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Mathematics

New submissions

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New submissions for Fri, 1 Mar 24

[1]  arXiv:2402.18580 [pdf, other]
Title: A short note on deformations of (strongly) Gorenstein-projective modules over the dual numbers
Comments: arXiv admin note: substantial text overlap with arXiv:2109.08015
Subjects: Representation Theory (math.RT)

Let $\mathbf{k}$ be a field of arbitrary characteristic, and let $\Lambda$ be a finite dimensional $\mathbf{k}$-algebra. In this short note we prove that if $V$ is a finitely generated strongly Gorenstein-projective left $\Lambda$-module whose stable endomorphism ring $\underline{\mathrm{End}}_{\Lambda}(V)$ is isomorphic to $\mathbf{k}$, then $V$ has an universal deformation ring $R(\Lambda,V)$ isomorphic to the ring of dual numbers $\mathbf{k}[\epsilon]$ with $\epsilon^2=0$. As a consequence, we obtain the following result. Assume that $Q$ is a finite connected acyclic quiver, let $\mathbf{k} Q$ be the corresponding path algebra and let $\Lambda = \mathbf{k} Q[\epsilon] = \mathbf{k} Q\otimes_{\mathbf{k}} \mathbf{k}[\epsilon]$. If $V$ is a finitely generated Gorenstein-projective left $\Lambda$-module with $\underline{\mathrm{End}}_{\Lambda}(V)=\mathbf{k}$, then $V$ has an universal deformation ring $R(\Lambda,V)$ isomorphic to $\mathbf{k}[\epsilon]

[2]  arXiv:2402.18585 [pdf, ps, other]
Title: The algebraic entropies of the Leavitt path algebra and the graph algebras agree
Subjects: Rings and Algebras (math.RA)

In this note we prove that the algebras $L_K(E)$ and $KE$ have the same entropy. Entropy is always referred to the standard filtrations in the corresponding kind of algebra. The main argument leans on (1) the holomorphic functional calculus; (2) the relation of entropy with suitable norm of the adjacency matrix; and (3) the Cohn path algebras which yield suitable bounds for the algebraic entropies.

[3]  arXiv:2402.18586 [pdf, ps, other]
Title: K-Primitivity : A Literature Survey
Authors: Monimala Nej
Comments: 19 pages
Subjects: History and Overview (math.HO); Combinatorics (math.CO)

A nonnegative matrix A is said to be primitive if there exists a positive integer m such that entries in A^m are positive and smallest such m is called the exponent of A: Primitive matrices are useful in the study of finite Markov chains theory. In 1998, in the context of finite Markov chains, Ettore Fornasini and Maria Elena Valcher [6] extended the notion of primitivity for a nonnegative matrix pair (A;B) by considering a positive discrete homogeneous two-dimensional (2D) state model. Further generalization to this notion of primitivity for k-tuple (A1;A2;...;Ak) of nonnegative matrices A1;A2;...;Ak is quite natural and known as k-primitivity. In this paper we present various results on k-primitivity given by different researchers from time to time.

[4]  arXiv:2402.18660 [pdf, other]
Title: Versatile mixed methods for compressible flows
Comments: 28 pages, 4 figures, 4 tables
Subjects: Numerical Analysis (math.NA)

Versatile mixed finite element methods were originally developed by Chen and Williams for isothermal incompressible flows in "Versatile mixed methods for the incompressible Navier-Stokes equations," Computers & Mathematics with Applications, Volume 80, 2020. Thereafter, these methods were extended by Miller, Chen, and Williams to non-isothermal incompressible flows in "Versatile mixed methods for non-isothermal incompressible flows," Computers & Mathematics with Applications, Volume 125, 2022. The main advantage of these methods lies in their flexibility. Unlike traditional mixed methods, they retain the divergence terms in the momentum and temperature equations. As a result, the favorable properties of the schemes are maintained even in the presence of non-zero divergence. This makes them an ideal candidate for an extension to compressible flows, in which the divergence does not generally vanish. In the present article, we finally construct the fully-compressible extension of the methods. In addition, we demonstrate the excellent performance of the resulting methods for weakly-compressible flows that arise near the incompressible limit, as well as more strongly-compressible flows that arise near Mach 0.5.

[5]  arXiv:2402.18666 [pdf, other]
Title: Linear shrinkage for optimization in high dimensions
Subjects: Optimization and Control (math.OC); Statistics Theory (math.ST)

In large-scale, data-driven applications, parameters are often only known approximately due to noise and limited data samples. In this paper, we focus on high-dimensional optimization problems with linear constraints under uncertain conditions. To find high quality solutions for which the violation of the true constraints is limited, we develop a linear shrinkage method that blends random matrix theory and robust optimization principles. It aims to minimize the Frobenius distance between the estimated and the true parameter matrix, especially when dealing with a large and comparable number of constraints and variables. This data-driven method excels in simulations, showing superior noise resilience and more stable performance in both obtaining high quality solutions and adhering to the true constraints compared to traditional robust optimization. Our findings highlight the effectiveness of our method in improving the robustness and reliability of optimization in high-dimensional, data-driven scenarios.

[6]  arXiv:2402.18670 [pdf, other]
Title: The inverse eigenvalue problem for probe graphs
Subjects: Combinatorics (math.CO)

In this paper, we initiate the study of the inverse eigenvalue problem for probe graphs. A probe graph is a graph whose vertices are partitioned into probe vertices and non-probe vertices such that the non-probe vertices form an independent set. In general, a probe graph is used to represent the set of graphs that can be obtained by adding edges between non-probe vertices. The inverse eigenvalue problem for a graph considers a family of matrices whose zero-nonzero pattern is defined by the graph and asks which spectra are achievable by matrices in this family. We ask the same question for probe graphs. We start by establishing bounds on the maximum nullity for probe graphs and defining the probe graph zero forcing number. Next, we focus on graphs of two parallel paths, the unique family of graphs whose (standard) zero forcing number is two. We partially characterize the probe graph zero forcing number of such graphs and prove some necessary structural results about the family. Finally, we characterize probe graphs whose minimum rank is $0, 1, 2, n-2,$ and $n-1$.

[7]  arXiv:2402.18676 [pdf, other]
Title: Determining surfaces by short curves and applications
Subjects: Geometric Topology (math.GT)

The goal of this work is to give new quantitative results about the distribution of semi-arithmetic hyperbolic surfaces in the moduli space of closed hyperbolic surfaces. We show that two coverings of genus $g$ of a fixed arithmetic surface $S$ are $P(\frac{1}{g})$ apart from each other with respect to Teichmuller metric, where $P$ is a polynomial depending only on $S$ whose degree is universal. We also give a super-exponential upper bound for the number of semi-arithmetic hyperbolic surfaces with bounded genus, stretch and degree of invariant trace field, generalizing for this class similar well known bounds for arithmetic hyperbolic surfaces. In order to get these results we establish, for any closed hyperbolic surface $S$ with injectivity radius at least $s$, a parametrization of Teichmuller space by length functions whose values on $S$ are bounded by a linear function (with constants depending only on $s$) on the logarithmic of the genus of $S.$

[8]  arXiv:2402.18683 [pdf, other]
Title: Integrated Sensing and Communication Meets Smart Propagation Engineering: Opportunities and Challenges
Comments: 7 pages, 5 figures, submitted to IEEE journal for possible publication
Subjects: Information Theory (cs.IT); Signal Processing (eess.SP)

Both smart propagation engineering as well as integrated sensing and communication (ISAC) constitute promising candidates for next-generation (NG) mobile networks. We provide a synergistic view of these technologies, and explore their mutual benefits. First, moving beyond just intelligent surfaces, we provide a holistic view of the engineering aspects of smart propagation environments. By delving into the fundamental characteristics of intelligent surfaces, fluid antennas, and unmanned aerial vehicles, we reveal that more efficient control of the pathloss and fading can be achieved, thus facilitating intrinsic integration and mutual assistance between sensing and communication functionalities. In turn, with the exploitation of the sensing capabilities of ISAC to orchestrate the efficient configuration of radio environments, both the computational effort and signaling overheads can be reduced. We present indicative simulation results, which verify that cooperative smart propagation environment design significantly enhances the ISAC performance. Finally, some promising directions are outlined for combining ISAC with smart propagation engineering.

[9]  arXiv:2402.18692 [pdf, ps, other]
Title: $T$-polynomial convexity and holomorphic convexity
Subjects: Complex Variables (math.CV)

We compare the $T$-polynomial convexity of Guedj with holomorphic convexity away from the support of $T$. In particular we show an Oka--Weil theorem for $T$-polynomial convexity, as well as present a situation when the notions of $T$-polynomial convexity and holomorphic convexity of $X\setminus\text{Supp }T$ coincide in the context of complex projective algebraic manifolds.

[10]  arXiv:2402.18693 [pdf, ps, other]
Title: Symbolic Powers of Classical Varieties
Comments: 19 pages, comments, and suggestions are welcome
Subjects: Commutative Algebra (math.AC)

Let $R=\mathbb{K}[x_1,\dots,x_n]$ and $\mathfrak{a}_1,\dots,\mathfrak{a}_m$ are homogeneous ideals satisfying certain properties, which includes a description of the Noetherian symbolic Rees algebra. Then, we compute the Waldschmidt constant and resurgence and show that it exhibits a stronger version of the Chudnovsky and Demailly-type bounds. We further show that these properties are satisfied for classical varieties such as the generic determinantal ideals, minors of generic symmetric matrices, generic extended Hankel matrices, and ideal of pfaffians of skew-symmetric matrices.

[11]  arXiv:2402.18704 [pdf, ps, other]
Title: Square-difference factor absorbing ideals of a commutative ring
Comments: 18 pages
Subjects: Commutative Algebra (math.AC)

Let $R$ be a commutative ring with $1 \neq 0$. A proper ideal $I$ of $R$ is a {\it square-difference factor absorbing ideal} (sdf-absorbing ideal) of $R$ if whenever $a^2 - b^2 \in I$ for $0 \neq a, b \in R$, then $a + b \in I$ or $a - b \in I$. In this paper, we introduce and investigate sdf-absorbing ideals.

[12]  arXiv:2402.18706 [pdf, ps, other]
Title: The porous medium equation on noncompact manifolds with nonnegative Ricci curvature: a Green function approach
Subjects: Analysis of PDEs (math.AP); Differential Geometry (math.DG); Functional Analysis (math.FA)

We consider the porous medium equation (PME) on complete noncompact manifolds $M$ of nonnegative Ricci curvature. We require nonparabolicity of the manifold and construct a natural space $X$ of functions, strictly larger than $L^1$, in which the Green function on $M$ appears as a weight, such that the PME admits a solution in the weak dual (i.e. potential) sense whenever the initial datum $u_0$ is nonnegative and belongs to $X$. Smoothing estimates are also proved to hold both for $L^1$ data, where they take into account the volume growth of Riemannian balls giving rise to bounds which are shown to be sharp in a suitable sense, and for data belonging to $X$ as well.

[13]  arXiv:2402.18712 [pdf, ps, other]
Title: Equivariant Chern Classes of Toric Vector Bundles over a DVR and Bruhat--Tits Buildings
Comments: Comments welcome!
Subjects: Algebraic Geometry (math.AG)

We define equivariant Chern classes of a toric vector bundle over a proper toric scheme over a DVR. We provide a combinatorial description of them in terms of piecewise polynomial functions on the polyhedral complex associated to the toric scheme, which factorize through to an extended Bruhat--Tits building. We further motivate this definition from an arithmetic perspective, connecting to the non-Archimedean Arakelov theory of toric varieties.

[14]  arXiv:2402.18716 [pdf, other]
Title: Temperley-Lieb Crystals
Subjects: Combinatorics (math.CO); Quantum Algebra (math.QA); Rings and Algebras (math.RA); Representation Theory (math.RT)

Elements of Lusztig's dual canonical bases are Schur-positive when evaluated on (generalized) Jacobi-Trudi matrices. This deep property was proved by Rhoades and Skandera, relying on a result of Haiman, and ultimately on the (proof of) Kazhdan-Lusztig conjecture. For a particularly tractable part of the dual canonical basis - called Temperley-Lieb immanants - we give a generalization of Littlewood-Richardson rule: we provide a combinatorial interpretation for the coefficient of a particular Schur function in the evaluation of a particular Temperley-Lieb immanant on a particular Jacobi-Trudi matrix. For this we introduce shuffle tableaux, and apply Stembridge's axioms to show that certain graphs on shuffle tableaux are type $A$ Kashiwara crystals.

[15]  arXiv:2402.18717 [pdf, ps, other]
Title: Finiteness of the Arithmetic Casas-Alvero Scheme
Authors: Soham Ghosh
Subjects: Algebraic Geometry (math.AG); Commutative Algebra (math.AC)

The Casas-Alvero conjecture (2001) predicts that the $n^{th}$ "arithmetic Casas-Alvero scheme" $X_n\rightarrow \operatorname{Spec}(\mathbb{Z})$ (constructed by von Bothmer-Labs-Schicho-de Woestline) does not have any $\mathbb{K}$-rational points for any characteristic zero field $\mathbb{K}$ and $n\geq 2$. In this paper, we show that for any field $\mathbb{K}$ (of any characteristic), $X_n$ has at most finitely many $\mathbb{K}$-rational points for all $n\geq 2$. We reformulate the conjecture as a complete intersection problem, which enables us to interpret the finiteness of rational points on $X_n$, as an almost complete intersection phenomenon. Furthermore, we prove several rigidity results towards the conjecture.

[16]  arXiv:2402.18721 [pdf, other]
Title: A collocation method for nonlinear tensor differential equations on low-rank manifolds
Authors: Alec Dektor
Comments: 19 pages, 4 figures
Subjects: Numerical Analysis (math.NA); Computational Physics (physics.comp-ph)

We present a new method to compute the solution to a nonlinear tensor differential equation with dynamical low-rank approximation. The idea of dynamical low-rank approximation is to project the differential equation onto the tangent space of a low-rank tensor manifold at each time. Traditionally, an orthogonal projection onto the tangent space is employed, which is challenging to compute for nonlinear differential equations. We introduce a novel interpolatory projection onto the tangent space that is easily computed for many nonlinear differential equations and satisfies the differential equation at a set of carefully selected indices. To select these indices, we devise a new algorithm based on the discrete empirical interpolation method (DEIM) that parameterizes any tensor train and its tangent space with tensor cross interpolants. We demonstrate the proposed method with applications to tensor differential equations arising from the discretization of partial differential equations.

[17]  arXiv:2402.18725 [pdf, other]
Title: Leveraging the turnpike effect for Mean Field Games numerics
Comments: 24 pages, 12 figures
Subjects: Optimization and Control (math.OC); Probability (math.PR)

Recently, a deep-learning algorithm referred to as Deep Galerkin Method (DGM), has gained a lot of attention among those trying to solve numerically Mean Field Games with finite horizon, even if the performance seems to be decreasing significantly with increasing horizon. On the other hand, it has been proven that some specific classes of Mean Field Games enjoy some form of the turnpike property identified over seven decades ago by economists. The gist of this phenomenon is a proof that the solution of an optimal control problem over a long time interval spends most of its time near the stationary solution of the ergodic solution of the corresponding infinite horizon optimization problem. After reviewing the implementation of DGM for finite horizon Mean Field Games, we introduce a ``turnpike-accelerated'' version that incorporates the turnpike estimates in the loss function to be optimized, and we perform a comparative numerical analysis to show the advantages of this accelerated version over the baseline DGM algorithm. We demonstrate on some of the Mean Field Game models with local-couplings known to have the turnpike property, as well as a new class of linear-quadratic models for which we derive explicit turnpike estimates.

[18]  arXiv:2402.18737 [pdf, ps, other]
Title: Localization of Random Surfaces with Monotone Potentials and an FKG-Gaussian Correlation Inequality
Authors: Mark Sellke
Subjects: Probability (math.PR); Mathematical Physics (math-ph)

The seminal 1975 work of Brascamp-Lieb-Lebowitz initiated the rigorous study of Ginzberg-Landau random surface models. It was conjectured therein that fluctuations are localized on $\mathbb Z^d$ when $d\geq 3$ for very general potentials, matching the behavior of the Gaussian free field. We confirm this behavior for all even potentials $U:\mathbb R\to\mathbb R$ satisfying $U'(x)\geq \min(\varepsilon x,\frac{1+\varepsilon}{x})$ on $x\in \mathbb R^+$. Given correspondingly stronger growth conditions on $U$, we show power or stretched exponential tail bounds on all transient graphs, which determine the maximum field value up to constants in many cases. Further extensions include non-wired boundary conditions and iterated Laplacian analogs such as the membrane model. Our main tool is an FKG-based generalization of the Gaussian correlation inequality, which is used to dominate the finite-volume Gibbs measures by mixtures of centered Gaussian fields.

[19]  arXiv:2402.18739 [pdf, ps, other]
Title: Decomposability of regular graphs to $4$ locally irregular subgraphs
Authors: Jakub Przybyło
Comments: 16 pages
Subjects: Combinatorics (math.CO)

A locally irregular graph is a graph whose adjacent vertices have distinct degrees. It was conjectured that every connected graph is edge decomposable to $3$ locally irregular subgraphs, unless it belongs to a certain family of exceptions, including graphs of small maximum degrees, which are not decomposable to any number of such subgraphs. Recently Sedlar and \v{S}krekovski exhibited a counterexample to the conjecture, which necessitates a decomposition to (at least) $4$ locally irregular subgraphs. We prove that every $d$-regular graph with $d$ large enough, i.e. $d\geq 54000$, is decomposable to $4$ locally irregular subgraphs. Our proof relies on a mixture of a numerically optimized application of the probabilistic method and certain deterministic results on degree constrained subgraphs due to Addario-Berry, Dalal, McDiarmid, Reed, and Thomason, and to Alon and Wei, introduced in the context of related problems concerning irregular subgraphs.

[20]  arXiv:2402.18740 [pdf, other]
Title: Sixth-order parabolic equation on an interval: Eigenfunction expansion, Green's function, and intermediate asymptotics for a finite thin film with elastic resistance
Comments: 20 pages, 6 figures
Subjects: Numerical Analysis (math.NA); Fluid Dynamics (physics.flu-dyn)

A linear sixth-order partial differential equation (PDE) of ``parabolic'' type describes the dynamics of thin liquid films beneath surfaces with elastic bending resistance when deflections from the equilibrium film height are small. On a finite domain, the associated sixth-order Sturm--Liouville eigenvalue value problem is self-adjoint for the boundary conditions corresponding to a thin film in a closed trough, and the eigenfunctions form a complete orthonormal set. Using these eigenfunctions, we derive the Green's function for the governing sixth-order PDE on a finite interval and compare it to the known infinite-line solution. Further, we propose a Galerkin spectral method based on the constructed sixth-order eigenfunctions and their derivative expansions. The system of ordinary differential equations for the time-dependent expansion coefficients is solved by standard numerical methods. The numerical approach is applied to versions of the governing PDE with a second-order derivative (in addition to the sixth-order one), which arises from gravity acting on the film. In the absence of gravity, we demonstrate the self-similar intermediate asymptotics of initially localized disturbances on the film surface, at least until the disturbances ``feel'' the finite boundaries, and show that the derived Green's function is the global attractor for such solutions. In the presence of gravity, we use the proposed spectral numerical method to demonstrate that self-similar behavior persists, albeit for shortened intervals of time, even for large values of the gravity-to-bending ratio.

[21]  arXiv:2402.18770 [pdf, ps, other]
Title: Rational Cherednik Algebras and Torus Knot Invariants
Authors: Xinchun Ma
Comments: Comments are welcome!
Subjects: Representation Theory (math.RT); Combinatorics (math.CO)

The HOMFLY polynomial of the $(m,n)$ torus knot $T_{m,n}$ can be extracted from the doubly graded character of the finite-dimensional representation $\mathrm{L}_{\frac{m}{n}}$ of the type $A_{n-1}$ rational Cherednik algebra as observed by Gorsky, Oblomkov, Rasmussen and Shende. It is furthermore conjectured that one can obtain the triply-graded Khovanov-Rozansky homology of $T_{m,n}$ by considering a certain filtration on $\mathrm{L}_{\frac{m}{n}}$. In this paper, we show that two of the proposed candidates, the algebraic filtration and the inductive filtration, are equal.

[22]  arXiv:2402.18772 [pdf, ps, other]
Title: Sparsity of stable primes for dynamical sequences
Authors: Joachim König
Subjects: Number Theory (math.NT); Dynamical Systems (math.DS); Group Theory (math.GR)

We show that a dynamical sequence $(f_n)_{n\in \mathbb{N}}$ of polynomials over a number field whose set of stable primes is of positive density must necessarily have a very restricted, and in particular ``near-solvable" dynamical Galois group. Together with existing heuristics, our results suggest moreover that a polynomial $f$ all of whose iterates are irreducible modulo a positive density subset of the primes must necessarily be a composition of linear functions, monomials and Dickson polynomials.

[23]  arXiv:2402.18782 [pdf, other]
Title: Ivrii's conjecture for some cases in outer and symplectic billiards
Comments: 10 pages, 8 figures
Subjects: Dynamical Systems (math.DS); Symplectic Geometry (math.SG)

We give a proof for $(2n + 1,n)$ and $(2n, n-1)$-periodic Ivrii's conjecture for planar outer billiards. We also give new simple geometric proofs for the 3 and 4-periodic cases for outer and symplectic billiards, and generalize for higher dimensions in case of symplectic billiards.

[24]  arXiv:2402.18793 [pdf, other]
Title: An Adaptive Orthogonal Basis Method for Computing Multiple Solutions of Differential Equations with polynomial nonlinearities
Subjects: Numerical Analysis (math.NA)

This paper presents an innovative approach, the Adaptive Orthogonal Basis Method, tailored for computing multiple solutions to differential equations characterized by polynomial nonlinearities. Departing from conventional practices of predefining candidate basis pools, our novel method adaptively computes bases, considering the equation's nature and structural characteristics of the solution. It further leverages companion matrix techniques to generate initial guesses for subsequent computations. Thus this approach not only yields numerous initial guesses for solving such equations but also adapts orthogonal basis functions to effectively address discretized nonlinear systems. Through a series of numerical experiments, this paper demonstrates the method's effectiveness and robustness. By reducing computational costs in various applications, this novel approach opens new avenues for uncovering multiple solutions to differential equations with polynomial nonlinearities.

[25]  arXiv:2402.18795 [pdf, ps, other]
Title: Towards Large-scale Probabilistic Set Covering Problem: An Efficient Benders Decomposition Approach
Comments: 13 pages, accepted for publication in IOS 2024
Subjects: Optimization and Control (math.OC)

In this paper, we investigate the probabilistic set covering problems (PSCP) in which the right-hand side is a random vector {\xi} and the covering constraint is required to be satisfied with a prespecified probability. We consider the case arising from sample average approximation (or finite discrete distributions). We develop an effective Benders decomposition (BD) algorithm for solving large-scale PSCPs, which enjoys two key advantages: (i) the number of variables in the underlying Benders reformulation is independent of the scenario size; and (ii) the Benders cuts can be separated by an efficient combinatorial algorithm. For the special case that {\xi} is a combination of several independent random blocks/subvectors, we explicitly take this kind of block structure into consideration and develop a more efficient BD algorithm. Numerical results on instances with up to one million scenarios demonstrate the effectiveness of the proposed BD algorithms over a black-box MIP solver's branch-and-cut and automatic BD algorithms and a state-of-the-art algorithm in the literature.

[26]  arXiv:2402.18799 [pdf, other]
Title: Allen-Cahn equation and degenerate minimal hypersurface
Comments: 15 pages, 2 figures. Comments are welcome!
Subjects: Differential Geometry (math.DG); Analysis of PDEs (math.AP)

In this short note, we present new observations and examples concerning the existence and rigidity of solutions to the Allen-Cahn equation with degenerate minimal hypersurfaces as their limit interfaces.

[27]  arXiv:2402.18810 [pdf, ps, other]
Title: The numeraire e-variable
Subjects: Statistics Theory (math.ST); Methodology (stat.ME)

We consider testing a composite null hypothesis $\mathcal{P}$ against a point alternative $\mathbb{Q}$. This paper establishes a powerful and general result: under no conditions whatsoever on $\mathcal{P}$ or $\mathbb{Q}$, we show that there exists a special e-variable $X^*$ that we call the numeraire. It is strictly positive and for every $\mathbb{P} \in \mathcal{P}$, $\mathbb{E}_\mathbb{P}[X^*] \le 1$ (the e-variable property), while for every other e-variable $X$, we have $\mathbb{E}_\mathbb{Q}[X/X^*] \le 1$ (the numeraire property). In particular, this implies $\mathbb{E}_\mathbb{Q}[\log(X/X^*)] \le 0$ (log-optimality). $X^*$ also identifies a particular sub-probability measure $\mathbb{P}^*$ via the density $d \mathbb{P}^*/d \mathbb{Q} = 1/X^*$. As a result, $X^*$ can be seen as a generalized likelihood ratio of $\mathbb{Q}$ against $\mathcal{P}$. We show that $\mathbb{P}^*$ coincides with the reverse information projection (RIPr) when additional assumptions are made that are required for the latter to exist. Thus $\mathbb{P}^*$ is a natural definition of the RIPr in the absence of any assumptions on $\mathcal{P}$ or $\mathbb{Q}$. In addition to the abstract theory, we provide several tools for finding the numeraire in concrete cases. We discuss several nonparametric examples where we can indeed identify the numeraire, despite not having a reference measure. We end with a more general optimality theory that goes beyond the ubiquitous logarithmic utility. We focus on certain power utilities, leading to reverse R\'enyi projections in place of the RIPr, which also always exists.

[28]  arXiv:2402.18814 [pdf, other]
Title: New topological subsystem codes from semi-regular tessellations
Subjects: Information Theory (cs.IT)

In this work, we present new constructions for topological subsystem codes using semi-regular Euclidean and hyperbolic tessellations. They give us new families of codes, and we also provide a new family of codes obtained through an already existing construction, due to Sarvepalli and Brown. We also prove new results that allow us to obtain the parameters of these new codes.

[29]  arXiv:2402.18822 [pdf, ps, other]
Title: Hausdorff dimensions of affine multiplicative subshifts
Subjects: Dynamical Systems (math.DS)

We calculate the Minkowski and Hausdorff dimensions of affine multiplicative subshifts on $\mathbb{N}$.

