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Acknowledgements. <|MaskedSetence|> He is very grateful to the Radcliffe Institute for providing excellent working conditions. TT is supported by NSF Research Award DMS-0649473, the NSF Waterman award and a grant from the MacArthur Foundation. <|MaskedSetence|> All three authors are very grateful to the University of Verona for allowing them to use classrooms at Canazei during a week in July 2009. This work was largely completed during that week.
3. <|MaskedSetence|> | **A**: Basic notation.
**B**: TZ is supported by ISF grant 557/08, an Alon fellowship and a Landau fellowship of the Taub foundation.
**C**: BG was, for some of the period during which this work was carried out, a fellow of the Radcliffe Institute at Harvard.
| CBA | CBA | CBA | ABC | Selection 2 |
<|MaskedSetence|> <|MaskedSetence|> Doing this directly using (1.10) amounts
to solving a system of 123123123123 equations and 123 variables. Due to this we take an
alternative strategy. <|MaskedSetence|> We do not provide proofs for. | **A**: The first part of this procedure is similar to the one used
in [GGV1]*Section 8, and is inspired by [M].
**B**: been already be verified independently in [H] and [GGV2].
**C**: We will verify the
case (m,n)=(50,75)𝑚𝑛5075(m,n)=(50,75)( italic_m , italic_n ) = ( 50 , 75 ).
| BCA | BCA | BCA | BCA | Selection 3 |
To deal with the case where G=F4𝐺subscript𝐹4G=F_{4}italic_G = italic_F start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT and ⟨v⟩delimited-⟨⟩𝑣\langle v\rangle⟨ italic_v ⟩ has stabiliser of type D4subscript𝐷4D_{4}italic_D start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT, we perform a similar calculation with 𝔤[t]𝔤delimited-[]𝑡\mathfrak{g}[t]fraktur_g [ italic_t ] and working with matrices over ℤ[t]ℤdelimited-[]𝑡\mathbb{Z}[t]blackboard_Z [ italic_t ]. Let v=e1+te2𝑣subscript𝑒1𝑡subscript𝑒2v=e_{1}+te_{2}italic_v = italic_e start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_t italic_e start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT be a generic element in the 00-weight space of Vminsubscript𝑉minV_{\mathrm{min}}italic_V start_POSTSUBSCRIPT roman_min end_POSTSUBSCRIPT. <|MaskedSetence|> <|MaskedSetence|> It also turns out that for t≠0,1𝑡01t\neq 0,1italic_t ≠ 0 , 1, the resulting 26×24262426\times 2426 × 24-matrix has, for any choice of t𝑡titalic_t, a single non-zero entry in each row, with no two non-zero entries in a common column. <|MaskedSetence|> | **A**: Thus the rank of the matrix is 24242424 in all characteristics for all choices of t≠0,1𝑡01t\neq 0,1italic_t ≠ 0 , 1, and we are done.
∎
.
**B**: It turns out that 28282828 of these rows are identically zero and so the rank will not change after they are removed.
**C**: Form M𝑀Mitalic_M as before to get a 26×52265226\times 5226 × 52 matrix.
| CBA | CAB | CBA | CBA | Selection 4 |
The referee’s contribution is significant in this paper. Some explicitly appear in the text, but others got hidden in the revision process. The contributions of the latter nature include the following. (1) The author originally worked exclusively with motivic cohomology with compact supports. It was the referee who explained how to develop the axiomatic framework as in Section 2, and then apply it to the case of motives. <|MaskedSetence|> These simplified the author’s previous arguments, particularly the proofs of Propositions 2.4 and 2.10, and unified cases that had been treated separately. (2) The author originally worked only over an algebraically closed base field. It was the referee who suggested working over a perfect (sometimes even non-perfect) field whenever possible and explained how to do it. This turned out fruitful in the study of algebraic part, and we obtained the current version of Propositions 3.2 and 3.4. <|MaskedSetence|> Finally, Subsection 3.5 would have been nonexistent, had it not been for the referee’s explicit question on the topic. The referee also directed the author to the paper [ACMV17a]. For this subsection, we would also like to thank Charles Vial for pointing out independently that the method in ibid. is applicable. <|MaskedSetence|> Subsection 3.2 should be attributed to him or her. The idea of unpointed regular homomorphisms enables us to state the relation to Ayoub and Barbieri-Viale’s work (Remark 3.11). It also made the comparison with Serre-Ramachandran’s Albanese varieties simpler and conceptual (Subsection 3.3).
Convention. From now on, schemes are assumed separated and of finite type over a base field k,𝑘k,italic_k , and morphisms of schemes are those over the base field except in Subsection 3.5. Group schemes are also assumed to be defined over k.𝑘k.italic_k .. | **A**: Most importantly, Definition 2.3 and Axiom 2.6 are due to the referee.
**B**: The use of the symmetric power construction and [Mil86, Theorem 3.13] in the proof of Proposition 3.4 is also due to the referee.
**C**: (3) The definition of unpointed regular homomorphisms is due to the referee (Definition 3.7).
| ABC | ABC | ABC | BCA | Selection 1 |
1.4. Some observations
There are several applications of Inou-Shishikura’s invariant class. The first remarkable application is that Buff and Chéritat used it as one of the main tools to prove the existence of Julia sets of quadratic polynomials with positive area [BC12]. <|MaskedSetence|> <|MaskedSetence|> <|MaskedSetence|> For examples, the Feigenbaum Julia sets with positive area (which is different from the examples in [BC12]) have been found in [AL22], the Marmi-Moussa-Yoccoz conjecture for rotation numbers of high type has been proved in [CC15], the local connectivity of the Mandelbrot set at some infinitely satellite renormalizable points was proved in [CS15], some statistical properties of the quadratic polynomials was depicted in [AC18], the topological structure and the Hausdorff dimension of high type irrationally indifferent attractors were characterized in [Che22a] and [CDY20] respectively.
. | **A**: In [Che13] and [Che19], Cheraghi developed several elaborate analytic techniques based on Inou-Shishikura’s results.
**B**: Recently, Cheraghi and his collaborators have found several other important applications.
**C**: The tools in [Che13] and [Che19] have led to part of the recent major progresses on the dynamics of quadratic polynomials.
| BAC | ABC | BAC | BAC | Selection 1 |
The paper is organized as follows. <|MaskedSetence|> In Section 3 we prove the local index theorem for families of ∂¯¯\bar{\partial}over¯ start_ARG ∂ end_ARG-operators on Riemann orbisurfaces that are factors of the hyperbolic plane by the action of finitely generated cofinite Fuchsian groups. <|MaskedSetence|> In Section 4.1 we find a simple formula for a local Kähler potential of the elliptic metric, and in Section 4.2 we show that in the limit when the order of the elliptic element
tends to ∞\infty∞ the elliptic metric coincides with the corresponding cusp metric. <|MaskedSetence|> | **A**: Finally, in Section 4.3 we give a simple example of a relation between the elliptic metric and special values of Selberg zeta
function for Fuchsian groups of signature (0;1;2,2,2)..
**B**: Section 2 contains the necessary background material.
**C**: Specifically,
we show that the contribution to the local index formula from elliptic elements of Fuchsian groups is given by the symplectic form of a Kähler metric on the moduli space of orbisurfaces.
Since the cases of smooth (both compact and punctured) Riemann surfaces have been well understood by us quite a while ago [15, 10], in Section 3.2 we mainly emphasize the computation of the contribution from conical points corresponding to elliptic elements.
| BCA | BCA | ABC | BCA | Selection 1 |
<|MaskedSetence|> Realizing Harmonic Oscillators With Coupled Supersymmetry
As previously noted, the quantum mechanical harmonic oscillator is a specific instance of a coupled supersymmetry, albeit a somewhat trivial case in which the two coupled SUSY equations are identical. This is not the only manner in which the two are connected. <|MaskedSetence|> <|MaskedSetence|> If one takes γ=−δ𝛾𝛿\gamma=-\deltaitalic_γ = - italic_δ, then the coupled SUSY equations become
. | **A**:
7.
**B**: they satisfy the same Lie algebra and by virtue of Stone-von Neumann, may be realized in some way as harmonic oscillators.
**C**: Indeed, a special class of coupled SUSYs may be realized as harmonic oscillator-like systems, i.e.
| ACB | ACB | BCA | ACB | Selection 4 |
Throughout the text G𝐺Gitalic_G will denote a group, usually abelian and often ordered, ℭℭ\mathfrak{C}fraktur_C will denote a sufficiently saturated model of Th(G)Th𝐺\mathrm{Th}(G)roman_Th ( italic_G ). <|MaskedSetence|> We will need a few results from [27]. Since this text is not readily available, we try to keep the present work as self contained as possible, referring to more accessible sources whenever we are aware of such. <|MaskedSetence|> <|MaskedSetence|> Ordered abelian groups. | **A**: The next sub-section is dedicated to a quick overview of (parts) of the language we are using, and to the basic properties of definable sets.
2.1.
**B**: By definable we will mean definable with parameters.
**C**: In particular, for the study of ordered abelian groups we chose the language of [4], rather than the language used by Schmitt.
| CBA | BCA | BCA | BCA | Selection 3 |
<|MaskedSetence|> <|MaskedSetence|> Here in this paper, in the presence of rough coefficients, spectral techniques are employed to overcome such hurdle, and by solving local eigenvalue problems we define a space where the exponential decay of solutions is insensitive to high-contrast coefficients. Additionally, the spectral techniques remove
macro-elements corner singularities that occur in LOD methods based on
mixed finite elements. <|MaskedSetence|> Here, we propose eigenvalue problems based on edges of macro element removing the dependence. | **A**: We note the proposal in [CHUNG2018298] of generalized multiscale finite element methods based on eigenvalue problems inside the macro elements, with basis functions with support weakly dependent of the log of the contrast.
**B**: When LOD or VMS methods are considered, high-contrast coefficients might slow down the exponential decay of the solutions, making the method not so practical.
**C**:
One difficulty that hinders the development of efficient methods is the presence of high-contrast coefficients [MR3800035, MR2684351, MR2753343, MR3704855, MR3225627, MR2861254].
| CBA | BCA | CBA | CBA | Selection 4 |
The seminal work of Artzner et al. (1999) has bestowed upon the field of risk assessment a set of four pivotal axioms that stand as the cornerstones of coherence for any reputable risk measure. Building upon this foundational framework, Föllmer and Schied (2002), in tandem with the pioneering efforts of Frittelli and Rosazza-Gianin (2002) expanded the purview of risk measures. They introduced the broader class which are convex risk measures by dropping one of the coherency axioms. Song and Yan (2009) gave an overview of representation theorems for various static risk measures. <|MaskedSetence|> Zuo and Yin (2022) considered the multivariate tail covariance for generalized skew-elliptical distributions. Cai et al. <|MaskedSetence|> While these advancements have introduced sophisticated risk measures, it’s important to highlight that their theoretical foundation frequently exists within a static framework. The conventional depiction of these theories operates within a fixed temporal frame, offering a foundational understanding of risk.
Over the past two decades, not only has the study of static risk measures flourished, but also dynamic theories of risk measurement have developed into a thriving and mathematically refined area of research. Dynamic risk measures represent a sophisticated and evolving field within risk management, extending the analysis beyond static frameworks to account for temporal changes in risk. Unlike traditional static risk measures that provide a snapshot assessment, dynamic risk measures recognize the fluid nature of financial markets and aim to capture how risk evolves over time. Introduced by Riedel (2004), dynamic coherent risk measures offer a framework that allows for a more nuanced understanding of risk dynamics. This advancement enables a comprehensive assessment of risk in the context of changing market conditions and evolving investment portfolios. <|MaskedSetence|> | **A**: Additionally, the introduction of dynamic convex risk measures by Detlefsen and Scandolo (2005) further enriched the field, providing insights into the time consistency properties of risk measures over different time horizons..
**B**: (2022) defined a new multivariate conditional Value-at-Risk
risk measure based on the minimization of the expectation of a multivariate loss function.
**C**: Shushi and Yao (2020) proposed two multivariate risk measures based on conditional expectation and derived explicit formulae for exponential dispersion models.
| ACB | CBA | CBA | CBA | Selection 2 |
Let q𝑞qitalic_q be any point in π−1(p)superscript𝜋1𝑝\pi^{-1}(p)italic_π start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( italic_p ) of ramification index eqsubscript𝑒𝑞e_{q}italic_e start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT. <|MaskedSetence|> If any σ∈Γq𝜎subscriptΓ𝑞\sigma\in\Gamma_{q}italic_σ ∈ roman_Γ start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT exchanges two branches then the point p=π(q)𝑝𝜋𝑞p=\pi(q)italic_p = italic_π ( italic_q ) is smooth in Σ¯¯Σ\bar{\Sigma}over¯ start_ARG roman_Σ end_ARG, which contradicts
the assumption that p𝑝pitalic_p is a nodal point. Thus, ΓqsubscriptΓ𝑞\Gamma_{q}roman_Γ start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT must preserve branches. <|MaskedSetence|> <|MaskedSetence|> | **A**: Therefore, by the condition (5.1), we can choose a formal coordinate system z′,z′′superscript𝑧′superscript𝑧′′z^{\prime},z^{\prime\prime}italic_z start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_z start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT around the nodal point q𝑞qitalic_q such that 𝒪^Σ,q≃ℂ[[z′,z′′]]/(z′z′′)similar-to-or-equalssubscript^𝒪Σ𝑞ℂdelimited-[]superscript𝑧′superscript𝑧′′superscript𝑧′superscript𝑧′′\hat{\mathscr{O}}_{{\Sigma},q}\simeq\mathbb{C}[[z^{\prime},z^{\prime\prime}]]/%
(z^{\prime}z^{\prime\prime})over^ start_ARG script_O end_ARG start_POSTSUBSCRIPT roman_Σ , italic_q end_POSTSUBSCRIPT ≃ blackboard_C [ [ italic_z start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_z start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ] ] / ( italic_z start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_z start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ), and a generator σqsubscript𝜎𝑞\sigma_{q}italic_σ start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT of ΓqsubscriptΓ𝑞\Gamma_{q}roman_Γ start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT such that
.
**B**: In particular, since no nontrivial element of ΓΓ\Gammaroman_Γ fixes pointwise any irreducible component of ΣΣ\Sigmaroman_Σ, ΓqsubscriptΓ𝑞\Gamma_{q}roman_Γ start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT is cyclic.
**C**: Since π−1(p)superscript𝜋1𝑝\pi^{-1}(p)italic_π start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( italic_p ) consists of nodal points by assumption, there are two branches in the formal neighborhood of q𝑞qitalic_q.
| CBA | BAC | CBA | CBA | Selection 3 |
The second question regards how a subsystem of anyons in a many-anyon system may be characterized by certain good quantum numbers. This is an important question because often (e.g., in topological quantum computing) we may be concerned about only the entanglement between such subsystems and ignore what is inside each subsystem. <|MaskedSetence|> <|MaskedSetence|> If we can find the complete set of observables (giving rise to good quantum numbers) of the system, we may take their eigenvectors to form the basis of the Hilbert space. As to be seen in this paper, the clarification of the physical meaning of pseudo-species turns out to be the answer to this question too. <|MaskedSetence|> The dependency coefficients are called the statistics parameters. In terms of these extended statistics parameters, we extend Wu’s formula for statistical weight, which lays the foundation of the statistical mechanics of anyons on systems with gapped boundaries. Clearly, such exclusion statistics put the boundary components and bulk anyons on an equal footing. We shall dub this kind of exclusion statistics by extended anyonic exclusion statistics.. | **A**: Recall that the Hilbert space of a fermionic or bosonic system is taken as a Fock space, which is the tensor product of the local Hilbert spaces of single-particle states.
**B**: To study the Hilbert space structure of a system of non-Abelian anyons, however, there demands a new formulation of the basis, as many-anyon states do not have an obvious Fock space analogy formed by the tensor product of local single-anyon Hilbert spaces.
**C**: More precisely, the good quantum numbers of a subsystem are the eigenvalues of the observables of the relevant pseudo-species.
