Mathematics
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 [1] arXiv:2406.12852 [pdf, other]

Title: Analyzing Dynamical Systems Inspired by Montgomery's Conjecture: Insights into Zeta Function Zeros and Chaos in Number TheoryComments: 37pages,14figure,4tablesSubjects: General Mathematics (math.GM); Chaotic Dynamics (nlin.CD)
In this study, we delve into a novel dynamic system inspired by Montgomery's pair correlation conjecture in number theory. The dynamic system is intricately designed to emulate the behavior of the nontrivial zeros of the Riemann zeta function. Our exploration encompasses bifurcation analysis and Lyapunov exponents to scrutinize the system's behavior and stability, offering insights into both small and large initial conditions. Our efforts extend to unveiling the probability distribution characterizing the dynamics for varying initial conditions. The dynamic system unfolds intricate behaviors, displaying sensitivity to initial conditions and revealing complex bifurcation patterns. Small deviations in the initial conditions unveil significantly different trajectories, reminiscent of chaotic systems. Lyapunov exponents become our lens into understanding stability and chaos within the system. A comparative analysis between the dynamic system's approximate solutions and the actual nontrivial zeros of the Riemann zeta function enhances our comprehension of model accuracy and its potential implications for number theory.
This research illuminates the versatility of dynamic systems as analogs for studying complex mathematical phenomena. It provides fresh perspectives on the pair correlation conjecture, establishing connections with nonlinear dynamics and chaos theory. Notably, we delve into the boundedness of solutions for both small and large initial conditions, unraveling the distinctive probability distribution governing the dynamics in each scenario. Furthermore, we introduce an indepth analysis of the entropy of our dynamic system for both small and large initial conditions. The entropy study enhances our understanding of the predictability and stability of the system, shedding light on its behavior in different parameter regimes.  [2] arXiv:2406.12853 [pdf, other]

Title: Review of the analytical prediction method of surfriding threshold in following sea, and its relation to IMO secondgeneration intact stability criteriaSubjects: General Mathematics (math.GM)
In highspeed maritime operations, the broaching phenomenon can pose a significant risk when navigating in following/quartering seas. The occurrence of this phenomenon can result in a violent yaw motion, regardless of the steering effort, which, in turn, cause the resulting centrifugal force to capsize a vessel. A necessary condition for the occurrence of broaching is the surfriding phenomenon. Therefore, the International Maritime Organization (IMO) has set up criteria to include theoretical formulas for estimating the occurrence of surfriding phenomena. The theoretical equation used in the IMO's secondgeneration intact stability criteria (SGISC) to estimate the surfriding threshold is based on Melnikov's method. This paper presents nonlinear equations describing the forward and backward motions of a ship. However, such equations cannot be directly solved; therefore, we proposed the use of and explain various approximate solution methods, including Meknikov's method. Subsequently, the relationship between the theoretical prediction method of the surfriding threshold rooted in Melnikov's method and the IMO's SGISC is determined.
 [3] arXiv:2406.12854 [pdf, other]

Title: Time and band limiting for exceptional polynomialsJournalref: Applied and Computational Harmonic Analysis, Volume 68, 2024Subjects: Spectral Theory (math.SP); Functional Analysis (math.FA); Operator Algebras (math.OA)
The "timeandband limiting" commutative property was found and exploited by D. Slepian, H. Landau and H. Pollak at Bell Labs in the 1960's, and independently by M. Mehta and later by C. Tracy and H. Widom in Random matrix theory. The property in question is the existence of local operators with simple spectrum that commute with naturally appearing global ones.
Here we give a general result that insures the existence of a commuting differential operator for a given family of exceptional orthogonal polynomials satisfying the "bispectral property". As a main tool we go beyond bispectrality and make use of the notion of Fourier Algebras associated to the given sequence of exceptional polynomials.
We illustrate this result with two examples, of Hermite and Laguerre type, exhibiting also a nice Perline's form for the commuting differential operator.  [4] arXiv:2406.12855 [pdf, other]

Title: Moving frame and spin field representations of submanifolds in flat spaceComments: 20 pages, 1 figureSubjects: General Mathematics (math.GM); General Relativity and Quantum Cosmology (grqc); High Energy Physics  Theory (hepth); Mathematical Physics (mathph)
We introduce a spin field approach, that is compatible with the Cartan moving frame method, to describe the submanifold in a flat space. In fact, we consider a kind of spin field $\psi$, that satisfies a Killing spin field equation (analogous to a Killing spinor equation) written in terms of the Clifford algebra, and we use the spin field to locally rotate the orthonormal basis $\{\hat{e}_\mathtt{I}\}$. Then, the deformed orthonormal frame $\{\tilde{\psi}\hat{e}_\mathtt{I}\psi\}$ can be seen as the moving frame of a submanifold. We find some solutions to the Killing spin field equation and demonstrate an explicit example. Using the product of the spin fields, one can easily generate a new immersion submanifold, and this technique should be useful for studies in geometry and physics. Through the spin field, we find a linear relation between the connection and the extrinsic curvature of the submanifold. We propose a conjecture that any solution of the Killing spin field equation can be locally written as the product of the solutions we find.
 [5] arXiv:2406.12856 [pdf, other]

Title: Dynamics of a Model of Polluted Lakes via FractalFractional Operators with Two Different Numerical AlgorithmsComments: This is a preprint of a paper whose final and definite form is published Open Access in 'Chaos Solitons Fractals' at [this https URL]Journalref: Chaos Solitons Fractals 181 (2024), Art. 114653, 21 ppSubjects: Dynamical Systems (math.DS)
We employ MittagLeffler type kernels to solve a system of fractional differential equations using fractalfractional (FF) operators with two fractal and fractional orders. Using the notion of FFderivatives with nonsingular and nonlocal fading memory, a model of three polluted lakes with one source of pollution is investigated. The properties of a nondecreasing and compact mapping are used in order to prove the existence of a solution for the FFmodel of polluted lake system. For this purpose, the LeraySchauder theorem is used. After exploring stability requirements in four versions, the proposed model of polluted lakes system is then simulated using two new numerical techniques based on AdamsBashforth and Newton polynomials methods. The effect of fractalfractional differentiation is illustrated numerically. Moreover, the effect of the FFderivatives is shown under three specific input models of the pollutant: linear, exponentially decaying, and periodic.
 [6] arXiv:2406.12857 [pdf, other]

Title: Transformations preserving the effective spectral radius of a matrixComments: arXiv admin note: text overlap with arXiv:2103.10330Subjects: Spectral Theory (math.SP)
We discuss transformations on matrices that preserve the effective spectrum and/or the effective spectral radius.
 [7] arXiv:2406.12858 [pdf, other]

Title: Tutorial on onedimensional $q$Fourier transformsComments: 44 pagesSubjects: Quantum Algebra (math.QA)
This paper is an introductory text to the theory of $q$deformed Fourier transforms, as first discussed by Rogov and Olshanetsky. We derive the wellknown results in detail, present them in a format that suits our needs, and include some new findings on specific aspects of the theory.
 [8] arXiv:2406.12859 [pdf, other]

Title: Cohomologies of Reynolds LieYamaguti algebras of any weight and applicationsSubjects: Rings and Algebras (math.RA)
The purpose of the present paper is to investigate cohomologies of Reynolds LieYamaguti algebras of any weight and provide some applications. First, we introduce the notion of Reynolds LieYamaguti algebras and give some new examples. Moreover, cohomologies of Reynolds operators and Reynolds LieYamaguti algebras with coefficients in a suitable representation are established. Finally, formal deformations and abelian extensions of Reynolds LieYamaguti algebras are characterized in terms of lower degree cohomology groups.
 [9] arXiv:2406.12860 [pdf, other]

Title: Permanence and Uniform Asymptotic Stability of Positive Solutions of SAIQH Models on Time ScalesComments: 11 page, 6 figuresSubjects: Dynamical Systems (math.DS)
A susceptible, asymptomatic, infectious, quarantined, and hospitalized (SAIQH) compartmental model on time scales is introduced and a suitable Lyapunov function is defined. Main results include: the proof that the system is permanent; proof of existence of solution; and sufficient conditions implying the dynamic system to have a unique almost periodic solution that is uniformly asymptotically stable. An example is presented supporting the obtained results.
 [10] arXiv:2406.12861 [pdf, other]

Title: Isomorphisms between lattices of hyperinvariant subspacesSubjects: Rings and Algebras (math.RA)
Given two nilpotent endomorphisms, we determine when their lattices of hyperinvariant subspaces are isomorphic. The study of the lattice of hyperinvariant subspaces can be reduced to the nilpotent case when the endomorphism has a JordanChevalley decomposition; for example, it occurs if the underlying field is the field of complex numbers.
 [11] arXiv:2406.12862 [pdf, other]

Title: Kato's chaos of multiple mappings and its continuous selfmapsSubjects: Dynamical Systems (math.DS)
In 2016, Hou and Wang introduced the concept of multiple mappings based on iterated function system, which is an important branch of fractal theory. In this paper, we introduce the definitions of sensitivity, accessibility, and Kato's chaos of multiple mappings from a setvalued perspective. We show that multiple mappings and its continuous selfmaps do not imply each other in terms of sensitivity and accessibility. While a sufficient condition for multiple mappings to be sensitive, accessible and Kato's chaotic is provided, respectively. And the sensitivity, accessibility, and Kato's chaos of multiple mappings are preserved under topological conjugation.
 [12] arXiv:2406.12863 [pdf, other]

Title: Derivation of Chaotic Dynamics from Montgomery Conjecture and Its Interpretation as an Electrical SystemComments: 25 pages ,13 figuresSubjects: General Mathematics (math.GM)
In this paper, we present an innovative derivation of chaotic dynamics rooted in the Montgomery conjecture, specifically addressing the pair correlation of Riemann zeta function zeros. Our exploration unveils a recursive relation inspired by the conjecture, manifesting chaotic behavior. Remarkably, we interpret this derived chaotic dynamics as a unique representation of an electrical system, providing a novel perspective within the domain of electrical engineering. Beyond this groundbreaking derivation, our study delves into the potential applications of chaos theory, bifurcation analysis, and entropy within the framework of this electrical system. We scrutinize the implications of chaos for signal processing, conduct stability analysis through bifurcation studies, and investigate the role of entropy in quantifying the randomness or predictability of electrical signals. Additionally, we explore energy distribution aspects within the electrical system, shedding light on how chaotic dynamics influence energy dissipation and allocation. In the course of our research, a new finding emerged, contributing to our understanding of the derived chaotic dynamics. This discovery enhances the applicability of our framework within electrical engineering, paving the way for innovative applications and deeper explorations in the realm of system dynamics. Moreover, we draw comparisons with the dynamics presented in our recent paper published in the European Physical Journal, specifically addressing Yitang Zhang's contributions. This comparative analysis further underscores the unique characteristics and potential applications of the derived chaotic dynamics. This study not only elucidates the intricate connection between chaotic dynamics and number theory but also offers a transformative perspective that extends the boundaries of conventional electrical system analysis
 [13] arXiv:2406.12864 [pdf, other]

Title: Flatvirtual knot: introduction and some invariantsSubjects: Geometric Topology (math.GT)
The motivation for this work is to construct a map from classical knots to virtual ones. What we get in the paper is a series of maps from knots in the full torus (thickened torus) to flatvirtual knots. We give definition of flatvirtual knots and presents Alexanderlike polynomial and (picturevalued) Kauffman bracket for them.
 [14] arXiv:2406.12865 [pdf, other]

Title: Properties of minimal charts and their applications X: charts of type $(5,2)$Comments: 39 pages, 30 figuresSubjects: Geometric Topology (math.GT)
Charts are oriented labeled graphs in a disk. Any simple surface braid (2dimensional braid) can be described by using a chart. Also, a chart represents an oriented closed surface embedded in 4space. In this paper, we investigate embedded surfaces in 4space by using charts. Let $\Gamma$ be a chart, and we denote by $\Gamma_m$ the union of all the edges of label $m$. A chart $\Gamma$ is of type $(5,2)$ if there exists a label $m$ such that $w(\Gamma)=7$, $w(\Gamma_m\cap\Gamma_{m+1})=5$, $w(\Gamma_{m+1}\cap\Gamma_{m+2})=2$ where $w(G)$ is the number of white vertices in $G$. In this paper, we investigate a minimal chart of type $(5,2)$.
 [15] arXiv:2406.12866 [pdf, other]

Title: Skewsupersymmetric solution of the super Malcev YangBaxter equation and PreMalcev superalgebrasSubjects: Rings and Algebras (math.RA)
The purpose of this paper is to introduce the notion of preMalcev superalgebras as the algebraic structure behind the super $\mathcal{O}$operators on Malcev superalgebras. Moreover, the relations among Malcev superalgebras, preMalcev superalgebras and prealternative superalgebras are established. Then, we study the operator forms of the classical YangBaxter equation (CYBE) in Malcev superalgebras and give their relationship with super $\mathcal{O}$operators. There are close relationships between the CYBE in Malcev superalgebras and preMalcev superalgebras which can be interpreted through the super $\mathcal{O}$operators.
 [16] arXiv:2406.12867 [pdf, other]

Title: Classification of quasiaffine Generalized Dynkin Diagrams with Rank $3$ and Rank $2$Comments: 338 pagesSubjects: Quantum Algebra (math.QA)
All quasiaffine connected Generalized Dynkin Diagram with rank $= 3$ and $2$ are found. All quasiaffine Nichols (Lie braided) algebras with rank $ 3$ and $2$ are also found.
 [17] arXiv:2406.12868 [pdf, other]

Title: Triple products of eigenfunctions and spectral geometryComments: 6 pages, no figuresSubjects: Spectral Theory (math.SP); Mathematical Physics (mathph); Functional Analysis (math.FA)
Using elementary techniques from Geometric Analysis, Partial Differential Equations, and Abelian $C^*$ Algebras, we uncover a novel, yet familiar, global geometric invariant  namely the indexed set of integrals of triple products of eigenfunctions of the LaplaceBeltrami operator, to precisely characterize which isospectral closed Riemannian manifolds are isometric.
 [18] arXiv:2406.12871 [pdf, html, other]

Title: Weighted differential ($q$tri)dendriform algebrasComments: 26 pages. arXiv admin note: substantial text overlap with arXiv:2305.19609Subjects: Rings and Algebras (math.RA)
In this paper, we first introduce a weighted derivation on algebras over an operad $\cal P$, and prove that for the free $\cal P$algebra, its weighted derivation is determined by the restriction on the generators. As applications, we propose the concept of weighted differential ($q$tri)dendriform algebras and study some basic properties of them. Then Novikov(tri)dendriform algebras are initiated, which can be induced from differential ($q$tri) dendriform of weight zero. Finally, the corresponding free objects are constructed, in both the commutative and noncommutative contexts.
 [19] arXiv:2406.12872 [pdf, html, other]

Title: On the third cohomology of the Lie algebra of vector fields on weighted densities on RComments: 11 pagesSubjects: Rings and Algebras (math.RA)
Let Vect($\mathbb{R}$) be the Lie algebra of smooth vector fields on $\mathbb{R}$ and $\mathbb{F}_{\lambda}$ be the space of $\lambda$densities on $\mathbb{R}$.
Vect($\mathbb{R}$) acts on $\mathbb{F}_{\lambda}$ by Lie derivative. In this paper, we compute the third differential cohomology of the Lie algebra Vect($\mathbb{R}$) with coeffcients in the space $\mathbb{F}_{\lambda}.$ Explicit cocycles spanning these cohomology spaces are given.  [20] arXiv:2406.12884 [pdf, html, other]

Title: Automorphisms of free metabelian Lie algebrasComments: 33 pagesSubjects: Rings and Algebras (math.RA); Group Theory (math.GR)
We show that every automorphism of a free metabelian Lie algebra $M_n$ of rank $n\geq 4$ over an arbitrary field $K$ is almost tame, that is, it is a product of socalled Chein automorphisms (or onerow transformations). Moreover, we show that the group of all automorphisms $\mathrm{Aut}(M_n)$ of $M_n$ of rank $n\geq 4$ over a field $K$ of characteristic $\neq 3$ is generated by all linear automorphisms, as well as one quadratic and one cubic automorphism. The same result holds for fields of any characteristic if $n\geq 5$.
We also show that all Chein automorphisms of lower degree $\geq 4$ and all exponential automorphisms of lower degree $\geq 5$ are tame, which contradicts the results of \cite{BN,OE}.  [21] arXiv:2406.12890 [pdf, other]

Title: The conductor ideals of maximal subrings in noncommutative ringsSubjects: Rings and Algebras (math.RA)
Let $R$ be a maximal subring of a ring $T$, and $(R:T)$, $(R:T)_l$ and $(R:T)_r$ denote the greatest ideal, left ideal and right ideal of $T$ which are contained in $R$, respectively. It is shown that $(R:T)_l$ and $(R:T)_r$ are prime ideals of $R$ and $Min_R((R:T))\leq 2$. We prove that if $T_R$ has a maximal submodule, then $(R:T)_l$ is a right primitive ideal of $R$. We investigate that when $(R:T)_r$ is a completely prime (right) ideal of $R$ or $T$. If $R$ is integrally closed in $T$, then $(R:T)_l$ and $(R:T)_r$ are prime onesided ideals of $T$. We observe that if $(R:T)_lT=T$, then $T$ is a finitely generated left $R$module and $(R:T)_l$ is a finitely generated right $R$module. We prove that $Char(R/(R:T)_l)=Char(R/(R:T)_r)$, and if $Char(T)$ is neither zero or a prime number, then $(R:T)\neq 0$. If $Min(R)\geq 3$, then $(R:T)$ and $(R:T)_l(R:T)_r$ are nonzero ideals. Finally we study the Noetherian and the Artinian properties between $R$ and $T$.
 [22] arXiv:2406.12891 [pdf, other]

Title: Classical ideals theory of maximal subrings in noncommutative ringsSubjects: Rings and Algebras (math.RA)
Let $R$ be a maximal subring of a ring $T$. In this paper we study relation between some important ideals in the ring extension $R\subseteq T$. In fact, we would like to find some relation between $Nil_*(R)$ and $Nil_*(T)$, $Nil^*(R)$ and $Nil^*(T)$, $J(R)$ and $J(T)$, $Soc({}_RR)$ and $Soc({}_RT)$, and finally $Z({}_RR)$ and $Z({}_RT)$; especially, in certain cases, for example when $T$ is a reduced ring, $R$ (or $T$) is a left Artinian ring, or $R$ is a certain maximal subring of $T$. We show that either $Soc({}_RR)=Soc({}_RT)$ or $(R:T)_r$ (the greatest right ideal of $T$ which is contained in $R$) is a left primitive ideal of $R$. We prove that if $T$ is a reduced ring, then either $Z({}_RT)=0$ or $Z({}_RT)$ is a minimal ideal of $T$, $T=R\oplus Z({}_RT)$, and $(R:T)=(R:T)_l=(R:T)_r=ann_R(Z({}_RT))$. If $T=R\oplus I$, where $I$ is an ideal of $T$, then we completely determine relation between Jacobson radicals, lower nilradicals, upper nilradicals, socle and singular ideals of $R$ and $T$. Finally, we study the relation between previous ideals of $R$ and $T$ when either $R$ or $T$ is a left Artinian ring.
 [23] arXiv:2406.12900 [pdf, html, other]

Title: Factor Graph Optimization of ErrorCorrecting Codes for Belief Propagation DecodingSubjects: Information Theory (cs.IT); Artificial Intelligence (cs.AI); Machine Learning (cs.LG)
The design of optimal linear block codes capable of being efficiently decoded is of major concern, especially for short block lengths. As near capacityapproaching codes, LowDensity ParityCheck (LDPC) codes possess several advantages over other families of codes, the most notable being its efficient decoding via Belief Propagation. While many LDPC code design methods exist, the development of efficient sparse codes that meet the constraints of modern short code lengths and accommodate new channel models remains a challenge. In this work, we propose for the first time a datadriven approach for the design of sparse codes. We develop locally optimal codes with respect to Belief Propagation decoding via the learning on the Factor graph (also called the Tanner graph) under channel noise simulations. This is performed via a novel tensor representation of the Belief Propagation algorithm, optimized over finite fields via backpropagation coupled with an efficient linesearch method. The proposed approach is shown to outperform the decoding performance of existing popular codes by orders of magnitude and demonstrates the power of datadriven approaches for code design.
 [24] arXiv:2406.12920 [pdf, html, other]

Title: CrossDimensional Mathematics: A Foundation For STP/STASubjects: Rings and Algebras (math.RA); Optimization and Control (math.OC)
When investigating the set of matrices and vectors with mixed dimensions (MVMD), which are posed by crossdimensional operators: semitensor product (STP) and semitensor addition (STA), the existing mathematical tools seem not very efficient. A new mathematical structure, called the mixdimensional mathematics (MDM), is proposed. The MDM considered in this paper consists of three parts: hyper algebra, hyper geometry, and hyper Lie group/Lie algebra. Hyper algebra proposes some new algebraic structures such as hyper group, hyper ring, and hyper module over MVMDs. They have sets of classical groups, rings, and modules as their components and crossdimensional connections among their components. Their basic properties are investigated. Hyper geometry starts from mixed dimensional Euclidian space, and hyper vector space. Then the hyper topological vector space, hyper inner product space, and hyper manifold are constructed. They have a joined crossdimensional geometric structure. Finally, hyper metric space, topological hyper group and hyper Lie algebra are built gradually, and finally, their corresponding hyper Lie group is introduced. All these concepts are built over MVMDs, and to reach our purpose in addition to using existing STPs and STAs, a couple of most general STP and STA are introduced. Some existing structures/results about STPs/STAs have also been resumed and integrated into this MDM.
 [25] arXiv:2406.12941 [pdf, html, other]

Title: Metered Parking FunctionsSubjects: Combinatorics (math.CO)
We introduce a generalization of parking functions called $t$metered $(m,n)$parking functions, in which one of $m$ cars parks among $n$ spots per hour then leaves after $t$ hours. We characterize and enumerate these sequences for $t=1$, $t=m2$, and $t=n1$, and provide data for other cases. We characterize the $1$metered parking functions by decomposing them into sections based on which cars are unlucky, and enumerate them using a Lucas sequence recursion. Additionally, we establish a new combinatorial interpretation of the numerator of the continued fraction $n1/(n1/\cdots)$ ($n$ times) as the number of $1$metered $(n,n)$parking functions. We introduce the $(m,n)$parking function shuffle in order to count $(m2)$metered $(m,n)$parking functions, which also yields an expression for the number of $(m,n)$parking functions with any given first entry. As a special case, we find that the number of $(m2)$metered $(m, m1)$parking functions is equal to the sum of the first entries of classical parking function of length $m1$. We enumerate the $(n1)$metered $(m,n)$parking functions in terms of the number of classical parking functions of length $n$ with certain parking outcomes, which we show are periodic sequences with period $n$. We conclude with an array of open problems.
 [26] arXiv:2406.12955 [pdf, other]

Title: An EndtoEnd Coding Scheme for DNABased Data Storage With NanoporeSequenced ReadsLorenz Welter, Roman Sokolovskii, Thomas Heinis, Antonia WachterZeh, Eirik Rosnes, Alexandre Graell i AmatSubjects: Information Theory (cs.IT)
We consider errorcorrecting coding for deoxyribonucleic acid (DNA)based storage using nanopore sequencing. We model the DNA storage channel as a sampling noise channel where the input data is chunked into $M$ short DNA strands, which are copied a random number of times, and the channel outputs a random selection of $N$ noisy DNA strands. The retrieved DNA reads are prone to stranddependent insertion, deletion, and substitution (IDS) errors. We construct an indexbased concatenated coding scheme consisting of the concatenation of an outer code, an index code, and an inner code. We further propose a lowcomplexity (linear in $N$) maximum a posteriori probability decoder that takes into account the stranddependent IDS errors and the randomness of the drawing to infer symbolwise a posteriori probabilities for the outer decoder. We present MonteCarlo simulations for informationoutage probabilities and frame error rates for different channel setups on experimental data. We finally evaluate the overall system performance using the read/write cost tradeoff. A powerful combination of tailored channel modeling and soft information processing allows us to achieve excellent performance even with errorprone nanoporesequenced reads outperforming stateoftheart schemes.%
 [27] arXiv:2406.12982 [pdf, html, other]

Title: Hyperbolic actions of Thompson's group $F$ and generalizationsComments: 60 pages, 4 figuresSubjects: Group Theory (math.GR)
We study the poset of hyperbolic structures on Thompson's group $F$ and its generalizations $F_n$ for $n \geq 2$. The global structure of this poset is as simple as one would expect, with the maximal nonelementary elements being two quasiparabolic actions corresponding to wellknown ascending HNNextension expressions of $F_n$. However, the local structure turns out to be incredibly rich, in stark contrast with the situation for the $T$ and $V$ counterparts. We show that the subposet of quasiparabolic hyperbolic structures consists of two isomorphic posets, each of which contains uncountably many subposets of \emph{lamplike} structures, which can be described combinatorially in terms of certain hyperbolic structures on related lamplighter groups. Moreover, each of these subposets, as well as intersections and complements thereof, is very large, in that it contains a copy of the power set of the natural numbers. We also prove that these uncountably many uncountable subposets are not the entire picture, indeed there exists a copy of the power set of the natural numbers consisting entirely of nonlamplike structures. We also prove that this entire vast array of hyperbolic structures on $F_n$ collapses as soon as one takes a natural semidirect product with $\mathbb{Z}/2\mathbb{Z}$. These results are all proved via a detailed analysis of confining subsets, and along the way we establish a number of fundamental results in the theory of confining subsets of groups.
 [28] arXiv:2406.12988 [pdf, html, other]

Title: Solutions for fourth order anisotropic nonlinear Schr\"odinger equations in $\R^2$Comments: 22 pagesSubjects: Analysis of PDEs (math.AP)
In this paper, we consider solutions to the following fourth order anisotropic nonlinear Schrödinger equation in $\R \times \R^2$, $$ \left\{ \begin{aligned} &\textnormal{i}\partial_t\psi+\partial_{xx} \psi\partial_{yyyy} \psi +\psi^{p2} \psi=0, \\ &\psi(0)=\psi_0 \in H^{1,2}(\R^2), \end{aligned} \right. $$ where $p>2$. First we prove the local/global wellposedness and blowup of solutions to the Cauchy problem for the anisotropic nonlinear Schrödinger equation. Then we establish the existence, axial symmetry, exponential decay and orbital stability/instability of standing waves to the anisotropic nonlinear Schrödinger equation. The pictures are considerably different from the ones for the isotropic nonlinear Schrödinger equations. The results are easily extendable to the higher dimensional case.
 [29] arXiv:2406.12989 [pdf, html, other]

Title: The isoperimetric peak of complete treesSubjects: Combinatorics (math.CO)
We give exact values and bounds on the isoperimetric peak of complete trees, improving on known results. For the complete $q$ary tree of depth $d$, if $q\ge 5$, then we find that the isoperimetric peak equals $d$, completing an open problem. In the case that $q$ is 3 or 4, we determine the value up to three values, and in the case $q=2$, up to a logarithmic additive factor. Our proofs use novel compression techniques, including left, down, and aeolian compressions. We apply our results to show that the vertex separation number and the isoperimetric peak of a graph may be arbitrarily far apart as a function of the order of the graph and give new bounds on the pathwidth and pursuitevasion parameters on complete trees.
 [30] arXiv:2406.12991 [pdf, html, other]

Title: Variational multirate integratorsSubjects: Numerical Analysis (math.NA)
The simulation of systems that act on multiple time scales is challenging. A stable integration of the fast dynamics requires a highly accurate approximation whereas for the simulation of the slow part, a coarser approximation is accurate enough. With regard to the general goals of any numerical method, high accuracy and low computational costs, a popular approach is to treat the slow and the fast part of a system differently. Embedding this approach in a variational framework is the keystone of this work. By paralleling continuous and discrete variational multirate dynamics, integrators are derived on a time grid consisting of macro and micro time nodes that are symplectic, momentum preserving and also exhibit good energy behaviour. The choice of the discrete approximations for the action determines the convergence order of the scheme as well as its implicit or explicit nature for the different parts of the multirate system. The convergence order is proven using the theory of variational error analysis. The performance of the multirate variational integrators is demonstrated by means of several examples.
 [31] arXiv:2406.12993 [pdf, html, other]

Title: Navigating the Noise: A CBF Approach for Nonlinear Control with Integral ConstraintsComments: 7 pages, 7 figures, Conference on Decision and Control(2024)Subjects: Dynamical Systems (math.DS)
Many physical phenomena involving mobile agents involve timevarying scalar fields, e.g., quadrotors that emit noise. As a consequence, agents can influence and can be influenced by various environmental factors such as noise. This paper delves into the challenges of controlling such agents, focusing on scenarios where we would like to prevent excessive accumulation of some quantity over select regions and extended trajectories. We use quadrotors that emit noise as a primary example, to regulate the trajectory of such agents in the presence of obstacles and noise emitted by the aerial vehicles themselves. First, we consider constraints that are defined over accumulated quantities, i.e functionals of the entire trajectory, as opposed to those that depend solely on the current state as in traditional Higher order Control Barrier Functions (HOCBF). Second, we propose a method to extend constraints from individual points to lines and sets by using efficient overapproximations. The efficacy of the implemented strategies is verified using simulations. Although we use quadrotors as an example, the same principles can equally apply to other scenarios, such as light emission microscopy or vehicle pollution dispersion. The technical contribution of this paper is twofold.
 [32] arXiv:2406.12994 [pdf, html, other]

Title: Interpolation theorems for conjugations and applicationsSubjects: Functional Analysis (math.FA); Operator Algebras (math.OA)
Let $H$ be a separable complex Hilbert space. A conjugatelinear map $C:H\to H$ is a conjugation if it is an involutive isometry. In this paper, we consider the following interpolation problems: Let $\{x_i\}_{i\in I}$ and $\{y_i\}_{i\in I}$ be two orthogonal sets of vectors in $H$, and let $N$ and $\{N_k\}_{k\in K}$ be normal operators such that the $N_k$'s mutually commute. Then, under which conditions does there exist a conjugation $C$ on $H$ such that
(a) $Cx_i=y_i$ and $CN_kC=N_k^*$ for all $i\in I$ and $k\in K$; or
(b) $Cx_i=y_i$, for every $i\in I$, and $CNC=N^*$.
We provide complete answers to problems (a) and (b). As a consequence of our results, we give necessary and sufficient conditions for the existence of solutions of some equations in $L^{\infty}(\mu)$.  [33] arXiv:2406.13004 [pdf, html, other]

Title: Residuality of Dynamical Morphisms for Amenable Group ActionsSubjects: Dynamical Systems (math.DS)
We extend the classical Baire category approach, used in proving the finite generator theorem of Krieger, the homomorphism theorem of Sinai and the isomorphism theorem of Ornstein, applying a similar reasoning to the case of actions of countably infinite amenable groups. In principle we follow the lines of the paper by Burton, Keane and Serafin (\cite{BKS}), showing that measures defining homomorphisms or isomorphisms form residual subsets in suitably chosen spaces of joinings.
 [34] arXiv:2406.13013 [pdf, html, other]

Title: An uniform lower bound for classical Kloosterman sums and an applicationComments: 11 pagesSubjects: Number Theory (math.NT)
We present an elementary uniform lower bound for the classical Kloosterman sum $S(a,b;c)$ under the condition of its nonvanishing and $(ab,c)=1$, with $c$ being an odd integer. We then apply this lower bound for Kloosterman sums to derive an explicit lower bound in the Petersson's trace formula, subject to a pertinent condition. Consequently, we achieve a modified version of a theorem by Jung and Sardari, wherein the parameters $k$ and $N$ are permitted to vary independently.
 [35] arXiv:2406.13014 [pdf, html, other]

Title: Stable polynomials and admissible numerators in product domainsComments: 15 pagesSubjects: Complex Variables (math.CV); Algebraic Geometry (math.AG); Classical Analysis and ODEs (math.CA)
Given a polynomial $p$ with no zeros in the polydisk, or equivalently the polyupper halfplane, we study the problem of determining the ideal of polynomials $q$ with the property that the rational function $q/p$ is bounded near a boundary zero of $p$. We give a complete description of this ideal of numerators in the case where the zero set of $p$ is smooth and satisfies a nondegeneracy condition. In three variables, we give a description of the ideal in terms of an integral closure when $p$ has an isolated zero on the distinguished boundary. Constructions of multivariate stable polynomials are presented to illustrate sharpness of our results and necessity of our assumptions.
 [36] arXiv:2406.13023 [pdf, other]

Title: Stackelberg Games with $k$Submodular Function under Distributional RiskReceptiveness and RobustnessSubjects: Optimization and Control (math.OC); Machine Learning (cs.LG)
We study submodular optimization in adversarial context, applicable to machine learning problems such as feature selection using data susceptible to uncertainties and attacks. We focus on Stackelberg games between an attacker (or interdictor) and a defender where the attacker aims to minimize the defender's objective of maximizing a $k$submodular function. We allow uncertainties arising from the success of attacks and inherent data noise, and address challenges due to incomplete knowledge of the probability distribution of random parameters. Specifically, we introduce Distributionally RiskAverse $k$Submodular Interdiction Problem (DRA $k$SIP) and Distributionally RiskReceptive $k$Submodular Interdiction Problem (DRR $k$SIP) along with finitely convergent exact algorithms for solving them. The DRA $k$SIP solution allows riskaverse interdictor to develop robust strategies for realworld uncertainties. Conversely, DRR $k$SIP solution suggests aggressive tactics for attackers, willing to embrace (distributional) risk to inflict maximum damage, identifying critical vulnerable components, which can be used for the defender's defensive strategies. The optimal values derived from both DRA $k$SIP and DRR $k$SIP offer a confidence intervallike range for the expected value of the defender's objective function, capturing distributional ambiguity. We conduct computational experiments using instances of feature selection and sensor placement problems, and Wisconsin breast cancer data and synthetic data, respectively.
 [37] arXiv:2406.13033 [pdf, html, other]

Title: Arrival of information at a target set in a networkSubjects: Combinatorics (math.CO); Mathematical Physics (mathph); Dynamical Systems (math.DS)
We consider labelings of a finite regular tree by a finite alphabet subject to restrictions specified by a nonnegative transition matrix, propose an algorithm for determining whether the set of possible configurations on the last row of the tree is independent of the symbol at the root, and prove that the algorithm succeeds in a bounded number of steps, provided that the dimension of the tree is greater than or equal to the maximum row sum of the transition matrix. (The question was motivated by calculation of topological pressure on trees and is an extension of the idea of primitivity for nonnegative matrices.)
 [38] arXiv:2406.13053 [pdf, html, other]

Title: Tree independence number III. Thetas, prisms and starsSubjects: Combinatorics (math.CO)
We prove that for every $t\in \mathbb{N}$ there exists $\tau=\tau(t)\in \mathbb{N}$ such that every (theta, prism, $K_{1,t}$)free graph has tree independence number at most $\tau$ (where we allow "prisms" to have one path of length zero).
 [39] arXiv:2406.13063 [pdf, html, other]

Title: Rings of almost everywhere defined functionsComments: 21 pagesSubjects: Rings and Algebras (math.RA); Operator Algebras (math.OA)
We prove the following representation theorem: A partially ordered commutative ring $R$ is a subring of a ring of almost everywhere defined continuous realvalued functions on a compact Hausdorff space $X$ if and only if $R$ is archimedean and localizable. Here we assume that the positive cone of $R$ is closed under multiplication and stable under multiplication with squares, but we also show that one of these assumptions implies the other. An almost everywhere defined function on $X$ is one that is defined on a dense open subset of $X$. These functions can be added and multiplied pointwise so that the result is again almost everywhere defined. A partially ordered commutative ring $R$ is archimedean if the underlying additive partially ordered abelian group is archimedean, and $R$ is localizable essentially if its order is compatible with the construction of a localization with sufficiently large, positive denominators. As application we obtain several more specific representation theorems: representations by continuous realvalued functions on some topological space if $R$ is $\sigma$bounded, and representation of latticeordered commutative rings ($f$rings), of partially ordered fields, and of commutative operator algebras.
 [40] arXiv:2406.13077 [pdf, html, other]

Title: An extension of Gauss's arithmeticgeometric mean (AGM) to three variables iteration schemeComments: 10 pages, 0 figuresSubjects: Classical Analysis and ODEs (math.CA); Number Theory (math.NT)
Gauss's arithmeticgeometric mean (AGM) which is described by two variables iteration $(a_n, b_n)\rightarrow (a_{n+1}, b_{n+1})$ by $a_{n+1}=(a_n+b_n)/2,\ b_{n+1}=\sqrt{a_nb_n}$. We extend it to three variables iteration $(a_n, b_n, c_n)\rightarrow (a_{n+1}, b_{n+1}, c_{n+1})$ which reduces to Gauss's AGM when $c_0=0$. Our iteration starting from $a_0>b_0>c_0>0$ with further restriction $a_0>b_0+c_0$ converges to $a_\infty=b_\infty=M(a_0, b_0, c_0)$ and $c_\infty=0$. The limit $M(a_0, b_0, c_0)$ is expressed by Appell's hypergeometric function $F_1(1/2, \{1/2, 1/2\}, 1; \kappa, \lambda)$ of two variables $(\kappa, \lambda)$ which are determined by $(a_0, b_0, c_0)$. A relation between two hypergeometric functions (Gauss's and Appell's) is found as a byproduct.
 [41] arXiv:2406.13088 [pdf, html, other]