[30]  arXiv:2402.18832 [pdf, ps, other]
Title: A necessary and sufficient condition for bound on the sum of a list of real numbers and its applications
Authors: Xiwu Yang
Comments: 34 pages, 9 figures
Subjects: Combinatorics (math.CO)

Let $x_1,...,x_n$ be a list of real numbers, let $s :=\sum_{i=1}^{n}x_i$, and let $h:\mathbb{N} \rightarrow \mathbb{R}$ be a function. We gave a necessary and sufficient condition for $s>h(n)$ (respectively, $s<h(n)$). Let $G=(V,E)$ be a graph, let $\{H_1,...,H_n\}$ and $\{V_1,...,V_n\}$ be a decomposition and a partition of $G$, respectively. $G$ is \emph{generalized periodic} or \emph{partition-transitive} if for each pair of integers $(i,j), 1\leq i\leq j\leq n$, there exists an automorphism $\theta_{i,j}$ of $G$ such that $\theta_{i,j}(H_{i+k})=H_{j+k}$ or $\theta_{i,j}(V_{i+k})=V_{j+k}$ for all $k$, $1\leq k\leq n$, respectively, where subscripts are taken modulo $n$. Let $f:E \rightarrow \mathbb{R}$ and $g:V \rightarrow \mathbb{R}$ be mappings, and let the \emph{weight} of $f$ or $g$ on $G$ be $\Sigma_{e\in E}f(e)$ or $\Sigma_{v\in V}g(v)$, respectively. Suppose that parameters $\lambda$ and $\xi$ of $G$ can be expressed as the minimum or maximum weight of specified $f$ and $g$, respectively. Then our conditions imply a necessary and sufficient condition for $\lambda(G_1)=h(n)$ (respectively, $\xi(G_2)=h(n)$), where $G_1$ is generalized periodic and $G_2$ is partition-transitive. For example, $\textrm{cr}(\odot(T^n))=h(n)$, where $\textrm{cr}(\odot(T^n))$ is the crossing number of a periodic graph $\odot(T^n)$. As applications, we determined the crossing number of the circulant $C(4n;\{1,4\})$, the paired domination number of $C_5\Box C_n$ and the upper total domination number of $C_4\Box C_n$.

[31]  arXiv:2402.18843 [pdf, other]
Title: A variation of parameters formula for nonautonomous linear impulsive differential equations with piecewise constant arguments of generalized type
Comments: 25 pages, 6 figures, 1 table
Subjects: Dynamical Systems (math.DS); Functional Analysis (math.FA)

In this work, we give a variation of parameters formula for nonautonomous linear impulsive differential equations with piecewise constant arguments of generalized type. We cover several cases of differential equations with deviated arguments investigated before as particular cases. We also give some examples showing the applicability of our results.

[32]  arXiv:2402.18845 [pdf, other]
Title: Angle Parametrization of Teichmüller space
Comments: 7 pages
Subjects: Geometric Topology (math.GT)

Let $S_g$ be a closed orientable surface of genus $g \geq 3$, and let $\mathcal{T}_g$ be the Teichm\"{u}ller space of $S_g$. In this work, we have proved that $\mathcal{T}_g$ can be parametrized by $6g-5$ angle parameters.

[33]  arXiv:2402.18847 [pdf, other]
Title: Flexible Precoding for Multi-User Movable Antenna Communications
Subjects: Information Theory (cs.IT); Signal Processing (eess.SP)

This letter rethinks traditional precoding in multi-user wireless communications with movable antennas (MAs). Utilizing MAs for optimal antenna positioning, we introduce a sparse optimization (SO)-based approach focusing on regularized zero-forcing (RZF). This framework targets the optimization of antenna positions and the precoding matrix to minimize inter-user interference and transmit power. We propose an off-grid regularized least squares-based orthogonal matching pursuit (RLS-OMP) method for this purpose. Moreover, we provide deeper insights into antenna position optimization using RLS-OMP, viewed from a subspace projection angle. Overall, our proposed flexible precoding scheme demonstrates a sum rate that exceeds more than twice that of fixed antenna positions.

[34]  arXiv:2402.18857 [pdf, ps, other]
Title: Arithmetic and birational properties of linear spaces on intersections of two quadrics
Comments: 26 pages, comments are welcome
Subjects: Algebraic Geometry (math.AG)

We study rationality questions for Fano schemes of linear spaces on smooth complete intersections of two quadrics, especially over non-closed fields. Our approach is to study hyperbolic reductions of the pencil of quadrics associated to $X$. We prove that the Fano schemes $F_r(X)$ of $r$-planes are birational to symmetric powers of hyperbolic reductions, generalizing results of Reid and Colliot-Th\'el\`ene--Sansuc--Swinnerton-Dyer, and we give several applications to rationality properties of $F_r(X)$.
For instance, we show that if $X$ contains an $(r+1)$-plane over a field $k$, then $F_r(X)$ is rational over $k$. When $X$ has odd dimension, we show a partial converse for rationality of the Fano schemes of second maximal linear spaces, generalizing results of Hassett--Tschinkel and Benoist--Wittenberg. When $X$ has even dimension, the analogous result does not hold, and we further investigate this situation over the real numbers. In particular, we prove a rationality criterion for the Fano schemes of second maximal linear spaces on these even-dimensional complete intersections over $\mathbb R$; this may be viewed as extending work of Hassett--Koll\'ar--Tschinkel.

[35]  arXiv:2402.18860 [pdf, ps, other]
Title: Error estimation for finite element method on meshes that contain thin elements
Subjects: Numerical Analysis (math.NA)

In an error estimation of finite element solutions to the Poisson equation, we usually impose the shape regularity assumption on the meshes to be used. In this paper, we show that even if the shape regularity condition is violated, the standard error estimation can be obtained if "bad" elements (elements that violate the shape regularity or maximum angle condition) are covered virtually by "good" simplices. A numerical experiment confirms the theoretical result.

[36]  arXiv:2402.18869 [pdf, other]
Title: Evaluating the Gilbert-Varshamov Bound for Constrained Systems
Comments: 27 Pages, 5 figures, submitted to Entropy
Subjects: Information Theory (cs.IT); Discrete Mathematics (cs.DM); Combinatorics (math.CO)

We revisit the well-known Gilbert-Varshamov (GV) bound for constrained systems. In 1991, Kolesnik and Krachkovsky showed that GV bound can be determined via the solution of some optimization problem. Later, Marcus and Roth (1992) modified the optimization problem and improved the GV bound in many instances. In this work, we provide explicit numerical procedures to solve these two optimization problems and hence, compute the bounds. We then show the procedures can be further simplified when we plot the respective curves. In the case where the graph presentation comprise a single state, we provide explicit formulas for both bounds.

[37]  arXiv:2402.18872 [pdf, other]
Title: Semistatic robust utility indifference valuation and robust integral functionals
Authors: Keita Owari
Comments: 20 pages
Subjects: Functional Analysis (math.FA); Probability (math.PR); Mathematical Finance (q-fin.MF)

We consider a discrete-time robust utility maximisation with semistatic strategies, and the associated indifference prices of exotic options. For this purpose, we introduce a robust form of convex integral functionals on the space of bounded continuous functions on a Polish space, and establish some key regularity and representation results, in the spirit of the classical Rockafellar theorem, in terms of the duality formed with the space of Borel measures. These results (together with the standard Fenchel duality and minimax theorems) yield a duality for the robust utility maximisation problem as well as a representation of associated indifference prices, where the presence of static positions in the primal problem appears in the dual problem as a marginal constraint on the martingale measures. Consequently, the resulting indifference prices are consistent with the observed prices of vanilla options.

[38]  arXiv:2402.18896 [pdf, other]
Title: On the maximum size of variable-length non-overlapping codes
Authors: Geyang Wang, Qi Wang
Subjects: Information Theory (cs.IT); Combinatorics (math.CO)

Non-overlapping codes are a set of codewords such that the prefix of each codeword is not a suffix of any codeword in the set, including itself. If the lengths of the codewords are variable, it is additionally required that every codeword is not contained in any other codeword as a subword. Let $C(n,q)$ be the maximum size of $q$-ary fixed-length non-overlapping codes of length $n$. The upper bound on $C(n,q)$ has been well studied. However, the nontrivial upper bound on the maximum size of variable-length non-overlapping codes of length at most $n$ remains open. In this paper, by establishing a link between variable-length non-overlapping codes and fixed-length ones, we are able to show that the size of a $q$-ary variable-length non-overlapping code is upper bounded by $C(n,q)$. Furthermore, we prove that the average length of the codewords in a $q$-ary variable-length non-overlapping codes is lower bounded by $\lceil \log_q \tilde{C} \rceil$, and is asymptotically no shorter than $n-2$ as $q$ approaches $\infty$, where $\tilde{C}$ denotes the cardinality of $q$-ary variable-length non-overlapping codes of length up to $n$.

[39]  arXiv:2402.18903 [pdf, other]
Title: An Adaptive Hybrid Genetic and Large Neighborhood Search Approach for Multi-Attribute Vehicle Routing Problems
Subjects: Optimization and Control (math.OC)

Known for its dynamic utilization of destroy and repair operators, the Adaptive Large Neighborhood Search (ALNS) seeks to unearth high-quality solutions and has thus gained widespread acceptance as a meta-heuristic tool for tackling complex Combinatorial Optimization Problems (COPs). However, challenges arise when applying uniform parameters and acceptance criteria to diverse instances of the same COP, resulting in inconsistent performance outcomes. To address this inherent limitation, we propose the Adaptive Hybrid Genetic Search and Large Neighborhood Search (AHGSLNS), a novel approach designed to adapt ALNS parameters and acceptance criteria to the specific nuances of distinct COP instances. Our evaluation focuses on the Multi-Attribute Vehicle Routing Problem, a classical COP prevalent in real-world semi-automated storage and retrieval robotics systems. Empirical findings showcase that AHGSLNS not only competes effectively with ALNS under varying parameters but also exhibits superior performance in terms of convergence and stability. In alignment with our dedication to research transparency, the implementation of the proposed approach will be made publicly available.

[40]  arXiv:2402.18907 [pdf, ps, other]
Title: Boundary estimates and Green function's expansion for elliptic systems with random coefficients
Authors: Li Wang, Qiang Xu
Comments: Readers' comments and suggestions are welcome; 36 pages
Subjects: Analysis of PDEs (math.AP)

Focused on elliptic operators with stationary random coefficients of integrable correlations, stemming from stochastic homogenization theory, this paper primarily aims to investigate boundary estimates. As practical applications, we establish decay estimates for Green functions in both the quenched and annealed senses, as well as, some useful annealed estimates (including CLT-scaling) for boundary correctors. By extending Bella-Giunti-Otto's lemma in \cite{Bella-Giunti-Otto17} (to the version with a boundary condition), we ultimately obtain an error estimate for two-scale expansions of Green functions at mixed derivatives level, and thereby build connections to other interesting fields.

[41]  arXiv:2402.18914 [pdf, ps, other]
Title: Smooth Structures on $M^n\times\mathbb{S}^k$
Subjects: Algebraic Topology (math.AT)

This paper explores various differentiable structures on the product manifold $M \times \mathbb{S}^k$, where $M$ is either a 4-dimensional closed oriented manifold or a simply connected 5-dimensional closed manifold. We identify the possible stable homotopy types of $M$ and use it to calculate the concordance inertia group and the concordance structure set of $M\times\mathbb{S}^k$ for $1\leq k\leq 10.$ These calculations enable us to further classify all manifolds that are homeomorphic to $\mathbb{C}P^2\times\mathbb{S}^k$, up to diffeomorphism, for each $4\leq k\leq 6$.

[42]  arXiv:2402.18916 [pdf, ps, other]
Title: The JSJ-decomposition of the 3-manifold obtained by 0-surgery along a classical pretzel knot of genus one
Authors: Nozomu Sekino
Comments: 19 pages, 19 figures
Subjects: Geometric Topology (math.GT)

We consider the JSJ-decomposition of the 3-manifold obtained by 0-surgery along a classical pretzel knot of genus one. We use the classification of exceptional fillings of minimally twisted five-chain links by B. Martelli, C. Petronio and F. Roukema.

[43]  arXiv:2402.18921 [pdf, other]
Title: Semi-Supervised U-statistics
Subjects: Statistics Theory (math.ST); Methodology (stat.ME); Machine Learning (stat.ML)

Semi-supervised datasets are ubiquitous across diverse domains where obtaining fully labeled data is costly or time-consuming. The prevalence of such datasets has consistently driven the demand for new tools and methods that exploit the potential of unlabeled data. Responding to this demand, we introduce semi-supervised U-statistics enhanced by the abundance of unlabeled data, and investigate their statistical properties. We show that the proposed approach is asymptotically Normal and exhibits notable efficiency gains over classical U-statistics by effectively integrating various powerful prediction tools into the framework. To understand the fundamental difficulty of the problem, we derive minimax lower bounds in semi-supervised settings and showcase that our procedure is semi-parametrically efficient under regularity conditions. Moreover, tailored to bivariate kernels, we propose a refined approach that outperforms the classical U-statistic across all degeneracy regimes, and demonstrate its optimality properties. Simulation studies are conducted to corroborate our findings and to further demonstrate our framework.

[44]  arXiv:2402.18928 [pdf, ps, other]
Title: Global well-posedness and long time behavior of 2D MHD equations with partial dissipation in half space
Comments: arXiv admin note: text overlap with arXiv:2401.10456
Subjects: Analysis of PDEs (math.AP)

In this paper, we obtain the low order global well-posedness and the asymptotic behavior of solution of 2D MHD problem with partial dissipation in half space with non-slip boundary condition. When magnetic field equal zero, the system be reduced to partial dissipation Navier-Stokes equation, so this result also implies the stabilizing effects of magnetic field in electrically conducting fluids. We use the resolvent estimate method to obtain the long time behavior for the solution of weak diffusion system, which is not necessary to prove global well-posedness.

[45]  arXiv:2402.18931 [pdf, ps, other]
Title: On the discrete analogues of Appell function $F_4$
Subjects: Classical Analysis and ODEs (math.CA)

In this paper, we study the Appell function $F_4$ from discrete point of view. In particular, we obtain regions of convergence, difference-differential equations, finite and infinite summation formulas and a list of recursion relations satisfied by the discrete analogues of Appell function $F_4$.

[46]  arXiv:2402.18937 [pdf, other]
Title: Equivalence of ADER and Lax-Wendroff in DG / FR framework for linear problems
Subjects: Numerical Analysis (math.NA)

ADER (Arbitrary high order by DERivatives) and Lax-Wendroff (LW) schemes are two high order single stage methods for solving time dependent partial differential equations. ADER is based on solving a locally implicit equation to obtain a space-time predictor solution while LW is based on an explicit Taylor's expansion in time. We cast the corrector step of ADER Discontinuous Galerkin (DG) scheme into an equivalent quadrature free Flux Reconstruction (FR) framework and then show that the obtained ADER-FR scheme is equivalent to the LWFR scheme with D2 dissipation numerical flux for linear problems. This also implies that the two schemes have the same Fourier stability limit for time step size. The equivalence is verified by numerical experiments.

[47]  arXiv:2402.18939 [pdf, ps, other]
Title: Positive values of non-homogeneous quadratic forms of type (1,4): A conjecture of Bambah, Dumir and Hans-Gill
Comments: 128 pages
Subjects: Number Theory (math.NT)

Let $Q(x_1, \cdots,x_n)$ be a real indefinite quadratic form of the type $(r,s)$, $n=r+s$, signature $\sigma=r-s$ and determinant $D\neq 0$. Let $\Gamma_{r,n-r}$ denote the infimum of all numbers $\Gamma$ such that for any real numbers $c_1, c_2 ,\cdots, c_n$ there exist integers $x_1, x_ 2,\cdots, x_n$ satisfying $$0< Q(x_1+c_1,x_2+c_2,\cdots,x_n+c_n)\leq (\Gamma |D|)^{1/n}.$$ All the values of $\Gamma_{r,n-r}$ are known except for $\Gamma_{1,4}$. Earlier it was shown that $8\leq \Gamma_{1,4}<12$. It is conjectured that $\Gamma_{1,4}=8$. Here we shall prove that $\Gamma_{1,4}=8$, when (i) $c_2 \not \equiv 0 \pmod 1$, (ii) $c_2 \equiv 0 \pmod 1$, $a\geq \frac{1}{2}$, where $a$ is minima of positive definite ternary quadratic forms with determinant $4|D|$, and (iii) in some cases of $c_2 \equiv 0 \pmod 1$, $a< \frac{1}{2}$. We also obtain six critical forms for which the constant 8 is attained. In the remaining cases we prove that $\Gamma_{1,4}< \frac{32}{3}$.

[48]  arXiv:2402.18948 [pdf, other]
Title: Towards the boundary of the fine curve graph
Comments: 24 pages, 2 figures
Subjects: Geometric Topology (math.GT); Dynamical Systems (math.DS); Group Theory (math.GR)

The fine curve graph was introduced as a geometric tool to study the homeomorphisms of surfaces. In this paper we study the Gromov boundary of this space and the local topology near points associated with minimal measurable foliations. We then give several applications including finding explicit elements with positive stable commutator length, and proving a Tits alternative for subgroups of $\textrm{Homeo}(S)$ containing a pseudo-Anosov map, generalizing a result of Hurtado-Xue.

[49]  arXiv:2402.18952 [pdf, ps, other]
Title: A classification of 2-dimensional endo-commutative straight algebras of type II_1
Comments: 33 pages
Subjects: Rings and Algebras (math.RA)

In this paper, we present a complete classification of 2-dimensional endo-commutative straight algebras of type II$_1$ over any field. An endo-commutative algebra is a non-associative algebra in which the square mapping preserves multiplication. A 2-dimensional straight algebra satisfies the condition that there exists an element $x$ such that $x$ and $x^2$ are linearly independent. The term type II$_1$ denotes a distinguishing characteristic of its structure matrix, which has rank 2. We provide multiplication tables for these algebras, listing them up to isomorphism.

[50]  arXiv:2402.18955 [pdf, other]
Title: Santaló Geometry of Convex Polytopes
Comments: 25 pages, 4 figures, comments welcome
Subjects: Algebraic Geometry (math.AG); Optimization and Control (math.OC)

The Santal\'o point of a convex polytope is the interior point which leads to a polar dual of minimal volume. This minimization problem is relevant in interior point methods for convex optimization, where the logarithm of the dual volume is known as the universal barrier function. When translating the facet hyperplanes, the Santal\'o point traces out a semi-algebraic set. We describe and compute this geometry using algebraic and numerical techniques. We exploit connections with statistics, optimization and physics.

[51]  arXiv:2402.18961 [pdf, ps, other]
Title: A new interacting Fock space, the Quon algebra with operator parameter and its Wick's theorem
Authors: Yungang Lu
Subjects: Mathematical Physics (math-ph)

Motivated by the creation-annihilation operators in a newly defined interacting Fock space, we initiate the introduction and the study of the Quon algebra. This algebra serves as an extension of the conventional quon algebra, where the traditional constant parameter $q$ found in the $q$--commutation relation is replaced by a specific operator. Importantly, our investigation aims to establish Wick's theorem in the Quon algebra, offering valuable insights into its properties and applications.

[52]  arXiv:2402.18966 [pdf, ps, other]
Title: Vector Valued Gårding Inequality for pseudo-differential operators on compact homogeneous manifolds
Subjects: Analysis of PDEs (math.AP)

We obtain sufficient conditions in order to obtain a sharp G\r{a}rding inequality for pseudo-differential operators acting on vector-valued functions on compact Lie groups. As a consequence, we obtain a sharp G\r{a}rding inequality for compact homogeneous vector bundles and compact homogeneous manifolds. The sharp G\r{a}rding inequality is the strongest lower bound estimate known to hold for systems on $\mathbb{R}^n$, and the aim of this paper is to extend this property to systems on compact Lie groups and compact homogeneous manifolds. As an application, we establish existence and uniqueness of solution to a class of systems of initial value problems of pseudo-differential equations on compact Lie groups and compact homogeneous manifolds.

[53]  arXiv:2402.18980 [pdf, ps, other]
Title: Helper Data Schemes for Coded Modulation and Shaping in Physical Unclonable Functions
Subjects: Information Theory (cs.IT)

In this paper, we consider the generation and utilization of helper data for physical unclonable functions (PUFs) that provide real-valued readout symbols. Compared to classical binary PUFs, more entropy can be extracted from each basic building block (PUF node), resulting in longer keys/fingerprints and/or a higher reliability. To this end, a coded modulation and signal shaping scheme that matches the (approximately) Gaussian distribution of the readout has to be employed. A new helper data scheme is proposed that works with any type of coded modulation/shaping scheme. Compared to the permutation scheme from the literature, less amount of helper data has to be generated and a higher reliability is achieved. Moreover, the recently proposed idea of a two-metric helper data scheme is generalized to coded modulation and a general S-metric scheme. It is shown how extra helper data can be generated to improve decodability. The proposed schemes are assessed by numerical simulations and by evaluation of measurement data. We compare multi-level codes using a new rate design strategy with bit-interleaved coded modulation and trellis shaping with a distribution matcher. By selecting a suitable design, the rate per PUF node that can be reliably extracted can be as high as 2~bit/node.

[54]  arXiv:2402.18982 [pdf, other]
Title: Splitting integrators for linear Vlasov equations with stochastic perturbations
Subjects: Numerical Analysis (math.NA); Probability (math.PR)

We consider a class of linear Vlasov partial differential equations driven by Wiener noise. Different types of stochastic perturbations are treated: additive noise, multiplicative It\^o and Stratonovich noise, and transport noise. We propose to employ splitting integrators for the temporal discretization of these stochastic partial differential equations. These integrators are designed in order to preserve qualitative properties of the exact solutions depending on the stochastic perturbation, such as preservation of norms or positivity of the solutions. We provide numerical experiments in order to illustrate the properties of the proposed integrators and investigate mean-square rates of convergence.

[55]  arXiv:2402.18983 [pdf, other]
Title: Free energy expansions of a conditional GinUE and large deviations of the smallest eigenvalue of the LUE
Comments: 40 pages, 7 figures
Subjects: Mathematical Physics (math-ph); Complex Variables (math.CV); Probability (math.PR)

We consider a planar Coulomb gas ensemble of size $N$ with the inverse temperature $\beta=2$ and external potential $Q(z)=|z|^2-2c \log|z-a|$, where $c>0$ and $a \in \mathbb{C}$. Equivalently, this model can be realised as $N$ eigenvalues of the complex Ginibre matrix of size $(c+1) N \times (c+1) N$ conditioned to have deterministic eigenvalue $a$ with multiplicity $cN$. Depending on the values of $c$ and $a$, the droplet reveals a phase transition: it is doubly connected in the post-critical regime and simply connected in the pre-critical regime. In both regimes, we derive precise large-$N$ expansions of the free energy up to the $O(1)$ term, providing a non-radially symmetric example that confirms the Zabrodin-Wiegmann conjecture made for general planar Coulomb gas ensembles. As a consequence, our results provide asymptotic behaviours of moments of the characteristic polynomial of the complex Ginibre matrix, where the powers are of order $O(N)$. Furthermore, by combining with a duality formula, we obtain precise large deviation probabilities of the smallest eigenvalue of the Laguerre unitary ensemble. Our proof is based on a refined Riemann-Hilbert analysis for planar orthogonal polynomials using the partial Schlesinger transform.

[56]  arXiv:2402.18984 [pdf, ps, other]
Title: Graph Burning: Bounds and Hardness
Comments: 22 pages, 6 figures
Subjects: Combinatorics (math.CO); Discrete Mathematics (cs.DM)

The burning number of a graph $G$, denoted by $b(G)$, is the minimum number of steps required to burn all the vertices of a graph where in each step the existing fire spreads to all the adjacent vertices and one additional vertex can be burned as a new fire source. In this paper, we study the burning number problem both from an algorithmic and a structural point of view. The decision problem of computing the burning number of an input graph is known to be NP-Complete for trees with maximum degree at most three and interval graphs. Here, we prove that this problem is NP-Complete even when restricted to connected proper interval graphs and connected cubic graphs. The well-known burning number conjecture asserts that all the vertices of any graph of order $n$ can be burned in $\lceil \sqrt{n}~\rceil$ steps. In line with this conjecture, upper and lower bounds of $b(G)$ are well-studied for various special graph classes. Here, we provide an improved upper bound for the burning number of connected $P_k$-free graphs and show that the bound is tight up to an additive constant $1$. Finally, we study two variants of the problem, namely edge burning (only edges are burned) and total burning (both vertices and edges are burned). In particular, we establish their relationship with the burning number problem and evaluate the complexity of these variants.

[57]  arXiv:2402.18987 [pdf, ps, other]
Title: The Catalan's triangle system, the Catalan's trapezoids and (q,2)--Fock space
Authors: Yungang Lu
Subjects: Combinatorics (math.CO)

We provide an explicit formulation for the solution to the Catalan's triangle system using Catalan's trapezoids and a specified boundary condition. Additionally, we study this system with various boundary conditions obtained by utilizing different types of Fock spaces.

[58]  arXiv:2402.18999 [pdf, ps, other]
Title: Mixing Times for the Facilitated Exclusion Process
Comments: 32 pages, 9 figures
Subjects: Probability (math.PR); Statistical Mechanics (cond-mat.stat-mech)

The facilitated simple exclusion process (FEP) is a one-dimensional exclusion process with a dynamical constraint. We establish bounds on the mixing time of the FEP on the segment, with closed boundaries, and the circle. The FEP on these spaces exhibits transient states that, if the macroscopic density of particles is at least $1/2$, the process will eventually exit to reach an ergodic component. If the macroscopic density is less than $1/2$ the process will hit an absorbing state. We show that the symmetric FEP (SFEP) on the segment $\{1,\ldots,N\}$, with $k>N/2$ particles, has mixing time of order $N^{2}\log(N-k)$ and exhibits the pre-cutoff phenomenon. For the asymmetric FEP (AFEP) on the segment, we show that there exists initial conditions for which the hitting time of the ergodic component is exponentially slow in the number of holes $N-k$. In particular, when $N-k$ is large enough, the hitting time of the ergodic component determines the mixing time. For the SFEP on the circle of size $N$, and macroscopic particle density $\rho \in(1/2,1)$, we establish bounds on the mixing time of order $N^{2}\log N$ for the process restricted to its ergodic component. We also give an upper bound on the hitting time of the ergodic component of order $N^{2}\log N$ for a large class of initial conditions. The proofs rely on couplings with exclusion processes (both open and closed boundaries) via a novel lattice path (height function) construction of the FEP.

[59]  arXiv:2402.19000 [pdf, ps, other]
Title: A note on subgroups whose quotients are quasi-lines
Subjects: Group Theory (math.GR)

We study finitely generated pairs of groups $H \leq G$ such that the Schreier graph of $H$ is quasi-isometric to a line. Under this hypothesis, we show that $H$ is a virtual fiber subgroup if and only if $G$ contains infinitely many double cosets of $H$. Along the way, we prove that if a group acts essentially on a finite dimensional CAT(0) cube complex with no facing triples then it virtually surjects onto the integers with kernel commensurable to a hyperplane stabiliser.

[60]  arXiv:2402.19003 [pdf, ps, other]
Title: GVZ-groups with two character degrees
Subjects: Group Theory (math.GR)

We investigate the finite groups $G$ for which $\chi(1)^{2}=|G:Z(\chi)|$ for all characters $\chi \in Irr(G)$ and $|cd(G)|=2$. We obtain some alternate characterizations of these groups and we obtain some information regarding the structure of these groups.

