$$ the $n$-sheet Riemann surface can be mapped to a disc $|z|\leqslant 1$ which is smooth surface. The boundary x=0 corresponds to $|z|=1$. Furthermore, we can map disc to the upper half plane (UHP) $t \geqslant 0$ with the other conformal map $$\begin{aligned} \xi=-i\frac{z+1}{z-1}\label{Transformation2}.\end{aligned}$$ After two conformal maps, one can make use of well known results of 2D CFT on UHP. Finally, the variation of the Rényi entropy (\[Mainformula\]) is $$\begin{aligned} \label{result1} \Delta S_{A}^{(n)}=\frac{1}{1-n}\log \Big[\prod_{k=1}^{2n} (\frac{d\omega_k}{d\xi_k})^{-h}(\frac{d\omega_k}{d\xi_k})^{-\bar h}\frac{\langle O^{\dag}(\xi_1,\bar \xi_1)O(\xi_2,\bar \xi_2)...O^{\dag}(\xi_{2n-1},\bar \xi_{2n-1})O(\xi_{2n},\bar \xi_{2n})\rangle_{UHP}}{(\langle O^{\dag}(\omega_1,\bar \omega_1)O(\omega_2,\bar \omega_2)\rangle_{B_1})^n}\Big].\nonumber\\ \\end{aligned}$$ This is the key formula in the remainder of this paper. As a simple example, we consider the 2nd Rényi entropy firstly. These coordinates on UHP can be expressed by the original spacetime coordinate as follows. $$\begin{aligned} \label{Transformationpoint} \xi_1=-i\frac{(\frac{\omega_1+l}{\omega_1-l})^{1/2}+1}{(\frac{\omega_1+l}{\omega_1-l})^{1/2}-1},&& \xi_2=-i\frac{(\frac{\omega_2+l}{\omega_2-l})^{1/2}+1}{(\frac{\omega_2+l}{\omega_2-l})^{1/2}-1},\nonumber \\ \xi_3=-i\frac{-(\frac{\omega_1+l}{\omega_1-l})^{1/2}+1}{-(\frac{\omega_1+l}{\omega_1-l})^{1/2}-1},&& \xi_4=-i\frac{-(\frac{\omega_2+l}{\omega_2-l})^{1/2}+1}{-(\frac{\omega_2+l}{\omega_2-l})^{1/2}-1},\nonumber \\ \bar \xi_1=i\frac{(\frac{\bar \omega_1+l}{\bar \omega_1-l})^{1/2}+1}{(\frac{\bar \omega_1+l}{\bar \omega_1-l})^{1/2}-1},&& \bar \xi_2=i\frac{(\frac{\bar \omega_1+l}{\bar \omega_1-l})^{1/2}+1}{(\frac{\bar \omega_1+l}{\bar \omega_1-l})^{1/2}-1},\nonumber \\ \bar \xi_3=i\frac{-(\frac{\bar \omega_1+l}{\bar \omega_1-l})^{1/2}+1}{-(\frac{\bar \omega_1+l}{\bar \omega_1-l})^{1/2}-1},&& \bar \xi_4=i\frac{-(\frac{\bar \omega_1+l}{\bar \omega_1-l})^{1/2}+1}{-(\frac{\bar \omega_1+l}{\bar \omega_1-l})^{1/2}-1}.\end{aligned}$$ One can analyze the above formula carefully in two different time regions. When $0<t<L-l$ or $t>l+L$ with $\epsilon \rightarrow 0$, $$\begin{aligned} \label{Varianceofpoint} \xi_1-\xi_2 \simeq \frac{2\xi_1\epsilon}{\sqrt{\omega_1+l}\sqrt{\omega_1-l}},\nonumber \\ \xi_3-\xi_4 \simeq \frac{2\xi_3\epsilon}{\sqrt{\omega_1+l}\sqrt{\omega_1-l}},\nonumber \\ \bar \xi_1-\bar \xi_2 \simeq \frac{2\bar \xi_1\epsilon}{\sqrt{\bar \omega_1-l}\sqrt{\bar \omega_1+l}},\nonumber \\ \bar \xi_3-\bar \xi_4 \simeq \frac{2\bar \xi_3\epsilon}{\sqrt{\bar \omega_1-l}\sqrt{\bar \omega_1+l}},\end{aligned}$$ and the Jacobi factor of conformal transformation, $$\begin{aligned} \frac{d\xi_1}{d\omega_1}&\simeq& \frac{d\xi_2}{d\omega_2}\simeq{1\over 2 i\epsilon}\xi_{1 2},\text{ }\text{}\frac{d\bar{\xi}_1}{d\bar{\omega}_1}\simeq \frac{d\bar{\xi}_2}{d\bar{\omega}_2}\simeq -{1\over 2 i\epsilon}\bar{\xi}_{1 2}\label{DerivitiveGeneralzationfour12}\\ \frac{d\xi_3}{d\omega_3}&\simeq& \frac{d\xi_4}{d\omega_4}\simeq {1\over 2 i\epsilon}\xi_{34}, \text{ }\text{ }\frac{d\bar{\xi}_3}{d\bar{\omega}_3}\simeq \frac{d\bar{\xi}_4}{d\bar{\omega}_4}\simeq -{1\over 2 i\epsilon}\bar{\xi}_{34}\label{DerivitiveGeneralzationfour34}\\\end{aligned}$$ When $L-l<t<L+l$ with $\epsilon \rightarrow 0$, $$\begin{aligned} \label{Varianceofpoitmiddletime} \xi_4-\xi_1 \simeq \frac{2\xi_1\epsilon}{\sqrt{\omega_1+l}\sqrt{\omega_1-l}},\nonumber \\ \xi_2-\xi_3 \simeq \frac{2\xi_2\epsilon}{\sqrt{\omega_1+l}\sqrt{\omega_1-l}},\nonumber \\ \bar \xi_1-\bar \xi_2 \simeq -\frac{2\bar \xi_1\epsilon}{\sqrt{\bar \omega_1-l}\sqrt{\bar \omega_1+l}},\nonumber \\ \bar \xi_3-\bar \xi_4 \simeq -\frac{2\bar \xi_3\epsilon}{\sqrt{\bar \omega_1-l}\sqrt{\bar \omega_1+l}},\end{aligned}$$ and the Jacobi factor of conformal transformation, $$\begin{aligned} \frac{d{\xi}_1}{d{\omega}_1}&\simeq& \frac{d{\xi}_4}{d{\omega}_4}={1\over 2 i\epsilon}{\xi}_{1 4}, \text{ }\text{ }\frac{d\bar{\xi}_1}{d\bar{\omega}_1}\simeq \frac{d\bar{\xi}_2}{d\bar{\omega}_2}\simeq-{1\over 2 i\epsilon}\bar{\xi}_{1 2}\label{DerivitiveGeneralzationfour21}\\ \frac{d\xi_2}{d\omega_2}&\simeq& \frac{d\xi_3}{d\omega_3}\simeq{1\over 2 i\epsilon}\xi_{23},\text{ }\text{ }\frac{d\bar{\xi}_3}{d\bar{\omega}_3}\simeq \frac{d\bar{\xi}_4}{d\bar{\omega}_4}\simeq -{1\over 2 i\epsilon}\bar{\xi}_{34}\label{DerivitiveGeneralzationfour22}\end{aligned}$$ 2nd Rényi entropy for free boson -------------------------------- We will focus on the following local operators in the free scalar field firstly, $$\begin{aligned} \label{OperatorBoson} O_1=e^{i\phi/2}, \ \ O_2=\frac{1}{\sqrt{2}}(e^{i\phi/2}+e^{-i\phi/2}).\end{aligned}$$ The time evolution of Rényi entropy for such operators have already been studied in [@He:2014mwa] in 2D CFT on the complex plane. There are two kinds of boundary conditions for 2D free scalar field theory. One is $\frac{\partial \phi}{\partial n}|_B=0$ called Neumann boundary condition, the other is $\phi|_B=0$ called Dirichlet boundary condition. Since the boundary condition is homogenous, it is invariant under the conformal transformation.\ The image method [@CFT] [^2] is an efficient way to obtain the correlation function on UHP from correlation function on the full complex plane. The two kinds of boundary conditions correspond to different parity transformation in image method. Due to the presence of boundary, there are constraints on local conformal transformation, the anti-holomorphic and the holomorphic sectors in correlation function are no longer independent. More precisely, the correlation function on the upper half plane can be expressed by holomorphic part of conformal block on the full complex plane with including the ‘images’ of the holomorphic coordinates. That is to say, $$\begin{aligned} \langle \phi(z_1,\bar z_1)\phi(z_2,\bar z_2)...\phi(z_n,\bar z_n)\rangle_{UHP}\end{aligned}$$ equals to $$\begin{aligned} \label{4ninR2} \langle \phi(z_1)\bar \phi( {z_1}^*)\phi(z_2)\bar \phi( {z_2}^*)...\phi(z_n)\bar \phi(z_n^*)\rangle_{R^2},\end{aligned}$$ where $\phi$ and $\bar \phi$ refer to the holomorphic and anti-holomorphic part of the field $\phi$ [[^3]]{}. After a parity transformation the anti-holomorphic part become a holomorphic field with conformal dimension of the original anti-holomorphic part. In terms of image method, we should introduce parity transformation. For the free boson the parity transformation is $$\begin{aligned} \label{ParityBoson} \phi(z,\bar z)=\eta \phi(\bar z,z),\ \ \eta=\pm 1,\end{aligned}$$ $\eta=1,-1$ corresponds to the Neumann boundary condition and Dirichlet boundary condition respectively. In terms of (\[result1\]), to obtain Rényi entropy, it is necessary to know the two-point and four-point correlation function on the UHP. ### Local excitation $O_1$ {#3.2.1} Let’s consider the operator $O_1$ firstly. Using the image method, we could get the two-point function $$\begin{aligned} \label{twopointfunction} \langle O_1^{\dag}(\omega_1,\bar \omega_1)O_1(\omega_2,\bar \omega_2)\rangle_{B_1}&=&\prod_{i=1}^2(\frac{d\omega_i}{d\xi_i^{'}})^{-h}(\frac{d\bar \omega_i} {d\bar \xi_i^{'}})^{-\bar h}\langle O_1^{\dag}(\xi_1^{'},\bar \xi_1^{'})O_1(\xi_2^{'},\bar \xi_2^{'})\rangle_{UHP}\nonumber \\&=&\prod_{i=1}^2(\frac{d\omega_i}{d\xi_i^{'}})^{-h}(\frac{d\bar \omega_i}{d\bar \xi_i^{'}})^{-\bar h}\Big[\langle O_1^{\dag}(\xi_1^{'},\bar \xi_1^{'})O_1(\xi_2^{'},\bar \xi_2^{'})\tilde{O_1}^\dag (\xi_3^{'},\bar \xi_3^{'})\tilde{O_1}(\xi_4^{'},\bar \xi_4^{'})\rangle_{R^2}\Big]_{\text{holo}}\nonumber \\ &\simeq &\frac{1}{(\xi^{'}_{12}\xi^{'}_{34})^{1/4}}=\frac{1}{(4\epsilon^2)^{1/4}}.\end{aligned}$$ where $\xi_3^{'}\equiv \xi_1^{'*}$,$\xi_4\equiv \xi_2^{'*}$,$\xi_i^{'}=i\omega_i$, $h=\bar h=1/8$ and $\tilde{O_1}$ is the field with parity transformation. The subindex ‘holo’ means that we only keep the holomorphic part of the correlation and set the anti-holomorphic part to be constant [^4] which is determined by boundary condition in general. For two-point function one could normalize the field and take constant to be $1$. The 4-point correlation function could be obtained by similar procedure. $$\begin{aligned} \label{Fourpoitfunction} &&\langle O_1^{\dag}(\omega_1,\bar \omega_1)O_1(\omega_2,\bar \omega_2)O_1^{\dag}(\omega_3,\bar \omega_3)O_1(\omega_4,\bar \omega_4)\rangle_{B_2}\nonumber \\ &&=\prod_{i=1}^4(\frac{d\omega_i}{d\xi_i})^{-h}(\frac{d\bar \omega_i}{d\bar \xi_i})^{-\bar h}\langle O_1^{\dag}(\xi_1,\bar \xi_1)O_1(\xi_2,\bar \xi_2)O_1^{\dag}(\xi_3,\bar \xi_3)O_1(\xi_4,\bar \xi_4)\rangle_{UHP}\nonumber \\ &&= \prod_{i=1}^4(\frac{d\omega_i}{d\xi_i})^{-h}(\frac{d\bar \omega_i}{d\bar \xi_i})^{-\bar h}\Big[\langle O_1^{\dag}(\xi_1,\bar \xi_1)...O_1(\xi_4,\bar \xi_4)\tilde{O_1}^\dag (\xi_5,\bar \xi_5)...\tilde{O_1}(\xi_8,\bar \xi_8)\rangle_{R^2}\Big]_{\text{holo}},\nonumber \\ &&\end{aligned}$$ where $\xi_5=\bar \xi_1$, $\xi_6=\bar \xi_2$, $\xi_7=\bar \xi_3$, $\xi_8=\bar \xi_4$, the operator $\tilde{O}_1=e^{i\eta \phi}$ on $R^2$ with parity transformation. In the region $0<t<L-l$ or $t>l+L$, as (\[Varianceofpoint\]) shows, the correlation function (\[Fourpoitfunction\]) can be factorized as $$\begin{aligned} \label{factorized} &&\langle O_1^{\dag}(\xi_1,\bar \xi_1)...O_1(\xi_4,\bar \xi_4)\tilde{O_1}^\dag (\xi_5,\bar \xi_5)...\tilde{O_1}(\xi_8,\bar \xi_8)\rangle_{R^2}\nonumber \\ &&\propto\langle O_1^{\dag}(\xi_1,\bar \xi_1)O_1(\xi_2,\bar \xi_2)\rangle_{R^2}...\langle \tilde{O_1}^{\dag}(\xi_7,\bar \xi_7)\tilde{O_1}(\xi_8,\bar \xi_8)\rangle_{R^2}\end{aligned}$$ in $\epsilon\rightarrow 0$ limit. The image method leaves us with a constant C. To fix the constant C, we take the limit $\xi_1\to \xi_2$, $\bar \xi_1 \to \bar \xi_2$, $\xi_3\to \xi_4$ and $\bar \xi_3 \to \bar \xi_4$ in (\[Fourpoitfunction\]), one could find $$\begin{aligned} &&\langle O_1^{\dag}(\xi_1,\bar \xi_1)O_1(\xi_2,\bar \xi_2)O_1^{\dag}(\xi_3,\bar \xi_3)O_1(\xi_4,\bar \xi_4)\rangle_{UHP}\nonumber \\ &\simeq& \langle O_1^{\dag}(\xi_1,\bar \xi_1)O_1(\xi_2,\bar \xi_2)\rangle_{UHP} \langle O_1^{\dag}(\xi_3,\bar \xi_3)O_1(\xi_4,\bar \xi_4)\rangle_{UHP}.\label{blockCC}\end{aligned}$$ In terms of image method, we can also obtain $$\begin{aligned} &&\langle O_1^{\dag}(\xi_1,\bar \xi_1)O_1(\xi_2,\bar \xi_2)O_1^{\dag}(\xi_3,\bar \xi_3)O_1(\xi_4,\bar \xi_4)\rangle_{UHP}\nonumber \\ &=&\Big[\langle O_1^{\dag}(\xi_1,\bar \xi_1)...O_1(\xi_4,\bar \xi_4)O(\xi_2,\bar \xi_2)\tilde{O_1}^\dag (\xi_5,\bar \xi_5)...\tilde{O_1}(\xi_8,\bar \xi_8)\rangle_{R^2}\Big]_{\text{holo}}\nonumber\\ &\simeq& \frac{C }{(\xi_{12}\xi_{34}\xi_{56}\xi_{78})^{1/4}},\label{blockC}\end{aligned}$$ where $C$ is constant. Comparing (\[blockCC\]) with (\[blockC\]), we can fix $C=1$ which is consistent with the normalization of the two point correlation function. For $n=2$, the variation of the Rényi entropy (\[result1\]) is $$\begin{aligned} \label{O1result1} \Delta S^{(2)}_A&=&-\log \Big[\prod_{k=1}^{4} (\frac{d\omega_k}{d\xi_k})^{-h}(\frac{d\omega_k}{d\xi_k})^{-\bar h}\frac{\langle O^{\dag}(\xi_1,\bar \xi_1)O(\xi_2,\bar \xi_2)O^{\dag}(\xi_{3},\bar \xi_{3})O(\xi_{4},\bar \xi_{4})\rangle_{UHP}}{(\langle O^{\dag}(\omega_1,\bar \omega_1)O(\omega_2,\bar \omega_2)\rangle_{B_1})^2}\Big]\nonumber\\ &=& -\log \Big[(\frac{(\xi_{12}\xi_{34}\xi_{56}\xi_{78})^{2}}{(2\epsilon)^{8}})^{1/8}\frac{(4\epsilon^2)^{1/2}}{(\xi_{12}\xi_{34}\xi_{56}\xi_{78})^{1/4}}\Big]=0,\end{aligned}$$ where we have used (\[twopointfunction\]), (\[Fourpoitfunction\]),(\[blockC\]) and Jacobi factor (\[DerivitiveGeneralzationfour12\])(\[DerivitiveGeneralzationfour34\]). In the other region $L-l<t<L+l$, we can not factorize the correlation function as (\[factorized\]) directly. For 2D free scalar theory, the situation becomes much simpler. The correlation function could be expressed as [@CFT] $$\begin{aligned} \label{Bosoncorrelation} \langle e^{i\alpha_1\phi}...e^{i\alpha_n \phi}\rangle=\prod_{i<j}[z_{ij}^{\alpha_i\alpha_j}][\bar z_{ij}^{\alpha_i\alpha_j}],\end{aligned}$$ with the neutral condition $\alpha_1+\alpha_2+...\alpha_n=0$ and $z_{ij}=z_i-z_j, \bar{z}_{ij}=\bar{z}_i-\bar{z}_j $. Thus $$\begin{aligned} \label{8pointfunction} &&\langle O_1^{\dag}(\xi_1,\bar \xi_1)...O_1(\xi_4,\bar \xi_4)\rangle_{UHP} \nonumber\\ &=&\Big[\langle O_1^{\dag}(\xi_1,\bar \xi_1)...O_1(\xi_4,\bar \xi_4)O(\xi_2,\bar \xi_2)\tilde{O_1}^\dag (\xi_5,\bar \xi_5)...\tilde{O_1}(\xi_8,\bar \xi_8)\rangle_{R^2}\Big]_{\text{holo}}\nonumber\\ &\simeq& \frac{1}{(\xi_{41}\xi_{23}\xi_{56}\xi_{78})^{1/4}},\end{aligned}$$ where we have fix the constant to be 1. To get $\Delta S^{(2)}_A$ we only need to change $2\leftrightarrow 4$ in (\[O1result1\]). Using (\[Varianceofpoitmiddletime\])(\[twopointfunction\])(\[8pointfunction\]) and Jacobi factor (\[DerivitiveGeneralzationfour21\])(\[DerivitiveGeneralzationfour22\]), we get $$\begin{aligned} \Delta S^{(2)}_A= -\log \Big[(\frac{(\xi_{14}\xi_{32}\xi_{56}\xi_{78})^{2}}{(2\epsilon)^{8}})^{1/8}\frac{(4\epsilon^2)^{1/2}}{(\xi_{14}\xi_{32}\xi_{56}\xi_{78})^{1/4}}\Big]=0,\end{aligned}$$ in $L-l<t<L+l$. To close this subsection, one more thing should be noted that $\Delta S^{(2)}_A$ does not depend on the choice of parity transformation. Actually in our above calculation we do not use the value of $\eta$, which is related to the boundary condition. The leading order of the correlation function is same. $\Delta S^{(2)}_A$ is always zero for operator $O_1$. To see the effect of the boundary, we should consider more complicated example. ### Local excitation $O_2$ $O_2$ is a linear combination between $O_1$ and $O^{\dag}_1$. We can calculate $\Delta S^{(2)}_A$ with following the logic in previous section. The two-point correlation function of $O_2$, $$\begin{aligned} \label{2pointfunctionO2} &&\langle O^{\dag}_2(\xi_1^{'},\bar \xi_1^{'}) O_2(\xi_2^{'},\bar \xi_2^{'})\rangle_{UHP}= \frac{1}{2}\Big[\langle O^{\dag}_1(\xi_1^{'},\bar \xi_1^{'}) O^{\dag}_1(\xi_2^{'},\bar \xi_2^{'})\rangle_{UHP} +\langle O_1(\xi_1^{'},\bar \xi_1^{'})O_1(\xi_2^{'},\bar \xi_2^{'})\rangle_{UHP} \nonumber \\ &&+\langle O_1(\xi_1^{'},\bar \xi_1^{'}) O^{\dag}_1(\xi_2^{'},\bar \xi_2^{'})\rangle_{UHP}+\langle O^{\dag}_1(\xi_1^{'},\bar \xi_1^{'}) O_1(\xi_2^{'},\bar \xi_2^{'})\rangle_{UHP} \Big]\nonumber \\ &\simeq& \frac{1}{(\xi_{12}^{'})^{1/4}(\xi_{34}^{'})^{1/4}}= \frac{1}{(4\epsilon^2)^{1/4}},\end{aligned}$$ in the limit $\epsilon \to 0$. Here we have used the parity transformation (\[ParityBoson\]) related to the boundary condition. We have set anti-holomorphic parts to be $1$. The four-point correlation function on UHP is $$\begin{aligned} \label{O2operator4pointfunction} \langle O_2^{\dag}(\xi_1,\bar \xi_1)...O_2(\xi_4,\bar \xi_4)\rangle_{UHP} &&=\frac{1}{4}\Big[\langle O_1(\xi_1,\bar \xi_1)O^{\dag}_1(\xi_2,\bar \xi_2) O_1(\xi_3,\bar \xi_3)O^{\dag}_1(\xi_4,\bar \xi_4) \rangle_{UHP}\nonumber \\ &&+\langle O_1(\xi_1,\bar \xi_1)O_1(\xi_2,\bar \xi_2) O^{\dag}_1(\xi_3,\bar \xi_3)O^{\dag}_1(\xi_4,\bar \xi_4) \rangle_{UHP}\nonumber \\ &&+\langle O^{\dag}_1(\xi_1,\bar \xi_1)O_1(\xi_2,\bar \xi_2) O_1(\xi_3,\bar \xi_3)O^{\dag}_1(\xi_4,\bar \xi_4) \rangle_{UHP}\nonumber \\ &&+\langle O_1(\xi_1,\bar \xi_1)O^{\dag}_1(\xi_2,\bar \xi_2) O^{\dag}_1(\xi_3,\bar \xi_3)O_1(\xi_4,\bar \xi_4) \rangle_{UHP}\nonumber \\ &&+\langle O^{\dag}_1(\xi_1,\bar \xi_1)O_1(\xi_2,\bar \xi_2) O^{\dag}_1(\xi_3,\bar \xi_3)O_1(\xi_4,\bar \xi_4) \rangle_{UHP}\nonumber \\ &&+\langle O^{\dag}_1(\xi_1,\bar \xi_1)O^{\dag}_1(\xi_2,\bar \xi_2) O_1(\xi_3,\bar \xi_3)O_1(\xi_4,\bar \xi_4) \rangle_{UHP}\nonumber \\ &&+\langle O^{\dag}_1(\xi_1,\bar \xi_1)O^{\dag}_1(\xi_2,\bar \xi_2) O^{\dag}_1(\xi_3,\bar \xi_3)O^{\dag}_1(\xi_4,\bar \xi_4) \rangle_{UHP}\nonumber \\ &&+\langle O_1(\xi_1,\bar \xi_1)O^{\dag}_1(\xi_2,\bar \xi_2) O^{\dag}_1(\xi_3,\bar \xi_3)O^{\dag}_1(\xi_4,\bar \xi_4) \rangle_{UHP}\nonumber \\ &&+\langle O^{\dag}_1(\xi_1,\bar \xi_1)O^{\dag}_1(\xi_2,\bar \xi_2) O_1(\xi_3,\bar \xi_3)O^{\dag}_1(\xi_4,\bar \xi_4) \rangle_{UHP}\nonumber \\ &&+\langle O^{\dag}_1(\xi_1,\bar \xi_1)O_1(\xi_2,\bar \xi_2) O^{\dag}_1(\xi_3,\bar \xi_3)O^{\dag}_1(\xi_4,\bar \xi_4) \rangle_{UHP}\nonumber \\ &&+\langle O_1(\xi_1,\bar \xi_1)O_1(\xi_2,\bar \xi_2) O_1(\xi_3,\bar \xi_3)O^{\dag}_1(\xi_4,\bar \xi_4) \rangle_{UHP} \nonumber \\ &&+\langle O^{\dag}_1(\xi_1,\bar \xi_1)O^{\dag}_1(\xi_2,\bar \xi_2) O^{\dag}_1(\xi_3,\bar \xi_3)O_1(\xi_4,\bar \xi_4) \rangle_{UHP}\nonumber \\ &&+\langle O_1(\xi_1,\bar \xi_1)O_1(\xi_2,\bar \xi_2) O^{\dag}_1(\xi_3,\bar \xi_3)O_1(\xi_4,\bar \xi_4) \rangle_{UHP}\nonumber \\ &&+\langle O_1(\xi_1,\bar \xi_1)O^{\dag}_1(\xi_2,\bar \xi_2) O_1(\xi_3,\bar \xi_3)O_1(\xi_4,\bar \xi_4) \rangle_{UHP} \nonumber \\ &&+\langle O^{\dag}_1(\xi_1,\bar \xi_1)O_1(\xi_2,\bar \xi_2) O_1(\xi_3,\bar \xi_3)O_1(\xi_4,\bar \xi_4) \rangle_{UHP}\nonumber \\ &&+\langle O_1(\xi_1,\bar \xi_1)O_1(\xi_2,\bar \xi_2) O_1(\xi_3,\bar \xi_3)O_1(\xi_4,\bar \xi_4) \rangle_{UHP}\Big].\end{aligned}$$ We could use the result of four-point function of $O_1$ which has been studied in section (\[3.2.1\]). From (\[ParityBoson\])and (\[4ninR2\]) one could see that the four-point function is dependent on the boundary conditions or parity transformations. We will calculate $\Delta S^{(2)}_A$ with Neumann and Dirichlet boundary condition respectively. 1.  For the Neumann boundary condition, i.e., $\eta=1$. In (\[O2operator4pointfunction\]), terms containing equal number of $O_1$ and $O^{\dag}_1$ will survive due to the neutrality condition. Thus there are 6 terms making contribution to the 4-point correlation function. In the region $0<t<L-l$ or $t>L+l$, (\[O2operator4pointfunction\]) can be expressed by factorized form on the $R^2$. For example, the first term in (\[O2operator4pointfunction\]) as a leading term is $$\begin{aligned} &&\langle O_1(\xi_1,\bar \xi_1)O^{\dag}_1(\xi_2,\bar \xi_2) O_1(\xi_3,\bar \xi_3)O^{\dag}_1(\xi_4,\bar \xi_4) \rangle_{UHP}\nonumber \\ &=&\Big[\langle O_1(\xi_1,\bar \xi_1)O^{\dag}_1(\xi_2,\bar \xi_2) O_1(\xi_3,\bar \xi_3)O^{\dag}_1(\xi_4,\bar \xi_4)O_1(\xi_5,\bar \xi_5) O^{\dag}_1(\xi_6,\bar \xi_6)O_1(\xi_7,\bar \xi_7) O^{\dag}_1(\xi_8,\bar \xi_8)\rangle\Big]_{\text{holo}}\nonumber \\ &\simeq& \Big[\langle O_1(\xi_1,\bar \xi_1)O^{\dag}_1(\xi_2,\bar \xi_2)\rangle \langle O_1(\xi_3,\bar \xi_3)O^{\dag}_1(\xi_4,\bar \xi_4) \rangle \langle O_1(\xi_5,\bar \xi_5)O_1(\xi_6,\bar \xi_6)\rangle\nonumber \\ &&\langle O^{\dag}_1(\xi_7,\bar \xi_7)O^{\dag}_1(\xi_8,\bar \xi_8)\rangle\Big]_{\text{holo}}=\frac{1}{(\xi_{12}\xi_{34}\xi_{56}\xi_{78})^{1/4}}\sim \frac{1}{\epsilon},\end{aligned}$$ in the limit $\epsilon \to 0$. The second terms as sub-leading term is $$\begin{aligned} &&\langle O_1(\xi_1,\bar \xi_1)O_1(\xi_2,\bar \xi_2) O^{\dag}_1(\xi_3,\bar \xi_3)O^{\dag}_1(\xi_4,\bar \xi_4)\rangle_{UHP}=\Big[\langle O_1(\xi_1,\bar \xi_1)O_1(\xi_2,\bar \xi_2) \nonumber \\ &&O^{\dag}_1(\xi_3,\bar \xi_3)O^{\dag}_1(\xi_4,\bar \xi_4)O_1(\xi_5,\bar \xi_5)O_1(\xi_6,\bar \xi_6) O^{\dag}_1(\xi_7,\bar \xi_7)O^{\dag}_1(\xi_8,\bar \xi_8)\rangle_{R_2}\Big]_{\text{holo}} \nonumber \\ &\simeq& (\frac{\xi_{12}\xi_{34}...\xi_{56}\xi_{78}}{\xi_{13}\xi_{14}...\xi_{57}\xi_{58}})^{1/4}\sim \epsilon,\end{aligned}$$ in the limit $\epsilon \to 0$. Totally, there are four terms that are of order $O(\epsilon^{-1})$ in (\[O2operator4pointfunction\]). Using (\[result1\]), we get $\Delta S_A^{(n)}$ for $n=2$, $$\begin{aligned} \Delta S^{(2)}_A&\simeq &-\log \Big[(\frac{(\xi_{12}\xi_{34}\xi_{56}\xi_{78})^{2}}{(2\epsilon)^{8}})^{1/8} \frac{\langle O_2^{\dag}(\xi_1,\bar \xi_1)...O_2(\xi_4,\bar \xi_4)\rangle_{UHP}}{(\langle O^{\dag}_2(\xi_1,\bar \xi_1) O_2(\xi_2,\bar \xi_2)\rangle_{UHP})^2}\Big]\nonumber \\ &=&-\log \Big[(\frac{(\xi_{12}\xi_{34}\xi_{56}\xi_{78})^{2}}{(2\epsilon)^{8}})^{1/8}\frac{(4\epsilon^2)^{1/2}}{(\xi_{12}\xi_{34}\xi_{56}\xi_{78})^{1/4}}\Big]=0,\end{aligned}$$ In $L-l<t<L+l$ with $\epsilon \to 0$, (\[Varianceofpoitmiddletime\]) shows $\xi_{14}\sim \xi_{23}\sim \epsilon$. Terms making non-vanishing contribution to 4-point correlation function will change in the limit $\epsilon \to 0$. For example the third term in (\[O2operator4pointfunction\]) as a sub-leading term is, $$\begin{aligned} &&\langle O^{\dag}_1(\xi_1,\bar \xi_1)O_1(\xi_2,\bar \xi_2) O_1(\xi_3,\bar \xi_3)O^{\dag}_1(\xi_4,\bar \xi_4) \rangle_{UHP}\nonumber \\ &=&\Big[\langle O^{\dag}_1(\xi_1,\bar \xi_1)O_1(\xi_2,\bar \xi_2) O_1(\xi_3,\bar \xi_3)O^{\dag}_1(\xi_4,\bar \xi_4)O^{\dag}_1(\xi_5,\bar \xi_5)O_1(\xi_6,\bar \xi_6) O_1(\xi_7,\bar \xi_7)O^{\dag}_1(\xi_8,\bar \xi_8) \rangle_{R^2}\Big]_{\text{holo}}\nonumber \\ &=&(\frac{\xi_{14}\xi_{23}...\xi_{58}\xi_{67}}{\xi_{12}\xi_{34}...\xi_{56}\xi_{78}})^{1/4}\sim O(1),\end{aligned}$$ where ‘...’ stands for the terms that are $O(1)$ in the limit $\epsilon \to 0$. One could count the leading contribution term by term in (\[O2operator4pointfunction\]), there are only two terms that are of $O(\epsilon^{-1})$. Thus the variation of the Rényi entropy $\Delta S^{(n)}_{A}$ for $n=2$ is $$\begin{aligned} \Delta S^{(2)}_A&\simeq &-\log \Big[(\frac{(\xi_{14}\xi_{32}\xi_{56}\xi_{78})^{2}}{(2\epsilon)^{8}})^{1/8} \frac{\langle O_2^{\dag}(\xi_1,\bar \xi_1)...O_2(\xi_4,\bar \xi_4)\rangle_{UHP}}{(\langle O^{\dag}_2(\xi_1,\bar \xi_1) O_2(\xi_2,\bar \xi_2)\rangle_{UHP})^2}\Big]\nonumber \\ &=&-\log \Big[(\frac{(\xi_{14}\xi_{32}\xi_{56}\xi_{78})^{2}}{(2\epsilon)^{8}})^{1/8}\frac{(4\epsilon^2)^{1/2}}{2(\xi_{14}\xi_{23}\xi_{56}\xi_{78})^{1/4}}\Big]=\log 2,\end{aligned}$$ 2.  For the Dirichlet boundary condition, i.e., $\eta =-1$.\ All the terms in (\[O2operator4pointfunction\]) are non-vanishing in term of the neutrality condition. However, there are still four different terms in (\[O2operator4pointfunction\]) contributing to the leading order in the region $L+l<t$ or $t<L-l$ in $\epsilon \rightarrow 0$ limit. Thus the ratio (\[result1\]) is 1 and $\Delta S^{(2)}_A=0$. In the region $L-l<t<L+l$, there are [two]{} terms in (\[O2operator4pointfunction\]) with $\epsilon \rightarrow 0$ limit, which is the same as the situation of Neumann boundary condition. Thus the ratio (\[result1\]) is ${1\over 2}$ and $\Delta S^{(2)}_A=\log 2$. In this example we could see that $\Delta S^{(2)}_A$ does not depend on the choice of boundary conditions. n-th Rényi Entropy for free boson --------------------------------- In this subsection, we would like to generalize our studies to n-th Rényi entropy which involves in the the 2n-point correlation function on $B_n$. The conformal transformation (\[Transformation1\]) (\[Transformation2\]) can map $B_n$ to UHP. Finally one can calculate the 2n-point correlation function on UHP by the ‘method of images’ in terms of 4n-point correlation function on $R_2$. The points $\xi_1,\xi_2...\xi_{2n}$ on $B_n$ are $$\begin{aligned} \label{Npointaftermapping} &&\xi_{2k+1}=-i \frac{e^{2i\pi k/n}(\frac{\omega_1+l}{\omega_1-l})^{1/n}+1}{e^{2i\pi k/n}(\frac{\omega_1+l}{\omega_1-l})^{1/n}-1},\ \ \ \ \xi_{2k+2}=-i \frac{e^{2i\pi k/n}(\frac{\omega_2+l}{\omega_2-l})^{1/n}+1}{e^{2i\pi k/n}(\frac{\omega_2+l}{\omega_2-l})^{1/n}-1},\nonumber \\ &&\bar \xi_{2k+1}=-i \frac{e^{2i\pi k/n}(\frac{\bar \omega_1+l}{\bar \omega_1-l})^{1/n}+1}{e^{2i\pi k/n}(\frac{\bar \omega_1+l}{\bar \omega_1-l})^{1/n}-1},\ \ \ \ \bar \xi_{2k+2}=-i \frac{e^{2i\pi k/n}(\frac{\bar \omega_2+l}{\bar \omega_2-l})^{1/n}+1}{e^{2i\pi k/n}(\frac{\bar \omega_2+l}{\bar \omega_2-l})^{1/n}-1},\end{aligned}$$ where $0\le k\le n-1$. In the region $t>L+l$ or $t<L-l$, one can obtain $$\begin{aligned} \label{varianceofpoint1} &&\xi_{2k+1}-\xi_{2k+2}\simeq \frac{8ie^{2i\pi k/n} l \epsilon}{n(\omega_1-l)^{1-1/n}(\omega_1+l)^{1-1/n}[e^{2i\pi k/n}(\omega_1-l)^{1/n}-(\omega_1+l)^{1/n}]^2}\nonumber \\ &&\bar \xi_{2k+1}-\bar \xi_{2k+2}\simeq \frac{-8ie^{2i\pi k/n} l \epsilon}{n(\bar \omega_1-l)^{1-1/n}(\bar \omega_1+l)^{1-1/n}[e^{2i\pi k/n}(\bar \omega_1-l)^{1/n}-(\bar \omega_1+l)^{1/n}]},\nonumber \\ \\end{aligned}$$ and the Jacobi factor of conformal transformation, $$\begin{aligned} \frac{d\xi_{2k+1}}{d\omega_{2k+1}}&\simeq&\frac{d\xi_{2k+2}}{d\omega_{2k+2}}\simeq{1\over 2 i\epsilon}\xi_{2k+1, 2k+2}\label{DerivitiveGeneralzation11}\\ \frac{d\bar {\xi}_{2k+1}}{d\bar{\omega}_{2k+1}}&\simeq&\frac{d\bar {\xi}_{2k+2}}{d\bar{\omega}_{2k+2}}\simeq-{1\over 2 i\epsilon}\bar{\xi}_{2k+1, 2k+2}\label{DerivitiveGeneralzation12}\end{aligned}$$ In $L-l<t<L+l$, one could find $$\begin{aligned} \label{varianceofpoint2} &&\xi_{2k}-\xi_{2k+1} \simeq \frac{8 i e^{2i\pi k/n} l \epsilon }{n (-l+\omega_1)^{1-1/n}(l+\omega_1)^{1-1/n}[e^{2i\pi k/n}(\omega_1-l)^{1/n}-(\omega_1+l)^{1/n}]^2}\nonumber \\ &&\bar \xi_{2k+1}-\bar \xi_{2k} \simeq \frac{-8ie^{2i\pi k/n} l \epsilon}{n(\bar \omega_1-l)^{1-1/n}(\bar \omega_1+l)^{1-1/n}[e^{2i\pi k/n}(\bar \omega_1-l)^{1/n}-(\bar \omega_1+l)^{1/n}]}\end{aligned}$$ and the Jacobi factor of conformal transformation, $$\begin{aligned} \frac{d\xi_{2k}}{d\omega_{2k}}&\simeq&\frac{d\xi_{2k+1}}{d\omega_{2k+1}}\simeq{1\over 2 i\epsilon}\xi_{2k, 2k+1}\label{DerivitiveGeneralzation21}\\ \frac{d\bar{\xi}_{2k}}{d\bar{\omega}_{2k}}&\simeq&\frac{d\bar{\xi}_{2k+1}}{d\bar{\omega}_{2k+1}}\simeq-{1\over 2 i\epsilon}\bar{\xi}_{2k, 2k+1}\label{DerivitiveGeneralzation22}.\end{aligned}$$ ### Local excitation $O_1$ {#local-excitation-o_1} We consider the operator $O_1$ firstly. In the region $L-l<t$ or $t>L+l$ with $\epsilon \rightarrow 0$, the 2n-point correlation function of $O_1$ is $$\begin{aligned} \label{BosonGeneraliztion2nO1} &&\langle O_1^{\dag}(\xi_1,\bar \xi_1)O_1(\xi_2,\bar \xi_2)...O_1(\xi_{2n},\bar \xi_{2n})\rangle_{UHP}\nonumber \\ &&=\Big[\langle O_1^{\dag}(\xi_1,\bar \xi_1)...O_1(\xi_{2n},\bar \xi_{2n})\tilde{O}_1^{\dag}(\xi_{2n+1},\bar \xi_{2n+1})...\tilde{O}_1(\xi_{4n},\bar \xi_{4n})\rangle_{R_2}\Big]_{\text{holo}}\nonumber \\ &&\simeq \Big[\langle O_1^{\dag}(\xi_1,\bar \xi_1)O_1(\xi_{2},\bar \xi_{2})\rangle... \langle \tilde{O}_1^{\dag}(\xi_{4n-1},\bar \xi_{4n-1})\tilde{O}_1(\xi_{4n},\bar \xi_{4n})\rangle_{R_2}\Big]_{\text{holo}}\nonumber \\ &&=\frac{1}{(\xi_{12}...\xi_{2n-1,2n})^{1/4}(\xi_{2n+1,2n+2}...\xi_{4n-1,4n})^{1/4}},\end{aligned}$$ where $\tilde{O}_1$ is defined as $e^{-i\eta \phi/2}$ with parity transformation and $\xi_{2n+1}=\bar \xi_{1}$,...,$\xi_{4n}=\bar \xi_{2n}$. Using (\[result1\]), the variation of the $n$-th Rényi entropy can be obtained as follows. $$\begin{aligned} \Delta S^{(n)}_{A}&=&\frac{1}{1-n}\log \Big[\prod_{k=1}^{2n} (\frac{d\omega_k}{d\xi_k})^{-h}(\frac{d\omega_k}{d\xi_k})^{-\bar h}\frac{\langle O^{\dag}(\xi_1,\bar \xi_1)O(\xi_2,\bar \xi_2)...O^{\dag}(\xi_{2n-1},\bar \xi_{2n-1})O(\xi_{2n},\bar \xi_{2n})\rangle_{UHP}}{(\langle O^{\dag}(\omega_1,\bar \omega_1)O(\omega_2,\bar \omega_2)\rangle_{B_1})^n}\Big]\nonumber\\ &=&\frac{1}{1-n}\log \Big[ \Big(\frac{(\xi_{12}...\xi_{2n-1,2n})^{2}(\xi_{2n+1,2n+2}...\xi_{4n-1,4n})^{2}}{(2\epsilon)^{4n}}\Big)^{1/ 8} \nonumber\\&& \frac{(4\epsilon^2)^{{n/ 4}}}{(\xi_{12}...\xi_{2n-1,2n})^{1/4}(\xi_{2n+1,2n+2}...\xi_{4n-1,4n})^{1/4}}\Big]=0,\end{aligned}$$ where we have used (\[twopointfunction\])(\[varianceofpoint1\])(\[BosonGeneraliztion2nO1\]), and Jacobi factor (\[DerivitiveGeneralzation11\])(\[DerivitiveGeneralzation12\]). In the region $L-l<t<L+l$ with $\epsilon\rightarrow 0$, the $2n$-point correlation function on UHP is $$\begin{aligned} \label{BosonGeneralizationO1two} &&\langle O_1^{\dag}(\xi_1,\bar \xi_1)O_1(\xi_2,\bar \xi_2)...O_1(\xi_{2n},\bar \xi_{2n})\rangle_{UHP}\nonumber \\ &\simeq& \frac{1}{(\xi_{23}...\xi_{2n,1})^{1/4}(\xi_{2n+1,2n+2}...\xi_{4n-1,4n})^{1/4}}.\end{aligned}$$ Then, $$\begin{aligned} \Delta S^{(n)}_{A}&=&\frac{1}{1-n}\log \Big[ \Big(\frac{(\xi_{23}...\xi_{2n,1})^{2}(\xi_{2n+1,2n+2}...\xi_{4n-1,4n})^{2}}{(2\epsilon)^{4n}}\Big)^{1/ 8} \nonumber\\&& \frac{(4\epsilon^2)^{{n/ 4}}}{(\xi_{23}...\xi_{2n,1})^{1/4}(\xi_{2n+1,2n+2}...\xi_{4n-1,4n})^{1/4}}\Big]=0,\end{aligned}$$ where we have made use of (\[BosonGeneralizationO1two\]) and Jacobi factor (\[DerivitiveGeneralzationfour21\])(\[DerivitiveGeneralzationfour22\]). ### Local excitation $O_2$ For operator $O_2$, there are $2^{2n}$ terms making contribution to the correlation function like the ones in (\[O2operator4pointfunction\]). Firstly let us consider the case with Neumman boundary condition. These terms with equal number of $O_1$ and $O^{\dag}_1$ can survive in the limit $\epsilon \rightarrow 0$. The $2n$-point correlation function of $O_1$ on $B_n$ can be expressed by $4n$-point correlation function on $R^2$, $$\begin{aligned} \label{2npointfunction} &&\langle O_1^{\dag}(\xi_1,\bar \xi_1)O_1(\xi_2,\bar \xi_2)...O_1^{\dag}(\xi_{2n-1},\bar \xi_{2n-1})O_1(\xi_{2n},\bar \xi_{2n})\rangle_{UHP}\nonumber \\ &=& \Big[\langle O_1^{\dag}(\xi_1,\bar \xi_1)...O_1(\xi_{2n},\bar \xi_{2n})\tilde{O}^{\dag}_1(\xi_{2n+1},\bar \xi_{2n+1})...\tilde{O}_1(\xi_{4n},\bar \xi_{4n})\rangle\Big]_{\text{holo}}.\end{aligned}$$ To be convenient, we use the symbol $+1$ referring to $O^\dag_1$ and $-1$ referring to $O_1$ in the correlation function to simplify our analysis. Then the $2n$-point correlation function on UHP can be formally written as $$\begin{aligned} \label{Key2nBosonO22} &&\langle O_2^{\dag}(\xi_1,\bar \xi_1)O_2(\xi_2,\bar \xi_2)...O_2^{\dag}(\xi_{2n-1},\bar \xi_{2n-1})O_2(\xi_{2n},\bar \xi_{2n})\rangle_{UHP}\nonumber \\ &=&\frac{1}{2^n}\sum_{\substack{i_1,i_2,...,i_{2n}=\pm1 \\ i_{2n+1}=i_1,...i_{4n}=i_{2n}} }\langle i_1,i_2,...,i_{2n},i_{2n+1}...,i_{4n} \rangle_{R^2}, \end{aligned}$$ where we have made use of the image method in the second line and $i_j=\pm 1$ stands for the operator $O_1$ or $O^{\dag}_1$ with the coordinate ($\xi_j$,$\bar \xi_j$). The constraints ${i_{2n+j}}={i_{j}}$ with $ 1\le j \le 2n$ corresponds to the Neumann boundary condition in terms of image method. For the correlation function in the Neumann boundary condition case, the non-zero terms in (\[Key2nBosonO22\]) should satisfy neural condition $$\begin{aligned} \label{Neumannconstraint} i_1+i_2+...+i_{2n}+...i_{4n}=0.\end{aligned}$$ In the region $t<L-l$ or $t>L+l$ with the limit $\epsilon\rightarrow 0$, due to (\[varianceofpoint1\]), the leading contribution of the 2n-point correlation function (\[Key2nBosonO22\]) are $$\begin{aligned} \label{free1st2n} &&\sum_{\substack {i_1+i_2=0,i_3+i_4=0 \\...i_{4n-1}+i_{4n}=0 \\ {i_{2n+1}}={i_{1}}...{i_{4n}}={i_{2n}}}} \langle i_1,i_2,...,i_{2n},i_{2n+1}...,i_{4n} \rangle_{R^2}\nonumber \\ &&\simeq 2^n \langle i_1,i_2\rangle_{R^2}...\langle i_{2n-1},i_{2n}\rangle_{R^2} \langle i_{2n+1},i_{2n+2}\rangle_{R^2} ...\langle i_{4n-1},i_{4n} \rangle_{R^2},\end{aligned}$$ there are $2^n$ terms that are leading divergence after considering the constraints, and these terms are all equal to each other. Equivalently, (\[Key2nBosonO22\]) can be written by the notation $O_{1}$ and $O^{\dag}_1 $ as $$\begin{aligned} &&2^n \langle O_1^\dag(\xi_{1}) O_1(\xi_{2})\rangle_{R^2} ...\langle O_1^\dag(\xi_{2n-1}) O_1(\xi_{2n})\rangle_{R^2}...\langle O_1^\dag(\xi_{4n-1}) O_1(\xi_{4n})\rangle_{R^2}\nonumber \\ &=&\frac{2^n}{(\xi_{12}...\xi_{2n-1,2n})^{1/4}(\xi_{2n+1,2n+2}...\xi_{4n-1,4n})^{1/4}}.\end{aligned}$$ Using (\[Key2nBosonO22\]), the $2n$-point correlation function on $B_{2n}$ is $$\begin{aligned} &&\langle O_2^{\dag}(\xi_1,\bar \xi_1)O_2(\xi_2,\bar \xi_2)...O_2^{\dag}(\xi_{2n-1},\bar \xi_{2n-1})O_2(\xi_{2n},\bar \xi_{2n})\rangle_{UHP}\nonumber \\ &\simeq& \frac{1}{(\xi_{12}...\xi_{2n-1,2n})^{1/4}(\xi_{2n+1,2n+2}...\xi_{4n-1,4n})^{1/4}},\end{aligned}$$ which is same as the $2n$-point correlation function of $O_{1}$ in $t<L-l$ or $L+l<t$. Thus the variation of the $n$-th Rényi entropy in the CFT with Neumman boundary condition is $$\begin{aligned} \label{EarlytimeresultinO2General} \Delta S_{A}^{(n)}=0.\end{aligned}$$ In $L-l<t<L+l$ with the limit $\epsilon\rightarrow 0$, due to (\[varianceofpoint2\]) the leading contribution of (\[Key2nBosonO22\]) should satisfy following constraints $$\begin{aligned} \label{constraint1} i_{2n+1}+i_{2n+2}=0,\ \ i_{2n+3}+i_{2n+4}=0,...,\ \ i_{4n-1}+i_{4n}=0.\end{aligned}$$ Combining with ${i_{2n+j}}={i_{j}}$ and (\[constraint1\]), one can obtain $$\begin{aligned} \label{constraint2} i_{1}+i_{2}=0,\ \ i_{3}+i_{4}=0,...,\ \ i_{2n-1}+i_{2n}=0.\end{aligned}$$ In terms of (\[varianceofpoint2\]), the leading terms of (\[Key2nBosonO22\]) should also satisfy following constraints $$\begin{aligned} \label{constraint3} i_{2}+i_3=0,\ \ i_{4}+i_5=0,...,\ \ i_{2n}+i_{1}=0.\end{aligned}$$ With these constraints (\[constraint1\])(\[constraint2\])(\[constraint3\]), the correlation function is $$\begin{aligned} &&\langle O_2^{\dag}(\xi_1,\bar \xi_1)O_2(\xi_2,\bar \xi_2)...O_2^{\dag}(\xi_{2n-1},\bar \xi_{2n-1})O_2(\xi_{2n},\bar \xi_{2n})\rangle_{UHP}\nonumber \\ &\simeq& \frac{2}{2^{n}}\frac{1}{(\xi_{23}...\xi_{2n,1})^{1/4}(\xi_{2n+1,2n+2}...\xi_{4n-1,4n})^{1/4}}\label{freen}.\end{aligned}$$ Putting (\[freen\]) into (\[result1\]) with considering Jacobi factor (\[DerivitiveGeneralzationfour21\])(\[DerivitiveGeneralzationfour22\]), the variation of the $n$-th Rényi entropy is $$\begin{aligned} \label{LatertimeresultO2General} \Delta S_{A}^{(n)}=\log 2.\end{aligned}$$ In CFT with the Dirichlet boundary, the only difference with the Neumann boundary condition is the constraints $i_{2n+j}=i_{j}$ $\to$ $i_{2n+j}=-i_{j}$. One could check that the leading order correlation function is same as the Neumann boundary condition. Thus the $n$-th Rényi entropy is not dependent on the choice of boundary condition. We do not repeat here. Rényi Entropy in Ising model ---------------------------- It is natural to ask how about the Ising model which is simplest unitary minimal model. There are three kinds of primary operators, i.e., the identity $\bm{\mathbb{I}}$, the spin operator $\bm{\sigma}$ and the energy operator $\bm{\epsilon}$. There are also two kinds of parity transformation which involve in two kinds of boundary conditions, which correspond to two different parity transformations. One of the parity transformation [@CFT] is $$\begin{aligned} \label{parityIsing1} \sigma(z,\bar z)\to \sigma(\bar z,z),\ \ \ \ \ \mu(z,\bar z)\to \mu(\bar z,z),\end{aligned}$$\[parityIsing2\] where $\bm{\mu}$ is the disorder operator. The other parity transformation [@CFT] is $$\begin{aligned} \sigma(z,\bar z)\to \mu(\bar z,z),\ \ \ \ \ \mu(z,\bar z)\to \sigma(\bar z,z).\end{aligned}$$ We would like to study the local excitation by the spin operator $\bm{\sigma}$ with conformal dimension ($h=\frac{1}{16}$, $\bar h=\frac{1}{16}$) in the same setup given in section (\[3.1\]). The variation of the Rényi entropy for the subsystem $-l<x<0$ is given by (\[result1\]). The two-point correlation function $$\begin{aligned} &&\langle\sigma^{\dag}(\omega_1,\bar \omega_1)\sigma(\omega_2,\bar \omega_2)\rangle_{B_1}=\prod_{i=1}^2(\frac{d\omega_i}{d\xi_i^{'}})^{-h}(\frac{d\bar \omega_i} {d\bar \xi_i^{'}})^{-\bar h}\langle \sigma^{\dag}(\xi_1^{'},\bar \xi_1^{'})\sigma(\xi_2^{'},\bar \xi_2^{'})\rangle_{UHP}\nonumber \\ &=&\prod_{i=1}^2(\frac{d\omega_i}{d\xi_i^{'}})^{-h}(\frac{d\bar \omega_i} {d\bar \xi_i^{'}})^{-\bar h}\langle \sigma^{\dag}(\xi^{'}_1) \sigma(\xi^{'}_2) \tilde{\sigma}^{\dag}(\xi^{'}_3) \tilde{\sigma}(\xi^{'}_4) \rangle_{R_2}\end{aligned}$$ where $\xi_3^{'}\equiv \xi_1^{'*}$,$\xi_4\equiv \xi_2^{'*}$,$\xi_i^{'}=i\omega_i$, $\tilde{\sigma}$ is the field with parity transformation. The 2-point correlation functions on UHP have already obtained in literature, e.g., [@Cardy1][@Dotsenko:1984nm][@Dotsenko:1984ad], $$\begin{aligned} \label{IsingTwopointfunction} &&\langle \sigma^{\dag}(\xi^{'}_1) \sigma(\xi^{'}_2) \sigma^{\dag}(\xi^{'}_3) \sigma(\xi^{'}_4) \rangle_{R_2}= (\frac{\xi^{'}_{13}\xi^{'}_{24}}{\xi^{'}_{12}\xi^{'}_{23}\xi^{'}_{14}\xi^{'}_{34}})^{\frac{1}{8}}F_{+}(x^{'}),\nonumber \\ &&\langle \sigma^{\dag}(\xi^{'}_1) \sigma(\xi^{'}_2) \mu^{\dag}(\xi^{'}_3) \mu(\xi^{'}_4) \rangle_{R_2}= (\frac{\xi^{'}_{13}\xi^{'}_{24}}{\xi^{'}_{12}\xi^{'}_{23}\xi^{'}_{14}\xi^{'}_{34}})^{\frac{1}{8}}F_{-}(x^{'}),\end{aligned}$$ with conformal blocks [@BPZ] $$\begin{aligned} \label{F} &&F_{+}(x^{'})=\sqrt{\sqrt{1-x^{'}}+1}+\sqrt{\sqrt{1-x^{'}}-1},\text{ }\text{ }\text{and}\nonumber \\ &&F_{-}(x^{'})=\sqrt{\sqrt{1-x^{'}}+1}-\sqrt{\sqrt{1-x^{'}}-1},\end{aligned}$$ where $x^{'}$ is the conformal cross ratio $x^{'}=\xi^{'}_{12}\xi^{'}_{34}/\xi^{'}_{13}\xi^{'}_{24}$. $F_{+}(x^{'})$ and $F_{-}(x^{'})$ correspond to different boundary conditions respectively. The leading behavior of the 2-point correlation function in $\epsilon \rightarrow 0$ is $$\begin{aligned} \label{2pointising} \langle\sigma^{\dag}(\omega_1,\bar \omega_1)\sigma(\omega_2,\bar \omega_2)\rangle_{B_1}\simeq \frac{\sqrt{2}}{(4\epsilon^2)^{1/8}}.\end{aligned}$$ (\[2pointising\]) for both boundary conditions. In $t>L+l$ or $t<L-l$ with $\epsilon \rightarrow 0$, the leading behavior of 4-point correlation function is $$\begin{aligned} \label{4pointIsing} \langle \sigma^{\dag}(\xi_1,\bar \xi_1)\sigma(\xi_2,\bar \xi_2) \sigma^{\dag}(\xi_3,\bar \xi_3)\sigma(\xi_4,\bar \xi_4) \rangle_{UHP}&\simeq& \langle \sigma^{\dag}(\xi_1,\bar \xi_1)\sigma(\xi_2,\bar \xi_2)\rangle_{UHP}\langle \sigma^{\dag}(\xi_3,\bar \xi_3)\sigma(\xi_4,\bar \xi_4) \rangle_{UHP}\nonumber\\ &=& \frac{2}{(\xi_{12}\xi_{56})^{1/8}(\xi_{34}\xi_{78})^{1/8}},\nonumber\\\end{aligned}$$ where we have used the 2-point function of Ising model on UHP (\[IsingTwopointfunction\]). In terms of (\[result1\]), $\Delta S_{A}^{(2)}$ is $$\begin{aligned} \label{Isingresultone1} \Delta S_{A}^{(2)}&\simeq& -\log \Big[(\frac{(\xi_{12}\xi_{34}\xi_{56}\xi_{78})^{2}}{(2\epsilon)^{4}})^{1/16} \frac{\langle \sigma_2^{\dag}(\xi_1,\bar \xi_1)...\sigma_2(\xi_4,\bar \xi_4)\rangle_{UHP}}{(\langle \sigma^{\dag}_2(\xi_1,\bar \xi_1) \sigma_2(\xi_2,\bar \xi_2)\rangle_{UHP})^2}\Big]\nonumber \\ &=&-\log \Big[(\frac{(\xi_{12}\xi_{34}\xi_{56}\xi_{78})^{2}}{(2\epsilon)^{4}})^{1/16} \frac{(4\epsilon^2)^{1/8}}{(\xi_{12}\xi_{34}\xi_{56}\xi_{78})^{1/8}}\Big]=0,\end{aligned}$$ In $L-l<t<L+l$ with the limit $\epsilon \rightarrow 0$, the 4-point correlation function (\[4pointIsing\]) on UHP can not be factorized directly. We also use the image method, which states that the 4-point correlation function on UHP can be expressed as linear combination of the holomorphic part of conformal blocks of the 8-point correlation function on the full complex plane, with coordinates $\xi_1...\xi_8$. As we know in 2 dimension full complex plane, there are $(n-3)$ independent cross ratios for $n$-point correlation function. In our case, there are 5 independent cross ratios. Thus the 4-point correlation function on UHP can be expressed by conformal blocks [@BPZ] $$\begin{aligned} \label{UHP-Ising} &&\langle \sigma^{\dag}(\xi_1,\bar \xi_1)\sigma(\xi_2,\bar \xi_2) \sigma^{\dag}(\xi_3,\bar \xi_3)\sigma(\xi_4,\bar \xi_4) \rangle_{UHP}\nonumber\\&=&(\frac{\xi_{13}\xi_{24}}{\xi_{12}\xi_{23}\xi_{14}\xi_{34}})^{\frac{1}{8}}\sum_b A^bC^b \mathcal{F}[b;x,x_1,x_2,...,x_4],\end{aligned}$$ where we define the 5 independent conformal ratios $x,x_1,...,x_4$, i.e. $x=\xi_{12} \xi_{34}/\xi_{13}\xi_{24}$, $x_1=\xi_{14} \xi_{\bar{1}\bar{2}}/\xi_{1\bar{1}}\xi_{4\bar{2}},x_2=\xi_{1\bar{1}} \xi_{\bar{2}\bar{3}}/\xi_{1\bar{2}}\xi_{\bar{1}\bar{3}},x_3=\xi_{2\bar{1}} \xi_{\bar{2}\bar{4}}/\xi_{2\bar{2}}\xi_{\bar{1}\bar{4}}, x_4=\xi_{3\bar{2}} \xi_{\bar{3}\bar{4}}/\xi_{3\bar{3}}\xi_{\bar{2}\bar{4}}$, $b$ in conformal blocks $\mathcal{F}$ runs over all the intermediate conformal families and the coefficients $A^p$ are determined by boundary conditions. Here we define $\xi_{\bar{i}}=\bar{\xi}_{i}$. The conformal blocks [@BPZ] satisfy the fusion transformation under the braiding operation [@Moore:1988uz][@Verlinde:1988sn][@Lewellen:1991tb], i.e., $$\begin{aligned} \mathcal{F}[b;x,x_1,x_2,...,x_4]=F_{bc} \mathcal{F}[c;1-x,x_1,x_2,x_3,x_4],\end{aligned}$$ $F_{bc}$ is the fusion matrix. Making the fusion is equal to $\xi_{2}\leftrightarrow \xi_4$. In summary, the leading divergence of the correlation function is related to $c=0$ as identity as follows $$\begin{aligned} (\ref{UHP-Ising})=\sum_{b,c} F_{bc}(\frac{\xi_{13}\xi_{42}}{\xi_{14}\xi_{43}\xi_{12}\xi_{23}})^{\frac{1}{8}}A^b C^b \mathcal{F}[c;1-x,x_1,x_2,...,x_4].\end{aligned}$$ In terms of (\[Varianceofpoitmiddletime\]), one can find leading contribution of express (\[UHP-Ising\]) as following form in $L-l<t<L+l$ with $\epsilon \rightarrow 0$ $$\begin{aligned} (\ref{UHP-Ising})\simeq F_{00}\langle \sigma^{\dag}(\xi_1,\bar \xi_1)\sigma(\xi_4,\bar \xi_2) \rangle_{UHP} \langle \sigma^{\dag}(\xi_3,\bar \xi_3)\sigma(\xi_2,\bar \xi_4) \rangle_{UHP},\end{aligned}$$ where $0$ stands for the identity operator. Then the variation of the Rényi entropy for $n=2$ $$\begin{aligned} \Delta S^{(2)}_{A}=-\log F_{00}=\log \sqrt{2},\end{aligned}$$ where we have used the fact $F_{00}=\frac{1}{\sqrt{2}}$ [@Moore:1988ss] in two-dimensional Ising model. Note that the Rényi entropy does not depend on the choice of boundary condition in the Ising model, since the leading behavior of $F_{+}$ is the same as one of $F_{-}$ with $\epsilon \rightarrow 0$. One alternative way to understand the phenomenon is to make use of diagram representation of conformal block as fig.\[\[f4-T-4\]\]. =8.5 cm =1.5 cm Since the behavior of coordinates $\bar \xi_j$ does not change when $L-l<t<L+l$, we only need one time fusion transformation, which is different with [@He:2014mwa]. In the fig.\[\[f4-T-4\]\], $$\begin{aligned} F_{00}[\sigma]=F_{00}\left[ \begin{array}{cc} \sigma & \sigma \\ \sigma & \sigma \\ \end{array} \right].\end{aligned}$$ In $L-l<t<L+l$ with $\epsilon \rightarrow 0$, the leading contribution of (\[UHP-Ising\]) originates from the above conformal block involving in identity operator. Because the fusion factor $F_{00}$ can not be canceled in the ratio (\[result1\]), $\Delta S^{(2)}_{A}=-\log F_{00}=\log \sqrt{2}$ [@Moore:1988ss]. Rényi Entropy in General Rational CFTs -------------------------------------- In this subsection, we would like to generalize the analysis to the rational CFTs with a boundary in our previous set-up shown in subsection (\[3.1\]). In terms of (\[result1\]), we should know the 2-point correlation on $B_1$ and $2n$-point correlation fuction on UHP as usual. In generic rational CFTs with a boundary, the 2-point correlation can be expressed as $$\begin{aligned} \label{2pointfucntiongeneral} &&\langle O^{\dag}(\omega_1,\bar \omega_1)O(\omega_2,\bar \omega_2)\rangle_{B_1}=\langle O^{\dag}(\xi_1^{'},\bar \xi^{'}_1)O(\xi^{'}_2,\bar \xi^{'}_2)\rangle_{UHP}\nonumber \\ &=&\frac{1}{(\xi^{'}_{13}\xi^{'}_{24})^{2h}}\sum_b A^b C^b \mathcal{F}[b;\xi^{'}],\end{aligned}$$ where $A^b$ are constants which are determined by the boundary condition, $\xi^{'}_3$, $\xi^{'}_4$, $\xi^{'}$ are given in (\[twopointfunction\]). 4-point correlation function [@Dotsenko:1984nm][@Dotsenko:1984ad] of a primary function $O$ on the $R^2$, which can be expressed by $$\begin{aligned} \label{genericfourpoint} \langle O^{\dag}(z_1,\bar z_1) O(z_2,\bar z_2) O^{\dag} (z_3,\bar z_3)O (z_4,\bar z_4) \rangle_{R_2}=\sum_b\frac{1}{(z_{13}z_{24})^{2h}} C^b \mathcal{F}[b;z]\times c.c.\end{aligned}$$ In the limit $z\to 0$ $$\begin{aligned} \label{Blocksim} \mathcal{F}[b;z]=z^{h_b-2h}+...,\end{aligned}$$ where “..." stands for higher order terms and $z={z_{12}z_{34}\over z_{13}z_{24}}$ and $h_b$ is conformal dimension of primary class $b$. In terms of (\[Blocksim\]), (\[2pointfucntiongeneral\]) with taking $\xi^{'}\to 0$ is $$\begin{aligned} \langle O^{\dag}(\omega_1,\bar \omega_1)O(\omega_2,\bar \omega_2)\rangle_{B_1}&=&\frac{1}{(\xi^{'}_{13}\xi^{'}_{24})^{2h}} A^0 C^0 {\xi^{'}}^{-2h} \simeq \frac{A^0 C^0}{(4\epsilon^2)^{2h}},\end{aligned}$$ In $t>L+l$ or $t<L-l$ with $\epsilon \rightarrow 0$, the 2n-point correlation function on UHP is $$\begin{aligned} \label{2npointfunction} &&\langle O^{\dag}(\xi_1,\bar \xi_1)O(\xi_2,\bar \xi_2)...O^{\dag}(\xi_{2n-1},\bar \xi_{2n-1})O(\xi_{2n},\bar \xi_{2n})\rangle_{UHP}\nonumber \\ &\simeq& \langle O^{\dag}(\xi_1,\bar \xi_1)O(\xi_2,\bar \xi_2)\rangle_{UHP} ...\langle O^{\dag}(\xi_{2n-1}, \bar \xi_{2n-1})O(\xi_{2n},\bar \xi_{2n})\rangle_{UHP}.\end{aligned}$$ The 2-point correlation on UHP is $$\begin{aligned} \label{2pointfunctionafterfactor} \langle O^{\dag}(\xi_{2k-1},\bar \xi_{2k-1})O(\xi_{2k},\bar \xi_{2k})\rangle_{UHP}\simeq \frac{A^0C^0}{(\xi_{2k-1,2k}\xi_{2n+2k-1,2n+2k})^{2h}},\end{aligned}$$ where $1\le k\le n$, $\xi_{2n+2k-1}\equiv \bar \xi_{2k-1}$. Taking (\[2npointfunction\])(\[2pointfunctionafterfactor\]) (\[varianceofpoint1\])(\[2pointfucntiongeneral\])and Jacobi factor (\[DerivitiveGeneralzation11\])(\[DerivitiveGeneralzation12\])into (\[result1\]), one can obtain $$\begin{aligned} \label{genralresult1} \Delta S_{A}^{(n)}=0.\end{aligned}$$ In $L-l<t<l+L$ with $\epsilon \rightarrow 0$, the $2n$-point correlation function on UHP could be written as a linear combination of the holomorphic conformal blocks of the $4n$-point correlation function. We make the following fusions [@Moore:1988uz][@Verlinde:1988sn][@Lewellen:1991tb] $$\begin{aligned} &&(\xi_1,\xi_2)(\xi_3,\xi_4)...(\xi_{4n-1},\xi_{4n})\to(\xi_2,\xi_3)(\xi_1,\xi_4)...(\xi_{4n-1},\xi_{4n})\nonumber \\ &\to& (\xi_2,\xi_3)(\xi_4,\xi_5)(\xi_1,\xi_{6})...\to...\to (\xi_2,\xi_3)...(\xi_1,\xi_{2n})...(\xi_{4n-1},\xi_{4n}),\end{aligned}$$ where $\bar \xi_{2n+i}=\xi_i$, $0\le i\le 2n$. With using $n-1$ times fusion transformation as the one shown in fig.\[\[f4-T-4\]\], we get $$\begin{aligned} &&\langle O^{\dag}(\xi_1,\bar \xi_1)O(\xi_2,\bar \xi_2)...O^{\dag}(\xi_{2n-1},\bar \xi_{2n-1})O(\xi_{2n},\bar \xi_{2n})\rangle_{UHP}\nonumber \\ &\simeq& F_{00}[O]^{n-1} \langle O^{\dag}(\xi_1,\bar \xi_1)O(\xi_{2n},\bar \xi_2)\rangle_{UHP} ...\langle O^{\dag}(\xi_{2n-2}, \bar \xi_{2n-1})O(\xi_{2n-1},\bar \xi_{2n})\rangle_{UHP},\end{aligned}$$ We could calculate $\Delta S^n_A$ by using (\[result1\]). The variation of the $n$-Rényi entropy is $$\begin{aligned} \label{genralresult2} \Delta S_{A}^{(n)}=-\log F_{00}=\log d_{O},\end{aligned}$$ where $d_O$ is the quantum dimension [@Moore:1988ss] of operator $O$. (\[genralresult1\]) and (\[genralresult2\]) are same as the case in rational CFTs living on the full complex plane. We apply the fusion rule of conformal block of $2n$-point function on UHP to obtain the $\Delta S^{(n)}_A$. As we know, the $2n$-point correlation function on UHP can be expressed by linear combination of chiral part of conformal block associated with $4n$-point correlation on full plane. In this subsection, we do not make use of parity transformation containing the boundary information. The boundary data has been encoded in the coefficient of conformal block in this subsection, i.e., $A^p$ in (\[genericfourpoint\]). We could see that $\Delta S^n_A$ also does not depend on the choice of boundary. Conclusion and Discussion ========================= In this paper, we have investigated two kinds of effects on the locally excited states with time evolution. In the first case, we study the locally excited states with thermal effect in low temperature system. We have figured out the thermal correlation which is the same as [@Cardy:2014jwa] for the short interval limit. The Rényi entropy is equal to summation over the logarithmic of quantum dimension and thermal entropy in low temperature during the time $L-l<t<L+l$. In this paper, we just only confirm that such kind of sum rule is only true for the short interval $l$ in the low temperature limit. We make use of different approach [@Herzog:2012bw] to obtain the thermal correction to Rényi entropy which can be reduced to [@Caputa:2014eta] in low temperature. One can also calculate the Rényi entropy in the large interval[@Cardy:2014jwa] limit, the higher temperature limit[@Chen:2014hta] as well as beyond the leading order of the perturbation (\[Expand\]). We expected the sum rule relation is still hold in those cases. But one should note that we actually do not consider the back-reaction of the locally excited states to the thermal environment. When the energy of the local excitation is much lower than the thermal environment, it is safe to ignore the back-reaction. But in some special situation we expect the sum rule will break down. It is an interesting topic to consider in the future. In the second case, we have studied the Rényi entropy of local excited states in 2 dimensional CFTs with a boundary. For 2D CFTs with a boundary, to obtain Rényi entropy can be converted to obtain the correlation function on UHP by using of conformal transformation technique. As a warm up, the Rényi entropy has been calculated with help of image method in the 2D free field theory with a boundary. The Rényi entropy is vanishing for operator $O_2$ (\[OperatorBoson\]) in $t>L+l$ or $t<L-l$ and $\log 2$ in $L-l<t<l+L$, which is the same as previous study [@He:2014mwa] in full complex plane without boundary. To confirm this fact, the Rényi entropy have been calculated in Ising model and more generic rational CFTs. Although the correlation function and conformal blocks in 2D CFTs with boundary are totally different from the ones in 2D CFT without boundary, we get a same maximal value of the Rényi entropy for the rational CFTs without a boundary[@He:2014mwa]. In generic 2D rational CFTs with a boundary, we confirm that the maximal value of Rényi entropy is the same as the one in 2D rational CFTs without boundary. [@Jackson:2014nla] also try to understand the fact which is not contract with that the left- and right-moving chiral sectors are decoupled. [@Jackson:2014nla] generalize the result in [@He:2014mwa] to irrational CFT, for example, Liouvile CFT. They found that the left-right entanglement entropy saturates the Cardy entropy. In terms of standard view, the Cardy entropy counts the microscopic entropy of actual CFT spectrum. The Cardy entropy seems to suggest two chiral sectors are decoupled. The authors in [@Jackson:2014nla] proposed a pragmatic point of view to reconcile [@He:2014mwa] with the fact that there should also be a comparably large EE between the two chiral sectors of CFT. For example, Non-chiral local operators will be left-right entangled. In BCFTs, the two chiral sectors are no longer independent. We have shown some additional examples to confirm the pragmatic perspective. For general rational CFTs in 2D, the Rényi entropy highly relys on the conformal blocks of the theory. The $n$-point correlation functions in 2D CFTs with boundary are related to the holomorphic part of conformal blocks of the $2n$-point correlation functions on the 2D full complex plane. This relationship had been studied by the image method [@Cardy1][@Cardy2] very well. More precise relation is that an $n$-point function in the UHP, which is a function of the coordinates $(z_1,,z_n; \bar{z}_1,...,\bar{z}_n)$ behaves under conformal transformations in the same way as the holomorphic factor of a $2n$-point function in the full plane which depends on $(z_1,...,z_n; z_1^*,...,z_n^*)$, analytically continued to $z_j^*=\bar{z_j}$. In [@He:2014mwa], the time evolution of Rényi entropy highly depends on the holomorphic part of conformal block. In 2D CFTs with a boundary, the boundary changes the evolution of the Rényi entropy but does not change the value of the Rényi entropy, which is closely related to fusion constants in the bulk. Because the behavior of the ‘image’ coordinates (anti-holomorphic coordinates) does not change as the holomorphic coordinates when $L-l<t<L+l$, we get the same Rényi entropy as [@He:2014mwa]. The boundary introduced here works as the infinite potential barrier for the time evolution of the entangled quasi-particles pairs [@Jean-Marie; @Stphan][@Nozaki:2014hna][@He:2014mwa] triggered by local excitation as shown in fig.\[\[fig2\]\]. =12.0 cm =4.0 cm The the Rényi entropy measures the entanglement between the quasi-particles generated by local excitation. After entangled pairs are created at $-L$, the two quasi-particles will propagate in two opposite directions, i.e., left-moving and right moving. When the right-moving particle enter the interval $-l<x<0$ denoted by $A$, the Rényi entropy takes maximal value due to entanglement between two entangled particles. In fig.\[\[fig2\]\], the blue wave lines mimic the entanglement of two entangled quasi-particles. When the right-moving quasi-particle reaches the boundary, the quasi-particle will be reflected by the boundary without losing energy. As the calculation in [@Caputa:2014vaa] shows the locally excited states carry the energy of $O(\epsilon^{-1})$. The conformal transformation, $z\to z+\epsilon(z)$ and $\bar z\to \bar z+\bar \epsilon(\bar z)$, should keep the boundary conformal invariant, which lead to the constraint $T=\bar T$ on the boundary. In the Cartesian coordinates, the constraint becomes $T_{xt}=0$, which means that no energy can flow across the boundary. This is main reason the quasi-particle must be reflected no matter what is the conformal invariant boundary condition. In this sense, the boundary change the time evolution of Rényi entropy.In our paper we take the scale $\epsilon$ as the minimal scale, and keep the leading order of $\epsilon$ in the calculation. So we must miss some information when $t\sim L$, i.e., the quasi-particle is close to the boundary. 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Here we only want to express that the $n$-point correlation functions on UHP which are dependent on the holomorphic conformal blocks of $2n$-point correlation functions on the full complex plane. In 2D free field theory, the conformal blocks of the operator $O_1$ is trivial. In this paper, we will also show the image method in 2D Ising model and other generic 2D CFTs. [^4]: We can use this rule in 2D free scalar field theory, since the conformal blocks related $O_1$ are actually trivial. We would like to appreciate communication with Cardy on this point.

--- abstract: 'Gamma-ray burst (GRB) afterglows are well described by synchrotron emission originating from the interaction between a relativistic blast wave and the external medium surrounding the GRB progenitor. We introduce a code to reconstruct spectra and light curves from arbitrary fluid configurations, making it especially suited to study the effects of fluid flows beyond those that can be described using analytical approximations. As a check and first application of our code we use it to fit the scaling coefficients of theoretical models of afterglow spectra. We extend earlier results of other authors to general circumburst density profiles. We rederive the physical parameters of GRB 970508 and compare with other authors.' author: - | H.J. van Eerten$^{1}$[^1] and R.A.M.J. Wijers$^{1}$\ $^{1}$Astronomical Institute ’Anton Pannekoek’, Kruislaan 403, 1098 SJ Amsterdam, the Netherlands date: 'Accepted . Received ; in original form ' title: 'Gamma-Ray Burst afterglow scaling coefficients for general density profiles' --- \[firstpage\] gamma rays: bursts – gamma rays: theory – plasmas – radiation mechanisms: nonthermal – shockwaves Introduction ============ In the fireball model, Gamma-Ray Burst (GRB) afterglows are thought to be the result of synchrotron radiation generated by electrons during the interaction of a strongly collimated relativistic jet from a compact source with its environment (for recent reviews, see @Piran2005 [@Meszaros2006]). Initially the resulting spectra and light curves have been modelled using only the shock front of a spherical explosion and a simple power law approximation for the synchrotron radiation (e.g. @Wijers1997 [@Meszaros1997; @Sari1998; @Rhoads1999]). One or more spectral and temporal breaks were used to connect regimes with different power law slopes. For the dynamics the self similar approximation of a relativistic explosion was used [@Blandford1976]. These models have been refined continuously. More details of the shock structure were included (e.g. @Granot1999 [@Gruzinov1999]), more accurate formulae for the synchrotron radiation were used (e.g. @Wijers1999) and efforts have been made to implement collimation using various analytical approximations to the jet structure and lateral spreading behaviour (see @Granot2005 for an overview). On top of that, there have been studies focussing on arrival time effects (e.g. @Huang2007) and some numerical simulations (e.g. @Salmonson2003 [@Granot2001; @Nakar2007]). The aim of this paper is twofold. The first aim is to introduce a new method to derive light curves and spectra by post-processing relativistic hydrodynamic (RHD) jet simulations of arbitrary dimension, properly taking into account all beaming and arrival time effects, as well as the precise shape of the synchrotron spectrum and electron cooling (in this paper we will ignore self-absorption, although it can in principle be included in our method). This is done in sections \[peak\_section\] and \[cooling\_section\]. The second aim is to present a set of scaling coefficients for the slow-cooling case for a density profile $\rho = \rho_0 \cdot (R/R_0)^{-k}$ for general values of $k$. Fits to afterglow data using $k$ as a free fitting parameter have yielded values markedly different from both $k=0$ and $k=2$ [@Starling2007], although with error bars not excluding either option. The scaling coefficients have been obtained by application of our post-process code not to a full hydrodynamic simulation but to an emulation of this. From the spherical Blandford & McKee (BM) analytical solution for the blast wave for the impulsive energy injection scenario, snapshots containing the state of the fluid at given emission times were constructed and stored to provide the input for the post-process code. The use of the BM solution provides us with an opportunity to check the results and the consistency of the code in an environment where we already have a lot of analytical control and understanding. The scaling coefficients are presented in section \[coefficients\_section\]. They can be used by observers to obtain the physical parameters for the blast wave (e.g. explosion energy and circumburst density) from the values for the peak flux and break frequencies that have been obtained from fits to the data. Readers interested only in the coefficients can skip ahead to this point. The fluxes in the transitional regions between the different power law regimes have often been described using heuristic equations that smoothly change from one dominant power law to the next. The abruptness of this change depends on a sharpness parameter $s$. Using the detailed results from our simulations, in section \[smoothPL\_section\] we provide equations for $s$ in terms of two fit parameters: the slope of the accelerated particle distribution $p$ and the aforementioned $k$ that describes the circumburst density structure. In section \[application\_section\] we apply our results to GRB 970508. We discuss our results in section \[discussion\_section\]. Some cumbersome equations and derivations have been deferred to appendices. Description of the post-processing code {#peak_section} ======================================= The code takes as input a series of snapshots of relativistic hydrodynamics configurations on a (in this paper, one-dimensional) grid. Although we will treat only the analytical Blandford-McKee solution [@Blandford1976] for the blast wave dynamics put on a grid here, the code is written with the intention to interact with the AMRVAC adaptive mesh refinement code [@Meliani2007] and will read from file the following conserved variables: $$D = \gamma \rho', \quad \vec{S} = \gamma^2 h' \vec{v}, \quad \tau = \gamma^2 h' - p' - \gamma \rho' c^2,$$ with $\gamma$ the Lorentz factor, $\rho'$ the proper density, $h'$ the relativistic (i.e. including rest mass) enthalpy density, $\vec{v}$ the proper velocity, $p'$ the pressure and $c$ the speed of light. From the conserved values we can reconstruct all hydrodynamical quantities using the equation of state $$p' = ( \Gamma_{ad} - 1 ) e_{th}',$$ where $\Gamma_{ad}$ the adiabatic index that is kept fixed and $e_{th}'$ the thermal energy density. In the entire paper, all comoving quantities will be primed. The grids represent a spherically symmetric fluid configuration and all grid cells are assumed to emit a fraction of their energy as radiation. This fraction of course has to be small enough not to affect the dynamics, since the post-processing approach does not allow for feedback. For the time being we restrict ourselves to the optically thin case. Four ignorance parameters are provided to the code at runtime: $p$, $\xi_N$, $\epsilon_E$ and $\epsilon_B$, denoting respectively the slope of the relativistic particle distribution, the fraction of particles accelerated to this relativistic distribution at any given time, the fraction of thermal energy that is carried by the relativistic electrons and the fraction of thermal energy that resides in the (tangled-up) magnetic field. To be precise: the fractions $\epsilon_E$ and $\epsilon_B$ are fractions of $e_{th}'$, which is strictly speaking the sum of the thermal energy of the protons and non-accelerated electrons plus the energy of the accelerated electrons plus the magnetic field energy. Since we consider fully relativistic gases, the adiabatic indices of the electrons and protons are both at $\Gamma_{ad} = 4/3$. Also, if the magnetic flux enclosed by the surface of any arbitrary fluid element is an adiabatic invariant, we find that $B^2 \propto \rho^{4/3}$, which tells us that the behaviour of the magnetic energy density $B^2/8\pi$ is identical to that of the thermal energy. Or in other words, $\epsilon_B$ retains a constant value away from the shock front. The fraction of shock-accelerated particles $\xi_N$ is often set to one, but we have already kept it explicit in our calculations. At late times (i.e. when the fluid flow is no longer relativistic) $\xi_N$ has te be lower than unity in order to have enough energy per accelerated particle for synchrotron emission. In this work we consider synchrotron radiation only. All grid cells contain a macroscopic number of radiating particles and the radiation from these particle distributions is calculated following @Sari1998 and @Rybicki, but with two important differences: the transition to the lab frame is postponed as long as possible and no assumption about the dynamics of the system is used anywhere as this should be provided by the snapshot files. For clarity of presentation we will ignore the effect of electron cooling in this section. For the emitted power per unit frequency of a typical electron we have $$\begin{aligned} \frac{ {\ensuremath{\,\mathrm{d}}}P'_{<e>}}{{\ensuremath{\,\mathrm{d}}}\nu'} (\nu') & = & \frac{p-1}{2} \cdot \frac{\sqrt{3}{q_e}^3B'}{m_ec^2} \cdot Q \left( \frac{\nu'}{\nu'_{cr,m}}\right). \label{ensemble_electronpower_equation}\end{aligned}$$ Here $q_e$ denotes the electron charge, $m_e$ the electron mass (later on we will also encounter the proton mass $m_p$) and $B'$ the local magnetic field strength. The function $Q$ contains the shape of the spectrum. It shows the expected limiting behaviour: $Q(x) \propto x^{1/3}$ for $x \ll 1$ and $Q(x) \propto x^{(1-p)/2}$ for $x \gg 1$. It incorporates an integration over all pitch angles between electron velocities and the local magnetic field and an integration over the accelerated particle distribution. We use a power law particle distribution with a lower cut-off Lorentz factor $\gamma_m$. Equation (\[ensemble\_electronpower\_equation\]), the critical frequency $\nu'_{cr,m}$ and the full shape of $Q$ are derived in appendix \[distro\_section\]. Assuming isotropic radiation in the comoving frame, we arrive at $$\frac{ {\ensuremath{\,\mathrm{d}}}^2 P_{<e>}' } {{\ensuremath{\,\mathrm{d}}}\nu' {\ensuremath{\,\mathrm{d}}}\Omega' } ( \nu' ) = \frac{1}{4 \pi} \frac{ {\ensuremath{\,\mathrm{d}}}P'_{<e>} }{{\ensuremath{\,\mathrm{d}}}\nu'}(\nu')$$ per solid angle $\Omega'$. To get to the *received* power per unit volume in the lab frame, we have to apply the correct beaming factors, Doppler shift the frequency and multiply the above result for a single particle with the lab frame particle density: $$\frac{ {\ensuremath{\,\mathrm{d}}}^2 P_V }{{\ensuremath{\,\mathrm{d}}}\nu {\ensuremath{\,\mathrm{d}}}\Omega}( \nu' (\nu) ) = \frac{\xi_N n}{\gamma^3 (1-\beta \mu)^3} \cdot \frac{ {\ensuremath{\,\mathrm{d}}}^2 P_{<e>}' } {{\ensuremath{\,\mathrm{d}}}\nu' {\ensuremath{\,\mathrm{d}}}\Omega' } ( \nu \gamma (1 - \beta \mu) ),$$ with $\mu$ now denoting the cosine of the angle between the fluid velocity and the observer (unprimed, so measured in the lab frame), $\beta$ the fluid velocity in units of $c$ and $n$ the number density. Finally, the flux the observer receives at a given observer time is given by $$F(\nu) = \frac{1}{r_{obs}^2} \int \frac{ {\ensuremath{\,\mathrm{d}}}^2 P_V }{{\ensuremath{\,\mathrm{d}}}\nu {\ensuremath{\,\mathrm{d}}}\Omega}( \nu' (\nu) ) ( 1 - \beta \mu ) c {\ensuremath{\,\mathrm{d}}}A {\ensuremath{\,\mathrm{d}}}t_e. \label{integral}$$ Here $r_{obs}$ is the observer distance[^2], approximately the same for all fluid cells (though the differences in arrival times *are* taken into account). The area $A$ denotes the *equidistant surface*. For every emitting time $t_e$ a specific intersecting (with the radiating volume) surface exists from which radiation arrives exactly at $t_{obs}$. The integration over the emission times $t_e$ (represented in the different snapshot files) requires an extra beaming factor and a factor of $c$ to transform the total integral to a volume integral. To perform the surface integrals, the post-processing code uses a Monte Carlo integration algorithm with both importance and stratified sampling, using the pseudo-random Sobol’ sequence.[^3] For the integral over emission times, a combination of modified midpoint integration and Richardson extrapolation is used (the latter allowing us to occasionally skip a snapshot if the desired convergence is already reached). All methods are explained in detail in @Press1992. A minor complication is here that not every $t_e$ probed has a corresponding snapshot file available and interpolation between snapshot files may be needed. The boundaries for surface $A$ are analytically known conic sections and depend on the jet opening angle and observer angle. Two useful consistency checks are observing a spherical explosion from different angles and calculating the volume of a grid snapshot via integration over different observer times while setting the emissivity to one. When creating snapshot files directly from the BM solution we found that sufficient convergence (below the cooling break) was obtained during the post-processing even for modest grid resolutions.[^4] For spherical explosions we used jets with an opening angle of 180 degrees, which makes no noticeable difference for the resulting signal because of relativistic beaming. It is worth emphasizing that it is our method that allows for the modest grid resolution and keeps calculation time short. This is because instead of binning the output from all grid cells, it takes an observer time as the starting point and then probes the appropriate contributing grid cells only (resolving the structure within the cell by including neighbouring cells in the interpolation). We have checked our results by increasing the accuracy (e.g. larger number of grid cells, more snapshot files, smaller step sizes in the integrals etc.) and by replacing the Monte Carlo integration routine with a nested one-dimensional Bulirsch-Stoer algorithm. These consistency checks are in addition to the two mentioned earlier. Finally we have checked the grid interpolation and snapshot I/O routines by comparing the results of our post-process code with those of a code that does not read profiles from disc but calculates the BM solution at run time. The inclusion of electron-cooling {#cooling_section} ================================= The code as described so far is purely a post-process code that in principle can be applied directly to the output of any RHD simulation. If we want to include electron cooling however, we can no longer reconstruct the electron energy distribution from the conserved quantities alone. In the particle distribution function, in addition to the lower boundary $\gamma_m$ , we will also have an upper boundary $\gamma_M$ beyond which all electrons have cooled. The time evolutions of both the lower cut-off Lorentz factor $\gamma'_m$ and the upper cut-off Lorentz factor $\gamma'_M$ (that we have tacitly kept at infinity in the previous section) of this distribution are no longer dictated by adiabatic cooling alone but also by radiation losses. This implies that when running an RHD simulation we need to keep track of at least one extra quantity (at least $\gamma'_M$, although in practice we will trace both). With the introduction of a second critical frequency $\nu'_{cr,M}$, the equation describing the total emitted power now becomes, $$\frac{ {\ensuremath{\,\mathrm{d}}}P'_{<e>}}{{\ensuremath{\,\mathrm{d}}}\nu'} (\nu') = \frac{p-1}{2} \cdot \frac{\sqrt{3}{q_e}^3B'}{m_ec^2} \cdot \mathcal{Q} \left( \frac{\nu'}{\nu'_{cr,M}}, \frac{\nu'}{\nu'_{cr,m}}\right),$$ instead of eq. (\[ensemble\_electronpower\_equation\]). The function $\mathcal{Q}(x_M, x_m)$ and $\nu'_{cr,M}$ are derived and described in appendix \[cooling\_details\_section\]. For $\gamma_M$ at infinity we have $\mathcal{Q}( 0, x_m ) \to Q(x_m)$. The particle distribution that lies beneath the derivation of this new function $\mathcal{Q}$ is no longer a simple power law, but drops off sharply for particle Lorentz factors approaching the peak value of $\gamma'_M$. A subtlety worth noting here is that the critical frequency $\nu'_{cr,M}$ corresponding to $\gamma'_M$ is *not* the cooling frequency, but a frequency beyond which the signal will drop exponentially. Since we put $\gamma'_M$ at infinity directly behind the shock, we will not directly observe $\nu'_{cr,M}$. The actual cooling frequency is found between $\nu'_{cr,m}$ and $\nu'_{cr,M}$, at the point where the shape of the particle distribution ceases to be characterized by a power law but starts to be characterized by the strong drop towards $\gamma'_M$. We will discuss the distinction between the cooled and uncooled region in appendix \[hot\_region\_section\]. A consequence of electron cooling is that the amount of energy in the shock-accelerated electrons is no longer a constant fraction of the thermal energy. $\epsilon_E$ now refers to the fraction of thermal energy in the shock accelerated electrons *directly* behind the shock front instead and the further evolution of the available energy is traced via $\gamma_m$ and $\gamma_M$. Scaling coefficients {#coefficients_section} ==================== ![different possible spectra[]{data-label="twospectra"}](spectrum1.eps "fig:"){width="20.00000%"} ![different possible spectra[]{data-label="twospectra"}](spectrum5.eps "fig:"){width="20.00000%"} Especially for high Lorentz factors, the shape of the spectrum is dominated by the radiation coming from a very thin slab right behind the shock front. So we expect the flux to scale as $$F \propto (p-1) \cdot N_{tot} \cdot (\frac{ {\ensuremath{\,\mathrm{d}}}\mu}{(1-\beta \mu)^3 \gamma^3} \cdot B' \cdot \mathcal{Q} \left( \frac{\nu}{\nu_{cr,M}}, \frac{ \nu }{\nu_{cr,m}} \right). \label{F_scaling_equation}$$ Here $N_{tot}$ is the total number of radiating particles and ${\ensuremath{\,\mathrm{d}}}\mu$ reflects the increasing visible size (due to decrease of beaming) of the slab. The two possible spectra that the code can generate are shown in fig. \[twospectra\], where we used the labelling from @Granot2002 to distinguish the different power law regimes. In tables (\[scalings\_table\]) and (\[frequencies\_table\]) we give the expressions for the absolute scalings in the different regimes $D$, $E$, $F$, $G$, $H$ and the critical frequencies. Scaling coefficients aside, these equations are similar to those given in @vanderHorst2008. The flux in regime $D$ is denoted by $F_D$, the critical peak frequency in spectrum 1 is denoted by $\nu_{m,1}$, the critical cooling frequency in spectrum 1 by $\nu_{c,1}$ and so on. $$\begin{aligned} \hline \hline F_D & = & C_D(p,k) \cdot \frac{\xi_N}{r_{obs,28}^2} \cdot \left( \frac{\epsilon_E}{\xi_N} \right)^{-2/3} \cdot \epsilon_B^{1/3} \cdot n_0^{\frac{2}{4-k}} \cdot E_{52}^{\frac{10-4k}{3(4-k)}} \cdot t_{obs,d}^{\frac{2-k}{4-k}} \cdot (1+z)^{\frac{10-k}{3(4-k)}} \cdot \nu^{1/3} \textrm{ mJy,} \nonumber \\ \hline F_E & = & C_E(p,k) \cdot \frac{ \xi_N }{ r_{obs,28}^2 } \cdot \epsilon_B \cdot n_0^{\frac{10}{3(4-k)}} \cdot E_{52}^{\frac{-6k+14}{3(4-k)}} \cdot t_{obs,d}^{\frac{2-3k}{3(4-k)}} \cdot (1+z)^{\frac{14-k}{3(4-k)}} \cdot \nu^{1/3} \textrm{ mJy.} \\ \hline F_F & = & C_F(p,k) \cdot \frac{ \xi_N }{ r_{obs,28}^2 } \cdot \epsilon_B^{-1/4} E_{52}^{3/4} \cdot t_{obs,d}^{-1/4} \cdot (1+z)^{\frac{3}{4}} \cdot \nu^{-1/2} \textrm{ mJy.} \\ \hline F_G & = & C_G(p,k) \cdot \frac{\xi_N}{r_{obs,28}^2} \cdot \left( \frac{\epsilon_E}{\xi_N} \right)^{p-1} \cdot \epsilon_B^{(p+1)/4} \cdot n_0^{2/(4-k)} \cdot E_{52}^{\frac{-kp-5k+4p+12}{4(4-k)}} \cdot t_{obs,d}^{\frac{3kp-5k-12p+12}{4(4-k)}} \\ \\ & & \cdot (1+z)^{\frac{12-k+4p-kp}{4(4-k)}} \cdot \nu^{-(p-1)/2} \textrm{ mJy.} \\ \hline F_H & = & C_H(p,k) \cdot \frac{\xi_N}{ r_{obs,28}^2 } \cdot \left( \frac{\epsilon_E}{\xi_N} \right)^{p-1} \cdot \epsilon_B^{(p-2)/4} \cdot E_{52}^{(p+2)/4} \cdot t_{obs,d}^{(2-3p)/4} \cdot (1+z)^{\frac{2+p}{4}} \cdot \nu^{-p/2} \textrm{ mJy.} \\ \hline \hline\end{aligned}$$ \[scalings\_table\] $$\begin{aligned} \hline \hline \nu_{m,1} & = & \left( \frac{C_G}{C_D} \right)^{6/(3p-1)} \cdot \left( \frac{\epsilon_E}{\xi_N} \right)^2 \cdot \epsilon_B^{1/2} \cdot E_{52}^{1/2} \cdot t_{obs,d}^{-3/2} \cdot (1+z)^{1/2} \textrm{ Hz.} \\ \hline \nu_{c,1} & = & \left( \frac{ C_H }{C_G} \right)^2 \cdot\epsilon_B^{-3/2} \cdot n_0^{\frac{-4}{4-k}} \cdot E_{52}^{\frac{3k-4}{2(4-k)}} \cdot t_{obs,d}^{\frac{-4+3k}{2(4-k)}} \cdot (1+z)^{-\frac{4+k}{2(4-k)}} \textrm{ Hz.} \\ \hline \nu_{c,5} & = & \left( \frac{C_F}{C_E} \right)^{6/5} \cdot \epsilon_B^{-3/2} \cdot n_0^{-4/(4-k)} \cdot E_{52}^{\frac{3k-4}{2(4-k)}} \cdot t_{obs,d}^{\frac{-4+3k}{2(4-k)}} \cdot (1+z)^{-\frac{4+k}{2(4-k)}} \textrm{ Hz.} \\ \hline \nu_{m,5} & = & \left( \frac{C_H}{C_F} \right)^{2/(p-1)} \cdot \left( \frac{ \epsilon_E }{\xi_N} \right)^2 \cdot \epsilon_B^{1/2} E_{52}^{1/2} \cdot t_{obs,d}^{-3/2} \cdot (1+z)^{1/2} \textrm{ Hz.} \\ \hline \hline\end{aligned}$$ \[frequencies\_table\] The equations in the tables introduce a number of symbols that need an explanation. The cosmic redshift is given by $z$, while the luminosity distance $r_{obs,28}$ is measured in units of $10^{28}$ cm. $E_{52}$ is the explosion energy $E$ in units of $10^{52}$ erg. The observer time in days is denoted by $t_{obs,d}$. The characteristic distance $R_0$ we put at $10^{17}$ cm and $\rho_0$ and $n_0$ are related via the proton mass: $\rho_0 = m_p n_0$. The scaling coefficients $C_D$, $C_E$ etc. contain a number of numerical constants (determined by fitting to output from our code) and some explicit dependencies on $k$ and $p$ and are further explained in appendix \[scaling\_derivation\_section\]. Before the cooling break the scaling behaviour is dictated by the asymptotic behaviour of $Q(\nu'/\nu'_{cr,m})$. The steepening of the spectrum beyond the cooling breaks and the corresponding changes in the scaling behaviour are due to the fact that beyond the cooling break frequency the region behind the shock that still significantly contributes to the total flux (i.e. the *hot region*) becomes noticably smaller than the shock width. The changes in the scalings reflects the change in the size of region. The hot region is discussed separately in appendix \[hot\_region\_section\]. sharpness of broken power law {#smoothPL_section} ============================= In simple power law model fits, the gradual transition between regimes is often handled by a free parameter, the sharpness factor $s$. In more detailed calculations like those done here the gradual transitions are included automatically and we can use this to provide the correct dependence of $s$ on $p$ and $k$. This eliminates $s$ as a free parameter, simplifying the fit to the data and allowing the shape of the transition to help determine whether a particular model fits the data or not. For spectrum 1, we use the following equation to describe the flux density near the peak break $\nu_{m,1}$: $$F(\nu) = F_{m,1} \cdot \left[ \left( \frac{\nu}{\nu_{m,1}} \right)^{\textstyle -\frac{s_{m,1}}{3}} + \left( \frac{\nu}{\nu_{m,1}} \right)^{\textstyle -\frac{s_{m,1}(1-p)}{2}} \right]^{\textstyle -\frac{1}{s_{m,1}}}, \label{Fsmooth_m1_equation}$$ where $F_{m,1}$ denotes the flux at the critical frequency $\nu_{m,1}$ for infinite sharpness $s_{m,1}$ (i.e. the meeting point of the asymptotic power laws). When we switch off cooling in our simulation, we can determine $s_{m,1}$ from fitting against the resulting spectrum while keeping the other parameters in equation (\[Fsmooth\_m1\_equation\]) fixed. The sharpness is a function mainly of $p$ and to a lesser extent of $k$ and the other simulation input parameters. Rather than attempting to include all secondary dependencies when formulating a description for $s_{m,1}$, we find that the following approximation for $s_{m,1}$ is always valid up to a few percent: $$s_{m,1} = 2.2 - 0.52 p.$$ When we switch on electron cooling, the flux is best approximated by $$\begin{aligned} F(\nu) & = & F_{m,1} \nonumber \\ & & \cdot \left[ \left( \frac{\nu}{\nu_{m,1}} \right)^{\textstyle -\frac{s_{m,1}}{3}} + \left( \frac{\nu}{\nu_{m,1}} \right)^{\textstyle -\frac{s_{m,1}(1-p)}{2}} \right]^{\textstyle -\frac{1}{s_{m,1}}} \nonumber \\ & & \cdot \left[ 1 + \left( \frac{\nu}{\nu_{c,1}} \right)^{s_{c,1}/2} \right]^{\textstyle -\frac{1}{s_{c,1}}}. \label{F1smooth_cooling_equation}\end{aligned}$$ If we fit this function against simulation output using $s_{c,1}$ as a fitting parameter we find that the results are described (up to a few percent) by $$s_{c,1} = 1.6 - 0.38 p -0.16 k + 0.078 pk.$$ A simultaneous fit using both $s_{m,1}$ and $s_{c,1}$ yields the same results. For spectrum 5 the order of the breaks is reversed and the smooth power law for both breaks is given by $$\begin{aligned} F(\nu) & = & F_{c,5} \cdot \left[ \left( \frac{\nu}{\nu_{c,5}} \right)^{\textstyle -\frac{s_{c,5}}{3}} + \left( \frac{\nu}{\nu_{c,5}} \right)^{\textstyle \frac{s_{c,5}}{2}} \right]^{\textstyle -\frac{1}{s_{c,5}}} \nonumber \\ & & \cdot \left[ 1 + \left( \frac{\nu}{\nu_{m,5}} \right)^{\textstyle s_{m,5}\cdot \frac{p-1}{2}} \right]^{\textstyle -\frac{1}{s_{m,5}}},\end{aligned}$$ where $F_{c,5}$ denotes the peak flux for infinite sharpness $s_{c,5}$ and the prescriptions for the sharpness are $$s_{c,5} = 0.66 - 0.16k,$$ and $$s_{m,5} = 3.7 - 0.94p + 3.64k - 1.16pk.$$ Once again valid up to a few percent. Given their accuracies, all sharpness prescriptions are consistent with [@Granot2002]. application to GRB 970508 {#application_section} ========================= Various authors have used flux scaling equations to derive the physical properties of GRB 970508 from afterglow data [@Galama1999; @Granot2002; @Yost2003; @vanderHorst2008]. This provides us with a context to illustrate the scaling laws derived in section \[coefficients\_section\]. We will use the fit parameters obtained from broadband modeling by @vanderHorst2008. They have fit simultaneously in time and frequency while keeping $k$ as a fitting parameter. Because the only model dependencies that have been introduced by this approach are the scalings of $t$ and $\nu$ (and no scaling coefficients), their fit results are still fully consistent with our flux equations. Using the cosmology $\Omega_M = 0.27$, $\Omega_\Lambda=0.73$ and Hubble parameter $H_0=71$ km s$^{-1}$ Mpc$^{-1}$, they have $r_{obs,28} = 1.635$ and $z =0.835$ [@Metzger1997], leading, at $t_{obs,d} = 23.3$ days, to $\nu_{c,1} = 9.21\cdot10^{13}$ Hz, $\nu_{m,1} = 4.26\cdot10^{10}$ Hz, $F_{m,1} = 0.756$ mJy, $p = 2.22$ and $k = 0.0307$. Both @vanderHorst2008 and @Galama1999 take for the hydrogen mass fraction of the circumburst medium $X=0.7$, which in our flux equations is mathematically equivalent (though conceptually different) to setting $\xi_N = (1+X)/2 = 0.85$. Unfortunately this still leaves us with four variables to determine ($\epsilon_B$, $\epsilon_E$, $E_{52}$, $n_0$) and only three constraints (peak flux, cooling and peak frequency). From a theoretical study of the microstructure of collisionless shocks @Medvedev2006 arrives at the following constraint: $$\epsilon_E \backsim \sqrt{\epsilon_B}.$$ We include this constraint to have a closed set of equations. For the values quoted above we obtain: $E_{52} = 0.155$, $n_0 = 1.28$, $\epsilon_B = 0.1057$, $\epsilon_E = 0.325$. In figures \[GRB970508\_spectrum\_figure\] and \[GRB970508\_lightcurves\_figure\] we have plotted a comparison between the spectrum generated by using these values as input parameters for the BM solution and the spectrum as it is represented by applying the results of the broadband fit of @vanderHorst2008 for the values of the critical frequencies and the peak flux to equation (\[F1smooth\_cooling\_equation\]). Our scaling coefficients were fixed for arbitrary $k$ and for comparison we also give results for $k = 0$ and $k=2$. The ISM case is virtually identical to $k=0.0307$ and yields: $E_{52} = 0.155$, $n_0 = 1.23$, $\epsilon_B = 0.106$, $\epsilon_E = 0.325$. The stellar wind case yields: $E_{52} = 0.161$, $n_0 = 6.45$, $\epsilon_B = 0.0957$, $\epsilon_E = 0.309$. The quantity $n_0$ (the particle number density at the characteristic distance $1\cdot10^{17}$ cm) is affected most. Also for comparison we give some of the values obtained by other authors. @Galama1999 obtain for the ISM case: $E_{52} = 3.5$, $n_0 = 0.03$, $\epsilon_B = 0.09$, $\epsilon_E = 0.12$. @Granot2002 obtain for the ISM case: $E_{52} = 0.12$, $n_0 = 22$, $\epsilon_B = 0.012$, $\epsilon_E = 0.57$. Both use $p=2.2$. Finally @vanderHorst2008 obtain for $k=0.0307$: $E_{52} = 0.435$, $n_0 = 0.0057$, $\epsilon_B = 0.103$, $\epsilon_E = 0.105$. The large differences between the various results illustrate the importance of using the correct scaling coefficients to derive physical parameters of GRBs and provide a strong motivation for this work. Because the error on $\epsilon_B$ in particular is rather large for the quoted authors, who have used the self absorption critical frequency to provide a fourth constraint, the constraint from @Medvedev2006 can not be rejected based on their fit results. The extension of our code to include self-absorption will yield an alternative and can be used to further study the applicability of Medvedev’s constraint. Summary and discussion {#discussion_section} ====================== In this paper we have introduced an approach to reconstruct light curves and spectra from hydrodynamic simulations. The central idea is that we do not start from simulation snapshots and bin the output of each grid cell, but that for representative snapshots we integrate over the intersecting surface that contains all points where radiation is generated that is due to arrive at a given observer time and frequency. When performing these integrations we interpolate within and between grid cells. While in the context of this paper we have used only snapshots that contain mimicked RHD output using the BM solution, first results using real simulations have been obtained and will be discussed in a later paper. An important thing to note here is that, even though the post-process code only required a very modest resolution, the underlying hydrodynamics code usually does not. @Meliani2007-2 used 1200 base level cells and 15 refinement levels to simulate the evolution of the blast wave (earlier, when they were putting their code to the test they even used 30,000 base level cells at one point, see @Meliani2007). This means that, in general, parallel computer systems are required to run these simulations, something for which the RHD code that our post-process code interacts with (AMRVAC) was explicitly designed. In our code we included synchrotron radiation and electron cooling. We use a parametrisation of the accelerated particle distribution in terms of $\gamma_M$ and $\gamma_m$. Thermal radiation from the particles not accelerated to a power law distribution can be included in a straightforward manner. The code can also be extended to include self-absorption and since the outgoing synchrotron radiation from a grid cell is independent of the incoming radiation, this can be done without expanding to a full radiative transfer code including scattering. Effectively, all that is needed is to postpone the integration over the intersecting surfaces until after the integration over emission times, while in the meantime diminishing the output from earlier surfaces according to the column densities in the lines of sight, which amounts to solving linear transport equations only. As a consistency check and a first application of the code we calculated the scaling coefficients of the flux scaling equations for GRB afterglow spectra for arbitrary values of $k$ with unprecedented accuracy. These results can be used to obtain the physical parameters of the burst from fits to afterglow data. For the ISM and stellar wind scenario’s the results have been checked against the results of @Granot2002 and are found to be fully consistent. The motivation for the choice of arbitrary $k$ is that various authors have now used $k$ as a fitting parameter (e.g. @vanderHorst2008 [@Yost2003]). Values of $k$ other than 0 or 2 reflect the structure of a circumburst medium altered by shock interactions or more complicated stellar wind structures. We have used GRB970508 to illustrate the effect of using our scaling coefficients to deduce the physical properties of a GRB. Here we have used an additional constraint by @Medvedev2006 to obtain a closed set of equations in the absence of a full description for the self-absorption. Acknowledgments {#acknowledgments .unnumbered} =============== This research was supported by NWO Vici grant 639.043.302 (RAMJW) and NOVA project 10.3.2.02 (HJvE). HJvE wishes to thank Atish Kamble and Alexander van der Horst for useful discussion. We are indebted to the anonymous referee for pointing out a numerical error in our original submission (the origin of this type of error is discussed in appendix \[hot\_region\_section\]). [99]{} R.D. Blandford and C.F. McKee. 1976, Phys. Fluids 19, 8 T.J. Galama, R.A.M.J. Wijers, P.M. Vreeswijk et al. 1999, A&A Suppl. S. 138, 451 J. Granot, T. Piran and R. Sari. 1999, ApJ, 513, 679 (astro-ph/9806192) J. Granot et al. 2001, Gamma-Ray Bursts in the Afterglow Era: Proceedings of the International Workshop Held in Rome, Italy, 17-20 October 2000, ESO ASTROPHYSICS SYMPOSIA. ISBN 3-540-42771-6. Edited by E. Costa, F. Frontera, and J. Hjorth. Springer-Verlag, 2001, p. 312 (astro-ph/0103038) J. Granot and R. Sari. 2002, ApJ, 568, 820 (astro-ph/0108027) J. Granot. 2005, Rev. Mex. A&A accepted (astro-ph/0610379) A. Gruzinov and E. Waxman. 1999, ApJ, 511, 852 (astro-ph/9807111) A.J. van der Horst. R.A.M.J. 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Frail. 2003, ApJ, 597: 459 (astro-ph/0307056) Derivation of emitted power per electron {#distro_section} ======================================== For each electron Lorentz factor $\gamma_e$ we define two critical frequencies $\nu_{cr,e, \alpha}'$ and $\nu_{cr,e}$: $$\nu_{cr,e, \alpha}' = \frac{3}{4\pi} \gamma_{e}'^2 \frac{q_e B'}{ m_e c } \sin \alpha \equiv \nu_{cr,e}' \sin \alpha, \label{nume}$$ where $q_e$ denotes the electron charge, $m_e$ the electron mass and $\alpha$ the pitch angle between field and velocity. It is around (but not exactly *at*) these values that the spectrum peaks and we will find them useful as integration variables later on. The power per unit frequency emitted by an electron is ([@Rybicki]): $$\frac{ {\ensuremath{\,\mathrm{d}}}P'_{e,\alpha} }{{\ensuremath{\,\mathrm{d}}}\nu'} (\nu') = \frac{\sqrt{3} {q_e}^3 B' \sin \alpha}{m_e c^2} F(\frac{\nu'}{\nu'_{cr,e,\alpha}}),$$ where $$F(x) \equiv x \int_x^\infty K_{\frac{5}{3}}(\xi) {\ensuremath{\,\mathrm{d}}}\xi,$$ with $K_{\frac{5}{3}}$ a modified Bessel function of fractional order. $F(x)$ behaves as follows in the limits of small and large $x$: $$F(x) \backsim \frac{4\pi}{\sqrt{3}\Gamma(\frac{1}{3})}\left(\frac{x}{2} \right)^{1/3} \left(1 - \frac{\Gamma(\frac{1}{3})}{2^{5/3}} x^{2/3} + \frac{3}{16} x^2 \right), \qquad x \ll 1,$$ $$F(x) \backsim \sqrt{ \frac{\pi}{2} } x^{1/2} e^{-x} \left( 1 + \frac{55}{72} \frac{1}{x} - \frac{10151}{10368} \frac{1}{x^2} \right), \qquad x \gg 1,$$ where $\Gamma(x)$ is the gamma function of argument $x$. For the mean power averaged over all pitch angles while assuming an isotropic pitch angle distribution we obtain: $$\frac{ {\ensuremath{\,\mathrm{d}}}P'_{e} }{{\ensuremath{\,\mathrm{d}}}\nu'} (\nu') = \frac{\sqrt{3} {q_e}^3 B'}{m_e c^2} \mathcal{P} (\frac{\nu'}{\nu'_{cr,e}}), \label{Pe}$$ where $$\mathcal{P}(x) \equiv \frac{1}{2} \int_0^{\pi} (\sin \alpha)^2 F(\frac{x}{\sin \alpha}) {\ensuremath{\,\mathrm{d}}}\alpha.$$ In the limit of small and large $x$, $P(x)$ behaves as follows: $$\mathcal{P}(x) \backsim \frac{2^{2/3} \cdot \pi^{3/2} \cdot \sqrt{3}}{9 \Gamma(\frac{11}{6})} x^{1/3} - \frac{\pi}{\sqrt{3}}x - \frac{5 \cdot \pi \cdot \sqrt{3} }{ 48 \cdot 2^{1/3} \Gamma( \frac{11}{6} )} x^{7/3}, \qquad x \ll 1,$$ $$\mathcal{P}(x) \backsim \frac{\pi}{2} e^{-x}, \qquad x \gg 1.$$ The effective lower cut-off Lorentz factor of a collection of electrons $\gamma_m'$ can be expressed in terms of local fluid quantities. The integrated power law particle distribution $C \gamma_e'^{-p} {\ensuremath{\,\mathrm{d}}}\gamma_e'$ ($C$ is a constant of proportionality) must yield the total number density of particles: $$\int_{\gamma_m'}^{\infty} C (\gamma_e')^{-p} {\ensuremath{\,\mathrm{d}}}\gamma_e' = \xi_N n' \to C = - \frac{1-p}{(\gamma_m')^{1-p}} \xi_N n'. \label{electron_power_equation}$$ Similarly the integrated particle energies must yield the total energy: $$\frac{\int_{\gamma_m'}^\infty C \gamma_e'^{-p} \ \gamma_e' m_e c^2 \ d \gamma_e'}{\int_{\gamma_m'}^{\infty} C \gamma_e'^{-p} d\gamma_e'} = \frac{ \epsilon_E e_{th}' + \xi_N n' m_e c^2 }{\xi_N n'}.$$ Combining these equations and dropping the rest mass term in the energy equation (it will be negligible for relativistic electrons), we obtain $$\gamma_m' = \left( \frac{2-p}{1-p} \right) \cdot \left( \frac{\epsilon_E}{\xi_N} \frac{e_{th}'}{n'} \frac{1}{m_e c^2} \right). \label{gamma_m_equation}$$ If we integrate (\[Pe\]) over the particle distribution and divide the result by the total electron density, we obtain the emitted power per ensemble electron[^5]: $$\frac{ {\ensuremath{\,\mathrm{d}}}P'_{<e>}}{{\ensuremath{\,\mathrm{d}}}\nu'} (\nu') = \frac{p-1}{2} \cdot \frac{\sqrt{3}{q_e}^3B'}{m_ec^2} \cdot Q \left( \frac{\nu'}{\nu'_{cr,m}}\right).$$ Here $\nu'_{cr,m}$ denotes the resulting value of $\nu'_{cr,e}$ when we substitute $\gamma'_m$ for $\gamma'_e$ in equation (\[nume\]). It surfaces when we switch integration variables from $\gamma'_e$ to $\nu'_{cr,e}$. The auxiliary function $Q$ is defined as $$Q ( x ) \equiv x^{\frac{1-p}{2}} \int_0^x y^{\frac{p-3}{2}} \mathcal{P} (y) {\ensuremath{\,\mathrm{d}}}y.$$ In the limit of small and large $x$, $Q(x)$ behaves as follows: $$Q(x) \backsim \frac{2^{5/3} \sqrt{3\pi} \Gamma( \frac{1}{6} )}{5(3p-1)} x^{1/3} - \frac{2\pi}{\sqrt{3}(p+1)} x + \frac{3 \sqrt{3\pi} \Gamma( \frac{1}{6} )}{2^{1/3}(88+24p)} x^{7/3}, \qquad x \ll 1, \label{Q_small_equation}$$ $$Q(x) \backsim \sqrt{\pi} \frac{\Gamma(\frac{5}{4}+\frac{p}{4})}{ \Gamma(\frac{7}{4}+\frac{p}{4}) } \cdot \frac{2^{\frac{p-1}{2}}}{p+1} \cdot \Gamma( \frac{p}{4}+\frac{19}{12} ) \Gamma( \frac{p}{4} - \frac{1}{12})\cdot x^{\frac{1-p}{2}} -\frac{\pi}{2} \frac{e^{-x}}{x}, \qquad x \gg 1. \label{Q_large_equation}$$ In practice, the computer code uses lookup tables for $F(x)$, $\mathcal{P}(x)$ and $Q(x)$. The three functions have been plotted in figure (\[FPQplot\]) ($Q$ for both $p = 2.2$ and $p = 2.8$), allowing for comparison between the spectra from a single electron, an angle-averaged electron and an ensemble electron. Emitted power with electron cooling {#cooling_details_section} =================================== If the only processes that are of importance are synchrotron emission and adiabatic cooling, the evolution of the Lorentz factor of a single electron is described by $$\frac{{\ensuremath{\,\mathrm{d}}}\gamma_e}{{\ensuremath{\,\mathrm{d}}}t'} = - \frac{\sigma_T (B')^2}{6\pi m_e c} \gamma_e^2 + \frac{\gamma_e}{3n'} \frac{{\ensuremath{\,\mathrm{d}}}n'}{{\ensuremath{\,\mathrm{d}}}t'}, \label{evolution_equation}$$ where $\sigma_T$ denotes the Thomson cross section. In [@Granot2002] this differential equation is applied to the BM solution by expressing it in terms of the self-similar variable and solving it analytically. In our case we can use eq. (\[gamma\_m\_equation\]) to establish $\gamma'_m$ directly behind the shock front and initially put $\gamma'_M$, the upper cut-off Lorentz factor due to cooling, at a sufficiently large value (instead of infinity). Sufficiently large for example can be taken such that $$\left| \frac{ \int_{\gamma'_m}^{\gamma'_M} \gamma_e^{1-p} - \int_{\gamma'_m}^{\infty} \gamma_e^{1-p} }{ \int_{\gamma'_m}^{\infty} \gamma_e^{1-p} } \right| \le \epsilon,$$ with $\epsilon$ some tolerance for the error in the energy. The real $\gamma'_M$ will quickly catch up with the approximated $\gamma'_M$, as can be seen from equation (\[evolution\_equation\]). The analytical solution for the particle distribution in the BM case is given by $$N_e (\gamma_e') = C \gamma_e'^{-p} \cdot ( 1 - \frac{\gamma_e'}{\gamma_M'})^{p-2},$$ where the factor $C$ now stands for $$C = (p-1) \xi n' \gamma_m'^{p-1} \cdot ( 1 - \frac{\gamma_m'}{\gamma_M'} )^{1-p}. \label{norm_equation}$$ We take this to hold for the output of real RHD simulations as well, so that we have an approximate parametrisation for the particle distribution in any grid cell in terms of $\gamma'_m$ and $\gamma'_M$ alone. A more complete treatment of the particle distribution (e.g. [@Peer2005]) would effectively introduce an additional dimension to the RHD simulation and slow down the calculations accordingly. Via reasoning completely analogous to the non-cooling case (where we use eq. (\[nume\]) with $\gamma'_M$ instead of $\gamma'_e$ to obtain $\nu'_{cr,M}$) we arrive at an auxiliary function $\mathcal{Q}$ given by $$\mathcal{Q}(y_M, y_m) = y_m^{\frac{1-p}{2}}\cdot(1-(\frac{y_M}{y_m})^{1/2})^{1-p} \cdot \int_{y_M}^{y_m} y^{\frac{p-3}{2}} \cdot (1 - (\frac{y_M}{y} )^{1/2} )^{p-2} \mathcal{P}(y) {\ensuremath{\,\mathrm{d}}}y, \label{Q_cooling_equation}$$ ocurring in $$\frac{ {\ensuremath{\,\mathrm{d}}}P'_{<e>}}{{\ensuremath{\,\mathrm{d}}}\nu'} (\nu') = \frac{p-1}{2} \cdot \frac{\sqrt{3}{q_e}^3B'}{m_ec^2} \cdot \mathcal{Q} \left( \frac{\nu'}{\nu'_{cr,M}}, \frac{\nu'}{\nu'_{cr,m}}\right).$$ Since this is a function of two variables instead of one, its limiting behaviour is more complicated. If $y_M \ll 1$, $\mathcal{Q}(y_M, y_m )$ approximately reduces to $$\mathcal{Q}(y_M, y_m ) \propto (1 - ( \frac{y_M}{y_m} )^{1/2} )^{1-p} \cdot Q( y_m ), \qquad y_M \ll 1,$$ which can be obtained by approximating $y_M$ by zero in the integration limits and integrand of equation (\[Q\_cooling\_equation\]). If $y_m / y_M \to 1$, the approximate result is $$\mathcal{Q}( y_M, y_m ) \propto \mathcal{P}(y_m), \qquad \frac{y_m}{y_M} \to 1,$$ which follows from approximating the integral from (\[Q\_cooling\_equation\]) by its value at $y_m$ times the integration domain. If $y_m \ll 1$ as well, we can use the first term of the lower limit series expansion for $\mathcal{P}$ in the integral and solve it to refine the approximate result to $$\mathcal{Q}( y_M, y_m ) \propto \mathcal{P} (y_m ) / (p - 1), \qquad \frac{y_m}{y_M} \to 1, \quad y_m \ll 1. \label{Q_cool_equation}$$ On the other hand, if $y_m / y_M \gg 1$, we can approximate the result in terms of $\mathcal{Q}( y_M, y'_m )$ for a smaller value $y'_m$ (i.e. the last tabulated value): $$\mathcal{Q}( y_M, y_m ) \backsim \mathcal{Q}( y_M, y'_m ) \cdot \left( \frac{y_m}{y'_m} \right)^{\frac{1-p}{2}} + Q( y_m ) - Q( y'_m ) \left( \frac{y_m}{y'_m} \right)^{\frac{1-p}{2}}, \qquad \frac{y_m}{y_M} \gg 1,$$ which further reduces to $$\mathcal{Q}( y_M, y_m ) \backsim \mathcal{Q}( y_M, y'_m ) \cdot \left( \frac{y_m}{y'_m} \right)^{\frac{1-p}{2}} - \frac{\pi}{2} \frac{ e^{-y_m}}{y_m} + \frac{\pi}{2} \frac{e^{-y'_m}}{y'_m} \cdot \left( \frac{y_m}{y'_m} \right)^{\frac{1-p}{2}}, \qquad \frac{y_m}{y_M} \gg 1,$$ for sufficiently high values of $y'_m$ and $y_m$. Finally, for $y_M \gg 1$ we find from fitting to tabulated values that $\mathcal{Q}( y_M, y_m )$ is best described by $$\mathcal{Q}( y_M, y_m ) \propto \left( \frac{ y_m }{y_M} \right)^{\frac{1-p}{2}} \cdot (1 - (\frac{y_M}{y_m})^{1/2} )^{(1-p)} \cdot e^{-y_M} \cdot y_M^{p-1}, \qquad y_M \gg 1. \label{Q_drop_equation}$$ In practice the code uses a two-dimensional table with numerically calculated values in addition to the analytical expressions above. The contribution from the region where $y_M \gg 1$ is effectively zero due to the exponential term $e^{-y_M}$. Derivation of Scaling coefficients {#scaling_derivation_section} ================================== We summarize the equations for the scaling coefficients in table (\[coefficients\_table\]). Aside from some explicit dependencies on $p$ and $k$ these equations also contain truly numerical constants with values that have been determined by fitting to output of our code. For example $C_D(p,k)$ contains the constants $C_{D0}$, $C_{Dk}$ and $C_{Dkk}$ (with $C_{Dk}^k$ denoting $C_{Dk}$ to the power $k$ etc.). Their numerical values are listed in table (\[coefficient\_values\_table\]). Instead of incorporating these numerical constants in the total flux formula as we have done here we could also have used a fitting polynomial, but this approach more closely reflects the $k$ and $p$ dependencies. $$\begin{aligned} \hline \hline C_D & \equiv & (p-1) \cdot \left( C_{D0} C_{Dk}^k C_{Dkk}^{k^2} \right)^{1/(4-k)} \cdot \frac{1}{3-k} \cdot \left( \frac{p-2}{p-1} \right)^{-2/3} \\ & & \cdot (17-4k)^{\frac{10-4k}{3(4-k)}} \cdot (4-k)^{\frac{2-k}{4-k}} \cdot Q_- \\ \hline C_E & \equiv & (p-1) \cdot \left( C_{E0} C_{Ek}^k C_{Ekk}^{k^2} \right)^{1/(4-k)} \cdot \frac{1}{3-k} \\ & & \cdot (17-4k)^{\frac{-6k+14}{3(4-k)}} \cdot (4-k)^{\frac{2-3k}{3(4-k)}} \cdot Q_{cool} \\ \hline C_F & \equiv & (p-1) \cdot \left( C_{F0} C_{Fk}^k C_{Fkk}^{k^2} \right)^{1/(4-k)} \cdot \frac{1}{3-k} \\ & & \cdot (17-4k)^{3/4} \cdot (4-k)^{-1/4} \cdot Q_{cool} \\ \hline C_G & \equiv & (p-1) \cdot \left( C_{G0} C_{Gk}^k C_{Gkk}^{k^2} C_{Gp}^p C_{Gpk}^{pk} C_{Gpkk}^{pk^2} C_{Gpp}^{p^2} C_{Gppk}^{p^2k} C_{Gppkk}^{p^2k^2} \right)^{1/(4-k)} \cdot \frac{1}{3-k} \cdot \left( \frac{p-2}{p-1} \right)^{p-1} \\ & & \cdot (17-4k)^{\frac{-kp-5k+4p+12}{4(4-k)}} \cdot (4-k)^{\frac{3kp-5k-12p+12}{4(4-k)}} \cdot Q_+ \\ \hline C_H & \equiv & (p-1) \cdot \left( C_{H0} C_{Hk}^k C_{Hkk}^{k^2} C_{Hp}^p C_{Hpk}^{pk} C_{Hkk}^{pk^2} C_{Hpp}^{p^2} C_{Hppk}^{p^2k} C_{Hppkk}^{p^2k^2} \right)^{1/(4-k)} \cdot \frac{1}{3-k} \cdot \left( \frac{p-2}{p-1} \right)^{p-1} \cdot \\ & & (17-4k)^{\frac{p+2}{4}} \cdot (4-k)^{\frac{2-3p}{4}} \cdot Q_+ \\ \hline \hline\end{aligned}$$ \[coefficients\_table\] The first term $(p-1)$ in these equations is also the first term in eq. (\[F\_scaling\_equation\]). From the contribution of $N_{tot}$ we obtain a contribution $1/(3-k)$ via $$N_{tot} = \xi_N 4 \pi \int_0^R r^2 n_0 \left( \frac{r}{R_0} \right)^{-k} {\ensuremath{\,\mathrm{d}}}r = \xi_N \frac{4 \pi n_0 }{3-k} \left( \frac{R}{R_0} \right)^{3-k}.$$ The origin of the combination $(p-2)/(p-1)$ can be traced to equation (\[gamma\_m\_equation\]) in appendix \[distro\_section\] of this paper ([@Granot2002]) actually absorb it into $\epsilon_E$). The term $(17-4k)$ is linked to the energy $E_{52}$ and the two will always occur with the same power as can be seen from comparing tables \[coefficients\_table\] and \[scalings\_table\]. It enters our calculations via equation (69) from [@Blandford1976]. The term $(4-k)$ is likewise linked to the observer time $t_{obs,d}$. The extra term is a result from the transition from emission time in the grid lab frame to observer time. For the shock front the two are related via $$t_{e} = ( 2 (4-k) t_{obs} )^{1/(4-k)} \left( \frac{E (17-4k)}{8\pi \rho_0 c^{5-k} R_0^k } \right)^{1/(4-k)}. \label{t_e_front_equation}$$ The final terms are different for the different power law regimes. They are contributed by the leading order terms of the various approximations of $\mathcal{Q}$. $Q_+$ is given by (see eqn. (\[Q\_large\_equation\])): $$Q_+ \equiv \frac{\Gamma(\frac{5}{4}+\frac{p}{4}) \Gamma ( \frac{p}{4}+\frac{19}{12} ) \Gamma ( \frac{p}{4}-\frac{1}{12} )}{ \Gamma(\frac{7}{4}+\frac{p}{4})(p+1)}.$$ For low uncooled frequencies we have $$Q_- \equiv \frac{1}{3p-1},$$ as can be seen from equation (\[Q\_small\_equation\]). When cooling plays a role we find that equation (\[Q\_cool\_equation\]) provides us with $$Q_{cool} \equiv \frac{1}{p-1}.$$ Note that the effect of $Q_{cool}$ in $C_E$ and $C_F$ is to cancel out the first $(p-1)$ term -we only kept both terms for clarity of presentation. D G H F E -------- ---------------------- ----------------------- ---------------------- ---------------------- ----------------------- 0 $5.12\cdot10^{-17}$ $2.78 \cdot 10^{-31}$ $5.68 \cdot 10^{-1}$ $1.16 \cdot 10^{30}$ $2.95 \cdot 10^{-16}$ $k$ $1.18 \cdot 10^4$ $4.54 \cdot 10^{7}$ $6.94 \cdot 10^{-1}$ $1.36 \cdot 10^{-8}$ $2.04 \cdot 10^4$ $kk$ $9.01 \cdot 10^{-1}$ $8.95 \cdot 10^{-1}$ $9.27 \cdot 10^{-1}$ $1.01$ $9.41 \cdot 10^{-1}$ $p$ $2.25 \cdot 10^{32}$ $5.40 \cdot 10^{30}$ $pk$ $7.27 \cdot 10^{-9}$ $1.65 \cdot 10^{-8}$ $pkk$ $9.41 \cdot 10^{-1}$ $1.06$ $pp$ $1.77$ $2.99$ $ppk$ $8.07 \cdot 10^{-1}$ $7.01 \cdot 10^{-1}$ $ppkk$ $1.03$ $1.01$ \[coefficient\_values\_table\] The hot region {#hot_region_section} ============== For any given observer time and observer frequency there is a region behind the shock front where the emitting electrons have not yet cooled below the observer frequency. Although, when we set $\gamma_M$ initially at infinity, the size of this region never becomes zero, it can become very small, even when compared to the analytical error on the BM solution. The size of the hot region also determines the slope of the spectrum beyond the cooling break. We calculate its properties below. The BM solution is obtained by a change of variables from $t$ and $r$ to $\chi$ and $1 / \Gamma^2$, where the fact that the latter becomes very small is continually used to simplify the dynamic equations using first order approximations. The $\chi$ coordinate of a fluid element is given by $$\chi = [1 + 2(4-k)\Gamma^2] ( 1 - \frac{r}{ct} ),$$ which is 1 at the shock front and increases roughly one order in magnitude until the back of the shock. The radiation received at a given observer time is obtained by integrating over equidistant surfaces that have a one-on-one correspondence to emission times. To obtain an order of magnitude estimate for the size of the hot region we look solely at the jet axis, where each emission time and hence each equidistant surface corresponds to a given position $\chi$, via $$\chi( r, t ) = \chi( c (t_e - t_{obs}), t_e ) \approx \frac{ t_{obs}}{t_e} \cdot 2(4-k)\Gamma^2. \label{chi_and_t_e_equation}$$ We define the boundary of the hot region $\chi_{hot}$ at the point where $\nu'_{cr,M} = \nu'$ (i.e. when the second argument of $\mathcal{Q}( \frac{\nu'}{\nu'_{cr,m}}, \frac{\nu'}{\nu'_{cr,M}} )$ is equal to one). The critical frequency $\nu'_{cr,M}$ is related to $\gamma'_M$ via the usual relation (see eqn. \[nume\]), and an expression for $\gamma'_M$ in terms of the self-similar parameter $\chi$ can be found in [@Granot2002]: $$\gamma'_M(\chi) = \frac{2(19-2k)\pi m_e c \gamma}{\sigma_T B^2 t_e^2} \frac{1}{\chi^{(19-2k)/3(4-k)}-1}.$$ Using the above we can find an expression for $\chi_{hot}$ -or equivalently $t_{e,hot}$, since the two are related via eqn. (\[chi\_and\_t\_e\_equation\]). To first order in $\chi_{hot} - 1$ we find $$\chi_{hot} - 1 \approx \left( \frac{ 27 q_e m_e ( 4-k)^2}{ \nu \sigma_T^2 128 \sqrt{2 \pi} c^2 \epsilon_B^{3/2} \rho_0^{3/2} R_0^{3k/2} } \right)^{1/2} \cdot \left( \frac{E (17-4k)t_{obs} 2 (4-k)}{8 \pi \rho_0 R_0^k c^{5-k} } \right)^{\frac{4-3k}{4(k-4)}},$$ $$t_{e,front} - t_{e,hot} \approx \frac{1}{4-k} \cdot \left( \frac{ 27 q_e m_e ( 4-k)^2}{ \nu \sigma_T^2 128 \sqrt{2 \pi} c^2 \epsilon_B^{3/2} \rho_0^{3/2} R_0^{3k/2} } \right)^{1/2} \cdot \left( \frac{E (17-4k)t_{obs} 2 (4-k)}{8 \pi \rho_0 R_0^k c^{5-k} } \right)^{\frac{-3k}{4(k-4)}},$$ where $t_{e,front}$ is the emission time of the shock front (see equation \[t\_e\_front\_equation\]). From this we can draw a number of conclusions. The size of the hot region is dependent on the observer frequency via $\nu^{-1/2}$. For observer frequencies beyond the cooling break, it is effectively this region alone that contributes to the observed flux, since the contribution from the cool region drops exponentially (see equation \[Q\_drop\_equation\]). A steepening of the spectral slope by -1/2 is therefore expected: $(1-p)/2 \to -p/2$. This results from multiplying the pre-cooling break flux by the fraction of the total emitting region that consists of the hot region -which is given by $$\frac{t_{e,front} - t_{e,hot}}{t_{e,front}} \approx \frac{1}{4-k} \cdot \left( \frac{ 27 q_e m_e ( 4-k)^2}{ \nu \sigma_T^2 128 \sqrt{2 \pi} c^2 \epsilon_B^{3/2} \rho_0^{3/2} R_0^{3k/2} } \right)^{1/2} \cdot \left( \frac{E (17-4k)t_{obs} 2 (4-k)}{8 \pi \rho_0 R_0^k c^{5-k} } \right)^{\frac{4-3k}{4(k-4)}}$$ Note that from this equation all post-cooling break scalings (e.g. $\epsilon_B$, $E_{52}$ etc.) can be derived by multiplying with the relevant pre-cooling flux. Another important issue is that the size of the hot region can become smaller than the analytical error inherent in the BM solution, which cuts off beyond $1/\Gamma^2$. This happens at late times, when $\Gamma$ has dropped significantly. This puts a practical limit on a direct numerical implementation of the BM solution in our radiation code, ironically not due to numerical limitations of the code but because of the upper limit on the accuracy of the analytical solution that we have used to generate our grid files[^6]. On can however still extrapolate the heuristic description of the spectra and light curves that we have obtained for arbitrary $k$ to late times -this is completely consistent with the canonical approach to light curve and spectrum modelling. \[lastpage\] [^1]: E-mail: H.J.vanEerten@uva.nl (HJvE); R.A.M.J.Wijers@uva.nl (RAMJW) [^2]: For cosmological distances $r_{obs}$ denotes the luminosity distance and redshift terms $(1+z)$ need to be inserted in the appropiate places in the equations. [^3]: But if symmetry allows (e.g. the observer is on the jet axis), we just do a straightforward Bulirsch-Stoer integration [^4]: On the order of 120 base cells with 8 levels of refinement (an increase in refinement means a local increase of resolution by a factor of two) for a region $\backsim 10^{17}$ cm to $\backsim 10^{18}$ cm and a relatively small number of snapshots ($\backsim 1000$) to go from $\Gamma \backsim 100$ down to $\Gamma \backsim 2$. Unfortunately, the resolution will eventually be dictated by that required by RHD simulations, which will be much higher. [^5]: an ensemble electron contribution is therefore constructed as the total of all electron contributions divided by the number of electrons. [^6]: In general, when post-processing grid files from simulations the issue does not occur because we use the AMR structure of the grid to set the local integration accuracy. If the hot region becomes very small, then this will be dealt with at the earlier stage of the RHD simulation. Also, when directly integrating the flux equations for the BM solution by first expressing everything in terms of the self-similar coordinate and sticking to that frame, the issue is largely avoided as well.