Title: Stratification of Derived Categories of Tate MotivesSubjects: Algebraic Geometry (math.AG); Category Theory (math.CT); KTheory and Homology (math.KT)
We classify the localizing tensor ideals of the derived categories of mixed Tate motives over certain algebraically closed fields. More precisely, we prove that these categories are stratified in the sense of Barthel, Heard and Sanders. A key ingredient in the proof is the development of a new technique for transporting stratification between categories by means of BrownAdams representability, which may be of independent interest.
 [42] arXiv:2406.13089 [pdf, html, other]

Title: Uniqueness and CLT for the Ground State of the Disordered MonomerDimer Model on $\mathbb{Z}^{d}$Subjects: Mathematical Physics (mathph); Probability (math.PR)
We prove that the disordered monomerdimer model does not admit infinite volume incongruent ground states in $\mathbb{Z}^d$ which can be obtained as a limit of finite volume ground states. Furthermore, we also prove that these ground states are stable under perturbation of the weights in a precise sense.
As an application, we obtain a CLT for the ground state weight for a growing sequence of tori. Our motivation stems from a similar and long standing open question for the short range EdwardsAnderson spin glass model.  [43] arXiv:2406.13091 [pdf, other]

Title: Convergence Analysis of Ensemble Filters for Linear Stochastic Systems with PoissonSampled ObservationsComments: Accepted for presentation at MTNS 2024. To appear in IFACPapersOnLineSubjects: Optimization and Control (math.OC)
For continuoustime linear stochastic dynamical systems driven by Wiener processes, we consider the problem of designing ensemble filters when the observation process is randomly timesampled. We propose a continuousdiscrete McKeanVlasov type diffusion process with additive Gaussian noise in observation model, which is used to describe the evolution of the individual particles in the ensemble. These particles are coupled through the empirical covariance and require less computations for implementation than the optimal ones based on solving Riccati differential equations. Using appropriate analysis tools, we show that the empirical mean and the sample covariance of the ensemble filter converges to the mean and covariance of the optimal filter if the mean sampling rate of the observation process satisfies certain bounds and as the number of particles tends to infinity.
 [44] arXiv:2406.13095 [pdf, html, other]

Title: Explicit formulas for the Grothendieck class of $\overline{\mathcal M}_{0,n}$Comments: 14 pagesSubjects: Algebraic Geometry (math.AG)
We obtain closed form expressions for the class in the Grothendieck group of varieties of the moduli space of genus $0$ stable curves with $n$ marked points. This information is equivalent to the Poincaré polynomial, so it implies explicit expressions for the Betti numbers of the moduli space in terms of Stirling numbers or, alternatively, Bernoulli numbers.
The result is proved as a consequence of explicit expressions for the generating function for the Grothendieck class, which we prove by solving a differential equation characterizing this generating function by work of E. Getzler and Y. Manin from the 1990s. Our proof reduces the solution to two combinatorial identities which follow from applications of Lagrange series.  [45] arXiv:2406.13108 [pdf, html, other]

Title: Distinguishing Martin's axiom from its restrictionsSubjects: Logic (math.LO)
We introduce an iteration of forcing notions satisfying the countable chain condition with minimal damage to a strong coloring. Applying this method, we prove that Martin's axiom is strictly stronger than its restriction to forcing notions satisfying the countable chain condition in all finite powers. Our method shows also the finer distinction, that Martin's axiom is strictly stronger than its restriction to forcing notions whose squares satisfy the countable chain condition.
 [46] arXiv:2406.13110 [pdf, html, other]

Title: DenjoyCarleman solvability of Vekuatype periodic operatorsComments: 28 pagesSubjects: Analysis of PDEs (math.AP)
This paper explores the solvability and global hypoellipticity of Vekuatype differential operators on the ndimensional torus, within the framework of DenjoyCarleman ultradifferentiability. We provide the necessary and sufficient conditions for achieving these global properties in the case of constantcoefficient operators, along with applications to classical operators. Additionally, we investigate a class of variable coefficients and establish conditions for its solvability.
 [47] arXiv:2406.13120 [pdf, html, other]

Title: A different approach to positive traces on generalized qWeyl algebrasComments: 6 pages, slightly different version to appear in the Proceedings of the 15th International Workshop "Lie Theory and Its Applications in Physics" (LT15), 1925 June 2023, Varna, BulgariaSubjects: Representation Theory (math.RT); High Energy Physics  Theory (hepth); Mathematical Physics (mathph)
Positive twisted traces are mathematical objects that could be useful in computing certain parameters of superconformal field theories. The case when $\mathcal{A}$ is a $q$Weyl algebra and $\rho$ is a certain antilinear automorphism of $\mathcal{A}$ was considered in arXiv:2105.12652. Here we consider more general choices of $\rho$. In particular, we show that for $\rho$ corresponding to a standard Schur index of a fourdimensional gauge theory a positive trace is unique.
 [48] arXiv:2406.13148 [pdf, html, other]

Title: Data Value in Distribution System OperationsSubjects: Optimization and Control (math.OC)
The rise of advanced data technologies in electric power distribution systems enables operators to optimize operations but raises concerns about data security and consumer privacy. Resulting data protection mechanisms that alter or obfuscate datasets may invalidate the efficacy of datadriven decisionsupport tools and impact the value of these datasets to the decisionmaker. This paper derives tools for distribution system operators to enrich datadriven operative decisions with information on data quality and, simultaneously, assess data usefulness in the context of this decision. To this end, we derive an AC optimal power flow model for radial distribution systems with datainformed stochastic parameters that internalize a data quality metric. We derive a tractable reformulation and discuss the marginal sensitivity of the optimal solution as a proxy for data value. Our model can capture clustered data provision, e.g., from resource aggregators, and internalize individual data quality information from each data provider. We use the IEEE 33bus test system, examining scenarios with varying photovoltaic penetration, to demonstrate the application of our approach and discuss the relationship between data quality and its value.
 [49] arXiv:2406.13160 [pdf, html, other]

Title: Global bases for Bosonic extensions of quantum unipotent coordinate ringsComments: 37pagesSubjects: Representation Theory (math.RT)
In the paper, we establish the global basis theory for the bosonic extension $\widehat{\mathcal{A}}$ associated with an arbitrary generalized Cartan matrix. When $\widehat{\mathcal{A}}$ is of simplylaced finite type, it is isomorphic to the quantum Grothendieck ring of the HernandezLeclerc category over a quantum affine algebra. In this case, we show that the $(t,q)$characters of simple modules in the HernandezLeclerc category correspond to the normalized global basis of $\widehat{\mathcal{A}}$.
 [50] arXiv:2406.13168 [pdf, other]

Title: Stochastic Multiobjective Multitrip AMR Routing Problem with Time WindowsSubjects: Optimization and Control (math.OC)
In recent years, with the rapidly aging population, alleviating the pressure on medical staff has become a critical issue. To improve the work efficiency of medical staff and reduce the risk of infection, we consider the multitrip autonomous mobile robot (AMR) routing problem with the stochastic environment to find the solution to minimizing the total expected operating cost and maximizing the total service quality of patients so that each route violates the vehicle capacity and the time window with only a very small probability. The travel time of AMRs is stochastic affected by the surrounding environment, the demand for each ward is unknown until the AMR reaches the ward, and the service time is linearly related to the actual demand. We develop a populationbased tabu search algorithm (PTS) that combines the genetic algorithm with the tabu search algorithm to solve the problem. Extensive numerical experiments were conducted on the modified Solomon instances to show that the PTS algorithm the efficient and reveals the impacts of the confidence level on the optimal solution, providing insights for the decisionmaker to devise delivery schemes that tradeoff the operating cost for patient satisfaction.
 [51] arXiv:2406.13172 [pdf, other]

Title: Affine and cyclotomic websComments: 46 pages, many figuresSubjects: Representation Theory (math.RT)
Generalizing the polynomial web category, we introduce a diagrammatic $\Bbbk$linear monoidal category, the affine web category, for any commutative ring $\Bbbk$. Integral bases consisting of elementary diagrams are obtained for the affine web category and its cyclotomic quotient categories. Connections between cyclotomic web categories and finite $W$algebras are established, leading to a diagrammatic presentation of idempotent subalgebras of $W$Schur algebras introduced by BrundanKleshchev. The affine web category will be used as a basic building block of another $\Bbbk$linear monoidal category, the affine Schur category, formulated in a sequel.
 [52] arXiv:2406.13174 [pdf, html, other]

Title: Multilinear paraproducts on Sobolev spacesSubjects: Classical Analysis and ODEs (math.CA); Analysis of PDEs (math.AP); Functional Analysis (math.FA)
Paraproducts are a special subclass of the multilinear CalderónZygmund operators, and their Lebesgue space estimates in the full multilinear range are characterized by the $\mathrm{BMO}$ norm of the symbol. In this note, we characterize the Sobolev space boundedness properties of multilinear paraproducts in terms of a suitable family of TriebelLizorkin type norms of the symbol. Coupled with a suitable wavelet representation theorem, this characterization leads to a new family of Sobolev space $T(1)$type theorems for multilinear CalderónZygmund operators.
 [53] arXiv:2406.13176 [pdf, html, other]

Title: A spectral Erd\H{o}sFaudreeRousseau theoremComments: 30 pages. At the end of the paper, we proposed many spectral extremal graph problems for readers. Any comments are welcomeSubjects: Combinatorics (math.CO)
A wellknown theorem of Mantel states that every $n$vertex graph with more than $\lfloor n^2/4\rfloor $ edges contains a triangle. An interesting problem in extremal graph theory studies the minimum number of edges contained in triangles among graphs with a prescribed number of vertices and edges. Erdős, Faudree and Rousseau (1992) showed that a graph on $n$ vertices with more than $\lfloor n^2/4\rfloor $ edges contains at least $2\lfloor n/2\rfloor +1$ edges in triangles. Such edges are called triangular edges. In this paper, we present a spectral version of the result of Erdős, Faudree and Rousseau. Using the supersaturationstability and the spectral technique, we prove that every $n$vertex graph $G$ with $\lambda (G) \ge \sqrt{\lfloor n^2/4\rfloor}$ contains at least $2 \lfloor {n}/{2} \rfloor 1$ triangular edges, unless $G$ is a balanced complete bipartite graph. The method in our paper has some interesting applications. Firstly, the supersaturationstability can be used to revisit a conjecture of Erdős concerning with the booksize of a graph, which was initially proved by Edwards (unpublished), and independently by Khadžiivanov and Nikiforov (1979). Secondly, our method can improve the bound on the order $n$ of a graph by dropping the condition on $n$ being sufficiently large, which is obtained from the triangle removal lemma. Thirdly, the supersaturationstability can be applied to deal with the spectral extremal graph problems on counting triangles, which was recently studied by Ning and Zhai (2023).
 [54] arXiv:2406.13182 [pdf, html, other]

Title: The saddlepoint approximation factors over sample paths of recursively compounded processesComments: 17 pagesSubjects: Probability (math.PR)
This paper presents an identity between the multivariate and univariate saddlepoint approximations applied to sample path probabilities for a certain class of stochastic processes. This class, which we term the recursively compounded processes, includes branching processes and other models featuring sums of a random number of i.i.d. terms; and compound Poisson processes and other Lévy processes in which the additive parameter is itself chosen randomly. For such processes, $\hat{f}_{X_1,\dotsc,X_N  X_0=x_0}(x_1,\dots,x_N) = \prod_{n=1}^N \hat{f}_{X_n  X_0=x_0,\dots,X_{n1}=x_{n1}}(x_n),$ where the lefthand side is a multivariate saddlepoint approximation applied to the random vector $(X_1,\dots,X_N)$ and the righthand side is a product of univariate saddlepoint approximations applied to the conditional onestep distributions given the past. Two proofs are given. The first proof is analytic, based on a changeofvariables identity linking the functions that arise in the respective saddlepoint approximations. The second proof is probabilistic, based on a representation of the saddlepoint approximation in terms of tilted distributions, changes of measure, and relative entropies.
 [55] arXiv:2406.13192 [pdf, html, other]

Title: Recovery of rational functions via Hankel pencil method and sensitivities of the polesSubjects: Numerical Analysis (math.NA)
In the paper, we develop a new method for the recovery of rational functions. Our idea is based on the property that Fourier coefficients of rational functions have the exponential structure and reconstruction of this exponential structure with the ESPRIT method in the frequency domain. Further we present sensitivity analysis for poles of rational functions reconstructed with our method in case of unstructured and structured perturbations. Finally, we consider several numerical experiments and, using sensitivities, explain the recovery errors for poles.
 [56] arXiv:2406.13202 [pdf, html, other]

Title: Finite Abelian Groups with Toroidal Subgroup LatticesSubjects: Combinatorics (math.CO); Group Theory (math.GR)
In this paper, we determine the genus of the subgroup lattice of several families of abelian groups. In doing so, we classify all finite abelian groups whose subgroup lattices can be embedded into the torus.
 [57] arXiv:2406.13204 [pdf, html, other]

Title: Normalized solutions to Schr\"{o}dinger systems with potentialsSubjects: Analysis of PDEs (math.AP)
In this paper, we study the normalized solutions of the Schrödinger system with trapping potentials \begin{equation}\label{eq:diricichlet} \begin{cases} \Delta u_1+V_1(x)u_1\lambda_1 u_1=\mu_1 u_1^3+\beta u_1u_2^{2}+\kappa u_2~\hbox{in}~ \mathbb{R}^3,\\ \Delta u_2+V_2(x)u_2\lambda_2 u_2=\mu_2 u_2^3+\beta u_1^2u_2+\kappa u_1~\hbox{in}~ \mathbb{R}^3,
u_1\in H^1(\mathbb{R}^3), u_2\in H^1(\mathbb{R}^3),\nonumber \end{cases} \end{equation} under the constraint \begin{equation} \int_{\mathbb{R}^3} u_1^2=a_1^2,~\int_{\mathbb{R}^3} u_2^2=a_2^2\nonumber, \end{equation} where $\mu_1,\mu_2,a_1,a_2,\beta>0$, $\kappa\in\mathbb{R}$, $V_1(x)$ and $V_2(x)$ are trapping potentials, and $\lambda_1,\lambda_2$ are lagrangian multipliers, this is a typical $L^2$supercritical case in $\mathbb{R}^3$. We obtain the existence of solutions to this system by minimax theory on the manifold for $\kappa=0$ and $\kappa\neq 0$ respectively.  [58] arXiv:2406.13207 [pdf, html, other]

Title: Generalized Metric Subregularity with Applications to HighOrder Regularized Newton MethodsComments: 33 pages (including appendix) and 2 figuresSubjects: Optimization and Control (math.OC)
This paper pursues a twofold goal. First, we introduce and study in detail a new notion of variational analysis called generalized metric subregularity, which is a fargoing extension of the conventional metric subregularity conditions. Our primary focus is on examining this concept concerning firstorder and secondorder stationary points. We develop an extended convergence framework that enables us to derive superlinear and quadratic convergence under the generalized metric subregularity condition, broadening the widely used KL convergence analysis framework. We present verifiable sufficient conditions to ensure the proposed generalized metric subregularity condition and provide examples demonstrating that the derived convergence rates are sharp. Second, we design a new highorder regularized Newton method with momentum steps, and apply the generalized metric subregularity to establish its superlinear convergence. Quadratic convergence is obtained under additional assumptions. Specifically, when applying the proposed method to solve the (nonconvex) overparameterized compressed sensing model, we achieve global convergence with a quadratic local convergence rate towards a global minimizer under a strict complementarity condition.
 [59] arXiv:2406.13212 [pdf, html, other]

Title: A Comparison of Takai and Treumann DualitiesComments: 35 pagesSubjects: KTheory and Homology (math.KT); Algebraic Topology (math.AT); Operator Algebras (math.OA)
We prove a comparison result between two duality statements  Takai duality, which is implemented by the crossed product functor $ \rtimes G: KK^{G} \to KK^{\hat G}$ on equivariant Kasparov categories; and Treumann duality, which asserts the existence of an exotic equivalence of stable $\infty$categories $\text{Mod}(KU_p[G])^{ft} \simeq \text{Mod}(KU_p[\hat G])^{ft}$ given by tensoring with a particular $(G,\hat G)$bimodule $M_E$ and $p$completing.
 [60] arXiv:2406.13226 [pdf, html, other]

Title: Optimizing Inventory Management through Multiobjective Reverse Logistics with Environmental ImpactComments: 24 pages, 16 figuresSubjects: Optimization and Control (math.OC)
We present novel mathematical models for inventory management within a reverse logistics system. Technological advancements, sustainability initiatives, and evolving customer behaviours have significantly increased the demand for repaired products. Our models account for varying demand levels for newly produced and repaired items. To optimize overall costs with constrained scenarios, we formulated mixed integer programming problems. Solution procedures for the proposed problems are introduced, and the accuracy of these solutions has been validated through numerical experiments. Additionally, we address the cost of waste disposal as an environmental concern. This paper develops a multiobjective mathematical model and provides an algorithm for the Pareto solution. Various scalarization techniques are utilized to identify the Pareto front, and a comparison of these techniques is presented.
 [61] arXiv:2406.13240 [pdf, other]

Title: $\omega$weak equivalences between weak $\omega$categoriesComments: 25 pages. Comments welcome!Subjects: Category Theory (math.CT)
We study $\omega$weak equivalences between weak $\omega$categories in the sense of BataninLeinster. Our $\omega$weak equivalences are strict $\omega$functors satisfying essential surjectivity at every dimension, and when restricted to those between strict $\omega$categories, they coincide with the weak equivalences in the model category of strict $\omega$categories defined by Lafont, Métayer, and Worytkiewicz. We show that the class of $\omega$weak equivalences has the 2outof3 property. We also consider a generalisation of $\omega$weak equivalences, defined as weak $\omega$functors (in the sense of Garner) satisfying essential surjectivity, and show that this class also has the 2outof3 property.
 [62] arXiv:2406.13241 [pdf, html, other]

Title: Achirality of Sol 3Manifolds, Stevenhagen Conjecture and Shimizu's LseriesComments: 19 pagesSubjects: Geometric Topology (math.GT); Number Theory (math.NT)
A closed orientable manifold is {\em achiral} if it admits an orientation reversing homeomorphism. A commensurable class of closed manifolds is achiral if it contains an achiral element, or equivalently, each manifold in $\CM$ has an achiral finite cover.
Each commensurable class containing nonorientable elements must be achiral.
It is natural to wonder how many
commensurable classes are achiral and how many achiral classes have nonorientable elements.
We study this problem for Sol 3manifolds. Each commensurable class $\CM$ of Sol 3manifold has a complete topological invariant $D_{\CM}$, the discriminant of $\CM$. Our main result is:
(1) Among all commensurable classes of Sol 3manifolds, there are infinitely many achiral classes; however ordered by discriminants, the density of achiral commensurable classes is 0.
(2) Among all achiral commensurable classes of Sol 3manifolds, ordered by discriminants, the density of classes containing nonorientable elements is $1\rho$,
where $$\rho:=\prod_{j=1}^\infty \left(1+2^{j}\right)^{1} = 0.41942\cdots.$$  [63] arXiv:2406.13243 [pdf, html, other]

Title: Abelian Group Codes for Classical and ClassicalQuantum Channels: Oneshot and Asymptotic Rate BoundsComments: 41 pagesSubjects: Information Theory (cs.IT)
We study the problem of transmission of information over classical and classicalquantum channels in the oneshot regime where the underlying codes are constrained to be group codes. In the achievability part, we introduce a new input probability distribution that incorporates the encoding homomorphism and the underlying channel law. Using a random coding argument, we characterize the performance of group codes in terms of hypothesis testing relativeentropic quantities. In the converse part, we establish bounds by leveraging a hypothesis testingbased approach. Furthermore, we apply the oneshot result to the asymptotic stationary memoryless setting, and establish a singleletter lower bound on group capacities for both classes of channels. Moreover, we derive a matching upper bound on the asymptotic group capacity.
 [64] arXiv:2406.13244 [pdf, other]

Title: The rigidity of filtered colimits of ncluster tilting subcategoriesSubjects: Representation Theory (math.RT)
Let $\Lambda$ be an artin algebra and $\mathcal{M}$ be an ncluster tilting subcategory of $\Lambda$mod with $n\ge 2$. From the viewpoint of higher homological algebra, a question that naturally arose in [17] is when $\mathcal{M}$ induces an ncluster tilting subcategory of $\Lambda$Mod. In this paper, we answer this question and explore its connection to Iyama's question on the finiteness of ncluster tilting subcategories of $\Lambda$mod. In fact, our theorem reformulates Iyama's question in terms of the vanishing of Ext; and highlights its relation with the rigidity of filtered colimits of $\mathcal{M}$. Also, we show that Add$(\mathcal{M})$ is an ncluster tilting subcategory of $\Lambda$Mod if and only if Add$(\mathcal{M})$ is a maximal nrigid subcategory of $\Lambda$Mod if and only if $\lbrace X\in \Lambda$Mod$~~ {\rm Ext}^i_{\Lambda}(\mathcal{M},X)=0 ~~~ {\rm for ~all}~ 0<i<n \rbrace \subseteq {\rm Add}(\mathcal{M})$ if and only if $\mathcal{M}$ is of finite type if and only if ${\rm Ext}_{\Lambda}^1({\underrightarrow{\lim}}\mathcal{M}, {\underrightarrow{\lim}}\mathcal{M})=0$. Moreover, we present several equivalent conditions for Iyama's question which shows the relation of Iyama's question with different subjects in representation theory such as purity and covering theory.
 [65] arXiv:2406.13248 [pdf, html, other]

Title: Overlay SpaceAirGround Integrated Networks with SWIPTEmpowered Aerial CommunicationsComments: 36 pages, 14 figures, This work has been submitted to the IEEE for possible publication. Copyright may be transferred without notice, after which this version may no longer be accessibleSubjects: Information Theory (cs.IT); Signal Processing (eess.SP)
In this article, we consider overlay spaceairground integrated networks (OSAGINs) where a low earth orbit (LEO) satellite communicates with ground users (GUs) with the assistance of an energyconstrained coexisting airtoair (A2A) network. Particularly, a nonlinear energy harvester with a hybrid SWIPT utilizing both powersplitting and timeswitching energy harvesting (EH) techniques is employed at the aerial transmitter. Specifically, we take the random locations of the satellite, ground and aerial receivers to investigate the outage performance of both the satellitetoground and aerial networks leveraging the stochastic tools. By taking into account the ShadowedRician fading for satellite link, the Nakagami\emph{m} for ground link, and the Rician fading for aerial link, we derive analytical expressions for the outage probability of these networks. For a comprehensive analysis of aerial network, we consider both the perfect and imperfect successive interference cancellation (SIC) scenarios. Through our analysis, we illustrate that, unlike linear EH, the implementation of nonlinear EH provides accurate figures for any target rate, underscoring the significance of using nonlinear EH models. Additionally, the influence of key parameters is emphasized, providing guidelines for the practical design of an energyefficient as well as spectrumefficient future nonterrestrial networks. Monte Carlo simulations validate the accuracy of our theoretical developments.
 [66] arXiv:2406.13255 [pdf, html, other]

Title: Padic Poissonian Pair Correlations via the Monna MapSubjects: Number Theory (math.NT)
Although the existence of sequences in the padic integers with Poissonian pair correlations has already been shown, no explicit examples had been found so far. In this note we discuss how to transfer real sequences with Poissonian pair correlations to the padic setting by making use of the Monna map.
 [67] arXiv:2406.13263 [pdf, other]

Title: Wellposedness of the Euler equations in a stably stratified ocean in isopycnal coordinatesThéo Fradin (IMB)Subjects: Analysis of PDEs (math.AP); Atmospheric and Oceanic Physics (physics.aoph); Fluid Dynamics (physics.fludyn)
This article is concerned with the wellposedness of the incompressible Euler equations describing a stably stratified ocean, reformulated in isopycnal coordinates. Our motivation for using this reformulation is twofold: first, its quasi2D structure renders some parts of the analysis easier. Second, it closes a gap between the analysis performed in the paper by Bianchini and Duch{ê}ne in 2022 in isopycnal coordinates, with shear velocity but with a regularizing term, and the analysis performed in the paper by Desjardins, Lannes, Saut in 2020 in Eulerian coordinates, without any regularizing term but without shear velocity. Our main result is an energy estimate in Sobolev spaces on the system in isopycnal coordinates, with shear velocity, without any regularizing term. With additional assumptions, it is uniform in the shallowwater parameter. The main difficulty consists in transposing to the isopycnal reformulation the symmetric structure of the system which is more straightforward in Eulerian coordinates.
 [68] arXiv:2406.13277 [pdf, html, other]

Title: On areaminimizing subgraphs in integer latticesComments: 39 pages, 52 figuresSubjects: Combinatorics (math.CO); Differential Geometry (math.DG)
We introduce areaminimizing subgraphs in an infinite graph via the formulation of functions of bounded variations initiated by De Giorgi. We classify areaminimizing subgraphs in the twodimensional integer lattice up to isomorphisms, and prove general geometric properties for those in highdimensional cases.
 [69] arXiv:2406.13278 [pdf, html, other]

Title: Mean Values of the auxiliary functionComments: 9 pages, 1 figureSubjects: Number Theory (math.NT)
Let $\mathop{\mathcal R}(s)$ be the function related to $\zeta(s)$ found by Siegel in the papers of Riemann. In this paper we obtain the main terms of the mean values \[\frac{1}{T}\int_0^T \mathop{\mathcal R}(\sigma+it)^2\Bigl(\frac{t}{2\pi}\Bigr)^\sigma\,dt, \quad\text{and}\quad \frac{1}{T}\int_0^T \mathop{\mathcal R}(\sigma+it)^2\,dt.\] Giving complete proofs of some result of the paper of Siegel about the Riemann Nachlass. Siegel follows Riemann to obtain these mean values. We have followed a more standard path, and explain the difficulties we encountered in understanding Siegel's reasoning.
 [70] arXiv:2406.13279 [pdf, html, other]

Title: Various Representation Dimensions associated with a Finite GroupSubjects: Representation Theory (math.RT)
To a finite group $G$, one can associate several notions of dimensions (or degrees). In this survey, we attempt to bring together some of the notions of dimensions or degrees defined using representations of the group in General Linear Groups and permutation groups. These are embedding degree, minimal faithful irreducible character degree, minimal faithful permutation representation degree, minimal faithful quasipermutation representation degree and essential dimension. We briefly present the progress in understanding these notions and the related problems.
 [71] arXiv:2406.13285 [pdf, html, other]

Title: The extremal problem for weighted combined energy and $\rho$Nitsche type inequalityComments: 14 pagesSubjects: Analysis of PDEs (math.AP); Complex Variables (math.CV)
Let $A_1$ and $A_2$ be two circular annuli and let $\rho$ be a radial metric defined in the annuli $A_2$. We study the existence and uniqueness of the extremal problem for weighted combined energy between $A_1$ and $A_2$, and obtain that the extremal mapping is a certain radial mapping. In fact, this extremal mapping generalizes the $\rho$harmonic mapping and satisfies equation (2.7) obtained by mean of variation for weighted combined energy. Meanwhile, we get a $\rho$Nitsche type inequality. This extends the results of Kalaj (J. Differential Equations, 268(2020)) and YTF (Arch. Math., 122(2024)), where they considered the case $\rho=1$ and $\rho=\frac{1}{h^{2}}$, respectively.
Moreover, in the course of proving the extremal problem for weighted combined energy we also investigate the extremal problem for the weighted combined distortion (see Theorem 4.1). This extends the result obtained by Kalaj (J. London Math. Soc., 93(2016)).  [72] arXiv:2406.13288 [pdf, html, other]

Title: Asymptotics of twodimensional hydroelastic waves: The zero mass, zero bending limitComments: 40 pagesSubjects: Analysis of PDEs (math.AP)
We consider twodimensional hydroelastic waves, in which a free fluid surface separates two fluids of infinite vertical extent. Elastic effects are accounted for at the interface, with a parameter measuring the elastic bending force and another parameter measuring the mass of the elastic sheet. In prior work, the authors have demonstrated wellposedness of this initial value problem in Sobolev spaces. We now take the limit as these two parameters vanish. Since the size of the time interval of existence given by this prior theory vanishes as the mass and bending parameters go to zero, we now establish estimates which are uniform with respect to these parameters. We may then make an additional estimate which demonstrates that the solutions form a Cauchy sequence as the parameters go to zero, so that the limit may be taken. This demonstrates that the vortex sheet with surface tension is the zero mass, zero bending limit of hydroelastic waves in two spatial dimensions.
 [73] arXiv:2406.13291 [pdf, html, other]

Title: On completely alternating sequences generated by rational functionsComments: 8 pagesSubjects: Functional Analysis (math.FA)
For any $k \in \mathbb N$, we discuss the problem of determining completely alternating sequence of rational functions of the form $\frac{p(x)}{\prod_{i=1}^{k}(x+b_i)}$, where $p$ is a polynomial of degree atmost $k+1$ and $b_1,\ldots,b_k$ are distinct positive real numbers. In particular, we also prove that the sequences corresponding to the rational functions $\frac{\prod_{i=1}^{k}(x+a_i)}{\prod_{i=1}^{k}(x+b_i)}$ and $\frac{\prod_{i=1}^{k+1}(x+a_i)}{\prod_{i=1}^{k}(x+b_i)}$ are completely alternating if it satisfies $0< a_1 < b_1 < a_2 < b_2 < \ldots < a_k < b_k $ and $0< a_1 < b_1 < a_2 < b_2 < \ldots < a_k < b_k < a_{k+1}$, respectively. Moreover, we also show a characterization of a class of joint completely monotone nets in two variables with the help of certain completely alternating sequences.
 [74] arXiv:2406.13293 [pdf, html, other]

Title: Existence of traveling wave solutions in continuous OV modelsComments: 26 pages, 4 figuresSubjects: Dynamical Systems (math.DS); Analysis of PDEs (math.AP)
In traffic flow, selforganized wave propagation, which characterizes congestion, has been reproduced in macroscopic and microscopic models. Hydrodynamic models, a subset of macroscopic models, can be derived from microscopiclevel carfollowing models, and the relationship between these models has been investigated. However, most validations have relied on numerical methods and formal analyses; therefore, analytical approaches are necessary to rigorously ensure their validity. This study aims to investigate the relationship between macroscopic and microscopic models based on the properties of the solutions corresponding to congestion with sparse and dense waves. Specifically, we demonstrate the existence of traveling wave solutions in macroscopic models and investigate their properties.
 [75] arXiv:2406.13298 [pdf, html, other]

Title: On certain analytic functions defined by differential inequalitySubjects: Complex Variables (math.CV)
For the family of analytic functions $f(z)$ in the open unit disk $\mathbb{D}$ with $f(0)=f'(0)1=0$, satisfying the differential equation \begin{equation*} zf'(z)  f(z) = \dfrac{1}{2} z^2 \phi(z), \quad \phi(z) \leq 1, \end{equation*} we obtain radii of convexity, starlikeness, and closetoconvexity of partial sums of $f(z)$. We also study the generalization of this family having the form \begin{equation*} zf'(z)f(z) = \lambda z^2 \phi(z), \quad \phi(z) \leq 1, \end{equation*} where $\lambda > 0,$ and obtain some useful properties of these functions.
 [76] arXiv:2406.13309 [pdf, html, other]

Title: The Powell Conjecture in genus fourComments: 17 pages, 12 figuresSubjects: Geometric Topology (math.GT)
The Powell Conjecture states that four specific elements suffice to generate the Goeritz group of the Heegaard splitting of the $3$sphere. We show that this conjecture is true when the genus of the splitting is four.
 [77] arXiv:2406.13311 [pdf, html, other]

Title: Certain subclass of harmonic functions associated with univalent functionsSubjects: Complex Variables (math.CV)
In this paper, we define a subclass of sensepreserving harmonic functions associated with a class of analytic functions satisfying a differential inequality. We then establish a close relation between both subclasses. Further, we obtain some characteristic properties, including radius properties, convolution, coefficient estimates and their properties, growth estimates, and convex combination for the functions in the defined subclass. At last, we produce conditions for some special functions as well as harmonic univalent polynomials to belong to the defined subclass of harmonic functions.
 [78] arXiv:2406.13318 [pdf, html, other]

Title: IBIS primitive groups of almost simple typeComments: 43 pagesSubjects: Group Theory (math.GR); Combinatorics (math.CO)
Let $G$ be a finite permutation group on $\Omega$. An ordered sequence
$(\omega_1\ldots,\omega_\ell)$ of elements of $\Omega$ is an irredundant base for $G$ if the pointwise stabilizer is trivial and no point is fixed by the stabilizer of its predecessors. The minimal cardinality of a base is said to be the base size of $G$. If all irredundant bases of $G$ have the same cardinality, $G$ is said to be an IBIS group.
In this paper, we classify the finite almost simple primitive IBIS groups whose base size is at least $6$.  [79] arXiv:2406.13321 [pdf, html, other]

Title: On dualABABfree and related hypergraphsSubjects: Combinatorics (math.CO); Computational Geometry (cs.CG)
Geometric motivations warranted the study of hypergraphs on ordered vertices that have no pair of hyperedges that induce an alternation of some given length. Such hypergraphs are called ABAfree, ABABfree and so on. Since then various coloring and other combinatorial results were proved about these families of hypergraphs. We prove a characterization in terms of their incidence matrices which avoids using the ordering of the vertices. Using this characterization, we prove new results about the dual hypergraphs of ABABfree hypergraphs. In particular, we show that dualABABfree hypergraphs are not always proper $2$colorable even if we restrict ourselves to hyperedges that are larger than some parameter $m$.
 [80] arXiv:2406.13328 [pdf, html, other]

Title: Radii for sections of functions convex in one directionSubjects: Complex Variables (math.CV)
Let $\mathcal{G}(\alpha)$ denote the family of functions $ f(z)$ in the open unit disk $\mathbb D :=\{z\in\mathbb{C}: z<1\}$ that satisfy $ f(0)=0= f'(0)=1$ and \[\Re \left(1+ \dfrac{z f''(z)}{ f'(z)}\right)<1+\dfrac{\alpha}{2} , \quad z\in \mathbb D.\] We determine the disks $z<\rho_n$ in which sections $ s_n(z; f)$ of $ f(z)$ are convex, starlike, and closetoconvex of order $\beta\;(0\le \beta< 1)$. Further, we obtain certain inequalities of sections in the considered class of functions.
 [81] arXiv:2406.13333 [pdf, html, other]

Title: Functions of unitaries with $\mathcal{S}^p$perturbations for non continuously differentiable functionsComments: 31 pagesSubjects: Functional Analysis (math.FA)
Consider a function $f : \mathbb{T} \to \mathbb{C}$, $n$times differentiable on $\mathbb{T}$ and such that its $n$th derivative $f^{(n)}$ is bounded but not necessarily continuous. Let $U : \mathbb{R} \to \mathcal{U}(\mathcal{H})$ be a function taking values in the set of unitary operators on some separable Hilbert space $\mathcal{H}$. Let $1<p<\infty$ and let $\mathcal{S}^p(\mathcal{H})$ be the Schatten class of order $p$ on $\mathcal{H}$. If $\tilde{U}:t\in\mathbb{R} \mapsto U(t)U(0)$ is $n$times $\mathcal{S}^p$differentiable on $\mathbb{R}$, we show that the operator valued function $\varphi : t\in \mathbb{R} \mapsto f(U(t))  f(U(0)) \in \mathcal{S}^p(\mathcal{H})$ is $n$times differentiable on $\mathbb{R}$ as well. This theorem is optimal and extends several results related to the differentiability of functions of unitaries. The derivatives of $\varphi$ are given in terms of multiple operator integrals and a formula and $\mathcal{S}^p$estimates for the Taylor remainders of $\varphi$ are provided.
 [82] arXiv:2406.13334 [pdf, html, other]

Title: On Telhcirid's theorem on arithmetic progressionsComments: 28 pagesSubjects: Number Theory (math.NT)
In this paper, we study the distribution of the digital reverses of prime numbers, which we call the "reversed primes". We prove the infinitude of reversed primes in any arithmetic progression satisfying straightforward necessary conditions provided the base is sufficiently large. We indeed prove an effective SiegelWalfisz type result for reversed primes, which has a larger admissible level of modulus than the classical case.
 [83] arXiv:2406.13341 [pdf, html, other]

Title: Bootstrap percolation on the highdimensional Hamming graphSubjects: Combinatorics (math.CO); Probability (math.PR)
In the random $r$neighbour bootstrap percolation process on a graph $G$, a set of initially infected vertices is chosen at random by retaining each vertex of $G$ independently with probability $p\in (0,1)$, and "healthy" vertices get infected in subsequent rounds if they have at least $r$ infected neighbours. A graph $G$ \emph{percolates} if every vertex becomes eventually infected. A central problem in this process is to determine the critical probability $p_c(G,r)$, at which the probability that $G$ percolates passes through one half. In this paper, we study random $2$neighbour bootstrap percolation on the $n$dimensional Hamming graph $\square_{i=1}^n K_k$, which is the graph obtained by taking the Cartesian product of $n$ copies of the complete graph $K_k$ on $k$ vertices. We extend a result of Balogh and Bollobás [Bootstrap percolation on the hypercube, Probab. Theory Related Fields. 134 (2006), no. 4, 624648. MR2214907] about the asymptotic value of the critical probability $p_c(Q^n,2)$ for random $2$neighbour bootstrap percolation on the $n$dimensional hypercube $Q^n=\square_{i=1}^n K_2$ to the $n$dimensional Hamming graph $\square_{i=1}^n K_k$, determining the asymptotic value of $p_c\left(\square_{i=1}^n K_k,2\right)$, up to multiplicative constants (when $n \rightarrow \infty$), for arbitrary $k \in \mathbb N$ satisfying $2 \leq k\leq 2^{\sqrt{n}}$.
 [84] arXiv:2406.13353 [pdf, html, other]