[61]  arXiv:2402.19015 [pdf, other]
Title: Fractional material derivative: pointwise representation and a finite volume numerical scheme
Subjects: Numerical Analysis (math.NA)

The fractional material derivative appears as the fractional operator that governs the dynamics of the scaling limits of L\'evy walks - a stochastic process that originates from the famous continuous-time random walks. It is usually defined as the Fourier-Laplace multiplier, therefore, it can be thought of as a pseudo-differential operator. In this paper, we show that there exists a local representation in time and space, pointwise, of the fractional material derivative. This allows us to define it on a space of locally integrable functions which is larger than the original one in which Fourier and Laplace transform exist as functions.
We consider several typical differential equations involving the fractional material derivative and provide conditions for their solutions to exist. In some cases, the analytical solution can be found. For the general initial value problem, we devise a finite volume method and prove its stability, convergence, and conservation of probability. Numerical illustrations verify our analytical findings. Moreover, our numerical experiments show superiority in the computation time of the proposed numerical scheme over a Monte Carlo method applied to the problem of probability density function's derivation.

[62]  arXiv:2402.19018 [pdf, ps, other]
Title: Tangle free permutations and the Putman-Wieland property of Random covers
Subjects: Geometric Topology (math.GT); Metric Geometry (math.MG)

Let $\Sigma^p_g$ denote a surface of genus $g$ and with $p$ punctures. Our main result is that the fraction of degree $n$ covers of $\Sigma^p_g$ which have the Putman-Wieland property tends to $1$ as $n\to \infty$. In addition, we show that the monodromy of a random cover of $\Sigma^p_g$ is asymptotically almost surely tangle free.

[63]  arXiv:2402.19031 [pdf, ps, other]
Title: A closure theorem for $Γ$-convergence and H-convergence with applications to non-periodic homogenization
Subjects: Analysis of PDEs (math.AP)

In this work we examine the stability of some classes of integrals, and in particular with respect to homogenization. The prototypical case is the homogenization of quadratic energies with periodic coefficients perturbed by a term vanishing at infinity, which has been recently examined in the framework of elliptic PDE. We use localization techniques and higher-integrability Meyers-type results to provide a closure theorem by $\Gamma$-convergence within a large class of integral functionals. From such result we derive stability theorems in homogenization which comprise the case of perturbations with zero average on the whole space. The results are also extended to the stochastic case, and specialized to the $G$-convergence of operators corresponding to quadratic forms. A corresponding analysis is also carried on for non-symmetric operators using the localization properties of $H$-convergence. Finally, we treat the case of perforated domains with Neumann boundary condition, and their stability.

[64]  arXiv:2402.19032 [pdf, ps, other]
Title: Effective Results in The Metric Theory of Quantitative Diophantine Approximation
Comments: 48 pages, 0 figures
Subjects: Number Theory (math.NT)

Many results related to quantitative problems in the metric theory of Diophantine approximation are asymptotic, such as the number of rational solutions to certain inequalities grows with the same rate almost everywhere modulo an asymptotic error term. The error term incorporates an implicit constant that varies from one point to another. This means that applications of these results does not give concrete bounds when applied to, say a finite sum, or when applied to counting the number of solutions up to a finite point for a given inequality. This paper addresses this problem and makes the tools and their results effective, by making the implicit constant explicit outside of an exceptional subset of Lebesgue measure at most $\delta>0$, an arbitrarily small constant chosen in advance. We deduce from this the fully effective results for Schmidt's Theorem, quantitative Koukoulopoulos-Maynard Theorem and quantitative results on $M_{0}$-sets; we also provide effective results regarding statistics of normal numbers and strong law of large numbers.

[65]  arXiv:2402.19036 [pdf, other]
Title: Empirical Bayes in Bayesian learning: understanding a common practice
Subjects: Statistics Theory (math.ST)

In applications of Bayesian procedures, even when the prior law is carefully specified, it may be delicate to elicit the prior hyperparameters so that it is often tempting to fix them from the data, usually by their maximum likelihood estimates (MMLE), obtaining a so-called empirical Bayes posterior distribution. Although questionable, this is a common practice; but theoretical properties seem mostly only available on a case-by-case basis. In this paper we provide general properties for parametric models. First, we study the limit behavior of the MMLE and prove results in quite general settings, while also conceptualizing the frequentist context as an unexplored case of maximum likelihood estimation under model misspecification. We cover both identifiable models, illustrating applications to sparse regression, and non-identifiable models - specifically, overfitted mixture models. Finally, we prove higher order merging results. In regular cases, the empirical Bayes posterior is shown to be a fast approximation to the Bayesian posterior distribution of the researcher who, within the given class of priors, has the most information about the true model's parameters. This is a faster approximation than classic Bernstein-von Mises results. Given the class of priors, our work provides formal contents to common beliefs on this popular practice.

[66]  arXiv:2402.19050 [pdf, ps, other]
Title: The Shigesada-Kawasaki-Teramoto model: conditional symmetries, exact solutions and their properties
Subjects: Mathematical Physics (math-ph); Exactly Solvable and Integrable Systems (nlin.SI)

We study a simplification of the well-known Shigesada-Kawasaki-Teramoto model, which consists of two nonlinear reaction-diffusion equations with cross-diffusion. A complete set of Q-conditional (nonclassical) symmetries is derived using an algorithm adopted for the construction of conditional symmetries. The symmetries obtained are applied for finding a wide range of exact solutions, possible biological interpretation of some of which being presented. Moreover, an alternative application of the simplified model related to the polymerisation process is suggested and exact solutions are found in this case as well.

[67]  arXiv:2402.19053 [pdf, other]
Title: Geometric approach for the identification of Hamiltonian systems of quasi-Painlevé type
Comments: 39 pages, 20 figures
Subjects: Classical Analysis and ODEs (math.CA); Mathematical Physics (math-ph); Algebraic Geometry (math.AG); Complex Variables (math.CV)

Some new Hamiltonian systems of quasi-Painlev\'e type are presented and their Okamoto's space of initial conditions computed. Using the geometric approach that was introduced originally for the identification problem of Painlev\'e equations, comparing the irreducible components of the inaccessible divisors arising in the blow-up process, we find bi-rational, symplectic coordinate changes between some of these systems that give rise to the same global symplectic structure. This scheme thus gives a method for identifying Hamiltonian systems up to bi-rational symplectic maps, which is performed in this article for systems of quasi-Painlev\'e type having singularities that are either square-root type algebraic poles or ordinary poles.

[68]  arXiv:2402.19056 [pdf, other]
Title: Recovering the Polytropic Exponent in the Porous Medium Equation: Asymptotic Approach
Subjects: Numerical Analysis (math.NA)

In this paper we consider the time dependent Porous Medium Equation, $u_t = \Delta u^\gamma$ with real polytropic exponent $\gamma>1$, subject to a homogeneous Dirichlet boundary condition. We are interested in recovering $\gamma$ from the knowledge of the solution $u$ at a given large time $T$. Based on an asymptotic inequality satisfied by the solution $u(T)$, we propose a numerical algorithm allowing us to recover $\gamma$. An upper bound for the error between the exact and recovered $\gamma$ is then showed. Finally, numerical investigations are carried out in two dimensions.

[69]  arXiv:2402.19057 [pdf, ps, other]
Title: Locally approximable CR functions, a sharp maximum modulus principle and holomorphic extension
Subjects: Complex Variables (math.CV)

We introduce a notion of locally approximable continuous CR functions on locally closed subsets of reduced complex spaces, generalizing both holomorphic functions and CR functions on CR submanifolds. Under additional assumptions of set-theoretical weak pseudoconcavity we prove optimal maximum modulus principles for these functions. Restricting to real submanifolds (possibly with CR singularities) of complexmanifolds, we generalize results on holomorphic extension known for CR submanifolds.

[70]  arXiv:2402.19065 [pdf, other]
Title: Spline-Based Rotor and Stator Optimization of a Permanent Magnet Synchronous Motor
Subjects: Optimization and Control (math.OC)

This work features the optimization of a Permanent Magnet Synchronous Motor using 2D nonlinear simulations in an Isogeometric Analysis framework. The rotor and stator designs are optimized for both geometric parameters and surface shapes via modifications of control points. The scaling laws for magnetism are employed to allow for axial and radial scaling, enabling a thorough optimization of all critical machine parameters for multiple operating points. The process is carried out in a gradient-based fashion with the objectives of lowering motor material cost, torque ripple and losses. It is shown that the optimization can be efficiently conducted for many optimization variables and all objective values can be reduced.

[71]  arXiv:2402.19067 [pdf, ps, other]
Title: Norm attaining operators into locally asymptotically midpoint uniformly convex Banach spaces
Authors: Audrey Fovelle
Comments: 5 pages
Subjects: Functional Analysis (math.FA)

We prove that if $Y$ is a locally asymptotically midpoint uniformly convex Banach space which has either a normalized, symmetric basic sequence that is not equivalent to the unit vector basis in $\ell_1$, or a normalized sequence with upper p-estimates for some $p>1$, then $Y$ does not satisfy Lindenstrauss' property B.

[72]  arXiv:2402.19070 [pdf, ps, other]
Title: Sharp interface limit for $1$D stochastic Allen-Cahn equation in full small noise regime
Subjects: Probability (math.PR); Analysis of PDEs (math.AP)

We study the sharp interface limit for the $1$D stochastic Allen-Cahn equation, and extend earlier work by Funaki to the full small noise regime. The main new idea is the construction of a series of functional correctors, which are designed to recursively cancel potential divergences.
In addition, in order to show these correctors are well-behaved, we develop a systematic decomposition of functional derivatives of the deterministic Allen-Cahn flow of all orders. This decomposition is of its own interest, and may be useful in other situations as well.

[73]  arXiv:2402.19074 [pdf, ps, other]
Title: Dynamical Systems on Compact Metrizable Groups
Authors: Binghui Xiao
Subjects: Dynamical Systems (math.DS)

This paper mainly studies the theory of dynamical systems on compact metrizable groups, with a special focus on extending the maximal entropy theorem and the ergodicity of invariant measure convolution from the isomorphic case to the surjective homomorphism case. It explores in detail the dynamical systems under surjective homomorphisms, especially the variation of measure entropy. Let $G_{2}$ be a compact metric group acting freely on $G_{1}$, and the continuous mapping $T_{2}$ $G_{2}-$commutes with $T_{1}$, where $T_{2}: G_{2} \to G_{2}$ is a surjective homomorphism. If $\mu_{0} \in M(T_{0})$, we prove that when $\mu \in M(T_{1}, \mu_{0})$, the measure entropy $h(T_{1}, \mu_{0}')$ is always greater than or equal to $h(T_{1}, \mu)$; if $\mu_{0}'$ is ergodic with respect to $T_{1}$, $\mu \ne \mu_{0}'$, the Haar measure $m$ on $G_{2}$ is ergodic with respect to $T_{2}$, and $h(T_{1}, \mu_{0}') < \infty$, it can be concluded that the measure entropy $h(T_{1}, \mu_{0}')$ strictly exceeds $h(T_{1}, \mu)$.
Finally, this paper also specifically discusses the ergodicity problem of measure convolution. Let $T$ be a surjective homomorphism on $G$. If $(G, T, \mathcal{F}, \mu)$ and $(G, T, \mathcal{F}, \nu)$ are disjoint ergodic dynamical systems, then $\mu * \nu$ is ergodic. Through a proof by contradiction, the study demonstrates that under the condition that $T$ is a surjective homomorphism on $G$, the measure convolution of two disjoint ergodic dynamical systems also maintains ergodicity. These results extend Kenneth R. Berg's findings on the maximal entropy theorem and the ergodicity of measure convolution to the case of surjective homomorphisms.

[74]  arXiv:2402.19077 [pdf, ps, other]
Title: The operad of Latin hypercubes
Subjects: Combinatorics (math.CO); Algebraic Topology (math.AT); Representation Theory (math.RT)

We show that the sets of $d$-dimensional Latin hypercubes over a non-empty set $X$, with $d$ running over the positive integers, determine an operad which is isomorphic to a sub-operad of the endomorphism operad of $X$. We generalise this to categories with finite products, and briefly discuss requirements for lifting this operad to certain Cartesian closed monoidal categories.

[75]  arXiv:2402.19081 [pdf, ps, other]
Title: Finitely generated weakly monotone C*-algebra
Subjects: Operator Algebras (math.OA)

We consider the $C^*$-algebra generated by finitely many annihilation operators acting on the weakly monotone Fock space, and we call it weakly monotone $C^*$-algebra. We give an abstract representation for this algebra, showing that it is isomorphic to a suitable quotient of a Cuntz-Krieger $C^*$-algebra $\mathcal{O}_A$ corresponding to a suitable matrix $A$. Furthermore, we show that the diagonal subalgebra of the weakly monotone $C^*$-algebra is a MASA and we give the detailed description of its Gelfand spectrum.

[76]  arXiv:2402.19084 [pdf, other]
Title: High multiplicity of positive solutions in a superlinear problem of Moore-Nehari type
Subjects: Analysis of PDEs (math.AP); Classical Analysis and ODEs (math.CA); Numerical Analysis (math.NA)

In this paper we consider a superlinear one-dimensional elliptic boundary value problem that generalizes the one studied by Moore and Nehari in [43]. Specifically, we deal with piecewise-constant weight functions in front of the nonlinearity with an arbitrary number $\kappa\geq 1$ of vanishing regions. We study, from an analytic and numerical point of view, the number of positive solutions, depending on the value of a parameter $\lambda$ and on $\kappa$.
Our main results are twofold. On the one hand, we study analytically the behavior of the solutions, as $\lambda\downarrow-\infty$, in the regions where the weight vanishes. Our result leads us to conjecture the existence of $2^{\kappa+1}-1$ solutions for sufficiently negative $\lambda$. On the other hand, we support such a conjecture with the results of numerical simulations which also shed light on the structure of the global bifurcation diagrams in $\lambda$ and the profiles of positive solutions.
Finally, we give additional numerical results suggesting that the same high multiplicity result holds true for a much larger class of weights, also arbitrarily close to situations where there is uniqueness of positive solutions.

[77]  arXiv:2402.19087 [pdf, other]
Title: Semiclassical expansion for exactly solvable differential operators
Subjects: Classical Analysis and ODEs (math.CA)

Below we study a linear differential equation $\MM (v(z,\eta))=\eta^M{v(z,\eta)}$, where $\eta>0$ is a large spectral parameter and $\MM=\sum_{k=1}^{M}\rho_{k}(z)\frac{d^k}{dz^k},\; M\ge 2$ is a differential operator with polynomial coefficients such that the leading coefficient $\rho_M(z)$ is a monic complex-valued polynomial with $\dgr{\rho_M }=M$ and other $\rho_k(z)$'s are complex-valued polynomials with $\dgr{\rho_k }\leq k$. We prove the Borel summability of its WKB-solutions in the Stokes regions. For $M=3$ under the assumption that $\rho_M$ has simple zeros, we give the full description of the Stokes complex (i.e. the union of all Stokes curves) of this equation. Finally, we show that for the Euler-Cauchy equations, their WKB-solutions converge in the usual sense.

[78]  arXiv:2402.19092 [pdf, ps, other]
Title: A two spaces extension of Cauchy-Lipschitz Theorem
Authors: Charles Bertucci (CMAP), Pierre Louis Lions (CEREMADE, CdF (institution))
Subjects: Analysis of PDEs (math.AP)

We adapt the classical theory of local well-posedness of evolution problems to cases in which the nonlinearity can be accurately quantified by two different norms. For ordinary differential equations, we consider $\dot{x} = f(x,x)$ for a function $f: V\times E \to E$ where $E$ is a Banach space and $V \hookrightarrow E$ a normed vector space. This structure allows us to distinguish between the two dependencies of $f$ in $x$ and allows to generalize classical results. We also prove a similar results for partial differential equations.

[79]  arXiv:2402.19093 [pdf, other]
Title: Mathematical and computational framework for moving and colliding rigid bodies in a Newtonian fluid
Authors: Céline Van Landeghem (IRMA), Luca Berti (IRMA), Vincent Chabannes (IRMA), Christophe Prud'Homme (IRMA), Agathe Chouippe (ICube), Yannick Hoarau (ICube), Laëtitia Giraldi (CALISTO)
Subjects: Analysis of PDEs (math.AP)

We studied numerically the dynamics of colliding rigid bodies in a Newtonian fluid. The finite element method is used to solve the fluid-body interaction and the fluid motion is described in the Arbitrary-Lagrangian-Eulerian framework. To model the interactions between bodies, we consider a repulsive collision-avoidance model, defined by R. Glowinski. The main emphasis in this work is the generalization of this collision model to multiple rigid bodies of arbitrary shape. Our model first uses a narrow-band fast marching method to detect the set of colliding bodies. Then, collision forces and torques are computed for these bodies via a general expression, which does not depend on their shape. Numerical experiments examining the performance of the narrow-band fast marching method and the parallel execution of the collision algorithm are discussed. We validate our model with literature results and show various applications of colliding bodies in two and three dimensions. In these applications, the bodies move due to forces such as gravity, a fluid flow, or their own actuation. Finally, we present a tool to create arbitrarily shaped bodies in discretized fluid domains, enabling conforming body-fluid interface and allowing to perform simulations of fluid-body interactions with collision treatment in these realistic environments. All simulations are conducted with the Feel++ open source library.

[80]  arXiv:2402.19098 [pdf, ps, other]
Title: Symmetries and exact solutions of the diffusive Holling-Tanner prey-predator model
Subjects: Mathematical Physics (math-ph)

We consider the classical Holling-Tanner model extended on 1D space by introducing the diffusion term. Making a reasonable simplification, the diffusive Holling-Tanner system is studied by means of symmetry based methods. Lie and Q-conditional (nonclassical) symmetries are identified. The symmetries obtained are applied for finding a wide range of exact solutions, their properties are studied and a possible biological interpretation is proposed. 3D plots of the most interesting solutions are drown as well.

[81]  arXiv:2402.19104 [pdf, ps, other]
Title: The Author of a Quotation Goethe Adduced Against Newton
Authors: Hubert Kalf
Comments: 7 pages
Subjects: History and Overview (math.HO); History and Philosophy of Physics (physics.hist-ph)

The hitherto unknown author of a citation by Goethe in his History of Colours is identified as J. E. Montucla and the context of Montucla's quotation is discussed.

[82]  arXiv:2402.19125 [pdf, other]
Title: Highly efficient Gauss's law-preserving spectral algorithms for Maxwell's double-curl source and eigenvalue problems based on eigen-decomposition
Subjects: Numerical Analysis (math.NA)

In this paper, we present Gauss's law-preserving spectral methods and their efficient solution algorithms for curl-curl source and eigenvalue problems in two and three dimensions arising from Maxwell's equations. Arbitrary order $H(curl)$-conforming spectral basis functions in two and three dimensions are firstly proposed using compact combination of Legendre polynomials. A mixed formulation involving a Lagrange multiplier is then adopted to preserve the Gauss's law in the weak sense. To overcome the bottleneck of computational efficiency caused by the saddle-point nature of the mixed scheme, we present highly efficient solution algorithms based on reordering and decoupling of the resultant linear algebraic system and numerical eigen-decomposition of one dimensional mass matrix. The proposed solution algorithms are direct methods requiring only several matrix-matrix or matrix-tensor products of $N$-by-$N$ matrices, where $N$ is the highest polynomial order in each direction. Compared with other direct methods, the computational complexities are reduced from $O(N^6)$ and $O(N^9)$ to $O(N^3)$ and $O(N^4)$ with small and constant pre-factors for 2D and 3D cases, respectively, and can further be accelerated to $O(N^{2.807})$ and $O(N^{3.807})$, when boosted with the Strassen's matrix multiplication algorithm. Moreover, these algorithms strictly obey the Helmholtz-Hodge decomposition, thus totally eliminate the spurious eigen-modes of non-physical zero eigenvalues. Extensions of the proposed methods and algorithms to problems in complex geometries with variable coefficients and inhomogeneous boundary conditions are discussed to deal with more general situations. Ample numerical examples for solving Maxwell's source and eigenvalue problems are presented to demonstrate the accuracy and efficiency of the proposed methods.

[83]  arXiv:2402.19127 [pdf, ps, other]
Title: Hankel Determinants of convoluted Catalan numbers and nonintersecting lattice paths: A bijective proof of Cigler's Conjecture
Authors: Markus Fulmek
Subjects: Combinatorics (math.CO)

In recent preprints, Cigler considered certain Hankel determinants of convoluted Catalan numbers and conjectured identities for these determinants. In this note, we shall give a bijective proof of Cigler's Conjecture by interpreting determinants as generating functions of nonintersecting lattice paths: This proof employs the reflection principle, the Lindstr\"om-Gessel-Viennot-method and a certain construction involving reflections and overlays of nonintersecting lattice paths.

[84]  arXiv:2402.19130 [pdf, ps, other]
Title: Maps preserving ascent/descent of triple Jordan product
Subjects: Functional Analysis (math.FA); Operator Algebras (math.OA)

Let $\mathcal{X}$ be a real or complex Banach space with $ \dim \mathcal{X}\geq 3$. We give a complete description of surjective mappings on $\mathcal{B(X)}$ that preserve the ascent of Jordan triple product of operators or, preserve the descent of Jordan triple product of operators.

[85]  arXiv:2402.19132 [pdf, ps, other]
Title: Weighted least $\ell_p$ approximation on compact Riemannian manifolds
Comments: 23 pages
Subjects: Numerical Analysis (math.NA)

Given a sequence of Marcinkiewicz-Zygmund inequalities in $L_2$ on a compact space, Gr\"ochenig in \cite{G} discussed weighted least squares approximation and least squares quadrature. Inspired by this work, for all $1\le p\le\infty$, we develop weighted least $\ell_p$ approximation induced by a sequence of Marcinkiewicz-Zygmund inequalities in $L_p$ on a compact smooth Riemannian manifold $\Bbb M$ with normalized Riemannian measure (typical examples are the torus and the sphere). In this paper we derive corresponding approximation theorems with the error measured in $L_q,\,1\le q\le\infty$, and least quadrature errors for both Sobolev spaces $H_p^r(\Bbb M), \, r>d/p$ generated by eigenfunctions associated with the Laplace-Beltrami operator and Besov spaces $B_{p,\tau}^r(\Bbb M),\, 0<\tau\le \infty, r>d/p $ defined by best polynomial approximation. Finally, we discuss the optimality of the obtained results by giving sharp estimates of sampling numbers and optimal quadrature errors for the aforementioned spaces.

[86]  arXiv:2402.19137 [pdf, ps, other]
Title: Global well-posedness for 2D generalized Parabolic Anderson Model via paracontrolled calculus
Comments: 19 pages
Subjects: Analysis of PDEs (math.AP); Probability (math.PR)

This article revisits the problem of global well-posedness for the generalized parabolic Anderson model on $\mathbb{R}^+\times \mathbb{T}^2$ within the framework of paracontrolled calculus \cite{GIP15}. The model is given by the equation:
\begin{equation*}
(\partial_t-\Delta) u=F(u)\eta
\end{equation*}
where $\eta\in C^{-1-\kappa}$ with $1/6>\kappa>0$, and $F\in C_b^2(\mathbb{R})$. Assume that $\eta\in C^{-1-\kappa}$ and can be lifted to enhanced noise, we derive new a priori bounds. The key idea follows from the recent work
\cite{CFW24} by A.Chandra, G.L. Feltes and H.Weber to represent the leading error term as a transport type term, and our techniques encompass the paracontrolled calculus, the maximum principle, and the localization approach (i.e. high-low frequency argument).

[87]  arXiv:2402.19138 [pdf, other]
Title: Generalized Pentagon Equations
Comments: 19 pages, 3 figures
Subjects: Quantum Algebra (math.QA)

Drinfeld defined the Knizhinik--Zamolodchikov (KZ) associator $\Phi_{\rm KZ}$ by considering the regularized holonomy of the KZ connection along the {\em droit chemin} $[0,1]$. The KZ associator is a group-like element of the free associative algebra with two generators, and it satisfies the pentagon equation.
In this paper, we consider paths on $\mathbb{C}\backslash \{ z_1, \dots, z_n\}$ which start and end at tangential base points. These paths are not necessarily straight, and they may have a finite number of transversal self-intersections. We show that the regularized holonomy $H$ of the KZ connection associated to such a path satisfies a generalization of Drinfeld's pentagon equation. In this equation, we encounter $H$, $\Phi_{\rm KZ}$, and new factors associated to self-intersections, to tangential base points, and to the rotation number of the path.

[88]  arXiv:2402.19147 [pdf, other]
Title: Efficient quaternion CUR method for low-rank approximation to quaternion matrix
Subjects: Numerical Analysis (math.NA)

The low-rank quaternion matrix approximation has been successfully applied in many applications involving signal processing and color image processing. However, the cost of quaternion models for generating low-rank quaternion matrix approximation is sometimes considerable due to the computation of the quaternion singular value decomposition (QSVD), which limits their application to real large-scale data. To address this deficiency, an efficient quaternion matrix CUR (QMCUR) method for low-rank approximation is suggested, which provides significant acceleration in color image processing. We first explore the QMCUR approximation method, which uses actual columns and rows of the given quaternion matrix, instead of the costly QSVD. Additionally, two different sampling strategies are used to sample the above-selected columns and rows. Then, the perturbation analysis is performed on the QMCUR approximation of noisy versions of low-rank quaternion matrices. Extensive experiments on both synthetic and real data further reveal the superiority of the proposed algorithm compared with other algorithms for getting low-rank approximation, in terms of both efficiency and accuracy.

[89]  arXiv:2402.19151 [pdf, other]
Title: Approximations of symbolic substitution systems in one dimension
Authors: Lior Tenenbaum
Comments: 12 pages, 4 figures, written for the proceedings of ICQ 15 and submitted to the Israel Journal of Chemistry
Subjects: Dynamical Systems (math.DS); Mathematical Physics (math-ph); Spectral Theory (math.SP)

Periodic approximations of quasicrystals are a powerful tool in analyzing spectra of Schr\"odinger operators arising from quasicrystals, given the known theory for periodic crystals. Namely, we seek periodic operators whose spectra approximate the spectrum of the limiting operator (of the quasicrystal). This naturally leads to study the convergence of the underlying dynamical systems. We treat dynamical systems which are based on one-dimensional substitutions. We first find natural candidates of dynamical subsystems to approximate the substitution dynamical system. Subsequently, we offer a characterization of their convergence and provide estimates for the rate of convergence. We apply the proposed theory to some guiding examples.

[90]  arXiv:2402.19152 [pdf, ps, other]
Title: Banach lattices with upper $p$-estimates: free and injective objects
Comments: 37 pages
Subjects: Functional Analysis (math.FA)

We study the free Banach lattice $FBL^{(p,\infty)}[E]$ with upper $p$-estimates generated by a Banach space $E$. Using a classical result of Pisier on factorization through $L^{p,\infty}(\mu)$ together with a finite dimensional reduction, it is shown that the spaces $\ell^{p,\infty}(n)$ witness the universal property of $FBL^{(p,\infty)}[E]$ isomorphically. As a consequence, we obtain a functional representation for $FBL^{(p,\infty)}[E]$. More generally, our proof allows us to identify the norm of any free Banach lattice over $E$ associated with a rearrangement invariant function space.
After obtaining the above functional representation, we take the first steps towards analyzing the fine structure of $FBL^{(p,\infty)}[E]$. Notably, we prove that the norm for $FBL^{(p,\infty)}[E]$ cannot be isometrically witnessed by $L^{p,\infty}(\mu)$ and settle the question of characterizing when an embedding between Banach spaces extends to a lattice embedding between the corresponding free Banach lattices with upper $p$-estimates. To prove this latter result, we introduce a novel push-out argument, which when combined with the injectivity of $\ell^p$ allows us to give an alternative proof of the subspace problem for free $p$-convex Banach lattices. On the other hand, we prove that $\ell^{p,\infty}$ is not injective in the class of Banach lattices with upper $p$-estimates, elucidating one of many difficulties arising in the study of $FBL^{(p,\infty)}[E]$.