--- author: - 'Jens Niklas Eberhardt, Shane Kelly' bibliography: - 'main.bib' title: Mixed Motives and Geometric Representation Theory in Equal Characteristic ---

--- abstract: 'Absolute separable states is a kind of separable state that remain separable under the action of any global unitary transformation. These states may or may not have quantum correlation and these correlations can be measured by quantum discord. We find that the absolute separable states are useful in quantum computation even if it contains infinitesimal quantum correlation in it. Thus to search for the class of two-qubit absolute separable states with zero discord, we have derived an upper bound for $Tr(\varrho^{2})$, where $\varrho$ denoting all zero discord states. In general, the upper bound depends on the state under consideration but if the state belong to some particular class of zero discord states then we found that the upper bound is state independent. Later, it is shown that among these particular classes of zero discord states, there exist sub-classes which are absolutely separable. Then we construct a ball for $2\otimes d$ quantum system described by $Tr(\rho^{2})\leq Tr(X^{2})+2\lambda_{min}(X)\lambda_{min}(Z)+Tr(Z^{2})$, where the $2\otimes d$ quantum system is described by the density operator $\rho$ which can be expressed by $d\otimes d$ block matrices $X,Y$ and $Z$ with $X,Z\geq 0$ and $\lambda_{min}(X)$, $\lambda_{min}(Z)$ denoting the minimum eigenvalues of the block matrices $X$ and $Z$ respectively. In particular, we show that the newly constructed ball contain class of absolute separable states described by two-qubit density operator $\rho$ that are lying outside the ball described by $Tr(\rho^{2})\leq \frac{1}{3}$. Also we have discussed an example of a class of absolute separable states in $2\otimes 3$ system where we find that most of the absolute separable states are residing inside the new ball and few of them are lying outside the ball.' author: - Satyabrata Adhikari title: 'Constructing a ball of separable and absolutely separable states for $2\otimes d$ quantum system' --- \[theorem\][Lemma]{} \[theorem\][Corollary]{} \[theorem\][Proposition]{} \[theorem\][Conjecture]{} \[theorem\][Definition]{} Introduction ============ Quantum correlation can be considered as a necessary ingredient for the development of quantum information theory and quantum computation. A remarkable application of quantum correlation can be found in different areas of quantum communication such as quantum teleportation [@bennett], quantum dense coding [@wiesner], quantum remote state preparation [@pati] , quantum cryptography [@gisin] etc. Till few years ago, it has been thought that this non-local feature in terms of quantum correlation only exist in the entangled state and responsible for the computational speed-up in the known quantum algorithms [@ekert]. Later, Lloyd [@lloyd] showed that there are quantum search methods which does not require entanglement to provide a computational speed-up over classical methods. In this modern line of research, Ahn et al. [@ahn] have shown that instead of entanglement, quantum phase is an essential ingredient for the computational speed-up in the Grover’s quantum search algorithm [@grover]. Meyer [@meyer] was able to reduce the number of queries in a quantum search compared to classical search of a database using only interference, not entanglement. Gottesman-Knill theorem also declare the fact that entanglement is not only a factor to for a quantum computers to outperform classical computers [@nielsen]. In 2004, E. Biham et.al. [@biham] then conclude that entangled state is not a compulsory ingredient for quantum computing and discovered that there exist quantum state lying arbitrarily close to the maximally mixed states, which are enough to increase the computational speed-up in quantum algorithms. To characterize the nature of mixed density matrices lying in the sufficiently small neighbourhood of maximally mixed state, it has been shown that all such states are separable states [@Zyczkowski; @braunstein; @vidal].\ Separable states can be defined as the mixture of locally indistinguishable states. Mathematically, a bipartite separable state described by the density operator $\rho$ in a composite Hilbert space $H_{1}\otimes H_{2}$ can be expressed as $$\begin{aligned} \rho=\sum_{k} p_{k} \rho_{1}^{k}\otimes \rho_{2}^{k},~~0\leq p_{k}\leq 1 \label{sepstate}\end{aligned}$$ where $\rho_{1}$ and $\rho_{2}$ represents two density operators in two Hilbert spaces $H_{1}$ and $H_{2}$ respectively. These states can be prepared using local quantum operation and classical communication (LOCC). Thus prescription of its preparation is different from entangled states, which cannot be prepared with the help of LOCC. Since separable states are prepared by performing quantum operation within the structure of LOCC on quantum bit so they can exhibit quantum correlation [@modi]. Therefore, it can be inferred that not only entaglement but also this non-classical feature exhibited by some separable states.\ The entanglement measures [@wootters; @vidal1] cannot quantify the quantum correlation present in the separable state due to the reason that any entanglement measure gives value zero for all separable states. Thus, Ollivier and Zurek [@ollivier] proposed a measure for quantum correlation which can be defined as the difference between the quantum mutual information and the measurement-induced quantum mutual information. This measure is commonly known as quantum discord. If a separable state has no quantum correlation then they are termed as zero discord state. The quantum discord of two-qubit maximally mixed marginals and two qubit X-state has been calculated in [@luo; @ali]. It has been found that quantum correlation plays a vital role in mixed-state quantum computation speed-up and it is due to the correlation present in the separable states [@dutta2005; @dutta2007].\ We should note an important fact that there exist separable states (with or without quantum correlation) that can be converted to entangled state under the action of global unitary operation. The class of separable states that remain separable state after performing global unitary operation are known as absolutely separable states [@kus]. The necessary and sufficient condition for the absolute separability of a state in $2\otimes 2$ system described by the density operator $\sigma$ is given by [@verstraete] $$\begin{aligned} \lambda_{1}\leq \lambda_{3}+2\sqrt{\lambda_{2}\lambda_{4}} \label{abssepcond}\end{aligned}$$ where $\lambda_{i},i=1,2,3,4$ denoting the eigenvalues of $\sigma$ arranged in descending order as $\lambda_{1}\geq \lambda_{2}\geq\lambda_{3}\geq\lambda_{4}$. Further, Johnston [@johnston] generalize the absolute separability condition for $H_{2}\otimes H_{d}$ system and showed that a state $\sigma \in H_{2}\otimes H_{d}$ is absolute separable if and only if $$\begin{aligned} \lambda_{1}\leq \lambda_{2d-1}+2\sqrt{\lambda_{2d-2}\lambda_{2d}} \label{abssepcondgen}\end{aligned}$$ Since the absolute separability conditions (\[abssepcond\]) and (\[abssepcondgen\]) depends on the eigenvalues of the state under investigation so sometimes it is also known as separability from spectrum. Recently, the absolutely separable states are detected and characterized in [@nirman; @halder].\ This work is motivated by two earlier results and they are described as follows: Firstly, we can observe that the state used in solving the Deutsch-Jozsa (DJ) problem [@deutsch] is a pseudo-pure state (PPS) [@gershenfeld] which can be expressed as $$\begin{aligned} \rho_{PPS}^{(2)}=\epsilon |\psi\rangle\langle \psi|+\frac{1-\epsilon}{4}I_{4}, 0\leq \epsilon \leq 1 \label{pps}\end{aligned}$$ where $|\psi\rangle$ is any two-qubit pure state. If $|\psi\rangle$ represent any two-qubit pure maximally entangled state then it reduces to the two-qubit Werner state. The sufficient condition that the state $\rho_{PPS}^{(2)}$ is separable whenever [@braunstein] $$\begin{aligned} \epsilon < \frac{1}{9} \label{sepcondpps}\end{aligned}$$ The quantum algorithm of Deustch and Jozsa solves DJ problem with a single query while classical algorithm uses 3 queries if initially two-qubit PPS with parameter $\epsilon$ $(0\leq \epsilon \leq 1)$ is used in the algorithm. As the number of qubit increases in the initial PPS, the number of queries increases exponentially for classical algorithm while the quantum algorithm of Deustch and Jozsa still require single query and this provide the quantum advantage over classical algorithm. It has been shown that if $\epsilon \leq \frac{1}{33}$ then the initial separable PPS with which the computation has started remain separable throughout the entire computation [@biham]. This may imply that the initial PPS is absolutely separable for $0\leq \epsilon \leq \frac{1}{33}$. To verify this statement, let us consider the PPS described by the density operator $\rho_{PPS}$ given by $$\begin{aligned} \rho_{PPS}=\epsilon |\psi^{-}\rangle\langle \psi^{-}|+\frac{1-\epsilon}{4}I_{4}, 0\leq \epsilon \leq \frac{1}{33} \label{newpps}\end{aligned}$$ where $|\psi^{-}\rangle=\frac{1}{\sqrt{2}}(|01\rangle-|10\rangle)$.\ The eigenvalues of $\rho_{PPS}$ can be arranged in descending order as $\mu_{1}\geq \mu_{2}\geq \mu_{3}\geq \mu_{4}$, where $$\begin{aligned} \mu_{1}=\frac{1+3\epsilon}{4} ,\mu_{2}=\frac{1-\epsilon }{4}, \mu_{3}=\frac{1-\epsilon }{4} , \mu_{4}=\frac{1-\epsilon }{4} \label{eigvalnew}\end{aligned}$$ We find that the state $\rho_{PPS}$ is absolutely separable if and only if $$\begin{aligned} 0\leq \epsilon\leq \frac{1}{33} \label{abssepex}\end{aligned}$$ Again, the quantum discord of the state $\rho_{PPS}$ is given by [@luo] $$\begin{aligned} D(\rho_{PPS})&=&\frac{1-\epsilon}{4}log_{2}(1-\epsilon)-\frac{1+\epsilon}{2}log_{2}(1+\epsilon)\nonumber\\&&+ \frac{1+3\epsilon}{4}log_{2}(1+3\epsilon) \label{discordluo}\end{aligned}$$ In particular, if we choose the value of the parameter $\epsilon$ very close to zero, say $\epsilon=0.001$ then $D(\rho_{PPS})\simeq 1.441255\times 10^{-6}$. Therefore, the discord $D(\rho_{PPS})$ is very negligible and can be approximated to zero. Although the quantum correlation of the absolute separable state $\rho_{PPS}$ measured by quantum discord is very near to zero but still it is useful in quantum computation. This fact motivate us to search for the class of absolute separable state with zero discord.\ Secondly, the largest ball constructed for $d\otimes d$ quantum system is given by $Tr(\rho^{2})\leq \frac{1}{d^{2}-1}$, where the density operator $\rho$ representing either separable or absolutely separable states [@Zyczkowski; @gurvit]. In particular, it was shown that the largest ball for $2\otimes 2$ quantum system centered at maximally mixed state neither contain all separable states nor absolutely separable states [@kus]. To illustrate this, let us consider a state described by the density operator $\sigma_{1}$ given by $$\begin{aligned} \sigma_{1}=(\frac{1}{5}|0\rangle\langle 0|)-\frac{4}{5}|1\rangle\langle 1|)\otimes \frac{1}{2}I_{2} \label{abs1}\end{aligned}$$ The eigenvalues of $\sigma_{1}$ are given by $\frac{1}{10}, \frac{1}{10}, \frac{2}{5}, \frac{2}{5}$. It can be easily verified that the eigenvalues of $\sigma_{1}$ satisfy the condition (\[abssepcond\]). Thus the state $\sigma_{1}$ is absolutely separable state. Next, our task is to verify whether the state $\sigma_{1}$ satisfies the inequality $Tr(\sigma_{1}^{2})\leq \frac{1}{3}$. We find that $Tr(\sigma_{1}^{2})=\frac{17}{50}$ which is greater than $\frac{1}{3}$. This implies that the state $\sigma_{1}$ lies outside the ball described by $Tr(\rho^{2})\leq \frac{1}{3}$. To deal with this problem, we construct a ball for $2\otimes d$ quantum system and in particular, we have shown that the constructed ball contain almost all absolute separable states in $2\otimes 2$ dimensional Hilbert space.\ This work is organised as follows: In section-II, we derive the upper bound of $Tr(\varrho^{2})$, where the two-qubit zero discord states are described by the density operator $\varrho$. It is shown that the upper bound is state independent for certain classes of two-qubit zero discord state. These specific classes of two-qubit zero discord states satisfy the condition for the separability from spectrum. In particular, we unearth the class of two-qubit product states which are absolutely separable. Also we find that there exist states from the class of absolute separable zero discord states are not lying within and on a ball described by $Tr(\rho^{2})\leq \frac{1}{3}$. In section-III, we have constructed a ball of separable as well as absolute separable state in $2\otimes d$ dimensional system. In section-IV, we give few examples to support that the newly constructed ball is larger in size. It is evident from the fact that in $2\otimes 2$ dimensional system, it contain two-qubit absolute separable states which are lying not only inside but also outside the ball described by $Tr(\rho^{2})\leq \frac{1}{3}$. Further, we provided the example of absolute separable states in $2\otimes 3$ quantum system. In section-V, we end with concluding remarks. Identification of a class of absolutely separable states that does not contain quantum correlation ================================================================================================== In this section, we first derive the upper bound of $Tr(\rho_{ZD}^{2})$, where $\rho_{ZD}$ represent the zero discord state and thereby constructing a ball in which the zero discord state is lying. Then we show that there exist a class of zero discord state residing in the region within the ball, which is separable from spectrum. Construction of a ball that contain zero discord state ------------------------------------------------------ We construct a ball of zero discord state and to accomplish this task we derive an inequality in terms of $Tr(\rho_{ZD}^{2})$, where the density matrix $\rho_{ZD}$ denoting the zero discord state lying in $2\otimes 2$ dimensional Hilbert space. In general, the derived upper bound of the inequality is state dependent but we found some particular class of zero discord state for which the upper bound is independent of the state.\ To start with, let us consider a $2\otimes 2$ dimensional zero discord state $\rho_{ZD}$ that can be expressed as [@dakic] $$\begin{aligned} \rho_{ZD}= p|\psi\rangle\langle\psi|\otimes \rho_{1}+(1-p)|\psi_{\perp}\rangle\langle\psi_{\perp}|\otimes \rho_{2},~~0\leq p\leq 1 \label{zdstate}\end{aligned}$$ where the pure states $|\psi\rangle$ and $|\psi_{\perp}\rangle$ are orthogonal to each other i.e. $\langle\psi|\psi_{\perp}\rangle=0$. The single qubit density operator $\rho_{i} (i=1,2)$ are given by $$\begin{aligned} \rho_{i}= \frac{1}{2}I_{2}+\vec{r}_{i}.\vec{\sigma},~~i=1,2 \label{singlequbitdensity}\end{aligned}$$ $I_{2}$ represent a $2\times 2$ identity matrix, $\vec{r}_{i}=(r_{i1},r_{i2},r_{i3})$ denote the Bloch vector and the component of $\vec{\sigma}=(\sigma_{1},\sigma_{2},\sigma_{3})$ are usual Pauli matrices.\ **Theorem-1:** A two-qubit zero discord state $\rho_{ZD}$ satisfies the inequality $$\begin{aligned} Tr(\rho_{ZD}^{2})\leq min\{\frac{1}{2}+2|\vec{r}_{1}|^{2},\frac{1}{2}+2|\vec{r}_{2}|^{2}\} \label{th1}\end{aligned}$$ **Proof:** Let us start with the expression of $Tr(\rho_{ZD}^{2})$, which is given by $$\begin{aligned} Tr(\rho_{ZD}^{2})= p^{2}Tr(\rho_{1}^{2})+(1-p)^{2}Tr(\rho_{2}^{2}) \label{zdexpression}\end{aligned}$$ It can be seen that the value of $Tr(\rho_{ZD}^{2})$ is changing by varying the values of the parameter $p$ in the range $0\leq p \leq 1$, and the block vectors $\vec{r_{i}},i=1,2$ satisfying $|\vec{r_{i}}|^{2}\leq 1$. Thus one can ask for the upper bound of $Tr(\rho_{ZD}^{2})$. To probe this question, we assume that the zero discord state $\rho_{ZD}$ satisfies the inequality given by $$\begin{aligned} Tr(\rho_{ZD}^{2})\leq \alpha(\vec{r}_{i}),~~i=1,2 \label{assumption}\end{aligned}$$ regardless of the parameter $p$, where $\alpha(\vec{r}_{i})$ denote the parameter depend on the state parameter $\vec{r}_{i},i=1,2$.\ Our task is to find $\alpha(\vec{r}_{i})$. To search for $\alpha(\vec{r}_{i})$, we need to combine (\[zdexpression\]) and (\[assumption\]). Thus, we obtain $$\begin{aligned} &&p^{2}Tr(\rho_{1}^{2})+(1-p)^{2}Tr(\rho_{2}^{2})\leq \alpha(\vec{r}_{i}) \nonumber\\&& \Rightarrow p^{2}(Tr(\rho_{1}^{2})+Tr(\rho_{2}^{2}))-2pTr(\rho_{2}^{2})+\nonumber\\&&(Tr(\rho_{2}^{2})-\alpha(\vec{r}_{i}))\leq 0 \label{step1}\end{aligned}$$ Solving the inequality (\[step1\]) for the parameter $p$, we get $$\begin{aligned} a\leq p\leq b \label{step2a}\end{aligned}$$ where $a$ and $b$ are given by $$\begin{aligned} a=\frac{Tr(\rho_{2}^{2})-\sqrt{\alpha(\vec{r}_{i})(Tr(\rho_{1}^{2})+Tr(\rho_{2}^{2}))-Tr(\rho_{1}^{2})Tr(\rho_{2}^{2})}}{Tr(\rho_{1}^{2})+Tr(\rho_{2}^{2})} \label{step2b}\end{aligned}$$ and $$\begin{aligned} b=\frac{Tr(\rho_{2}^{2})+\sqrt{\alpha(\vec{r}_{i})(Tr(\rho_{1}^{2})+Tr(\rho_{2}^{2}))-Tr(\rho_{1}^{2})Tr(\rho_{2}^{2})}}{Tr(\rho_{1}^{2})+Tr(\rho_{2}^{2})} \label{step2c}\end{aligned}$$ We impose the condition on $a$ and $b$ in such a way so that $0\leq p\leq 1$ is satisfied. The required conditions are given below $$\begin{aligned} a\geq 0 \label{acond1}\end{aligned}$$ $$\begin{aligned} b\leq 1 \label{bcond2}\end{aligned}$$ The first condition (\[acond1\]) gives $$\begin{aligned} &&\frac{Tr(\rho_{2}^{2})-\sqrt{\alpha(\vec{r}_{i})(Tr(\rho_{1}^{2})+Tr(\rho_{2}^{2}))-Tr(\rho_{1}^{2})Tr(\rho_{2}^{2})}}{Tr(\rho_{1}^{2})+Tr(\rho_{2}^{2})}\geq 0\nonumber\\&& \Rightarrow \alpha(\vec{r}_{i})\leq Tr(\rho_{2}^{2}) \label{cond1}\end{aligned}$$ The second condition (\[bcond2\]) gives $$\begin{aligned} &&\frac{Tr(\rho_{2}^{2})+\sqrt{\alpha(\vec{r}_{i})(Tr(\rho_{1}^{2})+Tr(\rho_{2}^{2}))-Tr(\rho_{1}^{2})Tr(\rho_{2}^{2})}}{Tr(\rho_{1}^{2})+Tr(\rho_{2}^{2})}\leq 1\nonumber\\&& \Rightarrow \alpha(\vec{r}_{i})\leq Tr(\rho_{1}^{2}) \label{cond2}\end{aligned}$$ Therefore, (\[cond1\]) and (\[cond2\]) can be expressed jointly as $$\begin{aligned} \alpha(\vec{r}_{i}) \leq min\{Tr(\rho_{1}^{2}),Tr(\rho_{2}^{2})\} \label{condcomp}\end{aligned}$$ Now we calculate $Tr(\rho_{i}^{2})$ by recalling (\[singlequbitdensity\]), and it is given by $$\begin{aligned} Tr(\rho_{i}^{2})=\frac{1}{2}+2|\vec{r}_{i}|^{2}, ~~~i=1,2 \label{trrho}\end{aligned}$$ Using (\[condcomp\]) and (\[trrho\]), we get $$\begin{aligned} \alpha(\vec{r}_{i})\leq min\{\frac{1}{2}+2|\vec{r}_{1}|^{2},\frac{1}{2}+2|\vec{r}_{2}|^{2}\} \label{redineq}\end{aligned}$$ Combining the inequalities (\[assumption\]) and (\[redineq\]), we arrive at the required result given by $$\begin{aligned} Tr(\rho_{ZD}^{2}) \leq min\{\frac{1}{2}+2|\vec{r}_{1}|^{2},\frac{1}{2}+2|\vec{r}_{2}|^{2}\} \label{th1a}\end{aligned}$$ Geometrically, the inequality given by (\[th1a\]) represent a region within and on a ball containing zero discord state. From (\[th1a\]), it can be easily seen that the upper bound of $Tr(\rho_{ZD}^{2})$ depends on the local bloch vector $\vec{r}_{i}$ and hence the upper bound is state dependent. The state independent bound of $Tr(\rho_{ZD}^{2})$ can be obtained for particular classes of zero discord state and it is given in the corollary below.\ **Corollary-1:** The density operators $\rho_{ZD}^{(1)}$ and $\rho_{ZD}^{(2)}$ satisfy the inequality $$\begin{aligned} Tr([\rho_{ZD}^{(i)}]^{2})\leq \frac{1}{2},~~i=1,2 \label{cor1}\end{aligned}$$ where $\rho_{ZD}^{(1)}$ and $\rho_{ZD}^{(2)}$ denote the particular class of zero discord state given by $$\begin{aligned} \rho_{ZD}^{(1)}= p|\psi\rangle\langle\psi|\otimes \frac{1}{2}I_{2}+(1-p)|\psi_{\perp}\rangle\langle\psi_{\perp}|\otimes \rho_{2},\nonumber\\ \rho_{ZD}^{(2)}= p|\psi\rangle\langle\psi|\otimes \rho_{1}+(1-p)|\psi_{\perp}\rangle\langle\psi_{\perp}|\otimes \frac{1}{2}I_{2}, \label{zdstate1}\end{aligned}$$ $0\leq p\leq 1$.\ **Proof:** To prove it, consider the following two cases: (i) $|\vec{r}_{1}|^{2}\leq |\vec{r}_{2}|^{2}$, (ii) $|\vec{r}_{2}|^{2}\leq |\vec{r}_{1}|^{2}$.\ **Case-I:** If $|\vec{r}_{1}|^{2}\leq |\vec{r}_{2}|^{2}$ then theorem-1 gives $$\begin{aligned} Tr(\rho_{ZD}^{2}) \leq \frac{1}{2}+2|\vec{r}_{1}|^{2} \label{ineq1}\end{aligned}$$ In particular, the inequality (\[ineq1\]) holds even if we take the minimum value of the expression $\frac{1}{2}+2|\vec{r}_{1}|^{2}$ over all $\vec{r}_{1}$. Therefore, we have $$\begin{aligned} Tr(\rho_{ZD}^{2}) \leq min_{\vec{r}_{1}}[\frac{1}{2}+2|\vec{r}_{1}|^{2}] \label{ineq2}\end{aligned}$$ We obtain $min_{\vec{r}_{1}}[\frac{1}{2}+2|\vec{r}_{1}|^{2}]=\frac{1}{2}$ and the minimum value is attained when $\vec{r}_{1}=\vec{0}$. Thus, the minimum value is obtained when the state $\rho_{ZD}$ reduces to $\rho_{ZD}^{(1)}$. Hence the inequality (\[ineq2\]) reduces to $$\begin{aligned} Tr([\rho_{ZD}^{(1)}]^{2})\leq \frac{1}{2} \label{cor1a}\end{aligned}$$ **Case-II:** If $|\vec{r}_{2}|^{2}\leq |\vec{r}_{1}|^{2}$ then we can proceed in a similar way as in case-I and obtain $Tr([\rho_{ZD}^{(2)}]^{2})\leq \frac{1}{2}$.\ Therefore, we have obtained the particular classes of zero discord states described by the density operators $\rho_{ZD}^{(i)} (i=1,2)$ given in (\[zdstate1\]) satisfy the inequality $Tr([\rho_{ZD}^{(i)}]^{2})\leq \frac{1}{2} (i=1,2)$. Thus, the upper bound does not depend on the state $\rho_{ZD}^{(i)} (i=1,2)$. Class of zero discord state which is separable from spectrum ------------------------------------------------------------ Let us consider a class of zero discord state either described by the density operator $\rho_{ZD}^{(1)}$ or $\rho_{ZD}^{(2)}$ given in (\[zdstate1\]). Recalling $\rho_{ZD}^{(1)}= p|\psi\rangle\langle\psi|\otimes \frac{1}{2}I_{2}+(1-p)|\psi_{\perp}\rangle\langle\psi_{\perp}|\otimes \rho_{2}$ with the single qubit density operator $\rho_{2}$ given by (\[singlequbitdensity\]) and a pair of orthogonal pure states $|\psi\rangle$ and $|\psi_{\perp}\rangle$, where $|\psi\rangle=\alpha|0\rangle+\beta|1\rangle$ and $|\psi_{\perp}\rangle=\beta|0\rangle-\alpha|1\rangle$. We assume that the parameters $\alpha$ and $\beta$ are real number satisfying $\alpha^{2}+\beta^{2}=1$. Therefore, the density matrix for $\rho_{ZD}^{(1)}$ is given by $$\begin{aligned} \rho_{ZD}^{(1)}= \begin{pmatrix} a_{11} & a_{12} & a_{13} & a_{14} \\ a_{12}^{*} & a_{22} & a_{23} & a_{24} \\ a_{13}^{*} & a_{23}^{*} & a_{33} & a_{34} \\ a_{14}^{*} & a_{24}^{*} & a_{34}^{*} & a_{44} \end{pmatrix}, \sum_{i=1}^{4}a_{ii}=1\end{aligned}$$ where $$\begin{aligned} &&a_{11}=p\frac{\alpha^{2}}{2}+(1-p)\beta^{2}(\frac{1}{2}+r_{23}),\nonumber\\&& a_{12}=(1-p)\beta^{2}(r_{21}-ir_{22}),\nonumber\\&& a_{13}=p\frac{\alpha\beta}{2}-(1-p)\alpha\beta(\frac{1}{2}+r_{23}),\nonumber\\&& a_{14}=-(1-p)\alpha\beta(r_{21}-ir_{22}), \nonumber\\&& a_{22}=p\frac{\alpha^{2}}{2}+(1-p)\beta^{2}(\frac{1}{2}-r_{23}),\nonumber\\&& a_{23}=-(1-p)\alpha\beta(r_{21}+ir_{22}),\nonumber\\&& a_{24}=p\frac{\alpha\beta}{2}-(1-p)\alpha\beta(\frac{1}{2}-r_{23}),\nonumber\\&& a_{33}=p\frac{\beta^{2}}{2}+(1-p)\alpha^{2}(\frac{1}{2}+r_{23}),\nonumber\\&& a_{34}=(1-p)\alpha^{2}(r_{21}-ir_{22}),\nonumber\\&& a_{44}=p\frac{\beta^{2}}{2}+(1-p)\alpha^{2}(\frac{1}{2}-r_{23}) \label{matrix}\end{aligned}$$ The eigenvalues of $\rho_{ZD}^{(1)}$ are given by $$\begin{aligned} &&\lambda_{1}=\frac{1-p}{2}(1+2|\vec{r}_{2}|),\lambda_{2}=\frac{1-p}{2}(1-2|\vec{r}_{2}|)\nonumber\\&& \lambda_{3}=\lambda_{4}=\frac{p}{2} \label{eigval}\end{aligned}$$ The state $\rho_{ZD}^{(1)}$ satisfy the positive semi-definiteness property if $$\begin{aligned} |\vec{r}_{2}|\leq \frac{1}{2} \label{possem}\end{aligned}$$ Now our task reduces to the following; (i) verify whether the class of states $\rho_{ZD}^{(1)}$ satisfy the condition of separability from spectrum and (ii) if the class of states verified as absolute separable states then find out whether they lying within the ball described by $Tr([\rho_{ZD}^{(1)}]^{2})\leq\frac{1}{3}$. In this context, a table is constructed by taking different ranges of the parameter $p$ and some values of $|\vec{r}_{2}|$ for which we find that the zero discord state described by the density operator $\rho_{ZD}^{(1)}$ satisfy the inequality (\[abssepcond\]). This means that there exist classes of two-qubit zero discord states that are absolutely separable also. We call these classes of two-qubit states as Absolutely Separable Zero Discord Class $(ASZDC)$. Further, we have constructed another table which reveals the fact that whether the class of states given by $ASZDC$ satisfies the inequality $Tr([\rho_{ASZDC}]^{2})\leq\frac{1}{3}$. Without any loss of generality, we have verified the above two tasks by considering the values of the parameter $p$ in $[0,\frac{1}{2}]$ and taking few values of $|\vec{r_{2}}|$. Similar analysis can be done for other range the parameter $p\in [\frac{1}{2},1]$ and other values of $|\vec{r_{2}}|\leq \frac{1}{2}$.\ ------------------- ---------------- ----------------------------------- -------------------- $Parameter$ $Parameter$ $\lambda_{1}-\lambda_{3}$ $Nature~of~state$ $(|\vec{r}_{2}|)$ (p) $-2\sqrt{\lambda_{2}\lambda_{4}}$ 0 \[0, 0.15) positive Separable 0 \[0.15, 0.5\] Negative Absolute separable 0.1 \[0, 0.213) positive Separable 0.1 \[0.213, 0.5\] Negative Absolute separable 0.2 \[0, 0.291) positive Separable 0.2 \[0.291, 0.5\] Negative Absolute separable 0.3 \[0, 0.38) positive Separable 0.3 \[0.38, 0.5\] Negative Absolute separable 0.4 \[0, 0.483) positive Separable 0.4 \[0.483, 0.5\] Negative Absolute separable 0.5 \[0, 0.5\] positive Separable ------------------- ---------------- ----------------------------------- -------------------- : Table verifying whether the state $\rho_{ZD}^{(1)}$ satisfy (\[abssepcond\]) ------------------- ------------------- -------------------------- ------------ $Parameter$ $Parameter$ $Tr([\rho_{ASZDC}]^{2})$ $State$ $(|\vec{r}_{2}|)$ (p) $\leq\frac{1}{3}$ $residing$ 0 \[0.15, 0.211) Violated Outside 0 \[0.211, 0.5\] Satisfied Inside 0.1 \[0.213, 0.2325) Violated Outside 0.1 \[0.2325, 0.5\] Satisfied Inside 0.2 \[0.291, 0.29205) Violated Outside 0.2 \[0.29205, 0.5\] Satisfied Inside 0.3 \[0.38, 0.38056) Violated Outside 0.3 \[0.38056, 0.5\] Satisfied Inside 0.4 \[0.483, 0.49) Violated Outside 0.4 \[0.49, 0.5\] Satisfied Inside ------------------- ------------------- -------------------------- ------------ : Table shows that whether the absolute separable state given in Table-I is residing inside or outside the ball described by $Tr([\rho_{ASZDC}]^{2})\leq\frac{1}{3}$ Since the maximal ball described by $Tr([\rho_{ASZDC}]^{2})\leq\frac{1}{3}$ does not contain all states from the class ASZDC and such states lying outside the ball so we investigate in the next section that whether it is possible to increase the size of the maximal ball. Constructing the bigger ball of separable as well as absolutely separable states around maximally mixed state ============================================================================================================= In this section, we will show that it is possible to construct a ball which is larger than the earlier constructed ball described by $Tr(\rho^{2})\leq\frac{1}{3}$ where the state $\rho$ represent either separable or absolutely separable states around maximally mixed state. This means that there is a possibility for the new ball, constructed in this work, to contain those separable as well as absolute separable states which are lying outside the ball described by $Tr(\rho^{2})\leq\frac{1}{3}$. A Few Definitions and Results ----------------------------- Firstly, we recapitulate a few definitions and earlier obtained results which are required to construct a new ball.\ **Definition-1:** *p*-norm of a matrix $A$ is defined as $$\begin{aligned} (\|A\|_{p})^{p}=Tr(A^{\dagger}A)^{\frac{p}{2}} \label{pnorm}\end{aligned}$$ In particular for $p=2$ and $A=\rho$, where $\rho$ denoting a quantum state, we have $$\begin{aligned} (\|\rho\|_{2})^{2}=Tr(\rho^{2}) \label{pnorm1}\end{aligned}$$ **Definition-2: [@hilderbrand]** A quantum state $\rho \in H_{2}\otimes H_{d}$ is absolutely separable if $U\rho U^{\dagger}$ remain a separable state for all global unitary operator $U \in U(2d)$.\ If we denote $\rho'=U\rho U^{\dagger}$ then it can be easily shown that $Tr[(\rho')^{2}]=Tr[(\rho)^{2}]$, i.e. $Tr[(\rho)^{2}]$ is invariant under unitary transformation.\ **Result-1 [@king]:** Let $M$ be a $2d\times 2d$ positive semi-definite matrix expressed in the block form as $$\begin{aligned} M= \begin{pmatrix} A & C \\ C^{\dag} & B \end{pmatrix}\end{aligned}$$ where $A,B,C$ are $d\times d$ matrices.\ If we define the $2\times 2$ matrix as $$\begin{aligned} m= \begin{pmatrix} \|A\|_{p} & \|C\|_{p}\\ \|C\|_{p} & \|B\|_{p} \end{pmatrix}\end{aligned}$$ then the following inequalities hold:\ (a) for $1\leq p \leq 2$,\ $$\begin{aligned} \|M\|_{p}\geq \|m\|_{p} \label{inequalitya}\end{aligned}$$ (b) for $2\leq p < \infty$,\ $$\begin{aligned} \|M\|_{p}\leq \|m\|_{p} \label{inequalityb}\end{aligned}$$ Thus for $p=2$, we have $$\begin{aligned} \|M\|_{2}= \|m\|_{2} \label{inequality}\end{aligned}$$ **Result-2 [@johnston]:** Let us choose $d\times d$ matrices $A,B,C$ such that $A$ and $B$ are positive semi-definite matrices. Then the block matrix $$\begin{aligned} X= \begin{pmatrix} A & C \\ C^{\dag} & B \end{pmatrix}\end{aligned}$$ is separable if $\|C\|_{2}^{2}\leq \lambda_{min}(A)\lambda_{min}(B)$, where $\lambda_{min}(A)$ and $\lambda_{min}(B)$ denoting the minimum eigenvalue of the matrices $A$ and $B$ respectively.\ Construction of a new ball that contain separable as well as absolutely separable states ---------------------------------------------------------------------------------------- Let us consider a quantum state described by the density matrix $\rho \in H_{2}\otimes H_{d}$. The density matrix can be written in the block form as $$\begin{aligned} \rho= \begin{pmatrix} X & Y \\ Y^{\dagger} & Z \end{pmatrix}\end{aligned}$$ where $X,Y,Z$ denoting $d\times d$ matrices with $X,Z\geq 0$.\ Using Result-1, we have $$\begin{aligned} \|\rho\|_{2}= \|\begin{pmatrix} X & Y\\ Y^{\dagger} & Z \end{pmatrix}\|_{2}=\|\begin{pmatrix} \|X\|_{2} & \|Y\|_{2}\\ \|Y\|_{2} & \|Z\|_{2} \end{pmatrix}\|_{2}\end{aligned}$$ Let us now calculate the value of $Tr(\rho^{2})$. It is given by $$\begin{aligned} Tr(\rho^{2})&=&\|\rho\|_{2}^{2}= \|X\|_{2}^{2}+2\|Y\|_{2}^{2}+\|Z\|_{2}^{2}\nonumber\\&=& Tr(X^{2})+2\|Y\|_{2}^{2}+Tr(Z^{2})\nonumber\\&\leq& Tr(X^{2})+2\lambda_{min}(X)\lambda_{min}(Z)\nonumber\\&+&Tr(Z^{2}) \label{trace}\end{aligned}$$ where $\lambda_{min}(X)$ and $\lambda_{min}(Z)$ denoting the minimum eigenvalues of the block matrices $X$ and $Z$ respectively. The last inequality follows from Result-2. Therefore, the state $\rho$ is separable if $$\begin{aligned} Tr(\rho^{2})\leq Tr(X^{2})+2\lambda_{min}(X)\lambda_{min}(Z)+Tr(Z^{2}) \label{trace1}\end{aligned}$$ The state described by the density operator $\rho$ is absolutely separable if for any global unitary transformation $U \in U(2d)$, the inequality $$\begin{aligned} Tr[(U\rho U^{\dagger})^{2}]=Tr(\rho^{2})&\leq& Tr(X^{2})+2\lambda_{min}(X)\lambda_{min}(Z)\nonumber\\&+&Tr(Z^{2}) \label{abscond}\end{aligned}$$ holds.\ It can be observed that the upper bound of the inequality (\[abscond\]) depends on the parameter of the state under consideration. Thus the upper bound is state dependent and it can be maximized over the given range of the parameter of the state. We grasp this idea to show that there is a possibility to increase the size of the ball that contains more separable as well as absolutely separable state compared to $Tr(\rho^{2})\leq \frac{1}{3}$. Illustrations ============= In this section, we will show with examples that the new ball constructed in this work described by (\[abscond\]) contains more two-qubit absolutely separable states than the ball descibed by $Tr(\rho^{2})\leq \frac{1}{3}$. Also, we discuss about the absolute separable states in $2\otimes 3$ quantum system. ### Two-qubit class of sates from ASZDC Let us consider a subclass of the two-qubit quantum state belong to ASZDC described by the density operator $\rho^{(1)}$ as $$\begin{aligned} \rho^{(1)}&=& (p|0\rangle\langle 0|-(1-p)|1\rangle\langle 1|)\otimes \frac{1}{2}I_{2},\nonumber\\&& 0\leq p\leq 1 \label{class1a}\end{aligned}$$ where $I_{2}$ represent the identity matrix of order 2. The state $\rho^{(1)}$ is a product state and thus separable for  $0\leq p\leq 1$.\ The matrix representation of $\rho^{(1)}$ is given by $$\begin{aligned} \rho^{(1)}= \begin{pmatrix} X & Y \\ Y^{\dagger} & Z \end{pmatrix}\end{aligned}$$ where $Y$ is a null matrix and the matrices $X$ and $Z$ are given by $$\begin{aligned} &&X= \begin{pmatrix} \frac{p}{2}& 0 \\ 0 & \frac{p}{2} \end{pmatrix},\nonumber\\&& Z=\begin{pmatrix} \frac{1-p}{2}& 0 \\ 0 & \frac{1-p}{2} \end{pmatrix}\end{aligned}$$ The eigenvalues of $\rho^{(1)}$ are given by $\frac{p}{2}, \frac{p}{2},\frac{1-p}{2},\frac{1-p}{2}$.\ Case-I: When the parameter $p$ is lying in the interval $[0,\frac{1}{2}]$ then the eigenvalues are arranged in descending order as $\lambda_{1}\geq \lambda_{2}\geq\lambda_{3}\geq\lambda_{4}$, where $$\begin{aligned} \lambda_{1}=\frac{1-p}{2}, \lambda_{2}=\frac{1-p}{2},\lambda_{3}=\frac{p}{2},\lambda_{4}=\frac{p}{2} \label{eigenvalue1}\end{aligned}$$ The state $\rho^{(1)}$ is separable from spectrum if $$\begin{aligned} p+\sqrt{p(1-p)}\geq \frac{1}{2} \label{eigenvalue11}\end{aligned}$$ The inequality (\[eigenvalue11\]) holds if $\frac{3}{20}\leq p\leq 1/2$. Therefore, the state $\rho^{(1)}$ is absolutely separable for $p \in [\frac{3}{20},\frac{1}{2}]$.\ Now, $Tr[(\rho^{(1)})^{2}]$ can be calculated as $$\begin{aligned} Tr[(\rho^{(1)})^{2}]=\frac{p^{2}}{2}+\frac{(1-p)^{2}}{2} \label{trace11}\end{aligned}$$ From Fig.1, it can be seen that there exist absolutely separable states for $p \in [\frac{3}{20}, \frac{21}{100}]$ that are lying outside the ball described by $Tr[(\rho^{(1)})^{2}]\leq\frac{1}{3}$. Thus, it is interesting to see whether the newly constructed ball contain all the absolutely separable states for $p \in [\frac{3}{20}, \frac{21}{100}]$. To probe this, we calculate the upper bound of $Tr[(\rho^{(1)})^{2}]$ using the inequality (\[abscond\]). The upper bound is given by $$\begin{aligned} &&Tr(X^{2})+2\lambda_{min}(X)\lambda_{min}(Z)+Tr(Z^{2})\nonumber\\&&=\frac{1}{2}[1-p(1-p)] \label{ub}\end{aligned}$$ Again, Fig.1 shows that the newly constructed ball described by $Tr[(\rho^{(1)})^{2}]\leq \frac{1}{2}[1-p(1-p)]$ contains all absolutely separable belong to the class described by the density operator $\rho^{(1)}$. ![Plot of $Tr[(\rho^{(1)})^{2}]$ versus the state parameter $p$ ](figure-1.pdf) Case-II: In a similar fashion, the case where $p \in [\frac{1}{2},1]$ can be analyzed. ### $2\times 2$ isotropic state Let us consider a $2\otimes 2$ isotropic state given by $$\begin{aligned} \rho_{2\otimes 2}^{(iso)}(f)= \begin{pmatrix} \frac{1+2f}{6} & 0 & 0 & \frac{4f-1}{6} \\ 0 & \frac{1-f}{3} & 0 & 0 \\ 0 & 0 & \frac{1-f}{3} & 0 \\ \frac{4f-1}{6} & 0 & 0 & \frac{1+2f}{6} \end{pmatrix}, 0\leq f\leq 1\end{aligned}$$ It is known that the state described by the density operator $\rho_{2\otimes 2}^{(iso)}$ is separable for $0\leq f\leq \frac{1}{2}$. Further, it can be easily verified that all separable states in the class represented by $\rho_{2\times 2}^{(iso)}$ are also absolute separable states.\ The matrix of $2\otimes 2$ isotropic state can be re-expressed in terms of block matrices of order $2\times 2$ as $$\begin{aligned} \rho_{2\otimes 2}^{(iso)}(f)= \begin{pmatrix} X & Y \\ Y^{\dagger} & Z \end{pmatrix}\end{aligned}$$ where $2\times 2$ block matrices $X,Y$ and $Z$ are given by $$\begin{aligned} X= \begin{pmatrix} \frac{1+2f}{6} & 0\\ 0 & \frac{1-f}{3} \end{pmatrix}, Y=\begin{pmatrix} 0 & \frac{4f-1}{6}\\ 0 & 0 \end{pmatrix}, Z=\begin{pmatrix} \frac{1-f}{3} & 0\\ 0 & \frac{1+2f}{6} \end{pmatrix}\end{aligned}$$ The minimum eigenvalue of the block matrices $X$ and $Z$ are given by $$\begin{aligned} \lambda_{min}(X)=\lambda_{min}(Z)&=&\frac{1+2f}{6},~~0\leq f\leq \frac{1}{4}\nonumber\\&=& \frac{1-f}{3},~~\frac{1}{4}\leq f\leq \frac{1}{2} \label{mineigen}\end{aligned}$$ We now discuss two cases based on different ranges of the parameter $f$.\ **Case-I:** When $0\leq f \leq \frac{1}{4}$ $$\begin{aligned} Tr[(\rho_{2\otimes 2}^{(iso)}(f))^{2}]\leq \frac{2f^{2}+1}{3} \label{trace1}\end{aligned}$$ Since $f \in [0,\frac{1}{4}]$ so (\[trace1\]) can be re-expressed as $$\begin{aligned} Tr[(\rho_{2\otimes 2}^{(iso)}(f))^{2}]\leq Max_{0\leq f\leq\frac{1}{4}}\frac{2f^{2}+1}{3} \label{trace11}\end{aligned}$$ Since $\frac{2f^{2}+1}{3}$ is an increasing function of the parameter $f$ so its maximum value is attained at $f=\frac{1}{4}$. Therefore, $$\begin{aligned} Max_{0\leq f\leq\frac{1}{4}}\frac{2f^{2}+1}{3}=\frac{3}{8} \label{max1}\end{aligned}$$ Thus, the state $\rho_{2\otimes 2}^{(iso)}(f)$ satisfies the inequality given by $$\begin{aligned} Tr[(\rho_{2\otimes 2}^{(iso)}(f))^{2}]\leq \frac{3}{8} \label{case1}\end{aligned}$$ **Case-II:** When $\frac{1}{4}\leq f \leq \frac{1}{2}$ $$\begin{aligned} Tr[(\rho_{2\otimes 2}^{(iso)}(f))^{2}]\leq \frac{4f^{2}-4f+3}{6} \label{trace2}\end{aligned}$$ Since $f \in [\frac{1}{4},\frac{1}{2}]$ so (\[trace2\]) can be reexpressed as $$\begin{aligned} Tr[(\rho_{2\otimes 2}^{(iso)}(f))^{2}]\leq Max_{\frac{1}{4}\leq f\leq\frac{1}{2}}\frac{4f^{2}-4f+3}{6} \label{trace22}\end{aligned}$$ Since $\frac{4f^{2}-4f+3}{6}$ is a decreasing function of the parameter $f$ so its maximum value is attained at $f=\frac{1}{4}$. Therefore, $$\begin{aligned} Max_{\frac{1}{4}\leq f\leq\frac{1}{2}}\frac{4f^{2}-4f+3}{6}= \frac{3}{8} \label{max2}\end{aligned}$$ Thus, in this case also the state $\rho_{2\otimes 2}^{(iso)}(f)$ obey the inequality given by $$\begin{aligned} Tr[(\rho_{2\otimes 2}^{(iso)}(f))^{2}]\leq \frac{3}{8} \label{case2}\end{aligned}$$ Combining the above two cases, it can be concluded that the state $\rho_{2\otimes 2}^{(iso)}(f)$ satisfy the inequality $$\begin{aligned} Tr[(\rho_{2\otimes 2}^{(iso)}(f))^{2}]\leq\frac{3}{8}, 0\leq f\leq\frac{1}{2} \label{abssep1}\end{aligned}$$ Since $\frac{3}{8}>\frac{1}{3}$ so the new ball described by (\[abssep1\]) is bigger in size compared to the ball described by $Tr[(\rho_{2\otimes 2}^{(iso)}(f))^{2}]\leq \frac{1}{3}$ and hence the new ball contains more absolutely separable state. ### Class of states in $2\otimes 3$ quantum system Let us consider a class of states in $2\otimes 3$ quantum system parameterized with two parameters $\alpha$ and $\gamma$, which is given by [@chi] $$\begin{aligned} &&\rho_{\alpha,\gamma}^{2\otimes 3}=\alpha(|02\rangle\langle 02|+|12\rangle\langle 12|)+\frac{4\gamma+2\alpha-1}{3} |\psi^{-}\rangle\langle \psi^{-}|+\nonumber\\&& \frac{1-\gamma-2\alpha}{3}(|00\rangle\langle 00|+|01\rangle\langle 01|+|10\rangle\langle 10|+|11\rangle\langle 11|),~~\nonumber\\&& 0\leq \alpha \leq \frac{1}{2},~~ 0\leq \gamma \leq 1 \label{state23}\end{aligned}$$ where $|\psi^{-}\rangle=\frac{1}{\sqrt{2}}(|01\rangle-|10\rangle)$. The state is separable if and only if $\alpha+\gamma\leq \frac{1}{2}$.\ To simplify the calculation, let us choose $\gamma=\frac{1}{3}$. For this particular case, the state $\rho_{\alpha,\frac{1}{3}}^{2\otimes 3}$ is separable if and only if $0\leq \alpha\leq \frac{1}{6}$. Therefore, with this chosen value of $\gamma$, we can re-express the state $\rho_{\alpha,\frac{1}{3}}^{2\otimes 3}$ in terms of block matrices as $$\begin{aligned} \rho_{\alpha,\frac{1}{3}}^{2\otimes 3}= \begin{pmatrix} X_{1} & Y_{1} \\ Y_{1}^{\dagger} & Z_{1} \end{pmatrix}\end{aligned}$$ where $3\times 3$ block matrices $X_{1},Y_{1}$ and $Z_{1}$ are given by $$\begin{aligned} &&X_{1}= \begin{pmatrix} \frac{2-6\alpha}{9} & 0 & 0\\ 0 & \frac{5-6\alpha}{18} & 0\\ 0 & 0 & \alpha \end{pmatrix}, Y_{1}=\begin{pmatrix} 0 & -\frac{1+6\alpha}{18} & 0\\ 0 & 0 & 0\\ 0 & 0 & 0 \end{pmatrix},\nonumber\\&& Z_{1}=\begin{pmatrix} \frac{5-6\alpha}{18} & 0 & 0\\ 0 & \frac{2-6\alpha}{9} & 0\\ 0 & 0 & \alpha \end{pmatrix}, 0\leq \alpha \leq \frac{1}{6}\end{aligned}$$ The eigenvalues of the state $\rho_{\alpha,\frac{1}{3}}^{2\otimes 3}$ arranged in descending order $(\varepsilon_{1}\geq \varepsilon_{2} \geq \varepsilon_{3}\geq \varepsilon_{4}\geq \varepsilon_{5}\geq \varepsilon_{6})$ for different ranges of $\alpha$ as\ (i) When $0\leq \alpha \leq 0.134$ $$\begin{aligned} \varepsilon_{1}=\frac{1}{3}, \varepsilon_{2}=\varepsilon_{3}=\varepsilon_{4}=\frac{2-6\alpha}{9}, \varepsilon_{5}=\varepsilon_{6}=\alpha \label{eigenvsl23a}\end{aligned}$$ (ii) When $0.134 \leq \alpha \leq \frac{1}{6}$ $$\begin{aligned} \varepsilon_{1}=\frac{1}{3}, \varepsilon_{2}=\varepsilon_{3}=\alpha, \varepsilon_{4}=\varepsilon_{5}=\varepsilon_{6}=\frac{2-6\alpha}{9} \label{eigenvsl23b}\end{aligned}$$ It can be easily verified using (\[abssepcondgen\]) that the state $\rho_{\alpha,\frac{1}{3}}^{2\otimes 3}$ represent absolute separable state for $0.019\leq \alpha \leq \frac{1}{6}$.\ Further, we find that the class of absolute separable states described by the density operator $\rho_{\alpha,\frac{1}{3}}^{2\otimes 3}$ lying within the ball described by (\[abscond\]) for $0.084\leq \alpha \leq \frac{1}{6}$. Thus, the new ball contain most of the absolute separable states but does not contain all absolute separable states. Conclusion ========== To summarize, we have characterize the absolute separable states in terms of quantum correlation which can be measured by quantum discord. We found an instance of absolute separable states with such negligible amount of quantum correlation that can be approximated to zero but still it is useful in quantum algorithm to solve Deutsch-Jozsa problem. Since these absolute separable states have approximately zero quantum correlation so we expect that it can be prepared in the experiment easily and not only that these states give quantum advantage over classical with respect to the running time of the algorithms. This prompted us to investigate about the structure of the class of absolute separable states with zero discord. 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--- abstract: 'Tunable beam splitter (TBS) is a fundamental component which has been widely used in optical experiments. We realize a polarization-independent orbital-angular-momentum-preserving TBS based on the combination of modified polarization beam splitters and half-wave plates. Greater than 30 dB of the extinction ratio of tunableness, lower than $6\%$ of polarization dependence and more than 20 dB of the extinction ratio of OAM preservation show the relatively good performance of the TBS. In addition, the TBS can save about 3/4 of the optical elements compared with the existing scheme to implement the same function[@yang2016experimental], which makes it have great advantages in scalable applications. Using this TBS, we experimentally built a Sagnac interferometer with the mean visibility of more than $99\%$, which demonstrates its potential applications in quantum information process, such as quantum cryptography.' author: - 'Ya-Ping Li' - 'Fang-Xiang Wang' - Wei Chen - 'Guo-Wei Zhang' - 'Zhen-Qiang Yin' - 'De-Yong He' - Shuang Wang - 'Guang-Can Guo' - 'Zheng-Fu Han' title: 'A resource-saving realization of the polarization-independent orbital-angular-momentum-preserving tunable beam splitter' --- Orbital angular momentum (OAM) has recently attracted a growing interest as a high-dimensional resource for quantum information. Beams of OAM-carrying photons have an azimuthal phase dependence in the form of $e^{il\phi}$, where topological charge $l$ can take any integer value[@franke2008advances][@yao2011orbital][@willner2015optical]. Due to its unique property, OAM light can be applied to many fields, such as quantum entanglement[@weihs2001entanglement][@leach2010quantum], quantum simulation[@cardano2015quantum][@luo2015quantum] and quantum communication[@molina2004triggered][@vallone2014free]. However, dedicated techniques are necessary for manipulating and transmitting OAM of photons. Up to now, researchers have designed many optical elements for the translation and manipulation of OAM light, such as the OAM fiber[@ramachandran2009generation][@bozinovic2013terabit]and Q-plate[@marrucci2006optical][@d2012deterministic]. Meanwhile, tailored optical devices which can achieve fundamental optical functions are necessary as well. Among these devices, tunable beam splitter (TBS) is an essential element to compose complex optical structures[@higgins2007entanglement][@ma2011quantum]. There are three major methods to implement TBS. The first common realization of TBS is using the combination of a polarization beam splitter (PBS) and a half-wave plate (HWP), which has high extinction ratio, while it is polarization-dependent[@marcikic2003long]. Another type of polarization-independent TBS employs Mach-Zehender interferometers (MZIs) with high-speed modulators[@ma2011high]. Although this method can realize high-speed modulation, it is sensitive to external environment disturbance in different light path, such as the vibrations and temperature variations. Moreover, it has a relative low extinction ratio in some specific high-precision applications[@ma2011high]. Recently, Yang et al. realizes a polarization-independent TBS using the MZI composed of beam displacers (BD) and HWPs, in which the TBS has a relatively high polarization independence and high interference visibility. But it is sensitive to the phase in different paths and it has a complicated construction[@yang2016experimental]. Here, we propose a polarization-independent OAM-preserving TBS based on HWPs and modified PBSs. The relatively low polarization dependence combining high extinction ratio of the TBS can save about 3/4 of the number of optical elements compared with the work in [@yang2016experimental]. The realization of Sagnac interferometer with interference visibility of above $99\%$ based on the TBS demonstrates its relatively good performance. ![Schematic diagrams of PBS and TBS. HWP, half wave plate; PBS, polarization beam splitter[]{data-label="fig:PBS_TBS"}](PBS_TBS.eps) The structure of the TBS is shown in Fig. \[fig:PBS\_TBS\] (a). The TBS is composed of two modified PBSs and three HWPs, in which $HWP_{\uppercase\expandafter{\romannumeral2}}$ is used for adjusting the splitting ratio and $HWP_{\uppercase\expandafter{\romannumeral1}}$, $HWP_{\uppercase\expandafter{\romannumeral3}}$ are used to eliminate polarization dependence of the TBS. For a common cubic beam splitter (BS) or PBS, it changes the sign of topological charge $l$ of OAM light after a reflection. Therefore, the transmission matrix of BS or PBS is no longer a unitary matrix for OAM-carrying light, which may introduce inconvenience in some experiments[@mafu2013higher][@zhang2016engineering]. The modified PBS used in the TBS is composed of two rhombic prisms (Sunlight Technology Co., H-K9L material), as shown in Fig. \[fig:PBS\_TBS\] (b). While a horizontal (vertical) polarization state enters into modified PBS from $Port_1$, it will exit from $Port_4$ ($Port_3$) after twice reflections and once transmission (only twice reflections). Therefore when photons that carry an OAM of $l\hbar$ enter into the PBS, the topological charge $l$ is preserved. Meanwhile the HWP does not change the topological charge. Thus, the TBS will be OAM-preserving. The property of polarization-independent of the TBS will be discussed in detail below. In a word, when an OAM photon with a quantum number of $l$ and arbitrary polarization state enters into the TBS from $Port_1$ or $Port_2$, it will exit from $Port_5$ and $Port_6$, preserving the original polarization state carrying OAM as shown in Fig. \[fig:PBS\_TBS\] (a). The transmission matrix of the TBS can be specified using Dirac notation. When photons enter into $PBS_{\uppercase\expandafter{\romannumeral1}}$ from $Port_1$, and exit from $Port_3$ and $Port_4$, as shown in Fig. \[fig:PBS\_TBS\] (b). The operator of the $PBS_{\uppercase\expandafter{\romannumeral1}}$ can be described as: $$M_{PBS_{\uppercase\expandafter{\romannumeral1}}} = \Ket{h\otimes l,4}\Bra{h\otimes l,1}+\Ket{v\otimes l,3}\Bra{v\otimes l,1}$$ where 1 (3 and 4) means the $Port_1$ ($Port_3$ and $Port_4$) of the PBS. $h\otimes l$ ($v\otimes l$) denotes the incident horizontal (vertical) polarization state with OAM of $l\hbar$. Then the light enters into $HWP_{\uppercase\expandafter{\romannumeral2}}$ from $Port_3$ ($Port_4$) of $PBS_{\uppercase\expandafter{\romannumeral1}}$, whose operator can be described as: $$\begin{aligned} M_{HWP_{\uppercase\expandafter{\romannumeral2}}-3} &= cos2\theta\Ket{h\otimes l,3}\Bra{h\otimes l,3}+sin2\theta\Ket{h\otimes l,3}\Bra{v\otimes l,3} \\ &+sin2\theta\Ket{v\otimes l,3}\Bra{h\otimes l,3}-cos2\theta\Ket{v\otimes l,3}\Bra{v\otimes l,3} \end{aligned}$$ $$\begin{aligned} M_{HWP_{\uppercase\expandafter{\romannumeral2}}-4} &= cos2\theta\Ket{h\otimes l,4}\Bra{h\otimes l,4}+sin2\theta\Ket{h\otimes l,4}\Bra{v\otimes l,4} \\ &+sin2\theta\Ket{v\otimes l,4}\Bra{h\otimes l,4}-cos2\theta\Ket{v\otimes l,4}\Bra{v\otimes l,4} \end{aligned}$$ where $\theta$ is the angle between the fast axis of HWP and the horizontal axis. $HWP_{\uppercase\expandafter{\romannumeral2}-3}$ ($HWP_{\uppercase\expandafter{\romannumeral2}-4}$) means the light enters into $HWP_{\uppercase\expandafter{\romannumeral2}}$ from $Port_3$ ($Port_4$). When the light enters into the $PBS_{\uppercase\expandafter{\romannumeral2}}$ from $Port_3$ and $Port_4$, the operator of $PBS_{\uppercase\expandafter{\romannumeral2}}$ can be described as: $$\begin{aligned} M_{PBS_{\uppercase\expandafter{\romannumeral2}}} = \Ket{h\otimes l,6}\Bra{h\otimes l,3}-\Ket{v\otimes l,5}\Bra{v\otimes l,3} +\Ket{h\otimes l,5}\Bra{h\otimes l,4}+\Ket{v\otimes l,6}\Bra{v\otimes l,4} \end{aligned}$$ where the minus of the second item is due to the film coated on the left prism (here we assume the light reflected by coated film will introduce phase $\pi$). It is noted that the film is coated on the right prism in $PBS_{\uppercase\expandafter{\romannumeral1}}$. The operator of $HWP_{\uppercase\expandafter{\romannumeral3}}$ with $\theta$ equals to $45^\circ$ will be $$M_{HWP_{\uppercase\expandafter{\romannumeral3}}} = \Ket{h\otimes l,6}\Bra{v\otimes l,6} + \Ket{v\otimes l,6}\Bra{h\otimes l,6}$$ Thus, we can deduce the operator of the TBS: $$\begin{aligned} M_{TBS}&=(1+M_{HWP_{\uppercase\expandafter{\romannumeral3}}})M_{PBS_{\uppercase\expandafter{\romannumeral2}}}(M_{HWP_{\uppercase\expandafter{\romannumeral2}-3}}+M_{HWP_{\uppercase\expandafter{\romannumeral2}-4}})M_{PBS_{\uppercase\expandafter{\romannumeral1}}} \\ &=cos2\theta\Ket{h\otimes l,5}\Bra{h\otimes l,1}+sin2\theta\Ket{h\otimes l,6}\Bra{h\otimes l,1}\\ &-cos2\theta\Ket{v\otimes l,5}\Bra{v\otimes l,1}+sin2\theta\Ket{v\otimes l,6}\Bra{v\otimes l,1} \end{aligned}$$ When an arbitrary polarization state carrying OAM of $l\hbar$ enters into the TBS from $Port_1$, the incident state can be described as: $$\begin{aligned} \Ket{In}=\alpha\Ket{h\otimes l,1}+\beta\Ket{v\otimes l,1}\end{aligned}$$ where $\alpha$ and $\beta$ are complex constants, and $\|{\alpha}\|^2+\|{\beta}\|^2=1.$ Thus, the output state should be: $$\begin{aligned} \Ket{Out} & =M_{TBS}\Ket{In} =cos2\theta(\alpha\Ket{h\otimes l,5}+\beta\Ket{v\otimes l,5}) +sin2\theta(\alpha\Ket{h\otimes l,6}+\beta\Ket{v\otimes l,6}) \label{out}\end{aligned}$$ According to Eq. (\[out\]), the output states from $Port_5$ and $Port_6$ remain their original polarization and OAM states. The splitting ratio is determined by the $\theta$ of $HWP_{\uppercase\expandafter{\romannumeral2}}$. The same conclusion can be obtained by using a similar process when light enters into the TBS from $Port_2$. ![(a) The experimental setup to measure the splitting ratio and the polarization independence. (b) 4F optical system for process tomography (figure not to scale). (c) The structure of the Sagnac interferometer. LD, laser diode; D, optical power meter; M, mirror; SLM, spatial light modulator; Lens, plano-vertex lens[]{data-label="fig:Exp_Setup0321"}](experiment_setup_20170928.eps){width="\linewidth"} To verify the performance of the TBS, three experiments are conducted respectively as shown in Fig. \[fig:Exp\_Setup0321\]. The experimental setup to test the split-ratio tunableness of the TBS is shown in Fig. \[fig:Exp\_Setup0321\] (a). A horizontal polarized light beam emits from a continuous-wave laser with the wavelength of 780 nm enters into $HWP_{0}$ driven by a DC servo motor (PR50CC, Newport Co.), which can generate arbitrary linear polarization states. Here, we test the split-ratio tunableness when $\Ket{H}$ and $\Ket{V}$ enter into the TBS, respectively. For a fixed incident polarization state, we detect the light intensity by optical power meters $D_1$ and $D_2$ in the sample rate of 100 Hz when rotating the $HWP_{\uppercase\expandafter{\romannumeral2}}$ in the range of zero to $180^\circ$ with the precision of $0.1^\circ$. The intensities of the output light changed with the angle of $HWP_{\uppercase\expandafter{\romannumeral2}}$ are shown in Fig. \[fig:Spli\_Ratio\]. When the angle of $HWP_{\uppercase\expandafter{\romannumeral2}}$ is $0^\circ$ in Fig. \[fig:Spli\_Ratio\] (a), the intensity detected by $D_1$ and $D_2$ are maximum and minimum, respectively. It is a balanced beam splitter when the angle of $HWP_{\uppercase\expandafter{\romannumeral2}}$ is $22.5^\circ$, as shown in the intersection of the two curves. The criterion for evaluating the performance of the split-ratio tunableness of TBS is the extinction ratio (ER), which is defined as: $$ER=\frac{I_{max}}{I_{min}} \label{ER}$$ where $I_{max}$ and $I_{min}$ refer to the maximum and minimum intensity of the output ports respectively. According to Eq. (\[ER\]), the mean, maximum and minimum ERs of tunableness of the TBS are 34 dB, 30.7 dB and 40.1 dB, respectively. ![The tunable capacity of the TBS when the states (a) $\Ket{H}$ and (b) $\Ket{V}$ are prepared in $LD_1$, respectively. The states (c) $\Ket{H}$ and (d) $\Ket{V}$ are corresponding to the states prepared in $LD_2$.[]{data-label="fig:Spli_Ratio"}](Spli_Ratio.eps){width="\linewidth"} To verify the polarization independence of the TBS, we test the variation of the splitting ratio with the incident polarization states as shown in Fig. \[fig:Exp\_Setup0321\] (a). In the tests, various linear polarization states are prepared by rotating the $HWP_{0}$ in the range of zero to $180^\circ$ with the accuracy of $0.1^\circ$, and the light intensities are detected by optical power meters in the sample rate of 100 Hz. The results are shown in Fig. \[fig:DOPD\]. To evaluate the performance of the polarization independence, we define the splitting ratio (SR) and polarization dependence (PD) as follows: $$SR=\frac{R}{R+T}$$ $$PD= \frac{|SR_{exp}-SR_{th}|}{SR_{th}} \label{DOPD}$$ where R (T) denotes the intensity of reflected (transmission) light, and $SR_{exp}$ means the experimental splitting ratio with a maximum deviation from theoretical value, which is the worst case. $SR_{th}$ is the theoretical splitting ratio. In the experiment, we set the reflected light to be the weak light when TBS is in the unbalanced beam splitting. Therefore, the range of $SR_{th}$ will be 0.1 to 0.5 in our tests. The smaller the value of PD, the better the performance of polarization independence. ![The polarization dependence of the TBS. The splitting ratios change with $HWP_0$ when light from (a) $LD_1$ and (b) $LD_2$.[]{data-label="fig:DOPD"}](DOPD.eps){width="\linewidth"} According to Eq. (\[DOPD\]), the measurement results of the polarization dependence in different splitting ratios are shown in Tab. \[splitting ratio\]. The worst case of the polarization dependence is lower than $6\%$ with a triple standard deviation of below $0.2\%$. With the decrease of SRs, PDs are gradually rising according to the experimental results. This is due to the more obvious influence of the intensity fluctuation to weak light. $SR$ $50: 50$ $60: 40$ $70: 30$ $80: 20$ $90: 10$ -------- -------------------- -------------------- -------------------- -------------------- -------------------- $PD_1$ $1.73\%\pm 0.06\%$ $2\%\pm 0.10\%$ $2.63\%\pm 0.10\%$ $3.68\%\pm 0.19\%$ $5.75\%\pm 0.15\%$ $PD_2$ $2\%\pm 0.07\%$ $2.42\%\pm 0.14\%$ $2.88\%\pm 0.10\%$ $3.69\%\pm 0.21\%$ $5.35\%\pm 0.14\%$ : **Polarization dependence in different splliting ratios** \[splitting ratio\] To verify the OAM preservation performance of the TBS, a process tomography based on a 4F optical system has been proposed as shown in Fig. \[fig:Exp\_Setup0321\] (b). A 780 nm continuous-wave diode laser illuminates a spatial light modulator ($SLM_1$) to generate kinds of OAM light. A 4F optical system consisting of two plano-convex lenses with the focal length of 750 mm and a spatial filter with the aperture of 8 mm are employed to isolate the first order of the beam diffracted by the $SLM_1$. After transmitting through the PBS and TBS, OAM light is demodulated by $SLM_2$ and coupled into a single mode fiber (SMF) to be detected. If the forked hologram loaded by $SLM_2$ is identical to the $SLM_1$ ($l=l_1-l_2=0$), the OAM light will totally be converted to Gaussian beam theoretically[@mair2001entanglement]. However, imperfect devices will lead to the crosstalk from other OAM modes which can be seen as a criterion to evaluate the performance of OAM preservation of optical elements. In the process tomography, we prepare the OAM light with $l=0$, $\pm1$, $\pm2$, $\pm3$, $\pm4$ respectively, and then measure it with the same sets. The experimental results of the PBS and TBS are shown in Fig. \[fig:OAM\]. In order to assess the OAM preservation performance, we define the ER of OAM preservation as follows: $$ER_{OAM}=\frac{I_i}{\sum_{k \ne i} I_k} \label{EXT_OAM}$$ where $I_i$ and $I_k$ refer to the detected light intensity when the hologram loaded by $SLM_2$ is the identical order and other orders of OAM light, respectively. According to Eq. (\[EXT\_OAM\]), the ERs of more than 20 dB are obtained, which are close to the situations without PBS or TBS. These results demonstrate the PBS and TBS can preserve the OAM light well. ![The process tomography of (a) the PBS and (b) the TBS in the 4F optical system, respectively.[]{data-label="fig:OAM"}](OAM.eps){width="\linewidth"} . Interferometer is a kind of important element in optical systems. Therefore, we built a Sagnac interferometer using this TBS to evaluate its performance in practical applications, as shown in Fig. \[fig:Exp\_Setup0321\] (c). In the Sagnac interferometer, the light emitted from $Port_{5}$ ($Port_{6}$) will be reflected by the mirrors $M_1$ and $M_2$ and returns to the TBS from $Port_{6}$ ($Port_{5}$), in which the two components throughing the same path form a closed loop. We test the visibility of the Sagnac interferometer during 30 minutes with the different incident polarization states. The interference visibility is defined as: $$V=\frac{I_{max}-I_{min}}{I_{max}+I_{min}}$$ where $I_{max}$ and $I_{min}$ are the maximum intensity and minimum intensity of the output ports, respectively. As shown in Fig. \[fig:Interference5050\] (a), the interference visibility of above $99\%$ with the triple standard deviations of below $0.2\%$ is obtained, which prove its good stability when the light enters into $Port_1$ and $Port_2$, individually. Then we test the interference visibility of different incident polarization state by rotating $HWP_0$ in a period from $0^\circ$ to $90^\circ$ with a step of $0.1^\circ$. The mean interference visibility for two input ports are all greater than $99\%$ as shown in Fig. \[fig:Interference5050\] (b). The visibility of the incident polarization states $\Ket{H}$ and $\Ket{V}$ are closed, while the interference visibility reaches its maximum and minimum value when the incident polarization states are $\Ket{H+V}$ and $\Ket{H-V}$, respectively, as whown in Fig. \[fig:Interference5050\] (b). This is due to the insertion loss of different polarization states in different ports and extinction ratio of PBS. The other curve of incident light from $Port_2$ is reflexive associative with the curve from $Port_1$, which is reasonable since the state $\Ket{H+V}$ from $Port_1$ is corresponding to the state $\Ket{H-V}$ from $Port_2$. ![(a) The test of stability of the Sagnac interferometer. (b) The test of polarization independence of Sagnac interferometer.[]{data-label="fig:Interference5050"}](Interference5050.eps){width="\linewidth"} In conclusion, we realize a polarization-independent OAM-preserving TBS which needs much less optical devices comparing with the existing schemes. We experimentally evaluated the key parameters of the scheme, and demonstrated that the ERs of tunableness are greater than 30 dB, polarization dependence are lower than $6\%$ and ERs of OAM preservation are more than 20 dB. Using this TBS, we built a Sagnac interferometer with the mean visibility of above $99\%$, which makes it have potential to be utilized in kinds of quantum information processing. This resource-saving structure has the potential advantage to simplify the optical systems and be applied to the scalable applications[@wang2016scalable][@krenn2017entanglement]. It is noted that the scheme can be implemented more compactly with emerging techniques such as integrated optics. 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--- abstract: 'In this article, we study optimal control problems of spiking neurons whose dynamics are described by a phase model. We design minimum-power current stimuli (controls) that lead to targeted spiking times of neurons, where the cases with unbounded and bounded control amplitude are considered. We show that theoretically the spiking period of a neuron, modeled by phase dynamics, can be arbitrarily altered by a smooth control. However, if the control amplitude is bounded, the range of possible spiking times is constrained and determined by the bound, and feasible spiking times are optimally achieved by piecewise continuous controls. We present analytic expressions of these minimum-power stimuli for spiking neurons and illustrate the optimal solutions with numerical simulations.' author: - Isuru Dasanayake - 'Jr-Shin Li' bibliography: - 'SingleNeuron\_PRE.bib' nocite: '[@*]' title: 'Optimal Design of Minimum-Power Stimuli for Spiking Neurons' --- Introduction {#sec:intro} ============ Control of neurons and hence the nervous system by external current stimuli (controls) has received increased scientific attention in recent years for its wide range of applications from deep brain stimulation to oscillatory neurocomputers [@uhlhaas06; @osipov07; @Izhikevich99]. Conventionally, neuron oscillators are represented by phase-reduced models, which form a standard nonlinear system [@Brown04; @Winfree01]. Intensive studies using phase models have been carried out, for example, on the investigation of the patterns of synchrony that result from the type and architecture of coupling [@Ashwin92; @Taylor98] and on the response of large groups of oscillators to external stimuli [@Moehlis06; @Tass89], where the inputs to the neuron systems were initially defined and the dynamics of neural populations were analyzed in detail. Recently, control theoretic approaches have been employed to design external stimuli that drive neurons to behave in a desired way. For example, a multilinear feedback control technique has been used to control the individual phase relation between coupled oscillators [@kano10]; a nonlinear feedback approach has been employed to engineer complex dynamic structures and synthesize delicate synchronization features of nonlinear systems [@Kiss07]; and our recent work has illustrated controllability of a network of neurons with different natural oscillation frequencies adopting tools from geometric control theory [@Li_NOLCOS10]. There has been an increase in the demand for controlling not only the collective behavior of a network of oscillators but also the behavior of each individual oscillator. It is feasible to change the spiking periods of oscillators or tune the individual phase relationship between coupled oscillators by the use of electric stimuli [@Schiff94; @kano10]. Minimum-power stimuli that elicit spikes of a neuron at specified times close to the natural spiking time were analyzed [@Moehlis06]. Optimal waveforms for the entrainment of weakly forced oscillators that maximize the locking range have been calculated, where first and second harmonics were used to approximate the phase response curve [@kiss10]. These optimal controls were found mainly based on the calculus of variations, which restricts the optimal solutions to the class of smooth controls and the bound of the control amplitude was not taken into account. In this paper, we apply the Pontryagin’s maximum principle [@Pontryagin62; @Stefanatos10] to derive minimum-power controls that spike a neuron at desired time instants. We consider both cases when the available control amplitude is unbounded and bounded. The latter is of practical importance due to physical limitations of experimental equipment and the safety margin for neurons, e.g., the requirement of a mild brain stimulations in neurological treatments for Parkinson’s disease and epilepsy. This paper is organized as follows. In Section \[sec:phase\_model\], we introduce the phase model for spiking neurons and formulate the related optimal control problem. In Section \[sec:minpower\_control\], we derive minimum-power controls associated with specified spiking times in the absence and presence of control amplitude constraints, in which various phase models including sinusoidal PRC, SNIPER PRC, and theta neuron models are considered. In addition, we present examples and simulations to demonstrate the resulting optimal control strategies. Optimal Control of Spiking Neurons {#sec:phase_model} ================================== A periodically spiking or firing neuron can be considered as a periodic oscillator governed by the nonlinear dynamical equation of the form $$\label{eq:phasemodel} \frac{d\theta}{dt}=f(\theta)+Z(\theta)I(t),$$ where $\theta$ is the phase of the oscillation, $f(\theta)$ and $Z(\theta)$ are real-valued functions giving the neuron’s baseline dynamics and its phase response, respectively, and $I(t)$ is an external current stimulus [@Brown04]. The nonlinear dynamical system described in is referred to as the phase model for the neuron. The assumption that $Z(\theta)$ vanishes only on isolated points and that $f\left(\theta\right)>0$ are made so that a full revolution of the phase is possible. By convention, neuron spikes occur when $\theta=2n\pi$, where $n\in\mathbb{N}$. In the absence of any input $I(t)$, the neuron spikes periodically at its natural frequency, while the periodicity can be altered in a desired manner by an appropriate choice of $I(t)$. In this article, we study optimal design of neural inputs that lead to the spiking of neurons at a specified time $T$ after spiking at time $t=0$. In particular, we find the stimulus that fires a neuron with minimum power, which is formulated as the following optimal control problem, $$\begin{aligned} \label{eq:opt_con_pro} \min_{I(t)} \quad & \int_0^T I(t)^2\,dt\\ {\rm s.t.} \quad & \dot{\theta}=f(\theta)+Z(\theta)I(t), \nonumber\\ &\theta(0)=0, \quad \theta(T)=2\pi \nonumber\\ &|I(t)|\leq M, \ \ \forall\ t, \nonumber\end{aligned}$$ where $M>0$ is the amplitude bound of the current stimulus $I(t)$. Note that instantaneous or arbitrarily delayed spiking of a neuron is possible if $I(t)$ is unbounded, i.e., $M=\infty$; however, the range of feasible spiking periods of a neuron described as in is restricted with a finite $M$. We consider both unbounded and bounded cases. Minimum-Power Stimulus for Specified Firing Time {#sec:minpower_control} ================================================ We consider the minimum-power optimal control problem of spiking neurons as formulated in for various phase models including sinusoidal PRC, SNIPER PRC, and theta neuron. Sinusoidal PRC Phase Model {#sec:sine_prc} -------------------------- Consider the sinusoidal PRC model, $$\label{eq:sin_model} \dot{\theta}=\omega+z_d\sin\theta\cdot I(t),$$ where $\omega$ is the natural oscillation frequency of the neuron and $z_d$ is a model-dependent constant. The neuron described by this phase model spikes periodically with the period $T=2\pi/\omega$ in the absence of any external input, i.e., $I(t)=0$. ### Spiking Neurons with Unbounded Control {#sec:unbounded_control_sine} The optimal current profile can be derived by Pontryagin’s Maximum Principle [@Pontryagin62]. Given the optimal control problem as in , we form the control Hamiltonian $$\label{eq:hamiltonian} H=I^2+\lambda(\omega+z_d\sin\theta\cdot I),$$ where $\lambda$ is the Lagrange multiplier. The necessary optimality conditions according to the Maximum Principle give $$\begin{aligned} \label{eq:lambda_dot} \dot{\lambda}=-\frac{\partial H}{\partial \theta}=-\lambda z_d I\cos\theta,\end{aligned}$$ and $\frac{\partial H}{\partial I}=2I+\lambda z_d\sin\theta=0$. Hence, the optimal current $I$ satisfies $$\begin{aligned} \label{eq:I} I=-\frac{1}{2}\lambda z_d\sin\theta.\end{aligned}$$ Substituting into and , the optimal control problem is then transformed to a boundary value problem, which characterizes the optimal trajectories of $\theta(t)$ and $\lambda(t)$, $$\begin{aligned} \label{eq:theta_p2} \dot{\theta} &= \omega-\frac{z_d^2\lambda}{2} \sin^2 \theta,\\ \label{eq:lambda_p2} \dot{\lambda} &= \frac{z_d^2\lambda^2}{2} \sin\theta\cos\theta,\end{aligned}$$ with boundary conditions $\theta(0)=0$ and $\theta(T)=2\pi$ while $\lambda(0)$ and $\lambda(T)$ are unspecified. Additionally, since the Hamiltonian is not explicitly dependent on time, the optimal triple $(\lambda,\theta, I)$ satisfies $H(\lambda,\theta,I)=c$, $\forall\, 0\leq t\leq T$, where $c$ is a constant. Together with , this yields $$\label{eq:quadratic} -\frac{z_d^2}{4}\sin^2\theta\lambda^2+\omega\lambda=c,$$ and the constant $c=\omega\lambda_0$ is obtained by the initial conditions $\theta(0)=0$ and $\lambda(0)=\lambda_0$, which is undetermined. Then, the optimal multiplier can be found by solving the above quadratic equation , which gives $$\label{eq:lambda} \lambda=\frac{2\omega\pm2\sqrt{\omega^2-\omega\lambda_0z_d^2\sin^2\theta}}{z_d^2\sin^2\theta},$$ and the optimal trajectory of $\theta$ follows $$\label{eq:theta} \dot{\theta}=\mp\sqrt{\omega^2-\omega\lambda_0 z_d^2\sin^2 \theta},$$ by plugging $\lambda$ in into . Integrating by separation of variables, we find the spiking time $T$ with respect to the initial condition $\lambda_0$, $$\label{eq:T} T=\frac{1}{\omega}F\left(\scriptstyle{2\pi,\sqrt{\frac{\lambda_0}{\omega}}z_d}\right)=\int_0^{2\pi}{\frac{1}{\scriptstyle{\sqrt{\omega^2-\omega\lambda_0 z_d^2\sin^2\theta}}}}d\theta,$$ where $F$ denotes the elliptic integral. Note that we choose the positive sign in since the negative velocity indicates the backward phase evolution. Therefore, given a desired spiking time $T$ of the neuron, the initial value, $\lambda_0$, corresponding to the optimal trajectory of the multiplier can be found via the one-to-one relation in . Consequently, the optimal trajectories of $\theta$ and $\lambda$ can be easily computed by evolving and forward in time. Plugging into , we obtain the optimal feedback law $$\label{eq:I*} I^*= \frac{-\w+\sqrt{\w^2-\w\lambda_0 z_d^2\sin^2\t}}{z_d\sin\t},$$ which drives the neuron from $\theta(0)=0$ to $\theta(T)=2\pi$ with minimum power. The feasibility of spiking the neuron at a desired time $T$ largely depends on the initial value of the multiplier, $\lambda_0$. It is clear from and that a complete $2\pi$ revolution is impossible when $\lambda_0>\omega/z_d^2$. This fact also can be seen from FIG. \[fig:vectorfield\], where the system evolution defined by and for $z_d=1$ and $\omega=1$ with respect to different $\lambda_0$ values ($\t=0$ axis) is illustrated. When $\lambda_0=0$, according to equation , the spiking period is equal to the natural spiking period, $2\pi/\omega$, and no external stimulus needs to be applied, i.e., $I^*(t)=0$, $\forall t\in[0,2\pi/\w]$. Since, from , $T$ is a monotonically increasing function of $\lambda_0$ for fixed $\w$ and $z_d$ and, from , the average phase velocity decreases when $\lambda_0$ increases, the spiking time $T>2\pi/\omega$ for $\lambda_0>0$ and $T<2\pi/\omega$ for $\lambda_0<0$. FIG. \[fig:T\_vs\_lambda0\_sine\] shows variation of the spiking time $T$ with the $\lambda_0$ corresponding to the optimal trajectories for different $\omega$ values with $z_d=1$. ![Extremals of sinusoidal PRC system with $z_d=1$ and $\w=1$[]{data-label="fig:vectorfield"}](fig1_vectorfield.eps) ![Variation of the spiking time, $T$, with respect to the initial multiplier value, $\lambda_0$, leading to optimal trajectories, with different values of $\omega$ and $z_d=1$ for sinusoidal PRC model.[]{data-label="fig:T_vs_lambda0_sine"}](fig2_terminal_time_vs_lambda0_sine.eps) \ The relation between the spiking time $T$ and required minimum power $E=\min\int_0^{T}{{I}^2(t)}dt$ is evident via a simple sensitivity analysis [@Bryson75]. Since a small change in the initial condition, $d\theta$, and a small change in the initial time, $dt$, result in a small change in power according to $dE=\lambda(t) d\theta-H(t)d\theta$, it follows that [@Bryson75] $$\label{eq:energy} -\frac{\partial E}{\partial t}=H=c=\w\lambda_0.$$ This implies that $E$ increases with initial time $t$ for $\lambda_0<0$ and decreases for $\lambda_0>0$. Since the increment of the initial time is equivalent to the decrement of spiking time $T$, $\partial E/\partial T=\omega\lambda_0$. Since $\lambda_0<0$ ($\lambda_0>0$) corresponds to $T<2\pi/\omega$ ($T>2\pi/\omega$), we see that the required minimum power increases if we move away from the natural spiking time. The minimum-power stimulus $I^*$ as in plotted with respect to time and phase for various spiking times $T=3,5,10,12$ with $\w=1$ and $z_d=1$ are shown in FIG. \[fig:control\_vs\_time\_sine\] and \[fig:control\_vs\_theta\_sine\], respectively. The respective optimal trajectories of $\lambda(\t)$ and $\t(t)$ for these spiking times are presented in FIG. \[fig:lambda\_vs\_theta\_sine\] and \[fig:theta\_vs\_time\_sine\]. ### Spiking Neurons with Bounded Control {#sec:bounded_control_sine} In practice, the amplitude of stimuli in physical systems are limited, so we consider spiking the sinusoidal neuron with bounded control amplitude, namely, in the optimal control problem , $|I(t)|\leq M<\infty$ for all $t\in[0,T]$, where $T$ is the desired spiking period. In this case, there exists a range of feasible spiking periods depending on the value of $M$, in contrast to the previous case where any desired spiking time is feasible. We first observe that given this bound $M$, the minimum time it takes to spike a neuron can be achieved by choosing the control that keeps the phase velocity $\dot{\t}$ maximum over $t\in[0,T]$. Such a time-optimal control, for $z_d>0$, can be characterized by a switching, i.e., $$\label{sin_Tmin_control} I^*_{Tmin}= \left\{\begin{array}{c} M \quad \mathrm{for} \quad 0\leq \theta <\pi\\ -M \quad \mathrm{for} \quad \pi \leq \theta <2\pi \end{array} \right..$$ Consequently, the spiking time with $I^*_{Tmin}$ can be computed using and , which yields $$T^{M}_{min}= \frac{2\pi-4\tan^{-1}\left\{z_dM/\sqrt{-z_d^2M^2+\omega^2}\right\}}{\sqrt{-z_d^2M^2+\omega^2}}. \label{eq:sin_min_T}$$ It follows that $I^*$, derived in , is the minimum-power stimulus that spikes the neuron at a desired spiking time $T$ if $|I^*|\leq M$ for all $t\in [0,T]$. However, there exists a shortest possible spiking time by $I^*$ given the bound $M$. Simple first and second order optimality conditions applied to find that the maximum value of $I^*$ occurs at $\theta=\pi/2$ for $\lambda_0<0$ and at $\theta=3\pi/2$ for $\lambda_0>0$. Therefore, the $\lambda_0$ for the shortest spiking time with control $I^*$ satisfying $|I^*(t)|\leq M$ can be calculated by substituting $I^*=M$ and $\theta=\pi/2$ to the equation , and then from we obtain this shortest spiking period $$\label{eq:min_time_UBC} T^{I^*}_{min}=\int_0^{2\pi}\frac{1}{\sqrt{\omega^2+z_dM(z_dM+2\omega)\sin^2(\theta)}}.$$ Note that $T^{M}_{min}<T^{I^*}_{min}$. According to when $M\geq\omega/z_d$, arbitrarily large spiking times can be achieved by making $\dot{\theta}$ arbitrary close to zero. Therefore we consider two cases for $M\geq\omega/z_d$ and $M<\omega/z_d$. ![Variation of the maximum value of $I^*$ with spiking time $T$ for sinusoidal PRC model with $\w=1$ and $z_d=1$.[]{data-label="fig:max_I_vs_T"}](fig4_max_I_vs_T.eps) *Case I: $M\geq\omega/z_d$*. Since $I^*$ takes the maximum value at $\t=3\pi/2$ for $\lambda_0>0$, we have $|I^*|\leq(\omega-\sqrt{\omega^2-\omega\lambda_0z_d^2})/z_d$, which leads to $|I^*|<\w/z_d\leq M$ for $\lambda_0>0$. This implies that $I^*$ is the minimum-power control for any desired spiking time $T>2\pi/\w$ when $M\geq\w/z_d$, and hence for any spiking time $T\geq T^{I^*}_{min}$. Variation of the maximum value of the control $I^*$ with spiking time $T$ for $\omega=1$ and $z_d=1$ is depicted in FIG. \[fig:max\_I\_vs\_T\]. Shorter spiking times $T\in[T^{M}_{min}, T^{I^*}_{min})$ are feasible but, due to the bound $M$, can not be achieved by $I^*$ since it requires a control with amplitude greater than $M$ for some $t\in[0,T]$. However, these spiking times can be optimally achieved by applying controls switching between $I^*$ and $I^*_{Tmin}$. Let the desired spiking time $T\in[T^{M}_{min}, T^{I^*}_{min})$. Then, there exist two angles $\theta_1=\sin^{-1}[-2M\omega/(z_dM^2+z_d\omega \lambda_0)]$ and $\theta_2=\pi-\theta_1$ where $I^*$ meets the bound $M$. When $\theta\in(\theta_1,\theta_2)$, $I^*>M$ and we take $I(\theta)=M$ for $\theta\in[\theta_1,\theta_2]$. The Hamiltonian of the system when $\theta\in[\theta_1,\theta_2]$ is then, from , $H=M^2+\lambda(\omega+z_d\sin\theta\,M)$. If the triple $(\lambda,\t,M)$ is optimal, then $H$ is a constant, which gives $\lambda=(H-M^2)/(\omega+z_dM\sin\theta)$. This multiplier satisfies the adjoint equation , and therefore $I(\theta)=M$ is optimal for $\theta\in[\theta_1, \theta_2]$. Similarly, by symmetry, $I^*<-M$ when $\t\in[\t_3,\t_4]$, where $\theta_3=\pi+\theta_1$ and $\theta_4=2\pi-\theta_1$, if the desired spiking time $T\in[T^{M}_{min}, T^{I^*}_{min})$. It can be easily shown by the same fashion that $I(\t)=-M$ is optimal in the interval $\t\in[\t_3,\t_4]$. Therefore, the minimum-power optimal control that spikes the neuron at $T\in[T^{M}_{min}, T^{I^*}_{min})$ can be characterized by four switchings between $I^*$ and $M$, i.e., $$\begin{aligned} \label{eq:I1*} I^*_1=\left\{\begin{array}{ll} I^* & \ 0\leq \theta < \theta_1 \\ M & \ \theta_1\leq\theta\leq\theta_2 \\ I^* & \ \theta_2 < \theta < \theta_3 \\ -M & \ \theta_3\leq\theta\leq\theta_4\\ I^* & \ \theta_4 < \theta \leq 2\pi. \end{array}\right.\end{aligned}$$ The initial value of the multiplier, $\lambda_0$, resulting in the optimal trajectory, can then be found according to the desired spiking time $T\in[T^{M}_{min}, T^{I^*}_{min})$ through the relation $$T=\int_0^{\theta_1}{\frac{4}{\sqrt{\scriptstyle{\omega^2-\omega\lambda_0z_d^2\sin^2\theta}}}}d\theta +\int_{\theta_1}^{\frac{\pi}{2}}{\frac{4}{\omega+z_dM\sin\left(\theta\right)}}d\theta.$$ FIG. \[fig:T\_vs\_lambda0\_sine\_for\_I1\] shows the relation between $\lambda_0$ and $T$ by $I^*_1$ for $M=2.5,\ z_d=1,$ and $ \omega =1$. From the minimum possible spiking time with this control bound $M=2.5$ is $T^{M}_{min}=2.735$ and from the minimum spiking time by $I^*$ is $T^{I^*}_{min}=3.056$. Thus, in this example, any desired spiking time $T>3.056$ can be optimally achieved by $I^*$ whereas any $T\in[2.735,3.056)$ can be optimally obtained by $I^*_1$ as in . FIG. \[fig:I1\_and\_I\_sine\] illustrates the bounded and unbounded optimal controls that fire the neuron at $T=2.8$, where $I^*$ is the minimum-power stimulus when the control amplitude is not limited and $I^*_1$ is the minimum-power stimulus when the bound $M=2.5$. $I^*$ drives the neuron from $\t(0)=0$ to $\t(2.8)=2\pi$ with 13.54 units of power whereas $I^*_1$ requires 14.13 units. *Case II : $M<\omega/z_d$*. In contrast with Case I in the previous section, achieving arbitrarily large spiking times is not feasible with a bound $M<\omega/z_d$. In this case, the longest possible spiking time is achieved by $$I^*_{Tmax}= \left\{\begin{array}{ll} -M & \quad \text{for} \ \ 0\leq \theta <\pi, \\ M & \quad \text{for} \ \ \pi\leq\theta<2\pi.\end{array}\right.$$ The spiking time of the neuron under this control is, $$T^M_{max}= \frac{2\pi+4\tan^{-1}\Big[z_dM/\sqrt{-z_d^2M^2+\omega^2}\Big]}{\sqrt{-z_d^2M^2+\omega^2}}, \label{eq:max_T_for_M<w/s}$$ and the longest spiking time feasible with control $I^*$ is given by $$\label{eq:max_time_UBC} T^{I^*}_{max}=\int_0^{2\pi}\frac{1}{\sqrt{\omega^2+z_dM(z_dM-2\omega)\sin^2(\theta)}}.$$ Then, by similar analysis for Case I, any spiking time $T\in[T^M_{min},T^{I^*}_{min})$ for a given $M<\w/z_d$ can be achieved with the minimum-power control $I^*_1$ as given in , any $T\in[T^{I^*}_{min},T^{I^*}_{max}]$ can be achieved with minimum power by $I^*$ in , and moreover any $T\in(T^{I^*}_{max},T^M_{max}]$ can be obtained by switching between $I^*$ and $I^*_{max}$. The corresponding switching angles are $\theta_5=\sin^{-1}[2M\omega/(z_dM^2+z_d\omega \lambda_0)],\theta_6=\pi-\theta_5,\theta_7=\pi+\theta_5$ and $\theta_8=2\pi-\theta_5$, and the minimum-power optimal control for $T\in(T^{I^*}_{max},T^M_{max}]$ is characterized by $$\begin{aligned} \label{eq:I2*} I^*_2=\left\{\begin{array}{ll} I^* & \ 0\leq \theta < \theta_5 \\ -M & \ \theta_5\leq\theta\leq\theta_6 \\ I^* & \ \theta_6 < \theta < \theta_7 \\ M & \ \theta_7\leq\theta\leq\theta_8\\ I^* & \ \theta_8 < \theta \leq 2\pi. \end{array}\right.\end{aligned}$$ The $\lambda_0$ resulting in the optimal trajectory by $I_2^*$ can be calculated according to the given $T\in(T^{I^*}_{max},T^M_{max}]$ via the relation $$T=\int_0^{\theta_5}{\frac{4}{\sqrt{\scriptstyle{\omega^2-\omega \lambda_0z_d^2\sin^2\theta}}}}d\theta+\int_{\theta_5}^{\frac{\pi}{2}}{\frac{4}{\omega-z_dM\sin\theta}}d\theta.$$ FIG. \[fig:T\_vs\_lambda0\_sine\_for\_I2\] shows the relation between $\lambda_0$ and $T$ by $I^*_2$ for $M=0.55$, $z_d=1$, and $\omega=1$. From the maximum possible spiking time with $M=0.55$ is $T^M_{max}=10.312$ and from the maximum spiking time feasible by $I^*$ is $I^{I^*}_{max}=9.006$. Therefore, in this example, any desired spiking time $T\in(9.006,10.312]$ can be obtained with minimum power by the use of $I^*_2$. FIG. \[fig:I2\_and\_I\_sine\] illustrates the bounded and unbounded optimal controls that spike the neuron at $T=10$, where $I^*$ is the minimum-power stimulus when the control amplitude is not limited and $I^*_2$ is the minimum-power stimulus when $M=0.55$. $I^*$ drives the neuron from $\t(0)=0$ to $\t(10)=2\pi$ with 2.193 units of power whereas $I^*_2$ requires 2.327 units. A summary of the optimal (minimum-power) spiking scenarios for a prescribed spiking time of the neuron governed by the sinusoidal phase model is illustrated in FIG. \[fig:scenarios\]. \ SNIPER PRC and Theta Neuron Phase Models {#sec:sniper_prc} ---------------------------------------- We now consider the SNIPER PRC model in which $f(\theta)=\omega$ and $Z(\theta)=z_d(1-\cos\theta)$, where $z_d>0$ and $\omega>0$. That is, $$\label{eq:model_sniper} \dot{\theta}=\omega+z_d(1-\cos\theta)I(t).$$ The minimum-power stimuli for spiking neurons modeled by this phase model can be easily derived with analogous analysis described previously in \[sec:unbounded\_control\_sine\] and \[sec:bounded\_control\_sine\] for the sinusoidal PRC phase model. ### Spiking Neurons with Unbounded Control {#sec:unbounded_control_sniper} Employing the maximum principle as in \[sec:unbounded\_control\_sine\], the minimum-power stimulus that spikes the SNIPER neuron at a desired time $T$ can be derived and given by $$\label{eq:I*sniper} I^*=\frac{-\omega+\sqrt{\omega^2-\omega \lambda_0 z_d^2(1-\cos\theta)^2}}{z_d(1-\cos\theta)},$$ where $\lambda_0$ corresponding to the optimal trajectory is determined through the integral relation with $T$, $$T=\int_0^{2\pi}{\frac{1}{\sqrt{\omega^2-\omega \lambda_0z_d^2(1-\cos \theta)^2}}}d\theta.$$ \ The minimum-power stimuli $I^*$ plotted with respect to time and phase for various spiking times $T=3,5,10,12$ with parameter values $z_d$=1 and $\omega=1$ are illustrated in FIG. \[fig:control\_vs\_time\_sniper\] and \[fig:control\_vs\_theta\_sniper\], respectively. The corresponding optimal trajectories of $\lambda(\t)$ and $\t(t)$ for these spiking times are displayed in FIG. \[fig:lambda\_vs\_theta\_sniper\] and \[fig:theta\_vs\_time\_sniper\]. ### Spiking Neurons with Bounded Control {#sec:bounded_control_sniper} When the amplitude of the available stimulus is limited, i.e., $|I(t)|\leq M$, the control that achieves the shortest spiking time for the SNIPER neuron modeled in is given by $I^*_{Tmin}=M>0$ for $0\leq\theta\leq 2\pi$, since $1-\cos\theta\geq 0$ for all $\theta\in [0,2\pi]$,. As a result, the shortest possible spiking time with this control is $T^M_{min}=2\pi/\sqrt{\omega^2+2z_d\omega M}$. Also, the shortest spiking time achieved by the control $I^*$ in given the bound $M$ is given by $$\label{eq:min_time_UBC_sniper} T^{I^*}_{min}=\int_0^{2\pi}\frac{1}{\sqrt{\omega^2+z_dM(z_dM+\omega)(1-\cos\theta)^2}}.$$ Similar to the sinusoidal PRC case, the longest possible spiking time of the neuron varies with the control bound $M$. If $M\geq\omega/(2z_d)$, an arbitrarily large spiking time is achievable, however, if $M<\omega/(2z_d)$ there exists a maximum spiking time. *Case I: $M\geq\omega/(2z_d)$*. Any spiking time $T\in[T^{I^*}_{min},\infty)$ is possible with control $I^*$ but a shorter spiking time $T\in[T^M_{min},T^{I^*}_{min})$ requires switching between $I^*$ and $I^*_{Tmin}$, which is characterized by two switchings, $$\begin{aligned} \label{eq:I1*sniper} I^*_1=\left\{\begin{array}{ll} I^*,&\quad 0\leq \theta < \theta_1\\ M,&\quad \theta_1\leq\theta\leq 2\pi-\theta_1 \\ I^*,& \quad 2\pi-\theta_1 < \theta \leq 2\pi \end{array}\right.\end{aligned}$$ where $\theta_1=\cos^{-1}\left[1+2\omega M/(z_dM^2+z_d\omega \lambda_0)\right]$. The initial value $\lambda_0$ which results in the optimal trajectory is given by, $$T=\int_0^{\theta_1}{\frac{2}{\sqrt{\scriptstyle{\omega^2-\omega \lambda_0z_d^2(1-\cos\theta)^2}}}}d\theta+\int_{\theta_1}^{\pi}{\frac{2}{\scriptstyle{\omega+z_dM(1-\cos\theta)}}}d\theta.$$ FIG. \[fig:T\_vs\_lambda0\_sniper\_for\_I1\] illustrates the relation between $\lambda_0$ and $T\in[T^M_{min},T^{I^*}_{min})$ by $I_1^*$ for $M=2$, $z_d=1$, and $\omega=1$. In this case, the shortest feasible spiking time is $T^{M}_{min}=2.09$ and the shortest with the control $I^*$ is $T^{I^*}_{min}=3.18$. Any spiking time in the interval $(2.09,3.18]$ is achievable by $I^*_1$ in with minimum-power. FIG. \[fig:I1\_and\_I\_sniper\] illustrates the unbounded and bounded, with $M=2$, optimal stimuli that fire the neuron at $T=3$ with minimum-power. *Case II: $M<\omega/(2z_d)$*. In this case there exists a longest possible spiking time which is achieved by $I_{max}^*=-M$ for all $\theta\in [0,2\pi]$. The longest spiking time feasible with the control $I^*$ as in is given by $$T^{I^*}_{max}=\int_0^{2\pi}\frac{1}{\sqrt{\omega^2+z_dM(z_dM-2\omega)(1-\cos\theta)^2}}.$$ Therefore, any spiking time $T\in[T^M_{min},T^{I^*}_{min})$ for a given $M<\w/z_d$ can be achieved with the minimum-power control $I^*_1$ as given in , any $T\in[T^{I^*}_{min},T^{I^*}_{max}]$ can be achieved with minimum power by $I^*$ in , and moreover any $T\in(T^{I^*}_{max},T^M_{max}]$ can be obtained by switching between $I^*$ and $I^*_{max}$, that is, $$\begin{aligned} I^*_2=\left\{\begin{array}{ll} I^*,& \quad 0\leq \theta < \theta_2 \\ -M,& \quad \theta_2\leq\theta\leq 2\pi-\theta_2 \\ I^*,&\quad 2\pi-\theta_2 < \theta < 2\pi \end{array}\right.\end{aligned}$$ where $\theta_2=\cos^{-1}\left[1-2\omega M/(z_dM^2+z_d\omega \lambda_0)\right]$. The $\lambda_0$ associated with the optimal trajectory is determined via the relation with the desired spiking time $T$, $$T=\int_0^{\theta_1}{\frac{2}{\sqrt{\scriptstyle {\omega^2-\omega \lambda_0z_d^2(1-\cos \theta)^2}}}}d\theta+\int_{\theta_1}^{\pi}{\frac{2}{\scriptstyle {\omega-z_dM(1-\cos\theta)}}}d\theta.$$ FIG. \[fig:T\_vs\_lambda0\_sniper\_for\_I2\] illustrates the relation between $\lambda_0$ and $T\in(T^{I^*}_{max},T^M_{max}]$ by $I_2^*$ for $M=0.3$, $z_d=1$, and $\omega=1$. In this case, the longest feasible spiking time is $T^M_{max}=9.935$ and the longest with the control $I^*$ is $T^{I^*}_{max}=8.596$. The unbounded and bounded, with $M=0.3$, optimal stimuli that fire the neuron at $T=9.8$ with minimum-power are illustrated in FIG. \[fig:I2\_and\_I\_sniper\]. A summary of the optimal (minimum-power) spiking scenarios for a prescribed spiking time of the neuron governed by the SNIPER PRC model in can be illustrated analogously to FIG. \[fig:spiking\_time\_line\_case1\] and \[fig:spiking\_time\_line\_case2\] for $M\geq \omega/(2z_d)$ and $M<\omega/(2z_d)$, respectively. The theta neuron phase model is described by the dynamical equation $$\label{eq:model_theta} \dot{\theta}=1+\cos\theta+z_d(1-\cos\theta)(I(t)+I_b),$$ where $I_b>0$ is the baseline current and the natural frequency $\omega$ of the theta neuron is given by $2\sqrt{I_b}$. In fact, by a coordinate transformation, $\theta(\phi)=2\tan^{-1}\left[\sqrt{I_b}\tan\left((\phi-\pi)/2\right)\right]+\pi$, the theta neuron model can be transformed to a SNIPER PRC with $z_d=\w/2$ in . Therefore, all of the results developed for the SNIPER PRC can be directly applied to the theta neuron phase model. Conclusion and Future Work\[sec:conclusion\] ============================================ In this paper, we studied various phase-reduced models that describe the dynamics of a neuron system. We considered the design of minimum-power stimuli for spiking a neuron at a specified time instant and formulated this as an optimal control problem. We investigated both cases when the control amplitude is unbounded and bounded, for which we found analytic expressions of optimal feedback control laws. In particular for the bounded control case, we characterized the ranges of possible spiking periods in terms of the control bound. Moreover, minimum-power stimuli for steering any nonlinear oscillators of the form as in between desired states can be derived following the steps presented in this article. In addition, the charge-balanced constraint [@Nabi09] can be readily incorporated into this framework as well. The optimal control of a single neuron system investigated in this work illustrates the fundamental limit of spiking a neuron with external stimuli and provides a benchmark structure that enables us to study optimal control of a spiking neural network with different individual oscillation frequencies. Our recent work [@Li_NOLCOS10] proved that simultaneous spiking of a network of neurons is possible; however, optimal control of such a spiking neural network has not been studied. We finally note that although one-dimensional phase models are reasonably accurate to describe the dynamics of neurons, studying higher dimensional models such as that of Hodgkin-Huxley is essential for more accurate computation of optimal neural inputs.

--- author: - | Zhinus Marzi,  Dinesh Ramasamy  and Upamanyu Madhow, \ `{zh_marzi, dineshr, madhow}@ece.ucsb.edu` [^1] bibliography: - 'references.bib' title: Compressive channel estimation and tracking for large arrays in mm wave picocells --- Compressive, mm wave, 60 GHz, picocells, RF beamforming Acknowledgement {#acknowledgement .unnumbered} =============== This work was supported by the National Science Foundation through the grant CNS-1317153, by the Institute for Collaborative Biotechnologies through the grant W911NF-09-0001 from the U.S. Army Research Office and by the Systems on Nanoscale Information fabriCs (SONIC), one of six centers supported by the STARnet phase of the Focus Center Research Program (FCRP), a Semiconductor Research Corporation program sponsored by MARCO and DARPA. The content of the information does not necessarily reflect the position or the policy of the Government, and no official endorsement should be inferred [^1]: Z. Marzi, D. Ramasamy and U. Madhow are with the Department of Electrical and Computer Engineering, University of California Santa Barbara, Santa Barbara, CA