Title: Dynamics of Fuchsian meromorphic connections with real periodsSubjects: Complex Variables (math.CV); Dynamical Systems (math.DS)
In this paper, we study the dynamics of geodesics of Fuchsian meromorphic connections with real periods, giving a precise characterization of the possible $\omega$limit sets of simple geodesics in this case. The main tools are the study of the singular flat metric associated to the meromorphic connection, an explicit description of the geodesics nearby a Fuchsian pole with real residue larger than $1$ and a farreaching generalization to our case of the classical Teichmüller lemma for quadratic differentials.
 [85] arXiv:2406.13355 [pdf, html, other]

Title: Linear codes in the folded Hamming distance and the quasi MDS propertySubjects: Information Theory (cs.IT)
In this work, we study linear codes with the folded Hamming distance, or equivalently, codes with the classical Hamming distance that are linear over a subfield. This includes additive codes. We study MDS codes in this setting and define quasi MDS (QMDS) codes and dually QMDS codes, which attain a more relaxed variant of the classical Singleton bound. We provide several general results concerning these codes, including restriction, shortening, weight distributions, existence, density, geometric description and bounds on their lengths relative to their field sizes. We provide explicit examples and a binary construction with optimal lengths relative to their field sizes, which beats any MDS code.
 [86] arXiv:2406.13370 [pdf, other]

Title: Computing the invariant distribution of McKeanVlasov SDEs by ergodic simulationSubjects: Probability (math.PR); Numerical Analysis (math.NA)
We design a fully implementable scheme to compute the invariant distribution of ergodic McKeanVlasov SDE satisfying a uniform confluence property. Under natural conditions, we prove various convergence results notably we obtain rates for the Wasserstein distance in quadratic mean and almost sure sense.
 [87] arXiv:2406.13373 [pdf, html, other]

Title: Virtual knots and links with unknotting index (n,m)Subjects: Geometric Topology (math.GT)
In [13], K. Kaur, S. Kamada et al. posed a problem of finding a virtual knot, if exists, with an unknotting index (n,m), for any pair of nonnegative integers (n,m). In this paper, we address this question by providing infinite families of virtual knots with unknotting index (n,m), for a given pair of nonnegative integers (n,m). Additionally, we extend our result for virtual links also.
 [88] arXiv:2406.13379 [pdf, html, other]

Title: Numerical Methods for Shape Optimal Design of FluidStructure Interaction ProblemsSubjects: Optimization and Control (math.OC); Numerical Analysis (math.NA)
We consider the method of mappings for performing shape optimization for unsteady fluidstructure interaction (FSI) problems. In this work, we focus on the numerical implementation. We model the optimization problem such that it takes several theoretical results into account, such as regularity requirements on the transformations and a differential geometrical point of view on the manifold of shapes. Moreover, we discretize the problem such that we can compute exact discrete gradients. This allows for the use of general purpose optimization solvers. We focus on an FSI benchmark problem to validate our numerical implementation. The method is used to optimize parts of the outer boundary and the interface. The numerical simulations build on FEniCS, dolfinadjoint and IPOPT. Moreover, as an additional theoretical result, we show that for a linear special case the adjoint attains the same structure as the forward problem but reverses the temporal flow of information.
 [89] arXiv:2406.13398 [pdf, other]

Title: Nonadditive derived functorsComments: 43 pagesSubjects: Category Theory (math.CT)
Let $F\colon \mathcal{C} \to \mathcal{E}$ be a functor from a category $\mathcal{C}$ to a homological (BorceuxBourn) or semiabelian (JanelidzeMárkiTholen) category $\mathcal{E}$. We investigate conditions under which the homology of an object $X$ in $\mathcal{C}$ with coefficients in the functor $F$, defined via projective resolutions in $\mathcal{C}$, remains independent of the chosen resolution. Consequently, the left derived functors of $F$ can be constructed analogously to the classical abelian case.
Our approach extends the concept of chain homotopy to a nonadditive setting using the technique of imaginary morphisms. Specifically, we utilize the approximate subtractions of BournJanelidze, originally introduced in the context of subtractive categories. This method is applicable when $\mathcal{C}$ is a pointed regular category with finite coproducts and enough projectives, provided these projectives are closed under protosplit subobjects, a new condition introduced in this article and naturally satisfied in the abelian context. We further assume that the functor $F$ meets certain exactness conditions: for instance, it may be protoadditive and preserve proper morphisms and binary coproducts  conditions that amount to additivity when $\mathcal{C}$ and $\mathcal{E}$ are abelian categories.
Within this framework, we develop a basic theory of derived functors, compare it with the simplicial approach, and provide several examples.  [90] arXiv:2406.13401 [pdf, html, other]

Title: Semidirect Product of Loops with GroupsSubjects: Group Theory (math.GR)
In this paper, we have studied the loops which are the semidirect products of a loop and a group. Also, the cummutant, nuclei and the center of such loops are studied.
 [91] arXiv:2406.13402 [pdf, html, other]

Title: When $t$intersecting hypergraphs admit bounded $c$strong colouringsComments: 13 pagesSubjects: Combinatorics (math.CO)
The $c$strong chromatic number of a hypergraph is the smallest number of colours needed to colour its vertices so that every edge sees at least $c$ colours or is rainbow. We show that every $t$intersecting hypergraph has bounded $(t + 1)$strong chromatic number, resolving a problem of Blais, Weinstein and Yoshida. In fact, we characterise when a $t$intersecting hypergraph has large $c$strong chromatic number for $c\geq t+2$. Our characterisation also applies to hypergraphs which exclude sunflowers with specified parameters.
 [92] arXiv:2406.13418 [pdf, html, other]

Title: Filtrations of torsion classes in proper abelian subcategoriesComments: 12 pages, comments are welcomeSubjects: Representation Theory (math.RT)
In an abelian category $\mathscr{A}$, we can generate torsion pairs from tilting objects of projective dimension $\leq 1$. However, when we look at tilting objects of projective dimension $2$, there is no longer a natural choice of an associated torsion pair. Instead of trying to generate a torsion pair, Jensen, Madsen and Su generated a triple of extension closed classes that can filter any objects of $\mathscr{A}$. We generalize this result to proper abelian subcategories.
 [93] arXiv:2406.13421 [pdf, html, other]

Title: The triangulantComments: 16 pagesSubjects: Algebraic Geometry (math.AG); Rings and Algebras (math.RA)
We introduce the triangulant of two matrices, and relate it to the existence of orthogonal eigenvectors. We also use it for a new characterization of mutually unbiased bases. Generalizing the notion, we introduce higher order triangulants of two matrices, and relate them to the existence of nontrivially intersecting invariant subspaces of complementary dimensions.
 [94] arXiv:2406.13422 [pdf, html, other]

Title: Formal deformations and extensions of `twisted' Lie algebrasComments: 14Subjects: Rings and Algebras (math.RA)
The interplay between derivations and algebraic structures has been a subject of significant interest and exploration. Inspired by Yau's twist and the Leibniz rule, we investigate the formal deformation of twisted Lie algebras by invertible derivations, herein referred to as "InvDer Lie". We define representations of InvDer Lie, elucidate cohomology structures of order 1 and 2, and identify infinitesimals as 2cocycles. Furthermore, we explore central extensions of InvDer Lie, revealing their intricate relationship with cohomology theory.
 [95] arXiv:2406.13428 [pdf, html, other]

Title: Petty projection inequality on the sphere and on the hyperbolic spaceSubjects: Metric Geometry (math.MG)
Using gnomonic projection and Poincaré model, we first define the spherical projection body and hyperbolic projection body in spherical space $\mathbb{S}^n$ and hyperbolic space $\mathbb{H}^n$, then define the spherical Steiner symmetrization and hyperbolic Steiner symmetrization, finally prove the spherical projection inequality and hyperbolic projection inequality.
 [96] arXiv:2406.13432 [pdf, html, other]

Title: Ramanujan vector fieldSubjects: Algebraic Geometry (math.AG); Number Theory (math.NT)
In this article we prove that for all primes $p\not=2,3$, the Ramanujan vector field has an invariant algebraic curve and then we give a moduli space interpretation of this curve in terms of Cartier operator acting on the de Rham cohomology of elliptic curves. The main ingredients of our study are due to Serre, SwinnertonDyer and Katz in 1973. We aim to generalize this for the theory of CalabiYau modular forms, which includes the generating function of genus $g$ GromovWitten invariants. The integrality of $q$expansions of such modular forms is still a main conjecture which has been only established for special CalabiYau varieties, for instance those whose periods are hypergeometric functions. For this the main tools are Dwork's theorem. We present an alternative project which aims to prove such integralities using modular vector fields and GaussManin connection in positive characteristic.
 [97] arXiv:2406.13437 [pdf, html, other]

Title: MsFEM for advectiondominated problems in heterogeneous media: Stabilization via nonconforming variantsSubjects: Numerical Analysis (math.NA)
We study the numerical approximation of advectiondiffusion equations with highly oscillatory coefficients and possibly dominant advection terms by means of the Multiscale Finite Element Method. The latter method is a now classical, finite element type method that performs a Galerkin approximation on a problemdependent basis set, itself precomputed in an offline stage. The approach is implemented here using basis functions that locally resolve both the diffusion and the advection terms. Variants with additional bubble functions and possibly weak interelement continuity are proposed. Some theoretical arguments and a comprehensive set of numerical experiments allow to investigate and compare the stability and the accuracy of the approaches. The best approach constructed is shown to be adequate for both the diffusion and advectiondominated regimes, and does not rely on an auxiliary stabilization parameter that would have to be properly adjusted.
 [98] arXiv:2406.13438 [pdf, other]

Title: Computing the center of a fusion categorySubjects: Representation Theory (math.RT); Mathematical Physics (mathph); Category Theory (math.CT); Quantum Algebra (math.QA)
We present a (Las Vegas) algorithm for explicitly computing the simple objects of the categorical (Drinfeld) center of a spherical fusion category. Our approach is based on decomposing the images of simple objects under the induction functor from the category to its center. We have implemented this algorithm in a generalpurpose software framework TensorCategories.jl for tensor categories that we develop within the opensource computer algebra system OSCAR. While the required computations are still too heavy to investigate standard examples whose center is not yet known up to some equivalence, our algorithm has the advantage of determining explicit halfbraidings for the simple central objects, avoiding abstract equivalences and the like. Furthermore, it also works over not necessarily algebraically closed fields, and this yields new explicit examples of nonsplit modular categories.
 [99] arXiv:2406.13447 [pdf, other]

Title: Highprobability minimax lower boundsComments: 37 pages, 3 figuresSubjects: Statistics Theory (math.ST); Information Theory (cs.IT); Machine Learning (cs.LG); Machine Learning (stat.ML)
The minimax risk is often considered as a gold standard against which we can compare specific statistical procedures. Nevertheless, as has been observed recently in robust and heavytailed estimation problems, the inherent reduction of the (random) loss to its expectation may entail a significant loss of information regarding its tail behaviour. In an attempt to avoid such a loss, we introduce the notion of a minimax quantile, and seek to articulate its dependence on the quantile level. To this end, we develop highprobability variants of the classical Le Cam and Fano methods, as well as a technique to convert local minimax risk lower bounds to lower bounds on minimax quantiles. To illustrate the power of our framework, we deploy our techniques on several examples, recovering recent results in robust mean estimation and stochastic convex optimisation, as well as obtaining several new results in covariance matrix estimation, sparse linear regression, nonparametric density estimation and isotonic regression. Our overall goal is to argue that minimax quantiles can provide a finergrained understanding of the difficulty of statistical problems, and that, in wide generality, lower bounds on these quantities can be obtained via userfriendly tools.
 [100] arXiv:2406.13449 [pdf, html, other]

Title: Consensus analysis of a twostep communication opinion dynamics model with group pressure and selfconfidenceComments: 7 pages, 15 figuresSubjects: Information Theory (cs.IT)
This paper considers the consensus problem of a novel opinion dynamics model with group pressure and selfconfidence. Different with the most existing paper, the influence of friends of friends in a social network is taken into account, which is modeled to be twostep communication. Based on this consideration, the neighbors of agents are classified into direct neighbors and indirect neighbors. Accordingly, the communication between agents and their neighbors is classified into onestep communication and twostep communication. By applying matrix analytic theory and graph theory, it is shown that the opinion consensus can be achieved. Moreover, the exactly consensus value of the opinion is obtained for three cases of the group pressure. Finally, simulation examples are provided to demonstrate the validity of the conclusions drawn in the paper.
 [101] arXiv:2406.13451 [pdf, other]

Title: Bifurcations in planar, quadratic massaction networks with few reactions and low molecularitySubjects: Dynamical Systems (math.DS)
In this paper we study bifurcations in massaction networks with two chemical species and reactant complexes of molecularity no more than two. We refer to these as planar, quadratic networks as they give rise to (at most) quadratic differential equations on the nonnegative quadrant of the plane. Our aim is to study bifurcations in networks in this class with the fewest possible reactions, and the lowest possible product molecularity. We fully characterise generic bifurcations of positive equilibria in such networks with up to four reactions, and product molecularity no higher than three. In these networks we find fold, AndronovHopf, BogdanovTakens and Bautin bifurcations, and prove the nonoccurrence of any other generic bifurcations of positive equilibria. In addition, we present a number of results which go beyond planar, quadratic networks. For example, we show that massaction networks without conservation laws admit no bifurcations of codimension greater than $m2$, where $m$ is the number of reactions; we fully characterise quadratic, rankone massaction networks admitting fold bifurcations; and we write down some necessary conditions for AndronovHopf and cusp bifurcations in massaction networks. Finally, we draw connections with a number of previous results in the literature on nontrivial dynamics, bifurcations, and inheritance in massaction networks.
 [102] arXiv:2406.13454 [pdf, html, other]

Title: Unifying nonlinearly constrained nonconvex optimizationComments: Submitted to Mathematical Programming Computation journalSubjects: Optimization and Control (math.OC)
Derivativebased iterative methods for nonlinearly constrained nonconvex optimization usually share common algorithmic components, such as strategies for computing a descent direction and mechanisms that promote global convergence. Based on this observation, we introduce an abstract framework based on four common ingredients that describes most derivativebased iterative methods and unifies their workflows. We then present Uno, a modular C++ solver that implements our abstract framework and allows the automatic generation of various strategy combinations with no programming effort from the user. Uno is meant to (1) organize mathematical optimization strategies into a coherent hierarchy; (2) offer a wide range of efficient and robust methods that can be compared for a given instance; (3) enable researchers to experiment with novel optimization strategies; and (4) reduce the cost of development and maintenance of multiple optimization solvers. Uno's software design allows user to compose new customized solvers for emerging optimization areas such as robust optimization or optimization problems with complementarity constraints, while building on reliable nonlinear optimization techniques. We demonstrate that Uno is highly competitive against stateoftheart solvers filterSQP, IPOPT, SNOPT, MINOS, LANCELOT, LOQO, and CONOPT on a subset of 429 small problems from the CUTEst collection. Uno is available as opensource software under the MIT license at this https URL .
 [103] arXiv:2406.13455 [pdf, html, other]

Title: Johnson graphs as slices of a hypercube and an algebra homomorphism from the universal Racah algebra into $U(\mathfrak{sl}_2)$Subjects: Combinatorics (math.CO); Representation Theory (math.RT)
From the viewpoint of Johnson graphs as slices of a hypercube, we derive a novel algebra homomorphism $\sharp$ from the universal Racah algebra $\Re$ into $U(\mathfrak{sl}_2)$. We use the Casimir elements of $\Re$ to describe the kernel of $\sharp$. By pulling back via $\sharp$ every $U(\mathfrak{sl}_2)$module can be viewed as an $\Re$module. We show that for any finitedimensional $U(\mathfrak{sl}_2)$module $V$, the $\Re$module $V$ is completely reducible and three generators of $\Re$ act on every irreducible $\Re$submodule of $V$ as a Leonard triple. In particular, Leonard triples can be constructed in terms of the second dual distance operator of the hypercube $H(D,2)$ and a decomposition of the second distance operator of $H(D,2)$ induced by Johnson graphs.
 [104] arXiv:2406.13456 [pdf, html, other]

Title: On the Schatten exponent in orthonormal Strichartz estimate for the Dunkl operatorsSubjects: Analysis of PDEs (math.AP); Functional Analysis (math.FA)
In \cite{PRA} and \cite{SSM} the orthonormal Strichartz estimates for the Schrödinger equation associated with the Dunkl Laplacian and the DunklHermite operator are obtained. In this article, we prove a necessary condition on the Schatten exponent for the above orthonormal Strichartz estimates, which turns out to be optimal for the Schrödinger equations associated with Laplacian and Hermite operator as a special case.
 [105] arXiv:2406.13460 [pdf, html, other]

Title: Generalized multiple BorelCantelli Lemma in dynamics and its applicationsSubjects: Dynamical Systems (math.DS)
Multiple BorelCantelli Lemma is a criterion that characterizes the occurrence of multiple rare events on the same time scale. We generalize the multiple BorelCantelli Lemma in dynamics established by Dolgopyat, Fayad and Liu [J. Mod. Dyn. 18 (2022) 209289], broadening its applications to encompass several nonsmooth systems with absolute continuous measures. Utilizing this generalization, we derive multiple Logarithm Law for hitting time and recurrence of dispersing billiard maps and piecewise expanding maps under some regular conditions, including tent map, Lorentzlike map and Gauss map.
 [106] arXiv:2406.13465 [pdf, other]

Title: Equivariant free boundary minimal discs and annuli in ellipsoidsComments: 15 pages, 3 figuresSubjects: Differential Geometry (math.DG)
We employ equivariant variational methods to construct new examples of nonplanar free boundary minimal discs in ellipsoids. We also prove that every ellipsoid contains at least three distinct embedded free boundary minimal annuli with dihedral symmetry.
 [107] arXiv:2406.13466 [pdf, other]

Title: Unveiling Covert Semantics: Joint SourceChannel Coding Under a Covertness ConstraintComments: Submitted to GLOBECOM'2024 for reviewSubjects: Information Theory (cs.IT)
The fundamental limit of Semantic Communications (joint sourcechannel coding) is established when the transmission needs to be kept covert from an external warden. We derive informationtheoretic achievability and matching converse results and we show that source and channel coding separation holds for this setup. Furthermore, we show through an experimental setup that one can train a deep neural network to achieve covert semantic communication for the classification task. Our numerical experiments confirm our theoretical findings, which indicate that for reliable joint sourcechannel coding the number of transmitted source symbols can only scale as the squareroot of the number of channel uses.
 [108] arXiv:2406.13475 [pdf, html, other]

Title: The Kummer distribution in free probability, and its characterizationsSubjects: Operator Algebras (math.OA)
We study the analogue of Kummer distribution in free probability. We prove characterization of freeKummer and free Poisson distributions by freeness properties together with some assumptions about conditional moments. Our main tools are subordination of free multiplicative convolution and Boolean cumulants.
 [109] arXiv:2406.13477 [pdf, other]

Title: An extension of the lowrank Lyapunov ADI to nonzero initial values and its applicationsSubjects: Numerical Analysis (math.NA)
We derive the AlternatingDirection Implicit (ADI) method based on a commuting operator split and apply the results to the continuous time algebraic Lyapunov equation with lowrank constant term and approximate solution. Previously, it has been mandatory to start the lowrank ADI (LRADI) with an allzero initial value. Our approach retains the known efficient iteration schemes of lowrank increments and residual to arbitrary lowrank initial values for the LRADI method. We further generalize some of the known properties of the LRADI for Lyapunov equations to larger classes of algorithms or problems.
We investigate the performance of arbitrary initial values using two outer iterations in which LRADI is typically called. First, we solve an algebraic Riccati equation with the Newton method. Second, we solve a differential Riccati equation with a firstorder Rosenbrock method. Numerical experiments confirm that the proposed new initial value of the alternatingdirections implicit (ADI) can lead to a significant reduction in the total number of ADI steps, while also showing a 17% and 8x speedup over the zero initial value for the two equation types, respectively.  [110] arXiv:2406.13484 [pdf, html, other]

Title: Unitary Easy Quantum Groups Arising as Quantum Symmetry of Graph C*algebrasComments: Preliminary version of the article, Comments are welcomeSubjects: Operator Algebras (math.OA)
In 2009, T. Banica and R. Speicher introduced the orthogonal easy quantum group which can be completely determined using the combinations of set partitions. Later, P. Tarrago and M. Weber extended this formulation to another special class of compact matrix quantum groups, known as unitary easy quantum groups, which are quantum subgroups of the free unitary quantum group ($U_n^+$) containing $S_n$. On the other hand, the quantum symmetry of graph $C^{*}$algebras has been explored by several mathematicians within different categories in the past few years. In this article, we establish that there are exactly three families of unitary easy quantum groups that can be achieved as the quantum symmetry of graph $C^*$algebra $C^*(\Gamma)$ associated with a finite, connected, directed graph $\Gamma$ in the category introduced by Joardar and Mandal. Moreover, we demonstrate that there does not exist any graph $C^*$algebra associated with a finite, connected, directed graph $\Gamma$ having $A_{U^t}(F^{\Gamma})$ as the quantum automorphism group of $C^*(\Gamma)$ for nonscalar matrix $F^{\Gamma}$.
 [111] arXiv:2406.13485 [pdf, html, other]

Title: Fra\"iss\'e's conjecture, partial impredicativity and wellordering principles, part ISubjects: Logic (math.LO)
Fraïssé's conjecture (proved by Laver) is implied by the $\Pi^1_1$comprehension axiom of reverse mathematics, as shown by Montalbán. The implication must be strict for reasons of quantifier complexity, but it seems that no better bound has been known. We locate such a bound in a hierarchy of Suzuki and Yokoyama, which extends Towsner's framework of partial impredicativity. Specifically, we show that Fraïssé's conjecture is implied by a principle of pseudo $\Pi^1_1$comprehension. As part of the proof, we introduce a cofinite version of the $\Delta^0_2$Ramsey theorem, which may be of independent interest. We also relate pseudo $\Pi^1_1$comprehension to principles of pseudo $\beta$model reflection (due to Suzuki and Yokoyama) and reflection for $\omega$models of transfinite induction (studied by Rathjen and ValenciaVizcaíno). In a forthcoming companion paper, we characterize pseudo $\Pi^1_1$comprehension by a wellordering principle, to get a transparent combinatorial bound for the strength of Fraïssé's conjecture.
 [112] arXiv:2406.13503 [pdf, html, other]

Title: Integrable $\mathbb{Z}_2^2$graded Extensions of the Liouville and SinhGordon TheoriesComments: 25 pagesSubjects: Mathematical Physics (mathph); High Energy Physics  Theory (hepth); Exactly Solvable and Integrable Systems (nlin.SI)
In this paper we present a general framework to construct integrable $\mathbb{Z}_2^2$graded extensions of classical, twodimensional Toda and conformal affine Toda theories. The scheme is applied to define the extended Liouville and SinhGordon models; they are based on $\mathbb{Z}_2^2$graded color Lie algebras and their fields satisfy a parabosonic statististics. The mathematical tools here introduced are the $\mathbb{Z}_2^2$graded covariant extensions of the Lax pair formalism and of the Polyakov's soldering procedure. The $\mathbb{Z}_2^2$graded SinhGordon model is derived from an affine $\mathbb{Z}_2^2$graded color Lie algebra, mimicking a procedure originally introduced by BabelonBonora to derive the ordinary SinhGordon model. The color Lie algebras under considerations are: the $6$generator $\mathbb{Z}_2^2$graded $sl_2$, the $\mathbb{Z}_2^2$graded affine ${\widehat{sl_2}}$ algebra with two central extensions, the $\mathbb{Z}_2^2$graded Virasoro algebra obtained from a Hamiltonian reduction.
 [113] arXiv:2406.13510 [pdf, html, other]

Title: Conic bundle threefolds differing by a constant Brauer class and connections to rationalityComments: 18 pagesSubjects: Algebraic Geometry (math.AG)
A double cover $Y$ of $\mathbb{P}^1 \times \mathbb{P}^2$ ramified over a general $(2,2)$divisor will have the structure of a geometrically standard conic bundle ramified over a smooth plane quartic $\Delta \subset \mathbb{P}^2$ via the second projection. These threefolds are rational over algebraically closed fields, but over nonclosed fields, including over $\mathbb{R}$, their rationality is an open problem. In this paper, we characterize rationality over $\mathbb{R}$ when $\Delta(\mathbb{R})$ has at least two connected components (extending work of M. Ji and the second author) and over local fields when all odd degree fibers of the first projection have nonsquare discriminant.
We obtain these applications by proving general results comparing the conic bundle structure on $Y$ with the conic bundle structure on a wellchosen intersection of two quadrics. The difference between these two conic bundles is encoded by a constant Brauer class, and we prove that this class measures a certain failure of Galois descent for the codimension 2 Chow group of $Y$.  [114] arXiv:2406.13513 [pdf, html, other]

Title: Sharp oracle inequalities and universality of the AIC and FPEComments: 89 pages, 3 figuresSubjects: Statistics Theory (math.ST); Probability (math.PR)
In two landmark papers, Akaike introduced the AIC and FPE, demonstrating their significant usefulness for prediction. In subsequent seminal works, Shibata developed a notion of asymptotic efficiency and showed that both AIC and FPE are optimal, setting the stage for decadeslong developments and research in this area and beyond. Conceptually, the theory of efficiency is universal in the sense that it (formally) only relies on secondorder properties of the underlying process $(X_t)_{t\in \mathbb{Z}}$, but, so far, almost all (efficiency) results require the much stronger assumption of a linear process with independent innovations. In this work, we establish sharp oracle inequalities subject only to a very general notion of weak dependence, establishing a universal property of the AIC and FPE. A direct corollary of our inequalities is asymptotic efficiency of these criteria. Our framework contains many prominent dynamical systems such as random walks on the regular group, functionals of iterated random systems, functionals of (augmented) Garch models of any order, functionals of (Banach space valued) linear processes, possibly infinite memory Markov chains, dynamical systems arising from SDEs, and many more.
 [115] arXiv:2406.13524 [pdf, html, other]

Title: Koebe uniformization for infinitely connected attracting Fatou domainsComments: 13 pagesSubjects: Complex Variables (math.CV); Dynamical Systems (math.DS)
This paper works on the structure of infinitely connected Fatou damains of rational maps in terms of Koebe uniformization. Due to the complicated boundary behavior, the existing uniformization results are failed to apply in general. We proved that if the rational map is geometrically finite, then its infinitely connected attracting Fatou damain is conformally homeomorphic to a circle domain.
 [116] arXiv:2406.13525 [pdf, html, other]

Title: On energydissipative finite element approximations for ratetype viscoelastic fluids with stress diffusionSubjects: Numerical Analysis (math.NA)
We study a fully discrete finite element approximation of a model for unsteady flows of ratetype viscoelastic fluids with stress diffusion in two and three dimensions. The model consists of the incompressible NavierStokes equation for the velocity, coupled with a diffusive variant of a combination of the OldroydB and the Giesekus model for the left CauchyGreen tensor. The discretization of the model is chosen such that an energy inequality is preserved at the fully discrete level. Thus, unconditional solvability and stability for the discrete system are guaranteed and the discrete CauchyGreen tensor is positive definite. Moreover, subsequences of discrete solutions converge to a globalintime weak solution, as the discretization parameters tend to zero. In the end, we present numerical convergence tests.
 [117] arXiv:2406.13528 [pdf, html, other]

Title: Enumeration of maps with tight boundaries and the Zhukovsky transformationComments: 62 pages, 7 figuresSubjects: Combinatorics (math.CO); Mathematical Physics (mathph); Probability (math.PR)
We consider maps with tight boundaries, i.e. maps whose boundaries have minimal length in their homotopy class, and discuss the properties of their generating functions $T^{(g)}_{\ell_1,\ldots,\ell_n}$ for fixed genus $g$ and prescribed boundary lengths $\ell_1,\ldots,\ell_n$, with a control on the degrees of inner faces. We find that these series appear as coefficients in the expansion of $\omega^{(g)}_n(z_1,\ldots,z_n)$, a fundamental quantity in the EynardOrantin theory of topological recursion, thereby providing a combinatorial interpretation of the Zhukovsky transformation used in this context. This interpretation results from the socalled trumpet decomposition of maps with arbitrary boundaries. In the planar bipartite case, we obtain a fully explicit formula for $T^{(0)}_{2\ell_1,\ldots,2\ell_n}$ from the ColletFusy formula. We also find recursion relations satisfied by $T^{(g)}_{\ell_1,\ldots,\ell_n}$, which consist in adding an extra tight boundary, keeping the genus $g$ fixed. Building on a result of Norbury and Scott, we show that $T^{(g)}_{\ell_1,\ldots,\ell_n}$ is equal to a paritydependent quasipolynomial in $\ell_1^2,\ldots,\ell_n^2$ times a simple power of the basic generating function $R$. In passing, we provide a bijective derivation in the case $(g,n)=(0,3)$, generalizing a recent construction of ours to the non bipartite case.
 [118] arXiv:2406.13537 [pdf, html, other]

Title: Feller's test for explosions of stochastic Volterra equationsSubjects: Probability (math.PR)
This paper provides a Feller's test for explosions of onedimensional continuous stochastic Volterra processes of convolution type. The study focuses on dynamics governed by nonsingular kernels, which preserve the semimartingale property of the processes and introduce memory features through a pathdependent drift. In contrast to the classical pathindependent case, the sufficient condition derived in this study for a Volterra process to remain in the interior of an interval is generally more restrictive than the necessary condition. The results are illustrated with three specifications of the dynamics: the Volterra squareroot diffusion, the Volterra Jacobi process and the Volterra powertype diffusion. For the Volterra squareroot diffusion, also known as the Volterra CIR process, the paper presents a detailed discussion on the approximation of the singular fractional kernel with a sum of exponentials, a method commonly employed in the mathematical finance literature.
 [119] arXiv:2406.13540 [pdf, html, other]

Title: Polyhedral products in abstract and motivic homotopy theoryComments: 37 pages, comments are welcomeSubjects: Algebraic Topology (math.AT); Algebraic Geometry (math.AG)
We introduce polyhedral products in an $\infty$categorical setting. We generalize a splitting result by Bahri, Bendersky, Cohen, and Gitler that determines the stable homotopy type of the a polyhedral product. We also introduce a motivic refinement of momentangle complexes and use the splitting result to compute cellular $\mathbb{A}^1$homology, and $\mathbb{A}^1$Euler characteristics.
 [120] arXiv:2406.13546 [pdf, html, other]

Title: Dual of the Geometric Lemma and the Second Adjointness Theorem for $p$adic reductive groupsComments: 20 pages, comments welcomeSubjects: Representation Theory (math.RT); Number Theory (math.NT)
Let $P,Q$ be standard parabolic subgroups of a $p$adic reductive group $G$. We study the smooth dual of the filtration on a parabolically induced module arising from the geometric lemma associated to the cosets $P\setminus G/Q$. We prove that the dual filtration coincides with the filtration associated to the cosets $P\setminus G/Q^$ via the BernsteinCasselman canonical pairing from the second adjointness of parabolic induction. This result generalizes a result of BezrukavnikovKazhdan on the explicit description in the second adjointness. Along the way, we also study some group theoretic results.
 [121] arXiv:2406.13562 [pdf, html, other]

Title: NonWeight modules over affine NappiWitten Lie algebrasComments: Comments are welcomeSubjects: Representation Theory (math.RT)
In this paper, we study the representation theory of affine NappiWitten Lie algebra $\widehat{H_4}$ corresponding to the NappiWitten Lie algebra $H_4$. We completely classify all Cartanfree modules of rank one for the NappiWitten Lie algebra $H_4$. With the help of Cartan free $H_4$ modules we classify all Cartanfree modules of rank one over affine Nappi Witten Lie algebra. We also give a necessary and sufficient condition for these modules to be irreducible. Finally as an application we classify Cartan free modules of rank one for affineVirasoro NappiWitten Lie algebras.
 [122] arXiv:2406.13566 [pdf, html, other]

Title: Parametric finite element approximation of twophase NavierStokes flow with viscoelasticitySubjects: Numerical Analysis (math.NA)
In this work, we present a parametric finite element approximation of twophase NavierStokes flow with viscoelasticity. The free boundary problem is given by the viscoelastic NavierStokes equations in the two fluid phases, connected by jump conditions across the interface. The elasticity in the fluids is characterised using the OldroydB model with possible stress diffusion. The model was originally introduced to approximate fluidstructure interaction problems between an incompressible Newtonian fluid and a hyperelastic neoHookean solid, which are possible limit cases of the model. We approximate a variational formulation of the model with an unfitted finite element method that uses piecewise linear parametric finite elements. The twophase NavierStokesOldroydB system in the bulk regions is discretised in a way that guarantees unconditional solvability and stability for the coupled bulkinterface system. Good volume conservation properties for the two phases are observed in the case where the pressure approximation space is enriched with the help of an XFEM function. We show the applicability of our method with some numerical results.
 [123] arXiv:2406.13567 [pdf, html, other]

Title: Galerkin Neural NetworkPOD for Acoustic and Electromagnetic Wave Propagation in Parametric DomainsSubjects: Numerical Analysis (math.NA)
We investigate reducedorder models for acoustic and electromagnetic wave problems in parametrically defined domains. The parametertosolution maps are approximated following the socalled Galerkin PODNN method, which combines the construction of a reduced basis via proper orthogonal decomposition (POD) with neural networks (NNs). As opposed to the standard reduced basis method, this approach allows for the swift and efficient evaluation of reducedorder solutions for any given parametric input.
As is customary in the analysis of problems in random or parametrically defined domains, we start by transporting the formulation to a reference domain. This yields a parameterdependent variational problem set on parameterindependent functional spaces. In particular, we consider affineparametric domain transformations characterized by a highdimensional, possibly countably infinite, parametric input. To keep the number of evaluations of the highfidelity solutions manageable, we propose using lowdiscrepancy sequences to sample the parameter space efficiently. Then, we train an NN to learn the coefficients in the reduced representation. This approach completely decouples the offline and online stages of the reduced basis paradigm.
Numerical results for the threedimensional Helmholtz and Maxwell equations confirm the method's accuracy up to a certain barrier and show significant gains in online speedup compared to the traditional Galerkin POD method.  [124] arXiv:2406.13580 [pdf, html, other]

Title: Wilf's question in numerical semigroups $S_3$ revisited and inequalities for higher generaComments: 9 pages, 1 FigureSubjects: Commutative Algebra (math.AC)
We consider numerical semigroups $S_3 = \langle d_1,d_2,d_3 \rangle$, minimally generated by three positive integers. We revisit the Wilf question in $S_3$ and, making use of identities for degrees of syzygies of such semigroups, give a short proof of existence of an affirmative answer. We find also the lower bound for Frobenius numbers of $S_3$ and upper and lower bounds for higher genera.
 [125] arXiv:2406.13581 [pdf, html, other]

Title: Optimal constants in concentration inequalities on the sphere and in the Gauss spaceComments: 29 pages, 6 figuresSubjects: Probability (math.PR); Functional Analysis (math.FA)
We show several variants of concentration inequalities on the sphere stated as subgaussian estimates with optimal constants. For a Lipschitz function, we give onesided and twosided bounds for deviation from the median as well as from the mean. For example, we show that if $\mu$ is the normalized surface measure on $S^{n1}$ with $n\geq 3$, $f : S^{n1} \to \mathbb{R}$ is $1$Lipschitz, $M$ is the median of $f$, and $t >0$, then $\mu\big(f \geq M +t\big) \leq \frac 12 e^{nt^2/2}$. If $M$ is the mean of $f$, we have a twosided bound $\mu\big(f  M \geq t\big) \leq e^{nt^2/2}$. Consequently, if $\gamma$ is the standard Gaussian measure on $\mathbb{R}^n$ and $f : \mathbb{R}^{n} \to \mathbb{R}$ (again, $1$Lipschitz, with the mean equal to $M$), then $\gamma \big(f  M \geq t\big) \leq e^{t^2/2}$. These bounds are slightly better and arguably more elegant than those available elsewhere in the literature.
 [126] arXiv:2406.13582 [pdf, html, other]

Title: Endomorphism rings of simple modules and block decompositionComments: 7 pagesSubjects: Rings and Algebras (math.RA)
A left and right noetherian semiperfect ring R is known to be indecomposable if and only if its factor by the second power of Jacobson radical is. This characterisation is used to study simple Rmodules in terms of their Ext groups. It is shown that if R is indecomposable, all its simple modules are either finite or have the same infinite cardinality and their endomorphism rings have the same characteristics. The results are further strengthened in the case when R is quasiFrobenius.
 [127] arXiv:2406.13592 [pdf, html, other]

Title: The Alexander and Markov theorems for strongly involutive linksComments: 60 pages, 61 figures, 1 table. Comments are welcome!Subjects: Geometric Topology (math.GT)
The Alexander theorem (1923) and the Markov theorem (1936) are two classical results in knot theory that show respectively that every link is the closure of a braid and that braids that have the same closure are related by a finite number of operations called Markov moves. This paper presents specialized versions of these two classical theorems for a class of links in S3 preserved by an involution, that we call strongly involutive links. When connected, these links are known as strongly invertible knots, and have been extensively studied. We develop an equivariant closure map that, given two palindromic braids, produces a strongly involutive link. We demonstrate that this map is surjective up to equivalence of strongly involutive links. Furthermore, we establish that pairs of palindromic braids that have the same equivariant closure are related by an equivariant version of the original Markov moves.
 [128] arXiv:2406.13595 [pdf, html, other]