[91]  arXiv:2402.19153 [pdf, ps, other]
Title: Asympotic bounds for Bombieri's inequality on products of homogeneous polynomials
Comments: 14 pages
Subjects: Analysis of PDEs (math.AP)

Let $P$ be a fixed homogeneous polynomial. We present a sharp condition on $P$ guaranteeing the existence of asymptotically larger bounds in Bombieri's inequality, so for every homogeneous polynomial $q_m$ of degree $m$ we have \begin{equation*} \left\Vert P q_{m}\right\Vert _{a}\geq C_{P} m^{l\left( P\right) /2}\left\Vert q_{m}\right\Vert _{a}, \end{equation*} where $\| \cdot \| _{a}$ denotes the apolar norm. Explicit estimates for $C_P > 0$ and $l(P) > 0$ are given.

[92]  arXiv:2402.19154 [pdf, other]
Title: Bialy-Mironov type rigidity for centrally symmetric symplectic billiards
Comments: 11 pages, 3 figures
Subjects: Dynamical Systems (math.DS)

The aim of the present paper is to establish a Bialy-Mironov type rigidity for centrally symmetric symplectic billiards. For a centrally symmetric $C^2$ strongly-convex domain $D$ with boundary $\partial D$, assume that the symplectic billiard map has a (simple) continuous invariant curve $\delta \subset \mathcal{P}$ of rotation number $1/4$ (winding once around $\partial D$) and consisting only of $4$-periodic orbits. If one of the parts between $\delta$ and each boundary of the phase-space is entirely foliated by continuous invariant closed (not null-homotopic) curves, then $\partial D$ is an ellipse. The differences with Birkhoff billiards are essentially two: it is possible to assume the existence of the foliation in one of the parts of the phase-space detected by the curve $\delta$, and the result is obtained by tracing back the problem directly to the totally integrable case.

[93]  arXiv:2402.19156 [pdf, ps, other]
Title: A rigorous approach to the sharp interface limit for phase-field models of tumor growth
Subjects: Analysis of PDEs (math.AP)

In this paper we consider two diffuse interface models for tumor growth coupling a Cahn-Hilliard type equation for the tumor phase parameter to a reaction-diffusion type equation for the nutrient. The models are distinguished by the presence of two different coupling source terms. For such problems, we address the question of the limit, as the diffuse interface parameter tends to zero, from diffuse interface models to sharp interface ones, justifying rigorously what was deduced via formal asymptotics in [18]. The resulting evolutions turn out to be varifold solutions to Mullins-Sekerka type flows for the tumor region suitably coupled with the equation for the nutrient.

[94]  arXiv:2402.19164 [pdf, other]
Title: Horizontal semiconcavity for the square of Carnot-Carathéodory distance on ideal Carnot groups and applications to Hamilton-Jacobi equations
Subjects: Analysis of PDEs (math.AP)

We show that the square of Carnot-Carath\'eodory distance from the origin, in ideal Carnot groups, enjoys the horizontal semiconcavity (h-semiconcavity) everywhere in the group including the origin. We apply this property to show h-semiconcavity for the solutions of a class of non-coercive evolutive Hamilton-Jacobi equations, by using the associated Hopf-Lax solutions.

[95]  arXiv:2402.19169 [pdf, ps, other]
Title: Improved bounds for skew corner-free sets
Authors: Adrian Beker
Comments: 15 pages
Subjects: Combinatorics (math.CO); Number Theory (math.NT)

We construct skew corner-free subsets of $[n]^2$ of size $n^2\exp(-O(\sqrt{\log n}))$, thereby improving on recent bounds of the form $\Omega(n^{5/4})$ obtained by Pohoata and Zakharov. In the other direction, we prove that any such set has size at most $O(n^2(\log n)^{-c})$ for some absolute constant $c > 0$. This improves on the previously best known upper bound, coming from Shkredov's work on the corners theorem.

[96]  arXiv:2402.19174 [pdf, ps, other]
Title: On an $n$-ary generalization of the Lie representation
Comments: 18 pages; Notation as in arXiv:1710.00376 [math.CO] and arXiv:2307.00587 [math.CO]
Subjects: Combinatorics (math.CO); Representation Theory (math.RT)

We continue our study, initiated in our prior work with Richard Stanley, of the representation of the symmetric group on the multilinear component of an $n$-ary generalization of the free Lie algebra known as the free Fillipov $n$-algebra with $k$ brackets. Our ultimate aim is to determine the multiplicities of the irreducible representations in this representation. This had been done for the ordinary Lie representation ($n=2$ case) by Kraskiewicz and Weyman. The $k=2$ case was handled in our prior work, where the representation was shown to be isomorphic to $S^{2^{n-1}1}$. In this paper, for general $n$ and $k$, we obtain decomposition results that enable us to prove that in the $k=3$ case, the representation is isomorphic to $S^{3^{n-1}1} \oplus S^{3^{n-2}21^2}$.

[97]  arXiv:2402.19175 [pdf, other]
Title: The coarse flag Hilbert-Poincaré series of the braid arrangement
Comments: 10 pages
Subjects: Combinatorics (math.CO)

The paper concerns the coarse flag Hilbert-Poincar\'e series of Maglione-Voll in the case of the braid arrangement associated to the symmetric group. We explicitly construct a companion statistic $\operatorname{ino} : \mathfrak{S}_{n+1} \times \operatorname{Sym}(n) \rightarrow \mathbb{N}$ for the descent statistic on $\operatorname{Sym}(n)$ using reverse $(P,\omega)$-partitions and quasisymmetric functions.

[98]  arXiv:2402.19176 [pdf, other]
Title: Proximal Dogleg Opportunistic Majorization for Nonconvex and Nonsmooth Optimization
Authors: Yiming Zhou, Wei Dai
Subjects: Optimization and Control (math.OC); Signal Processing (eess.SP)

We consider minimizing a function consisting of a quadratic term and a proximable term which is possibly nonconvex and nonsmooth. This problem is also known as scaled proximal operator. Despite its simple form, existing methods suffer from slow convergence or high implementation complexity or both. To overcome these limitations, we develop a fast and user-friendly second-order proximal algorithm. Key innovation involves building and solving a series of opportunistically majorized problems along a hybrid Newton direction. The approach directly uses the precise Hessian of the quadratic term, and calculates the inverse only once, eliminating the iterative numerical approximation of the Hessian, a common practice in quasi-Newton methods. The algorithm's convergence to a critical point is established, and local convergence rate is derived based on the Kurdyka-Lojasiewicz property of the objective function. Numerical comparisons are conducted on well-known optimization problems. The results demonstrate that the proposed algorithm not only achieves a faster convergence but also tends to converge to a better local optimum compare to benchmark algorithms.

[99]  arXiv:2402.19179 [pdf, ps, other]
Title: Boundary Regularity of Harmonic maps from $RCD(K,N)$-space to $CAT(0)$-space
Comments: 14 pages. 0 figures. We will submit to a journal in Chinese
Subjects: Differential Geometry (math.DG)

We establish the boundary regularity of harmonic maps from $RCD(K, N)$ metric measure spaces into $CAT(0)$ metric spaces.

[100]  arXiv:2402.19181 [pdf, other]
Title: Structure of Periodic Orbit Families in the Hill Restricted 4-Body Problem
Comments: Submitted to SIAM Journal on Applied Dynamical Systems
Subjects: Dynamical Systems (math.DS); Chaotic Dynamics (nlin.CD)

The Hill Restricted 4-Body Problem (HR4BP) is a coherent time-periodic model that can be used to represent motion in the Sun-Earth-Moon (SEM) system. Periodic orbits were computed in this model to better understand the periodic orbit family structures that exist in these types of systems. First, periodic orbits in the Circular Restricted 3-Body Problem (CR3BP) representation of the Earth-Moon (EM) system were identified. A Melnikov-type function was used to identify a set of candidate points on the EM CR3BP periodic orbits to start a continuation algorithm. A pseudo-arclength continuation scheme was then used to obtain the corresponding periodic orbit families in the HR4BP when including the effect of the Sun. Bifurcation points were identified in the computed families to obtain additional orbit families.

[101]  arXiv:2402.19182 [pdf, ps, other]
Title: Quantitative homogenization for log-normal coefficients via Malliavin calculus: the one-dimensional case
Subjects: Analysis of PDEs (math.AP); Probability (math.PR)

The quantitative analysis of stochastic homogenization problems has been a very active field in the last fifteen years. Whereas the first results were motivated by applied questions (namely, the numerical approximation of homogenized coefficients), the more recent achievements in the field are much more analytically-driven and focus on the subtle interplay between PDE analysis (and in particular elliptic regularity theory) and probability (concentration, stochastic cancellations, scaling limits). The aim of this article is threefold. First we provide a complete and self-contained analysis for the popular example of log-normal coefficients with possibly fat tails in dimension $d=1$, establishing new results on the accuracy of the two-scale expansion and characterizing fluctuations (in the perspective of uncertainty quantification). Second, we work in a context where explicit formulas allow us to by-pass analytical difficulties and therefore mostly focus on the probabilistic side of the theory. Last, the one-dimensional setting gives intuition on the available results in higher dimension (provided results are correctly reformulated) to which we give precise entries to the recent literature.

[102]  arXiv:2402.19183 [pdf, ps, other]
Title: Prime isogenous discriminant ideal twins
Comments: 35 pages
Subjects: Number Theory (math.NT)

Let $E_{1}$ and $E_{2}$ be elliptic curves defined over a number field $K$. We say that $E_{1}$ and $E_{2}$ are discriminant ideal twins if they are not $K$-isomorphic and have the same minimal discriminant ideal and conductor. Such curves are said to be discriminant twins if, for each prime $\mathfrak{p}$ of $K$, there are $\mathfrak{p}$-minimal models for $E_{1}$ and $E_{2}$ whose discriminants are equal. This article explicitly classifies all prime-isogenous discriminant (ideal) twins over $\mathbb{Q}$. We obtain this classification as a consequence of our main results, which constructively gives all $p$-isogenous discriminant ideal twins over number fields where $p\in\left\{ 2,3,5,7,13\right\}$, i.e., where $X_0(p)$ has genus $0$. In particular, we find that up to twist, there are finitely many $p$-isogenous discriminant ideal twins if and only if $K$ is $\mathbb{Q}$ or an imaginary quadratic field. In the latter case, we provide instructions for finding the finitely many pairs of $j$-invariants that result in $p$-isogenous discriminant ideal twins. We prove our results by considering the local data of parameterized $p$-isogenous elliptic curves.

[103]  arXiv:2402.19191 [pdf, other]
Title: An asymptotic-preserving method for the three-temperature radiative transfer model
Subjects: Numerical Analysis (math.NA); Mathematical Physics (math-ph)

We present an asymptotic-preserving (AP) numerical method for solving the three-temperature radiative transfer model, which holds significant importance in inertial confinement fusion. A carefully designedsplitting method is developed that can provide a general framework of extending AP schemes for the gray radiative transport equation to the more complex three-temperature radiative transfer model. The proposed scheme captures two important limiting models: the three-temperature radiation diffusion equation (3TRDE) when opacity approaches infinity and the two-temperature limit when the ion-electron coupling coefficient goes to infinity. We have rigorously demonstrated the AP property and energy conservation characteristics of the proposed scheme and its efficiency has been validated through a series of benchmark tests in the numerical part.

[104]  arXiv:2402.19201 [pdf, other]
Title: Boom and bust cycles due to pseudospectra of matrices with unimodular spectra
Comments: 10 pages, 5 figures
Subjects: Mathematical Physics (math-ph); Statistical Mechanics (cond-mat.stat-mech); Strongly Correlated Electrons (cond-mat.str-el); Quantum Physics (quant-ph)

We discuss dynamics obtained by increasing powers of non-normal matrices that are roots of the identity, and therefore have all eigenvalues on the unit circle. Naively, one would expect that the expectation value of such powers cannot grow as one increases the power. We demonstrate that, rather counterintuitively, a completely opposite behavior is possible. In the limit of infinitely large matrices one can have an exponential growth. For finite matrices this exponential growth is a part of repeating cycles of exponential growths followed by exponential decays. The effect can occur if the spectrum is different than the pseudospectrum, with the exponential growth rate being given by the pseudospectrum. We show that this effect appears in a class of transfer matrices appearing in studies of two-dimensional non-interacting systems, for a matrix describing the Ehrenfest urn, as well as in previously observed purity dynamics in a staircase random circuit.

[105]  arXiv:2402.19203 [pdf, ps, other]
Title: On non-negative solutions of stochastic Volterra equations with jumps and non-Lipschitz coefficients
Subjects: Probability (math.PR); Mathematical Finance (q-fin.MF)

We consider one-dimensional stochastic Volterra equations with jumps for which we establish conditions upon the convolution kernel and coefficients for the strong existence and pathwise uniqueness of a non-negative c\`adl\`ag solution. By using the approach recently developed in arXiv:2302.07758, we show the strong existence by using a nonnegative approximation of the equation whose convergence is proved via a variant of the Yamada--Watanabe approximation technique. We apply our results to L\'evy-driven stochastic Volterra equations. In particular, we are able to define a Volterra extension of the so-called alpha-stable Cox--Ingersoll--Ross process, which is especially used for applications in Mathematical Finance.

[106]  arXiv:2402.19211 [pdf, ps, other]
Title: Classifying pseudo-ovals, translation generalized quadrangles, and elation Laguerre planes of small order
Subjects: Combinatorics (math.CO)

We provide classification results for translation generalized quadrangles of order less or equal to $64$, and hence, for all incidence geometries related to them. The results consist of the classification of all pseudo-ovals in $PG(3n-1,2)$, for $n=3,4$, and that of the pseudo-ovals in $PG(3n-1,q)$, for $n=5,6$, such that one of the associated projective planes is Desarguesian.

[107]  arXiv:2402.19212 [pdf, ps, other]
Title: Deep Reinforcement Learning: A Convex Optimization Approach
Authors: Ather Gattami
Subjects: Optimization and Control (math.OC); Machine Learning (cs.LG)

In this paper, we consider reinforcement learning of nonlinear systems with continuous state and action spaces. We present an episodic learning algorithm, where we for each episode use convex optimization to find a two-layer neural network approximation of the optimal $Q$-function. The convex optimization approach guarantees that the weights calculated at each episode are optimal, with respect to the given sampled states and actions of the current episode. For stable nonlinear systems, we show that the algorithm converges and that the converging parameters of the trained neural network can be made arbitrarily close to the optimal neural network parameters. In particular, if the regularization parameter is $\rho$ and the time horizon is $T$, then the parameters of the trained neural network converge to $w$, where the distance between $w$ from the optimal parameters $w^\star$ is bounded by $\mathcal{O}(\rho T^{-1})$. That is, when the number of episodes goes to infinity, there exists a constant $C$ such that \[\|w-w^\star\| \le C\cdot\frac{\rho}{T}.\] In particular, our algorithm converges arbitrarily close to the optimal neural network parameters as the time horizon increases or as the regularization parameter decreases.

[108]  arXiv:2402.19213 [pdf, other]
Title: Classification of permanence and impermanence for a Lotka-Volterra model of three competing species with seasonal succession
Authors: Lei Niu, Xizhuang Xie
Subjects: Dynamical Systems (math.DS)

In this paper, we are concerned with the permanence of a Lotka-Volterra model of three competing species with seasonal succession. Based on the existence of a carrying simplex, that is a globally attracting hypersurface of codimension one, we provide a complete classification of the permanence and impermanence in terms of inequalities on the parameters of this model. Moreover, we numerically show that invariant closed curves can occur in the permanent classes, which means that the positive fixed point of the associated Poincare map in the permanent classes is not always globally asymptotically stable.

[109]  arXiv:2402.19214 [pdf, other]
Title: A Bayesian approach with Gaussian priors to the inverse problem of source identification in elliptic PDEs
Authors: Matteo Giordano
Comments: 16 Pages. The reproducible code is available at: this https URL
Subjects: Statistics Theory (math.ST); Methodology (stat.ME)

We consider the statistical linear inverse problem of making inference on an unknown source function in an elliptic partial differential equation from noisy observations of its solution. We employ nonparametric Bayesian procedures based on Gaussian priors, leading to convenient conjugate formulae for posterior inference. We review recent results providing theoretical guarantees on the quality of the resulting posterior-based estimation and uncertainty quantification, and we discuss the application of the theory to the important classes of Gaussian series priors defined on the Dirichlet-Laplacian eigenbasis and Mat\'ern process priors. We provide an implementation of posterior inference for both classes of priors, and investigate its performance in a numerical simulation study.

[110]  arXiv:2402.19219 [pdf, other]
Title: Transition of the semiclassical resonance widths across a tangential crossing energy-level
Comments: 2 figures
Subjects: Mathematical Physics (math-ph); Spectral Theory (math.SP)

We consider a 1D $2\times 2$ matrix-valued operator \eqref{System0} with two semiclassical Schr\"odinger operators on the diagonal entries and small interactions on the off-diagonal ones. When the two potentials cross at a turning point with contact order $n$, the corresponding two classical trajectories at the crossing level intersect at one point in the phase space with contact order $2n$. We compute the transfer matrix at this point between the incoming and outgoing microlocal solutions and apply it to the semiclassical distribution of resonances at the energy crossing level. It is described in terms of a generalized Airy function. This result generalizes \cite{FMW1} to the tangential crossing and \cite{AFH1} to the crossing at a turning point.

[111]  arXiv:2402.19228 [pdf, ps, other]
Title: A generalised block decomposition theorem
Authors: Jan-Paul Lerch
Comments: 10 pages
Subjects: Representation Theory (math.RT); Combinatorics (math.CO)

We discuss a class of linear representations of the product poset of totally ordered sets $P= T_1 \times \cdots \times T_n$ which decompose into interval representations for block intervals. These can be characterised in terms of a homological property which is called middle exactness.

[112]  arXiv:2402.19234 [pdf, ps, other]
Title: Broadcast independence number of oriented circulant graphs
Comments: arXiv admin note: text overlap with arXiv:2102.04094
Subjects: Combinatorics (math.CO); Discrete Mathematics (cs.DM)

In 2001, D. Erwin \cite{Erw01} introduced in his Ph.D. dissertation the notion of broadcast independence in unoriented graphs. Since then, some results but not many, are published on this notion, including research work on the broadcast independence number of unoriented circulant graphs \cite{LBS23}. In this paper, we are focused in the same parameter but of the class of oriented circulant graphs. An independent broadcast on an oriented graph $\overrightarrow{G}$ is a function $f: V\longrightarrow \{0,\ldots,\diam(\overrightarrow{G})\}$ such that $(i)$ $f(v)\leq e(v)$ for every vertex $v\in V(\overrightarrow{G})$, where $\diam(\overrightarrow{G})$ denotes the diameter of $\overrightarrow{G}$ and $e(v)$ the eccentricity of vertex $v$, and $(ii)$ $d_{\overrightarrow{G}}(u,v) > f(u)$ for every distinct vertices $u$, $v$ with $f(u)$, $f(v)>0$, where $d_{\overrightarrow{G}}(u,v)$ denotes the length of a shortest oriented path from $u$ to $v$. The broadcast independence number $\beta_b(\overrightarrow{G})$ of $\overrightarrow{G}$ is then the maximum value of $\sum_{v \in V} f(v)$, taken over all independent broadcasts on $\overrightarrow{G}$. The goal of this paper is to study the properties of independent broadcasts of oriented circulant graphs $\overrightarrow{C}(n;1,a)$, for any integers $n$ and $a$ with $n>|a|\geq 1$ and $a \notin \{1,n-1\}$. Then, we give some bounds and some exact values for the number $\beta_b(\overrightarrow{C}(n;1,a))$.

[113]  arXiv:2402.19235 [pdf, ps, other]
Title: Measurement Schemes in AQFT, Contextuality and the Wigner's Friend Gedankenexperiment
Comments: Dissertation submitted in partial fulfillment of the requirements for the degree of Master of Science in Physics at the university of S\~ao Paulo (Brazil). xvi+225 pages, 16 figures, 7 tables
Subjects: Mathematical Physics (math-ph); Operator Algebras (math.OA); Quantum Physics (quant-ph)

Measurements have historically presented a problem for the consistent description of quantum theories, be it in non-relativistic quantum mechanics or in quantum field theory. Drawing on a recent surge of interest in the description of measurements in Algebraic Quantum Field theory, it was decided that this dissertation would be focused on trying to close the gap between the description of measurements proposed by K. Hepp in the 70's, considering decoherence of states in quasilocal algebras and the new framework of generally covariant measurement schemes proposed recently by C. Fewster and R. Verch. Another recent result that we shall also consider is the Frauchinger-Renner Gedankenexperiment, that has taken inspiration on Hepp's article about decoherence based measurements to arrive at a no-go result about the consistency of quantum descriptions of systems containing rational agents, we shall seek to provide a closure for the interpretation of this result. In doing so we naturally arrive at the study of the contextual properties of measurement setups.

[114]  arXiv:2402.19256 [pdf, ps, other]
Title: Collet-Eckmann maps in the unicritical family
Subjects: Dynamical Systems (math.DS)

In this paper we study perturbations of complex unicritical polynomials satisfying the Collet-Eckmann condition. We show that Collet-Eckmann parameters are Lebesgue density points of the complement of the Mandelbrot set (i.e. the connectedness locus).

[115]  arXiv:2402.19266 [pdf, ps, other]
Title: Cauchy-completions and the rule of unique choice in relational doctrines
Subjects: Category Theory (math.CT); Logic in Computer Science (cs.LO); Logic (math.LO)

Lawvere's generalised the notion of complete metric space to the field of enriched categories: an enriched category is said to be Cauchy-complete if every left adjoint bimodule into it is represented by an enriched functor. Looking at this definition from a logical standpoint, regarding bimodules as an abstraction of relations and functors as an abstraction of functions, Cauchy-completeness resembles a formulation of the rule of unique choice. In this paper, we make this analogy precise, using the language of relational doctrines, a categorical tool that provides a functorial description of the calculus of relations, in the same way Lawvere's hyperdoctrines give a functorial description of predicate logic. Given a relational doctrine, we define Cauchy-complete objects as those objects of the domain category satisfying the rule of unique choice. Then, we present a universal construction that completes a relational doctrine with the rule of unique choice, that is, producing a new relational doctrine where all objects are Cauchy-complete. We also introduce relational doctrines with singleton objects and show that these have the minimal structure needed to build the reflector of the full subcategory of its domain on Cauchy-complete objects. The main result is that this reflector exists if and only if the relational doctrine has singleton objects and this happens if and only if its restriction to Cauchy-complete objects is equivalent to its completion with the rule of unique choice. We support our results with many examples, also falling outside the scope of standard doctrines, such as complete metric spaces, Banach spaces and compact Hausdorff spaces in the general context of monoidal topology, which are all shown to be Cauchy-complete objects for appropriate relational doctrines.

[116]  arXiv:2402.19268 [pdf, ps, other]
Title: Extremal quantiles of intermediate orders under two-way clustering
Subjects: Statistics Theory (math.ST); Econometrics (econ.EM)

This paper investigates extremal quantiles under two-way cluster dependence. We demonstrate that the limiting distribution of the unconditional intermediate order quantiles in the tails converges to a Gaussian distribution. This is remarkable as two-way cluster dependence entails potential non-Gaussianity in general, but extremal quantiles do not suffer from this issue. Building upon this result, we extend our analysis to extremal quantile regressions of intermediate order.

[117]  arXiv:2402.19271 [pdf, other]
Title: Coloring locally sparse graphs
Comments: 28 pages, 1 figure
Subjects: Combinatorics (math.CO)

A graph $G$ is $k$-locally sparse if for each vertex $v \in V(G)$, the subgraph induced by its neighborhood contains at most $k$ edges. Alon, Krivelevich, and Sudakov showed that for $f > 0$ if a graph $G$ of maximum degree $\Delta$ is $\Delta^2/f$-locally-sparse, then $\chi(G) = O\left(\Delta/\log f\right)$. We introduce a more general notion of local sparsity by defining graphs $G$ to be $(k, F)$-locally-sparse for some graph $F$ if for each vertex $v \in V(G)$ the subgraph induced by the neighborhood of $v$ contains at most $k$ copies of $F$. Employing the R\"{o}dl nibble method, we prove the following generalization of the above result: for every bipartite graph $F$, if $G$ is $(k, F)$-locally-sparse, then $\chi(G) = O\left( \Delta /\log\left(\Delta k^{-1/|V(F)|}\right)\right)$. This improves upon results of Davies, Kang, Pirot, and Sereni who consider the case when $F$ is a path. Our results also recover the best known bound on $\chi(G)$ when $G$ is $K_{1, t, t}$-free for $t \geq 4$, and hold for list and correspondence coloring in the more general so-called ''color-degree'' setting.

[118]  arXiv:2402.19283 [pdf, ps, other]
Title: The higher fixed point theorem for foliations. Applications to rigidity and integrality
Subjects: Differential Geometry (math.DG)

We give applications of the higher Lefschetz theorems for foliations of [BH10], primarily involving Haefliger cohomology. These results show that the transverse structures of foliations carry important topological and geometric information. This is in the spirit of the passage from the Atiyah-Singer index theorem for a single compact manifold to their families index theorem, involving a compact fiber bundle over a compact base. For foliations, Haefliger cohomology plays the role that the cohomology of the base space plays in the families index theorem.
We obtain highly useful numerical invariants by paring with closed holonomy invariant currents. In particular, we prove that the non-triviality of the higher A-hat genus of the foliation in Haefliger cohomology can be an obstruction to the existence of non-trivial leaf-preserving compact connected group actions. We then construct a large collection of examples for which no such actions exist. Finally, we relate our results to Connes' spectral triples, and prove useful integrality results.

[119]  arXiv:2402.19291 [pdf, ps, other]
Title: A Dold-Kan Equivalence for Crossed Simplicial Groups
Subjects: Algebraic Topology (math.AT); Category Theory (math.CT)

Using coinduction and restriction functors, we construct a Dold-Kan type equivalence for crossed simplicial groups. We first show that the original Dold-Kan equivalence is given by an adjoint pair of coinduction and restriction functors. Then we show that the category of simplicial vector spaces and the category of representations of a crossed simplicial group are similarly equivalent as model categories.