--- author: - | Zbigniew Jaskólski[^1]\ Institute of Theoretical Physics, Wroc[ł]{}aw University, Wroc[ł]{}aw, Poland\ - | Krzysztof A. Meissner[^2]\ International Centre for Theoretical Physics, Trieste, Italy date: 'January, 1994' title: | \  \ FIRST QUANTIZED NONCRITICAL\ RELATIVISTIC POLYAKOV STRING --- = 16truecm = 24truecm = -1.4truecm = -2truecm Introduction ============ In the present paper we address the question whether the Polyakov sum over random surfaces [@pol] yields in the range $1<d<25$ a consistent relativistic quantum mechanics of 1-dim extended objects. Within the Feynman path integral formulation the problem is to compute the Polyakov sum over all string trajectories starting and ending at prescribed string configurations and then to analyse the quantum mechanical content of the object obtained. Our motivation for considering the problem of the first quantized noncritical Polyakov string in this form is twofold. First the problem has its own interest as a nontrivial example of application of the covariant functional quantization techniques in the case of anomalous gauge theory. Secondly a well developed first quantized string in the functional formulation may shed new light on the long-standing problem of the interacting string theory in the physical dimension. The latter problem has recently received a considerable attention partly stimulated by the great progress which has been achieved over the last few years in the noncritical Polyakov string in dimensions $d \leq 1$ and $d=2$ [@gm]. Although there are some interesting attempts [@ag; @num] to go beyond the $c=1$ barrier still very little is known about the Polyakov model in the most interesting range $1<d<25$. Since the existence of the consistent quantum models of the free noncritical relativistic string is an old and well known fact [@b; @ct] it is probably in order to explain what kind of new insight one can get considering the path integral quantization of the free Polyakov string in the position representation. For that purpose we shall briefly consider two possible strategies of constructing noncritical string theory based on two equivalent but conceptually different formulations of the critical theory. For the sake of simplicity in all further considerations the open bosonic string in the flat Minkowski space is assumed. In the so called Polyakov covariant approach the interacting critical string theory is formulated in terms of on-shell Euclidean amplitudes given by the Polyakov sum over surfaces with appropriate vertex insertions. The relativistic S-matrix in this approach is defined by analytic continuation in the momenta of external states. In the conformal gauge the amplitudes can be expressed as correlators of the 2-dim conformal field theory (the tensor product of 26 copies of the free scalar field) integrated over moduli spaces of corresponding Riemann surfaces [@dhph]. In this form the Polyakov formulation can be seen as a modern covariant version of the old dual model construction [@dual] and allows for the straightforward generalization to arbitrary 2-dim conformal field theory with the central charge $c=26$. In contrast to the commonly used terminology we will call this formulation the critical Polyakov dual model. The second independent approach starts with the quantum mechanics of the relativistic free string. The full interacting theory is based on the simple picture of local “joining-splitting” interaction. For example, the open strings can interact by joining their end points and merging into a single one or by splitting a single string into two. It is assumed that no particular interaction occurs at joining or splitting points. The string amplitude for a given process is defined as a sum over all possible evolutions of the system. Each evolution in this sum is represented by a world-sheet in the Minkowski target space describing causally ordered processes of joining and splitting and contributing the factor ${\rm e}^{iS}$ where S is the classical string action. In contrast to the critical Polyakov dual model we call this formulation the critical relativistic string theory. The second approach is in fact known only in the case of the Nambu-Goto critical string in the light-cone gauge [@m]. The advantage of this gauge is that all restrictions imposed by the causality and locality principles of the relativistic quantum theory can be easily implemented in the path integral representation of the amplitudes and the unitarity of the S-matrix is manifest. The string amplitudes obtained within this approach are Lorentz covariant at the critical dimension and reproduce the amplitudes of the old dual models. The main difference in these approaches consists in their fundamental organizing principles. While in the case of the Polyakov dual model this is the principle of conformal invariance, in the case of the relativistic string theory one starts with the fundamental principles of the quantum mechanics of 1-dim extended relativistic systems. The equivalence between the critical Polyakov dual model and the Mandelstam light cone critical relativistic string theory, conjectured for a long time, has been proved few years ago [@gw; @dg]. This is one of the deepest and probably not fully appreciated results of the modern string theory. Strictly speaking this equivalence is the only known way by which one can give the stringy interpretation to the critical Polyakov dual model and prove the unitarity of its relativistic S-matrix. Let us stress that in spite of the suggestive picture of Riemann surfaces in the Euclidean target space (emphasized in almost every introductory text on string theory) the relation with the world-sheets of relativistic 1-dim extended objects interacting by joining and splitting is far from being obvious. The equivalence of the two conceptually different methods of constructing amplitudes in the critical theory is based on two facts. First of all due to the conformal invariance, the light cone diagram can be regarded as a special uniformization of the corresponding punctured Riemann surface. This in particular means that the singularities of the world-sheet in the Minkowski target space corresponding to the joining or splitting points are inessential. Secondly the parameters of this diagram yield a unique cover of the corresponding moduli space [@gw]. Note that the construction of the light cone diagram as well as the range of its characteristic parameters are uniquely determined by the causal propagation of joining and splitting strings in the Minkowski target space. It is one of the fundamental and nontrivial features of the critical string theory that the basic postulates of the relativistic quantum theory can be cast in a compact form of modular invariance in the Euclidean formulation. With the two equivalent formulation of the critical string theory there are two possible ways to go beyond d=26. The first one is to construct an appropriate generalization of the critical Polyakov dual model taking into account the conformal anomaly. In the covariant continuous formulation developed in [@ddk] it yields the noncritical Polyakov dual model given by some 2-dim conformal field theory with the central charge $c$ coupled to the conformal Liouville theory with the central charge $26-c$. The scheme of constructing dual amplitudes is the same as in the critical theory - they are given by correlators of vertex operators with the conformal weight 1. Vertices with this property are built from the conformal operators of the matter sector by gravitational dressing. By construction these amplitudes are conformally invariant. The basic problem of the conventional approach sketched above is the famous $c=1$ barrier which manifests itself in the appearance of complex critical exponents in the range $1<c<25$. This is commonly interpreted as a manifestation of the tachyon instability [@sei; @ku], although a precise mechanism of this phenomenon in the continuum approach is still unknown. In the case of the matter sector given by d-copies of the free scalar fields the noncritical Polyakov dual model can be seen as a theory of random surfaces in the Euclidean d-dim target space. With this interpretation it can be analysed by random triangulation techniques [@random]. The numerical simulations suggest the branched polymer phase [@num] which partly justifies the results of the continuum approach. Also the matrix model constructions designed to capture the range $1<c<25$ [@ag] indicate the polymerization of surfaces. The second possible way to construct the interacting string theory in the range $1<d<25$ is to follow the basic idea of the critical relativistic string theory (in the sense assumed in this paper). According to the brief description given above it consists of two steps : the relativistic quantum mechanics of 1-dim extended objects and the derivation of scattering amplitudes from the simple geometrical picture of joining-splitting interaction. Up to our knowledge this possibility remains completely unexplored. In fact all the recent achievements in noncritical string theory rely entirely on the noncritical Polyakov dual model. Taking the risk of missing important results in the rapidly developing field of research one can summarize the current state of affairs in the following diagram . (400,360)(-10,0) (70,0)[(80,50)]{} (75,-5)[(1,0)[80]{}]{} (75,-5)[(0,1)[5]{}]{} (155,45)[(-1,0)[5]{}]{} (155,45)[(0,-1)[50]{}]{} (250,0)[(80,50)]{} (160,80)[(80,50)]{} (0,230)[(80,50)]{} (5,225)[(1,0)[80]{}]{} (5,225)[(0,1)[5]{}]{} (85,275)[(-1,0)[5]{}]{} (85,275)[(0,-1)[50]{}]{} (160,200)[(80,50)]{} (165,195)[(1,0)[80]{}]{} (165,195)[(0,1)[5]{}]{} (245,245)[(-1,0)[5]{}]{} (245,245)[(0,-1)[50]{}]{} (70,310)[(80,50)]{} (250,310)[(80,50)]{} (200,335)[(-1,0)[40]{}]{} (200,335)[(1,0)[40]{}]{} (110,145)[(0,-1)[85]{}]{} (100,180)[(-3,4)[28]{}]{} (120,180)[(3,2)[28]{}]{} (200,160)[(0,1)[25]{}]{} (200,160)[(0,-1)[25]{}]{} (100,180)[(0,1)[125]{}]{} (110,175)[(0,1)[130]{}]{} (120,180)[(0,1)[125]{}]{} (30,150)[(1,0)[165]{}]{} (30,170)[(1,0)[165]{}]{} (30,150)[(0,1)[20]{}]{} (30,150)[(165,20)[$c=1$ barrier]{}]{} (205,150)(10,0)[16]{}[(1,0)[5]{}]{} (205,170)(10,0)[16]{}[(1,0)[5]{}]{} (292.5,305)(0,-10)[24]{}[(0,-1)[5]{}]{} (292.5,65)[(0,-1)[10]{}]{} (170,25)[(-1,0)[10]{}]{} (235,25)[(1,0)[10]{}]{} (175,25)(10,0)[6]{}[(1,0)[5]{}]{} The part of the diagram drawn in solid lines corresponds to the existing and well developed models and relations. Shadows of some boxes indicate the matrix model versions of the corresponding continuous models. We have also included the so called critical 2-dim string models arising from the interpretation of the Liouville field in the $c=1$ noncritical Polyakov dual models as the space-time coordinate in the target space [@c1]. With this interpretation they could be placed somewhere between the noncritical Polyakov dual model and the noncritical relativistic string theory. There is a number of interesting questions making the problem of constructing the missing part of the diagram above worth pursuing. One of them is whether the results indicating strong instability of the noncritical Polyakov dual model (or branched polymer nature of random surfaces) apply to the noncritical relativistic string theory. Clearly the answer depends on whether the equivalence between the Polyakov dual model and the relativistic string theory holds in noncritical dimensions. The necessary condition for the positive answer is the conformal invariance of the noncritical string amplitudes. If this condition is not satisfied or more generally if the equivalence does not hold there is still room for a consistent relativistic string theory in the physical dimensions, although it is hard to expect that the amplitudes of such theory will be dual. On the other hand if the equivalence holds it would provide the relativistic string interpretation of the noncritical Polyakov dual model, justifying commonly used stringy terminology which up to now is merely based on the equivalence in the critical dimension. Our main motivation to the present paper was to provide the first step toward the construction of the noncritical relativistic string theory. The path integral quantization of the Polyakov string in the position representation is especially convenient for this purpose. In fact it is the most suitable formalism for implementing the idea of “joining-splitting” interaction. As we shall see this approach allows for constructing the quantum theory without the assumption of the conformal symmetry and the equivalence mentioned above is a nontrivial and well posed problem. The first quantized noncritical BDHP string is essentially the problem of quantization of the anomalous theory. Within the Feynman path integral approach the idea is to formulate the quantum theory entirely in terms of the path integral over trajectories in the configuration space without referring to the canonical phase space analysis. In the case of model without anomaly such formulation must of course reproduce other methods of quantization. Applying this scheme to the model with anomaly one may hope that the resulting path integrals still can be given some meaning so it will make sense to ask about the consistency of the quantum theory derived in this way. In particular, in the case under consideration the conformal anomaly manifests itself in in the appearance of the effective action for the conformal factor. The functional measure in the resulting path integral can be dealt with in a similar way as in the noncritical Polyakov dual model [@mm]. The difference (and also the complication) is that we are considering the sum over rectangular-like surfaces connecting prescribed string configurations in the d-dim target space which brings new effects related with the boundary conditions. The main result of the present paper is that in the case of the open bosonic string described by the BDHP action the method sketched above yields a consistent relativistic quantum mechanics of 1-dim extended objects. The resulting theory coincides with the 20 years old Fairlie-Chodos-Thorn (FCT) model of the free massive string [@ct]. In the radial gauge the massive string is given by the following realization of the Virasoro algebra .3cm $$\begin{aligned} L_n&=& \frac{1}{2}\sum_{m=-\infty}^{\infty} : \alpha_m\cdot \alpha_{n-m} : +\frac{1}{2}\sum_{m=-\infty}^{\infty} : \beta_m\beta_{n-m} : +i(n+1) Q\beta_n \label{bcharge}\\ \left[\beta_m,\beta_n\right]&=& m\delta_{m,-n}\;\;\;,\;\;\; \left[\alpha^{\mu}_m,\alpha^{\nu}_n\right]\;=\; m\eta^{\mu\nu} \delta_{m,-n}\;\;\;,\;\;\; \eta^{\mu\nu}= diag(-1,+1,..,+1)\;\;\;.\nonumber\end{aligned}$$ .4cm In the Liouville sector this is the standard imaginary background charge realization commonly used in the noncritical Polyakov dual models [@pb]. The main difference consist in the hermicity properties of the operators involved. In contrast to the standard construction of the Feigin-Fuchs modules [@ff] one has $$\begin{aligned} \left(\alpha^{\mu}_k\right)^+ = \alpha^{\mu}_{-k}\;\;&,&\;\; \left(\beta_k\right)^+ = \beta_{-k}\;\;\;\;\;\;k\ne 0\;\;\;, \nonumber \\ \left(\alpha^{\mu}_0\right)^+ = \alpha^{\mu}_0\;\;&,&\;\; \left(\beta_0 +iQ \right)^+ = \beta_0 + iQ\;\;\;\;. \label{herm}\end{aligned}$$ As a consequence the structure of the tachyonic states is similar to that of the critical string - there are no excited tachyonic states in the spectrum of the free noncritical Polyakov string. There are some points in our derivation we would like to emphasize. First of all, in contrast to the noncritical Polyakov dual model, the derivation is independent of the techniques of 2-dim conformal field theory and does not rely on the principle of the conformal invariance. The key point of our derivation is the exact calculation of the transition amplitude between two arbitrary string configurations. This is done by an appropriate extension of the space of states and and then by translating the problem into an operator language. The Virasoro algebra of constraints arises as a set of consistency conditions of this method. As far as the theory of random surfaces is concerned there is no reason for introducing the full set of constraints. However if we assume the relativistic interpretation of the model at hand the additional constraints acquire a physical meaning - they can be seen as a consequence of the general kinematical requirement which must be satisfied by wave functionals of any quantum mechanics of relativistic 1-dim extended system. This indicates a difference between random surfaces and relativistic string theory. The general (Euclidean) path integral over surfaces with fixed boundaries is given by the model with hopelessly complicated boundary interaction. In a sense the relativistic string theory requires a very special type of this integral. The general theory of random surfaces with fixed boundaries is much more complicated and yet to be solved problem. The organization of the paper is as follows. In Section 2 we review the full scheme of the covariant functional quantization of the critical string in the Schrödinger representation. The material included has introductory character and serves as an illustration of the methods used in the following. The reason for a rather lengthy form of this section is twofold. First of all, although the main idea of Feynman’s quantization is well known, we are not aware of a self-contained presentation of this technique in the case of gauge models with reparameterization invariance, in particular in the form suitable for the quantization of the relativistic string. By self-contained we mean not only path integral representation of the transition amplitude (which is well known in the case of the critical string propagator [@propa; @vw]) but also the construction of the Hilbert space of states and the derivation of the physical state conditions from the “classical data”: the space of trajectories and the variational principle given by the classical action. Secondly a large part of the analysis given in Sect.2 applies without changes in the case of noncritical string which is of our main interest. This concerns in particular the geometry of the space of trajectories (Subsect.2.1), the construction of the space of states (Subsect.2.2) and the proper choice of boundary conditions in the matter and in the conformal factor sectors. The last issue is crucial for the consistent path integral representation of the transition amplitude (the open string propagator) constructed in Subsect.2.3. In Subsect.2.4 we derive the part of physical state conditions related to the constraints linear in momenta. Within the presented approach they are given by generators of the unitary realization of the residual gauge symmetry. In Subsect.2.5 the transition amplitude between the states satisfying constraints linear in momenta is calculated and the second part of the physical state conditions related to the constraints quadratic in momenta is derived. One of these constraints - the on-mass-shell condition is encoded in the transition amplitude. The rest can be regarded as a consequence of the general kinematical requirement mentioned above. Finally the full set of the physical state conditions can be expressed in terms of the familiar Virasoro constraints of the old covariant formulation of the first quantized critical string. In Section 3 the functional formalism developed in Section 2 is applied in the case of noncritical Polyakov string in the range $1<d<25$. The lower bound results from the relativistic interpretation of the string which breaks down for $d<2$ while the upper one from the coefficient in front of the effective Liouville action proportional to $(25-d)$. As it was mentioned above the first few steps of the quantization procedure proceed as in the case of the critical string. The main difference consists in the different symmetry properties of the noncritical string, briefly described in Subsect.3.1.. In Subsect.3.2 the transition amplitude for the noncritical string is constructed. As a result of the conformal anomaly and our choice of boundary conditions it involves the path integral over conformal factor satisfying the homogeneous Neumann boundary conditions. As a simple consequence of the Gauss-Bonnet theorem, the resulting theory is stable in the case of rectangle-like world-sheets if the bulk and the boundary cosmological constants vanish. The Liouville sector couples to the “matter” sector via boundary conditions for the $x$-variables which depend in a complicated way on the boundary value of the conformal factor. Even with the vanishing cosmological constant the resulting path integral cannot be directly calculated. Our method to overcome this difficulty is to express the transition amplitude as a matrix element of a simple operator in a suitably extended space of states. This is done by means of the generalized Forman formula. In Subsect.3.3 we derive the full set of the physical state conditions in the extended space. It consists of the constraints linear in momenta related to the extension itself, the on-mass-shell condition encoded in the transition amplitude and the set of quadratic in momenta constraints arising by the mechanism similar to that of the critical string theory. The resulting algebra of constraints yields the FCT massive string model [@ct]. Finally in Subsect.2.4 we provide the explicit DDF construction of the physical states of this model, which gives a simple proof of the no-ghost theorem. Section 4 contains the discussion of the results obtained. A comparison with the noncritical Polyakov dual model is given and the open problems of the first quantized noncritical relativistic string are reviewed. We conclude this section by a brief discussion of the choice of vanishing cosmological constant in the interacting theory. The paper contains three appendices. In Appendix A we gather some basic facts concerning the corner conformal anomaly. In Appendix B the 1-dim “conformal anomaly” is calculated. The proof of the generalized Forman formula is given in Appendix C. Functional quantization of critical Polyakov string =================================================== Space of trajectories --------------------- In the Euclidean formulation a trajectory of the open Polyakov string is given by a triplet $(M,g,x)$ where $M$ is a rectangle-like 2-dim manifold with distinguished “initial” $\partial M_i$ and “final” $\partial M_f$ opposite boundary components, $g$ is a Riemannian metric on $M$, and $x$ is a map from $M$ into the Euclidean target space ${\bf R}^d$ satisfying the boundary conditions $$n^a_g \partial_a x_{|\partial M_t} = 0 \label{bcxends}$$ along the “timelike” boundary components $\partial M_t = \partial M \setminus (\partial M_i \cup \partial M_f)$. In the formula above $n_g$ denotes the normal direction along $\partial M$ with respect to the metric $g$. In this section $d$ will be equal to 26 but will be sometimes left as $d$ to emphasize the dependence of quantities on the number of dimensions. The BDHP action functional $$S[M,g,x] = \int\limits_M \sqrt{g}\, d^2z\, g^{ab}\partial_a x^{\mu} \partial_b x_{\mu}$$ is invariant with respect to the Weyl rescaling of the internal metric as well as the general diffeomorphisms $f:M \rightarrow M'$ preserving the initial and final boundary components and their orientations. The latter invariance can be partly restricted by fixing a model manifold $M$ and a normal direction along the boundary $\partial M$. This can be seen as a partial gauge fixing and has important consequences for all further constructions. Let us note that other gauge fixings are also possible, although they are much more difficult to deal with [@r]. Let ${\cal M}_M^n$ be the space of all Riemannian metrics on $M$ with the normal direction $n_g = n$ and ${\cal E}^n_M$ the space of all maps from $M$ into the target space satisfying the boundary conditions (\[bcxends\]). In the $(M,n)$-gauge the space of trajectories is the Cartesian product $ {\cal M}_M^n \times {\cal E}^n_M$. The gauge transformations form the semidirect product ${\cal W}_M \odot {\cal D}_M^n$ of the additive group ${\cal W}_M$ of scalar functions on $M$ and the group ${\cal D}_M^n$ of diffeomorphisms of $M$ preserving corners and the normal direction $n$. The action of ${\cal W}_M \odot {\cal D}_M^n$ on $ {\cal M}_M^n \times {\cal E}^n_M$ is given by $${\cal M}_M^n \times {\cal E}^n_M \ni (g,x) { \begin{picture}(90,0)(0,0) \put(5,3){\vector(1,0){80}} \put(15,10){$\scriptstyle (\varphi,f) \in {\cal W}_M \odot {\cal D}_M^n $ } \end{picture}} ({\rm e}^{\varphi}f^*g,f^*x) \in {\cal M}_M^n \times {\cal E}^n_M\;\;\;$$ Space of states --------------- Within the covariant functional approach to quantization in the Schrödinger representation the space of states consists of wave functionals defined on the Cartesian product ${\cal C}\times {\bf R}$ where ${\cal C}$ is a suitably chosen space of boundary conditions (half of the Cauchy data for the classical trajectory in the case of nondegenerate Lagrangian) and ${\bf R}$ is the time axis. For gauge systems with the reparameterization invariance the inner time evolution is generated by a constraint quadratic in momenta and in the subspace of physical states it is simply given by the identity operator. In the covariant functional approach this feature manifests itself in the inner time independent formulation of variational principle. In consequence one can describe the space of states in terms of inner time independent wave functionals. The choice of ${\cal C}$ itself is slightly more complicated. In order to explain the intricacies involved we consider the space related to the reduced “position” representation in the $(M,n)$-gauge [@jas]. For every string trajectory $(g,x)\in {\cal M}_M^n \times {\cal E}^n_M$ we define the initial $(e_i,x_i)$ and the final $(e_f,x_f)$ boundary conditions $$\begin{aligned} e^2_i & = & g_{ab}t^at^b (dt)^2_{|\partial M_i} \;\;\;, \;\;\;(i \rightarrow f)\;\;, \label{bc} \\ x_i & = & x_{|\partial M_i} \;\;\;, \;\;\;(i \rightarrow f)\;\;\;; \nonumber\end{aligned}$$ where $t$ denotes a vector tangent to the boundary and $e_i, e_f$ are einbeins induced on the initial and the final boundary component respectively. All possible boundary conditions form the space $${\cal P}_i = {\cal M}_i \times {\cal E}_i\;\;\;;$$ where ${\cal M}_i$ consists of all einbeins on $\partial M_i$ and ${\cal E}_i$ is the space of maps $x_i : \partial M_i \rightarrow {\bf R}^d $ satisfying the Neumann boundary conditions at the ends of $\partial M_i$. Similarly the final boundary conditions form the space $${\cal P}_f = {\cal M}_f \times {\cal E}_f\;\;\;.$$ The interpretation of the transition amplitude as an integral kernel of some operator requires an identification of the spaces ${\cal P}_i$ and ${\cal P}_f$. It can be done by introducing a model interval $L$ and the space $${\cal P}_L = {\cal M}_L \times {\cal E}_L\;\;\; \label{sbc}$$ together with the isomorphisms $$\Gamma_i: {\cal P}_i \longrightarrow {\cal P}_L\;\;\;,\;\;\; (i \rightarrow f) \;\;\; , \label{isomor}$$ induced by some arbitrary chosen diffeomorphisms $$\gamma_i: L \longrightarrow \partial M_i \;\;\;,\;\;\; (i \rightarrow f)\;\;\;.$$ In the covariant functional approach, the part of the canonical analysis concerning constraints linear in momenta can be recovered by considering classes of gauge equivalent boundary conditions. We say that $p, p' \in {\cal P}_L$ are gauge equivalent if there exist two ${\cal W}_M \odot {\cal D}^n_M$ equivalent string trajectories with a common final boundary condition and starting at $p$ and $p'$ respectively. In our case the equivalence classes can be described as orbits of the group ${\cal W}_L \odot {\cal D}_L$, where ${\cal W}_L$ is the additive group of real functions on $L$ satisfying, as we shall see in the following, the Neumann boundary conditions at the ends of $L$, and ${\cal D}_L$ is the group of orientation preserving diffeomorphisms of $L$. The action of ${\cal W}_L \odot {\cal D}_L$ on ${\cal P}_L$ is given by $${\cal P}_L \ni (e_i,x_i) { \begin{picture}(92,0)(0,0) \put(5,3){\vector(1,0){82}} \put(13,10){$\scriptstyle (\widetilde{\varphi},\gamma) \in {\cal W}_M \odot {\cal D}_M^n $ } \end{picture}} ({\rm e}^{{\widetilde{\varphi}}\over 2}\gamma^*e_i,x_i \circ \gamma) \in {\cal P}_L\;\;.$$ The first factor in ${\cal W}_L \odot {\cal D}_L$ corresponds to the Weyl invariance in the space of trajectories while the second one – to the ${\cal D}^n_M$-invariance. In the space of states ${\cal H}({\cal P}_L)$ consisting of string wave functionals defined on ${\cal P}_L$ the “physical” states are ${\cal W}_L \odot {\cal D}_L$ invariant functionals. The comparison with the canonical quantization shows that the generators of this symmetry form an operator realization of the constraints linear in momenta. In particular the choice of the quotient $${\cal K}_L = \frac{ {\cal P}_L }{ {\cal W}_L \odot {\cal D}_L }$$ corresponds to a formulation in which all constraints linear in momenta are solved. As far as the constraints linear in momenta are concerned one can construct the space of states on an arbitrary quotient between ${\cal P}_L$ and ${\cal K}_L$. It turns out however that the consistency requirements concerning the path integral representation of the transition amplitude yield strong restrictions on possible choices [@jas]. In general these requirements depend on the gauge fixing used to calculate (=define) the transition amplitude and for each particular choice of the space ${\cal C}$ boil down to some regularity conditions concerning the gauge group action on the space ${\cal T}[c_i,c_f]$ of trajectories starting and ending at two fixed points $c_i,c_f \in {\cal C}$. The analysis of the geometry of this action together with the Faddeed-Popov method of calculating path integral can be seen as a covariant counterpart of the symplectic reduction in the phase space approach. In the case of the open Polyakov string in the conformal gauge there are only two admissible choices [@jas] $$\begin{aligned} {\cal C}_L & = & \frac{ {\cal P}_L}{{\cal D}_L } \;\;\;, \label{c} \\ {\cal C'}_L & = & \frac{ {\cal P}_L}{{\bf R}_+ \times {\cal D}_L } \;\;\;,\nonumber\end{aligned}$$ where ${\bf R}_+$ denotes the 1-dim group of constant rescalings acting on ${\cal M}_L$. Note that in the both cases above the identifications (\[isomor\]) factor out to the canonical $\gamma$-independent isomorphisms $${\cal C}_i \;= \;{\cal C}_f \;=\;{\cal C}_L \;\;\;,\;\;\; {\cal C'}_i\; = \;{\cal C'}_f \;=\;{\cal C'}_L \;\;\;.$$ The quotients ${\cal C}_L,{\cal C'}_L$ correspond to the situation in which the constraints related to the Weyl invariance are represented on the quantum level by generators of some symmetry group acting on ${\cal C}_L$ (${\cal C'}_L$) while all the constraints related to the ${\cal D}^n_M$-invariance are completely solved. The only difference between the spaces ${\cal C}_L$ and ${\cal C'}_L$ consist in a different treatment of the constraint related to the Weyl rescaling of metric by conformal factor constant along the (initial) boundary. It is more convenient to realize this constraint on the quantum level which corresponds to the choice of ${\cal C}_L$. Our final task is to introduce an inner product in the space ${\cal H}({\cal C}_L)$ of string wave functionals on ${\cal C}_L$. To this end let us observe that the space ${\cal M}_L \times {\cal E}_L$ carries the ultralocal ${\cal D}_L$-invariant structure and the scalar product in ${\cal H}({\cal C}_L)$ is given by the path integral $$\langle\Psi | \Psi'\rangle = \int\limits_{{\cal M}_L \times {\cal E}_L} {\cal D}^ee\;{\cal D}^e\widetilde{x} \left( {\rm Vol}_e {\cal D}_L \right)^{-1} \overline{\Psi} \Psi'\;\;\;. \label{product}$$ In order to parameterize the quotient (\[c\]) it is convenient to use the 1-dim conformal gauge $\hat{e} = {\rm const}$ which yields the isomorphism $${\cal C}_L =_{\widehat{e}} {\bf R}_+ \times {\cal E}_L\;\;\;.$$ Using the Faddeev-Popov method in this gauge one gets $$\langle \Psi | \Psi' \rangle = \int\limits^{\infty}_0 d\alpha \int\limits_{{\cal E}_L} {\cal D}^{\alpha \hat{e} } \widetilde{x} \; \overline{\Psi[\alpha,\widetilde{x}]} \Psi'[\alpha,\widetilde{x}]\;\;\;. \label{gproduct}$$ Transition amplitude -------------------- The central object of the covariant functional quantization is the path integral representation of the transition amplitude. In the case of critical Polyakov string and with the choice of ${\cal C}_L$ as a space of boundary conditions it takes the following form $$P[c_i,c_f] = \int\limits_{{\cal F}[c_i,c_f]} {\cal D}^gg {\cal D}^gx \left(\mbox{Vol}_g {\cal W}_M \right)^{-1} \left(\mbox{Vol}_g {\cal D}_M^n \right)^{-1} \exp \left(- {\scriptstyle \frac{1}{4\pi \alpha '}} S[g,x]\right)\;\;\;. \label{pro}$$ $ {\cal F}[c_i,c_f] \subset {\cal M}_M^n \times {\cal E}_M^n$ in the formula above consists of all string trajectories starting and ending at $c_i,c_f \in {\cal C}_L$ respectively, i.e. satisfying the boundary conditions $$[(e_i,x_i)] = c_i \;\;\;, \;\;\; (i\rightarrow f) \;\;\;, \label{bcx}$$ where $e_i,x_i$ are given by (\[bc\]) and $[(e_i,x_i)]$ denotes the ${\cal D}_L$-orbit of the element $(e_i,x_i) \in {\cal M}_L\times {\cal E}_L$;$ (i\rightarrow f)$. For every $g \in {\cal M}_M^n$ the conditions (\[bcx\]) are $g$-dependent ${\cal D}_M^n$-invariant Dirichlet boundary conditions for $x$ on $\partial M_i \cup \partial M_f$. With the boundary conditions (\[bcxends\]),(\[bcx\]) the integration over $x$ in (\[pro\]) is Gaussian and yields ${\cal D}_M^n$-invariant functional on ${\cal M}_M^n$. This allows for application of the F-P procedure with respect to the group ${\cal D}_M^n$. The consistency conditions for this method uniquely determine boundary conditions for the metric part of a string trajectory [@ja]. The space ${\cal M}^{n*}_M$ of all metrics $g \in {\cal M}_M^n$ satisfying these conditions can be described as follows. Let ${\cal M}^{n0}_M$ be the space of all metrics from ${\cal M}_M^n$ with the scalar curvature $R_g = 0$, such that all boundary components are geodesic and meet orthogonally. Then ${\cal M}^{n*}_M$ consists of all metrics of the form $\exp (\varphi) g_0$, where $g_0 \in {\cal M}^{n0}_M$ and $\varphi$ satisfies the homogeneous Neumann boundary condition $$n^a \partial_a \varphi = 0 \label{bcphi}$$ on all boundary components of $M$. An important consequence of this result is that ${\cal C}_L$ is the largest space of boundary conditions for which there exists a consistent path integral representation of the transition amplitude. Note that in every conformal gauge in ${\cal M}^{n*}_M$ the conformal factor satisfies the boundary condition (\[bcphi\]). It follows that the gauge group ${\cal W}_M \odot {\cal D}^n_M$ must be restricted to the group ${\cal W}_M^n \odot {\cal D}^n_M$, where ${\cal W}^n_M$ consists of all $\varphi \in {\cal W}_M$ satisfying the boundary conditions (\[bcphi\]). This justifies our definition of the induced gauge transformations given in the previous subsection. The boundary conditions (\[bcx\]) yield the restrictions on the internal length of the initial and final boundary components which can be implemented by appropriate delta function insertions in the path integral representation (\[pro\]). With the space ${\cal F}[k_i,k_f]$ consisting of all string trajectories $(g,x) \in {\cal M}_M^{n*} \times {\cal E}_M^d$ such that $x$ satisfies the boundary conditions (\[bcxends\]),(\[bcx\]) the calculations of the integral (\[pro\]) proceeds along the standard lines. In the conformal gauge $$\begin{aligned} (\widehat{M}_t,\widehat{g}_t)& = &([0,t]\times[0,1], \left( \parbox{16pt}{ \scriptsize 1 \makebox[1pt]{} 0 \\ 0 \makebox[1pt]{} 1 } \right) ) \;\;\;, \label{gauge}\\ (\widehat{L} = \partial \widehat{M}_{ti} = \partial \widehat{M}_{tf},\widehat{e} ) & = & ([0,t],1)\;\;\;, \label{bgauge}\end{aligned}$$ the $x$-integration yields $$\int {\cal D}^{{\rm e}^{\varphi}\hat{g}_t}x \exp (- {\textstyle \frac{1}{4\pi \alpha'}}S[{\rm e}^{\varphi}\widehat{g}_t,x] ) = \left( \det {\cal L}_{{\rm e}^{\varphi}\hat{g}_t} \right)^{-{d \over 2}} \exp (- {\textstyle \frac{1}{4\pi \alpha'}} S[\widehat{g}_t,\varphi,\widetilde{x}_i,\widetilde{x}_f] )$$ where $$S[\widehat{g}_t,\varphi,\widetilde{x}_i,\widetilde{x}_f ] = \int^t_0 dz^0 \int^1_0 dz^1 \left(({\partial}_0 x_{cl})^2 + ({\partial}_1 x_{cl})^2 \right) \label{xaction}$$ and $x_{cl}: \widehat{M}_t \rightarrow R^d$ is the solution of the boundary value problem $$\begin{aligned} \left( {\partial}^2_0 + {\partial}^2_1 \right) x_{cl} & = & 0 \;\;,\nonumber\\ x_{cl}(0,z^1) &=& \widetilde{x}_i \circ \gamma[\widetilde{\varphi}_i](z^1) \;\;, \label{xbvp}\\ x_{cl}(t,z^1) &=& \widetilde{x}_f \circ \gamma[\widetilde{\varphi}_f](z^1) \;\;,\nonumber\\ \partial_1 x_{cl}(z^0,0) &=& \partial_1 x_{cl}(z^0,1) = 0 \;\;.\nonumber\end{aligned}$$ The functions $\widetilde{x}_i ,\widetilde{x}_f:[0,1] \rightarrow R^d$ are representants of $c_i , c_f$ in the 1-dim conformal gauge (\[bgauge\]) (i.e.$ [(\alpha_i,\widetilde{x}_i)] = c_i, [(\alpha_f,\widetilde{x}_f)] = c_f $ ), and the diffeomorphisms $\gamma[\widetilde{\varphi}_i],\gamma[\widetilde{\varphi}_f] : [0,1] \rightarrow [0,1]$ are uniquely determined by the equations $$\begin{aligned} \frac{d}{dz^1} \gamma[\widetilde{\varphi}_i](z^1) & \propto & \exp {\scriptstyle \frac{1}{2}} \widetilde{\varphi}_i(z^1) \;\;\;, \;\;\; \widetilde{\varphi}_i(z^1) \equiv \varphi(0,z^1) \;\;,\nonumber\\ \frac{d}{dz^1} \gamma[\widetilde{\varphi}_f](z^1) & \propto & \exp {\scriptstyle \frac{1}{2}} \widetilde{\varphi}_f(z^1) \;\;\;, \;\;\; \widetilde{\varphi}_f(z^1) \equiv\varphi(t,z^1) \;\;. \label{difeq}\end{aligned}$$ Solving the boundary value problem (\[xbvp\]) and inserting solution into the action (\[xaction\]) one has $$\begin{aligned} S[\widehat{g}_t,\varphi,\widetilde{x}_i , \widetilde{x}_f ] & = & \frac{\left( X_{i0} - X_{f0} \right)^2}{t} \\ & + & \frac{1}{2} \sum\limits_{m > 0} \frac{ \pi m}{ \sinh \pi mt } \left[ ( {X_{im}}^2 + {X_{fm}}^2 ) \cosh \pi mt - 2 X_{im}X_{fm} \right] \;\;\;, \nonumber\end{aligned}$$ where $$\begin{aligned} X^{\mu}_{i0} & = & \int\limits^1_0 dz^1 \widetilde{x}_i \circ \gamma[\widetilde{\varphi}_i](z^1) \;\;\;\;\;\;\;\;\;\;\;\;\; (i\rightarrow f)\;\;, \nonumber \\ X^{\mu}_{im} & = &2 \int\limits^1_0 dz^1 \widetilde{x}_i \circ \gamma[\widetilde{\varphi}_i](z^1) \cos \pi mz^1 \;\;\; (i\rightarrow f)\;\;. \nonumber\end{aligned}$$ Note that the functional $S[\widehat{g}_t,\varphi,\widetilde{x}_i , \widetilde{x}_f ]$ depends only on the boundary values $\widetilde{\varphi}_i, \widetilde{\varphi}_f$ of the conformal factor. Applying the F-P method to the resulting path integral over ${\cal M}^{n*}_M$ [@jas] and using the heat kernel method [@mm] to find out the $\varphi$-dependence hidden in the functional measure and in the volume of ${\cal W}_M^n$ one gets $$\begin{aligned} P[\alpha_f,\widetilde{x}_f;\alpha_i,\widetilde{x}_i] = \int\limits_0^{\infty}\!\! & dt &\!\! {\eta (t)}^{1 - {d \over 2}} t^{-{d \over 2}} \int\limits_{{\cal W}^n}\!\! {\cal D}^{\widehat{g}_t}\varphi \left(\mbox{Vol}_{\widehat{g}_t} {\cal W}_M^n \right)^{-1} \nonumber \\ &\times&\exp{\left(- {26-d \over 48\pi} S_L[\widehat{g}_t,\varphi]\right) } \nonumber \\ &\times& \exp{\left( {6-d \over 32} \sum\limits_{\mbox{\scriptsize corners}} \varphi(z_i) \right)} \label{propa}\\ &\times& \exp{\left(- {1 \over 4 \pi \alpha'} S[\widehat{g}_t,\varphi,\widetilde{x}_i,\widetilde{x}_f ] \right)} \;\;\;,\nonumber\\ &\times& \delta \left( \alpha_f - \int_0^1 dz^1 {\rm e}^{ \frac{1}{2} \widetilde{\varphi}_f} \right) \; \delta \left( \alpha_i - \int_0^1 dz^1 {\rm e}^{ \frac{1}{2} \widetilde{\varphi}_i} \right) \;\;\;,\nonumber\end{aligned}$$ where the Liouville action is given by $$S_L[g,\varphi ] = \int\limits_{M}\!\!\! \sqrt{g}\, d^2z\; \left( {1\over 2} g^{ab} \partial_a \varphi \partial_b \varphi + R_g\varphi + {\mu\over 2} {\rm e}^{ \varphi} \right) + \lambda \int\limits_{\partial M}\!\! e\,ds\; {\rm e}^{ \frac{\widetilde{\varphi}}{2}} \;\;\;, \label{liouville}$$ and $$\eta (t) = {\rm e}^{-\frac{\pi t}{12}}\prod_{n=1}^{\infty}\left( 1-{\rm e}^{-2\pi n t}\right)\;\;\;.$$ In contrast to the expression for an on-shell open string amplitude the conformal factor does not decouple in $d=26$. As we shall see, the decoupling takes place for the transition amplitudes between states in ${\cal H}({\cal C}_L)$ satisfying the constraints linear in momenta. Constraints linear in momenta ----------------------------- The next step in the quantization procedure is to determine the subspace of physical states in ${\cal H}({\cal C}_L)$. There are three groups of interrelated physical state conditions: the constraints linear in momenta given by the generators of the residual induced gauge symmetry in ${\cal C}_L$, the constraint quadratic in momenta encoded in the transition amplitude, and the kinematical constraints following from the interpretation of the string as a 1-dim extended relativistic system. In this subsection we will discuss the first group of physical state conditions consisting of constraints linear in momenta. The residual gauge symmetry in the space ${\cal C}_L$ can be described by $${\cal C}_L \ni [(e,\widetilde{x})] { \begin{picture}(50,0)(0,0) \put(5,3){\vector(1,0){40}} \put(15,10){$\scriptstyle \widetilde{\varphi} \in {\cal W}_L $ } \end{picture}} [({\rm e}^{{\widetilde{\varphi} \over 2}} e,\widetilde{x})] \in {\cal C}_L\;\;\;. \label{itrans}$$ In the 1-dim conformal gauge (\[bgauge\]) the transformation (\[itrans\]) takes the form $${\bf R}_+ \times {\cal E}_L \ni (\alpha,\widetilde{x}) { \begin{picture}(90,0)(0,0) \put(5,3){\vector(1,0){80}} \put(15,10){$\scriptstyle (\lambda,\gamma) \in {\bf R}_+ \times {\cal D}_L $ } \end{picture}} (\lambda[\widetilde{\varphi}]\alpha,\widetilde{x} \circ \gamma[\widetilde{\varphi}]) \in \mbox{ \boldmath R}_+ \times {\cal E}_L\;\;\;, \label{iitrans}$$ where $$\lambda[\widetilde{\varphi}] = \int\limits_0^1 {\rm e}^{{\widetilde{\varphi} \over 2}} dz^1\;\;\;,$$ and $\gamma[\widetilde{\varphi}]:[0,1] \rightarrow [0,1]$ is uniquely determined by the equation $$\frac{d}{dz^1} \gamma[\widetilde{\varphi}] = \left( \lambda[\widetilde {\varphi}] \right)^{-1} {\rm e}^{{\widetilde{\varphi} \over 2}}\;\;\;.$$ It follows that all induced gauge transformations form the group ${\bf R}_+ \times {\cal D}_L$ acting on ${\cal C}_L$ by (\[iitrans\]). Note that this group structure as well as the group action are consequences of the ${\cal D}^n_M$-invariant formulation and are independent of a gauge fixing used to parameterize the quotient (\[c\]). This remark also applies to all further considerations where for the sake of simplicity the conformal gauges (\[gauge\],\[bgauge\]) will be used. According to the discussion in Subsect.1.2 the wave functionals corresponding to physical states are invariant with respect to the induced gauge transformations represented in ${\cal H}({\bf R}_+ \times {\cal E}_L)$ by $$\Psi[\alpha,\widetilde{x}] { \begin{picture}(90,0)(0,0) \put(5,3){\vector(1,0){80}} \put(15,10){$\scriptstyle (\lambda,\gamma) \in {\bf R}_+ \times {\cal D}_L $ } \end{picture}} \Psi[\lambda\alpha,\widetilde{x}\circ \gamma] \label{naive}$$ The problem with the representation above is that the scalar product (\[gproduct\]) is not invariant with respect to (\[iitrans\]). As a result the subspace ${\cal H}_{\scriptstyle \rm inv}({\bf R}_+ \times {\cal E}_L)$ of invariant wave functionals does not have a well defined scalar product and, as explained below, the transformation (\[naive\]) should be modified. As far as the ${\bf R}_+$ symmetry is concerned this noninvariance is not important due to nondynamical nature of the $\alpha$ variable. As we shall see the ambiguity in the choice of the scalar product on the subspace ${\cal H}({\cal E}_L)$ of $\alpha$-independent wave functionals can be hidden in the overall normalization factor. For this reason we can restrict our considerations to the ${\cal D}_L$ gauge symmetry in the space ${\cal H}^{\alpha\widehat{e}}({\cal E}_L)$ defined as the space ${\cal H}({\cal E}_L)$ endowed with the scalar product $$\langle \Psi | \Psi' \rangle = \int\limits_{{\cal E}_L} {\cal D}^{\alpha \hat{e} } \widetilde{x} \; \overline{\Psi[\widetilde{x}]} \Psi'[\widetilde{x}]\;\;\;. \label{ggproduct}$$ Let us recall that the functional measure ${\cal D}^e \widetilde{x}$ in (\[ggproduct\]) is formally defined as the Riemannian volume element related to the following (weak) Riemannian structure on ${\cal E}_L$ $${ E^{e}}_{\widetilde{x}}(\delta \widetilde{x}, \delta \widetilde{x}') = \int\limits_0^1 e\;ds \;\delta \widetilde{x}^{\mu}(s) \delta \widetilde{x}'_{\mu}(s)\;\;\;.$$ The pull-back of the Riemannian structure $E^{e}$ by the gauge transformation $$F_{\gamma} : {\cal E}_L \ni \widetilde{x} \longrightarrow \widetilde{x} \circ \gamma \in {\cal E}_L\;\;\;,$$ is given by $${F_{\gamma}^*E^{e}}_{\widetilde{x}} (\delta \widetilde{x}, \delta \widetilde{x}') = \int\limits_0^1 e\;ds \;\delta \widetilde{x} \circ \gamma (s) \delta \widetilde{x}'\circ \gamma(s) = {E^{(\gamma^{-1})^*e}}_{\widetilde{x}}(\delta \widetilde{x}, \delta \widetilde{x}')\;\;\;.$$ For $e=\alpha\widehat{e}$ where $\widehat{e}$ is a constant (as it is in the 1-dim conformal gauge (\[bgauge\])) we have $$(\gamma^{-1})^* e = ( \gamma^{-1})'\alpha\widehat{e}\;\;\;.$$ Then, calculating the 1-dim “conformal anomaly” (Appendix B, (\[conano\])) and choosing the (1-dim) bulk renormalization constant equal zero one gets $${\cal D}^{\alpha\widehat{e}} \widetilde{x} \circ \gamma = {\rm e}^{-{d\over 8}(\log \gamma'(0) + \log \gamma'(1) )} {\cal D}^{\alpha\widehat{e}} \widetilde{x} \;\;\;.$$ It follows that in order to get the unitary ${\cal D}_L$-action on ${\cal H}({\cal E}_L)$ the “naive” representation (\[naive\]) of the induced gauge transformations must be replaced by $${\cal H}({\cal E}_L) \ni \Psi[\widetilde{x}] { \begin{picture}(50,0)(0,0) \put(5,3){\vector(1,0){40}} \put(15,10){$\scriptstyle \gamma \in {\cal D}_L $ } \end{picture}} \rho [\gamma] \Psi[\widetilde{x}\circ \gamma] \in {\cal H}({\cal E}_L)\;\;\;, \label{mtrans}$$ where $$\rho[\gamma] \equiv {\rm e}^{-{d\over 16}(\log \gamma'(0) + \log \gamma'(1) )} \;\;\;,\;\;\;\rho[\gamma \circ \delta] = \rho[\gamma]\rho[\delta]\;\;\;; \;\;\; \gamma,\delta \in {\cal D}_L \;\;\;.$$ Indeed in this case we have $$\begin{aligned} \int\limits_{{\cal E}_L} {\cal D}^{\alpha \hat{e} } \widetilde{x} \; \rho[\gamma]\overline{\Psi[\widetilde{x}\circ \gamma]} \rho[\gamma]\Psi'[\widetilde{x}\circ \gamma]&=& \int\limits_{{\cal E}_L} {\cal D}^{\alpha \hat{e} } \widetilde{x}\circ \gamma^{-1} \; \rho[\gamma]^2 \overline{\Psi[\widetilde{x}]} \Psi'[\widetilde{x}]\\ &=&\int\limits_{{\cal E}_L} {\cal D}^{\alpha \hat{e} } \widetilde{x} \; \overline{\Psi[\widetilde{x}]} \Psi'[\widetilde{x}]\;\;\;.\end{aligned}$$ Let us note that the modified ${\cal D}_L$-action is independent of $\alpha$. The fully covariant with respect to the choice of $\widehat{e}$ description of the modified action requires more general geometrical framework and will not be discussed here. According to (\[mtrans\]) the space ${\cal H}_{\scriptstyle \rm inv}( {\cal E}_L)$ consists of all functionals satisfying $$\Psi[\widetilde{x}] = {\rm e}^{-{d\over 16}(\log \gamma'(0) + \log \gamma'(1) )} \Psi[\widetilde{x}\circ \gamma]\;\;\;,\;\;\;\gamma \in {\cal D}_L\;\;\;.$$ The space ${\cal H}_{\scriptstyle \rm inv}( {\cal E}_L)$ can be also characterized in terms of constraints linear in momenta given by the generators of the representation (\[mtrans\]) ($k\geq 1$) $$\begin{aligned} V_k^x \; \equiv & -& \!\!\!i\int\limits_0^1 ds \; \sin \pi ks (\widetilde{x}^{\mu})'(s) \frac{\delta}{\delta \widetilde{x}^{\mu}(s) } - i \int\limits_0^1 ds \sin \pi ks {\frac{\delta}{\delta \gamma(s)} \rho(\gamma) }_{|\gamma = {\scriptstyle \rm id}_L } \label{vk}\\ = & -& {\pi\over 2}\sum\limits_{n = 1}^k n x_n^{\mu} p_{\mu (k-n)} + {\pi \over 2} \sum\limits_{n=1}^{\infty} \left( n x_k^{\mu}p_{\mu(n+k)} - (n+k)x_{n+k}^{\mu}p_{\mu n} \right) + i\pi {d\over 8}p(k)k \;\;\;;\nonumber\end{aligned}$$ where $$\begin{aligned} p_{\mu}(s) &\equiv & -i\frac{\delta}{\delta \widetilde{x}^{\mu}(s)} = p_{\mu 0} + 2\sum\limits_{k = 1}^{\infty} p_{\mu k} \cos \pi ks \;\;\;,\\ \widetilde{x}^{\mu}(s) & = & x^{\mu}_0 + \sum\limits_{k=1}^{\infty} x_k^{\mu } \cos \pi ks\;\;\;,\end{aligned}$$ and $$p(k) = \left\{ \begin{array}{ll} 1 & \mbox{for $k$ even} \\ 0 & \mbox{for $k$ odd} \end{array} \right.\;\;\;.$$ Let us note that the constraints $V^x_k$ are formally self-adjoint operators in the Hilbert space ${\cal H}^{\alpha\widehat{e}}({\cal E}_L)$ which is in agreement with the path integral derivation of the unitary ${\cal D}_L$-action given above. Another interesting property is that $V^x_k$ are normally ordered, a feature which is required in the canonical quantization on different grounds. Constraints quadratic in momenta -------------------------------- In the covariant formulation of the first quantized relativistic particle the Euclidean transition amplitude is interpreted as a matrix element of the constraint quadratic in momenta. Inverting this operator and performing the Wick rotation one gets the on-mass-shell condition simply given by the Klein-Gordon wave equation. As we shall see a similar interpretation is valid in the case of Polyakov string. Due to the presence of residual induced gauge symmetry it is enough to consider the transition amplitude between states $|\Psi\rangle \in {\cal H}({\cal C}_L)$ described by $\alpha$-independent, ${\cal D}_L$-invariant string wave functionals $\Psi[\widetilde{x}] $. According to (\[gproduct\]) and (\[propa\]), for $d=26$ the transition amplitude between states $\Psi,\Psi' \in {\cal H}_{\rm \scriptstyle inv} ({\cal E}_L) \subset {\cal H}({\cal C}_L)$ is given by $$\begin{aligned} \langle \Psi | P | \Psi' \rangle &=& \int\limits^{\infty}_0 d\alpha_f \int\limits_{{\cal E}_L} {\cal D}^{\alpha_f \hat{e} } \widetilde{x}_f \; \int\limits^{\infty}_0 d\alpha_i \int\limits_{{\cal E}_L} {\cal D}^{\alpha_i \hat{e} } \widetilde{x}_i \; \overline{\Psi[\widetilde{x}_f]} P[\alpha_f,\widetilde{x}_f; \alpha_i,\widetilde{x}_i] \Psi'[\widetilde{x}_i] \\ &=& \int\limits_0^{\infty}dt \; {\eta (t)} \int\limits_{{\cal W}^n}\!\! {\cal D}^{\widehat{g}_t}\varphi \left(\mbox{Vol}_{\widehat{g}_t} {\cal W}_M^n \right)^{-1} {\rm e}^{ \frac{6-d}{ 32}\!\!\!\! \sum\limits_{\mbox{ \scriptsize corners}} \!\!\!\!\varphi(z_i) }\\ &\times& \int\limits^{\infty}_0 d\alpha_f \; \delta ( \alpha_f - {\textstyle \int {\rm e}^{ \frac{1}{2} \widetilde{\varphi}_f} }) \; \int\limits_{{\cal E}_L} {\cal D}^{\alpha_f \hat{e} } \widetilde{x}_f \; \int\limits^{\infty}_0 d\alpha_i\; \delta ( \alpha_i - {\textstyle \int {\rm e}^{ \frac{1}{2} \widetilde{\varphi}_i} }) \; \int\limits_{{\cal E}_L} {\cal D}^{\alpha_i \hat{e} } \widetilde{x}_i \;\\ &\times& \overline{\Psi[\widetilde{x}_f]} \langle \widetilde{x}_f\circ \gamma[\widetilde{\varphi_f}] | {\rm e}^{-tH_0^x} | \widetilde{x}_i\circ \gamma[\widetilde{\varphi}_i]\rangle \Psi'[\widetilde{x}_i] \;\;\;;\end{aligned}$$ where $$H_0^x = {\pi\over 2} \left[ {1\over 2M_x} p^2_0 + \sum\limits_{k=1}^{\infty} \left( {1\over M_x} p^2_k + M_x k^2 x^2_k \right) \right] \;\;\;;\;\;\; M_x = {1\over 4\alpha'}\;\;. \label{ho}$$ Changing variables $$\widetilde{x}_i \longrightarrow \widetilde{x}_i \circ \gamma[\widetilde{\varphi}_i]^{-1} \;\;\;\;,\;\;\;(i\rightarrow f)\;\;\;,$$ and using the relations (valid for $\alpha_i = \int {\rm e}^{ \frac{1}{2} \widetilde{\varphi}_i}$; see Appendix B, (\[conano\])) $${\cal D}^{\alpha_i\widehat{e}} \widetilde{x}_i^{\mu} \circ \gamma[\widetilde{\varphi}_i]^{-1} = {\rm e}^{ \frac{d}{16} (\widetilde{\varphi}_i(0) + \widetilde{\varphi}_i(1))} {\cal D}^{\widehat{e}} \widetilde{x}_i^{\mu} \;\;\;,\;\;\;(i\rightarrow f)\;\;\;, \label{relations}$$ one gets for ${\cal D}_L$-invariant states $$\begin{aligned} \langle \Psi |P|\Psi'\rangle &=& \int\limits_0^{\infty}dt \; {\eta (t)} \int\limits_{{\cal W}^n_M}\!\! {\cal D}^{\widehat{g}_t}\varphi \left(\mbox{Vol}_{\widehat{g}_t} {\cal W}_M^n \right)^{-1} {\rm e}^{ \frac{ 3}{ 16}\!\!\!\! \sum\limits_{{ \scriptstyle {\rm corners}}} \!\!\!\!\varphi(z_i) } \left({\textstyle \int {\rm e}^{{1\over 2}\varphi_i} \int {\rm e}^{{1\over 2}\varphi_f} } \right)^{{13 \over 4}} \nonumber\\ &\times& \int\limits_{{\cal E}_L} {\cal D}^{ \hat{e} } \widetilde{x}_f \; \int\limits_{{\cal E}_L} {\cal D}^{ \hat{e} } \widetilde{x}_i \; \overline{\Psi[\widetilde{x}_f]} \langle \widetilde{x}_f| {\rm e}^{-tH_0^x} | \widetilde{x}_i\rangle \Psi'[\widetilde{x}_i]\;\;\;.\nonumber\end{aligned}$$ In the formula above the integration over conformal factor decouples yielding an overall divergent factor independent of the states $\Psi, \Psi' \in {\cal H}_{\rm \scriptstyle inv} ({\cal C}_L)$. It follows that one can restrict oneself to the space ${\cal H}({\cal E}_L) \subset {\cal H}({\cal C}_L)$ of $\alpha$-independent states endowed with the scalar product (\[ggproduct\]) with $e= \widehat{e}$. Then the (regularized) transition amplitude between ${\cal D}_L$-invariant states in ${\cal H}^{\widehat{e}}({\cal E}_L)$ takes the following simple form $$\langle\Psi|P_R|\Psi'\rangle = \langle\Psi|\int\limits_0^{\infty} dt\; \eta(t) {\rm e}^{-tH_0^x}|\Psi'\rangle \;\;\;. \label{propagat}$$ Let us note that the path integral representation (\[propa\]) of the transition amplitude is well defined only on the subspace ${\cal H}_{\rm \scriptstyle inv} ({\cal E}_L) \subset {\cal H}^{\widehat{e}}({\cal E}_L)$. In order to get a well defined representation in the whole Hilbert space ${\cal H}^{\widehat{e}}({\cal E}_L)$ one has to restrict the space of trajectories in (\[pro\]) such that the conformal factor already decouples in the formula (\[propa\]). This restriction depends on some additional geometrical data which in the conformal gauges (\[gauge\],\[bgauge\]) consist of fixed parameterizations of the initial and final boundary components. Any particular choice of this data leads to some extension of the formula (\[propagat\]) to the space ${\cal H}^{\widehat{e}}({\cal E}_L)$ and can be regarded as a choice of gauge in the second quantized theory [@jas; @jask]. For the sake of simplicity we will use the simplest extension $$P_R[\widetilde{x}_f,\widetilde{x}_i] = \langle \widetilde{x}_f|P_R|\widetilde{x}_i\rangle = \langle \widetilde{x}_f| \int\limits_0^{\infty} dt\; \eta(t) {\rm e}^{-tH_0^x} |\widetilde{x}_i\rangle \;\;\;. \label{pr}$$ It should be stressed that all further considerations are independent of this choice. In order to derive the on-mass-shell condition one would like to invert the operator $P_R$. The problem is that, due to the $\eta$-function insertion, $P_R$ is not invertible on the whole space ${\cal H}^{\widehat{e}}({\cal E}_L)$. Moreover the formula (\[pr\]) does not describe any operator on the subspace ${\cal H}_{\rm \scriptstyle inv} ({\cal E}_L)$ of ${\cal D}_L$-invariant states. The largest subspace of ${\cal H}_{\rm \scriptstyle inv} ({\cal E}_L)$ on which $P_R[\widetilde{x}_f,\widetilde{x}_i]$ can be regarded as an integral kernel of a well defined operator is characterized by the equations $$H_k^x |\Psi\rangle = 0\;\;\;,\;\;\;V_k^x |\Psi\rangle = 0\;\;\; ;k=1,...\;\;\;, \label{phsc}$$ where $$\begin{aligned} H_k^x& \equiv& -{i\over \pi k} [H_0^x,V_k^x] \label{hk} \\ &=& { \pi\over 2} \left[ {1\over 2M_x}p_k \cdot p_0 + {1\over 2}\sum\limits_{n=1}^{k} \left( {1\over M_x} p_n \cdot p_{k-n} -M_x n(k-n) x_n \cdot x_{k-n} \right) \right. \nonumber \\ &+& \left. \sum\limits_{n=1}^{\infty} \left( {1\over M_x} p_n \cdot p_{k+n} + M_x n(k+n)x_n \cdot x_{k+n} \right) \right]\;\;\;. \nonumber\end{aligned}$$ In order to analyse the integrability conditions of the equations (\[phsc\]) it is convenient to introduce the operators $$L^x_{\pm k} \equiv {1\over \pi} (H_k^x \pm iV_k^x)\;\;\;, \;\;\;k=1,...\;;$$ which are just the standard representations of the Virasoro generators : $$\begin{aligned} L^x_k &=& {1\over 2} \sum\limits_{-\infty}^{+\infty} \alpha_{-n} \cdot \alpha_{k+n} \;\;\;,\;\;\;k=\pm 1,\pm 2,...\label{lxk}\\ \alpha_0^{\mu} &\equiv& {1\over \sqrt{2M_x}} p_{\mu 0}\;\;\;,\nonumber\\ \alpha_n^{\mu } &\equiv & {1\over \sqrt{2}} \left( {1\over \sqrt{M_x}}p_{\mu n}-i \sqrt{M_x} nx_n^{\mu} \right) \;\;\;,\;\;\;n=\pm 1,\pm 2,...\;\;\;;\nonumber\end{aligned}$$ $$\left[\alpha_n^{\mu},\alpha_m^{\nu}\right] = n \delta^{\mu \nu} \delta_{n,-m}\;\;\;\;\; ,\;\;\;\;\;\alpha_n^{\mu } = \alpha_{-n}^{\mu}\;\;\;.$$ In terms of $L^x_k$ the integrability conditions take the form $$\left[ L^x_n ,L^x_m \right] = (n-m) L^x_{n+m} + {26 \over 12} \delta_{n,-m}(n^3-n)\;\;\;,$$ where $$L^x_0 \equiv {1\over 2}\alpha_0 \cdot \alpha_0 + \sum\limits_{n=1}^{\infty} \alpha_{-n} \cdot \alpha_n\;\;\;. \label{lo}$$ In order to obtain nontrivial solutions to the equations (\[phsc\]) one has to relax their strong form. Since $L_k^{x+} = L^x_{-k}$, this leads to the familiar conditions for the off-mass-shell physical states $$L^x_k|\Psi\rangle = 0\;\;\;,\;\;\; k\geq1\;\;\;. \label{oph}$$ The derivation of the physical state conditions presented above requires an explanation. The additional constraints $H_k^x$ have been introduced for technical reasons. There is however a physical motivation for these constraints stemming from the fact that the string is an extended relativistic system. This implies that the intersection of the string world sheet by an equal time hyperplane provides a half of the Cauchy data for string trajectory. It follows that for each particular choice of reference system in the Minkowski space-time the string wave functional should be independent of string fluctuations in the time direction. This kinematical requirement can be regarded as a manifestation of the general locality and causality principles of relativistic quantum theory and indicates a fundamental difference between the theory of relativistic string and the theory of random surfaces. Note that in the canonical quantization the kinematical requirement simply means that all negative norm states must decouple. The way in which the $H_k^x$ constraints yield exactly the missing part of the physical state conditions necessary and sufficient to satisfy the kinematical requirement is not quite straightforward. The rough counting of the degrees of freedom shows that two sets of constraints $\left\{ V_k\right\}$ and $\left\{ H_k \right\}$ reduce by 2 the number of physical direction. However, as it was mentioned above one can impose only a half of these constraints as conditions for physical states, which explicitly removes only one direction. The mechanism which ensures the decoupling of the second direction is that of the null states which are among solutions to the equations (\[oph\]) [@re; @sch]. As the null states decouple from all other solutions the space of physical states is effectively given by the space of equivalence classes. One possible way to describe the corresponding quotient is to introduce the Euclidean “quantum” version of the light-cone gauge conditions $$\alpha^+_k|\Psi\rangle \equiv \left( i\alpha_k^0 + \alpha_k^{25} \right) | \Psi\rangle = 0\;\;\;,\;\;\;k\geq 1\;\;\;. \label{lc}$$ The conditions above can be seen as an explicit implementation of the kinematical requirement in the limit case of the light-like direction. It is an interesting open question whether the Virasoro algebra of constraints is the only solution to the kinematical requirement of the quantum theory of 1-dim extended relativistic systems. The subspace ${\cal H}_{\scriptstyle \rm ph}^{\scriptstyle \rm off}({\cal E}_L)$ of solutions to the equations (\[oph\],\[lc\]) can be explicitly constructed by means of the (extended) DDF method [@ddf]. One can check that the operators $\left\{ \widetilde{x}^{\perp}(s) \right\} \equiv \left\{ \widetilde{x}^k(s) \right\}_{k=1,...,d-2}$ form in ${\cal H}_{\scriptstyle \rm ph}^{\scriptstyle \rm off}({\cal E}_L)$ a complete set of commuting “observables”. In particular every state $|\Psi_{\scriptstyle \rm ph}\rangle \in {\cal H}_{\scriptstyle \rm ph}^{\scriptstyle \rm off}({\cal E}_L)$ is completely described by the wave functional $$\Psi_{\scriptstyle \rm ph}[\widetilde{x}^{\perp}] = \langle \widetilde{x}^{\perp} |\Psi_{\scriptstyle \rm ph} \rangle \;\;\;. \label{st}$$ The transition amplitude between off-mass-shell physical states $\Psi_{\scriptstyle \rm ph},\Psi_{\scriptstyle \rm ph}' \in {\cal H}_{\scriptstyle \rm ph}^{\scriptstyle \rm off}({\cal E}_L)$ described by the wave functionals (\[st\]) takes the following form $$\begin{aligned} \langle\Psi_{\scriptstyle \rm ph}|P_R|\Psi_{\scriptstyle \rm ph}'\rangle &=& \int\limits_{{\cal E}_L} {\cal D}^{ \hat{e} } \widetilde{x}_f \; \int\limits_{{\cal E}_L} {\cal D}^{ \hat{e} } \widetilde{x}_i \; \overline{\Psi_{\scriptstyle \rm ph}[\widetilde{x}^{\perp}_f]} \langle \widetilde{x}_f| \int\limits_0^{\infty}\!\! dt\; \eta(t) {\rm e}^{-tH_0^x}| \widetilde{x}_i\rangle \Psi_{\scriptstyle \rm ph}'[\widetilde{x}^{\perp}_i] \nonumber \\ &=& \langle\Psi_{\scriptstyle \rm ph}| \int\limits_0^{\infty} \!\!dt\; {\rm e}^{-t\pi (K^x_0 -1)} |\Psi_{\scriptstyle \rm ph}' \rangle_{\scriptstyle \rm ph}^{\scriptstyle \rm off} \;\;\;, \label{matrix}\end{aligned}$$ where $K^x_0$ denotes the restriction of the operator $L^x_0$ to the subspace ${\cal H}_{\scriptstyle \rm ph}^{\scriptstyle \rm off}({\cal E}_L)$ and $\langle...|...\rangle_{\scriptstyle \rm ph}^{\scriptstyle \rm off}$ is the scalar product in ${\cal H}_{\scriptstyle \rm ph}^{\scriptstyle \rm off}({\cal E}_L)$ regarded as a Hilbert subspace of ${\cal H}^{\widehat{e}}({\cal E}_L)$. It follows that the operator defined by the matrix elements (\[matrix\]) can be inverted on the subspace ${\cal H}_{\scriptstyle \rm ph}^{\scriptstyle \rm off}({\cal E}_L)$. Then the on-mass-shell condition in ${\cal H}_{\scriptstyle \rm ph}^{\scriptstyle \rm off}({\cal E}_L)$ is given by $$(K^x_0 - 1) |\Psi\rangle = 0\;\;\;,$$ which is equivalent to the condition $$(L^x_0 - 1) |\Psi\rangle = 0\;\;\;. \label{mshell}$$ in the space ${\cal H}^{\widehat{e}}({\cal E}_L)$. Performing the Wick rotation in the physical state conditions (\[oph\],\[mshell\]) one gets the familiar equations of the so called old covariant approach. All further steps of quantization proceed along the standard lines [@re; @sch]. The noncritical Polyakov string =============================== Gauge symmetry --------------- In this section we will present the covariant functional quantization of the Polyakov string in the flat target space of dimension $d$ in the range $1<d<25$. The first steps of this quantization procedure – the description of the space of trajectories in the configuration space, the construction of the space of states, and the path integral representation of the transition amplitude – are almost the same as in the case of the critical string. The main difference consists in the symmetry requirements imposed on the quantum theory. Since, in the range $1<d<25$, the conformal anomaly breaks the Weyl invariance completely, the gauge symmetry in the space of trajectories reduces to the group ${\cal D}_M^n$ acting on ${\cal M}_M^n \times {\cal E}_M^n$ by $${\cal M}_M^n \times {\cal E}^n_M \ni (g,x) { \begin{picture}(60,0)(0,0) \put(5,3){\vector(1,0){50}} \put(15,10){$\scriptstyle f \in {\cal D}_M^n $ } \end{picture}} (f^*g,f^*x) \in {\cal M}_M^n \times {\cal E}^n_M\;\;\;.$$ Accordingly, the induced gauge transformations in the space ${\cal P}_L$ (\[sbc\]) of boundary conditions take the form $${\cal P}_L \ni (e_i,x_i) { \begin{picture}(50,0)(0,0) \put(5,3){\vector(1,0){40}} \put(13,10){$\scriptstyle \gamma \in {\cal D}_L $ } \end{picture}} (\gamma^*e_i,x_i \circ \gamma) \in {\cal P}_L\;\;.$$ Repeating the reasoning of Subsect.2.2 one gets the space of states ${\cal H}({\cal C}_L)$ endowed with the scalar product (\[product\]). Because of the restricted gauge group there is no residual gauge invariance in ${\cal H}({\cal C}_L)$. Transition amplitude -------------------- According to the different symmetry requirements, the path integral representation (\[pro\]) for the transition amplitude gets slightly modified $$P[c_f,c_i] = \int\limits_{{\cal F}[c_f,c_i]} {\cal D}^gg {\cal D}^gx \left(\mbox{Vol}_g {\cal D}_M^n \right)^{-1} \exp \left(- {\scriptstyle \frac{1}{4\pi \alpha '}} S[g,x]\right)\;\;\;.$$ In the conformal gauges (\[gauge\],\[bgauge\]) one has $$\begin{aligned} P[\alpha_f,\widetilde{x}_f;\alpha_i,\widetilde{x}_i] = \int\limits_0^{\infty}\!\! & dt &\!\! {\eta (t)}^{1 - {d \over 2}} t^{-{d \over 2}} \int\limits_{{\cal W}^n_M}\!\! {\cal D}^{\widehat{g}_t}\varphi \nonumber \\ &\times&\exp{\left(- {25-d \over 48\pi} S_L[\widehat{g}_t,\varphi]\right) } \nonumber \\ &\times& \exp{\left( {7-d \over 32} \sum\limits_{\mbox{\scriptsize corners}} \varphi(z_i) \right)} \label{npropa}\\ &\times& \exp{\left(- {1 \over 4 \pi \alpha'} S[\widehat{g}_t,\varphi,\widetilde{x}_i,\widetilde{x}_f ] \right)} \;\;\;,\nonumber\\ &\times& \delta \left( \alpha_f - \int_0^1 dz^1 {\rm e}^{ \frac{1}{2} \varphi_f} \right) \; \delta \left( \alpha_i - \int_0^1 dz^1 {\rm e}^{ \frac{1}{2} \varphi_i} \right) \;\;\;.\nonumber\end{aligned}$$ Lets us note that in contrast to the formula (\[propa\]) the $\varphi$-dependence of the functional measure ${\cal D}^{{\rm e}^{\varphi}\widehat{g}_t}\varphi$ is not canceled by the similar $\varphi$-dependence in the volume factor $\mbox{Vol}_{{\rm e}^{\varphi}\widehat{g}_t} {\cal W}_M^n$. As a result, one gets the different coefficients in front of the Liouville action and the corner anomaly term. Some remarks concerning the formula (\[npropa\]) are in order. First of all one has to choose some values of the renormalization constants $\mu,\lambda$ appearing in the Liouville action (\[liouville\]). In the following we will restrict ourselves to the simplest choice $$\mu = \lambda = 0\;\;\;. \label{cosm}$$ This is well justified in the free theory. First, let us observe that the nonvanishing boundary cosmological constant is incompatible with the boundary conditions (\[bcphi\]). Secondly, using the Gauss-Bonnet theorem on the rectangle one can easily show that for $\widehat{g}_t \in {\cal M}^{n*}_M$ the Liouville equation $$\Delta_{\widehat{g}_t} \varphi + \mu {\rm e}^{\varphi} =0 \;\;\;,$$ does not have any solution in the space ${\cal W}_M^n$. It follows that the variational problem given by the Liouville action (\[liouville\]) is well posed and has a solution in ${\cal W}_M^n$ if and only if the equations (\[cosm\]) are satisfied. Let us stress that the conclusion above is not necessarily valid for more complicated world sheet topologies. In particular in the case of hyperbolic hexagon the classical solution exists only for $\mu > 0$. An independent motivation for the choice (\[cosm\]) in the free string theory stems from the semiclassical calculations of the static potential [@jm], where one obtains the same result for the vanishing and for the positive bulk cosmological constant. Under the assumption (\[cosm\]) and in the conformal gauge (\[gauge\]) the Liouville action is just the free field action $$S[\widehat{g}_t,\varphi ] = {1 \over 2} \int^t_0 dz^0 \int^1_0 dz^1 \left(({\partial}_0 \varphi)^2 + ({\partial}_1 \varphi)^2 \right) \;\;.$$ Even with this simplification the formula (\[npropa\]) still contains the complicated nonlocal interaction which prevents calculations of the functional integral in all but few special cases of $x$-boundary conditions [@jm]. Our idea to overcome this difficulty is to regard the transition amplitude (\[npropa\]) as a matrix element of some simple operator between special states in an extended space. The method to find such extension of ${\cal H}({\cal C}_L)$ is based on the simple observation that the interaction terms depend only on the boundary values of the conformal factor. Thus it should be possible to replace the integration over fields satisfying the homogeneous Neumann boundary condition by the integral over fields with a fixed nonhomogeneous Dirichlet condition (which is Gaussian) and then an integral over all possible boundary values. In the case under consideration one can expect the following formula $$\int\limits_{{\cal W}^n_M} \!\!\! {\cal D}^{\widehat{g}_t}\varphi\; {\rm e}^{ - {25-d \over 48\pi} S_L[\widehat{g}_t,\varphi] } F[\widetilde{\varphi}_f,\widetilde{\varphi}_i] = \eta^{-{1\over2}} t^{-{1\over 2}} \int\limits_{{\cal W}_L} \!\!\! {\cal D}^{\widehat{e}} \widetilde{\varphi}_f \int\limits_{{\cal W}_L} \!\!\! {\cal D}^{\widehat{e}} \widetilde{\varphi}_i \;{\rm e}^{ - {25-d \over 48\pi} S_L[\widehat{g}_t,\widetilde{\varphi}_f,\widetilde{\varphi}_i] } F[\widetilde{\varphi}_f,\widetilde{\varphi}_i]\;\;, \label{formula}$$ where $$S_L[\widehat{g}_t,\widetilde{\varphi}_f,\widetilde{\varphi}_i] \equiv S_L[\widehat{g}_t,\varphi_{cl}]\;\;\;,$$ and $\varphi_{cl}:\widehat{M}_t \rightarrow R^d$ is the solution of the boundary value problem $$\begin{aligned} \left( {\partial}^2_0 + {\partial}^2_1 \right) \varphi_{cl} & = & 0 \;\;,\nonumber\\ \varphi_{cl}(0,z^1) &=& \widetilde{\varphi}_i(z^1) \;\;,\nonumber\\ \varphi_{cl}(t,z^1) &=& \widetilde{\varphi}_f(z^1) \;\;,\nonumber\\ \partial_1 \varphi_{cl}(z^0,0) &=& \partial_1 \varphi_{cl}(z^0,1) = 0 \;\;.\nonumber\end{aligned}$$ The relation (\[formula\]) is well known for Gaussian integrals (as a formula for determinants [@f]) and is supposed to be valid in general situation [@car]. Since we are not aware of any proof in the case of boundary interactions, a simple derivation of the formula (\[formula\]) is given in the Appendix B. Using (\[formula\]) the transition amplitude between arbitrary states $|\Psi\rangle,|\Psi'\rangle \in$ ${\cal H}( \mbox{{\boldmath R}}_+\times {\cal E}_L) =_{\widehat{e}} {\cal H}({\cal C}_L)$ can be rewritten in the following form $$\begin{aligned} \langle \Psi | P | \Psi' \rangle &=& \int\limits^{\infty}_0 d\alpha_f \int\limits_{{\cal E}_L} {\cal D}^{\alpha_f \hat{e} } \widetilde{x}_f \; \int\limits^{\infty}_0 d\alpha_i \int\limits_{{\cal E}_L} {\cal D}^{\alpha_i \hat{e} } \widetilde{x}_i \; \overline{\Psi[\alpha_f,\widetilde{x}_f]} P[\alpha_f,\widetilde{x}_f; \alpha_i,\widetilde{x}_i] \Psi'[\alpha_i,\widetilde{x}_i] \\ &=& \int\limits_{{\cal W}_L} {\cal D}^{\widehat{e}}\widetilde{\varphi}_f \int\limits_{{\cal E}_L} {\cal D}^{\alpha_f \widehat{e} } \widetilde{x}_f \; \int\limits_{{\cal W}_L} {\cal D}^{\widehat{e}}\widetilde{\varphi}_i \int\limits_{{\cal E}_L} {\cal D}^{\alpha_i \widehat{e} } \widetilde{x}_i \;\\ &\times& \overline{\Psi[\alpha_f,\widetilde{x}_f]} \ {\rm e}^{ \frac{7-d}{ 32}(\widetilde{\varphi}_f(0) + \widetilde{\varphi}_f(1))}\\ &\times& \langle \widetilde{\varphi}_f | \otimes \langle \widetilde{x}_f\circ \gamma[\widetilde{\varphi}_f]| \int\limits_0^{\infty}dt \; {\eta (t)} {\rm e}^{-t(H_0^x + H^{\varphi}_0)} | \widetilde{\varphi}_i \rangle \otimes | \widetilde{x}_i\circ \gamma [ \widetilde{\varphi}_i] \rangle \\ &\times& \Psi'[\alpha_i,\widetilde{x}_i] {\rm e}^{ \frac{7-d}{ 32}( \widetilde{\varphi}_i(0) + \widetilde{\varphi}_i(1))} \;\;\;,\end{aligned}$$ where $\alpha_i = \int {\rm e}^{ \frac{1}{2} \widetilde{\varphi}_i}, (i\rightarrow f)$. In the formula above $H_0^x + H_0^{\varphi}$ is regarded as an operator on the space ${\cal H}({\cal W}_L \times {\cal E}_L)$ with the $x$-part defined by (\[ho\]) and with the $\varphi$-part given by $$H_0^{\varphi} \equiv {\pi\over 2} \left[ {1\over 2M_{\varphi}} \pi^2_0 + \sum\limits_{k=1}^{\infty} \left( {1\over M_{\varphi}} \pi^2_k + M_{\varphi} k^2 {\varphi}^2_k \right) \right] \ \ \ \ \ \ \ \ M_{\varphi} = {25 - d \over 96}\;\;;$$ where $$\begin{aligned} \pi(s) &\equiv & -i\frac{\delta}{\delta \widetilde{\varphi}(s)} = \pi_{0} + 2\sum\limits_{k = 1}^{\infty} \pi_{ k} \cos \pi ks \;\;\;, \nonumber\\ \widetilde{\varphi}(s) & = & \varphi_0 + \sum\limits_{k=1}^{\infty} \varphi_k \cos \pi ks\;\;\;. \label{modes}\end{aligned}$$ Changing variables $$\widetilde{x}_i \longrightarrow \widetilde{x}_i \circ \gamma[\widetilde{\varphi}_i]^{-1} \;\;\;\;,\;\;\;(i\rightarrow f)\;\;\;,$$ and using the relations (\[relations\]) one gets the representation required $$\langle \Psi |P|\Psi'\rangle = \langle \widetilde{\Psi} | \int\limits_0^{\infty}dt \; {\eta (t)} {\rm e}^{-t(H_0^x + H_0^{\varphi})} | \widetilde{\Psi}'\rangle\;\;\;, \label{npropag}$$ where for each state $|\Psi\rangle \in {\cal H}({\bf R}_+\times{\cal E}_L)$ the state $|\widetilde{\Psi}\rangle \in {\cal H}({\cal W}_L \times {\cal E}_L)$ is given by the wave functional $$\widetilde{\Psi}[\widetilde{\varphi},\widetilde{x}] \equiv {\rm e}^{ \frac{7+d}{ 32}(\widetilde{\varphi}(0) + \widetilde{\varphi}(1))} \Psi[{\textstyle \int {\rm e}^{ \frac{1}{2} \widetilde{\varphi}}}, \widetilde{x}\circ \gamma[\widetilde{\varphi}]^{-1}] \;\;\;. \label{nphyst}$$ and the scalar product $$\langle \Psi | \Psi' \rangle = \int\limits_{{\cal W}_L} {\cal D}^{\widehat{e}}\widetilde{\varphi} \int\limits_{{\cal E}_L} {\cal D}^{ \widehat{e} } \widetilde{x} \; \overline{\Psi[\widetilde{\varphi},\widetilde{x}]} \Psi'[\widetilde{\varphi}, \widetilde{x}] \;\;\; \label{gggproduct}$$ is used on the r.h.s of (\[npropag\]). Physical state conditions ------------------------- Due to the simple representation (\[npropag\]) of the transition amplitude it is convenient to analyse the physical state conditions in the extended space ${\cal H}({\cal W}_L \times {\cal E}_L)$. We start with the discussion of the conditions related to the extension itself. Their role is to select in the extended space the image of the original space of states under the extension map $${\rm Ext}: {\cal H}({\bf R}_+ \times {\cal E}_L) \ni \Psi \longrightarrow \widetilde{\Psi} \in {\cal H}({\cal W}_L \times {\cal E}_L) \;\;\;,$$ where $\widetilde{\Psi}$ is given by the formula (\[nphyst\]). Using the equations (\[difeq\]) one can show that the functionals $\Phi[\widetilde{\varphi},\widetilde{x}] \in {\cal H}({\cal W}_L \times {\cal E}_L)$ of the form (\[nphyst\]) can be uniquely characterized as functionals invariant with respect to the following ${\cal D}_L$ action $${\cal H}({\cal W}_L \times {\cal E}_L) \ni \Phi [\widetilde{\varphi},\widetilde{x}] { \begin{picture}(55,0)(0,0) \put(5,3){\vector(1,0){45}} \put(13,10){$\scriptstyle \gamma \in {\cal D}_L $ } \end{picture}} \widetilde{\rho}[\gamma] \Phi [\widetilde{\varphi} \circ \gamma + 2\log \gamma', \widetilde{x} \circ \gamma] \in {\cal H}({\cal W}_L \times {\cal E}_L)\;\;\;, \label{ntrans}$$ where $$\widetilde{\rho}[\gamma] \equiv {\rm e}^{-{7+d\over 16}(\log \gamma'(0) + \log \gamma'(1) )} \;\;\;,\;\;\;\widetilde{\rho}[\gamma \circ \delta] = \widetilde{\rho}[\gamma]\widetilde{\rho}[\delta]\;\;\;; \;\;\; \gamma,\delta \in {\cal D}_L \;\;\;. \label{nrho}$$ Within the old covariant approach the ${\cal D}_L$-action (\[ntrans\]) has to be modified in order to meet the requirement of unitary realization of the residual symmetry. The analysis of the transformation properties of the scalar product (\[gggproduct\]) with respect to the transformations $${\cal W}_L \times {\cal E}_L \ni (\widetilde{\varphi},\widetilde{x}) { \begin{picture}(55,0)(0,0) \put(5,3){\vector(1,0){45}} \put(13,10){$\scriptstyle \gamma \in {\cal D}_L $ } \end{picture}} (\widetilde{\varphi} \circ \gamma + 2\log \gamma', \widetilde{x} \circ \gamma) \in {\cal W}_L \times {\cal E}_L\;\;\;,$$ leads to the following unitary ${\cal D}_L$-action $${\cal H}({\cal W}_L \times {\cal E}_L) \ni \Phi [\widetilde{\varphi},\widetilde{x}] { \begin{picture}(55,0)(0,0) \put(5,3){\vector(1,0){45}} \put(13,10){$\scriptstyle \gamma \in {\cal D}_L $ } \end{picture}} \rho[\gamma] \Phi [\widetilde{\varphi} \circ \gamma + 2\log \gamma', \widetilde{x} \circ \gamma] \in {\cal H}({\cal W}_L \times {\cal E}_L)\;\;\;, \label{nnntrans}$$ with $$\rho[\gamma] \equiv {\rm e}^{-{1+d\over 16}(\log \gamma'(0) + \log \gamma'(1) )} \;\;\;,\;\;\;\rho[\gamma \circ \delta] =\rho[\gamma]\rho[\delta]\;\;\;; \;\;\; \gamma,\delta \in {\cal D}_L \;\;\;. \label{nnrho}$$ The discrepancy between $\widetilde{\rho}$ (\[nrho\]) derived from the representation (\[npropag\]) and $\rho$ (\[nnrho\]) obtained from the unitarity requirement is related to the fact that in the old covariant approach one disregards the ghost sector. The detailed discussion of this point requires the full BRST formulation which is beyond the scope of the present paper. Let us only mention that $\widetilde{\rho}$ given by the formula (\[nrho\]) leads to a unitary realization of the ${\cal D}_L$-symmetry in the BRST extended space. The generators of the representation (\[nnntrans\]) take the form $$\begin{aligned} V_k \; \equiv & -& \!\!\!i\int\limits_0^1 ds \; \sin \pi ks (\widetilde{x}^{\mu})'(s) \frac{\delta}{\delta \widetilde{x}^{\mu}(s) } \nonumber \\ &-&\!\!\!i\int\limits_0^1 ds \; \sin \pi ks (\widetilde{\varphi})'(s) \frac{\delta}{\delta \widetilde{\varphi}(s) } - i2\pi k \int\limits_0^1 ds \; \cos \pi ks \frac{\delta}{\delta \widetilde{\varphi}(s) } \nonumber \\ &-&\!\!\! i \int\limits_0^1 ds \sin \pi ks {\frac{\delta}{\delta \gamma(s)} \rho[\gamma] }_{|\gamma = {\scriptstyle \rm id}_L } \;\;\;,\;\;\;k\geq 1 \nonumber\\ = & & \!\!\!\!\!\! V^x_k + V^{\varphi}_k \;\;\;;\nonumber\end{aligned}$$ where $V^x_k$ is given by (\[vk\]) and $$\begin{aligned} V_k^{\varphi} & = & - {\pi\over 2}\sum\limits_{n = 1}^k n \varphi_n \pi_{ (k-n)} + {\pi \over 2} \sum\limits_{n=1}^{\infty} \left( n \varphi_n\pi_{n+k} - (n+k)\varphi_{n+k}\pi_{ n} \right) \nonumber\\ && + 2\pi k\pi_k + i\pi {1\over 8}p(k)k\;\;\;,\;\;\;k\geq 1\;\;\;. \nonumber\end{aligned}$$ Note that $V_k$ are hermitian with respect to the scalar product (\[gggproduct\]) and normally ordered. The subspace ${\cal H}_{\scriptstyle \rm inv}({\cal W}_L \times {\cal E}_L)$ of states invariant with respect to the action (\[nnntrans\]) is determined by the equations $$V_k |\Phi\rangle = 0\;\;\;,\;\;\;k\geq 1\;\;.$$ The rest of the physical state conditions can be derived using the method discussed in Subsect.2.4. The subspace of $ {\cal H}_{\scriptstyle \rm inv}({\cal W}_L \times {\cal E}_L)$ on which $$\int\limits_0^{\infty}dt \; {\eta (t)} {\rm e}^{-t(H_0^x + H_0^{\varphi})}$$ reduces to a well defined operator is given by the equations $$H_k |\Psi\rangle = 0\;\;\;, \;\;\;V_k |\Psi\rangle = 0\;\;\;;k=1,...\;\;\;, \label{nphsc}$$ where $$H_k \equiv -{i\over \pi k} [H_0^x + H^{\varphi}_0,V_k] \;=\; H^x_k + H^{\varphi}_k$$ and $$\begin{aligned} H_k^{\varphi}& \equiv& -{i\over \pi k} [H_0^{\varphi},V_k^{\varphi}] \nonumber\\ &=& { \pi\over 2} \left[ {1\over 2M_{\varphi}}\pi_k\pi_0 + {1\over 2}\sum\limits_{n=1}^{k} \left( {1\over M_{\varphi}} \pi_n \pi_{k-n} -M_{\varphi} n(k-n){\varphi}_n {\varphi}_{k-n} \right) \right. \nonumber \\ &+& \left. \sum\limits_{n=1}^{\infty} \left( {1\over M_{\varphi}} \pi_n \pi_{k+n} + M_{\varphi} n(k+n){\varphi}_n {\varphi}_{k+n} \right) \;+\;4M_{\varphi} k^2 \varphi_k \right]\;\;\;. \nonumber\end{aligned}$$ The analysis of the integrability conditions of the equations (\[nphsc\]) can be simplified by introducing $$L_{\pm k} \equiv {1\over \pi} (H_k \pm i V_k)\;=\; L^x_{\pm k} +L^{\varphi}_{\pm k} \;\;\;, \;\;\;k=1,...$$ where the operators $L^x_k$ are given by (\[lxk\]) and $L^{\varphi}_k$ are just the Virasoro generators in the FCT representation [@ct; @t] with the central charge $c = 26-d$ $$\begin{aligned} L^{\varphi}_k &\equiv& {1\over 2} \sum\limits_{-\infty}^{+\infty} \beta_{-n} \beta_{k+n} +ikQ\beta_k \;\;\;,\;\;\;k=\pm 1,\pm 2,...\\ Q &\equiv & 2\sqrt{2 M_{\varphi}} = \sqrt{{25 -d \over 12}}\;\;\;,\\ \beta_0^{\mu} &\equiv& {1\over \sqrt{2M_{\varphi}}} \pi_0\;\;\;,\\ \beta_n^{\mu } &\equiv & {1\over \sqrt{2}} \left( {1\over \sqrt{M_{\varphi}}}\pi_{ n}- i \sqrt{M_{\varphi}} n \varphi_n \right) \;\;\;,\;\;\;n=\pm 1,\pm 2,...\;\;\;;\end{aligned}$$ $$\left[\beta_n,\beta_m\right] = n \delta_{n,-m}\;\;\;\;\; ,\;\;\;\;\;\beta_n^+ = \beta_{-n}\;\;\;.$$ The integrability conditions take the following form $$\left[ L_n ,L_m \right] = (n-m) L_{n+m} + {26 \over 12} \delta_{n,-m}(n^3-n)\;\;\;,$$ where $$\begin{aligned} L_0& \equiv & L_0^x + L^{\varphi}_k\;\;\;, \nonumber\\ L^{\varphi}_0 &\equiv& {1\over 2}\beta_0^2 + \sum\limits_{n=1}^{\infty} \beta_{-n}\beta_n + \frac{Q^2 }{2}\;\;\;, \nonumber\end{aligned}$$ and $L_0^x$ is given by (\[lo\]). As in the case of critical string relaxing the strong form of the equations (\[nphsc\]) one gets the conditions for the off-mass-shell physical states $$L_k|\Psi\rangle = 0\;\;\;,\;\;\;k\geq 1\;\;\;. \label{noph}$$ The structure of the space ${\cal H}_{\scriptstyle \rm ph}^{\scriptstyle \rm off}({\cal W}_L \times {\cal E}_L)$ of solutions to the equations (\[noph\]) is similar to that of critical string theory. As we shall see in the next subsection, the quotient space of the off-mass-shell physical states modulo null states can be uniquely characterized as the space of solutions of (\[noph\]) satisfying the “quantum” light-cone gauge conditions $$\alpha^+_k|\Psi\rangle \equiv \left(i \alpha_k^0 + \alpha_k^{d-1} \right) | \Psi\rangle = 0\;\;\;,\;\;\;k\geq 1\;\;\;.$$ In particular repeating the considerations presented in Subsect.2.4. one can derive the on-mass-shell condition which in the space ${\cal H}({\cal W}_L\times{\cal E}_L)$ takes the following form $$(L_0 -1)|\Psi\rangle = 0\;\;\;. \label{nmshell}$$ DDF construction and no-ghost theorem ------------------------------------- In this subsection we present an explicit construction of physical states of the relativistic theory. Performing the Wick rotation in the conditions (\[noph\]), (\[nmshell\]) one gets $$L_k|\Psi\rangle = 0\;\;\;,\;\;\;k\geq 1\;\;\;, \;\;\;(L_0 -1)|\Psi\rangle = 0\;\;\;, \label{nph}$$ where $$\begin{aligned} L_n&=& \frac{1}{2}\sum_{m=-\infty}^{\infty}\alpha_m\cdot \alpha_{n-m} +\frac{1}{2}\sum_{m=-\infty}^{\infty}\beta_m\beta_{n-m}+inQ\beta_n \ \ \ \ \ n\ne 0\\ L_0&=& \frac{1}{2}\alpha_0\cdot\alpha_0+\frac{1}{2}\beta_0^2+ \sum_{m=1}^{\infty}\alpha_{-m}\cdot \alpha_m +\sum_{m=1}^{\infty}\beta_{-m}\beta_m+\frac{Q^2}{2}\end{aligned}$$ and $$\left[\beta_m,\beta_n\right]= m\delta_{m,-n}\;\;\;,\;\;\; \left[\alpha^{\mu}_m,\alpha^{\nu}_n\right]= m\eta^{\mu\nu} \delta_{m,-n}\;\;\;,\;\;\; \eta^{\mu\nu}= diag(-1,+1,..,+1)\;\;\;.$$ Following the DDF approach [@ddf] we introduce the operators of “position” and “momentum” : $$\begin{aligned} X^{\mu}(\theta)&=& x^{\mu}_0+\alpha^{\mu}_0\theta+ \sum_{k\ge 1} \frac{i}{k}\left(\alpha_k^{\mu}{\rm e}^{-ik\theta}- \alpha_k^{\mu +}{\rm e}^{ik\theta}\right)\;\;\;,\\ P^{\mu}(\theta)&=& \alpha^{\mu}_0+ \sum_{k\ge 1} \left(\alpha_k^{\mu}{\rm e}^{-ik\theta}+ \alpha_k^{\mu +}{\rm e}^{ik\theta}\right)\;\;\;,\\ \Phi(\theta)&=& \varphi_0+\beta_0\theta+ \sum_{k\ge 1} \frac{i}{k}\left(\beta_k{\rm e}^{-ik\theta}- \beta_k^{+}{\rm e}^{ik\theta}\right)\;\;\;,\\ \Pi(\theta)&=& \beta_0+ \sum_{k\ge 1} \left(\beta_k{\rm e}^{-ik\theta}+ \beta_k^{+}{\rm e}^{ik\theta}\right) \;\;\;.\end{aligned}$$ Using the relations $$\begin{aligned} \left[L_n,\alpha^{\mu}_m\right]&=& -m\alpha^{\mu}_{m+n}\;\;\;,\\ \left[L_n,\beta_m\right]&=& -m\beta_{m+n}+iQn^2\delta_{n,-m}\;\;\;,\\ \left[L_n,L_m\right]&=& (n-m)L_{n+m}+\frac{26}{12}(n^3-n)\delta_{n,-m} \;\;\;,\end{aligned}$$ one gets $$\begin{aligned} \left[X^{\mu}(\theta),X^{\nu}(\theta')\right]&=& -i\pi\eta^{\mu\nu}{\rm sgn} (\theta-\theta')\;\;\;,\nonumber\\ \left[P^{\mu}(\theta),P^{\nu}(\theta') \right]&=& 2i\pi\eta^{\mu\nu}\delta'(\theta- \theta')\;\;\;,\nonumber\\ \left[L_n,P^{\mu}(\theta)\right]&=& -i\frac{d}{d\theta}\left(P^{\mu} {\rm e}^{in\theta}\right)\;\;\;,\nonumber\\ \left[L_n,\Pi(\theta)\right]&=& -i\frac{d}{d\theta}\left(\Pi^{\mu} {\rm e}^{in\theta}\right) +iQn^2 {\rm e}^{in\theta} \;\;\;. \label{nonhom}\end{aligned}$$ Let us consider the state $|p_L,p^{\mu}\rangle $ satisfying $$\begin{aligned} \alpha_n^{\mu}\vert p_L,p^{\mu}\rangle &=& \delta_{n0} p^{\mu}\vert p_L,p^{\mu}\rangle \;\;\;,\nonumber\\ \beta_n \vert p_L,p^{\mu}\rangle &=& \delta_{n0} p_L \vert p_L,p^{\mu}\rangle\;\;\;, \label{vacuum}\\ (L_0 -1)|p_L,p^{\mu}\rangle&=&0\;\;\;.\nonumber\end{aligned}$$ For $p^{\mu} \ne 0$ there exists a vector $k$ such that $k^{\mu}k_{\mu}=0$ and $k^{\mu}p_{\mu}=1$. The construction of vertex operators, which acting on the state (\[vacuum\]) generate positive norm physical states with transverse excitations is the same as in the case of Nambu-Goto string. One gets the operators $$A^i_n= \frac{1}{2\pi} \int^{2\pi}_0 d\theta\ P^i(\theta){\rm e}^{ink\cdot X(\theta)} \;\;\;,\;\;\; i=1,...,d-2\;\;,\;\;n\geq 1\;\;,$$ satisfying the relations $$\begin{aligned} \left[A^i_m,A^j_n\right]&=& m\delta^{ij}\delta_{m,-n}\;\;\;,\nonumber\\ \left[L_k,A^i_m\right]&=& 0\;\;\;,\;\;\;k\geq 0\label{altr}\\ A^{i+}_n&=& A^i_{-n}\;\;\;.\nonumber\end{aligned}$$ Due to the $n^2$-term in the commutation relation (\[nonhom\]) the construction of the vertex operator generating states with excitation in the Liouville direction is slightly more complicated. In order to compensate this term one can use a modification introduced by Brower in his construction of the vertex generating longitudinal excitations in the Nambu-Goto string [@b] and write $$A^L_n= \frac{1}{2\pi} \int^{2\pi}_0 d\theta\ \left(\Pi(\theta)-Q(k\cdot\dot P(\theta)) (k\cdot P(\theta))^{-1}\right){\rm e}^{ink\cdot X(\theta)}\;\;\;,\;\;\; n\geq 1\;\;\;.$$ In contrast to Brower’s longitudinal vertex the operator above satisfies the relations analogous to $A_n^i$: $$\begin{aligned} \left[A^L_m,A^L_n\right]&=&m\delta_{m,-n}\;\;\;,\nonumber\\ \left[L_k,A^L_m\right]&=& 0\;\;\;,\;\;\;k\geq 0\;\;\;, \label{allo}\\ \left[A^i_m,A^L_n\right]&=& 0 \;\;\;,\nonumber\\ A^{L+}_n&=&A^L_{-n}\;\;\;.\nonumber\end{aligned}$$ All the states generated by the operators $A^i_n, A^L_m$ from the states $|p_L,p^{\mu}\rangle$ with $p^{\mu}\ne 0$ we call the DDF states. The commutation relations (\[altr\]),(\[allo\]) imply that the DDF states are physical states with positive norm. The inverse statement can be formulated as follows [**Theorem.**]{} [*Any solution of the equations [(\[nph\])]{} in the Hilbert space ${\cal H}({\cal W}_L\times {\cal E}_L)$ is of the form $$|\Phi\rangle = |\Psi\rangle + |ns\rangle\;\;\;,$$ where $|\Psi\rangle$ is either a DDF state or one of the states $$\vert p_L=\pm Q',p^{\mu}=0\rangle \;\;\;,\;\;\; Q'=\sqrt{\frac{d-1}{12}}\;\;\;, \label{ovacuum}$$ and $|ns\rangle$ is a null spurious state, i.e. $|ns\rangle$ is orthogonal to all physical states and is a linear combination of states of the form $L_{-n}|\chi\rangle , n\geq 1$.*]{} A counterpart of the theorem above for the standard free field realization of the Virasoro algebra with the central charge $c=26$ and $\alpha_0 =1$ has been proved long time ago by Goddard and Thorn [@gt]. Since the first of the two proofs given in [@gt] is based only on the algebraic properties of the DDF operators it applies without modification in the present case. In fact for a fixed kinematical configuration given by a state (\[vacuum\]) with $p^{\mu}\ne 0$ one can introduce the operators $$K_n=k\cdot \alpha_n$$ satisfying the algebra $$\left[K_m,K_n\right]=0\ \ \ \left[L_m,K_n\right]=-nK_{n+m} \ \ \ K_n^+=K_{-n}$$ The algebra of $A$’s, $K$’s and $L$’s is exactly the same as in [@gt]. The only difference is the number of positive-norm directions ($d-1$ instead of $24$) which however does not alter the reasoning given in [@gt]. To complete the proof of the present version of the theorem let us observe that in the space ${\cal H}({\cal W}_L\times {\cal E}_L)$ the operator $\beta_0$ is self-adjoint and therefore has a real spectrum. Consequently the only physical states with all components of the spacetime momenta equal zero are the lowest states given by (\[ovacuum\]) and all excited positive norm physical states can be achieved by the DDF construction. As a simple consequence of the theorem above and the algebra (\[altr\]), (\[allo\]), one gets the no-ghost theorem for the model given by the equations (\[nph\]). The physical content of the model can be easily inferred using the DDF construction. One can show that the positive norm physical states could be uniquely characterized in terms of the space-time spin and momenta along with an additional internal quantum number represented by the operator $\beta_0$. For each particular eigenvalue $p_L$ of $\beta_0$ the physical states satisfying $\beta_0|\psi\rangle = p_L|\Psi\rangle$ form the Hilbert space ${\cal H}(p_L)$ describing a free noncritical string with the intercept $\alpha_0 = {d-1\over 24}- {p_L^2\over 2}$. For $p_L^2 < {d-1\over 24}$ the lowest states in the space ${\cal H}(p_L)$ are tachyons. Let us note that the theory described by the Hilbert space ${\cal H}(p_L)$ differs from that with the same intercept and obtained by the dimension reduction from the critical Nambu-Goto string [@sch]. In fact, if $T^d(N)$ is the number of states on the level $N$ generated by the d-component oscillators, then the numbers of the positive norm physical states on the level $N$ is $T^{d-1}(N)$ in the Polyakov string while in the reduced Nambu-Goto string one gets $T^{d-1}(N)-T^{d-1}(N-1)$ [@b]. Conclusions =========== The main result of the present paper is that the Polyakov path integral over surfaces does lead to a free quantum theory of 1-dim extended relativistic system in the range $1<d<25$. The resulting theory is equivalent to the FCT “massive” string model. As far as the free theory is concerned this model can be directly compared with the noncritical Polyakov dual model in the range $1<d<25$. In the commonly used radial gauge the “massive” string is given by the realization (\[bcharge\]),(\[herm\]) of the Virasoro algebra. From this point of view the FCT string can be regarded as a special version of the 2-dim Liouville model coupled to d-copies of the free scalar conformal field theory , characterized by the equations $$\begin{aligned} \mu& =&0 \;\;\;, \label{mu} \\ (\beta_0 + iQ)^+& =& \beta_0 + iQ \;\;\;. \label{her}\end{aligned}$$ Within the Polyakov dual model approach the equations (\[mu\], \[her\]) are just a special (in a sense trivial) choices of free “parameters”: the cosmological constant and the scalar product in the space of states. In the present approach the first equation is an assumption partly justified by the requirement of stability while the second one uniquely follows from the interpretation of the model as a quantum mechanics of 1-dim extended relativistic system. Since the equations are crucial for the physical interpretation of the model we shall briefly discuss their origin within the present approach. First of all let us stress that our whole derivation is based on the particular choice of boundary conditions for the string trajectories. In the “matter” sector these boundary conditions have been first introduced in [@dop]. More recently it was shown that they are relevant for constructing the off-shell-critical string amplitudes [@jas]. The derivation of the boundary conditions in the metric sector, based on the geometry of the space of trajectories has been presented in [@ja]. The outcome of this analysis is that as far as the interpretation of the Polyakov path integral as a sum over bordered surfaces is assumed the boundary conditions in this sector are uniquely determined. Finally the relevance of these boundary conditions in the noncritical Polyakov string theory has been confirmed in our previous paper [@jm] concerning the quasiclassical calculation of the static potential in the range $1<d<25$. All these results along with the considerations of Sect.2 show that within the Feynman functional quantization scheme in the $(M,n)$-gauge the choice of boundary conditions is unique. The second important point in our paper is the assumption concerning the vanishing cosmological constant. As it was discussed in Subsect.3.2 in the case of rectangle and for $\mu \neq 0$ there is no classical solutions of the Liouville equation of motion in the space of conformal factors over which one has to integrate in the path integral representation of the transition amplitude. If we interpret the absence of classical extremum as an indication of instability of the system the only consistent choice is $\mu = 0$. Whether or not this conclusion is fully justified is still an open problem. One possible approach is to assume that the generalized Forman formula still holds in the case of the bulk exponential interaction (which can be justified to some extent by means of the perturbation expansion [@car]) and then to analyse the resulting theory in the extended space. Whatever the final understanding of the model with $\mu > 0$ would be, the simplest case $\mu = 0$ yields a consistent free theory and it is a nontrivial and interesting problem to investigate the “joining-splitting” interaction in this model. While the equation (\[mu\]) is an assumption more or less justified by our choice of boundary conditions, the second equation (\[her\]) uniquely follows from them. In fact the central technical point of our approach – the generalized Forman formula - yields not only the simple expression for the transition amplitude in the extended space but also the inner product in the conformal factor sector. The relation of the FCT realization of the Virasoro algebra with the Polyakov path integral over surfaces has been known since the first attempts to quantize the Liouville theory [@cct]. More recently it became a standard tool in analysing the physical states [@pb] and correlators [@dof; @corel] in the Liouville gravity coupled to the conformal matter. The question arises what we have learned from lengthy derivation of the particular realization characterized by the equation (\[her\]). The most important lesson is the path integral formulation of the FCT string. As it was emphasized in the introduction it paves a way for analysing the “joining-splitting” interaction in this string model. Note that the lack of the path integral formulation was a basic obstacle for developing the interacting theory of the “massive” string twenty years ago [@ct]. A related issue is the target space interpretation of the free Polyakov model in the range $1<d<25$. As far as this interpretation is concerned there are no physical states with imaginary (or complex) Liouville momentum in the model. This is in contrast with the Polyakov noncritical dual model interpreted as the 2-dim Euclidean gravity coupled to the conformal matter. This interpretation concentrates on the word sheet physics bringing all the questions of the theory of 2-dim statistical systems. In particular one of the prominent observables in these framework is the area operator getting complex for the central charge of the matter sector in the range $1<d<25$. The physical states with the imaginary Liouville momenta are therefore indispensable within this interpretation and lead to the “unstable” critical behaviour [@ddk; @sei]. The derivation of the FCT free string from the Polyakov path integral over bordered surfaces given in Sect.3 sheds new light on the role the conformal factor plays in the Polyakov string model. The noncritical string model we have started with was originally described in terms of the variables $\{ \alpha, x^{\mu}(\sigma) \}_{\sigma \in [0,1]}$. Roughly speaking the dynamics is given by an on-mass-shell condition (the string wave equation) and the kinematical requirement. It means that in a fixed frame in the Minkowski target space all nonzero modes of the $x^+$ variable are unphysical and there is a relation for the momenta conjugate to the zero modes $\{x^{\mu}_0\}_{\mu=0,...,d-1}$. It follows that the set $\{\alpha,x^{\mu}_0, x^-_k,x^i_k\}_{k=1,...}^{\mu =0,...,d;i=1,...,d-1}$ is a complete system of commuting physical micro-observables. We have used the prefix micro- in order to distinguish them from the “true” physical macro-observables which are given by the generators of the Poincare group in the Minkowski target space. Actually the spectra of the macro-observables are of the main interest in the free theory as they provide a relativistic particle interpretation of the string physical states. In terms of the physical micro-observables $\{\alpha,x^{\mu}_0, x^-_k,x^i_k\}_{k=1,...}^{\mu =0,...,d;i=1,...,d-1}$ the geometrical content of the model is clear - it is a theory of the free parametrized string with internal length. In this formulation however the macro-observables are not diagonal. Moreover we do not know how to derive the on-mass-shell condition in these variables from the complicated form of the transition amplitude. The idea of extension we applied to deal with this problem was to introduce an auxiliary variable $\varphi(\sigma)$ along with some constraints ensuring the equivalence with the original theory. This allows for expressing the original set of physical micro-observables in terms of a new one $\left\{\varphi_0,x^{\mu}_0,\varphi_k,x^i_k \right\}^{\mu =0,...,d;i=1,...,d-1}_{k=1,...}$. As follows from the DDF construction given in Subsect.3.4 in the new variables the macro-observables are diagonal and the relativistic particle content of the model can be easily inferred. The role of the conformal factor is therefore to express in a convenient way the influence of the nontrivial dynamics of the internal length $\alpha$ and the longitudinal excitations $\{x^-_k\}_{k\geq1}$ on the particle spectrum of the noncritical relativistic string model. In this sense the Liouville theory describes the dynamics of the longitudinal modes. Although the FCT model satisfies all the consistency conditions of formal relativistic quantum mechanics its physical content is not quite satisfactory. First of all it contains an internal quantum number entering the on-mass-shell condition which entails an undesirable continuous range of intercepts. Secondly for some values of this quantum number one gets tachyonic states on the lowest level. In the free string theory the zero mode of the “Liouville momenta” is conserved and the theory can be truncated at any real value of $p_L$. On the other hand the appearance of this additional internal degree of freedom is a consequence of our choice of boundary conditions for string trajectories involving the internal string length $\alpha$. The relation between $\alpha$ and $p_L$ is a part of the relation between two sets of physical micro-observables discussed above $$\left\{\alpha,x^{\mu}_0, x^-_k,x^i_k\right\}_{k=1,...}^{\mu =0,...,d;i=1,...,d-1} \leftarrow \! \! \rightarrow \left\{\varphi_0,x^{\mu}_0,\varphi_k,x^i_k \right\}^{\mu =0,...,d;i=1,...,d-1}_{k=1,...}\;\;\;. \label{rela}$$ Using the constraint equations one can easily express the set of micro-observables $\left\{\alpha,x^{\mu}_0, x^-_k,x^i_k\right\}_{k=1,...}^{\mu =0,...,d;i=1,...,d-1}$ in terms of $\left\{\varphi_0,x^{\mu}_0,\varphi_k,x^i_k \right\}^{\mu =0,...,d;i=1,...,d-1}_{k=1,...}$ The opposite relation is however very complicated and we have not found any convincing method of removing the $\alpha$-dependence within the first quantized theory. It seems that the truncation is essentially the problem of the interacting theory where $\alpha$ plays the role similar to that of the “length” parameter in the “covariantized” light cone formulation of the critical string field theory [@hata]. Let us only mention that the results concerning noncritical Polyakov string with fixed ends [@jm] and the naive consideration of the “joining-splitting” interaction suggest a consistent truncation at $p_L = 0$. The second problem with the physical interpretation of the FCT massive string is the presence of tachyons in its spectrum. Since the structure of the model is similar to that of the critical string, one may expect that the problem can be solved in the fermionic noncritical Polyakov string by a counterpart of GSO projection [@gso]. The crucial issue here is an appropriate choice of boundary conditions for the fermionic string trajectories. Since the geometry of the fermionic path integral is far less understood than that of the bosonic one (the infinite-dimensional supergeometry virtually does not exist [@nb]) the methods we have used to determine the boundary conditions in the bosonic case are not available. This makes the problem of a “super” generalization of our approach more difficult than the construction of the supersymmetric noncritical Polyakov dual model [@super]. The considerations of the present paper are entirely devoted to the so called “old” covariant formulation of the free open bosonic string. This leaves a number of interesting questions concerning the first quantized theory. [*Closed string*]{}. In contrast to the open string there is no 1-dim conformal anomaly and the natural scalar product is diff-invariant. The issue of the normal ordering of the Virasoro generators appears if one tries to separate the left and right movers. The additional complication is that this decomposition is not invariant with respect to the residual $S^1$-symmetry. For these reasons the free closed noncritical string is an interesting problem and would lead to a better insight into the relation between the geometry of the scalar product and the ordering problem. [*BRST formulation*]{}. Within the path integral approach of this paper the idea is in a sense opposite to that of the “old” covariant formulation. One starts with the same path integral representation of the transition amplitude in the extended space of boundary conditions. However instead of restricting oneself to a subspace of states on which the corresponding operator is well defined and invertible, we are looking for yet another (BRST) extension which allows to represent the transition amplitude as a special matrix element of an invertible operator. In the case of the critical string this idea can be easily realized [@jask] leading to the well known covariant BRST formulation. In the case of noncritical string the problem gets complicated due to nontrivial coupling of the conformal factor to the zero modes of the ghost sector. In particular the two extension procedures do not commute. Note that the complete BRST formulation of the free theory is interesting at least for two reasons. First it should provide a clarification of the discrepancy between the realization of the residual ${\cal D}_L$-symmetry calculated from the extension procedure and the unitary one in the “old” covariant approach. Secondly, it paves the way for the field theory formulation which gives new tools for investigating the interacting theory. In particular the problem of the joining-splitting interaction vertex can be posed as the problem of the BRST-invariant extension of the corresponding functional delta function in the variables $\{ \alpha, x^{\mu}(\sigma) \}_{\sigma \in [0,1]}$. Note that the role of $\alpha$ in the interacting theory is especially clear in this formulation. The values of $\alpha$ determine the way in which two parametrized strings form a third one. [*Light-cone formulation.*]{} The idea of this approach is to explicitly implement the basic kinematical requirement of the relativistic quantum mechanic of 1-dim extended objects by constructing the transition amplitude in a fixed reference system as a sum over causal string trajectories in the Minkowski target space. The main difficulty with respect to the critical string consists in the fact that the constraints appear in the process of quantization and one cannot describe relevant string trajectories in terms of “true” classical variables. A related problem is an appropriate choice of the space of internal metrics corresponding to causal string trajectories in the Minkowski target space. The form of the DDF-states given in Subsect.3.4 suggest however that this formulation has the structure very similar to that of the critical string theory. [*Operator - states correspondence.*]{} The basic tool in calculating the on-shell critical string amplitudes is the so called operator formalism based on the possibility of reproducing string wave functionals corresponding to the physical states by functional integral over half-disc with a local operator insertion. In the present case this equivalence still holds, as can be inferred from the construction of DDF states. The operator-state correspondence along with the solution of the previous problem form basic ingredients of the Mandelstam method [@m] of constructing the on-mass-shell “massive” string amplitudes. As it was emphasized in the introduction the most interesting open question is whether the “joining-splitting” interaction leads to a consistent interacting theory of the FCT string. We have already mentioned two possible approaches to solve this problem: the BRST and the light-cone formulations. We conclude this section by a brief discussion of only one aspect of the interacting theory which is to some extent independent of a particular formulation. The path integral formulation of the free FCT string derived in this paper involves the assumption that the cosmological constant vanishes. The question arises whether this assumption can be maintained also in the interacting theory. To analyse this problem let us consider the three point off-shell noncritical string amplitude. As in the case of the critical string theory [@jas] it is given by the path integral over string trajectories connecting three prescribed string configurations. On the tree level one has to sum over trajectories of the topology of hexagon. The construction of the path integral representation of the corresponding off-shell amplitude is exactly the same as in the case of rectangle-like trajectories relevant for the string propagator. Choosing the hyperbolic hexagon as a model manifold one gets in the conformal gauge an expression involving the path integral over all conformal factors satisfying the homogeneous Neumann boundary conditions. The reasoning we have used in the case of rectangle to derive the condition $\mu=0$ now leads to a positive cosmological constant. This makes the resulting path integral prohibitively complicated. Up to now the only method to deal with the resulting theory is to impose the requirement of the conformal invariance and then to translate the problem into the language of 2-dim conformal field theory. This leads however to the complex Liouville action – yet another manifestation of the $c=1$ barrier. (400,200)(30,0) (155,40)[(20,10)\[t\][(a)]{}]{} (290,40)[(20,10)\[t\][(b)]{}]{} (220,10)[(20,10)[Fig.1.]{} ]{} (260,110)[(1,0)[40]{}]{} (260,110)[(0,1)[60]{}]{} (260,170)[(1,0)[90]{}]{} (350,170)[(0,-1)[100]{}]{} (350,70)[(-1,0)[50]{}]{} (300,70)[(0,1)[70]{}]{} =cmr10 at 10pt (895,474) (891,472.92) ------------------------------------------------------------------------ (893.00,471.92)(-2.000,2.000)[2]{} ------------------------------------------------------------------------ (887,474.92) ------------------------------------------------------------------------ (889.00,473.92)(-2.000,2.000)[2]{} ------------------------------------------------------------------------ (883,476.92) ------------------------------------------------------------------------ (885.00,475.92)(-2.000,2.000)[2]{} ------------------------------------------------------------------------ (879,479.42) ------------------------------------------------------------------------ (881.00,477.92)(-2.000,3.000)[2]{} 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------------------------------------------------------------------------ (509,427.42) ------------------------------------------------------------------------ (509.00,428.92)(1.500,-3.000)[2]{} ------------------------------------------------------------------------ (512,424.42) ------------------------------------------------------------------------ (512.00,425.92)(1.500,-3.000)[2]{} ------------------------------------------------------------------------ (514.42,421) ------------------------------------------------------------------------ (512.92,423.00)(3.000,-2.000)[2]{} ------------------------------------------------------------------------ (516.92,418) ------------------------------------------------------------------------ (515.92,419.50)(2.000,-1.500)[2]{} ------------------------------------------------------------------------ (520,414.42) ------------------------------------------------------------------------ (520.00,415.92)(1.500,-3.000)[2]{} ------------------------------------------------------------------------ (522.42,411) ------------------------------------------------------------------------ (520.92,413.00)(3.000,-2.000)[2]{} ------------------------------------------------------------------------ (524.92,407) ------------------------------------------------------------------------ (523.92,409.00)(2.000,-2.000)[2]{} ------------------------------------------------------------------------ (528,403.42) ------------------------------------------------------------------------ (528.00,404.92)(1.500,-3.000)[2]{} ------------------------------------------------------------------------ (529.92,400) ------------------------------------------------------------------------ (528.92,402.00)(2.000,-2.000)[2]{} ------------------------------------------------------------------------ (533,396.42) ------------------------------------------------------------------------ (533.00,397.92)(1.500,-3.000)[2]{} ------------------------------------------------------------------------ (534.92,393) ------------------------------------------------------------------------ (533.92,395.00)(2.000,-2.000)[2]{} ------------------------------------------------------------------------ (536.92,389) ------------------------------------------------------------------------ (535.92,391.00)(2.000,-2.000)[2]{} ------------------------------------------------------------------------ (538.92,385) ------------------------------------------------------------------------ (537.92,387.00)(2.000,-2.000)[2]{} ------------------------------------------------------------------------ (540.92,382) ------------------------------------------------------------------------ (539.92,383.50)(2.000,-1.500)[2]{} ------------------------------------------------------------------------ (542.92,378) ------------------------------------------------------------------------ (541.92,380.00)(2.000,-2.000)[2]{} ------------------------------------------------------------------------ (544.92,374) ------------------------------------------------------------------------ (543.92,376.00)(2.000,-2.000)[2]{} ------------------------------------------------------------------------ (546.42,370) ------------------------------------------------------------------------ (545.92,372.00)(1.000,-2.000)[2]{} ------------------------------------------------------------------------ (547.92,366) ------------------------------------------------------------------------ (546.92,368.00)(2.000,-2.000)[2]{} ------------------------------------------------------------------------ (549.92,362) ------------------------------------------------------------------------ (548.92,364.00)(2.000,-2.000)[2]{} ------------------------------------------------------------------------ (551.42,358) ------------------------------------------------------------------------ (550.92,360.00)(1.000,-2.000)[2]{} ------------------------------------------------------------------------ (552.42,354) ------------------------------------------------------------------------ (551.92,356.00)(1.000,-2.000)[2]{} ------------------------------------------------------------------------ (553.92,350) ------------------------------------------------------------------------ (552.92,352.00)(2.000,-2.000)[2]{} ------------------------------------------------------------------------ (555.42,346) ------------------------------------------------------------------------ (554.92,348.00)(1.000,-2.000)[2]{} ------------------------------------------------------------------------ (556.42,341) ------------------------------------------------------------------------ (555.92,343.50)(1.000,-2.500)[2]{} ------------------------------------------------------------------------ (557.42,337) ------------------------------------------------------------------------ (556.92,339.00)(1.000,-2.000)[2]{} ------------------------------------------------------------------------ (558.42,333) ------------------------------------------------------------------------ (557.92,335.00)(1.000,-2.000)[2]{} ------------------------------------------------------------------------ (455.0,471.0) (559.42,325) ------------------------------------------------------------------------ (558.92,327.00)(1.000,-2.000)[2]{} ------------------------------------------------------------------------ (560.42,321) ------------------------------------------------------------------------ (559.92,323.00)(1.000,-2.000)[2]{} ------------------------------------------------------------------------ (561.0,329.0) (561.42,312) ------------------------------------------------------------------------ (560.92,314.00)(1.000,-2.000)[2]{} ------------------------------------------------------------------------ (563.0,316.0) ------------------------------------------------------------------------ (564,296.42) ------------------------------------------------------------------------ (564.00,296.92)(1.000,-1.000)[2]{} ------------------------------------------------------------------------ (566,296.92) ------------------------------------------------------------------------ (566.00,295.92)(2.000,2.000)[2]{} ------------------------------------------------------------------------ (570,298.92) ------------------------------------------------------------------------ (570.00,297.92)(2.000,2.000)[2]{} ------------------------------------------------------------------------ (574,300.92) ------------------------------------------------------------------------ (574.00,299.92)(2.500,2.000)[2]{} ------------------------------------------------------------------------ (579,302.92) ------------------------------------------------------------------------ (579.00,301.92)(2.000,2.000)[2]{} ------------------------------------------------------------------------ (583,304.92) ------------------------------------------------------------------------ (583.00,303.92)(2.000,2.000)[2]{} ------------------------------------------------------------------------ (587,306.42) ------------------------------------------------------------------------ (587.00,305.92)(2.000,1.000)[2]{} ------------------------------------------------------------------------ (591,307.92) ------------------------------------------------------------------------ (591.00,306.92)(2.500,2.000)[2]{} ------------------------------------------------------------------------ (596,309.42) ------------------------------------------------------------------------ (596.00,308.92)(2.000,1.000)[2]{} ------------------------------------------------------------------------ (600,310.92) ------------------------------------------------------------------------ (600.00,309.92)(2.000,2.000)[2]{} ------------------------------------------------------------------------ (604,312.42) ------------------------------------------------------------------------ (604.00,311.92)(2.500,1.000)[2]{} ------------------------------------------------------------------------ (609,313.42) ------------------------------------------------------------------------ (609.00,312.92)(2.000,1.000)[2]{} ------------------------------------------------------------------------ (613,314.92) ------------------------------------------------------------------------ (613.00,313.92)(2.500,2.000)[2]{} ------------------------------------------------------------------------ (618,316.42) ------------------------------------------------------------------------ (618.00,315.92)(2.000,1.000)[2]{} ------------------------------------------------------------------------ (622,317.42) ------------------------------------------------------------------------ (622.00,316.92)(2.500,1.000)[2]{} ------------------------------------------------------------------------ (564.0,299.0) ------------------------------------------------------------------------ (631,318.42) ------------------------------------------------------------------------ (631.00,317.92)(2.500,1.000)[2]{} ------------------------------------------------------------------------ (636,319.42) ------------------------------------------------------------------------ (636.00,318.92)(2.000,1.000)[2]{} ------------------------------------------------------------------------ (640,320.42) ------------------------------------------------------------------------ (640.00,319.92)(2.500,1.000)[2]{} ------------------------------------------------------------------------ (627.0,320.0) (650,321.42) ------------------------------------------------------------------------ (650.00,320.92)(2.000,1.000)[2]{} ------------------------------------------------------------------------ (645.0,323.0) ------------------------------------------------------------------------ (696,321.42) ------------------------------------------------------------------------ (696.00,321.92)(2.500,-1.000)[2]{} ------------------------------------------------------------------------ (654.0,324.0) ------------------------------------------------------------------------ (705,320.42) ------------------------------------------------------------------------ (705.00,320.92)(2.500,-1.000)[2]{} ------------------------------------------------------------------------ (710,319.42) ------------------------------------------------------------------------ (710.00,319.92)(2.000,-1.000)[2]{} ------------------------------------------------------------------------ (714,318.42) ------------------------------------------------------------------------ (714.00,318.92)(2.500,-1.000)[2]{} ------------------------------------------------------------------------ (701.0,323.0) (723,317.42) ------------------------------------------------------------------------ (723.00,317.92)(2.500,-1.000)[2]{} ------------------------------------------------------------------------ (728,316.42) ------------------------------------------------------------------------ (728.00,316.92)(2.000,-1.000)[2]{} ------------------------------------------------------------------------ (732,314.92) ------------------------------------------------------------------------ (732.00,315.92)(2.500,-2.000)[2]{} ------------------------------------------------------------------------ (737,313.42) ------------------------------------------------------------------------ (737.00,313.92)(2.000,-1.000)[2]{} ------------------------------------------------------------------------ (741,312.42) ------------------------------------------------------------------------ (741.00,312.92)(2.500,-1.000)[2]{} ------------------------------------------------------------------------ (746,310.92) ------------------------------------------------------------------------ (746.00,311.92)(2.000,-2.000)[2]{} ------------------------------------------------------------------------ (750,309.42) ------------------------------------------------------------------------ (750.00,309.92)(2.500,-1.000)[2]{} ------------------------------------------------------------------------ (755,307.92) ------------------------------------------------------------------------ (755.00,308.92)(2.000,-2.000)[2]{} ------------------------------------------------------------------------ (759,306.42) ------------------------------------------------------------------------ (759.00,306.92)(2.000,-1.000)[2]{} ------------------------------------------------------------------------ (763,304.92) ------------------------------------------------------------------------ (763.00,305.92)(2.000,-2.000)[2]{} ------------------------------------------------------------------------ (767,302.92) ------------------------------------------------------------------------ (767.00,303.92)(2.500,-2.000)[2]{} ------------------------------------------------------------------------ (772,300.92) ------------------------------------------------------------------------ (772.00,301.92)(2.000,-2.000)[2]{} ------------------------------------------------------------------------ (776,298.92) ------------------------------------------------------------------------ (776.00,299.92)(2.000,-2.000)[2]{} ------------------------------------------------------------------------ (780,296.92) ------------------------------------------------------------------------ (780.00,297.92)(2.000,-2.000)[2]{} ------------------------------------------------------------------------ (784,296.42) ------------------------------------------------------------------------ (784.00,295.92)(1.000,1.000)[2]{} ------------------------------------------------------------------------ (719.0,320.0) (784.42,308) ------------------------------------------------------------------------ (783.92,308.00)(1.000,2.000)[2]{} ------------------------------------------------------------------------ (786.0,299.0) ------------------------------------------------------------------------ (785.42,321) ------------------------------------------------------------------------ (784.92,321.00)(1.000,2.000)[2]{} ------------------------------------------------------------------------ (786.42,325) ------------------------------------------------------------------------ (785.92,325.00)(1.000,2.000)[2]{} ------------------------------------------------------------------------ (787.0,312.0) ------------------------------------------------------------------------ (787.42,333) ------------------------------------------------------------------------ (786.92,333.00)(1.000,2.000)[2]{} ------------------------------------------------------------------------ (788.42,337) ------------------------------------------------------------------------ (787.92,337.00)(1.000,2.000)[2]{} ------------------------------------------------------------------------ (789.42,341) ------------------------------------------------------------------------ (788.92,341.00)(1.000,2.500)[2]{} ------------------------------------------------------------------------ (790.92,346) ------------------------------------------------------------------------ (789.92,346.00)(2.000,2.000)[2]{} ------------------------------------------------------------------------ (792.42,350) ------------------------------------------------------------------------ (791.92,350.00)(1.000,2.000)[2]{} ------------------------------------------------------------------------ (793.42,354) ------------------------------------------------------------------------ (792.92,354.00)(1.000,2.000)[2]{} ------------------------------------------------------------------------ (794.92,358) ------------------------------------------------------------------------ (793.92,358.00)(2.000,2.000)[2]{} ------------------------------------------------------------------------ (796.42,362) ------------------------------------------------------------------------ (795.92,362.00)(1.000,2.000)[2]{} ------------------------------------------------------------------------ (797.92,366) ------------------------------------------------------------------------ (796.92,366.00)(2.000,2.000)[2]{} ------------------------------------------------------------------------ (799.42,370) ------------------------------------------------------------------------ (798.92,370.00)(1.000,2.000)[2]{} ------------------------------------------------------------------------ (800.92,374) ------------------------------------------------------------------------ (799.92,374.00)(2.000,2.000)[2]{} ------------------------------------------------------------------------ (802.92,378) ------------------------------------------------------------------------ (801.92,378.00)(2.000,2.000)[2]{} ------------------------------------------------------------------------ (804.92,382) ------------------------------------------------------------------------ (803.92,382.00)(2.000,1.500)[2]{} ------------------------------------------------------------------------ (806.92,385) ------------------------------------------------------------------------ (805.92,385.00)(2.000,2.000)[2]{} ------------------------------------------------------------------------ (808.92,389) ------------------------------------------------------------------------ (807.92,389.00)(2.000,2.000)[2]{} ------------------------------------------------------------------------ (811.42,393) ------------------------------------------------------------------------ (809.92,393.00)(3.000,2.000)[2]{} ------------------------------------------------------------------------ (813.92,397) ------------------------------------------------------------------------ (812.92,397.00)(2.000,1.500)[2]{} ------------------------------------------------------------------------ (815.92,400) ------------------------------------------------------------------------ (814.92,400.00)(2.000,2.000)[2]{} ------------------------------------------------------------------------ (819,403.42) ------------------------------------------------------------------------ (819.00,401.92)(1.500,3.000)[2]{} ------------------------------------------------------------------------ (820.92,407) ------------------------------------------------------------------------ (819.92,407.00)(2.000,2.000)[2]{} ------------------------------------------------------------------------ (823.42,411) ------------------------------------------------------------------------ (821.92,411.00)(3.000,2.000)[2]{} ------------------------------------------------------------------------ (827,414.42) ------------------------------------------------------------------------ (827.00,412.92)(1.500,3.000)[2]{} ------------------------------------------------------------------------ (828.92,418) ------------------------------------------------------------------------ (827.92,418.00)(2.000,1.500)[2]{} ------------------------------------------------------------------------ (831.42,421) ------------------------------------------------------------------------ (829.92,421.00)(3.000,2.000)[2]{} ------------------------------------------------------------------------ (835,424.42) ------------------------------------------------------------------------ (835.00,422.92)(1.500,3.000)[2]{} ------------------------------------------------------------------------ (838,427.42) ------------------------------------------------------------------------ (838.00,425.92)(1.500,3.000)[2]{} ------------------------------------------------------------------------ (841,430.42) ------------------------------------------------------------------------ (841.00,428.92)(1.500,3.000)[2]{} ------------------------------------------------------------------------ (844,433.42) ------------------------------------------------------------------------ (844.00,431.92)(2.000,3.000)[2]{} ------------------------------------------------------------------------ (848,436.42) ------------------------------------------------------------------------ (848.00,434.92)(1.500,3.000)[2]{} ------------------------------------------------------------------------ (851,439.42) ------------------------------------------------------------------------ (851.00,437.92)(1.500,3.000)[2]{} ------------------------------------------------------------------------ (854,442.42) ------------------------------------------------------------------------ (854.00,440.92)(2.000,3.000)[2]{} ------------------------------------------------------------------------ (858,445.42) ------------------------------------------------------------------------ (858.00,443.92)(1.500,3.000)[2]{} ------------------------------------------------------------------------ (861,448.42) ------------------------------------------------------------------------ (861.00,446.92)(2.000,3.000)[2]{} ------------------------------------------------------------------------ (865,451.42) ------------------------------------------------------------------------ (865.00,449.92)(1.500,3.000)[2]{} ------------------------------------------------------------------------ (868,453.92) ------------------------------------------------------------------------ (868.00,452.92)(2.000,2.000)[2]{} ------------------------------------------------------------------------ (872,456.42) ------------------------------------------------------------------------ (872.00,454.92)(2.000,3.000)[2]{} ------------------------------------------------------------------------ (876,458.92) ------------------------------------------------------------------------ (876.00,457.92)(1.500,2.000)[2]{} ------------------------------------------------------------------------ (879,461.42) ------------------------------------------------------------------------ (879.00,459.92)(2.000,3.000)[2]{} ------------------------------------------------------------------------ (883,463.92) ------------------------------------------------------------------------ (883.00,462.92)(2.000,2.000)[2]{} ------------------------------------------------------------------------ (887,465.92) ------------------------------------------------------------------------ (887.00,464.92)(2.000,2.000)[2]{} ------------------------------------------------------------------------ (891,467.92) ------------------------------------------------------------------------ (891.00,466.92)(2.000,2.000)[2]{} ------------------------------------------------------------------------ (789.0,329.0) The disappointing conclusion above is based on the assumption that the model manifold for a string trajectory describing the elementary process of joining or splitting is given by a hexagon with smooth “timelike” boundaries. If we consider such a process in the Minkowski target space the boundary of the corresponding word sheet has a corner at the interaction point. It follows that if we choose the hyperbolic hexagon (Fig.1.a) as a model manifold the causal string trajectories will be described by singular (at the interaction point) functions. An equivalent description in terms of regular $x$-functions can be achieved if we choose the “light-cone” model manifold (Fig.1.b). Note that in the “light-cone” model manifold the internal angle of the “interaction” corner equals $2\pi$. This is related to the assumption that no particular interaction occurs at the joining point. Applying the reasoning based on the Gauss-Bonnet theorem in the case of “light-cone” model manifold one gets as in the free theory the vanishing cosmological constant. It should be stressed that as far as the critical string theory is concerned the choice of the model manifold is irrelevant. In fact due to the decoupling of the conformal factor the critical on-shell amplitude is invariant with respect to the conformal transformations of the model manifold. As it was mentioned in the introduction this is one of the features crucial for the equivalence of the Polyakov dual model with the relativistic string theory in the critical dimension. The situation in the noncritical string theory is different. Roughly speaking due to the fact that the boundary conditions for the conformal factor are fixed the Liouville action “hears” the shape of the model manifold and the theories based on the hyperbolic hexagon and on the “light-cone” model manifold are different. It follows from the considerations above that the condition $\mu = 0$ can be consistently imposed in the interacting theory provided that we restrict ourself to the “light-cone-like” model manifolds. There are some interesting consequences of this choice. First of all due to the corner conformal anomaly one gets the operator insertion at the interaction point of the form $\exp \gamma\phi(z_i)$. Note that in contrast to the noncritical Polyakov dual model the operator $:\exp \gamma\phi(z_i) :$ has the conformal weight different from 1. In fact the coefficient $\gamma$ is a finite constant uniquely determined by the corner conformal anomaly. Let us also stress that within the present approach there is no reason to interpret the integral of this operator over the world-sheet as the “volume of the universe”. Assuming the same general form of the transition from the off-shell to the on-shell amplitudes as in the critical string theory one can expect that the on-shell amplitudes can be expressed in terms of correlators of the 2-dim conformal field theory involving the vertex operators corresponding to the DDF states and the insertions. This is a very promising feature of the model – one can use the techniques of the conformal field theory to analyse the FCT string amplitudes. The complete analysis of the interacting FCT string theory requires solutions of a number of technical and conceptual problems, which are far beyond the scope of the present paper. We hope however that the covariant functional integral formulation of the free FCT string makes the program of constructing the noncritical relativistic string theory more promising than twenty years ago and still worth pursuing. Acknowledgements {#acknowledgements .unnumbered} ================ We would like to thank Professor Abdus Salam, the International Atomic Energy Agency and UNESCO for support during our stay at the International Centre of Theoretical Physics where the large part of this work was carried out. [99]{} A.M.Polyakov, Phys.Lett. 103B (1981) 207. For review with references to the literature see P.Ginsparg and G.Moore, “Lectures on 2D Gravity and 2D String Theory”, YCTP-P23-92 (LA-UR-92-3479; hep-th/9304011), (1992/93) L.Alvarez-Gaumé, J.L.F.Barbón, and Č.Crnković, Nucl.Phys. B394 (1993) 383. S.M.Catterall, J.B.Kogut, and R.L.Renken, Phys.Lett. B292 (1992) 277;\ J.Ambj[ø]{}rn, B.Durhuus, T.Jónsson, and G.Thorleifsson, Nucl.Phys. B398 (1993) 568. R.C.Brower, Phys.Rev. D6 (1972) 1655 A.Chodos and C.B.Thorn, Nucl.Phys. B72 (1974) 509 E.D’Hoker and D.H.Phong, Rev.Mod.Phys. 60 (1988) 917 V.Alessandrini, D.Amati, F.Le Bellac, and D.Olive, Phys.Rep. C1 (1971) 269;\ J.H.Schwarz, Phys.Rep. C8 (1973) 269 S.Mandelstam, Phys.Rep. C13 (1974) 259 S.B.Giddings and S.A.Wolpert, Commun.Math.Phys. 109 (1987) 177 E.D’Hoker and S.B.Giddings, Nucl.Phys B291 (1987) 90 A.Polyakov, Mod.Phys.Lett. A2 (1987) 893;\ V.Knizhnik, A.Polyakov, and A.Zamolodchikov, Mod.Phys.Lett. A3 (1988) 819;\ F.David, Mod.Phys.Lett. A3 (1988) 1651;\ J.Distler and K.Kawai, Nucl.Phys. B321 (1989) 509 N.Seiberg, “Notes on Quantum Liouville Theory and Quantum Gravity”, in [*Common Trends in Mathematics and Quantum Field Theory*]{}, Proc. of the 1990 Yukawa International Seminar, Prog.Theor.Phys.Suppl. 102 (1991) D.Kutasov, “Some Properties of (Non) Critical Strings”, in [*String Theory and Quantum Gravity ’91*]{}, J.Harvey, R.Iengo, K.S.Narain, S.Randjbar-Daemi, H.Verlinde (eds), World Scientific 1992. J.Ambj[ø]{}rn, B.Durhuus, and J.Frölich, Nucl.Phys. B257\[FS14\] (1985) 433;\ V.A.Kazakov, I.K.Kostov, and A.A.Migdal, Phys.Lett. 157B (1985) 295;\ F.David, Nucl.Phys. B257\[FS14\] (1985) 543;\ J.Ambj[ø]{}rn, B.Durhuus, J.Frölich, and P.Orland, Nucl.Phys. B270\[FS16\] (1986) 457 J.Polchinski, Nucl.Phys. B324 (1989) 123;\ S.Das, S.Naik, and S.Wadia, Mod.Phys.Lett. A4 (1989) 1033;\ for a recent review see [@gm] N.Mavromatos and J.Miramontes, Mod.Phys.Lett. A4 (1989) 1849;\ E.D’Hoker and P.S.Kurzepa, Mod.Phys.Lett. A5 (1990) 1411 B.H.Lian and G.J.Zuckerman, Phys.Lett. B254 (1991) 417;\ P.Bouwknegt, J.McCarthy, and K.Pilch, Lett.Math.Phys. 23 (1991) 193;\ Commun.Math.Phys. 145 (1992) 541;\ A.Bilal, Phys.Lett. B282 (1992) 309 B.L.Feigin and D.B.Fuchs, Funct.Anal.Prilozhen. 16 (1982) 47 A.Cohen, G.Moore, P.Nelson, and J.Polchinski, Nucl.Phys. B267 (1986) 143;\ D.Birmingham and C.G.Torre, Phys.Lett. 205B (1988) 289 C.Varughese and W.I.Weisberger, Phys.Rev. D37 (1988) 981 A.N.Redlich, Phys.Lett. 205B (1988) 295; Nucl.Phys. B304 (1988) 129 Z.Jaskólski, Commun.Math.Phys. 128 (1990) 285 Z.Jaskólski, Int.J.Mod.Phys. A8 (1993) 1041 Z.Jaskólski, Commun.Math.Phys. 147 (1992) 343 C.Rebbi, Phys.Rep. C12 (1974) 1 J.Scherk, Rev.Mod.Phys. 47 (1975) 123 E.Del Giudice, P.Di Vecchia, and S.Fubini, Ann.Phys.(N.Y.) 70 (1972) 378 Z.Jaskólski and K.A.Meissner, “Static quark potential from the Polyakov sum over surfaces”, ICTP preprint IC/93/33 (1993), to appear in Nucl.Phys. B R.Forman, Invent.Math. 88 (1987) 447 S.Carlip, M.Clements, S.DellaPietra, and V.DellaPietra, Commun.Math.Phys. 127 (1990) 263 C.B.Thorn, Nucl.Phys. B248 (1984) 551 P.Goddard and C.B.Thorn, Phys.Lett. 40B (1972) 235 B.Durhuus, P.Olesen, and J.L.Petersen, Nucl.Phys. B232 (1984) 291 T.L.Curtright and C.B.Thorn, Phys.Rev.Lett. 48 (1982) 1309 V.I.Dotsenko and V.A.Fateev, Nucl.Phys. B240\[FS21\] (1984) 312; B251 (1985) 691 A.Gupta, S.P.Trivedi, and M.B.Wise, Nucl.Phys. B340 (1990) 85;\ M.Bershadsky and I.R.Klebanov, Phys.Rev.Lett. 65 (1990) 3088;\ P.D.Francesco and D.Kutasov, Nucl.Phys. B342 (1990) 589 ; Phys.Lett. B261 (1991) 385;\ M.Goulian and M.Li, Phys.Rev.Lett. 66 (1991) 2051;\ Y.Kitazawa, Int.J.Mod.Phys. A7 (1992) 3403;\ N. Sakai and Y. Tanii, Progr.Theor.Phys. 86 (1991) 547;\ Vl.I.Dotsenko, Mod.Phys.Lett. A6 (1991) 3601 ;\ K.Aoki and E.D’Hoker, Mod.Phys.Lett. A7 (1992) 235 H.Hata, K.Itoh, T.Kugo, H.Kunimoto, and K.Ogawa, Phys.Rev. D34 (1986) 2360 F.Gliozzi, J.Scherk, and D.Olive, Phys.Lett. 65B (1976) 282; Nucl.Phys. B122 (1977) 253 M.Batchelor, “In search for the graded manifolds of maps between graded manifolds” in [*Complex Differential Geometry and Suprmanifolds in String and Fields*]{}, P.J.M.Bongaarts and R.Martini (eds), Springer Lecture Notes in Physics 311 (1988);\ P.Nelson, Int.J.Mod.Phys. A3 (1988) 585 J.Distler, Z.Hlousek, and H.Kawai, Int.J.Mod.Phys. A5 (1990) 391 M.Kac, Amer.Math.Monthly 73, part II (1966) 1 Appendix A {#appendix-a .unnumbered} ========== In this Appendix we gather the results for the corner conformal anomaly in the case of Laplace-Beltrami operator acting on scalar functions with Dirichlet and Neumann boundary conditions. The corner anomaly appears in the expansion ($f$ is a scalar function) $${\rm Tr(e}^{t\Delta}f)=\frac{k_{-1}}{t}+\frac{k_{-1/2}}{t^{1/2}}+ k_0+O(t^{1/2})$$ in the t independent part (functional $k_0$) and sums the values of the function $f$ in the corners with the appropriate coefficient. As it was shown by Kac [@kac] the contribution from each corner is independent of the global geometry of the surface. It can be estimated by mapping a neighborhood of the corner to the wedge on the plain with the same opening angle. In the case of Dirichlet boundary conditions on both arms of the corner with opening angle $\gamma$ Kac derived the following formula for the corner conformal anomaly $$A_{DD}(\gamma)=\frac{\pi^2-\gamma^2}{24\pi\gamma}f(0) \;\;\;.$$ Using this result one can easily infer the corner conformal anomaly for the Dirichlet-Neumann and the Neumann-Neumann boundary conditions. Doubling the corner one gets the following relations $$\begin{aligned} A_{DD}(\gamma)+A_{DN}(\gamma)&=& A_{DD}(2\gamma)\;\;\;,\\ A_{DN}(\gamma)+A_{NN}(\gamma)&=& A_{NN}(2\gamma)\;\;\;.\end{aligned}$$ Using the result of Kac one has $$\begin{aligned} A_{DN}(\gamma)&=& -\ \frac{\pi^2+2\gamma^2}{48\pi\gamma}f(0)\;\;\;,\\ A_{NN}(\gamma)&=& \frac{\pi^2-\gamma^2}{24\pi\gamma}f(0)\;\;\;.\end{aligned}$$ In the problems discussed in this paper we have only right angles so we quote the results for $\gamma=\pi/2$: $$\begin{aligned} A_{DD}(\pi/2)&=& A_{NN}(\pi/2)\;=\;\frac{f(0)}{16}\;\;\;,\\ A_{DN}(\pi/2)&=& -\ \frac{f(0)}{16}\;\;\;.\end{aligned}$$ As far as the rectangle with the standard flat metric is concerned the results above are enough to derive the corner conformal anomaly for the operators $P^+P, PP^+$ acting on the vector fields and symmetric traceless tensors, respectively. In this case the corresponding operators act separately on every component of vector or tensor fields. Since the components satisfy independent boundary conditions the problem can be reduced to the scalar one. The corner conformal anomaly for these operators has been first derived in [@vw]. Appendix B {#appendix-b .unnumbered} ========== In this appendix we shall calculate the 1-dim conformal anomaly. Let $e$ be an einbein on the interval $[0,1]$ and $$\Delta_e \equiv -{1\over e}{d\over dt}{1\over e}{d\over dt}\;\;\;,$$ the 1-dim Laplace operator acting on the space ${\cal S}_N$ of scalar functions $\psi(t)$ on $[0,1]$ satisfying Neumann boundary conditions at the ends of $[0,1]$. Let us denote by ${\cal D}^e\psi$ the functional measure related to the scalar product on ${\cal S}_N$: $$\langle \psi | \psi'\rangle = \int\limits_0^1 e\, dt\, \overline{\psi(t)}\psi'(t) \;\;\;.$$ The 1-dim conformal anomaly $J[\varphi,\widehat{e}]$ is defined by the relation $${\cal D}^{{\rm e}^{{\varphi \over 2}}\widehat{e}}\psi = J[\varphi,\widehat{e}]\, {\cal D}^{\widehat{e}}\psi\;\;\; \label{conan}$$ between the functional measures corresponding to the einbeins $e={\rm e}^{{\varphi \over 2}} \widehat{e}$ and $e=\widehat{e}$. Using the method proposed in the context of the Liouville measure in 2 dimensions [@mm] one can derive the following regularized formula for variation $$\delta \log J_N[\varphi,\widehat{e}] = {1\over 4} \lim_{\epsilon \to 0} \int\limits_0^1 e\,dt\, {\rm e}^{-\epsilon\Delta_{e}}(t,t)\delta \psi(t) \;\;\;,$$ where $e = {\rm e}^{{\varphi\over 2}}\widehat{e}$. In terms of normalized eigenfunctions $\left\{\psi_m\right\}_{m\geq 0}$ of the operator $\Delta_e$ the formula above takes the form $$\delta \log J_N[\varphi,\widehat{e}] ={1\over 4} \lim_{\epsilon \to 0} \int\limits_0^1 dt \left( \sum\limits_{m\geq 0} {\rm e}^{-\epsilon {m^2\pi^2\over \alpha^2}} {\psi_m(t)}^2 \delta\varphi(t) \right)\;\;\;,$$ where $\alpha = \int_0^1 {\rm e}^{{\varphi \over 2}} \widehat{e} dt$. Proceeding to the parameterization $s=s(t)$ of $[0.1]$ in which ${\rm e}^{{\varphi \over 2}} \widehat{e} = {\rm const} = \alpha$ and using the expansion $$\delta \varphi(s) = \sum\limits_{n\geq 0}\delta\varphi_n \cos\pi ns\;\;\;,$$ one gets $$\begin{aligned} \delta \log J_N[\varphi,\widehat{e}] & =& {1\over 4} \lim_{\epsilon \to 0} \int\limits_0^1 ds \left( \sum\limits_{\parbox{26pt}{\scriptsize $\makebox[1pt]{}n\geq 0$ \\ $m\geq 1$}} {\rm e}^{-\epsilon {m^2\pi^2\over \alpha^2}} \delta\varphi_n 2\cos\pi ns \cos^2\pi ms + \sum\limits_{n\geq 0} \delta\varphi_n \cos\pi ns \right) \;\;\; \\ &=& {1\over 16} \left( \delta\varphi(0) + \delta\varphi(1) \right) + \lim_{\epsilon \to 0} {1\over 8\sqrt{\pi\epsilon}} \int\limits_0^1 {\rm e}^{{\varphi \over 2}} \widehat{e} dt \,\delta\varphi(t)\;\;\;. \end{aligned}$$ Integrating with respect to $\varphi$ one has $$\log J_N[\varphi,\widehat{e}] = {1\over 16} \left( \varphi(0) + \varphi(1)\right) + {1\over 4\sqrt{\pi\epsilon}} \int\limits_0^1 {\rm e}^{{\varphi \over 2}} \widehat{e} dt\;\;\;. \label{jacob}$$ The corresponding result in the case of Dirichlet boundary conditions takes the form $$\log J_D[\varphi,\widehat{e}] = - {1\over 16} \left( \varphi(0) + \varphi(1)\right) + {1\over 4\sqrt{\pi\epsilon}} \int\limits_0^1 {\rm e}^{{\varphi \over 2}} \widehat{e} dt\;\;\;.$$ Inserting (\[jacob\]) into (\[conan\]) one gets the formula used in the main text $${\cal D}^{{\rm e}^{{\varphi \over 2}}\widehat{e}}\psi = \exp \left[+{1\over 16} \left( \varphi(0) + \varphi(1) \right) + {1\over 4\sqrt{\pi\epsilon}} \int\limits_0^1 {\rm e}^{{\varphi \over 2}} \widehat{e} dt \right] {\cal D}^{\widehat{e}}\psi\;\;\;. \label{conano}$$ Appendix C {#appendix-c .unnumbered} ========== In this appendix we shall prove the formula (\[formula\]) of Subsect.3.2. Consider the mode expansion of the conformal factor $$\varphi = {2\over \sqrt{t} } \sum\limits_{m,n \geq 0} \varphi_{nm} \cos{\pi n z^0 \over t} \cos \pi m z^1$$ and the change of variables $$\begin{aligned} \psi_{00} &=& \varphi_{00}\;\;\;;\nonumber\\ \psi^+_{k0} &=& \sum\limits_{l\geq k}\varphi_{(2l)0}\;\;\;\;\;,\;\;\; \psi^-_{k0} \;=\; \sum\limits_{l\geq k}\varphi_{(2l-1)0} \;\;,\;\;k\geq 1\;\;\;;\label{varia}\\ \psi^-_{km} &=& \sum\limits_{l\geq k}\varphi_{(2l+1)m}\;\;\;,\;\;\; \psi^-_{km} \;=\; \sum\limits_{l\geq k}\varphi_{(2l+1)m} \;\;,\;\;k\geq 0,m\geq 1\;\;\;. \nonumber\end{aligned}$$ The modes $\widetilde{\varphi}_{im},\widetilde{\varphi}_{fm}$ of the boundary values $\widetilde{\varphi}_i,\widetilde{\varphi}_f$ of $\varphi$ (\[modes\]) can be expressed in terms of the variables (\[varia\]) as follows $$\begin{aligned} \widetilde{\varphi}_{i0} &=& {2\over \sqrt{t}} \left( \psi_{00} + \psi^+_{10} + \psi^-_{10} \right)\;\;\;,\\ \widetilde{\varphi}_{f0} &=& {2\over \sqrt{t}} \left( \psi_{00} + \psi^+_{10} - \psi^-_{10} \right)\;\;\;,\\ \widetilde{\varphi}_{im} &=& {2\over \sqrt{t}} \left( \psi^+_{0m} + \psi^-_{0m} \right)\;\;\;,\;\;\;m\geq 1\;\;\;,\\ \widetilde{\varphi}_{fm} &=& {2\over \sqrt{t}} \left( \psi^+_{0m} - \psi^-_{0m} \right)\;\;\;,\;\;\;m\geq 1\;\;\;.\end{aligned}$$ In terms of the variables (\[varia\]) the l.h.s. of the formula (\[formula\]) can be written as the iterated integral $$\begin{aligned} Z&=& {\rm const} \int d\psi_{00} \int \prod\limits_{k\geq 1} d\psi^+_{k0}d\psi^-_{k0} \nonumber\\ &\times& \exp \left[- {b\over 2t^2} \sum\limits_{k\geq1} 4k^2 \left(\psi^+_{k0} -\psi^+_{(k+1)0}\right)^2 \right] \label{inte} \\ &\times& \exp \left[- {b\over 2t^2} \sum\limits_{k\geq1} (2k+1)^2 \left(\psi^-_{k0} -\psi^-_{(k+1)0}\right)^2 \right] \times Z_1\;\;\;,\nonumber\end{aligned}$$ where $$\begin{aligned} Z_1 &=& \int \prod\limits_{\parbox{26pt}{\scriptsize \makebox[1pt]{} $k\geq 0$ \\ $m\geq 1$}} d\psi^+_{km}d\psi^-_{km} \exp \left[ -b \sum\limits_{m\geq1} m^2 \left( \psi^+_{0m} -\psi^+_{1m} \right)^2 \right] \nonumber\\ &\times& \exp \left[ -{b\over 2} \sum\limits_{\parbox{26pt}{\scriptsize \makebox[1pt]{} $k\geq 1$ \\ $m\geq 1$}} \left( {4k^2\over t^2} +m^2 \right) \left( \psi^+_{km} - \psi^+_{(k+1)m} \right)^2 \right] \label{inte2}\\ &\times& \exp \left[ - {b\over 2} \sum\limits_{\parbox{26pt}{\scriptsize \makebox[1pt]{} $k\geq 0$ \\ $m\geq 1$}} \left( \frac{(2k+1)^2}{t^2} + m^2 \right) \left( \psi^-_{km} - \psi^-_{(k+1)m} \right)^2 \right] \\ &\times& F[\widetilde{\varphi}_i,\widetilde{\varphi}_f] \;\;\;,\nonumber\end{aligned}$$ end $$b= {25-d\over 48}\pi\;\;\;.$$ Let us note that the functional $F[\widetilde{\varphi}_i,\widetilde{\varphi}_f]$ is independent of the modes $\left\{ \psi^+_{km},\psi^-_{km} \right\}_{k \neq 0}$ and $Z_1\equiv Z_1(\widetilde{\varphi}_{i0},\widetilde{\varphi}_{f0})$ is a function only of the zero modes of $\widetilde{\varphi}_i,\widetilde{\varphi}_f$. The integrals $Z,Z_1$ involve chains of Gaussian integrals which can be exactly calculated by means of the formula ($\lambda_{N+1} =0$) $$\begin{aligned} \int \prod\limits_{k\geq 1}^N \exp \!\!\!\!\!\!\!\!& &\!\!\!\!\!\left[ - {\beta \over 2} \lambda_0 (x_0 -x_1)^2 -{\beta \over 2} \sum\limits_{k\geq 1}^N \lambda_k (x_k - x_{k+1})^2 \right] =\nonumber\\ &=& \left( \prod\limits_{k\geq 0}^N \frac{\beta \lambda_k}{2\pi} \right)^{-{1\over 2}} \left( {\beta\over 2\pi A} \right)^{{1\over 2}} \exp \left(-{\beta \over 2\pi A} x_0^2 \right) \;\;\;,\label{gaus}\end{aligned}$$ where $$A = \sum\limits_{k\geq 0}^N {1 \over \lambda_k}\;\;\;\;\;.$$ Applying the formula (\[gaus\]) to the integral $Z$ one obtains $$\begin{aligned} Z\;=\;&{\rm const}& \left[ \prod\limits_{k\geq 1} {4k^2\over t^2}\frac{(2k+1)^2}{t^2} \right]^{-{1\over 2}} \int d\psi_{00} \int \frac{d\psi^+_{10}}{t\sqrt{A^+_0}} \int \frac{d\psi^-_{10}}{t\sqrt{A^-_0}} \\ &\times& \exp \left[ -\frac{b}{t^2A^+_0}{\psi^+_{10}}^2 -\frac{b}{t^2A^-_0}{\psi^-_{10}}^2 \right] \\ &\times& Z_1\left({2\over \sqrt{t}} \left( \psi_{00} + \psi^+_{10} + \psi^-_{10} \right), {2\over \sqrt{t}} \left( \psi_{00} + \psi^+_{10} - \psi^-_{10} \right)\right) \;\;\;,\end{aligned}$$ where $$\begin{aligned} A^+_0 &=& \sum\limits_{k\geq 1} {1\over 4k^2} \;=\;{\pi^2\over 24} \;\;\;,\\ A^-_0 &=& \sum\limits_{k\geq 0} {1\over (2k+1)^2} \;=\;{\pi^2\over 8} \;\;\;.\end{aligned}$$ Changing variables $$\begin{aligned} \widetilde{\varphi}_{i0} &=& {2\over \sqrt{t}} \left( \psi_{00} +\psi^+_{10} +\psi^-_{10} \right)\;\;\;,\\ \widetilde{\varphi}_{f0} &=& {2\over \sqrt{t}} \left( \psi_{00} +\psi^+_{10} -\psi^-_{10} \right)\;\;\;,\\ \varphi_{00} &=& \psi_{00}\;\;\;,\end{aligned}$$ and integrating over $\varphi_{00}$ one finally gets $$Z= {\rm const}\; t^{-{1\over 2}} \int d\widetilde{\varphi}_{i0} d\widetilde{\varphi}_{f0} \exp \left[ - {b\over 2\pi^2}\frac{\left( \widetilde{\varphi}_{f0} - \widetilde{\varphi}_{i0} \right)^2}{t} \right] Z_1(\widetilde{\varphi}_{i0},\widetilde{\varphi}_{f0})\;\;. \label{zet}$$ In the case of the integral $Z_1$ the formula (\[gaus\]) yields $$\begin{aligned} Z_1 \;=\;&{\rm const}& \left[ \prod\limits_{\parbox{26pt}{\scriptsize \makebox[1pt]{} $k\geq 0$ \\ $m\geq 1$}} \left( {4k^2\over t^2} + m^2\right) \left(\frac{(2k+1)^2}{t^2} +m^2\right) \right]^{-{1\over 2}} \int \prod\limits_{m\geq 1} \frac{d\psi^+_{0m}}{t\sqrt{A^+_m}} \frac{d\psi^-_{0m}}{t\sqrt{A^-_m}} \\ &\times& \exp \left[ -{b\over 2} \sum\limits_{m\geq 1} \left( \frac{{\psi^+_{0m}}^2}{A^+_m} + \frac{{\psi^-_{0m}}^2}{A^-_m} \right) \right] \times F[\widetilde{\varphi}_i,\widetilde{\varphi}_f] \;\;\;,\end{aligned}$$ where $$\begin{aligned} A^+_m &=& {1\over 2m^2} + \sum\limits_{k\geq 1} \frac{t^2}{4k^2 + t^2m^2}\;=\;{\pi t\over 4m} \coth {\pi mt\over 2}\;\;\;,\\ A^-_m &=& \sum\limits_{k\geq 1} \frac{t^2}{(2k+1)^2 + t^2m^2}\;=\;{\pi t\over 4m} \tanh {\pi mt\over 2}\;\;\;.\end{aligned}$$ Changing variables $$\begin{aligned} \widetilde{\varphi}_{im} &=& {2\over \sqrt{t}} \left(\psi^+_{0m} +\psi^-_{0m} \right)\;\;\;,\\ \widetilde{\varphi}_{fm} &=& {2\over \sqrt{t}} \left(\psi^+_{0m} -\psi^-_{0m} \right)\;\;\; ,\end{aligned}$$ and using the formula $$\eta(t) = \prod\limits_{\parbox{26pt}{\scriptsize \makebox[1pt]{} $k\geq 0$ \\ $m\geq 1$}} \left({k^2\over t^2} +m^2 \right)\;\;\;,$$ one has $$\begin{aligned} Z_1 \;=\;&{\rm const}& \eta(t)^{-{1\over 2}} \int \prod\limits_{m\geq 1} d\widetilde{\varphi}_{im}\; d\widetilde{\varphi}_{fm} \nonumber\\ &\times& \exp \left[ -{b\over 4\pi^2} \sum\limits_{m\geq 1}\frac{\pi m}{\sinh \pi mt} \left[ \left( \widetilde{\varphi}_{im}^2 + \widetilde{\varphi}_{fm}^2 \right) \cosh \pi mt - 2\widetilde{\varphi}_{im}\widetilde{\varphi}_{fm} \right]\right]\nonumber\\ &\times& F[\widetilde{\varphi}_i,\widetilde{\varphi}_f]\;\;\;. \label{zett}\end{aligned}$$ Substituting (\[zett\]) to (\[zet\]) one gets the generalized Forman formula (\[formula\]). [^1]: Institute of Theoretical Physics, Wroc[ł]{}aw University, pl. Maxa Borna 9, 50-204 Wroc[ł]{}aw, Poland; E-mail: jaskolsk@plwruw11.bitnet . [^2]: Permanent address: Institute of Theoretical Physics, Warsaw University, Hoża 69, 00-681 Warszawa, Poland; E-mail: meissner@fuw.edu.pl .

--- abstract: 'The equation of state of pure QCD, obtained from lattice QCD, is discussed for temperatures ranging from $0.9{T_{{\mbox{\scriptsize c}}}}$ to $4{T_{{\mbox{\scriptsize c}}}}$, as well as results on screening masses, the chiral condensate, and the pion decay constant close to the deconfinement phase transition in the confined phase of QCD. The equation of state differs significantly from that of a free gas. There is little evidence of a temperature dependence in the chiral condensate or the meson properties, but perhaps some for the nucleon screening mass. Above the phase transition one sees non-perturbative effects, even though hadron correlators show the existence of deconfined quarks.' --- hep-lat/9503011 BI-TP 95/12\ [**DECONFINEMENT AND HOT HADRONS\ IN CRAYS AND QUADRICS [^1]\ **]{} [G. BOYD\ ]{} [Fakultät für Physik, Universität Bielefeld\ Postfach 100131, D-33501 Bielefeld, Germany]{}\ Introduction ============ The study of hadronic matter at high temperature and / or density is receiving increasing attention, both experimentally and theoretically. It is important to gain a good understanding of the predictions of QCD in this regime as soon as possible. There are many approaches to this task, some of which are presented elsewhere in these proceedings. Each is advantageous for some aspect of the problem. One needs to combine them all to understand what QCD predicts over the entire range of temperatures and densities relevant for heavy ion collisions. Recent results from one of the major theoretical players, the lattice regulation of QCD at finite temperature, will be presented below. An introduction to lattice QCD may be found, for example, in [@MONTMUEN; @FKHWA]. When one is close to the phase transition, many of the traditional approaches to QCD run into problems, as both low temperature and high temperature expansions need to be extrapolated into regions far from the regions in which they are known to be secure. The lattice may well be the only satisfactory approach in this region. The equation of state, both for nuclear matter just below the deconfinement transition, and for the quark-gluon plasma above the phase transition, is required information for, amongst others, hydrodynamic models attempting to describe the evolution of the quark gluon plasma. Since the couplings and masses are not small below $2{T_{{\mbox{\scriptsize c}}}}$ the lattice approach is the obvious method to use. Despite extensive studies of the QCD phase transition [@FKHWA; @BP92KRH93] little is known with certainty about the changes in the excitation spectrum. In particular, the properties of the (quasi)-particle spectrum in the confined phase close to ${T_{{\mbox{\scriptsize c}}}}$ require a more thorough understanding, as the temperature dependence of hadron masses and other hadronic parameters will lead to observable consequences in current, and especially forthcoming, heavy ion collisions. The temperature dependence of various hadronic properties has been addressed using a number of different approaches. These calculations yield different predictions for the behaviour of some quantities [@LEUT90AND]. Since these approaches have limited applicability at high temperatures, one may hope that a lattice calculation of the variation with temperature of any of these quantities below the phase transition may shed light on the discrepancies. Another important quantity is the ratio of the chromo electric to chromo magnetic screening length. This is used, for example, in estimates of the colour transparency of the plasma [@GYU93]. One promising approach is to calculate it directly from measurements of the gluon self energy on the lattice. Recent results on the equation of state for pure gluon SU(3) theory and for QCD with two light flavours will be presented in section \[sec:eos\]. The results from a study of hadronic properties in quenched QCD at temperatures between $0.75{T_{{\mbox{\scriptsize c}}}}$ and $0.92{T_{{\mbox{\scriptsize c}}}}$ are presented in section \[sec:hadrons\]. Section \[sec:gluon\] contains the results pertaining to the chromo electric and magnetic screening masses. Equation of State {#sec:eos} ================= Recent work [@ENGELS90EKR94] has enabled the calculation of the energy density and pressure on the lattice using an entirely non-perturbative technique for the first time. This requires detailed a knowledge of $dg^{-2}/da$, ie. one needs to know the QCD beta function $\beta_{{\mbox{\scriptsize f}}}$ relating changes in physical quantities with changes in the coupling. This has been calculated by the TARO collaboration [@TARO93]. Given the beta function one calculates the interaction measure, the difference between the energy density and thrice the pressure, as follows: $$\label{eq.ta} \frac{\epsilon-3p}{T^4} = T \frac{\partial}{\partial T} \left( \frac{p}{T^4} \right) = a \left( \frac{d g^{-2}}{d a} \right) 6N_c N_{\tau}^4 \{ P_{\sigma} + P_{\tau} - 2 P_0 \}.$$ The pressure is given by $$\frac{p}{T^4} = \left. \frac{-f}{T^4} \right|^{\beta}_{\beta_0} = 3 N_{\tau}^4\int_{\beta_0}^{\beta} d\tilde \beta \, \{ P_{\sigma}(\tilde \beta) + P_{\tau}(\tilde \beta) - 2 P_0(\tilde \beta) \} \label{eq.fint}$$ The zero temperature plaquette is given by $P_0$, the spatial and temporal plaquettes at finite temperature are given by $P_{\sigma}$ and $P_{\tau}$ respectively. Both the energy density and the pressure are normalised to zero at zero temperature. The results obtained for the interaction measure, divided by $T^{4}$, are shown in figure \[fig:interaction\]. The peak in the interaction measure shortly after the first order deconfinement phase transition shows clearly the strong deviation from ideal gas behaviour, $\epsilon = 3P$. The energy density rises sharply, whilst the pressure lags behind until one has reached temperatures around $3{T_{{\mbox{\scriptsize c}}}}$. This is evident again in figure \[fig:ep\] where the energy density and pressure, scaled with $T^{4}$ are plotted together. Note that the interaction measure is related to the value of the gluon condensate at finite temperature [@MILLER94; @LEUT92] via $$\langle G^{2} \rangle_{T} = \langle 0 | G^{2} | 0 \rangle - (\epsilon - 3P)_{T},$$ where $\langle G^{2} \rangle_{T}$ and $\langle 0 | G^{2} | 0 \rangle$ are the condensates at zero and finite temperature respectively. The gluon field strength $G^2$ is given by $G^{2} = -[\beta_{{\mbox{\scriptsize f}}}/(2g^{3})]G^{\mu\nu}_{a}G^{a}_{\mu\nu}$. Taking $\langle 0 | G^{2} | 0 \rangle=0.0135$GeV$^{4}$ [@LEUT92] one finds that $\langle G^{2} \rangle_{T}$ drops to zero immediately after the phase transition, and then goes negative according to the above equation. The energy density for different values of the temporal extent is plotted in figure \[fig:evsesb\], compared to the Stefan-Boltzmann limit on this size lattice. If one takes the continuum limit one obtains a value around 83% of the Stefan-Boltzmann value for temperatures above $2{T_{{\mbox{\scriptsize c}}}}$ up to at least $3{T_{{\mbox{\scriptsize c}}}}$. It is clear that even at these temperatures the interactions and masses of the excitations cannot be neglected. A similar rapid rise in the energy density, and slower rise in the pressure, is seen in the case of two flavour QCD [@BKT94]. It is clear that, if there is a jump in the energy density, the pressure will lag behind in a thermodynamic system, from the relation $P/T = P_{0}/T + \int dT \epsilon / T^{2}$. The deviation from the ideal gas behaviour may be understood in terms of plasma consitituents that are massive, and have a strong interaction [@RIS92]. The speed of sound in a medium is another important parameter. This is shown in figure \[fig:sound\] for the high temperature phase of pure gauge theory. The speed of sound also approaches the ideal gas value ($1/\sqrt{3}$), but is only at around 80% of it even at $2{T_{{\mbox{\scriptsize c}}}}$. Hadronic Properties {#sec:hadrons} =================== Chiral Sector {#sec:chiral} ------------- The quark condensate, the quantity from which other approaches obtain much of the effect of temperature on the hadrons, is the logical place to start. It is known to change drastically at the deconfinement transition in quenched QCD. The chiral condensate is shown in figure \[fig:ccvsmq\](a) as a function of bare quark mass at $T=0.92{T_{{\mbox{\scriptsize c}}}}$. It may be calculated directly from the trace of the fermion matrix ($\langle{\bar{\psi}}\psi\rangle_{\rm SE}$) and by using the pion and sigma propagators, which corrects for the linear dependence on the quark mass [@KPS87]. One can see quite clearly that both methods extrapolate to the same value at zero quark mass. The same holds true for lower temperatures. The temperature dependence of the chiral condensate is shown in figure \[fig:chcond\](b), where the ratio of the finite temperature chiral condensate to that at $T=0$ is plotted. These values have been extrapolated to zero quark mass, which is the reason why the error bars are so large. If one does not extrapolate, the ratio remains the same, but with errors a factor of ten smaller. For comparison results for full QCD, including dynamical fermions, have been included. Since none of these results are at the physical quark mass, the effect of temperature is probably underestimated. It is clear that the chiral condensate does not change significantly until one is extremely close to the critical temperature. The pion mass displays the behaviour expected of a Goldstone particle, $m_{\pi}^{2} = A_{\pi}{m_{{\mbox{\scriptsize q}}}}$, at all temperatures. There is also no sign of a dependence on temperature in the pion mass, or in the slope $A_{\pi}$. Another quantity of considerable interest for understanding the temperature dependence of the chiral sector of QCD is the pion decay constant. It can be determined directly from the relevant matrix element on the lattice [@KPS87]. At zero temperature $f_\pi$ is related to the chiral condensate and the pion mass through the Gell-Mann–Oakes–Renner relation, which is expected to be valid as long as chiral symmetry remains spontaneously broken. $$m_\pi^2 f_\pi^2\;=\;{m_{{\mbox{\scriptsize q}}}}\langle \bar{\psi}\psi \rangle_{({m_{{\mbox{\scriptsize q}}}}=0)} ~~. \label{eq:gmor}$$ The temperature independence of the amplitude, $A_\pi$, and the chiral condensate up to $T=0.92{T_{{\mbox{\scriptsize c}}}}$ indicates that the pion decay constant will also be temperature independent. A direct calculation shows that it is, and that the GMOR relation holds, for temperatures up to at least $T=0.92{T_{{\mbox{\scriptsize c}}}}$. Nucleon and meson masses {#sec:nucrho} ------------------------ One obtains hadron screening masses by fitting the exponential decrease of the correlator $C_{{\mbox{\scriptsize H}}}(z)$ to, for example, an hyperbolic cosine. One obtains either local masses, if only two points are fitted to, or an estimate of the lowest state if one fits to the long distance part of the correlator. The lowest mass which can be extracted from the fit, $E_{{\mbox{\scriptsize H}}}$, is related to the screening mass for fermionic states via $$E_{{\mbox{\scriptsize H}}}^{2} = m_{{\mbox{\scriptsize H}}}^{2} + k \sin^{2}\left( \pi/N_{\tau} \right) , \label{eq:mn}$$ due to the contribution from the non-vanishing Matsubara energy, $p_0 = \pi T$. Note that for bosons $E_{{\mbox{\scriptsize H}}}^{2} = m_{{\mbox{\scriptsize H}}}^{2}$. The local rho screening masses and the value obtained from a fit to the full propagator at $T=0.92{T_{{\mbox{\scriptsize c}}}}$ and at zero temperature are shown in figure \[fig:mrholoc\](a). It is clear that the rho screening mass does not show any significant dependence on temperature. The nucleon has been examined using wall sources at a quark mass of 0.01 in lattice units. The local screening masses for $T=0.92{T_{{\mbox{\scriptsize c}}}}$, and the screening mass obtained from a full fit, are shown in figure \[fig:mnloc\](b) along with the result obtained at zero temperature. The increase in temperature clearly has a dramatic effect on the nucleon. However, most of this can be understood purely in terms of the fact that the lowest momentum of the nucleon is not zero, but $\pi T$, as discussed above. The screening mass at $T=0.92{T_{{\mbox{\scriptsize c}}}}$, extracted using eqn. \[eq:mn\] with the assumption that $k=1$, indicates that the mass rises slightly: $m_{{\mbox{\scriptsize n}}}(0.92{T_{{\mbox{\scriptsize c}}}}) = (1.1\pm 0.03)m_{{\mbox{\scriptsize n}}}(T=0)$. We note, however, that there is also the possibility of a modification of the energy dispersion relation at finite temperature, which may lead to a deviation of $k$ from unity [@LES90]. The error given above is from the statistical error alone, as we cannot determine the systematic error which may be introduced by our assuming that eq. \[eq:mn\] with $k = 1$ is applicable. Finally, one would like to know the properties of correlators with hadronic quantum numbers in the deconfined phase. These can be studied using correlators in the spatial direction, as above. One finds that the nucleon then approches thrice the lowest Matsubara frequency, and all mesons other than the pion approach twice the lowest Mastubara frequency (see [@MTC90] and refernces therein). This is what one expects if the quarks become deconfined. The pion, though, acquires a screening mass half as large as expected for a free meson. As it is the particle associated with chiral symmetry, one may expect it to have a special role even above the phase transition [@HAK85]. In order to understand whether the pion is indeed deconfined as well, with a different interaction between the two quarks, the meson correlators were examined using different techniques [@SPAT94]. In one of these, temporal correlators generated with wall sources were examined. Temporal correlators show the true mass, and not the screening mass. Wall sources are used in order to project out only the contribution from the lowest state. The results for the local masses are shown in Figure \[fig:tempwall\], along with results for the effective quark mass. One sees here that both the rho and the pion masses tend towards twice the effective quark mass at high temperatures. This, combined with other investigations on the lattice, indicate that the hadrons (including the pions) become deconfined above ${T_{{\mbox{\scriptsize c}}}}$, with rather large channel dependent residual interactions. One does, in all cases, see the factor of two difference between the pion and rho masses or screening masses. There is clearly a difference between a quark–anti-quark pair carrying pion quantum numbers and one carrying rho quantum numbers. Explanations for the relative lightness of the pion have been put forward based on spin–spin interactions [@KOCH92] and residual gluon condensates [@IHAT94]. Electric and Magnetic Screening Masses {#sec:gluon} ====================================== The chromo electric screening mass can be obtained by examining, amongst others, the correlation between Polyakov loops or the pole of the gluon correlator. At high temperature in perturbation theory at next to leading order it has been shown that the two approaches yield the same result [@REB94; @BRN94]. The chromo magnetic mass cannot be determined in perturbation theory, due to infra-red divergences. Preliminary results from calculations in progress [@BIELWIP], in pure gauge theory, indicate that the magnetic and electric screening masses have the same order of magnitude above the phase transition, and for temperatures as high as $2{T_{{\mbox{\scriptsize c}}}}$. This result is consistent with the results obtained in another study in SU(2) pure gauge theory [@RANK]. This can be compared with the results from the potential, and the spatial string tension [@ED94] which indicate that the coupling remains large well above the phase transition, with $g(T=2{T_{{\mbox{\scriptsize c}}}})\approx 2$. Conclusions {#sec:conc} =========== The chiral condensate does not show any significant temperature dependence up to $T= 0.92{T_{{\mbox{\scriptsize c}}}}$. In view of this the observation that there is no sign of a temperature dependence in the pion decay constant, pion screening mass, sigma screening mass or the rho screening mass may not come as a surprise. There is some, albeit inconclusive, indication of a temperature dependence for the screening mass of the nucleon. The details of the temperature dependence may change in the case of QCD with two flavours, where the transition is expected to be second order rather than first order as in the the pure $SU(3)$ gauge theory, and with quark masses close to the physical quark mass. However, these differences will probably not show up for temperatures less than $0.9{T_{{\mbox{\scriptsize c}}}}$. The equation of state for pure gauge theory is now reasonably well understood. One finds clear evidence for a departure from the ideal gas at below $2{T_{{\mbox{\scriptsize c}}}}$, at which point the energy density reaches only 83% of the Stefan-Boltzmann limit. Thus it is clear that there are strong interactions, or equivalently a large contribution from non-perturbative effects, above the phase transition. This is supported by measurements of the spatial string tension, which indicate a coupling $g\approx 2$, results for the chromo electric and magnetic screening masses, as well as the effective quark mass in the Landau gauge. All states with hadronic quantum numbers appear to be deconfined above the phase transition. However, the coupling between the quark and anti-quark depends on the channel considered, leading to pion / sigma masses of around half the rho mass. Acknowledgements {#acknowledgements .unnumbered} ================ The results discussed here have been supported in part by the Stabsabteilung Internationale Beziehungen,Kernforschungszentrum Karlsruhe, a NATO research grant, contract number CRG 940451 and a DFG grant, DFG Pe-340/1, DFG Pe-340/6-1, and the HLRZ in Jülich. I would like to thank the organisers of the conference for the invitation and the support received. I would also like to thank my colleagues, J. Engels, S. Gupta, F. Karsch, E. Laermann, C. Legeland, M. 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[^1]: Summary of invited talks presented at the Hirschegg Workshop XXIII ‘Dynamical Properties of Hadrons in Nuclear Matter’, Jan. 16–21, Hirschegg, Austria and ‘Chiral Dynamics in Hadrons and Nuclei’, Feb. 6–10, Seoul, Korea

--- abstract: 'One of the most essential aspects of cuprate superconductors is a large pseudogap coexisting with a superconducting gap, then some anomalous properties can be understood in terms of the formation of the pseudogap. Within the kinetic energy driven superconducting mechanism, the effect of the pseudogap on the infrared response of cuprate superconductors in the superconducting-state is studied. By considering the interplay between the superconducting gap and pseudogap, the electron current-current correlation function is evaluated based on the linear response approach and it then is employed to calculate finite-frequency conductivity. It is shown that in the underdoped and optimally doped regimes, the transfer of the part of the low-energy spectral weight of the conductivity spectrum to the higher energy region to form a midinfrared band is intrinsically associated with the presence of the pseudogap.' author: - Ling Qin - Jihong Qin - Shiping Feng title: Effect of the pseudogap on the infrared response in cuprate superconductors --- The parent compounds of cuprate superconductors are identified as Mott insulators [@Kastner98], in which the lack of conduction arises from anomalously strong electron-electron repulsion. Superconductivity then is obtained by adding charge carriers to insulating parent compounds with the superconducting (SC) transition temperature $T_{\rm c}$ takes a domelike shape with the underdoped and overdoped regimes on each side of the optimal doping, where $T_{\rm c}$ reaches its maximum [@Tallon95]. However, a number of experimental probes [@Deutscher05; @Devereaux07; @Hufner08; @Puchkov96; @Timusk99; @Basov05] show that below a characteristic temperature $T^{*}$, which can be well above $T_{\rm c}$ in the underdoped and optimally doped regimes, the physical response of cuprate superconductors can be interpreted in terms of the formation of a normal-state pseudogap by which it means a suppression of the spectral weight of the low-energy excitation spectrum. In particular, this normal-state pseudogap crossover temperature $T^{*}$ decreases with increasing doping in the underdoped regime and since $T_{\rm c}$ rises with doping, then $T^{*}$ seems to merge with $T_{\rm c}$ in the overdoped regime, eventually disappearing together with superconductivity at the end of the SC dome [@Deutscher05; @Devereaux07; @Hufner08; @Puchkov96; @Timusk99; @Basov05]. After intensive investigations over more than two decades, it has become clear that the normal-state pseudogap is thought to be key to understanding the mechanism of superconductivity in cuprate superconductors. The infrared measurements of the reflectance are an ideal way to study the low-energy excitations of cuprate superconductors [@Puchkov96; @Timusk99; @Basov05; @Puchkov96a; @Basov96; @Lee05; @Hwang07; @Mirzaei13]. In particular, it has been shown in terms of the Kramers-Kronig analysis of the reflectance that the SC-state conductivity is rather universal within the whole cuprate superconductors [@Puchkov96; @Timusk99; @Basov05; @Puchkov96a; @Basov96; @Lee05; @Hwang07; @Mirzaei13], where a key feature is the two-component conductivity: a non-Drude-like narrow band centered around energy $\omega\sim 0$ followed by a broadband centered in the midinfrared region in the underdoped and optimally doped regimes. In particular, this two-component conductivity extends to the normal-state pseudogap boundary in the phase diagram at $T^{*}$ [@Lee05; @Hwang07; @Mirzaei13]. However, this anomalous structure of the SC-state conductivity spectrum can not be described by a simple Bardeen-Cooper-Schrieffer (BCS) formalism [@Schrieffer83]. There is mounting evidence that the unusual feature of the SC-state conductivity spectrum in the underdoped and optimally doped regimes is dominated by the normal-state pseudogap [@Puchkov96; @Timusk99; @Basov05; @Puchkov96a; @Basov96; @Lee05; @Hwang07; @Mirzaei13]. In our recent study [@Qin14], the doping and temperature dependence of the optical conductivity of cuprate superconductors in the normal-state has been discussed by considering the effect of the normal-state pseudogap on the optical response, and then the main feature of the optical measurements on cuprate superconductors in the normal-state is qualitatively reproduced. As a complement of our previous analysis of the optical conductivity in the normal-state, we in this paper discuss the conductivity of cuprate superconductors in the SC-state based on the kinetic energy driven SC mechanism [@Feng0306] by considering the interplay between the SC gap and normal-state pseudogap [@Feng12]. In particular, we show that the part of the low-energy spectral weight of the conductivity spectrum in the underdoped and optimally doped regimes is transferred to the higher energy region to form the unusual midinfrared band, however, the onset of the region to which the spectral weight is transferred is always close to the normal-state pseudogap. The single common element of cuprate superconductors is two-dimensional CuO$_{2}$ planes [@Kastner98]. As originally emphasized by Anderson [@Anderson87], the essential physics of the doped CuO$_{2}$ plane is properly captured by the $t$-$J$ model acting on the space with no doubly occupied sites. This $t$-$J$ model consists of two parts, the kinetic energy part includes the nearest-neighbor (NN) hopping term $t$ and next NN hopping term $t'$, while the magnetic energy part is described by a Heisenberg term with the NN spin-spin antiferromagnetic (AF) exchange $J$. The high complexity in the $t$-$J$ model comes mainly from the local constraint of no double electron occupancy, i.e., $\sum_{\sigma}C^{\dagger}_{l\sigma}C_{l\sigma}\leq 1$, which can be treated properly within the charge-spin separation (CSS) fermion-spin theory [@Feng0494], where the constrained electron operators $C_{l\uparrow}$ and $C_{l\downarrow}$ are decoupled as $C_{l\uparrow}=h^{\dagger}_{l\uparrow}S^{-}_{l}$ and $C_{l\downarrow}=h^{\dagger}_{l\downarrow}S^{+}_{l}$, respectively, with the spinful fermion operator $h_{l\sigma}=e^{-i\Phi_{l\sigma}}h_{l}$ that keeps track of the charge degree of freedom together with some effects of spin configuration rearrangements due to the presence of the doped hole itself (charge carrier), while the spin operator $S_{l}$ represents the spin degree of freedom, then the local constraint of no double electron occupancy is satisfied in analytical calculations. In this CSS fermion-spin representation, the magnetic energy term in the $t$-$J$ model is only to form an adequate spin configuration, while the kinetic energy is transferred as the interaction between charge carriers and spins, and therefore dominates the essential physics in cuprate superconductors. Based on the $t$-$J$ model in the CSS fermion-spin representation, we [@Feng0306] have developed a kinetic energy driven SC mechanism. In particular, the interplay between the normal-state pseudogap state and superconductivity has been discussed [@Feng12] within the framework of the kinetic energy driven SC mechanism, where the interaction between charge carriers and spins directly from the kinetic energy in the $t$-$J$ model by exchanging spin excitations induces the SC-state in the particle-particle channel and the pseudogap state in the particle-hole channel, then there is a coexistence of the SC gap and pseudogap in the whole SC dome. This pseudogap antagonizes superconductivity, and then $T_{\rm c}$ is suppressed to low temperatures. Following our previous discussions [@Feng12], the full charge carrier diagonal and off-diagonal Green’s functions of the $t$-$J$ model in the SC-state can be evaluated as, \[hole-Green-function\] $$\begin{aligned} g({\bf k},\omega)&=&{1\over \omega-\xi_{\bf k}-\Sigma^{({\rm h})}_{1}({\bf k},\omega)-\bar{\Delta}^{2}_{\rm h}({\bf k})/[\omega+\xi_{\bf k} +\Sigma^{({\rm h})}_{1}({\bf k}, -\omega)]},\label{hole-diagonal-Green's-function}\\ \Gamma^{\dagger}({\bf k},\omega)&=&-{\bar{\Delta}_{\rm h}({\bf k})\over [\omega-\xi_{\bf k}-\Sigma^{({\rm h})}_{1}({\bf k},\omega)][\omega+\xi_{\bf k} +\Sigma^{({\rm h})}_{1}({\bf k},-\omega)]-\bar{\Delta}^{2}_{\rm h}({\bf k})},\label{hole-off-diagonal-Green's-function}\end{aligned}$$ with the self-energy in the particle-hole channel, $$\begin{aligned} \label{self-energy} \Sigma^{({\rm h})}_{1}({\bf k},\omega)\approx {[2\bar{\Delta}_{\rm pg}({\bf k})]^{2}\over \omega+M_{\bf k}},\end{aligned}$$ where we use the same notations as in Ref. [@Feng12], and in particular, the mean-field charge carrier spectra $\xi_{\bf k}$, $M_{\bf k}$, the effective d-wave charge carrier pair gap $\bar{\Delta}_{\rm h}({\bf k})$, and the effective normal-state pseudogap $\bar{\Delta}_{\rm pg}({\bf k})$ have been given explicitly in Ref. [@Feng12]. This kinetic energy driven SC mechanism [@Feng0306; @Feng12] also indicates that the strong electron correlation favors superconductivity, since the main ingredient is identified into a charge carrier pairing mechanism not involving the phonon, the external degree of freedom, but the internal spin degree of freedom of electron. Through the standard linear response theory [@Mahan81], the finite-frequency conductivity of cuprate superconductors can be expressed as, $\sigma(\omega)=-{\rm Im} \Pi(\omega) /\omega$, where $\Pi(\tau-\tau')=-\langle T_{\tau}{\bf j}(\tau)\cdot {\bf j}(\tau')\rangle$ is the electron current-current correlation function. This electron current operator ${\bf j}(\tau)$ is obtained in terms of the electron polarization operator ${\bf P}$, which is a summation over all the particles and their positions [@Mahan81], and can be given as ${\bf P}= -e\sum\limits_{l\sigma}{\bf R}_{l}C^{\dagger}_{l\sigma}C_{l\sigma}$. The external magnetic field can be coupled to the electrons, which are now represented by $C_{l\uparrow}=h^{\dagger}_{l\uparrow}S^{-}_{l}$ and $C_{l\downarrow}= h^{\dagger}_{l\downarrow}S^{+}_{l}$ in the CSS fermion-spin representation [@Feng0494]. In this CSS fermion-spin representation, the electron current operator can be decoupled as the charge carrier and spin parts, respectively. In particular, we [@Qin14] have shown that there is no direct contribution to the electron current operator from the electron spin part, and then the main contribution for the electron current operator comes from the charge carriers (then the electron charge), however, the strong interplay between the charge carriers and spins has been considered through the spin’s order parameters entering in the charge carrier part of the contribution to the current-current correlation. In this case, the finite-frequency conductivity of cuprate superconductors in the SC-state can be evaluated as, $$\begin{aligned} \label{conductivity} \sigma(\omega) &=& \left ({Ze\over \hbar}\right )^{2}{1\over N}\sum_{\bf k}\gamma^{2}_{{\rm s}{\bf k}}\int^{\infty}_{-\infty}{{\rm d}\omega'\over 2\pi}[A_{g}({\bf k},\omega+\omega') A_{g}({\bf k},\omega')+A_{\Gamma}({\bf k},\omega+\omega')A_{\Gamma}({\bf k},\omega')]{n_{\rm F}(\omega')-n_{\rm F}(\omega+\omega')\over\omega},\end{aligned}$$ width $n_{\rm F}(\omega)$ is the fermion distribution function, while the spectral functions $A_{g}({\bf k},\omega)$ and $A_{\Gamma}({\bf k},\omega)$ are obtained in terms of the charge carrier diagonal and off-diagonal Green’s functions in Eq. (\[hole-Green-function\]) as $A_{g}({\bf k},\omega)=-2{\rm Im}g({\bf k},\omega)$ and $A_{\Gamma}({\bf k},\omega) =-2{\rm Im}\Gamma^{\dagger}({\bf k},\omega)$, respectively. ![The conductivity as a function of energy in $\delta=0.09$ with $T=0.002J$ for $t/J=2.5$ and $t'/t=0.3$. Inset: the corresponding experimental data of the underdoped Bi$_{2}$Sr$_{2}$CaCu$_{2}$O$_{8+\delta}$ taken from Ref. . \[fig1\]](fig1.eps) In Fig. \[fig1\], we plot the results of the SC-state conductivity (\[conductivity\]) as a function of energy in the underdoping $\delta=0.09$ for parameters $t/J=2.5$ and $t'/t=0.3$ with temperature $T=0.002J$, where the charge $e$ has been set as the unit. For comparison, the corresponding experimental result [@Hwang07] of the underdoped Bi$_{2}$Sr$_{2}$CaCu$_{2}$O$_{8+\delta}$ is also plotted in Fig. \[fig1\] (inset). Obviously, the experimental data of cuprate superconductors obtained from the conductivity measurements are qualitatively reproduced [@Puchkov96; @Timusk99; @Basov05; @Puchkov96a; @Basov96; @Lee05; @Hwang07; @Mirzaei13]. At the low temperatures, the SC-state conductivity $\sigma(\omega)$ extends over a broad energy range, and consists of a low-energy component and a higher energy band separated by a gap at $\omega\sim 0.2t$. This higher energy band, corresponding to the “midinfrared band”, shows a broad peak at $\omega\sim 0.38t$, and persists in the normal-state pseudogap phase [@Qin14]. The low-energy component forms a sharp peak at $\omega\sim 0$, however, it deviates strongly from the Drude behavior, and has a power-law decay form as $\sigma(\omega)\rightarrow 1/\omega$ (the non-Drude formula). In comparison with our previous results of the optical conductivity in the normal-state [@Qin14], we find that the spectral weight of the low-energy component has been further suppressed by the SC gap, however, there is no depletion of the spectral weight of the higher energy midinfrared band, which is consistent with the experimental observation on cuprate superconductors [@Lee05]. Moreover, as in the case of the normal-state [@Qin14], the onset of the region to which the spectral weight is transferred is close to the normal-state pseudogap $\bar{\Delta}_{\rm pg}$, reflecting a fact that due to the presence of the normal-state pseudogap in cuprate superconductors, the part of the low-energy spectral weight of the SC-state conductivity spectrum in the underdoped regime is transferred to the higher energy region to form the unusual midinfrared band. In the other words, the appearance of the higher energy midinfrared band is closely related to the effect of the normal-state pseudogap on the infrared response in cuprate superconductors [@Lee05; @Hwang07; @Mirzaei13; @Yu08; @Hwang08]. ![The conductivity as a function of energy in $\delta=0.09$ (solid line), $\delta=0.15$ (dashed line), and $\delta=0.25$ (dotted line) with $T=0.002J$ for $t/J=2.5$ and $t'/t=0.3$.\[fig2\]](fig2.eps) The spectral weight is proportional to the area under the conductivity curve in Fig. \[fig1\]. However, the weight and position of the midinfrared band are strongly doping dependent. To see this point clearly, we have made a series of calculations for the SC-state conductivity (\[conductivity\]) in the whole doping range from the underdoped to heavily overdoped, and the results of $\sigma(\omega)$ as a function of energy in the underdoping $\delta=0.09$ (solid line), the optimal doping $\delta=0.15$ (dashed line), and the heavy overdoping $\delta=0.25$ (dotted line) for $t/J=2.5$ and $t'/t=0.3$ with $T=0.002J$ are plotted in Fig. \[fig2\]. It is shown clearly that as the charge carrier doping increases, the overall conductivity increases. Within the framework of the kinetic energy driven SC mechanism [@Feng0306], the magnitude of the normal-state pseudogap $\bar{\Delta}_{\rm pg}$ is much larger than that of the SC gap in the underdoped regime, then it smoothly decreases upon increasing doping, eventually disappearing together with superconductivity at the end of the SC dome [@Feng12]. In corresponding to this evolution of the normal-state pseudogap with doping, the positions of the gap and midinfrared peak gradually shift to the lower energies with increasing doping, i.e., there is a tendency that with increasing doping, the magnitude of the gap decreases, then the higher energy midinfrared band moves towards to the low-energy non-Drude band. This tendency is particularly obvious in the overdoped regime. In particular, in the heavily overdoped regime, the normal-state pseudogap is negligible [@Feng12], i.e., $\bar{\Delta}_{\rm pg}\approx 0$, then the full charge carrier diagonal and off-diagonal Green’s functions in Eq. (\[hole-Green-function\]) are reduced as a simple d-wave BCS formalism. It is thus similar to the conventional superconductors [@Schrieffer83] except the d-wave symmetry. In this case, the low-energy non-Drude peak incorporates with the midinfrared band, and then the low-energy Drude type behavior ($\sigma(\omega)\rightarrow 1/\omega^{2}$) of the conductivity recovers in the heavily overdoped regime, in qualitative agreement with the corresponding experimental data of cuprate superconductors [@Puchkov96; @Timusk99; @Basov05; @Puchkov96a; @Basov96; @Lee05; @Hwang07; @Mirzaei13]. ![The conductivity as a function of energy in $\delta=0.09$ with $T=0.002J$ (solid line) and $T=0.146J$ (dashed line) for $t/J=2.5$ and $t'/t=0.3$.\[fig3\]](fig3.eps) Within the framework of the kinetic energy driven SC mechanism, the calculated [@Feng12] $T_{\rm c}\sim 0.06J$ and $T^{*}\sim 0.19J$ at doping $\delta=0.09$. The low-energy non-Drude component of the conductivity spectrum in Fig. \[fig1\] broadens as temperature increases from the temperature $T\ll T_{\rm c}$ to $T>T_{\rm c}$ and continues to grow even just above the normal-state pseudogap crossover temperature $T^{*}$. However, this broadness of the low-energy component of the conductivity spectrum is accompanied by a decrease of the weight of the higher energy midinfrared band, and then the low-energy Drude type behavior of the conductivity recovers at the temperature above $T^{*}$. To see this point clearly, in Fig. \[fig3\], we plot $\sigma(\omega)$ as a function of energy with $T=0.002J$ (solid line) in the SC-state and $T=0.146J$ (dashed line) in the normal-state pseudogap phase in doping $\delta=0.09$ for $t/J=2.5$ and $t'/t=0.3$. In particular, the results in Fig. \[fig3\] confirm again that in spite of the conductivity in the SC-state or the normal-state pseudogap phase [@Qin14], the onset of the region to which the spectral weight is transferred is always close to the normal-state pseudogap $\bar{\Delta}_{\rm pg}$ [@Lee05; @Hwang07; @Mirzaei13; @Yu08; @Hwang08], also in qualitative agreement with the corresponding experimental data of cuprate superconductors [@Puchkov96; @Timusk99; @Basov05; @Puchkov96a; @Basov96; @Lee05; @Hwang07; @Mirzaei13]. Although the separate normal-state pseudogap and SC gap are underlying the redistribution of the spectral weight in the SC-state conductivity spectrum, the essential physics of the higher energy midinfrared band in cuprate superconductors in the SC-state is the same as in the case of the normal-state, and can be attributed to the emergence of the normal-state pseudogap. This follows a fact that in Eq. (\[conductivity\]), there are two parts of the charge carrier quasiparticle contribution to the redistribution of the spectral weight in the SC-state conductivity: the contribution from the first term of the right-hand side in Eq. (\[conductivity\]) comes from the spectral function obtained in terms of the charge carrier diagonal Green’s function (\[hole-diagonal-Green’s-function\]), and therefore is closely associated with the normal-state pseudogap $\bar{\Delta}_{\rm pg}$ in the particle-hole channel, while the additional contribution from the second term of the right-hand side in Eq. (\[conductivity\]) originates from the spectral function obtained in terms of the charge carrier off-diagonal Green’s function (\[hole-off-diagonal-Green’s-function\]), and is closely related to the charge carrier pair gap $\bar{\Delta}_{\rm h}$ in the particle-particle channel. However, since $\bar{\Delta}_{\rm h}\ll\bar{\Delta}_{\rm pg}$ in the underdoped and optimally doped regimes [@Feng12], the SC gap only suppresses the spectral weight of the low-energy component, while the normal-state pseudogap related shift of the spectral weight from the low-energy to the higher energy midinfrared band in the SC-state conductivity spectrum becomes arrested. Since both the normal-state pseudogap state and superconductivity in cuprate superconductors are the result of the strong electron correlation within the framework of the kinetic energy driven SC mechanism [@Feng12], the transfer of the part of the low-energy spectral weight of the conductivity spectrum in the underdoped and optimally doped regimes to the higher energy region to form the unusual midinfrared band is a natural consequence of the strongly correlated nature in cuprate superconductors. In particular, this strong electron correlation induces a shift of the spectral weight from the low-energy to the higher energy midinfrared band in the conductivity spectrum has been confirmed by the early numerical simulations based on the $t$-$J$ model in the normal-state [@Stephan90; @Dagotto94] and the SC-state [@Haule07]. In an ordinary metal, the shape of the conductivity is normally well accounted for by the low-energy Drude formula ($\sigma(\omega)\rightarrow 1/\omega^{2}$) that describes the charge carrier contribution to the conductivity, then when the temperature $T<T_{\rm c}$, the spectral weight of the condensate in the SC-state comes from low energies [@Schrieffer83]. However, in cuprate superconductors, since the higher energy midinfrared band is taken from the low-energy band, and the onset of the region to which the spectral weight is transferred is close to the normal-state pseudogap, so that the large normal-state pseudogap in the underdoped and optimally doped regimes heavily reduces the fraction of the charge carriers that condense in the SC-state [@Homes04]. In conclusion, we have shown very clearly in this paper that if the interplay between the SC gap and normal-state pseudogap is taken into account in the framework of the kinetic energy driven SC mechanism, the SC-state conductivity of the $t$-$J$ model calculated based on the linear response approach per se can correctly reproduce the main feature found in the infrared response measurements on cuprate superconductor in the SC-state. 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--- abstract: 'We numerically study the effect of the quantum spill-out (QSO) on the plasmon mode indices of an ultra-thin metallic slab, using the Fourier modal method (FMM). To improve the convergence of the FMM results, a novel nonlinear coordinate transformation is suggested and employed. Furthermore, we present a perturbative approach for incorporating the effects of QSO on the plasmon mode indices, which agrees very well with the full numerical results. The perturbative approach also provides additional physical insight, and is used to derive analytical expressions for the mode indices using a simple model for the dielectric function. The analytical expressions reproduce the results obtained from the numerically-challenging spill-out problem with much less effort and may be used for understanding the effects of QSO on other plasmonic structures.' address: | Department of Materials and Production, Aalborg University, 9220 Aalborg [Ø]{}st, Denmark\ Center for Nanostructured Graphene (CNG), 9220 Aalborg [Ø]{}st, Denmark\ author: - 'Alireza Taghizadeh, and Thomas Garm Pedersen' bibliography: - 'Spillout.bib' title: 'Plasmons in ultra-thin gold slabs with quantum spill-out: Fourier modal method, perturbative approach, and analytical model' --- Introduction ============ Surface plasmon polaritons (SPPs) are transverse magnetic (TM) surface modes, which propagate at metal-dielectric interfaces [@Maier2007]. In the last two decades, there has been a tremendous interest in SPPs due to their ability to guide light in a volume much smaller than typical dielectric structures [@Takahara1997; @Zia2004]. They have found numerous applications in solar cells, surface-enhanced Raman spectroscopy, lasers, sensors, etc [@Barnes2003; @Schuller2010; @Gramotnev2010; @Zhang2012]. When a nm-thick metal slab is surrounded by dielectric media, the SPPs at the two metal-dielectric interfaces will strongly couple and form so-called short- and long-range SPPs [@Sarid1981]. In particular, the long-range SPPs have smaller propagation losses than the single interface SPPs [@Charbonneau2000], which make them attractive for several applications such as nonlinear optics [@Berini2009] and opto-electronic devices [@Leosson2006]. It is known that at the metal surface, the electron density has an exponential tail penetrating into the dielectric region due to the barrier tunneling [@Lang1970]. This quantum mechanical effect, which is typically referred to as quantum spill-out (QSO), has been shown to have significant impact on the optical properties of nano-scale metallic structures [@Zuloaga2010; @Esteban2012; @David2014; @Zhu2016; @Enok2018; @Enok2019]. For a metallic slab, the QSO effect softens the abrupt changes of the dielectric function at the interfaces and leads to a position-dependent dielectric function across the slab. Therefore, numerical tools are required for calculating the plasmon mode indices when the QSO is included. This is in contrast to a classical metallic slab with spatially constant dielectric response, where the short- and long-range plasmon modes are obtained analytically. For a nm-thick metal slab, solving the QSO problem is numerically challenging due to the two vastly different length scales: the slab thickness ($\sim$1 nm) and the mode exponential decay in the dielectric regions ($\sim$1000 nm), particularly for the long-range SPPs. Recently, we have examined the effect of QSO on plasmons propagating in ultra-thin gold films [@Enok2018; @Enok2019], using a transfer matrix method. In this approach, the plasmon mode indices are obtained by finding the poles of the transfer matrix, relating the fields to the left and right of the slab [@Enok2018]. In the transfer matrix method, a staircase approximation should be employed for modeling the dielectric function in the vicinity of the interface (slicing the region into a large number of segments). To eliminate the discretization errors, the segment width should be as thin as 10$^{-4}$ nm [@Enok2019], which makes the calculations cumbersome. Furthermore, the transfer matrix method requires reasonable initial guesses in order to find the poles efficiently, and one can only find a single plasmon at each run. In the present work, we propose an alternative numerical approach for finding the plasmon mode indices of a thin metallic slab with QSO, based on the Fourier modal method (FMM). In this approach, all plasmon mode indices are obtained simultaneously by solving an eigenvalue problem in reciprocal (Fourier) space. To obtain acceptable numerical results with the FMM, we have implemented the fast factorization rule for TM polarization [@Li1996] and a nonlinear coordinate transformation. Inspired by Ref. [@Hugonin2005], we suggest a coordinate transformation, which drastically improves the convergence performance of the FMM for long-range SPPs. Furthermore, since the QSO correction to the plasmon mode index is typically small, a perturbation technique is proposed, which reproduces the numerical results very accurately with much less effort. The perturbative approach requires only the classical solutions of the slab waveguide problem, which are obtained analytically. Moreover, using a simple model for the dielectric function, which mimics the dielectric function obtained from density-functional theory (DFT), analytical expressions are derived for the plasmon mode indices. We provide numerical results for thin gold slabs, and, by direct comparison, confirm the validity of the perturbative approach. This paper is organized as follows. In [Sec. \[sec:Theory\]]{}, we define the problem, and present its solution with or without QSO. In addition, we briefly review the FMM with a coordinate transformation and derive the perturbation expression for the plasmon mode index in this section. In [Sec. \[sec:Numerics\]]{}, the numerical results for thin gold slabs are provided, and the validity of our perturbative approach is demonstrated. Finally, a summary of the main findings is provided in [Sec. \[sec:Conclusion\]]{}. A set of appendices provide checks of the validity of the perturbative approach, present the derivation of analytical expressions and explain the details of the coordinate transformation. Theoretical framework \[sec:Theory\] ==================================== The plasmonic modes of a metallic slab waveguide, illustrated schematically in Fig. \[fig:Schematic\](a), are obtained by finding the solutions to Maxwell’s equations for TM polarization (p-polarization), written for the magnetic field ${\mathbf{H}}$. The propagation direction is taken to be the $z$-direction, whereas the finite slab thickness is along the $x$-direction. The magnetic field is of the form ${\mathbf{H}}=H_y(x)\exp(i\beta z) {\mathbf{e}}_y$ (${\mathbf{e}}_\alpha$ is the unit vector along the $\alpha$-direction with $\alpha=\{x,y,z\}$), where the plasmon mode index $\beta$ and its field profile $H_y(x)$ are obtained by solving the wave equation given by $$\label{eq:Maxwell} \epsilon_{xx}(x) \frac{{\mathrm{d}{}}}{{\mathrm{d}{x}}} \bigg[ \frac{1}{\epsilon_{zz}(x)} \frac{{\mathrm{d}{H_y(x)}}}{{\mathrm{d}{x}}} \bigg] + k_0^2 \epsilon_{xx}(x) H_y(x) = \beta^2 H_y(x) \, .$$ Here, $\epsilon_{zz}(x)$ and $\epsilon_{xx}(x)$ are the parallel and perpendicular parts of the position-dependent dielectric function of the waveguide, respectively, and $k_0=\omega/c$ is the vacuum wavevector. In the present work, we neglect the anisotropy of the dielectric function for simplicity, and assume that $\epsilon_{xx}(x)=\epsilon_{zz}(x)=\epsilon(x)$. The anisotropy of the dielectric function has a negligible effect on the plasmon mode indices of the slab [@Enok2019], but it can be added straightforwardly if required. For a general dielectric profile, this eigenvalue problem should be solved numerically. Note that the electric field can be readily obtained from the magnetic field as ${\mathbf{E}}=(E_x{\mathbf{e}}_x+E_z{\mathbf{e}}_z)\exp(i\beta z)$ with $E_z(x)=i[\omega\epsilon_0\epsilon(x)]^{-1}{\mathrm{d}{H_y}}/{\mathrm{d}{x}}$ and $E_x(x)=[\omega\epsilon_0\epsilon(x)]^{-1}\beta H_y(x)$. In addition, the $z$-component of the complex poynting vector is given by $S_z(x,z) \equiv2^{-1}({\mathbf{E}}\times{\mathbf{H}}^*)\cdot{\mathbf{e}}_z=2^{-1} E_x(x)H_y^*(x) \exp(-2\beta_iz) = [2\omega\epsilon_0\epsilon(x)]^{-1}\beta |H_y(x)|^2 \exp(-2\beta_iz)$, in which the propagation loss (related to the real part of $S_z$) originates from the imaginary part of the mode index, $\beta_i$. ![(a) Schematic of a slab waveguide with a thickness of $d$, with two mapped regions used for coordinate transformation $x=F(x')$, see the text. We choose the mapped regions to be far from the waveguide to avoid overlap with the regions affected by the QSO, [e.g. ]{}$d_1=d+20a$. (b) Real (solid) and imaginary (dashed) parts of the dielectric function $\epsilon(x)$ across the metal slab including the QSO ($d=2$ nm). The blue curve is a typical profile obtained from a DFT calculations (using a jellium model), whereas the red curve shows a fitted profile using Eq. (\[eq:Tanh\]) with $a=0.09$ nm. \[fig:Schematic\]](Schematic.pdf){width="80.00000%"} Classical solution \[sec:Classical\] ------------------------------------ In the classical case, where the QSO is neglected, the interfaces between different material regions are modeled by sharp boundaries. The dielectric function of the slab waveguide then reads $\epsilon^{(0)}(x) \equiv \epsilon_1+(\epsilon_m-\epsilon_1) \theta(d/2-|x|)+(\epsilon_2-\epsilon_1) \theta(x-d/2)$. Here, $\theta$ denotes the Heaviside step function, $d$ is the slab thickness, and $\epsilon_m$, $\epsilon_1$ and $\epsilon_2$ are the bulk metal, superstrate and substrate dielectric constants, respectively. Note that we label all classical variables by superscript “0” or subscript “0”. Also, we use the experimental values for the dielectric constant of bulk gold reported in Ref. [@Johnson1972] for $\epsilon_m$. For $\epsilon^{(0)}(x)$, Eq. (\[eq:Maxwell\]) can be solved analytically by finding the solutions in three regions, [i.e. ]{}slab, substrate and superstrate, and matching these solutions across the two interfaces, [i.e. ]{}requiring $H_y(x)$ and $E_z(x)\propto\epsilon^{-1}(x){\mathrm{d}{H_y}}/{\mathrm{d}{x}}$ to be continuous functions. Therefore, the bound modes are given by $$H_y^{(0)}(x) = \left\{ \begin{array}{lr} A_1 e^{\kappa_1 x} \qquad \qquad & x<-d/2 \\ B_1 \cos(q x) + B_2 \sin(q x) \qquad \qquad & |x|<d/2 \\ A_2 e^{-\kappa_2 x} \qquad \qquad & x>d/2 \, , \\ \end{array} \right.$$ where $q \equiv \sqrt{\epsilon_mk_0^2-\beta_0^2}$ and $\kappa_i \equiv \sqrt{\beta_0^2-\epsilon_i k_0^2}$. Matching the partial solutions across the interfaces, the dispersion relation is obtained as $q\epsilon_m(\epsilon_1 \kappa_1+\epsilon_2\kappa_2)=(\epsilon_1\epsilon_2q^2-\epsilon_m^2\kappa_1\kappa_2) \tan(qd)$. In a symmetric geometry, where $\epsilon_1=\epsilon_2$, two modes with even ($A_1=A_2$ and $B_2=0$) and odd ($A_1=-A_2$ and $B_1=0$) symmetries are formed. The dispersion relations for the even and odd modes read $\epsilon_m\kappa=\epsilon_1q\tan(qd/2)$ and $\epsilon_m \kappa=-\epsilon_1q\cot(qd/2)$, respectively. Even (odd) modes are typically referred to as long-range (short-range) SPPs, since the imaginary parts of their mode indices are smaller (larger) than the single metal-dielectric interface SPPs. Hereafter, we only consider symmetric geometries for simplicity, but our results can be readily generalized to include non-symmetric structures. Furthermore, without loss of generality, we set $A_1=\exp(\kappa_rd/2)$ with $\kappa_r$ the real part of $\kappa \equiv \kappa_1=\kappa_2$, which leads to $|H_y^{(0)}(\pm d/2)|=1$. Note that although determining $\beta_0$ from the dispersion relation involves finding roots numerically, we refer to the classical solution as the analytical one, since this numerical task is straightforward. Numerical solution: Fourier modal method ---------------------------------------- The FMM is a variant of the modal methods, in which the structure is discretized into layers and the eigenmodes of each layer are connected by mode matching at the interfaces [@Lavrinenko2018]. In the FMM, which is also referred to as rigorous coupled wave analysis (RCWA), the eigenmodes are expanded using a Fourier basis. The FMM has been used widely for investigating gratings [@Moharam1995; @Granet1996; @Taghizadeh2015; @Yoon2015; @Learkthanakhachon2016; @Orta2016], since the Fourier basis is particularly suitable for periodic structures. Nonetheless, by introducing perfectly matched layers (PMLs) at the boundaries of simulation domains, the FMM has been successfully employed for studying various non-periodic structures such as dielectric waveguides [@Hugonin2005; @Silberstein2001; @Ctyroky2010], photonic crystals waveguides [@Pissoort2004; @Lecamp2007], finite gratings [@Pisarenco2010], microdisks [@Armaroli2008; @Bigourdan2014], and vertical cavities [@Taghizadeh2016; @Taghizadeh2017]. For the waveguide problems, the PML is typically implemented as a nonlinear coordinate transformation [@Hugonin2005], in which the infinite space $x$ is mapped to a finite space $x'$ with a width of $\Lambda$, [i.e. ]{}$x=F(x')$. In the transformed space, the Fourier basis is then used for expanding the field $H_y(x')=\sum_n h_n \exp(i k_n x')$, dielectric function $\epsilon(x')=\sum_n \varepsilon_n \exp(i k_n x')$, its inverse $\epsilon^{-1}(x')=\sum_n \eta_n \exp(i k_n x')$ and the derivative of the coordinate transformation $f(x')={\mathrm{d}{F}}/{\mathrm{d}{x'}}=\sum_n f_n \exp(i k_n x')$. Here, $k_n \equiv 2\pi n/\Lambda$ and the summations over $n$ run from $-N$ to $N$ with a total number of $N_t=2N+1$ basis functions. Using the Fourier expansions, Eq. (\[eq:Maxwell\]) will be transformed to a matrix eigenvalue problem given by [@Taghizadeh2016] $$\label{eq:FMM} {\mathbf{A}}^{-1}(k_0^2{\mathbf{I}}-{\mathbf{F}}{\mathbf{K}} {\mathbf{E}}^{-1} {\mathbf{F}} {\mathbf{K}}) {\mathbf{h}} = \beta^2 {\mathbf{h}}$$ where ${\mathbf{K}}=[[k_n\delta_{nm}]]$ is a diagonal matrix with entries $k_n$ ($\delta_{nm}$ is the Kronecker delta), and ${\mathbf{h}}=[h_n]$ is a vector formed by $h_n$. In addition, ${\mathbf{A}}=[[\eta_{n-m}]]$, ${\mathbf{E}}=[[\varepsilon_{n-m}]]$ and ${\mathbf{F}}=[[f_{n-m}]]$ are the Toeplitz matrices associated with $\eta_n$, $\varepsilon_n$ and $f_n$, respectively. Note that we have implemented the FMM with special care for TM polarization, in order to obtain rapidly converging results as discussed in Refs. [@Li1996; @Granet1996]. This is due to the fact that the Fourier coefficients of products of two functions having jump discontinuities converge much faster if the inverse rule is applied for obtaining the Fourier coefficients [@Li1996]. In addition, we have modified the coordinate transformation suggested in Ref. [@Hugonin2005] as explained in Appendix \[sec:AppC\], in order to improve the convergence of our numerical results for the long-range SPP. Perturbative solution --------------------- Perturbation methods have been highly successful in quantum mechanics for investigating the effects of small modifications on various quantum systems [@Griffiths2005]. They provide not only effective numerical tools for computations but also means of gaining useful physical insight into the systems. Surprisingly, their use has been restricted in optics, presumably due to the vectorial nature of Maxwell’s equations and the boundary conditions at the interfaces [@Johnson2002]. Nonetheless, here, we formulate a perturbative approach for obtaining the effect of QSO on the plasmon mode indices. This is done by noticing that the magnetic field $H_y$ and transverse component of the electric field $E_z$ of the plasmon modes are only modified slightly, when the QSO is included, compared to the case without QSO, [i.e. ]{}$H_y(x) \approx H_y^{(0)}(x)$ and $\epsilon^{-1}(x){\mathrm{d}{H_y}}/{\mathrm{d}{x}} \approx [\epsilon^{(0)}(x)]^{-1}{\mathrm{d}{H_y^{(0)}}}/{\mathrm{d}{x}}$ as shown in Appendix \[sec:AppA\]. Subsequently, we consider the integral of the complex poynting vector across the waveguide (the complex energy flux) at $z=0$, [i.e. ]{}$P=\beta(2\omega\epsilon_0)^{-1} \int {\mathrm{d}{x}} |H_y(x)|^2[\epsilon(x)]^{-1}$. One can readily show that $\beta P \approx \beta_0 P^{(0)}$, since $$P \propto \beta\int {\mathrm{d}{x}} \frac{|H_y|^2}{\epsilon(x)} = \int {\mathrm{d}{x}} \frac{H_y^*}{\beta} \Bigg\{ \frac{{\mathrm{d}{}}}{{\mathrm{d}{x}}}\bigg[ \frac{1}{\epsilon(x)} \frac{{\mathrm{d}{H_y}}}{{\mathrm{d}{x}}} \bigg] + k_0^2 H_y(x) \Bigg\} \approx \frac{\beta_0^2}{\beta} \int {\mathrm{d}{x}} \frac{|H_y^{(0)}|^2}{\epsilon^{(0)}(x)} \propto \dfrac{\beta_0}{\beta} P^{(0)} \, .$$ Here, the pre-factor $(2\omega\epsilon_0)^{-1}$ is omitted in the first and last relations, and Eq. (\[eq:Maxwell\]) has been used in the second and third relations. Now, by replacing $H_y$ by $H_y^{(0)}$ in the original expression for $P$, the following simple approximation is derived for the plasmon mode index: $$\label{eq:Perturbation} \beta^2 \approx \beta_0^2 \dfrac{\int {\mathrm{d}{x}} |H_y^{(0)}(x)|^2[\epsilon^{(0)}(x)]^{-1} }{\int {\mathrm{d}{x}} |H_y^{(0)}(x)|^2\epsilon^{-1}(x) } = \beta_0^2 \dfrac{t_0}{t_0+\Delta t} \, .$$ Here, $t_0$ and $\Delta t$ are defined as $t_0 \equiv \int {\mathrm{d}{x}} |H_y^{(0)}(x)|^2[\epsilon^{(0)}(x)]^{-1}$ and $\Delta t \equiv \int {\mathrm{d}{x}} |H_y^{(0)}(x)|^2[1/\epsilon(x)-1/\epsilon^{(0)}(x)]$, respectively. As shown in Appendix \[sec:AppB\], $t_0$ can be calculated analytically, whereas $\Delta t$ should be computed numerically for a general dielectric profile $\epsilon(x)$. Note that the integrand in $\Delta t$ is only non-negligible in the vicinity of the interfaces, which makes its computation straightforward. It is expected that $|\Delta t| \ll |t_0|$ when the QSO is a small perturbation, and hence $\beta \approx \beta_0 (1-\Delta t/2t_0)$. Equation (\[eq:Perturbation\]) can be readily used to compute the effect of QSO on the mode indices with a impressive accuracy as shown in the following section. Numerical results \[sec:Numerics\] ================================== In this section, we study the performance of the FMM and perturbative approach for computing the plasmon mode indices of gold slabs. The dielectric function of the gold slab with QSO is calculated using DFT based on a jellium model as explained in details in Ref. [@Enok2019]. The jellium model provides the position-dependent electron density $n(x)$ due to the free electrons, which converges to the bulk electron density $n_0$ near the slab midpoint. The effect of bound electrons in the lower $d$-bands is added to retain the experimental value of the bulk dielectric constant of gold in the central part of the slab. Hence, the contribution of bound electrons in the dielectric constant is assumed to be $\epsilon_b=\epsilon_m-\epsilon_D$, where $\epsilon_D=1-\omega_{p0}^2/(\omega^2+i\omega\gamma)$ is the Drude response of the bulk gold ($\omega_{p0}=\sqrt{n_0e^2/(m\epsilon_0)} \approx 9.025$ eV and $\gamma=65.8$ meV [@Novotny2012]). Therefore, the position-dependent dielectric function $\epsilon(x)$ of the gold slab reads $$\epsilon(x) = 1-\frac{\omega_p^2(x)}{\omega(\omega+i \gamma)} + \epsilon_b \theta(d/2-|x|) + (\epsilon_1-1) \theta(|x|-d/2) \, ,$$ where $\omega_p(x)=\sqrt{n(x)e^2/(m\epsilon_0)}$. A typical calculated dielectric function for a 2 nm gold slab surrounded by glass (on both sides) is shown in Fig. \[fig:Schematic\](b). Note that the abrupt jumps ($\Delta\epsilon=\epsilon_b-\epsilon_1+1$) at two interfaces originate mainly from the bound electron term. ![Real (a,c) and imaginary (b,d) parts of the long-range (a,b) and short-range (c,d) SPP mode indices $\beta$ with QSO, normalized to the corresponding values without QSO, $\beta_0$, versus the spill-out parameter $a$. The calculations are performed for three waveguide widths: $d=1$ nm (blue), $d=2$ nm (red), $d=5$ nm (green). The results are obtained for a gold slab ($\epsilon_m=-21.995+ 1.363i$[@Johnson1972] and $\epsilon_b=8.778 + 0.056i$) surrounded by glass ($\epsilon_1=2.25$) at the wavelength of 775 nm, modeled by the dielectric function in Eq.(\[eq:Tanh\]). The solid-lines are obtained from the FMM by solving Eq. (\[eq:FMM\]) numerically, while the dashed-lines are the perturbation results using Eq. (\[eq:Perturbation\]). The stars show the analytical results from Eqs. (\[eq:Analytic0\]) and (\[eq:Analytic\]). For small values of $a$, the effect of QSO vanishes and hence, all curves converge toward one as expected, whereas for large values of $a$ the perturbative results deviate from the numerical solutions. \[fig:Pert\_Limit\]](Pert_Limit.pdf){width="80.00000%"} Using the DFT model, we have computed the dielectric function for various slab thicknesses, ranging from 0.3 nm to 200 nm. For various slab thicknesses, the dielectric function shows a similar pattern: a smooth variation of the response in $\sim0.5$ nm-width regions near the boundaries followed by multiple Friedel oscillations inside the metal close to the interfaces [@Enok2019]. After careful consideration, we conclude that the effect of QSO can be captured fairly accurately by using a toy dielectric function, $\epsilon_t(x)$, which can emulate the smooth behavior of the Drude part of the dielectric function at the interface (but not Friedel oscillations). This toy model can be used for quantitative understanding of the effect of QSO on plasmon mode indices and allows us investigate the performance of the perturbative approach systematically. Hence, we choose $\epsilon_t(x)$ to be $$\begin{aligned} \label{eq:Tanh} \epsilon_t(x) &= 1+\frac{\epsilon_D-1}{4}(1-e^{-2d/a}) \bigg[ \tanh\Big(\frac{x+d/2}{a}\Big)+1 \bigg] \bigg[ \tanh\Big(\frac{-x+d/2}{a}\Big)+1 \bigg] \nonumber \\ & \hspace{5cm} + \epsilon_b \theta(d/2-|x|) + (\epsilon_1-1) \theta(|x|-d/2) \, ,\end{aligned}$$ where $a$ is a spill-out parameter with dimensions of length, that captures the degree of QSO. Also, $1-\exp(-2d/a)$ is introduced in order to conserve the average of the dielectric function, [i.e. ]{}$\int \epsilon_t(x){\mathrm{d}{x}}=\int \epsilon^{(0)}(x){\mathrm{d}{x}}$. In Fig. \[fig:Schematic\](b), we compare $\epsilon_t(x)$ with the DFT calculated dielectric function for a gold slab surrounded by glass, where the spill-out parameter is fitted to be $a=0.09$ nm. The QSO affects the dielectric function mainly close to the interfaces in a region that depends on $a$ (approximately $d/2-3a<|x|<d/2+3a$). The toy dielectric function will converge toward the classical case by letting $a\rightarrow 0$, and the strength of the perturbation can be tuned by varying $a$. Furthermore, we can compute $\Delta t$ analytically as discussed in Appendix \[sec:AppB\], and derive a closed form expression for $\beta$. This expression provides a simple measure of the impact of QSO on plasmon mode indices. ![Real (red) and imaginary (blue) parts of the short-range (a) and long-range (b) SPPs with QSO, $\beta$, normalized to the corresponding values without QSO, $\beta_0$, versus the waveguide width $d$. Three different methods are used: numerical method (FMM), perturbation approach with the DFT-based dielectric profile (Pert.) and analytical solution (Ana.) The gold slab is surrounded by glass and the results are obtained at 775 nm wavelength, see caption of Fig. \[fig:Pert\_Limit\]. \[fig:Changed\]](Changed.pdf){width="80.00000%"} We compute both real and imaginary parts of the short/long-range mode indices, $\beta$, for the toy dielectric function and normalize them to those of the corresponding classical value, $\beta_0$, and plot them versus the spill-out parameter $a$ for three different slab widths $d=\{1,2,5\}$ nm in Fig. \[fig:Pert\_Limit\]. Three different approaches are employed for each set of calculations: (i) FMM by solving Eq. (\[eq:FMM\]) numerically, (ii) the perturbative approach by numerical evaluation of Eq. (\[eq:Perturbation\]), and (iii) the analytical approach by employing Eqs. (\[eq:Analytic0\]) and (\[eq:Analytic\]). Since the effect of QSO vanishes if $a \rightarrow 0$, all graphs approach one for small values of $a$. The results show that the perturbative solutions, either using the analytical expressions or numerical integrations, are remarkably accurate for $a<0.1$ nm in all cases. Since the value of $a$ is smaller than 0.1 nm for gold, one can expect that the perturbative approach performs very well for computing the effect of QSO also if the more realistic dielectric profile based on DFT is used as demonstrated below. Furthermore, the analytical model works as well as the numerical perturbation approach, which makes it very useful for predicting the dependence of mode indices on various parameters of the problem. For instance, the linear dependence of the perturbative results on $a$ follows directly from Eq. (\[eq:Analytic\]). Now we proceed to the more realistic dielectric function obtained from DFT, and compare its results with the toy model. We keep the frequency constant and vary the slab thickness from 0.3 to 200 nm. Both real and imaginary parts of plasmon mode indices for the short- and long-range modes are normalized to the corresponding values of the classical case and plotted in Figs. \[fig:Changed\](a) and \[fig:Changed\](b), respectively. Note that the numerical FMM results are identical to the results reported in Ref. [@Enok2019], obtained from the transfer matrix method. The real parts of the mode indices for both plasmon modes are approximately unaffected by the QSO as the ratios for the real parts are nearly one. In contrast, the imaginary parts change drastically for thin slabs due to the QSO, while it saturates to $\sim1.2$ for thick slabs. In particular, the effect of QSO is evident for very wide slabs, which proves its significant role even for commonly fabricated metallic slabs. Comparing the different methods for computing the graphs in Fig. \[fig:Changed\], we find that the numerical and perturbative results are in excellent agreement. Hence, the perturbation method can be used safely for computing plasmon mode indices, without using advanced numerical techniques. Furthermore, the analytical expressions (based on the toy dielectric function) capture basically all features of the graphs accurately. For instance, we derive closed form expressions for the saturation values at large $d$ in Eqs. (\[eq:InfinitedA\]) and (\[eq:InfinitedA\]), which agree fairly well with the full numerical results. Also, Eq. (\[eq:InfinitedA\]) shows that the QSO modifies the imaginary part of the plasmon modes by an amount that is linearly proportional to the electron density spill-out. Hence, the analytical expressions can be used to estimate quantitatively the effect of QSO on the plasmon mode indices and avoid solving the challenging problem of DFT in combination with Maxwell’s equations. Conclusion \[sec:Conclusion\] ============================= In summary, we have computed the plasmon mode indices of a metallic slab using the FMM, with inclusion of QSO. A perturbative approach is then proposed, which reproduces the numerical results very accurately with much less effort. Using a simple model for dielectric function with QSO, analytical expressions have been derived for the mode indices of the short- and long-range SPPs. The analytical results agree very well with the numerical solutions, and may be used to obtain more physical insight into the impact of QSO on other plasmonic structures. Appendices {#appendices .unnumbered} ========== Perturbation validity \[sec:AppA\] ================================== To demonstrate that the modifications of $H_y$ and $E_z$ due to QSO are negligible, we compare the field profiles of the short-range SPP obtained with or without QSO in Figs. \[fig:Conv\_Pert\](a) and \[fig:Conv\_Pert\](b) for the 2 nm-thick gold slab of Fig. \[fig:Schematic\](b). The results show that the parallel components of the fields, $H_y(x)$ and $E_z(x)$, change only slightly when QSO is included. Note that the perpendicular component of the electric field, $E_x(x)$, is drastically modified as shown in Fig. \[fig:Conv\_Pert\](c), which is the origin of the increased imaginary part of the plasmon mode index. The drastic change is due to the fact that the real part of the dielectric function is crossing zero at a point outside the slab, which leads to a large value for $E_x(x) \propto H_y(x)/\epsilon(x)$ as seen in Fig. \[fig:Conv\_Pert\](c). ![Normalized amplitude of (a) $H_y(x)$, (b) $E_z(x)$, (c) $E_x(x)$ profiles obtained for the short-range SPP of a 2 nm-thick gold slab, when QSO is included (red) or neglected (blue). QSO mainly modifies $E_x$ in a narrow region outside the slab and close to the interfaces where the field amplitude becomes very large. (d) The relative error of mode indices for the short-range (blue) and long-range (red) SPPs versus the number of Fourier functions, $N_t$, for two coordinate transformations: TR1 the transformation suggested in Ref. [@Hugonin2005], and TR2 the transformation in Eq. (\[eq:Transformation\]). The relative error is defined as $|(\beta-\beta_0)/\beta_0|$, where $\beta$ is obtained by using the FMM for the classical dielectric function, $\epsilon^{(0)}(x)$. \[fig:Conv\_Pert\]](Conv_Pert.pdf){width="80.00000%"} Analytical expressions \[sec:AppB\] =================================== The perturbation method allows us to derive analytical expressions, when the QSO is included, for simple geometries and some forms of dielectric function such as Eq. (\[eq:Tanh\]). For symmetric geometries, an analytical equation can be derived for $t_0$ as $$\label{eq:Analytic0} t_0 = \dfrac{1}{\epsilon_1 \kappa_r} + \dfrac{1}{\epsilon_m} \dfrac{\sinh(q_i d)/q_i \pm \sin(q_r d)/q_r}{\cosh(q_i d) \pm \cos(q_r d)}$$ where $r$ and $i$ subscripts denote the real and imaginary parts of the corresponding parameter, respectively, and $+$ ($-$) is used for the long-range (short-range) SPP. Regarding $\Delta t$, we can derive an analytical expression employing $\epsilon_t(x)$ in Eq. (\[eq:Tanh\]) by noticing that the integrand in $\Delta t$ is only non-zero for a narrow region in the vicinity of $|x|=d/2$. Moreover, the field $H_y^{(0)}$ does not vary considerably in this narrow region, and hence $$\begin{aligned} \label{eq:Analytic} \Delta t \approx & 2|H_y^{(0)}(d/2)|^2 \int_{d/2}^{\infty} {\mathrm{d}{x}} \bigg\{ \dfrac{2}{\epsilon_p+2\epsilon_1+\epsilon_p\tanh[(d/2-x)/a]}-\dfrac{1}{\epsilon_1} \bigg\} \nonumber \\ +&2|H_y^{(0)}(d/2)|^2 \int_{-\infty}^{d/2} {\mathrm{d}{x}} \bigg\{ \dfrac{2}{\epsilon_m+\epsilon_b+1+\epsilon_p\tanh[(d/2-x)/a]}-\dfrac{1}{\epsilon_m} \bigg\} \nonumber \\ =& a\epsilon_p \bigg[ \dfrac{ \log(2-\epsilon_p/\epsilon_m)}{\epsilon_m(\epsilon_m-\epsilon_p)} - \dfrac{\log(2+\epsilon_p/\epsilon_1)}{\epsilon_1(\epsilon_1+\epsilon_p)} \bigg] \, , $$ Here, $\epsilon_p\equiv\epsilon_D-1$ and it is assumed that $d/a \gg 1$, which simplifies the expression for $\epsilon_t$ by letting $\exp(-2d/a) \rightarrow 0$ and $\tanh[(d/2+x)/a] \rightarrow 1$. Equations (\[eq:Analytic0\]) and (\[eq:Analytic\]) can be employed to derive useful expressions for the saturation levels in Fig. \[fig:Changed\] by taking the limit $d\rightarrow\infty$ as $$\begin{aligned} \label{eq:InfinitedA} &\dfrac{\Re(\beta)}{\Re(\beta_0)} \bigg|_{d\rightarrow\infty} \approx 1+\dfrac{ak_0\epsilon_1}{2\sqrt{-\epsilon_{mr}}}\Re[\log(2+\epsilon_p/\epsilon_1)] \, , \\ \label{eq:InfinitedB} &\dfrac{\Im(\beta)}{\Im(\beta_0)} \bigg|_{d\rightarrow\infty} \approx \dfrac{\Re(\beta)}{\Re(\beta_0)}\bigg|_{d\rightarrow\infty} + \bigg(\dfrac{2\epsilon_{mr}^2}{\epsilon_1\epsilon_{mi}}\bigg) \dfrac{ak_0\epsilon_1}{2\sqrt{-\epsilon_{mr}}}\Im[\log(2+\epsilon_p/\epsilon_1)] \, . \end{aligned}$$ Here, $\epsilon_{mr}\equiv\Re(\epsilon_m)$, $\epsilon_{mi}\equiv\Im(\epsilon_m)$, and we assume that $|\epsilon_m|,|\epsilon_D| \gg \epsilon_1, \epsilon_{mi}$. Equations (\[eq:InfinitedA\]) and (\[eq:InfinitedB\]) show that the saturation levels will increase linearly with $a$, and confirm the non-negligible modification of the imaginary ratio due to the appearance of the $2\epsilon_{mr}^2/(\epsilon_1\epsilon_{mi})$ factor, which can be large. These expressions predict the saturation values to be 1.0004 and 1.171 for the real and imaginary parts, respectively, which are in reasonable agreement with the results in Fig. \[fig:Changed\]. Coordinate transformation for Fourier modal method \[sec:AppC\] =============================================================== A nonlinear coordinate transformation is suggested in Ref. [@Hugonin2005] (referred to as TR1 hereafter), which typically works well for dielectric waveguide structures. However, TR1 is not sufficiently accurate for the plasmonic waveguides in this work, in particular for long-range SPPs, since the plasmon modes decay very slowly in the dielectric regions. Therefore, we modify TR1 by introducing a second mapped region as shown in Fig. \[fig:Schematic\](a), which leads to improvement of the convergence rate. The transformation of the infinite space $x$ to the finite space $x'$ with the length $\Lambda$ is defined as $$\label{eq:Transformation} x = F(x') = \left\{ \begin{array}{ll} x' & |x'| \le d_1/2 \\ \dfrac{x'}{|x'|} \dfrac{d_1}{2} e^{2|x'|/d_1-1} & d_1/2 < |x'| \le d_2/2 \\ \dfrac{x'}{|x'|} e^{2d_2/d_1-1} \Bigg[ \dfrac{d_1}{2}+\dfrac{\Lambda-d_2}{2} \tan\left(\pi\dfrac{|x'|-d_2/2}{\Lambda-d_2}\right) \Bigg] \qquad & d_2/2 < |x'| \le \Lambda/2 \, . \\ \end{array} \right.$$ This transformation and its derivative are continuous functions, and the Fourier coefficients of its derivative $f(x')\equiv{\mathrm{d}{F}}/{\mathrm{d}{x'}}$ \[which are required in Eq. (\[eq:FMM\])\] can be obtained analytically. We refer to this transformation as TR2. There are three length parameters in the TR2 $\{d_1, d_2, \Lambda \}$, which are chosen to achieve the required convergence. In particular, we choose $d_1$ and $d_2$ such that $\exp(d_2/d_1-1)$ is larger than the mode extension inside the dielectric regions. Note that by setting $d_1=d_2$, TR1 can be recovered. To study the performance of the coordinate transformation, we compare the mode indices obtained for the short- and long-range SPPs of a gold slab (without QSO) for the two transformations (TR1 and TR2) as a function of $N_t$ in Fig. \[fig:Conv\_Pert\](d). The relative error is defined with respect to the exact solution of the metallic slab, see [Sec. \[sec:Classical\]]{}. The results show that using TR2 improves the convergence rate for the long-range SPP. Note that TR2 does not change the convergence performance of the short-range SPP in this example, but it can improve the convergence rate for thicker slabs. Funding {#funding .unnumbered} ======= Danish National Research Foundation (Project No. DNRF103). Disclosures {#disclosures .unnumbered} =========== The authors declare no conflicts of interest.