In this paper, we extend Haldane’s generalized exclusion principle to systems with gapped boundaries, by proposing that the number of available single particle states for additional particles linearly and mutually depends on the number of (every species of) existing anyons and the number of (every boundary type of) existing gapped boundaries.
| ABC | ABC | ABC | ACB | Selection 2 |
<|MaskedSetence|> We may assume that both Wλ,n0subscriptsuperscript𝑊0𝜆𝑛W^{0}_{\lambda,n}italic_W start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_λ , italic_n end_POSTSUBSCRIPT and Wλ,n0subscriptsuperscript𝑊0𝜆𝑛W^{0}_{\lambda,n}italic_W start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_λ , italic_n end_POSTSUBSCRIPT are nonempty (the proof in the case when at least one of Wλ,n0subscriptsuperscript𝑊0𝜆𝑛W^{0}_{\lambda,n}italic_W start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_λ , italic_n end_POSTSUBSCRIPT and Wλ,n0subscriptsuperscript𝑊0𝜆𝑛W^{0}_{\lambda,n}italic_W start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_λ , italic_n end_POSTSUBSCRIPT is empty is an easier variation of the proof in the other case). <|MaskedSetence|> We will assume that sup(Wλ,nϵ)supsubscriptsuperscript𝑊italic-ϵ𝜆𝑛\operatorname{sup}(W^{\epsilon}_{\lambda,n})roman_sup ( italic_W start_POSTSUPERSCRIPT italic_ϵ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_λ , italic_n end_POSTSUBSCRIPT ) has countable cofinality (the proof when Wλ,nϵsubscriptsuperscript𝑊italic-ϵ𝜆𝑛W^{\epsilon}_{\lambda,n}italic_W start_POSTSUPERSCRIPT italic_ϵ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_λ , italic_n end_POSTSUBSCRIPT is empty or has a maximum is an easier variant of the proof in the case that cf(sup(Wλ,nϵ))=ωcfsupsubscriptsuperscript𝑊italic-ϵ𝜆𝑛𝜔\operatorname{cf}(\operatorname{sup}(W^{\epsilon}_{\lambda,n}))=\omegaroman_cf ( roman_sup ( italic_W start_POSTSUPERSCRIPT italic_ϵ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_λ , italic_n end_POSTSUBSCRIPT ) ) = italic_ω). <|MaskedSetence|> We construct qλ,n,+0superscriptsubscript𝑞𝜆𝑛0q_{\lambda,n,+}^{0}italic_q start_POSTSUBSCRIPT italic_λ , italic_n , + end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT and qλ,n,+0superscriptsubscript𝑞𝜆𝑛0q_{\lambda,n,+}^{0}italic_q start_POSTSUBSCRIPT italic_λ , italic_n , + end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT as the greatest lower bound of (qλ,n,m0)m<ωsubscriptsuperscriptsubscript𝑞𝜆𝑛𝑚0𝑚𝜔(q_{\lambda,n,m}^{0})_{m<\omega}( italic_q start_POSTSUBSCRIPT italic_λ , italic_n , italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT ) start_POSTSUBSCRIPT italic_m < italic_ω end_POSTSUBSCRIPT and (qλ,n,m1)m<ωsubscriptsuperscriptsubscript𝑞𝜆𝑛𝑚1𝑚𝜔(q_{\lambda,n,m}^{1})_{m<\omega}( italic_q start_POSTSUBSCRIPT italic_λ , italic_n , italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ) start_POSTSUBSCRIPT italic_m < italic_ω end_POSTSUBSCRIPT, respectively, where qλ,n,00=qλ,n0superscriptsubscript𝑞𝜆𝑛00superscriptsubscript𝑞𝜆𝑛0q_{\lambda,n,0}^{0}=q_{\lambda,n}^{0}italic_q start_POSTSUBSCRIPT italic_λ , italic_n , 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT = italic_q start_POSTSUBSCRIPT italic_λ , italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT and qλ,n,01=qλ,n1superscriptsubscript𝑞𝜆𝑛01superscriptsubscript𝑞𝜆𝑛1q_{\lambda,n,0}^{1}=q_{\lambda,n}^{1}italic_q start_POSTSUBSCRIPT italic_λ , italic_n , 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT = italic_q start_POSTSUBSCRIPT italic_λ , italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT and where, for each m<ω𝑚𝜔m<\omegaitalic_m < italic_ω,
qλ,n,m+10=rλ,n,m0⊕qλ,n,m0superscriptsubscript𝑞𝜆𝑛𝑚10direct-sumsuperscriptsubscript𝑟𝜆𝑛𝑚0superscriptsubscript𝑞𝜆𝑛𝑚0q_{\lambda,n,m+1}^{0}=r_{\lambda,n,m}^{0}\oplus q_{\lambda,n,m}^{0}italic_q start_POSTSUBSCRIPT italic_λ , italic_n , italic_m + 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT = italic_r start_POSTSUBSCRIPT italic_λ , italic_n , italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT ⊕ italic_q start_POSTSUBSCRIPT italic_λ , italic_n , italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT and qλ,n,m+11=rλ,n,m1⊕qλ,n,m1superscriptsubscript𝑞𝜆𝑛𝑚11direct-sumsuperscriptsubscript𝑟𝜆𝑛𝑚1superscriptsubscript𝑞𝜆𝑛𝑚1q_{\lambda,n,m+1}^{1}=r_{\lambda,n,m}^{1}\oplus q_{\lambda,n,m}^{1}italic_q start_POSTSUBSCRIPT italic_λ , italic_n , italic_m + 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT = italic_r start_POSTSUBSCRIPT italic_λ , italic_n , italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ⊕ italic_q start_POSTSUBSCRIPT italic_λ , italic_n , italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT, where rλ,n,m0subscriptsuperscript𝑟0𝜆𝑛𝑚r^{0}_{\lambda,n,m}italic_r start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_λ , italic_n , italic_m end_POSTSUBSCRIPT, rλ,n,m1∈ℚβmsubscriptsuperscript𝑟1𝜆𝑛𝑚subscriptℚsubscript𝛽𝑚r^{1}_{\lambda,n,m}\in{\mathbb{Q}}_{\beta_{m}}italic_r start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_λ , italic_n , italic_m end_POSTSUBSCRIPT ∈ blackboard_Q start_POSTSUBSCRIPT italic_β start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT end_POSTSUBSCRIPT are conditions extending qλ,n,m0superscriptsubscript𝑞𝜆𝑛𝑚0q_{\lambda,n,m}^{0}italic_q start_POSTSUBSCRIPT italic_λ , italic_n , italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT and qλ,n,m1superscriptsubscript𝑞𝜆𝑛𝑚1q_{\lambda,n,m}^{1}italic_q start_POSTSUBSCRIPT italic_λ , italic_n , italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT, respectively, and such that
. | **A**: Let ϵ∈{0,1}italic-ϵ01\epsilon\in\{0,1\}italic_ϵ ∈ { 0 , 1 } be such that sup(Wλ,nϵ)=max{sup(Wλ,n0),sup(Wλ,n1)}supsubscriptsuperscript𝑊italic-ϵ𝜆𝑛supsubscriptsuperscript𝑊0𝜆𝑛supsubscriptsuperscript𝑊1𝜆𝑛\operatorname{sup}(W^{\epsilon}_{\lambda,n})=\max\{\operatorname{sup}(W^{0}_{%
\lambda,n}),\operatorname{sup}(W^{1}_{\lambda,n})\}roman_sup ( italic_W start_POSTSUPERSCRIPT italic_ϵ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_λ , italic_n end_POSTSUBSCRIPT ) = roman_max { roman_sup ( italic_W start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_λ , italic_n end_POSTSUBSCRIPT ) , roman_sup ( italic_W start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_λ , italic_n end_POSTSUBSCRIPT ) }.
**B**: Let (βm)m<ωsubscriptsubscript𝛽𝑚𝑚𝜔(\beta_{m})_{m<\omega}( italic_β start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_m < italic_ω end_POSTSUBSCRIPT be a strictly increasing sequence of ordinals in Mλsubscript𝑀𝜆M_{\lambda}italic_M start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT converging to sup(Wλ,nϵ)supsubscriptsuperscript𝑊italic-ϵ𝜆𝑛\operatorname{sup}(W^{\epsilon}_{\lambda,n})roman_sup ( italic_W start_POSTSUPERSCRIPT italic_ϵ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_λ , italic_n end_POSTSUBSCRIPT ).
**C**:
We of course have that |Wλ,nϵ|≤ℵ0subscriptsuperscript𝑊italic-ϵ𝜆𝑛subscriptℵ0|W^{\epsilon}_{\lambda,n}|\leq\aleph_{0}| italic_W start_POSTSUPERSCRIPT italic_ϵ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_λ , italic_n end_POSTSUBSCRIPT | ≤ roman_ℵ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT.
| CAB | BCA | CAB | CAB | Selection 1 |
1.6. <|MaskedSetence|> This research was carried out within the HSE University Basic Research Program and funded by the Russian Academic Excellence Project ’5-100’. <|MaskedSetence|> The work of both authors has also been supported in part by the Simons Foundation. <|MaskedSetence|> | **A**: Acknowledgements.
We thank the referees for the careful reading of the first version of the text and for many helpful remarks.
**B**: The results of Section 4 has been obtained under support of the RSF grant 19-11-00056.
**C**: The first author is a Young Russian Mathematics award winner and would like to thank its sponsors and jury.
.
| ABC | ABC | CBA | ABC | Selection 2 |
Let now nk+1=msubscript𝑛𝑘1𝑚n_{k+1}=mitalic_n start_POSTSUBSCRIPT italic_k + 1 end_POSTSUBSCRIPT = italic_m. We show that all conditions (0),(1),(2),(3),(4) will be valid for (rnk+1)subscriptsuperscript𝑟𝑘1𝑛(r^{k+1}_{n})( italic_r start_POSTSUPERSCRIPT italic_k + 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ). <|MaskedSetence|> <|MaskedSetence|> (3) holds by definition. <|MaskedSetence|> | **A**: The first step was to include ak+1subscript𝑎𝑘1a_{k+1}italic_a start_POSTSUBSCRIPT italic_k + 1 end_POSTSUBSCRIPT hence (4) is valid too.
.
**B**: (2) is valid because we keep all previous ni(1≤i≤k)subscript𝑛𝑖1𝑖𝑘n_{i}\ (1\leq i\leq k)italic_n start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( 1 ≤ italic_i ≤ italic_k ) and by (3) for (rnk)subscriptsuperscript𝑟𝑘𝑛(r^{k}_{n})( italic_r start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ).
**C**: (0) and (1) are obvious by definition.
| ABC | CBA | CBA | CBA | Selection 3 |
In [7], coarse proximity spaces were introduced to axiomatize the “at infinity” perspective of coarse geometry, providing general definitions of coarse neighborhoods (whose metric space specific definition was given by Dranishnikov in [3]), asymptotic disjointness, and closeness “at infinity.” Coarse proximity structures lie between metric spaces and coarse spaces (as defined by Roe in [12]) in a way similar to how proximity spaces relate to metric spaces and uniform spaces (see [8]). <|MaskedSetence|> In this paper, we generalize the notion of asymptotic inductive dimension to all coarse proximity spaces (whose definition agrees with Dranishnokov’s definition for proper metric spaces) and investigate both of Dranishnikov’s questions in this more general context. <|MaskedSetence|> <|MaskedSetence|> We also show that the answer to Dranishnikov’s first question (“Does the asymptotic inductive dimension of the proper metric space coincide with the covering dimension of its Higson corona?”) generalized to this broader context is negative. In section 4, we describe two classes of completely traceable coarse proximity spaces in which the answer to the second of Dranishnokov’s questions (“Does the asymptotic inductive dimension of a proper metric space coincide with the large inductive dimension of its Higson corona?”) generalized to this broader context is positive. Specifically, these are locally compact Hausdorff spaces that admit metrizable compactification and spaces admitting compactifications whose boundaries are Z𝑍Zitalic_Z-sets. These classes include well-known boundaries such as the Gromov and visual boundaries, as well as the boundaries of the “coarse-compactification,” described in [5].
2. Preliminaries. | **A**: In section 3, we define the asymptotic inductive dimension of coarse proximity spaces and show that it is an invariant within the category of coarse proximity spaces.
**B**: In [9], the authors construct a functor from the category of coarse proximity spaces to the category of compact Hausdorff spaces that assigns to each coarse proximity space a certain “boundary space.” This functor provides a common language for speaking of boundary spaces such as the Higson corona, the Gromov boundary, and other well-known boundary spaces.
**C**: In section 2, we review the necessary background information surrounding proximities as well as coarse proximities and their boundaries.
| BCA | BCA | BCA | BCA | Selection 2 |
point in the domain (i.e.,formulae-sequence𝑖𝑒i.e.,italic_i . <|MaskedSetence|> <|MaskedSetence|> <|MaskedSetence|> italic_e . , Σn=1Nnϕn(𝐫)=Σn=1Nnψn(𝐫)=1subscript𝑁𝑛𝑛1Σsubscriptitalic-ϕ𝑛𝐫subscript𝑁𝑛𝑛1Σsubscript𝜓𝑛𝐫1\overset{N_{n}}{\underset{n=1}{\Sigma}}\phi_{n}(\textbf{$\mathbf{r}$})=%
\overset{N_{n}}{\underset{n=1}{\Sigma}}\psi_{n}(\textbf{$\mathbf{r}$})=1start_OVERACCENT italic_N start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_OVERACCENT start_ARG start_UNDERACCENT italic_n = 1 end_UNDERACCENT start_ARG roman_Σ end_ARG end_ARG italic_ϕ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( bold_r ) = start_OVERACCENT italic_N start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_OVERACCENT start_ARG start_UNDERACCENT italic_n = 1 end_UNDERACCENT start_ARG roman_Σ end_ARG end_ARG italic_ψ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( bold_r ) = 1.
. | **A**: , at non-nodal locations, as well as at
nodal locations), is equal to one.
**B**: This property also hold for the
pyramid side functions, i.e.,formulae-sequence𝑖𝑒i.e.,italic_i .
**C**: italic_e .
| CAB | CAB | ACB | CAB | Selection 1 |
<|MaskedSetence|> <|MaskedSetence|> Let x∈ℳ×∩𝒱1𝑥superscriptℳsubscript𝒱1x\in{\mathscr{M}}^{\times}\cap{\mathscr{V}}_{1}italic_x ∈ script_M start_POSTSUPERSCRIPT × end_POSTSUPERSCRIPT ∩ script_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT. Then xv1v2=1𝑥subscript𝑣1subscript𝑣21xv_{1}v_{2}=1italic_x italic_v start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = 1 for some vi∈𝒱isubscript𝑣𝑖subscript𝒱𝑖v_{i}\in{\mathscr{V}}_{i}italic_v start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∈ script_V start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT. <|MaskedSetence|> Hence, the image of v2subscript𝑣2v_{2}italic_v start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT in Λ2/Λ0≅Λ/Λ1subscriptΛ2subscriptΛ0ΛsubscriptΛ1\Lambda_{2}/\Lambda_{0}\cong\Lambda/\Lambda_{1}roman_Λ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT / roman_Λ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ≅ roman_Λ / roman_Λ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT is 1111, and v2∈𝒱1∩Λ0=𝒱0subscript𝑣2subscript𝒱1subscriptΛ0subscript𝒱0v_{2}\in{\mathscr{V}}_{1}\cap\Lambda_{0}={\mathscr{V}}_{0}italic_v start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ∈ script_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∩ roman_Λ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = script_V start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT by Lemma 2.1.3. This shows v1v2∈𝒱1subscript𝑣1subscript𝑣2subscript𝒱1v_{1}v_{2}\in{\mathscr{V}}_{1}italic_v start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ∈ script_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT, and hence, x∈𝒱1×𝑥superscriptsubscript𝒱1x\in{\mathscr{V}}_{1}^{\times}italic_x ∈ script_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT × end_POSTSUPERSCRIPT.
. | **A**: Going to Λ/Λ1ΛsubscriptΛ1\Lambda/\Lambda_{1}roman_Λ / roman_Λ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT, we see that the image of v2subscript𝑣2v_{2}italic_v start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT in Λ/Λ1ΛsubscriptΛ1\Lambda/\Lambda_{1}roman_Λ / roman_Λ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT is 1111.
**B**: Claim 2.
**C**: ℳℳ{\mathscr{M}}script_M dominates 𝒱isubscript𝒱𝑖{\mathscr{V}}_{i}script_V start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT for i=1,2𝑖12i=1,2italic_i = 1 , 2.
Proof of Claim 2: We consider the case of 𝒱1subscript𝒱1{\mathscr{V}}_{1}script_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT.
| BAC | BCA | BCA | BCA | Selection 4 |
<|MaskedSetence|> <|MaskedSetence|> Such refinement of nefness was also given in a recent work [T22] by a different method. <|MaskedSetence|> We note that the approach of the current paper is not ‘similar’, as opposed to a mention in [T22]. . | **A**:
Remark.
**B**: Theorem 1.1 in particular recovers nefness of J𝐽Jitalic_J in [Ka98] by Proposition 2.1.
**C**: 222We mention that the current paper first appeared on arXiv and [T22] was received by a journal, both in October 2019, whereas a manuscript of the current paper was brought to the attention of the author of [T22] in March 2019.
| BAC | ABC | ABC | ABC | Selection 4 |
(2)⇒(3)⇒23(2)\Rightarrow(3)( 2 ) ⇒ ( 3 ). Fix M𝑀Mitalic_M and b𝑏bitalic_b as in (3). <|MaskedSetence|> <|MaskedSetence|> As N𝑁Nitalic_N is prime over Mb𝑀𝑏Mbitalic_M italic_b, we may assume N⪯N∗precedes-or-equals𝑁superscript𝑁N\preceq N^{*}italic_N ⪯ italic_N start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT. <|MaskedSetence|> | **A**: For the final sentence, let N∗superscript𝑁N^{*}italic_N start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT be any model dominated by b𝑏bitalic_b over M𝑀Mitalic_M.
**B**: But, as N∗superscript𝑁N^{*}italic_N start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT is minimal over
Mb𝑀𝑏Mbitalic_M italic_b by Proposition 5.14(2), N=N∗𝑁superscript𝑁N=N^{*}italic_N = italic_N start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT, so N∗superscript𝑁N^{*}italic_N start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT is constructible over Mb𝑀𝑏Mbitalic_M italic_b.
.
**C**: As M𝑀Mitalic_M is countable and the isolated types are dense, it follows from Vaught that a constructible model N𝑁Nitalic_N over Mb𝑀𝑏Mbitalic_M italic_b
exists.
| ACB | CAB | CAB | CAB | Selection 4 |
Outside of the finite type settings, the singularities of finite-dimensional Schubert varieties in the affine Grassmannian, in type A𝐴Aitalic_A as well as all types, have been widely studied, see e.g., [MOV05, KL+09, BM10, HR20]. For Schubert varieties in general Kac-Moody flag varieties, a criterion for smoothness is given in [Kum96].
On the other hand, the Gröbner geometry in the affine type settings has not been extensively studied by the literature. <|MaskedSetence|> Their work is related to but different from ours. One distinction is the finite dimensional versus codimensional conditions, which is not essentially different in the finite type case, but significantly different in the affine case. <|MaskedSetence|> <|MaskedSetence|> We thank the anonymous referees for their very careful reading and many helpful suggestions.. | **A**: In the thesis [Bru14], Brunson investigated (set-theoretic) equations for finite-dimensional Schubert conditions in certain matrix coordinates for the affine Grassmannian, extending the work of [KLMW04].
**B**: The authors would like to thank Shiliang Gao, Allen Knutson, Mark Shimozono, and Alex Yong for helpful conversations.
**C**: Our work shows that it is possible to work with finite-codimensional Schubert conditions explicitly via restrictions to finite-dimensional cells.
Acknowledgements.
| ACB | ACB | ABC | ACB | Selection 4 |
Since our identification will use a certain set of (framed) wild harmonic bundles, we remark that this set does not match any of the usual wild moduli spaces ℳHitsubscriptℳHit\mathcal{M}_{\text{Hit}}caligraphic_M start_POSTSUBSCRIPT Hit end_POSTSUBSCRIPT. <|MaskedSetence|> <|MaskedSetence|> On the other hand, in our set of wild harmonic bundles we will allow the simple pole of the Higgs field and the parabolic structure to vary. <|MaskedSetence|> Hence our moduli space must a priori be different from the usual moduli spaces of wild harmonic bundles.