Title: Topological representations for framevalued domains via $L$sobrietyGuojun Wu (1,2), Wei Yao (1,2), Qingguo Li (3) ((1) School of Mathematics and Statistics, Nanjing University of Information Science and Technology,(2) Applied Mathematics Center of Jiangsu Province, Nanjing University of Information Science and Technology,(3) School of Mathematics, Hunan University)Subjects: General Topology (math.GN)
With a frame $L$ as the truth value table, we study the topological representations for framevalued domains. We introduce the notions of locally supercompact $L$topological space and strong locally supercompact $L$topological space. Using these concepts, continuous $L$dcpos and algebraic $L$dcpos are successfully represented via $L$sobriety. By means of Scott $L$topology and specialization $L$order, we establish a categorical isomorphism between the category of the continuous (resp., algebraic) $L$dcpos with Scott continuous maps and that of the locally supercompact (resp., strong locally supercompact) $L$sober spaces with continuous maps. As an application, for a continuous $L$poset $P$, we obtain a categorical isomorphism between the category of directed completions of $P$ with Scott continuous maps and that of the $L$sobrifications of $(P, \sigma_{L}(P))$ with continuous maps.
 [129] arXiv:2406.13601 [pdf, html, other]

Title: Selfnormalized Sums in Free Probability TheoryComments: 33 pagesSubjects: Probability (math.PR); Operator Algebras (math.OA)
We show that the distribution of selfnormalized sums of free selfadjoint random variables converges weakly to Wigner's semicircle law under appropriate conditions and estimate the rate of convergence in terms of the Kolmogorov distance. In the case of free identically distributed selfadjoint bounded random variables, we retrieve the standard rate of order $n^{1/2}$ up to a logarithmic factor, whereas we obtain a rate of order $n^{1/4}$ in the corresponding unbounded setting. These results provide free versions of certain selfnormalized limit theorems in classical probability theory.
 [130] arXiv:2406.13608 [pdf, html, other]

Title: Wiretapped Commitment over Binary ChannelsComments: 13 Pages, 1 figureSubjects: Information Theory (cs.IT); Cryptography and Security (cs.CR)
We propose the problem of wiretapped commitment, where two parties, say committer Alice and receiver Bob, engage in a commitment protocol using a noisy channel as a resource, in the presence of an eavesdropper, say Eve. Noisy versions of Alice's transmission over the wiretap channel are received at both Bob and Eve. We seek to determine the maximum commitment throughput in the presence of an eavesdropper, i.e., wiretapped commitment capacity, where in addition to the standard security requirements for twoparty commitment, one seeks to ensure that Eve doesn't learn about the commit string.
A key interest in this work is to explore the effect of collusion (or lack of it) between the eavesdropper Eve and either Alice or Bob. Toward the same, we present results on the wiretapped commitment capacity under the socalled 1private regime (when Alice or Bob cannot collude with Eve) and the 2private regime (when Alice or Bob may possibly collude with Eve).  [131] arXiv:2406.13612 [pdf, html, other]

Title: On Computation of Approximate Solutions to LargeScale Backstepping Kernel Equations via Continuum ApproximationComments: 13 pages, 5 figures, submitted to Systems & Control LettersSubjects: Optimization and Control (math.OC); Systems and Control (eess.SY)
We provide two methods for computation of continuum backstepping kernels that arise in control of continua (ensembles) of linear hyperbolic PDEs and which can approximate backstepping kernels arising in control of a largescale, PDE system counterpart (with computational complexity that does not grow with the number of state components of the largescale system). In the first method, we identify a class of systems for which the solution to the continuum (and hence, also an approximate solution to the respective largescale) kernel equations can be constructed in closed form. In the second method, we provide explicit formulae for the solution to the continuum kernels PDEs, employing a (triple) power series representation of the continuum kernel and establishing its convergence properties. In this case, we also provide means for reducing computational complexity by properly truncating the power series (in the powers of the ensemble variable). We also present numerical examples to illustrate computational efficiency/accuracy of the approaches, as well as to validate the stabilization properties of the approximate control kernels, constructed based on the continuum.
 [132] arXiv:2406.13628 [pdf, html, other]

Title: Stability of extremal domains for the first eigenvalue of the Laplacian operatorComments: 21 pages. Comments welcome!Subjects: Differential Geometry (math.DG); Analysis of PDEs (math.AP)
In this paper, we compute the second variation of the first Dirichlet eigenvalue on extremal domains and establish criteria for stability. We classify the stable extremal domains in the 2sphere and in higherdimensional spheres when the boundary is minimal. We also prove topological bounds for stable domains with nonnegative total Gaussian curvature in a general compact Riemannian surface.
 [133] arXiv:2406.13630 [pdf, other]

Title: AGZTLectures on formal multiple zeta valuesComments: Comments are welcomeSubjects: Number Theory (math.NT)
Formal multiple zeta values allow to study multiple zeta values by algebraic methods in a way that the open question about their transcendence is circumvented. In this note we show that Hoffman's basis conjecture for formal multiple zeta values is implied by the free odd generation conjecture for the double shuffle Lie algebra. We use the concept of a postLie structure for a convenient approach to the multiplication on the double shuffle group. From this, we get a coaction on the algebra of formal multiple zeta values. This in turn allows us to follow the proof of Brown's celebrated and unconditional theorem for the same result in the context of motivic multiple zeta values. We need the free odd generation conjecture twice: at first it gives a formula for the graded dimensions and secondly it is a key to derive a lift of the Zagier formula to the formal context.
 [134] arXiv:2406.13644 [pdf, html, other]

Title: Kinetic Monte Carlo methods for threedimensional diffusive capture problems in exterior domainsComments: 32 pages, 10 figuresSubjects: Numerical Analysis (math.NA); Analysis of PDEs (math.AP); Biological Physics (physics.bioph); Quantitative Methods (qbio.QM)
Cellular scale decision making is modulated by the dynamics of signalling molecules and their diffusive trajectories from a source to small absorbing sites on the cellular surface. Diffusive capture problems are computationally challenging due to the complex geometry and the applied boundary conditions together with intrinsically long transients that occur before a particle is captured. This paper reports on a particlebased Kinetic Monte Carlo (KMC) method that provides rapid accurate simulation of arrival statistics for (i) a halfspace bounded by a surface with a finite collection of absorbing traps and (ii) the domain exterior to a convex cell again with absorbing traps. We validate our method by replicating classical results and in addition, newly developed boundary homogenization theories and matched asymptotic expansions on capture rates. In the case of nonspherical domains, we describe a new shielding effect in which geometry can play a role in sharpening cellular estimates on the directionality of diffusive sources.
 [135] arXiv:2406.13647 [pdf, html, other]

Title: Inner automorphisms as 2cellsJournalref: Theory and Applications of Categories, Vol. 42, No. 2, 2024, pp. 1940Subjects: Category Theory (math.CT)
Abstract inner automorphisms can be used to promote any category into a 2category, and we study twodimensional limits and colimits in the resulting 2categories. Existing connected colimits and limits in the starting category become twodimensional colimits and limits under fairly general conditions. Under the same conditions, colimits in the underlying category can be used to build many notable twodimensional colimits such as coequifiers and coinserters. In contrast, disconnected colimits or genuinely 2categorical limits such as inserters and equifiers and cotensors cannot exist unless no nontrivial abstract inner automorphisms exist and the resulting 2category is locally discrete. We also study briefly when an ordinary functor can be extended to a 2functor between the resulting 2categories.
 [136] arXiv:2406.13658 [pdf, html, other]

Title: Generalized Hamming weights and symbolic powers of StanleyReisner ideals of matroidsComments: 37 pages. Comments welcome!Subjects: Commutative Algebra (math.AC)
It is wellknown that the first generalized Hamming weight of a code, more commonly called \textit{the minimum distance} of the code, corresponds to the initial degree of the StanleyReisner ideal of the matroid of the dual code. Our starting point in this paper is a generalization of this fact  namely, the $r$th generalized Hamming weight of a code is the smallest degree of a squarefree monomial in the $r$th symbolic power of the StanleyReisner ideal of the matroid of the dual code (in the appropriate range for $r$). It turns out that the squarefree monomials in successive symbolic powers of the StanleyReisner ideal of a matroid suffice to describe all symbolic powers of the StanleyReisner ideal. This implies that generalized Hamming weights  which can be defined in a natural way for matroids  are fundamentally tied to the structure of symbolic powers of StanleyReisner ideals of matroids. We illustrate this by studying initial degree statistics of symbolic powers of the StanleyReisner ideal of a matroid in terms of generalized Hamming weights and working out many examples that are meaningful from a codingtheoretic perspective. Our results also apply to projective varieties known as matroid configurations introduced by GeramitaHarbourneMiglioreNagel.
 [137] arXiv:2406.13666 [pdf, html, other]

Title: Markov partitions for nontransitive expansive flowsComments: 9 pages, 1 figureSubjects: Dynamical Systems (math.DS)
In this note we construct Markov partitions for nontransitive expansive flows in dimension 3.
 [138] arXiv:2406.13670 [pdf, html, other]

Title: On squarefree powers of simplicial treesComments: Any comments are welcomeSubjects: Commutative Algebra (math.AC)
In this article, we study the squarefree powers of facet ideals associated with simplicial trees. Specifically, we examine the linearity of their minimal free resolution and their regularity. Additionally, we investigate when the first syzygy module of squarefree powers of a simplicial tree is linearly related. Finally, we provide a combinatorial formula for the regularity of the squarefree powers of $t$path ideals of path graphs.
 [139] arXiv:2406.13675 [pdf, html, other]

Title: AIAssisted Dynamic Port and Waveform Switching for Enhancing UL Coverage in 5G NRAlejandro VillenaRodríguez, Gerardo Gómez, Mari Carmen AguayoTorres, Francisco J. MartínVega, José OutesCarnero, F. Yak NgMolina, Juan RamiroMorenoComments: 5 pages, 3 figures, 4 tablesSubjects: Information Theory (cs.IT)
The uplink of 5G networks allows selecting the transmit waveform between cyclic prefix orthogonal frequency division multiplexing (CPOFDM) and discrete Fourier transform spread OFDM (DFTSOFDM), which is appealing for celledge users using highfrequency bands, since it shows a smaller peaktoaverage power ratio, and allows a higher transmit power. Nevertheless, DFTSOFDM exhibits a higher block error rate (BLER) which complicates an optimal waveform selection. In this paper, we propose an intelligent waveformswitching mechanism based on deep reinforcement learning (DRL). In this proposal, a learning agent aims at maximizing a function built using available throughput percentiles in real networks. Said percentiles are weighted so as to improve the celledge users' service without dramatically reducing the cell average. Aggregated measurements of signaltonoise ratio (SNR) and timing advance (TA), available in real networks, are used in the procedure. Results show that our proposed scheme greatly outperforms both metrics compared to classical approaches.
 [140] arXiv:2406.13682 [pdf, html, other]

Title: A variational perspective on the dissipative Hamiltonian structure of the VlasovFokkerPlanck equationComments: 51 pagesSubjects: Analysis of PDEs (math.AP); Metric Geometry (math.MG); Probability (math.PR)
The VlasovFokkerPlanck equation describes the evolution of the probability density of the position and velocity of particles under the influence of external confinement, interaction, friction, and stochastic force. It is wellknown that this equation can be formally seen as a dissipative Hamiltonian system in the Wasserstein space of probability measures. In order to better understand this geometric formalism, we introduce a timediscrete variational scheme, solutions of which converge to the solution of the VlasovFokkerPlanck equation as timestep vanishes. The implicit scheme is based on the symplectic Euler scheme, and updates the probability density at each iteration first in the velocity variable then in the position variable.
The algorithm leverages the geometric structure of the Wasserstein space, and has several desirable properties. Energy functionals involved in each variational problem are geodesicallyconvex, which implies the unique solvability of the problem. Furthermore, the correct dissipation of the Hamiltonian is observed at the discrete level up to higher order errors.  [141] arXiv:2406.13684 [pdf, html, other]

Title: Some nonArchimedean pluripotential theory on polarized affine conesComments: 26 pagesSubjects: Algebraic Geometry (math.AG); Complex Variables (math.CV)
We undertake a preliminary step towards studying nonArchimedean pluripotential theory on polarized affine cones over a trivially valued field. We study plurisubharmonic functions and the MongeAmpère operator defined on the finite energy class, partially generalizing a result of BoucksomJonsson on projective varieties.
 [142] arXiv:2406.13687 [pdf, html, other]

Title: Diffraction of the primes and other sets of zero densityComments: 33 pagesSubjects: Functional Analysis (math.FA); Number Theory (math.NT)
In this paper, we show that the diffraction of the primes is absolutely continuous, showing no bright spots (Bragg peaks). We introduce the notion of counting diffraction, extending the classical notion of (density) diffraction to sets of density zero. We develop the counting diffraction theory and give many examples of sets of zero density of all possible spectral types.
 [143] arXiv:2406.13696 [pdf, html, other]

Title: A nonlocal approximation of the area in codimension twoComments: 46 pagesSubjects: Differential Geometry (math.DG); Analysis of PDEs (math.AP); Optimization and Control (math.OC)
For $s\in (0,1)$ we introduce a notion of fractional $s$mass on $(n2)$dimensional closed, orientable surfaces in $\R^n$. Moreover, we prove its $\Gamma$convergence, with respect to the flat topology, and pointwise convergence to the $(n2)$dimensional area.
 [144] arXiv:2406.13699 [pdf, html, other]

Title: Classification results, rigidity theorems and semilinear PDEs on Riemannian manifolds: a Pfunction approachSubjects: Analysis of PDEs (math.AP)
We consider solutions to some semilinear elliptic equations on complete noncompact Riemannian manifolds and study their classification as well as the effect of their presence on the underlying manifold. When the Ricci curvature is nonnegative, we prove both the classification of positive solutions to the critical equation and the rigidity for the ambient manifold. The same results are obtained when we consider solutions to the Liouville equation on Riemannian surfaces. The results are obtained via a suitable Pfunction whose constancy implies the classification of both the solutions and the underlying manifold. The analysis carried out on the Pfunction also makes it possible to classify nonnegative solutions for subcritical equations on manifolds enjoying a Sobolev inequality and satisfying an integrability condition on the negative part of the Ricci curvature. Some of our results are new even in the Euclidean case.
 [145] arXiv:2406.13722 [pdf, html, other]

Title: Channel Charting in RealWorld Coordinates with Distributed MIMOComments: Submitted to a journal. arXiv admin note: substantial text overlap with arXiv:2308.14498Subjects: Information Theory (cs.IT); Signal Processing (eess.SP)
Channel charting is an emerging selfsupervised method that maps channelstate information (CSI) to a lowdimensional latent space (the channel chart) that represents pseudopositions of user equipments (UEs). While channel charts preserve local geometry, i.e., nearby UEs are nearby in the channel chart (and vice versa), the pseudopositions are in arbitrary coordinates and global geometry is typically not preserved. In order to embed channel charts in realworld coordinates, we first propose a bilateration loss for distributed multipleinput multipleoutput (DMIMO) wireless systems in which only the access point (AP) positions are known. The idea behind this loss is to compare the received power at pairs of APs to determine whether a UE should be placed closer to one AP or the other in the channel chart. Second, we propose a lineofsight (LoS) boundingbox loss that places the UE in a predefined LoS area of each AP that is estimated to have a LoS path to the UE. We demonstrate the efficacy of combining both of these loss functions with neuralnetworkbased channel charting using raytracingbased and measurementbased channel vectors. Our approach outperforms several baselines and maintains the selfsupervised nature of channel charting as it does not rely on geometrical propagation models or require groundtruth UE position information.
 [146] arXiv:2406.13723 [pdf, html, other]

Title: Distortion in groups of generalized piecewiselinear transformationsComments: 18 pages, 6 figuresSubjects: Group Theory (math.GR); Dynamical Systems (math.DS)
For each natural number $n$, we consider the subgroup $\mathcal{R}_n$ of Homeo$_+[0,1]$ made by the elements that are linear except for a subset whose CantorBendixson rank is less than or equal to $n$. These groups of generalized piecewiselinear transformations yield an ascending chain of groups as we increase $n$. We study how the notion of distorted element changes along this chain. Our main result establishes that for each natural number $n$, there exits an element that is undistorted of $\mathcal{R}_n$ yet distorted in $\mathcal{R}_{n+1}$. Actually, such an element is explicitly constructed.
 [147] arXiv:2406.13726 [pdf, html, other]

Title: Global Solutions to Master Equations for Continuous Time Heterogeneous Agent Macroeconomic ModelsSubjects: Optimization and Control (math.OC); Machine Learning (cs.LG); General Economics (econ.GN)
We propose and compare new global solution algorithms for continuous time heterogeneous agent economies with aggregate shocks. First, we approximate the agent distribution so that equilibrium in the economy can be characterized by a high, but finite, dimensional nonlinear partial differential equation. We consider different approximations: discretizing the number of agents, discretizing the agent state variables, and projecting the distribution onto a finite set of basis functions. Second, we represent the value function using a neural network and train it to solve the differential equation using deep learning tools. We refer to the solution as an Economic Model Informed Neural Network (EMINN). The main advantage of this technique is that it allows us to find global solutions to high dimensional, nonlinear problems. We demonstrate our algorithm by solving important models in the macroeconomics and spatial literatures (e.g. Krusell and Smith (1998), Khan and Thomas (2007), Bilal (2023)).
 [148] arXiv:2406.13728 [pdf, html, other]

Title: A Combinatorial Perspective on the Noncommutative Symmetric FunctionsComments: 34 pages, 4 figuresSubjects: Combinatorics (math.CO)
The noncommutative symmetric functions $\textbf{NSym}$ were first defined abstractly by Gelfand et al. in 1995 as the free associative algebra generated by noncommuting indeterminants $\{\boldsymbol{e}_n\}_{n\in \mathbb{N}}$ that were taken as a noncommutative analogue of the elementary symmetric functions. The resulting space was thus a variation on the traditional symmetric functions $\Lambda$. Giving noncommutative analogues of generating function relations for other bases of $\Lambda$ allowed Gelfand et al. to define additional bases of $\textbf{NSym}$ and then determine changeofbasis formulas using quasideterminants. In this paper, we aim for a selfcontained exposition that expresses these bases concretely as functions in infinitely many noncommuting variables and avoids quasideterminants. Additionally, we look at the noncommutative analogues of two different interpretations of changeofbasis in $\Lambda$: both as a product of a minimal number of matrices, mimicking Macdonald's exposition of $\Lambda$ in Symmetric Functions and Hall Polynomials, and as statistics on brick tabloids, as in work by Eğecioğlu and Remmel, 1990.
 [149] arXiv:2406.13729 [pdf, html, other]

Title: On the Saito number of plane curvesSubjects: Algebraic Geometry (math.AG); Complex Variables (math.CV)
In this work we study the \emph{Saito number} of a plane curve and we present a method to determine the minimal Saito number for plane curves in a given equisingularity class, that gives rise to an actual algorithm. In particular situations, we also provide various formulas for this number. In addition, if $\nu_0$ and $\nu_1$ are two coprime positive integers and $N>0$ then we show that for any $1\leq k\leq \left [\frac{N\nu_0}{2}\right ]$ there exits a plane curve equisingular to the curve $$y^{N\nu_0}x^{N\nu_1}=0$$ such that its Saito number is precisely $k$.
 [150] arXiv:2406.13730 [pdf, html, other]

Title: Prescribed exponential stabilization of a onelayer neural network with delayed feedback: Insights in seizure prevention and neural controlSubjects: Spectral Theory (math.SP); Dynamical Systems (math.DS); Optimization and Control (math.OC); Neurons and Cognition (qbio.NC)
This paper provides controloriented delaybased modelling of a onelayer neural network of Hopfieldtype subject to an external input designed as delayed feedback. The specificity of such a model is that it makes the considered neuron less susceptible to seizure caused by its inherent dynamic instability. This modelling exploits a recently set partial pole placement for linear functional differential equations, which relies on the coexistence of real spectral values, allowing the explicit prescription of the closedloop solution's exponential decay. The proposed framework improves some pioneering and scarce results from the literature on the characterization of the exact solution's exponential decay when a simple real spectral value exists. Indeed, it improves neural stability when the inherent dynamic is stable and provides insights into the design of a onelayer neural network that can be stabilized exponentially with delayed feedback and with a prescribed decay rate regardless of whether the inherent neuron dynamic is stable or unstable.
 [151] arXiv:2406.13759 [pdf, html, other]

Title: The Structure of Symbolic Powers of MatroidsSubjects: Commutative Algebra (math.AC)
We describe the structure of the symbolic powers $I^{(\ell)}$ of the StanleyReisner ideals, and cover ideals, $I$, of matroids. We (a) prove a structure theorem describing a minimal generating set for every $I^{(\ell)}$; (b) describe the (nonstandard graded) symbolic Rees algebra $\mathcal{R}_s(I)$ of $I$ and show its minimal algebra generators have degree at most ht $I$; (c) provide an explicit, simple formula to compute the largest degree of a minimal algebra generator of $\mathcal{R}_s(I)$; (d) provide algebraic applications, including formulas for the symbolic defects of $I$, the initial degree of $I^{(\ell)}$, and the Waldschmidt constant of $I$; (e) provide a new algorithm allowing fast computations of very large symbolic powers of $I$.
One of the byproducts is a new characterization of matroids in terms of minimal generators of $I^{(\ell)}$ for some $\ell\geq 2$. In particular, it yields a new, simple characterization of matroids in terms of the minimal generators of $I^{(2)}$. This is the first characterization of matroids in terms of $I^{(2)}$, and it complements a celebrated theorem by MinhTrung, Varbaro, and TeraiTrung which requires the investigation of homological properties of $I^{(\ell)}$ for some $\ell\geq 3$.  [152] arXiv:2406.13761 [pdf, html, other]

Title: Exponential time differencing for matrixvalued dynamical systemsSubjects: Numerical Analysis (math.NA); Machine Learning (cs.LG); Optimization and Control (math.OC)
Matrix evolution equations occur in many applications, such as dynamical Lyapunov/Sylvester systems or Riccati equations in optimization and stochastic control, machine learning or data assimilation. In many cases, their tightest stability condition is coming from a linear term. Exponential time differencing (ETD) is known to produce highly stable numerical schemes by treating the linear term in an exact fashion. In particular, for stiff problems, ETD methods are a method of choice. We propose an extension of the class of ETD algorithms to matrixvalued dynamical equations. This allows us to produce highly efficient and stable integration schemes. We show their efficiency and applicability for a variety of realworld problems, from geophysical applications to dynamical problems in machine learning.
 [153] arXiv:2406.13771 [pdf, html, other]

Title: Notes on the Cheeger and Colding version of the Reifenberg theorem for metric spacesSubjects: Metric Geometry (math.MG); Differential Geometry (math.DG)
The classical Reifenberg's theorem says that a set which is sufficiently well approximated by planes uniformly at all scales is a topological Hölder manifold. Remarkably, this generalizes to metric spaces, where the approximation by planes is replaced by the GromovHausdorff distance. This fact was shown by Cheeger and Colding in an appendix of one of their celebrated works on Ricci limit spaces [8]. Given the recent interest around this statement in the growing field of analysis in metric spaces, in this note we provide a self contained and detailed proof of the Cheeger and Colding result. Our presentation substantially expands the arguments in [8] and makes explicit all the relevant estimates and constructions. As a byproduct we also shows a biLipschitz version of this result which, even if folklore among experts, was not present in the literature. This work is an extract from the doctoral dissertation of the second author.
 [154] arXiv:2406.13772 [pdf, html, other]

Title: A new look at subrepresentation formulasSubjects: Classical Analysis and ODEs (math.CA)
We extend the subrepresentation formula $$ f(x)\le c_n\,I_1(\nabla f)(x) $$ in several ways. First, we consider more general $A_1$potential operators on the righthand side and prove local and global pointwise inequalities for these operators. Second, we show that we can improve the righthand side using fractional derivatives. Finally, we extend our results to rough singular integral operators, similar to the main result in [HMP1].
 [155] arXiv:2406.13773 [pdf, other]

Title: A rigorous approach to pattern formation for isotropic isoperimetric problems with competing nonlocal interactionsSubjects: Analysis of PDEs (math.AP); Mathematical Physics (mathph)
We introduce a rigorous approach to the study of the symmetry breaking and pattern formation phenomenon for isotropic functionals with local/nonlocal interactions in competition.
We consider a general class of nonlocal variational problems in dimension $d\geq 2$, in which an isotropic surface term favouring pure phases competes with an isotropic nonlocal term with power law kernel favouring alternation between different phases.
Close to the critical regime in which the two terms are of the same order, we give a rigorous proof of the conjectured structure of global minimizers, in the shape of domains with flat boundary (e.g., stripes or lamellae).
The natural framework in which our approach is set and developed is the one of calculus of variations and geometric measure theory.
Among others, we detect a nonlocal curvaturetype quantity which is controlled by the energy functional and whose finiteness implies flatness for sufficiently regular boundaries.
The power of decay of the considered kernels at infinity is $p\geq d+3$ and it is related to pattern formation in synthetic antiferromagnets. The decay $p=d+3$ is optimal to get the flatness of regular boundaries of finite energy in the critical regime.  [156] arXiv:2406.13774 [pdf, html, other]

Title: Chessboard and level sets of continuous functionsComments: 19 pagesSubjects: General Topology (math.GN); Combinatorics (math.CO)
We show the following result: Let $f \colon I^n \to \mathbb{R}^{n1}$ be a continuous function. Then, there exist $p \in \mathbb{R}^{n1}$ and compact subset $S \subset f^{1}\left[\left\{p\right\}\right]$ which connects some opposite faces of the $n$dimensional unit cube $I^n$. We give an example that shows it cannot be generalized to pathconnected sets. We also provide a discrete version of this result which is inspired by the $n$dimensional Steinhaus Chessboard Theorem. Additionally, we show that the latter one and the Brouwer Fixed Point Theorem are simple consequences of the main result.
 [157] arXiv:2406.13780 [pdf, html, other]

Title: On the maximum $F$free induced subgraphs in $K_t$free graphsComments: 14 pagesSubjects: Combinatorics (math.CO)
For graphs $F$ and $H$, let $f_{F,H}(n)$ be the minimum possible size of a maximum $F$free induced subgraph in an $n$vertex $H$free graph. This notion generalizes the Ramsey function and the ErdősRogers function. Establishing a container lemma for the $F$free subgraphs, we give a general upper bound on $f_{F,H}(n)$, assuming the existence of certain locally dense $H$free graphs. In particular, we prove that for every graph $F$ with $\mathrm{ex}(m,F) = O(m^{1+\alpha})$, where $\alpha \in [0,1/2)$, we have
\[
f_{F, K_3}(n) = O\left(n^{\frac{1}{2\alpha}}\left(\log n\right)^{\frac{3}{2
\alpha}}\right)
\quad
\textrm{and}
\quad
f_{F, K_4}(n) = O\left(n^{\frac{1}{32\alpha}}\left(\log n\right)^{\frac{6}{32\alpha}}\right).
\] For the cases where $F$ is a complete multipartite graph, letting $s = \sum_{i=1}^r s_i$, we prove that
\[
f_{K_{s_1,\ldots,s_r}, K_{r+2}}(n) = O \left( n^{\frac{2s 3}{4s 5}} (\log n)^{3} \right).
\]
We also make an observation which improves the bounds of $\mathrm{ex}(G(n,p),C_4)$ by a polylogarithmic factor.  [158] arXiv:2406.13790 [pdf, html, other]

Title: The extended reverse ultra logconcavity of transposed BorosMoll sequencesComments: 15 pagesSubjects: Combinatorics (math.CO)
The BorosMoll sequences $\{d_\ell(m)\}_{\ell=0}^m$ arise in the study of evaluation of a quartic integral. After the infinite logconcavity conjecture of the sequence $\{d_\ell(m)\}_{\ell=0}^m$ was proposed by Boros and Moll, a lot of interesting inequalities on $d_\ell(m)$ were obtained, although the conjecture is still open. Since $d_\ell(m)$ has two parameters, it is natural to consider the properties for the sequences $\{d_\ell(m)\}_{m\ge \ell}$, which are called the \emph{transposed BorosMoll sequences} here. In this paper, we mainly prove the extended reverse ultra logconcavity of the transposed BorosMoll sequences $\{d_\ell(m)\}_{m\ge \ell}$, and hence give an upper bound for the ratio ${d_\ell^2(m)}/{(d_\ell(m1)d_\ell(m+1))}$. A lower bound for this ratio is also established which implies a result stronger than the logconcavity of the sequences $\{d_\ell(m)\}_{m\ge \ell}$. As a consequence, we also show that the transposed BorosMoll sequences possess a stronger logconcave property than the BorosMoll sequences do. At last, we propose some conjectures on the BorosMoll sequences and their transposes.
 [159] arXiv:2406.13798 [pdf, html, other]

Title: Aubin Property and Strong Regularity Are Equivalent for Nonlinear SecondOrder Cone ProgrammingSubjects: Optimization and Control (math.OC)
This paper solves a fundamental open problem in variational analysis on the equivalence between the Aubin property and the strong regularity for nonlinear secondorder cone programming (SOCP) at a locally optimal solution. We achieve this by introducing a reduction approach to the Aubin property characterized by the Mordukhovich criterion and a lemma of alternative choices on cones to replace the Slemma used in Outrata and Ramírez [SIAM J. Optim. 21 (2011) 789823] and Opazo, Outrata, and Ramírez [SIAM J. Optim. 27 (2017) 21412151], where the same SOCP was considered under the strict complementarity condition except for possibly only one block of constraints. As a byproduct, we also offer a new approach to the wellknown result of Dontchev and Rockafellar [SIAM J. Optim. 6 (1996) 10871105] on the equivalence of the two concepts in conventional nonlinear programming.
 [160] arXiv:2406.13802 [pdf, html, other]

Title: Sextactic and type9 points on the Fermat cubic and associated objectsComments: 9 pagesSubjects: Algebraic Geometry (math.AG); Commutative Algebra (math.AC)
In this note we study line and conic arrangements associated to sextactic and type 9 points on the Fermat cubic $F$ and we provide explicit coordinates for each of the 72 type 9 points on $F$.
 [161] arXiv:2406.13834 [pdf, html, other]

Title: Optimizing Wireless Discontinuous Reception via MAC Signaling LearningComments: 9 pages, 10 figures, submitted to IEEE TMLCNSubjects: Information Theory (cs.IT); Machine Learning (cs.LG)
We present a Reinforcement Learning (RL) approach to the problem of controlling the Discontinuous Reception (DRX) policy from a Base Transceiver Station (BTS) in a cellular network. We do so by means of optimally timing the transmission of fast Layer2 signaling messages (a.k.a. Medium Access Layer (MAC) Control Elements (CEs) as specified in 5G New Radio). Unlike more conventional approaches to DRX optimization, which rely on finetuning the values of DRX timers, we assess the gains that can be obtained solely by means of this MAC CE signalling. For the simulation part, we concentrate on traffic types typically encountered in Extended Reality (XR) applications, where the need for battery drain minimization and overheating mitigation are particularly pressing. Both 3GPP 5G New Radio (5G NR) compliant and noncompliant ("beyond 5G") MAC CEs are considered. Our simulation results show that our proposed technique strikes an improved tradeoff between latency and energy savings as compared to conventional timerbased approaches that are characteristic of most current implementations. Specifically, our RLbased policy can nearly halve the active time for a single User Equipment (UE) with respect to a naïve MAC CE transmission policy, and still achieve near 20% active time reduction for 9 simultaneously served UEs.
 [162] arXiv:2406.13841 [pdf, html, other]

Title: KacMoody algebras in Deligne's CategorySubjects: Representation Theory (math.RT); Category Theory (math.CT); Quantum Algebra (math.QA)
We generalize the notion of a KacMoody Lie algebra to the setting of Deligne Categories. Then we derive the KacWeyl formula for the category $\mathcal{O}$ integrable representations for such an algebra. This paper generalizes results of A. Pakharev \cite{AP}.
 [163] arXiv:2406.13848 [pdf, html, other]

Title: Answers to questions about medial layer graphs of selfdual regular and chiral polytopesJournalref: In: Ars Mathematica Contemporanea (Dec. 2023)Subjects: Combinatorics (math.CO)
An abstract $n$polytope $\mathcal{P}$ is a partiallyordered set which captures important properties of a geometric polytope, for any dimension $n$. For even $n \ge 2$, the incidences between elements in the middle two layers of the Hasse diagram of $\mathcal{P}$ give rise to the medial layer graph of $\mathcal{P}$, denoted by $\mathcal{G} = \mathcal{G}(\mathcal{P})$. If $n=4$, and $\mathcal{P}$ is both highly symmetric and selfdual of type $\{p,q,p\}$, then a Cayley graph $\mathcal{C}$ covering $\mathcal{G}$ can be constructed on a group of polarities of $\mathcal{P}$. In this paper we address some open questions about the relationship between $\mathcal{G}$ and $\mathcal{C}$ that were raised in a 2008 paper by Monson and Weiss, and describe some interesting examples of these graphs. In particular, we give the first known examples of improperly selfdual chiral polytopes of type $\{3,q,3\}$, which are also among the very few known examples of highly symmetric selfdual finite polytopes that do not admit a polarity. Also we show that if $p=3$ then $\mathcal{C}$ cannot have a higher degree of $s$arctransitivity than $\mathcal{G}$, and we present a family of regular $4$polytopes of type $\{6,q,6\}$ for which the vertexstabilisers in the automorphism group of $\mathcal{C}$ are larger than those for $\mathcal{G}$.
 [164] arXiv:2406.13861 [pdf, html, other]

Title: Skew circuits and circumference in a binary matroidSubjects: Combinatorics (math.CO)
Let C_1 and C_2 be skew circuits in a binary matroid having circumference c. For any positive integer k there is a constant a_k such that if min { A ; C_1 \subset A \subset EA} > a_k, then C_1 + C_2 < 2c k.
 [165] arXiv:2406.13866 [pdf, html, other]

Title: Finite group actions on dg categories and Hochschild homologySubjects: KTheory and Homology (math.KT); Algebraic Geometry (math.AG); Category Theory (math.CT)
We prove a decomposition of the Hochschild homology groups of the equivariant dg category $\mathscr{C}^G$ associated to a small dg category $\mathscr{C}$ with direct sums on which a finite group $G$ acts. When the ground field is $\mathbb{C}$ this decomposition is related to a categorical action of $\text{Rep}(G)$ on $\mathscr{C}^G$ and the resulting action of the representation ring $R_\mathbb{C}(G)$ on $HH_\bullet(\mathscr{C}^G)$.
 [166] arXiv:2406.13867 [pdf, html, other]

Title: ErrorCorrecting Graph CodesComments: 27 pages, 3 figures, 1 tableSubjects: Information Theory (cs.IT); Discrete Mathematics (cs.DM)
In this paper, we define, study, and construct {\em ErrorCorrecting Graph Codes}. An errorcorrecting graph code of distance $\delta$ is a family $C$ of graphs, on a common vertex set of size $n$, such that if we start with any graph in $C$, we would have to modify the neighborhoods of at least $\delta n$ vertices in order to reach some other graph in $C$.
This is a natural graph generalization of the standard Hamming distance errorcorrecting codes for binary strings. We show:
1. Combinatorial results determining the optimal rate vs distance tradeoff nonconstructively.
2. A connection to rankmetric codes, enabling some simple and some involved constructions achieving certain positive rates and distances.
3. Graph code analogues of ReedSolomon codes and code concatenation, leading to positive distance codes for all rates and positive rate codes for all distances.
4. Graph code analogues of dualBCH codes, yielding large codes with distance $\delta = 1o(1)$. This gives an explicit "graph code of Ramsey graphs".
Several recent works, starting with the paper of Alon, Gujgiczer, Körner, Milojević, and Simonyi, have studied more general graph codes; where the symmetric difference between any two graphs in the code is required to have a desired property. Errorcorrecting graph codes are a particularly interesting instantiation of this concept.  [167] arXiv:2406.13872 [pdf, html, other]

Title: Least SQuares Discretizations (LSQD): a robust and versatile meshless paradigm for solving elliptic PDEsSubjects: Numerical Analysis (math.NA)
Searching for numerical methods that combine facility and efficiency, while remaining accurate and versatile, is critical. Often, irregular geometries challenge traditional methods that rely on structured or bodyfitted meshes. Meshless methods mitigate these issues but oftentimes require the weak formulation which involves defining quadrature rules over potentially intricate geometries. To overcome these challenges, we propose the Least Squares Discretization (LSQD) method. This novel approach simplifies the application of meshless methods by eliminating the need for a weak formulation and necessitates minimal numerical analysis. It offers significant advantages in terms of ease of implementation and adaptability to complex geometries. In this paper, we demonstrate the efficacy of the LSQD method in solving elliptic partial differential equations for a variety of boundary conditions, geometries, and data layouts. We monitor hP convergence across these parameters and construct an a posteriori builtin error estimator to establish our method as a robust and accessible numerical alternative.
 [168] arXiv:2406.13874 [pdf, html, other]

Title: Structure theorems for braided Hopf algebrasComments: Comments welcomeSubjects: Quantum Algebra (math.QA); Algebraic Topology (math.AT)
We develop versions of the PoincaréBirkhoffWitt and CartierMilnorMoore theorems in the setting of braided Hopf algebras. To do so, we introduce new analogues of a Lie algebra in the setting of a braided monoidal category, using the notion of a braided operad.
 [169] arXiv:2406.13888 [pdf, html, other]