[120]  arXiv:2402.19300 [pdf, other]
Title: $\mathrm{SL}_2$-like Properties of Matrices Over Noncommutative Rings and Generalizations of Markov Numbers
Comments: 40 pages, 10 figures. Comments welcome
Subjects: Rings and Algebras (math.RA); Number Theory (math.NT); Quantum Algebra (math.QA)

We study $2\times 2$ matrices over noncommutative rings with anti-involution, with a special focus on the symplectic group $\mathrm{Sp}_2(\mathcal{A},\sigma)$. We define traces and determinants of such matrices and use them to prove a Cayley Hamilton identity and trace relations which generalize well known relations for elements of $\mathrm{SL}_2(R)$ over a commutative ring. We compare the structure of elements of $\mathrm{Sp}_2(\mathcal{A},\sigma)$ with Manin matrices over general noncommutative rings; this naturally leads to a quantization $\mathrm{Sp}_2(\mathcal{A},\sigma)_q$. In contrast to the usual definition of the quantum group as a deformation of the ring of matrix functions on $\mathrm{SL}_2(R)$, this quantization produces a group of matrices over a new noncommutative ring with involution. We finish the comparison by constructing a generalization of a Hopf algebra structure on the noncommutative ring of matrix functions of our quantum group. Finally, we use the noncommutative surface-type cluster algebras of Berenstein and Retakh to give a geometric interpretation of our Hopf algebra structure and to produce noncommutative generalizations of Markov numbers over many rings with involution including the complex numbers, dual numbers, matrix rings, and group rings.

[121]  arXiv:2402.19301 [pdf, ps, other]
Title: Strong divisibility sequences and sieve methods
Comments: 23 pages; appendix by Sandro Bettin
Subjects: Number Theory (math.NT)

We investigate strong divisibility sequences and produce lower and upper bounds for the density of integers in the sequence which only have (somewhat) large prime factors. We focus on the special cases of Fibonacci numbers and elliptic divisibility sequences, discussing the limitations of our methods. At the end of the paper there is an appendix by Sandro Bettin on divisor closed sets, that we use to study the density of prime terms that appear in strong divisibility sequences.

[122]  arXiv:2402.19324 [pdf, other]
Title: Entropy of axial product of multiplicative subshifts
Subjects: Dynamical Systems (math.DS); Combinatorics (math.CO); Number Theory (math.NT)

We obtain the entropy and the surface entropy of the axial products on $\mathbb{N}^d$ and the $d$-tree $T^d$ of two types of systems: the subshift and the multiplicative subshift.

[123]  arXiv:2402.19331 [pdf, ps, other]
Title: Multiplicative Hitchin Fibration and Fundamental Lemma
Authors: X. Griffin Wang
Comments: 243 pages
Subjects: Algebraic Geometry (math.AG); Number Theory (math.NT)

Let $k$ be a finite field and let $G$ be a reductive group over $k[[\pi]]$. Suppose $\mathrm{char}(k)$ is larger than twice the Coxeter number of $G$, we prove the standard endoscopic fundamental lemma for the spherical Hecke algebra of $G$ using multiplicative Hitchin fibrations.

[124]  arXiv:2402.19332 [pdf, ps, other]
Title: Siegel Zeros and the Hardy-Littlewood Conjecture
Authors: Yunan Wang
Subjects: Number Theory (math.NT)

In 2016, Fei \cite{fei2016application} established a bound on the Siegel zeros for real primitive Dirichlet characters modulo $q$, assuming the weak Hardy-Littlewood conjecture. Building on Fei's work, Jia \cite{jia2022conditional} demonstrated the same bound using a stronger version of the Hardy-Littlewood conjecture. In this paper, we present a slightly simplified approach to reprove their results.

[125]  arXiv:2402.19335 [pdf, ps, other]
Title: On the structure of the character degree graphs having diameter three
Subjects: Group Theory (math.GR)

The structure of the character degree graphs $\Delta(G)$, i.e. the prime graphs on the set $\mathrm{cd}(G)$ of the irreducible character degrees of a finite group $G$, such that $G$ is solvable and $\Delta(G)$ has diameter three, remains an intriguing area of study. However, a comprehensive understanding of these structures remains elusive. In this paper, we prove some properties and provide an infinite series of examples of this class of graphs, building on the ideas of Mark Lewis.

[126]  arXiv:2402.19342 [pdf, ps, other]
Title: Integrate on a closed stratified surface
Authors: Xiao-Xue Wei
Subjects: Category Theory (math.CT)

We prove that the Drinfeld center centralized by a symmetric fusion category is a symmetric monoidal functor if we choose proper domain and codomain categories. We also compute the factorization homology of stratified surfaces with coefficients satisfying certain anomaly-free conditions.

[127]  arXiv:2402.19345 [pdf, other]
Title: Multi-frequency tracking via group-sparse optimal transport
Comments: 6 pages, 9 figures
Subjects: Optimization and Control (math.OC); Signal Processing (eess.SP)

In this work, we introduce an optimal transport framework for inferring power distributions over both spatial location and temporal frequency. Recently, it has been shown that optimal transport is a powerful tool for estimating spatial spectra that change smoothly over time. In this work, we consider the tracking of the spatio-temporal spectrum corresponding to a small number of moving broad-band signal sources. Typically, such tracking problems are addressed by treating the spatio-temporal power distribution in a frequency-by-frequency manner, allowing to use well-understood models for narrow-band signals. This however leads to decreased target resolution due to inefficient use of the available information. We propose an extension of the optimal transport framework that exploits information from several frequencies simultaneously by estimating a spatio-temporal distribution penalized by a group-sparsity regularizer. This approach finds a spatial spectrum that changes smoothly over time, and at each time instance has a small support that is similar across frequencies. To the best of the authors knowledge, this is the first formulation combining optimal transport and sparsity for solving inverse problems. As is shown on simulated and real data, our method can successfully track targets in scenarios where information from separate frequency bands alone is insufficient.

[128]  arXiv:2402.19351 [pdf, ps, other]
Title: Oriented trees in $O(k \sqrt{k})$-chromatic digraphs, a subquadratic bound for Burr's conjecture
Comments: 17 pages, 2 figures
Subjects: Combinatorics (math.CO); Discrete Mathematics (cs.DM)

In 1980, Burr conjectured that every directed graph with chromatic number $2k-2$ contains any oriented tree of order $k$ as a subdigraph. Burr showed that chromatic number $(k-1)^2$ suffices, which was improved in 2013 to $\frac{k^2}{2} - \frac{k}{2} + 1$ by Addario-Berry et al. We give the first subquadratic bound for Burr's conjecture, by showing that every directed graph with chromatic number $8\sqrt{\frac{2}{15}} k \sqrt{k} + O(k)$ contains any oriented tree of order $k$. Moreover, we provide improved bounds of $\sqrt{\frac{4}{3}} k \sqrt{k}+O(k)$ for arborescences, and $(b-1)(k-3)+3$ for paths on $b$ blocks, with $b\ge 2$.

[129]  arXiv:2402.19360 [pdf, other]
Title: Joint Chance Constrained Optimal Control via Linear Programming
Subjects: Optimization and Control (math.OC); Systems and Control (eess.SY)

We establish a linear programming formulation for the solution of joint chance constrained optimal control problems over finite time horizons. The joint chance constraint may represent an invariance, reachability or reach-avoid specification that the trajectory must satisfy with a predefined probability. Compared to the existing literature, the formulation is computationally tractable and the solution exact.

[130]  arXiv:2402.19363 [pdf, ps, other]
Title: Approximate controllability and Irreducibility of the transition semigroup associated with Convective Brinkman-Forchheimer extended Darcy Equations
Subjects: Probability (math.PR); Analysis of PDEs (math.AP); Optimization and Control (math.OC)

In this article, the following controlled convective Brinkman-Forchheimer extended Darcy (CBFeD) system is considered in a $d$-dimensional torus $\mathbb{T}^d$:
\begin{align*}
\frac{\partial\boldsymbol{y}}{\partial t}-\mu \Delta\boldsymbol{y}+(\boldsymbol{y}\cdot\nabla)\boldsymbol{y}+\alpha\boldsymbol{y}+\beta\vert \boldsymbol{y}\vert^{r-1}\boldsymbol{y}+\gamma\vert \boldsymbol{y}\vert ^{q-1}\boldsymbol{y}+\nabla p=\boldsymbol{g}+\boldsymbol{u},\ \nabla\cdot\boldsymbol{y}=0,
\end{align*}
where $d\in\{2,3\}$, $\mu,\alpha,\beta>0$, $\gamma\in\mathbb{R}$, $r,q\in[1,\infty)$ with $r>q\geq 1$ and $\boldsymbol{u}$ is the control. For the super critical ($r>3$) and critical ($r=3$ with $2\beta\mu>1$) cases, we first show the approximate controllability of the above system in the usual energy space (divergence-free $\mathbb{L}^2(\mathbb{T}^d)$ space). As an application of the approximate controllability result, we establish the irreducibility of the transition semigroup associated with stochastic CBFeD system perturbed by non-degenerate Gaussian noise in the usual energy space by exploiting the regularity of solutions, smooth approximation of the multi-valued map $\mathrm{sgn}(\cdot)$ a density argument and monotonicity properties of the linear and nonlinear operators.

[131]  arXiv:2402.19368 [pdf, ps, other]
Title: Orbifold Euler Characteristics of Compactified Jacobians
Authors: Sofia Wood
Comments: 18 pages; comments very welcome!
Subjects: Algebraic Geometry (math.AG)

We calculate the orbifold Euler characteristics of all the degree d fine universal compactified Jacobians (defined by Pagani and Tommasi) over the moduli space of stable curves of genus g with n marked points. We show that this orbifold Euler characteristic agrees with the Euler characteristic of the moduli space of stable, genus 0 curves with 2g+n markings up to a combinatorial factor, and in particular, is independent of the degree d and the choice of degree d universal fine compactified Jacobian. As a special case, we see that the universal fine compactified Jacobians defined by Kass and Pagani, depending on a universal polarization, have orbifold Euler characteristic independent of this choice of polarization.

[132]  arXiv:2402.19374 [pdf, ps, other]
Title: The $X$-semiprimeness of Rings
Comments: 19 pages, comments are welcome
Subjects: Rings and Algebras (math.RA)

For a subset $X$ of a ring $R$, the ring $R$ is called $X$-semiprime if, given $a\in R$, $aXa=0$ implies $a=0$. This provides with a proper class of semiprime rings. If $\mathit{Id}(R)$ denotes the set of all idempotents of $ R $, and $U(R)$ denotes the set of all units of $R$, $R$ is also called idempotent semiprime (resp. unit-semiprime) if $R$ is $\mathit{Id}(R)$ -semiprime (resp. $U(R)$-semiprime). It is proved that every idempotent semiprime ring is unit-semiprime. This gives a close link to a previous research of the first author. Given a Lie ideal $L$ of a ring $R$, we give a criterion for the $L$-semiprimeness of the ring $R$. We also give a complete characterization of Lie ideals $L$ of a prime ring $R$ such that it is $L$ -semiprime. In particular, matrix rings, semiprime rings, prime rings possessing a nontrivial idempotent, and regular rings are studied. The analogous notion of $X$-primeness is also defined and studied.

[133]  arXiv:2402.19381 [pdf, other]
Title: Optimized Bayesian Framework for Inverse Heat Transfer Problems Using Reduced Order Methods
Subjects: Numerical Analysis (math.NA)

A stochastic inverse heat transfer problem is formulated to infer the transient heat flux, treated as an unknown Neumann boundary condition. Therefore, an Ensemble-based Simultaneous Input and State Filtering as a Data Assimilation technique is utilized for simultaneous temperature distribution prediction and heat flux estimation. This approach is incorporated with Radial Basis Functions not only to lessen the size of unknown inputs but also to mitigate the computational burden of this technique. The procedure applies to the specific case of a mold used in Continuous Casting machinery, and it is based on the sequential availability of temperature provided by thermocouples inside the mold. Our research represents a significant contribution to achieving probabilistic boundary condition estimation in real-time handling with noisy measurements and errors in the model. We additionally demonstrate the procedure's dependence on some hyperparameters that are not documented in the existing literature. Accurate real-time prediction of the heat flux is imperative for the smooth operation of Continuous Casting machinery at the boundary region where the Continuous Casting mold and the molten steel meet which is not also physically measurable. Thus, this paves the way for efficient real-time monitoring and control, which is critical for preventing caster shutdowns.

[134]  arXiv:2402.19386 [pdf, ps, other]
Title: The viscous variational wave equation with transport noise
Authors: Peter H.C. Pang
Comments: 40 pages
Subjects: Analysis of PDEs (math.AP); Probability (math.PR)

This article considers the variational wave equation with viscosity and transport noise as a system of three coupled nonlinear stochastic partial differential equations. We prove pathwise global existence, uniqueness, and temporal continuity of solutions to this system in $L^2_x$. Martingale solutions are extracted from a two-level Galerkin approximation via the Skorokhod--Jakubowski theorem. We use the apparatus of Dudley maps to streamline this stochastic compactness method, bypassing the usual martingale identification argument. Pathwise uniqueness for the system is established through a renormalisation procedure that involves double commutator estimates and a delicate handling of noise and nonlinear terms. New model-specific commutator estimates are proven.

[135]  arXiv:2402.19396 [pdf, other]
Title: Machine Learning for Quantitative MR Image Reconstruction
Subjects: Optimization and Control (math.OC)

In the last years, the design of image reconstruction methods in the field of quantitative Magnetic Resonance Imaging (qMRI) has experienced a paradigm shift. Often, when dealing with (quantitative) MR image reconstruction problems, one is concerned with solving one or a couple of ill-posed inverse problems which require the use of advanced regularization methods. An increasing amount of attention is nowadays put on the development of data-driven methods using Neural Networks (NNs) to learn meaningful prior information without the need to explicitly model hand-crafted priors. In addition, the available hardware and computational resources nowadays offer the possibility to learn regularization models in a so-called model-aware fashion, which is a unique key feature that distinguishes these models from regularization methods learned in a more classical, model-agnostic manner. Model-aware methods are not only tailored to the considered data, but also to the class of considered imaging problems and nowadays constitute the state-of-the-art in image reconstruction methods. In the following chapter, we provide the reader with an extensive overview of methods that can be employed for (quantitative) MR image reconstruction, also highlighting their advantages and limitations both from a theoretical and computational point of view.

[136]  arXiv:2402.19409 [pdf, ps, other]
Title: $C_{10}$ has positive Turán density in the hypercube
Comments: 5 pages
Subjects: Combinatorics (math.CO)

The $n$-dimensional hypercube $Q_n$ is a graph with vertex set $\{0,1\}^n$ and there is an edge between two vertices if they differ in exactly one coordinate. For any graph $H$, define $\text{ex}(Q_n,H)$ to be the maximum number of edges of a subgraph of $Q_n$ without a copy of $H$. In this short note, we prove that for any $n \in \mathbb{N}$ $$\text{ex}(Q_n, C_{10}) > 0.024 \cdot e(Q_n).$$ Our construction is strongly inspired by the recent breakthrough work of Ellis, Ivan, and Leader, who showed that "daisy" hypergraphs have positive Tur\'an density with an extremely clever and simple linear-algebraic argument.

[137]  arXiv:2402.19434 [pdf, other]
Title: Digital Twin Aided Massive MIMO: CSI Compression and Feedback
Comments: Accepted in ICC 2024. Dataset and code files will be available soon on the DeepMIMO website this https URL
Subjects: Information Theory (cs.IT); Signal Processing (eess.SP)

Deep learning (DL) approaches have demonstrated high performance in compressing and reconstructing the channel state information (CSI) and reducing the CSI feedback overhead in massive MIMO systems. One key challenge, however, with the DL approaches is the demand for extensive training data. Collecting this real-world CSI data incurs significant overhead that hinders the DL approaches from scaling to a large number of communication sites. To address this challenge, we propose a novel direction that utilizes site-specific \textit{digital twins} to aid the training of DL models. The proposed digital twin approach generates site-specific synthetic CSI data from the EM 3D model and ray tracing, which can then be used to train the DL model without real-world data collection. To further improve the performance, we adopt online data selection to refine the DL model training with a small real-world CSI dataset. Results show that a DL model trained solely on the digital twin data can achieve high performance when tested in a real-world deployment. Further, leveraging domain adaptation techniques, the proposed approach requires orders of magnitude less real-world data to approach the same performance of the model trained completely on a real-world CSI dataset.

[138]  arXiv:2402.19438 [pdf, ps, other]
Title: Revealing the hidden beauty of finite groups with Cayley graphs
Authors: Matthew Macauley
Comments: Submitted to the 2024 Bridges Conference: Mathematics, Art, Music, Architecture, Culture. 8 pages, 10 figures
Subjects: History and Overview (math.HO)

Group theory involves the study of symmetry, and its inherent beauty gives it the potential to be one of the most accessible and enjoyable areas of mathematics, for students and non-mathematicians alike. Unfortunately, many students never get a glimpse into the more alluring parts of this field because "traditional" algebra classes are often taught in a dry axiomatic fashion, devoid of visuals. This article will showcase aesthetic pictures that can bring this subject to life. It will also leave the reader with some (intentionally) unanswered puzzles that undergraduate students, hobbyists, and mathematical artists can explore and answer, and even create new versions themselves.

[139]  arXiv:2402.19439 [pdf, ps, other]
Title: Global well-posedness for supercritical SQG with perturbations of radially symmetric data
Subjects: Analysis of PDEs (math.AP)

We study the global well-posedness of the supercritical dissipative surface quasi-geostrophic (SQG) equation, a key model in geophysical fluid dynamics. While local well-posedness is known, achieving global well-posedness for large initial data remains open. Motivated by enhanced decay in radial solutions, we aim to establish global well-posedness for small perturbations of potentially large radial data. Our main result shows that for small perturbations of radial data, the SQG equation admits a unique global solution.

[140]  arXiv:2402.19444 [pdf, other]
Title: The "spread" of Thompson's group $F$
Authors: Gili Golan
Comments: 21 pages, 1 figure. arXiv admin note: text overlap with arXiv:2210.03564
Subjects: Group Theory (math.GR)

Recall that a group $G$ is said to be $\frac{3}{2}$-generated if every non-trivial element $g\in G$ has a co-generator in $G$ (i.e., an element which together with $g$ generates $G$). Thompson's group $V$ was proved to be $\frac{3}{2}$-generated by Donoven and Harper in 2019. It was the first example of an infinite finitely presented non-cyclic $\frac{3}{2}$-generated group. In 2022, Bleak, Harper and Skipper proved that Thompson's group $T$ is also $\frac{3}{2}$-generated. Since the abelianization of Thompson's group $F$ is $\mathbb{Z}$, it cannot be $\frac{3}{2}$-generated. However, we recently proved that Thompson's group $F$ is "almost" $\frac{3}{2}$-generated in the sense that every element of $F$ whose image in the abelianization forms part of a generating pair of $\mathbb{Z}^2$ is part of a generating pair of $F$.
A natural generalization of $\frac{3}{2}$-generation is the notion of spread. Recall that the spread of a group $G$ is the supremum over all integers $k$ such that every $k$ non-trivial elements of $G$ have a common co-generator in $G$. The uniform spread of a group $G$ is the supremum over all integers $k$ for which there exists a conjugacy class $C\subseteq G$ such that every $k$ non-trivial elements of $G$ have a common co-generator which belongs to $C$. In this paper we study modified versions of these notions for Thompson's group $F$.

[141]  arXiv:2402.19445 [pdf, ps, other]
Title: Solution of the Diophantine equation $x^2 + p^k=y^n$
Authors: Arkabrata Ghosh
Comments: 12 pages
Subjects: Number Theory (math.NT)

The main aim of this article is to find all solutions of the Diophantine equation $x^2 + p^k=y^n$ where $p \equiv 1 \pmod 4$, $\frac{p-1}{3}$ is a perfect square and the class number of $\mathbb{Z}[\sqrt{-p}]$ is $2$. In this article, I used a method involving prime factorization and class numbers which is different from using congruent number argument which is widely used in this type of problem.

[142]  arXiv:2402.19447 [pdf, ps, other]
Title: Weighted Catalan convolution and $(q,2)$-Fock space
Authors: Yungang Lu
Subjects: Combinatorics (math.CO)

Motivated by the study of certain combinatorial properties of $(q,2)$-Fock space, we compute explicitly a sequence driven by the Catalan's convolution and parameterized by $1+q$. As an application of this explicit form, we calculate the number of pair partitions involved in the determination of the vacuum--moments of the field operator defined on the $(q,2)$-Fock space.

[143]  arXiv:2402.19461 [pdf, ps, other]
Title: Wreath-like products of groups and their von Neumann algebras III: Embeddings
Subjects: Operator Algebras (math.OA); Group Theory (math.GR)

For a class of wreath-like product groups with property (T), we describe explicitly all the embeddings between their von Neumann algebras. This allows us to provide a continuum of ICC groups with property (T) whose von Neumann algebras are pairwise non (stably) embeddable. We also give a construction of groups in this class only having inner injective homomorphisms. As an application, we obtain examples of group von Neumann algebras which admit only inner endomorphisms.

Cross-lists for Fri, 1 Mar 24

[144]  arXiv:2301.06836 (cross-list from gr-qc) [pdf, ps, other]
Title: Anisotropic solutions for $R^2$ gravity model with a scalar field
Comments: 13 pages, accepted for publication in Phys. Atom. Nuclei
Journal-ref: Phys. Atom. Nuclei 86 (2023) 1526
Subjects: General Relativity and Quantum Cosmology (gr-qc); Cosmology and Nongalactic Astrophysics (astro-ph.CO); Mathematical Physics (math-ph)

We study anisotropic solutions for the pure $R^2$ gravity model with a scalar field in the Bianchi I metric. The evolution equations have a singularity at zero value of the Ricci scalar $R$ for anisotropic solutions, whereas these equations are smooth for isotropic solutions. So, there is no anisotropic solution with the Ricci scalar smoothly changing its sign during evolution. We have found anisotropic solutions using the conformal transformation of the metric and the Einstein frame. The general solution in the Einstein frame has been found explicitly. The corresponding solution in the Jordan frame has been constructed in quadratures.

[145]  arXiv:2402.18377 (cross-list from cs.LG) [pdf, other]
Title: Out-of-Domain Generalization in Dynamical Systems Reconstruction
Subjects: Machine Learning (cs.LG); Artificial Intelligence (cs.AI); Dynamical Systems (math.DS); Chaotic Dynamics (nlin.CD)

In science we are interested in finding the governing equations, the dynamical rules, underlying empirical phenomena. While traditionally scientific models are derived through cycles of human insight and experimentation, recently deep learning (DL) techniques have been advanced to reconstruct dynamical systems (DS) directly from time series data. State-of-the-art dynamical systems reconstruction (DSR) methods show promise in capturing invariant and long-term properties of observed DS, but their ability to generalize to unobserved domains remains an open challenge. Yet, this is a crucial property we would expect from any viable scientific theory. In this work, we provide a formal framework that addresses generalization in DSR. We explain why and how out-of-domain (OOD) generalization (OODG) in DSR profoundly differs from OODG considered elsewhere in machine learning. We introduce mathematical notions based on topological concepts and ergodic theory to formalize the idea of learnability of a DSR model. We formally prove that black-box DL techniques, without adequate structural priors, generally will not be able to learn a generalizing DSR model. We also show this empirically, considering major classes of DSR algorithms proposed so far, and illustrate where and why they fail to generalize across the whole phase space. Our study provides the first comprehensive mathematical treatment of OODG in DSR, and gives a deeper conceptual understanding of where the fundamental problems in OODG lie and how they could possibly be addressed in practice.

[146]  arXiv:2402.18591 (cross-list from cs.LG) [pdf, ps, other]
Title: Stochastic contextual bandits with graph feedback: from independence number to MAS number
Subjects: Machine Learning (cs.LG); Computer Science and Game Theory (cs.GT); Statistics Theory (math.ST)

We consider contextual bandits with graph feedback, a class of interactive learning problems with richer structures than vanilla contextual bandits, where taking an action reveals the rewards for all neighboring actions in the feedback graph under all contexts. Unlike the multi-armed bandits setting where a growing literature has painted a near-complete understanding of graph feedback, much remains unexplored in the contextual bandits counterpart. In this paper, we make inroads into this inquiry by establishing a regret lower bound $\Omega(\sqrt{\beta_M(G) T})$, where $M$ is the number of contexts, $G$ is the feedback graph, and $\beta_M(G)$ is our proposed graph-theoretical quantity that characterizes the fundamental learning limit for this class of problems. Interestingly, $\beta_M(G)$ interpolates between $\alpha(G)$ (the independence number of the graph) and $\mathsf{m}(G)$ (the maximum acyclic subgraph (MAS) number of the graph) as the number of contexts $M$ varies. We also provide algorithms that achieve near-optimal regrets for important classes of context sequences and/or feedback graphs, such as transitively closed graphs that find applications in auctions and inventory control. In particular, with many contexts, our results show that the MAS number completely characterizes the statistical complexity for contextual bandits, as opposed to the independence number in multi-armed bandits.

[147]  arXiv:2402.18596 (cross-list from cs.GR) [pdf, other]
Title: Image-To-Mesh Conversion for Biomedical Simulations
Comments: 37 pages, 26 figures
Subjects: Graphics (cs.GR); Mathematical Software (cs.MS); Numerical Analysis (math.NA)

Converting a three-dimensional medical image into a 3D mesh that satisfies both the quality and fidelity constraints of predictive simulations and image-guided surgical procedures remains a critical problem. Presented is an image-to-mesh conversion method called CBC3D. It first discretizes a segmented image by generating an adaptive Body-Centered Cubic (BCC) mesh of high-quality elements. Next, the tetrahedral mesh is converted into a mixed-element mesh of tetrahedra, pentahedra, and hexahedra to decrease element count while maintaining quality. Finally, the mesh surfaces are deformed to their corresponding physical image boundaries, improving the mesh's fidelity. The deformation scheme builds upon the ITK open-source library and is based on the concept of energy minimization, relying on a multi-material point-based registration. It uses non-connectivity patterns to implicitly control the number of extracted feature points needed for the registration and, thus, adjusts the trade-off between the achieved mesh fidelity and the deformation speed. We compare CBC3D with four widely used and state-of-the-art homegrown image-to-mesh conversion methods from industry and academia. Results indicate that the CBC3D meshes (i) achieve high fidelity, (ii) keep the element count reasonably low, and (iii) exhibit good element quality.