--- abstract: 'Input-to-state stability (ISS) of switched systems is studied where the individual subsystems are connected in a serial cascade configuration, and the states are allowed to reset at switching times. An ISS Lyapunov function is associated to each of the two blocks connected in cascade, and these functions are used as building blocks for constructing ISS Lyapunov function for the interconnected system. The derivative of individual Lyapunov functions may be bounded by nonlinear decay functions, and the growth in the value of Lyapunov function at switching times may also be a nonlinear function of the value of other Lyapunov functions. The stability of overall hybrid system is analyzed by constructing a newly constructed ISS-Lyapunov function and deriving lower bounds on the average dwell-time. The particular case of linear subsystems and quadratic Lyapunov functions is also studied. The tools are also used for studying the observer-based feedback stabilization of a nonlinear switched system with event-based sampling of the output and control inputs. We design dynamic sampling algorithms based on the proposed Lyapunov functions and analyze the stability of the resulting closed-loop system.' address: - 'Department of Aerospace Engineering, University of California at Irvine, USA' - 'Laboratory for Analysis and Architecture of Systems, Toulouse, CNRS, France' author: - GuangXue Zhang - Aneel Tanwani bibliography: - 'localEventTrig.bib' title: 'ISS Lyapunov Functions for Cascade Switched Systems and Sampled-Data Control' --- = \[draw, rectangle, minimum height=3em, minimum width=6em\] = \[draw, circle, node distance=1cm\] = \[coordinate\] = \[coordinate\] = \[pin edge=[to-,thin,black]{}\] Switched systems ,input-to-state stability ,cascade connection ,multiple Lyapunov functions ,average dwell-time ,output feedback ,event-based control Introduction ============ Switched systems, or in general, hybrid dynamical systems provide a framework for modeling a large class of physical phenomenon and engineering systems which combine discrete and continuous dynamics. Due to their wide utility, such systems have been extensively studied in the control community over the past two decades; see the books by [@Libe03] and [@GoebSanf12] for comprehensive overview. This article addresses a robust stability problem for systems with switching vector fields and jump maps. In our setup, each subsystem has a two-stage serial cascade structure where the output of first block acts as an input to the second block, and the disturbances we consider are an exogenous input to the first block, see Figure \[fig:cascade\]. By proposing a novel construction for multiple Lyapunov functions for such configurations, we analyze the stability of the interconnected switched system by deriving lower bounds on average dwell-time between switching times. It is seen that such a configuration arises in the context of output feedback stabilization of switched systems with known switching signal where the inputs and outputs are time-sampled. The theoretical tools developed in the earlier part of this paper are then used to design sampling algorithms and analyze stability of the resulting sampled-data system. A preliminary version of the sampled-data problem, studied in the later part of this paper, has appeared in [@ZhanTanw18]. Stability of switched systems has been a topic of interest in control community for past two decades now. Depending on the class of switching signals, or the assumptions imposed on the continuous dynamics, different approaches have been adopted in the literature to study the convergence of the state trajectories. The book [@Libe03] provides an overview on this subject. For our purposes, the approach based on slow switching is more relevant. In this direction, the pioneering contribution comes from [@HespMorse99] where the lower bounds on average dwell-time are computed using multiple Lyapunov functions. Another result on slow switching, but with state-dependent average dwell-time, has appeared in [@DePeDeSa03]. The second fundamental tool, that we build on, relates to the robustness with respect to external disturbances, formalized by the notion of [*input-to-state stability*]{} (ISS) introduced in [@Sontag89]. Using these classical works as foundation, our article provides a certain construction of the ISS-Lyapunov functions for the switched systems in cascade configuration and develops lower bounds on the dwell-time that guarantee ISS property for the switched system. One of the first results on input-to-state stability of switched systems appears in [@VuChat07], where the authors associate an ISS-Lyapunov function to each subsystem with linear decay rate, and assume that the Lyapunov function for each subsystem can be linearly dominated by the Lyapunov function of another subsystem at switching times. Other relevant papers studying ISS property for systems with jump maps using dwell-time conditions are [@HespLibe08], [@DashMiron13]. Using the notion of ISS, tools such as small gain theorems [@JiangTeel94], or cascade principles [@SontTeel95] are developed to study different applications. The small gain theorems have in particular found utility in the stability analysis of interconnected systems [@Ito06], [@DashRuff10]. For hybrid systems, in general, the ISS results using Lyapunov functions appear in [@CaiTeel09], [@CaiTeel13]. Their utility is seen in analyzing stability of two interconnected hybrid systems in [@Sanf14] and [@LibeNesi14], where the later in particular focuses on small-gain theorems and their application in control over networks. The stability of interconnected switched systems based on small gain theorems also appears in [@YangLibe15]. The more recent article then generalizes the results on interconnections [@YangLibe16] while allowing for potentially unstable subsystems and jump dynamics. (1,0) node \[rectangle, rounded corners, draw, minimum height = 0.75cm, text centered\] (sys1) [$\begin{aligned}\dot e &= f_{o,\sigma}(e,d) \\ e^+ &= g_o(e,d)\end{aligned}$]{}; (4.5,0) node \[rectangle, rounded corners, draw, minimum height = 0.75cm, text centered\] (sys2) [$\begin{aligned}\dot x &= f_{c,\sigma}(x,e) \\ x^+ &= g_c(x,e)\end{aligned}$]{}; (sys1.east) – node\[anchor=south\][$e$]{} (sys2.west); (-1,0) node\[anchor=south\][$d$]{}– (sys1.west); (sys2.east) – node\[anchor=south\][$x$]{} (6.5,0); The first part of this article is also built on analyzing the stability of interconnected subsystems with continuous and discrete dynamics. However, we are interested in studying systems where the interconnection is described by a cascade configuration, see Fig. \[fig:cascade\]. Using the Lyapunov function construction in [@TanwTeel15], we construct the Lyapunov functions for this cascade interconnection. We then use the framework of hybrid systems to describe the overall system with jump maps, and switching signal with average dwell-time constraints. A novel Lyapunov function is constructed for this hybrid system and the corresponding analysis provides the lower bounds on average dwell-time which yield global asymptotic stability of a certain set. In our approach, we do not require the decay rates of the individual Lyapunov functions to be linear, and the upper bounds on the value of individual Lyapunov function at jump instants may be nonlinear functions of other Lyapunov functions. When studying linear systems as an example, even though we associate quadratic Lyapunov functions to individual subsystems, the Lyapunov function for the overall hybrid system involves a product of the exponential function with a non-quadratic function, which to the best of our knowledge is a novel observation. We then use these constructions to study the feedback stabilization of switched nonlinear systems when the output measurements and control inputs are time-sampled. Using an observer-based controller, where the estimation error dynamics and the closed-loop system (with static control) are ISS with respect to measurement errors, we rewrite the whole system in cascade configuration where the estimation error drives the state of the controlled plant. The measurement errors are introduced because we only send time-sampled outputs to the controller, and the controller only sends sampled control inputs to the plant. In both cases, the sampled measurements are subjected to a zero-order hold, and thus remain constant until the next sampling instant. Our goal is to derive algorithms to compute sampling algorithms which result in global asymptotic stability of the closed-loop system under the average dwell-time assumptions derived earlier. The event-based sampling strategy that we use is inspired from [@TanwTeel15], where the dynamic filters are introduced. The next sampling instant occurs when the difference between the current value of the output (resp. input) and its last sample is comparatively larger than the value of the dynamic filter’s state. Beyond the realm of periodic sampling, stabilization of dynamical systems has been studied subject to various sampling techniques, see for example [@HeemJoha12] and [@TanwChat18] for recent surveys. Among these methods, event-based control has received attention as an effective means of sampling and various variants of this problem have been studied over the past few years. However, this technique has not yet been studied for switched systems which is the second main contribution of this article. The remainder of the article is organized as follows: In Section \[sec:prelim\], we provide an overview of basic stability notions and existing results which will be used in this article. The system class of interest is introduced in Section \[sec:cascade\], where we develop the main theoretical results on construction of Lyapunov functions, and developing bounds for average dwell-time. These results are applied in Section \[sec:sampling\] to study dynamic feedback stabilization of switched nonlinear systems with sampled-data, and the second main results concerning the design of sampling algorithms and stability analysis of closed-loop system is developed in this section. As an illustration, we provide simulation results for an academic example in Section \[sec:example\], followed by some concluding remarks in Section \[sec:conc\]. Preliminaries {#sec:prelim} ============= In this section, we recall some basic notions of interest which relate to the stability of a hybrid system. For our purposes, it is useful to consider hybrid systems with inputs studied in [@CaiTeel13], which are described by following inclusions: $$\begin{aligned} \label{eq:genHybSys} \begin{cases} \dot \xi \in {\mathcal{F}}(\xi, d), \quad (\xi,d) \in {\mathcal{C}}, \\ \xi^+ \in {\mathcal{G}}(\xi,d), \quad (\xi,d) \in {\mathcal{D}}, \end{cases} \end{aligned}$$ where $\xi \in {\mathcal{X}}$ is the state and $d \in {\mathbb{R}}^{{\overline}d}$ is the external disturbance. The [*flow set*]{} ${\mathcal{C}}\subseteq {\mathcal{X}}\times {\mathbb{R}}^{{\overline}d}$, and the [*jump set*]{} ${\mathcal{D}}\subseteq {\mathcal{X}}\times {\mathbb{R}}^{{\overline}d}$ are assumed to be relatively closed in ${\mathcal{X}}\times {\mathbb{R}}^{{\overline}d}$. The set-valued map ${\mathcal{F}}: {\mathcal{C}}\rightrightarrows {\mathcal{X}}$ describes the continuous dynamics when $\xi$ belongs to the flow set ${\mathcal{C}}$. The mapping ${\mathcal{G}}:{\mathcal{D}}\rightrightarrows {\mathcal{X}}$ defines the state reset map, when $\xi$ belongs to the jump set ${\mathcal{D}}$. The solution of the hybrid system  is defined on a [*hybrid time domain*]{}. A set $E \subseteq {\mathbb{R}}_{\ge 0} \times {\mathbb{Z}}_{\ge 0}$ is called a compact hybrid time domain if $E = \cup_{j = 0}^J ([t_j,t_{j+1}],j)$ for some finite sequence $0 = t_0 \le t_1 \le \dots \le t_{J+1}$. We say that $E$ is a [*hybrid time domain*]{} if for each $(T,J)$ in $E$, the set $E \cap [0,T] \times \{0,1,\dots,J\}$ is a compact hybrid time domain. A function defined on a hybrid time domain is called a [*hybrid signal*]{}. In , the disturbance $d$ is a hybrid signal, so that $d(\cdot,j)$ is locally essentially bounded. A [*hybrid arc*]{} $\xi$ is a hybrid signal for which $\xi(\cdot,j)$ is locally absolutely continuous for each $j \in {\mathbb{Z}}_{\ge 0}$, and we use the notation $\xi(t,j)$ to denote the value of $\xi$ at time $t$ after $j$ jumps. For a given initial condition $\xi(0,0) \in {\mathcal{C}}\cup {\mathcal{D}}$, and a hybrid signal $d$, the solution $\xi$ to is a hybrid arc if $\operatorname{dom}\xi \subseteq \operatorname{dom}d$, and it holds that i) for every $j\in {\mathbb{Z}}_{\ge 0}$ and almost every $t \in {\mathbb{R}}_{\ge 0}$ such that $(t,j) \in \operatorname{dom}\xi$, we have $(\xi(t, j), d(t, j)) \in {\mathcal{C}}$ and $\dot \xi(t, j) \in {\mathcal{F}}(\xi(t, j), d(t, j))$; ii) for $(t, j) \in \operatorname{dom}\xi$ such that $(t,j + 1) \in \operatorname{dom}\xi$, we have $(\xi(t,j),d(t,j)) \in {\mathcal{D}}$ and $\xi(t,j + 1) \in {\mathcal{G}}(\xi(t, j), d(t, j))$. It is assumed that the quadruple $({\mathcal{F}},{\mathcal{G}},{\mathcal{C}},{\mathcal{D}})$ satisfies the basic assumptions listed in [@GoebSanf12 Assumption 6.5], so that system  has a well-defined solution $\xi$ in the space of hybrid arcs, for each hybrid signal $d$, but not necessarily unique. To study the stability notions of interest for hybrid arcs, we need some notation. A function $\alpha:{\mathbb{R}}_{\ge 0} \to {\mathbb{R}}_{\ge 0}$ is said to be of class ${\mathcal{K}}$ if it is continuous, strictly increasing, and $\chi(0) = 0$. If $\alpha$ is also unbounded, then it is said to be of class ${\mathcal{K}}_\infty$. A function $\beta: {\mathbb{R}}_{\ge 0} \times {\mathbb{R}}_{\ge 0} \to {\mathbb{R}}_{\ge 0}$ is said to be of class ${\mathcal{K}}{\mathcal{L}}$ if $\beta (\cdot,t)$ is of class ${\mathcal{K}}$ for each $t \in {\mathbb{R}}_{\ge 0}$ and $\beta(r,t) \to 0$ as $t \to \infty$ for each fixed $r \in {\mathbb{R}}_{\ge 0}$; see [@Khalil02 Chapter 4] for their use in formulation of common stability notions. In addition, we require class ${\mathcal{K}}{\mathcal{L}}{\mathcal{L}}$ function: A function $\beta:{\mathbb{R}}_{\ge 0} \times {\mathbb{R}}_{\ge 0} \times {\mathbb{R}}_{\ge 0} \to {\mathbb{R}}_{\ge 0}$ is a class ${\mathcal{K}}{\mathcal{L}}{\mathcal{L}}$ function if $\beta(\cdot,\cdot,j)$ is a class ${\mathcal{K}}{\mathcal{L}}$ function for each $j \ge 0$ and $\beta(\cdot,s,\cdot)$ is a class ${\mathcal{K}}{\mathcal{L}}$ function for each $s \ge 0$. For a compact set ${\mathcal{A}}\subset {\mathcal{X}}$ and $\xi \in {\mathcal{X}}$, $\vert \xi \vert_{{\mathcal{A}}} := \inf_{z\in{\mathcal{A}}} \vert \xi - z \vert$, where $\vert \cdot \vert$ denotes the usual Euclidean norm. Following [@CaiTeel09], for a hybrid signal $d$, we use the notation $\|d \|_{(t, j)}$ to denote the maximum between $\displaystyle \operatorname*{ess-sup}_{(\hat t, \hat j+1) \not \in \operatorname{dom}d, \hat t+ \hat j \le t + j} \vert d(\hat t, \hat j) \vert$ and $\displaystyle\sup_{(\hat t, \hat j+1) \in \operatorname{dom}d, \hat t+ \hat j\le t+ j} \vert d(\hat t, \hat j) \vert $. For positive-valued functions $\alpha, \chi$ over ${\mathbb{R}}_{\ge 0}$, we also use the Landau-notation to write $\alpha(s) = o(\chi(s))$ as $s \to c$ if $\lim_{s\to c} \frac{\alpha(s)}{\chi(s)} = 0$; similarly, we write $\alpha(s) = O(\chi(s))$ as $s \to c$ if $\lim_{s\to c} \frac{\alpha(s)}{\chi(s)} \le M$ for some $M > 0$. Input-to-State Stability ------------------------ We recall basic definitions and Lyapunov characterizations of ISS for hybrid systems from [@CaiTeel09]. System is [*ISS*]{} with respect to a compact set ${\mathcal{A}}\subset {\mathcal{X}}$ if there exist functions $\gamma \in {\mathcal{K}}$, and $\beta \in {\mathcal{K}}{\mathcal{L}}{\mathcal{L}}$ such that $$\label{eq:defISS} |\xi(t,j)|_{{\mathcal{A}}} \leq \beta(|\xi(0,0)|_{{\mathcal{A}}}, t,j) + \gamma \left( \|d\|_{(t,j)} \right),$$ for every $(t,j) \in \operatorname{dom}\xi$. \[ISS Lyapunov function\] A smooth function $V: {\mathcal{X}}\to {{\mathbb{R}}_{\geq 0}}$ is an ISS-Lyapunov function of the hybrid system w.r.t. a compact set ${\mathcal{A}}\subset {\mathcal{X}}$ if the following hold: - there exist ${\underline}\alpha, {\overline}\alpha \in {\mathcal{K}_\infty}$ such that $$\label{boundV} {\underline}\alpha(\vert \xi \vert_{{\mathcal{A}}}) \leq V(\xi) \leq {\overline}\alpha(\vert \xi \vert_{{\mathcal{A}}}), \quad \forall \, \xi \in {\mathcal{C}}\cup {\mathcal{D}}\cup {\mathcal{G}}({\mathcal{D}}),$$ - there exist $\widehat\alpha \in {\mathcal{K}}_\infty$ and $\widehat\gamma \in {\mathcal{K}}$ such that $$\label{eq:defIssVFlow} \vert\xi\vert_{{\mathcal{A}}} \geq \widehat\gamma(|d|) \Rightarrow \langle \nabla V(\xi), f \rangle \leq -\widehat\alpha(\vert\xi\vert_{{\mathcal{A}}}),$$ holds for every $(\xi,d) \in {\mathcal{C}}$ and $f \in {\mathcal{F}}(\xi,d)$. - the functions $\widehat \alpha$ and $\widehat \gamma$ also satisfy $$\vert\xi\vert_{{\mathcal{A}}} \geq \widehat\gamma(|d|) \Rightarrow V(g) - V(\xi) \leq -\widehat\alpha(\vert\xi\vert_{{\mathcal{A}}}),$$ for every $(\xi,d) \in {\mathcal{D}}$ and for every $g \in {\mathcal{G}}(\xi, d)$. The following result provides an alternate charaterization of ISS for system  by combining results given in [@CaiTeel09 Proposition 1] and [@LibeShim15 Theorem 1]. \[hybridISSprop\] Consider system  and a compact set ${\mathcal{A}}\subset {\mathcal{X}}$. A differentiable function $V: {\mathcal{X}}\to {{\mathbb{R}}_{\geq 0}}$ satisfying with ${\underline}\alpha, {\overline}\alpha \in {\mathcal{K}_\infty}$ is an ISS-Lyapunov function w.r.t. ${\mathcal{A}}$ if and only if there exist $\alpha_{\mathcal{C}}\in {\color{blue}{\mathcal{K}}_\infty}$, $\gamma_{\mathcal{C}}\in {\mathcal{K}}$ and a continuous nonnegative function $\varrho:{\mathbb{R}}_{\ge 0} \to {\mathbb{R}}_{\ge 0}$ such that $$\label{prop1Vflow} \langle \nabla V(\xi), f \rangle \leq -\alpha_{\mathcal{C}}(\vert\xi\vert_{{\mathcal{A}}})+ \varrho(\vert \xi \vert_{{\mathcal{A}}}) \, \gamma_{\mathcal{C}}(|d|)$$ holds for each $(\xi,d) \in {\mathcal{C}}$ and $f \in {\mathcal{F}}(\xi,d)$, there exist $\alpha_{\mathcal{D}}\in {\mathcal{K}_\infty}$, $\gamma_{\mathcal{D}}\in {\mathcal{K}}$ that satisfy $$\label{prop1Vjump} V(g) - V(\xi) \leq - \alpha_{\mathcal{D}}(\vert\xi\vert_{{\mathcal{A}}})+ \gamma_{\mathcal{D}}(|d|)$$ for each $(\xi,d) \in {\mathcal{D}}$ and $g \in {\mathcal{G}}(\xi, d)$, the functions $\alpha_{\mathcal{C}},\varrho$ satisfy the [*asymptotic ratio*]{} condition $$\label{eq:ratioCond} \limsup_{r \to \infty} \, \frac{\varrho(r)}{\alpha_{\mathcal{C}}(r)} = 0.$$ The inequality is different from the expression given in [@CaiTeel09 [Proposition 2.6]{}]. It can be shown that also implies . This implication is proved in a constructive manner, that is, the pair $(\widehat \alpha, \widehat \gamma)$ is constructed from the triplet $(\alpha_{\mathcal{C}},\varrho,\gamma_{\mathcal{C}})$, in [@LibeShim15 Theorem 1] using the condition , which appears in Remark 1 of that paper. Cascade Switched Systems ------------------------ The framework of is useful for modeling switched systems. We are interested in studying switched systems in cascade configuration which comprise a family of dynamical subsystems described by \[cascnonlinearsys\] $$\begin{aligned} &\dot x = f_{c,p}(x,e), \label{stage1p} \\ &\dot e = f_{o,p}(e,d), \label{stage2p}\end{aligned}$$ where $p$ belongs to a [*finite*]{} index set ${\mathcal{P}}$. The vector fields $f_{c,p} : {\mathbb{R}}^{n_c} \times {\mathbb{R}}^{n_o} \to {\mathbb{R}}^{n_c}$ and $f_{o,p}:{\mathbb{R}}^{n_o}\times {\mathbb{R}}^{n_d} \to {\mathbb{R}}^{n_o}$ are assumed to be continuous for each $p \in {\mathcal{P}}$. It is also assumed that $f_{c,p}(0,0) = 0$, and $f_{o,p} (0,0) = 0$, and the stability of the origin $\{0\} \in {\mathbb{R}}^{n_c + n_o}$ is the topic of interest in the sequel. The switched system generated by is \[cascswisys\] $$\begin{aligned} \dot x = f_{c,\sigma}(x,e)\\ \dot e = f_{o,\sigma}(e,d),\end{aligned}$$ where $\sigma:{\mathbb{R}}_{\ge 0} \to {\mathcal{P}}$ denotes the piecewise constant right-continuous switching signal. The function $\sigma$ changes its value at switching times which are denoted by $\{t_i\}_{i \in {\mathbb{N}}}$. At these switching times, we allow the state values to have jumps defined by the maps \[eq:jumpSwSys\] $$\begin{aligned} x^+ &= g_c(x,e) \\e^+ & = g_o(e,d),$$ so that $x(t_i^+) = \big(x(t_i)\big)^+$, and $e(t_i^+) = \big(e(t_i)\big)^+$ denote the value of the state variables just after the switching times. We say that the switching signal $\sigma$ has an average dwell-time $\tau_a$, denoted $\sigma \in {\mathcal{S}}_{\tau_a}$ if there exists $N_0 \ge 1$ such that for each $t > s \ge 0$, it holds that $$\label{defADT} N_{\sigma(t, s)} \leq N_0 + \frac{t-s}{\tau_a}$$ where $N_{\sigma(t, s)}$ is the number of switching in the interval $(s, t]$. The constant $N_0$ is called the [ *chatter bound*]{} giving the tolerance number of fast switchings. #### Problem 1 Given that each subsystem in (with $e$ as input) and (with $d$ as input) admits an ISS Lyapunov function w.r.t. the origin, how can we 1. compute the lower bound on $\tau_a$, and 2. construct an ISS Lyapunov function for the hybrid system -, such that, for each $\sigma \in {\mathcal{S}}_{\tau_a}$, we have $$\left\vert (x(t,j),e(t,j)) \right\vert \le \beta (\left\vert (x(0,0),e(0,0)) \right\vert, t, j) + \gamma \left( \|d\|_{(t,j)} \right)$$ for some $\beta \in {\mathcal{K}}{\mathcal{L}}{\mathcal{L}}$, and $\gamma \in {\mathcal{K}}$. Stability of Cascade System {#sec:cascade} =========================== To find a solution to the problem mentioned above, we proceed in several steps which allow us to arrive at the result given in Theorem \[thm:mainISS\]. Individual Lyapunov Functions ----------------------------- The first step is to formally state the stability assumptions imposed on the dynamical subsystem and which are formally listed below: \[Voprop\] For each $p \in {\mathcal{P}}$, there exists a continuously differentiable function $V_{o,p} : {\mathbb{R}}^{n_o} \to {{\mathbb{R}}_{\geq 0}}$, and there exist class ${\mathcal{K}}_\infty$ functions $\overline{\alpha}_{o,p}$, $\underline{\alpha}_{o,p}$, $\alpha_{o,p}$ and $\gamma_{o,p}$ such that $$\underline{\alpha}_{o,p}(|e|) \leq V_{o,p}(e) \leq \overline{\alpha}_{o,p}(|e|)$$ $$\left\langle\frac{\partial V_{o,p}}{\partial e}, f_{o,p}(e,d)\right\rangle \leq - \alpha_{o,p}(V_{o,p}(e)) + \gamma_{o,p}(|d|)$$ hold for every $(e,d) \in {\mathbb{R}}^{n_o} \times {\mathbb{R}}^{n_d}$. \[Vcprop\] For each $p \in {\mathcal{P}}$, there exists a continuously differentiable function $V_{c,p} : {\mathbb{R}}^{n_c} \to {{\mathbb{R}}_{\geq 0}}$, and there exist class ${\mathcal{K}}_\infty$ functions $\overline{\alpha}_{c,p}$, $\underline{\alpha}_{c,p}$, $\alpha_{c,p}$, and $\gamma_{c,p}$ such that $$\underline{\alpha}_{c,p}(|x|) \leq V_{c,p}(x) \leq \overline{\alpha}_{c,p}(|x|)$$ $$\left\langle\frac{\partial V_{c,p}}{\partial x} f_{c,p}(x,e) \right\rangle\leq - \alpha_{c,p}(V_{c,p}(x)) + \gamma_{c,p}(V_{o,p}(e)),$$ hold for every $(x,e) \in {\mathbb{R}}^{n_c} \times {\mathbb{R}}^{n_o}$. \[Assumq\] As $s \to 0^+$, we have $\gamma_{c,p}(s) = O(\alpha_{o,p}(s))$, that is, if we let $$\label{eq:defbarnu} {\overline}\nu_p(s) := \frac{\gamma_{c,p} (s)}{\alpha_{o,p}(s)}, \quad \text{for }s > 0,$$ then there exists a constant $M >0$, such that $$\lim_{s \to 0^+} {\overline}\nu_p(s) \leq M.$$ In addition, we introduce the following assumption[^1] on the jump maps introduced in . \[assJump\] For each $(x,e,d) \in {\mathbb{R}}^{n_c + n_o+n_d}$, the jump maps at switching times satisfy $$\begin{aligned} & | g_c(x,e)| \leq \widehat{\alpha}_c(|(x,e)|) \notag \\& | g_o(e,d)| \leq \widehat{\alpha}_o(|e|) + \widehat{\rho}_o(|d|) \notag \end{aligned}$$ for some class ${\mathcal{K}}_\infty$ functions $\widehat{\alpha}_c$, $\widehat{\alpha}_o$, $\widehat{\rho}_c$, and $\widehat{\rho}_o$. Using the assumptions introduced above, it is possible to construct a candidate Lyapunov function $V_p(x,e)$ for each subsystem $p\in {\mathcal{P}}$. This construction is primarily inspired from the work of [@TanwTeel15]. The assumption \[Assumq\] is introduced to provide a construction of the candidate Lyapunov function $V_p$ explicitly in terms of $V_{o,p}$ and $V_{c,p}$. One can always modify the function $V_{c,p}$ to $\widehat V_{c,p}$ such that the resulting $\widehat\gamma_{c,p}$ satisfies \[Assumq\]; this is a direct consequence of [@SontTeel95 Theorem 1] as we can choose $\widehat\gamma_{c,p}(s) = O(\alpha_{o,p}(s)) \cdot \underline{\alpha}_{o,p}(s)$ for $s$ sufficiently small in the neighborhood of origin. \[basicVp\] Consider the family of dynamical subsystems satisfying \[Voprop\], \[Vcprop\], and \[Assumq\], along with the jump dynamics satisfying \[assJump\]. For each $p \in {\mathcal{P}}$, introduce the continuously differentiable function $V_p$, $$\label{defVp} V_p(x,e) := \int_0^{V_{o,p}(e)} \nu_p(s) \, ds + V_{c,p}(x),$$ where $\nu_p:{\mathbb{R}}_{\ge 0} \to {\mathbb{R}}_{\ge 0}$ is a continuous and nondecreasing function with $\nu_p(s) \ge 4 {\overline}\nu_p(s)$, for each $s > 0$. It then holds that, for some ${\underline}\alpha_p, {\overline}\alpha_p \in {\mathcal{K}}_\infty$, $$\label{eq:VpBound} {\underline}\alpha_p(\vert(x,e)\vert) \le V_p(x,e) \le {\overline}\alpha_p(\vert(x,e)\vert), \quad \forall (x,e) \in {\mathbb{R}}^{n_c+n_o}.$$ There also exist $\alpha_p,\gamma_p\in {\mathcal{K}_\infty}$ such that $$\label{eq:VpDecayBound} \left\langle \nabla V_p(x,e), \begin{pmatrix} f_{c,p}(x,e)\\ f_{o,p}(e,d)\end{pmatrix}\right\rangle \le -\alpha_p(V_p(x,e)) + \gamma_p (|d|),$$ for every $(x,e,d) \in {\mathbb{R}}^{n_c+n_o+n_d}$. Moreover, there exist $\chi, \rho \in {\mathcal{K}_\infty}$ such that for each $(p,q) \in {\mathcal{P}}\times {\mathcal{P}}$, $q \neq p$, $$\label{eq:VpJumpBound} V_q (x^+,e^+) \le \chi (V_p(x,e)) + \rho (|d|),$$ for every $(x,e,d) \in {\mathbb{R}}^{n_c+n_o+n_d}$. Fix $p\in{\mathcal{P}}$. Introduce the class ${\mathcal{K}_\infty}$ function $\ell_p: {{\mathbb{R}}_{\geq 0}}\to {{\mathbb{R}}_{\geq 0}}$ as follows: $$\label{defl} \ell_p(s) = \int_{0}^{s} \nu_p(r) dr$$ where $\nu_p$ was introduced in , so that $$V_p(x,e) = (\ell_p \circ V_{o,p})(e) + V_{c,p}(x).$$ The bound is seen to hold since $\ell_p$ is a class ${\mathcal{K}}_\infty$ function. Using \[Voprop\], we now obtain $$\begin{gathered} \label{eq:dlVoTemp} \left\langle \nabla (\ell_p \circ V_{o,p})(e), f_{o,p}(e,d) \right\rangle \\ \le \nu_p(V_{o,p}(e)) ( - \alpha_{o,p}(V_{o,p}(e)) + \gamma_{o,p}(|d|)).\end{gathered}$$ To analyze the right-hand side of , first consider the case where $\gamma_{o,p}(\vert d \vert) \le \frac 12 \alpha_{o,p}(V_{o,p}(e))$, so that $$\left\langle \nabla (\ell_p \circ V_{o,p})(e), f_{o,p}(e,d) \right\rangle \le -\frac 12 \nu_p(V_{o,p}(e))\alpha_{o,p}(V_{o,p}(e));$$ else, by introducing $\theta_p(s):=\alpha_{o,p}^{-1}(2\gamma_{o,p} (s))$, $$\begin{aligned} \frac{1}{2} \alpha_{o,p}(V_{o,p}(e)) \le \gamma_{o,p}(\vert d \vert) \Leftrightarrow V_{o,p}(e) & \le \alpha_{o,p}^{-1}(2\gamma_{o,p}(\vert d \vert)) \\ & = \theta_p(\vert d \vert)\end{aligned}$$ so that $\nu_p(V_{o,p}(e)) \le \nu_p(\theta_{p}(\vert d \vert))$, because $\nu_p$ is by construction nondecreasing, and $$\begin{gathered} \left\langle \nabla (\ell_p \circ V_{o,p})(e), f_{o,p}(e,d) \right\rangle \le \\ - \nu_p(V_{o,p}(e))\alpha_{o,p}(V_{o,p}(e)) + \nu_p(\theta_p(\vert d \vert))\gamma_{o,p}(\vert d \vert).\end{gathered}$$ From these two cases, the inequality results in $$\begin{gathered} \label{eq:dlVo} \left\langle \nabla (\ell_p \circ V_{o,p})(e), f_{o,p}(e,d) \right\rangle \le \\ -\frac 12 \nu_p(V_{o,p}(e))\alpha_{o,p}(V_{o,p}(e)) + \nu_p(\theta_p(\vert d \vert))\gamma_{o,p}(\vert d \vert).\end{gathered}$$ Using \[Vcprop\], , and the fact that $\nu_p(s) \ge 4 {\overline}\nu_p(s)$, for each $s > 0$ with ${\overline}\nu_p$ given in , we can now derive as follows: $$\begin{aligned} & \left\langle \nabla V_p(x,e), \begin{pmatrix} f_{c,p}(x,e)\\ f_{o,p}(e,d)\end{pmatrix}\right\rangle \\ & \quad \le -\frac 12 \nu_p(V_{o,p}(e))\alpha_{o,p}(V_{o,p}(e)) + \nu_p(\theta_p(\vert d \vert))\gamma_{o,p}(\vert d \vert) \\ & \quad \quad - \alpha_{c,p}(V_{c,p}(x)) + \gamma_{c,p}(V_{o,p}(e))\\ & \quad \le - \gamma_{c,p}(V_{o,p}(e))) - \alpha_{c,p}(V_{c,p}(x)) + \nu_p(\theta_p(|d|)) \gamma_{o,p}(|d|) \\ & \quad \le -\alpha_p(V_p(x,e)) + \gamma_p(\vert d \vert),\end{aligned}$$ where we used the definitions $\gamma_p(s) := \nu_p(\theta_p(s)) \gamma_{o,p}(s)$, $$\label{defalphapinput} \alpha_p(s) := \min \left\{ \alpha_{c,p} \left({\frac{1}{2}}s\right), \gamma_{c,p} \left( {\frac{1}{2}}\ell_p{^{\raisebox{.2ex}{$\scriptscriptstyle-1$}}}(s)\right) \right\},$$ and the triangle-inequality type result from [@Kell14 Lemma 10] to derive the last inequality. Next, to derive , we observe that $$V_q(x^+,e^+) = (\ell_q \circ V_{o,q})(g_o(e,d)) + V_{c,q}(g_c(x,e)).$$ Using \[assJump\], it then follows that $$\begin{aligned} V_q(x^+,e^+) &\leq \ell_q \circ \overline{\alpha}_{o,q}(|g_o(e,d)|) + \overline{\alpha}_{c,q}(| g_c(x,e)|) \notag \\ & \leq \ell_q \circ \overline{\alpha}_{o,q} \left(2 \widehat{\alpha}_o(|(x,e)|) \right) + \ell_q \circ \overline{\alpha}_{o,q} (2 \widehat{\rho}_o(|d|) \notag \\ &\quad + \overline{\alpha}_{c,q}(\widehat{\alpha}_c(|(x,e)|)) \notag \\ &\leq \chi(V_p(x,e)) + \rho (|d|)\end{aligned}$$ where, recalling , we used the definitions $$\label{eq:defchi} \chi(s) := \max_{p,q \in {\mathcal{P}}} \left\{ \ell_q \circ \overline{\alpha}_{o,q} \left(2 \widehat{\alpha}_o \circ {\underline}\alpha_p{^{\raisebox{.2ex}{$\scriptscriptstyle-1$}}}(s)\right) + \overline{\alpha}_{c,q}(\widehat{\alpha}_c \circ {\underline}\alpha_p{^{\raisebox{.2ex}{$\scriptscriptstyle-1$}}}(s)) \right\}$$ and $$\label{eq:defrho} \rho(s) := \max_{q \in {\mathcal{P}}} \left\{ \ell_q \circ \overline{\alpha}_{o,q} (2 \widehat{\rho}_o(s)) \right\}, $$ which establishes the desired bound since both functions are class ${\mathcal{K}}_\infty$. Stability of Overall Hybrid System {#sec:plainHybSys} ---------------------------------- We now use the result of Proposition \[basicVp\] to compute lower bounds on the dwell-time which result in cascade switched system being globally ISS. To do so, we find it convenient to express the switched system in the framework of the hybrid system adopted in [@GoebSanf12]. This is done by introducing the augmented state variable $\xi := \left( x, e, p, \tau \right) \in {\mathcal{X}}:= {\mathbb{R}}^{n_c+n_o} \times {\mathcal{P}}\times [0,N_0]$, where $p$ is a discrete variable denoting a subsystem, and $\tau$ plays the role of a scaled timer. The hybrid model capturing the dynamics of the switched system driven by an external disturbance $d \in {\mathbb{R}}^{{\overline}d}$, and where the switching signals have an average dwell-time $\tau_a$, is \[eq:smallHybSys\] $$\begin{aligned} &(\xi,d) \in {\mathcal{C}}: \begin{cases} \dot x = f_{c,p}(x, e)\\ \dot e = f_{o,p}(e, d)\\ \dot p = 0 \\ \dot \tau \in [0, \frac{1}{\tau_a}]\\ \end{cases} \label{eq:smallHybSysa} \\ &(\xi,d) \in {\mathcal{D}}: \begin{cases} x^+ = g_c(x,e) \\ e^+ = g_o(e,d) \\ p^+ \in {\mathcal{P}}\setminus \{p\} \\ \tau^+ = \tau -1 \end{cases} \label{eq:smallHybSysb}\end{aligned}$$ where the flow set ${\mathcal{C}}:={\mathcal{X}}\times {\mathbb{R}}^{{\overline}d}$, and the jump set ${\mathcal{D}}:={\mathbb{R}}^{n_c+n_o} \times {\mathcal{P}}\times [1,N_0] \times {\mathbb{R}}^{{\overline}d}$. We denote the set-valued mapping on the right-hand side of by ${\mathcal{F}}(\xi,d)$, and the mapping on the right-hand side of is denoted by ${\mathcal{G}}(\xi,d)$. We are interested in studying the ISS property of the system  (driven by the disturbance $d$) with respect to the compact set $$\label{eq:defA0} {\mathcal{A}}_0 := \{0\}^{n_c+n_o} \times {\mathcal{P}}\times [0,N_0]$$ by finding an appropriate ISS Lyapunov function. To do so, we introduce the function $\varphi:{\mathbb{R}}_{\ge 0} \to {\mathbb{R}}_{\ge 0}$ defined as $$\label{eq:defPhi} \varphi(s) = \begin{cases} \exp \left( \int_{1}^{s} \frac{2{\color{blue}c_0}}{\psi(r)} \, dr \right), & s > 0 \\ 0, & s = 0 \end{cases}$$ where $\psi:{\mathbb{R}}_{\ge 0} \to {\mathbb{R}}_{\ge 0}$ is a continuously differentiable class ${\mathcal{K}}_\infty$ function, with $\psi'(0) = 0$, and $$\begin{aligned} & \psi(s) \le \min\{c_0s, \alpha_p(s) \, \vert \, p \in {\mathcal{P}}\}, \quad {\color{blue} s \ge 0},\\ {\color{blue}\text{\sout{and}}} \quad & {\color{blue}\hbox{\sout{$\psi(s) \le \min\{\alpha_p(s) \, \vert \, p \in {\mathcal{P}}\}, \quad s \ge c_1$}}},\end{aligned}$$ for some $c_0 > 0$. We recall that $\alpha_p \in {\mathcal{K}}_\infty$ were introduced as the decay function for $V_p$ in . The function $\varphi$ is now used in the following result: \[thm:mainISS\] Consider system and suppose that \[Voprop\], \[Vcprop\], \[Assumq\], \[assJump\] hold. Let $\chi \in {\mathcal{K}}_\infty$ be as in . If, for some ${\varepsilon}>0$, the average dwell-time $\tau_a$ satisfies $$\label{condADT} \tau_a > \zeta^* := \sup_{s \ge 0} \, \int_{s }^{(1+{\varepsilon})\chi(s)} \frac{1}{\psi(r)} \, dr,$$ then for each $\tau_a > \zeta > \zeta^*$, $$\label{eq:defWp} W (x,e,p,\tau) := \exp(2 \, {\color{blue}c_0} \, \zeta\tau)\varphi(V_p(x,e)),$$ is an ISS Lyapunov function for the hybrid system w.r.t. the compact set ${\mathcal{A}}_0$, and input $d$. To gain an insight about the constraints imposed by the stability condition on the system structure, we study particular instances where $\alpha(s) := \min\{\alpha_p(s)\, \vert \, p \in {\mathcal{P}}\}$ exhibits linear, super-linear and sub-linear growth. It can be seen that if the jump map $\chi$ does not grow too fast compared to $\alpha(s)$, then $\zeta^*$ in is finite. For the sake of simplicity, let $\alpha(s) := a s^k$, with $a,k > 0$, and choose $c_0 = a$ in the definition of $\psi$. - [*Linear decay:*]{} We first consider the case $k = 1$, so that $\alpha(s) = a s$, and we let $\psi (s) = a s$, for $s \ge 0$. This gives $\zeta^* = \sup_{s\ge 0}\frac{1}{a} \log{\frac{(1+{\varepsilon})\chi(s)}{s}}$, which is finite if $\lim_{s\to\infty} \chi (s) = O(s)$, and $\lim_{s \to 0} \chi (s) = O(s)$. If $\chi(s) = \mu s$, then $\zeta^* = \frac{1}{a}\log((1+{\varepsilon})\mu)$, which resembles the bound given in [@Libe03 Chapter 3] by taking ${\varepsilon}> 0$ arbitrarily small. - [*Super-linear decay:*]{} Next consider the case where $\alpha(s) = a s^k$ with $k > 1$. Choose $c_1 =1$, then we can let $\psi(s) = a s^k$, for every [$s \in [0,c_1]$. Thus, $\zeta^*$ is the maximum between $\frac{1}{k-1}\sup_{s\in [0,c_1]} \left[\frac{1}{as^{k-1}} - \frac{1}{a\chi(s)^{k-1}}\right]$ and $\sup_{s\ge 1}\frac{1}{a} \log{\frac{(1+{\varepsilon})\chi(s)}{s}}$]{}. With $\chi \in {\mathcal{K}}_\infty$, it is seen that $\zeta^*$ is finite and positive if $\lim_{s\to 0^+} \left[\frac{1}{as^{k-1}} - \frac{1}{a\chi(s)^{k-1}}\right] \le 0$, which holds if $\chi(s) = o(s)$ when $ s \to 0$, and [$\lim_{s\to\infty} \chi(s) = O(s)$.]{} - [*Sub-linear decay:*]{} Lastly, consider the case where $\alpha(s) = a s^k$ with $k < 1$. Choose $c_1 =1$, then there exist ${\underline}c < 1$, ${\overline}c >1$ and a continuously differentiable function $\psi$ such that $\psi(s) = as$, $s\in [0,{\underline}c]$, and $\psi(s) = a s^k$ for $s \ge {\overline}c$. With this choice of $\psi$, the lower bound $\zeta^*$ is a finite positive scalar, if $\lim_{s\to 0^+} \chi(s) = O(s)$, and $\lim_{s\to\infty} \chi(s) = o(s)$. \[rem:phi\] A function similar to $\varphi$ defined in also appears in [@PralWang96] to transform nonlinear decay rates to linear ones in inequalities associated with Lyapunov functions, while keeping the modified Lyapunov function differentiable. In the proof of Theorem \[thm:mainISS\], this function serves the same purpose. Here, the construction of $\psi$ is modified slightly. The proof is based on showing that $W$ satisfies the conditions listed in Proposition \[hybridISSprop\]. It is seen that $\varphi$ is differentiable (away from the origin) on ${\mathbb{R}}_{>0}$, and it is shown in [@PralWang96 Lemma 12] that the function $\varphi$ is also continuously differentiable in the neighborhood of the origin with $\varphi'(0) = 0$. Therefore, $W$ is also continuously differentiable. Since $\varphi$ is a class ${\mathcal{K}}_\infty$ function, one can easily verify that $${\underline}\alpha(\vert \xi \vert_{{\mathcal{A}}_0}) \le W(\xi) \le {\overline}\alpha(\vert \xi \vert_{{\mathcal{A}}_0})$$ for some functions ${\underline}\alpha$, ${\overline}\alpha$ of class ${\mathcal{K}}_\infty$. From the definition of ${\mathcal{A}}_0$ and the assumption that $f_{c,p}(0,0) = f_{o,p}(0,0) = 0$, $p \in {\mathcal{P}}$, it immediately follows that for each $\xi \in {\mathcal{A}}_0$ such that $(\xi,0) \in {\mathcal{D}}$, we have ${\mathcal{G}}(\xi,0) \in {\mathcal{A}}_0$. Also, along any continuous motion resulting from , with initial condition $\xi \in {\mathcal{A}}_0$ satisfying $(\xi,0)\in {\mathcal{C}}$, the system trajectory stays within ${\mathcal{A}}_0$. Hence, ${\mathcal{A}}_0$ is forward invariant with $d = 0$. Let $f$ be an element of ${\mathcal{F}}(\xi,d)$. When $(\xi,d) \in {\mathcal{C}}$, it follows from that $$\begin{gathered} \left\langle \nabla W(\xi), f \right\rangle \le \frac{2 {\color{blue}c_0}\zeta \exp({2 {\color{blue}c_0} \zeta \tau})}{\tau_a} \varphi(V_p(x,e))\\ + \exp(2 {\color{blue}c_0} \zeta \tau) \varphi(V_p(x,e)) \left[-\frac{2{\color{blue}c_0}\alpha_p(V_p(x,e))} {\psi(V_p)} + \frac{2{\color{blue}c_0} \gamma_p(\vert d \vert)} {\psi(V_p)} \right] \end{gathered}$$ and hence $$\left\langle \nabla W(\xi), f \right\rangle \le W(\xi) \left[ \frac{2 {\color{blue}c_0} \zeta}{\tau_a} - \frac{2 {\color{blue}c_0} \alpha_p (V_p)} {\psi(V_p) } + \frac{2 {\color{blue}c_0} \gamma_p(|d|)}{\psi(V_p)}\right].$$ Since $\psi(s) \le \alpha_p(s)$ by construction, and $\zeta$ is chosen to satisfy $\tau_a >\zeta$, we get an $a:= 2{\color{blue}c_0}(1 - \zeta/\tau_a) > 0$ such that $$\left\langle \nabla W(\xi), f \right\rangle \le - aW(\xi) + \frac{2 {\color{blue}c_0} W(\xi)}{\psi(V_p(x,e))} \gamma_p(|d|), \quad \xi \in {\mathcal{C}}$$ which is of the same form[^2] as . It is readily checked that the asymptotic ratio condition holds since $$\lim_{\vert(x,e)\vert \to \infty} \frac{1}{\psi(V_p(x,e))} = 0.$$ The next step is to show that holds under condition for the jump maps . Let $g$ denote an element of ${\mathcal{G}}(\xi,d)$. It is seen that for each $(\xi,d) \in {\mathcal{D}}$, that is, whenever $\tau \in [1,N_0]$, it follows from that $$\begin{aligned} \max_{g \in {\mathcal{G}}} W^+ &= \ \max_{g \in {\mathcal{G}}} \ \exp(2 {\color{blue}c_0} \zeta\tau^+) \varphi( V_{p^+}(x^+,e^+)) \notag \\ & \le \ \exp(2 {\color{blue}c_0} \zeta \tau - 2 {\color{blue}c_0}\zeta) \varphi \left( \chi (V_p(x,e)) + \rho(|d|)\right), $$ where $\chi$ and $\rho$ were introduced in and . Take an ${\varepsilon}> 0$ for which holds, then it follows from [@Kell14 Lemma 10] that: $$\begin{gathered} \varphi \left( \chi (V_p(x,e)) + \rho(|d|)\right) \le \varphi((1+{\varepsilon})\chi(V_p(x,e))) \\ + \varphi \left(\frac{1+{\varepsilon}}{{\varepsilon}}\rho(|d|)\right).\end{gathered}$$ For $(x,e) \neq 0$, the bound on the first term on the right-hand side is given by $$\begin{aligned} &\varphi((1+{\varepsilon})\chi(V_p(x,e))) = \exp\left(\int_{1}^{(1+{\varepsilon})\chi(V_p(x,e))} \frac{2 {\color{blue}c_0} dr}{\psi(r)} \right)\\ &\quad = \exp\left(\int_{V_p(x,e)}^{(1+{\varepsilon})\chi(V_p(x,e))} \frac{2 {\color{blue}c_0} dr}{\psi(r)}\right) \cdot \exp\left( \int_{1}^{V_p(x,e)} \frac{2 {\color{blue}c_0} dr}{\psi(r)}\right)\\ & \quad \le \exp(2 {\color{blue}c_0} \zeta^*) \, \varphi (V_p(x,e))\end{aligned}$$ where $\zeta^*$ is defined as in . Letting $$\label{eq:defrhotilde} \widetilde{\rho}(s) :=\exp{(2 {\color{blue}c_0} \zeta N_0 - 2 {\color{blue}c_0} \zeta)} \ \varphi \left(\frac{1+{\varepsilon}}{{\varepsilon}}\rho(s)\right),$$ and noting that, for $\tau \in [0,N_0]$, $$\widetilde \rho(s) \ge \exp{(2 {\color{blue}c_0} \zeta \tau -2 {\color{blue}c_0} \zeta)} \ \varphi \left(\frac{1+{\varepsilon}}{{\varepsilon}}\rho(s)\right),$$ we obtain $$\begin{aligned} \max_{g \in {\mathcal{G}}} W^+ & \leq \exp{(2 {\color{blue}c_0} \zeta^*-2 {\color{blue}c_0} \zeta)} \exp{(2{\color{blue}c_0} \zeta \tau)}\varphi(V_p(x,e)) + \widetilde{\rho}(|d|) \\ & \leq \exp(2 {\color{blue}c_0} \zeta^* - 2 {\color{blue}c_0} \zeta) W (\xi) + \widetilde{\rho}(|d|).\end{aligned}$$ Having chosen $\zeta$ such that $\zeta^*-\zeta < 0$, we see that holds. The result of Proposition \[hybridISSprop\] thus ensures that $W$ is an ISS Lyapunov function for w.r.t. the set ${\mathcal{A}}_0$, with input $d$. Linear Case ----------- We use the following linear example to illustrate the Theorem \[thm:mainISS\]. Assume that the system is linear \[casclinearsys\] $$\begin{aligned} \dot x = A_px + B_p e \label{linearsysx} \\ \dot e = F_p e + G_p d \label{linearsyse} \end{aligned}$$ with the matrices $A_p,F_p$ being Hurwitz. For the sake of simplicity, we assume that the states $(x,e)$ do not go through any jump dynamics, and remain unchanged at switching instances. This way, we let the maps in be $$g_c(x,e) = x, \quad \text { and } \quad g_o(e,d) = e.$$ We can now choose quadratic Lyapunov functions to satisfy \[Voprop\] and \[Vcprop\]. This is done by computing symmetric positive definite matrices $P_{o,p}, P_{c,p} > 0$ such that, for each $p \in {\mathcal{P}}$, $$\begin{aligned} F_p^\top P_{o,p} + P_{c,p} F_p & \le -Q_{o,p} \\ A_p^\top P_{c,p} + P_{c,p} A_p & \le -Q_{c,p}\end{aligned}$$ for some symmetric positive definite matrices $Q_{o,p}, Q_{c,p} > 0$. By letting $$V_{c,p}(x) = x^\top P_{c,p} x, \quad \text{ and } \quad \text V_{o,p} = e^\top P_{o,p} \, e$$ we get $${\underline{a}_{o,p}}|x|^2 \leq V_{o,p}(x) \leq {\overline{a}_{o,p}}|x|^2$$ with ${\underline{a}_{o,p}}= \lambda_{\min}(P_{o,p})$, and ${\overline{a}_{o,p}}= \lambda_{\max}(P_{o,p})$. Similarly, it holds that $${\underline{a}_{c,p}}|x|^2 \leq V_{c,p}(x) \leq {\overline{a}_{c,p}}|x|^2$$ with ${\underline{a}_{c,p}}= \lambda_{\min}(P_{c,p})$, and ${\overline{a}_{c,p}}= \lambda_{\max}(P_{c,p})$. It can be readily shown that $$\left\langle\nabla V_{o,p}, F_px + G_p e\right\rangle \le - a_{o,p} V_{o,p}(x) + {\overline{\gamma}_{o,p}}|d|^2$$ by letting $$a_{o,p} = \frac{\lambda_{\min}(Q_{o,p})}{2\lambda_{\max}(P_{o,p})} \quad \text{ and } \quad {\overline}\gamma_{o,p} = 2 \frac{\| P_{o,p}G_{p} \|^2}{\lambda_{\min}(Q_{o,p})}$$ and likewise $$\left\langle\nabla V_{c,p}, A_px + B_p e\right\rangle \le - a_{c,p} V_{c,p}(x) + {\overline{\gamma}_{c,p}}V_{o,p}(e)$$ with $$a_{c,p} = \frac{\lambda_{\min}(Q_{c,p})}{2\lambda_{\max}(P_{c,p})} \quad \text{ and } \quad {\overline{\gamma}_{c,p}}= 2 \frac{\| P_{c,p} B_{p} \|^2}{\lambda_{\min}(Q_{c,p})}.$$ For each $p \in {\mathcal{P}}$, the function ${\overline}\nu_p(s)$ in turns out to be a constant as $${\overline}\nu_p(s) = \frac{{\overline}\gamma_{c,p} \, s}{a_{o,p} \, s} = \frac{{\overline}\gamma_{c,p}}{a_{o,p}} =: {\overline}\nu_p.$$ Thus, we can choose the Lyapunov function in to be $$V_p(x,e) = 4 \, {\overline}\nu_p V_{o,p}(e) + V_{c,p} (x)$$ which leads to $$\left\langle \nabla V_p(x,e), \begin{pmatrix} A_p x + B_p e \\ F_p e + G_p d \end{pmatrix}\right\rangle \le -a_p \, V_p(x,e) + {\overline}\gamma_p |d|^2$$ in which $$\begin{aligned} &a_p = \min \left\{a_{c,p}, 0.75 \, a_{o,p} \right\},\\ &{\overline{\gamma}_p}= 4{\overline}\nu_p {\overline{\gamma}_{o,p}}.\end{aligned}$$ For the given jump maps at switching times, the maps in can be chosen such that $\rho \equiv 0$, and $$\label{eq:defchibar} \chi(s) = \overline{\chi} s, \quad \overline{\chi} = \max_{p,q \in {\mathcal{P}}} \left\{ \frac{{\overline}\nu_q}{{\overline}\nu_p} \frac{\lambda_{\max}(P_{o,q})}{\lambda_{\min}(P_{o,p})}, \frac{\lambda_{\max}(P_{c,q})}{\lambda_{\min}(P_{c,p})} \right\}.$$ To construct the Lyapunov function $W$ in , we let $$a:= \min_{p\in{\mathcal{P}}} \{a_p\}$$ so that $\psi(s) = a s$ satisfies the desired conditions with $c_0 = a$ and $c_1 > 0$ arbitrary. We now choose $W$ such that $W(x,e,p,\tau) = 0$ if $(x,e) = 0$, and for $(x,e) \neq 0$, $$\begin{aligned} W(x,e,p,\tau) &= \exp(2 {\color{blue}a} \zeta \tau) \exp \left( \int_{1}^{V_p(x,e)} \frac{2 {\color{blue}a}\,dr}{a\, r} \right) \\ & = \exp ({2{\color{blue}a}\zeta \tau}) V_p^{{\color{blue}2}}(x,e).\end{aligned}$$ This construction leads to the following result: The switched linear system with subsystems described by is input-to-state stable with respect to the origin and disturbance $d$ if $$\tau_a > \frac{1}{a} \ln(\overline \chi),$$ where $a = \min_{p \in {\mathcal{P}}}\{a_p\}$, and ${\overline}\chi$ is given in . Output Feedback Stabilization with Sampling {#sec:sampling} =========================================== In this section, we will use the theoretical results from the previous section to study the problem of feedback stabilization with dynamic output feedback for switched nonlinear systems with time-sampled measurements. Our starting point is a nominal setup where an output feedback controller is already designed for each subsystem. We assume this controller to be robust with respect to measurement errors, and this property is formalized using ISS notion. Next, we implement these controllers when the output measurements sent by the plant, and the control inputs sent by the controller are time-sampled and subjected to zero-order hold between two updates. Our goal is to design the sampling algorithms for control signals and measurements such that the resulting closed-loop system is asymptotically stable. Inspired by the work of [@TanwTeel15], we introduce the dynamic filters and event-based update rules to design the sampling algorithms. Problem Setup ------------- The switched nonlinear plant which we want to stabilize, is described by \[cascadeFam3\] $$\begin{aligned} &\dot x = f_{c,\sigma} (x, u) \label{eq:planta}\\ & y = h_\sigma(x),\end{aligned}$$ and the corresponding controller is described by \[samplingcontroller\] $$\begin{aligned} &\dot z = f_{o,\sigma}(z, u, y) \label{eq:obsa}\\ & u = k_\sigma(z).\label{eq:obsb}\end{aligned}$$ Here, for a finite index set ${\mathcal{P}}$, $\sigma:{\mathbb{R}}_{\ge 0} \to {\mathcal{P}}$ is the switching signal, and we emphasize that the controller is driven by the same switching signal as the plant. For each $p \in {\mathcal{P}}$, the mappings $f_{c,p}:{\mathbb{R}}^n \times {\mathbb{R}}^{n_u} \to {\mathbb{R}}^n$, $h_p:{\mathbb{R}}^{n} \to {\mathbb{R}}^{n_y}$, $f_{o,p}:{\mathbb{R}}^n \times {\mathbb{R}}^{n_u} \times {\mathbb{R}}^{n_y} \to {\mathbb{R}}^n$, and $k_p:{\mathbb{R}}^n \to {\mathbb{R}}^{n_u}$ are assumed to be continuous on their respective domains. The underlying working principle behind the controller is that the variable $z$ in acts as a full-state estimate for the variable $x$ governed by . The dynamics of the estimation error, denoted $e:=z-x$, are assumed to be ISS with respect to measurement errors, as described in \[spvop\] below. Also, the static feedback controller $k_p$ in , $p \in {\mathcal{P}}$, has the property that the corresponding subsystem $\dot x = f_{c,p}(x,k_p(x))$ is ISS with respect to errors in measurement of $x$, as stated in \[spvcp\]. With these properties, we can design sampling algorithms for output $y$ and input $u$, where the difference between the last updated value of the output (resp. input) and its current value acts as an error in measurement. In our setup of sampling, we use a zero-order-hold between sampling times so that the inputs and outputs stay constant between two successive updates. Thus, we introduce two additional states in our model, with piecewise constant trajectories, which keep track of the sampled values. These are $x_d:{\mathbb{R}}_{\ge 0} \to {\mathbb{R}}^{n}$ and $z_d:{\mathbb{R}}_{\ge 0} \to {\mathbb{R}}^{n}$, and are to be seen as the time-sampled versions of $x$ and $z$ respectively. We set $y_d= h_\sigma(x_d)$ to denote the sampled values of the output that are sent to the controller, and $u_d = k_\sigma(z_d)$ to denote the sampled control values, which are sent to the plant. Whenever a new sampled value of the output (resp. control input) is to be sent, we update $x_d^+ = x$ so that $y_d^+ = h_\sigma(x)$ (resp. $z_d^+=z$ so that $u_d=k_\sigma(z)$). We emphasize that the plant and controller use the same value of the switching signal $\sigma$ at all times. The overall schematic of the closed-loop system is given in Figure \[fig:sampledSys\]. We denote the measurement error due to sampling in the output by $d_y:= y_d - y$, and $d_z:= z_d - z$ denotes the error which appears in the plant dynamics due to sampling of the input. By constructing a Lyapunov function for the augmented system using the cascade principle, we next show that the state of system - converges to the origin for appropriately designed sampling algorithms. The assumptions imposed on the nominal system - are now listed below: \[alphah\] For each $p \in {\mathcal{P}}$, there exists a class ${\mathcal{K}}$ function $\alpha_{h,p}$ such that the function $h_p:{\mathbb{R}}^n \to {\mathbb{R}}^{n_y}$ satisfies: $$|y| = |h_p(x)| \leq \alpha_{h,p}(|x|), \quad \forall p \in {\mathcal{P}}, \forall x \in {\mathbb{R}}^n.$$ \[spvop\] There exist continuously differentiable functions $V_{o,p} : {\mathbb{R}}^n \to {{\mathbb{R}}_{\geq 0}}$, $p \in {\mathcal{P}}$, class ${\mathcal{K}}$ function $\alpha_{o,p}$, and class ${\mathcal{K}}_\infty$ functions $\overline{\alpha}_{o,p}$, $\underline{\alpha}_{o,p}$ such that, for every $(x,z,u,y,d_y) \in {\mathbb{R}}^{2n+n_u+2n_y}$, $$\begin{aligned} &\underline{\alpha}_{o,p}(|e|) \leq V_{o,p}(e) \leq \overline{\alpha}_{o,p}(|e|) \label{spvopbound} \\ &\left\langle\frac{\partial V_{o,p}}{\partial e}(e) , f_{c,p} (x, u) - f_{o,p}(z, u, y + d_y) \right\rangle \leq \notag \\ &\quad - \alpha_{o,p}(V_{o,p}(e)) + \gamma_{o,p}(|d_y|), \label{spdotvopbound}\end{aligned}$$ where $e = z - x$. \[spvcp\] There exist continuously differentiable functions $V_{c,p} : {\mathbb{R}}^n \to {{\mathbb{R}}_{\geq 0}}$, $p \in {\mathcal{P}}$, class ${\mathcal{K}}$ functions $\alpha_{c,p}$, $\gamma_{c,p}$, and class ${\mathcal{K}}_\infty$ functions $\overline{\alpha}_{c,p}$, $\underline{\alpha}_{c,p}$ such that $$\begin{aligned} &\underline{\alpha}_{c,p}(|x|) \leq V_{c,p}(x) \leq \overline{\alpha}_{c,p}(|x|)\label{spvcpbound} \\ &\left\langle \frac{\partial V_{c,p}}{\partial x} , f_{c,p}(x, k_p(x+e+d_z)) \right\rangle \leq - \alpha_{c,p}(V_{c,p}(x)) \notag \\ & \quad \quad + \gamma_{c,p}(V_{o,p}(e)) + \gamma_{c,p}(|d_z|),\label{spdotvcpbound} \end{aligned}$$ hold for every $(x,z,u,y,d_y) \in {\mathbb{R}}^{2n+n_u+2n_y}$. For the class of plants and controllers satisfying the aforementioned hypotheses, we are now interested in designing the sampling algorithms, and characterizing the class of switching signals which result in an overall asymptotically stable system. Sampling Algorithms ------------------- (-2,0) node \[rectangle, rounded corners, draw, minimum height =0.65cm, text centered\] (sys) [$ {\mathcal{P}}: \left\{ \begin{aligned}\dot x & = f_{c,p}(x,u_d) \\ y&= h_p(x)\end{aligned}\right .$]{}; (sys.east) node\[anchor=south west\] [$y$]{}; (sys.west) node\[anchor=south east\] [$u_d$]{}; (2,-2) node \[rectangle, rounded corners, draw, minimum height = 0.65cm, text centered\] (obs) [${\mathcal{C}}: \left\{ \begin{aligned} &\dot z = f_{o,p}(z,u_d,y_d)\\ &u_d= k_p(z_d)\end{aligned}\right .$]{}; (1.5,0) node \[rectangle, rounded corners, draw, minimum height = 0.65cm, text centered\] (zO) [$\dot \eta_o = \mathfrak{a}_p(\eta_o,y)$]{}; (-1.5,-2) node \[rectangle, rounded corners, draw, minimum height = 0.65cm, text centered\] (zC) [$\dot \eta_c = \mathfrak{b}_p(\eta_c,z)$]{}; (tr) at (\[xshift=1.7cm)\]zO.east); (br) at (\[yshift=-2cm)\]tr); (bl) at (\[xshift=-1.7cm)\]zC.west); (tl) at (\[yshift=2cm)\]bl); (sys.east)–(zO.west); (obs.west)–(zC.east); (zO.east) to \[make contact=[info’=[\[red\]$y_d$]{}]{}\] (tr); (tr)–(br); (br) – (obs.east); (zC.west) to \[make contact=[info’=[\[red\]$u_d$]{}]{}\] (bl); (bl)–(tl); (tl) – (sys.west); As mentioned in the introduction, we are interested in analyzing the stability of the closed-loop system under event-based sampling rules. To do so, the auxiliary signals $x_d$, $z_d$ are thus modeled as $$\begin{aligned} \begin{cases} \dot x_d = 0,\\ \dot z_d = 0, \end{cases} \qquad \begin{cases} x_d^+ = x, & \text{if ${\mathtt{event}}_1 = {\mathtt{true}}$}\\ z_d^+ = z, & \text{if ${\mathtt{event}}_2 = {\mathtt{true}}$} \end{cases}\end{aligned}$$ and by setting $y_d = h_\sigma(x_d)$ and $u_d = k_\sigma(z_d)$, the dynamics of the system with time-sampled inputs and outputs are given by $$\begin{aligned} \dot x & = f_{c,\sigma} (x, u_d) = f_{c,\sigma} (x, k_\sigma(z_d))\\ \dot z &= f_{o,\sigma}(z, u_d, y_d) = f_{o,\sigma}(z, k_\sigma(z_d), h_\sigma(x_d)).\end{aligned}$$ To define the events at which the outputs and inputs are updated, we introduce the following dynamic filters: \[dotetaoetac\] $$\begin{aligned} \dot \eta_o &:= -\beta_{o,p}(\eta_o) + \rho_{o,p}(|y|) + \gamma_{o,p}(|h_p(x) - h_p(x_d)|)\label{defdotetao} \\ \dot \eta_c &:= -\beta_{c,p}(\eta_c) + \rho_{c,p} \left(\frac{|z|}{2}\right) + \gamma_{c,p}(|z- z_d|)\end{aligned}$$ where, for each $p \in {\mathcal{P}}$, $\beta_{o,p},\beta_{c,p}, \rho_{o,p}, \rho_{o,p}, \gamma_{o,p}, \gamma_{c,p}$ are class ${\mathcal{K}_\infty}$ functions, and the initial conditions for $\eta_o$ and $\eta_c$ are chosen to be some positive numbers. We say that $$\label{eq:jumpCond} \begin{cases} {\mathtt{event}}_1 = {\mathtt{true}}& \text{if } \vert y - y_d\vert \ge \mu_{o,\sigma}(\eta_o)\\ {\mathtt{event}}_2 = {\mathtt{true}}& \text{if } \vert z - z_d \vert \ge \mu_{c,\sigma}(\eta_o), \end{cases}$$ for some class ${\mathcal{K}}_\infty$ functions $\mu_{o,p}, \mu_{c,p}$. Note that ${\mathtt{event}}_1$ and ${\mathtt{event}}_2$ may occur at different times, or simultaneously, and that each one corresponds to a different update rule. Hybrid Model ------------ Just as we did in Section \[sec:plainHybSys\], it is convenient to write the entire system with controlled plant dynamics, controller, and sampling algorithms using the framework of hybrid systems. To do so, we first introduce $$\xi := (x, z, x_d, z_d, \eta_o, \eta_c, p, \tau) \in {\mathbb{R}}^{4n+ 2} \times {\mathcal{P}}\times [0,N_0]$$ to describe the state the closed-loop system. The variables $u_d$ and $y_d$ are obtained by setting $u_d = k_p(z_d)$ and $y_d=h_p(x_d)$. The flow set is now described as $$\begin{aligned} {\mathcal{C}}&=\left\{\xi: |y-h_p(x_d)| \leq {\mu_{o,p}(\eta_o)}\right\} \cap \left\{\xi: |z-z_d| \leq {\mu_{c,p}(\eta_c)}\right\} \notag \\ &\quad \cap \left\{\xi: \eta_o \geq 0 \wedge \eta_c \geq 0\right\} \cap \left\{\xi: \tau \in [0,N_0] \right\}, \notag\end{aligned}$$ so that the state variables evolve according to a differential equation/inclusion when $\xi \in {\mathcal{C}}$. By construction, jumps in at least one of the state variables occur either due to switching, or when the condition for ${\mathtt{event}}_i$, $i = 1,2$, holds true. The jump set ${\mathcal{D}}$ therefore corresponds to the switching event, or the sampling event, and they may not occur at the same time. The jump set is defined as $${\mathcal{D}}= {\mathcal{D}}_{{\mathtt{sw}}} \cup {\mathcal{D}}_{u} \cup {\mathcal{D}}_y$$ where $$\begin{aligned} {\mathcal{D}}_{{\mathtt{sw}}} & := \left\{\xi: \tau \in [1,N_0]\right\} \\ {\mathcal{D}}_{u} &:= \left\{\xi: |z-z_d| \geq {\mu_{c,p}(\eta_c)}\right\} \\ {\mathcal{D}}_y &:= \left\{\xi: |y-h_p(x_d)| \geq {\mu_{o,p}(\eta_o)}\right\} \end{aligned}$$ so that the variables may get updated instantaneously when $\xi \in {\mathcal{D}}$. The evolution equations for the augmented variable $\xi$ can now be described as follows: \[eq:bigHybSys\] $$\begin{aligned} &\xi \in {\mathcal{C}}: \begin{cases} \dot x = f_{c,p}(x, k_p(z_d))\\ \dot z = f_{o,p}(z, k_p(z_d), h_p(x_d))\\ \dot x_d = 0 \\ \dot z_d =0 \\ \dot \eta_o = -\beta_{o,p}(\eta_o) + \rho_{o,p}(|y)|) + \gamma_{o,p}(\vert y-y_d\vert) \\ \dot \eta_c = -\beta_{c,p}(\eta_c) + \rho_{c,p}(\frac{|z|}{2}) + \gamma_{c,p}(|z- z_d|) \\ \dot p = 0 \\ \dot \tau \in [0, \frac{1}{\tau_a}]\\ \end{cases}\end{aligned}$$ where, in the description of $\eta_o$-dynamics, we recall that $y = h_p(x)$ and $y_d(x) = h_p(x_d)$. The jump maps for this system are: $$\begin{aligned} &\xi \in {\mathcal{D}}: \begin{cases} \xi^+ \in {\mathcal{G}}(\xi) = {\mathcal{G}}_{{\mathtt{sw}}} (\xi) \cup {\mathcal{G}}_u(\xi) \cup {\mathcal{G}}_y(\xi) \end{cases} $$ $${\mathcal{G}}_{{\mathtt{sw}}}(\xi) = \begin{bmatrix} x\\z\\ x \\ z \\ \eta_o\\ \eta_c\\ {\mathcal{P}}\setminus\{p\} \\ \tau-1\end{bmatrix}, {\mathcal{G}}_u(\xi) = \begin{bmatrix} x \\z \\ x_d \\ z\\ \eta_o\\ \eta_c\\ p \\ \tau\end{bmatrix}, {\mathcal{G}}_y(\xi) = \begin{bmatrix} x\\z \\ x \\ z_d \\ \eta_o\\ \eta_c\\ p \\ \tau\end{bmatrix}.$$ It is noted that this system satisfies the basic assumptions required for the existence of solutions [@GoebSanf09 Assumption 6.5]. We are interested in asymptotic stability of the compact target set ${\mathcal{A}}$ defined as $$\label{eq:defSetA} {\mathcal{A}}:= \left \{ 0 \right\}^{4n+2} \times {\mathcal{P}}\times [0,N_0]$$ for the hybrid system . Our design problem can thus be formulated as follows: [*Problem statement:*]{} For each $p \in {\mathcal{P}}$, find the design functions $\beta_{o,p},\beta_{c,p}, \rho_{o,p}, \rho_{c,p}, \gamma_{o,p}, \gamma_{c,p}, \mu_{o,p}, \mu_{c,p}$ appearing in , , and the lower bound on the average dwell-time $\tau_a$, such that the set ${\mathcal{A}}$ defined in is globally asymptotically stable for the hybrid system . Stability Analysis ------------------ To state the main result of this section on asymptotic stability of the set ${\mathcal{A}}$ for system , we introduce the design criteria that must be satisfied by the functions introduced in the sampling algorithms , . Recalling the function $\nu_p$ introduced in , and the the hypotheses \[spvop\], \[spvcp\], the following conditions are imposed on the functions $\beta_{o,p}$, $\beta_{c,p}$, $\mu_{o,p}$, $\mu_{c,p}$, $\rho_{o,p}$ and $\rho_{c,p}$, for each $p \in {\mathcal{P}}$: \[Dalphah\] $\beta_{o,p}$ and $\beta_{c,p}$ are differentiable functions of class ${\mathcal{K}}$. \[Dsigmaoc\] Let $\theta_p$ be a function of class ${\mathcal{K}_\infty}$ defined as: $$\theta_{p}(s) := \alpha_{o,p}{^{\raisebox{.2ex}{$\scriptscriptstyle-1$}}}(2\gamma_{o,p}(s)).$$ Choose the functions $\mu_{o,p}$ and $\mu_{c,p}$ such that, for some $\lambda \in (0,1)$, $$(\gamma_{o,p} \circ \mu_{o,p})(s) [1 + (\nu_p \circ \theta_{p} \circ \mu_{o,p})(s)] \leq (1- \lambda) \beta_{o,p}(s)$$ $$2(\gamma_{c,p} \circ \mu_{c,p}) (s) \leq (1 - \lambda)\beta_{c,p}(s).$$ \[Drhoc\] The functions $\rho_{o,p}$ and $\rho_{c,p}$ in are positive definite and are chosen such that for each $s \geq 0$: $$\rho_{o,p} \circ \alpha_{h,p} \circ \underline{\alpha}_{c,p}{^{\raisebox{.2ex}{$\scriptscriptstyle-1$}}}(s) \leq 0.5(1 - \lambda)\alpha_{c,p}(s)$$ $$\rho_{c,p}(s) \leq (1-\lambda) \min \left\{ \gamma_{c,p}(s), 0.5 \, \alpha_{c,p}(\underline{\alpha}_{c,p}(s)) \right\}.$$ It can be guaranteed that there always exists a solution to the inequalities in \[Dalphah\], \[Dsigmaoc\], \[Drhoc\] using the properties of ${\mathcal{K}}_\infty$ functions and the results given in [@GeisWirt14 Corollary 3.2] and [@Kell14]. To state the main result, we recall the definition of $\ell_p$ from and choose $\widetilde{\alpha}_p \in {\mathcal{K}}_\infty$ such that $\widetilde{\alpha}_p(s) = $ $$\min \left\{ \beta_{o,p}\left({\frac{1}{4}}s \right), \beta_{c,p}\left({\frac{1}{4}}s \right), \alpha_{c,p}\left({\frac{1}{4}}s \right), \gamma_{c,p} \left( {\frac{1}{4}}\ell_p {^{\raisebox{.2ex}{$\scriptscriptstyle-1$}}}(s) \right) \right\}.$$ The function $\psi$ is chosen to be a differentiable ${\mathcal{K}}_\infty$ function, with $\psi'(0) = 0$, and \[eq:psiSamp\] $$\begin{aligned} & \psi(s) \le \min\{c_0s, \widetilde \alpha_p(s) \, \vert \, p \in {\mathcal{P}}\}, \quad {\color{blue} s \ge 0},\\ {\color{blue}\text{\sout{and}}} \quad & {\color{blue}\hbox{\sout{$\psi(s) \le \min\{\widetilde \alpha_p(s) \, \vert \, p \in {\mathcal{P}}\}, \quad s \ge c_1$}}},\end{aligned}$$ for some $c_0> 0$. Finally, let $$\label{eq:chiSamp} \chi(s) := \max_{p,q \in {\mathcal{P}}} \left\{ \ell_q \circ \overline{\alpha}_{o,q} \circ {\underline}\alpha_p{^{\raisebox{.2ex}{$\scriptscriptstyle-1$}}}(s) + \overline{\alpha}_{c,q} \circ {\underline}\alpha_p{^{\raisebox{.2ex}{$\scriptscriptstyle-1$}}}(s) \right\}.$$ \[thm:stabSampSys\] Consider the system and assume that \[alphah\], \[Assumq\], \[spvop\], \[spvcp\] hold. Suppose that the sampling algorithms and are designed such that \[Dalphah\], \[Dsigmaoc\], \[Drhoc\] are satisfied for some $\lambda \in (0,1)$. If the average dwell-time $\tau_a$ satisfies: $$\label{condADTSamp} \lambda \tau_a > \zeta^*:= \sup_{s \ge 0} \, \int_{s }^{\chi(s)} \frac{dr}{\psi(r)}$$ for $\psi$ and $\chi$ given in and , then the set ${\mathcal{A}}$ given in is globally asymptotically stable for the system . The fundamental idea behind the proof is to first construct a weak Lyapunov function $W$ for system with respect to set ${\mathcal{A}}$ in . Using additional arguments based on cascade hybrid systems, it is then shown that the solutions along which the derivative of $W$ is possibly zero, also converge to the set ${\mathcal{A}}$. We start with the function $$\label{constwp2} W(\xi) := \exp{({\color{blue}2 c_0}\zeta \tau)} \varphi(V_p(x,e) + \eta_o + \eta_c)$$ where $\zeta \in (\zeta^* , \lambda \tau_a)$ with $\zeta^* $ given in , $\varphi$ is defined in , and the function $V_p$ is defined as in . It is noted that $W$ does not involve the variables $(x_d, z_d) \in {\mathbb{R}}^{2n}$, so we only have the bounds $$\label{eq:Wsamp} {\underline}\alpha (\vert \xi \vert_{{\mathcal{A}}_c}) \le W (\xi) \le {\overline}\alpha (\vert \xi \vert_{{\mathcal{A}}_c})$$ where ${\mathcal{A}}_c := \{0\}^{2n} \times {\mathbb{R}}^{2n} \times {\mathbb{R}}_{\ge 0}^2 \times {\mathcal{P}}\times [0,N_0]$, and ${\underline}\alpha$, ${\overline}\alpha$ are some class ${\mathcal{K}}_\infty$ functions. When $\xi \in {\mathcal{C}}$, we have the derivative of $W$: $$\label{eq:dotWpSamp} \dot W = W \left[2{\color{blue}c_0} \zeta \dot \tau + \frac{2{\color{blue}c_0}} {\psi(V_p + \eta_o + \eta_c) } \left(\dot V_p + \dot \eta_o + \dot \eta_c \right) \right].$$ To show that $\dot W$ is bounded by a negative semidefinite function, we compute bounds on $\dot V_p, \dot \eta_o$ and $\dot \eta_c$ in the flow set ${\mathcal{C}}$. Using \[Assumq\], \[spvop\], \[spvcp\] and the inequality derived in , we obtain $$\begin{gathered} \label{dotVpsample} \dot V_p \leq -{\frac{1}{2}}\nu_p(V_{o,p}(e)) \alpha_{o,p}(V_{o,p}(e))\\ + \nu_p(\theta_p(\vert y - y_d\vert))\gamma_{o,p}(|y - y_d|) \\ - \alpha_{c,p}(V_{c,p}(x)) + \gamma_{c,p} (V_{o,p}(e)) + \gamma_{c,p}(|z-z_d|).\end{gathered}$$ It follows from the definition of $\nu_p$, sampling condition , and that $$\begin{aligned} \label{spdotvp} \dot V_p &\leq - \gamma_{c,p} (V_{o,p}(e)) + \nu_p(\theta_p({\mu_{o,p}(\eta_o)})) \gamma_{o,p}({\mu_{o,p}(\eta_o)})\notag \\ &\quad - \alpha_{c,p}(V_{c,p}(x)) + \gamma_{c,p}({\mu_{c,p}(\eta_c)}).\end{aligned}$$ The derivative of $\eta_o$ is seen to satisfy $$\begin{aligned} \label{dotetao} \dot \eta_o &\leq -\beta_{o,p}(\eta_o) + \rho_{o,p} \circ \alpha_{h,p}(|x|) + \gamma_{o,p}({\mu_{o,p}(\eta_o)}) \notag \\ &\leq -\beta_{o,p}(\eta_o) + \rho_{o,p} \circ \alpha_{h,p} \circ \underline{\alpha}_{c,p}{^{\raisebox{.2ex}{$\scriptscriptstyle-1$}}}(V_{c,p}(x)) + \gamma_{o,p}({\mu_{o,p}(\eta_o)}).\end{aligned}$$ The derivative of $\eta_c$ can be bounded as follows: $$\begin{aligned} \label{dotetac} \dot \eta_c &\leq - \beta_{c,p}(\eta_c) + \rho_{c,p}\left( \frac{|x| + |e|}{2} \right) + \gamma_{c,p}({\mu_{c,p}(\eta_c)}) \notag \\ &\leq - \beta_{c,p}(\eta_c) + \rho_{c,p} \circ \underline{\alpha}_{c,p}{^{\raisebox{.2ex}{$\scriptscriptstyle-1$}}}(V_{c,p}(x)) \notag \\ &\quad + \rho_{c,p} \circ \underline{\alpha}_{o,p}{^{\raisebox{.2ex}{$\scriptscriptstyle-1$}}}(V_{o,p}(e)) + \gamma_{c,p}({\mu_{c,p}(\eta_c)}).\end{aligned}$$ Now combining , , , and using the inequalities given in \[Dalphah\], \[Dsigmaoc\], and \[Drhoc\], we get $$\begin{aligned} \dot V_p + \dot \eta_o + \dot \eta_c &\leq -\lambda [ \beta_{o,p}(\eta_o) + \beta_{c,p}(\eta_c) + \alpha_{c,p}(V_{c,p}(x)) \notag \\ & \quad + \gamma_{c,p} {^{\raisebox{.2ex}{$\scriptscriptstyle-1$}}}\circ \ell_p{^{\raisebox{.2ex}{$\scriptscriptstyle-1$}}}(\ell_p(V_{o,p}(e))) ] \\ &\le -\lambda \widetilde{\alpha}_p(V_p + \eta_o + \eta_c).\end{aligned}$$ Substituting this expression in , and using the definition of $\psi$ in , we obtain $$\begin{aligned} \dot W &\leq {\color{blue}c_0} W \left[ 2 \zeta \dot \tau - \frac{2 \lambda \widetilde{\alpha}_p(V_p + \eta_o + \eta_c)}{\psi(V_p + \eta_o + \eta_c)} \right] \\ &\le {\color{blue}c_0} W( 2 \zeta \dot \tau - 2 \lambda) \\ &\le {\color{blue}c_0} W \left(\frac{2 \zeta}{\tau_a} - 2 \lambda \right) \\ &= {\color{blue}c_0} W \left(\frac{2(\zeta - \lambda\tau_a )}{\tau_a} \right),\end{aligned}$$ which is the desired inequality for $\dot W$ over the flow set since we chose $\zeta < \lambda\tau_a$. When $\xi \in {\mathcal{D}}$, we calculate the maximum of $W(\xi^+)$, over the set ${\mathcal{G}}(\xi) \ni g = \xi^+$, as follows: $$\begin{aligned} \max_{g\in{\mathcal{G}}(\xi)} W(g) &= \max_{g\in{\mathcal{G}}(\xi)} \exp(2 {\color{blue}c_0} \zeta \tau^+) \varphi\left( V_{p^+}(x^+,e^+) + \eta_o^+ + \eta_c^+ \right) \\ &= \max_{p^+\in {\mathcal{P}}} \exp(2 {\color{blue}c_0} \zeta \tau - 2 {\color{blue}c_0} \zeta) \varphi\left(V_{p^+}(x,e) + \eta_o + \eta_c \right). \notag \\\end{aligned}$$ To get a bound on the right-hand side in terms of the value of $\xi$ just prior to the jump, we recall that the function $\chi$ introduced in satisfies $$V_q \leq \chi(V_p), \quad \forall p,q \in {\mathcal{P}}.$$ Moreover, from the definition of $\varphi$, it follows that $$\varphi\left(V_{p^+} + \eta_o + \eta_c \right) \leq \exp \left( \int_{1}^{\chi(V_p)+\eta_o + \eta_c} \frac{2 {\color{blue}c_0} \, dr}{\psi(r)} \right).$$ We then observe that $$\begin{aligned} \int_{1}^{\chi(V_p)+\eta_o + \eta_c} \frac{2 \,dr}{\psi(r)} &= \int_{V_p+\eta_o + \eta_c}^{\chi(V_p)+\eta_o + \eta_c} \frac{2 \, dr}{\psi(r)} + \int_{1}^{V_p+\eta_o + \eta_c} \frac{2 \,dr}{\psi(r)}. \notag \\\end{aligned}$$ Since $\eta_o + \eta_c \geq 0$ and $\frac{1}{\psi(r)}$ is decreasing, we have: $$\int_{V_p + \eta_o + \eta_c}^{\chi(V_p) + \eta_o + \eta_c} \frac{2 {\color{blue}c_0} \, dr}{\psi(r)} \leq \int_{V_p}^{\chi(V_p)} \frac{2 {\color{blue}c_0} \, dr}{\psi(r)} \le 2 {\color{blue}c_0} \zeta^*,$$ so that $$\int_{1}^{\chi(V_p)+\eta_o + \eta_c} \frac{2 {\color{blue}c_0} \, dr}{\psi(r)} \leq 2 {\color{blue}c_0} \zeta^* + \int_{1}^{V_p+\eta_o + \eta_c} \frac{2 {\color{blue}c_0} \,dr}{\psi(r)}.$$ Thus, the value of $W$ after each jump is bounded as $$\begin{aligned} \max_{g \in {\mathcal{G}}(\xi)} W(g) &\leq \exp(-2{\color{blue}c_0} (\zeta-\zeta^*))\exp(2 {\color{blue}c_0}\zeta \tau) \varphi(V_p + \eta_o + \eta_c) \notag \\ &\leq \exp(-2{\color{blue}c_0} (\zeta-\zeta^*)) W(\xi) \qquad \forall\,\xi \in {\mathcal{D}}.\end{aligned}$$ Because of the bounds in , it thus follows that $\xi$ converges asymptotically to the set ${\mathcal{A}}_c$. To conclude further that $(x_d,z_d)$ also converge to $\{0\}^{2n}$, one can invoke the arguments based on the LaSalle’s invariance principle [@GoebSanf12 Corollary 8.9(ii)], and cascaded hybrid systems [@GoebSanf09 Corollary 19]. Following the same recipe as in [@TanwTeel15 Proof of Theorem 1], we next show that the set ${\mathcal{A}}$ of the closed-loop system is a globally asymptotically stable (GAS) for system : [*Step 1 – Pre-GAS of $\{0\}$ for truncated systems:*]{} For a fixed initial condition, there exist compact set ${\mathcal{M}}_1 \subset {\mathbb{R}}^{2n}$, ${\mathcal{M}}_2 \subset {\mathbb{R}}^2 \times {\mathcal{P}}\times {\mathbb{R}}$ such that $(x,z) \in {\mathcal{M}}_1$ and $(\eta_o,\eta_c,p,\tau) \in {\mathcal{M}}_2$. Recalling that $z_d$ and $x_d$ remain constant during flows, and are reset to $z$ and $x$, which belong to a compact set, there exists a compact set ${\mathcal{M}}_d$ such that $(x_d, z_d) \in {\mathcal{M}}_d$. Consider the truncation of system  to the set ${\mathcal{M}}:={\mathbb{R}}^{2n} \times {\mathcal{M}}_d \times {\mathbb{R}}_{\ge 0}^2 \times {\mathcal{P}}\times [0,N_0]$, which has the flow set ${\mathcal{C}}_{{\mathcal{M}}}:={\mathcal{C}}\cap {\mathcal{M}}$, the jump set ${\mathcal{D}}_{{\mathcal{M}}}:= {\mathcal{D}}_{{\mathcal{M}}} \cap {\mathcal{M}}$. For this truncated system, it follows from the invariance principle [@GoebSanf12 Corollary 8.9(ii)] that the set ${\mathcal{A}}_1:= \{0\}^{2n} \times {\mathcal{M}}_d \times \{0\}^2 \times {\mathcal{P}}\times [0,N_0]$ is pre-GAS. We next invoke the stability result for cascaded hybrid systems [@GoebSanf09 Corollary 19] to claim that the set ${\mathcal{A}}$ in is pre-GAS for the truncated system. Indeed, for every system trajectory contained in ${\mathcal{A}}_1$, we have $\eta_o =\eta_c = 0$, and from the definition of the sets ${\mathcal{C}}$ and ${\mathcal{D}}$, we must then have $x_d = 0$ and $z_d = 0$. [*Step 2 – Bounded solutions and Pre-GAS of $\{0\}$ for :*]{} As shown in the first step, for each initial condition, there exist compact sets ${\mathcal{M}}_1$, ${\mathcal{M}}_2$ and ${\mathcal{M}}_d$ such that $\xi$ is contained in the compact set ${\mathcal{M}}_1 \times {\mathcal{M}}_d \times {\mathcal{M}}_2$ for all times. Boundedness of the solutions now allows us to conclude that ${\mathcal{A}}$ is pre-GAS for the original system . To see this, assume that there exists a solution for which $(x,z,x_d,z_d,\eta_o,\eta_c)$ does not converge to $\{0\}$. Since all solutions are bounded, there exists a compact set ${\mathcal{M}}_d$ such that this bounded solution eventually coincides with the solution of the system truncated to ${\mathbb{R}}^{2n} \times {\mathcal{M}}_d \times {\mathbb{R}}_{\ge 0}^2 \times {\mathcal{P}}\times [0,N_0]$. But, every solution of the truncated system must converge to ${\mathcal{A}}$. Hence, for , a bounded solution not converging to ${\mathcal{A}}$ cannot exist, proving that ${\mathcal{A}}$ is pre-GAS. [*Step 3 – $\{0\}$ is GAS for :*]{} To move from pre-asymptotic stability to asymptotic stability of the compact set ${\mathcal{A}}$, we next show that every solution of is forward complete. This is seen due to the fact that for each $\xi \in {\mathcal{C}}\setminus {\mathcal{D}}$, the solutions would always continue to flow. Moreover, after each jump the states are reset to the set ${\mathcal{C}}\cup {\mathcal{D}}$, making it possible to extend the time domain for the solutions either by jump or flow. Hence, each solution of the system is forward complete, proving that the set ${\mathcal{A}}$ is GAS. For implementation purposes, it is important to show that, for event-based sampling, there is a uniform lower bound on the minimal inter-sampling time between two consecutive sampling instants. For the algorithms employed in this article, such a lower bound has been obtained for nonswitched dynamical systems in [@TanwTeel15 Theorem 2] under certain additional assumptions on the functions appearing in the dynamic filters . For switched systems, when working under the slow switching assumption like dwell-time or average dwell-time, it suffices to have such a lower bound for an individual dynamical subsystem since this guarantees there will be no accumulations of jump events if the switching signal has no accumulation of switches. Example and Simulation Result {#sec:example} ============================= As an illustration of Theorem \[thm:stabSampSys\], we consider an academic example of a switched system with two modes. The first subsystem is described by linear dynamics as follows: $$\label{linearswisysexample} p = 1: \begin{cases} \dot x= A_1 x + B_1 u \\ y = C_1 x \end{cases}$$ The feedback controller related to this subsystem is: $$p = 1: \begin{cases} \dot z = A_1 z + B_1 u_d + L_1(y - C_1z) \\ u = - K_1 z, \end{cases}$$ where we choose $A_1 =\begin{bmatrix} 0.5 & -1\\ 0 & 0.5\end{bmatrix}$, $B_1 = \begin{bmatrix} 0 \\ 1\end{bmatrix} $, $C_1 = \begin{bmatrix}1 & 0\end{bmatrix}$, $L_1 = \begin{bmatrix} 3.5 \\ -3\end{bmatrix}$ and $K_1 =\begin{bmatrix}-1.5 & 2.5\end{bmatrix} $. The second subsystem has nonlinear dynamics described by: $$p = 2: \begin{cases} \dot x_{1} = x_{2} + 0.25 | x_{1} | \\ \dot x_{2} = \mbox{sat} (x_{1} )+u_d \\ y = x_{1}. \end{cases}$$ The notation $\mbox{sat}$ denotes the saturation function $\mbox{sat} (x_1) = \min \left\{ 1, \max \left\{ -1, x_{1}\right\} \right\}$. The corresponding feedback controller is: $$p = 2: \begin{cases} \dot z_{1} = z_{2} + 0.25 |y| + l_{1}\left(y - z_{1} \right) \\ \dot z_{2} = \mbox{sat}(y) + u_d + l_{2} \left( y - z_{1} \right) \\ u = \mbox{sat}(z_{1}) + K_2 z, \end{cases}$$ where we choose $K_2 = \begin{bmatrix} -2 & -2\end{bmatrix} $ and $L_2 =\begin{bmatrix} l_{1} \\ l_{2}\end{bmatrix}= \begin{bmatrix} 2 \\ 2\end{bmatrix}$. For both subsystems, we introduce the same form of Lyapunov function: $V_{c,p}(x) = x^T P_{c,p} x$ and $V_{o,p}(e) = e^T P_{o,p} e$. Since the controller is driven by the sampled output $y_{d}$, we have for each $p \in \left\{ 1, 2\right\}$: $$\dot V_{o,p} \leq -a_{o,p} V_{o,p}(e) + {\overline{\gamma}_{o,p}}\left| y - y_d \right|^2,$$ where $$a_{o,p} = \frac{\lambda_{\min}(Q_{o,p})}{2\lambda_{\max}(P_{o,p})} \quad \text{ and } \quad {\overline{\gamma}_{o,p}}= \frac{2\| P_{o,p}L_p \|^2}{\lambda_{\min}(Q_{o,p})}.$$ Similarly, for each $p \in \left\{ 1, 2\right\}$: $$\dot V_{c,p} \leq -a_{c,p} V_{c,p}(x) + {\overline{\gamma}_{c,p}}V_{o,p}(e)+ {\overline{\gamma}_{c,p}}\left| z- z_d \right|^2,$$ where $$a_{c,p} = \frac{\lambda_{\min}(Q_{c,p})}{2\lambda_{\max}(P_{c,p})},$$ and $${\overline{\gamma}_{c,p}}= \frac{4 \| P_{c,p} B_p K_p \|^2}{\lambda_{\min}(Q_{c,p})} \max\left\{ 1,\frac{1}{\lambda_{\min}(P_{o,p})}\right\}.$$ ![Simulation results: In the top plot, whenever $|y(t)-y_d(t)|$ reaches the sampling threshold $\overline \mu_{o,\sigma}\sqrt{\eta_o(t)}$, $y_d$ is updated. The middle plot shows $\overline \mu_{c,\sigma}\sqrt{\eta_c(t)}$ and $|z(t)-z_d(t)|$. The bottom plot shows the switching signal used in the simulations.[]{data-label="fig:simExSamp"}](simTemp.pdf "fig:"){width="48.00000%"} -0.5em For the dynamic filters, we choose $$\begin{aligned} &\dot \eta_o = -a_{o,p}\eta_o + {\overline{\rho}_{o,p}}|y|^2 + {\overline{\gamma}_{o,p}}|y-y_d|^2 \notag \\ & \dot \eta_c = -a_{c,p}\eta_c + {\overline{\rho}_{c,p}}\frac{|z|^2}{4} + {\overline{\gamma}_{c,p}}|z-z_d|^2, \notag\end{aligned}$$ where we let $$\begin{aligned} &{\overline{\rho}_{o,p}}:= \frac{(1-2{\varepsilon})\lambda_{\min} (P_{o,p})}{{\left\lVertC_p\right\rVert}^2},\\ &{\overline{\rho}_{c,p}}:= \min \left\{ (1-{\varepsilon}){\overline{\gamma}_{c,p}}, {\varepsilon}a_{c,p}\lambda_{\min} (P_{c,p}) \right\}\end{aligned}$$ for some small ${\varepsilon}\in (0,0.5)$. The jump set which describes the conditions when the sampled values get updated or when there is a switching event occurs, is defined as follows: $$\begin{gathered} {\mathcal{D}}= \left\{\xi : |y-y_d| \geq {\overline{\mu}_{o,p}}\sqrt{\eta_o} \right\} \cup \left\{ \xi : |z-z_d| \geq {\overline{\mu}_{c,p}}\sqrt{\eta_c}\right\} \\ \cup \left\{ \tau \in [1,N_0]\right\} \end{gathered}$$ where ${\overline{\mu}_{o,p}}:= \frac{(1-{\varepsilon}) \alpha_{o,p}}{(1+\overline{\nu}_p){\overline{\gamma}_{o,p}}}$ and ${\overline{\mu}_{c,p}}:= \frac{(1-{\varepsilon})\alpha_{c,p}}{2{\overline{\gamma}_{c,p}}}$, with $$\overline{\nu}_p = \frac{4\overline{\gamma}_{c,p}}{\lambda_{\min} (P_{o,p})}.$$ The function $\chi$ can thus be defined as $\chi (s)= {\overline}\chi s$, where $$\overline{\chi} = \max_{p,q \in \left\{ 1, 2 \right\}} \left\{ \frac{\overline{\nu}_p \lambda_{\max} (P_{o,p}) +\lambda_{\max} (P_{c,p})}{\overline{\nu}_q \lambda_{\min} (P_{o,q}) +\lambda_{\min} (P_{c,q}) } \right\}.$$ It follows from Theorem \[thm:stabSampSys\] that, if the average dwell-time $\tau_a$ satisfies that: $$\tau_a > \frac{\ln \overline{\chi}}{{\varepsilon}},$$ then we have the asymptotic stability of the origin for $(x,z)$ system. The simulation results reported in Figure \[fig:simExSamp\] indeed show the convergence of $(y,z,\eta_o,\eta_c)$ to the origin. Conclusion {#sec:conc} ========== In this article, the construction of ISS Lyapunov functions is considered for switched nonlinear systems in cascade configuration. The stability analysis for the resulting hybrid systems is carried out under an average dwell-time condition on the switching signal, and an asymptotic ratio condition for establishing ISS. The results pave the path for studying the stabilization of switched systems with dynamic output feedback. A Lyapunov function similar to the one used for non-sampled ISS system is constructed to design the sampling algorithms and for the analysis of the closed-loop hybrid systems with sampled measurements. The results are illustrated with the help of examples and simulations. One of the limitations of the dynamic output feedback problem considered in this paper is that the controller requires exact knowledge of the switching signal. It is of interest to develop theoretical tools when there is mismatch in the switching signal between the plant and the controller. One can also consider additional measurement errors, for example, due to quantization of output and input in space, as done in [@TanwPrie16]. One could also potentially study the affect of random uncertainties, on top of event-based samples, as has been recently proposed in [@TanwTeel17]. Acknowledgements {#acknowledgements .unnumbered} ================ The authors would like to thank Dr. Guosong Yang for his useful comments on an earlier version of this paper. [^1]: It is also possible to consider more general jump maps of the form $x^+ = g_c(x,e,d)$ and $e^+ = g_o(x,e,d)$, provided that the inequalities in \[assJump\] take the form $| g_c(x,e,d)| \leq \widehat{\alpha}_c(|(x,e)|) + \widehat{\rho}_c(|d|)$ and $| g_o(x,e,d)| \leq \widehat{\alpha}_o(|(x,e)|) + \widehat{\rho}_o(|d|)$. The results of this paper would carry just by changing the map $\rho$ in . [^2]: The function $\frac{2W(\xi)}{\psi(V_p(x,e))}$ is obviously nonnegative. The continuity follows from recalling that $\varphi$ is continuously differentiable with $\varphi'(0) = 0$, and observing that $\frac{2W(\xi)}{\psi(V_p(x,e))} = \exp{(2\zeta\tau)}\frac{\varphi(V_p(x,e))}{\psi(V_p(x,e))} = \exp{(2\zeta\tau)} \varphi'(V_p(x,e))$.

--- abstract: 'In simple fluids, such as water, invariance under parity and time-reversal symmetry imposes that the rotation of constituent “atoms” are determined by the flow and that viscous stresses damp motion. Activation of the rotational degrees of freedom of a fluid by spinning its atomic building blocks breaks these constraints and has thus been the subject of fundamental theoretical interest across classical and quantum fluids [@lenz_membranes_2003; @yeo_rheology_2010; @furthauer_active_2013; @nguyen_emergent_2014; @goto_purely_2015; @yeo_collective_2015; @climent_dynamic_2006; @avron_viscosity_1995; @wiegmann_anomalous_2014; @ariman_microcontinuum_1973; @rosensweig2013; @Rinaldi2014]. However, the creation of a model liquid which isolates chiral hydrodynamic phenomena has remained experimentally elusive. Here we report the creation of a cohesive two-dimensional chiral liquid consisting of millions of spinning colloidal magnets and study its flows. We find that dissipative viscous “edge pumping” is a key and general mechanism of chiral hydrodynamics, driving uni-directional surface waves and instabilities, with no counterpart in conventional fluids. Spectral measurements of the chiral surface dynamics reveal the presence of Hall viscosity, an experimentally long sought property of chiral fluids [@avron_viscosity_1995; @avron_odd_1998; @abanov2018; @banerjee_odd_2017]. Precise measurements and comparison with theory demonstrate excellent agreement with a minimal but complete chiral hydrodynamic model, paving the way for the exploration of chiral hydrodynamics in experiment.' author: - 'Vishal Soni\*' - 'Ephraim Bililign\*' - 'Sofia Magkiriadou\*' - Stefano Sacanna - Denis Bartolo - 'Michael J. Shelley' - 'William T. M. Irvine' title: 'The free surface of a colloidal chiral fluid: waves and instabilities from odd stress and Hall viscosity' --- Hydrodynamic theories describe the flow of systems as diverse as water, quantum electronic states [@bandurin_negative_2016], and galaxies [@pringle_astrophysical_2007] over decades in scale [@secchi_massive_2016]. Since hydrodynamic equations are built on symmetry principles and conservation laws alone, systems with similar symmetries have similar descriptions and flow in the same way. For example, symmetry under parity and time reversal – conditions met by all conventional fluids at thermal equilibrium – constrains both the stress and viscosity tensors to be symmetric. These constraints are in principle alleviated in collections of interacting units that are driven to rotate [@tsai_chiral_2005; @scaffidi_hydrodynamic_2017; @wiegmann_anomalous_2014; @banerjee_odd_2017; @van_zuiden_spatiotemporal_2016; @furthauer_active_2013; @avron_viscosity_1995; @snezhko_complex_2016; @kokot2017]. This seemingly innocent twist on an otherwise structureless fluid represents, however, an elemental change with rich hydrodynamic consequences common to quantum Hall fluids, vortex fluids, and chiral condensed matter. Collections of spinning particles offer a natural opportunity to engineer and study the properties of such chiral fluids; experimental examples include rotating bacteria [@petroff_fast-moving_2015], colloidal and millimeter-scale magnets [@grzybowski_dynamic_2000; @grzybowski_dynamic_2001; @grzybowski_dynamics_2002; @grzybowski_dynamic_2002; @belovs_hydrodynamics_2016; @yan_rotating_2014; @yan_jing_colloidal_2015], ferrofluids in rotating magnetic fields [@rosensweig2013; @Rinaldi2014], and shaken chiral grains [@tsai_chiral_2005; @scholz_rotating_2018]. Such systems have been shown to have non-trivial dynamics. For example, ferrofluids driven by AC fields can flow against external pressure [@Bacri1995] and small numbers of spinning particles self-assemble into dynamic crystalline clusters [@grzybowski_dynamic_2000; @grzybowski_dynamic_2001; @grzybowski_dynamics_2002; @grzybowski_dynamic_2002; @climent_dynamic_2006; @yan_rotating_2014; @yan_jing_colloidal_2015]. {width="\textwidth"} A colloidal chiral fluid ======================== We report the creation of a millimeter-scale cohesive chiral fluid (Fig. \[fig:chiralfluid\]a) by spinning millions of colloidal magnets with a magnetic field (Figs. \[fig:chiralfluid\]b, \[fig:chiralfluid\]c), and we track its flows over hours (see Supplementary Movies 1, 2). The macroscopic flow of our chiral fluid is reminiscent of free surface flows of Newtonian fluids: nearby droplets merge (Fig. \[fig:chiralfluid\]d and Supplementary Movie 3), fluid spreads on a surface under the influence of gravity (Fig. \[fig:chiralfluid\]e and Supplementary Movie 4), voids collapse (Fig. \[fig:chiralfluid\]f and Supplementary Movie 5), and thin streams go unstable, as revealed by flowing fluid past a solid object (Fig. \[fig:chiralfluid\]g and Supplementary Movie 6). We demonstrate that these seemingly familiar features are accompanied by unique free surface flows. We then exploit the odd interfacial dynamics of this prototypical chiral liquid to infer its material constants, which remain out of reach of conventional rheology. In contrast to Newtonian fluids, the surface of our fluid supports a spontaneous unidirectional edge flow in its rest state, as well as unusual morphological dynamics such as the rotation of asymmetric droplets illustrated in Supplementary Movie 3. Chiral surface waves and ‘edge pumping’ ======================================= To investigate these lively surface flows, we first look at surface excitations in a simple slab geometry, as shown in Fig. \[fig:spectrum\]a and Supplementary Movie 7. We measure the spectrum of surface fluctuations, $|h(k,\omega)|^2$, by tracing the height profile, $h(x,t)$, of the surface and Fourier-transforming it in space and time. We observe the spectrum to be peaked along a curve $\omega(k)$, revealing the existence of dispersive waves (see Fig. \[fig:spectrum\]b). The curve has only one branch with odd parity, meaning that the waves are unidirectional. This behavior contrasts that of conventional surface waves that propagate in all directions. {width="100.00000%"} These surface waves beg a hydrodynamic description. Chiral-fluid hydrodynamics follows from conservation of momentum and angular momentum, and thus includes both the spinning rate of individual fluid particles as well as the momentum and angular momentum of their flow [@furthauer_active_2013; @bonthuis_electrohydraulic_2009; @tsai_chiral_2005; @dahler_theory_1963; @huang_continuum_2010]. Because our colloids are birefringent, we are able to measure their individual spinning rate by imaging through crossed polarizers. We find that all particles rotate at the same rate, $\Omega$, which is set by the rotating magnetic field (see Fig. \[fig:droplets\]a and Supplementary Movie 8). From this it follows that the particles’ rotational inertia is negligible; the torque exerted on each particle by the magnetic field instantly adjusts to balance the frictional torques exerted by the neighboring particles and the solid substrate. This fast response enables the decoupling of the angular momentum equation from the momentum equation. Nonetheless a strong signature of the microscopic angular momentum manifests as an ‘odd’ stress. A minimal hydrodynamic theory then balances the force generated by viscous and odd hydrodynamic stresses, $\partial_j \sigma_{ij}$, against friction with the substrate, $\Gamma_{u} v_i$, and surface tension $\gamma$ at the fluid interface. In this theory, which has been used to capture the bulk flows of chiral granular fluids, the hydrodynamic stress tensor is given by: $$\begin{aligned} \sigma_{ij} &= -p \delta_{ij} + \eta \left(\partial_i v_j+\partial_j v_i\right) + \eta_{\rm R} \epsilon_{ij}\left(2\Omega-\omega\right).\end{aligned}$$ $\sigma_{ij}$ includes the pressure $p$ and ordinary viscous stress also present in Newtonian fluids with a shear viscosity $\eta$. The additional term containing the Levi-Civita symbol $\epsilon_{ij}$ and the rotational viscosity $\eta_R$, captures the rotational friction between neighboring particles [@dahler_theory_1963; @bonthuis_electrohydraulic_2009; @tsai_chiral_2005; @furthauer_active_2013]. Such an odd stress builds up as the local spinning rate $\Omega$ deviates from half the local fluid vorticity $\omega = \hat{z}\cdot (\nabla \times v)$. In torque-free fluids, angular momentum conservation constrains these two quantities to be equal: odd stresses are unique to chiral fluids. We note that there is no direct appearance of the magnetic field or its stresses in this hydrodynamic description unlike in conventional ferrofluids. In this respect, our colloidal chiral fluid can be seen as a special type of driven ferrofluid in which the only role of magnetic forces is to induce chirality. To make a quantitative comparison between our model and the flows we observe, we require a measurement of the hydrodynamic and friction coefficients $\eta$, $\eta_{\rm R}$, and $\Gamma_{u}$. Fortunately, the prominent effect of odd stress at the free surface of our chiral fluid can be effectively exploited to infer its bulk rheology. The homogeneous spinning motion of the colloidal particles gives rise to a net tangential edge flow even in the absence of pressure gradients. These tread-milling dynamics, characteristic of all chiral fluids [@tsai_chiral_2005; @nguyen_emergent_2014; @petroff_fast-moving_2015; @yan_rotating_2014; @van_zuiden_spatiotemporal_2016], are illustrated in circular droplets in Figs. \[fig:droplets\]b-e and Supplementary Movie 9. The tangential flow that is localized at the free surface is readily explained by expressing the hydrodynamic equation in terms of vorticity for an incompressible chiral fluid: $$\left(\nabla^2-{\delta^{-2}}\right)\omega=0 \label{Eq:vorticity}$$ where $\delta=\sqrt{{(\eta+\eta_{R})}/{\Gamma_{u}}}$. This Helmholtz equation indicates that the vorticity generated at the surface decays exponentially into the fluid, with a characteristic penetration depth $\delta$ (see Figs. \[fig:droplets\]c, d, g). In this model, the absence of substrate friction causes the penetration depth to diverge, resulting in rigid-body rotation of the entire fluid, as observed in ferrofluid droplets [@Bacri1994]. The magnitude of the vorticity at the free surface, $\omega_{\rm edge}=2\Omega\,\eta_{\rm R}/(\eta+\eta_{\rm R})$, is set by the stress-free boundary condition for a flat strip and expresses the competition between the odd and viscous stresses (see Supplementary Information). We point out that $\omega_{\rm edge}$ is directly proportional to $\eta_R$, which demonstrates the importance of odd stress for the dynamics. Comparison between experiment and prediction (Fig. \[fig:droplets\]d) yields the values of $\eta$ and $\eta_{R}$ in terms of $\Gamma_{u}$. The latter is then measured by tilting the substrate and measuring the sedimentation rate of droplets (see Fig. \[fig:droplets\]f, and Supplementary Information). Ultimately, we find $\eta= 4.9\pm 0.2 \times 10^{-8}\ \rm Pa\ m\ s$, $\eta_{R} = 9.1\pm 0.1\times 10^{-10}\ \rm Pa\ m\ s$, and $\Gamma_{u}=2.49\pm 0.03 \times 10^{3}\rm\ Pa\ s/m$. Equipped with the hydrodynamic coefficients we can now investigate the origin of the surface waves within our model. The mass flux in the tangential surface flow provides significant insight. This flow, sketched in Fig. \[fig:spectrum\]d and plotted in Figs. \[fig:spectrum\]e-f, is determined by the balance of the tangential odd stress at the boundary, the shear stress, and the substrate friction. In the presence of a perturbation that varies the curvature of the interface, resistance to flow due to the shear stress will be modulated. For a sinusoidal perturbation, there is enhanced flow in positively curved regions (top of the wave) and decreased flow in negatively curved regions (bottom of the wave). This ‘edge-pumping’ moves material away from curved regions towards the flat wave front, giving rise to uni-directional wave motion. A linear stability analysis of the hydrodynamic equations (see Supplementary Information for a detailed calculation) confirms this scenario and yields a prediction for the dispersion relation, dissipation rate, and flow fields of surface waves, which we plot in Fig. \[fig:spectrum\]b (red dashed curves). With no fitting parameters, our model shows excellent agreement with the experimentally measured dispersion relation. For surface waves $h\sim e^{i(k x + \omega t)} $ of long wavelength $k \ll 1/\delta$, the asymptotic dispersion relation is: $$\omega(k) = 2 \omega_{\rm edge} \frac{\eta_{\rm R}}{\eta + \eta_{\rm R}} (k \delta)^3 = 2 u_{\rm edge} \frac{\eta}{\Gamma_{u}} k^3. \label{Eq:dispersion}$$ where $u_{\rm edge}=2\frac{\eta_R}{\eta+\eta_R}\Omega \delta$. ![[**Characterization of a droplet of chiral spinner fluid.**]{} [**a,**]{} When viewed through crossed polarizers, the particles blink as they spin. This allows us to confirm that they all spin at the same frequency, set by the rotating magnetic field. [**b,**]{} By measuring the velocity of each particle within a cluster, we find a flow profile that is concentrated at the edge within a penetration layer $\delta$ shown in [**c, d,**]{} and [**g**]{}. [**c,**]{} A zoomed-in view of the flow streamlines, obtained by averaging several instantaneous velocity profiles such as the one shown in [**b**]{}. [**d,**]{} By measuring the flow profile, the edge current $\mathrm{u}_{\rm edge}$ and penetration depth $\delta$ are extracted. [**e, g,**]{} By measuring the flow profile $u(r)$ at a range of frequencies, we extract the shear viscosity, $\eta$, and rotational viscosity, $\eta_{R}$, in terms of the substrate friction, $\Gamma_{u}$. [**f,**]{} Finally, by tilting a sample and measuring the sedimentation velocity of a droplet, we extract the substrate friction. []{data-label="fig:droplets"}](fig3.pdf){width=".5\textwidth"} The wave dynamics are thus crucially sensitive to boundary layer flows. A natural avenue for investigation, then, is to seek to increase the thickness of the boundary in order to increase its relative role. We now show how a slight increase of the penetration depth of the boundary layer amplifies chiral effects and reveals a long sought-after source of stress, commonly referred to as Hall viscosity. Chiral wave damping and measurement of Hall Viscosity ===================================================== We reduce the surface friction by allowing our chiral liquid to sediment upon an air-water interface (Fig. \[fig:etao\]b), as opposed to a glass surface (Fig. \[fig:etao\]a). Due to the difficulty in maintaining a slab geometry in this regime, we examine surface fluctuations on circular droplets. As can be seen in Fig. \[fig:etao\]a-b and Supplementary Movie 10, the edge flow penetrates deeper into the chiral fluid as friction is reduced. The dispersion relations for high and low friction droplets display the same trend, although the range of accessible wave vectors normalized by the penetration length ($k\delta$) is larger in the low friction case. An extension of our theory to circular geometries (see Supplementary Information) again accurately captures the dispersion relations for high friction (Fig. \[fig:etao\]a) and low friction (Fig. \[fig:etao\]b). The remarkable agreement between experiment and theory is however challenged when investigating the damping dynamics of the chiral waves. Experimentally, the damping rate $\alpha$ of chiral waves of wave vector $k$ is given by fitting a Lorentzian to the width of the power spectrum (see Supplementary Information); the resulting damping rates are shown in Fig. \[fig:etao\]c-d. Our hydrodynamic theory predicts this damping rate to be proportional to surface tension. This is natural since surface tension flattens interfacial deformation: in the absence of inertia, the relaxation does not overshoot and capillary waves are overdamped. In the long wavelength limit ($k\delta\ll 1$), the damping rate $\alpha\sim (\gamma/\Gamma_u) |k|^3$ stems from the competition between surface tension and substrate friction. As seen in Fig. \[fig:etao\]c, in the high friction case we again find excellent agreement between theory and experiment, which provides a direct measurement of surface tension. The value we find, $\gamma = 2.3 \pm 0.2 \times 10^{-13}\rm\ N$, is consistent with an estimate based on magnetic interactions between rotating dipoles (see Supplementary Information). In the case of low surface friction, however, we observe a distinct new feature in the dissipation rate: a leveling off of the dissipation rate at short wavelengths which cannot be accounted for by the hydrodynamic theory discussed thus far, suggesting the presence of an additional mechanism for surface wave dissipation in our chiral fluid. Seeking a hydrodynamic description, we recall that isotropic chiral fluids can in principle possess an additional stress in their constitutive relation, known interchangeably as “anomalous viscosity", “odd viscosity" or “Hall viscosity" [@avron_viscosity_1995; @avron_odd_1998; @read_non-abelian_2009]. This non-dissipative, transverse stress is linked by Onsager relations to the breaking of time-reversal symmetry. Theoretically, odd viscosity has indeed been shown to arise in the hydrodynamics of plasmas and systems of spinning molecules, ‘gears’, as well as quantum Hall fluids and vortex fluids [@radin_lorentz_1972; @robinson_variational_1962; @pitaevskii_physical_1981; @wiegmann_anomalous_2014; @banerjee_odd_2017]. We therefore conjecture our chiral fluid to support an additional Hall stress $\sigma^{\rm o}_{ij}=\eta_o \left( \partial_i \epsilon_{jk}v_k + \epsilon_{ik}\partial_k v_j\right)$. In incompressible fluids such as the one considered here, the effect of odd viscosity can solely be seen at the edge. This is because in the bulk flow Hall stress is merely absorbed into the fluid pressure. The signature of odd viscosity in our chiral fluid is thus an additional boundary stress. The component normal to the interface $\sigma_{nn}$ is given by $$\sigma_{nn}=\eta_{\rm o}\left(\partial_sv_{n}+\frac{v_s}{R(s)}\right), \label{Eq:Hallstress}$$ where $v_{n}$ (resp. $v_s$) is the velocity normal (resp. tangential) to the surface (see Fig. \[fig:etao\]e), and $R(s)$ is the local radius of curvature. In our system, where odd stress powers a boundary-layer edge flow, we thus expect odd viscosity to flatten surface deformation in a manner akin to surface tension, $\sigma^{\rm o}\sim \eta_{\rm o}v_s/R$. The excellent agreement between our measurements and predictions from a full hydrodynamic theory confirms this simplified picture and establishes the presence of Hall viscosity in our colloidal chiral fluid (see Fig. \[fig:etao\]d, f-g). From the fit we obtain $\eta_{\rm o}=1.4\pm 0.1 \times 10^{-8}\ \rm Pa\ m\ s$. {width="100.00000%"} The clearly visible decrease in slope in the damping relation is the most visible signature of Hall viscosity in our data and can be understood on dimensional grounds. In the long wavelength limit, the wave relaxation time is controlled by the competition of either surface tension or Hall stress with substrate friction. Dimensionally this implies a scaling $\alpha \sim \vert k\vert^3$ since the ratios $\gamma/\Gamma_u$ and $\eta_{\rm o} v_s/\Gamma_u$ have dimension of volume per unit time. In contrast, in the short wavelength limit, surface friction plays no role and damping stems from the competition of surface tension or Hall stress and bulk viscosities alone. In this case dimensional analysis requires linear scaling with wavenumber in the case of surface tension, and wave-number independence in the case of Hall stress (see Supplementary Information). This change in wavenumber dependence brings about a visible rollover to a decreased slope in the wave damping rate. We note that for small ranges of $k\delta \sim [-1,1]$, characteristic of spectral measurements in the presence of high surface friction, the leveling off cannot be seen and the relative roles of Hall viscosity and surface tension become hard to separate. This is the case for the damping shown in Fig. \[fig:etao\]c which can be fit well by both a non-zero and zero value of Hall viscosity (see Supplementary Information). Having established the presence of Hall viscosity by examining wave damping, it follows to ask whether it has an effect on wave propagation. The first term in Eq.  suggests that Hall viscosity and surface tension could act together to support wave propagation. Surface tension acts on a sinusoidal surface deformation by pulling down peaks and pushing up troughs, generating an in-phase normal velocity component. The normal Hall stress $\partial_s v_n$ would then act out of phase on the inflection points of the sinusoidal perturbation to propagate it in a chiral fashion. Our full theory confirms that this additional wave-pumping mechanism indeed exists and generates waves even in the absence of edge currents. However, for our hydrodynamic parameters, their effect on the dispersion is minimal. An odd instability ================== In much of the phenomenology we have discussed, surface dynamics are essentially boundary layer dynamics. Another natural question, then, is what happens when two boundary layers meet? Draining fluid past a curved obstacle brings about the progressive thinning of a curved strip of chiral fluid, as shown in Fig. \[fig:chiralfluid\]g and Supplementary Movie 6. The flow is smooth until the strip thickness becomes comparable to the penetration depth $\delta$; at that point the flow goes unstable, resulting in the formation of circular droplets. We study this novel pearling mechanism in experiment by creating a sequence of strips of decreasing thickness, as shown in Fig. \[fig:instability\]a and Supplementary Movie 11. We find that over a period of 10 minutes the strips of chiral fluid are stable for thicknesses above $\sim 32\,\rm \mu m$ and unstable below. {width="\textwidth"} Although visually reminiscent of the Rayleigh-Plateau instability of a thin fluid cylinder jet [@eggers_nonlinear_1997], this instability is fundamentally different. In our two-dimensional system, surface tension is a purely stabilizing force, as seen in the wave analysis discussed above. Instead, the instability originates from the chiral surface dynamics of our fluid. A visual signature of this origin is the consistent offset in the phase between top and bottom perturbations at the moment the instability occurs in all strips: Fig. \[fig:instability\]b shows one such example. A linear stability analysis of a thin strip of chiral fluid quantitatively predicts the existence of unstable modes which consist of wave-like perturbations on the top and bottom surfaces that have a relative phase offset, as sketched in Fig. \[fig:instability\]d. These are accompanied by a stable mode with an opposite relative phase. The associated stability diagram is shown in Fig. \[fig:instability\]e, together with our experimental observations. As the Hall stress has little effect on the stability of modes for small $\delta$ (see Supplementary Information), here we set $\eta_o=0$. An intuitive picture for the mechanism driving the instability is illustrated in Fig. \[fig:instability\]d. The geometry of a thin slab with out-of-phase perturbations on the top and bottom surfaces can be approximated by a collection of elongated droplets of chiral fluid all canted in the same direction. Droplets of this kind rotate in the direction of the edge current, in this case clockwise (see Fig. \[fig:chiralfluid\]d and Supplementary Movie 3). Depending on the phase difference between the two interfaces, the rotation of these effective droplets will either increase the amplitude of the perturbation, resulting in the breakup of the strip (top); or decrease the amplitude of the perturbation and restore the flat interface (bottom). The consistent observation of this phase relation between the top and bottom perturbations across many experiments of strips going unstable (Fig. \[fig:instability\]c) further corroborates our theoretical picture of the instability. We have broken parity symmetry at the microscopic level in a colloidal chiral fluid, resulting in the emergence of an odd stress that in turn generates lively surface flows. Likewise, we have broken time reversal symmetry, giving rise to Hall viscosity, a dissipationless transport property which has thus far remained experimentally elusive. The combination of these features drives rich interfacial dynamics with no analogues in conventional fluids. These dynamics include the uni-directional propagation and anomalous attenuation of surface waves and an asymmetric pearling instability. In principle, these chiral phenomena can be tuned, for instance by altering the colloidal particles’ shape and their effective interactions. Beyond enabling the study of universal aspects of a new class of hydrodynamics, colloidal chiral fluids provide a platform for engineering active materials with so far untapped, ‘odd’ behaviors [@avron_odd_1998; @avron_viscosity_1995; @banerjee_odd_2017]. 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--- author: - 'L. De Maio' - 'F. Dunlop$^{\dagger}$' bibliography: - 'ldfd.bib' title: Sessile drop on oscillating incline --- [**u**]{} Introduction ============ Equilibrium of a drop pinned on an incline was studied by many authors, see [@DDH17] and references therein. Shape and motion of drops sliding down an inclined plane have also been studied, see [@LDL05] and references therein. Drops on vibrating horizontal surfaces have been the subject of much interest recently from experimental, theoretical or numerical points of view. The vibrations or oscillations of the substrate can be horizontal [@DCG05; @LLS04; @DCC06; @CK06], or vertical [@LLS06]. The effect of vibrations on hysteresis, pinning and depinning, was studied in particular by [@NBB04; @VSG07]. The effect of vibrations on the Cassie-Wenzel transition was studied in particular by [@BC09; @BPWE07]. A review of drop oscillations is given by [@MDCA14]. More recent experimental results and references are found in [@RE17]. Here we consider the case of an oscillating incline where the angle $\al(t)$ of the slope follows \[alphat\] (t)=[4]{}( t) while the circular basis of the drop remains fixed on the incline as a disc of radius $r$. We keep the frame of reference attached to the incline, so that the gravity vector oscillates: =( g((t))0-g((t)) ) We assume that the inertial pseudo-forces per unit volume, like the centrifugal force, are negligible with respect to gravity, which will be the case if $\om^2\,r\ll g$. The Bond number is the ratio between gravity and capillarity, and we define it precisely as Bo=[g r\^2]{} where $\Delta\rho$ is the density difference between the two fluids and $\sigma$ is the interface tension. We are interested in moderate but significant drop deformations, with Bond number of order one, as shown on Fig. \[fig:iso\]. All the simulations presented here will be with $Bo=0.22$. ![ Water drop at equilibrium pinned on incline of angle $\al=\pi/4$. Bond number $Bo=0.22$. The drop is surrounded by oil.[]{data-label="fig:iso"}](diapo5.png "fig:"){width="\columnwidth"}\ For bond number of order one and pulsation $\om$ not much larger than the natural pulsation of the drop, the fluid velocity will vary from 0 to about $\om r$ over a distance $r$. This motivates a Reynolds number defined as Re\_=[ r\^2]{} with $\rho=\rho_{\rm water}$ and $\eta=\eta_{\rm water}$. In the same regime the quadratic term in the Navier-Stokes equation (\[NS\]) will be of order $\rho\om^2\,r$, the same as the centrifugal force per unit volume. Therefore it will be consistent, and will save some computing time, to neglect it (Stokes flow). The equation remains non-linear due to the interfacial tension force. For pulsations $\om$ larger than the natural pulsation of the drop, the response of the drop and the actual velocity will be much smaller. A Reynolds number using the maximum measured velocity will always be less than 1 in our simulations. Viscosity plays an essential role in the present study, which does not allow short-cuts such as interface motion by curvature based on the Laplace-Young equation. Diffuse interface and level set method ====================================== The sharp interface between immiscible fluids is replaced by a diffuse interface spreading over a few mesh elements across the physical interface. A level set function $\phi$, inspired by van der Waals, goes smoothly from zero to one when crossing the interface from fluid 1 into fluid 2. The mixture obeys the Navier-Stokes equation for an incompressible fluid, +(u)u&=&&&1.5cm++[**f**]{}\_[st]{}\[NS\]\ u&=&0 where ${\bf I}$ is the identity matrix, and we neglect the quadratic term in (\[NS\]). The density and dynamic viscosity are functions defined by =(1-)\_1+\_2=(1-)\_1+\_2 The surface tension force per unit volume ${\bf f}_{st}$ is \[st\] [**f**]{}\_[st]{}=(([**I**]{}-(\^T))) where $\sigma$ is the interfacial tension, $\nphi=\na\phi/|\na\phi|$ is a normal vector also defined in the bulk, and $\delta=6|\na\phi|\,\phi(1-\phi)$ is a smooth Dirac delta function concentrated near the interface, which is the level set $\{\phi=0.5\}$. Formula (\[st\]), being the divergence of a flux, can be integrated by parts in the weak form of the partial differential equation, and then requires just one derivative of $\phi$. It was shown by [@LNSZ94] to be a smooth approximation to the usual Laplace force $\sigma H\nphi\delta({\rm interface})$ where $H$ is the mean curvature of the interface and $\delta({\rm interface})$ is a true Dirac delta function supported by the interface. The level set function $\phi$ obeys \[LS\] [t]{}+u=(-(1-)[||]{}) where $\ep$ in the diffusion term controls the interface thickness. It will be taken as $h/2$, half the mesh size. The parameter $\gamma$ is a constant with the dimension of a velocity, which we fix as $r\,\om/(2\pi)$ where $r$ is the initial radius of the drop. The level set method for two phase flow was developed in particular by [@OK05]. ![Setup.[]{data-label="fig:diapo3"}](diapo3.png "fig:"){width="\columnwidth"}\ Setup ===== The incline is designed with a circular hydrophilic patch of radius $r=2.5\,$mm and the remaining surface hydrophobic. The corresponding Young contact angles are set to 0 degree (perfectly hydrophilic) and 180 degrees (perfectly hydrophobic) respectively. A water drop of volume $2\pi r^3/3$ is deposited on the hydrophilic patch. The vessel is filled with oil, and closed with no air inside. In the absence of gravity, the drop is a hemisphere, with contact angle $\pi/2$. This will also be the initial configuration in our simulations. The vessel is intended to be large with respect to the water drop, so that friction occurs only near the drop. The Archimedes force, encapsulated in the pressure and gravity terms of the Navier-Stokes equation, does not depend upon the volume of the vessel. For simulation purposes, we have to use a simulation box of modest size. The effect of the box will be minimized if it has the symmetry of the problem at lowest order, hence a hemisphere with same center as the initial drop, and we choose its radius as four times the initial drop radius. On it we choose “slip” boundary conditions: impenetrable and frictionless, again to minimize the effect of having a relatively small simulation box. The center of the hydrophilic patch is chosen as origin of coordinates and the $z$-axis perpendicular to the incline. The incline then starts oscillating around the $y$-axis according to (\[alphat\]). The plane $\{y=0\}$ is a plane of symmetry, allowing to make the study in a quarter of a sphere, see Fig. \[fig:diapo3\]. In the stationary regime, the contact angles at the front ($x=r$) and at the back ($x=-r$) will oscillate between a minimum angle $\theta^{\rm min}$ and a maximum angle $\theta^{\rm max}$. So long as the maximum contact angle remains strictly less than 180 degrees, the contact line cannot move into the hydrophobic region. So long as the minimum contact angle remains strictly larger than 0 degree, the contact line cannot move into the hydrophilic region. The role of the substrate is to ensure pinning. For real substrates, the advancing angle $\theta^A$ of the hydrophobic material and the receding angle $\theta^R$ of the hydrophilic material will replace 180 degrees and zero degree respectively. The scope of our study is bounded by the conditions 0\^R&lt;\^[min]{}&lt;\^[max]{}&lt;\^A. It implies bounds on Bond number and slope angle $\al$, which are satisfied in the present study. It would be interesting to go beyond and also study depinning. This is left to future work. [|c|c|c|c|]{}\ & $\rho\,$\[kg/m$^3$\] & $\eta\,$\[Pas\] & $\sigma\,$\[N/m\]\ Engine oil & 888 & 0.079 &\ Water & 1000 & 0.001 &\ Interface & & & 0.031\ [*Comsol*]{} ============ We used the finite elements software [*Comsol*]{} (see https://www.comsol.com/) in mode [*Laminar Two-Phase Flow, Level Set*]{}, with fluid 1 as engine oil and fluid 2 as water, at 20$^\circ$C, see Table 1. ![Mesh.[]{data-label="fig:mesh"}](mesh.png "fig:"){width="\columnwidth"}\ The simulation box is a quarter of a sphere of radius $4r$. The outer sphere is not a physical boundary, and on it we choose [*slip*]{} boundary conditions: u=0&=(),&=(+u\^T)0.5cm where $K$ is the viscous stress vector upon an infinitesimal surface of normal $\n$. The symmetry plane $\{y=0\}$ obeys the same boundary conditions, with also =0 The hydrophilic patch is a [*wetted wall*]{} with contact angle $\theta_w=0$, meaning a boundary condition (-\_W)=uwhere $\beta$ is a slip length equal to the mesh size $h$. The remaining part of the incline is a [*wetted wall*]{} with contact angle $\theta_w=\pi$. The mesh is built as shown on Fig. \[fig:mesh\], with maximal mesh size $h=0.4\,$mm and $h\sim 0.1\,$mm in the region of the interface, leading to 24560 degrees of freedom. We impose at least one time step in every 1/40 of a period so as to be able to distinguish a sinusoidal response. With the chosen mesh, it turns out that [*Comsol*]{} does not need smaller time steps to satisfy its default tolerance. Each run for one value of $\om$ took about 20 hours with an Intel i7-3770CPU@3.40GHz x8. ![Trace of the drop on the symmetry plane $\{y=0\}$ at five times. $Bo=0.22$, $\om=0.1\,$s$^{-1}$.[]{data-label="fig:film"}](filmcontour.png "fig:"){width="\columnwidth"}\ Results ======= The trace of the drop on the symmetry plane $\{y=0\}$ at different times is shown on Fig. \[fig:film\]. ![Contact angles $\theta^r(t)$ at the front (red) and $\theta^{-r}(t)$ at the back (green), measured in degrees as (\[nnphi\]), for $Bo=0.22$, $\om=0.1\,$s$^{-1}$.[]{data-label="fig:thetarl"}](thetarl.png "fig:"){width="\columnwidth"}\ ![Normalized abscissa of water center of mass $\bar x(t)$, as Eq. (\[xt\]) (red +), and sinusoidal fit of permanent regime, $A\,\sin(\om\,t-\varphi)$ as (\[sin\]) (green, continuous), for $Bo=0.22$, $\om=5\,$s$^{-1}$.[]{data-label="fig:om5x10T"}](om5x10T.png "fig:"){width="\columnwidth"}\ The contact angles $\theta^r(t)$ and $\theta^{-r}(t)$ at the front and the back are shown on Fig. \[fig:thetarl\]. These contact angles are measured as \[nnphi\] =() at $(x,y,z)=(r,0,0)$ and $(x,y,z)=(-r,0,0)$ respectively. When the incline is set in motion, at $t=0$, the liquid drop does not follow instantaneously, whence a start below 90 degrees. In the stationary regime a noticeable feature is that the contact line spends more time near the minimum than near the maximum. Eq. (\[nnphi\]) is a measurement of the interface normal, pointing from oil into water, at a single point, which is a mesh vertex. Small numerical errors are clearly visible in Fig. \[fig:thetarl\]. A systematic error is also present: when the contact angle approaches $\theta^{\rm max}$, the contact line goes slightly into the hydrophobic region. Similarly when the contact angle approaches $\theta^{\rm min}$, the contact line goes slightly into the hydrophilic region. Measuring the contact angles at $x=\pm r$ underestimates the amplitude of oscillations. Measuring contact angles, experimentally or numerically, is subject to debate, especially in dynamics. Fitting individual images of a film is tedious and systematic deviations may also be present if the fit is over a length where gravity produces bending. In dynamics the bending effect of gravity cannot be computed exactly. We have therefore chosen to analyse the data in terms of the motion of the centre of mass of the drop, whose definition is obvious and whose statistics is optimal. The abscissa of the center of mass of water is recorded, normalized arbitrarily using the drop basis radius $r$ and the volume $\pi r^3/3$ of a quarter of a sphere of radius $r$: \[xt\] |x(t)=[dxdydz (x,y,z,t)xr\^4/3]{} The integral is over the simulation box, namely a quarter of a sphere of radius $4r$. After a transient, which lasts longer for larger $\om$, the system approaches a stable permanent regime, as shown on Figs. \[fig:om5x10T\], \[fig:om20x\]. The finite elements method does not conserve exactly the total mass of each fluid, and a small parasitic drift is often present in simulations, but it is not the case here. We then use the [*gnuplot*]{} fit, a nonlinear least-squares Marquardt-Levenberg algorithm, and search for an amplitude $A$ and a phase lag $\varphi$ such that \[sin\] |x(t)-A(t-)0 tResults are like the example shown on Fig. \[fig:om5x10T\], where the error measured by the $rms$ of residuals over one period falls below 1% after a few periods (after 7 periods in the example shown). The resulting incertainties over $A$ and $\varphi$ are also below 1%. A sinusoidal response with the same $\om$ as the incline angle was to be expected for a linear system. We used the Stokes equation with a non-linear surface tension force, and the transport equation (\[NS\]) is also non-linear. Results are listed in Table 2, where the case $\om=0$ is in fact the limit as $\om\to0$, namely the stationary case $\al(t)=\pi/4\ \forall t>0$. Plots of $A$ and $\varphi$ versus $\om$ are given in Fig. \[fig:xphi\] and Fig. \[fig:xphiphi\]. They look much like a driven over-damped linear oscillator, with a notable exception: the amplitude of the permanent oscillations behaves like $\om^{-1}$ at large $\om$ instead of $\om^{-2}$ for the driven damped linear oscillator. The phase lag $\varphi(\om)$ is proportional to $\om$ at small $\om$ like a driven damped linear oscillator. ![Normalized abscissa of water center of mass $\bar x(t)$, Eq. (\[xt\]), for $\om=20\,$s$^{-1}$.[]{data-label="fig:om20x"}](om20.png "fig:"){width="\columnwidth"}\ [|c|c|c|]{}\ $\om\,$\[s$^{-1}$\] & $A$ & $\varphi\,$\[rad\]\ 0 & 1.92 & 0\ 0.05 & 1.89 & 0.275\ 0.1 & 1.68 & 0.503\ 0.2 & 1.25 & 0.79\ 0.5 & 0.66 & 1.06\ 1 & 0.387 & 1.17\ 2 & 0.228 & 1.24\ 5 & 0.105 & 1.41\ 10 & 0.0564 & 1.55\ 20 & 0.0286 & 1.66\ ![Amplitude $A(\om)$ from (\[sin\]) with asymptote $0.58/\om$.[]{data-label="fig:xphi"}](xphi.pdf "fig:"){width="\columnwidth"}\ ![Phase lag $\varphi(\om)$ from (\[sin\]) with line at $\pi/2$ and tangent at the origin $\varphi=5.5\om$.[]{data-label="fig:xphiphi"}](xphiphi.pdf "fig:"){width="\columnwidth"}\ Heuristics for $\om\to\infty$. ============================== Let us first review the case of a driven solid oscillator, subject to a fluid friction force, obeying a differential equation of the form \[osc2\] x + [friction force]{} + [restoring force]{}=t As $\om\to\infty$ we don’t expect resonance. Therefore each term on the left-hand-side will be of order at most the order of the right-hand-side, namely $\OO(1)$. We expect a periodic permanent regime of period $T=2\pi/\om$. If the amplitude is $A$ then $\ddot x$ will be of order $A\om^2$, implying $A$ of order at most $\om^{-2}$. The restoring force will be $o(1)$, the velocity of order $A\om\sim\om^{-1}$ and the fluid friction force $o(1)$. Therefore, as $\om\to\infty$, the system tends to $\ddot x=\sin\om\,t$, leading to an amplitude $\om^{-2}$ and phase lag $\pi$, in agreement with the exact solution of the linear case. Another driven system may obey a first order differential equation of the form \[osc1\] x + [restoring force]{}=t Again we expect a periodic permanent regime of period $T=2\pi/\om$, and no resonance, so that each term on the left-hand-side will be of order at most $\OO(1)$. If the amplitude is $A$ then $\dot x$ will be of order $A\om$, implying $A$ of order at most $\om^{-1}$. The restoring force will be $o(1)$. Therefore, as $\om\to\infty$, the system tends to $\dot x=\sin\om\,t$, leading to an amplitude $\om^{-1}$ and phase lag $\pi/2$, in agreement with the exact solution of the linear case. We have studied a drop on an oscillating incline in a regime where the inertial forces, such as the centrifugal force, are negligible with respect to gravity and capillarity, both of same order for Bond number of order one. We thus have $\om^2r\ll g$. The acceleration term $\p u/\p t$ in the Navier-Stokes equation is of same order and therefore negligible. And the Reynolds number was always less than one so that the quadratic term in the Navier-Stokes equation could be neglected. Therefore a behaviour corresponding to (\[osc1\]) rather than (\[osc2\]) should be observed, leading to an amplitude $A\sim\om^{-1}$ rather than $A\sim\om^{-2}$ as $\om\to\infty$. In simple words: a liquid drop can deform in many different ways and will do so as far as the shear stress $\na \u$ remains bounded. If $A$ is the amplitude of the motion of the center of mass in the frame of reference of the incline, then $\na \u$ is of order $A\om/r$, giving $A\sim\om^{-1}$ as $\om\to\infty$. When $\om\to0$, acceleration is negligible in all cases, and both solid and liquid oscillators have an amplitude $\OO(1)$ and a phase lag $\OO(\om)$. Conclusion ========== A sessile millimetric droplet on an incline responds similarly to a driven damped linear oscillator to a sinusoidal oscillation of the angle of the incline. However, the amplitude of the drop deformation is proportional to $\om^{-1}$ at large $\om$, whereas a simple pendulum on an oscillating incline responds with an amplitude proportional to $\om^{-2}$ at large $\om$. The diffuse interface modelisation imply diffusion times larger than the true physical times, but the discrepancy should go to zero with finer and finer meshes. Also, because there is more space for water (originally in the small sphere) to diffuse into oil (originally in the large sphere), than conversely, the level set 0.5, considered as the interface, shrinks a little during the first seconds. This effect should also go to zero with finer and finer meshes. Beyond $\om\sim20\,$s$^{-1}$, in the oscillating frame of reference, one cannot neglect the inertial pseudo-forces. One can expect that including the centrifugal force in the Navier-Stokes equation would increase the drop deformation at large $\om$.