Title: Open Problem: Anytime Convergence Rate of Gradient DescentComments: COLT 2024 open problem; 5 pagesSubjects: Optimization and Control (math.OC); Machine Learning (cs.LG)
Recent results show that vanilla gradient descent can be accelerated for smooth convex objectives, merely by changing the stepsize sequence. We show that this can lead to surprisingly large errors indefinitely, and therefore ask: Is there any stepsize schedule for gradient descent that accelerates the classic $\mathcal{O}(1/T)$ convergence rate, at \emph{any} stopping time $T$?
 [170] arXiv:2406.13902 [pdf, html, other]

Title: Signed combinatorial interpretations in algebraic combinatoricsComments: 23 pagesSubjects: Combinatorics (math.CO); Discrete Mathematics (cs.DM)
We prove the existence of signed combinatorial interpretations for several large families of structure constants. These families include standard bases of symmetric and quasisymmetric polynomials, as well as various bases in Schubert theory. The results are stated in the language of computational complexity, while the proofs are based on the effective Möbius inversion.
 [171] arXiv:2406.13917 [pdf, html, other]

Title: Analytic Besov functions, preSchwarzian derivatives, and integrable Teichm\"uller spacesSubjects: Complex Variables (math.CV)
We consider the embedding of integrable Teichmüller spaces $T_p$ into analytic Besov spaces by using preSchwarzian derivatives. Unlike the case of the Bers embedding by Schwarzian derivatives, there is a big difference between the cases $p>1$ and $p=1$. In this paper, we focus on the case $p=1$ and generalize the previous results obtained for $p>1$. This gives a unified approach to the complex analytic theory of integrable Teichmüller spaces $T_p$ for all $p \geq 1$.
 [172] arXiv:2406.13922 [pdf, html, other]

Title: Explicit Performance Bound of Finite Blocklength Coded MIMO: TimeDomain versus Spatiotemporal Channel CodingComments: 9 pages, 5 figuresSubjects: Information Theory (cs.IT)
In the sixth generation (6G), ultrareliable lowlatency communications (URLLC) will further develop to achieve TKu extreme connectivity, and multipleinput multipleoutput (MIMO) is expected to be a key enabler for its realization. Since the latency constraint can be represented by the blocklength of a codeword, it is essential to analyze different coded MIMO schemes under finite blocklength regime. In this paper, we analyze the statistical characteristics of information density of timedomain coding and spatiotemporal coding MIMO, compute the channel capacity and dispersion, and present new explicit performance bounds of finite blocklength coded MIMO for different coding modes via normal approximation. As revealed by the analysis and simulation, spatiotemporal coding can effectively mitigate the performance loss induced by short blocklength by increasing the spatial degree of freedom (DoF). However, for timedomain coding, each spatial link is encoded independently, and the performance loss will be more severe with short blocklength under any spatial DoF. These results indicate that spatiotemporal coding can optimally exploit the spatial dimension advantages of MIMO systems compared with timedomain coding, and it has the potential to support URLLC transmission, enabling very low errorrate communication under stringent blocklength constraint.
 [173] arXiv:2406.13927 [pdf, html, other]

Title: Fully Nonlinear Elliptic Equations With Periodic DataSubjects: Analysis of PDEs (math.AP)
In this paper, we study solutions $u$ of fully nonlinear elliptic equations of the form $F(D^2u)=f$ in $\mathbb{R}^n$, where $f$ is periodic. We establish the existence and Liouville type results for entire quadratic polynomial growth solutions, that is, the solution is a quadratic polynomial plus a periodic function. As a consequence, we consider applications to $k$Hessian equations.
 [174] arXiv:2406.13931 [pdf, html, other]

Title: Dissipativeness of the hyperbolic quadrature method of moments for kinetic equationsComments: 23 pagesSubjects: Numerical Analysis (math.NA)
This paper presents a dissipativeness analysis of a quadrature method of moments (called HyQMOM) for the onedimensional BGK equation. The method has exhibited its good performance in numerous applications. However, its mathematical foundation has not been clarified. Here we present an analytical proof of the strict hyperbolicity of the HyQMOMinduced moment closure systems by introducing a polynomialbased closure technique. As a byproduct, a class of numerical schemes for the HyQMOM system is shown to be realizability preserving under CFLtype conditions. We also show that the system preserves the dissipative properties of the kinetic equation by verifying a certain structural stability condition. The proof uses a newly introduced affine invariance and the homogeneity of the HyQMOM and heavily relies on the theory of orthogonal polynomials associated with realizable moments, in particular, the moments of the standard normal distribution.
 [175] arXiv:2406.13943 [pdf, html, other]

Title: New QEC codes and EAQEC codes from repeatedroot cyclic codes of length $2^rp^s$Subjects: Information Theory (cs.IT)
Let $p$ be an odd prime and $r,s,m$ be positive integers. In this study, we initiate our exploration by delving into the intricate structure of all repeatedroot cyclic codes and their duals with a length of $2^rp^s$ over the finite field $\mathbb{F}_{p^m}$. Through the utilization of CSS and Steane's constructions, a series of new quantum errorcorrecting (QEC) codes are constructed with parameters distinct from all previous constructions. Furthermore, we provide all maximum distance separable (MDS) cyclic codes of length $2^rp^s$, which are further utilized in the construction of QEC MDS codes. Finally, we introduce a significant number of novel entanglementassisted quantum errorcorrecting (EAQEC) codes derived from these repeatedroot cyclic codes. Notably, these newly constructed codes exhibit parameters distinct from those of previously known constructions.
 [176] arXiv:2406.13944 [pdf, html, other]

Title: Generalization error of minnorm interpolators in transfer learningComments: 53 pages, 2 figuresSubjects: Statistics Theory (math.ST); Machine Learning (cs.LG); Methodology (stat.ME); Machine Learning (stat.ML)
This paper establishes the generalization error of pooled min$\ell_2$norm interpolation in transfer learning where data from diverse distributions are available. Minnorm interpolators emerge naturally as implicit regularized limits of modern machine learning algorithms. Previous work characterized their outofdistribution risk when samples from the test distribution are unavailable during training. However, in many applications, a limited amount of test data may be available during training, yet properties of minnorm interpolation in this setting are not wellunderstood. We address this gap by characterizing the bias and variance of pooled min$\ell_2$norm interpolation under covariate and model shifts. The pooled interpolator captures both early fusion and a form of intermediate fusion. Our results have several implications: under model shift, for low signaltonoise ratio (SNR), adding data always hurts. For higher SNR, transfer learning helps as long as the shifttosignal (SSR) ratio lies below a threshold that we characterize explicitly. By consistently estimating these ratios, we provide a datadriven method to determine: (i) when the pooled interpolator outperforms the targetbased interpolator, and (ii) the optimal number of target samples that minimizes the generalization error. Under covariate shift, if the source sample size is small relative to the dimension, heterogeneity between between domains improves the risk, and vice versa. We establish a novel anisotropic local law to achieve these characterizations, which may be of independent interest in random matrix theory. We supplement our theoretical characterizations with comprehensive simulations that demonstrate the finitesample efficacy of our results.
 [177] arXiv:2406.13952 [pdf, html, other]

Title: A general Liouvilletype theorem for the 3D steadystate MagneticB\'enard systemComments: 15 pagesSubjects: Analysis of PDEs (math.AP)
We establish a Liouvilletype theorem for the elliptic and incompressible MagneticBénard system defined over the entire threedimensional space. Specifically, we demonstrate the uniqueness of trivial solutions under the condition that they belong to certain local Morrey spaces. Our results generalize in two key directions: firstly, the MagneticBénard system encompasses other significant coupled systems for which the Liouville problem has not been previously studied, including the Boussinesq system, the MHDBoussinesq system, and the Bénard system. Secondly, by employing local Morrey spaces, our theorem applies to Lebesgue spaces, Lorentz spaces, Morrey spaces, and certain weightedLebesgue spaces.
 [178] arXiv:2406.13955 [pdf, html, other]

Title: A note on the threshold numbers of cyclesSubjects: Combinatorics (math.CO)
A graph $G=(V,E)$ is said to be a \textit{$k$threshold graph} with \textit{thresholds} $\theta_1<\theta_2<...<\theta_k$ if there is a map $r: V \longrightarrow \mathbb{R}$ such that $uv\in E$ if and only if $\theta_i\le r(u)+r(v)$ holds for an odd number of $i\in [k]$. The \textit{threshold number} of $G$, denoted by $\Theta(G)$, is the smallest positive integer $k$ such that $G$ is a $k$threshold graph. In this paper, we determine the exact threshold numbers of cycles by proving
\[ \Theta(C_n)=\begin{cases}
1 & if\ n=3,
2 & if\ n=4,
4 & if\ n\ge 5,
\end{cases}
\]
where $C_n$ is the cycle with $n$ vertices.  [179] arXiv:2406.13973 [pdf, html, other]

Title: The Unipotent Tropical Fundamental GroupComments: 48 pagesSubjects: Algebraic Geometry (math.AG); Combinatorics (math.CO)
We define the unipotent tropical fundamental group of a polyhedral complex in $\mathbb{R}^n$ as the Tannakian fundamental group of the category of unipotent tropical vector bundles with integrable connection. We show that it is computable in that it satisfies a SeifertVan Kampen theorem and has a description for fans in terms of a bar complex. We then review an analogous classical object, the unipotent de Rham fundamental group of a schön subvariety of a toric variety. Our main result is a correspondence theorem between classical and tropical unipotent fundamental groups: there is an isomorphism between the unipotent completion of the fundamental group of a generic fiber of a tropically smooth family over a disc and the tropical unipotent fundamental group of the family's tropicalization. This theorem is established using KatoNakayama spaces and a descent argument. It requires a slight enlargement of the relevant categories, making use of enriched structures and partial compactifications.
 [180] arXiv:2406.13974 [pdf, html, other]

Title: A Combinatorial Decomposition of Knapsack ConesComments: 22 pagesSubjects: Combinatorics (math.CO)
In this paper, we focus on knapsack cones, a specific type of simplicial cones that arise naturally in the context of the knapsack problem $x_1 a_1 + \cdots + x_n a_n = a_0$. We present a novel combinatorial decomposition for these cones, named \texttt{DecDenu}, which aligns with Barvinok's unimodular cone decomposition within the broader framework of Algebraic Combinatorics. Computer experiments support us to conjecture that our \texttt{DecDenu} algorithm is polynomial when the number of variables $n$ is fixed. If true, \texttt{DecDenu} will provide the first alternative polynomial algorithm for Barvinok's unimodular cone decomposition, at least for denumerant cones.
The \texttt{CTEuclid} algorithm is designed for MacMahon's partition analysis, and is notable for being the first algorithm to solve the counting problem for Magic squares of order 6. We have enhanced the \texttt{CTEuclid} algorithm by incorporating \texttt{DecDenu}, resulting in the \texttt{LLLCTEuclid} algorithm. This enhanced algorithm makes significant use of LLL's algorithm and stands out as an effective eliminationbased approach.  [181] arXiv:2406.13976 [pdf, html, other]

Title: Euler factors of equivariant $L$functions of Drinfeld modules and beyondSubjects: Number Theory (math.NT)
In \cite{FGHP}, the first author and his collaborators proved an equivariant Tamagawa number formula for the special value at $s=0$ of a Gosstype $L$function, equivariant with respect to a Galois group $G$, and associated to a Drinfeld module defined on $\Bbb F_q[t]$ and over a finite, integral extension of $\Bbb F_q[t]$. The formula in question was proved provided that the values at $0$ of the Euler factors of the equivariant $L$function in question satisfy certain identities involving Fitting ideals of certain $G$cohomologically trivial, finite $\Bbb F_q[t][G]$modules associated to the Drinfeld module. In \cite{FGHP}, we prove these identities in the particular case of the Carlitz module. In this paper, we develop general techniques and prove the identities in question for arbitrary Drinfeld modules. Further, we indicate how these techniques can be extended to the more general case of higher dimensional abelian $t$modules, which is relevant in the context of the proof of the equivariant Tamagawa number formula for abelian $t$modules given by N. Green and the first author in \cite{GreenPopescu}. This paper is based on a lecture given by the first author at ICMAT Madrid in May 2023 and builds upon results obtained by the second author in his PhD thesis \cite{Ramachandranthesis}.
 [182] arXiv:2406.13980 [pdf, html, other]

Title: On the complexity of matrix Putinar's PositivstellensatzComments: This manuscript has been submitted to a suitable journal for possible publication on May 7Subjects: Optimization and Control (math.OC)
This paper studies the complexity of matrix Putinar's Positivstellensatz on the semialgebraic set that is given by the polynomial matrix inequality. Under the archimedeanness, we prove a polynomial bound on degrees of terms appearing in the representation of matrix Putinar's Positivstellensatz. Estimates on the exponent and constant are given. As a byproduct, a polynomial bound on the convergence rate of matrix sumofsquares relaxations is obtained, which resolves an open question raised by Dinh and Pham. When the constraining set is unbounded (the archimedeanness fails), we also prove a similar bound for the matrix version of PutinarVasilescu's Positivstellensatz by exploiting homogenization techniques.
 [183] arXiv:2406.13994 [pdf, html, other]

Title: Stability analysis for a kinetic bacterial chemotaxis modelComments: 48 pagesSubjects: Analysis of PDEs (math.AP)
We perform stability analysis of a kinetic bacterial chemotaxis model of bacterial selforganization, assuming that bacteria respond sharply to chemical signals. The resulting discontinuous tumbling kernel represents the key challenge for the stability analysis as it rules out a direct linearization of the nonlinear terms. To address this challenge we fruitfully separate the evolution of the shape of the cellular profile from its global motion. We provide a full nonlinear stability theorem in a perturbative setting when chemical degradation can be neglected. With chemical degradation we prove stability of the linearized operator. In both cases we obtain exponential relaxation to equilibrium with an explicit rate using hypocoercivity techniques. To apply a hypocoercivity approach in this setting, we develop two novel and specific approaches: i) the use of the $H^1$ norm instead of the $L^2$ norm, and ii) the treatment of nonlinear terms. This work represents an important step forward in bacterial chemotaxis modeling from a kinetic perspective as most results are currently only available for the macroscopic descriptions, which are usually parabolic in nature. Significant difficulty arises due to the lack of regularization of the kinetic transport operator as compared to the parabolic operator in the macroscopic scaling limit.
 [184] arXiv:2406.13998 [pdf, html, other]

Title: Transversal Hamilton paths and cyclesComments: 33 pages, 10 figuresSubjects: Combinatorics (math.CO)
Given a collection $\mathcal{G} =\{G_1,G_2,\dots,G_m\}$ of graphs on the common vertex set $V$ of size $n$, an $m$edge graph $H$ on the same vertex set $V$ is transversal in $\mathcal{G}$ if there exists a bijection $\varphi :E(H)\rightarrow [m]$ such that $e \in E(G_{\varphi(e)})$ for all $e\in E(H)$. Denote $\delta(\mathcal{G}):=\operatorname*{min}\left\{\delta(G_i): i\in [m]\right\}$. In this paper, we first establish a minimum degree condition for the existence of transversal Hamilton paths in $\mathcal{G}$: if $n=m+1$ and $\delta(\mathcal{G})\geq \frac{n1}{2}$, then $\mathcal{G}$ contains a transversal Hamilton path. This solves a problem proposed by [Li, Li and Li, J. Graph Theory, 2023]. As a continuation of the transversal version of Dirac's theorem [Joos and Kim, Bull. Lond. Math. Soc., 2020] and the stability result for transversal Hamilton cycles [Cheng and Staden, arXiv:2403.09913v1], our second result characterizes all graph collections with minimum degree at least $\frac{n}{2}1$ and without transversal Hamilton cycles. We obtain an analogous result for transversal Hamilton paths. The proof is a combination of the stability result for transversal Hamilton paths or cycles, transversal blowup lemma, along with some structural analysis.
 [185] arXiv:2406.14000 [pdf, html, other]

Title: Robust nonlinear statefeedback control of secondorder systemsComments: 6 pages, 6 figuresSubjects: Optimization and Control (math.OC); Systems and Control (eess.SY)
This note proposes a novel nonlinear state feedback controller for perturbed secondorder systems. In analogy to a linear proportionalderivative (PD) output feedback control, the proposed nonlinear scheme uses the output state of interest and its time derivative for a robust finitetime regulation. The control has only one free design parameter, and the closedloop system is shown to be uniformly asymptotically stable in the presence of matched disturbances. We derive a strict Lyapunov function for the closed control loop with a bounded exogenous perturbation, and use it for both the control tuning and analysis of the finitetime convergence. Apart from the numerical results, a revealing experimental example is also shown in favor of the proposed control and in comparison with PD and suboptimal nonlinear damping regulators.
 [186] arXiv:2406.14007 [pdf, html, other]

Title: On canonical metrics of complex surfaces with split tangent and related geometric PDEsComments: 41 pagesSubjects: Differential Geometry (math.DG)
In this paper, we study biHermitian metrics on complex surfaces with split holomorphic tangent bundle and construct 2 types of metric cones. We introduce a new type of fully nonlinear geometric PDE on such surfaces and establish smooth solutions. As a geometric application, we solve the prescribed Bismut Ricci problem. In various settings, we obtain canonical metrics on 2 important classes of complex surfaces: primary Hopf surfaces and Inoue surfaces of type $\mathcal{S}_{M}$.
 [187] arXiv:2406.14010 [pdf, html, other]

Title: Average edge order of normal $3$pseudomanifoldsComments: 10 pagesSubjects: Combinatorics (math.CO); Geometric Topology (math.GT)
In their work [9], Feng Luo and Richard Stong introduced the concept of the average edge order, denoted as $\mu_0(K)$. They demonstrated that if $\mu_0(K)\leq \frac{9}{2}$ for a closed $3$manifold $K$, then $K$ must be a sphere. Building upon this foundation, Makoto Tamura extended similar results to $3$manifolds with nonempty boundaries in [10, 11]. In our present study, we extend these findings to normal $3$pseudomanifolds. Specifically, we establish that for a normal $3$pseudomanifold $K$ with singularities, $\mu_0(K)\geq\frac{30}{7}$. Moreover, equality holds if and only if $K$ is a onevertex suspension of $\mathbb{RP}^2$ with seven vertices. Furthermore, we establish that when $\frac{30}{7}\leq\mu_0(K)\leq\frac{9}{2}$, the $3$pseudomanifold $K$ can be derived from some boundary complexes of $4$simplices by a sequence of possible operations, including connected sums, bistellar $1$moves, edge contractions, edge expansions, vertex folding, and edge folding. In the end, we discuss some normal $3$pseudomanifolds exhibiting higher average edge orders.
 [188] arXiv:2406.14011 [pdf, html, other]

Title: PrimalDual Strategy (PDS) for Composite Optimization Over Directed graphsSubjects: Optimization and Control (math.OC); Signal Processing (eess.SP)
We investigate the distributed multiagent sharing optimization problem in a directed graph, with a composite objective function consisting of a smooth function plus a convex (possibly nonsmooth) function shared by all agents. While adhering to the network connectivity structure, the goal is to minimize the sum of smooth local functions plus a nonsmooth function. The proposed PrimalDual algorithm (PD) is similar to a previous algorithm \cite{b27}, but it has additional benefits. To begin, we investigate the problem in directed graphs, where agents can only communicate in one direction and the combination matrix is not symmetric. Furthermore, the combination matrix is changing over time, and the condition coefficient weights are produced using an adaptive approach. The strong convexity assumption, adaptive coefficient weights, and a new upper bound on stepsizes are used to demonstrate that linear convergence is possible. New upper bounds on stepsizes are derived under the strong convexity assumption and adaptive coefficient weights that are timevarying in the presence of both smooth and nonsmooth terms. Simulation results show the efficacy of the proposed algorithm compared to some other algorithms.
 [189] arXiv:2406.14031 [pdf, html, other]

Title: Model structure arising from one hereditary cotorsion pair on extriangulated categoriesComments: 23 pagesSubjects: Representation Theory (math.RT); Category Theory (math.CT)
Let $\mathcal{C}$ be a weakly idempotent complete extriangulated category. In contrast with the Hovey correspondence of admissible model structures on weakly idempotent complete exact categories from two complete cotorsion pairs, we give a construction of model structures on $\mathcal{C}$ from only one complete cotorsion pair. Our main result not only generalizes the work by BeligiannisReiten and CuiLuZhang, but also provides methods to construct model structures from silting objects of $\mathcal{C}$ and co$t$structures in triangulated categories.
 [190] arXiv:2406.14032 [pdf, html, other]

Title: Can the quadratrix truly square the circle?Comments: 18 pages, 7 figuresSubjects: Number Theory (math.NT); Metric Geometry (math.MG); Rings and Algebras (math.RA)
The quadratrix received its name from the circle quadrature, squaring the circle, but it only solves it if completed by taking a limit, as pointed out already in antiquity. We ask if it can square the circle without limits and restrict its use accordingly, to converting ratios of angles and segments into each other. The problem is then translated into algebra by analogy to straightedge and compass constructions, and leads to an open question in transcendental number theory. In particular, Lindemann's impossibility result no longer suffices, and the answer depends on whether $\pi$ belongs to the analog of Ritt's exponentiallogarithmic field with an algebraic base. We then derive that it does not from the wellknown Schanuel conjecture. Thus, the quadratrix so restricted cannot square the circle after all.
 [191] arXiv:2406.14041 [pdf, other]

Title: Discontinuous Galerkin method for a threedimensional coupled fluidporoelastic model with applications to brain fluid mechanicsComments: 22 pages, 8 figuresSubjects: Numerical Analysis (math.NA)
The modeling of the interaction between a poroelastic medium and a fluid in a hollow cavity is crucial for understanding, e.g., the multiphysics flow of blood and Cerebrospinal Fluid (CSF) in the brain, the supply of blood by the coronary arteries in heart perfusion, or the interaction between groundwater and rivers or lakes. In particular, the cerebral tissue's elasticity and its perfusion by blood and interstitial CSF can be described by Multicompartment Poroelasticity (MPE), while CSF flow in the brain ventricles can be modeled by the (Navier)Stokes equations, the overall system resulting in a coupled MPE(Navier)Stokes system. The aim of this paper is threefold. First, we aim to extend and verify in a threedimensional setting a discontinuous Galerkin method on polytopal grids recently presented for the MPEStokes problem. Second, we carry out the analysis of the method based on an extension of the proposed formulation so that physicsbased BeaversJosephSaffman conditions are taken into account at the interface: these conditions are essential to model the friction between the fluid and the porous medium. Finally, by a comparative numerical investigation, we assess the fluiddynamics effects of these boundary conditions and of employing either Stokes or NavierStokes equations to model the CSF flow. The semidiscrete numerical scheme for the coupled problem is proved to be stable and optimally convergent. Temporal discretization is obtained using Newmark's $\beta$method for the elastic wave equation and the $\theta$method for the remaining equations of the model. The theoretical error estimates are verified by numerical simulations on a test case with a manufactured solution, and a numerical investigation is carried out on a threedimensional geometry to assess the effects of interface conditions and fluid inertia on the system.
 [192] arXiv:2406.14053 [pdf, html, other]

Title: Generalization of Lyapunov Center Theorem for Hamiltonian systems via normal forms theoryComments: 22 pagesSubjects: Classical Analysis and ODEs (math.CA)
In this article we formulate and prove sufficient conditions for the existence of trajectories of nonstationary periodic solutions of autonomous Hamiltonian systems in a neighbourhood of equilibria. It is worth pointing out that assumptions of some wellknown theorems imply that of our main results. We obtain our results with the use of the theory of normal forms for Hamiltonian matrices and global bifurcation theory for autonomous Hamiltonian systems.
 [193] arXiv:2406.14055 [pdf, other]

Title: The Dynamics of OneDimensional QuasiAffine MapsComments: 15 pages, 4 figuresSubjects: Dynamical Systems (math.DS)
We study the dynamics of the onedimensional quasiaffine map $x\mapsto \left\lfloor \lambda x +\mu \right\rfloor$, providing a complete description of the map's periodic points, and of the limit points of every $x\in\mathbb{R}$ under the map, for all real parameter values. Specifically, we establish the existence of regions of parameter values for which the map possesses $n$ fixed points for all $n\in\mathbb{N}_0\cup \{\infty\}$, an explicit formula for the number of 2cycles possessed by the map, and the $\omega$limit set of any $x\in\mathbb{R}$ under the map, which, depending on the parameter values, is either a singleton of a fixed point, a 2cycle, $\{\infty,\infty\}$, $\{\infty\}$, or $\{\infty\}$.
 [194] arXiv:2406.14058 [pdf, html, other]

Title: Is Peirce's reduction thesis gerrymandered?Comments: 24 pages, 8 figuresJournalref: Transactions of the Charles S. Peirce Society, 58 (2022) no.4, 271300Subjects: Logic (math.LO)
We argue that traditional formulations of the reduction thesis that tie it to privileged relational operations do not suffice for Peirce's justification of the categories, and invite the charge of gerrymandering to make it come out as true. We then develop a more robust invariant formulation of the thesis by explicating the use of triads in any relational operations, which is immune to that charge. The explication also allows us to track how Thirdness enters the structure of higher order relations, and even propose a numerical measure of it. Our analysis reveals new conceptual phenomena when negation or disjunction are used to compound relations.
 [195] arXiv:2406.14060 [pdf, html, other]

Title: Distributed EventTriggered Bandit Convex Optimization with TimeVarying ConstraintsComments: 34 pages, 4 figures. arXiv admin note: text overlap with arXiv:2311.01957Subjects: Optimization and Control (math.OC)
This paper considers the distributed bandit convex optimization problem with timevarying inequality constraints over a network of agents, where the goal is to minimize network regret and cumulative constraint violation. Existing distributed online algorithms require that each agent broadcasts its decision to its neighbors at each iteration. To better utilize the limited communication resources, we propose a distributed eventtriggered online primaldual algorithm with twopoint bandit feedback. Under several classes of appropriately chosen decreasing parameter sequences and nonincreasing eventtriggered threshold sequences, we establish dynamic network regret and network cumulative constraint violation bounds. These bounds are comparable to the results achieved by distributed eventtriggered online algorithms with fullinformation feedback. Finally, a numerical example is provided to verify the theoretical results.
 [196] arXiv:2406.14063 [pdf, html, other]

Title: Global counterexamples to uniqueness for a Calder\'on problem with $C^k$ conductivitiesSubjects: Analysis of PDEs (math.AP); Mathematical Physics (mathph); Spectral Theory (math.SP)
Let $\Omega \subset R^n$, $n \geq 3$, be a fixed smooth bounded domain, and let $\gamma$ be a smooth conductivity in $\overline{\Omega}$. Consider a nonzero frequency $\lambda_0$ which does not belong to the Dirichlet spectrum of $L_\gamma = {\rm div} (\gamma \nabla \cdot)$. Then, for all $k \geq 1$, there exists an infinite number of pairs of nonisometric $C^k$ conductivities $(\gamma_1, \gamma_2)$ on $\overline{\Omega}$, which are close to $\gamma$ such that the associated DN maps at frequency $\lambda_0$ satisfy \begin{equation*}
\Lambda_{\gamma_1,\lambda_0} = \Lambda_{\gamma_2,\lambda_0}. \end{equation*}  [197] arXiv:2406.14064 [pdf, html, other]

Title: PAPR Reduction with Prechirp Selection for Affine Frequency Division MultipleSubjects: Information Theory (cs.IT); Signal Processing (eess.SP)
Affine frequency division multiplexing (AFDM) is a promising new multicarrier technique based on discrete affine Fourier transform (DAFT). By properly tuning prechirp parameter and postchirp parameter in the DAFT, the effective channel in the DAFT domain can completely avoid overlap of different paths, thus constitutes a full representation of delayDoppler profile, which significantly improves the system performance in high mobility scenarios. However, AFDM has the crucial problem of high peaktoaverage power ratio (PAPR) caused by phase randomness of modulated symbols. In this letter, an algorithm named grouped prechirp selection (GPS) is proposed to reduce the PAPR by changing the value of prechirp parameter on subcarriers group by group. Specifically, it is demonstrated first that the important properties of AFDM system are maintained when implementing GPS. Secondly, we elaborate the operation steps of GPS algorithm, illustrating its effect on PAPR reduction and its advantage in terms of computational complexity compared with the ungrouped approach. Finally, simulation results of PAPR reduction in the form of complementary cumulative distribution function (CCDF) show the effectiveness of the proposed GPS algorithm.
 [198] arXiv:2406.14065 [pdf, other]

Title: On onestep numerical schemes of weak convergence for SDEs with superlinear coefficientsSubjects: Numerical Analysis (math.NA); Probability (math.PR)
We consider weak convergence of onestep schemes for solving stochastic differential equations (SDEs) with onesided Lipschitz conditions. It is known that the superlinear coefficients may lead to a blowup of moments of solutions and their numerical solutions. When solutions to SDEs have all finite moments, weak convergence of numerical schemes has been investigated in [Wang et al (2023), Weak error analysis for strong approximation schemes of SDEs with superlinear coefficients, IMA Journal numerical analysis]. Some modified Euler schemes have been analyzed for weak convergence. In this work, we present a family of explicit schemes of first and secondorder weak convergence based on classical schemes for SDEs. We explore the effects of limited moments on these schemes. We provide a systematic but simple way to establish weak convergence orders for schemes based on approximations/modifications of drift and diffusion coefficients. We present several numerical examples of these schemes and show their weak convergence orders.
 [199] arXiv:2406.14074 [pdf, html, other]

Title: Strong existence and uniqueness of a calibrated local stochastic volatility modelSubjects: Probability (math.PR); Analysis of PDEs (math.AP); Mathematical Finance (qfin.MF)
We study a twodimensional McKeanVlasov stochastic differential equation, whose volatility coefficient depends on the conditional distribution of the second component with respect to the first component. We prove the strong existence and uniqueness of the solution, establishing the wellposedness of a twofactor local stochastic volatility (LSV) model calibrated to the market prices of European call options. In the spirit of [Jourdain and Zhou, 2020, Existence of a calibrated regime switching local volatility model.], we assume that the factor driving the volatility of the logprice takes finitely many values. Additionally, the propagation of chaos of the particle system is established, giving theoretical justification for the algorithm [Julien Guyon and HenryLabordère, 2012, Being particular about calibration.].
 [200] arXiv:2406.14076 [pdf, html, other]

Title: The limits of Kahler manifolds under holomorphic deformationsSubjects: Algebraic Geometry (math.AG)
With some mild assumptions on metric and topology of the central fiber, we prove that the limit of Kahler manifolds under holomorphic deformation is still Kahler.
 [201] arXiv:2406.14083 [pdf, html, other]

Title: Tight bounds for rainbow partial $F$tiling in edgecolored complete hypergraphsComments: 19 pages, 1 figues, comments are welcomeSubjects: Combinatorics (math.CO)
For an $r$graph $F$ and integers $n,t$ satisfying $t \le n/v(F)$, let $\mathrm{ar}(n,tF)$ denote the minimum integer $N$ such that every edgecoloring of $K_{n}^{r}$ using $N$ colors contains a rainbow copy of $tF$, where $tF$ is the $r$graphs consisting of $t$ vertexdisjoint copies of $F$. The case $t=1$ is the classical antiRamsey problem proposed by ErdősSimonovitsSós~\cite{ESS75}. When $F$ is a single edge, this becomes the rainbow matching problem introduced by Schiermeyer~\cite{Sch04} and ÖzkahyaYoung~\cite{OY13}. We conduct a systematic study of $\mathrm{ar}(n,tF)$ for the case where $t$ is much smaller than $\mathrm{ex}(n,F)/n^{r1}$. Our first main result provides a reduction of $\mathrm{ar}(n,tF)$ to $\mathrm{ar}(n,2F)$ when $F$ is bounded and smooth, two properties satisfied by most previously studied hypergraphs. Complementing the first result, the second main result, which utilizes gaps between Turán numbers, determines $\mathrm{ar}(n,tF)$ for relatively smaller $t$. Together, these two results determine $\mathrm{ar}(n,tF)$ for a large class of hypergraphs. Additionally, the latter result has the advantage of being applicable to hypergraphs with unknown Turán densities, such as the famous tetrahedron $K_{4}^{3}$.
 [202] arXiv:2406.14093 [pdf, html, other]

Title: Bridging bulk and surface: An interacting particle system towards the fieldroad diffusion modelSubjects: Analysis of PDEs (math.AP); Probability (math.PR)
We recover the socalled fieldroad diffusion model as the hydrodynamic limit of an interacting particle system. The former consists of two parabolic PDEs posed on two sets of different dimensions (a "field" and a "road" in a population dynamics context), and coupled through exchange terms between the field's boundary and the road. The latter stands as a Symmetric Simple Exclusion Process (SSEP): particles evolve on two microscopic lattices following a Markov jump process, with the constraint that each site cannot host more than one particle at the same time. The system is in contact with reservoirs that allow to create or remove particles at the boundary sites. The dynamics of these reservoirs are slowed down compared to the diffusive dynamics, to reach the reactions and the boundary conditions awaited at the macroscopic scale. This issue of bridging two spaces of different dimensions is, as far as we know, new in the hydrodynamic limit context, and raises perspectives towards future related works.
 [203] arXiv:2406.14094 [pdf, html, other]

Title: Logical reduction of relations: from relational databases to Peirce's reduction thesisComments: 31 pages, 4 figuresJournalref: Logic Journal of the IGPL, 2023Subjects: Logic (math.LO); Databases (cs.DB); Logic in Computer Science (cs.LO); Rings and Algebras (math.RA)
We study logical reduction (factorization) of relations into relations of lower arity by Boolean or relative products that come from applying conjunctions and existential quantifiers to predicates, i.e. by primitive positive formulas of predicate calculus. Our algebraic framework unifies natural joins and data dependencies of database theory and relational algebra of clone theory with the bond algebra of C.S. Peirce. We also offer new constructions of reductions, systematically study irreducible relations and reductions to them, and introduce a new characteristic of relations, ternarity, that measures their `complexity of relating' and allows to refine reduction results. In particular, we refine Peirce's controversial reduction thesis, and show that reducibility behavior is dramatically different on finite and infinite domains.
 [204] arXiv:2406.14104 [pdf, html, other]

Title: The Extreme Points of the Unit Ball of the James space $J$ and its dual spacesComments: 26 pagesSubjects: Functional Analysis (math.FA)
We provide a new proof of S. Bellenot's characterization of the extreme points of the unit ball $B_J$ of James quasireflexive space $J$. We also provide an explicit description of the norm of $J^{**}$ which yields an analogous characterization for the extreme points of $B_{J^{**}}$. In the last part of the paper we describe the set of all extreme points of $B_{J^*}$ and its norm closure. It is remarkable that the descriptions of the extreme points of $B_J$ and $B_{J^*}$ are closely connected.
 [205] arXiv:2406.14108 [pdf, html, other]

Title: Connected Vehicle Datadriven Robust Optimization for Traffic Signal Timing: Modeling Traffic Flow Variability and ErrorsComments: Accepted for podium session of the Conference in Emerging Technologies in Transportation Systems (TRC30)Subjects: Optimization and Control (math.OC)
Recent advancements in Connected Vehicle (CV) technology have prompted research on leveraging CV data for more effective traffic management. Despite the low penetration rate, such detailed CV data has demonstrated great potential in improving traffic signal performance. However, existing studies share a common shortcoming in that they all ignore traffic flow estimation errors in their modeling process, which is inevitable due to the sampling observation nature of CVs. This study proposes a CV datadriven robust optimization framework for traffic signal timing accounting for both traffic flow variability and estimation errors. First, we propose a general CV datadriven optimization model that can be widely applied to various signalized intersection scenarios including under/oversaturated and fixed/realtime. Then, we propose a novel datadriven uncertainty set of arrival rates based on the bounds information derived from CVs, which circumvents the errorprone arrival rate estimation process. Finally, a CV datadriven robust optimization model (CVRO) is formulated to explicitly handle arrival rate uncertainties. By means of the robust counterpart approach, this robust optimization problem can be equalized to a deterministic mixedinteger linear programming problem with an exact solution. The evaluation results highlight the superior performance of the CVRO model compared to the deterministic model and traditional methods across various scenarios: different penetration rates, traffic demands, and control types. Notably, the CVRO model demonstrates its excellence at lower CV penetration rates and in the presence of different traffic flow fluctuation levels, affirming its effectiveness and robustness.
 [206] arXiv:2406.14119 [pdf, html, other]

Title: Entropy stable hydrostatic reconstruction schemes for shallow water systemsComments: 28 pages, 10 figures, submitted to Journal of Computational PhysicsSubjects: Numerical Analysis (math.NA)
In this work, we develop a new hydrostatic reconstruction procedure to construct wellbalanced schemes for one and multilayer shallow water flows, including wetting and drying. Initially, we derive the method for a pathconservative finite volume scheme and combine it with entropy conservative fluxes and suitable numerical dissipation to preserve an entropy inequality in the semidiscrete case. We then combine the novel hydrostatic reconstruction with a collocated nodal splitform discontinuous Galerkin spectral element method, extending the method to highorder and curvilinear meshes. The highorder method incorporates an additional positivitylimiter and is blended with a compatible subcell finite volume method to maintain wellbalancedness at wet/dry fronts. We prove entropy stability, wellbalancedness, and positivitypreservation for both methods. Numerical results for the highorder method validate the theoretical findings and demonstrate the robustness of the scheme.
 [207] arXiv:2406.14125 [pdf, html, other]