. | **A**: In the usual story of moduli spaces of wild harmonic bundles over a punctured compact Riemann surface, one fixes the singular part of the Higgs field and the parabolic structure at the punctures.
**B**: Furthermore, we will have the additional data of a “framing”.
**C**: Under certain stability conditions, one obtains moduli spaces of these objects, with the natural hyperkähler metric [BB04].
| ACB | CBA | ACB | ACB | Selection 3 |
Broadly speaking, our work is related to a vast body of work on value-based reinforcement learning in tabular (Jaksch et al., 2010; Osband et al., 2014; Osband and Van Roy, 2016; Azar et al., 2017; Dann et al., 2017; Strehl et al., 2006; Jin et al., 2018) and linear settings (Yang and Wang, 2019b, a; Jin et al., 2019; Ayoub et al., 2020; Zhou et al., 2020), as well as nonlinear settings involving general function approximators (Wen and Van Roy, 2017; Jiang et al., 2017; Du et al., 2019b; Dong et al., 2019). In particular, our setting is the same as the linear setting studied by Ayoub et al. (2020); Zhou et al. (2020), which generalizes the one proposed by Yang and Wang (2019a). We remark that our setting differs from the linear setting studied by Yang and Wang (2019b); Jin et al. (2019). It can be shown that the two settings are incomparable in the sense that one does not imply the other (Zhou et al., 2020). <|MaskedSetence|> <|MaskedSetence|> (2019). <|MaskedSetence|> In particular, compared with the work of Yang and Wang (2019b, a); Jin et al. (2019); Ayoub et al. (2020); Zhou et al. (2020), which focuses on value-based reinforcement learning, OPPO attains the same T𝑇\sqrt{T}square-root start_ARG italic_T end_ARG-regret even in the presence of adversarially chosen reward functions. Compared with optimism-led iterative value-function elimination (OLIVE) (Jiang et al., 2017; Dong et al., 2019), which handles the more general low-Bellman-rank setting but is only sample-efficient, OPPO simultaneously attains computational efficiency and sample efficiency in the linear setting. Despite the differences between policy-based and value-based reinforcement learning, our work shows that the general principle of “optimism in the face of uncertainty” (Auer et al., 2002; Bubeck and Cesa-Bianchi, 2012) can be carried over from existing algorithms based on value iteration, e.g., optimistic LSVI, into policy optimization algorithms, e.g., NPG, TRPO, and PPO, to make them sample-efficient, which further leads to a new general principle of “conservative optimism in the face of uncertainty and adversary” that additionally allows adversarially chosen reward functions.
. | **A**: In comparison, we focus on policy-based reinforcement learning, which is significantly less studied in theory.
**B**: Also, our setting is related to the low-Bellman-rank setting studied by Jiang et al.
**C**: (2017); Dong et al.
| ABC | BCA | BCA | BCA | Selection 3 |
The conjecture has only been proven for various special cases in very specific settings. <|MaskedSetence|> <|MaskedSetence|> In this paper, we study the Ehrhart volume conjecture. <|MaskedSetence|> The main idea that goes into the disprove pertains to a certain construction of a ball in ℝnsuperscriptℝ𝑛\mathbb{R}^{n}blackboard_R start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT and the realization that after some little tweak of the internal structure, the ball satisfies the requirements of the conjecture but has too much volume, at least a volume beyond that postulated by Ehrhart. In particular, we prove the following lower bound
. | **A**: For instance, Ehrhart proved the conjecture in the two dimensional case and for simplices [11].
**B**: We show that the claimed inequality fails for some convex bodies, providing a counter example to the Ehrhart volume conjecture.
**C**: The conjecture has also been settled for a large class of rational polytopes [10].
| ACB | ABC | ACB | ACB | Selection 4 |
Plan of the paper
In §2, we discuss mirror symmetry for Fano varieties and give the definitions needed to state it precisely and recall basic definitions and constructions for quiver flag varieties. In §3, we introduce the class of Y𝑌Yitalic_Y-shaped quiver flag varieties, and describe a SAGBI basis for sections of certain line bundles on these quiver flag varieties, which produces a toric degeneration. <|MaskedSetence|> In the next section, §5, we introduce a generalization of the Przyjalkowski method and explain how to produce a Laurent polynomial from the data of the weights and divisors associated to a toric complete intersection. In the final section of the paper §6, we explain how to use this degeneration to find candidate Laurent polynomial mirrors for quiver flag zero loci with a convex partition. <|MaskedSetence|> In Appendix A, we discuss a connection to the Plücker coordinate mirror. <|MaskedSetence|> | **A**: Finally, we give one example of a degeneration and mirror outside of the family of Y𝑌Yitalic_Y-shaped quiver flag varieties: here, the degeneration is to a bound ladder quiver.
**B**: In §4, we define ladder diagrams and ladder quivers for Y𝑌Yitalic_Y-shaped quiver flag varieties, and prove that the SAGBI degeneration is precisely a degeneration to singular toric variety arising the from the ladder quiver.
**C**: In Appendix B, we give a table with the mirrors to 99 of the quiver flag zero loci found in [18], produced using the method outlined in the paper..
| BAC | BAC | BAC | BAC | Selection 2 |
For the curvature equations in classical geometry, the existence of hypersurfaces with prescribed Weingarten curvature was studied by Pogorelov [40], Caffarelli-Nirenberg-Spruck [4, 5], Guan-Guan [18], Guan-Ma [19] and the later work by Sheng-Trudinger-Wang [42]. The Hessian equation on Riemannian manifolds was also studied by Y.Y. Li [29], Urbas [51] and Guan [17]. Hessian type equations also appear in conformal geometry, which started from Viaclovsky [53], Chang-Gursky-Yang [6]. In Kähler geometry, the Hessian equation was studied by Hou-Ma-Wu [20] and Dinew-Kolodziej [12].
Meanwhile, the Neumann and oblique derivative problem of partial differential equations were widely studied. <|MaskedSetence|> <|MaskedSetence|> Especially for the mean curvature equation with prescribed contact angle boundary value problem, Ural’tseva [52], Simon-Spruck [43] and Gerhardt [16] got the boundary gradient estimates and the corresponding existence theorem. Recently in [39], the second author and J.J. <|MaskedSetence|> | **A**: For a priori estimates and the existence theorem of Laplace equation with Neumann boundary condition, we refer to the book [15].
**B**: Also, we recommend the recent book written by Lieberman [33] for the Neumann and the oblique derivative problems of linear and quasilinear elliptic equations.
**C**: Xu got the boundary gradient estimates and the corresponding existence theorem for the Neumann boundary value problem on mean curvature equation.
.
| ABC | ABC | BAC | ABC | Selection 4 |
2.2 Injective (Hyperconvex) metric spaces
A hyperconvex metric space is one where any collection of balls with non-empty pairwise intersections forces the non-empty intersection of all balls. <|MaskedSetence|> Isbell [52] proved that every metric space admits a smallest hyperconvex hull (cf. <|MaskedSetence|> <|MaskedSetence|> More recently, in [53] Joharinad and Jost considered relaxations of hyperconvexity and related it to a certain notion of curvature applicable to general metric spaces.. | **A**: These were studied by Aronszajn and Panitchpakdi [8] who showed that every hyperconvex space is an absolute 1-Lipschitz
retract.
**B**: the definition of tight span below).
**C**: Dress rediscovered this concept in [31] and subsequent work provided much development in the context of phylogenetics [77, 32].
| ABC | ACB | ABC | ABC | Selection 1 |
The subtlety which is often glossed over in the context of classical differential geometry, where all the mappings are assumed
smooth, or at least twice or thrice continuously differentiable, is that, due to nonlinearity, the passage from say a given set of first order equations, e.g. <|MaskedSetence|> <|MaskedSetence|> The above mentioned subtlety could be a source of confusion. Coming back to our example, it is often assumed that all isometric images of flat domains must be ruled, which is untrue. Already, at the dawn of modern differential geometry, no less figures than Lebesgue and Picard intuited the existence of a surface which could be flattened onto the plane with no distortion of relative distances, but nowhere contained a straight segment. <|MaskedSetence|> | **A**: the system of isometric immersion equations in our example, to the higher order equations requires a minimum of regularity.
**B**: There is no guarantee that in the absence of this regularity the geometric information hidden in the higher order equations would be accessible.
**C**: To quote Picard111[53, Page 555], translated and quoted in [7].:
.
| ABC | BAC | ABC | ABC | Selection 3 |
<|MaskedSetence|> Take V⊂Xm𝑉superscript𝑋𝑚V\subset X^{m}italic_V ⊂ italic_X start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT an open subset. Then we can write
V=U1×⋯×Um𝑉subscript𝑈1⋯subscript𝑈𝑚V=U_{1}\times\dots\times U_{m}italic_V = italic_U start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT × ⋯ × italic_U start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT for some Uj⊂Xsubscript𝑈𝑗𝑋U_{j}\subset Xitalic_U start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ⊂ italic_X, j=1,…,m𝑗1…𝑚j=1,\dots,mitalic_j = 1 , … , italic_m, open subsets. <|MaskedSetence|> <|MaskedSetence|> Moreover, it is not difficult to prove that
φσ(V)=Vσsubscript𝜑𝜎𝑉subscript𝑉𝜎\varphi_{\sigma}(V)=V_{\sigma}italic_φ start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( italic_V ) = italic_V start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT, then φσsubscript𝜑𝜎\varphi_{\sigma}italic_φ start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT is an open map.. | **A**: 3 φσsubscript𝜑𝜎\varphi_{\sigma}italic_φ start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT is an open map.
**B**: Take
Vσ=Uσ(1)×⋯×Uσ(m)subscript𝑉𝜎subscript𝑈𝜎1⋯subscript𝑈𝜎𝑚V_{\sigma}=U_{\sigma(1)}\times\dots\times U_{\sigma(m)}italic_V start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT = italic_U start_POSTSUBSCRIPT italic_σ ( 1 ) end_POSTSUBSCRIPT × ⋯ × italic_U start_POSTSUBSCRIPT italic_σ ( italic_m ) end_POSTSUBSCRIPT.
**C**: Vσsubscript𝑉𝜎V_{\sigma}italic_V start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT is open
because so they are Ujsubscript𝑈𝑗U_{j}italic_U start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT, j=1,…,m𝑗1…𝑚j=1,\dots,mitalic_j = 1 , … , italic_m.
| ABC | ABC | ABC | CBA | Selection 3 |
<|MaskedSetence|> only zero in this case) for a single choice of
parameters. In particular, in the case s1=12subscript𝑠112s_{1}=\tfrac{1}{2}italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = divide start_ARG 1 end_ARG start_ARG 2 end_ARG, the requirement −4V22−(1+2V3)2≥04superscriptsubscript𝑉22superscript12subscript𝑉320-4V_{2}^{2}-(1+2V_{3})^{2}\geq 0- 4 italic_V start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - ( 1 + 2 italic_V start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≥ 0 yields V2=0,V3=−1/2formulae-sequencesubscript𝑉20subscript𝑉312V_{2}=0,V_{3}=-1/2italic_V start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = 0 , italic_V start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT = - 1 / 2. Alternatively, in the case s2=−12subscript𝑠212s_{2}=-\tfrac{1}{2}italic_s start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = - divide start_ARG 1 end_ARG start_ARG 2 end_ARG from
equation −4V22−(1−2V3)≥04superscriptsubscript𝑉2212subscript𝑉30-4V_{2}^{2}-(1-2V_{3})\geq 0- 4 italic_V start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - ( 1 - 2 italic_V start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) ≥ 0 follows V2=0,V3=1/2formulae-sequencesubscript𝑉20subscript𝑉312V_{2}=0,V_{3}=1/2italic_V start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = 0 , italic_V start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT = 1 / 2. <|MaskedSetence|> <|MaskedSetence|> | **A**: Therefore, in this
algebra in fact there exist only isolated real square root of
𝖡=−𝐞3+𝐞12𝖡subscript𝐞3subscript𝐞12\mathsf{B}=-\mathbf{e}_{3}+\mathbf{e}_{12}sansserif_B = - bold_e start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + bold_e start_POSTSUBSCRIPT 12 end_POSTSUBSCRIPT..
**B**: made non-negative (i.e.
**C**: The both cases yield an isolated root ±12(1−𝐞3+𝐞12+𝐞123)plus-or-minus121subscript𝐞3subscript𝐞12subscript𝐞123\pm\frac{1}{2}(1-\mathbf{e}_{3}+\mathbf{e}_{12}+\mathbf{e}_{123})± divide start_ARG 1 end_ARG start_ARG 2 end_ARG ( 1 - bold_e start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + bold_e start_POSTSUBSCRIPT 12 end_POSTSUBSCRIPT + bold_e start_POSTSUBSCRIPT 123 end_POSTSUBSCRIPT ).
| BCA | ABC | BCA | BCA | Selection 4 |
Two major tools for studying stochastic controlled systems are Pontryagin’s maximum principle and Bellman’s dynamic programming. <|MaskedSetence|> Indeed, except for a very few specific cases, the determination of an optimal control (either exact or near) is a highly nontrivial problem to tackle.
In the Markovian case, a classical approach in solving stochastic control problems is given by the dynamic programming principle based on Hamilton-Jacobi-Bellman (HJB) equations. One popular approach is to employ verification arguments to check if a given solution of the HJB equation coincides with the value function at hand, and obtain as a byproduct the optimal control. <|MaskedSetence|> In this direction, several techniques based on Markov chain discretization schemes [33], Krylov’s regularization and shaking coefficient techniques (see e.g [31, 32]) and Barles-Souganidis-type monotone schemes [2] have been successfully implemented. <|MaskedSetence|> | **A**: While these two methods are known to be very efficient for establishing some key properties (e.g existence of optimal controls, smoothness of the value functional, sufficiency of subclasses of controls, etc), the problem of solving explicitly or numerically a given stochastic control problem remains a critical issue in the field of control theory.
**B**: Discretization methods also play an important role towards the resolution of the control problem.
**C**: We also refer the more recent probabilistic techniques on fully non-linear PDEs given by [20] and the randomization approach of [28, 29, 30]..
| ABC | BCA | ABC | ABC | Selection 1 |
There is a quite interesting evolution of constructions to present free groups in a self-similar way or even as automaton groups (see [15] for an overview). This culminated in constructions to present free groups of arbitrary rank as automaton groups where the number of states coincides with the rank [18, 17]. <|MaskedSetence|> While it is known that the free semigroup of rank one is not an automaton semigroup [4, Proposition 4.3], the free semigroups of higher rank can be generated by an automaton [4, Proposition 4.1]. In fact, the construction to generate these semigroups is quite simple [4, Proposition 4.1] (compare also to 3). <|MaskedSetence|> <|MaskedSetence|> On a side note, it is also worthwhile to point out that – although there does not seem to be much research on the topic – there are examples to generate the free inverse semigroup of rank one as a subsemigroup of an automaton semigroup [14, Theorem 25] and an adaption to present the free inverse monoid of rank one as an automaton semigroup [6, Example 2] (see also [8, Example 23]).
. | **A**: While these constructions and the involved proofs are generally deemed quite complicated, the situation for semigroups turns out to be much simpler.
**B**: Here, the main difference is that the free monoid in one generator can indeed be generated by an automaton: it is generated by the adding machine (see 1), which also generates the free group of rank one if inverses are added.
**C**: The same construction can also be used to generate free monoids as automaton semigroups or monoids.
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<|MaskedSetence|> In Appendix B we recall Knerr’s non-standard parabolic Schauder estimates. In Appendix C we prove that mean curvature flows with bounded curvature and controlled area ratios are unique in the class of Brakke flows. We prove Ilmanen’s localized avoidance principle in Appendix D. <|MaskedSetence|> In Appendix F we study weak set flows coming out of cones. We show that Brakke flows with sufficiently small singular set have connected regular part in Appendix G. <|MaskedSetence|> | **A**: Finally, in Appendix H we localize certain topological monotonicity results..
**B**: Appendix E recalls the non-compact Ecker-Huisken maximum principle.
**C**:
We apply this construction to the study of the mean curvature flow of generic low entropy hypersurfaces in Section 10 and to the study of the first non-generic time of the mean curvature flow of a generic surface in ℝ3superscriptℝ3\mathbb{R}^{3}blackboard_R start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT in Section 11.
In Appendix A we improve some decay estimates for asymptotically conical ends of shrinkers.
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<|MaskedSetence|> In [GiuntiHoefer19], the authors considered zero particle veolcities. <|MaskedSetence|> <|MaskedSetence|> This allows for many clusters of overlapping particles. A corresponding result for the Poisson equation has been obtained in [GiuntiHoferVelazquez18].
. | **A**: with only a (1+β)1𝛽(1+\beta)( 1 + italic_β ) moment bound.
**B**: The particle positions can be distributed to rather general stationary processes, and the radii are i.i.d.
**C**: For a more complete list and discussion of this literature, we refer the reader to [GiuntiHoferVelazquez18, GiuntiHoefer19].
In [GiuntiHoefer19, CarrapatosoHillairet20], the Brinkman equations have been derived under very mild assumptions on the particle configurations.
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<|MaskedSetence|> Yau for the constant support and encouragement. <|MaskedSetence|> <|MaskedSetence|> The calculations in Example 7.5 is done together with P. Bousseau at the time.
. | **A**: The author particularly want to thank Pierrick Bousseau for the discussions back in 2016 at MSRI.
**B**: The author would like to thank Man-Wai Cheung, Paul Hacking, Siu-Cheong Lau, Tsung-Ju Lee, Cheuk-Yu Mak for helpful discussions.
**C**: Acknowledgment
The author would like to thank S.T.