[148]  arXiv:2402.18637 (cross-list from hep-th) [pdf, other]
Title: Infrared finite scattering theory: Amplitudes and soft theorems
Comments: 40 pages + appendices
Subjects: High Energy Physics - Theory (hep-th); General Relativity and Quantum Cosmology (gr-qc); Mathematical Physics (math-ph)

Any non-trivial scattering with massless fields in four spacetime dimensions will generically produce an out-state with memory. Scattering with any massless fields violates the standard assumption of asymptotic completeness -- that all "in" and "out" states lie in the standard (zero memory) Fock space -- and therefore leads to infrared divergences in the standard $S$-matrix amplitudes. We define an infrared finite scattering theory valid for general quantum field theories and quantum gravity. The (infrared finite) "superscattering" map $\$$ is defined as a map between "in" and "out" states which does not require any a priori choice of a preferred Hilbert space. We define a "generalized asymptotic completeness" which accommodates states with memory in the space of asymptotic states. We define a complete basis of improper states on any memory Fock space (called "BMS particle" states) which are eigenstates of the energy-momentum -- or, more generally, the BMS supermomentum -- that generalize the usual $n$-particle momentum basis to account for states with memory. We then obtain infrared finite $\$$-amplitudes defined as matrix elements of $\$$ in the BMS particle basis. This formulation of the scattering theory is a key step towards analyzing fine-grained details of the infrared finite scattering theory. In quantum gravity, invariance of $\$$ under BMS supertranslations implies factorization of $\$$-amplitudes as the frequency of one of the BMS particles vanishes. This proves an infrared finite analog of the soft graviton theorem. Similarly, an infrared finite soft photon theorem in QED follows from invariance of $\$$ under large gauge transformations. We comment on how one must generalize this framework to consider $\$$-amplitudes for theories with collinear divergences (e.g., massless QED and Yang-Mills theories).

[149]  arXiv:2402.18684 (cross-list from quant-ph) [pdf, ps, other]
Title: Quantum State Compression with Polar Codes
Comments: Extended Version of ISIT 2024 Submission
Subjects: Quantum Physics (quant-ph); Information Theory (cs.IT)

In the quantum compression scheme proposed by Schumacher, Alice compresses a message that Bob decompresses. In that approach, there is some probability of failure and, even when successful, some distortion of the state. For sufficiently large blocklengths, both of these imperfections can be made arbitrarily small while achieving a compression rate that asymptotically approaches the source coding bound. However, direct implementation of Schumacher compression suffers from poor circuit complexity. In this paper, we consider a slightly different approach based on classical syndrome source coding. The idea is to use a linear error-correcting code and treat the message to be compressed as an error pattern. If the message is a correctable error (i.e., a coset leader) then Alice can use the error-correcting code to convert her message to a corresponding quantum syndrome. An implementation of this based on polar codes is described and simulated. As in classical source coding based on polar codes, Alice maps the information into the ``frozen" qubits that constitute the syndrome. To decompress, Bob utilizes a quantum version of successive cancellation coding.

[150]  arXiv:2402.18689 (cross-list from cs.LG) [pdf, other]
Title: The VOROS: Lifting ROC curves to 3D
Comments: 38 pages, 19 figures
Subjects: Machine Learning (cs.LG); Metric Geometry (math.MG); Statistics Theory (math.ST); Methodology (stat.ME)

The area under the ROC curve is a common measure that is often used to rank the relative performance of different binary classifiers. However, as has been also previously noted, it can be a measure that ill-captures the benefits of different classifiers when either the true class values or misclassification costs are highly unbalanced between the two classes. We introduce a third dimension to capture these costs, and lift the ROC curve to a ROC surface in a natural way. We study both this surface and introduce the VOROS, the volume over this ROC surface, as a 3D generalization of the 2D area under the ROC curve. For problems where there are only bounds on the expected costs or class imbalances, we restrict consideration to the volume of the appropriate subregion of the ROC surface. We show how the VOROS can better capture the costs of different classifiers on both a classical and a modern example dataset.

[151]  arXiv:2402.18697 (cross-list from stat.ML) [pdf, other]
Title: Inferring Dynamic Networks from Marginals with Iterative Proportional Fitting
Subjects: Machine Learning (stat.ML); Machine Learning (cs.LG); Social and Information Networks (cs.SI); Optimization and Control (math.OC); Statistics Theory (math.ST)

A common network inference problem, arising from real-world data constraints, is how to infer a dynamic network from its time-aggregated adjacency matrix and time-varying marginals (i.e., row and column sums). Prior approaches to this problem have repurposed the classic iterative proportional fitting (IPF) procedure, also known as Sinkhorn's algorithm, with promising empirical results. However, the statistical foundation for using IPF has not been well understood: under what settings does IPF provide principled estimation of a dynamic network from its marginals, and how well does it estimate the network? In this work, we establish such a setting, by identifying a generative network model whose maximum likelihood estimates are recovered by IPF. Our model both reveals implicit assumptions on the use of IPF in such settings and enables new analyses, such as structure-dependent error bounds on IPF's parameter estimates. When IPF fails to converge on sparse network data, we introduce a principled algorithm that guarantees IPF converges under minimal changes to the network structure. Finally, we conduct experiments with synthetic and real-world data, which demonstrate the practical value of our theoretical and algorithmic contributions.

[152]  arXiv:2402.18703 (cross-list from quant-ph) [pdf, ps, other]
Title: Zero-error communication, scrambling, and ergodicity
Comments: Preliminary version. Comments are welcome
Subjects: Quantum Physics (quant-ph); Information Theory (cs.IT); Operator Algebras (math.OA); Probability (math.PR)

The long term behaviour of a quantum channel under iterations (i.e. under repeated applications of itself) yields a plethora of interesting properties. These include ergodicity, mixing, eventual scrambling, becoming strictly positive, and the vanishing of its one-shot zero error capacities. We derive relations between these seemingly different properties and find novel bounds on indices which quantify the minimum number of iterations needed for the onset of some of these properties. We obtain a lower bound on the one-shot zero-error classical capacity of $n$ iterations of an ergodic channel (for any positive integer $n$) in terms of the cardinality of its peripheral spectrum. We also find upper bounds on the minimum number of iterations needed for the one-shot capacities of any channel to stabilize. We consider two classes of quantum channels, satisfying certain symmetries, for which upper bounds on the above indices are optimal, since they reduce to the corresponding indices for a stochastic matrix (for which the bounds are known to be optimal). As an auxiliary result, we obtain a trade-off relation between the one-shot zero error classical and quantum capacities of a quantum channel.

[153]  arXiv:2402.18714 (cross-list from quant-ph) [pdf, ps, other]
Title: A quantum algorithm for learning a graph of bounded degree
Comments: 15 pages
Subjects: Quantum Physics (quant-ph); Combinatorics (math.CO)

We are presented with a graph, $G$, on $n$ vertices with $m$ edges whose edge set is unknown. Our goal is to learn the edges of $G$ with as few queries to an oracle as possible. When we submit a set $S$ of vertices to the oracle, it tells us whether or not $S$ induces at least one edge in $G$. This so-called OR-query model has been well studied, with Angluin and Chen giving an upper bound on the number of queries needed of $O(m \log n)$ for a general graph $G$ with $m$ edges.
When we allow ourselves to make *quantum* queries (we may query subsets in superposition), then we can achieve speedups over the best possible classical algorithms. In the case where $G$ has maximum degree $d$ and is $O(1)$-colorable, Montanaro and Shao presented an algorithm that learns the edges of $G$ in at most $\tilde{O}(d^2m^{3/4})$ quantum queries. This gives an upper bound of $\tilde{O}(m^{3/4})$ quantum queries when $G$ is a matching or a Hamiltonian cycle, which is far away from the lower bound of $\Omega(\sqrt{m})$ queries given by Ambainis and Montanaro.
We improve on the work of Montanaro and Shao in the case where $G$ has bounded degree. In particular, we present a randomized algorithm that, with high probability, learns cycles and matchings in $\tilde{O}(\sqrt{m})$ quantum queries, matching the theoretical lower bound up to logarithmic factors.

[154]  arXiv:2402.18745 (cross-list from stat.ME) [pdf, other]
Title: Degree-heterogeneous Latent Class Analysis for High-dimensional Discrete Data
Subjects: Methodology (stat.ME); Statistics Theory (math.ST)

The latent class model is a widely used mixture model for multivariate discrete data. Besides the existence of qualitatively heterogeneous latent classes, real data often exhibit additional quantitative heterogeneity nested within each latent class. The modern latent class analysis also faces extra challenges, including the high-dimensionality, sparsity, and heteroskedastic noise inherent in discrete data. Motivated by these phenomena, we introduce the Degree-heterogeneous Latent Class Model and propose a spectral approach to clustering and statistical inference in the challenging high-dimensional sparse data regime. We propose an easy-to-implement HeteroClustering algorithm. It uses heteroskedastic PCA with L2 normalization to remove degree effects and perform clustering in the top singular subspace of the data matrix. We establish an exponential error rate for HeteroClustering, leading to exact clustering under minimal signal-to-noise conditions. We further investigate the estimation and inference of the high-dimensional continuous item parameters in the model, which are crucial to interpreting and finding useful markers for latent classes. We provide comprehensive procedures for global testing and multiple testing of these parameters with valid error controls. The superior performance of our methods is demonstrated through extensive simulations and applications to three diverse real-world datasets from political voting records, genetic variations, and single-cell sequencing.

[155]  arXiv:2402.18839 (cross-list from cs.LG) [pdf, other]
Title: Extended Flow Matching: a Method of Conditional Generation with Generalized Continuity Equation
Comments: 15 pages, 4 figures
Subjects: Machine Learning (cs.LG); Analysis of PDEs (math.AP); Functional Analysis (math.FA); Optimization and Control (math.OC); Probability (math.PR)

The task of conditional generation is one of the most important applications of generative models, and numerous methods have been developed to date based on the celebrated diffusion models, with the guidance-based classifier-free method taking the lead. However, the theory of the guidance-based method not only requires the user to fine-tune the "guidance strength," but its target vector field does not necessarily correspond to the conditional distribution used in training. In this paper, we develop the theory of conditional generation based on Flow Matching, a current strong contender of diffusion methods. Motivated by the interpretation of a probability path as a distribution on path space, we establish a novel theory of flow-based generation of conditional distribution by employing the mathematical framework of generalized continuity equation instead of the continuity equation in flow matching. This theory naturally derives a method that aims to match the matrix field as opposed to the vector field. Our framework ensures the continuity of the generated conditional distribution through the existence of flow between conditional distributions. We will present our theory through experiments and mathematical results.

[156]  arXiv:2402.18851 (cross-list from cs.LG) [pdf, other]
Title: Applications of 0-1 Neural Networks in Prescription and Prediction
Subjects: Machine Learning (cs.LG); Artificial Intelligence (cs.AI); Optimization and Control (math.OC); Machine Learning (stat.ML)

A key challenge in medical decision making is learning treatment policies for patients with limited observational data. This challenge is particularly evident in personalized healthcare decision-making, where models need to take into account the intricate relationships between patient characteristics, treatment options, and health outcomes. To address this, we introduce prescriptive networks (PNNs), shallow 0-1 neural networks trained with mixed integer programming that can be used with counterfactual estimation to optimize policies in medium data settings. These models offer greater interpretability than deep neural networks and can encode more complex policies than common models such as decision trees. We show that PNNs can outperform existing methods in both synthetic data experiments and in a case study of assigning treatments for postpartum hypertension. In particular, PNNs are shown to produce policies that could reduce peak blood pressure by 5.47 mm Hg (p=0.02) over existing clinical practice, and by 2 mm Hg (p=0.01) over the next best prescriptive modeling technique. Moreover PNNs were more likely than all other models to correctly identify clinically significant features while existing models relied on potentially dangerous features such as patient insurance information and race that could lead to bias in treatment.

[157]  arXiv:2402.18905 (cross-list from cs.LG) [pdf, other]
Title: On the Convergence of Differentially-Private Fine-tuning: To Linearly Probe or to Fully Fine-tune?
Subjects: Machine Learning (cs.LG); Artificial Intelligence (cs.AI); Cryptography and Security (cs.CR); Optimization and Control (math.OC)

Differentially private (DP) machine learning pipelines typically involve a two-phase process: non-private pre-training on a public dataset, followed by fine-tuning on private data using DP optimization techniques. In the DP setting, it has been observed that full fine-tuning may not always yield the best test accuracy, even for in-distribution data. This paper (1) analyzes the training dynamics of DP linear probing (LP) and full fine-tuning (FT), and (2) explores the phenomenon of sequential fine-tuning, starting with linear probing and transitioning to full fine-tuning (LP-FT), and its impact on test loss. We provide theoretical insights into the convergence of DP fine-tuning within an overparameterized neural network and establish a utility curve that determines the allocation of privacy budget between linear probing and full fine-tuning. The theoretical results are supported by empirical evaluations on various benchmarks and models. The findings reveal the complex nature of DP fine-tuning methods. These results contribute to a deeper understanding of DP machine learning and highlight the importance of considering the allocation of privacy budget in the fine-tuning process.

[158]  arXiv:2402.18963 (cross-list from eess.IV) [pdf, other]
Title: Quantification of Tracer Dilution Dynamics: An Exploration into the Mathematical Modeling of Medical Imaging
Subjects: Image and Video Processing (eess.IV); Spectral Theory (math.SP)

Convolution and deconvolution are essential techniques in various fields, notably in medical imaging, where they play a crucial role in analyzing dynamic processes such as blood flow. This paper explores the convolution and deconvolution of arterial and microvascular signals for determining impulse and residue functions from in vivo or simulated data and the derivation of the relationship between the residue function and perfusion metrics such as the Cerebral Blood Flow (CBF), Mean Transit Time (MTT) and Transit Time to Heterogeneity (TTH). The paper presents the spectral derivatives as a technique for recovering the impulse response function from the residue function, detailing the computational procedures involved and strategies for mitigating noise effects.

[159]  arXiv:2402.18997 (cross-list from gr-qc) [pdf, other]
Title: Motion of test particles in quasi anti-de Sitter regular black holes
Comments: 14 pages, 2 figures. Effort prepared for the Special Issue "20th Anniversary of IJGMMP"
Subjects: General Relativity and Quantum Cosmology (gr-qc); Mathematical Physics (math-ph)

We explore the characteristics of two novel regular spacetimes that exhibit a non-zero vacuum energy term, under the form of a (quasi) anti-de Sitter phase. Specifically, the first metric is spherical, while the second, derived by applying the generalized Newman-Janis algorithm to the first, is axisymmetric. We show that the equations of state of the effective fluids associated with the two metrics asymptotically tend to negative values, resembling quintessence. In addition, we study test particle motions, illustrating the main discrepancies among our models and more conventional metrics exhibiting non-vanishing anti-de Sitter phase.

[160]  arXiv:2402.19030 (cross-list from quant-ph) [pdf, other]
Title: A Faster Algorithm for the Free Energy in One-Dimensional Quantum Systems
Authors: Samuel O. Scalet
Subjects: Quantum Physics (quant-ph); Statistical Mechanics (cond-mat.stat-mech); Mathematical Physics (math-ph)

We consider the problem of approximating the free energy density of a translation-invariant, one-dimensional quantum spin system with finite range. While the complexity of this problem is nontrivial due to its close connection to problems with known hardness results, a classical subpolynomial-time algorithm has recently been proposed [Fawzi et al., 2022]. Combining several algorithmic techniques previously used for related problems, we propose an algorithm outperforming this result asymptotically and give rigorous bounds on its runtime. Our main techniques are the use of Araki expansionals, known from results on the nonexistence of phase transitions, and a matrix product operator construction. We also review a related approach using the Quantum Belief Propagation [Kuwahara et al., 2018], which in combination with our findings yields an equivalent result.

[161]  arXiv:2402.19047 (cross-list from cs.LG) [pdf, other]
Title: Theoretical Foundations of Deep Selective State-Space Models
Subjects: Machine Learning (cs.LG); Dynamical Systems (math.DS)

Structured state-space models (SSMs) such as S4, stemming from the seminal work of Gu et al., are gaining popularity as effective approaches for modeling sequential data. Deep SSMs demonstrate outstanding performance across a diverse set of domains, at a reduced training and inference cost compared to attention-based transformers. Recent developments show that if the linear recurrence powering SSMs allows for multiplicative interactions between inputs and hidden states (e.g. GateLoop, Mamba, GLA), then the resulting architecture can surpass in both in accuracy and efficiency attention-powered foundation models trained on text, at scales of billion parameters. In this paper, we give theoretical grounding to this recent finding using tools from Rough Path Theory: we show that when random linear recurrences are equipped with simple input-controlled transitions (selectivity mechanism), then the hidden state is provably a low-dimensional projection of a powerful mathematical object called the signature of the input -- capturing non-linear interactions between tokens at distinct timescales. Our theory not only motivates the success of modern selective state-space models such as Mamba but also provides a solid framework to understand the expressive power of future SSM variants.

[162]  arXiv:2402.19078 (cross-list from cs.LG) [pdf, other]
Title: Smooth Tchebycheff Scalarization for Multi-Objective Optimization
Subjects: Machine Learning (cs.LG); Artificial Intelligence (cs.AI); Neural and Evolutionary Computing (cs.NE); Optimization and Control (math.OC)

Multi-objective optimization problems can be found in many real-world applications, where the objectives often conflict each other and cannot be optimized by a single solution. In the past few decades, numerous methods have been proposed to find Pareto solutions that represent different optimal trade-offs among the objectives for a given problem. However, these existing methods could have high computational complexity or may not have good theoretical properties for solving a general differentiable multi-objective optimization problem. In this work, by leveraging the smooth optimization technique, we propose a novel and lightweight smooth Tchebycheff scalarization approach for gradient-based multi-objective optimization. It has good theoretical properties for finding all Pareto solutions with valid trade-off preferences, while enjoying significantly lower computational complexity compared to other methods. Experimental results on various real-world application problems fully demonstrate the effectiveness of our proposed method.

[163]  arXiv:2402.19096 (cross-list from cond-mat.stat-mech) [pdf, other]
Title: Structural Stability Hypothesis of Dual Unitary Quantum Chaos
Comments: 22 pages, 12 figures
Subjects: Statistical Mechanics (cond-mat.stat-mech); High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph); Chaotic Dynamics (nlin.CD); Quantum Physics (quant-ph)

Having spectral correlations that, over small enough energy scales, are described by random matrix theory is regarded as the most general defining feature of quantum chaotic systems as it applies in the many-body setting and away from any semiclassical limit. Although this property is extremely difficult to prove analytically for generic many-body systems, a rigorous proof has been achieved for dual-unitary circuits -- a special class of local quantum circuits that remain unitary upon swapping space and time. Here we consider the fate of this property when moving from dual-unitary to generic quantum circuits focussing on the \emph{spectral form factor}, i.e., the Fourier transform of the two-point correlation. We begin with a numerical survey that, in agreement with previous studies, suggests that there exists a finite region in parameter space where dual-unitary physics is stable and spectral correlations are still described by random matrix theory, although up to a maximal quasienergy scale. To explain these findings, we develop a perturbative expansion: it recovers the random matrix theory predictions, provided the terms occurring in perturbation theory obey a relatively simple set of assumptions. We then provide numerical evidence and a heuristic analytical argument supporting these assumptions.

[164]  arXiv:2402.19109 (cross-list from stat.ME) [pdf, other]
Title: Confidence and Assurance of Percentiles
Authors: Sanjay M. Joshi
Comments: 5 pages, 4 Figures
Subjects: Methodology (stat.ME); Information Theory (cs.IT)

Confidence interval of mean is often used when quoting statistics. The same rigor is often missing when quoting percentiles and tolerance or percentile intervals. This article derives the expression for confidence in percentiles of a sample population. Confidence intervals of median is compared to those of mean for a few sample distributions. The concept of assurance from reliability engineering is then extended to percentiles. The assurance level of sorted samples simply matches the confidence and percentile levels. Numerical method to compute assurance using Brent's optimization method is provided as an open-source python package.

[165]  arXiv:2402.19110 (cross-list from eess.SY) [pdf, other]
Title: Temporal-Aware Deep Reinforcement Learning for Energy Storage Bidding in Energy and Contingency Reserve Markets
Comments: 15 pages
Journal-ref: IEEE Transactions on Energy Markets, Policy and Regulation, 2024
Subjects: Systems and Control (eess.SY); Machine Learning (cs.LG); Optimization and Control (math.OC)

The battery energy storage system (BESS) has immense potential for enhancing grid reliability and security through its participation in the electricity market. BESS often seeks various revenue streams by taking part in multiple markets to unlock its full potential, but effective algorithms for joint-market participation under price uncertainties are insufficiently explored in the existing research. To bridge this gap, we develop a novel BESS joint bidding strategy that utilizes deep reinforcement learning (DRL) to bid in the spot and contingency frequency control ancillary services (FCAS) markets. Our approach leverages a transformer-based temporal feature extractor to effectively respond to price fluctuations in seven markets simultaneously and helps DRL learn the best BESS bidding strategy in joint-market participation. Additionally, unlike conventional "black-box" DRL model, our approach is more interpretable and provides valuable insights into the temporal bidding behavior of BESS in the dynamic electricity market. We validate our method using realistic market prices from the Australian National Electricity Market. The results show that our strategy outperforms benchmarks, including both optimization-based and other DRL-based strategies, by substantial margins. Our findings further suggest that effective temporal-aware bidding can significantly increase profits in the spot and contingency FCAS markets compared to individual market participation.

[166]  arXiv:2402.19163 (cross-list from cs.LG) [pdf, other]
Title: FedStruct: Federated Decoupled Learning over Interconnected Graphs
Comments: 10 pages plus 13 pages of appendices
Subjects: Machine Learning (cs.LG); Information Theory (cs.IT)

We address the challenge of federated learning on graph-structured data distributed across multiple clients. Specifically, we focus on the prevalent scenario of interconnected subgraphs, where inter-connections between different clients play a critical role. We present a novel framework for this scenario, named FedStruct, that harnesses deep structural dependencies. To uphold privacy, unlike existing methods, FedStruct eliminates the necessity of sharing or generating sensitive node features or embeddings among clients. Instead, it leverages explicit global graph structure information to capture inter-node dependencies. We validate the effectiveness of FedStruct through experimental results conducted on six datasets for semi-supervised node classification, showcasing performance close to the centralized approach across various scenarios, including different data partitioning methods, varying levels of label availability, and number of clients.

[167]  arXiv:2402.19172 (cross-list from eess.SP) [pdf, other]
Title: Point Processes and spatial statistics in time-frequency analysis
Comments: Submitted
Subjects: Signal Processing (eess.SP); Sound (cs.SD); Audio and Speech Processing (eess.AS); Probability (math.PR)

A finite-energy signal is represented by a square-integrable, complex-valued function $t\mapsto s(t)$ of a real variable $t$, interpreted as time. Similarly, a noisy signal is represented by a random process. Time-frequency analysis, a subfield of signal processing, amounts to describing the temporal evolution of the frequency content of a signal. Loosely speaking, if $s$ is the audio recording of a musical piece, time-frequency analysis somehow consists in writing the musical score of the piece. Mathematically, the operation is performed through a transform $\mathcal{V}$, mapping $s \in L^2(\mathbb{R})$ onto a complex-valued function $\mathcal{V}s \in L^2(\mathbb{R}^2)$ of time $t$ and angular frequency $\omega$. The squared modulus $(t, \omega) \mapsto \vert\mathcal{V}s(t,\omega)\vert^2$ of the time-frequency representation is known as the spectrogram of $s$; in the musical score analogy, a peaked spectrogram at $(t_0,\omega_0)$ corresponds to a musical note at angular frequency $\omega_0$ localized at time $t_0$. More generally, the intuition is that upper level sets of the spectrogram contain relevant information about in the original signal. Hence, many signal processing algorithms revolve around identifying maxima of the spectrogram. In contrast, zeros of the spectrogram indicate perfect silence, that is, a time at which a particular frequency is absent. Assimilating $\mathbb{R}^2$ to $\mathbb{C}$ through $z = \omega + \mathrm{i}t$, this chapter focuses on time-frequency transforms $\mathcal{V}$ that map signals to analytic functions. The zeros of the spectrogram of a noisy signal are then the zeros of a random analytic function, hence forming a Point Process in $\mathbb{C}$. This chapter is devoted to the study of these Point Processes, to their links with zeros of Gaussian Analytic Functions, and to designing signal detection and denoising algorithms using spatial statistics.

[168]  arXiv:2402.19225 (cross-list from physics.bio-ph) [pdf, other]
Title: Deterministic Molecular Assembly with a Finite Set of Building Blocks: Universal Assembly Kits for Backbone-Assisted and Sequence-Directed Abstract Tile Assembly Models
Subjects: Biological Physics (physics.bio-ph); Combinatorics (math.CO)

Backbone-assisted assembly processes -- such as protein folding -- allow the assembly of a large number of structures with high accuracy from only a small handful of fundamental building blocks. We aim to explore general principles underlying this phenomenon by studying variants of the temperature-1 abstract tile assembly model (aTAM). We consider the existence of finite sets of tile types that can deterministically assemble any shape producible by a given assembly model; we call such tile type sets universal assembly kits. Our first model, which we call the ``backboned aTAM", generates backbone-assisted assembly by forcing tiles to be be added to lattice positions neighbouring the immediately preceding tile, using a predetermined sequence of tile types. We demonstrate the existence of universal assembly kit for the backboned aTAM, and show that the existence of this set is maintained even under stringent restrictions to the rules of assembly. We hypothesise that the advantage of the backboned aTAM relative to the conventional aTAM -- which does not possess such a set -- is in part due to the specification of a sequence, and in part due to the geometric restrictions imposed by the backbone. To explore this intuition, we develop a second model call the ``sequenced aTAM". The sequenced aTAM uses a predetermined sequence of tiles, but does not constrain a tile to neighbour the immediately preceding tiles. We prove that this model has no universal assembly kit. The lack of such a kit is surprising, given that the number of tile sequences of length $N$ scales faster than the number of producible shapes of size $N$ for a sufficiently large -- but finite -- set of tiles.

[169]  arXiv:2402.19230 (cross-list from quant-ph) [pdf, other]
Title: A Simple and Efficient Joint Measurement Strategy for Estimating Fermionic Observables and Hamiltonians
Subjects: Quantum Physics (quant-ph); Mathematical Physics (math-ph)

We propose a simple scheme to estimate fermionic observables and Hamiltonians relevant in quantum chemistry and correlated fermionic systems. Our approach is based on implementing a measurement that jointly measures noisy versions of any product of two or four Majorana operators in an $N$ mode fermionic system. To realize our measurement we use: (i) a randomization over a set of unitaries that realize products of Majorana fermion operators; (ii) a unitary, sampled at random from a constant-size set of suitably chosen fermionic Gaussian unitaries; (iii) a measurement of fermionic occupation numbers; (iv) suitable post-processing. Our scheme can estimate expectation values of all quadratic and quartic Majorana monomials to $\epsilon$ precision using $\mathcal{O}(N \log(N)/\epsilon^2)$ and $\mathcal{O}(N^2 \log(N)/\epsilon^2)$ measurement rounds respectively, matching the performance offered by fermionic shadow tomography. In certain settings, such as a rectangular lattice of qubits which encode an $N$ mode fermionic system via the Jordan-Wigner transformation, our scheme can be implemented in circuit depth $\mathcal{O}(N^{1/2})$ with $\mathcal{O}(N^{3/2})$ two-qubit gates, offering an improvement over fermionic and matchgate classical shadows that require depth $\mathcal{O}(N)$ and $\mathcal{O}(N^2)$ two-qubit gates. We also benchmark our method on molecular Hamiltonians and observe performances comparable to those offered by fermionic classical shadows.