--- abstract: 'We study the dynamical response of extended systems, hosts, to smaller systems, satellites, orbiting around the hosts using extremely high-resolution $N$-body simulations with up to one billion particles. This situation corresponds to minor mergers which are ubiquitous in the scenario of hierarchical structure formation in the universe. According to [@1943ApJ....97..255C], satellites create density wakes along the orbit and the wakes cause a deceleration force on satellites, i.e. dynamical friction. This study proposes an analytical model to predict the dynamical response of hosts as reflected in their density distribution and finds not only traditional wakes but also mirror images of over- and underdensities centered on the host. Our controlled $N$-body simulations with high resolutions verify the predictions of the analytical model. We apply our analytical model to the expected dynamical response of nearby interacting galaxy pairs, the Milky Way - Large Magellanic Cloud system and the M31 - M33 system.' author: - | Go Ogiya$^{1,2,3}$[^1] and Andreas Burkert$^{1,2,3}$[^2]\ $^{1}$Max-Planck-Institut für extraterrestrische Physik, Postfach 1312, Giessenbachstraße, D-85741 Garching, Germany\ $^{2}$Universitäts-Sternwarte München, Scheinerstraße 1, D-81679 München, Germany\ $^{3}$Excellence Cluster Universe, Boltzmannstr. 2, D-85748, Garching, Germany bibliography: - './ref.bib' date: 'Accepted 2016 January 8. Received 2016 January 8; in original form October 27' title: Dynamical friction and scratches of orbiting satellite galaxies on host systems --- \[firstpage\] galaxies: Local Group – galaxies: Magellanic Clouds – galaxies: dwarf – galaxies: evolution – galaxies: formation Introduction {#sec:int} ============ [@1943ApJ....97..255C] first discussed a fundamental physical process called dynamical friction for collisionless systems. According to his calculation, a massive object moving through a sea of particles produces density enhancements, ‘wakes’ behind itself due to its gravitational force and the gravitationally induced wakes generate a decelerating force on the moving massive object. As a consequence, the massive object loses its orbital energy and angular momentum. [@1999ApJ...513..252O] later on studied the process of dynamical friction for gaseous systems. Dynamical friction arises in various astronomical phenomena [@2008gady.book.....B]. Considering galaxy mergers, satellite galaxies orbiting around larger ones lose orbital energy and angular momentum by the effects of dynamical friction and eventually fall into the center of their hosts [e.g. @1976MNRAS.174..467W; @1999ApJ...525..720C; @2001ApJ...559..716T]. A prominent example is the orbit decay of the Large Magellanic Cloud that has been studied in great details [e.g @1976ApJ...203...72T; @1980PASJ...32..581M; @1994MNRAS.266..567G; @2003ApJ...582..196H]. Dynamical friction also plays important roles for the formation and evolution of black hole binaries [@1980Natur.287..307B; @2002MNRAS.331..935Y; @2004ApJ...602...93M; @2011ApJ...728L..31T; @2013ApJ...777L..14F]. Studies based on numerical simulations reveal the validity but also problems of Chandrasekhar’s theory [@2006PASJ...58..743F; @2006MNRAS.373.1451R; @2009MNRAS.397..709I]. Important insight has been gained, applying the arguments of dynamical friction to flattened systems, such as galactic stellar discs. [@1977MNRAS.181..735B] proposed a correction of Chandrasekhar’s formulation of dynamical friction for systems with anisotropic velocity distributions. [@2004MNRAS.349..747P] confirmed the reliability of Binney’s treatment numerically. Dynamical friction generates the heating source to explain the thickness of stellar discs [e.g. @1999MNRAS.304..254V; @2008ApJ...674L..77M]. [@2002MNRAS.333..779P] and [@2008MNRAS.389.1041R] found that orbiting satellite galaxies around a larger galaxy with disc components are dragged into the disc plane by their dynamical friction. This may explain the distribution of satellite galaxies around the Milky Way [@2002MNRAS.333..779P] and predicts the existence of stable disc structures of dark matter which significantly boost capturing dark matter particles [@2008MNRAS.389.1041R; @2009MNRAS.397...44R]. Gravitational wakes were investigated in details analytically by early work . However, only a few studies have confirmed gravitational wakes directly by using numerical simulations in spite of the importance to get a better understanding of dynamical friction. @2002ApJ...580..627W [@2007MNRAS.375..425W] showed over- and underdensities induced by bars around the center of galaxies. [@2012ApJ...745...83A] illustrated the density response induced by black holes in galactic nuclei. For gaseous systems, [@2009ApJ...703.1278K] and [@2010ApJ...725.1069K] analysed density wakes by hydrodynamic simulations. Previous numerical studies were limited by particle resolution. The motivation of this study is to investigate the features of scratches induced by orbiting smaller systems (satellites) on larger ones (hosts) in great details adopting one billion particles. The structure of this paper is as follows. Section \[sec:ana\] provides an analytical description to predict the response of the hosts as reflected in their density distribution. We find not only traditional gravitational wakes, but also a mirror image of over- and underdensities around the center of hosts. In Section \[sec:sim\], we perform high-resolution $N$-body simulations to test the analytical prediction. The results of our simulations well match to the predictions of the analytical description. We use the analytical model for the galaxy pairs, the Milky Way - Large Magellanic Cloud system and the M31 - M33 system as applications in Section \[sec:app\]. Section \[sec:sum\] summarizes and discusses the results. Analytical model {#sec:ana} ================ The formula of dynamical friction proposed by [@1943ApJ....97..255C] can be written as $$\frac{d {\bf v}_{\rm sat}}{dt} = -4 \pi G^2 M_{\rm sat} \ln{\Lambda} \rho_{\rm host} \frac{v_{\rm sat}}{v_{\rm sat}^3}, \label{eq:df_ori}$$ where $M_{\rm sat}$ and $\rho_{\rm host}$ are the mass of the satellite and the density of the host, and ${\bf v}_{\rm sat}$ represents the velocity of the satellite in the rest frame of the host. $G$ and $\ln{\Lambda}$ are the gravitational constant and the Coulomb logarithm, respectively. Here, the host systems are assumed to have an isotropic velocity field in the initial equilibrium state. We suppose that the mass of a particle which belongs to the host is negligible compared with $M_{\rm sat}$ and assume that all particles in the host contribute to the force of dynamical friction for simplicity although only host particles which satisfy $v < |{\bf v}_{\rm sat}|$ cause dynamical friction in the original Chandrasekhar’s theory. These assumptions do not influence our conclusion as shown in Section \[sec:sim\]. In Chandrasekhar’s derivation of dynamical friction, an infinite homogeneous particle distribution is assumed. However, galaxies and dark haloes have finite sizes and gradients in their density structures. Thus the Coulomb logarithm may have a dependence on the satellite position within the host . One may modify Equation (\[eq:df\_ori\]) as follows, $$\frac{d {\bf v}_{\rm sat}}{dt} = -4 \pi G^2 M_{\rm sat} \ln{\Lambda({\bf r}_{\rm sat})} \rho_{\rm host}({\bf r}_{\rm sat}) \frac{v_{\rm sat}}{v_{\rm sat}^3}, \label{eq:df_mod}$$ where ${\bf r}_{\rm sat}$ is the position of the satellite in the frame whose origin is the center of the host. Hereafter, we study the evolution in this rest frame. Orbiting satellites are perturbers for hosts. Because of the gravitational force of the satellite, induced density perturbations will arise in the host. We label physical quantities of the host in the equilibrium state as ‘0’ and induced ones as ‘1’, respectively, i.e. $$\begin{aligned} \rho_{\rm host} = \rho_0 + \rho_1, \\ \Phi_{\rm host} = \Phi_0 + \Phi_1, \end{aligned}$$ where $\Phi$ is the gravitational potential. Poisson’s equation connects the gravitational potential with the density field. For the induced quantities, we get $${\bf \nabla}^2 \Phi_1 = -{\bf \nabla} \cdot {\bf g}_1 = 4 \pi G \rho_1, \label{eq:poisson}$$ where ${\bf g}_1$ is the specific gravitational force caused by the induced density perturbations. We can regard the deceleration force of the dynamical friction process as ${\bf g}_1$. Substituting Equation (\[eq:df\_mod\]) into Equation (\[eq:poisson\]), the induced density field is $$\begin{aligned} \rho_1({\bf r}, {\bf v}_{\rm sat}) = G M_{\rm sat} [ \rho_0({\bf r}) {\bf \nabla} \ln{\Lambda({\bf r})} + \ln{\Lambda({\bf r})} {\bf \nabla} \rho_0({\bf r})] \cdot \frac{{\bf v}_{\rm sat}}{v_{\rm sat}^3}. \label{eq:ind_dns}\end{aligned}$$ Here, we assume that the absolute values of induced quantities are much smaller than those in the equilibrium state. We need to provide the density distribution of the host in the equilibrium state, $\rho_0({\bf r})$ and the Coulomb logarithm, $\ln{\Lambda({\bf r})}$ to use Equation (\[eq:ind\_dns\]). The density distribution of the host galaxy may be expressed well by the following double power-law formula, $$\rho_0(r) = \frac{\rho_{\rm s}}{r^{\alpha} [1 + (r / r_{\rm s})]^{\beta}}, \label{eq:double_power_law}$$ where $\rho_{\rm s}$ and $r_{\rm s}$ are the scale density and length, respectively. The model of $\alpha=1, \beta=2$ is known as the Navarro-Frenk-White profile which well matches the density structure of cold dark matter haloes obtained in dissipationless cosmological simulations [@1997ApJ...490..493N]. The Hernquist profile [@1990ApJ...356..359H] which reproduces the de Vaucouleurs law [@1948AnAp...11..247D] and is frequently used to expess density structures of elliptical galaxies or bulges, corresponds to the model of $\alpha=1, \beta=3$ [see however e.g. @1995ApJ...447L..25B]. The derivative of Equation (\[eq:double\_power\_law\]) is given by $${\bf \nabla} \rho_0 (r) = -\rho_0(r) \biggl [ \frac{\alpha}{r} + \frac{\beta}{r + r_{\rm s}} \biggr] \frac{{\bf r}}{r}. \label{eq:dns_grad}$$ The Coulomb logarithm is defined as $\ln{\Lambda} \equiv \ln{(b_{\rm max} / b_{\rm min})}$ where $b_{\rm max}$ and $b_{\rm min}$ are the maximum and minimum impact parameters, respectively. The maximum impact parameter, $b_{\rm max}$ should depend on the vector pointing from the satelllite to given points, $ {\bf r}$, ${\bf d} = {\bf r} - {\bf r}_{\rm sat}$ since $b_{\rm max}$ determines the region affected by the gravitational force of the satellite. For simplicity, we suppose that $b_{\rm min}$ is a constant, $$b_{\rm min} = A l, \label{eq:bmin}$$ where $A$ and $l$ mean a constant and the size of the satellite, respectively [see also @1976ApJ...203...72T; @2003ApJ...582..196H]. We define the maximum impact parameter, $b_{\rm max}$ by the following procedure. Let us consider a plane which is perpendicular to the velocity vector of the satellite, ${\bf v}_{\rm sat}$. The vector, ${\bf v}_{\rm sat} \Delta t$ measures the distance between the satellite and plane and determines the plane on which a point is given by ${\bf r}_{\rm sat} - {\bf v}_{\rm sat} \Delta t$. Given points, ${\bf r}$ on the plane satisfy the condition, $$({\bf r} - {\bf r}_{\rm sat} + {\bf v}_{\rm sat} \Delta t) \cdot {\bf v}_{\rm sat} = 0. \label{eq:ip}$$ From Equation (\[eq:ip\]), $\Delta t$ is derived by $$\Delta t = -\frac{{\bf v}_{\rm sat} \cdot ({\bf r} - {\bf r}_{\rm sat})}{v_{\rm sat}^2} = -\frac{d \cos{\phi}}{v_{\rm sat}} \label{eq:dt}$$ where $\phi$ is the angle between the vectors, ${\bf d}$ and ${\bf v}_{\rm sat}$. We define $b_{\rm max}$ as the length of a free-fall motion in the time interval, $\Delta t$, i.e. $b_{\rm max} = (1/2) a \Delta t^2$, behind the satellite at which ${\bf v}_{\rm sat} \cdot {\bf d} < 0$ is satisfied. The absolute value of the gravitational acceleration in the perpendicular direction to ${\bf v}_{\rm sat}$, $a$ is $$a = \frac{G M_{\rm sat}}{[b_{\rm max}^2 + (v_{\rm sat} \Delta t)^2]^{3/2}} b_{\rm max}. \label{eq:a_para}$$ For a simple evaluation of the gravitational acceleration we assume that the satellite is located at ${\bf r}_{\rm sat}$ from $t=T - \Delta t$ to $t = T$. Solving Equation (\[eq:a\_para\]) for $b_{\rm max}$, we obtain $$b_{\rm max} = \sqrt{\biggl [ \frac{G^2 M_{\rm sat}^2 \Delta t^4}{4} \biggr ]^{1/3} - (v_{\rm sat} \Delta t)^2}. \label{eq:bmax}$$ for points at which ${\bf v}_{\rm sat} \cdot {\bf d} < 0$ is satisfied. Also, $$b_{\rm max} = B b_{\rm min} \label{eq:bmax2}$$ for points at which ${\bf v}_{\rm sat} \cdot {\bf d} \geq 0$ is satisfied. Here, $B$ is a constant. The derivative is $${\bf \nabla} \ln{\Lambda({\bf r})} = \frac{1}{b_{\rm max}^2} \biggl [ \biggl ( \frac{2 G^2 M_{\rm sat}^2 \cos^4{\phi}}{27 v_{\rm sat}^4} d \biggr )^{1/3} - d \cos^2{\phi} \biggr ] \frac{{\bf d}}{d} \label{eq:bmax_grad}$$ for points at which ${\bf v}_{\rm sat} \cdot {\bf d} < 0$ and $${\bf \nabla} \ln{\Lambda({\bf r})} = 0$$ for points at which ${\bf v}_{\rm sat} \cdot {\bf d} \geq 0$ is satisfied, respectively. When $b_{\rm max} < B b_{\rm min}$ in Equation (\[eq:bmax\]), we set $b_{\rm max} = B b_{\rm min}$ and ${\bf \nabla} \ln{\Lambda({\bf r})} = 0$. Hence, a constant, $B$ defines the minimum value of the Coulomb logarithm. Combined with Equations (\[eq:ind\_dns\]) and (\[eq:bmax\_grad\]), the first term in Equation (\[eq:ind\_dns\]) generates density perturbations around the satellite. Since Equation (\[eq:bmax\]) defines $b_{\rm max}$ on the back side of the satellite motion, the induced density arises only behind the satellite. Hence, the first term represents Chandrasekhar’s original gravitational wake. As indicated by combining Equations (\[eq:ind\_dns\]) and (\[eq:dns\_grad\]), the sign of the density fluctuations caused by the second term of Equation (\[eq:ind\_dns\]) depends on the angle between the vectors of the satellite velocity and the position. For spherical systems, the density fluctuations are symmetric with respect to the center of the host. Simulations {#sec:sim} =========== Set up {#sec:setup} ------ We perform controlled collisionless $N$-body simulations with a parallelized code optimized for graphic processing unit (GPU) clusters. The numerical code employs the tree algorithm proposed by [@1986Natur.324..446B] and the second-order Runge-Kutta scheme in time integration. Along the lines of [@2011arXiv1112.4539N], CPU cores construct tree structures of particles and GPU cards compute the gravitational force among particles through traversing the tree structures [@2013JPhCS.454a2014O]. We simulate mergers between two systems, the host and the satellite. In order to generate $N$-body systems which follow the NFW density profile, i.e. $\alpha=1, \beta=2$ in Equation (\[eq:double\_power\_law\]), in the equilibrium states, we use the method proposed by [@2006ApJ...641..647K] [see also @1916MNRAS..76..572E]. The phase-space distribution function is assumed to depend only on energy and the systems have an isotropic velocity dispersion initially. The distribution of particles is truncated at the virial radius, $R_{\rm v}$ that is related to the virial mass, $M_{\rm v}$ $$M_{\rm v} = \frac{4 \pi}{3} \Delta \rho_{\rm crit} (1+z)^3 R_{\rm v}^3, \label{eq:rv}$$ where $\rho_{\rm crit}$ and $z$ are the critical density of the universe and redshift, respectively. We adopt a conventional value of the overdensity, $\Delta = 200$ in this study. The concentration parameter, $c$ is defined by $c = R_{\rm v} / r_{\rm s}$. The host and satellite have $c = 10$ and $15$, respectively. The initial separation between the centers of the host and satellite is the virial radius of the host, $R_{\rm v,host}$. The orbit has circularity $\eta = 0.5$ initially. This value is very typical, as suggested by cosmological $N$-body simulations . The initial velocity of the satellite is $V_{\rm ini} = \eta V_{\rm c}(R_{\rm v,host}) = \eta (GM_{\rm v, host} / R_{\rm v,host})^{1/2}$. Here, $V_{\rm c}(r)$ is the circular velocity measured at $r$. In the coordinate system, centered on the host, the initial position and velocity vectors of the satellite are ${\bf X} = (R_{\rm v,host}, 0, 0)$ and ${\bf V} = (0, V_{\rm ini}, 0)$, respectively. The position and velocity of the host and satellite are determined by the following procedure. We calculate the bound mass of each system with the method proposed by [@1993PASJ...45..289F]. Particles initially belong either to the host or to the satellite system. At each snapshot, we compute the gravitational potential of each particle by particles which are still bound to the system at the previous snapshot. The bulk velocity of each system is determined by an iterative procedure [@2006PASJ...58..743F]. When the binding energy of a particle is positive, the particle is regarded as an escaper. We define the center of mass and bulk velocity of bound particles as the position and velocity of each system at given time. We construct the host systems which have the virial mass, $M_{\rm v, host}$ with $N_{\rm host}$ particles. For satellite systems with a virial mass, $M_{\rm v, sat}$, $N_{\rm sat} = (M_{\rm v, sat} / M_{\rm v, host}) N_{\rm host}$ particles are employed. Hence, all particles have equal masses in each simulation and the total number of particles is $N_{\rm tot} = (1 + M_{\rm v, sat} / M_{\rm v, host}) N_{\rm host}$. Table \[tab:run\] summarizes the simulations. Because we use sufficient numbers of particles, artificial effects such as two-body relaxation are negligible. In the simulations, the softening length is $\epsilon = 4 R_{\rm v, host} / \sqrt{N_{\rm host}}$ and the tolerance parameter of the tree algorithm is $\theta = 0.6$ [@2003MNRAS.338...14P]. We are free to scale the mass, length and timescales since our simulations only take into account gravitational effects. For Milky Way sized haloes with $M_{\rm v, host} = 10^{12} M_{\rm \odot}$ and cosmic redshifts, $z = 0$, the virial radius of the host halo is $R_{\rm v, host} \approx 211 {\rm kpc}$ and the dynamical time of the host, $T_{\rm d, host}$ defined by $$T_{\rm d,host} \equiv \frac{R_{\rm v,host}}{V_{\rm c}(R_{\rm v,host})} = \sqrt{\frac{R_{\rm v,host}^3}{G M_{\rm v,host}}}, \label{eq:td}$$ is $\approx 1.45 {\rm Gyr}$. Here, a Hubble constant of $H_0 = 67.5 {\rm km/s/Mpc}$ [@2015arXiv150201589P] is adopted. Simulation data are output every $0.05T_{\rm d, host}$. Results {#sec:res} ------- Figure \[fig:evo\] summarizes a typical case of galaxy merger. The satellite loses angular momentum due to dynamical friction and the orbit shrinks gradually (upper panel). In the meantime, the satellite is stripped by the tidal force of the host, especially when it approaches the pericenters. Eventually, it is completely destroyed at $t = 5.65 T_{\rm d,host}$ in run D and $6.0 T_{\rm d,host}$ in run E, respectively (lower panel). Figure \[fig:evo\] also tests the numerical convergence of the simulations. The orbital evolution is well converged. The evolution of the bound mass of the satellite for both resolutions deviates only at $t > 4T_{\rm d,host}$, at which most of its mass has already been stripped away and significant density scratches do not arise (see Figures \[fig:map1\] and \[fig:map2\]). Hence, the results of this paper do not depend on the number of particles, $N$. As shown in this figure, dynamical friction plays a key role during the merging process. Figures \[fig:map1\] and \[fig:map2\] illustrate that the orbiting satellite leaves significant scratches on the host. We find two kinds of scratches in its density distribution. The first one is the gravitational wake which has been found and discussed by many previous studies. The gravitational wake arises along the satellite orbit as imagined by [@1943ApJ....97..255C]. It can be found more clearly in the early phase of the merger process \[see panels (a) and (b)\] since it mixes with another type of scratch described below in the later phase. The second type of scratch caused by the gravitational field of the satellite is a pair of density enhancements and reductions around the center of the host. Similar results have been obtained analytically by [@1989MNRAS.239..549W]. This distribution of over- and underdensities is mirror image-like and the mirror plane is roughly perpendicular to the direction of the velocity vector of the satellite. The underdensity is located in the direction of the velocity vector of the satellite and the overdensity arises in the opposite direction. The directions are more visible in the early phase again due to the mixing of both, the wake and the central perturbation in the later phase. This central dipole scratch is caused by the motion of the location of the minimum of the potential, i.e. highest density point in the initial state. The tidal force of the satellite halo perturbs the position of the highest density point and it well matches the position of the highest overdensity. Hence, the motion of the minimum potential point well traces the overdensity around the center of the host in the center-of-mass frame (see thin black line in Figure \[fig:map1\]). Assuming that the satellite is a less massive particle in a two-body problem, the thin line looks like an orbit of the corresponding more massive particle. Because of mass conservation, the overdensity on one side causes a corresponding underdensity on the opposite side of the minimum potential point with respect to the center of mass in the host. Since there is a single point where the potential has its minimum in the host, the central over- and underdensities have a dipole structure. The effect does not affect the bulk structure of the host and it retains the initial spherical shape on the whole. Figures \[fig:map1\] and \[fig:map2\] also show that the amplitude of over- and underdensities decreases with time. This is because of the decreasing satellite mass as a result of tidal stripping \[see Equation (\[eq:ind\_dns\])\]. After tidal disruption of the satellite, little scratches remain for some time \[panels (k) and (l)\]. We study the relation between the amplitude of the scratches and the satellite mass. Figure \[fig:amp\] represents the maximum value of enhancement in column density as a function of the initial satellite mass, $M_{\rm v, sat}$ and shows that the maximum amplitude is proportional to $M_{\rm v, sat}$. The maximum enhancements are obtained when the satellite approaches the first pericenter ($t \sim 1.5 T_{\rm d, host}$) in all simulation runs. At that time, the gravitational wake merges with the central overdensity of the host system. In the analytical model, the satellite is regarded as a point mass and the amplitude in the density perturbation is proportional to the satellite mass at given points. As a consequence, the amplitude in the column density perturbation should also be proportional to the satellite mass. The results of our simulations validate the assumption in the analytical model. The comparison of run D and E (see Figure \[fig:amp\]) shows that the numerical simulations are well converged. Comparison with analytical predictions {#sec:comp} -------------------------------------- In order to test the validity of the analytical model described in Section \[sec:ana\] and to understand the simulation results, we compare our analytical model predictions with results of the simulation. A spherical system which follows an NFW density profile is assumed as the initial unperturbed state. This corresponds to assuming that the center of the system is the center of mass. Figure \[fig:comp\] demonstrates the predicted enhancement and reduction in the distribution of column density of the host system. Comparing the simulation results, panel (a) in Figure \[fig:map1\], with the top panel in Figure \[fig:comp\], the analytical prediction well reproduces the results of the simulation, not only the distribution of enhancement and reduction but also the amplitude. Middle and bottom panels show the contributions from the first and second terms in Equation (\[eq:ind\_dns\]). The results clearly indicate that the first and second terms make the gravitational wake and the mirror image of the over- and underdensities, respectively. Because most of previous studies have assumed homogeneous background density, the effects of the second term, the mirror image of the over- and underdensities have not been found and discussed. However the feature should arise in many astrophysical systems such as galaxies and galaxy clusters since their density distributions have gradients. As described above, the predictions well match the simulation results and the analytical model provides a clear understanding. However, there still remain small deviations between the simulation results and the analytical predictions. The direction of the mirror plane is slightly different. This is manly due to two effects. First of all, the stripped mass is not considered in the analytical model. The stripped mass is distributed along the satellite orbit in the simulation. On the other hand, the analytical model treats the satellite as a point mass and does not consider the stripped mass. This effect should become more important in the last phase of the merger process since more satellite mass has been stripped. The second effect is changes in the density distribution of the host. In the analytical model, we assume the initial density distribution of the host as the background field (physical quantities labeled ‘0’ in Section \[sec:ana\]). The density distribution of the host however changes with time. Hence, it is sensible to avoid applying the analytical model to systems in violently changing dynamical states. Application {#sec:app} =========== In this Section, we apply the analytical model to nearby interacting galaxy pairs, the Milky Way (MW) - Large Magellanic Cloud (LMC) system and the M31 - M33 system. Information about the position, velocity and mass of the satellites, the LMC and M33 and the background density field of the hosts, the MW and M31 are needed. We assume an NFW model with a virial mass, $M_{\rm v} = 1.26 \times 10^{12} M_{\rm \odot}$ and concentration parameter $c=10$ for the unperturbed density field of the MW and M31 [@2012ApJ...753....8V]. Actually, the observed density field of the hosts has been already perturbed, i.e. $\rho_{\rm obs} = \rho_{\rm host} = \rho_0 + \rho_1$. We assume here that $|\rho_1|$ is much smaller than $\rho_0$. The distance between the solar system and the Galactic Center is assumed to be $R_{\rm sol} = 8.5 {\rm kpc}$ [@1986MNRAS.221.1023K]. The parameters in Equations (\[eq:bmin\]) and (\[eq:bmax2\]) are the same as those used to plot Figure \[fig:comp\], $A = 3.0$ and $B = 1.5$. We take into account only a dark matter halo in the analysis for simplicity. Baryon components of host galaxies, such as bulges, discs and stellar haloes also react to the gravitational force of satellite galaxies and density scratches arise in them. Ongoing observations, e.g. Gaia and Subaru Hyper Suprime-Cam., may find not only density fluctuations, but also fluctuations in the velocity field caused by the induced density fields. Combining observational data with our analytical model might be interesting in order to constrain the the orbits and masses of the satellite galaxies. MW - LMC {#sec:mw_lmc} -------- It is useful to adopt a Cartesian coordinate system $(X, Y, Z)$, the so-called Galactocentric rest frame [e.g. @1994MNRAS.266..567G]. In this coordinate system, the origin corresponds to the Galactic Center and the $X-$, $Y-$ and $Z-$axes point in the direction from the solar system to the Galactic Center, in the direction of the Galactic Rotation of the solar system and towards the Galactic North Pole, respectively. The position of the solar system is given by ${\bf R}_{\rm sol} = (-R_{\rm sol}, 0, 0)$. [@2002AJ....124.2639V] provide the position of the LMC, $${\bf r}_{\rm LMC} = (-0.78, -41.55, -26.95) \ {\rm kpc},$$ and the relative volocity of the LMC with respect to the Galactic Center is obtained by [@2013ApJ...764..161K], $${\bf v}_{\rm LMC} = (-57 \pm 13, -226 \pm 15, 221 \pm 19) \ {\rm km \ s^{-1}}.$$ The total dynamical mass of the LMC is uncertain by a factor of 10. The enclosed mass within 8.7 kpc from the center of the LMC is $(1.7 \pm 0.7) \times 10^{10} M_{\rm \odot}$ [@2014ApJ...781..121V]. The total mass should be greater than this value. Determining the total mass of the LMC by using the abundance matching technique [@2010MNRAS.404.1111G], the upper mass limit of the LMC is $2.5 \times 10^{11} M_{\rm \odot}$ [@2013ApJ...764..161K]. This is consistent with the estimation by [@2015arXiv150703594P]. We assume that the mass and size of the LMC are $M_{\rm LMC} = 10^{11} M_{\rm \odot}$ and $l = 8.7 {\rm kpc}$, respectively. As shown in Figure \[fig:amp\], the amplitude of the density enhancement can be scaled by $\propto M_{\rm LMC}$. Figure \[fig:mw\_lmc\] demonstrates the predicted enhancement and reduction in the column density distribution of the MW. $XY$ and $ZX$ planes in the Galactocentric coordinate are good to find clear density scratches of the LMC. When one sees the south-side sky, the column density in the direction of the Galactic Rotation of the solar system (plus $Y$) is expected to be systematically greater than that in the opposite direction (upper panel). Also, the column density on the side of the Galactic North Pole (plus $Z$) should be systematically lower than that on the opposite side when one looks into the opposite direction of the the Galactic Rotation of the solar system (lower panel). M31 - M33 {#sec:m31_m33} --------- The position and velocity vectors of M31 and M33 are obtained by [@2012ApJ...753....8V]. We adopt a coordinate system, $(X_{\rm M31}, Y_{\rm M31}, Z_{\rm M31})$ in which the origin is the center of M31 and the $Z_{\rm M31}-$axis points in the direction from the solar system to the center of M31. $X_{\rm M31}-$ and $Y_{\rm M31}-$axes are perpendicular to the $Z_{\rm M31}-$axis and a right‐handed system is constructed. In the M31 rest frame, the position and velocity of M33 are $$\begin{aligned} {\bf r}_{\rm M33} &=& (140.5, 146.1, -2.3) \ {\rm kpc}, \nonumber \\ {\bf v}_{\rm M33} &=& (-147.4, -72.2, 117.9) \ {\rm km \ s^{-1}}. \end{aligned}$$ The total mass of M33 is uncertain by a factor of 10 similar to the LMC mass. [@2003MNRAS.342..199C] found that dark halo mass out to 17 kpc from the center of M33 is $\sim 5 \times 10^{10} M_{\rm \odot}$. [@2011ISRAA2011E...4S] obtained a virial mass of the dark halo surrounding M33 of $(2.2 \pm 0.1) \times 10^{11} M_{\rm \odot}$ from the HI rotation curve. We assume that the mass and size of M33 are $M_{\rm M33} = 10^{11} M_{\rm \odot}$ and $l = 17 {\rm kpc}$, respectively. The analytical model predicts a mirror image of the density enhancement and reduction around the center of M31 as shown in Figure \[fig:m31\_m33\]. The result appears to violate the basic assumption, $\rho_0 \gg |\rho_1|$, but the amplitude of density enhancement can be scaled by $\propto M_{\rm M33}$. Summary and discussion {#sec:sum} ====================== We have investigated the dynamical response of extended host systems to the gravitational force of orbiting satellite systems, ‘scratches’. The scratches are classified into two types: the first one is the gravitational wake along the orbit of satellites as discussed in [@1943ApJ....97..255C]. The second type is a mirror image of the over- and underdensities which become more evident in the center of the hosts. The mirror plane is perpendicular to the direction of the satellite velocity. We derive features analytically from Chandrasekhar’s formula of dynamical friction. Our $N$-body simulations validate the analytical predictions well. The scratches may be found in nearby interacting galaxies by observations. The dynamical mass including a dark halo of the satellite galaxies, the LMC and M33 is still uncertain by a factor of 10 [e.g. @2003MNRAS.342..199C; @2011ISRAA2011E...4S; @2013ApJ...764..161K; @2014ApJ...781..121V]. As indicated by Equation (\[eq:ind\_dns\]) and shown in Figure \[fig:amp\], the amplitude of the induced density is proportional to the satellite mass. Combining the analytical model with observations, new constraints for the satellite masses may be provided. The form of the Coulomb logarithm is important in order to determine the features and amplitudes of scratches. In this paper, we adopt a simple formula to provide the Coulomb logarithm as a function of position. A caveat is the constant minimum impact parameter, $b_{\rm min}$ in Equation (\[eq:bmin\]) and the parameter, $B$ in Equation (\[eq:bmax2\]). We determine them by fitting analytical predictions to the simulation result but they may vary from system to system. Actually, $b_{\rm min}$ may depend on the local density since it should have similar values as the typical distance between the satellite and nearby particles. More systematic studies can help to improve the form of the Coulomb logarithm. In a later step it also would be important to consider more realistic configurations of host systems with many satellite systems orbiting around them. The analytical arguments in this paper might also help to understand the dynamical phenomena in these more complexed systems. Acknowledgments {#acknowledgments .unnumbered} =============== We are grateful to the anonymous referee for providing many helpful comments and suggestions. We thank Alessandro Ballone, Manuel Behrendt, Masashi Chiba, Jorge Cuadra, Guinevere Kauffmann, Takanobu Kirihara, Lucio Mayer, Yohei Miki, Masao Mori, Daisuke Nagai, Simon White and Kohji Yoshikawa for fruitful discussions. Numerical simulations were performed with HA-PACS at the Center for Computational Sciences at University of Tsukuba. This work was supported by Grant-in-Aid for JSPS Fellows (25-1455 GO) and the DFG cluster of excellence ‘Origin and Structure of the Universe’ (www.universe-cluster.de). \[lastpage\] [^1]: E-mail:ogiya@mpe.mpg.de [^2]: Max–Planck Fellow

--- abstract: 'Robots rely on sensors to provide them with information about their surroundings. However, high-quality sensors can be extremely expensive and cost-prohibitive. Thus many robotic systems must make due with lower-quality sensors. Here we demonstrate via a case study how modeling a sensor can improve its efficacy when employed within a Bayesian inferential framework. As a test bed we employ a robotic arm that is designed to autonomously take its own measurements using an inexpensive LEGO light sensor to estimate the position and radius of a white circle on a black field. The light sensor integrates the light arriving from a spatially distributed region within its field of view weighted by its Spatial Sensitivity Function (SSF). We demonstrate that by incorporating an accurate model of the light sensor SSF into the likelihood function of a Bayesian inference engine, an autonomous system can make improved inferences about its surroundings. The method presented here is data-based, fairly general, and made with plug-and play in mind so that it could be implemented in similar problems.' author: - | Nabin K. Malakar\ W. B. Hanson Center for Space Sciences\ University of Texas at Dallas\ Richardson TX\ nabin.malakar@utdallas.edu\ - | Daniil Gladkov\ Department of Physics\ University at Albany (SUNY)\ Albany NY\ dgladkov@albany.edu\ - | Kevin H. Knuth\ Departments of Physics and Informatics\ University at Albany (SUNY)\ Albany NY\ kknuth@albany.edu bibliography: - 'references.bib' title: Modeling a Sensor to Improve its Efficacy --- Introduction ============ Robots rely on sensors to provide them with information about their surroundings. However, high-quality sensors can be cost-prohibitive and often one must make due with lower quality sensors. In this paper we present a case study which demonstrates how employing an accurate model of a sensor within a Bayesian inferential framework can improve the quality of inferences made from the data produced by that sensor. In fact, the quality of the sensor can be quite poor, but if it is known precisely how it is poor, this information can be used to improve the results of inferences made from the sensor data. To accomplish this we rely on a Bayesian inferential framework where a machine learning system considers a set of hypotheses about its surroundings and identifies more probable hypotheses given incoming sensor data. Such inferences rely on a likelihood function, which quantifies the probability that a hypothesized situation could have given rise to the data. The likelihood is often considered to represent the noise model, and this inherently includes a model of how the sensor is expected to behave when presented with a given stimulus. By incorporating an accurate model of the sensor, the inferences made by the system are improved. As a test bed we employ an autonomous robotic arm developed in the Knuth Cyberphysics Laboratory at the University at Albany (SUNY). The robot is designed to perform studies in autonomous experimental design [@knuth_arm_2007; @knuth_center2010]. In particular it performs autonomous experiments where it uses an inexpensive LEGO light sensor to estimate the position and radius of a white circle on a black field. The light sensor integrates the light arriving from a spatially distributed region within its field of view weighted by its Spatial Sensitivity Function (SSF). We consider two models of the light sensor. The naïve model predicts that the light sensor will return one value on average if it is centered over a black region, and another higher value on average if it is centered on a white region. The more accurate model incorporates information about the SSF of the light sensor to predict what values the sensor would return given a hypothesized surface albedo field. We demonstrate that by incorporating a more accurate model of the light sensor into the likelihood function of a Bayesian inference engine, a robot can make improved inferences about its surroundings. The efficacy of the sensor model is quantified by the average number of measurements the robot needs to make to estimate the circle parameters within a given precision. There are two aspects to this work. First is the characterization of the light sensor, and second is the incorporation of the light sensor model into the likelihood function of the robot’s machine learning system in a demonstration of improved efficacy. In the Materials and Methods Section we describe the robot, its light sensor, the experiment that it is designed to perform, and the machine learning system employed. We then discuss the methods used to collect data from the light sensor, the models used to describe the light sensor, their incorporation into the machine learning system, the methods used to estimate the model parameters and select the model order. The Results and Discussion Section describe the resulting SSF model and the results of the experiments comparing the naïve light sensor model to the more accurate SSF model. In the Conclusion we summarize our results which demonstrate how by incorporating a more accurate model of sensor, one can improve its efficacy. Materials and Methods ===================== In this section we begin by discussing various aspects of the robotic test bed followed by a discussion of the techniques used to characterize the light sensor. Robotic Arm Test Bed -------------------- The robotic arm is designed to perform studies in autonomous experimental design [@knuth_arm_2007; @knuth_center2010]. The robot itself is constructed using the LEGO NXT Mindstorms system (Figure \[fig:arm-sensor\]) [^1]. It employs one motor to allow it to rotate about a vertical axis indicated by the black line in top center of the figure, and two motors to extend and lower the arm about the joints located at the positions indicated by the short arrows. The motors are controlled directly by the LEGO brick, which is commanded via Bluetooth by a Dell Latitude laptop computer running the robot’s machine learning system, which is programmed in MatLab (Mathworks, Inc.). The LEGO light sensor is attached to the end of the arm (indicated by the long arrow in Figure \[fig:arm-sensor\] and displayed in the insight at the upper right). Based on commands issued by the laptop, the robotic arm can deploy the light sensor to any position within its reach on the playing field. The light sensor is lowered to an average height of 14mm above the surface before taking a measurement. The arm is designed using a trapezoidal construction that maintains the sensor’s orientation to be aimed at nadir, always normal to the surface despite the extension of the arm. The LEGO light sensor (LEGO Part 9844) consists of a photodiode-LED pair. The white circle is the photo diode, and the red circle is the illuminating LED. Note that they are separated by a narrow plastic ridge, which prevents the LED from shining directly into the photo diode. This ridge, along with the plastic lenses and the presence of the illuminating LED, each affect the spatial sensitivity of the sensor. When activated, the light sensor flashes for a brief instant and measures the intensity of the reflected light. The photo diode and its support circuitry are connected to the sensor port of the LEGO Brick (LEGO Part 9841), which runs on a 32-bit ARM7 ATMEL micro-controller. The measured intensities are converted by internal software running on the ATMEL micro-controller to a scale of $1$ to $100$, which we refer to as *LEGO units*. The light sensor integrates the light arriving from a spatially distributed region within its field of view. The spatial sensitivity of the sensor to light sources within its field of view is described by the Spatial Sensitivity Function (SSF). This function is unknown, but if it were known, one could weight the surface albedo field with the SSF and integrate to obtain an estimate of the recorded intensity of the reflected light. The Circle Characterization Experiment -------------------------------------- The robotic arm is designed to deploy the light sensor to take measurements of the surface albedo at locations within a playing field of dimensions approximately $131 \times 65$ LEGO distance units (1048mm x 520mm), within an automated experimental design paradigm [@knuth_arm_2007; @knuth_center2008; @knuth_center2010]. The robot is programmed with a set of hypotheses of what shapes it could find placed on the playing field. Instead of being programmed with a specific set of strategies for characterizing the hypothesized shapes, the robot utilizes a generalized Bayesian Inference Engine coupled to an Inquiry Engine to make inferences about the hypotheses based on the recorded data and to use uncertainties in its inferences to drive further exploration by autonomously selecting new measurement location that promise to provide the maximum amount of relevant information about the problem. In this experiment, the robotic arm is instructed that there is a white circle of unknown radius arbitrarily placed on the black field. Such an instruction is encoded by providing the robot with a model of the surface albedo consisting of three parameters: the center location $(x_o, y_o)$ and radius $r_o$, written jointly as $$\label{eq:circle_model} \mathbf{C} = \{(x_o, y_o), r_o\},$$ so that given a measurement location $(x_i, y_i)$ the albedo $S$ is expected to be $$S(x_i, y_i ; \mathbf{C}) = \\ \begin{cases} 1 & \text{if $D((x_i, y_i),(x_o, y_o)) \leq r_o$}\\ 0 & \text{if $D((x_i, y_i),(x_o, y_o)) > r_o$}, \end{cases}$$ where $$\label{eq:distance} D((x_i, y_i),(x_o, y_o)) = \sqrt{(x_i-x_o)^2 + (y_i-y_o)^2}$$ is the Euclidean distance between the measurement location $(x_i, y_i)$ and the center of the circle $(x_o, y_o)$, and an albedo of $1$ signifies that the surface is white, and 0 signifies that the surface is black. Precisely how these expectations are used to make inferences from data is explained in the next section. Keep in mind that while the circle’s precise radius and position is unknown, the robot has been provided with limited prior information about the allowable range of radii and positions. Again, it is important to note that the robot does not scan the surface to solve the problem, nor does it try to find three points along the edge of the circle. Instead, it employs a general system that works for any expected shape or set of shapes that autonomously and intelligently determines optimal measurement locations based both on what it knows and on what it does not know. The number of measurements needed to characterize all three circle parameters to within the desired accuracy is a measure of the efficiency of the experimental procedure. The Machine Learning System --------------------------- The machine learning system employs a Bayesian Inference Engine to make inferences about the circle parameters given the recorded light intensities, as well as an Inquiry Engine designed to use the uncertainties in the circle parameter estimates to autonomously select measurement locations that promise to provide the maximum amount of relevant information about the problem. The core of the Bayesian Inference Engine is centered around the computation of the *posterior probability* $Pr(\mathbf{C} | \mathbf{D}, I)$ of the albedo model parameters, $\mathbf{C}$ in (\[eq:circle\_model\]) given the light sensor recordings (data) $d_i$ recorded at locations $(x_i, y_i)$, which we write compactly as $$\mathbf{D} = \{(d_1, (x_1, y_1)), \ldots, (d_N, (x_N, y_N))\},$$ and any additional prior information $I$. Bayes’ Theorem allows one to write the posterior probability as a function of three related probabilities $$Pr(\mathbf{C} | \mathbf{D}, I) = Pr(\mathbf{C} | I) \frac{Pr(\mathbf{D} | \mathbf{C}, I)}{Pr(\mathbf{D} | I)},$$ where the right-hand side consists of the product of the *prior probability* of the circle parameters $Pr(\mathbf{C} | I)$, which describes what is known about the circle before any sensor data are considered, with a ratio of probabilities that are sensor data dependent. It is in this sense that Bayes’ Theorem represents a learning algorithm since what is known about the circle parameters before the data are considered (prior probability) is modified by the recorded sensor data resulting in a quantification of what is known about the circle parameters after the data are considered (posterior probability). The probability in the numerator on the right is the *likelihood* $Pr(\mathbf{D} | \mathbf{C}, I)$, which quantifies the probability that the sensor data could have resulted from the hypothesized circle parameterized by $\mathbf{C}$. The probability in the denominator is the *marginal likelihood* or the *evidence*, which here acts as a normalization factor. Later when estimating the SSF of the light sensor (which is a different problem), the evidence, which can be written as the integral of the product of the prior and the likelihood over all possible model parameters, will play a critical role. The likelihood term, $Pr(\mathbf{D} | \mathbf{C}, I)$, plays a critical role in the inference problem, since it essentially compares predictions made using the hypothesized circle parameters to the observed data. A naïve light sensor model would predict that if the sensor was centered on a black region, it would return a small number, and if the sensor was centered on a white region, it would return a large number. A more accurate light sensor model would take into account the SSF of the light sensor and perform an SSF-weighted integral of the hypothesized albedo field and compare this to the recorded sensor data. These two models of the light sensor will be discussed in detail in the next section. ![This figure illustrates the robot’s machine learning system’s view of the playing field using the naïve light sensor model. The axes label playing field coordinates in LEGO distance units. The previously obtained measurement locations used to obtain light sensor data are indicated by the black and white squares indicating the relative intensity with respect to the naïve light sensor model. The next selected measurement location is indicated by the green square. The blue circles represent the 50 hypothesized circles sampled from the posterior probability. The shaded background represents the entropy map, such that brighter areas indicate the measurement locations that promise to provide greater information about the circle to be characterized. Note that the low entropy area bounded by the white squares indicates that this region is probably inside the white circle and that measurements made here will not be as informative as measurements made elsewhere. The dark jagged edges at the bottom of colored high entropy region reflect the boundary between the playing field and the region that is outside of the robotic arm’s reach.[]{data-label="fig:entropy-map"}](entropy-map){width="90.00000%"} The robot not only makes inferences from data, but it designs its own experiments by autonomously deciding where to take subsequent measurements [@knuth_intelligent_2003; @knuth_arm_2007; @knuth_center2008; @knuth_center2010; @malakar_entropy-based_2010; @malakar_2011]. This can be viewed in terms of Bayesian experiment design [@lindley_measure_1956; @fedorov1972theory; @Chaloner95bayesianexperimental; @sebastiani_bayesian_1997; @sebastiani2000maximum; @Loredo03bayesianadaptive; @fischer2004bayesian] where the Shannon entropy [@shannon_mathematical_1949] is employed as the utility function used to decide where to take the next measurement. In short, the Bayesian Inference Engine samples 50 circles from the posterior probability using the nested sampling algorithm [@skilling_nested_2004; @sivia_data_2006]. Since these samples may consist of replications, they are further diversified by taking approximately 1000 Metropolis-Hastings steps [@Metropolis:1953; @Hastings:1970]. Figure \[fig:entropy-map\] illustrates the robot’s machine learning system’s view of the playing field after several measurements have been recorded. The 50 circles sampled from the posterior probability (blue circles) reflect what the robot knows about the white circle (not shown) given the light sensor data that it has collected. The algorithm then considers a fine grid of potential measurement locations on the playing field. At each potential measurement location, the 50 sampled circles are queried using the likelihood function to produce a sample of what measurement could be expected at that location given each hypothesized circle. This results in a set of potential measurement values at each measurement location. The Shannon entropy of the set of potential measurements at each location is computed, which results in an *entropy map*, which is illustrated in Figure \[fig:entropy-map\] as the copper-toned coloration upon which the blue circles lie. The set of measurement locations with the greatest entropy are identified, and the next measurement location is randomly chosen from that set. Thus the likelihood function not only affects the inferences about the circles made by the machine learning algorithm, but it also affects the entropy of the measurement locations, which guides further exploration. In the next section we describe the two models of the light sensor and their corresponding likelihood functions. The efficacy of the sensor model will be quantified by the average number of measurements the robot needs to make to estimate the circle parameters within a given precision. Models Describing a Light Sensor -------------------------------- In this section we discuss two models of a light sensor and indicate precisely how they are integrated into the likelihood function used by both the Bayesian Inference and Inquiry Engines. ### The Naïve Likelihood A naïve light sensor model would predict that if the sensor was centered on a black region (surface albedo of zero), the sensor would return a small number on average, and if it were centered on a white region (surface albedo of unity), it would return a large number on average. Of course, there are expected to be noise variations from numerous sources, such as uneven lighting of the surface, minor variations in albedo, and noise in the sensor itself. So one might expect that there is some expected squared deviation from the two mean sensor values for the white and black cases. For this reason, we model the expected sensor response with a Gaussian distribution with mean $\mu_B$ and standard deviation $\sigma_B$ for the black surface, and a Gaussian distribution with mean $\mu_W$ and standard deviation $\sigma_W$ for the white surface. The likelihood of a measurement $d_i$ at location $(x_i, y_i)$ corresponding to the naïve light sensor model, $Pr_{naive}(\{(d_i, (x_i, y_i))\} | \mathbf{C}, I)$, can be written compactly as: $$Pr_{naive}(\{(d_i, (x_i, y_i))\} | \mathbf{C}, I) = \begin{cases} (2\pi \sigma_W)^{-1/2} \exp{\Big[\frac{(\mu_W-d_i)^2}{2 \sigma_W^2}\Big]} & \text{for $D((x_i, y_i),(x_o, y_o)) \leq r_o$}\\ (2\pi \sigma_B)^{-1/2} \exp{\Big[\frac{(\mu_B-d_i)^2}{2 \sigma_B^2}\Big]} & \text{for $D((x_i, y_i),(x_o, y_o)) > r_o$}, \end{cases}$$ where $\mathbf{C} = \{(x_o, y_o), r_o\}$ represents the parameters of the hypothesized circle and $D((x_i, y_i),(x_o, y_o))$ is the Euclidean distance given in (\[eq:distance\]). The joint likelihood for $N$ independent measurements is found by taking the product of the $N$ single-measurement likelihoods. In a practical experiment, the means $\mu_B$ and $\mu_W$ and standard deviations $\sigma_B$ and $\sigma_W$ can be easily estimated by sampling known black and white regions several times using the light sensor. The SSF Likelihood {#sec:SSF-Likelihood} ------------------ A more accurate likelihood can be developed by taking into account the fact that the photo diode performs a weighted-integral of the light arriving from a spatially distributed region within its field of view, the weights being described by the Spatial Sensitivity Function (SSF) of the sensor. Since the SSF of the light sensor could be arbitrarily complex with many local peaks, but is expected to decrease to zero far from the sensor, we characterize it using a mixture of Gaussians (MoG) model, which we describe in this section. The sensor’s response situated a fixed distance above the point $(x_i, y_i)$ to a known black-and-white pattern can be modeled in the lab frame by $$\label{eq:modelI} M(x_i, y_i) = I_{min} + \left( I_{max} - I_{min}\right) R\left(x_i, y_i\right),$$ where $I_{min}$ and $I_{max}$ are observed intensities for a completely black surface (surface albedo of zero) and a completely white surface (surface albedo of unity), respectively, and $R$ is a scalar response function, varying between zero and one, that depends both on the SSF and the surface albedo [@malakar_SSF2009]. The minimum intensity $I_{min}$ acts as an offset, and $(I_{max} - I_{min})$ serves to scale the response to LEGO units. The response of the sensor is described by $R(x_i, y_i)$, which is a convolution of the sensor SSF and the surface albedo $S(x,y)$ given by $$\label{eq:RwithSSF} R(x_i,y_i) = \int dx ~dy ~SSF(x-x_i , y-y_i) ~S(x,y).$$ where the SSF is defined so that the convolution with a completely white surface results in a response of unity. In practice, we approximate this integral as a sum over a pixelated grid with 1mm square pixels $$\label{eq:discreteR} R(x_i,y_i) = \sum_{x,y}~SSF(x-x_i , y-y_i)~S(x,y).$$ We employ a mixture of Gaussians (MOG) as a parameterized model to describe the SSF in the sensor’s frame co-ordinates $(x', y') = (x-x_i , y-y_i)$ $$\begin{split} \label{Eq:ssf} SSF(x',y') = \frac{1}{K} & \sum_{n=1}^{N} a_n ~\times\\ & \exp\left[ -\left\{A_n (x' - u'_n)^2+ B_k (y' -v'_n)^2+ 2 C_n (x'-u'_n)(y'-v'_n)\right\}\right], \end{split}$$ where $(u'_n, v'_n)$ denotes the center of the $n^{th}$ two-dimensional Gaussian with amplitude $a_n$ and covariance matrix elements given by $A_n, B_n $ and $C_n$. The constant $K$ denotes a normalization factor, which ensures that the SSF integrates to unity [@malakar_SSF2009]. The model is sufficiently general so that one could vary the number of Gaussians to any number, although we found that in modeling this light sensor testing models with $N$ varying from $N=1$ to $N=4$ was sufficient. The MOG model results in a set of six parameters to be estimated for each Gaussian $$\theta_n = \left\{a_n , u'_n, v'_n, A_n, B_n, C_n \right\},$$ where the subscript $n$ is used to denote the $n^{th}$ Gaussian in the set. These must be estimated along with the two intensity parameters, $I_{min}$ and $I_{max}$ in (\[eq:modelI\]) so that an MOG model with $N$ Gaussians consists of $6 N + 2$ parameters to be estimated. We assign a Student-t distribution to the SSF model likelihood, which can be arrived at by assigning a Gaussian likelihood and integrating over the unknown standard deviation $\sigma$ [@sivia_data_2006]. By defining $\mathbf{D} = \{(d_1,(x_1,y_1)), (d_2,(x_2,y_2)), \ldots, (d_N,(x_N,y_N))\}$ and writing the hypothesized circle parameters as $\mathbf{C} = \{(x_o, y_o), r_o\}$, we have $$Pr_{SSF}(\mathbf{D} | \mathbf{C}, I) \propto \left[ \sum_{i=1}^{N}\left( M(x_i, y_i) - d_i \right)^2 \right] ^{-(N-1)/2} .$$ where $N$ is the number of measurements made by the robot, and the function $M$, defined in (\[eq:modelI\]), relies on both (\[eq:RwithSSF\]) and (\[Eq:ssf\]). Note that in practice, the MoG SSF model described in (\[Eq:ssf\]) is used to generate a discrete SSF matrix which is used in the convolution (\[eq:discreteR\]) to compute the likelihood function via the sensor response model $M$ in (\[eq:modelI\]). Data Collection for SSF Estimation ---------------------------------- In this section we describe the collection of the light sensor readings in the laboratory that were used to estimate the SSF of the light sensor. The SSF is not only a function of the properties of the photo diode, but also of the illuminating LED and the height above the surface. For this reason, measurements were made at a height of 14mm above a surface with a known albedo in a darkened room to avoid complications due to ambient light and to eliminate the possibility of shadows cast by the sensor or other equipment [@malakar_SSF2009]. ![(Bottom) This figure illustrates the laboratory surface, referred to as the *black-and-white boundary pattern*, with known albedo, which consisted of two regions: a black region on the left and a white region on the right separated by a sharp linear boundary. (Top) This illustrates the four sensor orientations used to collect data for the estimation of the SSF along with the symbols used to indicate the measured values plotted in Figure \[fig:data\]. The top row views the sensors as if looking up from the table so that the photodiode placement in the sensor package can be visualized. Below these the gray sensor shapes illustrate how they are oriented looking down at both the sensors and the albedo surface. Measurements were taken as the sensor was incrementally moved, with one millimeter steps, in a direction perpendicular to the boundary (as indicated by the arrow at the bottom of the figure) from a position of 5cm to the left of the boundary (well within the completely black region) to a position 5cm to the right of the boundary (well within the completely white region). This process was repeated four times with the sensor in each of the four orientations.[]{data-label="fig:orientation"}](sensor_Orientation-2){width="40.00000%"} The surface, which we refer to as the *black-and-white boundary pattern*, consisted of two regions: a black region on the left and a white region on the right separated by a sharp linear boundary as shown in Figure \[fig:orientation\] (bottom). Here the surface and sensor are illustrated as if viewing them by looking up at them from below the table surface. This is so the placement of the photodiode in the sensor package can be visualized. The lab frame was defined to be at the center of the black-white boundary so that the surface albedo, $S_{BW}(x,y)$, is given by $$S_{BW}(x,y) = \begin{cases} 1, & \text{for $ x > 0$}\\ 0, & \text{for $x \le 0$} \\ \end{cases}$$ Measurements were taken as the sensor was incrementally moved, with one millimeter steps, in a direction perpendicular to the boundary from a position of 5cm to the left of the boundary (well within the completely black region) to a position 5cm to the right of the boundary (well within the completely white region) resulting in 101 measurements. This process was repeated four times with the sensor in each of the four orientations (see Figure \[fig:orientation\] for an explanation), giving a total of 404 measurements using this pattern. The black-and-white boundary pattern does not provide sufficient information to uniquely infer the SSF, since the sensor may have a response that is symmetric about a line oriented at $45^o$ with respect to the linear boundary. For this reason, we employed four additional albedo patterns consisting of black regions with one white quadrant as illustrated in Figure \[four-extra\], which resulted in four more measurements. These have surface albedos defined by $$\begin{aligned} \label{eq:additional_albedos} S_a(x,y) & = \begin{cases} 1, & \text{for $ x > 0mm$ and $y > 0mm$}\\ 0, & \text{otherwise} \\ \end{cases}\\ S_b(x,y) & = \begin{cases} 1, & \text{for $ x < 0mm$ and $y > 0mm$}\\ 0, & \text{otherwise} \\ \end{cases}\\ S_c(x,y) & = \begin{cases} 1, & \text{for $ x > 4mm$ and $y > 4mm$}\\ 0, & \text{otherwise} \\ \end{cases}\\ S_d(x,y) & = \begin{cases} 1, & \text{for $ x < -4mm$ and $y > 4mm$}\\ 0, & \text{otherwise} \\ \end{cases}\end{aligned}$$ where the subscripts relate each albedo function to the pattern illustrated in Figure \[four-extra\]. ![Four additional symmetry-breaking albedo patterns were employed. In all cases, the sensor was placed in the $0^o$ orientation at the center of the pattern, indicated by $(0,0)$. In the two lower patterns, the center of the white square was shifted diagonally from the center by 4mm as indicated by the coordinates (white text) of the corner. The recorded intensity levels are displayed in the white albedo area.[]{data-label="four-extra"}](symmetry_breaking_intensity-2){width="30.00000%"} Estimating SSF MoG Model Parameters {#sec:estimating-SSF} ----------------------------------- In this section we describe the application of Bayesian methods to estimate the SSF MoG model parameters. Keep in mind that in this paper we are considering two distinct inference problems: the robot’s inferences about a circle and our inferences about the light sensor SSF. Both of these problems rely on making predictions about measured intensities using a light sensor. For this reason many of these equations will not only look similar to what we have presented above, but also depend on the same functions mapping modeled albedo fields to predicted sensor responses, which are collectively represented using the generic symbol $\mathbf{D}$. It may help to keep in mind that the robot is characterizing a circle quantified by model parameters represented jointly by the symbol $\mathbf{C}$ and we are estimating the SSF of a light sensor quantified by model parameters represented jointly by the symbol $\theta$ below. Aside from the evidence, represented by function $Z$ below (which is not used by the robot in this experiment), all of the probability functions contain the model parameters in their list of arguments making it clear to which inference problem they refer. The posterior probability for the SSF MoG model parameters, collectively referred to as $\theta = \{\theta_1, \theta_2, \ldots, \theta_N\}$, for a model consisting of $N$ Gaussians is given by $$\label{eq:bayes} Pr(\theta|\textbf{D}, I) = \frac{1}{Z} ~Pr(\theta | I) ~Pr(\textbf{D} |\theta, I),$$ where here $\mathbf{D}$ refers to the data collected for the SSF estimation experiment described in the previous section, $I$ refers to our prior information about the SSF (which is that it may have several local peaks and falls off to zero far from the sensor), and $Z$ refers to the evidence $Z = Pr(\textbf{D} |I)$, which can be found by $$\label{Evidence1} Z = \int d\theta Pr(\theta | I) Pr(\textbf{D} |\theta, I).$$ In our earlier discussion where Bayes’ theorem was used to make inferences about circles, the evidence played the role of a normalization factor. Here, since we can consider MoG models with different numbers of Gaussians and since we integrate over all of the possible values of the parameters, the evidence quantifies the degree to which the hypothesized model order $N$ supports the data. That is, the optimal number of Gaussians to be used in the MoG model of the SSF can be found by computing the evidence. All five sets of data $\mathbf{D}$ described in the previous section were used to compute the posterior probability. These are each indexed by the subscript $i$ where $i = 1,2,3,4$, so that $D_i$ refers to the data collected using each of the four orientations $\phi_i = \left\{0^o, 90^o, 180^o, 270^o \right\}$ to scan the black-and-white boundary pattern resulting in $N_i = 101$ measurements for $i = 1,2,3,4$, and where the value $i=5$ refers to the $N_i = 4$ measurements attained using the $0^o$ orientation with the set of four additional patterns. We assign uniform priors so that this is essentially a maximum likelihood calculation with the posterior being proportional to the likelihood. We assign a Student-t distribution to the likelihood, which can be arrived at by assigning a Gaussian likelihood and integrating over the unknown standard deviation $\sigma$ [@sivia_data_2006]. This can be written as $$\label{eq:studentT} Pr(D_i | ~\theta, I) \propto \left[ \sum_{j=1}^{N_i} \left( M_{ij}(x_{ij},y_{ij}) - D_i(x_{ij}, y_{ij}) \right)^2 \right] ^{-(N_i-1)/2} .$$ where $i$ denotes each of the five sets of data and $j$ denotes the $j^{th}$ measurement of that set, which was taken at position $(x_{ij},y_{ij})$. The function $M_{ij}(x,y)$ represents a predicted measurement value obtained from (\[eq:modelI\]) using (\[eq:discreteR\]) with the albedo function $S(x,y)$ defined using the albedo pattern $S_{BW}(x,y)$ with orientation $\phi_i$ for $i = 1,2,3,4$ and the albedo patterns $S_a, S_b, S_c, S_d$ for $i=5$ and $j=1,2,3,4$, respectively. As such the likelihood relies on a difference between predicted and measured sensor responses. The joint likelihood for the five data sets is found by taking the product of the likelihoods for each data set, since we expect that the standard deviations that were marginalized over to get the Student-t distribution could have been different for each of the five data sets as they were not all recorded at the same time $$Pr(\textbf{D} | ~\theta, I) = \prod_{i=1}^{5} Pr(D_i| ~\theta, I).$$ We employed nested sampling [@skilling_nested_2004; @sivia_data_2006] to explore the posterior probability since, in addition to providing parameter estimates, it is explicitly designed to perform evidence calculations, which we use to perform model comparison in identifying the most probable number of Gaussians in the MoG model. For each of the four MoG models (number of Gaussians varying from one to four) the nested sampling algorithm was initialized with 300 samples and iterated until the change in consecutive log-evidence values less than $10^{-8}$. Typically one estimates the mean parameter values by taking an average of the samples weighted by a quantity computed by the nested samp\[ling algorithm called the logWt in the references [@skilling_nested_2004; @sivia_data_2006]. Here we simply performed a logWt-weighted average of the sampled SSF fields computed using the sampled MoG model parameters (rather than the logWt-weighted average of the MoG model parameters themselves), so the result obtained using an MoG model consisting of a single Gaussian is not strictly a single two-dimensional Gaussian distribution. It is this discretized estimated SSF field matrix that is used directly in the convolution (\[eq:discreteR\]) to compute the likelihood functions as mentioned earlier in the last lines of Section \[sec:SSF-Likelihood\]. Results and Discussion ====================== In this section we present the SSF MoG light sensor model estimated from the laboratory data, and evaluate its efficacy by demonstrating a significant improvement of the performance in the autonomous robotic platform. Light Sensor SSF MoG Model -------------------------- ![This figure illustrates the intensity measurements, $D_1, D_2, D_3, D_4$ from the four sensor orientations, recorded from the sensor using the black-and-white boundary pattern. Figure \[fig:orientation\] shows the orientations of the sensor corresponding to the symbols used this figure.[]{data-label="fig:data"}](data-2){width="70.00000%"} The light sensor data collected using the black-and-white boundary pattern are illustrated in Figure \[fig:data\]. One can see that the intensity recorded by the sensor increases dramatically as the sensor crosses the boundary from the black region to the white region, but that the change is not a step function, which indicates the finite size of the surface area integrated by the sensor. It is this effect that is to be modeled by the SSF function. There is an obvious asymmetry between the $90^o$ and $270^o$ orientations due to the shift of the transition region. In addition, there is a significant different in slope of the transition region between the $0^o,180^o$ orientations and the $90^o,270^o$ orientations indicating a significant difference in the width of the SSF in those directions. Note also that the minimum recorded response is not zero, as the reflectance of the black surface is not completely zero. ![This figure illustrates the SSF obtained from the four MOG models along with their corresponding log-evidence values.[]{data-label="fig:mogSSF"}](FourSSF){width="60.00000%"} MoG Model Order Log-Evidence Number of Parameters ----------------- ----------------- ---------------------- -- 1 Gaussian $-665.5\pm 0.3$ 6 2 Gaussian $-674.9\pm 0.3$ 12 3 Gaussian $-671.9\pm 0.4$ 18 4 Gaussian $-706.1\pm 0.4$ 24 : A comparison of the tested MoG SSF Models and their respective Log-Evidence (in units of $data^{-408}$).[]{data-label="tableMoG"} The nested sampling algorithm produced the mean SSF fields for each of the MoG models tested, as well as the corresponding log-evidence. Table \[tableMoG\], which lists the log-evidence computed for each MoG model order, illustrates that the most probable model was obtained from the single two-dimensional Gaussian models by a factor of about $\exp(9)$, which means that it is about 8000 times more probable than the MoG consisting of two Gaussians. Figure \[fig:mogSSF\] shows the mean SSF fields described by the MOG models of different orders. In all cases, the center of the SSF is shifted slightly above the physical center of the sensor package due to the placement of the photodiode (refer to Figure \[fig:arm-sensor\]) as indicated by the data in Figure \[fig:data\]. In addition, as predicted, one sees that the SSF is wider along the $90^o-270^o$ axis than along the $0^o-180^o$ axis. Last, Figure \[fig:1mogprediction\] demonstrates that the predicted light sensor output shows excellent similarity to the recorded data for the black-and-white boundary pattern. ![A comparison of the observed data (red) with predictions (black) made by the SSF field estimated using the single two-dimensional Gaussian MoG model. []{data-label="fig:1mogprediction"}](intensity_compared1MoG-new){width="70.00000%"} In the next section, we demonstrate that explicit knowledge about how the light sensor integrates light arriving from within its field-of-view improves the inferences one can make from its output. Efficacy of Sensor Model ------------------------ The mean SSF field obtained using a single two-dimensional Gaussian model (Figure \[fig:mogSSF\], upper left), was incorporated into the likelihood function used by the robot’s machine learning system. Here we compare the robot’s performance in locating and characterizing a circle by observing the average number of measurements necessary to characterize the circle parameters to within a precision of 4mm (which is one-half of the spacing between the holes in the LEGO Technic Parts). ![(Left Column) These three panels illustrate the robot’s machine learning system’s view of the playing field using the naïve light sensor model as the system progresses through the first three measurements. The previously obtained measurement locations used to obtain light sensor data are indicated by the black and white squares indicating the relative intensity with respect to the naïve light sensor model. The next selected measurement location is indicated by the green square. The blue circles represent the 50 hypothesized circles sampled from the posterior probability. The shaded background represents the entropy map, which indicates the measurement locations that promise to provide maximal information about the circle to be characterized. Note that the low entropy area surrounding the white square indicates that the region is probably inside the white circle (not shown) and that measurements made there will not be as informative as measurements made elsewhere. The entropy map in Figure \[fig:entropy-map\] shows the same experiment at a later stage after seven measurements have been recorded. (Right Column) These three panels illustrate the robot’s machine learning system’s view of the playing field using the more accurate SSF light sensor model. Note that the entropy map reveals the circle edges to be highly informative. This is because it helps not only to identify whether the sensor is inside the circle (as is accomplished using the naïve light sensor model on the left), but also the extent to which the sensor is on the edge of the circle.[]{data-label="fig:entropy-map-scenes"}](entropy-map-scenes){width="90.00000%"} The three panels comprising the left column of Figure \[fig:entropy-map-scenes\] illustrate the robot’s machine learning system’s view of the playing field using the naïve light sensor model. The previously obtained measurement locations used to obtain light sensor data are indicated by the black and white squares indicating the relative intensity with respect to the naïve light sensor model. The next selected measurement location is indicated by the green square. The blue circles represent the 50 hypothesized circles sampled from the posterior probability. The shaded background represents the entropy map, which indicates the measurement locations that promise to provide maximal information about the circle to be characterized. The low entropy area surrounding the white square indicates that the region is probably inside the white circle (not shown) and that measurements made there will not be as informative as measurements made elsewhere. Note that the circles partition the plane, and that each partition has a uniform entropy. All measurement locations within that partition, or any other partition sharing the same entropy, all stand to be equally informative. Here it is the fact that the shape is known to be a circle that is driving the likelihood function. In contrast, the three panels comprising the right column of Figure \[fig:entropy-map-scenes\] illustrate the playing field using the more accurate SSF model. Here one can see that the entropy is higher along the edges of the sampled circles. This indicates that the circle edges promise to provide more information than the centers of the partitioned regions. This is because the SSF model enables one to detect not only whether the light sensor is situated above the circles edge but also how much of the SSF overlaps with the white circle. That is, it helps not only to identify whether the sensor is inside the circle (as is accomplished using the naïve light sensor model), but also the extent to which the sensor is on the edge of the circle. The additional information provided about the functioning of the light sensor translates directly into additional information about the albedo that results in the sensor output. This additional information can be quantified by observing how many measurements the robot is required to take to obtain estimates of the circle parameters to within the same precision in the cases of each light sensor model. Our experiments revealed that on average it takes $26\pm$ measurements using the naïve light sensor model compared to an average of $16\pm$ measurements for the more accurate SSF light sensor model. Conclusion ========== The quality of the inferences one makes from a sensor depend not only on the quality of the data returned by the sensor, but also on the information one possesses about the sensor’s performance. In this paper we have demonstrated via a case study, how more precisely modeling a sensor’s performance can improve the inferences one can make from its data. In this case, we demonstrated that one can achieve an almost 40% reduction of the number of measurements needed by a robot to make the same inferences by more precisely modeling its light sensor. This paper demonstrates how a machine learning system that employs Bayesian inference (and inquiry) relies on the likelihood function of the data given the hypothesized model parameters. Rather than simply representing a noise model, the likelihood function quantifies the probability that a hypothesized situation could have given rise to the recorded data. By incorporating more information about the sensors (or equipment) used to record the data, one naturally is incorporating this information into the posterior probability, which results in one’s inferences. This is made even more apparent by a careful study of the experimental design problem that this particular robotic system is designed to explore. For example, it is easy to show that by using the naïve light sensor model, the entropy distribution for a proposed measurement location depends solely on the number of sampled circles for which the location is in the interior of the circle, and the number of sampled circles for which the location is exterior to the circle. Given that we represented the posterior probability by sampling 50 circles, the maximum entropy occurs when the proposed measurement location is inside of 25 circles (and outside of 25 circles). As the robot’s parameter estimates converge, one can show that the system is simply performing a binary search by asking ‘yes’ or ‘no’ questions, which implies that each measurement results in one bit of information. However, in the case where the robot employs an SSF model of the light sensor, the question the robot is essentially asking is more detailed: ‘To what degree does the circle overlap the light sensor’s SSF?’ The answer to such a question tends to provide more information, which significantly improves system performance. One can estimate the information gain achieved by employing the SSF model. Consider that naïve model reveals that estimating the circle’s position and radius with a precision of 4mm given the prior information about the circle and the playing field requires 25 bits of information. The experiment using the SSF model requires on average 16 measurements, which implies that on average each measurement obtained using the SSF model provides about $25/16 = 1.56$ bits of information. One must keep in mind, however, that this is due to the fact that the SSF model is being used not only to infer the circle parameters from the data, but also to select the measurement locations. Because the method presented here is based on a very general inferential framework, these methods can easily be applied to other types of sensors and equipment in a wide variety of situations. If one has designed a robotic machine learning system to rely on likelihood functions, then sensor models can be incorporated in more or less a plug-and-play fashion. This not only promises to improve the quality of robotic systems forced to rely on lower quality sensors, but it also opens the possibility for calibration on the fly by updating sensor models as data are continually collected. Acknowledgements {#acknowledgements .unnumbered} ================ This research was supported in part by the University at Albany Faculty Research Awards Program (Knuth) and the University at Albany Benevolent Research Grant Award (Malakar). We would like to thank Scott Frasso and Rotem Guttman for their assistance in interfacing MatLab to the LEGO Brick. We would also like to thank Phil Erner and A.J. Mesiti for their preliminary efforts on the robotic arm experiments and sensor characterization. [^1]: The LEGO Mindstorm system was utilized in part to demonstrate that high-quality autonomous systems can be achieved when using lower-quality equipment if the machine learning and data analysis algorithms are handled carefully.

--- author: - 'A. Papitto' - 'A. D’Aì' - 'S. Motta' - 'A. Riggio$^{,}$' - 'L. Burderi' - 'T. Di Salvo' - 'T. Belloni' - 'R. Iaria' bibliography: - 'biblio.bib' title: The spin and orbit of the newly discovered pulsar --- Introduction ============ The dense environment of a globular cluster and the resulting frequent stellar encounters [@Mey97] make the production of binary systems hosting a compact object very efficient. Terzan 5 is probably one of the densest and metal-richest cluster in our Galaxy [@Chn02; @Ort07], as clearly indicated by the large number of rotation-powered millisecond pulsars discovered there ($\simgt30$, @Rns05 [@Hss06]). The cluster also contains at least 28 discrete X-ray sources, a substantial number of which can be identified as quiescent low-mass X-ray binaries [LMXB, @Hnk06]. According to the recycling scenario [see, e.g. @BhtvdH91], the population of rotation-powered millisecond pulsars and low-mass X-ray binaries (LMXB) share an evolutionary link, because the former are thought to be spun up by the accretion of mass and angular momentum in an LMXB. Accreting pulsars in LMXB are usually found with periods clustering in two distinct groups. So far, 13 sources have been discovered with spin periods lower than 10 ms and were therefore termed accretion-powered millisecond pulsars [see, e.g., @WijvdK98]. However, a smaller number of sources are found with longer periods and correspondingly higher estimates of the neutron star (NS) magnetic field. So far, the only bright transient LMXB known in the cluster Terzan 5 was the burster EXO 1745-248 [@Mks81]. The first detection of a new outburst of a source in this cluster was made on 2010 October 10.365 with [*INTEGRAL*]{} [@Brd10] and was tentatively attributed to EXO 1745-248. Follow-up [*Swift*]{} observations refined the source position, and a comparison with the position of sources previously known thanks to [*Chandra*]{} observations of the cluster suggested instead a different association [@Hnk10; @Knn10; @Poo10]. The X-ray transient is then considered a newly discovered source and named as ( in the following). A coherent signal at a period of 90.6 ms was detected thanks to observations performed with the [ in the following, @Str10]. A signal at the same period is present also during the several bursts that the source shows [@Alt10], while burst oscillations have never been observed from NS rotating at periods exceeding a few ms. A sudden decrease of the flux was tentatively attributed to an eclipse of the source by the companion [@Str10]. However, eclipses were not observed during subsequent observations, and the earlier flux decrease was identified with a lunar occultation [@Str10bis S10 in the following]. Below we present the first analysis of the properties of the coherent signal emitted by this source, using and [*Swift*]{} observations performed between 2010 October 10 and November 6, and give a refined orbital and timing solution of the pulsar with respect to those first proposed (@Ppt10, P10 in the following, S10). Observations and data analysis {#sec:obs} ============================== After the source discovery on 2010 October 10.365 [@Brd10], started monitoring the source on MJD 55482.010 (October 13.010; all the epochs reported in this paper are given with regard to the Barycentric Dynamical Time, TDB, system). We present an analysis of the observations performed until MJD 55506.359 (ObsId 95437-01-12-01), for a total exposure of 206 ks. In this time interval, a large number ($>$ 300) of X-ray bursts are observed with a recurrence time decreasing from $\simgt26$ to $\sim3$ min as the X-ray flux increases. The analysis of the bursts shown by the source, as well as of its aperiodic timing properties, will appear in a companion paper. The 2.5-25 keV light curve recorded by PCU2 of the Proportional Counter Array (PCA) on-board RXTE, with the burst intervals removed and background subtracted, is plotted in Fig.\[fig1\]. The count rate increases during the first days of the outburst, reaching a peak value of $\sim1700$ s$^{-1}$ at MJD 55487.5, and then decreases to a value of $\sim 800$ s$^{-1}$ with an exponential decay time scale of $\sim5$ d. Quite complex dipping-like structures also appear especially between MJD 55485 and 55490 as the flux shows sudden variations up to 75$\%$ on timescales of $\sim10$ min. The combined X-ray spectrum observed by the top layer of the PCU2 of the PCA (2.5–25 keV), and by Cluster A of the High Energy X-ray Timing Experiment (HEXTE, 22–50 keV), can be well modelled by a sum of a blackbody and a Comptonized component, which we model with `compps` [@PouSve96]. Throughout the observations considered here, the spectrum softens significantly as the source evolves towards higher luminosities. The unabsorbed total flux, extrapolated in the 0.1-100 keV band, rises from $0.47(3)\times 10^{-8}$ to a maximum level of $1.89(4)\times10^{-8}$ erg cm$^{-2}$ s$^{-1}$, observed during the observation of MJD 55487.5. All the uncertainties on the fluxes given here are quoted at a 90% confidence level. To analyse the spin and orbital properties of the source we consider data taken by the PCA in Event ($122\mu$s temporal resolution) and Good Xenon ($1\mu$s temporal resolution) packing modes. We discard 10s prior, and 100s after the onset of each type-I X-ray burst. The time series were also preliminarly barycentred with respect to the solar system barycentre using the available orbit files and assuming the best [*Chandra*]{} estimate of the source position, RA=17$^{h}$ 48$^{m}$ 4.831$^{s}$, DEC=-24$^{\circ}$ 46’ 48.87”, with an error circle of 0.06” (1$\sigma$ confidence level, @Hnk06 [@Poo10]). A coherent signal at a frequency of 11.045(1) Hz \[equivalent to a period of 90.539(8) ms\] is clearly detected in the power spectrum at a Leahy-normalised power of $\simeq 1.5\times10^4$. A first orbital solution is obtained modelling the observed modulation caused by the orbital motion, $$P(t)= P_0\left\{1+\frac{2\pi x}{P_{orb}} [\cos{m}+e\cos(2m-\omega)]\right\}.$$ Here $P_0$ is the barycentric spin period of the source, $x=a\sin{i}/c$ the semi-major axis of the NS orbit, $P_{orb}$ the orbital period, $m=2\pi(t-T*)/P_{orb}$ the mean anomaly, $T^*$ the epoch at which $m=0$, $e$ the eccentricity and $\omega$ the longitude of the periastron measured from the ascending node. The periods $P(t)$ are estimated by performing an epoch-folding search on 1.5 ks long data segments (for a total of 127) around the periodicity indicated by the power spectrum. The resulting variance profiles are fitted following @Leh87 and the uncertainties affecting the period estimates are evaluated accordingly. The best-fitting orbital solution we obtain with this technique is shown in the leftmost column of Table \[tab\]. The time series were then corrected for the orbital motion with these parameters and folded around the best estimate of the spin period over 300s data segments (for a total of 717). The pulse profiles could generally be modelled using up to three harmonics. The pulsed fraction is observed to greatly vary in between the first (MJD 55482.010 to 55482.043) and the other observations. During the former the pulse fractional amplitude of the first harmonic is very high \[$A_1\simeq0.252(2)$\], while the second harmonic is detected at a much lower amplitude \[$A_2=0.016(2)$\]. In subsequent observations the amplitude of the first harmonic drastically decreases to values between 0.02 and 0.04, whereas the second harmonic amplitude remains stable and a third harmonic is sometimes requested by the profile modelling. To show this, we plot in Fig. \[fig3\] the pulse profiles calculated over the observations performed during MJD 55482 (solid line) and MJD 55483 (dashed line), with the latter profile shown at a magnified scale. The pulsed fraction decrease is evident, as are the variation of the shape of the peak at rotational phase $\sim0.75$. To increase the accuracy of our timing solution we model the temporal evolution of the phases evaluated on the first and second harmonic of the pulse profiles with the relation: $$\label{eq:phases} \phi(t)=\phi_0+(\nu_0-\nu_{F})\;(t-T_{ref})+\frac{1}{2}{<\dot{\nu}>}(t-T_{ref})^2+R_{orb}(t).$$ Here $T_{ref}$ is the reference epoch for the timing solution, $\nu_0$ is the pulsar frequency at the reference epoch, $\nu_F=1/P_F$ is the folding frequency and $<\dot{\nu}>$ is the average spin frequency derivative. The term $R_{orb}(t)$ describes the phase residuals induced by a difference between the actual orbital parameters, namely $x$, $P_{orb}$, $T^*$, $e\sin{\omega}$ and $e\cos{\omega}$, and those used to correct the time series [see e.g. @DeeBoyPrv81]. Once a new set of orbital parameters is found, it is used to correct the time series, and the resulting phases are modelled again with Eq.(\[eq:phases\]). This procedure is iterated until the phase residuals are normally distributed around zero. Periods 1$^{st}$ harm. ph. 2$^{nd}$ harm. ph. --------------------------------- ------------ -------------------- -------------------- $\nu_0\:$-11.044885 ($\mu$Hz) $<5$ +0.64(1) +0.17(1) $<\dot{\nu}>$ ($10^{-12}$ Hz/s) $<16$ $1.22(1)$ $1.68(1)$ a$\sin{i}$/c (lt-s) 2.498(5) 2.4967(3) 2.4973(2) P$_{orb}$ (hr) 21.2744(8) 21.2745(1) 21.27454(8) T\* - 55481.0 (MJD) 0.7805(4) 0.78033(6) 0.78048(4) e $<0.02$ $<7\times10^{-4}$ $<6\times10^{-4}$ f($M_1$,$M_2$,i) (M$_{\odot}$) 0.0213(2) 0.0212587(8) 0.021275(5) $\chi^2$/dof 156/121 7879/662 3287/566 : Spin and orbital parameters of .\[tab\] Although no residual modulation at the orbital period is observed, the phases of the first harmonic are strongly affected by timing noise. The reduced $\chi^2$ we obtain modelling their evolution with Eq.(\[eq:phases\]) is extremely large ($\simeq 11.9$ over 662 d.o.f.). Such a behaviour is most probably caused by to pulse shape changes like the one shown in Fig \[fig3\]. The second harmonic phases appear to be less affected by timing noise, resulting in a reduced $\chi^2_r=5.8$ (566 d.o.f.). We argue that the second harmonic phases are better fitted with respect to the first harmonic because of the greater stability of this component [as already observed in some accreting millisecond pulsars, see, e.g., @Brd06; @Rgg08]. The best-fitting parameters calculated over the phase evolution of the first and second harmonic are quoted in the central and rightmost column of Table \[tab\]. In Fig. \[fig2\] we show the phases of both harmonics, when the observations corrected for the orbital motion of the source are folded around $P_F=90.539645$ ms. The phase evolution is clearly driven by at least a quadratic component. A consequence of timing noise is that the spin frequency and its derivative, estimated over the two harmonic components, are significantly different. We quote conservatively a value of $\nu_0=11.0448854(2)$ Hz that overlaps both frequency estimates, and use a spin frequency derivative between 1.2–1.7$\times10^{-12}$ Hz s$^{-1}$ in the discussion below. However, it is worthwhile to note that the orbital parameters are entirely consistent between the two harmonic solutions, which supports the reliability of these estimates. The solution we obtained is entirely compatible with, but more precise than, those proposed by P10[^1] and S10. Given the accuracy of the source position considered here (0.06”), the systematic uncertainties introduced by the position indetermination on the measured values of spin frequency and of its derivative [e.g. @Brd07] are $\sigma_{pos,\nu}\simlt3\times10^{-10}$ Hz and $\sigma_{pos,\dot{\nu}}\simlt6\times10^{-17}$ Hz s$^{-1}$, respectively. Finally, as the cluster moves towards the solar system at a velocity of $85\pm10$ km s$^{-1}$ [@Fer09], the measured value of the spin frequency is affected by a systematic offset of $\sim +3\times10^{-3}$ Hz, though this is unimportant when making conclusions about the source properties. In order to extend the range of fluxes at which the source was observed and pulsations were detected, we also analysed three [ *Swift*]{} observations (Obs. 00031841002, 00031841003 and 00031841004) in which the XRT observed in Windowed Timing (WT) mode, with a temporal resolution of 1.7ms. The [*Swift*]{} XRT started monitoring the source on MJD 55479.737, more than two days before , with the first observation in WT mode starting at MJD 55479.802 for 2 ks. After applying barycentric corrections for the satellite orbit, then correcting for the source orbital motion and selecting photons from a 50 pixel wide box around the source position, a pulsation is clearly detected at a period of $P_S=90.5395(2)$ ms by means of an epoch-folding search. The XRT signal is particularly strong and consistent with that seen by during its first observation, with a pulse profile modelled by a sinusoid of amplitude 0.23(2). Pulsations were also searched for in the subsequent $\sim1$ ks long XRT WT observations starting on MJD 55484.767 and 55485.357. Only a weak signal was detected in the latter at a fractional amplitude of 0.018(4), still compatible with that seen by at those later times. Discussion and conclusions ========================== We reported on the spin and orbital properties of the newly discovered accreting pulsar, . Its 90.6 ms period makes it the first confirmed accreting pulsar in the range 10–100 ms. Pulsations were detected in all observations performed by , as well as in two out of the three [*Swift*]{} observations performed in WT mode presented here.The pulsed fraction is observed to drastically change on a timescale of $\simlt 1$ d, after the observation performed on MJD 55482. While previously both [*Swift*]{} and observations revealed a strong signal dominated by a first harmonic component of fractional amplitude as large as $0.25$, later observations at higher fluxes performed by both satellites never detected an amplitude $\simgt 0.03$. Simultaneously the pulse shape changes and becomes more complex. This behaviour is suggestive of a change of the geometrical properties of the flow in the accretion columns above the NS hot spots. Because pulsations are detected throughout the observations shown here, an estimate of the NS magnetic field strength can be made. For accretion to proceed and pulsations to appear in the X-ray light curve, the inner disc radius, $R_{in}$, has to lie in between the NS radius, R$_{NS}$, and the corotation radius, $R_{C}$, defined as the distance from the NS at which the velocity of the magnetosphere equals the Keplerian velocity of the matter in the disc. For a larger accretion radius, accretion would be inhibited or severely reduced by the onset of a centrifugal barrier. For a pulsar spinning at 90.6 ms, $R_C=(GMP^2/4\pi^2)^{1/3}=338 m_{1.4}^{1/3}$ km, where $m_{1.4}$ is the NS mass in units of 1.4 M$_{\odot}$. Defining the inner disc radius in terms of the pressure equilibrium between the disc and the magnetosphere, one obtains $R_{in}\simeq 160 \; m_{1.4}^{1/7}\;R_6^{-2/7}\;L_{37}^{-2/7}\mu_{28}^{4/7}$ km [@Brd01], where $R_6$ is the NS radius in units of 10 km, $L_{37}$ the accretion luminosity in units of $10^{37}$ erg s$^{-1}$, and $\mu_{28}$ the magnetic dipole moment of the NS in units of $10^{28}$ G cm$^{3}$. Extrapolating the fluxes observed by to the 0.1–100 keV band, and assuming a distance of $d=5.5\pm0.9$ kpc to Terzan 5 [@Ort07], we estimate a maximum and minimum bolometric luminosities of $1.7(1)\times10^{37}$ d$_{5.5}^2$ erg s$^{-1}$ and $6.8(1)\times10^{37}$ d$_{5.5}^2$ erg s$^{-1}$, during the time covered by the observations considered here. Assuming that the X-ray luminosity is a good tracer of the accretion power and imposing $R_{NS}<R_{in}\simlt R_{C}$, we obtain $$0.02\: \: m_{1.4}^{-1/4}\: R_{6}^{9/4}\: d_{5.5} \simlt \mu_{28} \simlt 4.8 \:\: m_{1.4}^{1/3}\: R_{6}^{1/2}\: d_{5.5}.$$ The upper limit on the magnetic dipole can be reduced considering that pulsations are detected also in a Swift observation taking place $\sim2$ d earlier than the first observation. @Bzz10 estimated the source flux in that observation as $4.5(2)\times10^{-10}$ erg cm$^{-2}$ s$^{-1}$ (1–10 keV). This value is a factor $\sim4$ lower than the value obtained extrapolating the spectrum of the first observation in the same energy band. Assuming that this ratio holds also for the bolometric luminosity of the source, we get to an upper limit on the magnetic dipole moment of $\simeq 2.4\times10^{28}$ G cm$^{-3}$. The limits thus derived translate to a magnetic surface flux density between $\sim 2\times10^8$ and $\sim 2.4\times10^{10}$ G. The upper bound of this interval can be overestimated because the exact flux at which the pulsations appeared is unknown at present. Monitoring the presence of coherent pulsations as a function of the flux when the source fades will probably allow us to derive a tighter constraint. @Alt10bis have also reported the presence of a kHz QPO at $\sim815$ Hz (10–50 keV) during the observations performed on MJD 55487. Under the hypothesis that this feature originates in the innermost part of the accretion disc, it indicates an inner disc radius $R_{in}\simlt \:20 \:m_{1.4}^{1/3}$ km. As the luminosity we estimated during that day is $6.8(1)\times10^{37}$ d$_{5.5}^2$ erg s$^{-1}$, this would imply a magnetic field $\simlt7\times 10^8$ d$_{5.5}$ G if the disc is truncated at the magnetospheric radius. Despite the presence of timing noise, the analysis of the phase evolution over the $\sim24$ d time interval presented here clearly indicates the need for a quadratic component to model these phases. Interpreting this component as a tracer of the NS spin evolution, we thus conclude that the source spins up while accreting. Values of the spin-up rate between 1.2 and 1.7$\times10^{-12}$ Hz s$^{-1}$ are found, depending on the harmonic considered. This discrepancy is probably due to the effect of timing noise. These values are compatible with those expected for a NS accreting the Keplerian disc matter angular momentum given the observed luminosity, $\dot{\nu}\simeq1.5\times10^{-12}$ $(L_{37}/5)$ $(R_{in}/70km)^{1/2}$ $I_{45}^{-1}$ $R_6$ $m_{1.4}$ Hz s$^{-1}$. Here $I_{45}$ is the NS moment of inertia in units of $10^{45}$ g cm$^2$. The observed spin period and the magnetic field we estimated place this source between the population of “classical” ($B\simgt 10^{11}$G, $P\simgt 0.1$ s) and millisecond ($B\simeq 10^8$–$10^9$ G, $P\simeq 1.5$–$10$ms) rotation-powered pulsars. The observation of a significant spin-up at rates compatible with those predicted by the recycling scenario further supports the identification of this source as a slow, mildly recycled pulsar. We note that the only other two accreting pulsars with similar, though significantly different parameters, are GRO J1744-28 ($P_S=467ms$, $B\simeq2.4\times10^{11}$ G, @Cui97), and 2A 1822-371 ($P_S=590$ ms, $B\simlt 10^{11}$ G, @Jnk01). The orbital parameters we measured for the NS in allow us to derive constraints on the nature of its companion star. With a mass function of $f(M_2;M_1,i)\simeq 0.02$ M$_{\odot}$, a minimum mass for the companion can be estimated to be as low as 0.41 M$_{\odot}$ for an inclination of 90$^{\circ}$, and an NS mass of 1.4 M$_{\odot}$. Since the source shows no eclipses, the inclination is most probably $\simlt80^{\circ}$, [ and]{} the lower limit increases to $m_2=0.16+0.26 m_{1.4}$, where $m_2$ is the mass of the companion star in solar units. An upper limit can be obtained if the companion star [ is assumed not to]{} overfill its Roche lobe. Using the relation given by @Egg83 and the third Kepler law to relate the Roche Lobe radius to the orbital period and to the companion mass, $R_{L2}\simeq 0.55(GM_{\odot})^{1/3}(P_{orb}/2\pi)^{2/3}m_{1.4}\:q^{2/3}(1+q)^{1/3}/[0.6q^{2/3}+\log(1+q^{1/3})]$, where $q=M_2/M_1$, and assuming the companion star follows a main sequence mass-radius relation, $R_2/R_{\odot}\approx(M_2/M_{\odot})$, yields a maximum mass of 2.75 M$_{\odot}$ for the companion when $m_{1.4}=1$. This upper limit is indeed higher than the maximum mass expected for a main sequence star belonging to one of the two stellar populations found by @Fer09 in Terzan 5. One has in fact $m_2\simlt0.95$ if the companion of belongs to the older population [t=10 Gyr, @Dan10], while $m_2\simlt 1.2$ and $m_2\simlt 1.5$ if it belongs to a younger population of 6 and 4 Gyr, respectively (D’Antona, priv. comm.). We conclude that a reasonable upper limit for the companion-star mass is 1.5 M$_{\odot}$, possibly a main sequence or a slightly evolved star. This work is supported by the Italian Space Agency, ASI-INAF I/088/06/0 contract for High Energy Astrophysics, as well as by the operating program of Regione Sardegna (European Social Fund 2007-2013), L.R.7/2007, “Promotion of scientific research and technological innovation in Sardinia”. We thank F. D’Antona for providing the mass estimates of main sequence stars in Terzan 5, and the referee for the prompt reply and useful comments. [^1]: There is an offset between the values of frequency and epoch of mean longitude quoted by P10 and those presented here, as theirs were not referred to the TDB reference system.