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<|MaskedSetence|> In Section 3, we review Devoto’s equivariant elliptic cohomology. In Section 4, we recall the definition of twisted equivariant elliptic cohomology. In Section 5, we construct twisted quasi-elliptic cohomology. <|MaskedSetence|> <|MaskedSetence|> | **A**: In Section 5.3, we define a model of twisted loop space, with which we can construct twisted quasi-elliptic cohomology.
**B**: The theories are computed by applying the properties of quasi-elliptic cohomology theories and equivariant K-theories, especially the conclusions in Appendix A, which are corollaries of the decomposition formula in [ÁGU17, Theorem 3.6 and Corollary 3.7].
The interpretation of the results from the perspective of mathematical physics is not included in this paper.
In Section 2 we give a sketch of quasi-elliptic cohomology, including its definition, basic properties, and the loop space construction.
**C**: In Section 5.4.2, based on the Chern character of quasi-elliptic cohomology,.
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<|MaskedSetence|> It actually implies DF⟨1,1⟩=∑F˙2subscript𝐷𝐹11superscript˙𝐹2D_{F}\langle 1,1\rangle=\sum\dot{F}^{2}italic_D start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT ⟨ 1 , 1 ⟩ = ∑ over˙ start_ARG italic_F end_ARG start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT [24, Corollary 6.5 (2), p. <|MaskedSetence|> Indeed, if −1∈R(F)=∑F˙21𝑅𝐹superscript˙𝐹2-1\in R(F)=\sum\dot{F}^{2}- 1 ∈ italic_R ( italic_F ) = ∑ over˙ start_ARG italic_F end_ARG start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT one has DF⟨1,1⟩=F˙subscript𝐷𝐹11˙𝐹D_{F}\langle 1,1\rangle=\dot{F}italic_D start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT ⟨ 1 , 1 ⟩ = over˙ start_ARG italic_F end_ARG, by definition. <|MaskedSetence|> As examples, one has pseudo-real closed fields and formally real generalized Hilbert fields [17].
. | **A**: For a formally real quasi-Pythagorean field F𝐹Fitalic_F, Ware [32, Corollary 1] proved that GF(2)subscript𝐺𝐹2G_{F}(2)italic_G start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT ( 2 ) is the free pro-2222 product of a free pro-2222 group and a pro-2222 group generated by involutions, provided conditions that hold for (F˙:F˙2):˙𝐹superscript˙𝐹2(\dot{F}:\dot{F}^{2})( over˙ start_ARG italic_F end_ARG : over˙ start_ARG italic_F end_ARG start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) finite.
**B**: 452].
Unless that R(F)=F˙𝑅𝐹˙𝐹R(F)=\dot{F}italic_R ( italic_F ) = over˙ start_ARG italic_F end_ARG, F𝐹Fitalic_F is formally real.
**C**:
A field F𝐹Fitalic_F is called quasi-Pythagorean if R(F)=DF⟨1,1⟩𝑅𝐹subscript𝐷𝐹11R(F)=D_{F}\langle 1,1\rangleitalic_R ( italic_F ) = italic_D start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT ⟨ 1 , 1 ⟩.
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To address such an issue of divergence, nonlinear gradient TD (Bhatnagar et al., 2009) explicitly linearizes the value function approximator locally at each iteration, that is, using its gradient with respect to the parameter as an evolving feature representation. Although nonlinear gradient TD converges, it is unclear whether the attained solution is globally optimal. On the other hand, when the value function approximator in TD is an overparameterized multi-layer neural network, which is required to be properly scaled, such a feature representation stabilizes at the initial one (Cai et al., 2019), making the explicit local linearization in nonlinear gradient TD unnecessary. Moreover, the implicit local linearization enabled by overparameterization allows TD (and Q-learning) to converge to the globally optimal solution. However, such a required scaling, also known as the neural tangent kernel (NTK) regime (Jacot et al., 2018), effectively constrains the evolution of the induced feature presentation to an infinitesimal neighborhood of the initial one, which is not data-dependent.
Contribution. <|MaskedSetence|> <|MaskedSetence|> <|MaskedSetence|> | **A**: Moreover, in contrast to the NTK regime, the induced feature representation is able to deviate from the initial one and subsequently evolve into the globally optimal one, which corresponds to the global minimizer of the MSPBE.
**B**: Going beyond the NTK regime, we prove that, when the value function approximator is an overparameterized two-layer neural network, TD and Q-learning globally minimize the mean-squared projected Bellman error (MSPBE) at a sublinear rate.
**C**: We further extend our analysis to soft Q-learning, which is connected to policy gradient.
.
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1.7. Acknowledgements
During the writing of the paper, I was supported by the starter grant “Categorified Donaldson–Thomas theory” No. <|MaskedSetence|> <|MaskedSetence|> <|MaskedSetence|> Finally, I offer my heartfelt gratitude to Paul, Sophia, Sacha, Kristin and Nina, for their help and support throughout the writing of this paper.
. | **A**: I would like to thank Andrei Okounkov and Olivier Schiffmann for helpful conversations, Tristan Bozec for patiently explaining his work on crystals to me, Lucien Hennecart and Shivang Jindal for helpful comments regarding an earlier version of the paper, and an anonymous referee for a careful reading of the paper and many helpful suggestions.
**B**: I was also supported by a Royal Society university research fellowship.
**C**: 759967 of the European Research Council.
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furthermore B→C→𝐵𝐶B\to Citalic_B → italic_C. Apply [33, Corollary
5.14] to A𝐴Aitalic_A and B𝐵Bitalic_B. <|MaskedSetence|> <|MaskedSetence|> <|MaskedSetence|> This shows that φ𝜑\varphiitalic_φ is closed. | **A**: Finally, B~→B→C→~𝐵𝐵→𝐶\widetilde{B}\to B\to Cover~ start_ARG italic_B end_ARG → italic_B → italic_C, thus C⊧φmodels𝐶𝜑C\models\varphiitalic_C ⊧ italic_φ because φ𝜑\varphiitalic_φ is
closed under homomorphisms.
**B**: Then A~⊧φmodels~𝐴𝜑\widetilde{A}\models\varphiover~ start_ARG italic_A end_ARG ⊧ italic_φ because A→A~→𝐴~𝐴A\to\widetilde{A}italic_A → over~ start_ARG italic_A end_ARG and
φ𝜑\varphiitalic_φ is closed under homomorphisms.
**C**: Therefore B~⊧φmodels~𝐵𝜑\widetilde{B}\models\varphiover~ start_ARG italic_B end_ARG ⊧ italic_φ because A~~𝐴\widetilde{A}over~ start_ARG italic_A end_ARG and B~~𝐵\widetilde{B}over~ start_ARG italic_B end_ARG are
n𝑛nitalic_n-elementary equivalent.
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<|MaskedSetence|> <|MaskedSetence|> <|MaskedSetence|> For instance, employing Busemann functions on CAT(−1)1(-1)( - 1 ) spaces, Foertsch and Radke [10] characterized complete CAT(κ𝜅\kappaitalic_κ) spaces with κ<0𝜅0\kappa<0italic_κ < 0, with geodesic Hamenstädt boundary up to isometry. Moreover, Foertsch and Schroeder [12] investigated the relationship between Gromov hyperbolic spaces, CAT(-1) spaces and the Ptolemy inequality on their Gromov boundaries.
One of them is called a parabolic visual metric based on the vertical geodesic in some negatively curved solvable Lie groups in [24]. In [24, 26], Shanmugalingam and Xie proved that all self quasi-isometries of these groups are almost isometries. It should be noted that this parabolic visual metric was formerly named Euclid–Cygan metric by Hersonsky and Paulin [20] in the study of the rigidity of discrete isometry groups of negatively curved spaces. With the aid of this notion, Dymarz [8, 9] recently studied the quasi-isometric rigidity of mixed type locally compact amenable hyperbolic groups.. | **A**: The class of Hamenstädt metrics are defined by using Busemann functions, for related definitions and properties see [7, Section 3.3].
**B**: Roughly speaking, a Busemann function on a Gromov hyperbolic space is defined to be the distance function from a point on
the Gromov boundary.
**C**: This notion is very useful in many areas.
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<|MaskedSetence|> Within this generalised framework, the existence of multidimensional scale functions, known as ‘scale matrices’, was first discussed in [19] and were used to derive fluctuation identities and first passage results for continuous-time MAPs. <|MaskedSetence|> Further studies on MAPs and their exit/passage times can be found in [8], [4], [9], among others. <|MaskedSetence|> It is worth noting here that the authors in this work do discuss some of the corresponding results for the fully-discrete (time and space) MAP model considered in this paper, however, only a limited number of results are stated and a variety of important steps and proofs were omitted.
. | **A**: More recently, [17], derive and compare results for continuous-time MAPs with lattice (discrete-space) and non-lattice support.
**B**:
A natural generalisation of the above processes are the broad family of Markov Additive Processes (MAPs), which incorporate an externally influencing Markov environment, providing greater flexibility to the characteristics of the underlying process in terms of its claim frequency and severity distributions, see [1] (Chapter XI).
**C**: [15] extended the initial findings of [19] by providing the probabilistic construction of the scale matrices, identifying their transforms and considering an extensive study of exit problems including one-sided and two-sided exits, as well as exits for reflected processes via implementation of the occupation density formula.
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<|MaskedSetence|> has also stimulated substantial mathematical analysis of competition models involving two species. We mention the work of [30, 34, 42, 45] for passive dispersal, and [6, 9, 17, 16, 38, 39] for conditional dispersal. An interesting application concerns the evolution of dispersal in stream populations, which are subject to a uni-directional drift [51, 56]. It has been shown that in some circumstances, faster dispersal is sometimes selected for [46, 49]. See also [27, 40, 48]. We also mention the work [36] on the evolution of dispersal in phytoplankton populations, where individuals compete non-locally for sunlight.
Most of the existing results are restricted to the case when the number of species is equal to two. In this case, the theory of monotone dynamical systems [35, 41, 58] can be applied to determine the global dynamics of the competition system. Results for three or more competing species are relatively rare [8, 15, 22, 23, 25, 47, 52],
and the question of global dynamics remains an open and challenging problem. <|MaskedSetence|> <|MaskedSetence|> | **A**: In the following, we will address two conjectures of Dockery et al.
**B**: concerning a model involving N𝑁Nitalic_N competing species, which are identical except for the passive dispersal rates..
**C**: The work of Hastings and Dockery et al.
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<|MaskedSetence|> <|MaskedSetence|> Diagrammatically defined chains of algebras appear to have not been considered as objects whose representation category can be studied through the lens of representation stability. <|MaskedSetence|> The chain with respect to which one is considering representation stability there is of course still the chain of symmetric groups. Thus, representation stability with respect to a chain of diagrammatically defined algebras is not considered in [1] (Barter, Entova-Aizenbud, Heidersdorf).
. | **A**: It appears that much of the work in representation stability has focussed on algebraic objects which are either close to symmetric groups [5] [14] [8] (Wilson, Putman, Sam, Gunturkun, Snowden ) or are close to Lie groups [14] [17] (Sam, Snowden, Putman).
**B**:
To our knowledge, Temperley-Lieb algebras have not been studied in the representation stability literature, or within the broader context of representation stability and FIFI\operatorname{FI}roman_FI-modules.
**C**: Diagrammatics and representation stability have, however, been uttered in the same breadth, but in a different sense: in [1] (Barter, Entova-Aizenbud, Heidersdorf) the authors produce a functor from the category of FIFI\operatorname{FI}roman_FI-modules modulo finite length FIFI\operatorname{FI}roman_FI-modules to the abelian envelope of the Deligne category.
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<|MaskedSetence|> It states that the analysis of sites ("analysis situs") proposed by Gottfried Wilhelm Leibniz (1646-1716) is required to answer the question whether the seven bridges between the four city districts of Königsberg allow for a walk in which each bridge is passed exactly once. <|MaskedSetence|> A city quarter with an odd numbers of bridges prevents an Eulerian tour, since finally leaving it requires crossing an already used bridge. The worst prospect for an Eulerian tour is offered by the 3-regular Cayley graph (Fig.1b) in which each node has an odd number of three arrows, one incoming parent arrow and two outgoing arrows to the upward and leftward child.
The 3-regular Cayley graph can be transformed to a 4-regular graph middle pages graph (Fig.8) that offers each branching number an Eulerian tour, in line with classic 3-to-4-regular transformations. Plato (427-347bC) knew that a 3-regular cube with three edges at each node encapsulates a dual 4-regular octahedron with four edges at each node, and vice versa. René Descartes (1596-1650) knew that packaging an infinite number of cubes with three edges at each node gives a 3-dimensional universe of adjacent cubes. <|MaskedSetence|> | **A**: A tour, or round trip, in which each bridge, or arrow, is passed once and only once has become known as an Eulerian tour.
**B**: The article renowned as the first graph theoretical article [43] is the 1736 article by Leonhard Euler (1707-1783) on a walk over the seven bridges of Königsberg [45].
**C**: It can be projected to a 4-regular Cartesian coordinate system of adjacent squares.
.
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<|MaskedSetence|> Fmsubscript𝐹𝑚F_{m}italic_F start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT) is associated to the Chow group (or singular cohomology) (resp. K-theory). <|MaskedSetence|> We assume the equivariant cohomology theory 𝕙Tsubscript𝕙𝑇\mathbb{h}_{T}blackboard_h start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT is Chern-complete over the point for T𝑇Titalic_T, that is, the ring 𝕙T(pt)subscript𝕙𝑇pt\mathbb{h}_{T}(\operatorname{pt})blackboard_h start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT ( roman_pt ) is separated and complete with respect to the topology induced by the γ𝛾\gammaitalic_γ-filtration [5, Definition 2.2]. <|MaskedSetence|> | **A**:
For example, Fasubscript𝐹𝑎F_{a}italic_F start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT (resp.
**B**: In particular, this includes the completed equivariant Chow ring, the completed equivariant K-theory and equivariant algebraic cobordism.
Let S𝑆Sitalic_S be the formal group algebra defined in [4]:.
**C**: Both can be extended to the torus equivariant setting.
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Surprisingly, this is sharper than both our estimate in Corollary 3 and Sarnak and Xue’s estimate (but still weaker than Marshall’s estimate) for the compact case. The improvement results from our sharper injectivity radius estimates in Subsection 10.3. <|MaskedSetence|> <|MaskedSetence|> <|MaskedSetence|> The problem of estimating the number of cusps in terms of the volume on hyperbolic manifolds with cusps is a well-studied problem in geometric topology; see for example [Kel98], [Par98], [DD15], [BT18] among many other references. Nonetheless, here we need a different point of view on this problem: we consider the asymptotic behavior of the volume normalized number of cusps along a cofinal tower, and this point of view seems to be new. We refer to Section 11 for the background and details.
. | **A**: In the real hyperbolic case, our estimates extend Yeung’s estimates for cocompact lattices [Yeu94, Theorem 2.4.1] to noncocompact lattices.
Finally, we study the de Rham cohomology of complete finite volume hyperbolic manifolds with cusps along a cofinal tower.
**B**: [Zuc82]), and on an estimate of the number of cusps along such towers.
**C**: This relies on the topological interpretation of L2superscript𝐿2L^{2}italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT-cohomology of locally symmetric varieties (cf.
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We shall present in §3 a proof of Theorem 0.9, which is essentially a recast of the arguments in [9, §5]. <|MaskedSetence|> <|MaskedSetence|> Our intention here is to elucidate some issues on local vs. global isometric immersions in the literature. <|MaskedSetence|> | **A**: Note also that Theorem 0.9 (2) is a new result elusive in the existing literature.
.
**B**: Mardare [29], as well as applying Theorem 0.1 proved earlier in this note.
**C**: We utilise ideas from Tenenblat [44] and S.
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However, if a positive IRE is not a IET, i.e., its scheme contains at least one twisted cycle, then some of the beginning intervals in that cycle necessarily overlap, as well as do some of the ending intervals in this cycle (every point of the real axis is covered with the same number of beginning and ending intervals belonging to the same cycle). <|MaskedSetence|> The most natural approach here is to associate an IRE with a family of dynamical systems, which are effectively
IETs on trees. <|MaskedSetence|> <|MaskedSetence|> The resulting phase space will be a tree with disjoint components corresponding to the cycles in the IRE (some authors use to call a tree with disjoint components a “forest”, but we abstain). The tree obtained will be the union of all beginning intervals, as well as the union of all ending intervals, branching at the special points. The dynamical system on this tree will be determined by an “almost bijective” discontinuous map that shifts every beginning interval onto the corresponding ending interval, but the branching points have several (two, in a generic case) images and preimages.
. | **A**: The overlapping intervals will be joined at these points and disjoint beyond them.
**B**: In order to obtain a tree from every particular cycle of a positive IRE, one has to fix a set of special points numbered by the twist number of this cycle.
**C**: This configuration does not determine a dynamical system uniquely.
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<|MaskedSetence|> <|MaskedSetence|> We conclude the paper giving the natural generalisation of the Lie model Theorem for rough approximate subgroups. Applications of this result to the case of metric groups will be studied in a future paper.
We study in general piecewise hyperdefinable sets in Section 1, focussing on the properties of their logic topologies. The most important results of this section are given after the introduction of locally hyperdefinable sets in Section 1.4. Section 2 is the core of this paper and is devoted to the general study of piecewise hyperdefinable groups. The first fundamental result of the section is Theorem 2.22, in which we show that piecewise hyperdefinable groups satisfying a natural combinatorial condition are locally hyperdefinable. In Section 2.3, we define the model\hyptheoretic components for piecewise hyperdefinable groups, proving the existence of Gapsuperscript𝐺apG^{\mathrm{ap}}italic_G start_POSTSUPERSCRIPT roman_ap end_POSTSUPERSCRIPT in Theorem 2.32. Finally, we focus on the study of Lie cores, proving their existence (Theorem 2.34), the uniqueness of the minimal one (Theorem 2.36), giving a canonical representation of the minimal one in terms of the model\hyptheoretic components (Theorem 2.40) and showing its independence of expansions of the language (Corollary 2.41). <|MaskedSetence|> We conclude the paper stating the Rough Lie model Theorem 3.31 which generalises Hrushovski’s Lie Model Theorem to the case of rough approximate subgroups.. | **A**: The aim of this paper is to give the abstract basis for that kind of results, to find in the end possibly interesting applications to combinatorics.