Title: On Unique Error Patterns in the Levenshtein's Sequence Reconstruction ModelSubjects: Information Theory (cs.IT)
In the Levenshtein's sequence reconstruction problem a codeword is transmitted through $N$ channels and in each channel a set of errors is introduced to the transmitted word. In previous works, the restriction that each channel provides a unique output word has been essential. In this work, we assume only that each channel introduces a unique set of errors to the transmitted word and hence some output words can also be identical. As we will discuss, this interpretation is both natural and useful for deletion and insertion errors. We give properties, techniques and (optimal) results for this situation.
Quaternary alphabets are relevant due to applications related to DNAmemories. Hence, we introduce an efficient Las Vegas style decoding algorithm for simultaneous insertion, deletion and substitution errors in $q$ary Hamming spaces for $q \geq 4$.  [208] arXiv:2406.14138 [pdf, html, other]

Title: Classification of orientable torus bundles over closed orientable surfacesComments: 35 pages, 12 figuresSubjects: Geometric Topology (math.GT); Algebraic Topology (math.AT); Group Theory (math.GR)
Let $g$ be a nonnegative integer, $\Sigma _g$ a closed orientable surface of genus $g$, and $\mathcal{M}_g$ its mapping class group. We classify all the group homomorphisms $\pi _1(\Sigma _g)\to G$ up to the action of $\mathcal{M}_g$ on $\pi _1(\Sigma _g)$ in the following cases; (1) $G=PSL(2;\mathbb{Z})$, (2) $G=SL(2;\mathbb{Z})$. As an application of the case (2), we completely classify orientable $T^2$bundles over closed orientable surfaces up to bundle isomorphisms. In particular, we show that any orientable $T^2$bundle over $\Sigma _g$ with $g\geq 1$ is isomorphic to the fiber connected sum of $g$ pieces of $T^2$bundles over $T^2$. Moreover, the classification result in the case (1) can be generalized into the case where $G$ is the free product of finite number of finite cyclic groups. We also apply it to an extension problem of maps from a closed surface to the connected sum of lens spaces.
 [209] arXiv:2406.14140 [pdf, html, other]

Title: Nonparametric Jackknife Instrumental Variable Estimation and Confounding Robust Surrogate IndicesSubjects: Statistics Theory (math.ST)
Jackknife instrumental variable estimation (JIVE) is a classic method to leverage many weak instrumental variables (IVs) to estimate linear structural models, overcoming the bias of standard methods like twostage least squares. In this paper, we extend the jackknife approach to nonparametric IV (NPIV) models with many weak IVs. Since NPIV characterizes the structural regression as having residuals projected onto the IV being zero, existing approaches minimize an estimate of the average squared projected residuals, but their estimates are biased under many weak IVs. We introduce an IV splitting device inspired by JIVE to remove this bias, and by carefully studying this splitIV empirical process we establish learning rates that depend on generic complexity measures of the nonparametric hypothesis class. We then turn to leveraging this for semiparametric inference on average treatment effects (ATEs) on unobserved longterm outcomes predicted from shortterm surrogates, using historical experiments as IVs to learn this nonparametric predictive relationship even in the presence of confounding between short and longterm observations. Using splitIV estimates of a debiasing nuisance, we develop asymptotically normal estimates for predicted ATEs, enabling inference.
 [210] arXiv:2406.14143 [pdf, html, other]

Title: Using the Transport of Intensity and the Transport of Phase Equation for Phase RetrievalSubjects: Numerical Analysis (math.NA)
We investigate the transport of intensity equation (TIE) and the transport of phase equation (TPE) for solving the phase retrieval problem. Both the TIE and the TPE are derived from the paraxial Helmholtz equation and relate phase information to the intensity. The TIE is usually favored since the TPE is nonlinear. The main contribution of this paper is that we discuss situations in which it is possible to use the two equations in a hybrid manner: We show that 2dimensional phase information retrieved by the TIE can be used as initial data for the TPE, enabling the acquisition of 3dimensional phase information. The latter is solved using the method of characteristic and viscosity methods. Both the TIE and the viscosity method are numerically implemented with finite element methods.
 [211] arXiv:2406.14147 [pdf, html, other]

Title: A flexible polyhedron without selfintersections in Euclidean 3space, all of whose dihedral angles change during a flexComments: 21 pages, in Russian language, 6 figuresSubjects: Metric Geometry (math.MG)
We construct a spherehomeomorphic flexible selfintersection free polyhedron in Euclidean 3space such that all its dihedral angles change during some flex of this polyhedron. The constructed polyhedron has 26 vertices, 72 edges and 48 faces. To study its properties, we use symbolic computations in the Wolfram Mathematica software system.
 [212] arXiv:2406.14160 [pdf, html, other]

Title: Asymptotic bounds on the numbers of vertices of polytopes of polystochastic matricesComments: 8 pages, Section 2 is transferred from arXiv:2311.06905Subjects: Combinatorics (math.CO)
A multidimensional nonnegative matrix is called polystochastic if the sum of entries in each line is equal to $1$. The set of all polystochastic matrices of order $n$ and dimension $d$ is a convex polytope $\Omega_n^d$. In the present paper, we compare known bounds on the number of vertices of the polytope $\Omega_n^d$ and prove that the number of vertices of $\Omega_3^d$ is doubly exponential on $d$.
 [213] arXiv:2406.14168 [pdf, html, other]

Title: An Asymptotic Preserving and Energy Stable Scheme for the Euler System with Congestion ConstraintSubjects: Numerical Analysis (math.NA)
In this work, we design and analyze an asymptotic preserving (AP), semiimplicit finite volume scheme for the scaled compressible isentropic Euler system with a singular pressure law known as the congestion pressure law. The congestion pressure law imposes a maximal density constraint of the form $0\leq \varrho <1$, and the scaling introduces a small parameter $\varepsilon$ in order to control the stiffness of the density constraint. As $\varepsilon\to 0$, the solutions of the compressible system converge to solutions of the socalled freecongested Euler equations that couples compressible and incompressible dynamics. We show that the proposed scheme is positivity preserving and energy stable. In addition, we also show that the numerical densities satisfy a discrete variant of the constraint. By means of extensive numerical case studies, we verify the efficacy of the scheme and show that the scheme is able to capture the two dynamics in the limiting regime, thereby proving the AP property.
 [214] arXiv:2406.14175 [pdf, html, other]

Title: Disjointly strictly singular inclusions between variable Lebesgue spacesSubjects: Functional Analysis (math.FA)
Disjointly strictly singular inclusions between variable Lebesgue spaces $L^{p(\cdot)}(\mu)$ on finite measure are characterized. Suitable criteria in terms of the (bounded or unbounded) exponents are given. It is proved the equivalence of $L$weak compactness (also called almost compactness) and disjoint strict singularity for variable Lebesgue space inclusions. For infinite measure any inclusion $L^{p(\cdot)}(\mu) \hookrightarrow L^{q(\cdot)}(\mu)$ is not disjointly strictly singular. No restrictions on the exponent are imposed.
 [215] arXiv:2406.14211 [pdf, html, other]

Title: Optimization over boundedrank matrices through a desingularization enables joint global and local guaranteesSubjects: Optimization and Control (math.OC); Numerical Analysis (math.NA)
Convergence guarantees for optimization over boundedrank matrices are delicate to obtain because the feasible set is a nonsmooth and nonconvex algebraic variety. Existing techniques include projected gradient descent, fixedrank optimization (over the maximalrank stratum), and the LR parameterization. They all lack either global guarantees (the ability to accumulate only at critical points) or fast local convergence (e.g., if the limit has nonmaximal rank). We seek optimization algorithms that enjoy both.
Khrulkov and Oseledets [2018] parameterize the boundedrank variety via a desingularization to recast the optimization problem onto a smooth manifold. Building on their ideas, we develop a Riemannian geometry for this desingularization, also with care for numerical considerations. We use it to secure global convergence to critical points with fast local rates, for a large range of algorithms. On matrix completion tasks, we find that this approach is comparable to others, while enjoying better generalpurpose theoretical guarantees.  [216] arXiv:2406.14218 [pdf, other]

Title: Stability of nondegenerate ODE type blowup for the Fujita type heat equationSubjects: Analysis of PDEs (math.AP)
The asymptotic bahavior of blowup solutions to the Fujita type heat equation $u_t=\Delta u+u^{p1}u$ is studied. This equation admits the ODE type blowup given by $u(x,t)=(p1)^\frac{1}{p1}(Tt)^{\frac{1}{p1}}$. It is known that nondegenerate ODE type blowup is stable if $p\in(1,\frac{n+2}{n2})$ due to Fermanian KammererMerleZaag (2000). This paper extends their result to more general case.
 [217] arXiv:2406.14221 [pdf, html, other]

Title: On unsolvable equations of prime degreeComments: 1 figureSubjects: Number Theory (math.NT); History and Overview (math.HO)
Kronecker observed that either all roots or only one root of a solvable irreducible equation of odd prime degree with integer coefficients are real. This gives a possibility to construct specific examples of equations not solvable by radicals. A relatively elementary proof without using the full power of Galois theory is due to Weber. We give a rather short proof of Kronecker's theorem with a slightly different argument from Weber's. Several modern presentations of Weber's proof contain inaccuracies, which can be traced back to an error in the original proof. We discuss this error and how it can be corrected.
 [218] arXiv:2406.14223 [pdf, html, other]

Title: Chromatic number of randomly augmented graphsSubjects: Combinatorics (math.CO)
An extension of the ErdősRenyi random graph model $G_{n,p}$ is the model of perturbed graphs introduced by Bohman, Frieze and Martin (Bohman, Frieze, Martin 2003). This is a special case of the model of randomly augmented graphs studied in this paper. An augmented graph denoted by $pert_{H,p}$ is the union of a deterministic host graph and a random graph $G_{n,p}$. Among the first problems in perturbed graphs has been the question how many random edges are needed to ensure Hamiltonicity of the graph. This question was answered in the paper by Bohman, Frieze and Martin. The host graph is often chosen to be a dense graph. In recent years several papers on combinatorial problems in perturbed graphs were published, e.g. on the emergence of powers of Hamiltonian cycles (Dudek, Reiher, Ruciński, Schacht 2020), some positional games played on perturbed graphs (Clemens, Hamann, Mogge, Parczyk, 2020) and the behavior of multiple invariants e.g. fixed clique size (Bohman, Frieze, Krivelevich, Martin, 2004). In this paper we study the chromatic number of randomly augmented graphs. We concentrate on a host graph $H$ with chromatic number $o(n)$, augmented by a $G_{n,p}$ with $n^{\frac{1}{3} + \delta}\leq p(n) \leq 1\delta$ for some $\delta \in (0,1)$. Our main result is an upper bound for the chromatic number: we show that asymptotically almost surely $\chi(pert_{H,p}) \leq (1+o(1)) \cdot \frac{n \log(b)}{2 (\log(n)  \log(\chi(H))}$ where $b = (1p)^{1}$. This result collapses to the famous theorem of Bollobás (1988), when $H$ is the empty host graph, thus our result can be regarded as a generalization of the latter. Our proof is not constructive. Further, we give a constructive coloring algorithm, when the chromatic number of the host graph is at most $\frac{n}{\log(n)^{\alpha}},$ $\alpha>\frac{1}{2}.$
 [219] arXiv:2406.14233 [pdf, html, other]

Title: RegionSpecific Coarse Quantization with Check Node Awareness in 5GLDPC DecodingComments: This paper has been submitted to IEEE Transactions on CommunicationsSubjects: Information Theory (cs.IT)
This paper presents novel techniques for improving the error correction performance and reducing the complexity of coarsely quantized 5GLDPC decoders. The proposed decoder design supports arbitrary messagepassing schedules on a basematrix level by modeling exchanged messages with entryspecific discrete random variables. Variable nodes (VNs) and check nodes (CNs) involve compression operations designed using the information bottleneck method to maximize preserved mutual information between code bits and quantized messages. We introduce alignment regions that assign the messages to groups with aligned reliability levels to decrease the number of individual design parameters. Group compositions with degreespecific separation of messages improve performance by up to 0.4 dB. Further, we generalize our recently proposed CNaware quantizer design to irregular LDPC codes and layered schedules. The method optimizes the VN quantizer to maximize preserved mutual information at the output of the subsequent CN update, enhancing performance by up to 0.2 dB. A schedule optimization modifies the order of layer updates, reducing the average iteration count by up to 35 %. We integrate all new techniques in a ratecompatible decoder design by extending the alignment regions along a ratedimension. Our complexity analysis for 2bit decoding estimates up to 64 % higher throughput versus 4bit decoding at similar performance.
 [220] arXiv:2406.14237 [pdf, other]

Title: Finite Alphabet Fast List Decoders for Polar CodesComments: 6 pages, 7 figures, submitted to IEEE GLOBECOM 2024Subjects: Information Theory (cs.IT)
The socalled fast polar decoding schedules are meant to improve the decoding speed of the sequentialnatured successive cancellation list decoders. The decoding speedup is achieved by replacing various parts of the serial decoding process with efficient specialpurpose decoder nodes. This work incorporates the fast decoding schedules for polar codes into their quantized finite alphabet decoding. In a finite alphabet successive cancellation list decoder, the loglikelihood ratio computations are replaced with lookup operations on lowresolution integer messages. The lookup tables are designed using the information bottleneck method. It is shown that the finite alphabet decoders can also leverage the special decoder nodes found in the literature. Besides their inherent decoding speed improvement, the use of these special decoder nodes drastically reduces the number of lookup tables required to perform the finite alphabet decoding. In order to perform quantized decoding using lookup operations, the proposed decoders require up to 93% less unique lookup tables as compared to the ones that use the conventional successive cancellation schedule. Moreover, the proposed decoders exhibit negligible loss in error correction performance without necessitating alterations to the lookup table design process.
 [221] arXiv:2406.14241 [pdf, html, other]

Title: Zero sets of homogeneous polynomials containing infinite dimensional spacesComments: 11 pagesSubjects: Functional Analysis (math.FA)
Let $X$ be a (real or complex) infinite dimensional linear space. We establish conditions on a homogeneous polynomial $P$ on $X$ so that, if $W$ is any finite dimensional subspace of $X$ on which $P$ vanishes, then $P$ vanishes on an infinite dimensional subspace of $X$ containing $W$. In the complex case, this is a step beyond the classical result due to Plichko and Zagorodnyuk. Applications to the real case are also provided.
 [222] arXiv:2406.14244 [pdf, html, other]

Title: On simple matroids with a unique minimal tropical basisComments: 4 pagesSubjects: Combinatorics (math.CO)
In this note we characterize tropical bases as sets of circuits that by orthogonality determine the set of cocircuits of a simple matroid. Furthermore, we show that any circuit, which itself is closed, must be contained in any tropical basis. This yields a characterization of simple matroids which have a unique minimal tropical basis, giving a solution for a problem posted in the Matroid Union Blog.
 [223] arXiv:2406.14247 [pdf, html, other]

Title: Formal groups over noncommutative ringsSubjects: Algebraic Topology (math.AT)
We develop an extension of the usual theory of formal group laws where the base ring is not required to be commutative and where the formal variables need neither be central nor have to commute with each other.
We show that this is the natural kind of formal group law for the needs of algebraic topology in the sense that a (possibly noncommutative) complex oriented ring spectrum is canonically equipped with just such a formal group law. The universal formal group law is carried by the BakerRichter spectrum M{\xi} which plays a role analogous to MU in this noncommutative context.
As suggested by previous work of Morava the Hopf algebra B of "formal diffeomorphisms of the noncommutative line" of Brouder, Frabetti and Krattenthaler is central to the theory developed here. In particular, we verify Morava's conjecture that there is a representation of the Drinfeld quantumdouble D(B) through cohomology operations in M{\xi}.  [224] arXiv:2406.14249 [pdf, html, other]

Title: Sparse Subgaussian Random Projections for Semidefinite Programming RelaxationsSubjects: Optimization and Control (math.OC)
Random projection, a dimensionality reduction technique, has been found useful in recent years for reducing the size of optimization problems. In this paper, we explore the use of sparse subgaussian random projections to approximate semidefinite programming (SDP) problems by reducing the size of matrix variables, thereby solving the original problem with much less computational effort. We provide some theoretical bounds on the quality of the projection in terms of feasibility and optimality that explicitly depend on the sparsity parameter of the projector. We investigate the performance of the approach for semidefinite relaxations appearing in polynomial optimization, with a focus on combinatorial optimization problems. In particular, we apply our method to the semidefinite relaxations of MAXCUT and MAX2SAT. We show that for large unweighted graphs, we can obtain a good bound by solving a projection of the semidefinite relaxation of MAXCUT. We also explore how to apply our method to find the stability number of four classes of imperfect graphs by solving a projection of the second level of the Lasserre Hierarchy. Overall, our computational experiments show that semidefinite programming problems appearing as relaxations of combinatorial optimization problems can be approximately solved using random projections as long as the number of constraints is not too large.
 [225] arXiv:2406.14253 [pdf, html, other]

Title: A new formulation of regular singularitySubjects: Algebraic Geometry (math.AG)
We provide an alternative definition for the familiar concept of regular singularity for meromorphic connections. Our new formulation does not use derived categories, and it also avoids the necessity of finding a special good filtration as in the formulation due to KashiwaraKawai. Moreover, our formulation provides an explicit algorithm to decide the regular singularity of a meromorphic connection. An important intermediary result, interesting in its own right, is that taking associated graded modules with respect to (not necessarily canonical) $V$filtrations commutes with noncharacteristic restriction. This allows us to reduce the proof of the equivalence of our formulation with the classical concept to the onedimensional case. In that situation, we extend the wellknown onedimensional Fuchs criterion for ideals in the Weyl algebra to arbitrary holonomic modules over the Weyl algebra equipped with an arbitrary $(1,1)$filtration.
 [226] arXiv:2406.14254 [pdf, html, other]

Title: Shimura lift of RankinCohen brackets of eigenforms and theta seriesSubjects: Number Theory (math.NT)
The Shimura lift of a Hekce eigenform multiplied by a theta series is the square of the form. We extend this result by generalizing the product to the RankinCohen brackets. We prove that the Shimura lift of the RankinCohen bracket of an eigenform and a theta series is given by the RankinCohen bracket of the eigenform and itself.
 [227] arXiv:2406.14260 [pdf, html, other]

Title: On exact systems $\{t^{\alpha}\cdot e^{2\pi i nt}\}_{n\in\mathbb{Z}\setminus A}$ in $L^2 (0,1)$ which are not Schauder Bases and their generalizationsComments: 8 pagesSubjects: Functional Analysis (math.FA)
Let $\{e^{i\lambda_n t}\}_{n\in\mathbb{Z}}$ be an exponential Schauder Basis for $L^2 (0,1)$, for $\lambda_n\in\mathbb{R}$, and let $\{r_n(t)\}_{n\in\mathbb{Z}}$ be its dual Schauder Basis. Let $A$ be a nonempty subset of the integers containing exactly $M$ elements. We prove that for $\alpha >0$ the weighted system \[ \{t^{\alpha}\cdot r_n(t)\}_{n\in\mathbb{Z}\setminus A} \] is exact in the space $L^2 (0,1)$, that is, it is complete and minimal in $L^2 (0,1)$, if and only if \[ M\frac{1}{2}\le \alpha< M+\frac{1}{2}. \] We also show that such a system is not a Riesz Basis for $L^2 (0,1)$. In particular, the weighted trigonometric system $\{t^{\alpha}\cdot e^{2\pi i n t}\}_{n\in\mathbb{Z}\setminus A}$ is exact in $L^2 (0,1)$, if and only if $\alpha\in [M\frac{1}{2}, M+\frac{1}{2})$, but it is not a Schauder Basis for $L^2 (0,1)$.
 [228] arXiv:2406.14262 [pdf, other]

Title: On GinzburgKaplan gamma factors and BesselSpeh functions for finite general linear groupsComments: 69 Pages, comments are welcome!Subjects: Representation Theory (math.RT); Number Theory (math.NT)
We give a new construction of tensor product gamma factors for a pair of irreducible representations of $\operatorname{GL}_c\left(\mathbb{F}_q\right)$ and $\operatorname{GL}_k\left(\mathbb{F}_q\right)$. This construction is a finite field analog of a construction of doubling type due to Kaplan in the local field case and due to Ginzburg in the global case, and it only assumes that one of the representations in question is generic. We use this construction to establish a relation between special values of Bessel functions attached to Speh representations and exotic matrix Kloosterman sums. Using this relation, we establish various identities, including the multiplicativity identity of exotic matrix Kloosterman sums.
 [229] arXiv:2406.14269 [pdf, html, other]

Title: Concentration of a sparse Bayesian model with Horseshoe prior in estimating highdimensional precision matrixThe Tien MaiSubjects: Statistics Theory (math.ST); Machine Learning (stat.ML)
Precision matrices are crucial in many fields such as social networks, neuroscience, and economics, representing the edge structure of Gaussian graphical models (GGMs), where a zero in an offdiagonal position of the precision matrix indicates conditional independence between nodes. In highdimensional settings where the dimension of the precision matrix $p$ exceeds the sample size $n$ and the matrix is sparse, methods like graphical Lasso, graphical SCAD, and CLIME are popular for estimating GGMs. While frequentist methods are wellstudied, Bayesian approaches for (unstructured) sparse precision matrices are less explored. The graphical horseshoe estimate by \citet{li2019graphical}, applying the globallocal horseshoe prior, shows superior empirical performance, but theoretical work for sparse precision matrix estimations using shrinkage priors is limited. This paper addresses these gaps by providing concentration results for the tempered posterior with the fully specified horseshoe prior in highdimensional settings. Moreover, we also provide novel theoretical results for model misspecification, offering a general oracle inequality for the posterior.
 [230] arXiv:2406.14270 [pdf, html, other]

Title: Inverse optimal control problem in the non autonomous linearquadratic caseSubjects: Optimization and Control (math.OC)
Inverse optimal control problem emerges in different practical applications, where the goal is to design a cost function in order to approximate given optimal strategies of an expert. Typical application is in robotics for generation of human motions. In this paper we analyze a general class of non autonomous inverse linear quadratic problems. This class of problems is of particular interest because it arises as a linearization of a nonlinear problem around an optimal trajectory. The addressed questions are the injectivity of the inverse problem and the reconstruction. We show that the nonlinear problem admits the same characterization of the injectivity as the autonomous one. In the autonomous case we show moreover that the injectivity property is generic in the considered class. We also provide a numerical test of the reconstruction algorithm in the autonomous setting.
 [231] arXiv:2406.14271 [pdf, html, other]

Title: Pointwise convergence for the heat quation on tori $\mathbb T^n$ and waveguide manifold $\mathbb T^n \times \mathbb R^m$Comments: 27 pagesSubjects: Analysis of PDEs (math.AP)
We completely characterize the weighted Lebesgue spaces on the torus $\mathbb T^n$ and waveguide manifold $\mathbb T^n \times \mathbb R^m$ for which the solutions of the heat equation converge pointwise (as time tends to zero) to the initial data. In the process, we also characterize the weighted Lebesgue spaces for the boundedness of maximal operators on torus and waveguide manifold, which may be of independent interest.
 [232] arXiv:2406.14276 [pdf, html, other]

Title: (u,v)absorbing (prime) hyperideals in commutative multiplicative hyperringsSubjects: Commutative Algebra (math.AC)
In this paper, we will introduce the notion of (u,v)absorbing hyperideals in multiplicative hyperrings and we will show some properties of them. Then we extend this concept to the notion of (u,v)absorbing prime hyperideals and thhen we will give some results about them.
 [233] arXiv:2406.14279 [pdf, html, other]

Title: Uniqueness and modified Newton method for cracks from the far field patterns with a fixed incident directionSubjects: Analysis of PDEs (math.AP)
We consider the inverse cracks scattering problems from the far field patterns with a fixed incident direction. We firstly show that the soundsoft cracks can be uniquely determined by the multifrequency far field patterns with a fixed incident direction. The proof is based on a low frequency asymptotic analysis of the scattered field. One important feature of the uniqueness result is that the background can even be an unknown inhomogeneous medium. A modified Newton method is then proposed for the numerical reconstruction of the shapes and locations of the cracks. Compared to the classical Newton method, the modified Newton method relaxes the dependence of a good initial guess and can be applied for multiple cracks. Numerical examples in two dimensions are presented to demonstrate the feasibility and effectiveness of the modified Newton method. In particular, the quality of the reconstructions can be greatly improved if we use the measurements properly with two frequencies or two incident directions.
 [234] arXiv:2406.14280 [pdf, html, other]

Title: EichlerSelberg relations for singular moduliComments: 21 pagesSubjects: Number Theory (math.NT)
The EichlerSelberg trace formula expresses the trace of Hecke operators on spaces of cusp forms as weighted sums of HurwitzKronecker class numbers. We extend this formula to a natural class of relations for traces of singular moduli, where one views class numbers as traces of the constant function $j_0(\tau)=1$. More generally, we consider the singular moduli for the Hecke system of modular functions \[ j_m(\tau) := mT_m \left(j(\tau)744\right). \] For each $\nu\geq 0$ and $m\geq 1$, we obtain an EichlerSelberg relation. For $\nu=0$ and $m\in \{1, 2\},$ these relations are Kaneko's celebrated singular moduli formulas for the coefficients of $j(\tau).$ For each $\nu\geq 1$ and $m\geq 1,$ we obtain a new EichlerSelberg trace formula for the Hecke action on the space of weight $2\nu+2$ cusp forms, where the traces of $j_m(\tau)$ singular moduli replace HurwitzKronecker class numbers. These formulas involve a new term that is assembled from values of symmetrized shifted convolution $L$functions.
 [235] arXiv:2406.14286 [pdf, html, other]

Title: Trim Turnpikes for Optimal Control Problems with SymmetriesSubjects: Optimization and Control (math.OC)
Motivated by mechanical systems with symmetries, we focus on optimal control problems possessing symmetries. Following recent works, which generalized the classical concept of static turnpike to manifold turnpike, we extend the exponential turnpike property to the exponential trim turnpike for control systems with symmetries induced by abelian or nonabelian groups. Our analysis is mainly based on the geometric reduction of control systems with symmetries. More concretely, we first reduce the control system on the quotient space and state the turnpike theorem for the reduced problem. Then we use the group properties to obtain the trim turnpike theorem for the full problem. Finally, we illustrate our results on the Kepler problem and the Rigid body problem.
 [236] arXiv:2406.14299 [pdf, other]

Title: Symplectic Stiefel manifold: tractable metrics, secondorder geometry and Newton's methodsComments: 41 pages, 5 figures, 7 tablesSubjects: Optimization and Control (math.OC); Numerical Analysis (math.NA); Symplectic Geometry (math.SG); Quantum Physics (quantph)
Optimization under the symplecticity constraint is an approach for solving various problems in quantum physics and scientific computing. Building on the results that this optimization problem can be transformed into an unconstrained problem on the symplectic Stiefel manifold, we construct geometric ingredients for Riemannian optimization with a new family of Riemannian metrics called tractable metrics and develop Riemannian Newton schemes. The newly obtained ingredients do not only generalize several existing results but also provide us with freedom to choose a suitable metric for each problem. To the best of our knowledge, this is the first try to develop the explicit secondorder geometry and Newton's methods on the symplectic Stiefel manifold. For the Riemannian Newton method, we first consider novel operatorvalued formulas for computing the Riemannian Hessian of a~cost function, which further allows the manifold to be endowed with a weighted Euclidean metric that can provide a preconditioning effect. We then solve the resulting Newton equation, as the central step of Newton's methods, directly via transforming it into a~saddle point problem followed by vectorization, or iteratively via applying any matrixfree iterative method either to the operator Newton equation or its saddle point formulation. Finally, we propose a hybrid Riemannian Newton optimization algorithm that enjoys both global convergence and quadratic/superlinear local convergence at the final stage. Various numerical experiments are presented to validate the proposed methods.
 [237] arXiv:2406.14300 [pdf, other]

Title: The ConnesChamseddine Hochschild cocycle and the noncommutative integralSubjects: Differential Geometry (math.DG)
In [5], Connes and Chamseddine defined a Hochschild cocycle in the general framework of noncommutative geometry. They computed this Hochschild cocycle for the Dirac operator on 4dimensioanl manifolds. We propose a way to study the ConnesChamseddine Hochschild cocycle from the viewpoint of the noncommutative integral on 6dimensional manifolds in this paper. We compute several interesting noncommutative integral defined in [8] by the normal coodinated way on ndimensional manifolds. As a corollary, the ConnesChamseddine Hochschild cocycle on 6dimensional manifolds is obtained.
 [238] arXiv:2406.14304 [pdf, html, other]

Title: A Variational Characterization of $H$Mutual Information and its Application to Computing $H$CapacitySubjects: Information Theory (cs.IT)
$H$mutual information ($H$MI) is a wide class of information leakage measures, where $H=(\eta, F)$ is a pair of monotonically increasing function $\eta$ and a concave function $F$, which is a generalization of Shannon entropy. $H$MI is defined as the difference between the generalized entropy $H$ and its conditional version, including Shannon mutual information (MI), Arimoto MI of order $\alpha$, $g$leakage, and expected value of sample information. This study presents a variational characterization of $H$MI via statistical decision theory. Based on the characterization, we propose an alternating optimization algorithm for computing $H$capacity.
 [239] arXiv:2406.14311 [pdf, other]

Title: A HeegaardFloer TQFT for link cobordismsSubjects: Geometric Topology (math.GT); Algebraic Topology (math.AT); Symplectic Geometry (math.SG)
We introduce a HeegaardFloer homology functor from the category of oriented links in closed $3$manifolds and oriented surface cobordisms in $4$manifolds connecting them to the category of $\mathbb{F}[v]$modules and $\mathbb{F}[v]$homomorphisms between them, where $\mathbb{F}$ is the field with two elements. In comparison with previously defined TQFTs for decorated links and link cobordisms, the construction of this paper has the advantage of being independent from the decoration. Some of the basic properties of this functor are also explored.
 [240] arXiv:2406.14317 [pdf, html, other]

Title: Maximum principle preserving time implicit DGSEM for nonlinear scalar conservation lawsComments: 25 pages, 15 figures, 1 table, 55 referencesSubjects: Numerical Analysis (math.NA)
This work concerns the analysis of the discontinuous Galerkin spectral element method (DGSEM) with implicit time stepping for the numerical approximation of nonlinear scalar conservation laws in multiple space dimensions. We consider either the DGSEM with a backward Euler time stepping, or a spacetime DGSEM discretization to remove the restriction on the time step. We design graph viscosities in space, and in time for the spacetime DGSEM, to make the schemes maximum principle preserving and entropy stable for every admissible convex entropy. We also establish wellposedness of the discrete problems by showing existence and uniqueness of the solutions to the nonlinear implicit algebraic relations that need to be solved at each time step. Numerical experiments in one space dimension are presented to illustrate the properties of these schemes.
 [241] arXiv:2406.14321 [pdf, html, other]

Title: The motive of the Hilbert scheme of points in all dimensionsComments: 55 pages, comments welcomeSubjects: Algebraic Geometry (math.AG)
We prove a closed formula for the generating function $\mathsf Z_d(t)$ of the motives $[\mathrm{Hilb}^d(\mathbb A^n)_0] \in K_0(\mathrm{Var}_{\mathbb C})$ of punctual Hilbert schemes, summing over $n$, for fixed $d>0$. The result is an expression for $\mathsf Z_d(t)$ as the product of the zeta function of $\mathbb P^{d1}$ and a polynomial $\mathsf P_d(t)$, which in particular implies that $\mathsf Z_d(t)$ is a rational function. Moreover, we reduce the complexity of $\mathsf P_d(t)$ to the computation of $d8$ initial data, and therefore give explicit formulas for $\mathsf Z_d(t)$ in the cases $d \leq 8$, which in turn yields a formula for $[\mathrm{Hilb}^{\leq 8}(X)]$ for any smooth variety $X$. We perform a similar analysis for the Quot scheme of points, obtaining explicit formulas for the full generating function (summing over all ranks and dimensions) for $d \leq 4$. In the limit $n \to \infty$, we prove that the motives $[\mathrm{Hilb}^d(\mathbb A^n)_0]$ stabilise to the class of the infinite Grassmannian $\mathrm{Gr}(d1,\infty)$. Finally, exploiting our geometric methods, we conjecture (and partially confirm) a structural result on the 'error' measuring the discrepancy between the count of higher dimensional partitions and MacMahon's famous guess.
 [242] arXiv:2406.14332 [pdf, html, other]

Title: Sufficient conditions for closedtrailable in digraphsComments: 13 pagesSubjects: Combinatorics (math.CO)
A digraph $D$ with a subset $S$ of $V(D)$ is called $\boldsymbol{S}${\bf strong} if for every pair of distinct vertices $u$ and $v$ of $S$, there is a $(u, v)$dipath and a $(v, u)$dipath in $D$. We define a digraph $D$ with a subset $S$ of $V(D)$ to be $\boldsymbol{S}${\bf strictly strong} if there exist two nonadjacent vertices $u,v\in S$ such that $D$ contains a closed ditrail through the vertices $u$ and $v$; and define a subset $S\subseteq V(D)$ to be {\bf closedtrailable} if $D$ contains a closed ditrail through all the vertices of $S$. In this paper, we prove that for a digraph $D$ with $n$ vertices and a subset $S$ of $V(D)$, if $D$ is $S$strong and if $d(u) + d(v)\geq 2n 3$ for any two nonadjacent vertices $u,v$ of $S$, then $S$ is closedtrailable. This result generalizes the theorem of BangJensen et al. \cite{BaMa14} on supereulerianity. Moveover, we show that for a digraph $D$ and a subset $S$ of $V(D)$, if $D$ is $S$strictly strong and if $\delta^0(D\langle S\rangle)\geq\alpha'(D\langle S\rangle)>0$, where $\delta^0(D\langle S\rangle)$ is the minimum semidegree of $D\langle S\rangle$ and $\alpha'(D\langle S\rangle)$ is the matching number of $D\langle S\rangle$, then $S$ is closedtrailable. This result generalizes the theorem of Algefari et al. \cite{AlLa15} on supereulerianity.
 [243] arXiv:2406.14337 [pdf, html, other]

Title: Fast Convergence to SecondOrder Stationary Point through Random Subspace OptimizationComments: 39 pages, 1 figureSubjects: Optimization and Control (math.OC)
We propose the Random Subspace Homogenized Trust Region (RSHTR) method, which efficiently solves highdimensional nonconvex optimization problems by identifying descent directions within randomly selected subspaces. RSHTR provides the strongest theoretical guarantees among random subspace algorithms for nonconvex optimization, achieving an $\varepsilon$approximate firstorder stationary point in $O(\varepsilon^{3/2})$ iterations and converging locally at a linear rate. Furthermore, under rankdeficient conditions, RSHTR satisfies $\varepsilon$approximate secondorder necessary condition in $O(\varepsilon^{3/2})$ iterations and exhibits a local quadratic convergence.
 [244] arXiv:2406.14339 [pdf, html, other]

Title: Mixed multiquadratic splitting fieldsComments: 7 pagesSubjects: Number Theory (math.NT); Rings and Algebras (math.RA)
We study mixed multiquadratic field extensions as splitting fields for central simple algebras of exponent $2$ in characteristic $2$. As an application, we provide examples of nonexcellent mixed biquadratic field extensions.
 [245] arXiv:2406.14340 [pdf, other]

Title: Learning rate adaptive stochastic gradient descent optimization methods: numerical simulations for deep learning methods for partial differential equations and convergence analysesComments: 68 pages, 8 figuresSubjects: Optimization and Control (math.OC); Machine Learning (cs.LG); Numerical Analysis (math.NA)
It is known that the standard stochastic gradient descent (SGD) optimization method, as well as accelerated and adaptive SGD optimization methods such as the Adam optimizer fail to converge if the learning rates do not converge to zero (as, for example, in the situation of constant learning rates). Numerical simulations often use humantuned deterministic learning rate schedules or small constant learning rates. The default learning rate schedules for SGD optimization methods in machine learning implementation frameworks such as TensorFlow and Pytorch are constant learning rates. In this work we propose and study a learningrateadaptive approach for SGD optimization methods in which the learning rate is adjusted based on empirical estimates for the values of the objective function of the considered optimization problem (the function that one intends to minimize). In particular, we propose a learningrateadaptive variant of the Adam optimizer and implement it in case of several neural network learning problems, particularly, in the context of deep learning approximation methods for partial differential equations such as deep Kolmogorov methods, physicsinformed neural networks, and deep Ritz methods. In each of the presented learning problems the proposed learningrateadaptive variant of the Adam optimizer faster reduces the value of the objective function than the Adam optimizer with the default learning rate. For a simple class of quadratic minimization problems we also rigorously prove that a learningrateadaptive variant of the SGD optimization method converges to the minimizer of the considered minimization problem. Our convergence proof is based on an analysis of the laws of invariant measures of the SGD method as well as on a more general convergence analysis for SGD with random but predictable learning rates which we develop in this work.
 [246] arXiv:2406.14342 [pdf, html, other]

Title: Regularity properties and dispersive blowup for the fifth order Kortewegde Vries equation on the lineSubjects: Analysis of PDEs (math.AP)
In this work we prove that the initial value problem (IVP) for the fifth order Kortewegde Vries equation \begin{align*} \left. \begin{array}{rlr} u_t+\partial_x^5 u+u\partial_x u&\hspace{2mm}=0,&\quad x\in\mathbb R,\; t>0,\\ u(x,0)&\hspace{2mm}=u_0(x),& \end{array} \right\} \end{align*} has a unique local solution in time in the Bourgain spaces $X^{s,b}$ for appropriate values of $s$ and $b$. Besides we prove a regularity property concerning the nonlinear part of that solution. Finally, using the previous property we establish a dispersive blowup result for global in time solutions of this IVP.
 [247] arXiv:2406.14344 [pdf, html, other]