[170]  arXiv:2402.19242 (cross-list from cs.LG) [pdf, other]
Title: Derivative-enhanced Deep Operator Network
Subjects: Machine Learning (cs.LG); Computational Engineering, Finance, and Science (cs.CE); Numerical Analysis (math.NA)

Deep operator networks (DeepONets), a class of neural operators that learn mappings between function spaces, have recently been developed as surrogate models for parametric partial differential equations (PDEs). In this work we propose a derivative-enhanced deep operator network (DE-DeepONet), which leverages the derivative information to enhance the prediction accuracy, and provide a more accurate approximation of the derivatives, especially when the training data are limited. DE-DeepONet incorporates dimension reduction of input into DeepONet and includes two types of derivative labels in the loss function for training, that is, the directional derivatives of the output function with respect to the input function and the gradient of the output function with respect to the physical domain variables. We test DE-DeepONet on three different equations with increasing complexity to demonstrate its effectiveness compared to the vanilla DeepONet.

[171]  arXiv:2402.19247 (cross-list from quant-ph) [pdf, other]
Title: Noisy intermediate-scale quantum simulation of the one-dimensional wave equation
Comments: 10 pages, 7 figures, 1 table
Subjects: Quantum Physics (quant-ph); Mathematical Physics (math-ph); Applied Physics (physics.app-ph); Classical Physics (physics.class-ph); Computational Physics (physics.comp-ph)

We design and implement quantum circuits for the simulation of the one-dimensional wave equation on the Quantinuum H1-1 quantum computer. The circuit depth of our approach scales as $O(n^{2})$ for $n$ qubits representing the solution on $2^n$ grid points, and leads to infidelities of $O(2^{-4n} t^{2})$ for simulation time $t$ assuming smooth initial conditions. By varying the qubit count we study the interplay between the algorithmic and physical gate errors to identify the optimal working point of minimum total error. Our approach to simulating the wave equation can readily be adapted to other quantum processors and serve as an application-oriented benchmark.

[172]  arXiv:2402.19257 (cross-list from cs.DM) [pdf, ps, other]
Title: More algorithmic results for problems of spread of influence in edge-weighted graphs with and without incentives
Subjects: Discrete Mathematics (cs.DM); Combinatorics (math.CO)

Many phenomena in real world social networks are interpreted as spread of influence between activated and non-activated network elements. These phenomena are formulated by combinatorial graphs, where vertices represent the elements and edges represent social ties between elements. A main problem is to study important subsets of elements (target sets or dynamic monopolies) such that their activation spreads to the entire network. In edge-weighted networks the influence between two adjacent vertices depends on the weight of their edge. In models with incentives, the main problem is to minimize total amount of incentives (called optimal target vectors) which can be offered to vertices such that some vertices are activated and their activation spreads to the whole network. Algorithmic study of target sets and vectors is a hot research field. We prove an inapproximability result for optimal target sets in edge weighted networks even for complete graphs. Some other hardness and polynomial time results are presented for optimal target vectors and degenerate threshold assignments in edge-weighted networks.

[173]  arXiv:2402.19297 (cross-list from cond-mat.soft) [pdf, ps, other]
Title: Linear stability of cylindrical, multicomponent vesicles
Subjects: Soft Condensed Matter (cond-mat.soft); Mathematical Physics (math-ph); Fluid Dynamics (physics.flu-dyn)

Vesicles are important surrogate structures made up of multiple phospholipids and cholesterol distributed in the form of a lipid bilayer. Tubular vesicles can undergo pearling i.e., formation of beads on the liquid thread akin to the Rayleigh-Plateau instability. Previous studies have inspected the effects of surface tension on the pearling instabilities of single-component vesicles. In this study, we perform a linear stability analysis on a multicomponent cylindrical vesicle. We solve the Stokes equations along with the Cahn-Hilliard equations to develop the linearized dynamic equations governing the vesicle shape and surface concentration fields. This helps us show that multicomponent vesicles can undergo pearling, buckling, and wrinkling even in the absence of surface tension, which is a significantly different result from studies on single-component vesicles. This behaviour arises due to the competition between the free energies of phase separation, line tension, and bending for this multi-phospholipid system. We determine the conditions under which axisymmetric and non-axisymmetric modes are dominant, and supplement our results with an energy analysis that shows the sources for these instabilities. We further show that these trends qualitatively match recent experiments.

[174]  arXiv:2402.19309 (cross-list from eess.SY) [pdf, other]
Title: Closed-loop training of static output feedback neural network controllers for large systems: A distillation case study
Subjects: Systems and Control (eess.SY); Optimization and Control (math.OC)

The online implementation of model predictive control for constrained multivariate systems has two main disadvantages: it requires an estimate of the entire model state and an optimisation problem must be solved online. These issues have typically been treated separately. This work proposes an integrated approach for the offline training of an output feedback neural network controller in closed loop. Online this neural network controller computers the plant inputs cheaply using noisy measurements. In addition, the controller can be trained to only make use of certain predefined measurements. Further, a heuristic approach is proposed to perform the automatic selection of important measurements. The proposed method is demonstrated by extensive simulations using a non-linear distillation column model of 50 states.

[175]  arXiv:2402.19310 (cross-list from gr-qc) [pdf, ps, other]
Title: Some Remarks on Wang-Yau Quasi-Local Mass
Subjects: General Relativity and Quantum Cosmology (gr-qc); Mathematical Physics (math-ph)

We review Wang-Yau quasi-local definitions along the line of gravitational Hamiltonian. This makes clear the connection and difference between Wang-Yau definition and Brown-York or even global ADM definition. We make a brief comment on admissibility condition in Wang-Yau quasi-lcoal mass. We extend the positivity proof for Wang-Yau quasi-local energy to allow possible presence of strictly stable apparent horizons through establishing solvability of Dirac equation in certain 3-manifolds that possess cylindrical ends, as in the case of Jang's graph blowing up at marginally outer trapped surfaces.

[176]  arXiv:2402.19349 (cross-list from quant-ph) [pdf, other]
Title: Optimal Fermionic Joint Measurements for Estimating Non-Commuting Majorana Observables
Comments: 30 pages, 7 figures
Subjects: Quantum Physics (quant-ph); Mathematical Physics (math-ph)

An important class of fermionic observables, relevant in tasks such as fermionic partial tomography and estimating energy levels of chemical Hamiltonians, are the binary measurements obtained from the product of anti-commuting Majorana operators. In this work, we investigate efficient estimation strategies of these observables based on a joint measurement which, after classical post-processing, yields all sufficiently unsharp (noisy) Majorana observables of even-degree. By exploiting the symmetry properties of the Majorana observables, as described by the braid group, we show that the incompatibility robustness, i.e., the minimal classical noise necessary for joint measurability, relates to the spectral properties of the Sachdev-Ye-Kitaev (SYK) model. In particular, we show that for an $n$ mode fermionic system, the incompatibility robustness of all degree--$2k$ Majorana observables satisfies $\Theta(n^{-k/2})$ for $k\leq 5$. Furthermore, we present a joint measurement scheme achieving the asymptotically optimal noise, implemented by a small number of fermionic Gaussian unitaries and sampling from the set of all Majorana monomials. Our joint measurement, which can be performed via a randomization over projective measurements, provides rigorous performance guarantees for estimating fermionic observables comparable with fermionic classical shadows.

[177]  arXiv:2402.19362 (cross-list from gr-qc) [pdf, ps, other]
Title: Dark energy and dark matter configurations for wormholes and solitionic hierarchies of nonmetric Ricci flows and $F(R,T,Q,T_{m})$ gravity
Comments: latex23, 11pt, 37 pages, with table of content; published online by EPJC
Journal-ref: Eur. Phys. J. C 84 (2024) 211
Subjects: General Relativity and Quantum Cosmology (gr-qc); Cosmology and Nongalactic Astrophysics (astro-ph.CO); Mathematical Physics (math-ph)

We extend the anholonomic frame and connection deformation method, AFCDM, for constructing exact and parametric solutions in general relativity, GR, to geometric flow models and modified gravity theories, MGTs, with nontrivial torsion and nonmetricity fields. Following abstract geometric or variational methods, we can derive corresponding systems of nonmetric gravitational and matter field equations which consist of very sophisticated systems of coupled nonlinear PDEs. Using nonholonomic frames with dyadic spacetime splitting and applying the AFCDM, we prove that such systems of PDEs can be decoupled and integrated in general forms for generic off-diagonal metric structures and generalized affine connections. We generate new classes of quasi-stationary solutions (which do not depend on time like coordinates) and study the physical properties of some physically important examples. Such exact or parametric solutions are determined by nonmetric solitonic distributions and/or ellipsoidal deformations of wormhole hole configurations. It is not possible to describe the thermodynamic properties of such solutions in the framework of the Bekenstein-Hawking paradigm because such metrics do not involve, in general, certain horizons, duality, or holographic configurations. Nevertheless, we can always elaborate on associated Grigori Perelman thermodynamic models elaborated for nonmetric geometric flows. In explicit form, applying the AFCDM, we construct and study the physical implications of new classes of traversable wormhole solutions describing solitonic deformation and dissipation of non-Riemannian geometric objects. Such models with nontrivial gravitational off-diagonal vacuum are important for elaborating models of dark energy and dark matter involving wormhole configurations and solitonic-type structure formation.

[178]  arXiv:2402.19442 (cross-list from cs.LG) [pdf, other]
Title: Training Dynamics of Multi-Head Softmax Attention for In-Context Learning: Emergence, Convergence, and Optimality
Comments: 141 pages, 7 figures
Subjects: Machine Learning (cs.LG); Artificial Intelligence (cs.AI); Optimization and Control (math.OC); Statistics Theory (math.ST); Machine Learning (stat.ML)

We study the dynamics of gradient flow for training a multi-head softmax attention model for in-context learning of multi-task linear regression. We establish the global convergence of gradient flow under suitable choices of initialization. In addition, we prove that an interesting "task allocation" phenomenon emerges during the gradient flow dynamics, where each attention head focuses on solving a single task of the multi-task model. Specifically, we prove that the gradient flow dynamics can be split into three phases -- a warm-up phase where the loss decreases rather slowly and the attention heads gradually build up their inclination towards individual tasks, an emergence phase where each head selects a single task and the loss rapidly decreases, and a convergence phase where the attention parameters converge to a limit. Furthermore, we prove the optimality of gradient flow in the sense that the limiting model learned by gradient flow is on par with the best possible multi-head softmax attention model up to a constant factor. Our analysis also delineates a strict separation in terms of the prediction accuracy of ICL between single-head and multi-head attention models. The key technique for our convergence analysis is to map the gradient flow dynamics in the parameter space to a set of ordinary differential equations in the spectral domain, where the relative magnitudes of the semi-singular values of the attention weights determines task allocation. To our best knowledge, our work provides the first convergence result for the multi-head softmax attention model.

[179]  arXiv:2402.19449 (cross-list from cs.LG) [pdf, other]
Title: Heavy-Tailed Class Imbalance and Why Adam Outperforms Gradient Descent on Language Models
Subjects: Machine Learning (cs.LG); Computation and Language (cs.CL); Optimization and Control (math.OC); Machine Learning (stat.ML)

Adam has been shown to outperform gradient descent in optimizing large language transformers empirically, and by a larger margin than on other tasks, but it is unclear why this happens. We show that the heavy-tailed class imbalance found in language modeling tasks leads to difficulties in the optimization dynamics. When training with gradient descent, the loss associated with infrequent words decreases slower than the loss associated with frequent ones. As most samples come from relatively infrequent words, the average loss decreases slowly with gradient descent. On the other hand, Adam and sign-based methods do not suffer from this problem and improve predictions on all classes. To establish that this behavior is indeed caused by class imbalance, we show empirically that it persist through different architectures and data types, on language transformers, vision CNNs, and linear models. We further study this phenomenon on a linear classification with cross-entropy loss, showing that heavy-tailed class imbalance leads to ill-conditioning, and that the normalization used by Adam can counteract it.

[180]  arXiv:2402.19456 (cross-list from quant-ph) [pdf, other]
Title: Statistical Estimation in the Spiked Tensor Model via the Quantum Approximate Optimization Algorithm
Comments: 51 pages, 4 figures, 1 table
Subjects: Quantum Physics (quant-ph); Data Structures and Algorithms (cs.DS); Probability (math.PR); Statistics Theory (math.ST)

The quantum approximate optimization algorithm (QAOA) is a general-purpose algorithm for combinatorial optimization. In this paper, we analyze the performance of the QAOA on a statistical estimation problem, namely, the spiked tensor model, which exhibits a statistical-computational gap classically. We prove that the weak recovery threshold of $1$-step QAOA matches that of $1$-step tensor power iteration. Additional heuristic calculations suggest that the weak recovery threshold of $p$-step QAOA matches that of $p$-step tensor power iteration when $p$ is a fixed constant. This further implies that multi-step QAOA with tensor unfolding could achieve, but not surpass, the classical computation threshold $\Theta(n^{(q-2)/4})$ for spiked $q$-tensors.
Meanwhile, we characterize the asymptotic overlap distribution for $p$-step QAOA, finding an intriguing sine-Gaussian law verified through simulations. For some $p$ and $q$, the QAOA attains an overlap that is larger by a constant factor than the tensor power iteration overlap. Of independent interest, our proof techniques employ the Fourier transform to handle difficult combinatorial sums, a novel approach differing from prior QAOA analyses on spin-glass models without planted structure.