**B**:
Hrushovski already indicated in unpublished works that, using piecewise hyperdefinable groups, it should be possible to extend some of the results of [Hru12] to the context of metric groups.
**C**: Section 3 is devoted to the Stabilizer Theorem for piecewise hyperdefinable groups, which is divided over Theorems 3.23, 3.24 and 3.25 — Theorem 3.25 being the standard statement of the Stabilizer Theorem.
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•
In order to obtain a theory that more closely encapsulates some of the geometric features of finitary affine hyperplane arrangements, in Section 3 we state axioms for Finitary Affine Oriented Matroids (FAOMs). <|MaskedSetence|> A main theoretical feature of this restricted setting is that FAOMs are “orientations of finitary semimatroids”, i.e.: the zero sets of covectors of an FAOM constitute the geometric semilattice of flats of a finitary semimatroid
(e.g., in the sense of [17], generalizing the finite notion developed by Wachs and Walker [33] and by Ardila [3] and Kawahara [24]).
We carry out a basic study of tope graphs and covector posets of FAOMs (§3.1) and then we focus on topological properties. <|MaskedSetence|> <|MaskedSetence|> This allows us to single out a special class of FAOMs whose covector poset is affinely homeomorphic to Euclidean space ℝnsuperscriptℝ𝑛\mathbb{R}^{n}blackboard_R start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT (see Section 4).. | **A**: These are AOMs with some local cardinality restrictions.
**B**: Moreover, we derive some order-theoretic properties of the geometric parallelism relation in FAOMs (§3.6).
**C**: We prove that order complexes of covector posets of FAOMs are shellable (§3.2) and explicitly describe their homeomorphism type (§3.3).
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The first-named author was supported by the DFG project AN 1545 “Equivariant and weak orientations in the motivic homotopy theory”. The first-named and the second-named authors of the article were supported by the SPP 1786 “Homotopy theory and algebraic geometry” (DFG). <|MaskedSetence|> A part of this work was written when the second-named author was in St. <|MaskedSetence|> 075–15–2022–287.
The results of Section 4–5, 7 constitute the major part of the second author’s Ph. <|MaskedSetence|> Thesis [Lav, Chapter I, II], however, the proofs in this paper are simplified and the exposition is clarified.. | **A**: Petersburg University supported by the BASIS foundation grants “Young Russia Mathematics” and the Ministry of Science and Higher Education of the Russian Federation, agreement No.
**B**: D.
**C**: The third-named author was supported by the BASIS foundation grant “Young Russia Mathematics”.
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There are comprehensive survey papers that review the research on consensus protocols [19, 20, 21, 22]. <|MaskedSetence|> There is a large number of papers that propose consensus protocols with switching network topologies and convergence proofs of these algorithms are provided under various assumptions [27, 28, 29, 30, 31, 32].
In [27], a consensus protocol is proposed to solve the alignment problem of mobile agents, where the switching topology is assumed to be periodically connected.
This assumption means that
the union of networks over a finite time interval is strongly connected.
Another algorithm is proposed in [28] that assumes the underlying switching network topology is ultimately connected. <|MaskedSetence|> In [29], previous works are extended to solve the consensus problem on networks under limited and unreliable information exchange with dynamically changing interaction topologies. <|MaskedSetence|> | **A**: In many scenarios, the network topology of the consensus protocol is a switching topology due to failures, formation reconfiguration, or state-dependence.
**B**: This assumption means that the union of graphs over an infinite interval is strongly connected.
**C**: The convergence of the algorithm is provided under the ultimately connected assumption..
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Acknowledgement. <|MaskedSetence|> Walter Gubler for providing us with the main idea and helping us to see the possibility of the generalisation to normal bases over arbitary global fields. We are also indebted to Prof. Joseph Silverman for pointing the connection with [SilCall]. The first author is grateful to Prof. <|MaskedSetence|> <|MaskedSetence|> Yuri Bilu for introducing him to the very recent paper [dimitrov2020uniformity] as his potential master’s thesis and Ananyo Kazi for numerous fruitful discussions and for knowing Hartshorne by heart. The second author thanks Prof. Robin de Jong for the encouragement for the generalization of Silverman-Tate’s height inequality to the dynamical setting and for notifying the relation of Silverman’s specialization theorem and the uniform torsion conjecture, cf. [Holmes].
. | **A**: Qing Liu for answering questions regarding a generalisation.
**B**: The first author would also like to thank Prof.
**C**: We are indebted to Prof.
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<|MaskedSetence|> See, e.g.,
Detommaso et al. <|MaskedSetence|> (2018); Liu et al. (2019); Gong et al. (2019); Wang et al. (2019); Zhang et al. (2020); Ye et al. (2020)
and the references therein.
Departing from MCMC where independent stochastic particles are used, it leverages interacting deterministic particles to approximate the probability measure of interest. <|MaskedSetence|> | **A**: (2018); Han and Liu (2018); Chen et al.
**B**: In addition to gradient-based MCMC, variational transport also shares similarity with Stein variational gradient descent (SVGD) (Liu and Wang, 2016), which is a more recent particle-based algorithm for Bayesian inference.
Variants of SVGD have been subsequently proposed.
**C**: In the mean-field limit where the number of particles go to infinity, it can be viewed as the gradient flow of the KL-divergence with respect to a modified Wasserstein metric (Liu, 2017)..
| BAC | CBA | BAC | BAC | Selection 4 |
<|MaskedSetence|> <|MaskedSetence|> Moreover, in this paper we assume that the parameters of the model are known, but in many practical situations one is given a realization of the graph and the task is estimating the unknown parameters, see [10, 20, 21]. <|MaskedSetence|> This is an interesting open problem.
The outline of the paper is as follows: in Section 2 we define the model rigorously and we state our main result. In Section 3 we give the proof of our main result.. | **A**:
In practice, not all nodes that enter the network have the same degree, and thus it would be interesting to extend our result to the case of a random initial degree distribution.
**B**: If we consider a more general class of preferential attachment graphs, for which a model-free approach is used and therefore the exact distribution of the graph is not known (see for instance [11]), we expect that the techniques presented in this paper could be used to derive central limit theorem for all the degree counts.
**C**: Promising results on this model have been obtained in [7, 8].
| ACB | ACB | ACB | ABC | Selection 3 |
<|MaskedSetence|> This paper is organized as follows. <|MaskedSetence|> <|MaskedSetence|> In Section 4, we present the lower complexity bounds for saddle point problems without individual variables. Finally in Section 5, we show how the proposed algorithm can be applied to the problem computing Wasserstein barycenters .. | **A**:
Paper organization.
**B**: In Section 3, we provide the main algorithm of the paper to solve such kind of problems.
**C**: Section 2 presents a saddle point problem of interest along with its decentralized reformulation.
| ACB | ABC | ACB | ACB | Selection 1 |
<|MaskedSetence|> <|MaskedSetence|> This problem was formulated by Stepanec [7] and Zykov [8] for general graphs and by Hubicka and Syslo [9] in the strictly fundamental class context. In more concrete terms this problem is equivalent to finding the cycle basis with the sparsest cycle matrix. In [5] a unified perspective of the problem is presented. The authors show that the MCB problem is different in nature for each class. For example in [10] a remarkable reduction is constructed to prove that the MCB problem is NP-hard for the strictly fundamental class, while in [11] a polynomial time algorithm is given to solve the problem for the undirected class. <|MaskedSetence|> | **A**:
The length of a cycle is its number of edges.
**B**: The minimum cycle basis (MCB) problem is the problem of finding a cycle basis such that the sum of the lengths (or edge weights) of its cycles is minimum.
**C**: Some applications of the MCB problem are described in [5, 11, 10, 12].
.
| BCA | ABC | ABC | ABC | Selection 4 |
<|MaskedSetence|> When H=1/2𝐻12H=1/2italic_H = 1 / 2, one recovers standard Brownian motion. It is well known that fractional Brownian motion has stationary nonnegatively correlated increments for H≥1/2𝐻12H\geq 1/2italic_H ≥ 1 / 2 - see e.g. [16]. <|MaskedSetence|> <|MaskedSetence|> | **A**:
and variance function is V(t)=|t|2H𝑉𝑡superscript𝑡2𝐻V(t)=|t|^{2H}italic_V ( italic_t ) = | italic_t | start_POSTSUPERSCRIPT 2 italic_H end_POSTSUPERSCRIPT.
**B**: Therefore Proposition 2.3 applies and the optimal measure for BHsubscript𝐵𝐻B_{H}italic_B start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT on the interval [a,b]𝑎𝑏[a,b][ italic_a , italic_b ] is μ∗=δasuperscript𝜇normal-∗subscript𝛿𝑎\mu^{\ast}=\delta_{a}italic_μ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT = italic_δ start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT.
**C**: Therefore the rate function for the optimization problem σ∗2(a,b)superscriptsubscript𝜎normal-∗2𝑎𝑏\sigma_{\ast}^{2}(a,b)italic_σ start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_a , italic_b ) is
.
| ABC | ABC | BAC | ABC | Selection 4 |
∎
It is quite interesting, at this point, to ask whether there exist classes of surface bundles for which there is always virtual excessive homology, and hence virtual algebraic fibrations. <|MaskedSetence|> <|MaskedSetence|> In particular, it would be interesting to decide this case for the class of Kodaira fibrations, or of surface bundles of type III in Johnson’s trichotomy (injective monodromy). <|MaskedSetence|> [BHPV04].). | **A**: (The surface bundles discussed in Theorem 1 cannot be Kodaira fibrations, as these have strictly positive signature, see e.g.
**B**: [BHPV04].
**C**: For instance, this is the case when the fibration is a holomorphic bundle, see e.g.
| CBA | CBA | CBA | BCA | Selection 3 |
<|MaskedSetence|> In particular, the results of this paper are used in work of the author that uses the cohomology of Shimura varieties to deduce new formulas for the cohomology of Rapoport-Zink spaces ([Ber21]) and related work of the author and K.H. <|MaskedSetence|> In §2 we develop the abstract theory of the cohomology of ℰ2(K/F)subscriptℰ2𝐾𝐹\mathcal{E}_{2}(K/F)caligraphic_E start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_K / italic_F ) and ℰ1(K/F)subscriptℰ1𝐾𝐹\mathcal{E}_{1}(K/F)caligraphic_E start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_K / italic_F ), in particular constructing the maps and proving the commutativity of Diagram (1.1). In §3 we discuss the 𝐁(F,G)𝐁𝐹𝐺\mathbf{B}(F,G)bold_B ( italic_F , italic_G )-normalization of transfer factors for (G,H)𝐺𝐻(G,H)( italic_G , italic_H )-regular elements. We remark that to do this, we do not need the full strength of the theory developed in §2 because we need only work with the basic sets 𝐁i(F,G)bassubscript𝐁𝑖subscript𝐹𝐺bas\mathbf{B}_{i}(F,G)_{\mathrm{bas}}bold_B start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_F , italic_G ) start_POSTSUBSCRIPT roman_bas end_POSTSUBSCRIPT. However, the theory from §2 is used in Proposition 3.3, which is then used in Corollary 3.10. <|MaskedSetence|> | **A**:
The normalization of transfer factors for non-strongly regular elements is needed in the analysis of the trace formula for the cohomology of Shimura varieties.
**B**: We also use §2 to prove (before Corollary 3.10) that for a fixed pair (γ𝐇,γ)∈𝐇(F)(𝐆,𝐇)−reg×𝐆(F)superscript𝛾𝐇𝛾𝐇subscript𝐹𝐆𝐇reg𝐆𝐹(\gamma^{\mathbf{H}},\gamma)\in\mathbf{H}(F)_{(\mathbf{G},\mathbf{H})-\mathrm{%.
**C**: Nguyen proving the Kottwitz conjecture on the cohomology of Rapoport-Zink spaces for odd unramified unitary similitude groups ([BN23]).
Finally, we make some remarks about the organization of the paper.
| ACB | ACB | ACB | ACB | Selection 4 |
Proof.
Since 𝔐𝔐\mathfrak{M}fraktur_M is a Breuil-Kisin module over W(κ)𝑊𝜅W(\kappa)italic_W ( italic_κ ), 𝔐𝔐\mathfrak{M}fraktur_M is projective module over W(κ)[[u]]𝑊𝜅delimited-[]delimited-[]𝑢W(\kappa)[\![u]\!]italic_W ( italic_κ ) [ [ italic_u ] ]. <|MaskedSetence|> Further, 𝔐/u𝔐𝔐𝑢𝔐\mathfrak{M}/u\mathfrak{M}fraktur_M / italic_u fraktur_M is projective module over (ℛ⊗ℤpW(κ))[[u]]subscripttensor-productsubscriptℤ𝑝ℛ𝑊𝜅delimited-[]delimited-[]𝑢(\mathscr{R}\otimes_{\mathbb{Z}_{p}}W(\kappa))[\![u]\!]( script_R ⊗ start_POSTSUBSCRIPT blackboard_Z start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_W ( italic_κ ) ) [ [ italic_u ] ] if and only if it is projective over ℛ[[u]]ℛdelimited-[]delimited-[]𝑢\mathscr{R}[\![u]\!]script_R [ [ italic_u ] ]. On the other hand, the ℛ[[u]]ℛdelimited-[]delimited-[]𝑢\mathscr{R}[\![u]\!]script_R [ [ italic_u ] ]-module 𝔐𝔐\mathfrak{M}fraktur_M is finitely generated and projective once it is u𝑢uitalic_u-torsion free, u𝑢uitalic_u-adically complete and separated as well as 𝔐/u𝔐𝔐𝑢𝔐\mathfrak{M}/u\mathfrak{M}fraktur_M / italic_u fraktur_M is a finitely generated and projective ℛℛ\mathscr{R}script_R-module. <|MaskedSetence|> <|MaskedSetence|> | **A**: For the given ℤpsubscriptℤ𝑝\mathbb{Z}_{p}blackboard_Z start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT-algebra ℛℛ\mathscr{R}script_R, the (ℛ⊗ℤpW(κ))[[u]]subscripttensor-productsubscriptℤ𝑝ℛ𝑊𝜅delimited-[]delimited-[]𝑢(\mathscr{R}\otimes_{\mathbb{Z}_{p}}W(\kappa))[\![u]\!]( script_R ⊗ start_POSTSUBSCRIPT blackboard_Z start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_W ( italic_κ ) ) [ [ italic_u ] ]-module 𝔐𝔐\mathfrak{M}fraktur_M will be finitely generated and projective if and only if 𝔐/u𝔐𝔐𝑢𝔐\mathfrak{M}/u\mathfrak{M}fraktur_M / italic_u fraktur_M is finitely generated and projective as an ℛℛ\mathscr{R}script_R-module.
**B**: But it is equivalent to show that 𝔐/ui𝔐,i∈ℕ𝔐superscript𝑢𝑖𝔐𝑖ℕ\mathfrak{M}/u^{i}\mathfrak{M},\ i\in\mathbb{N}fraktur_M / italic_u start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT fraktur_M , italic_i ∈ blackboard_N is a projective ℛ[u]/uiℛdelimited-[]𝑢superscript𝑢𝑖\mathscr{R}[u]/u^{i}script_R [ italic_u ] / italic_u start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT-module.
**C**: For, if 𝔐𝔐\mathfrak{M}fraktur_M is u𝑢uitalic_u-adically complete, u𝑢uitalic_u-adically separated, and 𝔐/ui𝔐𝔐superscript𝑢𝑖𝔐\mathfrak{M}/u^{i}\mathfrak{M}fraktur_M / italic_u start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT fraktur_M is projective module over ℛ[u]/uiℛdelimited-[]𝑢superscript𝑢𝑖\mathscr{R}[u]/u^{i}script_R [ italic_u ] / italic_u start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT for each i𝑖iitalic_i, and if additionally 𝔐/u𝔐𝔐𝑢𝔐\mathfrak{M}/u\mathfrak{M}fraktur_M / italic_u fraktur_M is finitely generated and projective ℛℛ\mathscr{R}script_R-module, then 𝔐𝔐\mathfrak{M}fraktur_M is finitely generated and projective over ℛ[[u]]ℛdelimited-[]delimited-[]𝑢\mathscr{R}[\![u]\!]script_R [ [ italic_u ] ], shown as follows:.
| ABC | ABC | ABC | ABC | Selection 4 |
<|MaskedSetence|> <|MaskedSetence|> <|MaskedSetence|> A symmetric function is a formal power series f(𝒙)∈ℚ[[x1,x2,…]]𝑓𝒙ℚdelimited-[]subscript𝑥1subscript𝑥2…f(\bm{x})\in\mathbb{Q}[[x_{1},x_{2},\dots]]italic_f ( bold_italic_x ) ∈ blackboard_Q [ [ italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , … ] ] such that f(𝒙)𝑓𝒙f(\bm{x})italic_f ( bold_italic_x ) is of bounded-degree and
invariant under permuting the variables. Let ΛΛ\Lambdaroman_Λ be the ℚℚ\mathbb{Q}blackboard_Q-algebra of all symmetric functions in ℚ[[x1,x2,…]]ℚdelimited-[]subscript𝑥1subscript𝑥2…\mathbb{Q}[[x_{1},x_{2},\dots]]blackboard_Q [ [ italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , … ] ]. For k≥0𝑘0k\geq 0italic_k ≥ 0, let ΛksuperscriptΛ𝑘\Lambda^{k}roman_Λ start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT be the subspace of ℚ[[x1,x2,…]]ℚdelimited-[]subscript𝑥1subscript𝑥2…\mathbb{Q}[[x_{1},x_{2},\dots]]blackboard_Q [ [ italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , … ] ]. | **A**: Symmetric functions
Let 𝒙=(x1,x2,…)𝒙subscript𝑥1subscript𝑥2…\bm{x}=(x_{1},x_{2},\dots)bold_italic_x = ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , … ) be a set of formal variables, and 𝒙n=(x1,…,xn)subscript𝒙𝑛subscript𝑥1…subscript𝑥𝑛\bm{x}_{n}=(x_{1},\dots,x_{n})bold_italic_x start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT = ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_x start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ).