--- abstract: 'Polycyclic aromatic hydrocarbons (PAHs) are ubiquitous in astrophysical environments, as revealed by their pronounced emission features at 3.3, 6.2, 7.7, 8.6, 11.3, and 12.7${\,{\rm \mu m}}$ commonly ascribed to the C–H and C–C vibrational modes. Although these features have long been predicted to be polarized, previous searches for PAH polarization led to null or, at best, tentative detections. Here we report the definite detection of polarized PAH emission at 11.3${\,{\rm \mu m}}$ in the nebula associated with the Herbig Be star MWC 1080. We measure a polarization degree of 1.9$\pm$0.2%, which is unexpectedly high compared to models. This poses a challenge in the current understanding of the alignment of PAHs, which is required to polarize the PAH emission but thought to be substantially suppressed. PAH alignment with a magnetic field via a resonance paramagnetic relaxation process may account for such a high level of polarization.' author: - | Han Zhang, Charles M. Telesco, Thiem Hoang, Aigen Li, Eric Pantin, Christopher M. Wright, Dan Li, Peter Barnes\ [[*Accepted by [ApJ]{} June 5, 2017*]{}]{} title: Detection of Polarized Infrared Emission by Polycyclic Aromatic Hydrocarbons in the MWC 1080 Nebula --- Introduction ============ A distinctive series of infrared (IR) emission features generally attributed to the stretching and bending vibrational modes of planar polycyclic aromatic hydrocarbon (PAH) molecules are observed in most dusty astronomical objects at 3.3, 6.2, 7.7, 8.6, 11.3, and 12.7${\,{\rm \mu m}}$ . @1988prco.book..769L (1988, hereafter L88) first noted that these IR emission features, if due to PAHs, are expected to be polarized as a result of anisotropic illumination by a source of ultraviolet (UV) photons (e.g., stars). UV absorption is favored when the molecular plane faces the illuminating source. If the spinning of IR-emitting PAHs do not deviate significantly from their initial orientations at UV absorption, the IR emission will preferentially come from those PAHs that are facing the illuminating source at the time of UV absorption. Therefore, PAHs will emit polarized light, with the polarization direction of out-of-plane modes (11.3, 12.7${\,{\rm \mu m}}$) being along the source-molecule direction and that of in-plane modes (3.3, 6.2, 7.7, 8.6${\,{\rm \mu m}}$) being perpendicular to it. @2009ApJ...698.1292S (2009, hereafter SD09) revisited the L88 scheme by considering realistic rotational dynamics of PAHs as well as an arbitrary degree of internal alignment between the grain symmetry axis and its angular momentum. Using realistic estimates of rotational temperatures for a typical PAH molecule of 200 carbon atoms, SD09 derived a value for the polarization fraction of $0.53\%$ for the 11.3${\,{\rm \mu m}}$ feature in the case of the Orion Bar. performed the first systematic search for the polarization of the 3.3 and 11.3${\,{\rm \mu m}}$ PAH features in a number of astronomical sources, obtaining upper limits of 1% and 3% for the 3.3 and 11.3${\,{\rm \mu m}}$ features, respectively, in the Orion Bar. To test the PAH identification of these IR features and gain insight into the alignment of PAHs, we searched for linearly polarized PAH emission in the nebula associated with MWC 1080, a stellar cluster located at a distance of 2.2 kpc [@2008ApJ...673..315W]. The primary star, MWC 1080A, is classified as a B0e star and has a luminosity of $L/{L_\sun}\approx 10^{4}$, which, together with its stellar companions, illuminates the surrounding gas and dust in the adjacent molecular cloud. The mid-IR image of MWC 1080 at 11.2${\,{\rm \mu m}}$ resembles a pinwheel, with opposing arms curving off to the northwest and southeast (Fig.\[fig:1\]). The extended mid-IR emission resembling spiral arms or wings around MWC 1080 actually traces the internal surfaces of a biconical cavity created by the outflow from MWC 1080A [@2014ApJ...796...74L; @2007astro.ph..1215S]. The brightest part of the nebula (green rectangle in Fig.\[fig:1\]) lies 0.03${\,{\rm pc}}$ in projection to the northwest (hereafter NW nebula) of MWC1080A and extends $\sim$0.1${\,{\rm pc}}$ from the northeast to the southwest. That may well provide an optimal, Orion-Bar-like viewing geometry, i.e., a long column density along the line of sight through the photodissociation region and at an angle between the line of sight and illumination direction ($\alpha$$\approx$90) that is almost ideal for observing maximum polarization (SD09). Anomalous microwave emission (AME) is often ascribed to the rotational emission from rapidly-rotating nanoparticles [@1998ApJ...494L..19D; @1998ApJ...508..157D]. PAHs are often considered to be associated with the AME due to their abundances and small sizes [@1996ApJ...470..653K; @1997ApJ...486L..23L], although there now appears to be some doubt about the hypothesis [@2016ApJ...827...45H]. Nitrogen-substituted PAHs (i.e., nitrogen in place of carbon), with greater dipole moments, may also be important components of the carriers of the AME [@2005ApJ...632..316H; @2008ApJ...680.1243M]. Therefore, our observations have broader implications for determining the alignment and polarization of rapidly spinning PAHs, which bears on the quest for the cosmic microwave background (CMB) B-mode [@2013ApJ...779..152H]. Observation and Data Reduction ============================== CanariCam is the mid-IR (8-25${\,{\rm \mu m}}$) multi-mode facility spectrometer and camera on the 10.4 m Gran Telescopio CANARIAS (GTC) in La Palma, Spain [@2003SPIE.4841..913T]. It employs a 320$\times$240-pixel Raytheon detector array with a pixel scale of 0$\farcs$079, which provides a field of view of 26$\times$19with Nyquist sampling (two pixels per $\lambda$/D) of the diffraction-limited point-spread function at 8${\,{\rm \mu m}}$. Polarimetry is accomplished through the use of a Wollaston prism and a half-wave plate rotated to angles of 0, 22.5, 45, and 67.5. A Wollaston prism in the optical path divides incoming light into two beams (ordinary and extraordinary), which are recorded by the detector simultaneously. We obtained low-resolution (R$\equiv$$\lambda$/$\Delta\lambda$$\approx$50) spectropolarimetry observations of the NW nebula of MWC 1080 on 2015 July 31, August 5, and August 7 spanning the wavelength range 7.5–13.0${\,{\rm \mu m}}$. We made four separate measurements, as indicated in the observation log presented in Table \[tab:1\]. The spectroscopic observations of the NW nebula were interlaced with observations of the Cohen standard star HD 21330 [@1999AJ....117.1864C] for flux and point-spread-function (PSF) calibration, and the standard star AFGL 2591, selected from [@2000MNRAS.312..327S] to calibrate the polarization position angle. The standard mid-IR chop-nod technique was applied with an 11$\arcsec$ chop throw in the northwest-southeast direction. We positioned the 1$\farcs$04$\times$2$\farcs$08 slit with the slit’s longer axis oriented at 45$\degr$ from the North to intersect the brightest part of the nebula, enclosed by the green rectangle in Fig.\[fig:1\]. The 11.2${\,{\rm \mu m}}$ image of MWC 1080 is adopted from [@2014ApJ...796...74L], and the data were taken using Michelle, the facility mid-IR camera at Gemini North. The data were reduced using custom IDL software, as described in [@2014ascl.soft05014B] and [@2014ascl.soft11009L]. We extracted one-dimensional spectra by integrating the central 21 pixels (1$\farcs$6) along the slit direction. Wavelength calibration was done using several sky lines identified in the raw images. We computed normalized Stokes parameters *q* (*q=Q/I*) and *u* (*u=U/I*) using both the difference and ratio methods, with each providing consistent results [@2005aspo.book.....T]. The data were calibrated for instrumental efficiency and polarization. The estimated instrumental polarization was 0.6% as measured with HD 21330 and was subtracted from the observations of the NW nebula in the $Q-U$ plane. The degree of polarization $p=\sqrt{q^{2}+u^{2}-\sigma^{2}}$, where the last term (the ‘debias’ term) was introduced to remove a positive offset in the signal floor resulting from squared background noise. Debiasing may introduce negative values if the noise fluctuations are stronger than the signal. The polarization position angle was computed as $\theta=0.5{\rm arctan}(u/q)$. The uncertainties $\sigma_{q}$ and $\sigma_{u}$ associated with the normalized Stokes parameters were derived using a standard 3-sigma clipping algorithm [@1987igbo.conf.....R], and were then propagated through the analysis to obtain the polarization uncertainty $\sigma_{p}$ and position angle uncertainty $\sigma_{\theta}=\sigma_{p}/2p$ [@2006PASP..118..146P]. All the polarization position angles were calibrated east from north. We masked out the region between 9.2–10.0${\,{\rm \mu m}}$, which is dominated by the atmospheric ozone feature. The three-pixel (0.06${\,{\rm \mu m}}$) boxcar-smoothed intensity spectrum of the nebula (Stokes I) is presented in Fig.\[fig:2\]a. To further increase the signal-to-noise ratio (S/N) of the polarization measurements, we rebinned the ordinary and extraordinary ray spectra into 0.12${\,{\rm \mu m}}$ wavelength (6 pixels) bins (downsampling) and then applied an additional three-pixel (0.36${\,{\rm \mu m}}$) boxcar-smoothing to the data. That results in an equivalent spectral resolution of the polarization spectrum of R$\approx$32 (Fig.\[fig:2\]b). Stokes $u$ and $q$ are plotted in Figs.\[fig:3\]b and c, respectively. To emphasize the statistical significance of the measurements, we plot the S/N of the polarized intensity in Fig.\[fig:3\]a. Results ======= We present in Fig.\[fig:2\] mid-IR intensity and polarization spectra of the NW nebula covering the 8.0–13.0${\,{\rm \mu m}}$ wavelength range. The PAH emission features are clearly seen in Fig.\[fig:2\]a, including the in-plane C–H bending feature at 8.6${\,{\rm \mu m}}$ and the out-of-plane C–H bending features at 11.3 (solo-CH), 12.0 (duet-CH), and 12.7${\,{\rm \mu m}}$ (trio-CH) [@1989ApJS...71..733A]. The relative strengths of the PAH features depend on the size, structure, and charging of PAHs , and the physical conditions of the region where they are found (@1994ApJ...427..822B; @2001ApJ...548..296W). By fitting the spectrum of MWC 1080 obtained with the [*Infrared Space Observatory*]{} (ISO), which exhibits a more complete set of PAH emission features but mixed emission from both stars and the nebula, [@2017ApJ...835..291S] determine that the best fit of PAHs in this environment are mostly neutral and large. Exposed to the energetic photons from a B0e star with an effective temperature of $\sim$30,000${\,{\rm K}}$ [@2008ApJ...673..315W], small PAHs are probably unstable, and the only PAHs that survive are ones that are large and cata-condensed in structure. The polarization spectrum shown together with 1$\sigma$ error bars of the NW nebula is presented in Fig.\[fig:2\]b. Two spectral regions show significant polarization. One is in the range of 10.9–11.7${\,{\rm \mu m}}$, with p$_{11.3}$=1.9$\pm$0.2% and position angle of 77.2$\pm$3.2$\degr$, the latter indicated by the solid red lines superimposed on the intensity map of MWC 1080 in Fig.\[fig:1\]. There is good consistency among the four separate measurements listed in Table \[tab:1\]. The other significant polarization is at 10.0–10.7${\,{\rm \mu m}}$, peaking around 10.3${\,{\rm \mu m}}$, and with p$_{10.3}$=5.4$\pm$1.6% and position angle of 46.7$\pm$8.2. While the 10.3${\,{\rm \mu m}}$ polarization feature is seemingly higher than that at 11.3${\,{\rm \mu m}}$, the statistical significance of the results needs to be considered. We note here that the detection of polarization near 10.3${\,{\rm \mu m}}$ is only marginally significant, being barely 3$\sigma$, whereas the 11.3${\,{\rm \mu m}}$ detection is robust, being around 9$\sigma$ (Fig.\[fig:3\]a). The 3$\sigma$ upper limits for the polarization at 8.6, 12.0 and 12.7${\,{\rm \mu m}}$ are 1.9%, 2.0%, and 2.8%, respectively (Fig.\[fig:2\]b). While, theoretically, the emission at longer wavelengths is expected to have higher polarization (SD09), the 8.6, 12.0 and 12.7${\,{\rm \mu m}}$ PAH emission features are much weaker than the 11.3${\,{\rm \mu m}}$ feature and the lack of detection of the polarization in these regions is not surprising. Examining Stokes $u$ and $q$ explicitly (Fig.\[fig:3\]) demonstrates that the 10.3${\,{\rm \mu m}}$ polarization is apparent mainly in Stokes $u$, while the 11.3${\,{\rm \mu m}}$ polarization is mainly in Stokes $q$. This difference implies a different origin for the 10.3${\,{\rm \mu m}}$ if real and 11.3${\,{\rm \mu m}}$ features, almost certainly indicating that they originate in different dust populations. Discussion {#sec:dis} ========== The polarization feature between 10.9 and 11.7${\,{\rm \mu m}}$ is well correlated in wavelength position with the 11.3${\,{\rm \mu m}}$ PAH emission feature (Fig.\[fig:3\]), and we conclude that the polarized emission is indeed due to PAH molecules. The peak value $p_{11.3}\simeq2\%$ is consistent with the previous PAH polarization search by who established an upper limit of 3% on its polarization in the Orion Bar. The polarization position angle of the feature has an offset angle of $\sim$60from the position angle of the projected illumination direction from the star to the nebula ($\sim$315), as shown in Fig.1. Numerical Calculations using SD09 --------------------------------- We adopt the up-to-date SD09 models to model the degree of polarization in the environment of a reflection nebula. SD09 calculated the polarization for randomly oriented PAHs. They considered the intramolecular vibrational-rotation energy transfer (IVRET) process, which allows the efficient energy exchange between rotation and vibration. SD09 model the polarization of PAH emission by introducing two parameters, $\gamma_{\rm 0}=T_{\rm rot}/T_{\rm 0}$ and $\gamma_{\rm ir}=T_{\rm rot}/T_{\rm ir}$, where $T_{\rm rot}$ is the rotational temperature determined by gas-grain interactions, photon absorption and IR emission, $T_{\rm 0}$ is the vibrational grain temperature that determines internal alignment prior to UV absorption, and $T_{\rm ir}$ is the temperature during IR emission. Alignment by an external magnetic field is ignored in their model, and the polarization degree of PAH emission features is essentially determined by two parameters, $\gamma_{\rm ir}$ and $\gamma_{\rm 0}$. Since the internal alignment temperature $T_{\rm ia}\equiv T_{\rm ir}$ is fixed for the different PAHs, and $T_{\rm 0}$ is determined by the radiation field, the polarization degree is determined by $T_{\rm rot}$. Consequently, the knowledge of $T_{\rm rot}$ is critical for achieving realistic predictions for the polarization of PAH emission. Collisions with gas neutrals and ions, UV absorption and subsequent IR emission, and electric dipole emission contribute to the rotation of PAHs (see @1998ApJ...508..157D; @2010ApJ...715.1462H). We assume in a reflection nebula that the gas temperature $T_{\rm gas}$=100 ${\,{\rm K}}$, gas density $n_{\rm H}=10^{3}$ ${\rm cm}^{-3}$, and the radiation field parameter $U=10^{3}$. We consider the PAH geometry as in [@1998ApJ...508..157D] and a typical grain size of $a$=7.5$\,\rm \AA$ (200 carbon atoms) in which most of the PAH mass is concentrated as in the ISM (SD09). We then carry out simulations of PAH dynamics using the Langevin code [@2010ApJ...715.1462H] and calculate the degree of polarization. However, the values of polarization obtained using the SD09 model with the consideration of different hydrogen ionization fractions $x_{\rm H}$, i.e., $n_{\rm H^{+}}$/$n_{\rm H}$ ($n_{\rm H^{+}}$ is the number density of ionized hydrogen), are less than 0.5% (Table \[tab:2\]), much smaller than our measured value of $\sim$2%. It appears that other mechanisms that can enhance the PAH alignment need to be considered. Alignment with Magnetic Fields ------------------------------ SD09 discussed two possibilities that can significantly enhance the polarization of PAH emission, including suprathermal rotation (i.e., the PAH rotational temperature $T_{\rm rot}$ is much higher than the gas temperature $T_{\rm gas}$) and perfect internal alignment (i.e., the molecule principal axis is aligned with the angular momentum during both UV absorption and IR emission). The former is unlikely, since there are no obvious physical processes that can spin-up nanoparticles to suprathermal rotation rates (@1998ApJ...508..157D; @2010ApJ...715.1462H). On the other hand, perfect internal alignment can only be achieved at very low dust temperatures (T$_{\rm 0}$ of a few K) during the UV photon absorption, which requires very efficient energy exchange among the vibrational-rotational modes. This is also very unlikely due to quantum suppression that may occur in nanoparticles (@2016ApJ...831...59D). Alternatively, we recognize that the enhanced polarization may arise from external alignment, i.e., the partial alignment of the angular momentum with an ambient magnetic field. Therefore, we have repeated the modeling incorporating the mechanism of resonance paramagnetic relaxation to align nanoparticles [@2000ApJ...536L..15L]. Subject to an external magnetic field, protons in aromatic hydrocarbons in the laboratory are known to experience stronger shielding (or deshielding) effects than regular hydrocarbons, because the $\pi$-electrons are delocalized and are free to circulate. Astronomical magnetic fields can induce diamagnetic ring currents and polarizabilities in the $\pi$-electron clouds, resulting in coupling between the magnetic fields and two-dimensional PAHs, thereby forcing some alignment. In our models, the magnetic field strength is assumed to be B=100 $\mu$G . The calculation is described here briefly, with more details presented in Hoang (2017). We define $\hat{u}$-$\hat{v}$ to be the plane of the sky (see Hoang 2017). We define $F_{\rm u,v}^{\|,\perp}$ to be the in-plane ($\|$) and out-of-plane ($\perp$) emission by a PAH molecule with the electric field $\bE$ in the $\hat{u}$-$\hat{v}$ plane. Then $I_{\rm u,v}^{\|,\perp}$ is the emission intensity from the PAH. The flux $F_{\rm u,v}^{\|,\perp}$ depends on: 1) $f_{\rm LTE}(\theta,J)$, the probability distribution of the principal axis of the PAH plane being aligned with the grain angular momentum $\bJ$ (LTE stands for local thermal equilibrium); and 2) $f_{\rm J}$ ($\bJ$), the probability distribution of the angular momentum J being aligned with the direction of the magnetic field. The grain angular momentum $\bJ$ has spherical angles $\theta$ and $\phi$. $\beta$ is the nutation angle between $\bJ$ and the principal axis of the grain. The emission $I_{\rm w}$, with $w=(u,v)$, is obtained by integrating over the distribution functions $$\begin{aligned} I_{w}^{\|,\perp}(\alpha) &=& \int_{J} f_{J}(\bJ)d\bJ\int_{0}^{\pi}f_{LTE,0}(\theta_{0},J)d\theta_{0} \nonumber\\ &&\times \int_{0}^{\pi}f_{\rm LTE,ir}(\theta, J)d\theta A_{\star}(\beta,\theta_{0})F_{w}^{\|,\perp}(\beta, \phi,\theta,\alpha),\label{eq:Istar_uv}\end{aligned}$$ where $A_{\star}$ is the cross-section of UV absorption as given in SD09. $f_{\rm LTE,0}$ and $f_{\rm LTE,ir}$ describe the thermal fluctuations of the principal axis of PAHs before UV absorption and during IR emission. $f_{\rm LTE}(\theta,J)$ can be described by the Boltzmann distribution [@1997ApJ...484..230L] with $\int_{0}^{\pi}f_{\rm LTE}(\theta, J){\rm sin}\theta d\theta =1$. To simplify the calculation and derive the maximum value of polarization, we assume that the magnetic field is parallel to the stellar incident radiation direction. We simulate the distribution of angular momentum from the Langevin equation assuming the ergodic system approximation and compute numerically the intensity of radiation using Equation (\[eq:Istar\_uv\]). The resulting degree of polarization with the viewing angle $\alpha$ for in-plane and out-of-plane modes is $$p^{\|,\perp}(\alpha)=\frac{I_{u}^{\|,\perp}(\alpha)-I_{v}^{\|,\perp}(\alpha)}{I_{u}^{\|,\perp}(\alpha)+I_{v}^{\|,\perp}(\alpha)},\label{eq:pol}$$ We show the modeling results in Table \[tab:2\] for the reflection nebula, which includes the ratio of the rotational temperature $T_{\rm rot}$ and the gas temperature $T_{\rm gas}$, the degree of alignment of the angular momentum with the magnetic field $Q_{\rm J}$, and the estimated polarization for the different hydrogen ionization fractions $x_{\rm H}$ with a viewing angle $\alpha$=90. The polarization varies from $0.14\%$ to $0.34\%$ in model a, increasing to $0.87\%$–$2.1\%$ when the external alignment is taken into account as in model b. In Table \[tab:2\], the degrees of polarization $p$ computed for both models increase with the increasing hydrogen ionization fraction $x_{\rm H}$. The results suggest that the polarization of PAH emission is dominated by PAHs in regions with a higher fraction of hydrogen in ionic form. In regions with a higher $x_{\rm H}$ (i.e., higher $n_{\rm H^{+}}$/$n_{\rm H}$), there will be more electrons available to neutralize the PAH ions created by photoionization [@2005pcim.book.....T]. Indeed, as shown in [@2017ApJ...835..291S], the aromatic hydrocarbon emission features observed in MWC 1080 are best modeled in terms of a mixture of PAHs with ${\sim\,}$80% being neutral and ${\sim\,}$20% being ionized. As mentioned earlier, the delocalized $\pi$-electrons in neutral PAHs may play a crucial role in coupling neutral PAHs with the magnetic field. Partial alignment of PAHs with the magnetic field at a level of $Q_{\rm J}\simeq 0.08$–0.1 (the averaged degree of alignment of angular momentum $\bJ$ with ) is required to reproduce the observed $\sim$2% polarization fraction at 11.3${\,{\rm \mu m}}$. This scenario also accounts for the observation that the polarization angle is offset from the illumination direction. When PAHs are aligned with the magnetic field, even though only partially, the polarization direction of the out-of-plane mode emission is expected to be along the magnetic field (Hoang 2017). Relationship between Polarization Angles and the Ambient Magnetic Field ----------------------------------------------------------------------- Both L88 and SD09 predict that the polarization associated with emission arising from the out-of-plane vibration mode should be along the illumination direction, which contrasts with our observed polarization angle having a $\sim$60$\degr$ offset. We explore one possible explanation, namely, PAH alignment by an external magnetic field (Table \[tab:2\]). If magnetic alignment is important, we expect the polarization direction to match that of the ambient magnetic field lines. Indeed, we find that the optical polarimetry observations of background stars within a few degrees on the sky from MWC1080 (serveral hundred parsecs in distance), indicates a fairly uniform optical polarization position angle of $\sim$80$\degr$, which suggests a smooth interstellar magnetic field threads the whole region (@2001AJ....122.3453M; ; @2000AJ....119..923H). The value of the position angle agrees well with our measured polarization position angle of 77.2$\pm$3.2$\degr$ at 11.3${\,{\rm \mu m}}$, which supports the hypothesis that PAHs are at least partially aligned with the ambient interstellar magnetic field threading the nebula and its neighborhood. Nevertheless, the emission and alignment of PAHs depend on local astrophysical conditions and the detailed properties of PAHs, especially their sizes. Nanoparticles with radii $\lesssim$10$\,\rm \AA$ are thought to be negligibly polarized with the greatest quantum suppression of alignment [@2016ApJ...831...59D]. Based on our results, it seems that other physical processes such as Faraday rotation braking that facilitate the alignment of nanoparticles need to be considered [@2016MNRAS.457.1626P], since it is evident that the starlight anisotropy scheme alone in L88 is not sufficient to explain the measured high level of polarization. It is also worth noting that, given their abundances and small sizes, emission by rapidly spinning PAHs is widely believed to be the origin of AME in the 10–60 GHz frequency range [@1998ApJ...508..157D; @1998ApJ...494L..19D]. If true, the considerable alignment of PAHs as suggested by our detection, naturally produces polarized spinning dust emission for which the polarization level is proportional to the degree of alignment of the PAH angular momentum with the magnetic field at $\sim$GHz frequencies [@2013ApJ...779..152H]. It implies that polarized emission from spinning Galactic-foreground PAHs can indeed constitute an obstacle to the detection of the CMB B-mode signal. Marginally Detected 10.3${\,{\rm \mu m}}$ Polarization Feature -------------------------------------------------------------- As shown in Fig.\[fig:3\]a, we have a barely significant polarization detection at 10.3${\,{\rm \mu m}}$ (3$\sigma$). If real, the different behaviors of the 10.3 and 11.3${\,{\rm \mu m}}$ features in Stokes $u$ and $q$ (Fig.\[fig:3\]) suggest that they originate in different dust populations. Therefore, it does not affect our interpretation of the high S/N (9$\sigma$) polarization detection at 11.3${\,{\rm \mu m}}$, our main focus of this work. We do note, however, that there is no distinct Stokes I spectral fingerprint coinciding with the 10.3${\,{\rm \mu m}}$ polarization. It is unlikely that the well-known silicate feature or one of its variants can account for this polarization, since the silicate absorptive polarization profiles are broad, spanning the entire 8–13${\,{\rm \mu m}}$ region (e.g., ; @2000MNRAS.312..327S; @2017MNRAS.465.2983Z) rather than relatively narrow and sharp as the feature we see here. Other possibilities, including nanoparticles with silicate [@RevModPhys.85.1021] or metallic Fe compositions, e.g., hygrogenated iron nanoparticles [@2017MNRAS.466L..14B], might be worth investigating [@2016ApJ...821...91H] if further observations confirm and gain insight into the feature. Summary ======= We report the unambiguous detection of polarized PAH emission at 11.3${\,{\rm \mu m}}$ with a position angle of 77.2$\pm$3.2$\degr$ and polarization degree of 1.9$\pm$0.2%, which confirms the PAH hypothesis that PAH molecules can indeed emit polarized light. The detection of polarization indicates that the alignment of PAHs is considerable. We find that the starlight anisotropy scheme alone is not sufficient to account for this polarization. The PAHs are at least partially aligned by the ambient magnetic field threading this young stellar region and its neighborhood, a conclusion strongly supported by the fact that the measured polarization angle is identical to the large-scale interstellar magnetic field spanning this region. This observation could have important consequences for the accurate estimate of Galactic foreground polarization, a consideration relevant to current goals to detect the CMB B-mode signal. We expect future polarimetry observations, e.g., with SOFIA/HAWC+ and GTC/CanariCam, covering the complete suite of PAH emission features (e.g., the 6.2${\,{\rm \mu m}}$ band dominated by PAH cations and 3.3${\,{\rm \mu m}}$ band by small PAHs) and various astrophysical environments, will deepen our understanding of the properties and alignment of PAHs, e.g., the effects of their sizes and charge states. The authors are grateful for the anonymous referee’s inspiring comments, which helped to improve the manuscript. The authors are grateful to the GTC staff for their outstanding support of the commissioning and science operations of CanariCam. We thank Ion Ghiviriga, Bruce Draine and Uma Gorti for their helpful discussions. C.M.T. acknowledges support from NSF awards AST-0903672, AST-0908624 and AST-1515331. A.L. is supported in part by NSF AST-1109039 and NNX13AE63G. E.P. acknowledges the support from the AAS through the Chŕetien International Research Grant and the FP7 COFUND program- CEA through an enhanced-Eurotalent grant, and the University of Florida for its hosting through a research scholarship. 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--- abstract: | It is well-known that a subspace $Y$ of a separable metrizable space $X$ is separable, but without $X$ being assumed metrizable this is not true even in the case that $Y$ is a closed linear subspace of a topological vector space $X$. Early this century K.H. Hofmann and S.A. Morris introduced the class of *pro-Lie groups* which consists of projective limits of finite-dimensional Lie groups and proved that it contains all compact groups, all locally compact abelian groups, and all connected locally compact groups and is closed under the formation of products and closed subgroups. They defined a topological group $G$ to be *almost connected* if the quotient group of $G$ by the connected component of its identity is compact. We prove that an almost connected pro-Lie group is separable if and only if its weight is not greater than the cardinality $\cont$ of the continuum. It is deduced from this that an almost connected pro-Lie group is separable if and only if it is homeomorphic to a subspace of a separable Hausdorff space. It is also proved that a locally compact (even feathered) topological group $G$ which is a subgroup of a separable Hausdorff topological group is separable, but the conclusion is false if it is assumed only that $G$ is homeomorphic to a subspace of a separable Tychonoff space. It is shown that every precompact (abelian) topological group of weight less than or equal to $\cont$ is topologically isomorphic to a closed subgroup of a separable pseudocompact (abelian) group of weight $\cont$. This result implies that there is a wealth of closed non-separable subgroups of separable pseudocompact groups. An example is also presented under the Continuum Hypothesis of a separable countably compact abelian group which contains a non-separable closed subgroup. It is proved that the following conditions are equivalent for an $\omega$-narrow topological group $G$: (i) $G$ is homeomorphic to a subspace of a separable regular space; (ii) $G$ is a subgroup of a separable topological group; (iii) $G$ is a closed subgroup of a separable path-connected locally path-connected topological group. address: - 'Department of Mathematics, Ben-Gurion University of the Negev, Beer Sheva, P.O.B. 653, Israel' - 'Faculty of Science, Federation University Australia, P.O.B. 663, Ballarat, Victoria, 3353, Australia Department of Mathematics and Statistics, La Trobe University, Melbourne, Victoria, 3086, Australia' - 'Departamento de Matemáticas, Universidad Autónoma Metropolitana, Av. San Rafael Atlixco 186, Col. Vicentina, Del. Iztapalapa, C.P. 09340, México, D.F., Mexico' author: - 'Arkady G. Leiderman, Sidney A. Morris, and Mikhail G. Tkachenko' title: Density Character of Subgroups of Topological Groups --- Introduction {#intro} ============ All topological spaces and topological groups are assumed to be Hausdorff. The *weight* $w(X)$ of a topological space $X$ is defined as the smallest cardinal number of the form $|\mathcal{B}|$, where $\mathcal{B}$ is a base of topology in $X$. The *density character* $d(X)$ of a topological space $X$ is $\min\{|A|: A\,\,\mbox{is dense in} \,X \}$. If $d(X) \leq \aleph_0$, then we say that the space $X$ is *separable*. It is well-known that a subspace $S$ of a separable metrizable space $X$ is separable, but a closed subspace $S$ of a separable Hausdorff topological space $X$ is not necessarily separable. Even a closed linear subspace $S$ of a separable Hausdorff topological vector space $X$ can fail to be separable [@Lohman]. In several classes of topological groups, the situation improves notably. It has been proved by W. Comfort and G. Itzkowitz [@Comfort_Itzkowitz] that a closed subgroup $S$ of a separable locally compact topological group $G$ is separable. It is known also that a metrizable subgroup of a separable topological group is separable [@Vidossich p. 89] (see also [@Lohman]). In Section \[Positive\] of this article we look at conditions on the topological group $G$ which are sufficient to guarantee its separability if $G$ is a subspace of a separable Hausdorff or regular space $X$. En route we show in Theorem \[Theorem 10\] that an almost connected pro-Lie group (for example, a connected locally compact group or a compact group) is separable if and only if its weight is not greater than $\cont$. One cannot drop in the latter result since by Corollary \[Cor:subg\], there exists a separable *prodiscrete* abelian (hence pro-Lie) group which contains a closed non-separable subgroup. Let us note that a prodiscrete group is a complete group with a base at the identity consisting of open subgroups. Clearly closed subgroups of prodiscrete groups are prodiscrete. It is proved in Theorem \[Th:Main\] that if an almost connected pro-Lie group $G$ is a subspace of a separable space, then $G$ is also separable. In the rest of Section \[Positive\] we consider subgroups of separable topological groups. It is shown in Theorem \[Th:Fea\] that every feathered subgroup of a separable group is separable. The class of feathered groups contains both locally compact and metrizable groups, thus we obtain a generalization of aforementioned results of [@Comfort_Itzkowitz] and [@Vidossich], [@Lohman]. In Section \[Sec2\] we consider closed isomorphic embeddings into separable groups. We prove in Theorem \[Th:nA\] that a precompact topological group of weight $\leq\cont$ is topologically isomorphic to a closed subgroup of a separable pseudocompact group $H$ of weight $\leq\cont$. Since there is a wealth of non-separable pseudocompact groups of weight $\cont$, we conclude that closed subgroups of separable pseudocompact groups can fail to be separable. The proof of Theorem \[Th:nA\] is somewhat technical, so we find it convenient to supply the reader with a considerably easier proof of this theorem for the abelian case (see Theorem \[Th:ab\]). We also present in Proposition \[Pr:CoCo\], under the Continuum Hypothesis, an example of a separable countably compact abelian group $G$ which contains a non-separable closed subgroup. We do not know if such an example exists in $ZFC$ alone (see Question \[Example 3\]). A topological group which has a local base at the identity element consisting of open subgroups is called *protodiscrete*. A complete protodiscrete group is said to be *prodiscrete*. It is shown in Section \[Sec:Pd\] that closed subgroups of separable prodiscrete abelian groups can fail to be separable (see Corollary \[Cor:subg\]). Since prodiscrete abelian groups are pro-Lie, we see that closed subgroups of separable pro-Lie groups need not be separable either. In Section \[Sec:Emb\] we compare embeddings of topological groups in separable regular spaces and in separable topological groups. It is proved in Theorem \[Th:Emb\] that in the class of $\omega$-narrow topological groups, the difference between the two types of embeddings disappears, even if we require an embedding to be closed. In fact, we show that if an $\omega$-narrow topological group $G$ is *homeomorphic* to a subspace of a separable regular space, then $G$ is topologically isomorphic to a *closed* subgroup of a separable path-connected, locally path-connected topological group. Background Results {#Background} ================== In this section we collect several important well-known facts that will be frequently applied in the sequel. \[Th:1\][(De Groot, [@Hodel Theorem 3.3])]{} If $X$ is a separable regular space, then $w(X)\le \cont$. More generally, every regular space $X$ satisfies $w(X)\le 2^{d(X)}$. Compact dyadic spaces are defined to be continuous images of generalized Cantor cubes $\{0,1\}^\kappa$, where $\kappa$ is an arbitrary cardinal number. \[Pro:Eng\][(Engelking, [@Eng2 Theorem 10])]{} Let $\kappa$ be an infinite cardinal. A compact dyadic space $K$ with $w(K)\leq 2^\kappa$ satisfies $d(K)\leq\kappa$. In particular, if $w(K)\leq\cont$, then $K$ is separable. \[Theorem 3\][(Comfort, [@Comfort Theorem 3.1])]{} Every infinite compact topological group $G$ satisfies $|G|= 2^{w(G)}$ and $d(G)=\log w(G)$. In particular, if a compact group $G$ satisfies $w(G)\leq\cont$ then it is separable. Theorem \[Theorem 3\] is deduced in [@Comfort] from the fact that compact topological groups are dyadic. In fact as Comfort observes, if $G$ is a compact group of weight $\alpha\ge \aleph_0$, then there are continuous surjections as indicated below: $$\{0,1\}^\alpha \to G \to [0,1]^\alpha$$ and $|\{0,1\}^\alpha| = | [0,1]^\alpha| =2^\alpha$, and so $|G|= 2^\alpha$. A key result we shall need is the following one: \[HMP\][(Hewitt–Marczewski–Pondiczery, [@Hodel Theorem 11.2])]{} Let $\{X_i: i\in I\}$ be a family of topological spaces and $X=\prod_{i\in I}X_i$, where $|I|\le 2^\kappa$ for some cardinal number $\kappa\geq\omega$. If $d(X_i)\le \kappa$ for each $i\in I$, then $d(X)\le \kappa$. In particular, the product of no more than $\cont$ separable spaces is separable. In the following two definitions we introduce the main concepts of our study in Section \[Positive\]. \[almost connected\][(Hofmann–Morris, [@PROBOOK])]{} [A Hausdorff topological group $G$ is said to be *almost connected* if the quotient group $G/G_0$ is compact, where $G_0$ is the connected component of the identity in $G$.]{} Clearly the class of almost connected topological groups includes all compact groups, all connected topological groups, and their products. \[pro-Lie group\][(Hofmann–Morris, [@PROBOOK])]{} [A topological group is called a *pro-Lie* group if it is a projective limit of finite-dimensional Lie groups.]{} As shown in [@PROBOOK] the class of pro-Lie groups includes all locally compact abelian topological groups, all compact groups, all connected locally compact topological groups, and all almost connected locally compact topological groups. Further, every closed subgroup of a pro-Lie group is again a pro-Lie group and any finite or infinite product of pro-Lie groups is a pro-Lie group. \[Theorem 7\][(Hofmann–Morris, [@PROBOOK Theorem 12.81])]{} Let $G$ be a connected pro-Lie group. Then $G$ contains a maximal compact connected subgroup $C$ such that $G$ is homeomorphic to $C\times {{\mathbb{R}}}^\kappa$, for some cardinal $\kappa$. Almost connected pro-Lie groups also admit a topological characterization similar to the one in Theorem \[Theorem 7\]. \[Th\_HM\][(Hofmann–Morris, [@HM2 Corollary 8.9])]{} Every almost connected pro-Lie group $G$ is homeomorphic to the product ${\mathbb{R}}^\kappa \times \{0,1\}^\lambda \times B$, where $\{0,1\}$ is the discrete two-element group, $B$ is a compact connected group, and $\kappa,\lambda$ are cardinals. \[Remark 8\] [It is known that a topological group $G$ is separable if it contains a closed subgroup $H$ such that both $H$ and the quotient space $G/H$ are separable. In particular, separability is a three space property [@AT Theorem 1.5.23].]{} A topological group is said to be *$\omega$-narrow* [@AT Section 3.4] if it can be covered by countably many translations of every neighborhood of the identity element. It is known that every separable topological group is $\omega$-narrow (see [@AT Corollary 3.4.8]). The class of $\omega$-narrow groups is productive and hereditary with respect to taking arbitrary subgroups [@AT Section 3.4], so $\omega$-narrow groups need not be separable. In fact, $\omega$-narrow groups can have uncountable cellularity (see [@Usp] or [@AT Example 5.4.13]). The following theorem characterizes the class of $\omega$-narrow topological groups. \[Th:Gur\][(Guran, [@Gur])]{} A topological group $G$ is $\omega$-narrow if and only if $G$ is topologically isomorphic to a subgroup of a product of second countable topological groups. Separability of pro-Lie groups {#Positive} ============================== We prove in this section that an almost connected pro-Lie group $G$ is separable if and only if $w(G)\leq \cont$. This fact is then used to show that if an almost connected pro-Lie group $G$ is a subspace of a separable space, then $G$ is separable as well. We also prove in Corollary \[Theorem 14\] that a locally compact subgroup of a separable topological group is separable. \[Theorem 10\] An almost connected pro-Lie group $G$ is separable if and only if $w(G) \le \cont$. In particular this is the case if $G$ is a connected locally compact group. If $G$ is separable, then $w(G)\leq\cont$ by Theorem \[Th:1\], since Hausdorff topological groups are regular. Conversely, assume that $w(G)\leq\cont$. By Theorem \[Th\_HM\], the group $G$ is homeomorphic to the product ${\mathbb{R}}^\kappa\times\{0,1\}^\lambda\times B$, where $B$ is a compact connected topological group and $\kappa,\lambda$ are cardinals. It is clear that $w(G)=\kappa\cdot\lambda\cdot\mu\leq\cont$, where $\mu=w(B)$. Hence the spaces ${\mathbb{R}}^\kappa$ and $\{0,1\}^\lambda$ are separable by Theorem \[HMP\], while the separability of $B$ follows from Theorem \[Theorem 3\]. Hence $G$ is also separable as the product of three separable spaces. Since every connected locally compact group is $\sigma$-compact, the last part of Theorem \[Theorem 10\] admits the following slightly more general form which will be applied in the proof of Theorem \[Th:Main\_B\]: \[Le:PonGen\] Let $N$ be a closed subgroup of an $\omega$-narrow topological group $G$. If the quotient space $G/N$ is locally compact, then the inequality $w(G/N)\leq\cont$ is equivalent to the separability of $G/N$. Let $\pi\colon G\to G/N$ be the quotient mapping of $G$ onto the locally compact left coset space $G/N$. If $G/N$ is separable, then Theorem \[Th:1\] implies that $w(G/N)\leq\cont$. Assume therefore that $w(G/N)\leq\cont$. Take an open neighborhood $U$ of the identity element in $G$ such that the closure of the open set $\pi(U)$ in $G/N$ is compact. Since $G$ is $\omega$-narrow, there exists a countable set $C\subset G$ such that $G=CU$. Hence the compact sets $\overline{\pi(xU)}$, with $x\in C$, cover the space $G/N$. This proves that $G/N$ is $\sigma$-compact. Let $\{K_n: n\in\omega\}$ be a countable family of compact sets that covers $G/N$. Making use of the local compactness of $G/N$ we can find, for every $n\in\omega$, an open set $O_n$ with compact closure in $G/N$ such that $K_n\subset O_n$. Since the space $G/N$ is normal, there exists a closed $G_\delta$-set $F_n$ in $G/N$ such that $K_n\subset F_n\subset O_n$. It is clear that $F_n$ is compact for each $n\in\omega$. Summing up, each $F_n$ is a compact $G_\delta$-set in the quotient space $G/N$ of the $\omega$-narrow topological group $G$, so Theorem 2 of [@Usp1] implies that each $F_n$ is a dyadic compact space. As $w(F_n)\leq w(G/N)\leq\cont$ for each $n\in\omega$, it follows from Proposition \[Pro:Eng\] that each $F_n$ is separable. The inclusions $K_n\subset F_n$ with $n\in\omega$ imply that $G/N=\bigcup_{n\in\omega} F_n$, so the space $G/N$ is separable. \[Re:Corr\] [Theorem \[Theorem 10\] would not be valid in $ZFC$ if one replaced the condition $w(G)\leq\cont$ by the weaker one, $|G|\leq 2^\cont$. Indeed, the compact topological group $G=\{0,1\}^\kappa$ with $\kappa=\cont^+$ satisfies $w(G)=\kappa$, so $G$ is not separable by Theorem \[Theorem 3\]. However, it is consistent with $ZFC$ that $|G|=2^\kappa=2^\cont$ (see [@Kun Chap. VIII, Sect. 4]).]{} Theorem \[Theorem 10\] generalizes the second part of Theorem \[Theorem 3\]. It is also clear that Theorem \[Theorem 10\] is not valid for arbitrary locally compact groups—it suffices to take a discrete group of cardinality $\cont$. A family $\mathcal{N}$ of subsets of a topological space $Y$ is called a *network* for $Y$ if for every point $y \in Y$ and any neighbourhood $U$ of $y$ there exists a set $F \in\mathcal{N}$ such that $y \in F \subset U$. The *network weight* $nw(Y)$ of a space $Y$ is defined as the smallest cardinal number of the form $|\mathcal{N}|$, where $\mathcal{N}$ is a network for $Y$. \[Le:Dya\] If $L$ is a Lindelöf subspace of a separable Hausdorff space $X$, then $nw(L)\leq\cont$. Hence every compact subspace $K$ of a separable Hausdorff space satisfies $w(K)\leq\cont$. Denote by $D$ a countable dense subset of $X$. Let $$\mathcal{D}=\bigl\{\overline{C}: C\subseteq D\bigr\}\, \mbox{ and }\, \mathcal{N}=\bigl\{\,\bigcap\gamma: \gamma\subseteq\mathcal{D},\ |\gamma|\leq\omega\bigr\}.$$ Then $|\mathcal{D}|\leq\cont$, $|\mathcal{N}|\leq\cont^\omega=\cont$, and we claim that the family $\{N\cap L: N\in\mathcal{N}\}$ is a network for $L$. First we note the family $\mathcal{D}$ separates points of $X$. In other words, for every distinct elements $x,y\in X$, there exists $C\in\mathcal{D}$ such that $x\in C\not\ni y$. This is clear since the space $X$ is Hausdorff. Take a point $x\in L$ and an arbitrary open neighborhood $U$ of $x$ in $X$. Denote by $\mathcal{D}_x$ the family of all $C\in\mathcal{D}$ with $x\in C$. Since $\mathcal{D}$ separates points of $X$, we see that $\bigcap\mathcal{D}_x=\{x\}$. Using the Lindelöf property of $L$, we can find a countable subfamily $\gamma$ of $\mathcal{D}_x$ such that $L\cap\bigcap\gamma\subseteq L\cap U$. Then $F=\bigcap\gamma\in\mathcal{N}$ and $x\in L\cap F\subseteq L\cap U$. This proves that $\mathcal{N}$ is a network for $L$. If $K$ is a compact subset of $X$, then $w(K)=nw(K)$. Since $K$ is obviously Lindelöf, we conclude that $w(K)\leq\cont$. Combining Lemma \[Le:Dya\] and Theorem \[Theorem 3\], we deduce the following fact: \[Corollary 12\] If a compact group $G$ is a subspace of a separable Hausdorff space, then $G$ is separable. It turns out that the compactness of $G$ in Corollary \[Corollary 12\] cannot be weakened to $\sigma$-compactness: \[Example 2\] (See [@Comfort_Itzkowitz Lemma 3.1]) Let $X$ be any separable compact space which contains a closed non-separable subspace $Y$. The free abelian topological group $A(Y)$ naturally embeds into $A(X)$ as a closed subgroup. Then $A(X)$ is a separable $\sigma$-compact group, while $A(Y)$ is not separable— otherwise $Y$ would be separable. The conclusion of Corollary \[Corollary 12\] remains valid for connected pro-Lie groups. Later, in Theorem \[Th:Main\], we will show that can be weakened to . \[Le:pro-L\] If a connected pro-Lie group $G$ is a subspace of a separable Hausdorff space $X$, then $G$ is separable. By Theorem \[Theorem 7\], the connected pro-Lie group $G$ is homeomorphic to the product $C\times {{\mathbb{R}}}^\kappa$, where $C$ is a compact connected group and $\kappa$ is a cardinal. Since $C$ can be identified with a subspace of $G$, Lemma \[Le:Dya\] implies that $w(C)\leq\cont$. Further, ${{\mathbb{R}}}^\kappa$ contains a compact subspace homeomorphic with $\{0,1\}^\kappa$. Since the latter space has weight $\kappa$, we apply Lemma \[Le:Dya\] once again to conclude that $\kappa\leq\cont$. Hence $w(G)\leq\cont$. Therefore $G$ is separable, by Theorem \[Theorem 10\]. Since the class of connected pro-Lie groups is productive and contains connected locally compact groups [@PROBOOK], the next fact is immediate from Proposition \[Le:pro-L\]. \[Corollary 13\] Let $G$ be a product of connected locally compact groups. If $G$ is a subspace of a separable Hausdorff space $X$, then $G$ is separable. The next result, one of the main in this section, extends Corollary \[Corollary 12\] and Proposition \[Le:pro-L\] to almost connected pro-Lie groups. \[Th:Main\] Let $G$ be an almost connected pro-Lie group. If $G$ is homeomorphic to a subspace of a separable Hausdorff space, then it is separable as well. By Theorem \[Th\_HM\], the group $G$ is homeomorphic to ${\mathbb{R}}^\kappa \times \{0,1\}^\lambda \times B$, where $\{0,1\}$ is the discrete two-element group, $B$ is a compact connected group, and $\kappa,\lambda$ are cardinals. Assume that $G$ is homeomorphic to a subspace of a separable Hausdorff space. The connected pro-Lie group ${\mathbb{R}}^\kappa$ is separable by Proposition \[Le:pro-L\]. The compact group $K=\{0,1\}^\lambda\times B$ is separable according to Corollary \[Corollary 12\]. Therefore $G$ is separable as the product of two separable spaces, ${\mathbb{R}}^\kappa$ and $K$. (Let us note that $\kappa\cdot\lambda\leq\cont$.) In the following result we establish a simple relationship between almost connected pro-Lie groups and $\omega$-narrow groups. \[Le:Re\] Every almost connected pro-Lie group has countable cellularity and hence is $\omega$-narrow. By Theorem \[Th\_HM\], every almost connected pro-Lie group $G$ is homeomorphic to the product ${\mathbb{R}}^\kappa\times\{0,1\}^\lambda\times B$, where $B$ is a compact connected group and $\kappa,\lambda$ are cardinals. The space ${\mathbb{R}}^\kappa\times\{0,1\}^\lambda$ has countable cellularity as a product of separable spaces [@Eng Theorem 2.3.17]. The compact group $B$ also has countable cellularity by [@AT Corollary 4.1.8]. Hence the cellularity of the space ${\mathbb{R}}^\kappa\times\{0,1\}^\lambda\times B$ is countable (see Corollary 5.4.9 of [@AT]). Finally, every topological group of countable cellularity is $\omega$-narrow according to [@AT Theorem 3.4.7]. It turns out that Theorem \[Th:Main\] extends to a wider class of topological groups which contains both almost connected pro-Lie groups and locally compact $\sigma$-compact groups. First we need a simple auxiliary result which complements Pontryagin’s open homomorphism theorem (see [@Mor Theorem 3]). \[Le:aux\] Let $f\colon X\to Y$ be a continuous bijection. If the spaces $X$ and $Y$ are locally compact and $\sigma$-compact and $X$ is homogeneous, then $f$ is a homeomorphism. Let $U_0\subset X$ be a non-empty open set with compact closure. Take a non-empty open set $U$ in $X$ such that $\overline{U}\subset U_0$. Denote by $Homeo(X)$ the family of all homeomorphisms of $X$ onto itself. Since $X$ is homogeneous and $\sigma$-compact, there exists a countable subfamily $\mathcal{A}\subset Homeo(X)$ such that $X=\bigcup\{\alpha(U): \alpha\in\mathcal{A}\}$. Note that the closure of $\alpha(U)$ is compact, for each $\alpha\in\mathcal{A}$. The family $\{f(\alpha(U)): \alpha\in\mathcal{A}\}$ is a countable cover of $Y$. Since $Y$ is locally compact it has the Baire property. Hence there exists $\alpha\in\mathcal{A}$ such that the closure of $f(\alpha(U))$ has a non-empty interior in $Y$. Let $V$ be a non-empty open set in $Y$ contained in $\overline{f(\alpha(U))}=f(\alpha(\overline{U}))$. Since $f$ is one-to-one, we see that $f^{-1}(V)\subset \alpha(\overline{U})\subset \alpha(U_0)$, so $W=f^{-1}(V)$ is an open subset of $\alpha(U_0)$. The closure of $\alpha(U_0)$ in $X$ is the compact set $\alpha(\overline{U_0})$, so the restriction of $f$ to the open subset $W$ of $\alpha(\overline{U_0})$ is a homeomorphism of $W$ onto its image $V=f(W)$. Therefore, by the homogeneity argument, $f$ is a homeomorphism. \[Th:Main\_B\] Let $G$ be an $\omega$-narrow topological group which contains a closed subgroup $N$ such that $N$ is an almost connected pro-Lie group and the quotient space $G/N$ is locally compact. If $G$ is homeomorphic to a subspace of a separable Hausdorff space, then it is separable as well. Assume that $G$ is a subspace of a separable Hausdorff space $X$. Since $N\subset G\subset X$, it follows from Theorem \[Th:Main\] that the group $N$ is separable and, hence, $w(N)\leq\cont$ by Theorem \[Th:1\]. Let $\tau$ be the topology of $X$. The family of all regular open sets in $X$ constitutes a base for a weaker topology on $X$, say, $\sigma$. Since the topology $\tau$ is separable, the space $Y=(X,\sigma)$ has a base of the cardinality at most $\cont$. Indeed, let $S$ be a countable dense subset of $X$. Then the family $$\mathcal{B}=\{\operatorname{\rm Int}_X\overline{D}: D\subseteq S\}\setminus\{\emptyset\}$$ is a base for $Y$ and, clearly, $|\mathcal{B}|\leq\cont$. It is also clear that the space $Y$ is Hausdorff. We see in particular that the pseudocharacter of $Y$ is at most $\cont$, i.e. $\psi(Y)\leq\cont$. Since the identity mapping of $X$ onto $Y$ is continuous, it follows that $\psi(X)\leq\psi(Y)\leq\cont$. Hence the subspace $G$ of $X$ satisfies $\psi(G)\leq\cont$ as well. This is the first important property of the group $G$. We claim that there exists a continuous isomorphism (not necessarily a homeomorphism) $\pi$ of $G$ onto a Hausdorff topological group $H$ with the following properties: 1. $w(H)\leq\cont$; 2. the restriction of $\pi$ to $N$ is a topological isomorphism of $N$ onto the closed subgroup $K=\pi(N)$ of $H$; 3. the quotient space $H/K$ is locally compact. Indeed, it follows from [@AT Corollary 3.4.19] that for every neighborhood $U$ of the identity $e$ in $G$, there exists a continuous homomorphism $\pi_U$ of $G$ onto a second-countable Hausdorff topological group $H_U$ such that $\pi_U^{-1}(V)\subseteq U$, for some open neighborhood $V$ of the identity in $H_U$. Let $\mathcal{P}$ be a family of open neighborhoods of $e$ in $G$ such that $\{e\}=\bigcap\mathcal{P}$ and $|\mathcal{P}|\leq\cont$ (we use the fact that $\psi(G)\leq\cont$). Let also $\mathcal{Q}$ be a family of open neighborhoods of $e$ in $G$ such that $|\mathcal{Q}|\leq\cont$ and $\{W\cap N: W\in\mathcal{Q}\}$ is a local base for $N$ at $e$ (here we use the inequality $w(N)\leq\cont$). Then the cardinality of the family $\mathcal{R}=\mathcal{P}\cup\mathcal{Q}$ is not greater than $\cont$. For every element $U\in\mathcal{R}$, we take a continuous homomorphism $\pi_U$ of $G$ onto a second countable topological group $H_U$ as above. Further, since the space $G/H$ is locally compact, there exists an open neighborhood $U_0$ of $e$ in $G$ such that the closure of $\varphi_G(U_0)$ in $G/H$ is compact, where $\varphi_G\colon G\to G/H$ is the quotient mapping. Take a continuous homomorphism $p\colon G\to H_0$ to a second countable topological group $H_0$ such that $p^{-1}(V_0)\subset U_0$, for some open neighborhood $V_0$ of $p(e)$ in $H_0$. Let $\pi$ be the diagonal product of the family $\{\pi_U: U\in\mathcal{R}\}\cup\{p\}$. Then $\pi$ is a continuous homomorphism of $G$ to the product $P=H_0\times\prod_{U\in\mathcal{R}} H_U$ of second countable Hausdorff topological groups. It follows from our choice of $\mathcal{P}$ and the inclusion $\mathcal{P}\subseteq\mathcal{R}$ that $\pi$ is a monomorphism. Since $|\mathcal{R}|\leq\cont$, the group $P$ and its subgroup $H=\pi(G)$ have weight at most $\cont$. Denote by $p_0$ the projection of $P$ onto the factor $H_0$ and let $W_0=H\cap p_0^{-1}(V_0)$. Then $W_0$ is an open neighborhood of the identity in $H$ and since $p_0\circ\pi=p$, we see that $\pi^{-1}(W_0)=p^{-1}(V_0)\subset U_0$. Let us verify that $\pi(N)$ is closed in $H$. It follows from our choice of the family $\mathcal{Q}$ and the inclusion $\mathcal{Q}\subseteq\mathcal{R}$ that the restriction of $\pi$ to $N$ is a topological isomorphism of $N$ onto its image $\pi(N)$. Since the pro-Lie group $N$ is complete, so is the subgroup $K=\pi(N)$ of $H$. Hence $K$ is closed in $H$. In particular, the quotient space $H/K$ is Hausdorff. This proves our claim. Let $\varphi_G\colon G\to G/N$ and $\varphi_H\colon H\to H/K$ be the canonical quotient mappings onto the left coset spaces $G/N$ and $H/K$, respectively. We define a mapping $i\colon G/N\to H/K$ by letting $i(\varphi_G(x))=\varphi_H(\pi(x))$, for each $x\in G$. Since $K=\pi(N)$, this definition is correct. It follows from our definition of $i$ that the diagram below commutes. $$\xymatrix{G\ar@{>}[r]^{\varphi_G\,\,}\ar@{>}[d]_{\pi} &G/N \ar@{>}[d]^{i}\\ H \ar@{>}[r]^{\varphi_H\,\,} & H/K }$$ Since $\pi$ is algebraically an isomorphism and $K=\pi(N)$, we see that $i$ is a bijection. The continuity of the mapping $i$ follows from the facts that $\pi$ and $\varphi_H$ are continuous, while $\varphi_G$ is open and continuous. The set $\varphi_H(W_0)$ is an open neighborhood of the identity in $H/K$ and the closure of $\varphi_H(W_0)$ is compact, i.e. the space $H/K$ is locally compact. Indeed, it follows from $\pi^{-1}(W_0)\subset U_0$ that $\varphi_H(W_0)\subset i(\varphi_G(U_0))\subset i(\overline{\varphi_G(U_0)})$. Since $\overline{\varphi_G(U_0)}$ is compact, so are the sets $i(\overline{\varphi_G(U_0)})$ and $\overline{\varphi_H(W_0)}$. We have thus proved that the homomorphism $\pi\colon G\to G/H$ satisfies (i)–(iii). The group $H$ is $\omega$-narrow as a continuous homomorphic image of the $\omega$-narrow group $G$. Hence $H$ can be covered by countably many translations of the open set $W_0$. Since the set $\overline{\varphi_H(W_0)}$ is compact, it follows that the space $H/K$ is $\sigma$-compact. Similarly, since the group $G$ is $\omega$-narrow and the set $\overline{\varphi_G(U_0)}$ is compact, the space $G/N$ is also $\sigma$-compact. Therefore, both spaces $G/N$ and $H/K$ are locally compact, $\sigma$-compact, and homogeneous. Finally, it follows from Lemma \[Le:aux\] that the bijection $i\colon G/N\to H/K$ is a homeomorphism. It is clear that $w(H/K)\leq w(H)\leq\cont$, so we conclude that $w(G/N)=w(H/K)\leq\cont$. Hence Lemma \[Le:PonGen\] implies that the space $G/N$ is separable. Since the subgroup $N$ of $G$ is also separable, the separability of $G$ follows from Remark \[Remark 8\]. In the sequel we consider embeddings into separable topological groups. As one can expect, the situation improves notably when compared to embeddings into separable Hausdorff spaces. Let us recall that a topological group $G$ is called *feathered* if it contains a nonempty compact subset with a countable neighborhood base in $G$. Equivalently, $G$ is feathered if it contains a compact subgroup $K$ such that the quotient space $G/K$ is metrizable (see [@AT Section 4.3]). All metrizable groups and all locally compact groups are feathered. Notice that the class of feathered groups is countably productive according to [@AT Proposition 4.3.13]. \[Th:Fea\] Let a feathered topological group $G$ be a subgroup of a separable topological group. Then $G$ is separable. Assume that $G$ is a subgroup of a separable topological group $X$. By [@AT Corollary 3.4.8], the group $X$ is $\omega$-narrow. Hence, according to [@AT Theorem 3.4.4], the subgroup $G$ of $X$ is also $\omega$-narrow. Applying [@AT 4.3.A], we deduce that $G$ is Lindelöf. Take a compact subgroup $K$ of $G$ such that the quotient space $G/K$ is metrizable. Note that the space $G/K$ is Lindelöf as a continuous image of the Lindelöf space $G$. Hence $G/K$ is separable. Finally, the compact group $K$ is separable by Lemma \[Le:Dya\]. Hence the separability of $G$ follows from Remark \[Remark 8\]. Since all locally compact groups and all metrizable groups are feathered, the following two corollaries are immediate from Theorem \[Th:Fea\]; the second of them is well known. \[Theorem 14\] If a locally compact topological group $G$ is a subgroup of a separable topological group, then $G$ is separable. \[Cor:Met\] If a metrizable group $G$ is a subgroup of a separable topological group, then $G$ is separable. In fact, the conclusion of Corollary \[Cor:Met\] remains valid if $G$ is a subgroup of a topological group $X$ with countable cellularity. Indeed, the group $X$ is $\omega$-narrow by [@AT Theorem 3.4.7], and so is its subgroup $G$. Since $G$ is first countable, it follows from [@AT Proposition 3.4.5] that $G$ has a countable base and hence is separable. \[Re:Ark\] 1\) A discrete (hence locally compact and metrizable) group $G$ homeomorphic to a closed subspace of a separable Tychonoff space is not necessarily separable. Indeed, it suffices to consider the Niemytzki plane which contains a discrete copy of the real numbers, the $x$-axis. Therefore, Theorem \[Th:Fea\] and Corollaries \[Theorem 14\] and \[Cor:Met\] would not be valid if the group $G$ were assumed to be a subspace of a separable Hausdorff (or even Tychonoff) space. Neither is Corollary \[Theorem 14\] valid if $G$ is assumed only to be a pro-Lie group. We will present in Corollary \[Cor:subg\] an example of a separable prodiscrete abelian group which contains a closed non-separable subgroup. 2\) The separable connected pro-Lie group $G={\mathbb{R}}^\cont$ contains a closed non-separable subgroup. To see this, we consider the closed subgroup $\Z^\cont$ of $G$. By a theorem of Uspenskij [@Usp3], the group $\Z^\cont$ contains a subgroup $H$ of uncountable cellularity. The closure of $H$ in $G$, say, $K$ is a closed non-separable subgroup of $G$. By Proposition \[Le:pro-L\], the group $K$ cannot be connected. A natural question, after Proposition \[Le:pro-L\] and Corollary \[Cor:Met\], is whether a connected metrizable group must be separable if it is a subspace of a separable Hausdorff (or regular) space. Again the answer is . Indeed, consider an arbitrary connected metrizable group $G$ of weight $\cont$. For example, one can take $G=C(X)$, the Banach space of continuous real-valued functions on a compact space $X$ satisfying $w(X)=\cont$, endowed with the sup-norm topology. Since $w(G)=\cont$, the space $G$ is homeomorphic to a subspace of the Tychonoff cube $I^\cont$, where $I=[0,1]$ is the closed unit interval. Thus $G$ embeds as a subspace in a separable regular space, but both the density and weight of $G$ are equal to $\cont$. Embedding theorems {#Sec2} ================== The following result shows that there is a wealth of separable pseudocompact topological abelian groups with closed non-separable subgroups. Later, in Theorem \[Th:nA\], we will present a general version of this result, without restricting ourselves to the abelian case. \[Th:ab\] Every precompact abelian group of weight $\leq\cont$ is topologically isomorphic to a closed subgroup of a separable, connected, pseudocompact abelian group $H$ of weight $\leq\cont$. Let $G$ be a precompact abelian topological group satisfying $w(G)\leq\cont$. Denote by $\varrho{G}$ the Raĭkov completion of $G$. Then $\varrho{G}$ is a compact abelian topological group. Since $G$ is dense in $\varrho{G}$, we see that $\chi(\varrho{G})=\chi(G)$. Further, the character and weight of every precompact (more generally, $\omega$-narrow) topological group coincide (see [@AT Corollary 5.2.4]). Therefore, $w(\varrho{G})=w(G)\leq\cont$. Since $\varrho{G}$ is a compact abelian topological group of weight $\leq\cont$, it is topologically isomorphic to a subgroup of $\Pi=\mathbb{T}^\cont$, where $\mathbb{T}$ is the circle group with its usual topology inherited from the complex plane. In particular, we can identify $G$ with a subgroup of $\Pi$. We will construct the required group $H$ as a dense subgroup of $\Pi^\cont$. For every $\alpha<\cont$, let $\pi_\alpha$ be the projection of $\Pi^\cont$ to the $\alpha$th factor $\Pi_{(\alpha)}$. Let $$\Delta=\{x\in \Pi^\cont: \pi_\alpha(x)=\pi_\beta(x) \mbox{ for all } \alpha,\beta\in\cont \}$$ be the diagonal in $\Pi^\cont$. It is clear that $\Delta$ is a closed subgroup of $\Pi^\cont$. Further, we can identify $G$ with the subgroup $$\widetilde{G}=\{x\in\Delta: \pi_0(x)\in G\}$$ of $\Delta$. To define the subgroup $H$ of $\Pi^\cont$ we need some preliminary work. For every element $x\in\Pi^\cont$, let $$\operatorname*{\rm supp}(x)=\{\alpha\in\cont: \pi_\alpha(x)\neq e_\Pi\},$$ where $e_\Pi$ is the identity element of $\Pi$. Then $$\Sigma=\{x\in\Pi^\cont: |\operatorname*{\rm supp}(x)|\leq\omega\}$$ is a dense countably compact subgroup of $\Pi^\cont$ [@AT Corollary 1.6.34]. Hence $\Sigma$ is pseudocompact. It is clear that $\Pi^\cont\cong (\mathbb{T}^\cont)^\cont\cong \mathbb{T}^{\cont\times\cont}$. We define an element $x_0\in\mathbb{T}^{\cont\times\cont}$ as follows. Let $\{t_p: p\in\cont\times\cont\}$ be an independent system of elements of $\mathbb{T}$ having infinite order. In other words, if $p_1,\ldots,p_k$ are pairwise distinct elements of $\cont\times\cont$ and $n_1,\ldots,n_k$ are arbitrary integers, then the equality $t_{p_1}^{n_1}\cdots t_{p_k}^{n_k}=1$ in $\mathbb{T}$ implies that $n_1=\cdots=n_k=0$. Take an element $x_0\in\mathbb{T}^{\cont\times\cont}$ such that $x_0(p)=t_p$ for each $p=(\alpha,\beta)\in\cont\times\cont$. Then the cyclic subgroup $\hull{x_0}$ of $\Pi^\cont\cong\mathbb{T}^{\cont\times\cont}$ generated by $x_0$ is dense in $\Pi^\cont$ (see the proof of Corollary 25.15 in [@HR]). Finally we put $H=\widetilde{G}+\Sigma+\hull{x_0}$. Then $H$ is dense in $\Pi^\cont$ and separable since $H$ contains the countable dense subgroup $\hull{x_0}$ of $\Pi^\cont$. It is also clear that $H$ is pseudocompact since it contains the dense pseudocompact subgroup $\Sigma$. By [@AT Theorem 2.4.15], the subgroup $H$ of $\Pi^\cont \cong\mathbb{T}^{\cont\times\cont}$ is connected. It remains to verify that $$\label{Eq:1} H\cap \Delta=\widetilde{G}.$$ Indeed, since $\Delta$ is closed in $\Pi^\cont$, it follows from (\[Eq:1\]) that $\widetilde{G}\cong G$ is closed in $H$. To verify (\[Eq:1\]), we need the following fact: [**Claim.**]{} *If $\alpha<\beta<\cont$ and $n\neq 0$ is an integer, then $\pi_\alpha(x_0^n)\neq \pi_\beta(x_0^n)$*. We shall now use proof by contradiction. Suppose that $\pi_\alpha(x_0^n)=\pi_\beta(x_0^n)$ for distinct $\alpha,\beta\in\cont$ and an integer $n\neq 0$. Let $J=\{\alpha,\beta\}\subset \cont$ and denote by $\pi_J$ the projection of $\Pi^\cont$ onto $\Pi^J$. Since $\hull{x_0}$ is dense in $\Pi^\cont$, the cyclic subgroup $\hull{y_0}$ of $\Pi^J$ is dense $\Pi^J$, where $y_0=\pi_J(x_0)$. It follows from $\pi_\alpha(x_0^n)=\pi_\beta(x_0^n)$ that the element $z_0=(\pi_\alpha(x_0))^{-1}\cdot\pi_\beta(x_0)$ of $\Pi$ satisfies $z_0^n=e_\Pi$. Denote by $p$ the canonical homomorphism of $\Pi$ onto $\Pi/\hull{z_0}$. Let $p^2$ be the canonical homomorphism of $\Pi^J$ onto $\Pi/\hull{z_0}\times\Pi/\hull{z_0}$. Since $\pi_\beta(x_0)= z_0\cdot\pi_\alpha(x_0)$, we see that $p(\pi_\alpha(x_0))=p(\pi_\beta(x_0))$. Hence the coordinates of the element $p^2(y_0)=(p(\pi_\alpha(x_0)),p(\pi_\beta(x_0)))$ coincide and the diagonal of $\Pi/\hull{z_0}\times\Pi/\hull{z_0}$ contains the group $p^2(\hull{y_0})$. Since $\hull{y_0}$ is dense in $\Pi^J$ and the homomorphism $p^2$ is continuous, it follows that the diagonal in $\Pi/\hull{z_0}\times\Pi/\hull{z_0}$ is dense in $\Pi/\hull{z_0}\times\Pi/\hull{z_0}$. This contraction finishes the proof of the Claim. Let us turn back to the proof of the equality (\[Eq:1\]). Take $g\in\widetilde{G}$, $s\in\Sigma$, and $n\in\mathbb{Z}$ such that $d=g\cdot s\cdot x_0^n\in\Delta$. Since $g\in\Delta$, it follows that $d\cdot g^{-1}=s\cdot x_0^n\in\Delta$. The element $s\in\Sigma$ has al most countably many coordinates distinct from $e_\Pi$, so the last equality implies that all the coordinates of the element $x_0^n$, except for countably many of them, coincide. By the above Claim, this is possible only if $n=0$. Since $s=s\cdot x_0^n\in\Delta$, we see that $s\in\Sigma\cap\Delta$, so $s$ is the identity element of $\Pi^\cont$ and $d=g\in\widetilde{G}$. This proves (\[Eq:1\]) and completes our argument. Theorem \[Th:ab\] remains valid in the non-abelian case, but our argument becomes considerably more complicated. First we need an auxiliary lemma. \[Le:tec2\] There exists a sequence $\{\varphi_m: m\in\omega\}$ of mappings of $\omega$ to $\omega$ satisfying the following condition: For every integer $k\geq 1$ and every vector triple $(\overline{m},\overline{\jmath}_1, \overline{\jmath}_2)$, where $\overline{m}=(m_1,\ldots,m_k)$, $\overline{\jmath}_1 =(j_{1,1},\ldots,j_{k,1})$, and $\overline{\jmath}_2=(j_{1,2},\ldots,j_{k,2})$ are elements of $\omega^k$ and $m_1,\ldots,m_k$ are pairwise distinct, there exists $n\in\omega$ such that $\varphi_{m_i}(n)=j_{i,1}$ and $\varphi_{m_i}(n+1)=j_{i,2}$ for each $i$ with $1\leq i\leq k$. Let us enumerate the family of all triples $(\overline{m}, \overline{\jmath}_1,\overline{\jmath}_2)$ as in the lemma in such a way that if a triple has number $n$ and its first entry is $\overline{m}= (m_1,\ldots,m_k)$, then $m_1\ldots,m_k$ are less than or equal to $n$. Assume that at a stage $n$ of our inductive construction we have defined the values $\varphi_m(j)$ for all $m<n$ and $j\leq 2n$. We now consider the triple $(\overline{m},\overline{\jmath}_1,\overline{\jmath}_2)$ which has number $n$ in our enumeration and note that the coordinates $m_1,\ldots,m_k$ of $\overline{m}$ are less than or equal to $n$. According to the conclusion of the lemma, we have to put $\varphi_{m_i}(2n+1)=j_{i,1}$ and $\varphi_{m_i}(2n+2)=j_{i,2}$ for each $i$ with $1\leq i\leq k$. The rest of the values $\varphi_m(j)$ with $m\leq n$ and $j\leq 2n+2$ can be chosen arbitrarily. Continuing this way, we obtain the sequence $\{\varphi_m: m\in\omega\}$ satisfying the condition of the lemma. \[Th:nA\] Every precompact topological group of weight $\leq\cont$ is topologically isomorphic to a closed subgroup of a separable, connected, pseudocompact group $H$ of weight $\leq\cont$. It is known that every second countable compact topological group is topologically isomorphic to a subgroup of the group $\U=\prod_{n\in\N} U(n)$, where $U(n)$ is the group of unitary $n\times n$ matrices with complex entries for each $n\in\N$. Hence every compact topological group is topologically isomorphic to a subgroup of some power of the group $\U$. In particular, if $G$ is a precompact topological group and $w(G)\leq\cont$, then the Raĭkov completion of $G$, $\varrho{G}$, is a compact topological group of weight $\leq\cont$, so $\varrho{G}$ and $G$ are topologically isomorphic to subgroups of the compact group $\Pi=\U^\cont$. As $\U$ is connected, so is $\Pi$. Let us identify $G$ with a subgroup of $\Pi$. Since $\U$ is second countable, the group $\Pi$ is separable by the Hewitt–Marczewski–Pondiczery theorem. Let $S=\{s_n: n\in\omega\}$ be a dense subset of $\Pi$, where each $s_n$ is distinct from the identity element $e_\Pi$ of $\Pi$. First we are going to define a special countable dense subset of $\Pi^\cont$ modifying the original construction of Hewitt–Marczewski–Pondiczery. To this end, we replace the index set $\cont$ with ${\mathbb{R}}$, so we will work with $\Pi^{{\mathbb{R}}}$ in place of $\Pi^\cont$. Let $\mathcal{B}$ be the base for the usual topology on ${\mathbb{R}}$ which consists of the intervals $(a,b)$ with rational endpoints $a,b$. We are now in the position to define a special countable dense subset $D$ of $\Pi^{\mathbb{R}}$. Let $$\begin{aligned} \mathcal{A} = \{ & (U_1,\ldots,U_k, s_{i_1},\ldots,s_{i_k}): k\geq 1,\ U_1,\ldots,U_k \mbox{ are pairwise disjoint }\\ & \mbox{ elements of } \mathcal{B}, \mbox{ and } i_1,\ldots,i_k\in\omega \}.\end{aligned}$$ It is clear that $\mathcal{A}$ is countable. Let us enumerate this family as $\mathcal{A}=\{L_n: n\in\omega\}$. We also choose a pairwise disjoint sequence $\{V_m: m\in\omega\}$ of open unbounded subsets of ${\mathbb{R}}$. Let $\{\varphi_m: m\in\omega\}$ be a sequence of mappings of $\omega$ to $\omega$ satisfying the conclusion of Lemma \[Le:tec2\]. Given $n\in\omega$ and $L_n=(U_1,\ldots,U_k, s_{i_1},\ldots,s_{i_k})$ in $\mathcal{A}$, we define an element $a_n\in\Pi^{\mathbb{R}}$ as follows: $$a_n(r)=\begin{cases} s_{i_j} &\text{if $r\in U_j$ for some $j\leq k$};\\ s_{\varphi_n(j)} &\text{if $r\in V_j\setminus\bigcup_{i\leq k} U_k$ for some $j\in\omega$};\\ e_\Pi&\text{otherwise}, \end{cases}$$ where $r\in{\mathbb{R}}$. A standard argument shows that the countable set $D=\{a_n: n\in\omega\}$ is dense in $\Pi^\cont$. Let also $\Sigma$ be the $\Sigma$-product of continuum many copies of the group $\Pi$ considered as a subgroup of $\Pi^{\mathbb{R}}$. As in Theorem \[Th:ab\], $\Sigma$ is a dense pseudocompact subgroup of $\Pi^{\mathbb{R}}$. Denote by $\Delta$ the diagonal subgroup of $\Pi^{\mathbb{R}}$: $$\Delta=\{x\in\Pi^{\mathbb{R}}: x(r)=x(s) \mbox{ for all } r,s\in{\mathbb{R}}\}.$$ It is clear that $$G_0=\{x\in\Delta: x(0)\in G\}$$ is an isomorphic topological copy of $G$. Denote by $E$ the subgroup of $\Pi^{\mathbb{R}}$ generated by the set $G_0\cup D$. Finally we put $H=E\cdot\Sigma$. Then $H$ is a subgroup of $\Pi^{\mathbb{R}}$ since $\Sigma$ is an invariant subgroup of $\Pi^{\mathbb{R}}$. As in the proof of Theorem \[Th:ab\], it is easy to see that $H$ is a separable, dense, pseudocompact subgroup of $\Pi^{\mathbb{R}}\cong \U^{{\mathbb{R}}\times\cont}$. Since the group $\U$ is compact and metrizable, the subgroup $H$ of the connected group $\Pi^{\mathbb{R}}$ is also connected according to [@AT Theorem 2.4.15]. The main difficulty here is to verify the equality $H\cap\Delta=G_0$ analogous to (1) in the proof of Theorem \[Th:ab\]. Once this is done, we will immediately conclude that $G_0\cong G$ is closed in $H$. Assume that $d=w\cdot b\in\Delta$, where $w\in E$ and $b\in\Sigma$. We have to show that $d\in G_0$. Let $B=\operatorname*{\rm supp}(b)$. Then $w(x)=w(y)$ for all $x,y\in{\mathbb{R}}\setminus B$. Further, the element $w\in E$ has the form $w=g_1d_1^{\epsilon_1}\cdots g_nd_n^{\epsilon_n}g_{n+1}$, where $g_1,\ldots,g_n,g_{n+1}\in G_0$, $d_1,\ldots,d_n\in D$, and $\epsilon_1, \ldots,\epsilon_n\in\{1,-1\}$. Here some (or even all) of the elements $g_1,\ldots,g_n,g_{n+1}$ can be equal to the identity element $e$ of $\Pi^{\mathbb{R}}$ and some of $d_i$ can coincide. The word $w$ has the form $w=g_1d_1^{\epsilon_1}\cdots g_nd_n^{\epsilon_n}g_{n+1}$, where $n\geq 1$. Substituting $d_i$ with variables $z_j$ and assigning the same variable to $d_{i_1}$ and $d_{i_2}$ whenever $d_{i_1}=d_{i_2}$, we obtain the word $w[\overline{z}]=g_1z_1^{\epsilon_1}\cdots g_nz_p^{\epsilon_n}g_{n+1}$, where $\overline{z}=(z_1,\ldots,z_k)$, $k\leq n$, and $1\leq p\leq k$. For example, $k=1$ if and only if all the $d_i$ are equal, and $k=n$ if all the $d_i$ are pairwise distinct. Let $$R=\{w[\overline{z}]: \overline{z}=(z_1,\ldots,z_k)\in (\Pi^{\mathbb{R}})^k\}$$ be the range of $w[\,\cdot\,]$. We consider two cases. Case 1. All values of $w(\overline{z})$ coincide, i.e. $|R|=1$. Let us take $\overline{z}_0=(e,\ldots,e)$, i.e. $z_1=\cdots=z_k=e$. Then $w=w[\overline{z}_0]=g_1\cdots g_n\cdot g_{n+1}\in G_0$. It now follows from $d=w\cdot b\in\Delta$ that $b=w^{-1}d\in\Delta\cap\Sigma$, and we conclude that $b=e$ and $d=w\in G_0$. Case 2. $|R|\geq 2$. Then we choose $\overline{z}_1=(z_{1,1}, \ldots,z_{k,1})$ and $\overline{z}_2=(z_{1,2},\ldots,z_{k,2})$ in $(\Pi^{\mathbb{R}})^k$ such that $w[\overline{z}_1]\neq w[\overline{z}_2]$. We claim that there exist $x,y\in{\mathbb{R}}\setminus B$ such that $w(x)\neq w(y)$, which is a contradiction. Indeed, take $r\in{\mathbb{R}}$ such that $w[\overline{z}_1](r)\neq w[\overline{z}_2](r)$. Hence $$\begin{aligned} g_1(r)z_{1,1}(r)^{\epsilon_1}\cdots z_{p,1}(r)^{\epsilon_n}g_{n+1}(r) &\neq\\ g_1(r)z_{1,2}(r)^{\epsilon_1}\cdots z_{p,2}(r)^{\epsilon_n}g_{n+1}(r) &.\end{aligned}$$ Since multiplication and inversion in $\Pi$ are continuous, we can find an open symmetric neighborhood $W$ of $e_\Pi$ in $\Pi$ such that the sets $$O_\delta=g_1(r)\Bigl(z_{1,i}(r)^{\epsilon_1}W\Bigr)\cdots g_n(r)\Bigl(z_{n,i}(r)^{\epsilon_n}W\Bigr)g_{n+1}(r)$$ with $\delta=1,2$ are disjoint. Let us put $t_{i,\delta}=z_{i,\delta}(r)$ for all $i=1,\ldots,k$ and $\delta=1,2$. Since the set $S=\{s_j: j\in\omega\}$ is dense in $\Pi$, we can choose, for every $i\leq n$ and $\delta=1,2$, an element $s_{i,\delta}\in S\cap t_{i,\delta}W\cap Wt_{i,\delta}$. Then $s_{i,\delta}^\epsilon\in t_{i,\delta}^{\epsilon}W$ for each $\epsilon=\pm1$. Furthermore, according to our choice of the variables $z_i$, the elements $s_{i,\delta}$ can be chosen to satisfy $s_{i,\delta}=s_{l,\delta}$ whenever $d_i=d_l$, where $i,l\leq n$ and $\delta=1,2$. It now follows from $O_1\cap O_2=\emptyset$ that the elements $$\label{Eq:2} h_1=g_1(r)s_{1,1}^{\epsilon_1}\cdots g_i(r)s_{i,1}^{\epsilon_i}\cdots g_n(r)s_{n,1}^{\epsilon_n}g_{n+1}(r)\in O_1$$ and $$\label{Eq:3} h_2=g_1(r)s_{1,2}^{\epsilon_1}\cdots g_i(r)s_{i,2}^{\epsilon_i}\cdots g_n(r)s_{n,2}^{\epsilon_n}g_{n+1}(r)\in O_2$$ are distinct. Let us recall that each $d_i$ is in $D=\{a_m: m\in\omega\}$, so for every $i\leq n$ there exists $m\in\omega$ such that $d_i=a_m$. Since the set $\{d_i: 1\leq i\leq n\}$ contains exactly $k$ pairwise distinct elements, there are pairwise distinct non-negative integers $m_1,\ldots,m_k$ such that $\{d_i: 1\leq i\leq n\}=\{a_{m_1},\ldots,a_{m_k}\}$ and $a_{m_j}$ corresponds to the variable $z_j$ for $j=1,\ldots,k$. Similarly, choose integers $j_{i,\delta}$ for $1\leq i\leq k$ and $\delta=1,2$ such that $\{s_{i,\delta}: 1\leq i\leq n\}=\{s_{j_{1,\delta}},\ldots, s_{j_{k,\delta}}\}$ for each $\delta=1,2$, where both $s_{j_{i,1}}$ and $s_{j_{i,2}}$ correspond to the variable $z_i$, $1\leq i\leq k$. Consider the $k$-tuples $(m_1,\ldots,m_k)$, $(j_{1,1},\ldots,j_{k,1})$, and $(j_{1,2},\ldots,j_{k,2})$. By Lemma \[Le:tec2\], there exists $n_0\in\omega$ such that $\varphi_{m_i}(n_0)=j_{i,1}$ and $\varphi_{m_i}(n_0+1)= j_{i,2}$ for all $i=1,\ldots,k$. Let $L_{n_0}=(U_1,\ldots,U_l,s_{i_1}, \ldots,s_{i_l})$. Take $x\in V_p\setminus (B\cup\bigcup_{i\leq l} U_i)$ and $y\in V_{p+1}\setminus (B\cup\bigcup_{i\leq l} U_i)$, where $B=\operatorname*{\rm supp}(b)$. This choice of $x$ and $y$ is possible since the sets $V_p$ and $V_{p+1}$ are unbounded in ${\mathbb{R}}$. Then our definition of the elements $a_m$ implies that $a_{m_i}(x)=s_{j_{i,1}}$ and $a_{m_i}(y)=s_{j_{i,2}}$ for each $i=1,\ldots,k$. Therefore the elements $w(x)=h_1\in O_1$ and $w(y)=h_2\in O_2$ of $\Pi$ are distinct (see the equalities (\[Eq:2\]) and (\[Eq:3\])). This contradiction completes the proof of the theorem. Our next aim is to present an example of a countably compact separable abelian group with a closed non-separable subgroup. Our argument makes use of an *$\omega$-hereditarily finally dense* subgroup of $\Z(2)^{\omega_1}$ constructed in [@HJ] by A. Hajnal and I. Juhász under the assumption of the Continuum Hypothesis. Hence our example here also requires $CH$. The following lemma is almost evident. \[Le:pro\] Let $K$ and $L$ be subgroups of a topological abelian group $G$. If $K$ is countably compact and $L$ is $\omega$-bounded, then $K+L$ is a countably compact subgroup of $G$. It is known that the product of a countably compact space and an $\omega$-bounded space is countably compact (this follows from [@Vau Theorem 3.3]). Since $K+L$ is a continuous image of the countably compact space $K\times L$, the required conclusion is immediate. \[Pr:CoCo\] Under $CH$, there exists a separable countably compact topological abelian group $G$ which contains a closed non-separable subgroup. Let $P=\Z(2)^{\omega_1}$ and $\Pi=P^{\omega_1}$, where both groups carry the usual Tychonoff product topology; so $P$ and $\Pi$ are compact topological abelian groups. Since $\Pi=(\Z(2)^{\omega_1})^{\omega_1} \cong \Z(2)^{\omega_1\times\omega_1}\cong \Z(2)^{\omega_1}$, it follows from [@HJ Theorem 2.2] that under $CH$, $\Pi$ contains a dense countably compact *$\omega$-HFD* subgroup $H$ of the cardinality $\cont$. The latter means that for every infinite subset $S$ of $H$, there exists a countable set $C\subset\omega_1$ such that $\pi_{\omega_1\setminus C}(S)$ is dense in $P^{\omega_1\setminus C}$, where $\pi_J$ denotes the projection of $P^{\omega_1}$ onto $P^J$ for each non-empty set $J\subset\omega_1$. \[More precisely, the fact that $H$ is $\omega$-HFD appears on page 202 in the proof of Theorem 2.2 in [@HJ].\] Further, according to [@HJ Theorem 2.2], the group $H$ is hereditarily separable, i.e. every subspace of $H$ is separable. In particular, $H$ is separable. Denote by $\Delta$ the diagonal subgroup of $\Pi$, i.e. let $$\Delta=\{x\in\Pi: \pi_\alpha(x)=\pi_\beta(x) \mbox{ for all } \alpha,\beta\in\omega_1\},$$ where $\pi_\alpha\colon\Pi\to P_{(\alpha)}$ is the projection of $\Pi$ to the $\alpha$th factor. Then $\Delta$ is a closed subgroup of $\Pi$ topologically isomorphic to $P$—the projection $\pi_0$ of $\Delta$ to $P_{(0)}$ is a topological isomorphism. Let $\Sigma_P$ be the $\Sigma$-product of $\omega_1$ copies of the group $\Z(2)$ which is identified with the corresponding subgroup of $P=\Z(2)^{\omega_1}$. Then $\Sigma_P$ is a proper, dense, countably compact subgroup of $P$. In fact, $\Sigma_P$ is *$\omega$-bounded*, i.e. the closure in $\Sigma_P$ of every countable subset of $\Sigma_P$ is compact [@AT Corollary 1.6.34]. Further, since $\Sigma_P$ is not compact, it cannot be separable. Let $G=H+\Sigma_0$, where $$\Sigma_0=\{x\in\Delta: \pi_0(x)\in\Sigma_P\}$$ is the copy of the group $\Sigma_P$. Again, the groups $\Sigma_0$ and $\Sigma_P$ are topologically isomorphic. Let us note that by Lemma \[Le:pro\], $G$ is a countably compact subgroup of $\Pi$. It is clear that $G$ is separable since it contains a dense separable subgroup $H$. We claim that the intersection $K=G\cap\Delta$ satisfies $|K:\Sigma_0|<\omega$. It is clear that $\Sigma_0\subset K$, so it suffices to verify that $|H\cap\Delta|<\omega$. Indeed, since the projection of $\Delta$ to an arbitrary sub-product $P^J$, with $|J|\geq\omega$, is nowhere dense in $P^J$, we see that $H\cap \Delta$ is not finally dense in $\Pi$. Hence $H\cap\Delta$ is finite. We have thus proved that $|K:\Sigma_0|<\omega$. It is clear that $K$ is a closed subgroup of $G$. It remains to show that the group $K$ is not separable. Since $|K:\Sigma_0|<\omega$ and $|\Delta:\Sigma_0|>\omega$, we conclude that $K$ is a proper dense subgroup of $\Delta$. Further, there exists a finite subset $F$ of $K$ such that $K=\Sigma_0+F$. Since $\Sigma_0$ is $\omega$-bounded, so is $K$. Thus, if $K$ were separable it would be compact, contradicting the fact that $K$ is a proper dense subgroup of $\Delta$. Since we have used $CH$ in Proposition \[Pr:CoCo\], it is natural to ask whether a similar construction is possible in $ZFC$ alone: \[Example 3\] Does there exist in $ZFC$ a countably compact separable group $X$ which contains a non-separable closed subgroup? Prodiscrete groups {#Sec:Pd} ================== Let us recall that a topological group which has a local base at the identity element consisting of open subgroups is called *protodiscrete.* A complete protodiscrete group is said to be *prodiscrete.* Protodiscrete topological groups are exactly the totally disconnected pro-Lie groups [@PROBOOK Proposition 3.30]. We show in Corollary \[Cor:subg\] below that closed subgroups of separable prodiscrete abelian groups can fail to be separable. First we prove a general result on embeddings into separable topological groups. \[Pro:12\] A (protodiscrete abelian) topological group $H$ is topologically isomorphic to a subgroup of a separable (prodiscrete abelian) topological group if and only if $H$ is $\omega$-narrow and satisfies $w(H)\leq\cont$. Assume that a topological group $H$ is a subgroup of a separable topological group $G$. Then $G$ is $\omega$-narrow [@AT Corollary 3.4.8] and satisfies $w(G)\leq\cont$. Since subgroups of $\omega$-narrow topological groups are $\omega$-narrow [@AT Theorem 3.4.4], we see that $H$ is also $\omega$-narrow and satisfies $w(H)\leq\cont$. Conversely, assume that an $\omega$-narrow group $H$ satisfies $w(H)\leq\cont$. It follows from Theorem \[Th:Gur\] (see also [@AT Theorem 3.4.23]) that $H$ is topologically isomorphic to a subgroup of a topological product $\Pi=\prod_{i\in I} G_i$, where the index set $I$ has the cardinality at most $\cont$ and each factor $G_i$ is a second countable topological group. The group $\Pi$ is separable by the Hewitt–Marczewski–Pondiczery theorem. Further, if the group $H$ is protodiscrete, then all factors $G_i$ can be chosen countable and discrete. Indeed, let $\mathcal{N}(e)$ be a local base at the identity element $e$ of $H$ consisting of open subgroups and satisfying $|\mathcal{N}(e)|\leq\cont$. For every $N\in\mathcal{N}(e)$, denote by $\pi_N$ the canonical homomorphism of $H$ onto the discrete quotient group $H/N$. Then the diagonal product of the family $\{\pi_N: N\in\mathcal{N}(e)\}$ is a topological isomorphism of $H$ onto a subgroup of the product group $P=\prod_{N\in\mathcal{N}(e)} H/N$. Since the group $H$ is $\omega$-narrow, each quotient group $H/N$ is countable. Thus $H$ is a topological subgroup of $P$, a product of countable discrete groups. As $|\mathcal{N}(e)|\leq\cont$, the group $P$ is separable. Evidently, the group $P$ is prodiscrete. This completes our argument. \[Cor:subg\] Closed subgroups of separable prodiscrete abelian groups need not be separable. Let $D$ be a discrete space of the cardinality $\aleph_1$. Denote by $L=D\cup\{x_0\}$ the space which contains $D$ as a dense open subspace and in which the sets of the form $L\setminus C$, where $C$ is an arbitrary countable subset of $D$, constitute a local base at $x_0$ in $L$. The space $L$ is known as a *one-point Lindelöfication* of $D$. Denote by $H$ the free abelian topological group over $L$. Let us note that $L$ is a Lindelöf *$P$-space*, i.e. every $G_\delta$-set in $L$ is open. Since all finite powers of $L$ are Lindelöf, $H$ is Lindelöf as well. According to [@AT Proposition 7.4.7], $H$ is also a $P$-space. Hence $H$ is protodiscrete by [@AT Lemma 4.4.1]. Notice that $H$ cannot be separable as a non-discrete Hausdorff $P$-space. Every Lindelöf $P$-group is complete (see [@Tk04 Proposition 2.3]), so the group $H$ is prodiscrete. Clearly, the Lindelöf topological group $H$ is $\omega$-narrow. It is not difficult to verify that the topological character of $H$ (the minimum cardinality of a local base at the identity of $H$) equals $\aleph_1$—this follows, for example, from the proof of [@BGHT Lemma 5.1]. Hence $w(H)=\chi(H)=\aleph_1\leq\cont$ (see [@AT Lemma 5.1.5]). We now apply Proposition \[Pro:12\] to conclude that $H$ is topologically isomorphic to a subgroup of a separable prodiscrete abelian group $G$. Since $H$ is complete, it is closed in $G$. Thus $G$ contains a closed non-separable subgroup. Embeddings into topological groups vs embeddings into regular spaces {#Sec:Emb} ==================================================================== It is natural to compare the restrictions on a given topological group $G$ imposed by the existence of either a topological embedding of $G$ into a separable regular space or a topological isomorphism of $G$ onto a subgroup of a separable topological group. First we note that the first of the two classes of topological groups is strictly wider than the second one. Indeed, consider an arbitrary discrete group $G$ satisfying $\omega<|G|\leq\cont$. Then $G$ embeds as a *closed subspace* into the separable space $\mathbb{N}^\cont$ [@Eng], where $\mathbb{N}$ is the set of non-negative integers endowed with the discrete topology. However, $G$ does not admit a topological isomorphism onto a subgroup of a separable topological group. Indeed, every subgroup of a separable topological group is $\omega$-narrow by Proposition \[Pro:12\]. Since the discrete group $G$ is uncountable, it fails to be $\omega$-narrow. The above observation makes it natural to restrict our attention to $\omega$-narrow topological groups when considering embeddings into separable topological groups. It turns out that in the class of $\omega$-narrow topological groups, the difference between the two types of embeddings disappears, even if we require an embedding to be closed. The next result complements Proposition \[Pro:12\]. The advantage of Theorem \[Th:Emb\] compared to Theorem \[Th:nA\] is that in the former one, we manage to identify a wide class of topological groups with closed subgroups of separable path-connected, locally path-connected topological groups. It is not surprising therefore that the Hartman–Mycielski construction [@HM] comes into play. \[Th:Emb\] The following are equivalent for an arbitrary $\omega$-narrow topological group $G$: 1. $G$ is homeomorphic to a subspace of a separable regular space; 2. $G$ is topologically isomorphic to a subgroup of a separable topological group; 3. $G$ is topologically isomorphic to a closed subgroup of a separable path-connected, locally path-connected topological group. Since Hausdorff topological groups are regular, it is clear that (c) implies (b) and (b) implies (a). Hence it suffices to show that (a) implies (c). Assume that an $\omega$-narrow topological group $G$ is homeomorphic to a subspace of separable regular space $X$. Then $w(X)\leq\cont$ by Theorem \[Th:1\] and hence $w(G)\leq w(X)\leq\cont$. Applying Proposition \[Pro:12\], we find a separable topological group $H$ containing $G$ as a topological subgroup. Clearly $G$ can fail to be closed in $H$, so our next step is to construct another separable topological group containing $G$ as a closed subgroup. To this end we will use the path-connected, locally path-connected group $H^\bullet$ corresponding to $H$ and consisting of *step functions* from the semi-open interval $J=[0,1)$ to the group $H$ (for a description of $H^\bullet$, see Hartman and Mycielski [@HM] or [@AT Construction 3.8.1]). The group $H$ is canonically isomorphic to a closed subgroup of $H^\bullet$, the corresponding monomorphism $i\colon H\to H^\bullet$ assigns to each element $h\in H$ the constant function $i(h)=h^\bullet\in H^\bullet$ defined by $h^\bullet(x)=h$ for all $x\in J$. Let $e$ be the identity of $H$. Denote by $E$ the set of all step functions $f$ from $J$ to $H$ satisfying the following condition: 1. there exists $b\in [0,1)$ such that $f(x)=e$ for each $x$ with $b\leq x<1$. It is clear that $E$ is a subgroup of $H^\bullet$. Let $D$ be a countable dense subgroup of $H$. Denote by $E'$ the subgroup of $E$ consisting of all $f\in E$ satisfying the following condition: 1. there exist rational numbers $0=b_0<b_1<\cdots< b_{m-1}<b_m=1$ such that $f$ is constant on each subinterval $[b_k,b_{k+1})$ and $f(b_k)=g_k\in D$ for $k=0,1,\ldots,m-1$. Notice that $E'$ is countable. The argument given in the proof of [@AT Theorem 3.8.8, item e)] shows that $E'$ is dense in $H^\bullet$. Denote by $G_0$ the subgroup of $H^\bullet$ generated by the set $i(G)\cup E$. Since $i\colon H\to H^\bullet$ is a topological monomorphism, the group $G$ is topologically isomorphic to the subgroup $i(G)$ of $H^\bullet$. It also follows from $E'\subset E\subset G_0$ that the group $G_0$ is separable. Let us verify that $i(G)$ is closed in $G_0$. First we note that $i(H)$ is closed in $H^\bullet$ according to [@AT Theorem 3.8.3]. Hence the required conclusion about $i(G)$ will follow if we show that $G_0\cap i(H)=i(G)$. Assume that $f\in G_0\cap i(H)$. Then $f$ is a constant function on $J$ with a single value $h_0\in H$. We have to show that $h_0\in G$, i.e. $f\in i(G)$. As $f\in G_0$, we can write $f$ in the form $$f=i(g_1)^{m_1}t_1^{n_1}\cdots i(g_k)^{m_k}t_k^{n_k}i(g_{k+1})^{m_{k+1}},$$ where $g_1,\ldots,g_k,g_{k+1}\in G$, $t_1,\ldots,t_k\in E$, and $m_i,n_i\in\mathbb{Z}$. Item (i) of our definition of the group $E$ implies that there exists $b<1$ such that $t_i(b)=e$ for each $i=1,\ldots,k$. Hence $h_0=f(b)=g_1^{m_1}\cdots g_k^{m_k} g_{k+1}^{m_{k+1}}\in G$. Since $f$ is a constant function, we see that $f\in i(G)$. This implies the inclusion $G_0\cap i(H)\subset i(G)$. The inverse inclusion is evident. Therefore, $G\cong i(G)$ is closed in $G_0$. Now we have to check that the group $G_0$ is path-connected and locally path-connected. It is worth mentioning that $G_0$ is a proper dense subgroup of $H^\bullet$, but not *every* dense subgroup of $H^\bullet$ inherits these properties from $H^\bullet$. We start with the path-connectedness of $G_0$. Since $G_0$ is generated by the set $i(G)\cup E$, it suffices to verify that for every element $f\in i(G)\cup E$, there exist a path in $G_0$ connecting the identity $e^\bullet$ of $H^\bullet$ with $f$. Indeed, every element $f\in G_0$ is a product of finitely many elements $f_1,\ldots,f_n$ of $i(G)\cup E$ and, multiplying the paths connecting $f_1,\ldots,f_n$ with $e^\bullet$, we obtain a path in $G_0$ connecting $e^\bullet$ and $f$. So take an arbitrary element $f\in i(G)\cup E$. Case 1. $f\in i(G)$. Then $f=g^\bullet$ for some $g\in G$. For every $r\in [0,1]$, let $f_r$ be a step function from $J$ to $H$ defined by $f_r(x)=g$ if $x<r$ and $f_r(x)=e$ if $r\leq x<1$. It is clear that $f_r\in H^\bullet$ for each $r\in [0,1]$ and that the mapping $\varphi\colon [0,1]\to H^\bullet$, $\varphi(r)=f_r$, is continuous. Since $\varphi(0)=e^\bullet$, $\varphi(1)=f\in i(G)$, and $f_r\in E\subset G_0$ for each $r\in [0,1)$, this proves that $\varphi$ is a path in $G_0$ connecting $e^\bullet$ and $f$. Case 2. $f\in E$. Choose a partition $0=b_0<b_1<\cdots<b_m=1$ of $J$ such that $f$ is constant on each interval $[b_i,b_{i+1})$, where $0\leq i<m$, and $f(b_{m-1})=e$. Let us define a path $\varphi\colon [0,1]\to G_0$ connecting $e^\bullet$ with $f$ as follows. First we put $l_k=b_{k+1}-b_k$ for $k=0,\ldots,m-1$. For every $r\in [0,1]$ and every $x\in J$, let $$f_r(x)=\begin{cases} f(x),&\text{if $b_k\leq x<b_k+r\cdot l_k$ for some $k$ with $0\leq k<m$};\\ \,\,\,\,e,&\text{if $b_k+r\cdot l_k\leq x<b_{k+1}$ for some $k$ with $0\leq k<m$.} \end{cases}$$ It is easy to verify that $f_0=e^\bullet$, $f_1=f$, and $f_r(x)=e$ if $b_{m-1}\leq x<1$. Hence $f_r\in E\subset G_0$ for each $r\in [0,1]$. Again, the mapping $\varphi\colon [0,1]\to H^\bullet$ defined by $\varphi(r)=f_r$ for each $r\in [0,1]$ is continuous, so $\varphi$ is a path in $G_0$ connecting $e^\bullet$ and $f$. Summing up, the group $G_0$ is path-connected. Finally, we check that $G_0$ is locally path-connected. Every neighborhood of $e^\bullet$ in $H^\bullet$ contains an open neighborhood of the form $$O(U,\varepsilon)=\{f\in H^\bullet: \mu(\{x\in J: f(x)\notin U\})<\varepsilon\},$$ where $U$ is an open neighborhood of the identity $e$ in $H$, $\varepsilon>0$, and $\mu$ is the Lebesgue measure on $J$. Therefore, by the homogeneity of $G_0$, it suffices to verify that the intersections $G_0\cap O(U,\varepsilon)$ are path-connected. Take an arbitrary element $f\in G_0\cap O(U,\varepsilon)$, where $U$ is an open neighborhood of $e$ in $H$ and $\varepsilon>0$. Then $f=i(g_1)^{m_1}t_1^{n_1}\cdots i(g_k)^{m_k}t_k^{n_k}i(g_{k+1}) ^{m_{k+1}}$, where $g_1,\ldots,g_k,g_{k+1}\in G$, $t_1,\ldots,t_k\in E$, and $m_i,n_i\in\mathbb{Z}$. Our aim is to define a path $\Phi\colon [0,1]\to G_0\cap O(U,\varepsilon)$ connecting $e^\bullet$ with $f$. We cannot apply directly the formula from the above Case 1 since otherwise we lose control over the measure of the set $\{x\in J: f_r(x)\notin U\}$ for some $r\in (0,1)$, where $f_r$ is assumed to be $\Phi(r)$. Instead, we adjust the speed of changes of the elements $i(g_1),\ldots, i(g_k),i(g_{k+1})$ and $t_1,\ldots,t_k$ on the appropriately chosen subintervals of $J$. First we choose a partition $0=b_0<b_1<\cdots<b_{m-1}<b_m=1$ of $J$ such that $t_i$ is constant on $[b_j,b_{j+1})$ for all integers $i\leq k$ and $j<m$. Let also $l_j=b_{j+1}-b_j$, where $j=0,\ldots,m-1$. For every $i=1,\ldots,k,k+1$ we define a path $\varphi_i\colon [0,1]\to H^\bullet$ by $$\varphi_i(r,x)=\begin{cases} g_i,&\text{if $b_j\leq x<b_j+r\cdot l_k$ for some $j$ with $0\leq j<m$};\\ \,e,&\text{if $b_j+r\cdot l_j\leq x<b_{j+1}$ for some $j$ with $0\leq j<m$.} \end{cases}$$ Then $\varphi_i(0,x)=e$, $\varphi_i(1,x)=g_i$ for each $x\in J$ and $\varphi_i(r,\cdot)\in G_0$ for each $r\in [0,1]$. The path $\varphi_i$ is continuous and connects $e^\bullet$ with $g_i^\bullet$ in $G_0$. Similarly, we define a path $\psi_i\colon [0,1]\to H^\bullet$ for each $i=1,\ldots,k$ by $$\psi_i(r,x)=\begin{cases} t_i(x),&\text{if $b_j\leq x<b_j+r\cdot l_k$ for some $j$ with $0\leq j<m$};\\ \,\,\,\,e,&\text{if $b_j+r\cdot l_j\leq x<b_{j+1}$ for some $j$ with $0\leq j<m$.} \end{cases}$$ It is clear that $\psi_i(0,x)=e$, $\psi_i(1,x)=t_i(x)$ for each $x\in J$ and $\psi_i(r,\cdot)\in G_0$ for each $r\in [0,1]$. The path $\psi_i$ is continuous and connects $e^\bullet$ with $t_i$ in $G_0$. Finally we define a path $\Phi$ in $G_0$ connecting $e^\bullet$ with $f$ by letting $$\Phi(r,x)=\varphi_1(r,x)^{m_1}\cdot\psi_1(r,x)^{n_1}\cdots\varphi_k(r,x)^{m_k}\cdot\psi_k(r,x)^{n_k}\cdot\varphi_{k+1}(r,x)^{n_{k+1}},$$ where $r\in [0,1]$ and $x\in J$. The path $\Phi$ is continuous being a product of continuous paths $\varphi_i$ and $\psi_i$. The following Claim describes a basic property of the path $\Phi$: **Claim.** *For all $r\in [0,1]$ and $x\in J$, either $\Phi(r,x)=f(x)$ or $\Phi(r,x)=e$.* Indeed, let $r\in [0,1]$ and $x\in J$ be arbitrary. Choose an integer $j<m$ such that $b_j\leq x<b_{j+1}$. If $b_j\leq x<b_j+rl_j$, then $\varphi_i(r,x)=g_i$ and $\psi_i(r,x)=t_i(x)$ for all $i$, whence it follows that $\Phi(r,x)=f(x)$. If $b_j+r\cdot l_j\leq x<b_{j+1}$, then $\varphi_i(r,x)=e$ and $\psi_i(r,x)=e$ for all $i$, so $\Phi(r,x)=e$. This proves our Claim. Applying Claim we see that $$\{x\in J: \Phi(r,x)\notin U\}\subset \{x\in J: f(x)\notin U\},$$ for every $r\in [0,1]$. 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--- abstract: | The circumference $c(G)$ of a graph $G$ is the length of a longest cycle. By exploiting our recent results on resistance of snarks, we construct infinite classes of cyclically $4$-, $5$- and $6$-edge-connected cubic graphs with circumference ratio $c(G)/|V(G)|$ bounded from above by $0.876$, $0.960$ and $0.990$, respectively. In contrast, the dominating cycle conjecture implies that the circumference ratio of a cyclically $4$-edge-connected cubic graph is at least $0.75$. In addition, we construct snarks with large girth and large circumference deficit, solving Problem 1 proposed in \[J. Hägglund and K. Markström, On stable cycles and cycle double covers of graphs with large circumference, Disc. Math. 312 (2012), 2540–2544\]. author: - | Edita Máčajová${}^1$, Ján Mazák${}^2$\ \ [{macajova, mazak}@dcs.fmph.uniba.sk]{}\ ${}^1$ Univerzita Komenského, Mlynská dolina, 842 48 Bratislava, Slovakia\ ${}^2$ Trnavská univerzita, Priemyselná 4, 918 43 Trnava, Slovakia title: Cubic graphs with large circumference deficit --- [**Keywords:**]{} circumference, cubic graph, snark, girth [**Classification:**]{} 05C15, 05C38 Introduction ============ A cycle is one of the most basic structures in a graph, so it comes as no surprise that cycles have been analysed from the very beginnings of graph theory. This article focuses on longest cycles in cubic graphs. The [*circumference*]{} $c(G)$ of a graph $G$ is the length of a longest cycle. The [*circumference ratio*]{} is the ratio of circumference to order. The [*circumference deficit*]{} is the difference between order and circumference. A lot of attention was given to Hamiltonian graphs, that is, graphs with zero circumference deficit. Compared to the vast tomes written on hamiltonicity, non-Hamiltonian graphs appear rather neglected despite there is plenty of investigation to be done. The problem of determining the circumference of a given graph is NP-hard and even approximation is a very tough problem [@NP], so no simple characterisations are expected. It transpired in many areas that the most interesting cubic graphs are those with chromatic index four. Such graphs are called [*snarks*]{}; we will additionally require snarks to have girth at least five and cyclic edge-connectivity at least four. We will encounter them in Section \[snarks\] where we prove the existence of a snark of girth at least $g$ and circumference deficit at least $g$ for every integer $g$. Section \[ratio\] is devoted to upper bounds on circumference ratio; we provide linear bounds for certain classes of cyclically $4$-, $5$-, and $6$-edge-connected cubic graphs. These bounds serve as the best presently known upper bounds on general lower bounds on circumference ratio. Each subcubic graph can be transformed into a $3$-edge-colourable graph by removing sufficiently many edges. The least number of edges that need to be removed is the [*resistance*]{} of the graph. There are snarks with arbitrarily large resistance (see e.g. [@steffen; @oddness]). Most of our constructions are based on building blocks with large resistance; the usefulness of such blocks is demonstrated in Lemma \[lemma1\] which plays the key role in our proofs. Circumference ratio of cubic graphs {#ratio} =================================== Circumference ratio of cubic graphs strongly depends on connectivity. Since each vertex of a cubic graph is separated by three edges from the rest of the graph, the classical notions of vertex-connectivity and edge-connectivity are of limited use. A refined measure of connectivity, much more appropriate for our purpose, is provided by cyclic edge-connectivity. A graph is [*cyclically $k$-edge-connected*]{} if at least $k$ edges must be removed to disconnect it into components among which there are at least two containing a cycle. For cubic graphs, the notion of cyclic $k$-edge-connectivity coincides with $k$-vertex-connectivity and $k$-edge-connectivity for $k\in\{1,2,3\}$ [@atoms]. If we allow bridges in our graphs, there are infinitely many trivial cubic graphs with circumference $5$. What is more interesting, Bondy and Entringer [@be3] proved that every $2$-edge-connected cubic graph $G$ contains a cycle of length at least $4\log |V(G)|-4\log \log |V(G)|-20$. This bound is essentially best possible, as shown by Lang and Walther [@lw10]. Bondy and Simonovits [@bs5] conjectured the existence of a constant $c$ such that every $3$-connected cubic graph $G$ has circumference at least $|V(G)|^c$ and showed that $c \le \log_98\approx 0.946$. The conjecture was verified by Jackson [@jackson] for $c = \log (1+\sqrt5)-1\approx 0.694$; the constant $c$ has recently been improved to $0.753$ [@bbmy]. Bondy also conjectured the following. There exists a constant $c > 0$ such that every cyclically $4$-edge-connected cubic graph $G$ has circumference at least $c\,|V(G)|$. \[conj:bondy\] The dominating cycle conjecture [@fleischner] implies that $c\ge 0.75$ since a dominating cycle in a cubic graph $G$ has length at least $0.75|V(G)|$. In addition, Thomassen [@thomassen] conjectured that there exists an integer $k$ such that every cyclically $k$-edge-connected cubic graph is Hamiltonian. This would mean $c = 1$ for sufficiently connected cubic graphs. We summarize the currently known results together with our contributions in Table \[tabulka\]. The column LB displays a lower bound which holds for all graphs; the column UB shows an upper bound on circumference for a certain infinite class of graphs with required cyclic edge-connectivity. All the bounds are asymptotic; each of them is expressed as a function of order $n$. connectivity LB conjectured LB UB -------------- ------------- ---------------- ------------------ 2 $\log n$ $\log n$ 3 $n^{0.753}$ $n^{0.946}$ 4 $n^{0.753}$ $0.75n$ $0.875n$ $\star$ 5 $n^{0.753}$ $0.75n$ $0.960n$ $\star$ 6 $n^{0.753}$ $0.75n$ $0.990n$ $\star$ 7+ $n^{0.753}$ $0.75n$    $n$ $n$ : Summary of results on circumference (our contribution are marked by $\star$).[]{data-label="tabulka"} The crucial observation used in our construction of graphs with large circumference deficit is captured in the following lemma. \[lemma1\] Let $H$ be a subgraph of a bridgeless cubic graph $G$. If $H$ has resistance $k$, then any cycle of $G$ not contained in $H$ misses at least $k$ vertices of $H$. If a cycle $C$ of $G$ is not contained in $H$, then its intersection with $H$ is a union of vertex-disjoint paths. Take each of those paths in turn and alternately colour the edges along the path by colours $1$ and $2$. Colour all the remaining edges by $3$. There are two types of vertices in $H$: those that have their incident edges coloured by $1$, $2$, $3$ and those with all incident edges coloured by $3$. We call the vertices of the latter type [*bad*]{}. By removing all the bad vertices we obtain a $3$-edge-colourable graph from $H$. However, $H$ has resistance $k$, and thus there are at least $k$ bad vertices. Obviously, no bad vertex belongs to $C$, hence $C$ misses at least $k$ vertices of $H$. In order to construct infinite classes of graphs with circumference ratio promised in Table \[tabulka\], we employ cubic construction blocks described in [@oddness]. (We do not include construction details or proofs of their properties in this article; an interested reader can find them in [@oddness].) ![The building block $N_2$ with $26$ vertices and resistance $2$.[]{data-label="fig:N2"}](bloky-1) The cubic graph $N_2$ has order $26$, resistance $2$, and two pairs of dangling edges (see Figure \[fig:N2\] and [@oddness Section 7]). We create a graph $G$ by arranging $m\ge 2$ copies of $N_2$ along a circle and for each copy $K$, we connect one pair of dangling edges of $K$ to a pair of dangling edges of the following copy and the other pair of dangling edges to a pair in the previous copy. (The exact way of how we do it does not matter because we only need to preserve cyclic $4$-edge-connectivity.) According to Lemma \[lemma1\], a cycle of $G$ either belongs to one copy of $N_2$ or misses at least $2$ vertices in each of the $m$ copies of $N_2$, hence $G$ has circumference deficit at least $2m$ and circumference ratio at most $24m/26m = 12/13\approx 0.92$. A different idea used in Theorem \[thm7/8\] leads to a better upper bound for cyclically $4$-edge-connected graphs. The construction described in the previous paragraph can be repeated with the cyclically $5$-edge-connected building block $Z$ with order $25$ and resistance $1$ (see [@oddness Section 8 and Figure 5] for a description of $Z$; this block was also used by Steffen under the name $T$ [@steffen Theorem 2.3]). The graph $Z$ has seven dangling edges naturally split into two triples and one single dangling edge. We repeat the circular construction with $m$ copies of $Z$ instead of $N_2$; the role of the pairs of dangling edges is now played by the triples. The single dangling edges are joined to a cycle of length $m$ (one dangling edge to each vertex of the cycle). The resulting graph has circumference ratio at most $24/25 = 0.96$ and is cyclically $5$-edge-connected. A similar construction can also be used to construct cyclically $6$-edge-connected graphs; however, the details are more complicated. Section 9 of [@oddness] describes a cyclically $6$-edge-connected graph $M_r$ of order $99r$ with resistance at least $r$ for each even positive integer $r$, but there are no dangling edges in this graph, thus we cannot use it directly: we first have to cut a few suitable edges to obtain a block which would allow a construction of cyclically $6$-edge-connected graphs. The graph $M_r$ is obtained by symmetrically applying superposition to a graph $L_r$ composed of $r$ circularly arranged isomorphic copies of the block $P_3$ (that is, the Petersen graph with one vertex removed). Therefore, $M_r$ also contains $r$ isomorphic blocks $A_1, A_2, \dots, A_r$ arranged along a circle. Each two consecutive blocks of $M_r$ are joined by three edges. We cut all the edges between $A_1$ and $A_2$ to form a cubic graph $M_r'$ with two triples of dangling edges. The graph $M_r'$ has order $99r$ and resistance at least $r-3$. (According to the definition of resistance, the removal of an edge can decrease resistance by at most $1$. The resistance of $M_r'$ is actually $r$, but that would require a detailed proof; we will use the obvious lower bound of $r-3$ here because it is sufficient for our purpose.) Consequently, we can use $M_r'$ in place of $N_2$ in the above-described construction (with triples of dangling edges instead of pairs). The resulting cyclically $6$-edge-connected cubic graph $G$ has order $m\cdot 99r$ and circumference deficit at least $m(r-3)$, thus its circumference ratio is at most $1-(r-3)/99r$. By taking a sufficiently large $r$ we can make this ratio to be arbitrarily close to $98/99\approx 0.990$. For each integer $m$, there exists a cyclically $4$-edge-connected cubic graph with order $8m$ and circumference $7m+2$. \[thm7/8\] Let $u$ and $v$ be two adjacent vertices of the Petersen graph $P$. We remove the path $uv$, but keep the dangling edges incident to exactly one of its endvertices; the dangling edges incident to $u$ are [*input edges*]{} and the dangling edges incident to $v$ are [*output edges*]{}. We say that a path [*passes through*]{} $B$ if it starts with a vertex incident to an input edge and ends in a vertex incident to an output edge. We say that a cycle passes through $B$ if a portion of this cycle (a path) passes through $B$. The resulting graph $B$ has two properties interesting to us. First, if a path passes through $B$, it cannot pass through all the vertices of $B$: otherwise we would be able to extend this path by $u$ and $v$ to a Hamiltonian cycle of $P$, but $P$ has no such cycle. Second, if we take two disjoint paths passing through $B$, there is at least one vertex of $B$ missed by both of these paths. Otherwise, we can extend the first path by $u$, extend the other path by $v$, and then concatenate them together by adding two edges to form a Hamiltonian cycle of $P$ which is a contradiction. Let $G$ be the graph obtained from $m$ copies of $B$ arranged along a circle in such a way that the output edges of each copy are identified with the input edges of the following copy (see Figure \[fig:chain\]). The graph $G$ is cyclically $4$-edge-connected and has order $8m$. ![The building block $B$ and a sketch of the graph $G$.[]{data-label="fig:chain"}](obr-1 "fig:") 1.5cm Let $C$ be a cycle in $G$. Note that $C$ passes through each copy of $B$ at most twice. According to the two properties of $B$ proved above, no matter how many times $C$ passes through $B$, at least one vertex of $B$ is missed. The only possibility for $C$ to contain all vertices of $B$ is to enter by an input edge and then leave by the other input edge (of course, it can also both enter and leave by output edges, which is essentially the same situation). Consequently, the cycle $C$ misses at least one vertex in each of at least $m-2$ copies of $B$, and thus the circumference of $G$ is at most $8m-(m-2) = 7m+2$. Since $G$ contains a cycle of length $7m+2$, the derived upper bound on its circumference is tight. We propose the following strengthening of Conjecture \[conj:bondy\]. Every cyclically $4$-edge-connected cubic graph has circumference ratio at least $7/8$. Large girth and large circumference deficit {#snarks} =========================================== This section is motivated by the cycle double cover conjecture (CDCC). Huck [@huck] showed that the smallest possible counterexample to CDCC has girth at least $12$. Brinkmann et al. [@bbb; @hm] proved that if a bridgeless cubic graph $G$ has a cycle of length at least $|V(G)|-10$, then $G$ has a cycle double cover. Put together, smallest counterexamples to CDCC can only be found in the class of snarks with girth at least $12$ and with circumference deficit at least $11$. Since no such snark has been known before, the following problem is very relevant. [**Problem**]{} (Hägglund and Markström [@hm]). For each integer $g$, construct a snark of girth at least $g$ and circumference deficit at least $g$. We solve this problem in Theorem \[thm3\]. The construction used in the proof of Theorem \[thm3\] can be modified to produce snarks with arbitrarily large girth and linear circumference deficit. For every integer $g$ there exists a snark with girth at least $g$ and circumference deficit at least $g$. \[thm3\] We construct the desired snark for every integer $g\ge 5$ which is enough to prove the theorem. Let $H_0$ be a snark of girth at least $g$; the existence of such snarks has been proved by Kochol [@kochol]. Let $v_1$ and $v_2$ be two adjacent vertices of $H_0$ and let $e_i$ and $f_i$, for $i\in\{1,2\}$, be the two edges incident to $v_i$ and not incident to $v_{3-i}$. Let $H_1$ be the cubic graph obtained from $H_0$ by removing $v_1$ and $v_2$ while keeping the dangling edges $e_1$, $e_2$, $f_1$, $f_2$. The well-known parity lemma assures that the edges $e_1$ and $f_1$ have the same colour in every $3$-edge-colouring of $H_1$ (otherwise $H_0$ would be $3$-edge-colourable, but it is a snark). ![The graph $H$.[]{data-label="fig:H_1"}](obr-3) Take two copies of $H_1$ and join them as indicated in Fig. \[fig:H\_1\] (the edge $f_1$ of the first copy is identified with the edge $f_1$ of the second copy and the edges $e_1$ of both copies are attached to an additional vertex $v$). If $H$ was $3$-edge-colourable, then the colour of $e_1$ of the first copy of $H_1$ would be the same as the colour of $f_1$ and, in turn, the same as the colour of $e_1$ of the second copy, leading to a contradiction at $v$. Hence, $H$ is not colourable and has resistance at least $1$. Moreover, there is no cycle of length less than $g$ in $H$, and any path with endvertices incident to dangling edges of $H$ passes through at least $g-1$ vertices of $H$. Let $G$ be a cubic graph obtained from $g$ copies of $H$ arranged along a circuit in such a way that two dangling edges of a copy of $H$ are attached to the previous copy and two of them are attached to the next. The remaining $g$ edges can be joined to a cycle of length $g$ in an arbitrary way preserving maximum degree $3$. The graph $G$ clearly has girth at least $g$, is cyclically $4$-edge-connected and is not $3$-edge-colourable. Any cycle of $G$ not contained in $H$ misses at least one vertex in each copy of $H$ thanks to Lemma \[lemma1\], and thus $G$ has circumference deficit at least $g$. [**Acknowledgements.**]{} This work was supported from the APVV grants APVV-0223-10 and ESF-EC-0009-10 within the EUROCORES Programme EUROGIGA (project GReGAS) of the European Science Foundation. [99]{} M. Bilinski, B. Jackson, J. Ma, X. Yu, Circumference of 3-connected claw-free graphs and large Eulerian subgraphs of 3-edge-connected graphs, J. of Comb. Theory Ser. B 101 (2011), 214–236. J. A. Bondy and R. C. Entringer, Longest cycles in 2-connected graphs with prescribed maximum degree, Canad. J. Math. 32 (1980), 1325–1332. J. A. Bondy and M. Simonovits, Longest cycles in 3-connected cubic graphs, Canad. J. Math. 32 (1980), 987–992. G. Brinkmann, J. Goedgebeur, J. Hägglund, K. Markström, Generation and properties of snarks, J. Comb. Theory Ser. B 103 (4) (2013), 468–488. G. Chen, J. Xu, and X. Yu, Circumference of graphs with bounded degree, SIAM J. Comput. 33 (5) (2004), 1136–1170. H. Fleischner, Some blood, sweat, but no tears in Eulerian graph theory, Congr. Numer. 63 (1988), 8–48. J. Hägglund, K. Markström, On stable cycles and cycle double covers of graphs with large circumference, Disc. Math. 312 (2012) 2540–2544. A. Huck, Reducible configurations for the cycle double cover conjecture, in: Proceedings of the 5th Twente Workshop on Graphs and Combinatorial Optimization (Enschede, 1997), vol. 99 (2000), 71–90. B. Jackson, Longest cycles in 3-connected cubic graphs, J. Comb. Theory Ser. B 41 (1986), 17–26. M. Kochol, Snarks without small cycles, J. Combin. Theory Ser. B 67 (1996), 34–47. R. Lang and H. Walther, Über längste Kreise in regulären Graphen, in “Beitrage zur Graphenteorie, Kolloquium, Manebach 1967”, Teubner, Leipzig (1968), 91–98. R. Lukoťka, E. Máčajová, J. Mazák, and M. Škoviera, Small snarks with large oddness, [arXiv:1212.3641]{}. R. Nedela and M. Škoviera, Atoms of cyclic connectivity in cubic graphs, Math. Slovaca 45 (1995), 481–499. E. Steffen, Measurements of edge-uncolorability, Disc. Math. 280 (2004), 191–214. C. Thomassen, personal communication, a conference in Vienna (1991).

--- address: - ', , , ' - ', ' - ', , ' - ', , ' author: - 'Zeyu LIAO\*' - Ken HAYAMI - Keiichi MORIKUNI - 'Jun-Feng YIN' bibliography: - 'wileyNJD-VANCOUVER.bib' nocite: '[@*]' title: A Stabilized GMRES Method for Solving Underdetermined Least Squares Problems --- Introduction {#sec1} ============ Consider solving the inconsistent underdetermined least squares problem $$\label{eq3} \min_{x\in \mathbb{R}^n}\|b-Ax\|_2,\qquad A\in \mathbb{R}^{m\times n},\qquad b\in \mathbb{R}^{m},\qquad b\notin \rm{\mathcal{R}}(\it{A}),\qquad m<n,$$ where $A$ is ill-conditioned and may be rank-deficient. Here, $\rm{\mathcal{R}}$$(A)$ denotes the range space of $A$. Such problems may occur in ill-posed problems where $b$ is given by an observation which contains noise. The least squares problem $(\ref{eq3})$ is equivalent to the normal equations $$\label{eq4} A^{\mathsf{T}}Ax=A^{\mathsf{T}}b.$$ The standard direct method for solving the least squares problem $(\ref{eq3})$ is to use the QR decomposition. However, when $A$ is large and sparse, iterative methods become necessary. The CGLS [@HS] and LSQR [@lsqr] are mathmetically equivalent to applying the conjugate gradient (CG) method to $(\ref{eq4})$. The convergence of these methods deteriorates for ill-conditioned problems and they require reorthogonalization [@Hayami10] to improve the convergence. Here, we say $(\ref{eq3})$ is ill-conditioned if the condition number $\kappa_2(A)=\|A\|_2\|A^\dag\|_2\gg 1$, where $A^\dag$ is the pseudoinverse of $A$. The LSMR [@lsmr] applies MINRES [@paige1975] to $(\ref{eq4})$. Hayami et al. [@Hayami10] proposed preconditioning the $m\times n$ rectangular matrix $A$ of the least squares problem by an $n\times m$ rectangular matrix $B$ from the right and the left, and using the generalized minimal residual (GMRES) method[@saad1986] for solving the preconditioned least squares problems (AB-GMRES and BA-GMRES methods, respectively). For ill-conditioned problems, the AB-GMRES and BA-GMRES were shown to be more robust compared to the preconditioned CGNE and CGLS, respectively. Note here that the BA-GMRES works with Krylov subspaces in $n$-dimensional space, whereas the AB-GMRES works with Krylov subspaces in $m$-dimensional space. Since $m<n$ in the underdetermined case, the AB-GMRES works in a smaller dimensional space than the BA-GMRES and should be more computationally efficient compared to the BA-GMRES for each iteration. Moreover, the AB-GMRES has the advantage that the weight of the norm in $(\ref{eq3})$ does not change for arbitrary $B$. Thus, we mainly focus on using the AB-GMRES to solve the underdetermined least squares problem $(\ref{eq3})$. Morikuni [@Morikuni13] showed that the AB-GMRES may fail to converge to a least squares solution in finite-precision arithmetic for inconsistent problems. We will review this phenomenon. The GMRES applied to inconsistent problems was also studied in other papers[@bw; @cr; @REICHEL05; @Morikuni15; @MorikuniRozloznik2018SIMAX]. In this paper, we first analyze the deterioration of convergence of the AB-GMRES. To overcome the deterioration, we use the normal equations of the upper triangular matrix arising in the AB-GMRES to change the inconsistent subproblem to a consistent one. In finite precision arithmetic, forming the normal equations for the subproblem will not square its condition number as would be predicted by theory. In the ill-conditioned case, the tiny singular values are shifted upwards due to rounding errors. In finite precision arithmetic, applying the standard Cholesky decomposition to the normal equations will result in a well-conditioned lower triangular matrix, which will ensure that the forward and backward substitutions work stably, and overcome the problem. Numerical experiments on a series of ill-conditioned Maragal matrices[@florida] show that the proposed method converges to a more accurate approximation than the original AB-GMRES. The method can also be used to solve general inconsistent singular systems. The rest of the paper is organized as follows. In Section 2, we briefly review the AB-GMRES and a related theorem. In Section 3, we demonstrate and analyze the deterioration of the convergence. In Section 4, we propose and present a stabilized GMRES method and explain a regularization effect of the method based on the normal equations for ill-conditioned problems. In Section 5, numerical results for the underdetermined case and the square case are presented. In Section 6, we conclude the paper. All the experiments in this paper were done using MATLAB R2017b in double precision, unless specified otherwise (where we extended the arithmetic precision by using the Multiprecision Computing Toolbox for MATLAB [@mptfm]), and the computer uesd was Alienware 15 CAAAW15404JP with CPU Inter(R) Core(TM) i7-7820HK (2.90GHz). Deterioration of convergence of AB-GMRES for inconsistent problems ================================================================== In this section, we review previous results. First, we introduce the right-preconditioned GMRES (AB-GMRES), which is the basic algorithm in this paper. Then, we show the phenomenon that the convergence of the AB-GMRES deteriorates for inconsistent problems. Finally, we cite a related theorem to analyze the deterioration. AB-GMRES method --------------- AB-GMRES for least squares problems applies GMRES to $\min_{u\in \mathbb{R}^{m}}\|b-ABu\|_2$ with $x=Bu$, where $B\in \mathbb{R}^{n\times m}$. Let $x_0$ be the initial solution (in all our numerical experiments, we set $x_0=0$), and $r_0=b-Ax_0.$ Then, AB-GMRES searches for $u$ in the Krylov subspace $\mathcal{K}_i(AB,r_0)=\rm span\it\{r_0,ABr_0,\dots,(AB)^{i-1}r_0\}$. The algorithm is given in Algorithm \[AL1\][@Hayami10]. Here, $H_{i+1,i}=(h_{pq})\in \mathbb{R}^{(i+1)\times i}$ and $e_1=(1,0,\dots,0)^{\mathsf{T}}\in \mathbb{R}^{i+1}.$ Choose $x_0\in \mathbb{R}^{n}$,$r_0=b-Ax_0$,$v_1=r_0/\|r_0\|_2$ $w_i=ABv_i$ $h_{i,j}=w_i^{{\mathsf{T}}}v_j$, $w_i=w_i-h_{j,i}v_j$ $h_{i+1,i}=\|w_i\|_2$, $v_{i+1}=w_i/h_{i+1,i}$ Compute $y_i\in \mathbb{R}^i$ which minimizes $\|r_i\|_2=\|\|r_0\|_2e_1-H_{i+1,i}y_i\|_2$ $x_i=x_0+B[v_1, v_2, \dots, v_i]y_i$, $r_i=b-Ax_i$ stop \[AL1\] To find $y_i\in \mathbb{R}^i$ that minimizes $\|r_i\|_2=\|\|r_0\|_2e_1 -H_{i+1,i}y_i\|_2$ in Algorithm \[AB-GMRES method\], the standard approach computes the QR decomposition of $H_{i+1,i}$ $$\label{EQ3} H_{i+1,i}=Q_{i+1}R_{i+1,i},\qquad Q_{i+1}\in \mathbb{R}^{{(i+1)}\times{(i+1)}},\qquad R_{i+1,i}=\left( %左括号 \begin{array}{ccc} %该矩阵一共3列，每一列都居中放置 R_i\\ %第一行元素 0^{\mathsf{T}}\\ %第二行元素 \end{array} \right)\in \mathbb{R}^{{(i+1)}\times{i}}, \qquad R_{i}\in \mathbb{R}^{{i}\times{i}},$$ where $Q_{i+1}$ is an orthogonal matrix and $R_i$ is an upper triangular matrix. Then, backward substitution is used to solve a system with the coefficient matrix $R_i$ as follows $$\|r_i\|_2=\min_{y_i\in \mathbb{R}^i}\| Q_{i+1}^{\mathsf{T}}\beta e_1-R_{i+1,i}y_i\|_2,$$ where $$\beta=\|r_0\|_2,\qquad Q_{i+1}^{\mathsf{T}}\beta e_1=\left( %左括号 \begin{array}{ccc} %该矩阵一共3列，每一列都居中放置 t_i\\ %第一行元素 \rho_{i+1}\\ %第二行元素 \end{array} \right),\qquad t_i\in \mathbb{R}^i,\qquad \rho_{i+1}\in \mathbb{R},\qquad y_i=R_i^{-1}t_i,$$ $$x_i=V_iy_i=V_i(R_i^{-1}t_i),\qquad V_i=[v_1, v_2, \dots, v_i]\in \mathbb{R}^{n\times i},\qquad V_i^{\mathsf{T}}\it{V_i}=I,$$ where $\rm{I}$ is the identity matrix. Note the following theorem. \[th1\] (Corollary 3.8 of Hayami et al.[@Hayami10]) If $\rm{\mathcal{R}}$$(A)=\rm{\mathcal{R}}$$(B^{\mathsf{T}})$ and $\rm{\mathcal{R}}$$(A^{\mathsf{T}})=\rm{\mathcal{R}}$$(B)$, then AB-GMRES determines a least squares solution of $\min_{x\in \mathbb{R}^n}\|b-Ax\|_2$ for all $b\in \mathbb{R}^m$ and for all $x_0\in \mathbb{R}^n$ without breakdown. Here, breakdown means $h_{i+1, i}=0$ in Algorithm \[AB-GMRES method\]. See Appendix B of [@Morikuni15]. In fact, if $\rm{\mathcal{R}}$$(A^{\mathsf{T}})=\rm{\mathcal{R}}$$(B)$ and $x_0\in \rm{\mathcal{R}}$$(A^{\mathsf{T}})$, the solution is a minimum-norm solution since $x=Bu\in \rm{\mathcal{R}}$$(A^{\mathsf{T}})=\rm{\mathcal{N}}(A)^{\bot}$, where $\rm{\mathcal{N}}$$(A)$ is the null space of $A$. From now on, we use AB-GMRES to solve $(\ref{eq3})$ with $B=A^{\mathsf{T}}$ and $x=Bu$, which means using the Krylov subspace $\mathcal{K}_i(AA^{\mathsf{T}},r_0)=\langle r_0,AA^{\mathsf{T}}r_0,\dots,(AA^{\mathsf{T}})^{i-1}r_0\rangle$ to approximate $u$. Hence, Theorem \[th1\] guarantees the convergence in exact arithmetic even in the inconsistent case. However, in finite precision arithmetic, AB-GMRES may fail to converge to a least squares solution for inconsistent problems, as shown later. AB-GMRES for inconsistent problems ---------------------------------- In this section, we perform experiments to show that the convergence of AB-GMRES deteriorates for inconsistent problems. Experiments were done on the transpose of the matrix Maragal$\_3$ [@florida], denoted by Maragal$\_3$T etc. Table \[tb1\] gives the information on the Maragal matrices, including the density of nonzero entries, rank and condition number. Here, the rank and condition number were determined by using the MATLAB functions `spnrank` [@if] and `svd`, respectively. [r|r|r|r|r|r]{} & & & & &\ ------------------------------------------------------------------------ Maragal$\_$3T & 858& 1682 & 1.27 & 613& 1.10$\times 10^{3}$\ Maragal$\_$4T & 1027& 1964 & 1.32 & 801&9.33$\times 10^{6}$\ Maragal$\_$5T & 3296& 4654 & 0.61 & 2147& 1.19$\times 10^{5}$\ Maragal$\_$6T & 10144& 21251 & 0.25 & 8331 &2.91$\times 10^{6}$\ Maragal$\_$7T & 26525& 46845 & 0.10 & 20843&8.91$\times 10^{6}$\ \[tb1\] ----------------------------------------------------------------------------------------------------------------------------------------------------------- ![$\kappa_2$($R_i$) and relative residual norm versus the number of iterations for Maragal$\_$3T.[]{data-label="lllw1"}](lw1new.eps "fig:"){width="12cm"} ----------------------------------------------------------------------------------------------------------------------------------------------------------- Figure \[lllw1\] shows the relative residual norm $\|A^{\mathsf{T}}r_i\|_2/\|A^{\mathsf{T}}b\|_2$ and $\kappa_2$($R_i$) versus the number of iterations for AB-GMRES with $B=A^{\mathsf{T}}$ for Maragal$\_3$T, where $r_i=b-Ax_i$, and the vector $b$ was generated by the MATLAB function `rand` which returns a vector whose entries are uniformly distributed in the interval $(0,1)$. Here $\kappa_2$($R_i$)=$\kappa_2$($H_{i+1,i})$ holds from $(\ref{EQ3})$. The value of $\kappa_2(R_i$) was computed by the MATLAB function `cond`. The relative residual norm $\|A^{\mathsf{T}}r_i\|_2/\|A^{\mathsf{T}}b\|_2$ decreased to $10^{-8}$ until the 525th iteration, and then increased sharply. The value of `cond(R_i)` started to increase rapidly around iterations 450–550. This observation shows that $R_i$ becomes ill-conditioned before convergence. Thus, AB-GMRES failed to converge to a least squares solution. This phenomnenon was observed by Morikuni[@Morikuni13]. The reason why $R_i$ becomes ill-conditioned before convergence in the inconsistent case will be explained by a theorem in the next subsection. GMRES for inconsistent problems {#sec2333} ------------------------------- Brown and Walker [@bw] introduced an effective condition number to explain why GMRES fails to converge for inconsistent least squares problems $$\label{eq5} \it{\min_{x\in \mathbb{R}^m}\|b-\widetilde{A}x\|_2},$$ where $\it{\widetilde{A}\in\mathbb{R}^{m\times m}}$ is singular, in the following Theorem \[th2\]. Let $b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}}$ denote the orthogonal projection of $b$ onto $\rm{\mathcal{R}(\widetilde{\it{A}})}$. Assume $\rm{\mathcal{N}}$$(\widetilde{A})=$$\rm{\mathcal{N}}$$(\widetilde{\it{A}}^{\mathsf{T}})$ and grade$(\widetilde{A},b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}})=k$. Here, grade$(\widetilde{A},\widetilde{b})$ for $\widetilde{A}\in \mathbb{R}^{m\times m}$, $\widetilde{b}\in \mathbb{R}^m$ is defined as the minimum $k$ such that $\mathcal{K}_{k+1}(\widetilde{A}, \widetilde{b})=\mathcal{K}_{k}(\widetilde{A}, \widetilde{b})$. Then, dim($\mathcal{K}_k(\widetilde{\it{A}},b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}}))=$ dim($\mathcal{K}_{k+1}(\widetilde{\it{A}},b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}}))=$dim($\widetilde{\it{A}}\mathcal{K}_{k}(\widetilde{\it{A}},b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}}))=$dim($\widetilde{\it{A}}\mathcal{K}_{k+1}(\widetilde{\it{A}},b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}}))=k$ (See Appendix \[ap1\]). Since $\rm{\mathcal{N}}$$(\widetilde{\it{A}})=$$\rm{\mathcal{N}}$$(\widetilde{\it{A}}^{\mathsf{T}})$, we obtain $\widetilde{A}b|_{\mathcal{R}(\widetilde{A})}=\widetilde{A}b$ and dim($\widetilde{A}\mathcal{K}_{k+1}(\widetilde{A},b))=$dim$(\widetilde{A}\mathcal{K}_{k+1}(\widetilde{A},b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}}))=k$. If $b\notin \mathcal{R}(\widetilde{A})$ and dim($\widetilde{A}\mathcal{K}_{k}(\widetilde{A},b))=k$, dim($\mathcal{K}_{k+1}(\widetilde{A},b))=k+1$ (See Appendix \[ap2\]). Let $x_0$ be the initial solution and $r_0=b-\widetilde{A}x_0.$ In the inconsistent case, a least squares solution is obtained at iteration $k$, and at iteration $k+1$ breakdown occurs because of dim($\widetilde{A}\mathcal{K}_{k+1}(\widetilde{A},r_0))< $ dim($\mathcal{K}_{k+1}(\widetilde{A},r_0))$, i.e. rank deficiency of $\it{\min_{z\in \mathcal{K}_{k+1}(\widetilde{A},r_0)}\|b-\widetilde{A}(x_0+z)\|_2}=\it{\min_{z\in \mathcal{K}_{k+1}(\widetilde{A},r_0)}\|r_0-\widetilde{A}z\|_2}$[@bw]. This case is also called the hard breakdown[@REICHEL05]. However, even if $\rm{\mathcal{N}}$$(\widetilde{A})=$$\rm{\mathcal{N}}$$(\widetilde{A}^{\mathsf{T}})$, when (\[eq5\]) is inconsistent, the least squares problem $\it{\min_{z\in \mathcal{K}_{i}(\widetilde{A},r_0)}\|r_0-\widetilde{A}z\|_2}$ may become ill-conditioned as shown below. \[th2\][@bw] Assume $\rm{\mathcal{N}}$$(\widetilde{A})=$$\rm{\mathcal{N}}$$(\widetilde{A}^{\mathsf{T}})$, and denote the least squares residual of *(\[eq5\])* by $r^*$, the residual at the $(i-1)$st iteration by $r_{i-1}$. If $r_{i-1}\neq r^*$, then $$\kappa_2(A_i)\geq \frac{\|A_i\|_2}{\|\bar{A_i}\|_2}\frac{\|r_{i-1}\|_2}{\sqrt{\|r_{i-1}\|_2^2-\|r^*\|_2^2}},$$ where $A_i\equiv \widetilde{A}|_{\mathcal{K}_i(A,r_0)}$and $\bar{A_i}\equiv \widetilde{A}|_{\mathcal{K}_i(A,r_0)+\rm{span}\{\it{r}^*\}}$. Here, $\widetilde{A}|_S$ is the restriction of $\widetilde{A}$ to a subspace $S\subseteq \mathbb{R}^{m}$. Theorem \[th2\] implies that GMRES suffers ill-conditioning for $b\notin$ $\rm{\mathcal{R}}$$(\widetilde{A})$ as $\|r_i\|$ approaches $\|r^*\|$. We can apply Theorem \[th2\] to AB-GMRES for least-squares problems by setting $\widetilde{A}\equiv AA^{\mathsf{T}}$. Theorem \[th2\] also implies that even if we choose $B$ as $A^{\mathsf{T}}$, which satisfies the conditions in Theorem \[th1\], AB-GMRES still may not converge numerically because of the ill-conditioning of $R_i$, losing accuracy in the solution computed in finite-precision arithmetic when $r_{i-1}$ approaches $r^*$. Analysis of the deterioration of convergence {#sec3} ============================================ In this section, we illustrate, the deterioration of convergence of GMRES through numerical experiments. There are two points to note in this section. The first point is that the condition number of $R_i$ tends to become very large as the iteration proceeds for inconsistent problems. Due to $H_{i+1,i}=Q_{i+1}R_{i+1,i}$, the condition number of $H_{i+1,i}$ is the same as that of $R_i$, and will also become very large. The second point is as follows. Since $y_i=R_i^{-1}t_i$, $y_i$ is obtained by applying backward substitution to the triangular system $R_iy_i=t_i$. When the triangular system becomes ill-conditioned, backward substitution becomes numerically unstable, and fails to give an accurate solution $y_i$. Figure \[lllw1\] shows that at step 550 the relative residual norm suddenly increases. To understand this increase, observe the singular values of $R_{550}$. -------------------------------------------------------------------------------------------------------------------------------------------------------------------- ![Singular value distribution of $R_{550}$ for Maragal$\_$3T in double and quadruple precision arithmetic.[]{data-label="nd"}](lw3bignew.eps "fig:"){width="12cm"} -------------------------------------------------------------------------------------------------------------------------------------------------------------------- The left of Figure \[nd\] shows the singular values of $R_{550}$ which were computed in double precision arithmetic. The smallest singular value of $R_{550}$ is $3.21\times 10^{-14}$, which means that the triangular matrix $R_{550}$ is very ill-conditioned and nearly singular in double precision arithmetic. The right of Figure \[nd\] shows the singular values of $R_{550}$ which were computed in quadruple precision arithmetic using the Multiprecision Computing Toolbox for MATLAB [@mptfm]. The smallest singular value of $R_{550}$ is $5.39\times 10^{-15}$. Since quadruple precision is more accurate, from now on, we mainly show singular value distributions computed in quadruple precision. ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ ![$\kappa_2(R_i)$, $\|y_i\|_2$, and $\|t_i-R_iy_i\|_2/\|t_i\|_2$ versus the number of iterations for Maragal$\_$3T.[]{data-label="lllw5"}](lrenew1.eps "fig:"){width="12cm"} ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ Figure \[lllw5\] shows $\kappa_2(R_i)$, $\|y_i\|_2$, and the relative residual norm $\|t_i-R_iy_i\|_2/\|t_i\|_2$ versus the number of iterations for AB-GMRES. The relative residual norm increases only gradually when the condition number of $R_i$ is less than $10^{8}$. When the condition number of $R_i$ becomes larger than $10^{10}$, the relative residual norm starts to increase sharply. This observation shows that when the condition number of $R_i$ becomes very large, the backward substitution will fail to give an accurate $y_i$. As a result, we would not get an accurate $x_i$, and the convergence of AB-GMRES would deteriorate. Stabilized GMRES method ======================= In this section, we first propose and present a stabilized GMRES method. Then, we explain its regularization effect comparing it with other regularization techniques. The stabilized GMRES {#sec41} -------------------- In order to overcome the deterioration of convergence of GMRES for inconsistent systems, we propose solving the normal equations $$\label{eq419} R_i^{\mathsf{T}}R_iy_i=R_i^{\mathsf{T}}t_i$$instead of $ R_iy_i=t_i, $ which we will call the stabilized GMRES. This makes the system consistent, and stabilizes the process, as will be shown in the following. One may also consider using the normal equations of $H_{i+1,i}$. However, before breakdown, we use AB-GMRES, which means we do not have to store $H_{i+1,i}$. We only store $R_i$ and update it in each iteration, which is cheaper. ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ![Comparison of the standard AB-GMRES with stabilized and TSVD stabilized AB-GMRES with $\mu=10^{-8}$ for Maragal$\_$3T.[]{data-label="lllw2"}](lw2bignew.eps "fig:"){width="12cm"} ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- Figure \[lllw2\] shows the relative residual norm $\|A^{\mathsf{T}}r\|_2/\|A^{\mathsf{T}}r_0\|_2$ versus the number of iterations for the standard AB-GMRES and stabilized AB-GMRES with $B=A^{\mathsf{T}}$ for Maragal$\_3$T. The stabilized method reaches the relative residual norm level of $10^{-11}$ which improves a lot compared to the standard method. The method which we used for solving the normal equations (\[eq419\]) is the standard Cholesky decomposition. We replace line 8 of Algorithm \[AL1\] by Algorithm \[AL2\]. We first checked that the method works for the standard Cholesky decomposition coded by ourselves. Later we applied the backslash function of Matlab to (\[eq419\]) to speed up. We checked that in the backslash, the Cholesky decomposition method `chol` is used until the GMRES residual norm stagnates at a small level as seen in Figure \[lllw2\]. In order to continue with further GMRES iterations, the `chol` is automatically switched to the `ldl`, which works even for singular systems. Compute the QR decomposition of $H_{i+1,i}=Q_{k+1}R_{i+1,i}.$ $R_{i+1,i}=\left( %左括号 \begin{array}{ccc} %该矩阵一共3列，每一列都居中放置 R_i\\ %第一行元素 0^{\mathsf{T}}\\ %第二行元素 \end{array} \right)$, $Q_{i+1}^{\mathsf{T}}\beta e_1=\left( %左括号 \begin{array}{ccc} %该矩阵一共3列，每一列都居中放置 t_i\\ %第一行元素 \rho_{i+1}\\ %第二行元素 \end{array} \right)$,$\widetilde{R_i}=R_i^{\mathsf{T}}R_i,\qquad \widetilde{t_i}=R_i^{\mathsf{T}}t_i$. Compute the Cholesky decomposition of $\widetilde{R_i}=LL^{\mathsf{T}}$. Solve $Lz_i=\widetilde{t_i}$ by forward substitution. Solve $L^{\mathsf{T}}y_i=z_i$ by backward substitution. In spite of the above mentioned merits of stabilization, solving the normal equations in AB-GMRES is expensive. Actually, we only need the stabilized AB-GMRES when $R_i$ becomes ill-conditioned. Thus, we can speed up the process by switching AB-GMRES to stabilized AB-GMRES only when $R_i$ becomes ill-conditioned. The condition number of an incrementaly enlarging triangular matrix can be estimated by techniques in [@tebbens2014]. In this paper, we adopt the switching strategy by monitoring the relative residual norm $\|A^{\mathsf{T}}r_i\|_2/\|A^{\mathsf{T}}r_0\|_2$. Let ATR($i$)=$\|A^{\mathsf{T}}r_i\|_2/\|A^{\mathsf{T}}r_0\|_2$ for the $i$th iteration. When ATR($v$)/ $\min_{i=1, 2, \dots, v-1}$ATR($i$)&gt;10, we judge that a jump in relative residual norm has occured, and we switch AB-GMRES to stabilized AB-GMRES at the $v$th iteration. Motivated by the stabilized AB-GMRES, we also applied the truncated singular value decomposition (TSVD) stabilization method and compared it with the stabilized AB-GMRES. The method modifies $R_i$ by truncating singular values smaller than $\mu$. More specifically, let $R_{i}=U\Sigma V^{\mathsf{T}}$ be the SVD of $R_i$, where the columns of $U=[u_1, u_2, \dots, u_i]$ and $V=[v_1, v_2, \dots, v_i]$ are the left and right singular vectors, respectively, and the diagonal entries of $\Sigma=$ diag$(\sigma_1, \sigma_2, \dots, \sigma_i)$ are the singular values of $R_i$ in discending order $\sigma_1\geq \sigma_2\geq \cdots\geq \sigma_i$. Then, the TSVD approximates $R_i\simeq \sum_{j=1}^k \sigma_j u_j v_j^{\mathsf{T}}$ with $k$ such that $\sigma_{k+1}\leq \mu\sigma_1\leq \sigma_k$ and $y_i= R_i^{-1}t_i\simeq \sum_{j=1}^k\frac{1}{\sigma_j}v_ju_j^{\mathsf{T}}t_i$. When $\mu= 10^{-13}, 10^{-12}, \dots, 10^{-4}$, the method converges but when $\mu$ is smaller than $10^{-13}$ or larger than $10^{-4}$, it diverges and is similar to the original AB-GMRES. Numerical experiments showed that $\mu=\sqrt{\epsilon}\simeq 10^{-8}$, where $\epsilon$ is the machine epsilon (about $10^{-16}$ in double presion arithmetic), gave the best result among $\mu = 10^{-1}, 10^{-2},\dots, 10^{-16}$ in terms of the relavtive residual as shown in Figure \[lllw2\] for the problem Maragal$\_$3T. The convergence behaviour of the TSVD stabilization method is similar to the stabilized AB-GMRES method, which suggests that eliminating tiny singular values which are less than $10^{-8}$ is effctive for sovling problem (\[eq3\]). However, the TSVD method requires computing the truncated singular value decomposition of $R_i$, and requires choosing the value of the threshold parameter $\mu$, whereas the stabilized AB-GMRES does not require either of them. [c|rrrrr]{} matrix & Maragal$\_$3T & Maragal$\_$4T & Maragal$\_$5T & Maragal$\_$6T & Maragal$\_$7T\ ------------------------------------------------------------------------ iter. &531 & 465 & 1110 & 2440 & 1864\ standard AB-GMRES &1.05$\times 10^{-8}$ & 2.09$\times 10^{-7}$ & 5.35$\times 10^{-6}$ & 8.26$\times 10^{-6}$ & 4.53$\times 10^{-6}$\ ------------------------------------------------------------------------ iter. &552 & 598 & 1226 & 3002 & 2459\ stabilized AB-GMRES &5.99$\times 10^{-12}$ & 5.59$\times 10^{-8}$ & 4.22$\times 10^{-6}$ & 3.88$\times 10^{-6}$& 2.80$\times 10^{-7}$\ \[tb10\] Table \[tb10\] gives more results for the Maragal matrices. The table shows that the stabilized AB-GMRES is more accurate than the standard AB-GMRES. This seems paradoxical, since forming the normal equations whose coefficient matrix $R_i^{\mathsf{T}}R_i$ would square the condition number compared to $R_i$, which would make the ill-conditioned problem even worse. Why can the stabilized AB-GMRES give a more accurate solution? We will explain why the stabilized AB-GMRES works in the next subsection. Why the stabilized GMRES method works ------------------------------------- Consider solving $R_iy_i=t_i, R_i\in \mathbb{R}^{i\times i}, t_i\in \mathbb{R}^{i}$ by solving the normal equations (\[eq419\]), which, in theory, squares the condition number and makes the problem become harder to solve numerically. However, in finite precision arithmetic, the condition number of the normal equations is not neccessarily squared. We will continue to illustrate the phenomenon by using the example in Section \[sec3\]. We used the MATLAB function `svd` in quadruple precision arithmetic [@mptfm] to calculate the singular values. The smallest singular value of $R_{550}$ is $ 5.39\times 10^{-15}$, so its square is $2.91\times 10^{-29}$. ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ ![Singular values $\sigma_i($fl$_d(R_{550}^{\mathsf{T}}R_{550}))$, $i=1, 2, \dots, 550$ in quadruple precision arithmetic.[]{data-label="lll7"}](lll8big.eps "fig:"){width="12cm"} ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ Let fl$(\cdot)$ denote the evaluation of an expression in floating point arithmetic and fl$_d(\cdot)$ and fl$_q(\cdot)$ denote the result in double precision arithmetic and quadruple precision arithimetic, respectively. Figure \[lll7\] shows that, numerically, the smallest singular value of fl$_d(R_{550}^{\mathsf{T}}R_{550})$ is $ 7.21\times 10^{-14}$, which is much larger than $2.91\times 10^{-29}$. Further, the Cholesky factor $L$ of fl$_d(R_{550}^{\mathsf{T}}R_{550})~=~LL^{\mathsf{T}}$ computed in double precision precision arithmetic has the smallest singular value $3.50\times 10^{-7}$, which is also larger than $\sqrt{2.91\times 10^{-29}}=5.39\times 10^{-15}$. Thus, the triangular systems $Lz_i=\widetilde{t_i}$ and $L^{\mathsf{T}}y_i=z_i$ are better-conditioned than $R_iy_i=t_i$, which will ensure the stability of the forward and backward substitutions and succeeds in obtaining a much more accurate solution than the standard approach. ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ![Singular values $\sigma_i($fl$_d(R_{550}^{\mathsf{T}}R_{550}))$, $\sigma_i(R_{550})^2$, $\sigma_i$(fl$_d(R_{610}^{\mathsf{T}}R_{610}))$, and $\sigma_i(R_{610})^2$ in quadruple precision arithmetic.[]{data-label="lll8"}](lm2new.eps "fig:"){width="12cm"} ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- The left of Figure \[lll8\] compares the singular values $\sigma_i($fl$_d(R_{550}^{\mathsf{T}}R_{550}))$ and $\sigma_i(R_{550})^2, i=1, 2, \dots, 550$. The first to the 549th singular values of fl$_d(R_{550}^{\mathsf{T}}R_{550})$ and the corresponding $\sigma(R_{550})^2$ are almost the same, while the last one is different. What will happen when $R_i$ contains a cluster of small singular values? The upper triangular matrix $R_{610}$ contains a cluster of small singular values. The right of Figure \[lll8\] compares the singular values $\sigma_i($fl $_q(R_{610}^{\mathsf{T}}R_{610}))$ and $\sigma_i(R_{610})^2$. The larger singular values are the same as the ‘exact’ values, while the smaller singular values become larger than the ‘exact’ ones. Experiment results show that finite precision arithmetic has the effect of shifting the tiny singular value upwards. That is the reason why the normal equations (\[eq419\]) help to reduce the condition number and makes the problem become better-conditioned. Next, we computed the multiplication $R_{550}^{\mathsf{T}}R_{550}$ in quadruple precision arithmetic and observed that the smallest singular values of $R_{550}^{\mathsf{T}}R_{550}$ conincided with the squared singular values $\sigma_i(R_{550})^2$ (blue circle symbol) in the left of Figure \[lll8\], unlike in double precision computation. Since the maximum of the elements of | fl$_q(R_{550}^{\mathsf{T}}R_{550})$ $-$ fl$_d(R_{550}^{\mathsf{T}}R_{550})$ | is approximately $8.16\times 10^{-12}$ , double precision arithmetic contains error of the order of $10^{-12}$. Thus, double precision arithmetic has an effect of regularizing the matrix $R_{550}^{\mathsf{T}}R_{550}$, since double precision matrix multiplication is not accurate enough to keep all the information. Quadruple precision ------------------- ------------------------------------------------------------------------------------------------------------------------------------------- ![Effect of the stabilized method in quadruple precision arithmetic for Maragal$\_$3T.[]{data-label="un1"}](un1.eps "fig:"){width="12cm"} ------------------------------------------------------------------------------------------------------------------------------------------- In order to see the effect of the machine precision $\epsilon$ on the convergence of the AB-GMRES, we compared the stabilized AB-GMRES with the AB-GMRES in quadruple precision arithmetic for the problem Maragal$\_$3T in Figure \[un1\]. For both methods, the relative residual norm reached a smaller level of $10^{-16}$ compared to $10^{-12}$ and $10^{-8}$, respectively, for double precision arithmetic in Figure \[lllw2\]. The curve of the relative residual norm became smoother compared to double precision. As seen in Figure \[un1\], the relative residual norm of the AB-GMRES method jumped to $10^{-1}$ after reaching $10^{-16}$, whereas the relative residual norm of the stabilized GMRES stayed around $10^{-16}$. When the stabilized GMRES method works -------------------------------------- [\[sec43\]]{} Motivated by the Läuchli matrix [@Higham], we consider solving the following EP (equal projection) problem $A_3x=(1,0,0)^\mathsf{T}$, where $A_3$ is null space symmetric, that is $\rm{\mathcal{N}}$$(A_3)=$$\rm{\mathcal{N}}$$(A_3^{\mathsf{T}})$ with null space $\rm{\mathcal{N}}$$(A_3)=\rm{span}\it\{(1,-1,1)^\mathsf{T}\}.$ $$\label{exep} A_3x=\left( \begin{array}{ccccc} \frac{\sqrt{2}}{2} &\qquad & \frac{\sqrt{2}}{2}- \frac{\sqrt{6\epsilon}}{6} &\qquad&- \frac{\sqrt{6\epsilon}}{6}\\ \frac{\sqrt{2}}{2}&\qquad & \frac{\sqrt{2}}{2}+ \frac{\sqrt{6\epsilon}}{6}&\qquad& \frac{\sqrt{6\epsilon}}{6} \\ 0 &\qquad& \frac{\sqrt{6\epsilon}}{3} &\qquad&\frac{\sqrt{6\epsilon}}{3}\\ \end{array} \right)x= \left( \begin{array}{c} 1\\ 0\\ 0\\ \end{array} \right),$$ where $\epsilon$ is the machine epsilon. Apply GMRES with $x_0=0$ to (\[exep\]). Let $R_s\in \mathbb{R}^{s\times s}$ be the upper triangular matrix obtained at the $s$th iteration of GMRES. In the second iteration, after applying the Givens rotation to $H_{3, 2}$, we obtain the following: $$\label{excep1} R_2= \left( \begin{array}{cc} 1 & 1 \\ 0 & \sqrt{\epsilon} \\ \end{array} \right),\qquad R_2^{\mathsf{T}}R_2= \left( \begin{array}{cc} 1 & 1 \\ 1 & 1+\epsilon \\ \end{array} \right)\simeq \left( \begin{array}{cc} 1 & 1 \\ 1 & 1\\ \end{array} \right).$$ Thus, there is a risk that the stabilized GMRES will give a numerically singular matrix $R_2^{\mathsf{T}}R_2$ in finite precision arithmetic for nonsingular $R_2$. We will analyze this phenominon. We define the following. $O(\epsilon)$ denotes that there exists a constant $c$ independent of $\epsilon$, such that $-c\epsilon<O(\epsilon)<c\epsilon$. Also, let $$\mathbb{O}(\epsilon)=\left( \begin{array}{c} O(\epsilon)\\ O(\epsilon)\\ \vdots\\ O(\epsilon) \\ \end{array}\right)\in \mathbb{R}^n ,\quad \mathcal{O}(\epsilon)= [\mathbb{O}(\epsilon), \mathbb{O}(\epsilon), \cdots, \mathbb{O}(\epsilon)]\in \mathbb{R}^{n\times n}.$$ We assume that the basic arithmetic operations op $=$ $+, -, *, /$ satisfy fl$(x$ op $y)= (x $ op $y)(1+O(\epsilon))$ as in [@nh]. Note also that the following hold from [@nh]. Let $x, y\in \mathbb{R}^n, A,B\in \mathbb{R}^{n\times n},$ and $$|x|=\left( \begin{array}{c} |x_1|\\ |x_2|\\ \vdots\\ |x_n| \\ \end{array}\right)\quad \rm for\quad \it x=\left(\begin{array}{c} x_1\\ x_2\\ \vdots\\ x_n \\ \end{array}\right),$$ $$|A|=\left( \begin{array}{cccc} |a_{11}|&|a_{12}|&\cdots&|a_{1n}|\\ |a_{21}|& |a_{22}| & \cdots &|a_{2n}|\\ \vdots & \vdots & \ddots& \vdots \\ |a_{n1}|& |a_{n2}| & \cdots &|a_{nn}| \\ \end{array}\right)$$ for $A=(a_{pq})$. Then $$\begin{aligned} \rm f\!l\it(x^\mathsf{T}y)&=x^\mathsf{T}y+O(n\epsilon)|x|^{\mathsf{T}}|y|=x^{\mathsf{T}}y+O(n\epsilon),\\ \rm f\!l\it(Ax)&=Ax+\mathbb{O}(n\epsilon)|A||x|=Ax+\mathbb{O}(n\epsilon),\\ \rm f\!l\it(AB)&=AB+O(n\epsilon)|A||B|=AB+\mathcal{O}(n\epsilon).\end{aligned}$$ Note also that the following theorem holds from Theorem 8.10 of [@nh]. \[th333\] Let $T=(t_{pq})\in \mathbb{R}^{n\times n}$ be a triangular matrix and $b\in \mathbb{R}^{n}.$ Then, the computed solution $\hat{x}$ obtained from substitution applied to $Tx=b$ satisfies $$\hat{x} = x+O(n^2\epsilon)M(T)^{-1}|b|.$$ Here, $M(T)=(m_{ij})$ is the comparison matrix such that $$m_{ij}=\left\{ \begin{array}{cc} |t_{ij}|, & i=j, \\ -|t_{ij}|, & i\neq j. \\ \end{array} \right.$$ \ Further, we define the following. Assume $\|A\|_2=O(1).$ We say $A\in \mathbb{R}^{n\times n}$ is numerically nonsingular if and only if $$\label{EQ14} \rm f\!l\it (Ax)=\mathbb{O}(\epsilon)\quad\Rightarrow\quad x=\mathbb{O}(\epsilon).$$ Note that this definition of numerical nonsingularity agrees with that of numerical rank due to the following. Let the SVD of $A=U\Sigma V^{\mathsf{T}}$ where $U, V$ are orthogonal matrices and $\Sigma=\rm diag\it (\sigma_1, \sigma_2, \dots, \sigma_n).$ Here, $\|A\|_2=\sigma_1=O(1).$ If the numerical rank of $A$ is $r<n$, there is a $\sigma_i=O(\epsilon),$ $r+1\leq i\leq n.$ Then, $Ax=U\Sigma V^{\mathsf{T}}x=O(\epsilon)$ admits $x'=V^{\mathsf{T}}x=(x_1', x_2', \dots, x_n')^{\mathsf{T}}$ such that $x_i'=O(1)$, and hence $x=\mathbb{O}(1).$ Thus, $A$ is numerical singular. Then, the following theorem holds. \[th55\] Let $R_s=(r_{pq})\in \mathbb{R}^{s\times s}$ be an upper-triangular matrix and $$R_{s+1}=\left( \begin{array}{cc} R_s & d \\ 0^{\mathsf{T}} & r_{s+1,s+1} \\ \end{array} \right)\in \mathbb{R}^{(s+1)\times(s+1)}.$$ Assume $R_s$ is numerically nonsingular, and $R_s=\mathcal{O}(1), R_s^{-1}=\mathcal{O}(1), M(R_s)^{-1}=\mathcal{O}(1), d=\mathbb{O}(1)$ and $O(s)=O(s^2)=O(1).$ Then, the following holds: $$\rm f\!l\it(R_{s+1}^{\mathsf{T}}R_{s+1})\rm~is~numerically~nonsingular \it\quad\Longleftrightarrow\quad \rm f\!l \it (r^2_{s+1,s+1})> \rm f\!l \it(d^{\mathsf{T}}d)O(\epsilon).$$ See Appendix \[ap3\]. Theorem \[th55\] gives the necessary and sufficient condition so that the stabilized GMRES works at the $(s+1)$st iteration, i.e. $ R_{s+1}^{\mathsf{T}}R_{s+1}$ is numerically nonsingular. ----------------------------------------------------------------------------------------------------------------------------------------------- ![$r_{i,i}^2$ $(i=s+1)$ and $d^{\mathsf{T}}d$ in stabilized AB-GMRES for Maragal$\_$3T.[]{data-label="t4dr"}](lt4dr.eps "fig:"){width="12cm"} ----------------------------------------------------------------------------------------------------------------------------------------------- The difficulty in solving $R_iy_i=t_i$ by backward substitution is not because the diagonals of $R_i$ are tiny. The reason is that $R_i$ has tiny singular values. However, the exceptional example (\[excep1\]) exists where the stabilized AB-GMRES does not work. The condition fl$(r^2_{s+1, s+1})$ $>$ fl$(d^{\mathsf{T}}d)O(\epsilon)$ in Theorem \[th55\] excludes such exceptions. Figure \[t4dr\] shows $r^2_{s+1, s+1}$ and $d^{\mathsf{T}}d$ together with the convergence of the AB-GMRES and that of the stabilized AB-GMRE for Maragal$\_$3T. The figure shows that upto 613 iterations, the conditions in Theorem \[th55\] are satisfied, and $ R_{s+1}^{\mathsf{T}}R_{s+1}$ is numerically nonsingular, so that the stabilized AB-GMRES works. Comparison with Tikhonov regularization method ---------------------------------------------- Another approach to stabilize the AB-GMRES would be to apply Tikhonov regularization. There are two methods to implement it. The first method is to solve the following square system: $$\label{eq66} (R_i^{\mathsf{T}}R_i+\lambda I)y_i=R_i^{\mathsf{T}}t_i,\qquad \lambda\geq 0$$ using the Cholesky decomposition. The second method is to solve the regularized least suqares problem $$\label{eqz16} \min_{y_i\in\mathbb{R}^i}\left|\left| \left( \begin{array}{c} t_i\\ 0\\ \end{array}\right)-\left(\begin{array}{c} R_i\\ \sqrt{\lambda} I\\ \end{array} \right)y_i\right|\right|_2$$ using the QR decomposition. These two methods are equivalent mathematically. However, they are not equivalent numerically. The behavior of the first method is similar to the stabilized AB-GMRES. Table \[tb33\] shows that AB-GMRES combined with the first method converges better when $\lambda=10^{-16}$ than when $\lambda=10^{-14}$. This method can be used to shift upwards the small singular values, but is less acurrate compared to the stabilized AB-GMRES (cf. Table \[tb10\]).\ ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ![Relative residual norm for the regularized AB-GMRES using (\[eqz16\]) versus number of iterations for different $\lambda$ for Maragal$\_$3T.[]{data-label="llwre999"}](llwre999.eps "fig:"){width="12cm"} ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- Table \[tb33\] also shows that the second method is even more accurate compared with the stabilized AB-GMRES method. There is no need to form the normal equations, so that less information is lost due to rounding error. However, one needs to choose an appropriate value for the regularization parameter $\lambda$. Figure \[llwre999\] shows the relative residual norm $\| A^\mathsf{T} r_i \|_2 / \| A^\mathsf{T} r_0\|_2$ for the regularized AB-GMRES using (\[eqz16\]) versus the number of iterations for different values of $\lambda$ for Maragal$\_$3T. According to Figure \[llwre999\], $\lambda=10^{-16}$ was optimal among $10^{-12}, 10^{-14}, 10^{-16}, 10^{-18}$, so we recommend this value in practice. We here note the following. Let $\sigma_1\geq \sigma_2\geq\cdots\geq\sigma_i$ be the the singular values of $R_i$. Then, the singular values of $$R_i'=\left( \begin{array}{c} R_i\\ \sqrt{\lambda}I\\ \end{array}\right)$$ are given by $\sqrt{\sigma_1^2+\lambda}\geq\sqrt{\sigma_2^2+\lambda}\geq\cdots\geq\sqrt{\sigma_i^2+\lambda}.$ See Appendix \[ap4\]. Then, let $$\kappa\equiv\kappa_2(R_i)=\frac{\sigma_1}{\sigma_i},\quad \kappa'^2\equiv\kappa_2(R_i')^2=\frac{\sigma_1^2+\lambda}{\sigma_1^2/\kappa^2+\lambda}=1+\frac{\sigma_1^2(1-1/\kappa^2)}{\sigma_1^2/\kappa^2+\lambda}.$$ Since $\kappa\geq 1, \rm d\it \kappa'/\rm d\it \lambda \leq 0$ for $\lambda\geq 0$ and $\kappa'(\lambda=0)=\kappa, \kappa'(\lambda=+\infty)=1.$ Note also that $$\lambda=\frac{\sigma_1^2[1+(\kappa'/\kappa)^2]}{\kappa'^2-1}$$ Therefore, for instance, if $\kappa\gg 1$ and we want $\kappa'=\sqrt{\kappa}$, $$\lambda=\frac{\sigma_1^2(1+1/\kappa)}{\kappa-1}\simeq \frac{\sigma_1^2}{\kappa}.$$ For example, if $\kappa=10^{16}$ and we want $\kappa'=10^8$, we should choose $\lambda\simeq \sigma_1^2\times 10^{-16}.$ For Maragal$\_$3T, the largest singular value $\sigma_1$ is about 12.64, so that we can estimate a reasonable value of $\lambda \simeq 1.60 \times 10^{-14}$. However, this estimation assumes $\kappa'=\sqrt{\kappa}$, and needs an extra cost for computing $\sigma_1$. See [@brezinski2009error] for other estimation techniques for the regularization parameter. [c|rrrrr]{} matrix & Maragal$\_$3T & Maragal$\_$4T & Maragal$\_$5T & Maragal$\_$6T & Maragal$\_$7T\ ------------------------------------------------------------------------ iter. &552 & 597 & 1304 & 2440 & 1864\ method (\[eq66\]) $\lambda=10^{-14}$ &5.08$\times 10^{-11}$ & 5.57$\times 10^{-8}$ & 1.05$\times 10^{-5}$ & 8.26$\times 10^{-6}$ & 4.53$\times 10^{-6}$\ ------------------------------------------------------------------------ iter. &570 & 598 & 1226 & 2440 & 1864\ method (\[eq66\]) $\lambda=10^{-16}$ &5.80$\times 10^{-12}$ & 5.59$\times 10^{-8}$ & 4.22$\times 10^{-6}$ & 8.26$\times 10^{-6}$& 4.53$\times 10^{-6}$\ ------------------------------------------------------------------------ iter. &553 & 547 & 1261 & 2937 & 2475\ method (\[eqz16\]) $\lambda=1.6\times 10^{-14}$ &7.54$\times 10^{-11}$ & 5.59$\times 10^{-8}$ & 1.15$\times 10^{-5}$ & 9.12$\times 10^{-6}$& 2.78$\times 10^{-7}$\ ------------------------------------------------------------------------ iter. &551 & 547 & 1262 & 3037 & 2475\ method (\[eqz16\]) $\lambda=10^{-16}$ &3.37$\times 10^{-12}$ & 5.59$\times 10^{-8}$ & 5.64$\times 10^{-7}$ & 1.91$\times 10^{-6}$& 2.78$\times 10^{-7}$\ \[tb33\] Comparisons with other methods ============================== Underdetermined inconsistent least squares problems --------------------------------------------------- First, we compared the stabilized AB-GMRES with the range restricted AB-GMRES (RR-AB-GMRES) [@NEUMAN2], where the Krylov subspace for the RR-AB-GMRES with $B=A^{\mathsf{T}}$ is $\textit{K}_i(AA^{\mathsf{T}},AA^{\mathsf{T}}r_0)$, AB-GMRES with $B=A^{\mathsf{T}}$, BA-GMRES with $B=A^{\mathsf{T}}$, LSQR[@lsqr] and LSMR[@lsmr]. All programs for iterative methods were coded according to the algorithms in [@NEUMAN2; @Hayami10; @lsqr; @lsmr]. Each method was terminated at the iteration step which gives the minimum relative residual norm within $m$ iterations, where $m$ is the number of the rows of the matrix. No restarts were used for GMRES. Experiments were done for rank-deficient matrices whose information is given in Table 1. Here, we have deleted the zero rows and columns of the test matrices beforehand. The elements of $b$ were randomly generated using the MATLAB function `rand`. Each experiment was done 10 times for the same right hand side $b$ and the average of the CPU times are shown. Symbol - denotes that $\|A^{\mathsf{T}}r_i\|_2/\|A^{\mathsf{T}}r_0\|_2$ did not reach $10^{-8}$ within $20n$ iterations. [c|rrrrr]{} matrix & Maragal$\_$3T & Maragal$\_$4T & Maragal$\_$5T & Maragal$\_$6T & Maragal$\_$7T\ ------------------------------------------------------------------------ iter. &531 & 465 & 1110 & 2440 & 1864\ standard AB-GMRES &1.05$\times 10^{-8}$ & 2.09$\times 10^{-7}$ & 5.35$\times 10^{-6}$ & 8.26$\times 10^{-6}$ & 4.53$\times 10^{-6}$\ ------------------------------------------------------------------------ iter. &552 & 598 & 1226 & 3002 & 2459\ stabilized AB-GMRES &5.99$\times 10^{-12}$ & 5.59$\times 10^{-8}$ & 4.22$\times 10^{-6}$ & 3.88$\times 10^{-6}$& 2.80$\times 10^{-7}$\ ------------------------------------------------------------------------ iter. &553 & 565 & 1223 & 2374 & 2474\ RR-AB-GMRES &2.57$\times 10^{-11}$ & 5.59$\times 10^{-8}$ & 3.62$\times 10^{-6}$ & 1.63$\times 10^{-5}$& 2.78$\times 10^{-7}$\ ------------------------------------------------------------------------ iter. &562 & 626 & 1263 & 4373 & 5658\ BA-GMRES &2.88$\times 10^{-14}$ & 7.92$\times 10^{-11}$ & 2.29$\times 10^{-12}$ & 5.12$\times 10^{-11}$& 2.03$\times 10^{-10}$\ ------------------------------------------------------------------------ iter. &1682 & 2375 & 4576 & 151013 & 97348\ LSQR &5.64$\times 10^{-14}$ & 2.77$\times 10^{-10}$ & 1.11$\times 10^{-11}$ & 5.87$\times 10^{-10}$& 1.33$\times 10^{-9}$\ ------------------------------------------------------------------------ iter. &1654 &2308 & 4273 & 127450 & 70242\ LSMR &5.51$\times 10^{-14}$ & 3.00$\times 10^{-10}$ & 3.25$\times 10^{-11}$ & 4.16$\times 10^{-10}$& 9.95$\times 10^{-10}$\ \[tb22\] [c|rrrrr]{} matrix & Maragal$\_$3T & Maragal$\_$4T & Maragal$\_$5T & Maragal$\_$6T & Maragal$\_$7T\ ------------------------------------------------------------------------ iter. &- & - & - & - & -\ standard AB-GMRES &- & - & - & - & -\ ------------------------------------------------------------------------ iter. &546 (526) & - & - & - & -\ stabilized AB-GMRES &2.01 & - & - & -& -\ ------------------------------------------------------------------------ iter. &545 & - & - & - & -\ RR-AB-GMRES &1.84 & - & - & -& -\ ------------------------------------------------------------------------ iter. &530 & 608 & 1232 & 3623 & 5001\ BA-GMRES &2.10 & 3.19 & 4.25$\times 10^{1}$ &1.81$\times 10^{3}$& 9.20$\times 10^{3}$\ ------------------------------------------------------------------------ iter. &1465 & 2120 & 4032 & 101893 & 54444\ LSQR &1.27$\times 10^{-1}$ & 2.56$\times 10^{-1}$ & 1.49 & 2.93$\times 10^{2}$& 4.33$\times 10^{2}$\ ------------------------------------------------------------------------ iter. &1456 &1989 & 4013 & 54017 & 31206\ LSMR &1.25$\times 10^{-1}$ & 2.37$\times 10^{-1}$ & 1.49 & 1.50$\times 10^{2}$& 2.23$\times 10^{2}$\ \[ntb1196\] Table \[tb22\] shows that the stabilized AB-GMRES is generally more accurate than the RR-AB-GMRES. The stabilized AB-GMRES took more iterations to attain the same order of the smallest residual norm than the RR-AB-GMRES. Table \[tb22\] also shows that for the same underdetermined least squares problems, the BA-GMRES was the best in terms of the attainable smallest relative residual norm and that the LSQR and LSMR are comparable to the BA-GMRES, but require less CPU time according to Tabel \[ntb1196\]. Inconsistent systems with highly ill-conditioned square coefficient matrices ---------------------------------------------------------------------------- The stabilized AB-GMRES is not restricted to solving underdetermined problems but can also be applied to solving the least squares problem $\min_{x\in \mathbb{R}^n} \|b-Ax\|_2$, where $A\in \mathbb{R}^{n\times n}$ is a highly ill-conditioned square matrix. Thus, we also test on square matrices of different kinds. Table \[tb94\] gives the information of the matrices. These matrices are all numerically singular. We generated the right-hand side $b$ by the MATLAB function `rand`, so that the systems are generically inconsistent. We compared the stabilized AB-GMRES with the standard AB-GMRES, RR-AB-GMRES, BA-GMRES with $B=A^{\mathsf{T}}$, LSMR [@lsmr], and LSQR [@lsqr]. Table \[tb95\] gives the smallest relative residual norm and the number of iterations. Table \[tb695\] gives the CPU times in seconds required to obtain relative residual norm $\|A^{\mathsf{T}}r_i\|_2/\|A^{\mathsf{T}}r_0\|_2<10^{-8}$. The switching strategy which was introduced in Section \[sec41\] was used for the stabilized AB-GMRES when measuing CPU times. The number of iterations when switching occurred is in brackets. [r|r|r|r|r|r]{} & & & & &\ ------------------------------------------------------------------------ Harvard500 & 500 & 1.05 &170 &1.30$\times 10^2$ & web connectivity\ netz4504 & 1961 & 0.13 & 1342&3.41$\times 10^1$ & 2D/3D finite element problem\ TS & 2142 & 0.99 & 2140 &3.52$\times 10^3$ &counter example problem\ grid2$\_$dual & 3136 & 0.12 & 3134 &8.58$\times 10^3$ &2D/3D finite element problem\ uk & 4828 & 0.06 & 4814 & 6.62$\times 10^3$&undirected graph\ bw42 & 10000 & 0.05 & 9999 &2.03$\times 10^{3}$ & partial differential equation[@bw]\ \[tb94\] [c|rrrrrr]{} matrix & Harvard500 & netz4504 &TS & grid2$\_$dual & uk & bw42\ ------------------------------------------------------------------------ iter. &104 & 144 & 1487 & 3134 & 4620 &715\ standard AB-GMRES &9.38$\times 10^{-9}$ & 4.51$\times 10^{-10}$ &1.56$\times 10^{-9}$ &5.98$\times 10^{-10}$& 1.35$\times 10^{-9}$ & 8.06$\times 10^{-8}$\ ------------------------------------------------------------------------ iter. &175 & 201 & 1617 & 3135 & 4779& 788\ stabilized AB-GMRES &4.53$\times 10^{-14}$ & 1.51$\times 10^{-14}$ & 1.54$\times 10^{-9}$ & 1.14$\times 10^{-9}$&6.81$\times 10^{-10}$& 1.66$\times 10^{-7}$\ ------------------------------------------------------------------------ iter. &135 & 200 & 1652 & 3134 & 4706&1163\ RR-AB-GMRES &7.78$\times 10^{-14}$ & 3.36$\times 10^{-14}$ & 4.56$\times 10^{-9}$ & 6.52$\times 10^{-8}$& 8.33$\times 10^{-8}$&1.56$\times 10^{-5}$\ ------------------------------------------------------------------------ iter. &139 & 194 & 1628 & 3134 & 4724&1520\ BA-GMRES &1.91$\times 10^{-15}$ & 7.27$\times 10^{-16}$ & 8.43$\times 10^{-13}$ & 1.23$\times 10^{-13}$& 6.94$\times 10^{-14}$& 1.97$\times 10^{-11}$\ ------------------------------------------------------------------------ iter. &391 & 198 & 6047 & 12549 & 6249&1256\ LSQR &3.59$\times 10^{-15}$ & 5.86$\times 10^{-16}$ & 1.96$\times 10^{-12}$ & 2.51$\times 10^{-13}$& 6.56$\times 10^{-14}$&1.59$\times 10^{-11}$\ ------------------------------------------------------------------------ iter. &338 &195 & 6219 & 12497 & 6199& 1212\ LSMR &2.01$\times 10^{-15}$ & 5.97$\times 10^{-16}$ & 1.25$\times 10^{-12}$ & 2.34$\times 10^{-13}$& 7.35$\times 10^{-14}$&1.60$\times 10^{-11}$\ \[tb95\] [r|rr]{} & &\ ------------------------------------------------------------------------ standard GMRES & 147& 8.08$\times 10^{-9}$\ stabilized GMRES & 219& 1.94$\times 10^{-11}$\ RR-GMRES & 220& 3.13$\times 10^{-11}$\ \[tbbw42\] [c|rrrrrr]{} matrix & Harvard500 & netz4504 &TS & grid2$\_$dual & uk & bw42\ ------------------------------------------------------------------------ iter. &104 & 134 & 1411 & 3134 & 4583 &-\ standard AB-GMRES &4.72$\times 10^{-2}$ & 1.87$\times 10^{-1}$ &2.14$\times 10$ &2.16$\times 10^{2}$& 6.93$\times 10^{2}$ & -\ ------------------------------------------------------------------------ iter. &104 & 134 & 1531 (182) & 3134 & 4679 (4199)& -\ stabilized AB-GMRES &4.78$\times 10^{-2}$ & 1.89$\times 10^{-1}$ & 8.19$\times 10$ & 2.21$\times 10^{2}$&1.93$\times 10^{3}$& -\ ------------------------------------------------------------------------ iter. &114 & 153 & 1530 & - & -&-\ RR-AB-GMRES &6.42$\times 10^{-2}$ & 2.62$\times 10^{-1}$ & 2.68$\times 10$ & -& -&-\ ------------------------------------------------------------------------ iter. &103 & 131 & 1379 & 3134 & 4562&738\ BA-GMRES &5.48$\times 10^{-2}$ & 1.72$\times 10^{-1}$ & 2.06$\times 10$ & 2.44$\times 10^{2}$& 7.55$\times 10^{2}$& 2.33$\times 10$\ ------------------------------------------------------------------------ iter. &222 & 134 & 4239 & 11802 & 5948&913\ LSQR &5.63$\times 10^{-3}$ & 6.61$\times 10^{-3}$ & 7.86$\times 10^{-1}$ & 1.15& 8.65$\times 10^{-1}$&3.12$\times 10^{-1}$\ ------------------------------------------------------------------------ iter. &215 &132 & 3913 & 11746 & 5898& 655\ LSMR &5.34$\times 10^{-3}$ & 6.42$\times 10^{-3}$ & 7.04$\times 10^{-1}$ & 1.15& 8.42$\times 10^{-1}$& 2.32$\times 10^{-1}$\ \[tb695\] Table \[tb95\] shows that for most problems the BA-GMRES was the best in terms of accuracy of relative residual norm. The LSQR and LSMR are similar and are comparable to the BA-GMRES, beacuse they all change the inconsistent problem into a consistent problem. The LSQR and LSMR are more suitable for large and sparse problems compared to the BA-GMRES because they require less CPU time and memory. For Harvard500 and bw42, the AB-GMRES could only converge to the level of $10^{-9}$ regarding the relative residual norm, while the stabilized AB-GMRES converged to the level of $10^{-14}$. The stabilized AB-GMRES was robust in the sense that it could continue to compute even when the upper triangular matrix $R_i$ became seriously ill-conditioned, and the relative residual norm did not increase sharply towards the end, but just stagnated at a low level, just like for consistent problems. Comparing the CPU time in Tabel \[tb695\], LSMR was the fastest. The stabilized AB-GMRES was usually faster than BA-GMRES. Thus, our stabilization method also makes AB-GMRES stable for highly ill-conditioned inconsistent systems with square coefficient matrices. The coefficient matrix $A$ of bw42 is singular and satisfies $\rm{\mathcal{N}}$$(A)=$$\rm{\mathcal{N}}$$(A^{\mathsf{T}})$. The problem comes from a finite-difference discritization of a PDE with periodic boundary condition (Experiment 4.2 in Brown and Walker[@bw] with the original $b$). Since the matrix is range symmetric, the GMRES, RR-GMRES, and stabilized GMRES can be directly applied to $Ax=b$ (See [@bw] Theorem 2.4, [@hayami2011g] Theorem 2.7, and [@calvetti2000g] Theorem 3.2.) as shown in Table \[tbbw42\]. The stabilized GMRES gave the relative residual norm 1.94$\times 10^{-11}$ for bw42 at the 219th iteration, similar to the BA-GMRES. Concluding Remarks ================== We proposed a stabilized AB-GMRES method for ill-conditioned underdetermined and inconsistent least squares problems. It shifts upwards the tiny singular values of the upper triangular matrix appearing in AB-GMRES, making the process more stable, giving better convergence, and more accurate solutions compared to AB-GMRES. The method is also effective for making AB-GMRES stable for inconsistent least squares problems with highly ill-conditioned square coefficient matrices. Acknowledgments {#acknowledgments .unnumbered} =============== Ken Hayami was supported by JSPS KAKENHI Grant Number 15K04768.\ Keiichi Morikuni was supported by JSPS KAKENHI Grant Number 16K17639 and Hattori Hokokai Foundation.\ Jun-Feng Yin was supported by the National Natural Science Foundation of China (No. $11971354$). Proof of statement in section 2.3 {#ap1} ================================= \[lemma1\] Assume $\rm{\mathcal{N}}$$(\widetilde{A})$ $\cap$ $\rm{\mathcal{R}}$$(\widetilde{\it{A}})=\{0\}$, and grade$(\widetilde{A},b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}})=k$. Then, $\mathcal{K}_{k+1}(\widetilde{\it{A}},b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}})$ = $\widetilde{A}\mathcal{K}_{k}(\widetilde{\it{A}},b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}})$ holds. Note that $$\widetilde{\it{A}}\mathcal{K}_{k}(\widetilde{\it{A}},b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}})=\rm{span}\it\{\widetilde{\it{A}}b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}}, \widetilde{\it{A}}^2b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}}, \cdots,\widetilde{\it{A}}^k b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}}\}\subseteqq \rm{span}\it\{b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}},\widetilde{\it{A}}b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}},\cdots,\widetilde{A}^k b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}}\}=\mathcal{K}_{k+1}(\widetilde{\it{A}},b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}}).$$ grade$(\widetilde{\it{A}},b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}})=k$ implies that $$\mathcal{K}_{k+1}(\widetilde{\it{A}},b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}})=\mathcal{K}_{k}(\widetilde{\it{A}},b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}})= \rm{span}\it\{b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}},\widetilde{\it{A}}b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}},\cdots,\widetilde{\it{A}}^{k-1} b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}}\}.$$ Hence, $$\widetilde{\it{A}}^k b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}}=c_0b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}}+c_1\widetilde{\it{A}}b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}}+\cdots+c_{k-1}\widetilde{\it{A}}^{k-1} b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}},\quad c_i\in \mathbb{R}, i=0, 1, 2, \cdots, k-1.$$ If $c_0=0,$ $$\widetilde{\it{A}}^k b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}}=c_1\widetilde{\it{A}}b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}}+c_2\widetilde{\it{A}}^2b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}}+\cdots+c_{k-1}\widetilde{\it{A}}^{k-1} b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}}.$$ Hence, $$c_1\widetilde{\it{A}}b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}}+c_2\widetilde{\it{A}}^2b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}}+\cdots+c_{k-1}\widetilde{\it{A}}^{k-1} b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}}-\widetilde{\it{A}}^k b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}}=\widetilde{\it{A}}(c_1b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}}+\cdots+c_{k-1}\widetilde{\it{A}}^{k-2} b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}}-\widetilde{\it{A}}^{k-1} b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}})=0.$$ Hence, $$c_1b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}}+c_2\widetilde{\it{A}}^2b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}}+\cdots+c_{k-1}\widetilde{\it{A}}^{k-2} b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}}-\widetilde{\it{A}}^{k-1} b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}}\in \rm{\mathcal{N}}(\widetilde{\it{A}})\cap \rm{\mathcal{R}}(\widetilde{\it{A}}) = \{0\}.$$ which implies $$\widetilde{\it{A}}^{k-1} b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}}=c_1b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}}+c_2\widetilde{\it{A}}b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}}+\cdots+c_{k-1}\widetilde{\it{A}}^{k-2} b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}}.$$ Thus, $$\mathcal{K}_{k}(\widetilde{A},b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}})=\mathcal{K}_{k-1}(\widetilde{\it{A}},b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}}),$$ which contradicts with grade$(\widetilde{A},b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}})=k$. Hence, $c_0\neq 0$, and $$b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}}=d_1\widetilde{\it{A}}b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}}+d_2\widetilde{\it{A}}^2b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}}+\cdots+d_{k-1}\widetilde{\it{A}}^{k-1} b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}}+d_k\widetilde{\it{A}}^k b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}}.$$ Hence, $$\mathcal{K}_{k+1}(\widetilde{A},b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}})=\rm{span}\it\{b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}},\widetilde{A}b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}},\cdots,\widetilde{A}^k b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}}\}\subseteqq\rm{span}\it\{\widetilde{\it{A}}b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}}, \widetilde{\it{A}}^2b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}}, \cdots,\widetilde{A}^k b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}}\}=\widetilde{A}\mathcal{K}_{k}(\widetilde{A},b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}}).$$ Thus, $$\mathcal{K}_{k+1}(\widetilde{A},b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}})=\widetilde{A}\mathcal{K}_{k}(\widetilde{A},b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}}).$$ \[corollary1\] Assume $\rm{\mathcal{N}}$$(\widetilde{A})$ $=$ $\rm{\mathcal{N}}$$(\widetilde{\it{A}}^{\mathsf{T}})$, and grade$(\widetilde{A},b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}})=k.$ Then, $\mathcal{K}_{k+1}(\widetilde{A},b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}})=\widetilde{A}\mathcal{K}_{k}(\widetilde{A},b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}})$ holds. $\rm{\mathcal{N}}$$(\widetilde{A})$ $=$ $\rm{\mathcal{N}}$$(\widetilde{\it{A}}^{\mathsf{T}})$ implies that $$\rm{\mathcal{N}}(\widetilde{\it{A}})\cap \rm{\mathcal{R}}(\widetilde{\it{A}})=\rm{\mathcal{N}}(\widetilde{\it{A}}^{\mathsf{T}})\cap \rm{\mathcal{R}}(\widetilde{\it{A}})=\rm{\mathcal{R}}(\widetilde{\it{A}})^{\bot}\cap \rm{\mathcal{R}}(\widetilde{\it{A}})=\{0\}.$$ Hence, from Lemma \[lemma1\], Corollary \[corollary1\] holds. Proof of statement in section 2.3 {#ap2} ================================= \[lemma2\] Assume $\rm{\mathcal{N}}$$(\widetilde{A})$ $\cap$ $\rm{\mathcal{R}}$$(\widetilde{\it{A}})=\{0\}$, grade$(\widetilde{A},b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}})=k$, and $b\notin \mathcal{R}(\widetilde{A})$. Then, dim$(\mathcal{K}_{k+1}(\widetilde{A},b))=k+1$ holds.\ Let $c_0, c_1, \dots, c_k\in \mathbb{R}$ satisfy $$c_0b+c_1\widetilde{\it{A}}b+\cdots+c_{k}\widetilde{\it{A}}^{k} b=0.$$ Since $\rm{\mathcal{N}}$$(\widetilde{A})$ $\cap$ $\rm{\mathcal{R}}$$(\widetilde{\it{A}})=\{0\}$, $$b=b|_{\rm{\mathcal{R}}(\widetilde{\it{A}})}\oplus b|_{\rm{\mathcal{N}}(\widetilde{\it{A}})},$$ where $b|_{\rm{\mathcal{N}}(\widetilde{\it{A}})}$ denotes the orthogonal projection of $b$ onto $\mathcal{N}(\widetilde{\it{A}}).$ Hence, $$c_0b|_{\rm{\mathcal{N}(\widetilde{\it{A}})}}+c_0b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}}+c_1\widetilde{\it{A}}b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}}+\cdots+c_{k}\widetilde{\it{A}}^{k} b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}}=0.$$ If $c_0\neq 0$ $$b|_{\rm{\mathcal{N}(\widetilde{\it{A}})}}=-b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}}-\frac{c_1}{c_0}\widetilde{A}b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}}-\cdots-\frac{c_{k}}{c_0}\widetilde{A}^{k} b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}}\in \rm{\mathcal{R}(\widetilde{\it{A}})}.$$ Hence, $$b|_{\rm{\mathcal{N}(\widetilde{\it{A}})}}\in \rm{\mathcal{N}}(\widetilde{\it{A}})\cap \rm{\mathcal{R}}(\widetilde{\it{A}})=\{0\}.$$ Thus, $b|_{\rm{\mathcal{N}(\widetilde{\it{A}})}}=0,$ which contradicts $b\notin \mathcal{R}(\widetilde{\it{A}})$. Hence, we have $c_0= 0$, and $$c_1\widetilde{A}b+c_2\widetilde{A}^2b+\cdots+c_{k}\widetilde{A}^{k} b=c_1\widetilde{A}b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}}+c_2\widetilde{A}^2b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}}+\cdots+c_{k}\widetilde{A}^{k} b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}}=0.$$ But, since $$\rm dim( \rm{span}\it\{\widetilde{\it{A}}b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}}, \widetilde{A}^2b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}}, \cdots,\widetilde{A}^k b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}}\})=\rm dim( \rm\widetilde{\it{A}} {span}\it\{b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}},\widetilde{A}b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}}\cdots,\widetilde{A}^{k-1} b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}}\})=\rm dim(\it \widetilde{\it{A}}\mathcal{K}_{k}(\widetilde{A},b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}}))=k$$ holds from Lemma \[lemma1\], we have $c_1=c_2=\cdots=c_k=0,$ which implies $ \rm dim(\it \mathcal{K}_{k+1}(\widetilde{\it{A}},b))=k+1$. Assume $\rm{\mathcal{N}}$$(\widetilde{\it{A}})=$ $\rm{\mathcal{N}}$$(\widetilde{A}^{\mathsf{T}})$, grade$(\widetilde{A},b|_{\rm{\mathcal{R}(\widetilde{\it{A}})}})=k$, and $b\notin \mathcal{R}(\widetilde{A})$. Then, dim$(\mathcal{K}_{k+1}(\widetilde{A},b))=k+1$ holds.\ $\rm{\mathcal{N}}$$(\widetilde{\it{A}})=$ $\rm{\mathcal{N}}$$(\widetilde{A}^{\mathsf{T}})$ implies $\rm{\mathcal{N}}$$(\widetilde{A})$ $\cap$ $\rm{\mathcal{R}}$$(\widetilde{\it{A}})=\{0\}$. Hence, the corollary follows from Lemma \[lemma2\]. Proof of Theorem 4 {#ap3} ================== Note that $$R_{s+1}^{\mathsf{T}}R_{s+1}=\left( \begin{array}{cc} R_s & 0 \\ d^{\mathsf{T}} & r_{s+1,s+1} \\ \end{array} \right)\left( \begin{array}{cc} R_s & d \\ 0^{\mathsf{T}} & r_{s+1,s+1} \\ \end{array} \right)=\left( \begin{array}{cc} R_s^{\mathsf{T}}R_s & R_s^{\mathsf{T}}d \\ d^{\mathsf{T}}R_s & d^{\mathsf{T}}d+r_{s+1,s+1}^2 \\ \end{array} \right).$$ Assume fl$(r^2_{s+1,s+1})$ $\leq$ fl$(d^{\mathsf{T}}d)O(\epsilon).$ Then, since\ $$\begin{aligned} \rm f\!l\it (d^{\mathsf{T}}d)&=d^{\mathsf{T}}d+O(s\epsilon)d^{\mathsf{T}}d=(1+O(s\epsilon))d^{\mathsf{T}}d,\\ \rm f\!l\it(d^{\mathsf{T}}d+r^2_{s+1,s+1})&=(d^{\mathsf{T}}d+r^2_{s+1,s+1})(1+O(s\epsilon))=d^{\mathsf{T}}d(1+O(s\epsilon)),\end{aligned}$$ we have\ $$\label{eqapc1} \rm f\!l\it(R_{s+1}^{\mathsf{T}}R_{s+1})=\left( \begin{array}{cc} R_s^{\mathsf{T}}R_s +O(s\epsilon)|R_s|^{\mathsf{T}}|R_s|& R_s^{\mathsf{T}}d+O(s\epsilon)|R_s|^{\mathsf{T}}|d| \\ d^{\mathsf{T}}R_s+O(s\epsilon)|d|^{\mathsf{T}}|R_s| & d^{\mathsf{T}}d+O(s\epsilon)d^{\mathsf{T}}d\\ \end{array} \right)=\left( \begin{array}{c}R_s^{\mathsf{T}}\\ d^{\mathsf{T}}\end{array}\right)\left( \begin{array}{cc}R_s & d\end{array}\right)+\mathcal{O}(s\epsilon),$$ since $R_s=\mathcal{O}(1)$ and $d=\mathbb{O}(1).$ Note $$\left( \begin{array}{cc}R_s & d\end{array}\right)\left( \begin{array}{c}-R_s^{-1}d\\ 1\end{array}\right)=-R_sR_s^{-1}d+d=0,$$ since $R_s$ is nonsingular. Hence,\ $$\rm f\!l\it(\left( \begin{array}{cc}R_s & d\end{array}\right)\left( \begin{array}{c}-R_s^{-1}d\\ 1\end{array}\right))=\rm f\!l\it \{R_s \rm f\!l\it(-R_s^{-1}d)+d\} = [\rm f\!l \it \{R_s\rm f\!l\it(-R_s^{-1}d)\}+d]\{1+O(\epsilon)\}.$$ Note here that $$\rm f\!l\it \{R_s\rm f\!l\it (-R_s^{-1}d)\}=R_s\rm f\!l\it(-R_s^{-1}d)+O(s\epsilon)|R_s||R_s^{-1}d|,$$ and $$\label{eqapc2} \rm f\!l\it (-R_s^{-1}d)= -R_s^{-1}d+O(s^2\epsilon)M(R_s)^{-1}|d|$$ from Theorem \[th333\]. Hence, $$\rm f\!l\it (\left( \begin{array}{cc}R_s & d\end{array}\right)\left( \begin{array}{c}-R_s^{-1}d\\ 1\end{array}\right)) = O(s^2\epsilon)R_sM(R_s)^{-1}|d|+O(s\epsilon)|R_s||R_s^{-1}d| =\mathbb{O}(s^2\epsilon),$$ since $R_s^{-1}=\mathcal{O}(1)$ and $M(R_s)^{-1}=\mathcal{O}(1).$ Then, $$\rm f\!l\it(R_{s+1}^{\mathsf{T}}R_{s+1}\left( \begin{array}{c}-R_s^{-1}d\\ 1\end{array}\right)) = \rm f\!l\it(\{\left( \begin{array}{c}R_s^{\mathsf{T}}\\ d^{\mathsf{T}}\end{array}\right)\left( \begin{array}{cc}R_s & d\end{array}\right)+\mathcal{O}(s\epsilon)\}\left( \begin{array}{c}-R_s^{-1}d+\mathcal{O}(s^2\epsilon)M(R_s)^{-1}|d|\\ 1\end{array}\right))=\mathbb{O}(s^2\epsilon)=\mathbb{O}(\epsilon),$$ since (\[eqapc1\]), (\[eqapc2\]), and $O(s^2)=O(1).$ Since $\left(\begin{array}{c}-R_s^{-1}d\\ 1\end{array}\right)=\mathbb{O}(1),$ $R_{s+1}^{\mathsf{T}}R_{s+1}$ is numerically singular. By contraposition, ($\Leftarrow$) holds. Assume $R_{s+1}^{\mathsf{T}}R_{s+1}$ is not numerically singular. Then, there exists a vector $\left( \begin{array}{c} z \\ w \\ \end{array} \right)\in \mathbb{R}^{s+1}$ such that $\left|\left( \begin{array}{c} z \\ w \\ \end{array} \right)\right|>\mathbb{O}(\epsilon),$ and $$\begin{aligned} \rm f\!l\it \{R_{s+1}^{\mathsf{T}}R_{s+1}\left( \begin{array}{c} z \\ w \\ \end{array} \right)\}= R_{s+1}^{\mathsf{T}}\left(R_{s+1}\left( \begin{array}{c} z \\ w \\ \end{array} \right)+|R_{s+1}|\left|\left( \begin{array}{c} z \\ w \\ \end{array} \right)\right|O((s+1)\epsilon)\right)+\\ \left|R_{s+1}^{\mathsf{T}}\right|\left|R_{s+1}\left( \begin{array}{c} z \\ w \\ \end{array} \right)+|R_{s+1}|\left|\left( \begin{array}{c} z \\ w \\ \end{array} \right)\right|O((s+1)\epsilon)\right|O((s+1)\epsilon)=\mathbb{O}(\epsilon)\end{aligned}$$ assuming $O(s+1)=O(1).$\ Hence, $$\rm f\!l\it\{R_{s+1}^{\mathsf{T}}R_{s+1}\left( \begin{array}{c} z \\ w \\ \end{array} \right)\}=\left( \begin{array}{cc} R_s^{\mathsf{T}}R_s & R_s^{\mathsf{T}}d \\ d^{\mathsf{T}}R_s & d^{\mathsf{T}}d+r_{s+1,s+1}^2 \\ \end{array} \right)\left( \begin{array}{c} z \\ w \\ \end{array} \right)+\mathbb{O}(\epsilon)=\mathbb{O}(\epsilon).$$ Thus, $$\label{c1} R_s^{\mathsf{T}}R_sz+wR_s^{\mathsf{T}}d=\mathbb{O}(\epsilon),$$ $$\label{c2} d^{\mathsf{T}}R_sz+(d^{\mathsf{T}}d+r_{s+1,s+1}^2)w=\mathbb{O}(\epsilon).$$ (\[c1\]) can be expressed as $R_s^{\mathsf{T}}(R_sz+wd)=\mathbb{O}(\epsilon).$ From Lemma \[lemma3\], $R_s^\mathsf{T}$ is numerically nonsingular, so that $$\label{c3} R_sz+wd=\mathbb{O}(\epsilon).$$ Hence, from (\[c2\]), $d^{\mathsf{T}}R_sz+w(d^{\mathsf{T}}d+r_{s+1,s+1}^2)=d^{\mathsf{T}}(R_sz+wd)+wr_{s+1,s+1}^2=O(\epsilon).$ Thus, $wr_{s+1,s+1}^2=O(\epsilon)$. If $w=O(\epsilon),$ $R_sz=\mathbb{O}(\epsilon)$ from (\[c3\]). Since $R_s$ is numerically nonsingular, $z=\mathbb{O}(\epsilon),$ which contradicts with the assumption. Hence, $|w|>O(\epsilon),$ so that $r_{s+1,s+1}^2=O(\epsilon),$ which gives $$\rm f\!l\it(r_{s+1,s+1}^2) = O(\epsilon)\leq \rm f\!l\it( d^{\mathsf{T}}d)O(\epsilon).$$ \[lemma3\] Let $n=O(1)$. If $A\in \mathbb{R}^{n\times n}$ is numerically nonsingular, and $A^{-1}=\mathcal{O}(1)$, then $A^{\mathsf{T}}$ is numerically nonsingular. If $$\rm f\!l\it (A^{\mathsf{T}}x) = A^{\mathsf{T}}x+\mathbb{O}(n\epsilon)|A^{\mathsf{T}}||x|=\mathbb{O}(n\epsilon),$$ then $$\rm f\!l\it(x^{\mathsf{T}}A) = x^{\mathsf{T}}A+\mathbb{O}^{\mathsf{T}}(n\epsilon)=\mathbb{O}^{\mathsf{T}}(n\epsilon).$$ Thus, $$\rm f\!l(\it x^{\mathsf{T}}Ay) = \rm f\!l\it (x^{\mathsf{T}}A)y+O(n\epsilon)|\rm f\!l\it(x^{\mathsf{T}}A)||y|=O(n\epsilon)$$ holds for all $y=\mathbb{O}(1).$ For arbitrary $z=\mathbb{O}(1)\in \mathbb{R}^{n},$ let $$y=A^{-1}z=\mathbb{O}(1).$$ Then, $$\rm f\!l(\it Ay) = Ay+O(n\epsilon)|A||y|=z+O(n\epsilon)|A||y|.$$ Hence, $$z=\rm f\!l\it(Ay) + O(n\epsilon)|A||y| = \rm f\!l\it(Ay) + \mathbb{O}(n\epsilon).$$ Thus, we have $$\rm f\!l(\it x^{\mathsf{T}}z) = x^{\mathsf{T}}z + O(n\epsilon)|x|^{\mathsf{T}}|z| =\rm f\!l\it(x^{\mathsf{T}}Ay) + O(n\epsilon)= O(n\epsilon)$$ for arbitrary $z=\mathbb{O}(1)\in \mathbb{R}^n.$ Hence, $x=\mathbb{O}(\epsilon),$ so that $A^{\mathsf{T}}$ is numerically nonsingular. Proof of Theorem 5 in section 4.5 {#ap4} ================================= Let the singular value decomposition of $R_i$ be given by $R_i=U\Sigma V^{\mathsf{T}}\in \mathbb{R}^{i\times i},$ where $U, V$ are orthogonal matrices and $\Sigma=\rm diag\it(\sigma_1, \sigma_2, \dots, \sigma_i).$ Let $I_i\in \mathbb{R}^{i\times i}$ be the identity matrix. Then, we have $R_i'=\left( \begin{array}{c} R_i\\ \sqrt{\lambda}I_i\\ \end{array}\right)=U'\Sigma'V^{\mathsf{T}}$, where $U'=\left( \begin{array}{cc} U & 0\\ 0 & V \end{array}\right)$ and $\Sigma'=\left( \begin{array}{c} \Sigma\\ \sqrt{\lambda}I_i\\ \end{array}\right).$ Since $\Sigma'^{\mathsf{T}}\Sigma'=\Sigma^2+\lambda I_i=\rm diag\it (\sigma_1^2+\lambda, \sigma_2^2+\lambda, \dots, \sigma_i^2+\lambda),$ the singular values of $\left( \begin{array}{c} R_i\\ \sqrt{\lambda}I_i\\ \end{array}\right)$ are $\sqrt{\sigma_1^2+\lambda}\geq\sqrt{\sigma_2^2+\lambda}\geq\cdots\geq\sqrt{\sigma_i^2+\lambda}.$