Replacements for Fri, 1 Mar 24

[181]  arXiv:1804.07698 (replaced) [pdf, ps, other]
Title: On the non-inner automorphism conjecture of finite $p$-groups
Subjects: Group Theory (math.GR)
[182]  arXiv:1904.06607 (replaced) [pdf, ps, other]
Title: On conformal pseudo-subriemannian fundamental graded Lie algebras associated with pseudo $H$-type Lie algebras
Authors: Tomoaki Yatsui
Subjects: Differential Geometry (math.DG)
[183]  arXiv:2002.08706 (replaced) [pdf, ps, other]
Title: Lojasiewicz exponent of a surface: an intrinsic view
Subjects: Algebraic Geometry (math.AG)
[184]  arXiv:2002.10175 (replaced) [pdf, other]
Title: Courant-Dorfman algebras of differential operators and Dorfman connections of Courant algebroids
Comments: The latter version was split, this is the final form of the first part with some additions
Journal-ref: Journal of Geometry and Physics 199 (2024) 105142
Subjects: Differential Geometry (math.DG)
[185]  arXiv:2006.04681 (replaced) [pdf, ps, other]
Title: Exact results on generalized Erdős-Gallai problems
Comments: minor changes suggested by referees. arXiv admin note: text overlap with arXiv:2003.07943
Subjects: Combinatorics (math.CO)
[186]  arXiv:2008.01887 (replaced) [pdf, ps, other]
Title: Finite-time blow-up prevention by logistic source in parabolic-elliptic chemotaxis models with singular sensitivity in any dimensional setting
Subjects: Analysis of PDEs (math.AP); Dynamical Systems (math.DS)
[187]  arXiv:2009.06708 (replaced) [pdf, ps, other]
Title: Moduli of Langlands Parameters
Comments: 90 pages
Subjects: Number Theory (math.NT); Representation Theory (math.RT)
[188]  arXiv:2010.14932 (replaced) [pdf, ps, other]
Title: Walking to Infinity Along Some Number Theory sequences
Comments: 29 pages, from Walking to Infinity Polymath REU
Subjects: Number Theory (math.NT)
[189]  arXiv:2101.12593 (replaced) [pdf, ps, other]
Title: On the Symbol Length of Fields with finite Square Class Number
Subjects: Number Theory (math.NT)
[190]  arXiv:2102.09520 (replaced) [pdf, ps, other]
Title: Higgs fields, non-abelian Cauchy kernels and the Goldman symplectic structure
Comments: V2: 29 pages, figure added and more accurate review of literature. V3: 31 pages, minor revision
Journal-ref: Nonlinearity 2024
Subjects: Algebraic Geometry (math.AG); Mathematical Physics (math-ph); Symplectic Geometry (math.SG)
[191]  arXiv:2107.02743 (replaced) [pdf, ps, other]
Title: Submodular Order Functions and Assortment Optimization
Authors: Rajan Udwani
Comments: To appear in Management Science
Subjects: Optimization and Control (math.OC); Data Structures and Algorithms (cs.DS)
[192]  arXiv:2111.07697 (replaced) [pdf, ps, other]
Title: Spectral analysis of a viscoelastic tube conveying fluid with generalised boundary conditions
Comments: typos corrected
Subjects: Analysis of PDEs (math.AP); Mathematical Physics (math-ph); Functional Analysis (math.FA)
[193]  arXiv:2112.04965 (replaced) [pdf, ps, other]
Title: Permutations of counters on a table
Authors: Samuel Korsky
Comments: 7 pages
Subjects: Combinatorics (math.CO)
[194]  arXiv:2112.05478 (replaced) [pdf, other]
Title: Critical configurations for three projective views
Comments: 40 pages, 9 figures. This is a companion paper to arXiv:2112.05074. Accepted manuscript published in Mathematica Scandinavica
Subjects: Algebraic Geometry (math.AG); Computer Vision and Pattern Recognition (cs.CV)
[195]  arXiv:2203.01360 (replaced) [pdf, other]
Title: Neural Galerkin Schemes with Active Learning for High-Dimensional Evolution Equations
Journal-ref: Journal of Computational Physics, Volume 496, 2024
Subjects: Numerical Analysis (math.NA); Machine Learning (cs.LG); Machine Learning (stat.ML)
[196]  arXiv:2204.03754 (replaced) [pdf, ps, other]
Title: Higher uniformity of arithmetic functions in short intervals I. All intervals
Comments: 103 pages; Some typo fixes and a slight fix in proof of Proposition 2.14 compared to the published version, acknowledgment added
Journal-ref: Forum of Mathematics, Pi, 11 (2023), e29
Subjects: Number Theory (math.NT)
[197]  arXiv:2205.01032 (replaced) [src]
Title: Étale extensions of polynomial rings are faithfully flat
Comments: There is a gap in the last step of the proof of Theorem 1.1. The author would like to thank R. van Dobben de Bruyn and Sean Cotner for bringing this point to the author's attention
Subjects: Commutative Algebra (math.AC); Algebraic Geometry (math.AG)
[198]  arXiv:2205.05860 (replaced) [pdf, ps, other]
Title: Rigidity for Lorentzian metrics with the same length of null-geodesics
Authors: Gregory Eskin
Subjects: Analysis of PDEs (math.AP)
[199]  arXiv:2205.09734 (replaced) [pdf, other]
Title: Saturation and recurrence of quantum complexity in random local quantum dynamics
Comments: 57 pages, 6 figures, comments and suggestions are welcome, major changes in v2: new title, simplified reasoning based on global gaps, inclusion of SLH model of random evolutions, numerous narrative changes throughout the text, new author
Subjects: Quantum Physics (quant-ph); Strongly Correlated Electrons (cond-mat.str-el); High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph)
[200]  arXiv:2206.00899 (replaced) [pdf, other]
Title: Stability of Chandrasekhar's nonlinear force-free fields
Authors: Ken Abe
Comments: 85 pages, new remark 1.3 and Appendix B
Subjects: Analysis of PDEs (math.AP)
[201]  arXiv:2206.08682 (replaced) [pdf, ps, other]
Title: Spectral inequality with sensor sets of decaying density for Schrödinger operators with power growth potentials
Comments: 17 pages; section added discussing recent developments, fixed typos, and editorial changes
Subjects: Analysis of PDEs (math.AP)
[202]  arXiv:2206.10485 (replaced) [pdf, other]
Title: Tight bounds for the learning of homotopy à la Niyogi, Smale, and Weinberger for subsets of Euclidean spaces and of Riemannian manifolds
Comments: 75 pages, 29 figures
Subjects: Computational Geometry (cs.CG); Algebraic Topology (math.AT)
[203]  arXiv:2206.11949 (replaced) [pdf, ps, other]
Title: Tight closure, coherence, and localization at single elements
Authors: Neil Epstein
Comments: According to further suggestions from the referee, I introduced a new notion, that of a *p-system* of ideals, which generalizes both p-families and F-graded systems, and I redid Section 4 to be compatible with this general notion. Accepted for publication in AMV (Acta Mathematica Vietnamica). 18 pages
Subjects: Commutative Algebra (math.AC); Algebraic Geometry (math.AG)
[204]  arXiv:2206.13158 (replaced) [pdf, other]
Title: Sharp inequalities involving the Cheeger constant of planar convex sets
Comments: 38 pages, 20 figures
Subjects: Analysis of PDEs (math.AP)
[205]  arXiv:2208.13611 (replaced) [pdf, other]
Title: Characterizing Hitchin representations over real closed fields
Authors: Xenia Flamm
Comments: 34 pages, 6 figures, comments welcome! Second version: Minor correction of the statement of Theorem 1.1 (former Theorem 1.2) and its proof, which is now simplified (the appendix was removed); Section 9 was added
Subjects: Geometric Topology (math.GT); Group Theory (math.GR)
[206]  arXiv:2209.11257 (replaced) [pdf, ps, other]
Title: Linear actions of $\mathbb{Z}/p\times\mathbb{Z}/p$ on $S^{2n-1}\times S^{2n-1}$
Comments: 13 pages
Subjects: Geometric Topology (math.GT)
[207]  arXiv:2210.08582 (replaced) [pdf, ps, other]
Title: Free colimit completion in $\infty$-categories
Authors: Charles Rezk
Comments: 14 pages. Several changes since prior version, including a major change in terminology: "regular class" is now used in place of "filtering class", which term has been redeployed for another meaning. Includes changes suggested by reviewer, and some additional remarks on the examples of filtering and cofiltering classes
Subjects: Category Theory (math.CT); Algebraic Topology (math.AT)
[208]  arXiv:2211.00769 (replaced) [pdf, other]
Title: Instability of electroweak homogeneous vacua in strong magnetic fields
Subjects: Mathematical Physics (math-ph)
[209]  arXiv:2211.07303 (replaced) [pdf, other]
Title: Adaptive Federated Minimax Optimization with Lower Complexities
Comments: To appear in AISTATS 2024
Subjects: Machine Learning (cs.LG); Optimization and Control (math.OC)
[210]  arXiv:2211.09624 (replaced) [pdf, ps, other]
Title: Quasi-countable inverse semigroups as metric spaces, and the uniform Roe algebras of locally finite inverse semigroups
Subjects: Operator Algebras (math.OA); Group Theory (math.GR)
[211]  arXiv:2212.14516 (replaced) [pdf, other]
Title: A hypergraph analog of Dirac's Theorem for long cycles in 2-connected graphs
Comments: 23 pages, 2 figures
Subjects: Combinatorics (math.CO)
[212]  arXiv:2301.06297 (replaced) [pdf, other]
Title: Inference via robust optimal transportation: theory and methods
Subjects: Statistics Theory (math.ST); Machine Learning (stat.ML)
[213]  arXiv:2301.12584 (replaced) [pdf, other]
Title: Fast Exact Leverage Score Sampling from Khatri-Rao Products with Applications to Tensor Decomposition
Comments: The 37th Conference on Neural Information Processing Systems (Neurips'23). 28 pages, 10 figures, 6 tables
Subjects: Numerical Analysis (math.NA)
[214]  arXiv:2302.01613 (replaced) [pdf, ps, other]
Title: Classification of modular data of integral modular fusion categories up to rank 12
Comments: 27 pages. Rank 13 reduced to 2 types. Found 3 non-pointed MD for MNSD Rank 17 (oversight in literature?). Type criteria improved. More details in T-matrix section. Computational process simplified. Put codes and data on GitHub. Comments are welcome!
Subjects: Quantum Algebra (math.QA); Mathematical Physics (math-ph); Category Theory (math.CT); Rings and Algebras (math.RA)
[215]  arXiv:2302.03535 (replaced) [pdf, other]
Title: When will (game) wars end?
Comments: 10 pages, improved presentation, added background and examples
Subjects: Combinatorics (math.CO)
[216]  arXiv:2302.09454 (replaced) [pdf, ps, other]
Title: Realizability of Some Combinatorial Sequences
Authors: Geng-Rui Zhang
Comments: 25 pages
Journal-ref: J. Integer Sequences 27 (2024), Article 24.3.3
Subjects: Number Theory (math.NT); Combinatorics (math.CO); Dynamical Systems (math.DS)
[217]  arXiv:2302.09529 (replaced) [pdf, ps, other]
Title: Duality for coalgebras for Vietoris and monadicity
Comments: 29 pages. Comments are welcome!
Subjects: Logic (math.LO); Category Theory (math.CT)
[218]  arXiv:2302.12877 (replaced) [pdf, ps, other]
Title: Rigorous computation of solutions of semi-linear PDEs on unbounded domains via spectral methods
Subjects: Analysis of PDEs (math.AP); Dynamical Systems (math.DS)
[219]  arXiv:2302.13649 (replaced) [pdf, other]
Title: Locally homogeneous holomorphic geometric structures on projective varieties
Authors: Indranil Biswas (1), Benjamin McKay (2) ((1) Department of Mathematics, Shiv Nadar University, (2) University College Cork)
Comments: 30 pages, no figures
Subjects: Algebraic Geometry (math.AG); Differential Geometry (math.DG)
[220]  arXiv:2303.00696 (replaced) [pdf, other]
Title: Trust your source: quantifying source condition elements for variational regularisation methods
Subjects: Numerical Analysis (math.NA)
[221]  arXiv:2303.04786 (replaced) [pdf, ps, other]
Title: Quotients of abelian varieties by reflection groups
Authors: Eric M. Rains
Comments: 77 pages LaTeX, 7 tables. v2: clarifies the (lack of) dependence on Popov's classification; adds some references to recent results on special cases
Subjects: Algebraic Geometry (math.AG)
[222]  arXiv:2303.12402 (replaced) [pdf, other]
Title: Benders decomposition algorithms for minimizing the spread of harmful contagions in networks
Comments: New theoretical and computational results added
Subjects: Optimization and Control (math.OC); Discrete Mathematics (cs.DM)
[223]  arXiv:2303.12407 (replaced) [pdf, ps, other]
Title: Non-asymptotic analysis of Langevin-type Monte Carlo algorithms
Authors: Shogo Nakakita
Subjects: Statistics Theory (math.ST); Probability (math.PR); Machine Learning (stat.ML)
[224]  arXiv:2303.17674 (replaced) [pdf, other]
Title: Convex Hulls of Reachable Sets
Comments: 19 pages. Submitted to the IEEE Transactions on Automatic Control. Substantial extension of arXiv:2303.17674v2
Subjects: Optimization and Control (math.OC); Machine Learning (cs.LG); Robotics (cs.RO); Systems and Control (eess.SY)
[225]  arXiv:2304.04485 (replaced) [pdf, other]
Title: Formulas for the visual angle metric
Comments: 15 pages, 9 Figures
Subjects: Complex Variables (math.CV); Metric Geometry (math.MG)
[226]  arXiv:2304.06100 (replaced) [pdf, other]
Title: Tridiagonal and single-pair matrices and the inverse sum of two single-pair matrices
Authors: Sebastien Bossu
Comments: Working paper, 26 pages
Subjects: Rings and Algebras (math.RA)
[227]  arXiv:2304.07146 (replaced) [pdf, other]
Title: Energy cascade for the Klein-Gordon lattice
Authors: Stefano Pasquali
Comments: Changes with respect to previous version: more details and comments added to the introduction and to the proof of Proposition 5.3, changes in Section 6, some typos corrected. Comments are welcome. arXiv admin note: text overlap with arXiv:1911.12648
Subjects: Dynamical Systems (math.DS); Mathematical Physics (math-ph); Analysis of PDEs (math.AP)
[228]  arXiv:2304.08579 (replaced) [pdf, ps, other]
Title: Fake degrees of classical Weyl groups
Comments: 10 pages
Subjects: Representation Theory (math.RT); Combinatorics (math.CO)
[229]  arXiv:2305.15991 (replaced) [pdf, ps, other]
Title: Finite sample rates for logistic regression with small noise or few samples
Subjects: Statistics Theory (math.ST)
[230]  arXiv:2306.00710 (replaced) [pdf, ps, other]
Title: Set-Valued Analysis of Generalized Barycentric Coordinates and Their Geometric Properties
Authors: Fabio V. Difonzo
Subjects: Metric Geometry (math.MG)
[231]  arXiv:2306.10442 (replaced) [pdf, ps, other]
Title: Partial data inverse problem for hyperbolic equation with time-dependent damping coefficient and potential
Comments: arXiv admin note: text overlap with arXiv:1702.07974 by other authors
Subjects: Analysis of PDEs (math.AP)
[232]  arXiv:2306.11022 (replaced) [pdf, other]
Title: Hadamard's inequality in the mean
Subjects: Analysis of PDEs (math.AP); Optimization and Control (math.OC)
[233]  arXiv:2306.11690 (replaced) [pdf, ps, other]
Title: A unified approach to the small-time behavior of the spectral heat content for isotropic Lévy processes
Comments: To appear in Stochastic Processes and their Applications
Subjects: Probability (math.PR)
[234]  arXiv:2306.15388 (replaced) [pdf, ps, other]
Title: On reachability categories, persistence, and commuting algebras of quivers
Comments: 16 pages. Comments welcome!
Subjects: Rings and Algebras (math.RA); Combinatorics (math.CO); Category Theory (math.CT)
[235]  arXiv:2306.16379 (replaced) [pdf, ps, other]
Title: Topology and monoid representations
Comments: This paper will now be split into 2 separate papers. While splitting the paper I found some minor typos, etc., and so I have updated this version to correct these. Further updates will only occur for the new versions
Subjects: Representation Theory (math.RT); Group Theory (math.GR); Rings and Algebras (math.RA)
[236]  arXiv:2307.03997 (replaced) [pdf, other]
Title: Efficient Model-Free Exploration in Low-Rank MDPs
Subjects: Machine Learning (cs.LG); Optimization and Control (math.OC)
[237]  arXiv:2307.06678 (replaced) [pdf, ps, other]
Title: The Frobenius transform of a symmetric function
Authors: Mitchell Lee
Comments: 31 pages
Subjects: Combinatorics (math.CO); Representation Theory (math.RT)
[238]  arXiv:2307.07357 (replaced) [pdf, other]
Title: Inverse Optimization for Routing Problems
Subjects: Optimization and Control (math.OC); Machine Learning (cs.LG)
[239]  arXiv:2307.08683 (replaced) [pdf, other]
Title: Quantum Covariance Scalar Products and Efficient Estimation of Max-Ent Projections
Comments: 25 pages, 9 figures. Comments and suggestions are welcome
Journal-ref: Physical Review A, Vol. 109, Iss. 2, (02/01/2024)
Subjects: Quantum Physics (quant-ph); Statistical Mechanics (cond-mat.stat-mech); Mathematical Physics (math-ph)
[240]  arXiv:2307.15852 (replaced) [pdf, other]
Title: Dimensionless Policies based on the Buckingham $π$ Theorem: Is This a Good Way to Generalize Numerical Results?
Authors: Alexandre Girard
Journal-ref: Mathematics 2024, 12(5), 709
Subjects: Optimization and Control (math.OC); Artificial Intelligence (cs.AI); Robotics (cs.RO); Systems and Control (eess.SY)
[241]  arXiv:2308.00299 (replaced) [pdf, ps, other]
Title: Full Souslin trees at small cardinals
Comments: Relaxed the hypothesis of Theorem C. Added Proposition 2.3
Subjects: Logic (math.LO)
[242]  arXiv:2308.04075 (replaced) [pdf, ps, other]
Title: Boundary-preserving Lamperti-splitting schemes for some Stochastic Differential Equations
Authors: Johan Ulander
Comments: 30 pages, 6 figures
Subjects: Numerical Analysis (math.NA)
[243]  arXiv:2308.04510 (replaced) [pdf, ps, other]
Title: Gromov-Hausdorff convergence of metric pairs and metric tuples
Comments: 30 pages
Subjects: Metric Geometry (math.MG); Algebraic Topology (math.AT); Differential Geometry (math.DG)
[244]  arXiv:2308.05240 (replaced) [pdf, ps, other]
Title: Local solvability and dilation-critical singularities of supercritical fractional heat equations
Subjects: Analysis of PDEs (math.AP)
[245]  arXiv:2308.05500 (replaced) [pdf, ps, other]
Title: An Adaptive Algorithm Based on Stochastic Discontinuous Galerkin for Convection Dominated Equations with Random Data
Comments: 26 pages, 19 figures, 5 tables
Subjects: Numerical Analysis (math.NA)
[246]  arXiv:2308.06854 (replaced) [pdf, ps, other]
Title: Characteristic $p$ analogues of the Mumford--Tate and André--Oort conjectures for products of ordinary GSpin Shimura varieties
Authors: Ruofan Jiang
Comments: 49 pages. Included the proof for the product case. Simplified the other parts. Comments welcome!
Subjects: Number Theory (math.NT); Algebraic Geometry (math.AG)
[247]  arXiv:2308.07917 (replaced) [pdf, other]
Title: Degenerate Stability of the Caffarelli-Kohn-Nirenberg Inequality along the Felli-Schneider Curve
Comments: 33 pages; final version
Journal-ref: Calculus of Variations and PDE 63, 44 (2024)
Subjects: Analysis of PDEs (math.AP); Functional Analysis (math.FA)
[248]  arXiv:2308.13424 (replaced) [pdf, ps, other]
Title: AG codes have no list-decoding friends: Approaching the generalized Singleton bound requires exponential alphabets
Subjects: Information Theory (cs.IT); Discrete Mathematics (cs.DM); Combinatorics (math.CO)
[249]  arXiv:2308.16794 (replaced) [pdf, other]
Title: An exceptional property of the one-dimensional Bianchi-Egnell inequality
Authors: Tobias König
Comments: 24 pages, minor changes with respect to v1
Subjects: Analysis of PDEs (math.AP)
[250]  arXiv:2309.00799 (replaced) [pdf, other]
Title: An Elementary Construction of Modified Hamiltonians and Modified Measures of 2D Kahan Maps
Comments: 30 pages, 7 figures
Subjects: Numerical Analysis (math.NA)
[251]  arXiv:2309.01980 (replaced) [pdf, other]
Title: Local properties and augmented Lagrangians in fully nonconvex composite optimization
Comments: 36 pages
Subjects: Optimization and Control (math.OC)
[252]  arXiv:2309.05083 (replaced) [pdf, ps, other]
Title: Quantum automorphism groups of $2$-graphs
Comments: 16 PAGES
Subjects: Operator Algebras (math.OA)
[253]  arXiv:2309.05979 (replaced) [pdf, other]
Title: Measure preservation and integrals for Lotka--Volterra tree-systems and their Kahan discretisation
Comments: 17 pages, 3 figures
Journal-ref: Journal of Computational Dynamics, 2024
Subjects: Dynamical Systems (math.DS); Numerical Analysis (math.NA)
[254]  arXiv:2309.08807 (replaced) [pdf, other]
Title: Optimal Ensemble Control of Matter-Wave Splitting in Bose-Einstein Condensates
Subjects: Optimization and Control (math.OC)
[255]  arXiv:2309.10454 (replaced) [pdf, ps, other]
Title: Maximum principle for a Markovian regime switching system with partial information under model uncertainty
Comments: 34 Pages
Subjects: Optimization and Control (math.OC)
[256]  arXiv:2309.13180 (replaced) [pdf, other]
Title: Modulus of edge covers and stars
Comments: Fixed an error in Theorem 3.9
Subjects: Combinatorics (math.CO)
[257]  arXiv:2309.16505 (replaced) [pdf, ps, other]
Title: On categorical local langlands program for $GL_n$
Authors: Kieu Hieu Nguyen
Comments: We added the minuscule assumption in section 5 and fixed inaccuracies related to various twists in the manuscript. We removed section 9 about the geometry of $B^+_{dR}$-Grassmannian due to some misunderstandings
Subjects: Number Theory (math.NT); Algebraic Geometry (math.AG); Representation Theory (math.RT)
[258]  arXiv:2309.17155 (replaced) [pdf, other]
Title: Immersed figure-8 annuli and anyons
Authors: Bowen Shi
Comments: 15 figures, 20+10 pages
Subjects: Quantum Physics (quant-ph); Strongly Correlated Electrons (cond-mat.str-el); High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph)
[259]  arXiv:2310.00762 (replaced) [pdf, ps, other]
Title: A note on the stabilizer formalism via noncommutative graphs
Comments: Final version. To appear in "Quantum Information Processing''
Subjects: Information Theory (cs.IT); Operator Algebras (math.OA); Quantum Physics (quant-ph)
[260]  arXiv:2310.01325 (replaced) [pdf, ps, other]
Title: On the finiteness of Bernoulli polynomials whose derivative has only integral coefficients
Authors: Bernd C. Kellner
Comments: 9 pages, final revised version
Journal-ref: J. Integer Seq. 27 (2024), Article 24.2.8, 1-11
Subjects: Number Theory (math.NT)
[261]  arXiv:2310.02592 (replaced) [pdf, ps, other]
Title: A Faster Deterministic Approximation Algorithm for TTP-2
Comments: 27 pages, 42 figures
Subjects: Data Structures and Algorithms (cs.DS); Combinatorics (math.CO)
[262]  arXiv:2310.07333 (replaced) [pdf, ps, other]
Title: Computing approximate roots of monotone functions
Comments: We solved all the open cases, except the case when f has 3 or more dimensions, and satisfies all monotonicity conditions except one. Any ideas?
Subjects: Computer Science and Game Theory (cs.GT); Numerical Analysis (math.NA)
[263]  arXiv:2310.10334 (replaced) [pdf, ps, other]
Title: On eigenfunctions of the block graphs of geometric Steiner systems
Comments: arXiv admin note: text overlap with arXiv:2306.00391
Subjects: Combinatorics (math.CO)
[264]  arXiv:2310.10796 (replaced) [pdf, ps, other]
Title: Mixed Mode Oscillations in a Three-Timescale Coupled Morris-Lecar System
Subjects: Dynamical Systems (math.DS)
[265]  arXiv:2310.14086 (replaced) [pdf, ps, other]
Title: Entropic partial orderings of quantum measurements
Comments: 15 pages. v2, minor updates
Subjects: Quantum Physics (quant-ph); Mathematical Physics (math-ph)
[266]  arXiv:2310.15242 (replaced) [pdf, ps, other]
Title: Accessibility, planar graphs, and quasi-isometries
Authors: Joseph MacManus
Comments: 55 pages, 17 figures. Heavily revised. comments welcome!
Subjects: Group Theory (math.GR); Combinatorics (math.CO); Metric Geometry (math.MG)
[267]  arXiv:2310.18053 (replaced) [pdf, ps, other]
Title: Phylogenetic invariants: straightforward from the general Markov to equivariant models
Comments: 30 pages, 4 figures
Subjects: Populations and Evolution (q-bio.PE); Algebraic Geometry (math.AG)
[268]  arXiv:2311.01947 (replaced) [pdf, other]
Title: Lengths of divisible codes -- the missing cases
Authors: Sascha Kurz
Comments: 11 pages, 1 table
Subjects: Combinatorics (math.CO); Information Theory (cs.IT)
[269]  arXiv:2311.04027 (replaced) [pdf, other]
Title: Harmonic analysis of Gaussian multiplicative chaos on the circle
Comments: 23 pages
Subjects: Probability (math.PR); Mathematical Physics (math-ph); Functional Analysis (math.FA)
[270]  arXiv:2311.05181 (replaced) [pdf, other]
Title: Energy-efficient flocking with nonlinear navigational feedback
Subjects: Dynamical Systems (math.DS); Multiagent Systems (cs.MA)
[271]  arXiv:2311.05358 (replaced) [pdf, ps, other]
Title: The uniform structure of $\mathfrak{g}^{\otimes 4}$
Comments: 10 pages, f-la (1.6) is corrected
Subjects: Mathematical Physics (math-ph); High Energy Physics - Theory (hep-th)
[272]  arXiv:2311.05595 (replaced) [pdf, other]
Title: An efficient topology optimization algorithm for large-scale three-dimensional structures
Subjects: Optimization and Control (math.OC)
[273]  arXiv:2311.08168 (replaced) [pdf, other]
Title: Time-Uniform Confidence Spheres for Means of Random Vectors
Comments: 46 pages, 1 figure
Subjects: Statistics Theory (math.ST); Information Theory (cs.IT); Methodology (stat.ME); Machine Learning (stat.ML)
[274]  arXiv:2311.09314 (replaced) [pdf, other]
Title: Multimatroids and rational curves with cyclic action
Subjects: Combinatorics (math.CO); Algebraic Geometry (math.AG)
[275]  arXiv:2311.11949 (replaced) [pdf, ps, other]
Title: Rational cuboid. Searching for solution
Authors: Boris Safin
Comments: 10 pages
Subjects: General Mathematics (math.GM)
[276]  arXiv:2311.12451 (replaced) [pdf, other]
Title: A frame approach for equations involving the fractional Laplacian
Subjects: Numerical Analysis (math.NA)
[277]  arXiv:2311.14658 (replaced) [pdf, other]
Title: Convergence Analysis for Learning Orthonormal Deep Linear Neural Networks
Subjects: Machine Learning (cs.LG); Optimization and Control (math.OC)
[278]  arXiv:2312.00433 (replaced) [pdf, ps, other]
Title: Sobolev improvements on sharp Rellich inequalities
Comments: 20 pages; references have been added and minor corrections have been made; results unchanged; to appear in the J. Spectral Theory
Subjects: Analysis of PDEs (math.AP)
[279]  arXiv:2312.01141 (replaced) [pdf, other]
Title: On the Moser's Bernstein Theorem
Comments: The text was proofread. 33 pages and 2 figures
Subjects: Differential Geometry (math.DG); Algebraic Geometry (math.AG); Complex Variables (math.CV)
[280]  arXiv:2312.01534 (replaced) [pdf, other]
Title: Skeletal Cut Loci on Convex Polyhedra
Comments: 20 pages, 12 figures, 9 references. v2: Many minor improvements
Subjects: Computational Geometry (cs.CG); Metric Geometry (math.MG)
[281]  arXiv:2312.04141 (replaced) [pdf, ps, other]
Title: Distributed Approximate Computing with Constant Locality
Subjects: Information Theory (cs.IT)
[282]  arXiv:2312.04525 (replaced) [pdf, ps, other]
Title: Supersymmetric generalization of q-deformed long-range spin chains of Haldane-Shastry type and trigonometric GL(N|M) solution of associative Yang-Baxter equation
Authors: M. Matushko, A. Zotov
Comments: 20 pages, minor corrections
Subjects: Mathematical Physics (math-ph); Strongly Correlated Electrons (cond-mat.str-el); High Energy Physics - Theory (hep-th); Quantum Algebra (math.QA); Exactly Solvable and Integrable Systems (nlin.SI)
[283]  arXiv:2312.05413 (replaced) [pdf, ps, other]
Title: An iterative method for computing $π$ by argument reduction of the tangent function
Comments: 35 pages
Subjects: General Mathematics (math.GM)
[284]  arXiv:2312.06898 (replaced) [pdf, ps, other]
Title: Explicit unit distance graphs with exponential chromatic number and arbitrary girth
Comments: 10 pages
Subjects: Combinatorics (math.CO); Metric Geometry (math.MG)
[285]  arXiv:2312.07193 (replaced) [pdf, ps, other]
Title: $(σ,δ)$-polycyclic codes in Ore extensions over rings
Comments: 20 pages, no figure
Subjects: Information Theory (cs.IT)
[286]  arXiv:2312.08974 (replaced) [pdf, other]
Title: Multifractal analysis via Lagrange duality
Authors: Alex Rutar
Comments: 31 pages, 3 figures. Expository article. v2: slightly more general definition of uniform densities; removal of SSC assumption in Proposition 3.13; numbering changes and improvement of exposition
Subjects: Dynamical Systems (math.DS)
[287]  arXiv:2312.11830 (replaced) [pdf, ps, other]
Title: Hopf orbits and the first ECH capacity
Comments: 24 pages, minor corrections
Subjects: Symplectic Geometry (math.SG); Dynamical Systems (math.DS)
[288]  arXiv:2401.00985 (replaced) [pdf, ps, other]
Title: Spectrum in alternative topological algebras and a new look at old theorems
Comments: In contrast to Version 2, this version features significantly revised and improved results
Subjects: Rings and Algebras (math.RA)
[289]  arXiv:2401.02215 (replaced) [pdf, ps, other]
Title: On the Rainbow Ramsey theorem and the Canonical Ramsey Theorem for pairs without AC
Comments: Some new results were added
Subjects: Logic (math.LO); Combinatorics (math.CO)
[290]  arXiv:2401.06307 (replaced) [pdf, other]
Title: Consistency of minimizing movements with smooth mean curvature flow of droplets with prescribed contact-angle in $\mathbb R^3$
Comments: 19 pages, 1 figures
Subjects: Differential Geometry (math.DG); Analysis of PDEs (math.AP)
[291]  arXiv:2401.07223 (replaced) [pdf, ps, other]
Title: Lipschitz Functions on Sparse Graphs
Comments: 15 pages
Subjects: Combinatorics (math.CO)
[292]  arXiv:2401.10152 (replaced) [pdf, ps, other]
Title: Sums of square roots that are close to an integer
Subjects: Number Theory (math.NT)
[293]  arXiv:2401.10807 (replaced) [pdf, ps, other]
Title: Isometric pairs with compact + normal cross-commutator
Comments: 58 pages. Revised. A new rank formula for general isometric pairs
Subjects: Functional Analysis (math.FA); Complex Variables (math.CV); Operator Algebras (math.OA)
[294]  arXiv:2401.12796 (replaced) [pdf, ps, other]
Title: Well-posedness of low regularity solutions for 3D relativistic Euler equations
Authors: Huali Zhang
Comments: Welcome all comments!
Subjects: Analysis of PDEs (math.AP)
[295]  arXiv:2401.12898 (replaced) [pdf, other]
Title: Information scrambling and chaos induced by a Hermitian Matrix
Subjects: Quantum Physics (quant-ph); Mathematical Physics (math-ph); Chaotic Dynamics (nlin.CD)
[296]  arXiv:2401.13962 (replaced) [pdf, other]
Title: An inf-sup Approach to $C_0$-Semigroup Generation for An Interactive Composite Structure-Stokes PDE Dynamics
Subjects: Analysis of PDEs (math.AP)
[297]  arXiv:2401.14803 (replaced) [pdf, other]
Title: Distortion in graphs of groups and Rapid Decay classification of 3-manifold groups
Comments: 27 pages, 3 figures
Subjects: Group Theory (math.GR)
[298]  arXiv:2401.16900 (replaced) [pdf, ps, other]
Title: 2-classifiers via dense generators and Hofmann-Streicher universe in stacks
Authors: Luca Mesiti
Comments: We improved size considerations
Subjects: Category Theory (math.CT); Logic (math.LO)
[299]  arXiv:2402.04237 (replaced) [pdf, other]
Title: A note on graphs of $k$-colourings
Comments: 10 pages, 3 figures
Subjects: Combinatorics (math.CO)
[300]  arXiv:2402.05730 (replaced) [pdf, ps, other]
Title: Deriving two dualities simultaneously from a family of identities for multiple harmonic sums
Comments: 11 pages
Subjects: Number Theory (math.NT)
[301]  arXiv:2402.07185 (replaced) [pdf, other]
Title: Existence of an equilibrium with limited stock market participation and power utilities
Subjects: Mathematical Finance (q-fin.MF); Analysis of PDEs (math.AP)
[302]  arXiv:2402.07524 (replaced) [pdf, ps, other]
Title: Semialgebraicity of the convergence domain of an algebraic power series
Authors: Tobias Kaiser
Comments: Updated version
Subjects: Complex Variables (math.CV); Algebraic Geometry (math.AG)
[303]  arXiv:2402.07836 (replaced) [pdf, ps, other]
Title: MAD Families of Gowers' Infinite Block Sequences
Authors: Clement Yung
Comments: 9 pages
Subjects: Logic (math.LO); Combinatorics (math.CO)
[304]  arXiv:2402.09079 (replaced) [pdf, ps, other]
Title: Radial symmetry and sharp asymptotic behaviors of nonnegative solutions to $D^{1,p}$-critical quasi-linear static Schrödinger-Hartree equation involving $p$-Laplacian $-Δ_{p}$
Subjects: Analysis of PDEs (math.AP)
[305]  arXiv:2402.10246 (replaced) [pdf, ps, other]
Title: A Random-Player Game and Derangement Numbers
Subjects: Combinatorics (math.CO); Probability (math.PR)
[306]  arXiv:2402.10568 (replaced) [pdf, ps, other]
Title: Examples and cofibrant generation of effective Kan fibrations
Comments: Corrected typos and mild reorganisation
Subjects: Algebraic Topology (math.AT); Category Theory (math.CT)
[307]  arXiv:2402.11050 (replaced) [pdf, other]
Title: Adaptive Constellation Multiple Access for Beyond 5G Wireless Systems
Comments: 5 pages, 6 figures, Submission to an IEEE Journal
Subjects: Information Theory (cs.IT); Signal Processing (eess.SP)
[308]  arXiv:2402.11116 (replaced) [pdf, ps, other]
Title: Thermodynamically consistent Cahn-Hilliard-Navier-Stokes equations using the metriplectic dynamics formalism
Subjects: Mathematical Physics (math-ph)
[309]  arXiv:2402.11352 (replaced) [pdf, other]
Title: Unified Capacity Results for Free-Space Optical Communication Systems Over Gamma-Gamma Atmospheric Turbulence Channels
Subjects: Information Theory (cs.IT); Signal Processing (eess.SP)
[310]  arXiv:2402.11644 (replaced) [pdf, ps, other]
Title: Grothendieck's theory of fibred categories for monoids
Authors: Ilia Pirashvili
Subjects: Category Theory (math.CT)
[311]  arXiv:2402.13407 (replaced) [pdf, other]
Title: Einstein metrics on homogeneous spaces $H\times H/ΔK$
Comments: 32 pages, 11 tables, 1 figure. New version: references, remarks and a figure added
Subjects: Differential Geometry (math.DG)
[312]  arXiv:2402.14132 (replaced) [pdf, ps, other]
Title: The symmetric Dunkl-classical orthogonal polynomials revisited
Authors: Khalfa Douak
Subjects: Classical Analysis and ODEs (math.CA)
[313]  arXiv:2402.14284 (replaced) [pdf, ps, other]
Title: Almost rigidity results of Green functions with non-negative Ricci curvature
Authors: Shouhei Honda
Comments: 15 pages, to appear in Springer Tohoku Series in Mathematics
Subjects: Differential Geometry (math.DG); Metric Geometry (math.MG)
[314]  arXiv:2402.14421 (replaced) [pdf, other]
Title: Thurston obstructions and tropical geometry
Authors: Rohini Ramadas
Comments: 15 pages
Subjects: Dynamical Systems (math.DS); Algebraic Geometry (math.AG)
[315]  arXiv:2402.15747 (replaced) [pdf, ps, other]
Title: On Gauss-Kraitchik formula for cyclotomic polynomials via symmetric functions
Authors: Tomohiro Yamada
Comments: 10 pages, minor revisions (L^1-norm definition and (28))
Subjects: Number Theory (math.NT)
[316]  arXiv:2402.16004 (replaced) [pdf, ps, other]
Title: Unexpected properties of Markov chains with infinitely countable set of states
Comments: 6 pages. In this version, we generalized the statement of the theorem keeping the same proof. We added Definition and slightly reformulated the statement of the theorem. As well, we added some more detailed explanations
Subjects: Probability (math.PR)
[317]  arXiv:2402.16167 (replaced) [pdf, other]
Title: A note on homotopy and pseudoisotopy of diffeomorphisms of $4$-manifolds
Comments: 5 pages
Subjects: Geometric Topology (math.GT); Algebraic Topology (math.AT)
[318]  arXiv:2402.16360 (replaced) [pdf, ps, other]
Title: Remark on Estimates in Modulation Spaces for Schrödinger Evolution Operators with Sub-quadratic Potentials
Subjects: Analysis of PDEs (math.AP)
[319]  arXiv:2402.16745 (replaced) [pdf, ps, other]
Title: On an evolution equation in sub-Finsler geometry
Authors: Nicola Garofalo
Comments: A new section has been added (Section 3). The original file has been accordingly revised
Subjects: Analysis of PDEs (math.AP)
[320]  arXiv:2402.16783 (replaced) [pdf, ps, other]
Title: Conditional optimal sets and the quantization coefficients for some uniform distributions
Comments: arXiv admin note: text overlap with arXiv:2402.08036
Subjects: Probability (math.PR)
[321]  arXiv:2402.17002 (replaced) [pdf, other]
Title: Discovering Symmetry Group Structures via Implicit Orthogonality Bias
Authors: Dongsung Huh
Comments: 19 pages, 14 figures
Subjects: Machine Learning (cs.LG); Group Theory (math.GR); Representation Theory (math.RT)
[322]  arXiv:2402.17473 (replaced) [pdf, other]
Title: A new class of bi-transversal matroids
Authors: Mahdi Ebrahimi
Subjects: Combinatorics (math.CO)
[323]  arXiv:2402.17688 (replaced) [pdf, other]
Title: Novel spectral methods for shock capturing and the removal of tygers in computational fluid dynamics
Subjects: Numerical Analysis (math.NA); Computational Physics (physics.comp-ph)
[324]  arXiv:2402.17886 (replaced) [pdf, other]
Title: Zeroth-Order Sampling Methods for Non-Log-Concave Distributions: Alleviating Metastability by Denoising Diffusion
Comments: Figure 4 on page 13 corrected. Comments are welcome
Subjects: Machine Learning (stat.ML); Machine Learning (cs.LG); Probability (math.PR); Statistics Theory (math.ST); Methodology (stat.ME)
[325]  arXiv:2402.17995 (replaced) [pdf, ps, other]
Title: Improved Bounds for Szemerédi's Theorem
Comments: 13 pages
Subjects: Combinatorics (math.CO); Number Theory (math.NT)
[326]  arXiv:2402.18499 (replaced) [pdf, other]
Title: On the exact solution for the Schrödinger equation
Authors: Yair Mulian
Comments: 45 pages, 8 figures
Subjects: Quantum Physics (quant-ph); High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph); Nuclear Theory (nucl-th)
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