**B**: A symmetric polynomial is a polynomial f(𝒙n)∈ℚ[x1,…,xn]𝑓subscript𝒙𝑛ℚsubscript𝑥1…subscript𝑥𝑛f(\bm{x}_{n})\in\mathbb{Q}[x_{1},\dots,x_{n}]italic_f ( bold_italic_x start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) ∈ blackboard_Q [ italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_x start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ] that is invariant under permuting the
variables.
**C**: 2.2.
| CAB | CAB | CBA | CAB | Selection 2 |
<|MaskedSetence|> italic_μ and thus ℱ4(μ)=μsuperscriptℱ4𝜇𝜇\mathcal{F}\hskip 0.5pt^{4}(\mu)=\mucaligraphic_F start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT ( italic_μ ) = italic_μ due
to I2=idsuperscript𝐼2idI^{2}=\mathrm{id}italic_I start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = roman_id. This implies λ4=1superscript𝜆41\lambda^{4}=1italic_λ start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT = 1 as in the case of
functions, so λ𝜆\lambdaitalic_λ must be a fourth root of unity. <|MaskedSetence|> italic_μ = caligraphic_F start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_μ ) = italic_λ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_μ, for any of
the possible eigenvalues. <|MaskedSetence|> | **A**: hence ℱ2(μ)=I.μformulae-sequencesuperscriptℱ2𝜇𝐼𝜇\mathcal{F}\hskip 0.5pt^{2}(\mu)=I.\hskip 0.5pt\mucaligraphic_F start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_μ ) = italic_I .
**B**: This has an immediate consequence as.
**C**: Note that we
also get I.μ=ℱ2(μ)=λ2μformulae-sequence𝐼𝜇superscriptℱ2𝜇superscript𝜆2𝜇I.\hskip 0.5pt\mu=\mathcal{F}\hskip 0.5pt^{2}(\mu)=\lambda^{2}\muitalic_I .
| ACB | ACB | BCA | ACB | Selection 1 |
whose parameters (ρ,𝔲,T)𝜌𝔲𝑇(\rho,\mathfrak{u},T)( italic_ρ , fraktur_u , italic_T ) satisfy the compressible Euler system. <|MaskedSetence|> It was formally derived in Sone’s book [39] that for the Boltzmann equation with the complete diffusive boundary condition (1.3), the limiting Euler system would be imposed on the impermeable boundary condition (see also [37], for instance). <|MaskedSetence|> We first state the main result of this paper in an informal way. <|MaskedSetence|> | **A**: The goal of the this paper is to rigorously justify this limit by using the method of Hilbert expansion, in the context of short time smooth solutions.
**B**: The precise statement will be given in the later part of Introduction, after we introduce some required notations.
.
**C**: In particular, for the domain with boundary, the key (also the difficulty) is to determine the boundary condition for the compressible Euler system.
| CAB | CAB | BCA | CAB | Selection 1 |
The organization of the paper is as follows. In Section 2, we introduce our notation and provide the required definitions with the notion of weak solution used in the paper. <|MaskedSetence|> <|MaskedSetence|> <|MaskedSetence|> We conclude the paper with Appendix A where we state all the required a priori estimates from [26, Section 3] and state a compact embedding lemma which is helpful in the proof of the crucial sequential stability result, see Theorem 3.3.
. | **A**: In Section 3, we introduce the concept of a semiflow selection in terms of the two state variables: the velocity and the energy.
**B**: Section 4 is devoted to the existence of a random dynamical system, Theorem 4.4, which is a central result of
this paper.
**C**: We also analyze the properties (compactness, shift invariance and
continuation) of the solution set and prove the existence of a semiflow selection, refer to Theorem 3.9.
| ACB | ACB | CAB | ACB | Selection 2 |
<|MaskedSetence|> <|MaskedSetence|> (2014)],
[Pfister et al. (2018)], [Chakraborty and Zhang (2019)]), graphical modeling ([Lauritzen (1996)], [Gan, Narisetty and Liang (2019)]), linguistics ([Nguyen and Eisenstein (2017)]), clustering (Székely and Rizzo, 2005), dimension reduction (Fukumizu, Bach and Jordan, 2004; Sheng and Yin, 2016). The traditional approach for testing independence is based on Pearson’s correlation coefficient; for instance, refer to Binet and Vaschide (1897), Pearson (1920), Spearman (1904), Kendall (1938). However, its lack of robustness to outliers and departures from normality eventually led researchers to consider alternative nonparametric procedures.
To overcome such a problem, a natural approach is to consider the functional difference between
the empirical joint distribution and the product of the empirical marginal distributions, see Hoeffding (1948), Blum, Kiefer and Rosenblatt (1961) and Bouzebda (2011).
This approach can also use characteristic empirical functions; see Csörgő (1985). <|MaskedSetence|> | **A**: Inspired by the work of Blum, Kiefer and Rosenblatt (1961) and Dugué (1975), Deheuvels (1981) studied a test of multivariate independence based on the Möbius decomposition, generalized in Bouzebda (2014)..
**B**: Testing independence also has many applications, including causal inference ([Pearl (2009)], [Peters et al.
**C**: [Bach and Jordan (2003)], [Chen and Bickel (2006)], [Samworth and Yuan (2012)] and [Matteson and Tsay (2017)].
| ABC | CBA | CBA | CBA | Selection 3 |
<|MaskedSetence|> We say that ℒℒ\mathcal{L}caligraphic_L is relational if it only contains relation symbols. <|MaskedSetence|> For every A⊆ℕ𝐴ℕA\subseteq\mathbb{N}italic_A ⊆ blackboard_N we denote by ⟨A⟩𝒩subscriptdelimited-⟨⟩𝐴𝒩\langle A\rangle_{\mathcal{N}}⟨ italic_A ⟩ start_POSTSUBSCRIPT caligraphic_N end_POSTSUBSCRIPT the smallest substructure of 𝒩𝒩\mathcal{N}caligraphic_N containing every element of A𝐴Aitalic_A in its domain. We say that 𝒩𝒩\mathcal{N}caligraphic_N is ultrahomogeneous if for every bijection f:A→B:𝑓→𝐴𝐵f\colon A\to Bitalic_f : italic_A → italic_B between finite subsets of ℕℕ\mathbb{N}blackboard_N which induces an isomorphism from ⟨A⟩𝒩subscriptdelimited-⟨⟩𝐴𝒩\langle A\rangle_{\mathcal{N}}⟨ italic_A ⟩ start_POSTSUBSCRIPT caligraphic_N end_POSTSUBSCRIPT to ⟨B⟩𝒩subscriptdelimited-⟨⟩𝐵𝒩\langle B\rangle_{\mathcal{N}}⟨ italic_B ⟩ start_POSTSUBSCRIPT caligraphic_N end_POSTSUBSCRIPT, we have that f𝑓fitalic_f extends to an automorphism of 𝒩𝒩\mathcal{N}caligraphic_N. We denote by Age(𝒩)Age𝒩\mathrm{Age}(\mathcal{N})roman_Age ( caligraphic_N ) the class of all finitely generated ℒℒ\mathcal{L}caligraphic_L-structures which embed into 𝒩𝒩\mathcal{N}caligraphic_N. <|MaskedSetence|> We recall here some amalgamation properties of 𝒦𝒦\mathcal{K}caligraphic_K which relate to the dynamical properties of P𝑃Pitalic_P mentioned in the introduction such as P𝑃Pitalic_P being locally finite or n𝑛nitalic_n-free.
. | **A**: Let ℒℒ\mathcal{L}caligraphic_L be a first order language.
**B**: If 𝒩𝒩\mathcal{N}caligraphic_N is ultrahomogeneous, then 𝒦:=Age(𝒩)assign𝒦Age𝒩\mathcal{K}:=\mathrm{Age}(\mathcal{N})caligraphic_K := roman_Age ( caligraphic_N ) satisfies the amalgamation property, i.e., for every two embeddings f:𝒜→ℬ:𝑓→𝒜ℬf\colon\mathcal{A}\to\mathcal{B}italic_f : caligraphic_A → caligraphic_B, g:𝒜→𝒞:𝑔→𝒜𝒞g\colon\mathcal{A}\to\mathcal{C}italic_g : caligraphic_A → caligraphic_C, with 𝒜,ℬ,𝒞∈𝒦𝒜ℬ𝒞𝒦\mathcal{A},\mathcal{B},\mathcal{C}\in\mathcal{K}caligraphic_A , caligraphic_B , caligraphic_C ∈ caligraphic_K, there exists 𝒟∈𝒦𝒟𝒦\mathcal{D}\in\mathcal{K}caligraphic_D ∈ caligraphic_K and embeddings f′:ℬ→𝒟:superscript𝑓′→ℬ𝒟f^{\prime}\colon\mathcal{B}\to\mathcal{D}italic_f start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT : caligraphic_B → caligraphic_D, g′:𝒞→𝒟:superscript𝑔′→𝒞𝒟g^{\prime}\colon\mathcal{C}\to\mathcal{D}italic_g start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT : caligraphic_C → caligraphic_D so that g′∘g=f′∘fsuperscript𝑔′𝑔superscript𝑓′𝑓g^{\prime}\circ g=f^{\prime}\circ fitalic_g start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∘ italic_g = italic_f start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∘ italic_f.
When 𝒩𝒩\mathcal{N}caligraphic_N is ultrahomogeneous, then various strengthenings of the amalgamation property, as well as several other combinatorial properties of 𝒦𝒦\mathcal{K}caligraphic_K, often correspond to dynamical properties of P:=Aut(𝒩)assign𝑃Aut𝒩P:=\mathrm{Aut}(\mathcal{N})italic_P := roman_Aut ( caligraphic_N ).
**C**: If ℳℳ\mathcal{M}caligraphic_M is an ℒℒ\mathcal{L}caligraphic_L-structure then we write ℳ=(M,…)ℳ𝑀…\mathcal{M}=(M,\ldots)caligraphic_M = ( italic_M , … ) to indicate that the domain of ℳℳ\mathcal{M}caligraphic_M is the set M𝑀Mitalic_M.
Let 𝒩=(ℕ,…)𝒩ℕ…\mathcal{N}=(\mathbb{N},\ldots)caligraphic_N = ( blackboard_N , … ) be an ℒℒ\mathcal{L}caligraphic_L-structure.
| ACB | ACB | CAB | ACB | Selection 4 |
\emptyset&\textrm{otherwise},\end{cases}italic_p ( italic_F start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∩ blackboard_R start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT × { italic_t } ) = { start_ROW start_CELL bold_D start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_CELL start_CELL ( italic_t = italic_t start_POSTSUBSCRIPT 7 end_POSTSUBSCRIPT ) , end_CELL end_ROW start_ROW start_CELL over~ start_ARG italic_α start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT end_ARG end_CELL start_CELL ( ( italic_t start_POSTSUBSCRIPT 5 end_POSTSUBSCRIPT + italic_t start_POSTSUBSCRIPT 6 end_POSTSUBSCRIPT ) / 2 < italic_t < italic_t start_POSTSUBSCRIPT 7 end_POSTSUBSCRIPT ) , end_CELL end_ROW start_ROW start_CELL over~ start_ARG italic_α start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT end_ARG ∪ | italic_B start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT | end_CELL start_CELL ( italic_t = ( italic_t start_POSTSUBSCRIPT 5 end_POSTSUBSCRIPT + italic_t start_POSTSUBSCRIPT 6 end_POSTSUBSCRIPT ) / 2 ) , end_CELL end_ROW start_ROW start_CELL roman_p ( italic_F start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∩ blackboard_R start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT × { italic_t } ) end_CELL start_CELL ( ( italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_t start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) / 2 < italic_t < ( italic_t start_POSTSUBSCRIPT 5 end_POSTSUBSCRIPT + italic_t start_POSTSUBSCRIPT 6 end_POSTSUBSCRIPT ) / 2 ) , end_CELL end_ROW start_ROW start_CELL over~ start_ARG italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT end_ARG ∪ | italic_B start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT | end_CELL start_CELL ( italic_t = ( italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_t start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) / 2 ) , end_CELL end_ROW start_ROW start_CELL over~ start_ARG italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT end_ARG end_CELL start_CELL ( italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT < italic_t < ( italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_t start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) / 2 ) , end_CELL end_ROW start_ROW start_CELL bold_d start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_CELL start_CELL ( italic_t = italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) , end_CELL end_ROW start_ROW start_CELL ∅ end_CELL start_CELL otherwise , end_CELL end_ROW
where |B1−|superscriptsubscript𝐵1|B_{1}^{-}|| italic_B start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT | (resp. |B1+|superscriptsubscript𝐵1|B_{1}^{+}|| italic_B start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT |) is the union of the bands belonging to B1−superscriptsubscript𝐵1B_{1}^{-}italic_B start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT (resp. B1+superscriptsubscript𝐵1B_{1}^{+}italic_B start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT), and 𝐝1subscript𝐝1\mathbf{d}_{1}bold_d start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT (resp. <|MaskedSetence|> <|MaskedSetence|> 𝐃1subscript𝐃1\mathbf{D}_{1}bold_D start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT) is disjoint from |B1−|superscriptsubscript𝐵1|B_{1}^{-}|| italic_B start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT | (resp. <|MaskedSetence|> | **A**: 𝐃1subscript𝐃1\mathbf{D}_{1}bold_D start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT) is a union of mutually disjoint m𝑚mitalic_m 2222-disks in ℝ3superscriptℝ3\mathbb{R}^{3}blackboard_R start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT bounded by α1∗~~superscriptsubscript𝛼1\widetilde{\alpha_{1}^{*}}over~ start_ARG italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT end_ARG (resp.
**B**: α2∗~~superscriptsubscript𝛼2\widetilde{\alpha_{2}^{*}}over~ start_ARG italic_α start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT end_ARG) such that 𝐝1subscript𝐝1\mathbf{d}_{1}bold_d start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT (resp.
**C**: |B1+|superscriptsubscript𝐵1|B_{1}^{+}|| italic_B start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT |) as in the left of Figure 16 except for the attaching arcs of the bands, respectively..
| ABC | ABC | ABC | CBA | Selection 1 |
<|MaskedSetence|> Here, the challenge consists in considering nonlinear damping which are distributed everywhere in the domain and acting only on the rotational angles. <|MaskedSetence|> More precisely, we prove that the energy decay rate, as introduced in [2] for a nonlinearly damped hyperbolic system coupled by velocities, can be extended to our plate model. The proofs of our results are based on multiplier techniques, weighted nonlinear integral inequalities and the optimal-weight convexity method used in [1, 2]. <|MaskedSetence|> In the present work we assume a convexity assumption on the feedback as we will see later on, and we prove the asymptotic behavior of the energy in higher dimensions. For more details, we refer the reader to [5] for the wave equation and to [2] for the one-dimensional Timoshenko system.
. | **A**: Taking into account the work of Alabau-Boussouira [2, 4], we aim here to establish a general and explicit decay result for the energy associated with the system (1.5)–(1.7) below.
**B**: Indeed, the latter method is originally developed in [1] where the author completed the study carried out in [23] and improved the results in [25].
**C**:
It is worth mentioning that the above mentioned results precise some of the necessary conditions that lead to the exponential stabilization of the model (depending on the choice of the boundary conditions, the equality or non-equality of the wave speeds and the nature of damping terms).
| ACB | CAB | CAB | CAB | Selection 3 |
This system is far from perfect, and various refinements of it have been proposed.
Still, even in this basic form, we can appreciate how escrows turn the strategy ‘running away with the money/goods’ into a non-rational choice. <|MaskedSetence|> Similarly, in refusing to pay Bogdan, Aki will never get her money back. So, both Aki and Bogdan have no economic incentive in not upholding their side of the trade and may do so only if motivated by irrational motives such as pure malicious intent.222Whoever is familiar with the blockchain ecosystem –and with life over the Internet– knows that this is still a big problem. <|MaskedSetence|> For the sake of simplicity, we will not focus on this in this work. <|MaskedSetence|> | **A**: The basic structure of an escrow is depicted in Figure 2.
.
**B**: In practice, the introduction of time windows that release the escrow to the original owner if certain conditions are met can mitigate the problem.
**C**: In refusing to ship the goods, Bogdan doesn’t gain any money because Aki will never release the escrow.
| CBA | CBA | CBA | ACB | Selection 3 |
Self-concordant functions have received strong interest in recent years due to the attractive properties that they allow to prove for many statistical estimation settings [Marteau-Ferey et al., 2019, Ostrovskii & Bach, 2021]. The original definition of self-concordance has been expanded and generalized since its inception, as many objective functions of interest have self-concordant-like properties without satisfying the strict definition of self-concordance. <|MaskedSetence|> <|MaskedSetence|> <|MaskedSetence|> This was fully formalized in Sun & Tran-Dinh [2019], in which the concept of generalized self-concordant functions was introduced, along with key bounds, properties, and variants of Newton methods for the unconstrained setting which make use of this property.
. | **A**: [2015], in which more general properties of these
pseudo-self-concordant functions were established.
**B**: This was also the case in Ostrovskii & Bach [2021] and Tran-Dinh et al.
**C**: For example, the logistic loss function used in logistic regression is not strictly self-concordant, but it fits into a class of pseudo-self-concordant functions, which allows one to obtain similar properties and bounds as those obtained for self-concordant functions [Bach, 2010].
| CBA | CBA | ABC | CBA | Selection 1 |
The representation type of tensor product algebras has been studied in various contexts. In the 1970s, Bondarenko and Drozd [BD] considered the representation type of finite groups, while Auslander and Reiten [AR] dedicated their effort to the representation type of triangular matrix rings. Moving into the 1980s, mathematicians delved into the representation type of triangular matrix algebras over different classes of algebras, for example, see [S] for Nakayama algebras, [HM] for self-injective algebras, [L2] for radical square zero algebras, etc. <|MaskedSetence|> <|MaskedSetence|> <|MaskedSetence|> | **A**: The most recent progress in this field can be attributed to Leszczyn´´n\acute{\text{n}}over´ start_ARG n end_ARGski and Skowron´´n\acute{\text{n}}over´ start_ARG n end_ARGski, as clear in their series of papers [L1, LS1, LS2].