Title: On the homogenization of a Signorinitype problem in a domain with inclusionsSubjects: Analysis of PDEs (math.AP)
In this paper we investigate the effect of a Signorinitype interface condition on the asymptotic behaviour, as $\varepsilon$ tends to zero, of problems posed in $\varepsilon$periodic domains with inclusions. The Signorinitype condition is expressed in terms of two complementary equalities involving the jump of the solution on the interface and its conormal derivative via a parameter $\gamma$. Our problem models the heat exchange in a medium hosting an $\varepsilon$periodic array of thermal conductors in presence of impurities distributed on some regions of the interface. Different limit problems are obtained according to different values of $\gamma$.
 [248] arXiv:2406.14346 [pdf, html, other]

Title: Atomic Toposes with CoWellFounded Categories of AtomsSubjects: Category Theory (math.CT); Logic (math.LO)
The atoms of the Schanuel topos can be described as the pairs $(n,G)$ where $n$ is a finite set and $G$ is a subgroup of $\operatorname{Aut}(n)$. We give a general criterion on an atomic site ensuring that the atoms of the topos of sheaves on that site can be described in a similar fashion. We deduce that these toposes are locally finitely presentable. By applying this to the MalitzGregory atomic topos, we obtain a counterexample to the conjecture that every locally finitely presentable topos has enough points. We also work out a combinatorial property satisfied exactly when the sheaves for the atomic topology are the pullbackpreserving functors. In this case, the category of atoms is particularly simple to describe.
 [249] arXiv:2406.14353 [pdf, html, other]

Title: Isolated and parameterized points on curvesComments: 26 pagesSubjects: Number Theory (math.NT); Algebraic Geometry (math.AG)
We give a selfcontained introduction to isolated points on curves and their counterpoint, parameterized points, that situates these concepts within the study of the arithmetic of curves. In particular, we show how natural geometric constructions of infinitely many degree $d$ points on curves motivate the definitions of $\mathbb{P}^1$ and AVparameterized points and explain how a result of Faltings implies that there are only finitely many isolated points on any curve. We use parameterized points to deduce properties of the density degree set and review how the minimum density degree relates to the gonality. The paper includes several examples that illustrate the possible behaviors of degree $d$ points.
 [250] arXiv:2406.14356 [pdf, html, other]

Title: $\Gamma$convergence and stochastic homogenization of second order singular perturbations model for phase transitionsSubjects: Analysis of PDEs (math.AP)
We study the effective behavior of random, heterogeneous, anisotropic, second order phase transitions energies that arise in the study of pattern formations in physicalchemical systems. Specifically, we study the asymptotic behavior, as $\epsilon$ goes to zero, of random heterogeneous anisotropic functionals in which the second order perturbation competes not only with a double well potential but also with a possibly negative contribution given by the first order term. We prove that, under suitable growth conditions and under a stationarity assumption, the functionals $\Gamma$converge almost surely to a surface energy whose density is independent of the space variable. Furthermore, we show that the limit surface density can be described via a suitable cell formula and is deterministic when ergodicity is assumed.
 [251] arXiv:2406.14368 [pdf, html, other]

Title: Minimal Attached Primes of Local Cohomology Modules of Binomial Edge Ideals of Block GraphsComments: 16 pagesSubjects: Commutative Algebra (math.AC)
We calculate the minimal attached primes of the local cohomology modules of the binomial edge ideals of block graphs. In particular, we obtain a combinatorial characterisation of which of these modules are nonvanishing.
We further conjecture that every attached prime of these modules is minimal, and, more strongly, that the binomial edge ideals of block graphs are sequentially CohenMacaulay.  [252] arXiv:2406.14369 [pdf, html, other]

Title: Weakly porous sets and $A_1$ Muckenhoupt weights in spaces of homogeneous typeComments: 21 pagesSubjects: Classical Analysis and ODEs (math.CA); Metric Geometry (math.MG)
In this work we characterize the sets $E\subset X$ for which there is some $\alpha>0$ such that the function $d(\cdot,E)^{\alpha}$ belongs to the Muckenhoupt class $A_1(X,d,\mu)$, where $(X,d,\mu)$ is a space of homogeneous type, extending a recent result obtained by Carlos Mudarra in metric spaces endowed with doubling measures. In particular, generalizations of the notions of weakly porous sets and doubling of the maximal hole function are given and it is shown that these concepts have a natural connection with the $A_1$ condition of some negative power of its distance function. The proof presented here is based on Whitneytype covering lemmas built on balls of a particular quasidistance equivalent to the initial quasidistance $d$ and provided by Roberto Macías and Carlos Segovia in "A wellbehaved quasidistance for spaces of homogeneous type", Trabajos de Matemática 32, Instituto Argentino de Matemática, 1981, 118.
 [253] arXiv:2406.14371 [pdf, html, other]

Title: AGDA+: Proximal Alternating Gradient Descent Ascent Method With a Nonmonotone Adaptive StepSize Search For Nonconvex Minimax ProblemsSubjects: Optimization and Control (math.OC)
We consider doubleregularized nonconvexstrongly concave (NCSC) minimax problems of the form $(P):\min_{x\in\mathcal{X}} \max_{y\in\mathcal{Y}}g(x)+f(x,y)h(y)$, where $g$, $h$ are closed convex, $f$ is $L$smooth in $(x,y)$ and strongly concave in $y$. We propose a proximal alternating gradient descent ascent method AGDA+ that can adaptively choose nonmonotone primaldual stepsizes to compute an approximate stationary point for $(P)$ without requiring the knowledge of the global Lipschitz constant $L$. Using a nonmonotone stepsize search (backtracking) scheme, AGDA+ stands out by its ability to exploit the local Lipschitz structure and eliminates the need for precise tuning of hyperparameters. AGDA+ achieves the optimal iteration complexity of $\mathcal{O}(\epsilon^{2})$ and it is the first stepsize search method for NCSC minimax problems that require only $\mathcal{O}(1)$ calls to $\nabla f$ per backtracking iteration. The numerical experiments demonstrate its robustness and efficiency.
 [254] arXiv:2406.14375 [pdf, html, other]

Title: Simple modules over 3cyclic quantum Weyl Algebra at roots of unityComments: 23 pages. Comments are welcomedSubjects: Representation Theory (math.RT); Quantum Algebra (math.QA)
This article undertakes an exploration of simple modules of 3cyclic quantum Weyl algebra at roots of unity. Under the roots of unity assumption, the algebra becomes a Polynomial Identity algebra and the vector space dimension of the simple modules is bounded above by its PI degree. The article systematically classifies all potential simple modules and computes the algebra's center.
 [255] arXiv:2406.14376 [pdf, html, other]

Title: Multicoloured Hardcore Model: Fast Mixing and QueueingComments: 12 pages. To appear at Analysis of Algorithms (AofA) 2024. Minor typocorrection and improvements to that versionSubjects: Probability (math.PR); Discrete Mathematics (cs.DM)
We extend the hardcore model to a multicoloured version: a subset of vertices of a graph are coloured such that no vertex is adjacent to one of the same colour; uncoloured vertices do not constrain neighbours. This mathematically models multichannel resource sharing, such as fibreoptic routing.
We analyse certain simple Glaubertype dynamics on such configurations, and find conditions which ensure fast mixing. These dynamics model a queueing system: customers queue for service at vertices, who only serve customers whilst they are coloured in the underlying configuration; uncoloured vertices sit idle. The mixing estimates are applied to control queue lengths in equilibrium.  [256] arXiv:2406.14390 [pdf, html, other]

Title: Asymmetric mixing Poisson suspensionsComments: includes English and Russian versionsSubjects: Dynamical Systems (math.DS)
Simple Sidon automorphisms are asymmetric. Poisson suspensions over them are mixing and asymmetric.
 [257] arXiv:2406.14405 [pdf, html, other]

Title: Constrained $L^p$ Approximation of Shape Tensors and its Role for the Determination of Shape GradientsSubjects: Numerical Analysis (math.NA); Optimization and Control (math.OC)
This paper extends our earlier work [arXiv:2309.13595] on the $L^p$ approximation of the shape tensor by Laurain and Sturm. In particular, it is shown that the weighted $L^p$ distance to an affine space of admissible symmetric shape tensors satisfying a divergence constraint provides the shape gradient with respect to the $L^{p^\ast}$norm (where $1/p + 1/p^\ast = 1$) of the elastic strain associated with the shape deformation. This approach allows the combination of two ingredients which have already been used successfully in numerical shape optimization: (i) departing from the Hilbert space framework towards the Lipschitz topology approximated by $W^{1,p^\ast}$ with $p^\ast > 2$ and (ii) using the symmetric rather than the full gradient to define the norm. Similarly to [arXiv:2309.13595], the $L^p$ distance measures the shape stationarity by means of the dual norm of the shape derivative with respect to the abovementioned $L^{p^\ast}$norm of the elastic strain. Moreover, the Lagrange multiplier for the momentum balance constraint constitute the steepest descent deformation with respect to this norm. The finite element realization of this approach is done using the weakly symmetric PEERS element and its threedimensional counterpart, respectively. The resulting piecewise constant approximation for the Lagrange multiplier is reconstructed to a shape gradient in $W^{1,p^\ast}$ and used in an iterative procedure towards the optimal shape.
 [258] arXiv:2406.14410 [pdf, html, other]

Title: Dynamical Morse entropyComments: 23 pages. This paper has been published in 2004 but was never posted on the arXivJournalref: Modern dynamical systems and applications, 2744, Cambridge Univ. Press, Cambridge, 2004Subjects: Dynamical Systems (math.DS); Differential Geometry (math.DG)
We consider actions of a tileable amenable group $\Gamma$ on a topological space $X$. For a continuous function on $X$, we define the entropy of the number of homologically detectable critical point of the average of that function over $\Gamma$. This number is bounded below by the sum of the Betti number entropy. This result is thus a generalization of a standard Morse inequality in differential geometry to this setting.
 [259] arXiv:2406.14414 [pdf, html, other]

Title: Normal forms for ordinary differential operators, IComments: 67 pSubjects: Algebraic Geometry (math.AG); Mathematical Physics (mathph); Quantum Algebra (math.QA); Rings and Algebras (math.RA)
In this paper we develop the generalised Schur theory offered in the recent paper by the second author in dimension one case, and apply it to obtain two applications in different directions of algebra/algebraic geometry.
The first application is a new explicit parametrisation of torsion free rank one sheaves on projective irreducible curves with vanishing cohomology groups.
The second application is a commutativity criterion for operators in the Weyl algebra or, more generally, in the ring of ordinary differential operators, which we prove in the case when operators have a normal form with the restriction top line (for details see Introduction).
Both applications are obtained with the help of normal forms. Namely, considering the ring of ordinary differential operators $D_1=K[[x]][\partial ]$ as a subring of a certain complete noncommutative ring $\hat{D}_1^{sym}$, the normal forms of differential operators mentioned here are obtained after conjugation by some invertible operator ("Schur operator"), calculated using one of the operators in a ring. Normal forms of commuting operators are polynomials with constant coefficients in the differentiation, integration and shift operators, which have a finite order in each variable, and can be effectively calculated for any given commuting operators.  [260] arXiv:2406.14418 [pdf, html, other]

Title: WorstCase Learning under a Multifidelity ModelSubjects: Numerical Analysis (math.NA)
Inspired by multifidelity methods in computer simulations, this article introduces procedures to design surrogates for the input/output relationship of a highfidelity code. These surrogates should be learned from runs of both the highfidelity and lowfidelity codes and be accompanied by error guarantees that are deterministic rather than stochastic. For this purpose, the article advocates a framework tied to a theory focusing on worstcase guarantees, namely Optimal Recovery. The multifidelity considerations triggered new theoretical results in three scenarios: the globally optimal estimation of linear functionals, the globally optimal approximation of arbitrary quantities of interest in Hilbert spaces, and their locally optimal approximation, still within Hilbert spaces. The latter scenario boils down to the determination of the Chebyshev center for the intersection of two hyperellipsoids. It is worth noting that the mathematical framework presented here, together with its possible extension, seems to be relevant in several other contexts briefly discussed.
 [261] arXiv:2406.14428 [pdf, html, other]

Title: Blowup for a double nonlocal heat equationSubjects: Analysis of PDEs (math.AP)
We study the blowup question for the diffusion equation involving a nonlocal derivative in time defined by convolution with a nonnegative and nonincreasing kernel, and a nonlocal operator in space driven by a nonnegative radial Lévy kernel. We show that the existence of solutions that blow up in finite time or exist globally depends only on the behaviour of the spatial kernel at infinity. A main difficulty of the work stems from estimating the fundamental pair defining the solution through a Duhamel formula, due to the generality of the setting, which includes singular or not, at the origin, spatial kernels, that can be either positive or compactly supported.
As a byproduct we obtain that the Fujita exponent for the fractional type operators similar to the Caputo fractional derivative and the fractional Laplacian.  [262] arXiv:2406.14431 [pdf, html, other]

Title: Some remarks on a K\"unneth formula for foliated de Rham cohomologyComments: 22 pages. This paper was published in 2011 but has never been posted on the arXivJournalref: Pacific Journal of Math., Vol. 252 (2011), No. 2, 257274Subjects: Differential Geometry (math.DG)
The Künneth formula is one of the basic tools for computing cohomology. Its validity for foliated cohomology, that is, for the tangential de Rham cohomology of a foliated manifold, is investigated. The main difficulty encountered is the nonHausdorff nature of the foliated cohomology spaces, forbidding the completion of the tensor product. The results presented here are a Künneth formula when both factors have Hausdorff foliated cohomology, a Künneth formula when one factor has Hausdorff finitedimensional foliated cohomology and a counterexample to an alternative version of the Künneth formula. The proof of the second result involves a right inverse for the foliated de Rham differential.
 [263] arXiv:2406.14438 [pdf, html, other]

Title: Cycle conjectures and birational invariants over finite fieldsComments: 33 pagesSubjects: Algebraic Geometry (math.AG); Number Theory (math.NT)
We study a natural birational invariant for varieties over finite fields and show that its vanishing on projective space is equivalent to the Tate conjecture, the Beilinson conjecture, and the GrothendieckSerre semisimplicity conjecture for all smooth projective varieties over finite fields. We further show that the Tate, Beilinson, and 1semisimplicity conjecture in half of the degrees implies those conjectures in all degrees.
 [264] arXiv:2406.14439 [pdf, html, other]

Title: Invariant rings of the special orthogonal group have nonunimodal $h$vectorsSubjects: Commutative Algebra (math.AC)
For $K$ an infinite field of characteristic other than two, consider the action of the special orthogonal group $\operatorname{SO}_t(K)$ on a polynomial ring via copies of the regular representation. When $K$ has characteristic zero, Boutot's theorem implies that the invariant ring has rational singularities; when $K$ has positive characteristic, the invariant ring is $F$regular, as proven by Hashimoto using good filtrations. We give a new proof of this, viewing the invariant ring for $\operatorname{SO}_t(K)$ as a cyclic cover of the invariant ring for the corresponding orthogonal group; this point of view has a number of useful consequences, for example it readily yields the $a$invariant and information on the Hilbert series. Indeed, we use this to show that the $h$vector of the invariant ring for $\operatorname{SO}_t(K)$ need not be unimodal.
 [265] arXiv:2406.14447 [pdf, html, other]

Title: Pursuing Coxeter theory for KacMoody affine Hecke algebrasComments: 23 pages. Comments welcomeSubjects: Representation Theory (math.RT)
The KacMoody affine Hecke algebra $\mathcal{H}$ was first constructed as the IwahoriHecke algebra of a $p$adic KacMoody group by work of Braverman, Kazhdan, and Patnaik, and by work of BardyPanse, Gaussent, and Rousseau. Since $\mathcal{H}$ has a Bernstein presentation, for affine types it is a positivelevel variation of Cherednik's double affine Hecke algebra.
Moreover, as $\mathcal{H}$ is realized as a convolution algebra, it has an additional "$T$basis" corresponding to indicator functions of double cosets. For classical affine Hecke algebras, this $T$basis reflects the Coxeter group structure of the affine Weyl group. In the KacMoody affine context, the indexing set $W_{\mathcal{T}}$ for the $T$basis is no longer a Coxeter group. Nonetheless, $W_{\mathcal{T}}$ carries some Coxeterlike structures: a Bruhat order, a length function, and a notion of inversion sets.
This paper contains the first steps toward a Coxeter theory for KacMoody affine Hecke algebras. We prove three results. The first is a construction of the length function via a representation of $\mathcal{H}$. The second concerns the support of products in classical affine Hecke algebras. The third is a characterization of length deficits in the KacMoody affine setting via inversion sets. Using this characterization, we phrase our support theorem as a precise conjecture for KacMoody affine Hecke algebras. Lastly, we give a conjectural definition of a KacMoody affine Demazure product via the $q=0$ specialization of $\mathcal{H}$.  [266] arXiv:2406.14451 [pdf, html, other]

Title: Gradient Estimation via Differentiable MetropolisHastingsComments: 27 pages, 3 figuresSubjects: Statistics Theory (math.ST); Probability (math.PR); Computation (stat.CO)
MetropolisHastings estimates intractable expectations  can differentiating the algorithm estimate their gradients? The challenge is that MetropolisHastings trajectories are not conventionally differentiable due to the discrete accept/reject steps. Using a technique based on recoupling chains, our method differentiates through the MetropolisHastings sampler itself, allowing us to estimate gradients with respect to a parameter of otherwise intractable expectations. Our main contribution is a proof of strong consistency and a central limit theorem for our estimator under assumptions that hold in common Bayesian inference problems. The proofs augment the sampler chain with latent information, and formulate the estimator as a stopping tail functional of this augmented chain. We demonstrate our method on examples of Bayesian sensitivity analysis and optimizing a random walk Metropolis proposal.
 [267] arXiv:2406.14468 [pdf, html, other]

Title: On $k$uniform tight cycles: the Ramsey number for $C_{kn}^{(k)}$ and an approximate Lehel's conjectureComments: 19 pagesSubjects: Combinatorics (math.CO)
A $k$uniform tight cycle is a $k$graph with a cyclic ordering of its vertices such that its edges are precisely the sets of $k$ consecutive vertices in that ordering. We show that, for each $k \geq 3$, the Ramsey number of the $k$uniform tight cycle on $kn$ vertices is $(1+o(1))(k+1)n$. This is an extension to all uniformities of previous results for $k = 3$ by Haxell, Łuczak, Peng, Rödl, Ruciński, and Skokan and for $k = 4$ by Lo and the author and confirms a special case of a conjecture by the former set of authors.
Lehel's conjecture, which was proved by Bessy and Thomassé, states that every redblue edgecoloured complete graph contains a red cycle and a blue cycle that are vertexdisjoint and together cover all the vertices. We also prove an approximate version of this for $k$uniform tight cycles. We show that, for every $k \geq 3$, every redblue edgecoloured complete $k$graph on $n$ vertices contains a red tight cycle and a blue tight cycle that are vertexdisjoint and together cover $n  o(n)$ vertices.  [268] arXiv:2406.14471 [pdf, html, other]

Title: Sharp Estimates for the Optimal Matching Problem on the 2TorusSubjects: Analysis of PDEs (math.AP)
In this paper, we prove sharp estimates for the average cost of the optimal matching problem on the flat 2torus, using quantitative linearization and the method of trajectories.
 [269] arXiv:2406.14487 [pdf, html, other]

Title: Dynamical properties of critical exponent functionsSubjects: Dynamical Systems (math.DS)
An error in the statement and proof of Theorem 7 in [D.Corona, A. Della Corte. The critical exponent functions. Comptes Rendus Mathématique, 360(G4), 315332, 2024] has been identified. The theorem was used to prove analytical and dynamical properties of a class of interval maps there introduced, called the critical exponent functions. We show here that most of the properties of these maps established in [D.Corona, A. Della Corte. The critical exponent functions. Comptes Rendus Mathématique, 360(G4), 315332, 2024] can be recovered with minor adjustments. Moreover, we propose as a conjecture a weaker form of Theorem 7.
 [270] arXiv:2406.14493 [pdf, html, other]

Title: On the modular Plesken Lie algebraSubjects: Representation Theory (math.RT)
Let G be a finite group. The Plesken Lie algebra L[G] is a subalgebra of the complex group algebra C[G] and admits a directsum decomposition into simple Lie algebras based on the ordinary character theory of G. In this paper we review the known results on L[G] and related Lie algebras, as well as introduce a conjecture on a characteristic p analog L_p[G], with a focus on when p divides the order of G.
 [271] arXiv:2406.14499 [pdf, html, other]

Title: Finite Groups of Symplectic Automorphisms of Supersingular K3 surfaces in Odd CharacteristicsComments: Any comments are welcomeSubjects: Algebraic Geometry (math.AG); Group Theory (math.GR)
In 2009, DolgachevKeum classify finite groups of tame symplectic automorphisms of K3 surfaces in positive characteristics. They show all such groups are subgroups of the Mathieu group of degree 23. In this paper, we utilize latticetheoretic methods to investigate symplectic actions of finite groups G on K3 surfaces in odd characteristics. In the tame cases (i.e., the order of G is coprime with p) and all the superspecial cases, one can associate with the action a Leech pair which can be detected via HöhnMason's list. Notice that the superspecial case has been recently resolved by OhashiSchütt. If the K3 surface is supersingular with Artin invariant at least two, we develop a new machinery called proot pairs to detect possible symplectic finite group actions (without the assumption of tameness). The concept of proot pair is closely related to root systems and Weyl groups. In particular, we recover many results by DolgachevKeum with a different method and give an upper bound for the exponent of p in G.
 [272] arXiv:2406.14509 [pdf, html, other]

Title: Consistency and independence phenomena involving cellularLindelof spacesSubjects: General Topology (math.GN); Logic (math.LO)
The cellularLindelöf property is a common generalization of the Lindelöf property and the countable chain condition that was introduced by Bella and Spadaro in 2018. We solve two questions of Alas, GutierrezDominguez and Wilson by constructing consistent examples of a normal almost cellularLindelöf space which is neither cellularLindelöf nor weakly Lindelöf and a Tychonoff cellularLindelöf space of Lindelöf degree $\omega_1$ and uncountable weak Lindelöf degree for closed sets. We also construct a ZFC example of a space for which both the cellularLindelöf property and normality are undetermined in ZFC.
 [273] arXiv:2406.14513 [pdf, html, other]

Title: On fibered Burnside rings, fiber change maps and cyclic fiber groupsSubjects: Representation Theory (math.RT); KTheory and Homology (math.KT)
Fibered Burnside rings appear as Grothendieck rings of fibered permutation representations of a finite group, generalizing Burnside rings and monomial representation rings. Their species, primitive idempotents and their conductors are of particular interest in representation theory as they encode information related to the structure of the group. In this note, we introduce fiber change maps between fibered Burnside rings, and we present results on their functoriality and naturality with respect to biset operations. We present some advances on the conductors for cyclic fiber groups, and fully determine them in particular cases, covering a wide range of interesting examples.
 [274] arXiv:2406.14518 [pdf, html, other]

Title: Algebraic geometry of bubbling Kahler metricsComments: 44 pagesSubjects: Algebraic Geometry (math.AG); High Energy Physics  Theory (hepth); Differential Geometry (math.DG)
We give an algebrogeometric or nonarchimedean framework to study bubbling phenomena of Kahler metrics with Euclidean volume growth, after [DS17, Sun23, dBS23]. In particular, for any degenerating family to log terminal singularity, we algebraically construct a finite sequence of birational modifications of the family with milder degenerations, and compare with analytic bubbling constructions in loc.cit. We also provide approaches in terms of coordinates and valuations. Our discussion partially depends on the general framework of stability theory in our [Od24b] (arXiv:2406.02489) after [HL14, AHLH23].
 [275] arXiv:2406.14530 [pdf, html, other]

Title: Relative Group TrisectionsComments: 16 pages, 1 figureSubjects: Geometric Topology (math.GT); Algebraic Topology (math.AT)
Trisections of closed 4manifolds, first defined and studied by Gay and Kirby, have proved to be a useful tool in the systematic analysis of 4manifolds via handlebodies. Subsequent work of Abrams, Gay, and Kirby established a connection with the algebraic notion of a group trisection, which strikingly defines a onetoone correspondence. We generalize the notion of a group trisection to the nonclosed case by defining and studying relative group trisections. We establish an analogous onetoone correspondence between relative trisections and relative group trisections up to equivalence. The key lemma in the construction may be of independent interest, as it generalizes the classical fact that there is a unique handlebody extension of a surface realizing a given surjection. Moreover, we establish a functorial relationship between relative trisections of manifolds and groups, extending work of Klug in the closed case.
 [276] arXiv:2406.14533 [pdf, other]

Title: Local symmetries in partially ordered setsComments: 33 pages, 5 figures, 3 tablesSubjects: Combinatorics (math.CO); General Relativity and Quantum Cosmology (grqc); Mathematical Physics (mathph)
Partially ordered sets (posets) have a universal appearance as an abstract structure in many areas of mathematics. Though, even their explicit enumeration remains unknown in general, and only the counts of all partial orders on sets of up to 16 unlabelled elements have been calculated to date, see sequence A000112 in the OEIS.
In this work, we study automorphisms of posets in order to formulate a classification by local symmetries. These symmetries give rise to a division operation on the set of all posets and lead us to the construction of symmetry classes that are easier to characterise and enumerate. Additionally to the enumeration of symmetry classes, I derive polynomial expressions that count certain subsets of posets with a large number of layers (a large height). As an application in physics, I investigate local symmetries (or rather their lack of) in causal sets, which are discrete spacetime models used as a candidate framework for quantum gravity.  [277] arXiv:2406.14536 [pdf, html, other]

Title: Formulation of Chimera Gradient Flows for Chemotaxis Systems with Indirect Signal Production and Degenerate DiffusionSubjects: Analysis of PDEs (math.AP)
A parabolic system of three unknown functions, not expressible as gradient flows, is treated as three coupled gradient flows. For each unknown function, the minimizing movement scheme is used to construct a timediscrete approximate solution. Unlike standard minimizing movement scheme for gradient flows, the relative compactness of the timediscrete approximate solution with respect to the time step is not inherently guaranteed. However, the existence of a Lyapunov functional ensures this relative compactness, leading to the existence of timeglobal solutions.
 [278] arXiv:2406.14538 [pdf, html, other]

Title: On the Burau representation of $B_3$ modulo $p$Comments: 67 pagesSubjects: Geometric Topology (math.GT); Group Theory (math.GR)
We present an algorithm that, given a prime $p$ as input, determines whether or not the Burau representation of the 3strand braid group modulo $p$ is faithful. We also prove that the representation is indeed faithful when $p\le 13$. Additionally, we repose Salter's question on the Burau representation of $B_3$ over finite fields $\mathbb{F}_p$, and solve it for every $p$.
 [279] arXiv:2406.14543 [pdf, other]

Title: Equivariant Vector Bundles with Connection on Drinfeld Symmetric SpacesSubjects: Number Theory (math.NT); Representation Theory (math.RT)
For a finite extension $F$ of $\mathbb{Q}_p$ and $n \geq 1$, let $D$ be the division algebra over $F$ of invariant $1/n$ and let $G^0$ be the subgroup of $\text{GL}_n(F)$ of elements with norm $1$ determinant. We show that the action of $D^\times$ on the Drinfeld tower induces an equivalence of categories from finite dimensional smooth representations of $D^\times$ to $G^0$finite $\text{GL}_n(F)$equivariant vector bundles with connection on $\Omega$, the $(n1)$dimensional Drinfeld symmetric space.
 [280] arXiv:2406.14547 [pdf, html, other]

Title: A Mathematical Definition of Path Integrals on Symplectic ManifoldsComments: 20 pagesSubjects: Symplectic Geometry (math.SG); High Energy Physics  Theory (hepth); Mathematical Physics (mathph); Quantum Algebra (math.QA)
We give a mathematical definition of some path integrals, emphasizing those relevant to the quantization of symplectic manifolds (and more generally, Poisson manifolds) $\unicode{x2013}$ in particular, the coherent state path integral. We show that Kähler manifolds provide many computable examples.
 [281] arXiv:2406.14557 [pdf, html, other]

Title: Generalized upwind summationbyparts operators and their application to nodal discontinuous Galerkin methodsJan Glaubitz, Hendrik Ranocha, Andrew R. Winters, Michael SchlottkeLakemper, Philipp Öffner, Gregor GassnerSubjects: Numerical Analysis (math.NA)
There is a pressing demand for robust, highorder baseline schemes for conservation laws that minimize reliance on supplementary stabilization. In this work, we respond to this demand by developing new baseline schemes within a nodal discontinuous Galerkin (DG) framework, utilizing upwind summationbyparts (USBP) operators and flux vector splittings. To this end, we demonstrate the existence of USBP operators on arbitrary grid points and provide a straightforward procedure for their construction. Our method encompasses a broader class of USBP operators, not limited to equidistant grid points. This approach facilitates the development of novel USBP operators on LegendreGaussLobatto (LGL) points, which are suited for nodal discontinuous Galerkin (DG) methods. The resulting DGUSBP operators combine the strengths of traditional summationbyparts (SBP) schemes with the benefits of upwind discretizations, including inherent dissipation mechanisms. Through numerical experiments, ranging from onedimensional convergence tests to multidimensional curvilinear and underresolved flow simulations, we find that DGUSBP operators, when integrated with flux vector splitting methods, foster more robust baseline schemes without excessive artificial dissipation.
New submissions for Friday, 21 June 2024 (showing 281 of 281 entries )
 [282] arXiv:2406.08385 (crosslist from nlin.CD) [pdf, other]

Title: Exploring Geometrical Properties of Chaotic Systems Through an Analysis of the Rulkov Neuron MapsComments: 125 pages, 54 figuresSubjects: Chaotic Dynamics (nlin.CD); Dynamical Systems (math.DS); Neurons and Cognition (qbio.NC)
While extensive research has been conducted on chaos emerging from a dynamical system's temporal dynamics, the research documented in this paper examines extreme sensitivity to initial conditions in discretetime dynamical systems from a geometrical perspective. The heart of this paper focuses on two simple neuron maps developed by Nikolai F. Rulkov in the early 2000s and the complex geometrical structures that emerge from them. Beginning with a conversational introduction to the geometry of chaos, this paper integrates mathematics, physics, neurobiology, computational modeling, and electrochemistry to present original research that provides a novel perspective on how types of geometrical sensitivity to initial conditions appear in discretetime neuron systems.
This paper was developed in the Thomas Jefferson High School for Science and Technology Quantum Lab as part of a senior research project.  [283] arXiv:2406.12895 (crosslist from qbio.NC) [pdf, html, other]

Title: Temporal Complexity of a HopfieldType Neural Model in Random and ScaleFree GraphsSubjects: Neurons and Cognition (qbio.NC); Disordered Systems and Neural Networks (condmat.disnn); Mathematical Physics (mathph); Numerical Analysis (math.NA); Adaptation and SelfOrganizing Systems (nlin.AO)
The Hopfield network model and its generalizations were introduced as a model of associative, or contentaddressable, memory. They were widely investigated both as a unsupervised learning method in artificial intelligence and as a model of biological neural dynamics in computational neuroscience. The complexity features of biological neural networks are attracting the interest of scientific community since the last two decades. More recently, concepts and tools borrowed from complex network theory were applied to artificial neural networks and learning, thus focusing on the topological aspects. However, the temporal structure is also a crucial property displayed by biological neural networks and investigated in the framework of systems displaying complex intermittency. The IntermittencyDriven Complexity (IDC) approach indeed focuses on the metastability of selforganized states, whose signature is a powerdecay in the interevent time distribution or a scaling behavior in the related eventdriven diffusion processes. The investigation of IDC in neural dynamics and its relationship with network topology is still in its early stages. In this work we present the preliminary results of a IDC analysis carried out on a bioinspired Hopfieldtype neural network comparing two different connectivities, i.e., scalefree vs. random network topology. We found that random networks can trigger complexity features similar to that of scalefree networks, even if with some differences and for different parameter values, in particular for different noise levels.
 [284] arXiv:2406.12915 (crosslist from cs.LG) [pdf, html, other]

Title: GROD: Enhancing Generalization of Transformer with OutofDistribution DetectionSubjects: Machine Learning (cs.LG); Probability (math.PR)
Transformer networks excel in natural language processing (NLP) and computer vision (CV) tasks. However, they face challenges in generalizing to OutofDistribution (OOD) datasets, that is, data whose distribution differs from that seen during training. The OOD detection aims to distinguish data that deviates from the expected distribution, while maintaining optimal performance on indistribution (ID) data. This paper introduces a novel approach based on OOD detection, termed the Generate Rounded OOD Data (GROD) algorithm, which significantly bolsters the generalization performance of transformer networks across various tasks. GROD is motivated by our new OOD detection Probably Approximately Correct (PAC) Theory for transformer. The transformer has learnability in terms of OOD detection that is, when the data is sufficient the outlier can be well represented. By penalizing the misclassification of OOD data within the loss function and generating synthetic outliers, GROD guarantees learnability and refines the decision boundaries between inlier and outlier. This strategy demonstrates robust adaptability and general applicability across different data types. Evaluated across diverse OOD detection tasks in NLP and CV, GROD achieves SOTA regardless of data format. On average, it reduces the SOTA FPR@95 from 21.97% to 0.12%, and improves AUROC from 93.62% to 99.98% on image classification tasks, and the SOTA FPR@95 by 12.89% and AUROC by 2.27% in detecting semantic text outliers. The code is available at https://anonymous.4open.science/r/GRODOODDetectionwithtransformersB70F.
 [285] arXiv:2406.12949 (crosslist from qbio.QM) [pdf, html, other]

Title: Integrating timeresolved $nrf2$ geneexpression data into a full GUTS model as a proxy for toxicodynamic damage in zebrafish embryoSubjects: Quantitative Methods (qbio.QM); Dynamical Systems (math.DS); Applications (stat.AP)
The immense production of the chemical industry requires an improved predictive risk assessment that can handle constantly evolving challenges while reducing the dependency of risk assessment on animal testing. Integrating 'omics data into mechanistic models offers a promising solution by linking cellular processes triggered after chemical exposure with observed effects in the organism. With the emerging availability of timeresolved RNA data, the goal of integrating gene expression data into mechanistic models can be approached. We propose a biologically anchored TKTD model, which describes key processes that link the gene expression level of the stress regulator $nrf2$ to detoxification and lethality by associating toxicodynamic damage with $nrf2$ expression. Fitting such a model to complex datasets consisting of multiple endpoints required the combination of methods from molecular biology, mechanistic dynamic systems modeling and Bayesian inference. In this study we successfully integrate timeresolved gene expression data into TKTD models, and thus provide a method for assessing the influence of molecular markers on survival. This novel method was used to test whether, $nrf2$, can be applied to predict lethality in zebrafish embryos. With the presented approach we outline a method to successively approach the goal of a predictive risk assessment based on molecular data.
 [286] arXiv:2406.12962 (crosslist from condmat.strel) [pdf, html, other]

Title: Gauging modulated symmetries: KramersWannier dualities and noninvertible reflectionsComments: 67 pagesSubjects: Strongly Correlated Electrons (condmat.strel); High Energy Physics  Theory (hepth); Mathematical Physics (mathph); Quantum Physics (quantph)
Modulated symmetries are internal symmetries that act in a nonuniform, spatially modulated way and are generalizations of, for example, dipole symmetries. In this paper, we systematically study the gauging of finite Abelian modulated symmetries in ${1+1}$ dimensions. Working with local Hamiltonians of spin chains, we explore the dual symmetries after gauging and their potential new spatial modulations. We establish sufficient conditions for the existence of an isomorphism between the modulated symmetries and their dual, naturally implemented by lattice reflections. For instance, in systems of prime qudits, translation invariance guarantees this isomorphism. For nonprime qudits, we show using techniques from ring theory that this isomorphism can also exist, although it is not guaranteed by lattice translation symmetry alone. From this isomorphism, we identify new KramersWannier dualities and construct related noninvertible reflection symmetry operators using sequential quantum circuits. Notably, this noninvertible reflection symmetry exists even when the system lacks ordinary reflection symmetry. Throughout the paper, we illustrate these results using various simple toy models.
 [287] arXiv:2406.12973 (crosslist from quantph) [pdf, html, other]

Title: Tomography of clock signals using the simplest possible referenceComments: 7 pages + 10 pages appendix, 1 figureSubjects: Quantum Physics (quantph); Mathematical Physics (mathph)
We show that finite physical clocks always have wellbehaved signals, namely that every waitingtime distribution generated by a physical process on a system of finite size is guaranteed to be bounded by a decay envelope. Following this consideration, we show that one can reconstruct the distribution using only operationally available information, namely, that of the ordering of the ticks of one clock with the respect to those of another clock (which we call the reference), and that the simplest possible reference clock  a Poisson process  suffices.
 [288] arXiv:2406.12983 (crosslist from qfin.CP) [pdf, other]