--- abstract: 'Recently, a Dirac (particle-hole symmetric) description of composite fermions in the half-filled Landau level (LL) was proposed \[D. T. Son, Phys. Rev. X [**5**]{}, 031027 (2015)\], and we study its possible consequences on BCS (Cooper) pairing of composite fermions (CF’s). One of the main consequences is the existence of anisotropic states in single and bilayer systems, which was previously suggested in Ref. \[J. S. Jeong and K. Park, Phys. Rev. B [**91**]{}, 195119 (2015)\]. We argue that in the half-filled LL in the single layer case the gapped states may sustain anisotropy, because isotropic pairings may coexist with anisotropic ones. Furthermore, anisotropic pairings with addition of a particle-hole (PH) symmetry breaking mass term may evolve into rotationally symmetric states, i.e. Pfaffian states of Halperin-Lee-Read (HLR) ordinary CF’s. On the basis of the Dirac formalism, we argue that in the quantum Hall bilayer at total filling one, with decreasing distance between layers weak pairing of $p$-wave paired CF’s is gradually transformed from Dirac to ordinary, HLR - like, with concomitant decrease in the CF number. Global characterization of low-energy spectrum based on the Dirac CF’s agrees well with previous calculations performed by exact diagonalization on a torus. Finally, we discuss features of Dirac formalism when applied in this context.' author: - 'M. V. Milovanović$^{1}$, M. Dimitrijević Ćirić$^{2}$, and V. Juričić$^{3}$' title: Pairing instabilities of Dirac composite fermions --- Introduction ============ Composite fermions (CF’s) \[\] describe the physics of electrons in fractional quantum Hall (FQH) regime. At filling factor $\nu=1/2$, essentially they absorb the external flux, and make a metallic state \[\] with its own Fermi surface - Fermi surface of CF’s. By slightly modifying Read’s dipole construction of composite (neutral) fermions in the half-filled lowest Landau level \[\], an argument can be given for the accumulation of Berry phase equal to $\pi$ as a CF encircles its own Fermi surface \[\]. This has motivated a description of the CF’s in this setting in terms of Dirac fermions, which has been recently introduced in Ref. \[\], and have attracted some interest \[\]. The PH symmetric description of the half-filled LL is given in terms of a Dirac system of composite quasiparticles - Dirac CF’s at a finite chemical potential \[\] and in the presence of a gauge field. However, the implied existence of singularity at zero momentum in the CF spectrum was criticized \[\]. We may add that, due to the requirement of gauge invariance in two dimensions (2D), a small mass must be introduced into the Dirac theory (“parity anomaly") \[\]. This may be a way to heal and complete in the high-energy domain (“UV completion" \[\]) Dirac description of CF’s, and avoid singularity. Thus the description in terms of Dirac fermions may have capacity to capture essential, at least qualitative, aspects of the CF’s physics. To further examine this possibility in this work we consider BCS pairing of Dirac CF’s. First, in the framework of Dirac description of a single CF, we point out that, assuming Cooper pairing between spinor components, besides so-called PH symmetric Pfaffian, also anisotropic states can be realized. This is analogous to the ${^3}$He system in which both $B$ and $A$ (anisotropic) phases are possible \[\]. Next we discuss unconventional $p$-wave pairing of two kinds of Dirac CF’s, motivated by the situation in the quantum Hall bilayer (QHB) at total filling factor one, i.e. with each layer with half-filled lowest LL. In this system $p$-wave pairing between two kinds of non-relativistic Halperin-Lee-Read (HLR) composite fermions at intermediate interlayer distances was proposed in Ref. \[\], and, recently, this scenario was further substantiated by a detection of the topological signatures of the $p$-wave system in the torus geometry \[\]. Therefore, it is natural to ask how this picture may be modified if we take into account the description by two Dirac CF’s of the two half-filled LL monolayers, and consider their possible pairing. One of the main conclusions that we can draw by applying the Dirac CF formalism in the context of BCS pairing is that due to the Dirac two-component nature, isotropic (gapped) pairing states may coexist with anisotropic ones, and this is in the accordance with the results on PH symmetric, single and bilayer fractional quantum Hall systems obtained by employing exact diagonalization \[\], as well as with experimental findings \[\], in which anisotropy is probed by in-plane magnetic field. This may be a direct consequence of the dipole nature of CF’s that is captured by Dirac formalism. Anisotropic pairing states may serve as seed states for Pfaffian and anti-Pfaffian through a process in which PH asymmetry increases by introducing a mass term, while rotational symmetry gradually sets in. Furthermore, we find that the features, in particular low-energy spectrum, of the QHB at intermediate distances between the layers are better captured if we assume Dirac rather than HLR $p$-wave paired CF’s at large distances (decoupled layers). Already at the effective field theory level, modeling the evolution with the distance between layers by Dirac CFs, we can detect the main feature of CF-composite boson (CB) mixed states \[\]: the decrease in the number of CF’s with decreasing distance. The paper is organized as follows. In Section II , based on Dirac formalism, we discuss the single layer case and its pairing instabilities, including the situation when the PH symmetry is spoiled by a mass term. In Section III we discuss the pairing instabilities in the bilayer system when the PH symmetry inside each layer is intact. In Section IV we examine the evolution of low-energy properties of the QHB with distance between layers, by including a mass term with an opposite sign in the two layers. The last section, Section V, is devoted to discussion and conclusions. Mean-field analysis of the coexistence of the isotropic and anisotropic pairings is presented in the Appendices. Dirac composite fermion and Cooper pairing ========================================== We begin by considering a single Dirac fermion which was proposed to effectively describe half-filled lowest Landau level of electrons \[\], with $s$-wave pairing between spinor components. The $s$-wave pairing suggested in Ref. \[\], can be expressed by the following Bogoliubov - de Gennes Hamiltonian in the Nambu - Gorkov notation, $$\begin{aligned} \label{BdGsD} H &=&\frac{1}{2} \sum_{\bf k} \left[ \begin{array}{cc} \Psi^\dagger ({\bf k}) & \tilde{\Psi}(-{\bf k}) \\ \end{array} \right] \\ \nonumber &\times&\left[ \begin{array}{cc} {\cal D} ({\bf k}) & {\cal P}({\bf k}) \\ {\cal P}^\dagger ({\bf k}) & - {\cal D} (-{\bf k}) \\ \end{array} \right] \left[ \begin{array}{c} \Psi ({\bf k}) \\ \tilde{\Psi}^\dagger (-{\bf k}) \\ \end{array} \right],\end{aligned}$$ where $\Psi ({\bf k})$ denotes a two-component spinor with momentum ${\bf k}$, $$\Psi ({\bf k}) = \left[ \begin{array}{c} \Psi_{a} ({\bf k}) \\ \Psi_{b} ({\bf k}) \\ \end{array} \right], \,\,\,\, \tilde{\Psi} ({\bf k}) = \left[ \begin{array}{c} \Psi_{b} ({\bf k}) \\ \Psi_{a} ({\bf k}) \\ \end{array} \right],$$ and $${\cal D}({\bf k}) = \left[ \begin{array}{cc} - \mu & k_x - i k_y \\ k_x + i k_y & - \mu \\ \end{array} \right]=-\mu\sigma_0+k_x \sigma_x+k_y \sigma_y, \label{dmatrix}$$ and $2 \times 2$ matrix ${\cal P}({\bf k})$ describes Cooper pairing between $a$ and $b$ spinor components $${\cal P}({\bf k}) = \left[ \begin{array}{cc} \Delta_s & 0\\ 0 & - \Delta_s \\ \end{array} \right]=\Delta_s \sigma_z , \label{right_pairing_sD}$$ or more explicitly $$\delta {\cal H} = \sum_{\bf k} \{ - \Delta_s \Psi_{a}({\bf k}) \Psi_{b}(-{\bf k}) + h.c. \}. \label{rightH}$$ Here, $\sigma_0$ is the $2\times2$ identity matrix, while ${\bm \sigma}$ are the standard Pauli matrices. Throughout the paper we set $\hbar = 1$, and the Fermi velocity, $v_F = 1$. $\mu$ denotes a chemical potential equal to $\mu = \sqrt{B} = k_F$, where $B$ and $k_F$ are the external magnetic field and Fermi vector, respectively. Since the pairing matrix anticommutes with the free Dirac Hamiltonian at the zero chemical potential, the dispersion of Bogoliubons has the rotationally symmetric form $$E_k^2 = (k \pm \mu)^2 + \Delta_{s}^2,$$ where $k\equiv |{\bf k}|$. This construction is considered in the literature as a basis for a PH symmetric Pfaffian system. However, a different type of pairing is also possible with the pairing matrix $${\cal P}({\bf k}) = \left[ \begin{array}{cc} 0 & \alpha k_x + \beta k_y\\ \alpha k_x - \beta k_y & 0 \\ \end{array} \right],$$ or more explicitly $$\begin{aligned} \delta {\cal H}'& =&\sum_{{\bf k}} \alpha k_x \{ \Psi_{a}^\dagger ({\bf k}) \Psi_{a}^\dagger(-{\bf k}) + \Psi_{b}^\dagger({\bf k}) \Psi_{b}^\dagger(-{\bf k}) \} + h.c. \nonumber\\ &+&\sum_{\bf k} \beta k_y \{ \Psi_{a}^\dagger({\bf k}) \Psi_{a}^\dagger(-{\bf k}) - \Psi_{b}^\dagger({\bf k}) \Psi_{b}^\dagger(-{\bf k}) \} + h.c. \nonumber \\ \label{anisotropicH}\end{aligned}$$ where $\alpha$ and $\beta$ are, in general, allowed to be complex coefficients. The overall form of $ \delta {\cal H}'$ is fixed by the requirement of the $CP$ symmetry, which, as emphasized in Ref. \[\], is equivalent to the requirement of the PH symmetry in the real electron system. Namely, the $CP$ symmetry is a product of the charge conjugation, $C$, $$C \Psi({\bf k}) C^{-1} = \sigma_x \Psi^* ({\bf k}),$$ and a parity transformation, $P$, $$P \Psi({\bf k}) P^{-1} = \Psi^* ({\bf k}'),$$ where ${\bf k} = (k_x, k_y) \rightarrow {\bf k}' = (k_x, - k_y)$ under the parity transformation. Thus, $$CP \Psi({\bf k}) (CP)^{-1} = \sigma_x \Psi({\bf k}'). \label{CP}$$ The starting Dirac Hamiltonian (\[BdGsD\]) with ${\cal P}=0$, as well $ \delta {\cal H}'$ are both invariant under the $CP$ transformation (\[CP\]). Notice that (\[rightH\]) is invariant up to a sign change under $CP$ transformation. This is also a property of the small mass term that seems necessary to ensure the gauge invariance of the theory, and to avoid the singularity at ${\bf k} = 0$ \[\]. The BCS pairing terms as the one in (\[rightH\]) may accommodate the sign change by gauge transformations \[\]. Thus the theory is invariant under $CP$ transformation in a more general sense, allowing for terms that are invariant up to a change of the sign. This makes our choice for $p$ wave not unique. Indeed, other $p$ wave pairing order parameters are also possible, including one analogous to the A phase of ${^3}$He system that features two (gapless) Fermi points. This case can be analyzed analogously to the one considered here, and the main conclusions remain. In the following, we restrict our discussion to the $p$-wave case (\[anisotropicH\]) invariant under $CP$ transformation in the strict sense. We now consider the pairings given by Eq. (\[anisotropicH\]), recently also discussed in Ref. \[\], in light of the possibility of introducing an anisotropy. The choice $\alpha = \Delta$ and $\beta = - i \Delta $ yields the pairing matrix, $ {\cal P}({\bf k})$, proportional to the Dirac Hamiltonian, ${\cal D}({\bf k})$, at chemical potential $\mu = 0$, and thus explicitly rotationally invariant. (See also Sec. III for further analysis of the rotational symmetry.). In that case, the dispersion relation of Bogoliubons, $E_{\bf k}^2 = k^2(1+\Delta^2)+\mu^2 \pm 2k \sqrt{\mu^2 + k^2 \Delta^2}$, implies that the the pairing just renormalizes chemical potential. On the other hand, by choosing $\alpha = \Delta$ and $\beta = + i \Delta $, we obtain $$E_{\bf k}^2 = k^2 (1 + \Delta^2) + \mu^2 \pm 2 \sqrt{\mu^2 k^2 + \Delta^2 (k_x^2 - k_y^2)^2}. \label{andispersion}$$ This dispersion describes an [*anisotropic*]{} gapless system with four nodes at $$k_x = \pm \frac{\mu}{\sqrt{1 - \Delta^2}}, \;\;\;\; {\rm and}\;\;\; k_y = 0,$$ and $$k_y = \pm \frac{\mu}{\sqrt{1 - \Delta^2}}, \;\;\;\; {\rm and}\;\;\; k_x = 0.$$ The appearance of the four nodes related by the discrete $C_4$ symmetry is a consequence of the $C_4$ symmetry of the pairing (\[anisotropicH\]) with $\alpha = \Delta$ and $\beta = + i \Delta $. In fact, Eq. (\[anisotropicH\]) describes a whole family of gapless anisotropic solutions. If we consider both $s$-wave (\[rightH\]) and $p$-wave (\[anisotropicH\]) with $\alpha = \Delta$ and $\beta = + i \Delta $ pairings, the dispersion of the Bogoliubov quasiparticles is $$\tilde{E}_{\bf k}^2 = \Delta_s^2 + k^2 (1 + \Delta^2) + \mu^2 \pm 2 \sqrt{\mu^2 k^2 + \Delta^2 (k_x^2 - k_y^2)^2},$$ i.e. the dispersion (\[andispersion\]) simply acquired a shift of $\Delta_s^2$ in the presence of the $s$-wave pairing. This is a consequence of the anticommutation of the matrices corresponding to the two pairings, similarly to the situation in Refs. \[, \], which makes their coexistence likely at a finite chemical potential. Assuming a generic form of the two couplings driving the instabilities in the isotropic and anisotropic channels, in the presence of a small mass term, we show in Appendix A that the low energy description implies that the isotropic instability (\[rightH\]) may coexist with the anisotropic one. This is consistent with experimental \[\], and theoretical \[\] findings pointing out that gapped states at half-filled Landau level can sustain and even harbor anisotropy. In connection with the possible pairings given by Eq. (\[anisotropicH\]) when $\alpha = \Delta$ and $\beta = + i \; \Delta $, we may notice that if we break $CP$ (particle-hole symmetry) by a mass term of the form $\sim \Psi^\dagger({\bf k}) \sigma_3 \Psi({\bf k})$, one component, $a$ or $b$, of the Dirac field will remain in the low energy sector. The remaining fermion should correspond to HLR (spinless) fermion which in turn pairs in the manner of $p$-wave. This should correspond to Pfaffian and anti-Pfaffian states (that comprise possible $( k_x \pm i k_y)$ states), in the absence of PH symmetry, but with an emergent rotational symmetry. A closely related proposal for the existence of the Pfaffian (Moore-Read) state in the presence of an excitonic instability already appeared in the context of Dirac CF physics in graphene Ref. \[\]. To further understand the pairings in Eqs. (\[rightH\]) and (\[anisotropicH\]), we now consider the chirality operator $\frac{{\bm\sigma} \cdot{\bf k}}{|k|}$, and its eigenstates $$|+ \rangle = \frac{1}{\sqrt{2}} \left[ \begin{array}{c} 1 \\ \frac{k_+}{k} \\ \end{array} \right], \,\,\,\, |- \rangle = \frac{1}{\sqrt{2}} \left[ \begin{array}{c} - 1 \\ \frac{k_+}{k} \\ \end{array} \right].\label{chiral-eigenstates}$$ We can introduce Dirac operators with a definite chirality $$\Psi_+ ({\bf k}) = \frac{1}{\sqrt{2}}( \Psi_a ({\bf k}) + \frac{k_-}{k} \Psi_b ({\bf k})) ,$$ and $$\Psi_- ({\bf k}) = \frac{1}{\sqrt{2}}( - \Psi_a ({\bf k}) + \frac{k_-}{k} \Psi_b ({\bf k})) ,$$ to find that $$\Psi_a ({\bf k}) \Psi_b ( - {\bf k}) =- \frac{1}{2} \frac{k_+}{k} [\Psi_+ ({\bf k}) \Psi_+ ( - {\bf k}) + \Psi_- ({\bf k}) \Psi_- (-{\bf k})], \label{s_into_p}$$ with $k_\pm\equiv k_x\pm i k_y$. We can clearly see from Eq. (\[s\_into\_p\]) that in the chirality basis i.e. the eigenbasis of the non-interacting system, the pairing (\[rightH\]), in fact, describes a pairing in the odd ($p$-wave) channel. This can be understood as a consequence of the non-trivial Berry phase contributions, as discussed in Ref. \[\]; see also Ref. \[\] for the influence of the singularities (topological charges) on the vorticity of Cooper pairs. On the other hand, the anisotropic pairing (\[anisotropicH\]) is a combination of odd channel components in the chirality basis. We now analyze an alternative scenario for the coexistence with the $p$-wave pairing represented by the pairing matrix ${\cal P}({\bf k})=(\alpha k_x+\beta k_y)\sigma_x$ that features two Fermi points and does not require a mass for the coexistence with the isotropic state. In particular, as shown in Appendix B, a special anisotropic pairing with $$\label{anisotropicHH} {\cal P} \sim i k_y\sigma_x$$ can coexist with the isotropic pairing. Analogously, we can discuss pairing with ${\cal P}({\bf k})\sim(\gamma k_x+\delta k_y)\sigma_y$, where $\gamma$ and $\delta$ are, in general, allowed to be complex coefficients. The ensuing pairing is then given by $$\label{anisotropicHH2} {\cal P}({\bf k})\sim k_x \sigma_y.$$ Both these pairings are invariant up to a change of sign (up to a gauge transformation) under the $CP$ transformation. Each pairing on its own features two Fermi points, and is likely energetically advantageous over the pairing in (\[anisotropicH\]) that has four Fermi points. As we explicitly show in Appendix B, these pairings do not need a mass term to coexist with the isotropic state. Furthermore, in the presence of a mass term, they develop new components, and may thus evolve into the rotationally symmetric pairings of HLR fermions. These are the reasons that make states given by Eqs. (\[anisotropicHH\]) or (\[anisotropicHH2\]) likely present when considering pairing instabilities in the half-filled LL, consistent with the exact diagonalization results of Refs. \[\]. Finally, we point out that the Dirac based microscopic wave functions of pairing instabilities have not been proposed and tested yet. The effective field theory approach seems currently to be the most efficient tool for treating the Dirac-based pairing instabilities and their properties. Once the microscopic description is provided, most importantly for the case of PH Pfaffian, anisotropic modifications may be induced in the manner described and discussed in Refs. \[, \]. Dirac fermions and $p$-wave pairing =================================== We consider the following general form of the Bogoliubov-de Gennes Hamiltonian, motivated by the situation in a QHB system with each of the two layers at half filling, $$\begin{aligned} \label{BdG} H &=& \sum_{\bf k} \left[ \begin{array}{cc} \Psi_\uparrow^\dagger ({\bf k}) & \Psi_\downarrow (-{\bf k}) \\ \end{array} \right] \\ \nonumber &\times&\left[ \begin{array}{cc} {\cal D}_\uparrow ({\bf k}) & {\cal P}({\bf k}) \\ {\cal P}^\dagger ({\bf k}) & - {\cal D}_\downarrow (-{\bf k}) \\ \end{array} \right] \left[ \begin{array}{c} \Psi_\uparrow ({\bf k}) \\ \Psi_\downarrow^\dagger (-{\bf k}) \\ \end{array} \right],\end{aligned}$$ where $\Psi_\uparrow ({\bf k})$ and $\Psi_\downarrow ({\bf k})$ are two component spinors, $$\Psi_\uparrow ({\bf k}) = \left[ \begin{array}{c} \Psi_{a \uparrow} ({\bf k}) \\ \Psi_{b \uparrow} ({\bf k}) \\ \end{array} \right], \,\,\,\, \Psi_\downarrow ({\bf k}) = \left[ \begin{array}{c} \Psi_{b \downarrow} ({\bf k}) \\ \Psi_{a \downarrow} ({\bf k}) \\ \end{array} \right].$$ Matrices ${\cal D}_\uparrow ({\bf k})$ and ${\cal D}_\downarrow ({\bf k})$ describe two identical Dirac systems, ${\cal D}_\uparrow ({\bf k})={\cal D}_\downarrow ({\bf k})={\cal D} ({\bf k})$, with ${\cal D} ({\bf k})$ given by Eq. (\[dmatrix\]), while $2 \times 2$ matrix ${\cal P}({\bf k})$ describes Cooper pairing between the two systems $\uparrow$ and $\downarrow$. A triplet $p$-wave pairing between the same spinor components can be expressed as the following term in the Hamiltonian, $$\begin{aligned} \delta {\cal H} &=& \sum_{\bf k} \{ [ \Delta^*_{\bf k} \Psi_{a \downarrow}(-{\bf k}) \Psi_{a \uparrow}({\bf k}) \\ \nonumber &+&\Delta^*_{\bf k} \Psi_{b \downarrow}(-{\bf k}) \Psi_{b \uparrow}({\bf k}) ] \, + \, h.c. \},\end{aligned}$$ with a pairing function $ \Delta_{\bf k} = \Delta (k_x \pm i k_y)$. The corresponding pairing matrix in the Hamiltonian (\[BdG\]) is $${\cal P}({\bf k}) = \left[ \begin{array}{cc} 0 & \Delta_{\bf k} \\ \Delta_{\bf k} & 0 \\ \end{array} \right] = \Delta_{\bf k} \sigma_x.\label{anisotropic_pairing}$$ A rotation around $z$ axis by an angle $\phi$ in both subsystems $\uparrow$ and $\downarrow$ is represented by a matrix $ R = \exp (i \sigma_z \phi/2)$ so that $$\begin{aligned} R \sigma_x R^{-1} &=& \sigma_x \cos \phi - \sigma_y \sin \phi, \\ \nonumber R \sigma_y R^{-1} &=& \sigma_x \sin \phi + \sigma_y \cos \phi.\end{aligned}$$ It can be readily seen that $ \tilde{R} H({\bf k}) \tilde{R}^{-1} \neq H({\bf k}')$ where $ k_{x}^{'} = k_x\cos\phi - k_y\sin\phi $ and $ k_{y}^{'} = k_x\sin\phi + k_y\cos\phi $, and ${\tilde R}=\tau_0\otimes R$, with $\tau_0$ as the $2\times2$ unity matrix in the subsystem space. Therefore, the system with the pairing matrix ${\cal P}({\bf k}) = \Delta_{\bf k} \sigma_x$ is not rotationally invariant, and may lead to anisotropic behavior. In fact, the system is gapless, and supports two anisotropic Dirac cones at $ k_x^2 = \mu^2/(1 - \Delta^2) = k_0^2$ and $k_y = 0$. Expanding around $\pm k_0$ we obtain for $\Delta \ll 1$, $ E^2 \approx (1 - 2 \Delta^2) (\delta k_x)^2 + \Delta^2 (\delta k_y)^2$. We find similar results if we choose, $${\cal P}({\bf k}) = \left[ \begin{array}{cc} 0 & \Delta_{\bf k} \\ -\Delta_{\bf k} & 0 \\ \end{array} \right].\label{sign_anisotropic_pairing}$$ Therefore the systems that we considered by now do not possess quantum spin Hall effect, and due to anisotropy are likely to be fragile under disorder, and certainly can not represent stable phases in realistic circumstances. On the other hand, the system with the pairing matrix $${\cal P}({\bf k}) = \left[ \begin{array}{cc} \Delta_{\bf k} & 0\\ 0 & - \Delta_{\bf k} \\ \end{array} \right] = \Delta_{\bf k} \sigma_z , \label{right_pairing}$$ yields the dispersion relation of the Bogoliubons $$E_{\pm} = \sqrt{(k \pm \mu)^2 + |\Delta_{\bf k}|^2}, \label{disper_triplet}$$ and therefore resembles very closely $p$-wave pairing of ordinary fermions. We now express the pairing in the chirality basis to obtain $$\begin{aligned} &&\Delta^*_{\bf k} (\Psi_{b \downarrow}(-{\bf k}) \Psi_{a \uparrow}({\bf k}) - \Psi_{a \downarrow}(-{\bf k}) \Psi_{b \uparrow}({\bf k}))=- \Delta^*_{\bf k} \frac{1}{2} \frac{k_+}{k} \nonumber \\ &\times& (\Psi_{+ \downarrow}(-{\bf k}) \Psi_{+ \uparrow}({\bf k}) - \Psi_{- \downarrow}(-{\bf k}) \Psi_{- \uparrow}({\bf k})). \nonumber \\ \end{aligned}$$ Thus depending whether $ \Delta_{\bf k} = \Delta (k_x + i k_y)$ or $ \Delta_{\bf k} = \Delta (k_x - i k_y)$, we obtain $s$-wave or $d$-wave pairing, respectively, in the chirality basis. In this sense there is no surprise to find that the pairing matrix (\[right\_pairing\]) gives rise to a singlet state for $\uparrow$ and $\downarrow$ electrons. The choice for the pairing without the minus sign in Eq. (\[right\_pairing\]), i.e., ${\cal P}({\bf k})=\Delta_{\bf k}\sigma_0$, is not energetically favorable, since pairing just renormalizes chemical potential in that case. We now provide a topological characterization of pairing in Eq. (\[right\_pairing\]) through the (pseudo)spin Chern number, $C_s$. In fact we find that in this case is $C_s = 1$, if we use the low-energy theory with (\[right\_pairing\]) and $ \Delta_{\bf k} = \Delta (k_x + i k_y)$. We calculated the Chern number by taking the eigenvectors of the two lower Bogoliubov bands, $ |v_{-}({\bf k})\rangle$ and $ |v_{+}({\bf k})\rangle$, corresponding to eigenvalues $ - E_{-}({\bf k})$ and $ - E_{+}({\bf k})$, respectively. We first computed the Berry curvature of each vector, $$F^{\sigma}_{xy}({\bf k}) = i (\partial_x \langle v_{\sigma}({\bf k})| \partial_y |v_\sigma ({\bf k}) \rangle - \partial_y \langle v_{\sigma}({\bf k})| \partial_x |v_\sigma ({\bf k}) \rangle ),$$ and then the Chern number, $$C_s = \frac{1}{2 \pi} \sum_\sigma \int d{\bf k} F^{\sigma}_{xy}({\bf k}), \label{chern}$$ where the sum in (\[chern\]) is over the two lowest bands. Nevertheless, as discussed in the previous paragraph, and, also, due to the form of eigenvectors below, we expect that the real winding number is zero or two if a complete description is taken into account. To further characterize the pairing state, let us consider the four-component vectors of the Bogoliubov bands with positive energy, $ E_{-}({\bf k})$ and $ E_{+}({\bf k})$, $$\begin{aligned} u_{-}(k) &=& \frac{1}{2 \sqrt{E_{-}}} \left\{ - \sqrt{E_{-} - (\mu - k)} \left(1, \frac{k_{+}}{k}\right),\right. \nonumber\\ & & \left.\frac{\Delta \cdot k}{\sqrt{E_{-} - (\mu - k)}} \left(- \frac{k_{-}}{k},1\right) \right\},\end{aligned}$$ and $$\begin{aligned} u_{+}(k) &=& \frac{1}{2 \sqrt{E_{+}}} \left\{ \sqrt{E_{+} - (\mu + k)} \left(1, - \frac{k_{+}}{k}\right),\right.\nonumber\\ & &\left. \frac{\Delta \cdot k}{\sqrt{E_{+} - (\mu + k)}} \left(\frac{k_{-}}{k}, 1\right) \right\},\end{aligned}$$ where we regrouped components to appear with common factors. In each Bogoliubov eigenstate, the first two-component spinor, ( , ), is an eigenstate of the chirality operator, given by Eq. (\[chiral-eigenstates\]), while the second one is the eigenstate that is complex conjugated and with inverted components due to the ordering in the Nambu - Gorkov representation, and we fix $ \Delta_{\bf k} = \Delta (k_x + i k_y)$. From the coefficients in front of the fixed chirality states, we find the long-distance behavior of the pairing function ($g_{\bf k} \sim v_{\bf k}/u_{\bf k}$ in the usual BCS problem) in each band $$g(z) \sim 1/|z| ,$$ where $g(z)$ is the pairing function in the real space, and $z = x + i y$. Thus the pairing function has the characteristic $s$-wave feature. In this case the lowest gap is at Fermi surface, $\Delta E \sim \Delta \cdot k_F$, in contrast with ordinary $p$-wave pairing where the lowest gap is at zero momentum, and it is equal to $\Delta E \sim k_F$ \[\]. Quantum Hall bilayer and $p$-wave paired composite fermions =========================================================== In light of recent advance in understanding of each (isolated PH symmetric) half-filled quantum Hall monolayer based on Dirac CF’s it is quite natural to consider the physics of the bilayer, especially at the intermediate distances, in the same framework. It is important to take into account the $p$-wave pairing \[\] which was initially expressed in terms of ordinary HLR CF’s. The picture based on the ordinary CF’s does not have a clear answer for the lowest lying spectrum which appears nearly gapless (with small gap) or gapless when the system is put on a torus, while the topological $p$-wave pairing of ordinary fermions \[\] would likely produce a clear gap of the order $\mu$. However, even if we neglect possible insufficiencies with $p$-wave pairing of ordinary fermions, it is fundamentally important to address the problem of the QHB in terms of Dirac CF’s. First we may notice that the presence of the interlayer Coulomb interaction, which increases with decreasing distance between layers, spoils PH symmetry inside a layer. We incorporate this breaking of the PH symmetry by introducing a mass $r$ in the Dirac matrices ${\cal D}_\uparrow ({\bf k})$ and ${\cal D}_\downarrow ({\bf k})$, with opposite signs in each layer, $${\cal D}_\uparrow ({\bf k}) = \sigma_x k_x + \sigma_y k_y - \mu + r \sigma_z = {\cal D}_\downarrow ({\bf k}).$$ Second, the components of the spinors in different layers are inverted with respect to each other, and thus the mass term of the opposite sign in the two layers enters with the same sign in the matrices ${\cal D}_\uparrow ({\bf k})$ and ${\cal D}_\downarrow ({\bf k})$. The dispersion relation in this case acquires the form $$E_{\pm} = \sqrt{(\sqrt{k^2 + r^2} \pm \mu)^2 + |\Delta_{\bf k}|^2}. \label{disper_triplet_with_r}$$ The masses in the two layers are of the opposite sign, due to the requirement of the PH symmetry of the whole system. Namely, under the transformation in each layer masses change sign \[\], and if we, in addition, exchange layer index we recover the original Hamiltonian. There are two important things to notice regarding the evolution of the CF state with increasing mass $r$: \(a) The minimum of the lower Bogoliubov band shifts from a finite value at $k_F^2 = \mu^2/(1 + \Delta^2)^2 - r^2$ to $k = 0$, and this transition - without closing of the gap - occurs at $r = \mu/(1 + \Delta^2)$; \(b) Because $k_F^2 = \mu^2/(1 + \Delta^2)^2 - r^2$, the Fermi momentum decreases with the mass, and therefore the number of CF’s reduces as the distance between the layers decreases. Therefore the most important consequence of the assumed Dirac description of individual layers at large distances is that the number of CF’s decreases as the distance between the layers decreases. For large distances we may assume that the pairing is weak, the order parameter is small, and the pairing cannot be detected then due to finite temperature effects, for instance. In any case we may choose, $ \Delta_{\bf k} = \Delta (k_x + i k_y)$, so that there is no Hall drag (pseudospin Hall effect) at large distances, but it develops gradually as the interlayer distance decreases and reaches the quantized value in agreement with experiments \[\]. This choice of the order parameter agrees with Refs. \[\]. For smaller distances $(r \sim \mu$ but $ r < \mu)$ we may assume that the upper Bogoliubov band is pushed to high energies and an effective description in terms of quadratically dispersing CF’s paired via weak $p$-wave pairing emerges, implying an algebraically decaying Cooper pair wave function \[\]. The description of the system within this scenario then implies that at intermediate distances CB-CF mixture accounts for the total number of electrons \[, \]. As a consequence, composite bosons cannot have long range order, and likely have critical, algebraic pairwise correlations \[\]. If at intermediate distances solely a collection of $p$-wave paired composite fermions, quadratically dispersing as in Ref. \[\], were present, signals of a topological phase with a large gap, $\Delta E \sim \mu$ would appear. Instead, as detected on a torus in Ref. \[\], there is an abundance of various low-energy excitations. This is in accordance with the above physical picture that implies a small portion of CF’s at intermediate distances in a topological phase with a small gap $\Delta E \sim \mu - r$, and $\mu \simeq r$. As in the single layer case, anisotropic gapless solution (\[anisotropic\_pairing\]) is possible also for a bilayer. In the presence of a mass term $\sim r$ and in the case of the pairing (\[anisotropic\_pairing\]) we obtain two anisotropic Dirac cones at $ k_x^2 = \mu^2/(1 - \Delta^2) = k_0^2$ and $k_y = 0$. Expanding around $\pm k_0$ with $r \ll \mu$ we obtain $ E^2 \approx (1 - 2 \Delta^2 - \frac{r^2}{\mu^2}) (\delta k_x)^2 + \Delta^2 (\delta k_y)^2$. The absence of a gap suggests a non-topological behavior. On the other hand, topological signatures were detected at intermediate distances in Ref. \[\], in agreement with the characterization of isotropic weak $p$-wave pairing. Thus the presence of the isotropic pairing, which may be accompanied by anisotropic ones, seems crucial for the explanation of the properties at intermediate distances. Discussion and Conclusions ========================== The existence of anisotropic candidates for BCS paired states, in the case of monolayer (Sec. II), and bilayer (Sec. III and IV), is in agreement with the results in Ref. \[\]. In that paper, the physics of the PH symmetric case of half-filled second Landau level is studied by exact diagonalization on a torus. The main result of this numerical study is that the paired quantum Hall state in that case, as well closely related (by antisymmetrization) bilayer state, made of two kinds of electrons that each occupy quarter of the available single particle states in the second Landau level, are susceptible to anisotropic instabilities. By using the Dirac description of the dipole nature of CFs, we can identify the paired quantum Hall state of Ref. \[\] with PH Pfaffian, and its closeness to anisotropy as a sign of the relevance of anisotropic solutions discussed in Sec. II. On the other hand, the relevance of the anisotropy for the bilayer state at effective $\nu=1/2 = 1/4 + 1/4$ total filling factor \[\], may be again due to the composite - dipole nature of the CF’s at filling factor $\nu=1/4$. The Dirac description could be the easiest way to capture the dipole nature of a CF, despite the doubling of the fermionic degrees of freedom. In other words, we need particles and holes to describe dipoles \[\], and the Dirac formalism could be a way to achieve that even in the cases when CF’s have a Berry phase equal to $\pi/2$ (at quarter filling), with appropriate mass and chemical potential. If the Diracness is the cause of the anisotropic behavior, we can conclude that the Dirac formalism is equally applicable at $\nu=1/2$ and $\nu=1/4$. In this sense “nothing is special at $\nu=1/2$" (Ref. \[\]) since only PH symmetry singles out Dirac description. The PH symmetry is sufficient but not necessary to cause the Diracness at the filling equal to one half. If we restrict our discussion only to the case when CF’s possess Berry phase equal to $\pi$, and thus Dirac formalism seems appropriate for the bilayer case at total filling one, we demonstrated that the description by Dirac fermions is justified due to a global appearance and characterization of low-energy spectrum from the exact diagonalization on a torus \[\]. In fact, the Dirac CF in the bilayer changes its Berry phase from value $\pi$ at large distances, to value $\sim 0$, at small distances (HLR fermion), while retaining its fermionic character. The second important consequence, due to the use of the Dirac formalism, is that the number of CF’s is decreasing with the decreasing distance between layers. This is in in agreement with the necessity to use CF-CB mixed states to describe the bilayer at intermediate distances \[\]. Thus we can conclude that Dirac formalism can capture the basic phenomenology of the bilayer at $\nu=1$, and the nature of the gapped paired states in the single layer quantum Hall systems with half-filled LL. We therefore expect it to become an indispensable tool for further understanding of the paired states in this context. [*Note added:*]{} While this manuscript was in the final stage of preparation, Ref. \[\] appeared. It is a study of possible pairings, based on Dirac formalism, and their realization in the case of a single layer with the half-filled LL. Ref. \[\] considered pairings in the low-energy subspace of Dirac spectrum in the context of a specific pairing mechanism. In our work the low-energy projection is in place after the consideration of the pairing instabilities within the Dirac formalism. In this way we are able to account for the anisotropic pairings, with the consequences consistent with theoretical and experimental findings, as we already emphasized. We would like to thank S. Simon for a discussion. The work was supported by the Ministry of Education, Science, and Technological Development of the Republic of Serbia under projects ON171017 and ON171031. Coexistence of the $CP$ invariant $p$-wave and $s$-wave pairings: Mean-field analysis ===================================================================================== The lower Bogoliubov band of the quadratic Hamiltonian, Eqs. (\[BdGsD\]-\[right\_pairing\_sD\]) with the additional pairing in Eq.  (\[anisotropicH\]) with $\alpha = \Delta$ and $\beta = + i \Delta $, and a mass term $r \Psi^\dagger({\bf k}) \sigma_3 \Psi({\bf k})$, is $$\begin{aligned} E^2&=&\mu^2 + \Delta_s^2 + r^2 + k^2 + \Delta^2 k^2 \nonumber \\ &-& 2 \sqrt{\mu^2 (k^2 + r^2) + [\Delta (k_x^2 - k_y^2) + \Delta_s r]^2 }. \nonumber\end{aligned}$$ We analyze the pairing instabilities in the low-energy theory by introducing a cut-off $\Lambda$, so that relevant momenta from the interval around Fermi energy are defined by $\Lambda$, $ k\in (\mu - \Lambda , \mu + \Lambda)$. Also we assume that $ \mu \Delta \ll \Delta_s \ll \Lambda \ll \mu$, and, at zero temperature, estimate the free energy when both isotropic $(\Delta_s)$ and anisotropic $(\Delta)$ pairings are present. From the BCS mean field decoupling of effective attractive interactions we obtain terms proportional to the order parameters $\Delta^2$ and $\Delta_s^2$ (condensate energy) besides the contribution arising from the quasiparticles in the lower Bogoliubov band. (The upper band is assumed effectively a constant due to the constraint on the momenta.) The free energy density, ${\cal F}/{A}$, then reads $$\begin{aligned} \frac{{\cal F}}{A} & = & g_1 \Delta_s^2 + g_2 \Delta^2 - \frac{\mu}{4 \pi} \Lambda^2 \nonumber \\ &-& \frac{\mu}{4 \pi} \{(1 + \ln \frac{4 \Lambda^2}{\Delta_s^2}) {\cal M}\} , \nonumber \\ \label{M0A}\end{aligned}$$ where $${\cal M} = \Delta_s^2 + \frac{\Delta^2 \mu^2}{4} - \frac{r}{2} \Delta_s \Delta, \label{MA}$$ with $g_1$ and $g_2$ as positive coupling constants which drive the instabilities in the respective channels. Here, we assume $ r \ll \frac{\Delta_s}{\Lambda} ( \mu\Delta)$. We derive Eq. (\[M0A\]) with (\[MA\]) by expanding the square root for large $\mu$, and then performing the integral over $k$ (i.e. radial component of vector ${\bf k}$). Before the final angular integration, we further simplified the result of the $k$ integration by assuming the stated ordering of scales. In the BCS weak coupling limit, by minimizing the free energy i.e. the total ground state energy, we obtain $$\begin{aligned} \Delta_s & \approx & 2 \Lambda \exp\{- \frac{2 \pi g_1}{\mu}\}, \nonumber \\ \Delta & \approx & \frac{r \tilde{g_1}}{\tilde{g_1} \mu^2 - 4 g_2} \Delta_s ,\end{aligned}$$ where $ \tilde{g_1} = \frac{\mu}{4 \pi} + g_1$. Thus we can conclude that for $ \mu \Delta \ll \Delta_s \ll \Lambda \ll \mu$, and in the presence of small mass $r$, the isotropic instability can be accompanied by the anisotropic pairing. This is due to the cross term in ${\cal F}$ with $\Delta_s$ and $\Delta$ - see Eqs. (\[MA\]) and (\[M0A\]). This may also be understood from the fact that the matrices corresponding to the isotropic and anisotropic pairings anticommute. Coexistence of the $CP$ asymmetric $p$-wave and $s$-wave pairings: Mean-field analysis ====================================================================================== Here we discuss a pairing defined by $${\cal P}({\bf k}) = \left[ \begin{array}{cc} 0 & \alpha k_x + \beta k_y\\ \alpha k_x + \beta k_y & 0 \\ \end{array} \right] = (\alpha k_x+\beta k_y)\sigma_x , \label{AppBsecond_pairing_sD}$$ or in terms of the spinors, as a part of the complete Hamiltonian, $$\begin{aligned} \sum_{\bf k}(\alpha k_x + \beta k_y) \{ \Psi_{a}({\bf k}) \Psi_{a}(-{\bf k}) + \Psi_{b}({\bf k}) \Psi_{b}(-{\bf k}) \} + h.c. , \nonumber \\ \label{anisotropicHA}\end{aligned}$$ where $\alpha$ and $\beta$ are, in general, allowed to be complex coefficients. The lower Bogoliubov band of the quadratic Hamiltonian, Eqs. (\[BdGsD\]-\[right\_pairing\_sD\]) with the additional pairing in (\[anisotropicHA\]), is $$\begin{aligned} E^2&=&\mu^2 + \Delta_s^2 + k^2 (1 + f_1^2 + f_2^2) \nonumber \\ &-& 2k\sqrt{\mu^2 + \Delta_s^2 f_2^2 + k_x^2 (f_1^2 + f_2^2) - 2 \Delta_s f_2 \frac{k_y}{k} \mu }. \nonumber\end{aligned}$$ Here, $\alpha k_x + \beta k_y = k (f_1 + i f_2)$ where $ f_i = \alpha_i \cos \phi + \beta_i \sin \phi, i =1,2$ and $\alpha_1 , \alpha_2, \beta_1,$ and $ \beta_2 $ are real, and $\phi$ is the polar angle of the momentum vector. As in Appendix A, here we also analyze the pairing instabilities in the low-energy theory by introducing a cut-off $\Lambda$, so that relevant momenta from the interval around Fermi energy are defined by $\Lambda$, $ k\in (\mu - \Lambda , \mu + \Lambda)$. Also we assume that $ \mu \omega \ll \Delta_s \ll \Lambda \ll \mu$, where $\omega$ can be $\alpha_1$, $\alpha_2, \beta_1,$ or $ \beta_2$, and, at zero temperature, estimate the free energy when both, isotropic $(\Delta_s)$ and anisotropic $(f_1 , f_2)$ pairings are present. From the BCS mean field decoupling of effective attractive interactions we have terms proportional to $f_1^2 , f_2^2$ (averaged over angles) and $\Delta_s^2$ next to the contribution from the lower Bogoliubov band. (The upper band is assumed effectively a constant due to the constraint on the momenta.) The free energy density, ${\cal F}/{A}$, then reads $$\begin{aligned} \frac{{\cal F}}{A} & = & g_1 \Delta_s^2 + g_2 (\alpha_1^2 + \alpha_2^2 + \beta_1^2 + \beta_2^2) \nonumber \\ &-& \frac{1}{2} \frac{1}{(2\pi)^2} \{ \mu \; \Lambda^2 \; 2 \pi + (1 + \ln \frac{4 \Lambda^2}{\Delta_s^2}) \times \pi \mu {\cal M}\} \label{M0} \nonumber \\\end{aligned}$$ where $${\cal M} = \Delta_s^2 + \frac{1}{2} \Delta_s \beta_2 \mu + \frac{1}{4} (\alpha_1^2 + \alpha_2^2 + 3 \beta_1^2 + 3 \beta_2^2) \mu^2, \label{M}$$ with $g_1$ and $g_2$ as positive coupling constants which drive the instabilities in the respective channels. The last contribution of the quadratic order in anisotropic parameters, proportional to $ \sum_i (\alpha_i^2 + 3 \beta_i^2)$ was derived assuming $ \Lambda \ll \frac{\beta_2 \mu}{\Delta_s} \mu $. To find this result for the free energy density we applied the same set of approximations as in Appendix A. We derived Eq. (\[M0\]) with (\[M\]) by expanding the value of the square root for large $\mu$, and then performing the integral over $k$. Before the final angular integration, we further simplified the result of the integration over $k$ by assuming the stated ordering of scales. In the BCS weak coupling limit, by minimizing the free energy i.e. the total ground state energy, assuming $ \Delta_s \gg \omega \mu$ where $\omega$ can be $\alpha_1$, $\alpha_2, \beta_1,$ or $ \beta_2$, we obtain $$\begin{aligned} \Delta_s & \approx & 2 \Lambda \exp\{- \frac{4 \pi g_1}{\mu}\}, \nonumber \\ \beta_2 & \approx & \frac{\mu^2}{32 \pi} \frac{1}{g_2} \Delta_s (1 + \frac{8 \pi g_1}{\mu}), \nonumber \\ \alpha_1 & = & \alpha_2 = \beta_1 = 0.\end{aligned}$$ Thus we can conclude that for cut-off $\Lambda$, $ \mu \beta_2 \ll \Delta_s \ll \Lambda \ll \mu$, and in the presence of the isotropic instability $\Delta_s$ we can expect the presence of the anisotropic pairing with the order parameter $\sim i \beta_2 k_y$. This is due to the cross term in ${\cal F}$ with $\Delta_s$ and $\beta_2$ - see Eqs. (\[M\]) and (\[M0\]). In the presence of mass $r$ the dispersion of the Bogoliubons is modified as $$\begin{aligned} E^2&=&\mu^2 + r^2 + \Delta_s^2 + k^2 (1 + f_1^2 + f_2^2) \nonumber \\ &-& 2 \sqrt{ \mu^2 k^2 + \Delta_s^2 f_2^2 k^2 + k_x^2 (f_1^2 + f_2^2)k^2 + {\cal R}}, \nonumber\end{aligned}$$ where $$\begin{aligned} {\cal R}&=& r^2 (\Delta_s^2 + \mu^2) \nonumber \\ &+& 2 \Delta_s k (- k_y f_2 \mu + k_x f_1 r).\end{aligned}$$ We can notice that besides the cross term $ \sim \Delta_s f_2 k_y$ under square root in the above equation, we have, in the presence of a mass $r$, the term $ \sim \Delta_s f_1 k_x$. By performing the similar mean field analysis as before, we can find that this term will lead to the development of the real component proportional to $k_x$ in the anisotropic pairing, $\alpha k_x + \beta k_y = \alpha_1 k_x + i \beta_2 k_y$, with $ \alpha_1 /\beta_2 \sim r/\mu$ for $r \ll \mu$. 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--- abstract: 'We propose and study the use of photon-mediated interactions for the generation of long-range steady-state entanglement between $N$ atoms. Through the judicious use of coherent drives and the placement of the atoms in a network of Cavity QED systems, a balance between their unitary and dissipative dynamics can be precisely engineered to stabilize a long-range correlated state of qubits in the steady state. We discuss the general theory behind such a scheme, and present an example of how it can be used to drive a register of $N$ atoms to a generalized W state and how the entanglement can be sustained indefinitely. The achievable steady-state fidelities for entanglement and its scaling with the number of qubits are discussed for presently existing superconducting quantum circuits. While the protocol is primarily discussed for a superconducting circuit architecture, it is ideally realized in any Cavity QED platform that permits controllable delivery of coherent electromagnetic radiation to specified locations.' author: - Camille Aron - Manas Kulkarni - 'Hakan E. Türeci' bibliography: - 'SpinChain.bib' title: | Photon-mediated interactions: a scalable tool\ to create and sustain entangled states of $N$ atoms --- Introduction {#sec:intro} ============ Photon-mediated interactions are ubiquitous in nature. While the traditional formulation of quantum electrodynamics places equal emphasis on fields and sources, it is possible to take a point of view where the electromagnetic degrees of freedom are integrated out to reach an effective nonlocal field theory for matter only [@feynman_mathematical_1950]. A particularly beautiful example of this point of view, most closely related to the phenomena investigated here, is Schwinger’s formulation of the Casimir effect [@schwinger_casimir_1975]. Here, the electromagnetic degrees of freedom are integrated out and result in a photon-mediated retarded and nonlocal interaction between two conducting surfaces. A classic example where photon-mediated interactions are at play is the superradiance (and subradiance) of a cluster of dipoles, first pointed out by Dicke [@dicke_coherence_1954]. Here, photon-mediated interactions ultimately lead to generation of transient coherence between dipoles [@gross_superradiance_1982; @brandes_coherent_2005], which results in the emission of a powerful pulse whose intensity scales with $N^2$, where $N$ is the number of dipoles within a volume $V \sim \lambda^3$ ($\lambda$ is the wavelength of radiation). In free space, however, such interactions decay fast with interdipole distance, and it is challenging to engineer such interactions in a controlled way [@devoe_observation_1996; @eschner_light_2001]. Photon-mediated interactions are also at play in the self-organization transition of optically driven cold atoms in a cavity [@ritsch_cold_2013]. In these systems, cavity-mediated long-range interactions between atoms, tunable by the drive strength, lead to softening of a motional excitation mode recently observed in experiments [@baumann_dicke_2010; @mottl_roton-type_2012]. Certain aspects of the underlying critical behavior of this non-equilibrium phase transition can be described through photon-mediated interactions between the atoms constituting the condensate [@brennecke_real-time_2013; @strack_dicke_2011; @kulkarni_cavity-mediated_2013; @konya_damping_2014]. In recent years, we have seen the first attempts to use such photon-mediated interactions to generate strong coupling and possibly entanglement between artificial atoms in solid-state cavity QED systems. These approaches capitalize on strong light-matter interactions that can be generated in confined geometries such as resonators [@majer_coupling_2007; @filipp_multimode_2011] and waveguides [@loo_photon-mediated_2013]. In particular, Ref. [@loo_photon-mediated_2013] demonstrated coherent exchange interactions between two superconducting qubits separated by as much as a full wavelength (in that case $\lambda \sim 18.6$ mm) in an open quasi-1D transmission line. More recently, superradiance of two artificial atoms was observed and characterized in a controlled setting in a superconducting quantum circuit [@mlynek_observation_2014]. ![(color online) (a) One-dimensional array of cavity-qubit systems coupled by photon exchange and subject to one or several ac microwave drives, cavity decay at rate $\kappa$, qubit relaxation $\gamma$ and pure dephasing $\gamma_\phi$. (b) Implementation with superconducting transmon qubits embedded in interconnected microwave cavities and driven by external continuous-wave generators.[]{data-label="figschem1d"}](many1d.eps "fig:"){width="8.5cm"} ![(color online) (a) One-dimensional array of cavity-qubit systems coupled by photon exchange and subject to one or several ac microwave drives, cavity decay at rate $\kappa$, qubit relaxation $\gamma$ and pure dephasing $\gamma_\phi$. (b) Implementation with superconducting transmon qubits embedded in interconnected microwave cavities and driven by external continuous-wave generators.[]{data-label="figschem1d"}](irf.eps "fig:"){width="7.0cm"} The goal of this paper is to show how photon-mediated interactions between qubits embedded in an engineered electromagnetic environment can be harnessed to controllably generate a certain large-scale entangled state of $N$ qubits [*in the steady state*]{}. In the present work, the role of photons is twofold. First, they mediate a coherent coupling between distant qubits. Second, they provide a controllable dissipative mechanism which can be used to stabilize a long-range correlated many-body state of the qubits. By driving the system at a suitable frequency, one can achieve a transition that produces the desired many-body state while dumping energy into one of the electromagnetic modes. The dissipative mechanisms are key to make the scheme steady-state, in contrast to better-known alternatives, such as those based on Rabi cycling between a ground state and a desired excited state. We show how the balance of the unitary and dissipative contributions can be precisely tuned by the placement of qubits in an engineered photonic environment, and a set of coherent electromagnetic drives with specified amplitudes and frequencies. Engineered dissipative dynamics has recently been employed in superconducting circuits to cool a single qubit to a desired state on the Bloch sphere [@murch_cavity-assisted_2012], and two qubits that reside in a single cavity to a Bell state [@shankar_autonomously_2013]. Furthermore, a number of recent theoretical works have focused on generation of high-fidelity steady-state entanglement between two superconducting qubits [@leghtas_stabilizing_2013; @reiter_steady-state_2013; @huang_generation_2013; @aron_steady-state_2014]. We present the general theory underlying these phenomena, deriving and solving the dynamics of $N$ qubits residing in an arbitrary open electromagnetic environment, subject to coherent driving and losses. [The underlying new principle is based on using collective electromagnetic (EM) modes of a large structure (such as Bloch modes of a lattice of cavities) to dissipatively stabilize a collective state of spins. By “scalability” we refer to the fact that, for the specific protocol worked out here for W states, that the same protocol used for $N$ qubits can be used for $N+1$ qubits. Generally it is to be expected that the fidelity of stabilization degrades as $N$ is increased because of spectral crowding, which we discuss for the particular protocol we investigate here, and provide an estimate for the maximum number of qubits that can be stabilized reliably. Interestingly, spatial symmetries in the system (such as discrete translational invariance) can be used along with a [*spatially*]{} modulated coherent drive, to greatly enhance the spectral resolution. In one of the protocols we propose, this allows us to resolve W states in a way that the state selectivity is limited by only the finesse of the EM system, instead of the mean level spacing of the collective spin states.]{} In Sec. \[sec:1dmodel\], we lay out a particular architecture that we have in mind for a proof-of-principle demonstration: we consider of model of $N$ qubits residing in a one-dimensional array of cavities. We present the different layers of approximations that allow us to have an analytic handle over the problem. First, we show how cavity photons mediate effective qubit-qubit interactions through the collective electromagnetic modes of the array. In Sec. \[sec:cooling\], we show how the [*full Liouvillian*]{} describing the evolution of the reduced density matrix of the $N$ qubits can be engineered to drive the qubit subsystem to generalized W states, and the entanglement sustained as long as the drives are on. The fabrication tolerances of such a cavity array has recently been studied experimentally [@underwood_low-disorder_2012]. Thus, the controlled fabrication of such arrays is well within reach of the current superconducting circuit technology both in coplanar and 3D configurations [@underwood_thesis_2015]. We present the fidelities that can be expected in recently fabricated systems and analyze the fault tolerance of the method to phase and amplitude noise of the drive parameters, as well as the nonuniformity of qubit and cavity parameters. Finally, in Sec. \[sec:ph-mediated\], we generalize our scheme to arbitrary arrangements of qubits coupled to an engineered photonic backbone, going beyond the tight-binding approximation for the EM system used in previous sections. One-dimensional lattice model {#sec:1dmodel} ============================= Consider a one-dimensional array of $N$ identical microwave cavities with nominally equal frequencies $\omega_{{\rm c}}$, coupled to each other capacitively described by a nearest-neighbor tunneling matrix element $J$. Each cavity houses a superconducting qubit with splitting $\omega_{{\rm q}}$. We shall consider both the cases of open (*i.e.* non-periodic) and periodic boundary conditions (identifying site $N$ with site 0). Later, we will relax our assumptions on identical cavity resonance frequencies and consider the most general case, showing that the general approach to the dissipative stabilization of a generalized W state stands. Furthermore, each cavity shall be driven by a coherent monochromatic microwave source with frequency $\omega_{{\rm d}}$ and a site-dependent amplitude $\epsilon^{{{\rm d}}}_i$ and phase $\Phi_i$; see Fig. \[figschem1d\](b) for a 3D superconducting circuit architecture of the system Fig. \[figschem1d\](a). We work in a regime where $\omega_{{\rm c}}$, $\omega_{{\rm q}}$ and $\omega_{{\rm d}}$ are mutually far detuned from each other (typically on the order of GHz). The starting Hamiltonian is that of a one-dimensional driven Jaynes-Cummings lattice model studied before in Refs. [@knap_emission_2011; @nissen_nonequilibrium_2012; @grujic_non-equilibrium_2012; @hur_many-body_2015], $$\begin{aligned} \label{eq:H} H(t) = H_\sigma + H_{\sigma a} + H_a(t) \,, \end{aligned}$$ where $H_\sigma$, $H_{\sigma a}$, and $H_a(t)$ are respectively the qubit, the Jaynes-Cummings light-matter coupling, and the driven cavity Hamiltonians, $$\begin{aligned} H_\sigma =& \! \sum_i \omega_{{\rm q}}\frac{\sigma_i^z}{2},\, H_{\sigma a} = g \! \sum_i\! \left[ a_i^\dagger \sigma^-_i + \mbox{H.c.} \right] \,, \\ H_a(t) =& \! \sum_i\! \left[ \omega_{{\rm c}}a_i^\dagger a_i - J ( a_{i}^\dagger a_{i+1} + \mbox{H.c.} ) \right. \nonumber \\ & \qquad \left. + 2 \epsilon^{{{\rm d}}}_i \cos(\omega_{{\rm d}}t + \Phi_i) \left( a_i + a^\dagger_i \right)\right] \,. \label{eq:Ha}\end{aligned}$$ Here, $i$ runs from site $0$ to $N-1$. The qubits are two-level systems described by the usual Pauli pseudo-spin operators $\sigma_i^{x,y,z}$ and $\sigma_i^\pm \equiv (\sigma^x_i \pm {{\rm i}}\sigma_i^y)/2$. $\epsilon^{{\rm d}}_i$ and $\Phi_i$ are, respectively, the cavity-dependent amplitude and phase of the ac microwave drives. Thermal equilibrium is achieved by setting all drive amplitudes to zero, $\epsilon^{{\rm d}}_i=0$. Without loss of generality, we assume that the detuning between cavity and qubit frequency $\Delta \equiv \omega_{{\rm q}}- \omega_{{\rm c}}> 0$, and $J>0$. We operate in the dispersive regime corresponding to $g/\Delta \sim 10^{-1}$ and at sufficiently weak drive amplitudes to ensure the presence of very few photons in the cavities, so the Schrieffer-Wolff perturbation theory to be applied shortly is well justified. Below, we entirely integrate out the photonic degrees of freedom, resulting in (a) an effective qubit-qubit interaction as discussed on more general grounds in Sec. \[sec:ph-mediated\], (b) local Zeeman fields of the form $\epsilon^{{\rm d}}_i \sigma_i^{x}$ (for $\Phi_i=0$), and (c) a non-unitary evolution characterized by controllable transition rates. #### Intrinsic dissipation in the system. {#intrinsic-dissipation-in-the-system. .unnumbered} In addition to the unitary dynamics described by Eq. (\[eq:Ha\]), we assume the individual qubits are coupled to uncontrolled environmental degrees of freedom that give rise to single qubit spin-flip rate ($\gamma$), a single-qubit pure dephasing rate $\gamma_\phi$, and a cavity decay rate $\kappa$. #### Typical system parameters. {#typical-system-parameters. .unnumbered} In a recent experiment studying the $N=2$ case of the protocol described here [@ucbpaper] in a 3D superconducting circuit architecture, typical system parameters were: $\omega_{{\rm c}}\simeq 7, \omega_{{\rm q}}\simeq 6, g \simeq J \simeq 10^{-1}, \kappa \simeq 7 \times 10^{-4}, \gamma \simeq 4 \times 10^{-5}$ all in units of $2\pi$ GHz. Below, our analytical approach is performed assuming the hierarchy of energy scales $\Delta \gg g, J \gg \kappa \gg \gamma \gg \gamma_\phi$. #### Rotating wave approximation. {#rotating-wave-approximation. .unnumbered} We eliminate the time dependence in $H_a(t)$ by working in the frame rotating at $\omega_{{\rm d}}$ and dropping the non-secular terms. In the rest of the Hamiltonian (\[eq:H\]), this also amounts to replacing $\omega_{{\rm q}}$ by $\Delta_{{\rm q}}\equiv \omega_{{\rm q}}- \omega_{{\rm d}}$ and $\omega_{{\rm c}}$ by $-\Delta_{{\rm c}}\equiv \omega_{{\rm c}}- \omega_{{\rm d}}$. Once expressed in the eigenbasis of the (undriven) coupled cavity system, $H_{a}$ is given by $$\begin{aligned} \label{eq:Hak} H_{a} = \sum_k (\omega_k -\omega_{{\rm d}}) a^\dagger_k a_k + ( \epsilon^{{\rm d}}_k a^\dagger_k + \mbox{H.c.} )\,.\end{aligned}$$ Here, $k$ is the discrete Bloch wavevector, $a_k^\dagger = \sum_j \varphi_{k}^*(j) \, a_j^\dagger$ creates a photon in the $k^{\mathrm{th}}$ mode. The specific discrete values of $k$ and the corresponding mode profiles $\varphi_k(j)$ are to be fixed by the boundary conditions. For periodic boundary conditions, the set of quasi-momenta are $k= 2\pi\, n/N$, with $n = 0,\ldots, N-1$ and $\varphi_{k}(j) = {{\rm e}}^{-{{\rm i}}k j} / \sqrt{N}$. For open boundary conditions, $ k = \pi (n+1)/(N+1)$ and $\varphi_k(j) = \sqrt{2/(N+1)} \sin\left(k(j+1)\right)$. In Eq. (\[eq:Hak\]), we incorporated the phase $\Phi_j$ in $\epsilon^{{\rm d}}_j$, which can henceforth be complex and $\epsilon^{{\rm d}}_k = \sum_j \varphi_{k}(j) \, \epsilon_j^{{\rm d}}$. The eigenfrequencies are given by the photonic dispersion relation $$\begin{aligned} \omega_k = \omega_{{\rm c}}- 2 J \cos(k)\,.\end{aligned}$$ ![(color online) Engineered qubit many-body spectrum. The non-equilibrium drive and the resulting photon fluctuation bath are used to create a dominant transition from the trivial ground state to a target entangled W state in the 1-excitation manifold. The existence of a non-zero spontaneous decay $\gamma$ is critical to avoid populating higher-excited state manifolds.[]{data-label="fig:mediated"}](mechaN.eps){width="3.5cm"} #### Effective dissipative $XY$ model. {#effective-dissipative-xymodel. .unnumbered} We eliminate the light-matter interaction to second-order perturbation theory in $g/\Delta$ by means of a Schrieffer-Wolf (SW) transformation which maps $H \mapsto {{\rm e}}^{X} H {{\rm e}}^{X^\dagger}$ where $$\begin{aligned} X \equiv g \sum_k \left[ \frac{a_k \sigma_k^\dagger}{\omega_{{\rm q}}- \omega_k} -\mathrm{H.c.} \right]\,.\end{aligned}$$ Collecting all the terms, we obtain an isotropic $XY$ model subject to a magnetic field for the qubit subsystem ($H_\sigma$), weakly coupled to the photon fluctuations of the EM backbone ($H_{\sigma a}$): $$\begin{aligned} H_\sigma =& \! \sum_i \boldsymbol{h_i} \cdot \frac{\boldsymbol{\sigma}_i}{2} - \frac{J}{2} \left(\frac{g}{\Delta}\right)^2 \left[ \sigma^x_{i} \sigma^x_{i+1} + \sigma^y_{i} \sigma^y_{i+1}\right]\,, \label{eq:Hspinchain} \\ H_{\sigma a} &= \sum_i \left(\frac{g}{\Delta}\right)^2 \sigma_i^z( \Delta a_i^\dagger a_i + \epsilon^{{\rm d}}_i a_i^\dagger +{\epsilon^{{\rm d}}_i}^* a_i)\,. \label{eqtransXY}\end{aligned}$$ Here, ${h}^x_i = 2 \mbox{Re}(\epsilon^{{\rm d}}_i) (g/\Delta)$, $h^y_i= -2 \mbox{Im}(\epsilon^{{\rm d}}_i) (g/\Delta)$, $h^z_i = \Delta_{{\rm q}}+ \Delta (g/\Delta)^2 $. The effective magnetic field $\boldsymbol{h}_i$ is mainly oriented along the $z$ direction but we shall see that, while they break the integrability of the model, the $x$ and $y$ components of the emergent Zeeman field play a crucial role for the scheme below. We note that the SW transformation is also responsible for subleading corrections to the strength of the dissipative terms (*e.g.* the qubit decay $\gamma$ acquires a contribution from the cavity decay $\kappa$, corresponding to a Purcell contribution, and *vice versa*) the local contributions of which can be simply included by working with renormalized parameters $\kappa$, $\gamma$ and $\gamma_\phi$ at low photon occupations [@boissonneault_dispersive_2009], which is the regime studied here. These effective parameters can be extracted from experimental measurements. When the drives are off, $\epsilon^{{\rm d}}_i = 0$, the system simply thermalizes with its environment – this is just the physics of blackbody radiation in a coupled cavity system (including $N$ dipoles). Viewed from the perspective of the qubits, while $H_\sigma$ describes a quantum phase transition from a paramagnetic to a ferromagnetic phase when the magnitude of the transverse field is on the order of the nearest-neighbor coupling $J ( g/\Delta)^2$, this regime is never reached for realistic system parameters. For Raman-driven qubits the situation is more interesting, and the phase diagram was recently studied displaying various exotic attractors [@schiro_exotic_2015]. Henceforth, we only work in the experimentally achievable regime $\omega_{{\rm q}}\gg J(g/\Delta)^2$ and, to leading order, the ground state of $H_\sigma$ is simply the separable state $|\boldsymbol{0} \rangle \equiv | \! \downarrow \ldots \downarrow \rangle$. The low-energy spectral content of the qubit sector is depicted in Fig. \[fig:mediated\]: the first $N$-qubit excited manifolds are roughly separated by $\Delta_{{\rm q}}$ while the lifting of degeneracy within each manifold is controlled by $J (g/\Delta)^2$ From the point of view of equilibrium statistical mechanics, the many-body system of qubits described by $H_\sigma$ is not interesting because it would thermalize with the radiative reservoir, reaching a steady state $\rho_\sigma (t \rightarrow \infty) \propto {{\rm e}}^{-\beta H_\sigma} \sim | \boldsymbol{0} \rangle \langle \boldsymbol{0}|$, a collection of uncorrelated qubits in their ground state. We shall see that turning on the drives will change this situation dramatically. A careful choice of drive parameters (even for small amplitudes of drives) will be shown to enable the stabilization of a particular many-body state of qubits in the excited-state manifold. Stabilization of generalized W states {#sec:cooling} ===================================== Our goal in this section is to identify non-trivial entangled eigenstates of the spin chain $H_\sigma$ and design a protocol which, starting from the ground state $|\boldsymbol{0}\rangle$ that can be straightforwardly prepared, achieves the stabilization of an interesting excited state of choice. Below we discuss the details of a robust protocol for the stabilization of a generalized W state of qubits with [*minimal*]{} resources. #### Linearized photon spectrum. {#linearized-photon-spectrum. .unnumbered} We first address the non-linearities in $H_{\sigma a}$ by decomposing the photonic field into mean-field plus bosonic fluctuations: $$\begin{aligned} a_k \equiv \bar a_k + d_k \mbox{ with } \bar a_k = \frac{\epsilon^{{\rm d}}_k}{\omega_{{\rm d}}- \omega_k + {{\rm i}}\kappa/2}\;. \label{eqsmallfluct}\end{aligned}$$ We assume here that all Bloch modes have the same loss $\kappa$. This can be made more precise but it will not qualitatively change the results we present. Neglecting those terms that are quadratic in the fluctuations and that couple to the qubits, the light sector reduces to $$\begin{aligned} H_a \rightarrow H_d &= \sum_k (\omega_k - \omega_{{\rm d}}) d_k^\dagger d_k \,, \label{eq:bath1} \\ H_{\sigma a} \rightarrow H_{\sigma d} &= \left(\frac{g}{\Delta}\right)^2 \sum_i \sigma^z_i [( \Delta \bar{a}_i + \epsilon^{{\rm d}}_i ) d_i^\dagger + \mbox{H.c.} ]\,. \label{eq:bath2}\end{aligned}$$ #### Diagonalization of the matter sector. {#diagonalization-of-the-matter-sector. .unnumbered} The Hamiltonian Eq. (\[eq:Hspinchain\]) in the presence of non-zero drive terms is non-integrable. We therefore proceed by projecting it into the low-energy sector with a maximum one excitation: $$\begin{aligned} \label{eq:Hstrunc} H_\sigma =\sum_k E_k |k \rangle \langle k | + \left( \frac{g}{\Delta} \right) \left( \epsilon^{{\rm d}}_k |k\rangle\langle \boldsymbol{0} | + \mbox{H.c.} \right),\end{aligned}$$ We have set the energy of the ground state $|\boldsymbol{0} \rangle \equiv | \downarrow \ldots \downarrow \rangle$ to zero ($E_{\boldsymbol{0}} = 0$). Here the states $$|k\rangle = \sum_{i=0}^{N-1} \varphi_k^* (i) \ |i\rangle\,,$$ with $|i\rangle \equiv |\downarrow_0 \ldots\downarrow_{i-1} \, \uparrow_i \, \downarrow_{i+1} \ldots \downarrow_{N-1} \rangle$ indicating one excitation located at site $i$, are states that carry a single qubit excitation of quasi-momentum $k$, entangled over the entire chain. These are the eigenstates of the [*undriven*]{} spin chain (*i.e.* for $\epsilon^{{\rm d}}_i =0 \, \forall\, i$) with a dispersion relation $$\begin{aligned} E_k = \epsilon_k - \omega_{{\rm d}}, \; \epsilon_k = \omega_{{\rm q}}+ \delta\omega_{{\rm q}}- 2 J \left(\frac{g}{\Delta}\right)^2 \! \cos(k)\;.\end{aligned}$$ Here $\delta\omega_{{\rm q}}\simeq g^2/\Delta$ is the cavity-induced Stark shift. This truncation of the Hamiltonian holds if the higher-excitation manifolds are not significantly occupied during the dynamics. This can be checked a posteriori and we do so. Let us first discuss the case of open boundary conditions for which the absence of translational and space-reversal symmetry generically yields a fully non-degenerate spectrum. The effect of the drive term in Eq. (\[eq:Hstrunc\]), assumed to be small as stated before, can be taken into account through a perturbation theory and yields the following eigenstates of $H_\sigma$ to lowest order in $({g}/{\Delta}) (\epsilon_k^{{\rm d}}/\Delta_{{\rm q}})$: $$\begin{aligned} |\widetilde{\boldsymbol{0}} \rangle &\simeq |\boldsymbol{0} \rangle \!-\! \left(\frac{g}{\Delta}\right) \! \sum_{k} \! \frac{\epsilon^{{\rm d}}_k}{\Delta_{{\rm q}}} |k \rangle,\, \widetilde{E}_{\boldsymbol{0}} \simeq \!-\! \left(\frac{g}{\Delta}\right)^2 \! \sum_k \! \frac{|\epsilon^{{\rm d}}_k|^2}{\Delta_{{\rm q}}},\\ |\widetilde{k} \rangle & \simeq |k \rangle + \left(\frac{g}{\Delta}\right) \frac{{\epsilon^{{\rm d}}_k}^*}{\Delta_{{\rm q}}} |\boldsymbol{0} \rangle,\, \widetilde{E}_k \simeq E_k + \left(\frac{g}{\Delta}\right)^2 \frac{|\epsilon^{{\rm d}}_k|^2}{\Delta_{{\rm q}}}. \label{eqqbitcollen}\end{aligned}$$ The above corrections to the undriven eigenstates are crucial for the success of the two-photon cooling mechanism presented below. #### Transition rates. {#transition-rates. .unnumbered} By virtue of Eq. (\[eqsmallfluct\]), the coupling of the photonic fluctuations (on top of a classical part) to the spin-chain Eq. (\[eqtransXY\]) can be treated in perturbation theory. This permits us to integrate them out arriving at an effective master equation for qubits only. We note that the fluctuations of the collective photon modes of the lattice, described by spectral function per mode $q$ $$\begin{aligned} \rho_q(\omega) = - \frac{1}{\pi} \mbox{Im } \frac{1}{\omega -\omega_q+{{\rm i}}\kappa/2}\;,\end{aligned}$$ can be easily manipulated by the design of the cavity lattice. Assuming that the photon fluctuations, which couple to the spin degrees of freedom via [Eq. (\[eq:bath2\])]{}, thermalize with the radiative environment which in turn is taken to be at very low temperature, we arrive at the effective master equation for spins only $\rho_\sigma$ at steady state, $\rho_\sigma^{\rm NESS} \equiv \lim\limits_{t\to\infty} \rho_\sigma$: $$\begin{aligned} \label{eq:master} \partial_t \rho^{\rm NESS}_\sigma & \! \! = 0 = -{{\rm i}}\left[ H_\sigma,\rho^{\rm NESS}_\sigma \right] + \sum_k \Gamma_{\boldsymbol{0}\to k} \mathcal{D}[| \widetilde{k} \rangle \langle \widetilde{\boldsymbol{0}} | ] \rho^{\rm NESS}_\sigma \nonumber \\ & \hspace{-3em} + \gamma \sum_k \mathcal{D}[| \widetilde{\boldsymbol{0}} \rangle \langle \widetilde{k} | ] \rho^{\rm NESS}_\sigma + \frac{2\gamma_\phi}{N} \sum_{k\,q} \mathcal{D}[| \widetilde{q} \rangle \langle \widetilde{k} | ] \rho^{\rm NESS}_\sigma.\end{aligned}$$ The derivation of this master equation relies on the separation of time scales: the relaxation time scale of cavity fluctuations $d_k$, on the order of $1/\kappa$, is much shorter than that of the reduced density matrix of the spins $\rho_\sigma(t)$. This separation of time scales is perfect in the steady state [@afoot; @gsc; @amitra] which we are interested in here. The Lindblad-type operators are defined as $\mathcal{D}[X] \rho \equiv \left( X \rho X^\dagger - X^\dagger X \rho + \rm{H.c.} \right)/2$ and the $\Gamma_{\boldsymbol{0} \to k} $’s correspond to non-equilibrium transition rates between the ground state $|\widetilde{\boldsymbol{0}} \rangle$ and a given excited many-body state $| \widetilde{k} \rangle$ \[see Eq. (\[eqqbitcollen\])\] in the one-excitation manifold of $H_\sigma$, given in Eq. (\[eq:Hspinchain\]). These are found to be $$\begin{aligned} \label{eq:rate} \Gamma_{\boldsymbol{0} \to k} = 2 \pi \sum_q \Lambda_{kq}^2\rho_q(\omega_{{\rm d}}+\widetilde{E}_{\boldsymbol{0}} -\widetilde{E}_k)\,,\end{aligned}$$ where the transition matrix element is given by $$\begin{aligned} \Lambda_{kq} =& \left| \left(1+2\frac{\Delta}{\Delta_{{\rm c}}} \right) \!\! \frac{1}{\Delta_{{\rm q}}} \!\! \left(\frac{g}{\Delta}\right)^3 \!\!\! \sum_{k' k'' q'} \!\!\! f_{k k' k''} f_{q' q k'}^* \epsilon^{{\rm d}}_{k''} \epsilon^{{\rm d}}_{q'} \right|\,,\end{aligned}$$ with the tensor $f_{k k' k''} \equiv \sum_i \varphi_k(i) \varphi_{k'}^*(i) \varphi_{k''}^*(i)$. The integration over the photon-fluctuation degrees of freedom also yields Lamb-shift corrections of the energy levels, but this does not play any substantial role in our scheme, see discussion below Eq. (\[eq:wdopt\]). #### Dynamics. {#dynamics. .unnumbered} By virtue of having written the steady-state master equation Eq. (\[eq:master\]) in the eigenbasis of $H_{\sigma}$, all off-diagonal matrix elements of $\rho_\sigma^{\rm NESS}$ by construction vanish as the steady state is approached. Consequently, the dynamics can be faithfully described by effective rate equations for the populations of eigenstates, $n_{\boldsymbol{0}}$ and $n_k$: $$\begin{aligned} \frac{{{\rm d}}n_{\boldsymbol{0}}}{{{\rm d}}t} &= \gamma \, \sum_q n_q - \Gamma_{\boldsymbol{0}\to q} \, n_{\boldsymbol{0}} \label{eq:eom1} \\ \frac{{{\rm d}}n_k}{{{\rm d}}t} &= - \gamma \, n_k + \Gamma_{\boldsymbol{0}\to k} \, n_{\boldsymbol{0}} + \frac{2 \gamma_\phi}{N} \sum_{q} (n_q -n_k\, ). \label{eq:eom2}\end{aligned}$$ The terms in $\gamma$ correspond to qubit decay, flipping down the pseudo spins and relaxing the energy by $\Delta_{{\rm q}}$ (in the rotating frame). The terms in $\gamma_\phi$ correspond to pure dephasing processes, the action of which is to equalize the populations of the states in the one-excitation manifold. The emergent level structure and rates are summarized in Fig. \[fig:mediated\]. We note that, while the full dynamical evolution of the qubit-EM system (viz. Eq. (\[eq:H\]) in the presence of qubit and cavity decay) is clearly non-Markovian, the proper secularization of the equations around the operation frequency $\omega_d$ allows us to describe the qubit dynamics through the relatively transparent rate equations (\[eq:eom1\] and \[eq:eom2\]). Irrespective of the initial conditions, Eqs. (\[eq:eom1\] and \[eq:eom2\]) have a unique non-equilibrium steady-state solution and, after transient dynamics, the occupation of the state $|k\rangle$ is given by $$\begin{aligned} n_k^{\rm NESS} &= \frac{1}{1+2\gamma_\phi/\gamma} \frac{\Gamma_{\boldsymbol{0} \to k} + (2\gamma_\phi/N\gamma) \sum_q \Gamma_{\boldsymbol{0} \to q} }{\gamma + \sum_q \Gamma_{\boldsymbol{0} \to q}} \,. \label{eq:nqness}\end{aligned}$$ #### Stabilization protocol. {#stabilization-protocol. .unnumbered} Equation (\[eq:nqness\]), together with Eq. (\[eq:rate\]), transparently elucidates how to stabilize a given pure entangled state of qubits $|k\rangle$ in the steady state. The protocol requires the maximization of $\Gamma_{\boldsymbol{0}\to k}$, given in Eq. (\[eq:rate\]), to make it the largest of all rates among ($\{ \Gamma_{\boldsymbol{0}\to q} \}$, $\gamma$, $\gamma_\phi$). This is performed by optimally tuning the drive frequency $\omega_{{\rm d}}$ such that the sharply peaked photonic spectrum $\rho_q(\omega_{{\rm d}}+\widetilde{E}_{\boldsymbol{0}} -\widetilde{E}_k)$ in Eq. (\[eq:rate\]) reaches the maximum amplitude of the Lorentzian, which is on the order of $1/\kappa$. This is possible whenever there is at least one mode $q_0$ with $\Lambda_{k q_0} \neq 0$ and the optimum $\omega_{{\rm d}}$ is the solution of the energy-conservation equation $\omega_{{\rm d}}+\widetilde{E}_{\boldsymbol{0}} -\widetilde{E}_k = \omega_{q_0}$, *i.e.* $$\begin{aligned} \omega_{{\rm d}}= & \frac{\omega_{{\rm q}}+ \delta\omega_{{\rm q}}+\omega_{{\rm c}}}{2} - J \cos(q_0) \nonumber \\ & \quad + \left(\frac{g}{\Delta}\right)^2 \left[ - J \cos(k) + \frac12 \sum_{q\neq k} \frac{|\epsilon^{{\rm d}}_q|^2}{\Delta_{{\rm q}}} \right] \,. \label{eq:wdopt}\end{aligned}$$ This energy-matching condition describes a one-photon process in the rotating frame equivalent to a two-photon process in the laboratory frame. The corresponding Raman inelastic scattering process uses the energy of the two incoming drive photons to perform the qubit transition while simultaneously dumping a photon in one of the cavity modes [@ucbpaper]. We note that when Eq. (\[eq:wdopt\]) is satisfied, the Lamb-shift correction of $\widetilde{E}_q$ vanishes. ![\[fig:gamma\] (color online) Schematics of the pumping rate $\Gamma_{0 \to k}$ as a function of the drive frequency $\omega_{{\rm d}}$ for $N=5$. (a) Driving first cavity only: any of the five photon-fluctuation modes (responsible for the five peaks) can be used for dissipative stabilization. Driving to $|k\rangle$ while avoiding populating the nearest state $|k'\rangle$ necessitates $\kappa < \Delta \widetilde{E} \equiv |\widetilde{E}_k - \widetilde{E}_{k'}| \sim 2\pi J (g/\Delta)^2 /N$. (b) Driving all cavities equally: only one of the five modes can channel the mechanism and driving to the nearest state $|k'\rangle$ is avoided if $\kappa < \Delta \omega \equiv |\omega_k - \omega_{k'}| \sim 2\pi J / N$. ](Gamma2.eps){width="8.4cm"} #### [Scalability and limitations.]{} {#scalability-and-limitations. .unnumbered} Equation (\[eq:nqness\]) sets an upper bound on the fidelities, $n_{k}^{\rm NESS} \leq n^{\rm max} = (\gamma+2\gamma_\phi/N)/(\gamma+ 2 \gamma_\phi)$, which highlights the necessity of working with qubits that have a pure dephasing rate $\gamma_\phi$ much smaller than their relaxation rate $\gamma$. This upper bound is not tight but allows us to highlight the role of the pure dephasing mechanisms. The success of the protocol, [and its scalability to large N]{}, also relies, via the numerator of Eq. (\[eq:nqness\]), on the resolving power of the spectral width of the photon density of states ($\sim \kappa$) *i.e.* the precision with which the photon fluctuations can target the spin-chain state $|k\rangle$ without exciting other eigenstates close in energy. The limitations on the resolving power depend strongly on the drive spatial profile. This is illustrated here considering two extreme cases, one where only one cavity is driven ($\epsilon^{{\rm d}}_i = \epsilon^{{\rm d}}\delta_{i1}$), the other corresponding to a case where all cavities are driven with equal amplitude ($\epsilon^{{\rm d}}_i = \epsilon^{{\rm d}}\, \forall i$). In the first case \[Fig. \[fig:gamma\](a)\], the transition rates for every $0 \to k$ have multiple peaks. Thus the stabilization of $|k\rangle$ at the optimal frequency given by Eq. (\[eq:wdopt\]) while avoiding the population of the nearest state $|k'\rangle$ requires the condition $\kappa < \Delta \widetilde{E} \equiv |\widetilde{E}_k - \widetilde{E}_{k'}| \sim 2\pi J (g/\Delta)^2 /N$. On the other hand, the second case, driving each site identically yields rates that have a single peak, that for neighboring spin chain states $k$ and $k'$ are separated by the free spectral range (of the collective EM modes) of the cavity chain, $\Delta \omega \equiv |\omega_k - \omega_{k'}| \sim 2\pi J / N$. [This indicates that the protocol with uniform driving can be scaled up to a number of qubits on the order of $N_{\rm max} \lesssim 2 \pi J / \kappa$.]{} This situation is easily understood: the rates given in Eq. (\[eq:rate\]) contain the fulfillment of an energy-conservation condition, which can, in principle, be satisfied picking any collective EM mode $q$. The set of non-zero transition matrix elements in the sum Eq. (\[eq:rate\]) can, however, be substantially narrowed down by choosing a drive amplitude profile ($\epsilon^{{\rm d}}_{i}$) that is narrow in the momentum domain. For example, $\epsilon^{{\rm d}}_{p} = \delta_{pp_0}$, collapses the sum to a single term by imposing a quasi-momentum conservation condition between $p_0$, the target spin-chain state with momentum $k$, and $q_0$, the quasi-momentum of the collective EM mode picked for stabilization (note that conservation of momentum is strictly valid in periodic systems). This is given by $k = 2p_0 - q_0$ and is consistent with the interpretation of stabilization via a two-photon process. For such driving, the optimal frequency of the drive $\omega_{{\rm d}}$ is then given by Eq. (\[eq:wdopt\]). We note that this transparent criterion was used in Ref. [@schwartz_toward_2015] to selectively stabilize either the triplet or the singlet state of two transmon qubits. #### Effective master equation simulations. {#effective-master-equation-simulations. .unnumbered} To complement the analytic approach, we have performed full numerical simulations of the effective master equation (\[eq:master\]) where we (i) compute exactly the full spectrum of the spin chain $H_\sigma$ in Eq. (\[eq:Hspinchain\]) with $N=5$ qubits and open boundary conditions (ii) determine the rates between all the eigenstates and (iii) solve for the steady-state populations. In these calculations, sufficient number of higher-excitation manifolds of the spin chain were included to achieve convergence. Because the parameter space is fairly large, we performed our simulations for a presently existing fabricated system for $N=2$ [@schwartz_toward_2015]. These parameters are quoted in the caption of Fig. \[fig:firstsiteopen\], where the fidelities to achieve various spin-chain states ($k$) for $N=5$ are compared for a localized drive \[Fig. \[fig:firstsiteopen\](a)\] and a spatially uniform drive \[Fig. \[fig:firstsiteopen\](b)\]. We note that compared to the current state of the art [@shankar_autonomously_2013], these are remarkable fidelities. These fidelities can be significantly improved by reducing the ratio of the dephasing over the qubit relaxation rate. We have also tested the robustness of our protocol against site-to-site inhomogeneities of the different parameters and found no qualitative difference for $\delta\omega_{{\rm c}}/\omega_{{\rm c}}\sim 10^{-2}$, $\delta\omega_{{\rm q}}/\omega_{{\rm q}}\sim 10^{-4}$, $\delta g/g \sim 10^{-4}$ and $\delta J/J \sim 10^{-2}$. A more extensive analysis of achievable fidelities in the presence of site-to-site inhomogeneities will be presented in future work. #### Periodic boundary conditions. {#periodic-boundary-conditions. .unnumbered} Let us now consider the case of periodic boundary conditions for which the [*undriven*]{} system is space-translational and space-reversal invariant. Such a symmetry results in degeneracies between the eigenstates $|k\rangle$ and $|2\pi -k \rangle$ of the undriven spin chain (except for $k=0$ and $k=\pi$). A symmetry-breaking drive profile will generically lift the degeneracy of the spectrum and, importantly, the emergent eigenstates will strongly depend on the particular drive profile. To exemplify this point, let us start by driving the first cavity only: $\epsilon^{{\rm d}}_i = \epsilon^{{\rm d}}\delta_{i,0}$. Second-order degenerate perturbation theory in $({g}/{\Delta})(\epsilon^{{\rm d}}/\sqrt{N}\Delta_{{\rm q}})$ lifts the degeneracy in the subspaces spanned by $|k\rangle$ and $|2\pi - k \rangle$. To lowest order, the eigenstates are $$\begin{aligned} \label{eq:kpm} |k_\pm \rangle \equiv \frac{| k \rangle \pm | 2\pi - k \rangle}{\sqrt{2}}\end{aligned}$$ for all $k \in\, ]0,\pi[$ complemented with the W state $|0_+\rangle \equiv | 0 \rangle$ and $|\pi_+\rangle \equiv | \pi \rangle$ (for $N$ even), one obtains the rates $\Gamma_{\boldsymbol{0} \to k_\pm} = 2 \pi \Lambda_{k_\pm}^2 \sum_q \rho_{q}(\omega_{{\rm d}}+\widetilde{E}_{\boldsymbol{0}} -\widetilde{E}_{k_\pm})$ with $$\begin{aligned} \Lambda_{k_-} \!= 0 \mbox{ and } \Lambda_{k_+} \! = \!\left| \frac{\sqrt{2}}{N^2} \left(1+2\frac{\Delta}{\Delta_{{\rm c}}} \right) \left(\frac{g}{\Delta}\right)^3 \frac{{(\epsilon^{{\rm d}}})^2}{\Delta_{{\rm q}}} \right| \end{aligned}$$ for all $|k_-\rangle$ and $|k_+\rangle$ except for $|0\rangle$ or $|\pi\rangle$ in which case $\Lambda_{k_+}$ is reduced by a factor $\sqrt{2}$. Figure \[fig:firstsite\] shows that the $|k_+\rangle$ states can be obtained with substantial fidelities. It is worth noting that, in the case of a generic driving profile $\epsilon^{{\rm d}}_i$, instead of Eq. (\[eq:kpm\]), the emergent eigenstates are given by $$\begin{aligned} |k_\pm \rangle \equiv \frac{| k \rangle \pm \alpha_k | 2\pi - k \rangle}{\sqrt{1+\alpha_k^2}}\end{aligned}$$ where $\alpha_k$, the relative weight of $|k\rangle$ and $|2\pi-k\rangle$, is now controlled by the ratio $\epsilon^{{\rm d}}_k/\epsilon^{{\rm d}}_{2\pi-k}$. Therefore, such a non-equilibrium symmetry-breaking scenario offers highly flexible control over the target entangled state by simply engineering the drive profile $\epsilon^{{\rm d}}_k$. Photon-mediated interactions: General formulation ================================================= ![\[fig:3dEM\] (color online) Light-mediated interactions offer a highly versatile platform to design and control networks of interacting qubits. (a) The qubits (blue arrows) are embedded in an electromagnetic environment, which is the solution of the Maxwell equations in a given scattering geometry. (b) Integrating out the EM degrees of freedom yields effective interactions between the qubits, forming a network that can sustain large-scale entangled many-body states.](3dEM2.eps){width="7.5cm"} \[sec:ph-mediated\] In the previous sections, we focused on a particular geometry of the cavity-qubit system, namely a one-dimensional tight-binding lattice of photons. In this section, we show that our dissipative stabilization scheme via photon-mediated interactions is broadly applicable to any engineered EM environment. The situation we consider here is depicted in Fig. 6 where the two-level systems are now placed at arbitrary locations and interact with a general EM environment \[Fig. 6(a)\]. In practice, the latter is described by the solution of Maxwell equations in an arbitrary scattering geometry, characterized by a spectral problem with certain continuity and boundary conditions. In the simplest case, this would be a single resonator or a waveguide or, as in the specific example discussed previously, an array of evanescently (or capacitively) coupled cavities (transmission line cavities). After the derivation of the most general result, we show how the effective tight-binding result can be derived from first principles. The qubits and their coupling to the EM fluctuations are described by the Hamiltonian $$\begin{aligned} H = H_{\sigma} + H_{\sigma-{\rm EM}} + H_{\rm EM}\,,\end{aligned}$$ with ($\hbar = 1$) $$\begin{aligned} & H_{\sigma} = \sum_i \omega^{{\rm q}}_i \frac{\sigma^z_i}{2}, \quad H_{\rm EM} = \frac{\epsilon_0}{2} \int_\mathcal{V} \!\! {{\rm d}}^3 \boldsymbol{x} \, \left( \boldsymbol{E}^2 + c^2 \boldsymbol{B}^2 \right), \\ & H_{\sigma-{\rm EM}} = - \int_\mathcal{V} \!\! {{\rm d}}^3 \boldsymbol{x} \,\boldsymbol{P} \cdot \boldsymbol{E}\,.\end{aligned}$$ The integrals above run over the entire volume $\mathcal{V}$ of the scattering structure. We note that such a Hamiltonian can indeed be obtained for sub-gap electrodynamics in a general superconducting circuit architecture by a proper choice of normal modes [@malekakhlagh_origin_2015], starting from the parameters (position-dependent capacitances and inductances per unit length and the parameters of the qubits) of the underlying electrical circuit. For qubits residing at $\boldsymbol{x}_i$, each with a dipole moment strength $\mu$ (projected along a particular eigenpolarization of the electromagnetic medium), the collective atomic polarization operator can be written as $$\begin{aligned} P(\boldsymbol{x}) = \mu \sum_i \delta^3(\boldsymbol{x}-\boldsymbol{x}_i) \, \sigma_i^x\,.\end{aligned}$$ The electric field can generally be written in terms of a complete set of modes and corresponding eigenfrequencies $\{ \varphi_n , \omega_n \}$ specific to the chosen architecture $$\begin{aligned} E(\boldsymbol{x}) = \sum_{n} \varepsilon_n \varphi_n(\boldsymbol{x}) a_{n} + \mbox{H.c.}\;,\end{aligned}$$ so that $$\begin{aligned} \label{eq:Ha1} H_{\rm EM} = \sum_{n} \omega_n a_{n}^\dagger a_{n}\,.\end{aligned}$$ Here, $a^\dagger_n$ ($a_n$) creates (annihilates) a photon in the spatial mode $\varphi_n(\boldsymbol{x})$ with frequency $\omega_n$ and corresponding zero-point electric field $\varepsilon_n$. The modes are assumed to satisfy the completeness and orthogonality conditions $\sum_n \varphi_n(\boldsymbol{x}) \varphi_n ^* (\boldsymbol{x}') = \delta^3(\boldsymbol{x}-\boldsymbol{x}')$ and $ \int_\mathcal{V} \! {{\rm d}}^3 \boldsymbol{x} \, \varphi_n(\boldsymbol{x}) \varphi_m^*(\boldsymbol{x}) = \delta_{nm} $. With these normalization conditions, the zero-point fields are given by $\varepsilon_n \equiv \sqrt{{\omega_n}/{2\epsilon_0}}$. Neglecting counter-rotating terms, the light-matter coupling becomes $$\begin{aligned} H_{\sigma-{\rm EM}} = & \sum_{i,n} g_n \varphi_n(\boldsymbol{x}_i)\, a_n^\dagger \sigma_i^- + \mbox{H.c.}\,,\end{aligned}$$ where $\sigma_i^\pm \equiv (\sigma^x_i \pm {{\rm i}}\sigma_i^y)/2$ and $g_n \equiv - \mu \varepsilon_n$. We shall consider the regime in which the qubit frequencies $\omega^{{\rm q}}_i$ are far detuned from the photonic modes $\omega_n$ such that the light-matter coupling can be treated via second-order perturbation theory in $g_n/(\omega^{{\rm q}}_i - \omega_n)$. This is achieved by means of a Schrieffer-Wolf (SW) transformation [@schrieffer_relation_1966] which maps $H \mapsto {{\rm e}}^{X} H {{\rm e}}^{X^\dagger}$ where $$\begin{aligned} \label{eq:SWX} X \equiv \sum_{n,i} \left[ \frac{g_n \varphi_n(\boldsymbol{x}_i)}{\omega_i^{{\rm q}}- \omega_n} \sigma_i^+ a_n -\mathrm{H.c.} \right]. \end{aligned}$$ This yields the following Hamiltonian $H = H_\sigma + H_{\sigma a} + H_a$ to $\mathcal{O}(g^2/\Delta^2)$, with $$\begin{aligned} H_{\sigma} =& \sum_i \omega^{{\rm q}}_i \frac{\sigma^z_i}{2} + \frac{1}{2} \sum_{ij} \Sigma_{ij}(\omega^{{\rm q}}_i) \sigma_j^- \sigma_i^+ + \mbox{H.c.}\,, \label{eq:hsigma} \\ H_{\sigma a} =& \sum_{i, mn} \lambda_{i, mn}(\omega_i^{{\rm q}}) \,\frac{\sigma^z_i}{2} \, a_n a^\dagger_m + \mbox{H.c.} \,, \label{eq:sigmaa}\end{aligned}$$ and $H_a$ is still given by Eq. (\[eq:Ha1\]). In this low-energy Hamiltonian, the second term in $H_\sigma$ describes the qubit-qubit interactions mediated by virtual photons. These photons can be emitted into and absorbed from photonic channels at frequencies $\omega_n$ with spatial distribution $\varphi_n(\boldsymbol{x})$. This is precisely the story told by the coefficients $\Sigma_{ij}(\omega) = \sum_n |g_n|^2 \varphi_n(\boldsymbol{x}_i) \varphi_n^* (\boldsymbol{x}_j)/(\omega - \omega_n)$, which appears as a self-energy correction to the qubit sector. $\Sigma_{ij}(\omega) \sigma_j^x$ can be seen as the electric field generated at $\boldsymbol{x}_i$ by a dipole at $\boldsymbol{x}_j$, oscillating harmonically at frequency $\omega$. Note that the bare electromagnetic retarded Green’s function is given by $G^{\rm R} (\boldsymbol{x},\boldsymbol{x}'; \omega) = \sum_n \varphi_n(\boldsymbol{x}) \varphi_n^*(\boldsymbol{x}')/(\omega - \omega_n)$. This immediately implies that, in principle, all qubits interact with each other, to the extent that they can radiate EM radiation to each other \[see schematic in Fig. 6(b)\]. We note that realization-specific and restricted versions of this interaction vertex have been derived before [@majer_coupling_2007; @loo_photon-mediated_2013]. For what is proposed here, however, an equally important role is played by the term $H_{\sigma a}$ in Eq. (\[eq:sigmaa\]). This is the generalized version of the ac Stark-shift contribution to a qubit’s frequency that is well known in the dispersive regime of single-mode Cavity QED [@boissonneault_dispersive_2009], which can also be interpreted as a scattering term for photons generated by the interaction of the radiation field with qubits. This term expresses the spatial fluctuations of the effective index of refraction of the electromagnetic medium through the dynamically generated polarization fluctuations \[*i.e.* $P(\boldsymbol{x})$\] of the qubits. The interaction vertex here is again given by the resonant modes and their frequencies: $\lambda_{i,mn}(\omega) = g_n g_m^* \varphi_n (\boldsymbol{x}_i) \varphi_m^*(\boldsymbol{x}_i)/(\omega - \omega_n)$. We note that for a 1D tight-binding model of a cavity array with nearest-neighbor hopping, $\Sigma_{ij} (\omega_{{\rm q}}) \simeq (g/\Delta)^2 [ \Delta \delta_{i,j} - J\delta_{i,j\pm 1}] $ and $\lambda_{i,kq} (\omega_{{\rm q}}) \simeq \Delta (g/\Delta)^2 \varphi_k^*(i) \varphi_q(i)$ to lowest relevant order in $J/\Delta$ and $g/\Delta$. This result agrees with our direct derivation in Sec. \[sec:cooling\]. These results can be extended to a full-fledged stabilization protocol for a generalized W state $| W_n \rangle = \sum_{i=0}^{N-1} \varphi_n^* (\boldsymbol{x}_i) |\downarrow_0 \ldots\downarrow_{i-1} \, \uparrow_i \, \downarrow_{i+1} \ldots \downarrow_{N-1} \rangle$. Discussion and Conclusions {#sec:end} ========================== We proposed a general and scalable method based on photon-mediated interactions to drive a set of $N$ qubits to a desired generalized W state. The particular protocol discussed here for qubits embedded in a cavity array, amounts to the dissipative stabilization of a particular [*excited state*]{} of a many-body system (in the present case, a non-integrable variant of the XY model). This approach stands in contrast to cooling techniques employed for condensed-matter and cold atomic systems that, at least in principle, target the stabilization of the ground state. An interesting feature of the present approach compared to earlier approaches to dissipative engineering of entanglement [@kraus_preparation_2008; @diehl_quantum_2008; @cormick_dissipative_2013; @lee_emergence_2013; @rao_deterministic_2014; @reiter_scalable_2015] is the fact that [*both*]{} the unitary and the dissipative parts of the dynamics are adjusted through coupling to a common photonic bath. The Hamiltonian part provides the set of pure many-body states that can be reached in the steady state, while the dissipative part determines the occupation of those states. For cavities that are high-Q, the transitions can be made very selective. In the case of the dissipative stabilization of a W state of $N$ qubits, we discussed the scaling of the fidelity with the system size $N$. The fact that various properties of multi-qubit dynamics can be precisely adjusted by drive parameters provides a suitable platform for quantum simulation [@rotondo_dicke_2015] and computation [@verstraete_quantum_2009]. In scaling up CQED-based simulators [[@PhysRevLett.115.240501]]{} to larger architectures, one of the main obstacles is the uncontrolled site-to-site fluctuation of system parameters [@underwood_low-disorder_2012]. In the presented scheme, the [*dynamical tuning*]{} of effective spin-chain parameters, in fact both the unitary and the dissipative parameters, through the drive frequency and amplitude provides a promising route to realize large-scale quantum simulators. More generally, the method proposed here and its possible generalization to higher dimensional lattices holds promise for various quantum information applications, such as deterministic teleportation [@agrawal_perfect_2006; @wang_simple_2009]. The reduction of the collective dephasing mechanisms and the generation of entangled states in higher-excitation manifolds are important goals. Another interesting open question is the adaptation and extension of our protocol for targeted many-body state preparation in the photonic sector, scaling up recent approaches [@leghtas_confining_2015]. Acknowledgements {#sec:ack} ================ We are grateful to Mollie Schwartz, Leigh Martin, Emmanuel Flurin, Irfan Siddiqi, Liang Jiang, and Alexandre Blais for helpful discussions. This work has been supported by ARO Grant No. W911NF-15-1-0299 and NSF Grant No. DMR-1151810. M.K gratefully acknowledges support from the Professional Staff Congress of the City University of New York award No. 68193-0046.

--- abstract: 'Functional determinants of differential operators play a prominent role in theoretical and mathematical physics, and in particular in quantum field theory. They are, however, difficult to compute in non-trivial cases. For one dimensional problems, a classical result of Gel’fand and Yaglom dramatically simplifies the problem so that the functional determinant can be computed without computing the spectrum of eigenvalues. Here I report recent progress in extending this approach to higher dimensions (i.e., functional determinants of [*partial*]{} differential operators), with applications in quantum field theory.' address: | Institut für Theoretische Physik, Universität Heidelberg, 69120 Heidelberg, Germany\ Department of Physics, University of Connecticut, Storrs, CT 06269, USA author: - 'Gerald V. Dunne' title: 'Functional Determinants in Quantum Field Theory[^1]' --- Introduction : why are determinants important in QFT? {#sec-intro} ===================================================== This talk considers the question: [*what is the determinant of a partial differential operator, and how might one compute it?*]{} The physical motivation for this question comes from several areas of physics: - Effective actions, grand canonical potentials: These appear in relativistic and non-relativistic many-body theory and quantum field theory (QFT), and are generically given by an expression of the form $\Omega =\tr\,\ln\, G=\ln\, \det\, G$, where $G$ is a Green’s function, and the trace is over an infinite set of quantum states. - Tunneling and semiclassical physics: tunneling and nucleation processes can be analyzed in a semiclassical approach, wherein the leading exponential contribution involves the action evaluated on a classical solution, while the next-to-leading prefactor is the determinant of the fluctuation operator describing quantum fluctuations about the classical solution: e\^[-S\[\]]{} - Gap equations: in many-body theory and relativistic QFT, one can find ground states by solving self-consistent Hartree-Fock or Schwinger-Dyson equations, varying the grand canonical potential or effective action with respect to a condensate field. Symbolically, the problem is of the form (x)=where ${\mathcal D}$ is some differential operator. Clearly, to be able to vary the determinant efficiently, one needs a clever way to evaluate the determinant. - Lattice gauge theory: integrating out quarks leads to fermion determinant factors [D]{}| e\^[-d\^4x|]{} =(D-8pt / -m) which are needed for dynamical fermion computations in QCD. There are still fundamental challenges in evaluating such determinants within the Monte Carlo approach to lattice gauge theory, especially at finite density and temperature. - Faddeev-Popov determinants: the determinant of the Faddeev-Popov operator plays an important role in gauge fixing and confinement. - Mathematical physics: the determinant of a partial differential operator, such as a Dirac or Klein-Gordon operator, encodes interesting spectral information. What we know ============ Exact results for functional determinants exist only for certain special cases. For example, the gauge theory effective action \[the logarithm of the determinant of a Dirac or Klein-Gordon operator\] for background fields with (covariantly) constant field strength can be expressed as simple proper-time integrals, building on the seminal work of Heisenberg and Euler [@he; @schwinger; @gvd]. Mathematically, these expressions are closely related to the multiple zeta and gamma functions [@ruijsenaars]. This constant curvature case is soluble because we can choose a Fock-Schwinger gauge with $A_\mu=-\frac{1}{2}F_{\mu\nu}\,x_\nu$, in which case the spectral problem decomposes into a set of harmonic oscillator problems, for which the spectral information is very simple. Similarly, in gravitational problems, for special spaces such as harmonic spaces or symmetric spaces, the spectral problem is sufficiently separable and simple that the determinant can be evaluated [@camporesi]. For example, the determinant of Dirac or Klein-Gordon operators on spheres, cones and tori are known in quite explicit form [@camporesi; @dowker]. Remarkably, this type of analysis can also be extended to general Riemann surfaces [@dhoker; @sarnak], with important applications to worldsheet problems in string theory and with beautiful applications in number theory and differential geometry. Things become more interesting when including general non-constant curvatures \[either gauge or gravitational\]. Then there are a few especially symmetric cases that are still soluble, but they are all one-dimensional. For example, in gauge theory, if the field strength points in a particular fixed direction, and has a magnitude that is a function of only one coordinate \[say $x_\mu$\], and is of the form $F\,{\rm sech}^2\left(k \, x_\mu\right)$, then the corresponding \[Dirac or Klein-Gordon\] spectral problem is hypergeometric, and the determinant can be computed explicitly. Also, for massless quarks in an instanton background, the fermion determinant is known. Such a computation was pioneered by ’t Hooft for the single $su(2)$ instanton [@thooft], but has since been generalized to instantons of full ADHM form [@osborn], to finite temperature [@gross], and to calorons with nontrivial holonomy [@diakonov]. These computations all rely on the fact that the Green’s functions for massless particles in an instanton background are known explicitly. The generalization to nonzero masses is nontrivial, and is discussed below in Section \[idet-sec\]. Some general bounds are known, based on lattice regularizations and generalizations of Kato’s inequality for Dirac and Klein-Gordon operators. For example [@seiler; @vafawitten], | | 1 ; | | 1 . While very general, these bounds are still quite weak, and do not contain information about the phase of the determinant, which can be physically significant. Tighter bounds have been found recently in QED [@fry], but general results are still relatively limited. For Schrödinger operators with general potentials, or Dirac/Klein-Gordon operators with arbitrary gauge and/or gravitational backgrounds, we rely heavily on approximate methods. One such approximation is the heavy mass expansion. Define ==-\_0\^ e\^[-m\^2 s]{} {e\^[-s [D]{}]{}} Then an inverse mass expansion follows (after renormalization) from the small $s$ asymptotic expansion of the heat kernel operator [@vassilevich] {e\^[-s [D]{}]{}} \~ \_[k=0]{}\^s\^k a\_k. Here the $a_k\left[ {\mathcal D}\right]$ are known functionals of the potentials appearing in ${\mathcal D}$. Similarly, a derivative expansion can be derived by expanding the heat kernel trace $\tr\left\{e^{-s\, {\mathcal D}}\right\} $ about the soluble constant background case, in powers of derivatives of the background. This corresponds to resumming all non-derivative terms in the inverse mass expansion. While very general, these expansions are asymptotic, with higher terms becoming rapidly unwieldy, and so have a somewhat limited range of application. Another approach is to define the functional determinant via a zeta function (s)\_ , where the sum is over the spectrum of the relevant differential operator. Then, by the formal manipulations $\zeta^\prime(0)=-\sum_\lambda \ln\,\lambda=-\ln\left(\prod_{\lambda}\lambda\right)$, one [*defines*]{} the determinant as =e\^[-\^(0)]{} . The problem becomes one of analytically continuing $\zeta(s)$ from the region in which it converges \[typically $\Re (s) > d/2$\] to the neighbourhood of $s=0$. For example, in a radially separable problem, one might estimate the spectrum using WKB phase shifts \[and the corresponding spectral function $\rho(k)=\frac{1}{\pi}\frac{d\delta}{d k}$\], thereby obtaining information about the zeta function. The zeta function approach is complementary to the so-called [*replica method*]{}, where one defines the logarithm of an operator by considering [*positive*]{} integer powers $n$ of the operator, and then analytically continues to $n=0$. There are also non-trivial interesting one dimensional quantum mechanical problems for which the determinant can be computed from the zeta function [@voros]. There are several special features of one-dimensional problems that deserve mention. First, the issue of renormalization does not really enter in 1d. Second, the theory of inverse scattering allows us to characterize a potential in terms of the associated scattering data. This is especially useful for solving variational and gap equations, as the variational problem can then be reformulated in terms of the scattering data, which is more direct. Beautiful applications of this idea occur in interacting 1+1 dimensional QFT’s such as the Gross-Neveu model, Sine-Gordon model and their generalizations [@dhn; @thies]. \[Inverse scattering has been generalized in higher dimensions for radial problems [@avan]\]. A third special feature of 1d problems is the Gel’fand Yaglom theorem [@gy; @levit], which provides a way to compute the determinant of an ordinary differential operator without computing its eigenvalues. The basic idea is that in 1d one can define the Green’s function without using an eigenfunction expansion, simply from the product of two independent solutions. Gel’fand-Yaglom theorem: ordinary differential operators ======================================================== The simplest form of this result can be stated as follows [@gy; @levit; @coleman; @forman; @kappeler; @kleinert]. Consider a Schrödinger operator on the interval $x\in [0, L]$, with Dirichlet boundary conditions: (x)= (x); (0)=(L)=0 Then to compute the determinant it is in fact not necessary to compute the infinite discrete set of eigenvalues $\left\{\lambda_1, \lambda_2, \dots\right\}$. Instead, solve the related [*initial value problem*]{} (x)=0 ; (0)=0; \^(0)=1 \[ivp\] Then =(L). \[gy1\] It is worth pausing to appreciate the simplicity of this result, as well as its practical utility. It is straightforward to implement numerically. The determinant follows, without having to compute any eigenvalues, let alone multiply them all together! Actually, [(\[gy1\])]{} is not quite right, as we can only define the [*ratio*]{} of two determinants [@simon], and [(\[gy1\])]{} should be understood in this sense. Often in physical applications one is computing the determinant [*relative to*]{} the corresponding determinant for the free operator, for which the initial value problem is usually known in closed form \[in this case it is numerically better to solve directly the initial value problem for the [*ratio*]{}\]. As an illustration, consider the following simple example of the massive Helmholtz operator $\left[-\frac{d^2}{dx^2}+m^2\right]$. The Dirichlet spectrum is $\lambda_n=m^2+\left(\frac{n\,\pi}{L}\right)^2$, so from the eigenvalues: -1cm = \_[n=1]{}\^= \_[n=1]{}\^= \[1dh-1\] On the other hand, if we use the Gel’fand-Yaglom result, we solve the initial value problems: $\left[-\frac{d^2}{dx^2}+m^2\right]\phi=0$, to find $\phi(x)=\frac{\sinh(m\, x)}{m}$; and $-\frac{d^2}{dx^2} \phi_0=0$, to find $\phi_0(x)=x$ for the massless operator. Then [(\[gy1\])]{} implies -1cm == \[1dh-2\] These clearly agree, but the point is that if $m^2$ were replaced by a nontrivial potential $V(x)$, the first approach, from the eigenvalues, would be extremely difficult, while the Gel’fand-Yaglom approach is still easy. As another example, consider the Pöschl-Teller system with $H=[-\frac{d^2}{dx^2}+m^2-j(j+1)\,{\rm sech}^2\,x]$, with integer $j$. This is a reflectionless potential with $j$ bound states at $E=m^2-l^2$, $(l=1,\dots j$), and phase shift $\delta(k)=2\sum_{l=1}^j {\rm arctan}(l/k)$. The determinant follows directly from the spectrum as -2cm ( )&=&\_[l=1]{}\^j(m\^2-l\^2)+\_0\^(m\^2+k\^2)\ &=& On the other hand, the solutions to the Gel’fand-Yaglom initial value problems [(\[ivp\])]{} are -2cm (x)&=& (P\_j\^m(-[th]{} L) Q\_j\^m([th]{} x)- Q\_j\^m(-[th]{} L) P\_j\^m([th]{} x))\ -2cm \_[(j=0)]{}(x)&=&((m L) (m x)+ (m L) (m x)) on the interval $x\in [-L, L]$. The $P_j^m$ and $Q_j^m$ are Legendre functions. Then -2cm =\_[L]{} = Similarly, one can define a general reflectionless potential with $N$ bound states at $E=m^2-\kappa_i^2$, $(i=1,\dots N$), by $V(x)=m^2-2\frac{d^2}{dx^2}\ln\,\det \, A(x)$ where $A(x)$ is the $N\times N$ matrix $A_{ij}=\delta_{ij}+\frac{c_i\, c_j}{\kappa_i+\kappa_j}e^{-(\kappa_i+\kappa_j)x}$. The $c_i$ are (constant) moduli which affect the shape of the potential, but not the spectrum. From the phase shift $\delta(k)=2\sum_{i=1}^N {\rm arctan}(\kappa_i/k)$, one finds the determinant =It is instructive to verify this numerically for various potentials using [(\[ivp\])]{} and [(\[gy1\])]{}. The basic result [(\[gy1\])]{} generalizes in several ways. First, it generalizes to other boundary conditions [@forman; @kleinert; @kirsten], characterized by two $2\times 2$ matrices $M$ and $N$ as M()+N ()=0 Construct two independent solutions $u_{(1),(2)}$ of $\left[-\frac{d^2}{dx^2}+V(x)\right]\phi(x)=0$ such that u\_[(1)]{}(0)=1 &;& u\^\_[(1)]{}(0)=0\ u\_[(2)]{}(0)=0 &;& u\^\_[(2)]{}(0)=1 Then the infinite dimensional determinant reduces to a simple $2\times 2$ determinant = \_[22]{}For example, -1cm [Dirichlet]{}: M=(), N=() && =u\_[(2)]{}(L)\ -1cm [Neumann]{}: M=(), N=() && =u\_[(1)]{}\^(L)\ -1cm[Periodic(P)]{}: M=[**1**]{}, N=-[**1**]{} && =2-(u\_[(1)]{}(L)+u\^\_[(2)]{}(L))\ -1cm[Antiperiodic (AP)]{}: M=[**1**]{}, N=[**1**]{}&& =2+(u\_[(1)]{}(L)+u\^\_[(2)]{}(L))In the P/AP cases we recognize $(u_{(1)}(L)+u^\prime_{(2)}(L))$ as the Floquet-Bloch [*discriminant*]{}, which takes values $\pm 2$ for P/AP eigenfunctions. Second, these results generalize straightforwardly to coupled systems of ODE’s [@forman; @kirsten], and also to linear ODE’s of any order [@dym]. Third, they generalize to Sturm-Liouville problems [@kirsten]. For example, consider the radial operator in $d$ dim., with Dirichlet b.c.’s \_[(l)]{}-++V(r). \[radial-op\] Define the function $\phi_{(l)}(r)$ as the solution to the radial initial value problem \_[(l)]{} (r) =0; \_[(l)]{}(r)\~r\^[l+(d-1)/2]{} ,r0 . \[phi-l\] Then (this should also be understood as applying to [*ratios*]{} of determinants) =\_[(l)]{}(R). \[gyr\] As an explicit example, consider the 2d radial Helmholtz problem, with eigenvalues in terms of the zeros $j_{(l),n}$ of the $J_l$ Bessel function: $\lambda_{(l), n}=m^2+\left(\frac{j_{(l),n}}{R}\right)^2$. Then -1cm = \_[n=1]{}\^ = \_[n=1]{}\^\[radial-eig\] On the other hand, if we use the Gel’fand-Yaglom result [(\[gyr\])]{}, we readily solve the relevant initial value problems to find $\phi_{(l)}$ in terms of the modified Bessel function $I_l$: = . \[radial-bessel\] The agreement between [(\[radial-eig\])]{} and [(\[radial-bessel\])]{} expresses the product formula for $I_l(x)$. Gel’fand-Yaglom theorem: partial differential operators? ======================================================== In generalizing from [*ordinary*]{} to [*partial*]{} differential operators, it is natural to consider first the case of [*separable*]{} operators. But the naïve extension fails. Formally, we simply take a product over the radial determinants \[each of which is evaluated as above\] for all possible angular momenta, with the appropriate degeneracies. However, this product diverges [@forman]. For example, for the 2d radial Helmholtz system [(\[radial-bessel\])]{} ( ) =( ) \~O(), lThe degeneracy factor is $1$ for $l=0$, and $2$ for all $l\geq 1$, so the angular momentum sum over $l$ diverges. This should not be so surprising from a physical point of view, as we expect to require renormalization in dimension higher than 1. To formulate the problem more precisely, consider the radially separable operators $ {\mathcal M}=-\Delta+V(r)$, and ${\mathcal M}^{\rm free}=-\Delta $, where $\Delta$ is the Laplace operator in ${\mbox{${\rm I\!R }$}}^d$, and $V(r)$ is a radial potential vanishing at infinity as $r^{-2-\epsilon}$ for $d=2$ and $d=3$, and as $r^{-4-\epsilon}$ for $d=4$. Since $V=V(r)$, we can separate variables and consider the Schrödinger-like radial operator ${\mathcal M}_{(l)}$ in [(\[radial-op\])]{}. For dimension $d\geq 2$, the radial eigenfunctions $\psi_{(l)}$ have degeneracy (l ; d) . \[deg\] Formally, we write ()=\_[l=0]{}\^(l; d) () . \[formal-sum\] Each term in the sum is computed straightforwardly using [(\[gyr\])]{}, but the $l$ sum is divergent. However, this divergence can be understood physically, leading to a finite and renormalized determinant ratio [@dk]: -2cm ()|\_[d=2]{} & =& ()+ \_[l=1]{}\^2 {()-}\ && +\_0\^dr r V \[2d-result\]\ -2cm ()|\_[d=3]{} & =& \_[l=0]{}\^(2 l+ 1 ) { ()-} \[3d-result\]\ -2cm ()|\_[d=4]{}& =& -2cm\ & &-3cm \_[l=0]{}\^(l+1)\^2 { ()-+}\ &-2cm &-2cm -\_0\^dr r\^3 V(V+2m\^2) . \[4d-result\] Here $\gamma$ is Euler’s constant, and $\mu$ is a renormalization scale, which is essential for physical applications. A conventional renormalization choice is to take $\mu=m$ in [(\[4d-result\])]{}. In each of [(\[4d-result\])]{}, the sum over $l$ is convergent once the indicated subtractions are made. The function $\phi_{(l)}(r)$ is defined in [(\[phi-l\])]{}. Notice that the results [(\[4d-result\])]{} state once again that the determinant is determined by the boundary values of solutions of $\left[{\mathcal M}+m^2\right]\phi=0$, with the only additional information being a finite number of integrals involving the potential $V(r)$. We also stress the computational simplicity of [(\[4d-result\])]{}; no phase shifts or eigenvalues need be computed to evaluate the renormalized determinant. These results have been derived [@dk] using the zeta function formalism. Scattering theory [@taylor] and standard contour manipulations [@kirsten; @kirstenbook] yield an integral representation \[valid for $\Re (s) > d/2$\] of the zeta function in terms of the Jost function $f_l(ik)$: $$\begin{aligned} \zeta(s) ={\sin \pi s\over \pi} \sum_{l=0}^{\infty}{\rm deg}(l; d) \, \int\limits_{m}^{\infty}dk\,\, [k^2-m^2]^{-s}~\frac{\partial}{\partial k}\ln f_l (ik)\quad . \label{zeta-jost}\end{aligned}$$ The technical problem is the analytic continuation of [(\[zeta-jost\])]{} to a neighborhood about $s=0$. This analytic continuation relies on the uniform asymptotic behavior of the Jost function $f_l (i k)$, which follows from standard results in scattering theory [@taylor]: $$\begin{aligned} \hskip -2.5 cm \ln f_l (ik) = \int\limits_0^{\infty}dr\, r\, V(r) K_{\nu} (kr) I_{\nu} (kr)-\int\limits_0^{\infty}dr\,r\, V(r) K_{\nu}^2 (kr) \int\limits_0^rdr'\, r' \, V(r') I_{\nu}^2 (kr')+ {\cal O} (V^3) {\nonumber}\label{jost-expansion}\end{aligned}$$ where $\nu\equiv l+\frac{d}{2}-1$. Then the calculation of the asymptotics of the Jost function reduces to the known uniform asymptotics of the modified Bessel functions $K_\nu$ and $I_\nu$. Using these asymptotics, we can [*define*]{} $\ln f_l^{asym}(ik)$ as the $O(V)$ and $O(V^2)$ parts of this uniform asymptotic expansion, and then by construction, $$\begin{aligned} \zeta_f' (0) &\equiv & -\sum_{l=0}^{\infty}{\rm deg}(l; d) \left[\ln f_l (i m) -\ln f_l^{asym} (i m) \right] \quad . \label {zetaprime-f}\end{aligned}$$ is now convergent. Adding back the subtracted terms in a dimensional regularization scheme, leads [@dk] to the results in [(\[4d-result\])]{}. Comparison With Feynman Diagram Approach {#sec-feynman} ======================================== These determinants can also be derived in a Feynman diagrammatic approach, where the perturbative expansion in powers of the potential $V$ is () & & \_[k=1]{}\^ A\^[(k)]{}\ &=&- + +…, \[feynman\] Here the solid dots denote insertions of the potential $V$. If the potential is radial, then with dimensional regularization one can express the renormalized 4d determinant as [@baacke] -2.5cm \_[d=4]{} &=& \_[l=0]{}\^(l+1)\^2{ f\_l(i m) - \_0\^drr V(r) K\_[l+1]{}(m r) I\_[l+1]{}(m r).\ &&. -3cm +\_0\^dr r V(r) K\_[l+1]{}\^2(m r) \_0\^r dr\^r\^V(r\^) I\_[l+1]{}\^2(m r\^)}+A\^[(1)]{}\_[fin]{}- A\^[(2)]{}\_[fin]{} \[baacke-split\] where the finite contributions from the first and second order Feynman diagrams in the $\overline{MS}$ scheme are [@baacke] (note the small typo in equation (4.32) of [@baacke]): -1cm A\_[fin]{}\^[(1)]{}&=&-\_0\^dr r\^3 V(r)\ -1cm A\_[fin]{}\^[(2)]{}&=&\_0\^dq q\^3 | (q)|\^2 \[a-finite\] Here $\tilde{V}(q)$ is the 4d Fourier transform of the radial potential $V(r)$. These expressions [(\[baacke-split\])]{} and [(\[a-finite\])]{} look quite different from [(\[4d-result\])]{}, but in fact they can be shown to be completely equivalent, using some Bessel identities [@dk]. However, it is clear that [(\[4d-result\])]{} is much easier to evaluate, especially if (as often happens) $V(r)$ is known only numerically. Removing Zero Modes {#zero-modes} =================== In certain quantum field theory applications the determinant may have zero modes, and correspondingly one is actually interested in computing the determinant with these zero modes removed, together with a collective coordinate factor [@coleman]. For example, consider the case of a self-interacting scalar field theory in $d$ dim Euclidean space with Euclidean action $ S[\Phi]=\int d^d x\left[ \partial_\mu \Phi \partial_\mu \Phi +U(\Phi)\right] $, where there is a \[radially symmetric\] classical solution $\Phi_{cl}(r)$ solving $ \Delta \Phi=\frac{d U}{d\Phi}$, with $ \Phi^\prime (0)=0$, and $\Phi \to 0$ as $r\to \infty$. The fluctuation operator about this classical solution is a radial operator, with $V(r)=\left(\left[\frac{d^2 U}{d \Phi^2}\right]_{\Phi=\Phi_{cl}(r)} -m^2\right)$. In the $l=1$ sector the corresponding radial operator ${\mathcal M}_{(l=1)}$ has a $d$-fold degenerate zero mode associated with translational invariance: \_[zero]{}(r)&=& . \[zero-mode\] Including the collective coordinate contribution [@coleman] due to translational invariance, the required determinant factor is \[the prime denotes exclusion of zero mode(s)\] ()\^[d/2]{} ( )\^[-1/2]{} . \[coll\] Since the determinant is expressed in terms of a solution to an ODE, we can use simple ODE theory to show that (generalizing the $d=1$ [@mckane] and $d=4$ [@dunnemin] analyses) =\^d, \[removed\] Here, $ \psi_{\rm zero}^{\rm free}(r)=2^{d/2} \Gamma (\frac{d}{2}+1) \, I_{d/2}(r)/r^{d/2-1}$, and the classical solution behaves as $ \Phi_{cl}(R)\sim \Phi_{\infty} \frac{K_{d/2-1}(R)}{R^{d/2-1}}$, as $R\to \infty$, for some constant $\Phi_{\infty}$. Thus the $R\to\infty$ limit exists and is simple. Furthermore, from the virial theorem, $ \int_0^\infty dr\, r^{d-1}\, \psi_{\rm zero}^2(r)=\frac{\Gamma\left(\frac{d}{2}+1\right) S[\Phi_{cl}]}{\pi^{d/2}\, \left(\Phi_{cl}^{\prime\prime}(0)\right)^2} $, so we arrive at the simple expression for the net contribution of this $l=1$ mode: ()\^[d/2]{} ( )\^[-1/2]{}=\^[d/2]{} . \[zero-factor\] This expresses the zero mode factor solely in terms of the asymptotic properties of the classical solution $\Phi_{cl}$, which are already known numerically as part of the determination of $\Phi_{cl}$ and the fluctuation potential $V(r)$. No further computation is needed. Application 1: fluctuation determinant for false vacuum decay ============================================================= The phenomenon of nucleation drives first order phase transitions in many applications in physics. The semiclassical analysis of the rate of such a nucleation process was pioneered by Langer [@langer], who identified a semiclassical saddle point solution that gives the dominant exponential contribution to the rate, with a prefactor to the exponential given by the quantum fluctuations about this classical solution. The nucleation rate is given by the quantum mechanical rate of decay of a metastable “false” vacuum, $\Phi_-$, into the “true” vacuum, $\Phi_+$. Decay proceeds by the nucleation of expanding bubbles of true vacuum within the metastable false vacuum [@langer; @kobzarev; @stone; @coleman-fv; @wipf; @gorokhov]. The semiclassical prefactor requires computing the determinant of the differential operator associated with quantum fluctuations about the classical solution. This is a technically difficult problem, but it is ideally suited to the above results for determinants of radially separable operators, because the fluctuation operator is radial. The expression [(\[dm-answer2\])]{} below can be applied to [*any*]{} metastable field potential $U(\Phi)$ [@dunnemin], without relying on the “thin-wall” limit (the limit of degenerate vacua) . For definiteness, consider the specific (rescaled) quartic potential: $ U(\Phi)=\frac{1}{2}\Phi^2-\frac{1}{2}\Phi^3+\frac{\alpha}{8}\Phi^4 $, where $\alpha$ characterizes the shape of the potential \[the thin-wall limit is $\alpha\to 1$\]. The decay rate per unit volume and unit time is = ()\^2 ||\^[-1/2]{} e\^[-S\_[cl]{}\[\]-\_[ct]{}S\[\]]{}, \[rate\] where the prime on the determinant means that the zero modes (corresponding to translational invariance) are removed. Here $\phic$ is a classical solution known as the “bounce” solution [@coleman-fv], defined below, and the prefactor terms in (\[rate\]) correspond to quantum fluctuations about this bounce solution. The second term in the exponent, $\delta_{ct}S[\phic]$, denotes the counterterms needed for renormalizing the classical action $S_{\rm cl}$. The first step in computing the false vacuum decay rate $\gamma$ is to find the classical bounce solution, $\phic(r)$, which is a radially symmetric stationary point of the classical Euclidean action, interpolating between the false and true vacuum. The bounce $\phic(r)$ solves the nonlinear ordinary differential equation -\^ -\^+-\^2+ \^3=0 \[bounceeq\] with boundary conditions $ \phic^\prime(0)=0$, $ \phic(r)\to \Phi_-\equiv 0$, as $r\to\infty$. $\phic(r)$ must be computed numerically, for example by the method of shooting, adjusting the initial value $\phic(0)$ until the boundary conditions at $r=\infty$ are satisfied. Given the bounce solution, $\phic(r)$, the corresponding radial fluctuation potential is U\^((r)) = 1-3(r)+\^2(r). \[flucpot\] There are three different types of eigenvalue of the fluctuation operator, each having a different role physically and mathematically. (i) Negative Mode: The $l=0$ sector has a negative eigenvalue mode of the fluctuation operator, and is responsible for the instability leading to decay. This mode contributes a factor to the decay rate $\gamma$ related to the [*absolute value*]{} of the determinant of the $l=0$ fluctuation operator [@langer; @coleman-fv]. (ii) Zero Modes: In the $l=1$ sector, there is a four-fold degenerate zero eigenvalue of the fluctuation operator. Integrating over the corresponding collective coordinates produces the factors of $\frac{S_{\rm cl}}{2\pi}$ in (\[rate\]). In computing the rate $\gamma$, we need the determinant of the fluctuation operator with the zero mode removed [@langer; @coleman-fv; @wipf]. This can be evaluated simply using the formula [(\[zero-factor\])]{}, in terms of the asymptotic behavior of the classical bounce solution $\phic(r)$. (iii) Positive Modes: For $l\geq 2$, the fluctuation operator has positive eigenvalues, each of degeneracy $(l+1)^2$. For each $l$, the associated radial determinant is computed numerically using [(\[gyr\])]{}. ![Comparison as $\alpha\to 1$ of the exact numerical result \[dots and solid blue line\] from [(\[dm-answer2\])]{} with the analytic thin-wall limit result $\frac{9}{32}\, \left[1-\frac{2\pi}{9\sqrt{3}}\right]\approx 0.16788..$ of [@kr].[]{data-label="fig-alpha"}](figure1.eps) But the sum over $l$ is divergent and the answer must be renormalized. Using the $\overline{{\rm MS}}$ scheme, the renormalized effective action is [@dunnemin]: -2cm \_ &=& |T\_[(0)]{}()|-2\ -2cm && + \_[l=2]{}\^(l+1)\^2 { (T\_[(l)]{}())-+}\ -2cm&& -\_0\^dr r V(r) + \_0\^dr r\^3 V(V+2)(-\_E-) \[dm-answer2\] The first term is from the negative mode, the second from the zero mode(s) \[using [(\[zero-factor\])]{}\], the third term from the numerical computation of the higher modes, and the last term is the renormalization counterterm contribution. The expression [(\[dm-answer2\])]{} can be applied to [*any*]{} field potential $U(\Phi)$, without relying on the thin-wall limit. A comparison with the analytic thin-wall computation of Konoplich and Rubin [@kr] is plotted in Fig. \[fig-alpha\], and shows excellent agreement. This approach can also be extended to false vacuum decay in curved space, where the structure of the bounce solutions [@hackworth], and the corresponding fluctuation equations [@dw], is much richer. Application 2: determinant for quarks in instanton background {#idet-sec} ============================================================= The fermion determinant in an instanton background is known in the massless (chiral) limit [@thooft], and in the heavy quark limit [@nsvz]. Here we compute it for [*any*]{} $m$ [@idet]. The relevant part of the renormalized effective action, $\tilde{\Gamma}^{S}_{\rm ren}(m)$, behaves as $$\begin{aligned} \hskip-2cm \tilde{\Gamma}^{S}_{\rm ren}(m)&=& \begin{cases} {\alpha(1/2)+\frac{1}{2}\left(\ln m+\gamma-\ln 2\right)m^2 +\dots \quad , \quad m\rightarrow 0 \cr \displaystyle { -\frac{\ln m}{6}-\frac{1}{75 m^2}-\frac{17}{735 m^4}+\frac{232}{2835 m^6}-\frac{7916}{148225 m^8}+\cdots \quad , \quad m\rightarrow \infty}} \end{cases} \label{masslimit}\end{aligned}$$ where $\alpha(1/2)= \simeq 0.145873$. The small mass expansion [@thooft] is based on the known massless propagators in an instanton background, while the large mass expansion [@nsvz] follows most easily from the Schwinger-DeWitt or heat kernel expansion. Since the instanton is self-dual, we can simplify things slightly and work with the scalar Klein-Gordon operator. For a single $su(2)$ instanton, the spectral problem separates into partial waves [@thooft], with radial Schrödinger-like operators (take isospin $\frac{1}{2}$) $$\hskip -2cm {\cal H}_{(l,j)} \equiv \left[ - \frac{\partial^2}{\partial r^2}-\frac{3}{r}\frac{\partial}{\partial r}+\frac{4l(l+1)}{r^2}+\frac{4(j-l)(j+l+1)}{r^2+1}-\frac{3}{(r^2+1)^2} \right]\, , \label{insth}$$ Here $l=0, \frac{1}{2}, 1, \frac{3}{2}, \cdots\;$, and $j=| l \pm \frac{1}{2}|$, and there is a degeneracy factor of $(2l+1)(2j+1)$. Since the “potential” involves $l$ and $j$, we cannot directly use the result [(\[4d-result\])]{}. In order to compare with ’t Hooft’s massless quark analysis [@thooft], we use a Pauli-Villars regulator $\Lambda$, and define the renomalized effective action by charge renormalization. To extract the renormalized effective action we need to consider the $\Lambda\to\infty$ limit in conjunction with the infinite sum over $l$. Split the partial wave sum into two parts: $$\begin{aligned} \Gamma_\Lambda^S(m)&=& \sum_{l=0,\frac{1}{2},\dots}^L \Gamma_{\Lambda, (l)}^S(m) + \sum_{l=L+\frac{1}{2}}^\infty \Gamma_{\Lambda, (l)}^S(m) \label{actionsplit}\end{aligned}$$ where $L$ is a large but finite integer. The first sum involves low partial wave modes, and the second sum involves the high partial wave modes. For the low partial wave modes, we can remove the regulator and evaluate the (finite) sum using the (numerical) Gel’fand-Yaglom result [(\[gyr\])]{}. But this sum diverges quadratically with $L$. In the second sum in (\[actionsplit\]) we cannot take the large $L$ and large $\Lambda$ limits blindly, as each leads to a divergence. Radial WKB is a good approximation for the high $l$ modes. With WKB we can compute [*analytically*]{} the large $\Lambda$ and large $L$ divergences of the second sum in (\[actionsplit\]), using the WKB approximation for the corresponding determinants [@idet]: $$\begin{aligned} \hskip -2cm \sum_{l=L+\frac{1}{2}}^\infty \Gamma_{\Lambda, (l)}^S(A; m)&\sim& \frac{1}{6}\ln \Lambda+2 L^2 + 4 L-\left(\frac{1}{6}+\frac{m^2}{2}\right)\ln L \nonumber\\ && +\left[\frac{127}{72}-\frac{1}{3}\ln 2+\frac{m^2}{2}-m^2 \ln 2+\frac{m^2}{2}\ln m \right]+O\left(\frac{1}{L}\right) \label{div}\end{aligned}$$ It is important to identify the physical role of the various terms in (\[div\]). The first term is the expected logarithmic charge renormalization counterterm. The next three terms give quadratic, linear and logarithmic divergences in $L$. These divergences cancel [*exactly*]{} against corresponding divergences in the first sum in (\[actionsplit\]), which are found numerically. Combining the numerical results for low partial wave modes with the radial WKB results for the high partial wave modes the renormalized effective action $\tilde{\Gamma}^S_{\rm ren}(m)$ is $$\begin{aligned} \hskip-1cm \tilde{\Gamma}^S_{\rm ren}(m)&=&\lim_{L\to \infty}\left\{\sum_{l=0,\frac{1}{2},\dots}^L (2l+1)(2l+2) P(l) +2 L^2 + 4 L-\left(\frac{1}{6}+\frac{m^2}{2}\right)\ln L \right.\nonumber \\ &&\left. +\left[\frac{127}{72}-\frac{1}{3}\ln 2+\frac{m^2}{2}-m^2 \ln 2+\frac{m^2}{2}\ln m \right]\right\}. \label{answer2}\end{aligned}$$ This is finite for any mass $m$. Figure \[fig4\] shows these results for $\tilde{\Gamma}^{S}_{\rm ren}(m)$, compared with the analytic small and large mass expansions in (\[masslimit\]). The agreement is remarkable [@idet]. It is a highly nontrivial check on the WKB computation that the correct mass dependence rises to interpolate between the large mass and small mass regimes. ![Plot of our numerical results for $\tilde{\Gamma}^{S}_{\rm ren}(m)$ from (), compared with the analytic extreme small and large mass limits \[dashed curves\] from (). The dots denote numerical data points from (), and the solid line is a fit through these points. \[fig4\]](figure2.eps) As an interesting [*analytic*]{} check, the formula (\[answer2\]) also provides a simple computation [@idet] of ’t Hooft’s leading small mass result $\alpha(1/2)=-\frac{17}{72}-\frac{1}{6}\ln 2+\frac{1}{6}-2\zeta^\prime(-1)=0.145873... $. Concluding remarks ================== In this talk I have reviewed some of the reasons for considering determinants of partial differential operators in quantum field theory. Mathematically, much is known for [*ordinary*]{} differential operators, especially through the work of Gel’fand-Yaglom [@gy; @levit] and its extensions [@forman; @kleinert; @kirsten], but considerably less is known for [*partial*]{} differential operators with non-trivial potentials. The new mathematical results [@dk] reported here are the formulas in [(\[4d-result\])]{}, which provide simple expressions for the determinant of a radially separable partial differential operator of the form $-\Delta+m^2+V(r)$. This generalizes the (Dirichlet) Gel’fand-Yaglom result [(\[gy1\])]{} to higher dimensions. These results lead to many direct applications in quantum field theory, where they extend the class of solvable fluctuation determinant problems away from the restrictive class of constant background fields, or one dimensional background fields, to the more general class of separable higher dimensional background fields. This includes, for example, applications in quantum field theory to the study of quantum fluctuations in the presence of vortices, monopoles, sphalerons, instantons, domain walls (branes), etc ... A number of further generalizations could be made. First, it is practically important to include [*directly*]{} the matrix structure that arises from Dirac-like differential operators and from non-abelian gauge degrees of freedom. This has been done in various ways for certain applications [@baacke2; @carson; @baacke3; @strumia; @burnier], but a simpler unified formalism along the lines of the explicit formulas [(\[4d-result\])]{} is probably possible, as in one dimension [@waxman]. Second, a practical extension to finite temperature and density would be useful in various physical applications. Third, the biggest technical challenge is to ask if the restriction of separability can be relaxed. A promising current approach is the numerical worldline loop method [@gies]. Without symmetry, computing functional determinants analytically remains a difficult problem, and any progress will be interesting. Acknowledgments {#acknowledgments .unnumbered} =============== I thank the organizers, especially Mariano Del Olmo, for a stimulating conference, in a beautiful city. I acknowledge support from the DFG through the Mercator Guest Professor Program, and from the DOE through grant DE-FG02-92ER40716. .2 cm [12345]{} W. Heisenberg and H. Euler, “Consequences of Dirac’s theory of positrons,” Z. Phys.  [**98**]{}, 714 (1936); English translation at \[arXiv:physics/0605038\]. J. S. Schwinger, “On gauge invariance and vacuum polarization,” Phys. Rev.  [**82**]{}, 664 (1951). G. V. Dunne, “Heisenberg-Euler effective Lagrangians: Basics and extensions,” in [*Ian Kogan Memorial Collection, ’From Fields to Strings: Circumnavigating Theoretical Physics’*]{}, M. Shifman et al (ed.) vol. 1 pp 445-522 (World Scientific, 2005). \[arXiv:hep-th/0406216\]. 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--- abstract: 'We discuss three related models of scale-free networks with the same degree distribution but different correlation properties. Starting from the Barabasi-Albert construction based on growth and preferential attachment we discuss two other networks emerging when randomizing it with respect to links or nodes. We point out that the Barabasi-Albert model displays dissortative behavior with respect to the nodes’ degrees, while the node-randomized network shows assortative mixing. These kinds of correlations are visualized by discussig the shell structure of the networks around their arbitrary node. In spite of different correlation behavior, all three constructions exhibit similar percolation properties.' address: - '$^1$Institut für Physik, Humboldt Universität zu Berlin, Invalidenstraße 110, D-10115 Berlin, Germany' - '$^2$Theoretische Polymerphysik, Universität Freiburg, Hermann Herder Str. 3, D-79104 Freiburg, Germany' author: - 'R. Xulvi-Brunet$^1$, W. Pietsch$^1$, and I.M. Sokolov$^{1,2}$' title: 'Correlations in Scale-Free Networks: Tomography and Percolation' --- [2]{} Introduction {#introduction .unnumbered} ============ Scale-free networks, i.e. networks with power-law degree distributions, have recently been widely studied (see Refs. [@baretalb; @dorogov] for a review). Such degree distributions have been found in many different contexts, for example in several technological webs like the Internet [@int; @past], the WWW [@www2; @WWW], or electrical power grids [@Wa], in natural networks like the network of chemical reactions in the living cell [@Oltvai; @Fell; @Mason] and also in social networks, like the network of human sexual contacts [@sex], the science [@New1; @New2] and the movie actor [@Am; @Alb] collaboration networks, or the network of the phone calls [@phonecall]. The topology of networks is essential for the spread of information or infections, as well as for the robustness of networks against intentional attack or random breakdown of elements. Recent studies have focused on a more detailed topological characterization of networks, in particular, in the degree correlations among nodes [@past; @Newm; @Berg; @Egui; @Bo; @mas; @Vaz; @Goh; @Bog; @NE; @Serr]. For instance, many technological and biological networks show that nodes with high degree connect preferably to nodes with low degree [@past; @mas], a property referred to as disassortative mixing. On the other hand, social networks show assortative mixing [@Newm; @NE], i.e. highly connected nodes are preferably connected to nodes with high degree. In this paper we shall study some aspects of this topology, specifically the importance of the degree correlations, in three related models of scale-free networks and concentrate on the two important characteristics: the tomography of shell structure around an arbitrary node, and percolation. The Models {#the-models .unnumbered} ========== Our starting model is the one of Barabasi and Albert (BA) [@BA-model], based on the growth algorithm with preferential attachment. Starting from an arbitrary set of initial nodes, at each time step a new node is added to the network. This node brings with it $m$ proper links which are connected to $m$ nodes already present. The latter are chosen according to the preferential attachment prescription: The probability that a new link connects to a certain node is proportional to the degree (number of links) of that node. The resulting degree distribution of such networks tends to [@Redner; @degdis; @Kra]: $$P(k)=\frac{2m(m+1)}{k(k+1)(k+2)} \sim k^{-3}. \label{degdis}$$ Krapivsky and Redner [@Kra] have shown that in the BA-construction correlations develop spontaneously between the degrees of connected nodes. To assess the role of such correlations we shall randomize the BA-network. Recently Maslov and Sneppen [@mas] have suggested an algorithm radomitzing a given network that keeps the degree distribution constant. According to this algorithm at each step two links of the network are chosen at random. Then, one end of each link is selected randomly and the attaching nodes are interchanged. However, in case one or both of these new links already exits in the network, this step is discarded and a new pair of edges is selected. This restriction prevents the apparearance of multiple edges connecting the same pair of nodes. A repeated application of the rewiring step leads to a randomized version of the original network. We shall refer to this model as link-randomized (LR) model. The LR model can be compared with another model which is widely studied in the context of scale-free networks, namely with the configuration model introduced by Bender and Canfield [@cand; @mollreed]. It starts with a given number $N$ of nodes and assigning to each node a number $k_i$ of “edge stubs” equal to its desired connectivity. The stubs of different nodes are then connected randomly to each other; two connected stubs form a link. One of the limitations of this “stub reconnection” algorithm is that for broad distribution of connectivities, which is usually the case in complex networks, the algorithm generates multiple edges joining the same pair of hub nodes and loops connecting the node to itself. However, the cofiguration model and the LR model get equivalent as $ N\rightarrow \infty $. One can also consider a node-randomized (NR) counterpart of the LR randomize procedure. The only difference to the link-radomized algorithm is that instead of choosing randomly two links we choose randomly two nodes in the network. Then the procedure is the same as in the LR model. As we proceed to show, the three models have different properties with respect to the correlations between the degrees of connected nodes. While the LR (configuration) model is random, the genuine BA prescription leads to a network which is dissortative with respect to the degrees of connected nodes, and the NR model leads to an assortative network. This fact leads to considerable differences in the shell structure of the networks and also to some (not extremely large) differences in their percolation characteristics. We hasten to note that our simple models neglect many important aspects of real networks like geography [@Soki; @geog] but stress the importance to consider the higher correlations in the degrees of connected nodes. Tomography of the Networks {#shell structure .unnumbered} ========================== Referring to spreading of computer viruses or human diseases, it is necessary to know how many sites get infected on each step of the infection propagation. Thus, we examine the local structure in the network. Cohen et al. [@tomography] examined the shells around the node with the highest degree in the network. In our study we start from a node chosen at random. This initial node (the root) is assigned to shell number 0. Then all links starting at this node are followed. All nodes reached are assigned to shell number 1. Then all links leaving a node in shell 1 are followed and all nodes reached that don’t belong to previous shells are labelled as nodes of shell 2. The same is carried out for shell 2 etc., until the whole network is exhausted. We then get $N_{l,r}$, the number of nodes in shell $l$ for root $r$. The whole procedure is repeated starting at all $% N $ nodes in the network, giving $P_{l}(k)$, the degree distribution in shell $l$. We define $P_{l}(k)$ as: $$P_{l}(k)=\frac{\sum_{r} N_{l,r}(k)}{\sum_{k,r} N_{l,r}(k)}. \label{aa}$$ We are most interested in the average degree $\langle k\rangle _{l}=\sum_{k}kP_{l}(k)$ of nodes of the shell $l$. In the epidemiological context, this quantity can be interpreted as a disease multiplication factor after $l$ steps of propagation. It describes how many neighbors a node can infect on average. Note that such a definition of $P_{l}(k)$ gives us for the degree distribution in the first shell: $$P_{1}(k)=\frac{\sum_{r} N_{1,r}(k)}{\sum_{k,r} N_{1,r}(k)}= \frac{kN_k}{\sum_k kN_k}=\frac{kP(k)}{\langle k \rangle}, \label{bb}$$ where $P(k)$ and $N_k$ are the degree distribution and the number of nodes with degree $k$ in the network respectively. We bear in mind that every link in the network is followed exactly once in each direction. Hence, we find that every node with degree $k$ is counted exactly $k$ times. From Eq.($\ref{bb}$) follows that $\langle k\rangle _{1}=\langle k^{2}\rangle / \langle k\rangle$. This quatity, that plays a very important role in the percolation theory of networks [@cohetal], depends only on the first and second moment of the degree distribution, but not on the correlations. Of course $P_0(k)=P(k)$. Note that as $N\rightarrow \infty $ we have $\langle k^{2}\rangle \rightarrow \infty $: for our scale-free constructions the mean degree in shell 1 depends significantly on the network size determining the cutoff in the degree distribution. However, the values of $\langle k\rangle _{1}$ are the same for all three models: The first two shells are determined only by the degree distributions. In all other shells the three models differ. For the LR (configuration) model one finds for all shells in the thermodynamic limit $P_{l}(k)=P_{1}(k)$. However, since these distributions do not possess finite means, the values of $\langle k\rangle _{l}$ are governed by the finite-size cutoff, which is different in different shells, since the network is practically exhausted within the first few steps, see Fig.1. In what follows we compare the shell structure of the BA, the LR and the NR models. We discuss in detail the networks based on the BA-construction with $m=2$. For larger $m$ the same qualitative results were observed. In the present work we refrain from discussion of a peculiar case $m=1$. For $m=1$ the topology of the BA-model is distinct from one for $m\geq 2$ since in this case the network is a tree. This connected tree is destroyed by the randomization procedure and is transformed into a set of disconnected clusters. On the other hand, for $m\geq 2$ the creation of large separate clusters under randomization is rather unprobable, so that most of the nodes stay connected. Fig. \[fig1\] shows $\langle k\rangle $ as a function of the shell number $l$. Panel (a) corresponds to the BA model, panel (b) to the LR model, and panel (c) to the NR model. The different curves show simulations for different network sizes: $N=3,000$; $N=10,000$; $N=30,000$; and $N=100,000$. All points are averaged over ten different realizations except for those for networks of 100,000 nodes with only one simulation. In panel (d) we compare the shell-structure for all three models at $N=30,000$. The most significant feature of the graphs is the difference in $\langle k \rangle _{2}$. In the BA and LR models the maximum is reached in the first shell, while for the NR model the maximum is reached only in the second shell: $\langle k\rangle_{2,BA}<\langle k\rangle _{2,LR}<\langle k\rangle _{2,NR}$. This effect becomes more pronounced with increasing network size. In shells with large $l$ for all networks mostly nodes with the lowest degree $2$ are found. The inset in graph (a) of Fig. \[fig1\] shows the relation between average age $\eta$ of nodes with connectivy $k$ in the network as a function of their degree for the BA model. The age of a node $n$ and of any of its proper links is defined as $\eta (n)=(N-t_{n})/N$ where $t_{n}$ denotes the time of birth of the node. For the randomized LR and NR models age has no meaning. The figure shows a strong correlation between age and degree of a node. The reasons for these strong correlations are as follows: First, older nodes experienced more time-steps than younger ones and thus have larger probability to acquire non-proper bonds. Moreover, at earlier times there are less nodes in the network, so that the probability of acquiring a new link per time step for an individual node is even higher. Third, at later time-steps older nodes already tend to have higher degrees than younger ones, so the probability for them to acquire new links is considerably larger due to preferential attachment. The correlations between the age and the degree bring some nontrivial aspects into the BA model based on growth, which are erased when randomizing the network. Let us discuss the degree distribution in the second shell. In this case we find as that every link leaving a node of degree $k$ is counted $k-1$ times. Let $P(l|k)$ be a probability that a link leaving a node of degree $k$ enters a node with degree $l$. Neglecting the possibility of short loops (which is always appropriate in the thermodynamical limit $N \rightarrow \infty$) and the inherent direction of links (which may be not totally appropriate for the BA-model) we have: $$P_{2}(l)=\frac{\sum_k kP(k)(k-1)P(l|k)}{\sum_{k}kP(k)(k-1)}. \label{P2} %P_{2}(l)=\sum_{k}\left[ \frac{kP(k)(k-1)P(l|k)}{\sum_{k}kP(k)(k-1)}% %\right] . \label{P2}$$ [2]{} The value of $\langle k\rangle _{2}$ gives important information about the type of mixing in the network. To study mixing in networks one needs to divide the nodes into groups with identical properties. The only relevant characteristics of the nodes that is present in all three models, is their degree. Thus, we can examine the degree-correlations between neighboring nodes, which we compare with the uncorrelated LR model, where the probability that a link connects to a node with a certain degree is independent from whatever is attached to the other end of the link: $P(k | l)=kP(k)/\langle k\rangle =kP(k)/2m$. All other relations would correspond to assortative or disassortative mixing. Qualitatively, assortativity then means that nodes attach to nodes with similar degree more likely than in the LR-model: $P(k | l)>P(k | l)_{LR}=kP(k) / \langle k\rangle$ for $k\approx l$. Dissortativity means that nodes attach to nodes with very different degree more likely than in the LR-model: $P(k | l)> kP(k) / \langle k\rangle $ for $k\gg l$ or $l\gg k$. Inserting this in Eq.(\[P2\]), and calculating the mean, one finds qualitatively that $\langle k \rangle _{1}=\langle k\rangle _{2,LR}<\langle k\rangle _{2}$ for assortativity, and $\langle k\rangle _{1}>\langle k\rangle _{2}$ for dissortativity. In the following we show where the correlations of the BA and NR model originate. A consequence of the BA-algorithm is that there are two different types of ends for the links. Each node has exactly $m$ proper links attached to it at the moment of its birth and a certain number of links that are attached later. Since each node receives the same number of links at its birth, towards the proper nodes a link encounters a node with degree $k$ with probability $P(k)$. To compensate for this, in the other direction a node with degree $k$ is encountered with the probability $\frac{(k-m)P(k)}{m} =2\frac{kP(k)}{\langle k\rangle }-P(k)$, so that both distributions together yield $kP(k)/\langle k\rangle $. On one end of the link nodes with small degree are predominant: $P(k)<kP(k)/\langle k\rangle $ for small $k$. On the other end nodes with high degree are predominant: $(k-m)P(k)/m>kP(k)/2m$ for $k$ large. This corresponds to dissortativity. Actually the situation is somewhat more complex since in the BA model these probability distributions also depend on the age of the link. Assortativity of the NR model is a result of the node-randomizing process. Since the nodes with smaller degree are predominant in the node population, those links are preferably chosen that have on the end with the randomly chosen node a node with a smaller degree ($P(k)>kP(k)/\langle k\rangle $ for $k$ small). Then the randomization algorithm exchanges the links and connects those nodes to each other. This leads to assortativity for nodes with small degree, which is compensated by assortativity for nodes with high degree. Percolation {#percolation .unnumbered} =========== Percolation properties of networks are relevant when discussing their vulnerability to attack or immunization which removes nodes or links from the network. For scale-free networks random percolation as well as vulnerability to a deliberate attack have been studied by several groups [@cohetal; @ben; @je; @cohetal2; @callnew]. One considers the removal of a certain fraction of edges or nodes in a network. Our simulations correspond to the node removal model; $q$ is the fraction of removed nodes. Below the percolation threshold $q<q_{c}$ a giant component (infinite cluster) exists, which ceases to exist above the threshold. A giant component, and consequently $q_{c}$ is exactly defined only in the thermodynamic limit $ N\rightarrow \infty $: it is a cluster, to which a nonzero fraction of all nodes belongs. In [@mollreed] and [@cohetal] a condition for the percolation transition in random networks has been discussed: Every node already connected to the spanning cluster is connected to at least one new node. Ref. [@cohetal] gives the following percolation criterion for the configuration model: $$1-q_{c}=\frac{\langle k\rangle }{\langle k^{2}\rangle -\langle k\rangle }, \label{condperc}$$ where the means correspond to an unperturbed network ($q=0$). For networks with degree distribution Eq.($\ref{degdis}$), $\langle k^{2}\rangle $ diverges as $N \rightarrow \infty$. This yields for the random networks with a such degree distribution a percolation threshold $q_{c}=1$ in the thermodinamic limit, independent of the minimal degree $m$; in the epidemiological terms this corresponds to the absence of herd immunities in such systems. Crucial for this threshold is the power-law tail of the degree distribution with an exponent $\leq 3$. Moreover, Ref. [@ben] shows that the critical exponent $\beta $ governing the fraction of nodes $M_{\infty }$ of the giant component, $M_{\infty }\propto (q_{c}-q)^{\beta }$, diverges as the exponent of the degree distribution approaches $-3$. Therefore $M_{\infty }$ approaches zero with zero slope as $q\rightarrow 1$. In Fig. \[fig2\] we plotted for the three models discussed $M_{\infty }$ as a function of $q$. The behavior of all three models for a network size of $300,000$ nodes is presented in panel (a). In the inset the size of the giant component was measured in relation to the number of nodes remaining in the network $(1-q)N$ and not to their initial number $N$. Other panels show the percolation behavior of each of the models at different network sizes: Panel (b) corresponds to the BA model, (c) to the LR model, and (d) to the NR model. For the largest networks with $N=300,000$ nodes we calculated 5 realizations for each model, for those with $30,000$; $10,000$; and $3,000$ nodes averaging over 10 realization was performed. For all three models within the error bars the curves at different network sizes coincide. This shows that even the smallest network is already close to the thermodynamical limit. R. Albert et al. have found a similar behavior in a study of BA-networks [@je]. They analyze networks of sizes $N=1000, 5000$ and $20000$ concluding “that the overall clustering scenario and the value of the critical point is independent of the size of the system”. In the simulations we find two regimes: for moderate $q$ we find, that the sizes of the giant components of the BA, LR, and NR model obey the inequalities $M_{\infty ,BA}>M_{\infty ,LR}>M_{\infty ,NR}$ , while for $q$ close to unity the inequalities are reverted: $M_{\infty ,BA}<M_{\infty ,LR}<M_{\infty ,NR}$. However, in this regime the differences between $M_{\infty ,BA},M_{\infty ,LR}$ and $M_{\infty ,NR}$ are subtle and hardly resolved on the scales of Fig. 2. We note that similar situation was observed in Ref. [@Newm]. However, there the size of the giant cluster was measured not as a function of $q$ but of a scaling parameter in the degree distribution. The observed effects can be explained by the correlations in the network. For $q=0$ one has $M_{\infty,BA}=M_{\infty ,LR}=M_{\infty ,NR}$. Now, the probability that single nodes loose their connection to the giant cluster depends only on the degree distribution, and not on correlations. So, the difference in the $M_{\infty}$ must be explained by the break-off of clusters containing more than one node. The probability for such an event is smaller in the BA than in the LR model, since dissortativity implies that one finds fewer ’regions’, where only nodes with low degree are present. However, when we get to the region of large $q$, as nodes with low degree act as ’bridges’ between the nodes with high degree, the connections between the nodes with high degree are weaker in the case of the BA model than in the case of the LR model. So, the probability that nodes with high degree break off is higher for the BA model than for the LR model. 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--- author: - | A dissertation submitted\ for the degree of\ Doctor of Philosophy in Physics\ \ by\ \ Shi Pu\ \ \ \ \ \ \ \ Supervisor: Qun Wang\ \ 2011 bibliography: - 'bib/main.bib' title: | [UNIVERSITY OF SCIENCE AND TECHNOLOGY OF CHINA ]{}\ [Heifei, CHINA]{}****\ ****\ ****\ ****\ ****\ **Relativistic fluid dynamics in heavy ion collisions** --- Copyright by Shi Pu 2011 [All Rights Reserved]{} **Dedicated to my dear family**