**B**: However, it is still open to distinguish representation-finite cases and tame cases from non-wild cases.
.
**C**: These papers provide a complete description of non-wild tensor product algebras.
| ACB | ACB | ACB | ACB | Selection 3 |
For this case we present Algorithm 2. <|MaskedSetence|> Here, as in Algorithm 1, the proximal operator is computed inexactly. <|MaskedSetence|> <|MaskedSetence|> Hence, the problem (4) is solved by Fast Gradient Descent. Further, we note that the algorithm’s steps in lines 3, 6, and 7 are local and separable on each machine. The following theorem states the convergence rate of Algorithm 2 with Accelerated Gradient Descent.
. | **A**: The problem (4) is divided into two minimization subproblems, by X𝑋Xitalic_X, and by Y𝑌Yitalic_Y.
**B**: Note that we need to communicate with other devices only when we solve the problem (4) and need to multiply by the matrix W𝑊Witalic_W.
**C**: This algorithm is the Tseng method [44] with a resolvent/proximal operator calculation (4).
| CBA | CBA | CBA | CBA | Selection 3 |
<|MaskedSetence|> A brief history of the formality problem
This formality result has quite an involved history, which we try to summarise here. Firstly, there are many incarnations of the result for the Yoneda algebra of a semisimple representation of a 2CY algebra. As pointed out by Bocklandt, Galluzzi and Vaccarino [BGV16], this result follows from Koszul duality arguments and [VdB15, Theorem 11.2.1]; despite the apparent exclusion of the n=2𝑛2n=2italic_n = 2 case from Van den Bergh’s theorem on nCY algebras in [ibid]. <|MaskedSetence|> <|MaskedSetence|> For Higgs bundles and representations of fundamental groups, formality dates back to classical results [DGMS75, GM88, Sim92] and is a crucial tool in the subject (e.g. in Simpson’s isosingularity theorem).. | **A**:
4.6.4.
**B**: This observation, along with deformation-theoretic arguments, was used to understand formal neighbourhoods in the coarse moduli space of representations of 2-Calabi–Yau algebras in [BGV16], with fuller details (including but not limited to the étale neighbourhood theorem for the coarse moduli space) provided in [KS19].
**C**: These results on coarse moduli spaces in turn generalise the known results on coarse moduli spaces of representations of preprojective algebras due to Crawley–Boevey [CB03].
| ABC | ABC | ABC | CAB | Selection 3 |
<|MaskedSetence|> Let xi,jsubscript𝑥𝑖𝑗x_{i,j}italic_x start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT be the image of Xi,jsubscript𝑋𝑖𝑗X_{i,j}italic_X start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT in R𝑅Ritalic_R. Also, Fi,j¯=fi,j¯subscript𝐹𝑖𝑗subscript𝑓𝑖𝑗\overline{F_{i,j}}=f_{i,j}over¯ start_ARG italic_F start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT end_ARG = italic_f start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT and F¯=f¯𝐹𝑓\overline{F}=fover¯ start_ARG italic_F end_ARG = italic_f for some F,Fi,j∈S𝐹subscript𝐹𝑖𝑗𝑆F,F_{i,j}\in Sitalic_F , italic_F start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT ∈ italic_S. Moreover, we set βi:=∑1≤j≤i+1Xi,1⋯Xi,j^⋯Xi,i+1assignsubscript𝛽𝑖subscript1𝑗𝑖1subscript𝑋𝑖1⋯^subscript𝑋𝑖𝑗⋯subscript𝑋𝑖𝑖1\beta_{i}:=\sum_{1\leq j\leq i+1}X_{i,1}\dotsm\widehat{X_{i,j}}\dotsm X_{i,i+1}italic_β start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT := ∑ start_POSTSUBSCRIPT 1 ≤ italic_j ≤ italic_i + 1 end_POSTSUBSCRIPT italic_X start_POSTSUBSCRIPT italic_i , 1 end_POSTSUBSCRIPT ⋯ over^ start_ARG italic_X start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT end_ARG ⋯ italic_X start_POSTSUBSCRIPT italic_i , italic_i + 1 end_POSTSUBSCRIPT.
Suppose m𝑚mitalic_m is the largest integer for which some Xm,tsubscript𝑋𝑚𝑡X_{m,t}italic_X start_POSTSUBSCRIPT italic_m , italic_t end_POSTSUBSCRIPT appears in F𝐹Fitalic_F. Let v:S→S:𝑣→𝑆𝑆v:S\rightarrow Sitalic_v : italic_S → italic_S be the homomorphism that sets any Xi,jsubscript𝑋𝑖𝑗X_{i,j}italic_X start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT such that i≥m+1𝑖𝑚1i\geq m+1italic_i ≥ italic_m + 1 equal to 00. We note that each nonunit in R𝑅Ritalic_R is actually nilpotent and so for any A∈S𝐴𝑆A\in Sitalic_A ∈ italic_S with nonzero coefficient, A¯¯𝐴\overline{A}over¯ start_ARG italic_A end_ARG is a unit. <|MaskedSetence|> If follows that for a large enough i𝑖iitalic_i, v(F)=v(Fi,1)⋯v(Fi,ki)=0𝑣𝐹𝑣subscript𝐹𝑖1⋯𝑣subscript𝐹𝑖subscript𝑘𝑖0v(F)=v(F_{i,1})\dotsm v(F_{i,k_{i}})=0italic_v ( italic_F ) = italic_v ( italic_F start_POSTSUBSCRIPT italic_i , 1 end_POSTSUBSCRIPT ) ⋯ italic_v ( italic_F start_POSTSUBSCRIPT italic_i , italic_k start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT ) = 0 modulo v(I)𝑣𝐼v(I)italic_v ( italic_I ). So, F=v(F)∈v(I)⊆I1+I2+v(I3)𝐹𝑣𝐹𝑣𝐼subscript𝐼1subscript𝐼2𝑣subscript𝐼3F=v(F)\in v(I)\subseteq I_{1}+I_{2}+v(I_{3})italic_F = italic_v ( italic_F ) ∈ italic_v ( italic_I ) ⊆ italic_I start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_I start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT + italic_v ( italic_I start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ). <|MaskedSetence|> Therefore, f=βm¯𝑓¯subscript𝛽𝑚f=\overline{\beta_{m}}italic_f = over¯ start_ARG italic_β start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT end_ARG. On the other hand, β1¯=β2¯=⋯¯subscript𝛽1¯subscript𝛽2⋯\overline{\beta_{1}}=\overline{\beta_{2}}=\dotsbover¯ start_ARG italic_β start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG = over¯ start_ARG italic_β start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_ARG = ⋯ and so for every i∈ℕ𝑖ℕi\in\mathbb{N}italic_i ∈ blackboard_N, we have f=βi¯𝑓¯subscript𝛽𝑖f=\overline{\beta_{i}}italic_f = over¯ start_ARG italic_β start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG.. | **A**: Now v(I3)⊆I3+v(S)βm𝑣subscript𝐼3subscript𝐼3𝑣𝑆subscript𝛽𝑚v(I_{3})\subseteq I_{3}+v(S)\beta_{m}italic_v ( italic_I start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) ⊆ italic_I start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + italic_v ( italic_S ) italic_β start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT, so F−Lβm∈I𝐹𝐿subscript𝛽𝑚𝐼F-L\beta_{m}\in Iitalic_F - italic_L italic_β start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ∈ italic_I for some L∈v(S)𝐿𝑣𝑆L\in v(S)italic_L ∈ italic_v ( italic_S ), and we may assume L=1𝐿1L=1italic_L = 1 since the product of any variable in v(S)𝑣𝑆v(S)italic_v ( italic_S ) and βmsubscript𝛽𝑚\beta_{m}italic_β start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT is in I1+I2subscript𝐼1subscript𝐼2I_{1}+I_{2}italic_I start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_I start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT.
**B**: We may consider each Fi,jsubscript𝐹𝑖𝑗F_{i,j}italic_F start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT as a sum of proper subproducts of generators of I1subscript𝐼1I_{1}italic_I start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT since other terms become 00 in R𝑅Ritalic_R.
**C**:
Suppose on the contrary that there exist nonzero nonunit elements f𝑓fitalic_f and fi,jsubscript𝑓𝑖𝑗f_{i,j}italic_f start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT such that f=fi,1⋯fi,ki𝑓subscript𝑓𝑖1⋯subscript𝑓𝑖subscript𝑘𝑖f=f_{i,1}\dotsm f_{i,k_{i}}italic_f = italic_f start_POSTSUBSCRIPT italic_i , 1 end_POSTSUBSCRIPT ⋯ italic_f start_POSTSUBSCRIPT italic_i , italic_k start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT, where {ki}i∈ℕsubscriptsubscript𝑘𝑖𝑖ℕ\{k_{i}\}_{i\in\mathbb{N}}{ italic_k start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT } start_POSTSUBSCRIPT italic_i ∈ blackboard_N end_POSTSUBSCRIPT is not bounded.
| CBA | CBA | CBA | CBA | Selection 2 |
It is well known that, the notion of non-positive curvature spaces were mentioned by
J. Hadamard and E. <|MaskedSetence|> Busemann and A.D. Aleksandrov generalized the concept of geodesic metric spaces based on the concept of manifolds with a non-positive sectional curvature. <|MaskedSetence|> <|MaskedSetence|> | **A**: Gromov suggested the notation CAT(0)CAT0\mathrm{CAT}(0)roman_CAT ( 0 ) for a non-positive curvature geodesic metric space.
**B**: Cartan in the 1920’s.
In 1950, H.
**C**: The letters C, A and T in CAT(0)CAT0\mathrm{CAT}(0)roman_CAT ( 0 ) stand for Cartan, Aleksandrov and Toponogov, respectively.
.
| BAC | BAC | BAC | BAC | Selection 1 |
In this section we validate the efficacy of the proposed algorithm and we verify our theoretical results
on optimization problems that arise in machine learning applications. Specifically, in the first two sections we use the conventional SIGMA (uniform sampling) to solve the maximum likelihood estimation problem based on the Poisson and Logistic models, respectively. Full details about the experimental setup, objective functions and the datasets are given in Appendix C. <|MaskedSetence|> <|MaskedSetence|> <|MaskedSetence|> | **A**: In Remark 4 we discuss how to efficiently compute the reduced Hessian matrix for Generalized Linear Models.
.
**B**: Furthermore, in Section 4.3 we provide comparisons between the conventional SIGMA and SIGMA with the different sampling strategies of Section 2.5.
**C**: Moreover, we provide additional experiments and we test SIGMA with sub-sampling (Section C.2.2).
| BCA | BCA | BCA | CAB | Selection 3 |
Acknowledgements. We thank M. Solleveld for his careful reading and many useful comments, particularly regarding Section 11, and G. <|MaskedSetence|> Reeder for their helpful suggestions. We also thank the referees for the thorough checking of the paper, for the corrections and suggestions for improvement. This research was supported in part by the EPSRC grant EP/V046713/1 (2021). <|MaskedSetence|> <|MaskedSetence|> | **A**: thanks Université Paris Cité and Sorbonne Université for their hospitality while part of this work was completed.
.
**B**: Lusztig and M.
**C**: D.C.
| BCA | BCA | BCA | CAB | Selection 2 |
<|MaskedSetence|> In [FNQ18] and [KRS20] the authors proved the logarithmic Sobolev inequality and the fractional logarithmic Sobolev inequality on the Heisenberg group and on homogeneous groups, respectively. <|MaskedSetence|> In this paper, we prove logarithmic Sobolev inequalities on graded groups and weighted logarithmic Sobolev inequalities on general Lie groups. As applications of these inequalities we show Nash and weighted Nash inequalities on graded and general Lie groups, respectively. The log-Sobolev type inequalities with some specific orders of weights are also sometimes called the log-Hardy inequalities [DDFT10].
The subject of the logarithmic inequalities has been extensively investigated and it is impossible to give a reasonably complete review of the literature here. We refer to surveys [AB00, GZ03], to works on relations to other inequalities [BL00, MF93], coercive inequalities on Carnot groups [Bou21, BZ21a, BZ21b, BZ21c, HZ10], Nash [Nas58, Bec98, BDS20, OS18] and weighted Nash [BBGM12] inequalities. <|MaskedSetence|> | **A**: A fractional weighted version of (1.3) on homogeneous groups was proved in [KS20].
**B**: There are works on the fractional Laplacian [Bec12] as well as on the.
**C**:
In [Mer08], the author obtained a logarithmic Gagliardo-Nirenberg inequality.
| ACB | CAB | CAB | CAB | Selection 3 |
<|MaskedSetence|> <|MaskedSetence|> That proof was rather complicated and required the extensive use of techniques from the study of iterated reflection. In this note, we present a simpler proof of Theorem 1.2 that uses more traditional and accessible techniques from ordinal analysis, namely, cut-elimination for infinitary derivations. The cut-elimination proof strengthens the connections between these two complementary areas of proof theory. <|MaskedSetence|> In §2 we cover some preliminaries, including our treatment of reflection principles and infinitary derivations. In §3 we provide our new proof of Theorem 1.2.
. | **A**: In [4], the focus was on the iterated reflection side; indeed, Theorem 1.2 was derived from a Schmerl-style [6] conservation theorem for iterated reflection principles.
**B**: Moreover, it should make the result more accessible to proof-theorists who are familiar with cut-elimination techniques.
Here is our plan for the rest of the paper.
**C**: Theorem 1.2 connects two distinct topics in proof theory: iterated reflection and ordinal analysis.
| CAB | CAB | CBA | CAB | Selection 2 |
Our main tools originate in some recent work on the theory of stacks. Alper, Halpern-Leistner and Heinloth [AHLH23] have recently developed a theory which produces moduli spaces for Artin stacks, generalizing results of Keel-Mori on Deligne-Mumford stacks. This can be combined with the theory of ΘΘ\Thetaroman_Θ-stability and ΘΘ\Thetaroman_Θ-stratifications by Halpern-Leistner [HL14] to produce a powerful stack-theoretic approach to the construction of moduli spaces and canonical filtrations (see [HL14, §5] and [HLHJ21, §2.5] for more details). Halpern-Leistner defines in [HL14] a notion of filtration of an object in an Artin stack. Given certain cohomology classes on the stack, one can define a numerical invariant on filtrations. <|MaskedSetence|> <|MaskedSetence|> <|MaskedSetence|> | **A**: If the numerical invariant satisfies certain properties, then the open substack of semistable objects admits a good moduli space (as defined in [Alp13]).
**B**: This is an intrinsic way of constructing the moduli space, in the sense that we do not need to choose a parameter space nor the action of a group.
.
**C**: An object in the stack is called semistable if the numerical function is non-positive for all filtrations.
| CAB | CBA | CAB | CAB | Selection 1 |
Relative boundedness can be understood as an analogue of relative Lipschitz continuity for variational inequalities. <|MaskedSetence|> This fact plays an important role in considering relatively Lipschitz continuous Lagrange saddle point problems and their reduction to corresponding variational inequalities with the relatively bounded operator. Recently, in [12] the authors proposed an adaptive version of the Mirror Prox method (extragradient type method) for variational inequalities with a condition similar to relative smoothness. It should be noted that variational inequalities with relatively smooth operators are applicable to the resource sharing problem [2]. Also, in [16] there were introduced some non-adaptive switching subgradient algorithms for convex programming problems with relatively Lipschitz continuous functions. Recently, there was proposed a non-adaptive method for solving variational inequalities with the relatively bounded operator [17]. In this paper, we propose an adaptive algorithm for the corresponding class of problems.
The paper consists of the introduction and 6 main sections. In Sect. 2 we give some basic notations and definitions. <|MaskedSetence|> 3 we consider the Minty variational inequality with a relatively bounded operator and propose an adaptive algorithm for solving it. Sect. <|MaskedSetence|> In Sect. 5 we propose some universal algorithms for minimizing relatively smooth and relatively Lipschitz continuous functions. Sect. 6 is devoted to the numerical experiments which demonstrate the effectiveness of the proposed methods.. | **A**: In Sect.
**B**: 4 is devoted to adaptive algorithms for relatively smooth optimization problems.
**C**: It should be noted that the subgradient of a relatively Lipschitz continuous function satisfies the relative boundedness condition.
| CAB | CBA | CAB | CAB | Selection 4 |
<|MaskedSetence|> He would like to thank Professor Bernard Leclerc and Professor Erez Lapid for useful comments and suggestions; He really appreciates Professor Markus Reineke for his instruction and his proof for the set of irreducible components of quiver Grassmannians; He really thanks Bernard Keller for comments on the graded quiver varieties. He would like to thank Professor Cerulli Irelli for his help and guidance during this work. His work was finished during his invitation to the University of Sapienza. The author is supported by the China Scholarships Council. <|MaskedSetence|> <|MaskedSetence|> | **A**: NO.202006040123.
2.
**B**: Premise.
**C**: The author is grateful to Professor Qin Fan for many helpful discussions.
| CAB | CAB | CAB | CBA | Selection 1 |
The paper is organized as follows. In Sections 1, 2 and 3 we give the description of the lattice path model and formulate the main theorem. <|MaskedSetence|> In Section 5 we will reduce the problem of counting paths between the wall and the filter to a problem of counting paths between two lines. <|MaskedSetence|> <|MaskedSetence|> In Section 9 we hint at possible applications of considered lattice path models to representation theory of quantum groups at roots of unity.
. | **A**: In Section 4 we define wall and filter restrictions and recall the reflection principle.
**B**: In Sections 6, 7 we will prove theorems for path counting in the presence of two filters and two filters together with the wall.
**C**: The proof of the main theorem is given in Section 8.
| ABC | CBA | ABC | ABC | Selection 3 |
The first author introduced in [1] a relational framework for developing the key notions and results on hidden variables and non-locality, which can be seen as a relational variant of the probabilistic setting of [10]. He introduced what he called “relational empirical models” and used them to show that the basic results of the foundations of quantum mechanics, usually formulated in terms of probabilistic models, can be seen already on the level of mere (two-valued) relations. Our basic observation is that we can think of the relational empirical models of [1] as teams in the sense of team semantics. The basic quantum-theoretic properties of relational empirical models can then be defined in terms of the independence atoms of independence logic [19]. We observe that the relationships between quantum theoretic properties of relational models become instances of logical consequence of independence logic in its team semantics. <|MaskedSetence|> The no-go theorems become instances of failure of logical consequence between specific formulas of independence logic. This extends also to probabilistic models with independence logic replaced by the probabilistic independence logic of [15], capturing the probabilistic notions of [10].
Logical consequence in independence logic is, in general, non-axiomatizable. <|MaskedSetence|> This shows that the concept of logical consequence is here highly non-trivial and potentially quite complex. It should be emphasised that the logical consequences arising from the quantum theoretical examples are purely logical, i.e. <|MaskedSetence|> the theory of social choice or biology. On the other hand, the first author introduces in [1] a concept which in team semantics characterizes teams which arise (potentially) from a quantum-mechanical experiment. Presumably the most subtle relationships between quantum-mechanical concepts are particular to such quantum theoretic teams. We introduce to probabilistic independence logic, expanding on the example of [1], the concepts of being (finite-dimensionally) tensor-product quantum-mechanical, quantum-approximable and commuting-operator quantum-mechanical, and propose questions they give rise to.. | **A**: have a priori nothing to do with quantum mechanics, and hence they apply to any other field where independence plays a role, e.g.
**B**: In fact, the existential-positive-conjunctive fragment suffices.
**C**: Even on the level of atoms no finite axiomatization exists [39].
| BCA | ABC | BCA | BCA | Selection 4 |
However, in the free product case the appeal to surface theory is more delicate and less obviously valid. That is, realising a homotopy equivalence of a graph as a homeomorphism of a surface is a core part of the theory for free group automorphisms going back to [4], but it is not just that this analogue is absent in the free product or relative case, there is good reason to think that it isn’t entirely valid. <|MaskedSetence|> <|MaskedSetence|> <|MaskedSetence|> See also Remark 2.10 of [32] for a discussion of this.)
But this is no longer true for free products, since one can write examples of train track representative of irreducible automorphisms which are not fully irreducible (that is, some power is reducible) and whose transition matrices are not primitive (this cannot happen in the absolute case).
. | **A**: In [11] this is called ‘weakly clean’ and in that paper, Proposition B.2, it is shown that this implies clean, which means having a primitive transition matrix.
**B**: (In [2] it is proved that any irreducible automorphism has a locally connected Whitehead graph - see also section 5.
**C**: For instance, any irreducible automorphism of exponential growth of a free group has a train track representative whose transition matrix is primitive.
| CBA | CBA | BCA | CBA | Selection 1 |
base case Δ=3Δ3\Delta=3roman_Δ = 3. They proved that P∗superscript𝑃P^{*}italic_P start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT is the only HZ-graph with maximum degree Δ=3Δ3\Delta=3roman_Δ = 3, an alternative proof was given later by Král’, Sereny, and Stiebitz (see [13, pp. <|MaskedSetence|> <|MaskedSetence|> Our main goal in this paper
is to develop two new concepts, namely “pseudo-multifan” and “lollipop” that generalize previously known
adjacency lemmas associated with multifans and Kierstead paths. <|MaskedSetence|> Furthermore, we have applied. | **A**: These developments were used
to prove the Core Conjecture [2].
**B**: 67–63]).
**C**: The next case, Δ=4Δ4\Delta=4roman_Δ = 4, was recently solved by Cranston and Rabern [6], they proved that the only HZ-graph with maximum degree Δ=4Δ4\Delta=4roman_Δ = 4 is obtained from the graph K5subscript𝐾5K_{5}italic_K start_POSTSUBSCRIPT 5 end_POSTSUBSCRIPT with one edge removed.
The conjecture is wide open for Δ≥5Δ5\Delta\geq 5roman_Δ ≥ 5.
| BCA | CAB | BCA | BCA | Selection 4 |
where C0subscript𝐶0C_{0}italic_C start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT is a universal constant.
Let ϕ(x,t)=ψ(d(x,x0,t)r)italic-ϕ𝑥𝑡𝜓𝑑𝑥subscript𝑥0𝑡𝑟\phi(x,t)=\psi(\frac{d(x,x_{0},t)}{r})italic_ϕ ( italic_x , italic_t ) = italic_ψ ( divide start_ARG italic_d ( italic_x , italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_t ) end_ARG start_ARG italic_r end_ARG ). <|MaskedSetence|> <|MaskedSetence|> Then (x1,t1)∈Q2r,T(x0,T)subscript𝑥1subscript𝑡1subscript𝑄2𝑟𝑇subscript𝑥0𝑇(x_{1},t_{1})\in Q_{2r,T}(x_{0},T)( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) ∈ italic_Q start_POSTSUBSCRIPT 2 italic_r , italic_T end_POSTSUBSCRIPT ( italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_T ) with t1≠Tsubscript𝑡1𝑇t_{1}\neq Titalic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ≠ italic_T. By Calabi’s trick [2] we may assume that ϕFitalic-ϕ𝐹\phi Fitalic_ϕ italic_F is smooth at (x1,t1)subscript𝑥1subscript𝑡1(x_{1},t_{1})( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ). Let τ1=T−t1subscript𝜏1𝑇subscript𝑡1\tau_{1}=T-t_{1}italic_τ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_T - italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT. <|MaskedSetence|> | **A**: Assume that ϕFitalic-ϕ𝐹\phi Fitalic_ϕ italic_F achieves its positive maximum at the point (x1,t1)subscript𝑥1subscript𝑡1(x_{1},t_{1})( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ).
**B**: Suppose that the maximum of the function ϕFitalic-ϕ𝐹\phi Fitalic_ϕ italic_F is positive, otherwise the result follows trivially.
**C**: We compute at the point (x1,t1)subscript𝑥1subscript𝑡1(x_{1},t_{1})( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) using Lemma 2.1,.
| CBA | BAC | BAC | BAC | Selection 3 |
Throughout this paper, we work exclusively with 𝕜𝕜\Bbbkroman_𝕜 a commutative ring with global dimension zero. <|MaskedSetence|> First, it induces that every object is cofibrant and fibrant in chain complexes and simplicial modules, and the projective and injective model structures are equal. In particular, the model structure on comodules is left-induced from a nice monoidal model category. <|MaskedSetence|> This creates several issues to understand the induced homotopy theory on comodules. <|MaskedSetence|> This allows us to understand finite limits of comodules, which is essential for our main theorem.
Furthermore, the cotensor product of comodules is not a comodule unless the coalgebra is flat.
. | **A**: There are several reasons this condition is imposed.
**B**: Moreover, as every module is flat, the tensor product preserves finite limits.
**C**: In [HKRS17], it was shown that the model structures for comodules are left-induced from injective model structures which are in general not monoidal model categories.
| CAB | ACB | ACB | ACB | Selection 4 |
<|MaskedSetence|> Continuous limits
The purpose of this section is to prove Theorem 1.5. In Section 4.1 we recall the continuous β𝛽\betaitalic_β-corners processes from Section 1.1 and derive a few of their properties. <|MaskedSetence|> <|MaskedSetence|> Finally, in Section 4.4 we derive our continuous multi-level loop equations by combining our weak convergence result, Proposition 4.3 from Section 4.3, and Lemma 3.5.. | **A**: In Section 4.3 we derive the continuous measures from Section 4.1 as diffuse limits of the measures in (1.5).
**B**: In Section 4.2 we summarize some useful notation for Jack symmetric functions and explain how the latter relate to the measures in (1.5).
**C**:
4.
| CBA | ABC | CBA | CBA | Selection 3 |
<|MaskedSetence|> <|MaskedSetence|> Furthermore, we extend these results to the n𝑛nitalic_n-tuple saddle point problem in Section 3. <|MaskedSetence|> Generalizations to n𝑛nitalic_n-tuple cases are provided in Section 5. In Section 6, numerical experiments for a 3-field formulation of the Biot model are provided to justify the advantages of using positively stable preconditioners. Finally, concluding remarks are given in Section 7.
. | **A**:
The outline of the remainder of this paper is as follows.
**B**: In section 2, we briefly recall the classic saddle point problem and its Schur complement, and introduce the twofold saddle point problem and the form of Schur complement, we then construct and analyze the block-triangular and block-diagonal preconditioners based on Schur complement for twofold saddle point problems.
**C**: Some additive Schur complement based preconditioners are constructed and the corresponding known results in the literature are recalled in Section 4 for twofold saddle point problems.
| ABC | ABC | ABC | ABC | Selection 4 |
The authors would like to thank the handling editor and two referees for their very detailed comments.
Changxin Mo acknowledges support from the National Natural Science Foundation of China (Grant No. 12201092), the Natural Science Foundation Project of CQ CSTC (Grant No. <|MaskedSetence|> KJQN202200512), the Chongqing Talents Project (Grant No. cstc2022ycjh-bgzxm0040), and the Research Foundation of Chongqing Normal University (Grant No. <|MaskedSetence|> <|MaskedSetence|> of China.
Weiyang Ding’s research is supported by the Science and Technology. | **A**: CSTB2022NSCQ-MSX0896), the Science and Technology Research Program of Chongqing Municipal Education Commission
(Grant No.
**B**: 21XLB040), P.
**C**: R.
| ABC | ABC | ABC | CBA | Selection 1 |
<|MaskedSetence|> <|MaskedSetence|> <|MaskedSetence|> The city of Kharkiv has been devastated during the ongoing invasion of
Ukraine, and the Institute has been severely damaged [NDGP22]. We dedicate this paper to Profs. Eremenko and Lyubich,
to the people of Kharkiv, and to all victims of the invasion of Ukraine.
. | **A**: At that time, Alex Eremenko was based at the Institute of Low Temperature Physics and Engineering, and it was there that he formulated what is now known as Eremenko’s conjecture.
**B**: Their collaboration, which pioneered the use of approximation theory in complex dynamics, took place in the fall of 1983 in Kharkiv.
**C**: This work follows in the footsteps of two world-leading Ukrainian mathematicians, Alex Eremenko and Misha Lyubich.
| CBA | CBA | BCA | CBA | Selection 1 |
3. Correspondence: Parahoric vs. <|MaskedSetence|> A similar correspondence also works for Higgs bundles and local systems [2, 14]. <|MaskedSetence|> <|MaskedSetence|> | **A**: Although the correspondence is only given in characteristic zero, it can be naturally generalized to prime characteristic under some necessary conditions (tame weights).
**B**: In this section, we first give the correspondence of parahoric torsors and equivariant bundles in positive characteristic, which is a direct generalization of Balaji and Seshadri’s work, and then we give the correspondences for Higgs bundles and local systems..
**C**: Equivariant
Balaji and Seshadri gives the correspondence between parahoric torsors and equivariant bundles [3].
| BCA | CAB | CAB | CAB | Selection 2 |
This is modified from the conservative Belyi polynomial Bd,1subscript𝐵𝑑1B_{d,1}italic_B start_POSTSUBSCRIPT italic_d , 1 end_POSTSUBSCRIPT. <|MaskedSetence|> <|MaskedSetence|> <|MaskedSetence|> We may choose a𝑎aitalic_a such that a+dd−1𝑎𝑑𝑑1a+\frac{d}{d-1}italic_a + divide start_ARG italic_d end_ARG start_ARG italic_d - 1 end_ARG is also a fixed point for f𝑓fitalic_f, thus making f𝑓fitalic_f PCF. We claim that there is one such choice of a𝑎aitalic_a such that the resulting PCF polynomial has potential good reduction at p𝑝pitalic_p.
The a𝑎aitalic_a values that result in the critical orbit described above are the roots of the following polynomial in a𝑎aitalic_a:
. | **A**: Like Bd,1subscript𝐵𝑑1B_{d,1}italic_B start_POSTSUBSCRIPT italic_d , 1 end_POSTSUBSCRIPT, it has just two finite critical points at 00 and 1111.
**B**: The first critical point, 0, is preperiodic, as f(0)=dd−1𝑓0𝑑𝑑1f(0)=\frac{d}{d-1}italic_f ( 0 ) = divide start_ARG italic_d end_ARG start_ARG italic_d - 1 end_ARG, which is a fixed point.
**C**: The second critical point 1111 maps to a+dd−1𝑎𝑑𝑑1a+\frac{d}{d-1}italic_a + divide start_ARG italic_d end_ARG start_ARG italic_d - 1 end_ARG.
| ABC | ABC | BCA | ABC | Selection 1 |
To prove (i) assume that f𝑓fitalic_f is Baire measurable. <|MaskedSetence|> <|MaskedSetence|> Then Player II fixes any point a∈G∩W𝑎𝐺𝑊a\in G\cap Witalic_a ∈ italic_G ∩ italic_W and picks U0:=B(a,1)∩G∩Wassignsubscript𝑈0𝐵𝑎1𝐺𝑊U_{0}:=B(a,1)\cap G\cap Witalic_U start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT := italic_B ( italic_a , 1 ) ∩ italic_G ∩ italic_W. At the (n+1)𝑛1(n+1)( italic_n + 1 )-th move, Player I chooses xn+1∈Unsubscript𝑥𝑛1subscript𝑈𝑛x_{n+1}\in U_{n}italic_x start_POSTSUBSCRIPT italic_n + 1 end_POSTSUBSCRIPT ∈ italic_U start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT.
Then Player II plays Un+1:={xn+1}∪(B(a,1n+1)∩G∩W)assignsubscript𝑈𝑛1subscript𝑥𝑛1𝐵𝑎1𝑛1𝐺𝑊U_{n+1}:=\{x_{n+1}\}\cup(B(a,\frac{1}{n+1})\cap G\cap W)italic_U start_POSTSUBSCRIPT italic_n + 1 end_POSTSUBSCRIPT := { italic_x start_POSTSUBSCRIPT italic_n + 1 end_POSTSUBSCRIPT } ∪ ( italic_B ( italic_a , divide start_ARG 1 end_ARG start_ARG italic_n + 1 end_ARG ) ∩ italic_G ∩ italic_W ). <|MaskedSetence|> | **A**: When the game is finished, one of the two cases is possible:.
**B**: We will describe a winning strategy for Player II in the game GfBairesubscriptsuperscript𝐺Baire𝑓G^{\mathrm{Baire}}_{f}italic_G start_POSTSUPERSCRIPT roman_Baire end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPT.
Let G⊆X𝐺𝑋G\subseteq Xitalic_G ⊆ italic_X be a dense Gδsubscript𝐺𝛿G_{\delta}italic_G start_POSTSUBSCRIPT italic_δ end_POSTSUBSCRIPT set such that f↾G↾𝑓𝐺f\restriction Gitalic_f ↾ italic_G is continuous.
**C**: (See [14, Theorem 8.38].)
Let W∈Baire+𝑊superscriptBaireW\in\mathrm{Baire}^{+}italic_W ∈ roman_Baire start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT
be chosen by Player I at the first move.
| BAC | BCA | BCA | BCA | Selection 2 |
Matching book embeddings of bipartite circulants C𝐶Citalic_C are given where the page number is equal to the vertex degree Δ(C)Δ𝐶\Delta(C)roman_Δ ( italic_C ), supporting the conjecture in [2]. It can be shown that regular dispersable graphs must be bipartite [21]. A nonbipartite circulant is nearly dispersable if one extra page suffices [21]. <|MaskedSetence|> For the complete bipartite graph and the hypercube, see [4]; for complete graphs and other bipartite graphs, see [21]. Cartesian products of even cycles are dispersable; even times odd cycles are nearly dispersable [17]; and short odd (length at most 5) and arbitrary odd cycles have nearly dispersable product [15]. <|MaskedSetence|> <|MaskedSetence|> Some graphs which are not vertex transitive also are dispersable such as trees [21], Halin trees [24], and cubic planar bipartite graphs [2, 19].. | **A**: So far,
all nonbipartite circulants have been nearly dispersable and we conjecture here that nonbipartite, vertex-transitive graphs are nearly dispersable.
Previous results support both conjectures.
**B**: Other classes of vertex-transitive graph that are known to be nearly dispersable include the product of two arbitrary cycles and of cycles with complete graphs, see [23, 25], and some products of bipartite and nonbipartite graphs [22].
**C**: See also [27] and §8.
| ABC | ABC | ABC | ABC | Selection 4 |
<|MaskedSetence|> We shall establish convergence of the corresponding Laplace transforms and characterize the limit by a second order linear differential equation of singular type, related to the Bessel differential equation. <|MaskedSetence|> We note that a corresponding differential equation for characteristic functions is hardly available. <|MaskedSetence|> Thus, if one would like to overcome the assumption of non-negativity, a different approach seems to be needed.
. | **A**: This would require the existence of the second moment, which for the inverse ΓΓ\Gammaroman_Γ-distribution is in general not at disposal.
**B**:
Let us turn to the proof of Theorem 2.
**C**: This approach necessitates that the terms 𝖠𝖠\mathsf{A}sansserif_A and 𝖡𝖡\mathsf{B}sansserif_B are non-negative.
| BCA | ABC | BCA | BCA | Selection 3 |
<|MaskedSetence|> They contain all algebraic numbers, as well as their logarithms, and some transcendental numbers like π𝜋\piitalic_π; they are exceedingly commonplace however not well understood.
We do not know how to decide (26), but we point out to some work that might prove to be helpful. One is Conjecture 1 in [KZ01], that says that if one period has two different representations as integrals, one can obtain one from the other through three simple operations: additivity, change of variables and Stokes’s formula. It is not clear however, even if the conjecture were to be true, how one can calculate a sequence of such operations. <|MaskedSetence|> <|MaskedSetence|> More seems to be known about the special case of curves [HW18], but in this case, for our purposes, we can give a more satisfactory answer by simpler means.
. | **A**: See [Ayo14] for definitions and a discussion about these two conjectures.
**B**: The class of numbers that can be expressed as integrals of algebraic functions over semialgebraic sets are known as periods [KZ01].
**C**: A more direct conjecture is one made by Grothendieck that predicts the transcendence degree of field extension of ℚℚ\mathbb{Q}blackboard_Q that are generated by a finite set of periods.
| CAB | BCA | BCA | BCA | Selection 3 |
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