--- abstract: 'We consider a dilute two-component atomic fermion gas with unequal populations in a harmonic trap potential using the mean field theory and the local density approximation. We show that the system is phase separated into concentric shells with the superfluid in the core surrounded by the normal fermion gas in both the weak-coupling BCS side and near the Feshbach resonance. In the strong-coupling BEC side, the composite bosons and left-over fermions can be mixed. We calculate the cloud radii and compare axial density profiles systemically for the BCS, near resonance and BEC regimes.' address: - '$^1$Department of Physics, National Chung Cheng University, Chiayi 621, Taiwan' - '$^2$Institute of Physics, Academia Sinica, Nankang, Taipei 115, Taiwan' author: - 'C.-H. Pao$^1$ and S.-K. Yip$^2$' title: Asymmetric Fermi superfluid in a harmonic trap --- \[intro\]Introduction: ====================== The original BCS state for superconductors considers pairing between two species of fermions with equal populations. For a long time, theorists studied the fermion system with unequal species, or mismatched Fermi surfaces, and proposed this system may have different ground state [@FFLO], in particular the so called Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) phase. Experimentally, however, such superfluid states remain unclear because of the difficulty in preparing the magnetized superconductors. Experiments with ultra-cold atoms have opened a new era to study this fermion system with unequal populations. Through the Feshbach resonance [@Feshbach], the effective interaction between atoms can be varied over a wide range such that the ground state can be turned from a weak-coupling BCS superfluid to a strong-coupling Bose-Einstein condensation (BEC) regime. In the homogeneous system, theoretical studies [@Carlson05; @pao06; @Son05] of the unequal fermion species show that the phase transition must occur when the resonance is crossed, in contrast to the equal population case where a smooth crossover takes place [@EL; @SRE93]. Breached pair phase [@LW], phase separated states are also proposed [@Bedaque03; @Caldas04] in this system. Two recent experiments [@ketterle06; @hulet06] studied the trapped $^6$Li atoms with imbalanced spin populations and obtained the density profiles for various population differences. Both groups found the system contains a superfluid core surrounded by a normal fermions and provide evidence for phase separation near the crossover. In this paper, we study this imbalanced fermion system by mean field approximation and evaluate the density profiles for various coupling strengths from weak-coupling BCS superfluid to strong coupling BEC regime. In particular, we calculate the axial density profiles, superfluid and minority cloud radii, and distinguish between the phase separation and Bose-Fermi mixture regimes. This paper is organized as follows: In Sec. II we briefly review the mean-field approximation for the dilute fermion atoms with unequal populations. In Sec. III we present our results for various polarizations from weaking-coupling BCS superfluid to strong-coupling BEC side. We show that the axial density profiles are constant within the superfluid core and decrease beyond the phase boundary for the phase separations but are smoothly decreasing functions for entire trap for the mixtures. Finally, we conclude with a briefly summary in Sec. IV. While this work was in progress, several theoretical papers have also studied the same problem under similar approximations [@Yi; @deSilva; @Haque; @chevy06] or going beyond [@kinnunen05; @pieri05]. Basically, ref [@Yi; @deSilva; @Haque; @chevy06; @kinnunen05] also conclude the system is phase separated into concentric shells with superfluid in the center and surrounding by leftover fermions near the resonance. The strong-coupling BEC limit has also been studied [@Yi; @deSilva; @pieri05]. In this case, the composite bosons and unpaired fermions can mix. As the population difference increases the unpaired fermions can even penetrate into the superfluid core. Our paper provides a more systematic study of the entire BCS-BEC regimes for all polarizations. \[form\]Formalism: ================== Restricting ourselves to wide Feshbach resonance, the two-component fermion system can be described by an effective one-channel Hamiltonian $$H\ =\ \label{eqh} \sum_{{\bf k}, \sigma} \xi_\sigma ({\bf k}) c^\dagger_{{\bf k},\sigma} c_{{\bf k},\sigma}\ +\ g \sum_{\bf k,k^\prime, q} c^\dagger_{{\bf k+q},\uparrow} c^\dagger_{{\bf k^\prime -q},\downarrow} c_{{\bf k^\prime},\downarrow} c_{{\bf k},\uparrow}\ ,$$ where $\xi_{\sigma}({\bf k})\, =\, \hbar^2 k^2/2m - \mu_\sigma$ and the index $\sigma$ runs over the two spin components. Within the BCS mean field approximation at zero temperature, the excitation spectrum in a homogeneous system for each spin is (see e.g. [@WY03] for details) $$E_\sigma ({\bf k})\, = \, \frac{ \xi_{\sigma}({\bf k}) - \xi_{- \sigma}({\bf k})}{ 2}\ +\ \sqrt{ \left ( \frac{ \xi_{\sigma}({\bf k}) + \xi_{-\sigma} ({\bf k})} {2 } \right )^2\, + \, \Delta^2}\ , \label{eqdisp}$$ where $\xi_{\sigma}({\bf k})\, =\, \hbar^2 k^2/2m - \mu_\sigma$ are the quasi-particle excitation energies for normal fermions, and $-\uparrow \equiv \downarrow$. For an inhomogeneous system, $e.g.$ the system in a harmonic trap, a finite system sized effect should be included [@bruun99]. However, the system can be treated as homogeneous locally if the number of particles are sufficiently large. A local density approximation, or Thomas-Fermi approximation (TFA), is applied and the chemical potential for spin $\sigma$ is replaced by $$\mu_\sigma (r) \ =\ \mu^0_\sigma\ - \frac{1}{2} m \omega^2 r^2\ ,$$ with $\omega$ the isotropic trap frequency and $r$ the distance from the trap center. We shall show results explicitly only for the isotropic trap. In the local density approximation, the densities $n_\sigma ({\bf r})$ depends only on the local chemical potentials $\mu_\sigma({\bf r})$. Hence the density profile of an anisotropic trap can be related to an isotropic one by rescaling the spatial coordinates appropriately. We then introduce the average chemical potential $$\mu (r)\ \equiv\ \frac{1}{2} [\mu_\uparrow (r) + \mu_\downarrow (r)]\ =\ \mu_0\ -\ \frac{1}{2} m \omega^2 r^2\ ,$$ and the difference $h\ \equiv \ [\mu_\uparrow (r) - \mu_\downarrow (r) ]/2\, =\, (\mu^0_\uparrow - \mu^0_\downarrow)/2 $. The dispersion relation in Eq. (\[eqdisp\]) becomes $$E_{\uparrow,\downarrow} ({\bf k,r})\, = \, \sqrt{ \xi({\bf k,r})^2 + \Delta^2({\bf r})} \mp h \label{Ek}$$ where $\xi({\bf k,r}) \equiv \hbar^2 k^2/2m - \mu (r)$. We take spin up to be the majority species so that $h$ and $E_\uparrow$ are positive always. Then the density profiles in a harmonic trap are $$\begin{aligned} n_s(r)& = & n_\uparrow(r) + n_\downarrow (r)\ =\ \int { d^3 k \over (2 \pi)^3} \left [ 1 - {2 \xi({\bf k, r}) \over E_\uparrow + E_\downarrow} f(-E_\uparrow)\right ]\ , \label{eqns}\\ n_d (r) & = & n_\uparrow(r) - n_\downarrow (r)\ =\ \int { d^3 k \over (2 \pi)^3} f(E_\uparrow)\ , \label{eqnd}\end{aligned}$$ and the total number of particles $N\, = \, \int d^3 r n_s(r) $. Here $f$ is the Fermi function. The polarization of the system is defined as $$P \ \equiv\ {N_\uparrow - N_\downarrow \over N}\ =\ {1 \over N}\int d^3 r n_d(r)\ .$$ Now the pairing field $\Delta$ depends on position also. In the local density approximation, it obeys an equation similar to the homogeneous case [@pao06; @WY03]: $$- \frac{ m }{ 4 \pi a} \Delta (r)\, =\, \Delta(r) \int \frac{ d^3 k}{ (2\pi)^3} \left [ \frac{ 1 - f(E_\uparrow) - f(E_\downarrow) }{ E_\uparrow + E_\downarrow }\, -\, \frac{ m }{ \hbar^2 k^2} \right ]\ . \label{eqgap}$$ For a given scattering length $a$, we solve equations (\[eqns\]), (\[eqnd\]) and (\[eqgap\]) self-consistently for fixed total number of particles $N$ and polarization $P$. The solutions to the“gap equation” (\[eqgap\]) may not be unique. The physical solution is determined by the condition of minimum free energy among the multiple solutions. We describe the detail procedures in \[freeenergy\]. \[result\]Results and Discussions ================================= In this section, we investigate the density profiles for various polarizations and coupling strengths from positive detuning BCS superfluid to negative detuning BEC side. With the aid of density profiles, we evaluate the radii of the superfluid phase boundaries for various cases and compare to the current experimental results. We close this section with a discussion of axial density profiles. Phase separation versus Bose-Fermi mixture can be clarified through the axial density profiles of the population difference. In Fig. \[fig1\], we plot the radial density profiles for three different coupling strengths $1 / k_F a\, =\, -0.61, \, 0.03$, and $2.44$ (for different columns) with polarization $P\, =\, 0.2\, , 0.5$, and $0.9$ (for different rows). The total number of particles are fixed to $2 \times 10^{5}$. Here $k_F$ is the Fermi wavevector at the trap center for an ideal symmetric Fermi gas with the same total number of particles. In all these plots, the system shows a superfluid cloud surrounded by a normal Fermi gas except the case in Fig. \[fig1\](c) where the system is completely in the normal state with the polarization $P = 0.9$. It is consistent with the experimental observation [@ketterle06] that the superfluid is destroyed by a sufficiently large population difference. We remark here however that the critical population difference $P_c$ for the destruction of superfluidity obtained here is much larger than that found in the experiment [@ketterle06]. For $1/k_F a = -0.61$ here, $P_c > 0.6$ theoretically whereas an extrapolation of the data of [@ketterle06] gives $P_c < 0.3$ (see further discussions below). For less polarized system on the BCS side \[Fig. \[fig1\](a) and (b)\], it shows a clear phase separation between the superfluid and a normal Fermi gas. Note that within the superfluid cloud, the population difference is zero and the system is just like the typically unpolarized superfluid. Outside the superfluid cloud, both components of the fermions exist in the normal state which indicates a fraction of the fermions which are not paired-up even at the zero temperature. The density profile of the population difference $n_d(r)$ peaks at the superfluid phase boundary and decreases gradually toward the edge of the trap. Its value is equal to the density profile of the majority when $r \ge r_\downarrow$, the radii of the minority cloud. At the superfluid phase boundary, both the majority and minority density profiles exhibit discontinuous which has been observed also by the others [@Yi; @deSilva; @Haque]. Near resonance \[Fig. \[fig1\](d)-(f)\], similar phase separated states are observed. However, most of the minority are paired up in this regime such that the density profiles contains mainly the excess fermions outside the superfluid cloud at all $P$’s. This is consistent with the homogeneous normal phase boundary extends near resonance to large population difference [@pao06]. Near resonance, the superfluid core survives at $P=0.9$ in our calculations whereas experimentally [@ketterle06] it vanishes already at $P \approx 0.7$. On the BEC side \[Fig. \[fig1\](g)-(i)\], all of the minority are paired up and the excess fermions can penetrate into the superfluid core. The system contains a superfluid core for any $P$ (if sufficiently deep in the BEC regime, see Fig. \[fig4\] below). The system then contains three different phases: the purely superfluid, the mixture phase, and the normal fermions from the trap center to the edge of the trap. The mixture phase extends toward the trap center as the polarization increases. In Fig. \[fig1\](i) ($P = 0.9$), the system is highly polarized and the excess fermions extend deeply into the trap center. In \[BEClimit\], we give an analytic discussion of the density profiles in the BEC limit. The radius $r_s$ of the superfluid core is one of the most interesting quantities in current studies of the imbalanced fermion system [@hulet06; @deSilva; @Haque; @chevy06; @kinnunen05]. From Fig. \[fig1\], the density profiles of the population difference $n_d(r)$ have maxima at the phase boundaries between the superfluid and the normal fermions. $r_s$ is thus also the peak position of $n_d(r)$. In Fig. \[fig2\], we plot $r_s$ as function of $P$ for three different coupling strengths. $r_s$ behaves quite differently above and below the resonance. On the BCS side the superfluid is eliminated when the polarization reaches around 0.65 for $1 /(k_F a) = -0.61$ and the system becomes completely normal with mismatch Fermi surfaces beyond this critical polarization. For large coupling strengths, $r_s$ is finite except when $P$ is exactly $1$, since the superfluid is stable for any finite ($\ne 1$) polarization in the homogeneous case [@pao06]. Except for small ($P \le 0.1$) or large ($P \ge 0.9$) polarizations, the sizes of the superfluid clouds have maxima near the Feshbach resonance at fixed polarization. We also plot the radii ($r_\downarrow$) of the minority (spin down fermions) cloud in Fig. \[fig3\]. Below the Feshbach resonance, this radius is the same as the radius of the superfluid core because all the minority of fermions are paired up. However, this won’t be true above the Feshbach resonance where part of minority are not paired up in this regime. Unlike $r_s$, $r_\downarrow$ decreases monotonically as the coupling strength increases. These two radii become identical when the minority are paired up completely when the system reaches the BEC regime. Due to the experimental constraints, one can not measure the radial density profiles directly. Instead the axial density profiles are reported in [@hulet06]. In Fig. \[fig4\], we plot the normalized axial density profiles $n_a(z)$ $[ \equiv \int dx dy n_d (\vec r) ]$ of the population difference for different coupling strengths at three fixed polarization $P$. For the cases with phase separations \[Figs. \[fig1\](a), (b), and (d)-(f)\] , the corresponding $n_a(z)$ are constants for $z \le r_s$ and have a kink at the phase boundary (for $z = r_s$). This feature results from the population difference $n_d(r)$ being zero inside the superfluid cloud (see \[adensity\] in detail) such that $n_a (z)$ remains the same value as at the phase boundary $z= r_s$. For a system with a mixed phase region \[Figs. \[fig1\](g)-(i)\], $n_a (z)$ increases smoothly toward the trap center even as $z \le r_s$. It reaches a constant value within the cloud containing superfluid only \[ $e.g.$, the cases with solid lines in Figs. \[fig4\](a) and (b)\]. At large polarization \[solid line in Fig. \[fig4\](c)\], the excess fermions mix with superfluid entirely for $1 / k_F a\, =\, 2.44$ such that $n_a (z)$ increases monotonically toward the trap center. The completely different features for the axial density profiles of the population difference inside the superfluid cloud can help us to clarify whether the system is phase separation or mixture. \[conclusion\]Conclusion ======================== We have investigated the radial density profiles of the two-component fermion system with unequal spin-populations under Feshbach resonance. The system shows a superfluid cloud in the trap center surrounding by normal fermions. In the weak-coupling BCS side, the superfluid is destroyed completely at the polarization $P \, \gtrsim\, 0.65$ for $1 /(k_F a) = -0.61$. Near the Feshbach resonance, almost all the minority are paired up and the system is phase separated into superfluid and the normal fermions. In the strong-coupling BEC side, the excess fermions can mix with the superfluid and the system contains three different phases, the purely superfluid cloud, the mixture phase, and the normal fermions from the trap center to the edge of the trap. In particular, we emphasize the difference in the axial density difference profiles between the phase separation and Bose-Fermi mixture regimes. The former shows a constant for $z < r_s$ and has a kink at the phase boundary but the later are smoothly increasing toward the trap center. However, Ref. [@hulet06] reported positive slopes of the axial density different profiles inside the superfluid cloud that indicates the $n_d (\vec r)$ need to be negative somewhere within the superfluid cloud. We do not obtain this phenomenon within current local density approach. This research was supported by the National Science Council of Taiwan under grant numbers NSC94-2112-M-194-001 (CHP) and NSC94-2112-M-001-002 (SKY), with additional support from National Center for Theoretical Sciences, Hsinchu, Taiwan. \[freeenergy\]Free energy ========================= To determine the minimum free energy state when there are multiple solutions, we need an expression for the free energy. We obtain this as follows. First consider a system with fixed volume $V$ and particle numbers $N_{\sigma}$. With the Hamiltonian in Eq. (\[eqh\]), it is straight-forward to show that [@AGD] the energy $E( N_{\sigma}, g) $ of this system obeys $$\frac{\partial E}{\partial g} = \frac{V}{g^2} |\Delta|^2 \label{Eg} \ .$$ We need to eliminate $g$ in favor of the scattering length $a$, the physical parameter of the system. These two variables are related by the expression $$\frac{m}{4 \pi \hbar^2 a}\ =\ \frac{1}{g} + \frac{1}{V} \sum_{\vec k} \frac{1}{2 \epsilon_k}\ . \label{ag}$$ We thus get $$\frac{m}{4 \pi \hbar^2} d \left( \frac{1}{a} \right) = d \left( \frac{1}{g} \right) \ , \label{dadg}$$ and therefore $$\frac{\partial E}{\partial (1/a)} \vert_{N_{\sigma}} = - V \frac{m}{4 \pi \hbar^2} |\Delta(a) |^2 \ . \label{Ea}$$ Writing this derivative to be ${\cal E}'$, we then get the thermodynamic relation $$dE = \sum_{\sigma} \mu_{\sigma} d N_{\sigma} + {\cal E}' d (\frac{1}{a}) \ . \label{dE}$$ The free energy $\Omega(\mu_{\sigma}, a) \equiv E - \mu_{\uparrow} N_{\uparrow} - \mu_{\downarrow} N_{\downarrow} $ then obeys $$d \Omega = - \sum_{\sigma} N_{\sigma} d \mu_{\sigma} + {\cal E}' d (\frac{1}{a}) \ . \label{dOmega}$$ We thus conclude that, for volume $V$ and chemical potentials $\mu_{\sigma}$, $$\frac{\partial \Omega}{\partial (1/a)} \vert_{\mu_{\sigma}} = - V \frac{m}{4 \pi \hbar^2} |\Delta|^2 \ . \label{dOda}$$ Though this expression is already sufficient to determine and thus compare the free energies of the different solutions for given scattering length $a$, we can convert it to an even more convenient form. To do this, let us write $x \equiv |\Delta|^2$ and $y \equiv 1/a$. We get, up to an overall multiplying factor $$\frac{\partial \Omega}{\partial y} = - x \ .\label{dOdx}$$ Thus, when the solutions are plotted in the form of Fig. \[fig5\], the free energy $\Omega$ of a state can be related to the free energy $\Omega_0$ of another state along the same curve by, again up to an overall multiplicative constant, $$\Omega = \Omega_0 - \int x dy \ .\label{Oxy}$$ Since $\int x dy$ is the area between the curve and the y-axis, the states corresponding to the minimum free energy at a given scattering length $a$ (hence $y$) can be found via the same procedure as the usual Maxwell construction. \[BEClimit\]BEC limit ===================== Here we try to understand the behaviour of the density profile in the BEC ($ 1/k_F a \gg 1$) limit, for the special case of small number of excess fermions \[$e.g.$ Fig. \[fig1\](g)\]. For a bulk system, it is straight forward to perform a low density expansion of the mean-field equations and obtain an expansion of the chemical potentials $\mu_f \equiv \mu_{\uparrow}$ and $\mu_b \equiv \mu_{\uparrow} + \mu_{\downarrow}$ as a series in the densities $n_f = n_d$ and $n_b \equiv n_{\downarrow}$ (see [@Yip02] for the details of this calculation). In the BEC limit, $n_d$ can be interpreted as the density of the excess unpaired fermions and $n_b$ as the density of the bosons which represent the bound fermion pairs. $\mu_f$ and $\mu_b$ can be interpreted as the corresponding chemical potentials. Explicitly we have an expansion of the form $$\begin{aligned} \mu_f &=& A n_d^{2/3} + g_{bf} n_b \ , \label{muf} \\ \mu_b &=& - \epsilon_b + g_{bb} n_b + g_{bf} n_f \ , \label{mub}\end{aligned}$$ where we have dropped the terms higher order in the densities. These terms are smaller in the limit $n_f a^3 \ll 1$ and $n_b a^3 \ll 1$. We obtain [@Yip02] $A = (6 \pi^2)^{2/3} / 2 m$, $\epsilon_b = \hbar^2 / m a^2$, $g_{bb} = 4 \pi \hbar^2 a / m$, $g_{bf} = 8 \pi \hbar^2 a / m$. The value of $A$ is as expected for a free Fermi gas and $\epsilon_b$ is the binding energy of the fermion pair. The values of $g_{bb}$ and $g_{bf}$ here, obtained from the mean-field equations, differ from the correct values known. Exact three-body calculation [@Skorniakov] gives $g_{bf}^{\rm exact} = 3.6 \pi \hbar^2 a / m$, while a recent four-body calculation [@Petrov04] gives $g_{bb}^{\rm exact} = 1.2 \pi \hbar^2 a / m$. Alternatively, defining the scattering lengths in the usual manner, our low density expansion yields the effective values $a_{bf} = 8 a /3$ and $a_{bb} = 2 a$, whereas the exact values should be $a_{bf}^{\rm exact} = 1.2 a$ and $a_{bb}^{\rm exact} = 0.6a $. The difference between the mean-field and exact results is due of course to the approximate nature of the mean-field theory. We note however, here that (a) $a_{bf}$ and $a_{bb}$ are both positive and of order $a$, hence in the BEC limit we have necessarily $n_d a^3 \ll 1$, thus the fermions and the bosons can mix [@viverit00]; (b) $g_{bf}/ g_{bb} > 1/2$ for both mean-field and exact values. We shall explain the importance of this second relation below. In the trap, Eqs. (\[muf\]) and (\[mub\]) becomes $$\begin{aligned} \mu_f &=& A n_d (\vec r) ^{2/3} + g_{bf} n_b (\vec r) + V( \vec r) \ , \label{muft} \\ \mu_b &=& - \epsilon_b + g_{bb} n_b (\vec r) + g_{bf} n_f (\vec r)+ 2 V (\vec r) \ . \label{mubt}\end{aligned}$$ Here $V(\vec r)$ is the trap potential. We shall only consider the case where the potential increases from the center of the trap. To appreciate the implications of the relation (b), consider the limit of a small number of fermions. The density profile of the bosons can then be determined by first ignoring the fermion term in Eq. (\[muft\]), and we obtain the boson density profile $$n_b (\vec r) = (\mu_b + \epsilon_b - 2 V (\vec r))/g_{bb} \ ,\label{nb0}$$ if the R.H.S. is larger than zero, and $n_b = 0$ otherwise. For ease of reference, we shall call these two regions “inside" and “outside" below. Eq. (\[muft\]) can be rewritten in the form $$\mu_f = A n_d (\vec r)^{2/3} + V_{\rm eff} (\vec r) \ ,\label{eff}$$ where $V_{\rm eff} (\vec r) = V (\vec r) + g_{bf} n_b (\vec r)$ is the effective potential for the fermions. We obtain, for the inside region, $V_{\rm eff} (\vec r) = V (\vec r) + (g_{bf} /g_{bb}) (\mu_b + \epsilon_b - 2 V (\vec r))$ whereas for the outside $V_{\rm eff} (\vec r) = V (\vec r)$. In the inside region, the position dependence is therefore $ - (2 g_{bf}/g_{bb} -1 ) V(\vec r)$. From this, it is clear that if $g_{bf} > g_{bb} /2$, then the effective potential actually is larger near $\vec r = 0$ than near the edge of the boson cloud. It decreases from the center of the trap till it reaches the edge of the boson cloud, then it increases again due to the spatial dependence of $V (\vec r)$. Therefore, we conclude that for ${g_{bf}}/{g_{bb}} > 1/2$, the excess fermions lie near the edge of the boson cloud, at least for a small number of Fermions [@Carr04; @pieri05]. The excess fermions lie near the edge of the cloud \[Fig. \[fig1\](g)\]. Thus a peak in $n_d(\vec r)$ does [*not*]{} indicate phase separation. \[adensity\]Axial Density ========================= We here discuss some useful expressions governing the axial density within the local density approximation. Our results here are, to some extent, a further development of those obtained in [@deSilva]. In local density approximation, any quantity, say the density difference $n_d (\vec r)$, is a function entirely of the local chemical potential $\mu (\vec r)$ (note that $h$ is a constant). Hence, we can write $$n_d (\vec r) = g (\mu(\vec r)) \equiv g (\mu(\vec r), h) \label{g}$$ for some function $g$. We here note that any density must be identically zero for sufficiently large and negative chemical potential, and thus $g(\zeta)$ vanishes exactly for sufficiently large and negative argument $\zeta$. It will be convenient to define another function $G(\zeta)$ via $$G(\zeta) \equiv \int_{-\infty}^\zeta d\zeta' g(\zeta') \ .\label{G}$$ Consider now for example the radially integrated density (referred afterwards as axial density) difference: $$n_{a} (z) \equiv \int dx dy n_d (\vec r) \ .\label{Na}$$ The integral can be written as, for a trap that is cylindrically symmetric with respect to $z$, $\pi \int_0^{\infty} d \rho^2 g (\mu_0 - \frac{1}{2} \alpha_z z^2 - \frac{1}{2} \alpha_{\parallel} \rho^2 ) $ where $\rho^2 \equiv x^2 + y^2$. This integral can be expressed in terms of $G$ and thus $$n_{a}(z) = \frac{2 \pi}{\alpha_{\parallel}} G (\mu_0 - \frac{1}{2} \alpha_z z^2) \ .\label{NaG}$$ This relation can be used to obtain directly the axial density in terms of the function $g$, without first calculating the density profile in trap and then performing the integration. Moreover, it can also be used to deduce the (unintegrated) density profile once the axial density is given, for example from experiment. To see this, we differentiate Eq. (\[NaG\]) with respect to $z$, and notice that $G'(\zeta) = g(\zeta)$ evaluated at $\zeta = \mu_0 - \frac{1}{2} \alpha_z z^2 $ is directly related to the corresponding density at the coordinate $(0,0,z)$. Thus we have $$n_d(0,0,z) = - \frac{\alpha_{\parallel}}{2 \pi \alpha_z} \frac{1}{z} \frac{\partial n_{a}(z)}{\partial z}\ . \label{nz}$$ Therefore the actual density for a point on the $z$ axis can be obtained from the axial density. Under the local density approximation, the density at any given point in the trap is a function of the local chemical potential only. Hence we can obtain the actual density at any given point in space provided the axial density is given. The relation Eq. (\[nz\]) also shows that, since $n_d$ is non-negative, the axial density profile is strictly decreasing (increasing) with $z$ for $z\, >\, (<)\, 0$, a result pointed out in reference [@deSilva]. In a region where the density vanishes, then the axial density should be a constant, a result recognized in references [@deSilva] and [@Haque]. References {#references .unnumbered} ========== [10]{} Fulde P and Ferrell R A 1964 A550 Larkin A I and Ovchinnikov Y N 1964 [*Zh. Éksp. Teor. Fiz.*]{} [**47**]{} 1136 \[1965 [*Sov. 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--- abstract: 'We present a method to sample reactive pathways via biased molecular dynamics simulations in trajectory space. We show that the use of enhanced sampling techniques enables unconstrained exploration of multiple reaction routes. Time correlation functions are conveniently computed via reweighted averages along a single trajectory and kinetic rates are accessed at no additional cost. These abilities are illustrated analyzing a model potential and the umbrella inversion of NH$_3$. The algorithm allows a parallel implementation and promises to be a powerful tool for the study of rare events.' author: - 'Davide Mandelli$^{a}$, Barak Hirshberg$^{b,c}$, Michele Parrinello$^{a,b,c,*}$' nocite: '[@Salvalaglio2014]' title: Metadynamics of paths --- Molecular dynamics (MD) simulations have become an invaluable tool in numerous branches of science. While experiments generally access only spatially and time averaged quantities, atomically detailed MD simulations allow tracking in real time the microscopic mechanisms underlying complex phenomena. Nevertheless, there is a large class of problems where a straightforward application of MD simulations is impractical. Important examples are crystal nucleation, slow diffusion in solids, chemical reactions and conformational changes of large molecules. In all these cases, the presence of large free energy barriers leads to extremely long computational times. Therefore, it is necessary to design efficient algorithms able to accelerate phase space exploration. In biased MD simulations, external potentials are used to drive transitions between meta-stable states. Among these methods, metadynamics [@Laio2002] (MetaD) has proven to be particularly successful. Sampling efficiency comes at the cost of introducing unphysical forces. Modifying the potential energy surface affects the dynamics of the system and information about the correct (unbiased) time evolution is lost. Methods have been developed aiming at retrieving dynamical information from biased MD simulations [@Tiwary2013; @Wu2014; @Rosta2015; @Wu2016]. In a recent effort, Donati and coworkers [@Donati2017] introduced a scheme to compute time correlation functions from biased Langevin trajectories. The main advantage of this technique is that it applies also to time dependent perturbations [@Donati2018]. However, applications are limited to cases where significant speed-up of phase space exploration can be achieved while keeping the external potential within a range compatible with the reweighting procedure, which otherwise leads to numerical instabilities. Besides biased MD simulations, there exists an alternative class of techniques that rely on the definition of a functional on the space of all trajectories and tackle the problem of rare events directly in the space of paths [@Elber1987; @Olender1996; @Passerone2001; @Lee2017; @Pratt1986; @Dellago1998a; @Fujisaki2010]. These include the transition path sampling (TPS) algorithm of Chandler and collaborators [@Dellago1998a]. TPS is a Monte Carlo procedure that harvests dynamical RPs connecting two given states. Two practical difficulties arise in its application. First, the computation of rate constants is time consuming [@Dellago1998a; @VanErp2003; @Moroni2004]. Secondly, this approach poses problems in systems with multiple channels, where the presence of energy barriers can prevent exploration of all the relevant reaction routes [@Vlugt2001; @Borrero2016]. In this Letter, we explore the possibility of combining biased MD simulations with action-based algorithms for the sampling of RPs. Specifically, we apply MetaD to MD simulations performed directly in the space of paths. We show that MetaD can be used to harvest RPs without the need of imposing endpoint constraints, allowing exploration of multiple reaction routes in the same run. Furthermore, standard reweighting procedures can be applied straightforwardly and provide a convenient way of computing rate constants and time correlation functions. In the following, we briefly review the theory and formalism behind the algorithm and then present two applications. The problem of interest here is the time evolution of a molecular system coupled to a thermal bath. The latter acts by introducing random fluctuations in the forces governing the dynamics. In this context, Onsager and Machlup [@Onsager1953] derived an expression for the probability of observing a given path $q(t)$ starting at $q_0$ at $t=0$ and ending in $q_{\tilde t}$ at $t$=$\tilde t$ $$\label{eq.OM} P[q(t);q_0,q_{\tilde t}]\propto e^{ - \int_{0}^{\tilde t} {\mathcal L} (q(t),\dot{q}(t)){\rm d}t } = e^{-{\mathcal S}[q(t)]}.$$ Here, ${\mathcal L}$ and $\mathcal S$ define the Onsager-Machlup (OM) function and associated OM action. Similar in spirit to Hamilton’s principle of least action, equation provides a route to construct the most probable trajectory by minimizing ${\mathcal S}$ under given endpoints constraints. This observation motivated previous methods that solved the boundary valued problem adopting suitably discretized approximations of $\mathcal S$ [@Elber1987; @Olender1996; @Passerone2001; @Lee2017]. Alternatively, one notices that the integral $$\label{eq.Z} {\mathcal Z}_{\tilde t} = \int {\mathcal D}{q(t)} e^{-{\mathcal S}[q(t)]},$$ over all paths of given duration, defines a partition function in the space of trajectories so that each of them can be assigned a well-defined probability $$\label{eq.P} P_{\tilde t}[q(t)] = e^{-{\mathcal S}[q(t)]}/{\mathcal Z}_{\tilde t}.$$ Equation is the starting point of statistical path-sampling techniques [@Pratt1986; @Dellago1998a; @Fujisaki2010] and is central to the present work. \[fig1.2Dpot\] {width="0.8\columnwidth"} We will now specialize equation to the case of a system evolving according to Brownian motion discretized as $$\label{eq.BD} R^{n+1}_\alpha=R_\alpha^n+\frac{\Delta t}{m_\alpha\nu}F^n_\alpha+\delta r_{\alpha},$$ where ${\bf R}$ is the $D$-dimensional coordinate vector, $F_\alpha^n=-\nabla_{R_\alpha^n}U$ is the force acting on the $\alpha$-[*th*]{} degree of freedom ($\alpha$=1–$D$) at step $n$, $m_\alpha$ is its mass, $\nu$ is the damping coefficient, $\Delta t$ is the time step and $\delta r_\alpha$ is a normally distributed random displacement with variance $\sigma_\alpha^2=2k_BT\Delta t/m_\alpha\nu$ and zero mean. The probability of observing a given trajectory $\{{\bf R}^1 \rightarrow {\bf R}^2 \dots \rightarrow {\bf R}^N \}$ of duration $\tilde t$=$N\Delta t$ can be rewritten as the product $P_{\tilde t}=\prod_{n=1}^{N-1}p({\bf R}^n\rightarrow{\bf R}^{n+1})$ of single-step probabilities. Known analytical results [@Chandrasekhar1943] for $p$ yield the discretized OM action $$\label{eq.S} {\mathcal S}/\beta=U({\bf R}^1)+\sum_{n=1}^{N-1}\sum_{\alpha=1}^D\frac{m_\alpha\nu}{4\Delta t}\left(R^{n+1}_\alpha-R_\alpha^n-\frac{\Delta t}{m_\alpha\nu}F^n_\alpha\right)^2,$$ where $\beta=1/k_BT$ and the potential energy $U({\bf R}^1)$ has been introduced to account for the Boltzmann weight of the initial configuration. Using equation , equation becomes formally equivalent to the classical partition function of an open polymer. Hence, one can sample paths distributed according to equation adopting a Hamiltonian approach as done in path integral MD [@Parrinello1984]. Accordingly, we define the effective potential $\mathcal U = \mathcal S/\beta$, introduce auxiliary momenta and masses {${\rm P}_\alpha^n$, ${\rm M}_\alpha$} and solve Hamilton’s equations $$\begin{aligned} \label{eq4.Hp} \dot {\rm P}^{n}_\alpha&=&-\nabla_{{R}^{n}_\alpha} \mathcal{U}\\ \dot {R}^{n}_\alpha&=&\frac{{\rm P}^{n}_\alpha}{{\rm M}_\alpha} \label{eq4.Hr}\end{aligned}$$ coupled to a thermostat. Although this approach was suggested in Ref. [@Dellago1998a], it was not further pursued mainly because of its computational burden. One issue is the presence in equation  of terms containing second derivatives of the potential. This problem is circumvented by using a symmetric finite difference formula [@Putrino2000; @Kapil2016] $$\label{eq5.FD} \sum_{\alpha=1}^D\frac{\partial F_\alpha^n}{\partial R_k^n}\eta_\alpha^n\approx\frac{F_k^n(R_\alpha^n+\varepsilon \eta_\alpha^n)-F_k^n(R_\alpha^n-\varepsilon \eta_\alpha^n)}{2\varepsilon}$$ where $\eta_\alpha^n=R_\alpha^{n+1}-R_\alpha^n-\frac{\Delta t}{m_\alpha\nu}F_\alpha^n$ and $\varepsilon$ is a number small enough to guarantee energy conservation in microcanonical simulations. Equation  amounts to a modest but necessary increase in computational cost as it avoids direct implementation of the Hessian $\frac{\partial^2 U({\bf R})}{\partial R_\alpha \partial R_\beta}$. Adopting this method, one time step in path space involves 3$\times$$N$$\times$$D$ force evaluations. This has to be compared with the cost of $N$ MD steps in configurational space, which involve $N$$\times$$D$ force evaluations. However, while the standard approach is intrinsically serial, the path approach has the advantage that it can be made highly parallel. Specifically, here we adopt the hyper-parallel scheme of Calhhoun [*et al.*]{} [@Calhoun1996] and implement the algorithm in the LAMMPS [@Plimpton1995] suite of codes. \[fig2.trap\] {width="0.8\columnwidth"} {width="0.7\columnwidth"} As a first test case, we consider the diffusion of a particle in the two dimensional double-well potential of figure \[fig1.2Dpot\](a) [@Dellago1998a]. The latter provides a simple example of a system with multiple RPs connecting meta-stable states. Simulations in path space are carried out using a polymer of $N$=200 beads and auxiliary masses $M$=1. The time step, damping coefficient and mass at the Brownian level are set equal to $\Delta t$=0.15, $\nu$=1, $m$=1. Temperature is controlled via a Nosé-Hoover chains thermostat [@Martina1992] and the equations of motion ,  are propagated adopting a standard velocity-Verlet integrator with time step $\Delta t_{\rm MD}$=0.01. Results are reported as obtained from simulations with the above set of unitless parameters [@SM]. In the first set of simulations, we enforce harmonic constraints [@SM] to pin the initial ($n$=1) and final ($n$=N) replica, respectively close to the left and right minimum (A and B in figure \[fig1.2Dpot\](a)). This setup is equivalent to the dynamical TPS algorithm [@Dellago1998a] that focuses on sampling RPs connecting known initial and final states. We start from an equilibrated polymer configuration realizing a path crossing the lower saddle (see figure \[fig1.2Dpot\](b)) and run 5 unbiased simulations of duration 5$\times$10$^7$ MD steps at temperature $k_BT$=0.05, much smaller than the minimum energy barrier $\Delta E\sim1$ separating the two minima. With this setup, the central maximum of the potential prevents the polymer from visiting path configurations that connect A and B passing via the upper saddle. This is demonstrated in figure \[fig2.trap\](a), showing the histogram of the positions visited by all beads during simulation, averaged over the 5 independent runs. This is a typical example of “path trapping” in a local minimum of the OM action. MetaD is designed precisely to overcome this type of problems. Moreover, as we will show, it can be applied directly to address the general situation where the final configuration is not known a priori. To this end, we first seek a suitable collective variable (CV) to which the bias is applied. For the present example the choice is straightforward and we select the polymer end-to-end distance $d_{\rm e2e}$=$\lvert {\bf R}^{N}-{\bf R}^1\rvert$ (see figure \[fig1.2Dpot\](b)). Using the enhanced sampling library PLUMED [@Tribello2014], we perform well-tempered MetaD [@Barducci2008; @Dama2014] while constraining the initial replica to reside in the left basin [@SM]. All other beads are free. Also in this case, we perform 5 independent runs of duration 5$\times$10$^7$ MD steps each. Figure \[fig2.trap\](b) reports the histogram of the positions visited by all beads during simulation. While most of the time the polymer remains crumpled within the left basin, the bias enables sampling of elongated configurations corresponding to successful RPs (see also insets of figure \[fig3.FES2D\]). To further illustrate the results, in figure \[fig2.trap\](c) we compare the average value of the potential felt by each replica during the unbiased simulations (red curve) with that computed using the RPs harvested using MetaD (black curve), showing good agreement. This demonstrates that the latter provide a good representation of the transition path ensemble sampled by TPS, even when obtained in presence of the bias. We now turn to the computation of kinetic rates for transitions between two long-lived states A and B. These can be extracted from simulations computing the time correlation function [@Chandler1987] $$\label{eq6.fAB} f_{\rm AB}(t)=\frac{\langle I_{\rm A}(0)I_{\rm B}(t) \rangle}{\langle I_{\rm A}(0)\rangle}.$$ Here, we have introduced the characteristic function $I_{\rm X}(t)$, which takes a value of 1 if ${\bf R}(t)\in$X, and 0 otherwise. Averages, $\langle\cdot\rangle$, are intended over all possible unconstrained paths. Equation expresses the probability of finding the system in state B at time $t$, given that it was in state A at $t=0$. In absence of intermediate states and after a short transient, $f_{\rm AB}$ enters a linear regime and the phenomenological rate constant is given by its slope $$\label{eq.kAB} k_{\rm AB}=\frac{{\rm d} f_{\rm AB}(t)}{{\rm d} t}.$$ \[fig3.FES2D\] {width="0.8\columnwidth"} Our method allows computing $f_{\rm AB}(t)$ from a single MD simulation in path space performed without imposing any constraint to the polymer. Alternatively, one notices that equation can be equivalently rewritten as $f_{\rm AB}(t)$=$\langle I_{\rm B}(t) \rangle_A$, where averages are now taken exclusively over paths initiating in A. The latter, in turn, can be sampled adopting a MetaD setup similar to the one described in previous section, adding suitable constraint on the first bead. We first consider the two dimensional diffusion problem of figure \[fig1.2Dpot\]. Biased simulations are performed using the convergence variant of the recent OPES [@invernizzi2019] method, which turned out to perform better than MetaD. Instead of the bias, OPES focuses on reconstructing the probability distribution, leading to improved convergence speed. Moreover, the method offers a more straightforward reweighting scheme and an efficient importance sampling that avoids visiting uninteresting high free energy regions [@invernizzi2019]. We further exploit the possibility of targeting a specific probability distribution, which, for the problem at hand, we choose to be a uniform distribution of $d_{\rm e2e}$ between 0 and 3 [@SM]. Finally, we do not impose any constraint on the polymer and compute $f_{\rm AB}$ directly via equation . We adopt the same parameters reported above and run simulations in the range $0.125\leq k_BT\leq0.5$. Figure \[fig3.FES2D\] shows the obtained free energy curves as a function of the end-to-end distance. At all temperatures considered, we observe a global minimum near $d_{\rm e2e}\sim0$, whose population corresponds to paths that never leave the left basin (see left inset). A secondary minimum is found at $d_{\rm e2e}\sim2$, which is the distance between the two minima of the potential. Accordingly, the corresponding polymer configurations represent RPs that successfully reach B after having crossed one of the two equivalent saddles (see right inset). Increasing the temperature lowers the free energy barrier, which reflects the thermally enhanced probability for the particle to cross from A to B. At low temperatures, configurations corresponding to intermediate values of $d_{\rm e2e}\sim1$ are “failed attempts” of the polymer that stretches towards the right basin without reaching it. On the other hand, for $k_BT\geq$0.4 we observe contributions also from configurations recrossing from B to A. \[fig4.2Drate\] {width="0.8\columnwidth"} In figure \[fig4.2Drate\](a) we show the correlation function $f_{\rm AB}(t)$ obtained at various temperatures, displaying the expected short transient, followed by linear growth. Note that for $k_BT\geq$0.4, recrossings of the polymer from B to A are observed at times $t\ge$10, where the conditions for equation to hold cease to apply [@Chandler1987]. Nevertheless, we were able to extract the phenomenological rate constant $k_{\rm AB}$ in the whole range of temperatures by considering the initial linear regime. These are reported in the Arrhenius plot of figure \[fig4.2Drate\](b), showing the expected linear trend of $\log(k_{\rm AB})$. A linear fit of the data yielded an activation energy of $\Delta E_{\rm fit}=1.05\pm0.02$, in agreement with the exact value of the potential barrier $\Delta E\simeq1.08$. As a second example, we consider the umbrella inversion transformation of ammonia in vacuum. In this configurational change the nitrogen atom passes through the hydrogen plane to reach an equivalent and symmetric position. Thus, the process can be conveniently described in terms of the oriented height $h$ of the NH$_3$ tetrahedron. Here, we describe intra-molecular forces using an empirical model [@Weismiller2010]. Under normal conditions, $h$ takes values of $\sim\pm0.4$Å, the two equivalent configurations being separated by a large barrier of $\Delta E\simeq$120 kJ/mol$\sim$50 $k_BT_{\rm room}$ [@SM]. Simulations in path space are performed with $N$=100, $\Delta t$=0.1 fs and $\nu$=0.14 fs$^{-1}$. Auxiliary masses are set equal to those of the corresponding atom in the replica and the MD integration step is set to $\Delta t_{\rm MD}$=0.25 fs. Instructed by the previous study, we choose as CV the generalized end-to-end distance $\Delta h_{\rm e2e}=(h^{N}-h^{1})$. Biased simulations are performed using the convergence variant of OPES [@invernizzi2019] targeting a uniform distribution of $\Delta h_{\rm e2e}$ between -1Å and 1Å [@SM]. Also in this case, we do not impose any constraint on the polymer and compute $f_{\rm AB}$ directly via equation . \[fig5.FES-NH3\] {width="0.8\columnwidth"} Figure \[fig5.FES-NH3\] reports the free energy curves obtained at temperature $T$=300, 500, 700, 900 K. Overall, we observe similar features to those of the previous example, namely, a global minimum at $\lvert \Delta h_{\rm e2e}\rvert\sim0$ Å followed by a plateau and a shallow minimum at $\lvert \Delta h_{\rm e2e}\rvert\sim0.8$ Å. Insets show representative configurations corresponding respectively to a diffusive path where all replicas share the same orientation and a successful RP where ammonia flips between the $h$=$\pm0.4$ Å states. In figure \[fig6.rateNH3\](a) we report the time correlation function $f_{\rm AB}(t)$ extracted from simulations. For the chosen set of parameters, the polymer length adopted is sufficient to observe the onset of the linear regime, from which we extract the phenomenological rates. The latter are displayed in the Arrhenius plot of panel (b), along with results from unbiased MD simulations in configurational space [@SM]. In the latter case, rates are accessible only at high temperatures. At T=900 K, we obtain respectively $k_{\rm AB}^{\rm OPES}\simeq$2.4$\times$10$^{-4}$ and $k_{\rm AB}^{\rm direct}\simeq$4.0$\times$10$^{-4}$ ps$^{-1}$, in fair agreement. On the other hand, the activation barrier of $\Delta E_{\rm fit}$=$118\pm1$ kJ/mol estimated using the results from OPES perfectly matches the exact value of $\Delta E\simeq$120 kJ/mol of the adopted force field. To conclude, we have presented a method to sample RPs via biased MD simulations in path space. The use of enhanced sampling techniques enables unconstrained exploration of RPs, making this approach more robust against problems like path trapping. Time correlation functions can be computed via straightforward (reweighted) averages along a single MD trajectory. Dynamical information such as kinetic rates are easily accessible at no additional cost. At the same time, adding suitable contraints at the polymer endpoints allows identification of most probable paths via direct minimization of the discretized action, which makes this approach very flexible. The method promises to be a powerful tool with potential applications in many fields including the study of chemical reactions, via implementation within the Car-Parrinello MD approach [@Car1985], and of the kinetics of biological systems. \[fig6.rateNH3\] {width="0.8\columnwidth"} In the present work we have adopted MetaD and OPES as biasing schemes, but any other enhanced sampling schemes could be applied as well. As in all biased MD simulations, prior knowledge about the mechanisms underlying the transition of interest is needed in order to build successful CVs. This is crucial to speed up convergence. It is encouraging that in the cases studied here a suitably defined end-to-end distance performed well. This represents the most natural choice. Complex systems will definitely require more fine tuning. However, this should not pose any major problem as one is assisted in the task by the vast literature on the subject [@Valsson2016]. Finally, we note that the proposed path approach effectively realizes parallelization at the level of time [@Rosa-Raices2019]. This, in turn, allows a highly parallel implementation [@Calhoun1996] that takes full advantage of modern massively parallel computer architectures. Given the increasing availability of massive parallel computational resources, we believe that this method will find many successful applications. Supplemental Materials ====================== Diffusion in a two dimensional double-well potential ---------------------------------------------------- In the first application, we considered the problem of a particle diffusing in the two dimensional potential [@Dellago1998a] $$U(x,y)= 2 + \frac 4 3 x^4 - 2y^2 + y^4 + \frac {10} {3} x^2(y^2-1).$$ Simulations were performed implementing the corresponding Onsager-Machlup (OM) action (equation (5) of the main text) using the following set of parameters at the Brownian level: $m$=1, $\Delta t$=0.15, $\nu$=1. In all simulations, we considered a polymer of size $N$=200 beads inside a square cell of side $L$=10, centered in the origin. Exact analytical expressions of all terms appearing in the definition of the forces governing the dynamics of the polymer (equation (6) of main text) have been hard coded in the LAMMPS [@Plimpton1995] suite of codes. The equations of motion in path space were then solved adopting a standard velocity-Verlet integrator with a time step of $\Delta t_{\rm MD}$=0.01 and auxiliary masses set to $M$=1. Temperature was controlled via a Nosé-Hoover chains thermostat [@Martina1992]. The PLUMED [@Tribello2014] enhanced sampling library was used to introduce harmonic restraints and to perform well-tempered metadynamics [@Barducci2008; @Dama2014] (WT-MetaD) as well as OPES [@invernizzi2019] simulations. Here and in the main text, all quantities are reported as obtained from simulations with the above set of unitless parameters. In the first set of simulations, we investigated the ability of our metadynamics (MetaD) approach to sample different reactive pathways (RPs) in the same run, and we compared results with those obtained adopting the dynamical transition path sampling (TPS) algorithm of Ref. . TPS simulations were performed at temperature $k_BT$=0.05, applying harmonic constraints (spring constant $K$=100) to the two distances $R_{1,{\rm A}}$=$\lvert {\bf R}^1-{\bf R}_{\rm A} \rvert$ and $R_{N,{\rm B}}$=$\lvert {\bf R}^N-{\bf R}_{\rm B} \rvert$. Here, ${\bf R}^{1,N}$ are the positions of the first and last bead of the polymer, while ${\bf R}_{\rm A,B}\simeq(\mp1,0)$ mark the two minima of the potential. With this choice of parameters, $R_{1,{\rm A}}$ and $R_{N,{\rm B}}$ were bound to values $\leq$0.1. WT-MetaD simulations were performed at the same temperature, applying same harmonic constraint only to $R_{1,{\rm A}}$. The polymer end-to-end distance $d_{\rm e2e}$=$\lvert {\bf R}^1-{\bf R}^N\rvert$ was used as collective variable (CV). The bias potential was built using a bias factor $\gamma$=20, depositing Gaussian kernels (height=1, $\sigma$=0.1) every 2500 MD steps. In both cases, we performed simulations of 5$\times$10$^{7}$ MD steps, sampling configurations every 2500 steps. In the case of TPS, all polymer configurations sampled were used to compute the average potential profile (figure 2(c) of main text). In the case of MetaD, we considered only the second half of the trajectory (well within the asymptotic regime of WT-MetaD) and defined successful RPs those for which $R_{1,{\rm A}}$$\leq$0.1 and $R_{N,{\rm B}}$$\leq$0.1 at the same time. These were used to compute the average potential profile. Each experiment was repeated 5 times. The curves shown in figure 2(c) of the main text report the associated average values and standard deviations. \[fig.trace2D\] ![(a) Time evolution of $d_{\rm e2e}$ obtained from biased simulation at $k_BT$=0.125 using OPES [@invernizzi2019]. (b) The corresponding biased probability distribution.](fig_trace-and-prob-2Dpot.pdf "fig:"){width="0.9\columnwidth"} \[fig.converge2D\] {width="0.9\columnwidth"} In the second set of simulations, we adopted the convergence variant of OPES [@invernizzi2019] to compute the phenomenological kinetic rate, $k_{\rm AB}$, for transitions between A and B. Simulations were performed at temperature $k_BT$=0.125, 0.2, 0.3, 0.4 and 0.5, without applying any constraint to the polymer. We used a bias factor $\gamma$=0 and deposited Gaussian kernels every 2500 MD steps. The standard deviation of the kernels was set equal to $\sigma$=0.1 at $k_BT$=0.125 and to $\sigma$=0.2 at all other temperatures. In OPES, the height of the kernels is automatically adjusted during runtime. The last input parameter is an estimate of the free energy barrier to be overcome, which we set equal to $\Delta F$=1.5 for $k_BT$=0.125, 0.2, 0.3 and to $\Delta F$=1.2 for $k_BT$=0.4, 0.5. These values were estimated from preliminary WT-MetaD simulations. With this setup, we effectively targeted a flat probability distribution of $d_{\rm e2e}$ in the interval 0$\lesssim d_{\rm e2e}\lesssim$3 (see figure \[fig.trace2D\]). At each temperature, we performed a simulation of 5$\times$10$^{8}$ MD steps. The time correlation function $f_{\rm AB}$ (equation (9) of the main text) was computed defining the characteristic functions of the two basins of the potential as $I_{\rm A,B}({\bf R}^n)$=1 if $\lvert {\bf R}^n-{\bf R}_{\rm A,B}\rvert<$0.7 and zero otherwise. Time averages were computed using the reweighting scheme suggested in Ref. [@invernizzi2019], neglecting the initial 5$\times$10$^{4}$ steps (see figure \[fig.converge2D\]). Each experiment was repeated 4 times. Figures 3 and 4(a) of the main text report the associated average values and standard deviations. \[fig.pes\] {width="0.9\columnwidth"} Umbrella inversion of NH$_3$ in vacuum -------------------------------------- In the second application, we considered the umbrella inversion of ammonia in vacuum. Intra-molecular interactions were described using the ReaxFF force field of Ref. , neglecting electrostatics. We used a cubic box of side $L$=30 Å, larger than twice the cutoff of 14 Å adopted for long range van der Waals interactions, and we enforced periodic boundary conditions. Figure \[fig.pes\] shows the potential energy profile obtained via a sequence of geometry optimizations at fixed values of the oriented height, $h$, of the NH$_3$ tetrahedron, which we used to describe the transformation. The two symmetric minimum energy confingurations at $h$$\sim\pm$0.4 Å are separated by a barrier of $\sim$120 kJ/mol. \[fig2.trace-e2e\] ![(a) Time evolution of $\Delta h_{\rm e2e}$ obtained from biased simulation at $T$=500 K using OPES [@invernizzi2019]. (b) The corresponding biased probability distribution.](fig_trace-and-prob-NH3.pdf "fig:"){width="0.9\columnwidth"} \[fig.convergeNH3\] {width="0.9\columnwidth"} Simulations in path space were performed setting $\Delta t$=0.1 fs, $\nu$=0.14 fs$^{-1}$ in the OM action and using a polymer of length $N$=100 beads. The equations of motion have been implemented in LAMMPS [@Plimpton1995] and solved adopting a velocity-Verlet algorithm with an MD integration step of $\Delta t_{\rm MD}$=0.25 fs. The auxiliary masses $M_{\alpha}$ were set equal to those of the corresponding atom in the ammonia replica. Terms in the forces containing second derivatives of the potential energy were estimated adopting the finite difference expression of equation (8) of the main text. Biased MD simulations were performed adopting the convergence variant of OPES [@invernizzi2019], as implemented in PLUMED [@Tribello2014]. We considered temperatures of $T$=300, 500, 700, 900 K and we did not apply any constraint to the polymer. We chose as CV the generalized polymer end-to-end distance $\Delta h_{\rm e2e}=(h^N-h^1)$, equal to the difference between the oriented height of the last and first ammonia replica. We used a bias factor $\gamma$=0 and deposited Gaussian kernels every 500 MD steps. The standard deviation of the kernels was set equal to $\sigma$=0.0032, 0.0043, 0.0051, 0.0059 Å, respectively for $T$=300, 500, 700, 900 K. In OPES, the height of the kernels is automatically adjusted during runtime. The last input parameter is an estimate of the free energy barrier to be overcome. This was estimated from preliminary WT-MetaD simulations and set equal to $\Delta F$=140 kJ/mol. With this setup, we effectively targeted a flat probability distribution of $\Delta h_{\rm e2e}$ in the interval -1 Å$\lesssim \Delta h_{\rm e2e}\lesssim$1 Å (see figure \[fig2.trace-e2e\]). At each temperature, we ran a simulation of $\sim$3$\times$10$^8$ MD steps. The time correlation function $f_{\rm AB}(t)$ was computed defining the characteristic function of the two basins $I_{\rm A,B}(h)$=1 if $\lvert h\rvert>0.02$ Å and zero otherwise. Time averages were computed using the reweighting scheme suggested in Ref. [@invernizzi2019], skipping the initial $\sim$10$^5$ MD steps (see figure \[fig.convergeNH3\]). Each experiment was repeated 4 times. Figures 5 and 6(a) of the main text report the associated average values and standard deviations. 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--- abstract: 'We report an infrared reflection spectroscopy study of La$_{1/2}$Ca$_{1/2}$MnO$_3$ over a broad frequency range and temperature interval which covers the transitions from the high temperature paramagnetic to ferromagnetic and, upon further cooling, to antiferromagnetic phase. The structural phase transition, accompanied by a ferromagnetic ordering at T$_C$=234 K, leads to enrichment of the phonon spectrum. A charge ordered antiferromagnetic insulating ground state develops below the Néel transition temperature T$_N$=163 K. This is evidenced by the formation of charge density waves and opening of a gap with the magnitude of 2$\Delta_0$ = (320 $\pm$ 15) cm$^{-1}$ in the excitation spectrum. Several of the infrared active phonons are found to exhibit anomalous frequency softening. The experimental data suggest coexistence of ferromagnetic and antiferromangetic phases at low temperatures.' author: - 'A.P. Litvinchuk$^1$, M. N. Iliev$^1$, M. Pissas$^2$, and C.W. Chu$^{1,3,4}$' date: 'June 6, 2004' title: 'Charge and Lattice Dynamics of Ordered State in La$_{1/2}$Ca$_{1/2}$MnO$_3$: Infrared Reflection Spectroscopy Study' --- Manganite perovskites R$_{1-x}$A$_x$MnO$_3$ (where R is a trivalent rare earth and A is a divalent alkaline rare earth) exhibit rich phase diagram and a variety of intriguing properties due the delicate interplay of spin, charge, lattice, and orbital degrees of freedom (Ref. [@review1; @review2; @piss03] and references cited therein). Well defined anomalies of physical properties at commensurate carrier concentrations of $x$=N/8 (where N=1, 3, 4, 5, and 7) and $x=L/3$ (where L=1,2) were unambiguously established. Of particular interest is the phenomenon of charge and orbital ordering, most clearly pronounced for $x$=1/2 [@chen96; @rada97; @mori98; @huang; @papa00; @abra01; @nale02]. In this communication we report the results of an infrared reflection study of La$_{1/2}$Ca$_{1/2}$MnO$_3$ over broad frequency range and at temperatures, covering the transitions from the high temperature paramagnetic to ferromagnetic and, upon further cooling, to antiferromagnetic phase. By making use of Kramers-Kronig analysis we obtained the spectral dependence of conductivity. Its analysis yields information on the evolution of phonons and electronic excitations as a function of temperature. We found that additional phonon modes appear in the spectra in the ferromagnetic phase, a fact which implies the occurrence of a structural phase transition. Upon further cooling the conductivity spectrum shows development of a gap, which signals the formation of a charge ordered state at low temperatures. At the same time a Drude-like component of the conductivity does not vanish completely and suggests the coexistence of a metallic and insulating phases. The measurements were performed on a dense ceramic pellet of La$_{1/2}$Ca$_{1/2}$MnO$_3$, mechanically polished to optical quality. The material preparation technique is described in Ref. [@piss03; @abra01; @simo00]. The samples were intensively characterized by the X-ray scattering, neutron diffraction, magnetization measurements, Raman scattering as well as Mössbauer spectroscopy (doped with 1% Sn or Fe). These measurements unambiguously identified the transition from the paramagnetic to ferromagnetic phase at T$_C$=234 K, and further to antiferromagnetic phase at T$^c_N$=163 K (cooling cycle) or T$^h_N$=196 K (heating cycle), see Fig. 6 in Ref.[@piss03]. The reflection measurements were performed on a Bomem-DA8 Fourier-transform interferometer in the frequency range 50-8000 cm$^{-1}$ with the use of a liquid-helium-cooled bolometer, HgCdTe, and InSb detectors and appropriate beam splitters. Spectral resolution was set to 1 cm$^{-1}$. A gold mirror was used as a reference. The sample was attached to the cold finger of a helium flow cryostat and during measurements the temperature was stabilized to within 0.2 K. The reflectance spectra R($\omega$) were extrapolated in the low frequency range by either Hagen-Rubens relation (1-R) $\sim$ $\omega ^{1/2}$ or a constant (at low temperatures) and R $\sim \omega^{-4}$ for high frequencies up to 30000 cm$^{-1}$. The results of Kramers-Kronig analysis of the spectra show that variation of extrapolated reflection do not influence conductivity values in the frequency range of interest (50-3000 cm$^{-1}$). The reflectance spectra of La$_{1/2}$Ca$_{1/2}$MnO$_3$ during cooling and heating cycles are shown in Fig. 1 for several temperatures between 215 and 110 K. As expected, a hysteretic temperature behavior is clearly seen around T$_N$, typical for the first-order transition occurring in the CE-type magnetic structure[@woll55]. Indeed, upon cooling the low-frequency reflectance (below 200 cm$^{-1}$) shows a “metal-like” increase toward lower wavenumbers for the four upper curves on the left panel of Fig. 1, but becomes less frequency dependent for the two lower curves (i.e. below 145 K). Instead, during the heating cycle the low-frequency reflectance keeps its behavior from the low temperatures up to 190 K and becomes more “metal-like” at higher temperatures only. To obtain more specific information on phonons and the charge dynamics, we performed Kramers-Kronig analysis, which yields the spectral dependence of conductivity. Fig. 2 illustrates the data, obtained in the cooling cycle. The conductivity spectra below 650 cm$^{-1}$ are dominated by phonons, while at higher frequencies the “background” conductivity steadily increases toward higher wavenumbers with apparent slope becoming larger upon lowering temperature. At T=80 K (the lowest panel in Fig. 2) the extrapolation of this background to lower frequencies crosses zero conductivity at positive wavenumbers, i.e. shows zero contribution to the [*dc*]{} conductivity. It signals opening of a gap, related to the formation of a charge density wave in charge ordered state due to real-space ordering of Mn$^{3+}$ and Mn$^{4+}$ ions[@review1; @chen96; @calv98]. Theoretical consideration of a charge ordered system yields the following frequency dependence of conductivity[@lee74]: $$\sigma(\omega) \sim (\omega - 2\Delta)^{\alpha},$$ where $2\Delta$ is the magnitude of the charge gap and $\alpha~=~1/2$. Using this prediction we fitted the conductivity spectra at frequencies above the highest energy phonon (in the range 750 to 2800 cm $^{-1}$) using $2\Delta$ and $\alpha$ as parameters. For all temperatures the values of $\alpha$ are found to be between 0.51 and 0.54, in good agreements with the theory. The parameter $2\Delta$ increases in a linear manner from the negative value of $-2777$ cm$^{-1}$ at room temperature to $-171$ cm$^{-1}$ at T=170 K and becomes positive at 160 K ($2\Delta$ = 165 cm$^{-1}$), signaling opening of a “real” gap. The temperature, at which this gap opens, unambiguously identifies it as being due to the formation of a charge ordered state in La$_{1/2}$Ca$_{1/2}$MnO$_3$ because the antiferromagnetic transition temperature for the sample is T$^c_N$=163 K. The gap fully opens below 150 K, where it reaches the value of $2\Delta_0$=(320 $\pm$ 15) cm$^{-1}$ (Fig. 3). Earlier experimental study of the charge density waves in La$_{1/2}$Ca$_{1/2}$MnO$_3$ by optical transmission technique yielded the gap value of 710 cm$^{-1}$ [@calv98]. Even larger value of about 3600 cm$^{-1}$ was obtained from a reflection studies by Kim et al.[@kim02]. Unlike present measurements, which were performed on a bulk sample, the measurements in Ref. [@calv98] we carried out on pressed pellets of finely milled La$_{1/2}$Ca$_{1/2}$MnO$_3$, embedded into CsI host matrix. We believe that there are at least two factors, which contributed to an overestimation of $2\Delta$ in this latter case. First, pellets non-uniformity may introduce scattering of the transmitted light beam and this way increase apparent optical density of the sample. Second, the authors of Ref. [@calv98] performed fitting of the “background” in a very narrow frequency interval 710-900 cm$^{-1}$, which could generate significant error in determining the slope and, correspondingly, the value of $2\Delta$. As to the results of Ref. [@kim02], the measurements were performed in a wide frequency range extending up to 30 eV with emphasis on the analysis of an intense feature near 1 eV (due to an interatomic Mn$^{3+} \rightarrow$ Mn$^{4+}$ transitions[@jung98]); the authors overlooked evolution of the spectra at lower wavenumbers and lower conductivity values. Next, we turn to the analysis of phonons, which dominate conductivity at frequencies below 650 cm$^{-1}$. The spectra for several temperatures above and below charge ordering temperature T$^c_N$=163 K are shown in Fig. 4. Each spectrum is fitted by a set of Lorentzians, which correspond to phonons. The contribution of free carries is accounted for by an additional oscillator, centered at zero frequency. As it is seen, with just 5 phonons one can adequately describe the spectrum at 230 K. At 215 K two new bands, centered at about 505 and 285 cm$^{-1}$, appear. Intensity of these bands gradually increases upon lowering temperature down to 150 K, and then becomes weakly dependent on temperature upon further cooling (Fig. 5(a)). These new lines in the spectra could be a consequence of either the formation of a novel orthorhombic phase below T$_C$=234 K, which has the same symmetry (space group [*Pnma*]{}), but slightly different lattice parameters compared to the room temperature phase of La$_{1/2}$Ca$_{1/2}$MnO$_3$[@huang], or the appearance of a superstructure with doubled $a$ lattice parameter and the space group [*P2$_1$/m*]{}[@rada97]. The observed intensity increase over rather wide temperature interval below T$_C$ indicates that the volume fraction of the novel phase increases upon sample cooling, in agreement with[@huang]. Note that appearance of vibrational modes with very similar frequencies was reported in charge ordered (LaPrCa)MnO$_3$ [@lee02]. The number of phonon lines observed in the infrared spectra is small compared to what is predicted by a group theoretical analysis[@abra01; @smir99]: 25 for the room temperature phase of La$_{1/2}$Ca$_{1/2}$MnO$_3$ (space group [*Pnma*]{}) and 63 for the low-temperature charge ordered phase (space group [*P2$_1$/m*]{}). For the parent compound LaMnO$_3$, as shown in the upper panel of Fig. 4, one clearly identifies majority of theoretically predicted lines, as also reported by Paolone et al.[@paol00] and Quijada et al.[@quij01]. The small number of lines observed in La$_{1/2}$Ca$_{1/2}$MnO$_3$ is probably due to the effect of compositional cation disorder (La/Ca), which considerably shorten the phonon lifetime and, consequently, broaden phonon peaks. Indeed, the typical phonon line width in LaMnO$_3$ is 10-40 cm$^{-1}$ at room temperature, while it is as high as 60-80 cm$^{-1}$ in La$_{1/2}$Ca$_{1/2}$MnO$_3$. The effect of phonon line broadening is documented in a study of 8% Ca-doped LaMnO$_3$. [@paol00] Another interesting experimental finding is that several phonon lines exhibit pronounced frequency softening upon entering the charge ordered insulating antiferromagnetic state (Fig. 5 (c,d)). This could be due to the variation of relevant bond distances, documented in the neutron diffraction studies[@rada97; @huang] and/or the effect of magnetic order on corresponding force constants, similar to those reported for other magnetic materials[@home95; @litv04]. It is important to note that the dielectric response of La$_{1/2}$Ca$_{1/2}$MnO$_3$, as obtained from the reflection spectroscopy data, is consistent with the phase separation scenario for manganese oxides (see review [@dago01] and references cites therein). Indeed, even below T$_N$ the dielectric function is not typical of an insulator, but contains the Drude-like component, a signature for presence in the sample of a conducting phase. Fig. 5(b) shows the temperature dependence of integrated conductivity in the frequency range 50-650 cm$^{-1}$ after high-frequency “background” was subtracted. It contains contribution of phonons and low-frequency metallic response. As phonons are only weakly change with temperature (see Fig. 4), this dependence reflects primarily the fraction of conducting (ferromagnetic) phase in the sample: it sharply increases upon entering ferromagnetic state below T$_C$, reaches its maximum around T$_N$, but does not completely disappear at lower temperatures. This finding is in agreement with earlier reports on La$_{1/2}$Ca$_{1/2}$MnO$_3$ [@simo00; @smol00; @roy00; @frei02; @tong03]. We have to mention that the presence of a conducting phase has only minor effect on the gap magnitude of the charge ordered state, as determined above. This is due to the fact that the conducting phase affects the dielectric function at low frequencies (below 300 cm$^{-1}$), while the gap magnitude determination involves analysis of the spectra at higher frequencies (above 750 cm$^{-1}$). The different value of the gap compared to that of Ref. [@calv98] could, at least in part, be due to different size of charge ordered domains in the material. In conclusion, the infrared reflection study of La$_{1/2}$Ca$_{1/2}$MnO$_3$ revealed the occurrence of a structural phase transition at T$_C$, evidenced by appearance of additional phonon lines. The charge and orbital ordering below the antiferromagnetic transition temperature T$_N$ was found to drastically modify the carrier dynamics as charge density waves develop, leading to the creation of a gap in the excitation spectrum with magnitude of 2$\Delta_0$=(320 $\pm$ 15) cm$^{-1}$. The experimental data suggest coexistence of ferromagnetic and antiferromangetic phases at low temperatures. The work at the University of Houston is supported in part by the State of Texas through the Texas Center for Superconductivity and Advanced Materials, NSF grant No. DMR-9804325, the T.L.L. Temple Foundation, the J.J. and R. Moores Endowment. At LBNL we acknowledge the support by the Director, Office of Energy Research, Office of Basic Energy Sciences, Division of Materials Sciences of the US Department of Energy under contract No. DE-AC03-76SF00098. 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--- abstract: 'When the interval between a transient flash of light (a “cue" ) and a second visual response signal (a “target") exceeds at least 200 ms, responding is slowest in the direction indicated by the first signal. This phenomenon is commonly referred to as inhibition of return (IOR). The dynamic neural field model (DNF) has proven to have broad explanatory power for IOR, effectively capturing many empirical results. Previous work has used a short-term depression (STD) implementation of IOR, but this approach fails to explain many behavioral phenomena observed in the literature. Here, we explore a variant model of IOR involving a combination of STD and delayed direct collicular inhibition. We demonstrate that this hybrid model can better reproduce established behavioural results. We use the results of this model to propose several experiments that would yield particularly valuable insight into the nature of the neurophysiological mechanisms underlying IOR.' author: - 'Jason Satel[^1]' - Ross Story - 'Matthew D. Hilchey' - | \ Zhiguo Wang - 'Raymond M. Klein' title: Using a Dynamic Neural Field Model to Explore a Direct Collicular Inhibition Account of Inhibition of Return --- Introduction ============ Inhibition of return (IOR) commonly refers to an extended period (about 3 s, e.g., [@vaughan84; @samuel03]) of slowed responding toward and/or at the location of a spatially irrelevant visual signal soon after its onset [@posner85] (see [@klein00], for review). The phenomenon has been extensively studied in the spatial cueing paradigm [@posner80; @posner84]. In this paradigm, the interval between two visual onset signals is often manipulated. Conventionally, the first and second signals are referred to as the “cue" and “target", respectively, and the interval between them is called the cue-target onset asynchrony (CTOA). IOR is robust at CTOAs greater than 300 ms but can also be detected as early as 50 ms post-cue onset [@danziger99]. IOR plays an important role in visual search by biasing responding against previously inspected regions in space [@posner85; @klein99] and may have evolved to optimize foraging behaviors [@klein88]. Neural Origins of IOR --------------------- The relative contribution of cortical and subcortical oculomotor processing mechanisms to IOR have been the subject of intense debate and controversy. IOR exists primarily in spatiotopic or environmental coordinates when dissociated from retinotopic reference frames [@maylor85; @posner84], an important property of IOR if it is to function effectively as a foraging facilitator [@hilchey12NSL]. The contribution of cortical processes, in particular the posterior parietal lobes, to spatiotopic IOR are well-established [@sapir04; @vankoningsbruggen09; @mirpour09] and appear necessary given that low-level oculomotor circuitry is retinotopically-organized. Tipper and colleagues [@jordan98; @tipper91] provided the seminal demonstration that IOR could also exist in object-based coordinates in addition to having demonstrated the contribution of cortical processes to object-based IOR [@tipper97] (cf. [@vivas03], for evidence that the parietal lobe plays a role in object-based IOR). Work from Sumner and colleagues [@sumner04] has revealed that IOR can still be generated to stimuli which initially bypass the superior colliculus (SC), but only in manual and not oculomotor response conditions (see also [@bourseous12; @smith12], for non-collicular origins of IOR). On the other hand, a variety of studies have provided strong evidence for the central role of the SC in generating IOR [@friedrich85; @posner85; @sapir99; @sereno06]. Simply, even in tasks requiring only manual responding, IOR is abolished in patients with SC lesions or degenerative conditions disrupting normal reflexive oculomotor functioning. Moreover, evidence for IOR has been observed in the archerfish, in which cortical processes are markedly underdeveloped [@gabay13]. Converging evidence for the role of low-level oculomotor processes comes from single unit recording studies of the primate SC (cf. [@fecteau05]). These studies have identified the intermediate layers of the superior colliculus (iSC) as a probable locus for at least some of the underlying mechanisms of IOR [@dorris02; @fecteau05]. Dorris et al. [@dorris02] examined monkeys trained on a simple IOR task by recording from visual and visuomotor neurons residing in the superficial layers of the SC (sSC) and the iSC, respectively, while the task was performed. These researchers found that there was a reduction in target-elicited activativation at the cued location that was correlated with behaviorally measured saccadic reaction times (SRTs). In another experiment, instead of probing an oculomotor response with a visual target, on 25% of the trials Dorris et al. evoked an oculomotor response by delivering a train of microstimulation directly to visuomotor neurons. At the 200 ms CTOA, electrically stimulated oculomotor responses were faster to cued as compared to uncued locations whereas no statistical effect was observed at the 1100 ms CTOA. These findings suggested that the reduction in activity arises from upstream sensory afferents. However, their monkeys neither exhibited IOR at long CTOAs when oculomotor responses were electrically-evoked (as noted) nor when made to visual stimulation. As such, it is still possible that local inhibition in the SC arises at later times after cue onset. The STD and DS theory of IOR ---------------------------- There are a variety of hypothesized models of IOR but here we focus on two computationally explicit theories. The early sensory adaptation or short-term depression (STD) theory and the local inhibition or direct suppression (DS) theory of IOR. STD theory, as presented by Satel, Wang, Trappenberg, and Klein [@satel11], implements IOR as the result of input attenuation of target-elicited early sensory input signals to iSC. There is strong evidence that STD is a component of early IOR in monkeys from single unit recordings of the iSC during traditional cue-target tasks [@dorris02; @fecteau05]. The previously mentioned reduction in activity in the iSC at cued locations is correlated with SRTs, and can also be observed in the sSC which only receives input from early sensory areas. Thus, the STD theory of IOR predicts that target-elicited visual inputs to the iSC will be reduced in magnitude for some period when presented at a previously cued location, as a function of the time since previous stimulation. ![Schematic comparison of the A) STD and B) DS theories (adapted from [@hilcheyUR]). Exogenous target: peripheral onsets; Endogenous target: arrows at fixation.](fig1.png) However, one notable shortcoming of the STD model is its inability to account for IOR when the stimulus commanding an oculomotor response does not occur at a previously stimulated location [@taylor00; @hilchey12PBR]. Furthermore, it is unclear whether STD could operate in spatiotopic coordinates - a fundamental property of IOR. STD reduces the magnitude of neural responses to stimuli at previously stimulated locations. This reduction in input strength in turn increases the time required for the targeted iSC neuron to reach a firing rate sufficient to initiate a saccade. Yet, as noted, even when an oculomotor response is generated by left- or right- ward pointing arrows on the fovea, oculomotor responding toward the cued location is delayed. Since STD is not active with arrow stimuli, but IOR is still displayed, some other mechanism must be responsible for the increased buildup time required to generate a saccade to location targeted by a central arrow. The local inhibition or direct suppression (DS) theory explains IOR in terms of a delayed inhibitory signal centered on a previously stimulated area, reducing the baseline activation level and thus increasing SRTs. As demonstrated by Dorris et al. [@dorris02], such an inhibitory signal is not present in association with IOR in the iSC up to 200 ms post-cue. At longer CTOAs, as aforementioned, the monkeys did not exhibit IOR as measured by oculomotor responses to visual stimulation nor was there any evidence for direct inhibition as revealed by microstimulation. Whereas Dorris et al. [@dorris02] failed to observe behavioral evidence for IOR at late CTOAs, behavioral investigations on humans reliably demonstrate IOR at late CTOAs and that its magnitude is similar whether the cue and response signal occur at the same location in space or whether the oculomotor response is commanded by input at fixation[@taylor00; @hilchey12; @hilcheyUR]. Such findings suggest that, at least in humans, a direct inhibitory signal may arrive at the iSC but perhaps later than 200 ms post-cue [@hilcheyUR]. The DS theory alone is unable to explain IOR measured at short CTOAs, but a hybrid STD plus DS theory, predicts behavior well at all CTOAs - with both peripheral onset and central arrow stimuli - forming an effective theoretical framework for understanding many of the behavioral results in the IOR literature (cf [@tassinari93], for similar considerations regarding keypress-measured inhibitory cueing effects). IOR is an effect that often operates on much longer time scales than those explored in the previously mentioned work supporting STD theory, and recent experimental work by Hilchey and colleagues suggests a secondary effect arising somewhere between 500 and 700 ms is responsible for the longer duration effects of IOR [@hilcheyUR]. Hilchey et al. performed the experiment illustrated in Fig. 2, where the time course of the effects of a transient peripheral cue were measured by way of oculomotor responses toward the cued or uncued location as commanded by either peripheral onset or central arrows signals. This work demonstrated that IOR as revealed by central arrow signals arises somewhere between 450 and 1050 ms post-cue. Furthermore, at the longest CTOA tested (1050 ms), the magnitude of the IOR effects were statistically indistinguishable when measured with peripheral and central signals, inviting the possibility that a common neural mechanism underlies the IOR effects at long CTOAs. We hypothesize that direct local inhibition of the iSC, beginning approximately 600 ms after its appearance, is responsible for this effect. Mathematically explicit computational models are a valuable tool for generating experimentally verifiable predictions from these theories and exploring their dynamics. The dynamic neural field (DNF) model [@amari77; @wilson73] has proven effective at modeling the dynamics of the iSC as they relate to saccade generation in a variety of paradigms [@kopecz95; @trappenberg01; @wilimzig06]. DNF models have also proven valuable in modeling IOR as understood with STD theory [@satel11]. Incorporating the dynamics of a DS theory of IOR with the previously implemented STD model of IOR should generate results that match those found by previous investigations using central arrow as well as peripheral targets [@hilcheyUR], while also maintaining the ability to generate previous results. Here, we will use a one dimensional DNF model of the iSC to compare the simulated results of these theories of IOR. First, we will simulate the projected results of the original STD theory advocated by Satel et al. [@satel11] by examining the effects of peripheral cues on subsequent oculomotor responses to either peripheral onset or central arrow signals in the DNF. Second, we will test the hybrid DS theory of IOR by introducing an inhibitory signal, centered on cued locations, 600 ms after cue onset.  Methods ======= Experimental design ------------------- Following Hilchey et al. [@hilcheyUR], the experimental paradigm simulated is illustrated in Fig. 2. Trials begin with subjects maintaining central fixation until the appearance of the target. Spatially uninformative peripheral cues appear at various CTOAs before target onset. Targets are lateralized peripheral onset signals requiring oculomotor localization responses or left- or right-pointing central arrow stimuli commanding left- or right-ward oculomotor responses, respectively. Dynamic neural field model dynamics ----------------------------------- The iSC is responsible for the initiation of saccades to the contralateral visual field, determining direction and amplitude by the location of activity on a retinotopic motor map. A one dimensional model of the iSC was used here, where each node represents the aggregated activity of a cluster of neurons. The DNF model is characterized by short distance excitation of nearby clusters and long distance inhibition that captures the behavior of iSC neurons very well [@trappenberg01]. This model was initially developed to explain more general lateral interaction in neural systems [@amari77; @wilson73], but has been shown to be effective for modeling the dynamics of the iSC by deriving parameters from neurophysiological studies in monkeys [@arai99; @trappenberg01]. Here, we will expand this model to further explore the STD and DS theories of IOR. In this model, $n = 1001$ nodes were used to represent 5 mm of iSC tissue. A weighting matrix, $w_{ij} $, represents the magnitude and inhibitory or excitatory nature of the connection between two nodes, following the pattern of proximal excitation and distal inhibition according to the equation below: $$w_{ij}=a*\exp(\frac{-((i-j)\Delta x)^2} {2\sigma_a^2})-b*\exp(\frac{-((i-j)\Delta x)^2} {2\sigma_b^2}) -c \, ,$$ with $a = 72, b = 24, c = 6.4, \sigma_a = 0.6 mm$, and $\sigma_b = 1.8 mm$. At each time step the internal state of each node changes in accordance with the following relationship: $$\tau \frac{du_i(t)}{dt}=-u_i(t)+ \sum_{j}w_{ij} r_j(t) \Delta x + I_i(t) + u_0 \, ,$$ where [*$u_0 = 0$*]{} regulates the baseline resting activity of each node, $r_j$ is the activation level of node $j$ and is defined as follows: $$r_j(t) = \frac {1}{1+exp(-\beta u_j(t) + \theta)} \,,$$ with $\beta = 0.07$ and $\theta = 0$ used as parameters in the sigmoidal gain function. $I_k$ is the effect of the external input centered on node $i$. The iSC integrates exogenous (visual) and endogenous (goal-directed) signals and our model reflects this. Each input is represented as a Gaussian, where $d$ is the strength of the input and $\sigma_d$ is its width: $$I_k=d*exp(\frac {-((k-i) \Delta x)^2} {2\sigma_d^2}) \,.$$ Exogenous and endogenous input signals were modeled with a width of $\sigma_{exo} = 0.7 mm$, and fixation input signals with a width of $\sigma_{endo} = 0.3 mm$. Exogenous signals used a transient strength of $d = 40$, with a decay constant of $t_{eff} = 1/\tau$. A 70 ms delay was introduced to exogenous signals to simulate early sensory processing delays. When testing the STD theory, endogenous move signals were introduced after an onset delay of 50 ms and sustained until a saccade was triggered [@trappenberg01]. When testing the DS theory, an inhibitory input, $I_{inh}$, was applied at the cued location with a strength of $d = 0.5$, arising 600 ms after cue onset. Target-elicited endogenous eye movement signals, $I_{endo}$, were simulated as sustained input with a 120 ms delay, with a strength varying from $d = 8$ to $d = 12$ based on the CTOA (to represent increasing temporal predictability) [@satel11], and a width of $\sigma_{endo} = 0.7 mm$. SRTs were calculated as the difference between the time of target onset and the time when any node reaches 80% of its maximal firing rate. A further 20 ms was added to account for the time taken for motor signals initiating saccades to traverse the brainstem and reach ocular muscles.  Results ======= The STD and hybrid DS theories of IOR were tested in same (cued, valid; cue and target occur at the same location) and opposite (uncued, invalid; cue and target occur on opposite sides of the visual field) conditions with both central arrow and peripheral onset target types. The IOR scores were calculated by subtracting cued from uncued SRT for each target type condition (see Figure 4). Figure 3 illustrates the simulated SRTs for each condition, showing a general pattern that is similar to that observed in behavior. As shown in Fig. 4A, behaviorally, IOR is only observed at late CTOAs when measured with central arrow targets, but is observed at both short and long CTOAs when peripheral onset targets are used. At short CTOAs, central targets actually led to behavioral facilitation. In the model, facilitation at short CTOAs is the result of lingering cue-elicited activation that counteracts the STD when it is generated. The STD theory alone (see Fig. 4B) can only explain IOR with repeated peripheral stimulation, and the effect decays at long intervals even though IOR is still observed. By incorporating an additional mechanism of direct inhibition that arises 600 ms after cue onset (see Fig. 1B & 4C), the complete behavioral pattern of effects can be reproduced by the model, suggesting that STD theory alone is insufficient to explain IOR. ![A) Behavioral results (adapted from [@hilcheyUR]). B) Simulated cueing effects predicted by STD theory and C) STD+DS theory.](fig4.png) Discussion ========== The purpose of these computational models is to reach a better understanding of the dynamics of the system of interest by simulating its postulates, establishing the explanatory power of the theories with respect to currently published results, and to motivate further study by predicting behavior under unexplored conditions. Here, we have implemented a mathematically explicit model exploring the hybrid STD and DS theory of IOR. On its own, STD theory is unable to explain IOR effects without repeated stimulation, as in trials involving central arrow targets. However, the previous, unextended, STD theory of IOR [@satel11] is a valuable component of the updated hybrid DS model, effectively capturing cueing effects at relatively short CTOAs with repeated peripheral stimuli. The DS theory of IOR, whereby, after a processing delay, an inhibitory signal centered on the cued node reduces baseline node activity, directly increasing the stimulation required to elicit an eye movement. If it exists, as suggested by the behavioural results of Hilchey et al. [@hilcheyUR] (see Fig. 4A), direct collicular local inhibition must arise at some time after around 500 ms post-cue and likely modulates activity in one or more cortical regions capable of processing complex stimuli like arrows. This DS theory of IOR is capable of capturing the behavioral effects of IOR at long CTOAs, but without the incorporation of STD cannot generate an inhibitory cueing effect at short CTOAs when peripheral signals overlap in space. When combined in the hybrid STD plus DS model, simulated response times and cueing effects match the pattern of human behavioral results nicely. Suggestions for future research ------------------------------- Further neurophysiological direct stimulation studies similar to those performed by Fecteau and Munoz [@fecteau05] and Dorris et. al [@dorris02] at CTOAs between 500 and 1000 ms in monkeys who are displaying behavioral IOR would be extremely valuable in determining the neurophysiological mechanisms underlying the hypothesized inhibitory mechanism that behavioral evidence suggests is responsible for long term IOR. If these studies revealed direct inhibition at longer CTOAs when IOR is observed, it would effectively disprove the main neurophysiological argument in favor of the STD model - namely that no direct inhibition was detected at a 200 ms CTOA when IOR was observed [@dorris02]. Of additional interest would be neurophysiological recordings during trials with different combinations of central arrow cues and targets, to provide better understanding of the source and nature of IOR. In a similar vein, further behavioral studies to better bound the activation latency of this secondary inhibitory input would better allow costly and time-consuming neurophysiological studies, as well as computational work, to be focused upon critical periods. Behavioral analysis at CTOAs between 500 and 1000 ms would give us a much better idea of when this signal is generated and a general examination of more time points would improve our understanding of the temporal dynamics of the mechanisms underlying IOR. Conclusion ---------- Our simulations show that the STD model is incapable of explaining IOR on its own, as it shows serious discrepancies with established behavioral data in the literature. The STD model represents early sensory effects and, in combination with facilitation, captures cueing effects quite well at short CTOAs, but cannot explain long CTOA effects, or IOR when using central arrows. When combined with a model of direct collicular local inhibition arising from cortical structures, the model is better able to reproduce established behavioral results in different paradigms. With more data from the experiments outlined above, these models could be further refined and the true neurophysiological mechanisms underlying IOR could be identified. Acknowledgments =============== This work was supported by R. M. Klein’s NSERC Discovery Grant. Z. Wang’s participation in this project was supported by a grant from the Natural Science Foundation of Zhejiang Province, China (Grant No.: LY13C090007). [47]{} Amari, S.: Dynamics of pattern formation in lateral-inhibition type neural fields. Biol. Cybern., 27, 77-87 (1977) Arai, K., Das, S., Keller, E. 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--- abstract: 'A *hole* is a chordless cycle of length at least $4$. A graph is *even-hole-free* if it does not contain any hole of even length as an induced subgraph. In this paper, we study the class of even-hole-free graphs with no star cutset. We give the optimal upper bound for its chromatic number in terms of clique number and a polynomial-time algorithm to color any graph in this class. The latter is, in fact, a direct consequence of our proof that this class has bounded rank-width.' author: - 'Ngoc Khang Le [^1] [^2]' bibliography: - 'color\_even\_hole.bib' title: '****' --- Introduction {#S:1} ============ All graphs in this paper are finite, simple and undirected. Let $F$ be a graph, we say that a graph $G$ is *$F$-free* if it does not contain $F$ as an induced subgraph. Let $\mathcal{F}$ be a (possibly infinite) family of graphs. A graph $G$ is *$\mathcal{F}$-free* if it is $F$-free, for every $F\in\mathcal{F}$. A *hole* is a chordless cycle of length at least $4$. A hole is *even* (*odd*) if it contains an even (resp. odd) number of nodes. Let us first introduce perfect graphs, a graph class which has a very close relation with even-hole-free graphs and is the motivation to study this class. A graph $G$ is *perfect* if for every induced subgraph $H$ of $G$, $\chi(H)=\omega(H)$, where $\chi(H)$ denote the *chromatic number* of $H$, i.e. the minimum number of colors needed to color the vertices of $H$ so that no two adjacent vertices receive the same color, and $\omega(H)$ denotes the size of a largest clique in $H$, where a *clique* is a graph in which all the vertices are pairwise adjacent. The famous Strong Perfect Graph Theorem (conjectured by Berge [@B61], and proved by Chudnovsky, Robertson, Seymour and Thomas [@CRST2006]) states that a graph is perfect if and only if it does not contain an odd hole nor an odd antihole as an induced subgraph (where an *antihole* is a complement of a hole). The graphs that do not contain an odd hole nor an odd antihole as an induced subgraph are known as *Berge* graphs. The structure of even-hole-free graphs was first studied by Conforti, Cornuéjols, Kapoor and Vušković in [@CCKV2002] and [@CCKV20022]. They were focused on showing that even-hole-free graphs can be recognized in polynomial time, and their primary motivation was to develop techniques which can then be used in the study of perfect graphs. In [@CCKV2002], they obtained a decomposition theorem for even-hole-free graphs that uses 2-joins and star, double star and triple star cutsets, and in [@CCKV20022], they used it to obtain a polynomial time recognition algorithm for even-hole-free graphs. That decomposition technique is actually useful since the Strong Perfect Graph Conjecture was proved in [@CRST2006] by decomposing Berge graphs using skew cutsets, 2-joins and their complements. Soon after, the recognition of Berge graphs was shown to be polynomial by Chudnovsky, Cornuéjols, Liu, Seymour and Vušković in [@CCLSV2005]. A better decomposition theorem for even-hole-free graphs using only 2-joins and star cutsets was given in [@DV13]. Finding a maximum clique, a maximum independent set and an optimal coloring are all known to be polynomial for perfect graphs [@GLS81; @GLS2012]. However, these algorithms rely on the ellipsoid method, which is impractical. It is still an open question to find a combinatorial algorithm for these problems. On the other hand, the complexities of finding a maximum stable set and an optimal coloring are both open for even-hole-free graphs. Note that a maximum clique of an even-hole-free graphs can be found in polynomial time, since a graph without a hole of length $4$ has polynomial number of maximal cliques and one can list them all in polynomial time [@F89]. Therefore, we would like to see if the decomposition theorem can be used to design polynomial-time algorithms for all these combinatorial problems. The general answer should be impossible since there are some kinds of decomposition which do not seem to be friendly with these problems like star or skew cutsets. On the other hand, 2-joins look very promising. Indeed, in [@TV12], Trotignon and Vu[š]{}kovi[ć]{} already gave the polynomial algorithms to find a maximum clique and maximum independent set in the subclasses of even-hole-free and Berge graphs which are fully decomposable by only 2-joins (namely, even-hole-free graphs with no star cutset and perfect graphs with no balanced skew-partition, homogenous pair nor complement 2-join). In [@CTTV2015], they generalize the result for Berge graphs to perfect graph with no balanced skew-partitions. Note that an $O(n^k)$ algorithm that computes a maximum weighted independent set for a class of perfect graphs closed under complementation, yields also an $O(n^{k+2})$ algorithm that computes an optimal coloring for the same class (see for instance [@KS97; @S2003]). Hence, all three problems (clique, independent set and coloring) are solved for perfect graph with no balanced skew-partitions. However, the coloring problem for even-hole-free graphs with no star cutset remains open despite of its nice structure. In this paper, we prove that this class has bounded rank-width, a graph parameter which will be defined in the next section. This implies that it also has bounded clique-width (a parameter which is equivalent to rank-width in the sense that one is bounded if and only if the other is also bounded). Therefore, coloring is polynomial-time solvable for even-hole-free graphs with no star cutset by combining the two results: Kobler and Rotics [@KR2003] showed that for any constant $q$, coloring is polynomial-time solvable if a $q$-expression is given, and Oum [@O2008] showed that a $(8^p-1)$-expression for any $n$-vertex graph with clique-width at most $p$ can be found in $O(n^3)$. Note that our result is strong in the sense that it implies that every graph problem expressible in monadic second-order logic formula is solvable in polynomial-time for even-hole-free graphs with no star cutset (including also finding a maximum clique and a maximum independent set). We also know that even-hole-free graphs are $\chi$-bounded by the concept introduced by Gyárfás [@G87]: A class of graphs $\mathcal{G}$ is *$\chi$-bounded* with *$\chi$-bounding function $f$* if for every graph $G\in \mathcal{G}$, $\chi(G)\leq f(\omega(G))$. In [@ACHRS2008], it is proved that $\chi(G)\leq 2\omega(G)-1$ for every even-hole-free graph $G$. One might be interested in knowing whether this bound could be improved for the class that we are considering, even-hole-free graphs with no star cutset. Let *$\operatorname{rwd}(G)$* denote the rank-width of some graph $G$. The main results of our paper are the two following theorems: \[T1\] Let $G$ be a connected even-hole-free graph with no star cutset. Then $\chi(G)\leq \omega(G)+1$. \[T2\] Let $G$ be a connected even-hole-free graph with no star cutset. Then $\operatorname{rwd}(G)\leq 3$. The rest of our paper is organized as follows. In Section \[S:2\], we formally define every notion and mention all the results that we use in this paper. The proof of Theorem \[T1\] is presented in Section \[S:3\] and the proof of Theorem \[T2\] is given in Section \[S:4\]. Preliminaries {#S:2} ============= Let $G(V,E)$ be a graph. For $X\subseteq V(G)$, we denote by $G\setminus X$ the graph obtained from $G$ by removing all the vertices in $X$. In case $X=\{v\}$, we write $G\setminus v$ instead of $G\setminus \{v\}$. We also denote by $G[X]$ the subgraph of $G$ induced by some $X\subseteq V(G)$. For $v\in V(G)$, let $N_G(v)$ denote the set of neighbors of $v$ in $G$. For $X\subseteq V(G)$, let $N_G(X)$ denote the set of vertices in $V(G)\setminus X$ adjacent to a vertex in $X$. We also write $N(v)$ or $N(X)$ instead of $N_G(v)$ or $N_G(X)$ if there is no ambiguity. Let $A\subseteq V(G)$ and $b\in V(G)\setminus A$, we say that $b$ is *complete* to $A$ if $b$ is adjacent to every vertex in $A$. A *clique* in $G$ is a set of pairwise adjacent vertices. A *stable set*, or an *independent set* in $G$ is a set of pairwise non-adjacent vertices. A *path* $P$ is a graph with vertex-set $\{p_1,\ldots,p_k\}$ such that either $k=1$, or for $i, j\in\{1,\ldots,k\}$, $p_i$ is adjacent to $p_j$ iff $|i-j| = 1$. We call $p_1$ and $p_k$ the *ends* of the path, $\{p_2,\dots,p_{k-1}\}$ its *interior* and also call each vertex in $\{p_2,\dots,p_{k-1}\}$ *interior vertex*. Let $P^*$ denote the path obtained from $P$ by removing its two ends. A *flat* path in $G$ is a path such that all of the interior vertices are of degree $2$. A *hole* $H$ is a graph with vertex-set $\{h_1,\ldots, h_k\}$ such that $k\geq 4$ and for $i,j\in\{1,\ldots,k\}$, $h_i$ is adjacent to $h_j$ iff $|i-j|=1$ or $|i-j|=k-1$. The *length* of a path or a hole is the number of its edges. A hole of length $k$ is called a *$k$-hole* Note that a path may have length $0$. A graph is *even-hole-free* if it does not contain any hole of even length as an induced subgraph. Our proof heavily relies on the decomposition lemmas for even-hole-free graphs with no star cutset given by Trotignon and Vu[š]{}kovi[ć]{} in [@TV12]. Hence, in the next part of this section, the formal definitions needed to state these lemmas will be given. In a connected graph $G$, a subset of nodes is a *cutset* if its removal yields a disconnected $G$. A cutset $S\subseteq V(G)$ is a *star cutset* if $S$ contains a node $x$ adjacent to every node in $S\setminus x$. A cutset $S\subseteq V(G)$ is a *clique cutset* if $S$ is a clique. It is clear that clique cutset is a particular star cutset. The only vertex of a clique cutset of size $1$ is called the *cut-vertex*. A *$2$-join* in a graph $G$ is a partition $(X_1, X_2)$ of $V(G)$ with specified sets $(A_1, A_2, B_1, B_2)$ such that the followings hold: - $|X_1|, |X_2| \geq 3$. - For $i=1,2$, $A_i\cup B_i \subseteq X_i$, and $A_i$ and $B_i$ are nonempty and disjoint. - Every node of $A_1$ is adjacent to every node of $A_2$, every node of $B_1$ is adjacent to every node of $B_2$, and these are the only adjacencies between $X_1$ and $X_2$. - For $i=1,2$, the graph induced by $X_i$, $G[X_i]$, contains a path with one end in $A_i$ and the other in $B_i$. Furthermore, $G[X_i]$ does not induce a path. In this case, we call $(X_1, X_2, A_1, B_1, A_2, B_2)$ a *split* of $(X_1, X_2)$. We also denote by $C_i$ the set $X_i\setminus(A_i\cup B_i)$ for $i=1,2$. Since the goal of decomposition theorems is to break our graphs into smaller pieces that we can handle inductively, we need a way to construct them. *Blocks of decomposition* with respect to a $2$-join (will be defined below) are built by replacing each side of the $2$-join by a path of length at least $3$ and the next lemma shows that for even-hole-free graphs, there exists a unique way to choose the parity of that path. \[parity\] Let $G$ be an even-hole-free graph and $(X_1, X_2, A_1, B_1, A_2, B_2)$ be a split of a $2$-join of $G$. Then for $i = 1, 2$, all the paths with an end in $A_i$, an end in $B_i$ and interior in $C_i$ have the same parity. Let $G$ be an even-hole-free graph and $(X_1, X_2, A_1, B_1, A_2, B_2)$ be a split of a $2$-join of $G$. The *blocks of decomposition* of $G$ with respect to ($X_1$, $X_2$) are the two graphs $G_1$, $G_2$ built as follows. We obtain $G_1$ by replacing $X_2$ by a *marker path* $P_2$ of length $k_2$, from a vertex $a_2$ complete to $A_1$, to a vertex $b_2$ complete to $B_1$ (the interior of $P_2$ has no neighbor in $X_1$). We choose $k_2=3$ if the length of all the paths with an end in $A_2$, an end in $B_2$ and interior in $C_2$ is odd (they have the same parity due to Lemma \[parity\]), and $k_2=4$ otherwise. The block $G_2$ is obtained similarly by replacing $X_1$ by a marker path $P_1$ of length $k_1$ with two ends $a_1$, $b_1$. Now we present some definitions for the basic classes in the decomposition theorem for even-hole-free graphs. Let $x_1,x_2,x_3,y$ be four distinct nodes such that $x_1,x_2,x_3$ induce a triangle. A *pyramid* is a graph induced by three paths $P_{x_1y}=x_1\ldots y$, $P_{x_2y}=x_2\ldots y$, $P_{x_3y}=x_3\ldots y$ such that any two of them induce a hole. By the definition, at most one of these paths is of length $1$. A pyramid is *long* if all three paths are of length greater than $1$. Note that in an even-hole-free graph, the lengths of all these three paths have the same parity. An *extended nontrivial basic* graph $R$ is defined as follows: 1. $V(R)=V(L)\cup\{x,y\}$. 2. $L$ is the line graph of a tree $T$. 3. $x$ and $y$ are adjacent, $x,y\notin V(L)$. 4. Every maximal clique of size at least $3$ in $L$ is called an *extended* clique. $L$ contains at least two extended cliques. 5. The nodes of $L$ corresponding to the edges incident with vertices of degree one in $T$ are called *leaf nodes*. Each leaf node of $L$ is adjacent to exactly one of $\{x,y\}$, and no other node of $L$ is adjacent to $\{x,y\}$. 6. These are the only edges in $R$. Note that the definition of the extended nontrivial basic graph we give here is simplified compared to the one from the original paper [@DV13] (since they prove a decomposition theorem for a more general class, namely, $4$-hole-free odd-signable graphs), but it is all we need in our proof. The following property of $R$ is easy to observe in even-hole-free graphs with no star cutset: \[pr4\] $x$ $($and $y)$ has at most one neighbor in every extended clique. Furthermore, if $x$ has some neighbor in an extended clique $K$, then $N(y)\cap K=\emptyset$. If $x$ has two neighbors $a$, $b$ in some extended clique $K$, then $N(a)\setminus\{b\}=N(b)\setminus\{a\}$, implying that there is a star cutset $S=(\{a\}\cup N(a))\setminus \{b\}$ in $R$ separating $b$ from the rest of the graph, a contradiction. Also, if $x$ and $y$ both have a neighbor in a same extended clique, called $a$ and $b$, respectively, then $\{x,a,b,y\}$ induces a $4$-hole, a contradiction. An even-hole-free graph is *basic* if it is one of the following graphs: - a clique, - a hole, - a long pyramid, or - an extended nontrivial basic graph. Now, we are ready to state the decomposition theorem for even-hole-free graphs given by Da Silva and Vu[š]{}kovi[ć]{}. A connected even-hole-free graph is either basic or it has a $2$-join or a star cutset. By this theorem, we already know that even-hole-free graphs with no star cutset always have a $2$-join. But we might prefer something a bit stronger for our purpose. A $2$-join is called *extreme* if one of its block of decomposition is basic. The two following lemmas (which can be found in Sections 3 and 4 in [@TV12]) say that: our blocks of decomposition with respect to a $2$-join remain in the class and our class is fully decomposable by extreme $2$-joins. This is convenient for an inductive proof. \[remain\] Let $G$ be a connected even-hole-free graph with no star cutset and $(X_1,X_2)$ is a $2$-join of $G$. Let $G_1$ be a block of decomposition with respect to this $2$-join. Then $G_1$ is a connected even-hole-free graph with no star cutset. \[decomposition\] A connected even-hole-free graph with no star cutset is either basic or it has an extreme $2$-join. By Lemmas \[remain\] and \[decomposition\], we know that even-hole-free graphs with no star cutset can be fully decomposed into basic graphs using only extreme $2$-joins. However, we need a little more condition to avoid confliction between these $2$-joins, that is, every $2$-join we use is *non-crossing*, meaning that every marker path in the process always lies entirely in one side of every following $2$-joins (the edges between $X_1$ and $X_2$ do not belong to any marker path). Now we define the *$2$-join decomposition tree* for this purpose. Note that this definition we give here is not only for even-hole-free graphs with no star cutset, but also works in a more general sense. It is well defined for any graph class with its own basic graphs. Let $\mathbb{D}$ be a class of graphs and $\mathbb{B}\subseteq \mathbb{D}$ be the set of basic graphs in $\mathbb{D}$. Given a graph $G\in\mathbb{D}$, a tree $\mathbb{T}_G$ is a *$2$-join decomposition tree* for $G$ if: - Each node of $\mathbb{T}_G$ is a pair $(H,S)$, where $H$ is a graph in $\mathbb{D}$ and $S$ is a set of disjoint flat paths of $H$. - The root of $\mathbb{T}_G$ is $(G,\emptyset)$. - Each non-leaf node of $\mathbb{T}_G$ is $(G',S')$, where $G'$ has a $2$-join $(X_1,X_2)$ such that the edges between $X_1$ and $X_2$ do not belong to any flat path in $S'$. Let $S_1,S_2\subseteq S'$ be the set of the flat paths of $S'$ in $G'[X_1]$, $G'[X_2]$, respectively (note that $S'=S_1\cup S_2$). Let $G_1$, $G_2$ be two blocks of decomposition of $G'$ with respect to this $2$-join with marker paths $P_2$, $P_1$, respectively. The node $(G',S')$ has two children, which are $(G_1,S_1\cup \{P_2\})$ and $(G_2,S_2\cup \{P_1\})$. - Each leaf node of $\mathbb{T}_G$ is $(G',S')$, where $G'\in\mathbb{B}$. Note that by this definition, each set $S'$ in some node $(G',S')$ of $\mathbb{T}_G$ is properly defined in top-down order (from root to leaves). A $2$-join decomposition tree is called *extreme* if each non-leaf node of it has a child which is a leaf node. \[non-crossing\] Every connected even-hole-free graphs with no star cutset has an extreme $2$-join decomposition tree. \[ob:1\] Every block of decomposition with respect to a $2$-join of a connected even-hole-free graph with no star cutset which is basic is either a long pyramid or an extended nontrivial basic graph. Let us review the definition of rank-width, which was first introduced in [@OS2006]. For a matrix $M=\{ m_{ij}: i\in R,\, j\in C\}$ over a field $F$, let $\operatorname{rk}(M)$ denote its linear rank. If $X\subseteq R$, $Y\subseteq C$, then let $M[X,Y]$ be the submatrix $\{ m_{ij}: i\in X,\, j\in Y\}$ of $M$. We assume that adjacency matrices of graphs are matrices over $GF(2)$. Let $G$ be a graph and $A$, $B$ be disjoint subsets of $V(G)$. Let $M$ be the adjacency matrix of $G$ over $GF(2)$. We define the *rank* of $(A,B)$, denoted by $\operatorname{rk}_G(A,B)$, as $\operatorname{rk}(M[A,B])$. The *cut-rank* of a subset $A\subseteq V(G)$, denoted by $\operatorname{cutrk}_G(A)$, is defined by $$\operatorname{cutrk}_G(A)=\operatorname{rk}_G(A,V(G)\setminus A).$$ A *subcubic tree* is a tree such that the degree of every vertex is either one or three. We call $(T,L)$ a *rank-decomposition* of $G$ if $T$ is a subcubic tree and $L$ is a bijection from $V(G)$ to the set of leaves of $T$. For an edge $e$ of $T$, the two connected components of $T\setminus e$ correspond to a partition $(A_e,V(G)\setminus A_e)$ of $V(G)$. The *width* of $e$ of the rank-decomposition $(T,L)$ is $\operatorname{cutrk}_G(A_e)$. The *width* of $(T,L)$ is the maximum width over all edges of $T$. The *rank-width* of $G$, denoted by $\operatorname{rwd}(G)$, is the minimum width over all rank-decompositions of $G$ (If $|V(G)|\leq 1$, we define $\operatorname{rwd}(G)=0$). \[ob:2\] The rank-width of a clique is at most $1$ and the rank-width of a hole is at most $2$. $\chi$-bounding function {#S:3} ======================== Special graphs -------------- Recall that the bound of chromatic number for even-hole-free graphs ($\chi(G)\leq 2\omega(G)-1$) is obtained by showing that there is a vertex whose neighborhood is a union of two cliques [@ACHRS2008]. We would like to do the same things for our class. However, since our class is not closed under vertex-deletion, instead of showing that there exists a vertex whose neighborhood is “simple”, we have to show that there is an elimination order such that the neighborhood of each vertex is “simple” in the remaining graph. To achieve that goal, we introduce *special* graphs. In fact, this is just a way of labeling vertices for the sake of an inductive proof. A graph $G$ is *special* if it is associated with a pair $(C_G, F_G)$ such that: - $C_G\subseteq V(G)$, $F_G\subseteq V(G)$ and $C_G\cap F_G=\emptyset$. - Every vertex in $F_G$ has degree $2$. - Every vertex in $C_G$ has at least one neighbor in $F_G$. Note that any graph can be seen as a special graph with $C_G=F_G=\emptyset$. Suppose that $G$ has some split $(X_1, X_2, A_1, B_1, A_2, B_2)$ of a $2$-join. Due to this new notion of special graph, we want to specify the pairs $(C_{G_1}, F_{G_1})$ and $(C_{G_2}, F_{G_2})$ for the blocks of decomposition $G_1$, $G_2$ of $G$ with respect to this $2$-join to ensure that the two blocks we obtained are also special. Let $C_i=C_G\cap X_i$, $F_i=F_G\cap X_i$ ($i=1,2$), we choose the pair $(C_{G_1}, F_{G_1})$ as follows: - If $|A_1|=1$, the only vertex in $A_1$ is in $C_G$ and $A_2\cap F_G\neq \emptyset$, then set $C_a=\emptyset$, $F_a=\{a_2\}$. Otherwise set $C_a=\{a_2\}$, $F_a=\emptyset$. - If $|B_1|=1$, the only vertex in $B_1$ is in $C_G$ and $B_2\cap F_G\neq \emptyset$, then set $C_b=\emptyset$, $F_b=\{b_2\}$. Otherwise set $C_b=\{b_2\}$, $F_b=\emptyset$. - Finally, set $C_{G_1}=C_1\cup C_a\cup C_b$, $F_{G_1}=F_1\cup F_a\cup F_b\cup V(P_2^*)$. The pair $(C_{G_2}, F_{G_2})$ for block $G_2$ is chosen similarly. \[remains\] Let $G$ be a special connected even-hole-free graph with no star cutset associated with $(C_G, F_G)$ and $(X_1, X_2, A_1, B_1, A_2, B_2)$ be a split of a $2$-join of $G$. Let $G_1$ be a block of decomposition with respect to this $2$-join. Then $G_1$ is a special graph associated with $(C_{G_1}, F_{G_1})$. Remark that since $G$ is $4$-hole-free, one of $A_1$ and $A_2$ must be a clique (similar for $B_1$ and $B_2$). Now we prove that if one of $A_1$ and $A_2$ intersects $F_G$, then the other set is of size $1$. Suppose that $A_1\cap F_G\neq \emptyset$, we will prove that $|A_2|=1$. Indeed, since $f\in A_1\cap F_G$ has degree $2$, $|A_2|\leq 2$. If $|A_2|=2$ then $f$ is the only vertex in $A_1$ (otherwise, $A_2$ must be a clique and $N(f)$ is a clique cutset separating $f$ from the rest of $G$, a contradiction to the fact that $G$ has no star cutset). Therefore, $f$ has no neighbor in $X_1$, so there is no path from $A_1$ to $B_1$ in $G[X_1]$, a contradiction to the definition of a $2$-join. This proves that $|A_2|=1$. Now, $G_1$ is a special graph associated with $(C_{G_1}, F_{G_1})$ because: 1. Every vertex $f$ in $F_{G_1}$ has degree $2$. If $f\in F_1\setminus (A_1\cup B_1)$, then degree of $f$ remains the same in $G$ and $G_1$. If $f\in F_1\cap (A_1\cup B_1)$, say $f\in F_1\cap A_1$, from the above remark, $|A_2|=1$, therefore the degree of $f$ remains the same in $G$ and $G_1$. If $f\in F_a\cup F_b$ then $|A_1|=1$ by the way we choose $F_{G_1}$, so $f$ has degree $2$ in $G_1$. If $f\in P_2^*$, then it is an interior vertex of a flat path, therefore it has degree $2$. 2. Every vertex $c$ in $C_{G_1}$ has at least a neighbor in $F_{G_1}$. If $c\in C_1$ and its neighbor in $F_G$ is in $X_1$, then $c$ has a neighbor in $F_1$. If $c\in C_1$ and its neighbor in $F_G$ is in $A_2\cup B_2$, say $A_2$, then its neighbor in $F_{G_1}$ is $a_2$. If $c\in C_a\cup C_b$, then its neighbor in $F_{G_1}$ is one of the two ends of $P_2^*$. Elimination order ----------------- Let $G$ be a special graph associated with $(C_G, F_G)$. A vertex $v\in V(G)$ is *almost simplicial* if its neighborhood induces a clique or a union of a clique $K$ and a vertex $u$ such that $u\notin C_G$ ($u$ can have neighbor in $K$). An elimination order $v_1$,…, $v_k$ of vertices of $G\setminus F_G$ is *nice* if for every $1\leq i\leq k$, $v_i$ is *almost simplicial* in $G\setminus (F_G\cup \{v_1,\ldots,v_{i-1}\})$. The next lemma is the core of this section. \[L1\] Let $G$ be a special connected even-hole-free graph with no star cutset associated with $(C_G,F_G)$. Then $G\setminus F_G$ admits a nice elimination order. By setting $C_G=F_G=\emptyset$, we have the following corollary of Lemma \[L1\]: Let $G$ be a connected even-hole-free graph with no star cutset. Then $G$ admits a nice elimination order. Theorem \[T1\] follows immediately from the above corollary since we can greedily color $G$ in the reverse order of that nice elimination order using at most $\omega(G)+1$ colors. Therefore, the rest of this section is devoted to the proof of Lemma \[L1\]. \[L2\] Let $G$ be a special basic even-hole-free graph with no star cutset associated with $(C_G, F_G)$ and $G$ is neither a clique nor a hole. Let $P$ be a flat path of length at least $2$ in $G$. We denote by $u_1$, $u_2$ the two ends of $P$. - If $N(u_1)\setminus V(P)$ is a clique, set $K_1=N(u_1)\setminus V(P)$, otherwise set $K_1=\emptyset$. - If $N(u_2)\setminus V(P)$ is a clique, set $K_2=N(u_2)\setminus V(P)$, otherwise set $K_2=\emptyset$. Let $Q_P=(K_1\cup K_2 \cup V(P))\setminus F_G$. Then $G\setminus F_G$ admits a nice elimination order $v_1$,…, $v_k$, where $Q_P$ is in the end of this order $($i.e. $Q_P=\{v_{k-|Q_P|+1},\ldots,v_k\})$. We prove the lemma when $G$ is a long pyramid or an extended nontrivial basic graph. In fact, since the proof for a long pyramid can be treated almost similarly, we only show here the proof in the case where $G$ is an extended nontrivial basic graph. Suppose that $V(G)=V(H)\cup \{x,y\}$, where $H$ is the line graph of a tree. We may assume the followings: 1. \[as1\] $P$ is a maximal flat path in $G$ (two ends of $P$ are of degree $\geq 3$). If the lemma is true when $P$ is a maximal flat path then it is also true for all subpaths of $P$, because $Q_P$ admits a perfect elimination order (an order of vertices in which the neighborhood of a vertex induces a clique at the time it is eliminated) where a fixed subpath of $P$ is in the end of this order. 2. \[as2\] All the vertices in $G\setminus (F_G\cup Q_P\cup\{x,y\})$ are not in $C_G$. Observe that the neighborhood of every vertex $v$ in $G$, except $x$ and $y$, induces a union of two cliques. Therefore, if $v\in C_G$, it must have a neighbor of degree $2$ in $F_G$, then its neighborhood in $G\setminus F_G$ actually induces a clique and it can be eliminated at the beginning of our order. 3. \[as3\] Every vertex in $G\setminus (F_G\cup Q_P)$ has at most one neighbor in $C_G$. Indeed, by the assumption \[as2\], $C_G\subseteq \{x,y\}\cup Q_P$. If a vertex $v\in G\setminus (F_G\cup Q_P)$ has two neighbors in $C_G$, then it must have a neighbor $u\in (C_G\cap Q_P)\setminus\{x,y\}$. By the definition of $C_G$, $v$ must be a vertex in $F_G$ since it is the only neighbor of degree $2$ of $u$, a contradiction to the choice of $v$. Let us first forget about the flat path $P$ and the restriction of putting all the vertices of $Q_P$ in the end of the order. We will show how to obtain a nice elimination order for $G\setminus F_G$ in this case. We choose an arbitrary extended clique $K_R$ in $H$ and call it the *root* clique. For each other extended clique $K$ in $H$, there exists a vertex $v\in K$ whose removal separates the root clique from $K\setminus v$ in $H$, we call it *B-vertex*. We call a node *E-vertex* if it is adjacent to $x$ or $y$. Note that in each extended clique $K$, we have exactly one B-vertex and at most one E-vertex (by Lemma \[pr4\]). For the root clique $K_R$, we also add a new vertex $r$ adjacent to all the vertices of $K_R$, and let it be the B-vertex for $K_R$. Now, if we remove every edge in every extended clique, except the edges incident to its B-vertex, we obtain a tree $T_H$ rooted at $r$. Note that $V(T_H)=V(H)\cup \{r\}$. We specify the nice elimination order for $G$ where all the vertices in $V(K_R)$ are removed last (we do not care about the order of eliminating $r$, this vertex is just to define an order for $V(G)$ more conveniently). Let $O_T$ be an order of visiting $V(T_H)\setminus (V(K_R)\cup \{r\})$ satisfying: - A node $u$ in $T_H$ is visited after all the children of $u$. - If $u$ is a B-vertex of some extended clique $K$, the children of $u$ must be visited in an order where the E-vertex in $K$ (if it exists) is visited last. Let us introduce some notions with respect to orders first. Let $O_1=v_1,\ldots,v_k$ and $O_2=u_1,\ldots,u_t$ be two orders of two distinct sets of vertices. We denote by *$O_1\oplus O_2$* the order $v_1,\ldots,v_k,u_1,\ldots,u_t$. If $S$ is a subset of vertices of some order $O_1$, we denote by *$O_1\setminus S$* the order obtained from $O_1$ by removing $S$. Let $u$ be a vertex, we also denote by $u$ the order of one element $u$. Let $O_{K_R}$ be an arbitrary elimination order for the vertices in $K_R$. Now the elimination order for $G\setminus F_G$ is $O=(O_T\setminus F_G)\oplus x\oplus y\oplus O_{K_R}$. We prove that this elimination order is nice. Indeed, let $u$ be a vertex of order $O_T$, $u\notin F_G$. If $u$ is not an E-vertex, then its neighborhood at the time it is eliminated is either a subclique of some extended clique (if its parent is a B-vertex) or a single vertex which is its parent. If $u$ is an E-vertex, since it is eliminated after all its siblings (the nodes share the same parent), its neighborhood consists of only two vertices: its parent and $x$ (or $y$). And by assumption \[as3\], at most one of these two vertices is in $C_G$, so $u$ is almost simplicial. Now when $x$ is removed in this order, it has at most two neighbors: one is $y$ and one is possibly a vertex in $K_R$, also not both of them are in $C_G$, so $x$ is almost simplicial. Vertex $y$ has at most one neighbor at the time it is eliminated. And finally, $K_R$ is a clique so any eliminating order for $K_R$ at this point is nice. Now we have to consider the flat path $P$, and put all the vertices of $Q_P$ in the end of the elimination order. There are two cases: - $P$ is a flat path not containing $x$ and $y$. In this case, both $K_1$ and $K_2$ are non-empty. The graph obtained from $G$ by removing $V(P)\cup \{x,y\}$ contains two connected components $H_1$, $H_2$, where $H_i$ ($i=1,2$) is the line graph of a tree. By considering $K_i$ as the root clique of $H_i$, from the above argument, we obtain two elimination orders $O_1$, $O_2$ for $H_1\setminus K_1$ and $H_2\setminus K_2$. Now, all the vertices not yet eliminated in $G$ are in $\{x,y\}\cup Q_P$. We claim that at least one of $x$, $y$ has at most two neighbors in the remaining graph. Indeed, otherwise $x$ and $y$ both have at least three neighbors, implying that they both have neighbors in $K_1$ and $K_2$, which contradicts Lemma \[pr4\]. Suppose $x$ has at most two neighbors, in this case we eliminate $x$ first, then $y$. Note that $x$ and $y$ are both almost simplicial in this elimination order, since they have at most two neighbors (not both of them in $C_G$ according to assumption \[as3\]) at the time they were eliminated. Finally, choose for $Q_P$ a perfect elimination order $O_Q$. Now the nice elimination order for $G\setminus F_G$ is $O=((O_1\oplus O_2)\setminus F_G)\oplus x\oplus y\oplus O_Q$. - $x$ or $y$ is an end of $P$. W.l.o.g, suppose $x$ is an end of $P$, say $x=u_1$. In this case $K_1=\emptyset$, $K_2\neq \emptyset$. The graph $H'$ obtained by removing $V(P)\cup \{y\}$ from $G$ is the line graph of a tree. We consider $K_2$ as the root clique of this graph. From above argument, we have a nice elimination order $O_{H'}$ for $V(H')\setminus K_2$. Now all the vertices left in $G$ are in $\{y\}\cup Q_P$. Observe that $y$ has at most two neighbors in the remaining graph ($x$ and possibly a vertex in $K_2$), therefore $y$ is almost simplicial and can be eliminated. Note that $x$ has no neighbor in $K_2$, since if $u\in K_2$ is adjacent to $x$, then $\{u\}\cup N(u)$ is a star cutset in $G$ separating $P^*$ from the rest of the graph ($P^*$ is non-empty since the length of $P$ is at least two). Finally, choose for $Q_P$ a perfect elimination order $O_Q$. Now the nice elimination order for $G\setminus F_G$ is $O=(O_{H'}\setminus F_G)\oplus y\oplus O_Q$. By Lemma \[non-crossing\], $G$ has an extreme 2-join decomposition tree $\mathbb{T}_G$. Now, for every node $(G',S')$ of $\mathbb{T}_G$, $G'$ is a special connected even-hole-free graph with no star cutset associated with $(C_{G'},F_{G'})$ (by Lemmas \[remain\] and \[remains\]). Now we prove that for every node $(G',S')$ of $\mathbb{T}_G$, $G'$ satisfies Lemma \[L1\]. This implies the correctness of Lemma \[L1\] since the root of $\mathbb{T}_G$ corresponds to $G$. First, we show that for every leaf node $(G',S')$ of $\mathbb{T}_G$, $G'$ satisfies Lemma \[L1\]. If $G'$ is a clique then any elimination order of $G'\setminus F_{G'}$ is nice. If $G'$ is a hole, there exists a vertex $v$ such that one of its neighbors is not in $C_{G'}$, then $v$ can be eliminated first. The vertices in the remaining graph induce a subgraph of a path, therefore $G'\setminus v$ admits a nice elimination order. If $G'$ is a long pyramid or an extended nontrivial basic graph, we have a nice elimination order for $G'$ by Lemma \[L2\]. Now, let us prove that Lemma \[L1\] holds for $G'$, where $(G',S')$ is a non-leaf node of $\mathbb{T}_G$. Since $\mathbb{T}_G$ is extreme, $G'$ admits an extreme $2$-join with the split $(X_1, X_2, A_1, B_1, A_2, B_2)$ and let $G_1$, $G_2$ be the blocks of decomposition of $G'$ with respect to this $2$-join. We may assume that $G_1$ is basic and $G_2$ satisfies Lemma \[L1\] by induction. Note that $V(G')=(V(G_1)\setminus V(P_2))\cup (V(G_2)\setminus V(P_1))$. Now we try to specify a nice elimination order for $G'$ by combining the orders for $G_1$ and $G_2$. Since $G_1$ is basic, apply Lemma \[L2\] for $G_1$ with $P=P_2$, we obtain the nice elimination order $O_1$ for $G_1\setminus (F_{G_1}\cup Q_P)$. Remark that all the vertices in $O_1$ are in $V(G')$ since we have not eliminated $Q_P$. By induction hypothesis, we obtain also a nice elimination order $O_2$ for $G_2\setminus F_{G_2}$. We create an order $O_2'$ from $O_2$ for $V(G')$ as follows ($a_1$, $b_1$ are two ends of the marker path $P_1$): - If $a_1\in C_{G_2}$ and $A_1$ is a clique, $O_2'$ is obtained from $O_2$ by substituting $a_1$ in $O_2$ by all the vertices in $A_1$ (in any order), otherwise set $O_2'=O_2\setminus\{a_1\}$. - If $b_1\in C_{G_2}$ and $B_1$ is a clique, $O_2'$ is obtained from itself by substituting $b_1$ in $O_2'$ by all the vertices in $B_1$ (in any order), otherwise set $O_2'=O_2'\setminus\{b_1\}$. We claim that $O=O_1\oplus O_2'$ is a nice elimination order for $G'\setminus F_{G'}$. Let $N'_{G'}(u)$ ($N'_{G_1}(u)$, $N'_{G_2}(u)$) be the set of neighbors of $u$ in the remaining graph when it is removed with respect to order $O$ ($O_1$, $O_2$, respectively). - If $u$ is a vertex in $O_1$. - If $u\notin A_1$ and $B_1$, then $N_{G'}(u)=N_{G_1}(u)$, because $u$ is almost simplicial in $G_1$ then it is also almost simplicial in $G'$ at the time it was eliminated. - If $u\in A_1$ or $B_1$, w.l.o.g, suppose that $u\in A_1$, then $A_1$ is not a clique, because we do not eliminate $Q_P$ in $O_1$. Since one of $A_1$, $A_2$ must be a clique to avoid $4$-hole, $A_2$ is a clique. If $a_2\in F_{G_1}$, then $|A_1|=1$ and $A_1$ is a clique of size $1$, a contradiction. Then $a_2\in C_{G_1}$. Because $a_2$ was not eliminated at the time we remove $u$ in order $O_1$, $a_2\in N'_{G_1}(u)$. We can obtain $N'_{G'}(u)$ from $N'_{G_1}(u)$ by substituting $a_2$ by $A_2$, therefore $u$ remains almost simplicial in $G'$. - If $u$ is a vertex in $O_2'$. - If $u\in X_2\setminus (A_2\cup B_2)$, then $N_{G'}(u)=N_{G_2}(u)$, because $u$ is almost simplicial in $G_2$ then it is also almost simplicial in $G'$ at the time it was eliminated. - If $u\in A_2$ or $B_2$, w.l.o.g, suppose $u\in A_2$. We may assume that $A_1$ is a clique, since otherwise it was eliminated in $O_1$ before $u$, implying $N'_{G'}(u)\subseteq N'_{G_2}(u)$ and $u$ is almost simplicial in $G'$. - Suppose $a_1\in C_{G_2}$. If $u$ is eliminated after $a_1$, then $N'_{G'}(u)=N'_{G_2}(u)$ and $u$ is almost simplicial in $G'$. If $u$ is eliminated before $a_1$, we can obtain $N'_{G'}(u)$ from $N'_{G_2}(u)$ by substituting $a_1$ by $A_1$, therefore $u$ remains almost simplicial. - Suppose $a_1\in F_{G_2}$. Since $A_1$ is a clique and it contains a vertex $v\in F_{G'}$, $|A_1|\leq 2$. If $|A_1|=2$, then $N_{G'}(v)$ is a clique cutset of size $2$ (star cutset) separating $v$ from the rest of $G'$, a contradiction. Thus $A_1=\{v\}$ and $N'_{G'}(u)=N'_{G_2}(u)$ (since $v$ is the only vertex in $A_1$ and $v\notin G'\setminus F_{G'}$) and $u$ is almost simplicial in $G'$. - If $u\in A_1$ or $B_1$, w.l.o.g, suppose $u\in A_1$, then $A_1$ is a clique, since otherwise it was removed in $O_1$. We can obtain $N'_{G'}(u)$ from $N'_{G_2}(a_1)$ by creating a clique $K$, which is a subclique of $A_1$ ($K$ is actually the set of vertices of $A_1$ going after $u$ in $O_2'$), and make it complete to $N'_{G_2}(a_2)$. Therefore, $u$ remains almost simplicial in $G'$. The bound is tight ------------------ ![Graph $G_5$ with $\omega(G_5)=5$ and $\chi(G_5)=6$[]{data-label="F:1"}](kai_plus_one.pdf){width="11cm"} Now we show how to construct for any $k\geq 3$ an even-hole-free graph $G_k$ with no star cutset such that $\omega(G_k)=k$ and $\chi(G_k)=k+1$. The set of vertices of $G_k$: $V(G_k)=A\cup B\cup C\cup D\cup E\cup F$, where $A$, $C$, $E$ are cliques of size $(k-1)$; $B$, $D$ are independent sets of size $(k-1)$ and $F$ is an independent set of size $(k-2)$. The vertices in each set are labeled by the lowercase of the name of that set plus an index, for example $A=\{a_1,\ldots,a_{k-1}\}$. The edges of $G_k$ as follows: - $A$ is complete to $B$, $C$ is complete to $D$. - $b_i$ is adjacent to $c_i$, $d_i$ is adjacent to $e_i$ ($i=1,\ldots,k-1$). - $a_{k-1}$ is complete to $E$. - $d_1$ is complete to $F$. - $a_i$ is adjacent to $f_i$ ($i=1,\ldots,k-2$). Figure \[F:1\] is an example of $G_k$, where $k=5$. The fact that $G_k$ is an even-hole-free graph with no star cutset can be checked by hand. For every $k\geq 3$, $\omega(G_k)=k$ and $\chi(G_k)=k+1$. It is clear that $\omega(G_k)=k$. We will show that $G_k$ is not $k$-colorable. Suppose we have a $k$-coloring of $G$. Because in that coloring, every clique of size $k$ must be colored by all $k$ different colors, then all the vertices in $B$ must receive the same color $1$. Therefore, the clique $C$ must be colored by $(k-1)$ left colors, and all the vertices in $D$ must be colored by color $1$ also. Therefore, the $k$-clique $\{a_{k-1},e_1,\ldots,e_{k-1}\}$ is not colorable since all of them must have color different from $1$, a contradiction. Rank-width {#S:4} ========== Bounded rank-width ------------------ Recall that the definition of rank-width and rank-decomposition are given in the last part of Section \[S:2\]. Given a graph $G$ and some rank-decomposition $(T,L)$ of $G$, a subset $X$ of $V(G)$ is said to be *separated* in $(T,L)$ if there exists an edge $e_X$ of $T$ corresponding to the partition $(X,V(G)\setminus X)$ of $V(G)$. Let $d$ be an integer, we say that graph $G$ has *property $\mathbb{P}(d)$* if for every set $S$ of disjoint flat paths of length at least $3$ in $G$, there is a rank-decomposition $(T,L)$ of $G$ such that the width of $(T,L)$ is at most $d$ and every flat path $P\in S$ is separated in $(T,L)$. The next lemma shows the relation between $2$-join and rank-width. \[rank-2-join\] Let $\mathbb{D}$ be a class of graphs and $\mathbb{B}\subseteq\mathbb{D}$ be the set of its basic graphs such that every graph $G\in \mathbb{D}$ has a $2$-join decomposition tree. Furthermore, there exists an integer $d\geq 2$ such that every basic graph in $\mathbb{D}$ has property $\mathbb{P}(d)$. Then for every graph $G\in \mathbb{D}$, $\operatorname{rwd}(G)\leq d$. Let $G$ be a graph in $\mathbb{D}$ and $\mathbb{T}_G$ be its $2$-join decomposition tree. We prove that every node $(G',S')$ of $\mathbb{T}_G$ satisfies the following *property $\mathbb{P'}(d)$*: there is a rank-decomposition $(T,L)$ of $G'$ such that the width of $(T,L)$ is at most $d$ and every flat path $P\in S'$ is separated in $(T,L)$. Note that property $\mathbb{P'}(d)$ is weaker than property $\mathbb{P}(d)$ since it is not required to be true for every choice of the set of disjoint flat paths, but only for a particular set $S'$ associated with $G'$ in $\mathbb{T}_G$. Proving this property for each node in $\mathbb{T}_G$ implies directly the lemma since if the root of $\mathbb{T}_G$ has property $\mathbb{P'}(d)$, then $\operatorname{rwd}(G)\leq d$. It is clear that every leaf node of $\mathbb{T}_G$ has property $\mathbb{P'}(d)$ since every basic graph has property $\mathbb{P}(d)$ by the assumption. Now we only have to prove that every non-leaf node $(G',S')$ of $\mathbb{T}_G$ has property $\mathbb{P'}(d)$ assuming that its two children $(G_1,S_1)$ and $(G_2,S_2)$ already have property $\mathbb{P'}(d)$. For $i\in\{1,2\}$, let $(T_i,L_i)$ be the rank-decomposition of $G_i$ satisfying property $\mathbb{P'}(d)$. We show how to construct the rank-decomposition $(T,L)$ of $G'$ satisfying this property. Recall that by the definition of a $2$-join decomposition tree, $G_1$ and $G_2$ are two blocks of decomposition with respect to some $2$-join $(X_1,X_2)$ of $G'$ together with some marker paths $P_2\in S_1$, $P_1\in S_2$, respectively. For $i\in\{1,2\}$, since $(G_i,S_i)$ satisfies property $\mathbb{P'}(d)$, $P_{3-i}$ is separated in $(T_i,L_i)$ by some edge $e_i=u_iv_i$ of $T_i$. Let $C_i$, $D_i$ be the two connected components (subtrees) of $T_i\setminus e_i$ (the tree obtained from $T_i$ by removing the edge $e_i$), where the leaves of $C_i$ correspond to $V(G_i)\setminus V(P_{3-i})$ and the leaves of $D_i$ correspond to $V(P_{3-i})$. W.l.o.g, we may assume that $u_i$ is in $C_i$ and $v_i$ is in $D_i$. The tree $T$ is then constructed from $T_1[V(C_1)\cup\{v_1\}]$ and $T_2[V(C_2)\cup\{v_2\}]$ by identifying $u_1$ with $v_2$ and $u_2$ with $v_1$. Note that $T$ is a subcubic tree and the leaves of $T$ now correspond to $V(G)$. The mapping $L$ is the union of the two mappings $L_1$ and $L_2$ restricted in $X_1$ and $X_2$, respectively. Now the node $(G',S')$ satisfies property $\mathbb{P}'_d$ since: - Every flat path $P\in S'$ is separated in $(T,L)$. It is true since for $i\in\{1,2\}$, every path $P\in S_i$ is separated in $(T_i,L_i)$. - The width of $(T,L)$ is at most $d$. It is easy to see that the width of the identified edge $e=u_1v_1$ of $T$ is $2$, since it corresponds to the partition $(X_1,X_2)$ of $G'$. For other edge $e$ of $C_i$ (for $i=1$ or $2$), it corresponds to a cut of $G'$ separating a subset $Z$ of $X_i$ from $V(G')\setminus Z$, and we have $\operatorname{cutrk}_{G'}(Z)=\operatorname{cutrk}_{G_i}(Z)$ (since the rank of the corresponding matrix stays the same if we just add several copies of the columns corresponding to the two ends of the marker path $P_{3-i}$), which implies that $\operatorname{cutrk}_{G'}(Z)\leq d$. Thanks to Lemma \[rank-2-join\] and the existence of a $2$-join decomposition tree by Lemma \[non-crossing\], to prove that the rank-width of even-hole-free graphs with no star cutset is at most $3$, we are left to only prove that every basic even-hole-free graph with no star cutset has property $\mathbb{P}(3)$. Actually, by Observation \[ob:1\], we do not have to prove it for cliques and holes, since they never appear in the leaf nodes of any $2$-join decomposition tree of any graph in our class. Therefore, Theorem \[T2\] is a consequence of Observation \[ob:2\] and the following lemma: \[rank-basic\] Every basic even-hole-free graph with no star cutset, which is neither a clique nor a hole, has property $\mathbb{P}(3)$. Let $G$ be a basic even-hole-free graph with no star cutset, which is different from a clique and a hole. Since $G$ is basic and $G$ is neither a clique nor a hole, $G$ must be an extended nontrivial basic graph or a long pyramid. Since the case where $G$ is a long pyramid can be followed easily from the case where it is an extended nontrivial basic graph. We omit the details for long pyramids here. Let $G$ be an extended nontrivial basic graph, $V(G)=V(H)\cup \{x,y\}$, where $H$ is the line graph of a tree. Let $S$ be some set of flat paths of length at least $3$ in $G$. Now we show how to build the rank-decomposition of $G$ satisfying the lemma. First, we construct the *characteristic* tree $F_H$ for $H$. We choose an arbitrary extended clique in $H$ as a *root* clique. Let $E$ be the set of flat paths obtained from $H$ by removing all the edges of every extended clique in $H$. Now, we define the father-child relation between two flat paths in $E$. A path $B$ is the *father* of some path $B'$ if they have an endpoint in the same extended clique in $H$ and any vertex of $B$ is a cut-vertex in $H$ which separates $B'$ from the root clique. If $B$ is the father of $B'$ then we also say that $B'$ is a *child* of $B$. Any path in $E$ which has only one endpoint in an extended clique is called *leaf* path, otherwise it is called *internal* path. Now, we consider each path $B$ in $E$ as a vertex $v_B$ in the characteristic tree $F_H$, and associate with each node $v_B$ a set $S_{v_B}=V(B)$. Each leaf path corresponds to a leaf in $F_H$ and each internal path corresponds to an internal node in $F_H$, which reserves the father-child relation (if a path $B$ is the father of some path $B'$ then $v_B$ is the father of $v_{B'}$ in $F_H$). We also add a root $r$ for $F_H$, and the children of $r$ are all the vertices $v_B$, where $B$ is a path with an endpoint in the root clique, let $S_r=\emptyset$. Now, we add two special vertices $x$, $y$ to attain the *characteristic* tree $F_G$ for $G$. If $x$ (or $y$) is an endpoint of some flat path $P$ in $S$, then we set $S_v=S_v\cup\{x\}$ ($S_v=S_v\cup\{y\}$, respectively), where $v$ is the leaf in $F_H$ corresponding to the leaf path in $E$ which contains $P\setminus\{x\}$ ($P\setminus\{y\}$, respectively). Otherwise set $S_v=S_v\cup\{x\}$ ($S_v=S_v\cup\{y\}$), where $v$ is a leaf in $F_H$ corresponding to any path in $E$ having an endpoint adjacent to $x$ ($y$, respectively). Figures \[F:2\] and \[F:3\] are the example of an extended nontrivial basic graph $G$ and its characteristic tree $F_G$ (the bold edges are the edges of flat paths in $S$). Note that each node in $F_G$ corresponds to a subset $S_v$ of $V(G)$, they are all disjoint, each of them induces a flat path in $G$ and $V(G)=\cup_{v\in F_G}S_v$. ![An extended nontrivial basic graph $G$ with a set of flat paths.[]{data-label="F:2"}](EH_1.pdf){width="7cm"} ![The characteristic tree $F_G$ for graph $G$ in Figure \[F:2\].[]{data-label="F:3"}](EH_2.pdf){width="11cm"} Now, we show how to build the rank-decomposition of $G$ from its characteristic tree $F_G$. We first define a special rooted tree, called *$k$-caterpillar* ($k\geq 1$) to achieve that goal. For $k\geq 1$, a graph $I$ is called *$k$-caterpillar* if: - For $k=1$, $V(I)=\{a_1,l_1\}$, $E(I)=\{a_1l_1\}$ and $a_1$ is the root of $I$. - For $k\geq 2$, $V(I)=\{a_1,\ldots,a_{k-1}\}\cup \{l_1,\ldots,l_{k}\}$, $E(I)=\{a_ia_{i+1}|1\leq i\leq k-2\}\cup \{a_il_i|1\leq i\leq k-1\}\cup \{a_{k-1}l_k\}$ and $a_1$ is the root of $I$. Notice that in the following discussion, for the sake of construction, the rank-decomposition $(T,L)$ we build for our graph is not exactly the same as in the definition of a rank-decomposition mentioned in Section \[S:2\], since we allow vertex of degree $2$ in tree $T$, but it does not change the definition of rank-width. A flat path in $G$ is called *mixed* if it contains a flat path in $S$ but it is not a flat path in $S$. We start by constructing the rank-decomposition of a non-mixed flat path in $G$. For a non-mixed flat path $P=p_1\ldots p_k$, we create a $k$-caterpillar $T_P$ which has exactly $k$ leaves $l_1,\ldots,l_k$ as in the definition and a bijection $L_P$ maps each vertex in $P$ to a leaf of $T_P$ such that $L_P(p_i)=l_i$. Since a mixed path can always be presented as a union of vertex-disjoint non-mixed paths $P=\cup_{i=1}^kP_i$ (where one end of $P_i$ is adjacent to one end of $P_{i+1}$ for $1\leq i\leq k-1$), let $(T_i,L_i)$ be the rank-decomposition for each non-mixed path $P_i$ constructed as above, we can build the tree $T_P$ by creating a $k$-caterpillar $I$ which has exactly $k$ leaves $l_1,\ldots,l_k$ as in the definition and identify each root of $T_i$ with the leaf $l_i$ of $I$ for $1\leq i\leq k$. Also, let the mapping $L_P$ from $V(P)$ to the leaves of $T_P$ be the union of all the mappings $L_i$’s for $1\leq i\leq k$. Now, we build the rank-decomposition $(T_G,L_G)$ of $G$ from its characteristic tree $F_G$ by visiting each node in $F_G$ in an order where all the children of any internal node is visited before its father. For a vertex $v\in F_G$, denote by $C_v$ the union of all connected components of $F_G\setminus v$ that does not contain $r$. Let $F_G(v)=F_G[V(C_v)\cup\{v\}]$, $X_v=\cup_{u\in F_G(v)}S_u$. At each node $v$ of $F_G$, we build the rank-decomposition $(T_v,L_v)$ of the graph $G_v$ induced by the subset $X_v$ of $V(G)$ by induction: 1. If $v$ is a leaf of $F_G$, build the rank-decomposition $(T_v,L_v)$ for the flat path corresponding to $v$ like above argument for mixed and non-mixed paths. 2. If $v$ is an internal node of $F_G$ different from its root and $v_1,\ldots,v_k$ are its children. Let $(T,L)$ be the rank-decomposition of the flat path corresponding to $v$ (built by above argument for mixed and non-mixed paths) and $(T_i,L_i)$ ($i=1\ldots k$) be the rank-decomposition of $G[X_{v_i}]$. We build $T_v$ by constructing a $(k+1)$-caterpillar having exactly $(k+1)$ leaves $l_1,\ldots,l_{k+1}$ as in the definition and identify the root of $T$ with $l_1$, the root of $T_i$ with $l_{i+1}$ for $1\leq i\leq k$. Let the mapping $L_v$ from $X_{v_i}$ to the leaves of $T_v$ be the union of the mapping $L$ and all the mappings $L_i$’s for $1\leq i\leq k$. 3. If $v$ is the root of $F_G$ and $v_1,\ldots,v_k$ are its children. Let $(T_i,L_i)$ ($i=1\ldots k$) be the rank-decompositions of $G[X_{v_i}]$. We build $T_v$ by constructing a $k$-caterpillar having exactly $k$ leaves $l_1,\ldots,l_{k}$ as in the definition and identify the root of $T_i$ with $l_i$ for $1\leq i\leq k$. Let the mapping $L_v$ from $V(G)$ to the leaves of $T_v$ be the union of all the mappings $L_i$’s for $1\leq i\leq k$. ![The rank-decomposition for graph $G$ in Figure \[F:2\].[]{data-label="F:4"}](EH_3.pdf){width="12cm"} The rank-decomposition $(T_r,L_r)$ corresponding to the root $r$ of $F_G$ is the desired rank-decomposition $(T,L)$ for $G$ (see Figure \[F:4\]). Now we prove that this rank-decomposition construction for the extended nontrivial basic graphs $G$ satisfies the lemma. \[prop1\] Let $(T,L)$ be the above constructed rank-decomposition for $G$. Then, every flat path $P$ in $S$ is separated in $(T,L)$. It is trivially true, because $P$ is a non-mixed subpath of some flat path $B$ in $E$, so $V(P)$ is separated in the rank-decomposition of $B$. And each flat path $B$ of $G$ is also separated in the rank-decomposition of $G$ by our construction. So $V(P)$ is separated in $(T,L)$. \[prop2\] The above constructed rank-decomposition $(T,L)$ of $G$ has width at most $3$. We prove by the structure of the characteristic tree $F_G$ of $G$. For an internal node $v$ of $F_G$, let $v_1,\ldots,v_k$ be its children, in some sense, the decomposition tree $T_v$ for $X_v$ is obtained by “glueing” the decomposition tree for $G[S_v]$ and all the decomposition trees $T_i$ for $G[X_{v_i}]$ for $1\leq i\leq k$ along a cut-vertex. Therefore, we consider an edge $e$ of $T_v$ as an edge of $T$ as well. Our goal is to prove that the width of any edge $e$ with respect to the rank-decomposition $(T,L)$ of $G$ is at most 3. For the sake of induction, at each node $v$ of $F_G$, we prove that the width of any edge $e$ of $T_v$ is at most $3$ with respect to the rank-decomposition $(T,L)$ of $G$ (we mention $v$ here just to specify an edge in our tree $T$): 1. If $v$ is a leaf in $F_G$. Every edge $e$ of $T_v$ corresponds to a partition of $V(G)$ into two parts where one of them is a subpath of the flat path corresponding to $v$, so the width of $e$ is at most $2$. 2. If $v$ is an internal node in $F_G$ and $v_1,\ldots,v_k$ are its children. Let $(T_i,L_i)$ ($i=1\ldots k$) be the rank-decompositions of $G[X_{v_i}]$. Let $e$ be an edge of $T_v$. If $e$ is an edge of $T_i$ then the width of $e$ is at most $3$ by induction. Otherwise, $e$ corresponds to one of the following situations: - $e$ corresponds to a partition $(V(P),V(G)\setminus V(P))$ of $V(G)$, where $P$ is a subpath of the flat path $G[S_v]$. In this case, the width of $e$ is clearly at most $2$. - $e$ corresponds to a partition $(U,V(G)\setminus U)$ of $V(G)$, where $U$ is the union of several $X_{v_i}$’s. Let $K$ be the extended clique intersecting every $X_{v_i}$. In this case, there are only three types of neighborhood of vertices of $U$ in $G\setminus U$: - $K\setminus U$, - $x$ if $x\notin U$, or $N(x)\setminus U$ if $x\in U$, and - $y$ if $y\notin U$, or $N(y)\setminus U$ if $y\in U$. Therefore, the width of $e$ is at most $3$. Lemma \[rank-basic\] is true because of the Propositions \[prop1\] and \[prop2\]. An even-hole-free graph with no clique cutset and unbounded rank-width ---------------------------------------------------------------------- It is clear that clique cutset is a particular type of star cutset. However, the class of even-hole-free graph with no clique cutset (a super class of even-hole-free graph with no star cutset) does not have bounded rank-width. Since clique-width and rank-width are equivalent, now we show how to construct for every $k\geq 4$, $k$ even an even-hole-free graph $G_k$ with no clique cutset and $\operatorname{cwd}(G_k)\geq k$. The set of vertices of $G_k$: $V(G_k)=\cup_{i=0}^{k}A_i$, where each $A_i=\{a_{i,0},\ldots,a_{i,k}\}$ is a clique of size $(k+1)$. We also have edges between two consecutive sets $A_i$, $A_{i+1}$ ($i=0,\ldots,k$, the indexes are taken modulo $(k+1)$). They are defined as follows: $a_{i,j}$ is adjacent to $a_{i+1,l}$ iff $j+l\leq k$. For every $k\geq 4$, $k$ even, $G_k$ is an even-hole-free graph with no clique cutset. By the construction, there is no hole in $G_k$ that contains two vertices in some set $A_i$ and every hole must contain at least a vertex in each set $A_i$. Therefore, every hole in $G_k$ has exactly one vertex from each set $A_i$, so its length is $(k+1)$ (an odd number). Hence, $G_k$ is even-hole-free. We see that every clique in $G_k$ is contained in the union of some two consecutive sets $A_i$, $A_{i+1}$. Hence, its removal does not disconnect $G_k$. Therefore, $G_k$ has no clique cutset. For every $k\geq 4$, $k$ even, $\operatorname{cwd}(G_k)\geq k$. The graph obtained from $G_k$ by deleting all the vertices in $A_0\cup_{i=1}^{k} \{a_{i,0}\}$ is isomorphic to the permutation graph $H_k$ introduced in [@GR2000]. And because it was already proved in that paper that $\operatorname{cwd}(H_k)\geq k$, and clique-width of $G_k$ is at least the clique-width of any of its induced subgraph then $\operatorname{cwd}(G_k)\geq k$. Note that another example of an even-hole-free graph with no clique cuset and unbounded rank-width is also given in [@VTRMLA17]. [^1]: This work was performed within the framework of the LABEX MILYON (ANR-10-LABX-0070) of Université de Lyon, within the program “Investissements d’Avenir” (ANR-11- IDEX-0007) operated by the French National Research Agency (ANR). Partially supported by ANR project Stint under reference ANR-13-BS02-0007. [^2]: Email: ngoc-khang.le@ens-lyon.fr