Title: Reinforcement Learning for Corporate Bond Trading: A Sell Side PerspectiveComments: Working PaperSubjects: Computational Finance (qfin.CP); Machine Learning (cs.LG); Optimization and Control (math.OC)
A corporate bond trader in a typical sell side institution such as a bank provides liquidity to the market participants by buying/selling securities and maintaining an inventory. Upon receiving a request for a buy/sell price quote (RFQ), the trader provides a quote by adding a spread over a \textit{prevalent market price}. For illiquid bonds, the market price is harder to observe, and traders often resort to available benchmark bond prices (such as MarketAxess, Bloomberg, etc.). In \cite{Bergault2023ModelingLI}, the concept of \textit{Fair Transfer Price} for an illiquid corporate bond was introduced which is derived from an infinite horizon stochastic optimal control problem (for maximizing the trader's expected P\&L, regularized by the quadratic variation). In this paper, we consider the same optimization objective, however, we approach the estimation of an optimal bidask spread quoting strategy in a data driven manner and show that it can be learned using Reinforcement Learning. Furthermore, we perform extensive outcome analysis to examine the reasonableness of the trained agent's behavior.
 [289] arXiv:2406.13000 (crosslist from cs.DS) [pdf, html, other]

Title: Randomized Greedy Online Edge Coloring Succeeds for Dense and RandomlyOrdered GraphsSubjects: Data Structures and Algorithms (cs.DS); Discrete Mathematics (cs.DM); Combinatorics (math.CO)
Vizing's theorem states that any graph of maximum degree $\Delta$ can be properly edge colored with at most $\Delta+1$ colors. In the online setting, it has been a matter of interest to find an algorithm that can properly edge color any graph on $n$ vertices with maximum degree $\Delta = \omega(\log n)$ using at most $(1+o(1))\Delta$ colors. Here we study the naïve random greedy algorithm, which simply chooses a legal color uniformly at random for each edge upon arrival. We show that this algorithm can $(1+\epsilon)\Delta$color the graph for arbitrary $\epsilon$ in two contexts: first, if the edges arrive in a uniformly random order, and second, if the edges arrive in an adversarial order but the graph is sufficiently dense, i.e., $n = O(\Delta)$. Prior to this work, the random greedy algorithm was only known to succeed in trees.
Our second result is applicable even when the adversary is adaptive, and therefore implies the existence of a deterministic edge coloring algorithm which $(1+\epsilon)\Delta$ edge colors a dense graph. Prior to this, the best known deterministic algorithm for this problem was the simple greedy algorithm which utilized $2\Delta1$ colors.  [290] arXiv:2406.13036 (crosslist from stat.ML) [pdf, html, other]

Title: Sharp detection of lowdimensional structure in probability measures via dimensional logarithmic Sobolev inequalitiesSubjects: Machine Learning (stat.ML); Machine Learning (cs.LG); Probability (math.PR); Statistics Theory (math.ST); Computation (stat.CO)
Identifying lowdimensional structure in highdimensional probability measures is an essential preprocessing step for efficient sampling. We introduce a method for identifying and approximating a target measure $\pi$ as a perturbation of a given reference measure $\mu$ along a few significant directions of $\mathbb{R}^{d}$. The reference measure can be a Gaussian or a nonlinear transformation of a Gaussian, as commonly arising in generative modeling. Our method extends prior work on minimizing majorizations of the KullbackLeibler divergence to identify optimal approximations within this class of measures. Our main contribution unveils a connection between the \emph{dimensional} logarithmic Sobolev inequality (LSI) and approximations with this ansatz. Specifically, when the target and reference are both Gaussian, we show that minimizing the dimensional LSI is equivalent to minimizing the KL divergence restricted to this ansatz. For general nonGaussian measures, the dimensional LSI produces majorants that uniformly improve on previous majorants for gradientbased dimension reduction. We further demonstrate the applicability of this analysis to the squared Hellinger distance, where analogous reasoning shows that the dimensional Poincaré inequality offers improved bounds.
 [291] arXiv:2406.13041 (crosslist from cs.LG) [pdf, html, other]

Title: Accelerated Stochastic MinMax Optimization Based on Biascorrected MomentumSubjects: Machine Learning (cs.LG); Optimization and Control (math.OC)
Lowerbound analyses for nonconvex stronglyconcave minimax optimization problems have shown that stochastic firstorder algorithms require at least $\mathcal{O}(\varepsilon^{4})$ oracle complexity to find an $\varepsilon$stationary point. Some works indicate that this complexity can be improved to $\mathcal{O}(\varepsilon^{3})$ when the loss gradient is Lipschitz continuous. The question of achieving enhanced convergence rates under distinct conditions, remains unresolved. In this work, we address this question for optimization problems that are nonconvex in the minimization variable and strongly concave or PolyakLojasiewicz (PL) in the maximization variable. We introduce novel biascorrected momentum algorithms utilizing efficient Hessianvector products. We establish convergence conditions and demonstrate a lower iteration complexity of $\mathcal{O}(\varepsilon^{3})$ for the proposed algorithms. The effectiveness of the method is validated through applications to robust logistic regression using realworld datasets.
 [292] arXiv:2406.13047 (crosslist from condmat.suprcon) [pdf, html, other]

Title: Symplectic Representation of the GinzburgLandau TheoryComments: 9 pages, 1 figuresSubjects: Superconductivity (condmat.suprcon); Mathematical Physics (mathph)
In this work, the GinzburgLandau theory is represented on a symplectic manifold with a phase space content. The order parameter is defined by a quasiprobability amplitude, which gives rise to a quasiprobability distribution function, i.e., a Wignertype function. The starting point is the thermal group representation of Euclidean symmetries and gauge symmetry. Wellknown basic results on the behavior of a superconductor are rederived, providing the consistency of representation. The critical superconducting current density is determined and its usual behavior is inferred. The negativety factor associated with the quasidistribution function is analyzed, providing information about the nonclassicality nature of the superconductor state in the region closest to the edge of the superconducting material.
 [293] arXiv:2406.13071 (crosslist from condmat.statmech) [pdf, html, other]

Title: Structural analysis of Gibbs states and metastates in shortrange classical spin glasses: indecomposable metastates, dynamicallyfrozen states, and metasymmetryComments: 69 pages (apologies)Subjects: Statistical Mechanics (condmat.statmech); Disordered Systems and Neural Networks (condmat.disnn); Mathematical Physics (mathph)
We consider shortrange classical spin glasses, or other disordered systems, consisting of Ising spins. For a lowtemperature Gibbs state in infinite size in such a system, for given random bonds, it is controversial whether its decomposition into pure states will be trivial or nontrivial. We undertake a general study of the overall structure of this problem, based on metastates, which are essential to prove the existence of a thermodynamic limit. A metastate is a probability distribution on Gibbs states, for given disorder, that satisfies certain covariance properties. First, we prove that any metastate can be decomposed as a mixture of indecomposable metastates, and that all Gibbs states drawn from an indecomposable metastate are alike macroscopically. Next, we consider stochastic stability of a metastate under random perturbations of the disorder, and prove that any metastate is stochastically stable. Dynamicallyfrozen states play a role in the analysis of Gibbs states drawn from a metastate, either as states or as parts of states. Using a mapping into real Hilbert space, we prove results about Gibbs states, and classify them into six types. Any indecomposable metastate has a compact symmetry group, though it may be trivial; we call this a metasymmetry. Metastateaverage states are studied, and can be related to states arising dynamically at long times after a quench from high temperature, under some conditions. Many features that are permitted by general results are already present in replica symmetry breaking (RSB). Our results are for cases both with and without spinflip symmetry of the Hamiltonian and, technically, we use mixed $p$spininteraction models.
 [294] arXiv:2406.13166 (crosslist from cs.LG) [pdf, other]

Title: Enhancing supply chain security with automated machine learningComments: 22 pagesSubjects: Machine Learning (cs.LG); General Economics (econ.GN); Optimization and Control (math.OC)
This study tackles the complexities of global supply chains, which are increasingly vulnerable to disruptions caused by port congestion, material shortages, and inflation. To address these challenges, we explore the application of machine learning methods, which excel in predicting and optimizing solutions based on large datasets. Our focus is on enhancing supply chain security through fraud detection, maintenance prediction, and material backorder forecasting. We introduce an automated machine learning framework that streamlines data analysis, model construction, and hyperparameter optimization for these tasks. By automating these processes, our framework improves the efficiency and effectiveness of supply chain security measures. Our research identifies key factors that influence machine learning performance, including sampling methods, categorical encoding, feature selection, and hyperparameter optimization. We demonstrate the importance of considering these factors when applying machine learning to supply chain challenges. Traditional mathematical programming models often struggle to cope with the complexity of largescale supply chain problems. Our study shows that machine learning methods can provide a viable alternative, particularly when dealing with extensive datasets and complex patterns. The automated machine learning framework presented in this study offers a novel approach to supply chain security, contributing to the existing body of knowledge in the field. Its comprehensive automation of machine learning processes makes it a valuable contribution to the domain of supply chain management.
 [295] arXiv:2406.13245 (crosslist from condmat.disnn) [pdf, html, other]

Title: Free energy equivalence between meanfield models and nonsparsely diluted meanfield modelsComments: 9 pages, 0 figureSubjects: Disordered Systems and Neural Networks (condmat.disnn); Statistical Mechanics (condmat.statmech); Mathematical Physics (mathph); Probability (math.PR)
We studied nonsparsely diluted meanfield models that differ from sparsely diluted meanfield models, such as the VianaBray model. We prove that the free energy of nonsparsely diluted meanfield models coincides exactly with that of the corresponding meanfield models with different parameters in ferromagnetic and spinglass models composed of any discrete spin $S$ in the thermodynamic limit. Our results are a broad generalization of the results of a previous study [Bovier and Gayrard, J. Stat. Phys. 72, 643 (1993)], where the densely diluted meanfield ferromagnetic Ising model (diluted CurieWeiss model) was analyzed rigorously, and it was proven that its free energy was exactly equivalent to that of the corresponding meanfield model (CurieWeiss model) with different parameters.
 [296] arXiv:2406.13423 (crosslist from nlin.SI) [pdf, html, other]

Title: Lagrangian multiform structure of discrete and semidiscrete KP systemsComments: 25 pages, 1 figureSubjects: Exactly Solvable and Integrable Systems (nlin.SI); Mathematical Physics (mathph)
A variational structure for the potential AKP system is established using the novel formalism of a Lagrangian multiforms. The structure comprises not only the fully discrete equation on the 3D lattice, but also its semidiscrete variants including several differentialdifference equations asssociated with, and compatible with, the partial difference equation. To this end, an overview is given of the various (discrete and semidiscrete) variants of the KP system, and their associated Lax representations, including a novel `generating PDE' for the KP hierarchy. The exterior derivative of the Lagrangian 3form for the lattice potential KP equation is shown to exhibit a doublezero structure, which implies the corresponding generalised EulerLagrange equations. Alongside the 3form structures, we develop a variational formulation of the corresponding Lax systems via the square eigenfunction representation arising from the relevant direct linearization scheme.
 [297] arXiv:2406.13425 (crosslist from stat.ML) [pdf, html, other]

Title: Coupled InputOutput Dimension Reduction: Application to Goaloriented Bayesian Experimental Design and Global Sensitivity AnalysisSubjects: Machine Learning (stat.ML); Machine Learning (cs.LG); Statistics Theory (math.ST)
We introduce a new method to jointly reduce the dimension of the input and output space of a highdimensional function. Choosing a reduced input subspace influences which output subspace is relevant and vice versa. Conventional methods focus on reducing either the input or output space, even though both are often reduced simultaneously in practice. Our coupled approach naturally supports goaloriented dimension reduction, where either an input or output quantity of interest is prescribed. We consider, in particular, goaloriented sensor placement and goaloriented sensitivity analysis, which can be viewed as dimension reduction where the most important output or, respectively, input components are chosen. Both applications present difficult combinatorial optimization problems with expensive objectives such as the expected information gain and Sobol indices. By optimizing gradientbased bounds, we can determine the most informative sensors and most sensitive parameters as the largest diagonal entries of some diagnostic matrices, thus bypassing the combinatorial optimization and objective evaluation.
 [298] arXiv:2406.13459 (crosslist from nlin.SI) [pdf, html, other]

Title: The RiemannHilbert approach for the nonlocal derivative nonlinear Schr\"odinger equation with nonzero boundary conditionsSubjects: Exactly Solvable and Integrable Systems (nlin.SI); Mathematical Physics (mathph)
In this paper, the nonlocal reverse spacetime derivative nonlinear Schrödinger equation under nonzero boundary conditions is investigated using the RiemannHilbert (RH) approach. The direct scattering problem focuses on the analyticity, symmetries, and asymptotic behaviors of the Jost eigenfunctions and scattering matrix functions, leading to the construction of the corresponding RH problem. Then, in the inverse scattering problem, the Plemelj formula is employed to solve the RH problem. So the reconstruction formula, trace formulae, $\theta$ condition, and exact expression of the singlepole and doublepole solutions are obtained. Furthermore, darkdark solitons, brightdark solitons, and breather solutions of the reverse spacetime derivative nonlinear Schrödinger equation are presented along with their dynamic behaviors summarized through graphical simulation.
 [299] arXiv:2406.13486 (crosslist from qfin.MF) [pdf, html, other]

Title: MeanVariance Portfolio Selection in LongTerm Investments with Unknown Distribution: Online Estimation, Risk Aversion under Ambiguity, and Universality of AlgorithmsComments: 21 pages, working paper, first draft version (may contain errors)Subjects: Mathematical Finance (qfin.MF); Machine Learning (cs.LG); Probability (math.PR); Portfolio Management (qfin.PM)
The standard approach for constructing a MeanVariance portfolio involves estimating parameters for the model using collected samples. However, since the distribution of future data may not resemble that of the training set, the outofsample performance of the estimated portfolio is worse than one derived with true parameters, which has prompted several innovations for better estimation. Instead of treating the data without a timing aspect as in the common trainingbacktest approach, this paper adopts a perspective where data gradually and continuously reveal over time. The original model is recast into an online learning framework, which is free from any statistical assumptions, to propose a dynamic strategy of sequential portfolios such that its empirical utility, Sharpe ratio, and growth rate asymptotically achieve those of the true portfolio, derived with perfect knowledge of the future data.
When the distribution of future data has a normal shape, the growth rate of wealth is shown to increase by lifting the portfolio along the efficient frontier through the calibration of risk aversion. Since risk aversion cannot be appropriately predetermined, another proposed algorithm updating this coefficient over time forms a dynamic strategy approaching the optimal empirical Sharpe ratio or growth rate associated with the true coefficient. The performance of these proposed strategies is universally guaranteed under specific stochastic markets. Furthermore, in stationary and ergodic markets, the socalled Bayesian strategy utilizing true conditional distributions, based on observed past market information during investment, almost surely does not perform better than the proposed strategies in terms of empirical utility, Sharpe ratio, or growth rate, which, in contrast, do not rely on conditional distributions.  [300] arXiv:2406.13522 (crosslist from eess.SY) [pdf, html, other]

Title: Measuredstate conditioned recursive feasibility for stochastic model predictive controlSubjects: Systems and Control (eess.SY); Optimization and Control (math.OC)
In this paper, we address the problem of designing stochastic model predictive control (MPC) schemes for linear systems affected by unbounded disturbances. The contribution of the paper is twofold. First, motivated by the difficulty of guaranteeing recursive feasibility in this framework, due to the nonzero probability of violating chanceconstraints in the case of unbounded noise, we introduce the novel definition of measuredstate conditioned recursive feasibility in expectation. Second, we construct a stochastic MPC scheme, based on the introduction of ellipsoidal probabilistic reachable sets, which implements a closedloop initialization strategy, i.e., the current measuredstate is employed for initializing the optimization problem. This new scheme is proven to satisfy the novel definition of recursive feasibility, and its superiority with respect to openloop initialization schemes, arising from the fact that one never neglects the information brought by the current measurement, is shown through numerical examples.
 [301] arXiv:2406.13529 (crosslist from cs.DM) [pdf, other]

Title: GMSNP and Finite StructuresSubjects: Discrete Mathematics (cs.DM); Combinatorics (math.CO); Logic (math.LO)
Given an (infinite) relational structure $\mathbb S$, we say that a finite structure $\mathbb C$ is a minimal finite factor of $\mathbb S$ if for every finite structure $\mathbb A$ there is a homomorphism $\mathbb S\to \mathbb A$ if and only if there is a homomorphism $\mathbb{C} \to \mathbb{A}$. In this brief note we prove that if CSP($\mathbb S$) is in GMSNP, then $\mathbb S$ has a minimal finite factor $\mathbb C$, and moreover, CSP($\mathbb C$) reduces in polynomial time to CSP($\mathbb S$). We discuss two nice applications of this result. First, we see that if a finite promise constraint satisfaction problem PCSP($\mathbb A,\mathbb B$) has a tractable GMSNP sandwich, then it has a tractable finite sandwich. We also show that if $\mathbb G$ is a nonbipartite (possibly infinite) graph with finite chromatic number, and CSP($\mathbb G$) is in GMSNP, then CSP($\mathbb G$) in NPcomplete, partially answering a question recently asked by Bodirsky and GuzmánPro.
 [302] arXiv:2406.13533 (crosslist from cs.LG) [pdf, other]

Title: DRACO: Decentralized Asynchronous Federated Learning over Continuous RowStochastic Network MatricesComments: This paper has been submitted to a peerreviewed journal and is currently under reviewSubjects: Machine Learning (cs.LG); Information Theory (cs.IT); Networking and Internet Architecture (cs.NI)
Recent developments and emerging use cases, such as smart Internet of Things (IoT) and Edge AI, have sparked considerable interest in the training of neural networks over fully decentralized (serverless) networks. One of the major challenges of decentralized learning is to ensure stable convergence without resorting to strong assumptions applied for each agent regarding data distributions or updating policies. To address these issues, we propose DRACO, a novel method for decentralized asynchronous Stochastic Gradient Descent (SGD) over rowstochastic gossip wireless networks by leveraging continuous communication. Our approach enables edge devices within decentralized networks to perform local training and model exchanging along a continuous timeline, thereby eliminating the necessity for synchronized timing. The algorithm also features a specific technique of decoupling communication and computation schedules, which empowers complete autonomy for all users and manageable instructions for stragglers. Through a comprehensive convergence analysis, we highlight the advantages of asynchronous and autonomous participation in decentralized optimization. Our numerical experiments corroborate the efficacy of the proposed technique.
 [303] arXiv:2406.13569 (crosslist from cs.LG) [pdf, html, other]

Title: Bayes' capacity as a measure for reconstruction attacks in federated learningSayan Biswas, Mark Dras, Pedro Faustini, Natasha Fernandes, Annabelle McIver, Catuscia Palamidessi, Parastoo SadeghiSubjects: Machine Learning (cs.LG); Artificial Intelligence (cs.AI); Cryptography and Security (cs.CR); Information Theory (cs.IT)
Within the machine learning community, reconstruction attacks are a principal attack of concern and have been identified even in federated learning, which was designed with privacy preservation in mind. In federated learning, it has been shown that an adversary with knowledge of the machine learning architecture is able to infer the exact value of a training element given an observation of the weight updates performed during stochastic gradient descent. In response to these threats, the privacy community recommends the use of differential privacy in the stochastic gradient descent algorithm, termed DPSGD. However, DP has not yet been formally established as an effective countermeasure against reconstruction attacks. In this paper, we formalise the reconstruction threat model using the informationtheoretic framework of quantitative information flow. We show that the Bayes' capacity, related to the Sibson mutual information of order infinity, represents a tight upper bound on the leakage of the DPSGD algorithm to an adversary interested in performing a reconstruction attack. We provide empirical results demonstrating the effectiveness of this measure for comparing mechanisms against reconstruction threats.
 [304] arXiv:2406.13590 (crosslist from condmat.soft) [pdf, html, other]

Title: Chirality Effects in Molecular ChainmailComments: 18 pages, 12 figuresSubjects: Soft Condensed Matter (condmat.soft); Statistical Mechanics (condmat.statmech); Differential Geometry (math.DG); General Topology (math.GN)
Motivated by the observation of positive Gaussian curvature in kinetoplast DNA networks, we consider the effect of linking chirality in square lattice molecular chainmail networks using Langevin dynamics simulations and constrained gradient optimization. Linking chirality here refers to ordering of overunder versus underover linkages between a loop and its neighbors. We consider fully alternating linking, maximally nonalternating, and partially nonalternating linking chiralities. We find that in simulations of polymer chainmail networks, the linking chirality dictates the sign of the Gaussian curvature of the final state of the chainmail membranes. Alternating networks have positive Gaussian curvature, similar to what is observed in kinetoplast DNA networks. Maximally nonalternating networks form isotropic membranes with negative Gaussian curvature. Partially nonalternating networks form flat diamondshaped sheets which undergo a thermal folding transition when sufficiently large, similar to the crumpling transition in tethered membranes. We further investigate this topologycurvature relationship on geometric grounds by considering the tightest possible configurations and the constraints that must be satisfied to achieve them.
 [305] arXiv:2406.13624 (crosslist from hepth) [pdf, other]

Title: Generalized $ \widetilde{W} $ algebrasComments: 47 pages, 2 figuresSubjects: High Energy Physics  Theory (hepth); Mathematical Physics (mathph)
Recently, a new generalized family of infinitedimensional $ \widetilde{W} $ algebras, each associated with a particular element of a commutative subalgebra of the $ W_{1+\infty} $ algebra, was described. This paper provides a comprehensive account of the aforementioned association, accompanied by the requisite proofs and illustrative examples. This approach allows a derivation of Ward identities for selected WLZZ matrix models and the expansion of corresponding $ W $operators in terms of an infinite set of variables $ p_k $.
 [306] arXiv:2406.13633 (crosslist from cs.LG) [pdf, html, other]

Title: Reinforcement Learning for InfiniteHorizon AverageReward MDPs with Multinomial Logistic Function ApproximationSubjects: Machine Learning (cs.LG); Optimization and Control (math.OC)
We study modelbased reinforcement learning with nonlinear function approximation where the transition function of the underlying Markov decision process (MDP) is given by a multinomial logistic (MNL) model. In this paper, we develop two algorithms for the infinitehorizon average reward setting. Our first algorithm \texttt{UCRL2MNL} applies to the class of communicating MDPs and achieves an $\tilde{\mathcal{O}}(dD\sqrt{T})$ regret, where $d$ is the dimension of feature mapping, $D$ is the diameter of the underlying MDP, and $T$ is the horizon. The second algorithm \texttt{OVIFHMNL} is computationally more efficient and applies to the more general class of weakly communicating MDPs, for which we show a regret guarantee of $\tilde{\mathcal{O}}(d^{2/5} \mathrm{sp}(v^*)T^{4/5})$ where $\mathrm{sp}(v^*)$ is the span of the associated optimal bias function.
We also prove a lower bound of $\Omega(d\sqrt{DT})$ for learning communicating MDPs with MNL transitions of diameter at most $D$. Furthermore, we show a regret lower bound of $\Omega(dH^{3/2}\sqrt{K})$ for learning $H$horizon episodic MDPs with MNL function approximation where $K$ is the number of episodes, which improves upon the bestknown lower bound for the finitehorizon setting.  [307] arXiv:2406.13635 (crosslist from stat.ME) [pdf, html, other]

Title: Temporal label recovery from noisy dynamical dataComments: 20 pages, 4 figuresSubjects: Methodology (stat.ME); Statistics Theory (math.ST); Applications (stat.AP)
Analyzing dynamical data often requires information of the temporal labels, but such information is unavailable in many applications. Recovery of these temporal labels, closely related to the seriation or sequencing problem, becomes crucial in the study. However, challenges arise due to the nonlinear nature of the data and the complexity of the underlying dynamical system, which may be periodic or nonperiodic. Additionally, noise within the feature space complicates the theoretical analysis. Our work develops spectral algorithms that leverage manifold learning concepts to recover temporal labels from noisy data. We first construct the graph Laplacian of the data, and then employ the second (and the third) Fiedler vectors to recover temporal labels. This method can be applied to both periodic and aperiodic cases. It also does not require monotone properties on the similarity matrix, which are commonly assumed in existing spectral seriation algorithms. We develop the $\ell_{\infty}$ error of our estimators for the temporal labels and ranking, without assumptions on the eigengap. In numerical analysis, our method outperforms spectral seriation algorithms based on a similarity matrix. The performance of our algorithms is further demonstrated on a synthetic biomolecule data example.
 [308] arXiv:2406.13657 (crosslist from cs.LO) [pdf, html, other]

Title: The strength of the dominance ruleComments: To appear in the proceedings of the 27th International Conference on Theory and Applications of Satisfiability Testing (SAT 2024)Subjects: Logic in Computer Science (cs.LO); Logic (math.LO)
It has become standard that, when a SAT solver decides that a CNF $\Gamma$ is unsatisfiable, it produces a certificate of unsatisfiability in the form of a refutation of $\Gamma$ in some proof system. The system typically used is DRAT, which is equivalent to extended resolution (ER)  for example, until this year DRAT refutations were required in the annual SAT competition. Recently [Bogaerts et al.~2023] introduced a new proof system, associated with the tool VeriPB, which is at least as strong as DRAT and is further able to handle certain symmetrybreaking techniques. We show that this system simulates the proof system $G_1$, which allows limited reasoning with QBFs and forms the first level above ER in a natural hierarchy of proof systems. This hierarchy is not known to be strict, but nevertheless this is evidence that the system of [Bogaerts et al. 2023] is plausibly strictly stronger than ER and DRAT. In the other direction, we show that symmetrybreaking for a single symmetry can be handled inside ER.
 [309] arXiv:2406.13661 (crosslist from cs.LG) [pdf, html, other]

Title: Hitchhiker's guide on EnergyBased Models: a comprehensive review on the relation with other generative models, sampling and statistical physicsDavide Carbone (1 and 2) ((1) Dipartimento di Scienze Matematiche, Politecnico di Torino, Torino, Italy, (2) INFN, Sezione di Torino, Torino, Italy)Subjects: Machine Learning (cs.LG); Mathematical Physics (mathph); Applied Physics (physics.appph); Data Analysis, Statistics and Probability (physics.dataan)
EnergyBased Models (EBMs) have emerged as a powerful framework in the realm of generative modeling, offering a unique perspective that aligns closely with principles of statistical mechanics. This review aims to provide physicists with a comprehensive understanding of EBMs, delineating their connection to other generative models such as Generative Adversarial Networks (GANs), Variational Autoencoders (VAEs), and Normalizing Flows. We explore the sampling techniques crucial for EBMs, including Markov Chain Monte Carlo (MCMC) methods, and draw parallels between EBM concepts and statistical mechanics, highlighting the significance of energy functions and partition functions. Furthermore, we delve into stateoftheart training methodologies for EBMs, covering recent advancements and their implications for enhanced model performance and efficiency. This review is designed to clarify the often complex interconnections between these models, which can be challenging due to the diverse communities working on the topic.
 [310] arXiv:2406.13680 (crosslist from physics.fludyn) [pdf, html, other]

Title: Effects of settling on inertial particle slip velocity statistics in wall bounded flowsSubjects: Fluid Dynamics (physics.fludyn); Mathematical Physics (mathph); Atmospheric and Oceanic Physics (physics.aoph)
Developing reduced order models for the transport of solid particles in turbulence typically requires a statistical description of the particleturbulence interactions. In this work, we utilize a statistical framework to derive continuum equations for the moments of the slip velocity of inertial settling Lagrangian particles in a turbulent boundary layer. Using coupled EulerianLagrangian direct numerical simulations, we then identify the dominant mechanisms controlling the slip velocity variance, and find that for a range of St+, Sv+, and Re, the slip variance is primarily controlled by local differences between the "seen" variance and the particle velocity variance, while terms appearing due to the inhomogeneity of the turbulence are subleading until Sv+ becomes large. We also consider several comparative metrics to assess the relative magnitudes of the fluctuating slip velocity and the mean slip velocity, and we find that the vertical mean slip increases rapidly with Sv+, rendering the variance relatively small  an effect found to be most substantial for Sv+>1. Finally, we compare the results to a model of the acceleration variance Berk and Coletti (2021) based the concept of a response function described in Csanady (1963), highlighting the role of the crossing trajectories mechanism. We find that while there is good agreement for low Sv+, systematic errors remain, possibly due to implicit nonlocal effects arising from rapid particle settling and inhomogeneous turbulence. We conclude with a discussion of the implications of this work for modeling the transport of coarse dust grains in the atmospheric surface layer.
 [311] arXiv:2406.13701 (crosslist from nlin.PS) [pdf, html, other]

Title: Windwave interaction in finite depth: linear and nonlinear approaches, blowup and soliton breaking in finite time, integrability perspectivesSubjects: Pattern Formation and Solitons (nlin.PS); Mathematical Physics (mathph)
This work is an analytical investigation of the evolution of surface water waves in Miles and Jeffreys theories of wind wave interaction in water of finite depth. The present review is divided into two major parts. The first corresponds to the surface water waves in a linear regime and its nonlinear extensions. In this part, Miles theory of wave amplification by wind is extended to the case of finite depth. The dispersion relation provides a wave growth rate depending on depth. Our theoretical results are in good agreement with the data from the Australian Shallow Water Experiment and the data from the Lake George experiment. In the second part of this study, Jeffreys theory of wave amplification by wind is extended to the case of finite depth, where the SerreGreenNaghdi is derived. We find the solitary wave solution of the system, with an increasing amplitude under the action of the wind. This continuous increase in amplitude leads to the soliton breaking and blowup of the surface wave in finite time. The theoretical blowup time is calculated based on actual experimental data. By applying an appropriate perturbation method, the SGN equation yields Korteweg de Vries Burger equation (KdVB). We show that the continuous transfer of energy from wind to water results in the growth of the KdVB soliton amplitude, velocity, acceleration, and energy over time while its effective wavelength decreases. This phenomenon differs from the classical results of Jeffreys approach due to finite depth. Again, blowup and breaking occur in finite time. These times are calculated and expressed for soliton and windappropriate parameters and values. These values are measurable in usual experimental facilities. The kinematics of the breaking is studied, and a detailed analysis of the breaking time is conducted using various criteria. Finally, some integrability perspectives are presented.
 [312] arXiv:2406.13767 (crosslist from physics.fludyn) [pdf, html, other]

Title: A fully observercovariant formulation of the fluid dynamics of simple fluids: derivation, simple examples and a generalized OrrSommerfeld equationSubjects: Fluid Dynamics (physics.fludyn); Mathematical Physics (mathph)
We present a formalism to describe the motion of a fluid fully which is fully covariant with respect to arbitrary observers. To achieve fully covariance, we write prognostic equations for quantities that belong to the graded exterior algebra of the cotangent bundle of the manifold occupied by the fluid. With the new formalism, we consider problems of stability, and we derive a generalization of the OrrSommerfeld equation that describes the evolution of perturbations relative to an arbitrary observer. The latter is applied to cases where the observer is the Lagrangian observer comoving with the background flow.
 [313] arXiv:2406.13823 (crosslist from quantph) [pdf, other]

Title: Inevitable Negativity: Additivity Commands Negative Quantum Channel EntropyComments: 16 pages (main text) + 21 pages (appendix), 6 figures, comments are welcomeSubjects: Quantum Physics (quantph); Mathematical Physics (mathph)
Quantum channels represent a broad spectrum of operations crucial to quantum information theory, encompassing everything from the transmission of quantum information to the manipulation of various resources. In the domain of states, the concept of majorization serves as a fundamental tool for comparing the uncertainty inherent in both classical and quantum systems. This paper establishes a rigorous framework for assessing the uncertainty in both classical and quantum channels. By employing a specific class of superchannels, we introduce and elucidate three distinct approaches to channel majorization: constructive, axiomatic, and operational. Intriguingly, these methodologies converge to a consistent ordering. This convergence not only provides a robust basis for defining entropy functions for channels but also clarifies the interpretation of entropy in this broader context. Most notably, our findings reveal that any viable entropy function for quantum channels must assume negative values, thereby challenging traditional notions of entropy.
 [314] arXiv:2406.13824 (crosslist from cs.GT) [pdf, html, other]

Title: Symmetrically Fair Allocations of Indivisible GoodsSubjects: Computer Science and Game Theory (cs.GT); Optimization and Control (math.OC)
We consider allocating indivisible goods with provable fairness guarantees that are satisfied regardless of which bundle of items each agent receives. Symmetrical allocations of this type are known to exist for divisible resources, such as consensus splitting of a cake into parts, each having equal value for all agents, ensuring that in any allocation of the cake slices, no agent would envy another. For indivisible goods, one analogous concept relaxes envy freeness to guarantee the existence of an allocation in which any bundle is worth as much as any other, up to the value of a bounded number of items from the other bundle. Previous work has studied the number of items that need to be removed. In this paper, we improve upon these bounds for the specific setting in which the number of bundles equals the number of agents.
Concretely, we develop the theory of symmetrically envy free up to one good, or symEF1, allocations. We prove that a symEF1 allocation exists if the vertices of a related graph can be partitioned (colored) into as many independent sets as there are agents. This sufficient condition always holds for two agents, and for agents that have identical, disjoint, or binary valuations. We further prove conditions under which exponentiallymany distinct symEF1 allocations exist. Finally, we perform computational experiments to study the incidence of symEF1 allocations as a function of the number of agents and items when valuations are drawn uniformly at random.  [315] arXiv:2406.13879 (crosslist from quantph) [pdf, html, other]

Title: A Catalyst Framework for the Quantum Linear System Problem via the Proximal Point AlgorithmSubjects: Quantum Physics (quantph); Data Structures and Algorithms (cs.DS); Machine Learning (cs.LG); Optimization and Control (math.OC)
Solving systems of linear equations is a fundamental problem, but it can be computationally intensive for classical algorithms in high dimensions. Existing quantum algorithms can achieve exponential speedups for the quantum linear system problem (QLSP) in terms of the problem dimension, but even such a theoretical advantage is bottlenecked by the condition number of the coefficient matrix. In this work, we propose a new quantum algorithm for QLSP inspired by the classical proximal point algorithm (PPA). Our proposed method can be viewed as a metaalgorithm that allows inverting a modified matrix via an existing \texttt{QLSP\_solver}, thereby directly approximating the solution vector instead of approximating the inverse of the coefficient matrix. By carefully choosing the step size $\eta$, the proposed algorithm can effectively precondition the linear system to mitigate the dependence on condition numbers that hindered the applicability of previous approaches.
 [316] arXiv:2406.13928 (crosslist from cs.LG) [pdf, html, other]

Title: Optimal deep learning of holomorphic operators between Banach spacesSubjects: Machine Learning (cs.LG); Numerical Analysis (math.NA)
Operator learning problems arise in many key areas of scientific computing where Partial Differential Equations (PDEs) are used to model physical systems. In such scenarios, the operators map between Banach or Hilbert spaces. In this work, we tackle the problem of learning operators between Banach spaces, in contrast to the vast majority of past works considering only Hilbert spaces. We focus on learning holomorphic operators  an important class of problems with many applications. We combine arbitrary approximate encoders and decoders with standard feedforward Deep Neural Network (DNN) architectures  specifically, those with constant width exceeding the depth  under standard $\ell^2$loss minimization. We first identify a family of DNNs such that the resulting Deep Learning (DL) procedure achieves optimal generalization bounds for such operators. For standard fullyconnected architectures, we then show that there are uncountably many minimizers of the training problem that yield equivalent optimal performance. The DNN architectures we consider are `problem agnostic', with width and depth only depending on the amount of training data $m$ and not on regularity assumptions of the target operator. Next, we show that DL is optimal for this problem: no recovery procedure can surpass these generalization bounds up to log terms. Finally, we present numerical results demonstrating the practical performance on challenging problems including the parametric diffusion, NavierStokesBrinkman and Boussinesq PDEs.
 [317] arXiv:2406.13936 (crosslist from stat.ML) [pdf, html, other]

Title: CommunicationEfficient Adaptive Batch Size Strategies for Distributed Local Gradient MethodsSubjects: Machine Learning (stat.ML); Machine Learning (cs.LG); Optimization and Control (math.OC)
Modern deep neural networks often require distributed training with many workers due to their large size. As worker numbers increase, communication overheads become the main bottleneck in dataparallel minibatch stochastic gradient methods with periteration gradient synchronization. Local gradient methods like Local SGD reduce communication by only syncing after several local steps. Despite understanding their convergence in i.i.d. and heterogeneous settings and knowing the importance of batch sizes for efficiency and generalization, optimal local batch sizes are difficult to determine. We introduce adaptive batch size strategies for local gradient methods that increase batch sizes adaptively to reduce minibatch gradient variance. We provide convergence guarantees under homogeneous data conditions and support our claims with image classification experiments, demonstrating the effectiveness of our strategies in training and generalization.
 [318] arXiv:2406.13971 (crosslist from cs.LG) [pdf, html, other]

Title: Complex fractal trainability boundary can arise from trivial nonconvexityComments: 11 pages, 9 figures, preliminary testsSubjects: Machine Learning (cs.LG); Dynamical Systems (math.DS); Chaotic Dynamics (nlin.CD)
Training neural networks involves optimizing parameters to minimize a loss function, where the nature of the loss function and the optimization strategy are crucial for effective training. Hyperparameter choices, such as the learning rate in gradient descent (GD), significantly affect the success and speed of convergence. Recent studies indicate that the boundary between bounded and divergent hyperparameters can be fractal, complicating reliable hyperparameter selection. However, the nature of this fractal boundary and methods to avoid it remain unclear. In this study, we focus on GD to investigate the loss landscape properties that might lead to fractal trainability boundaries. We discovered that fractal boundaries can emerge from simple nonconvex perturbations, i.e., adding or multiplying cosine type perturbations to quadratic functions. The observed fractal dimensions are influenced by factors like parameter dimension, type of nonconvexity, perturbation wavelength, and perturbation amplitude. Our analysis identifies "roughness of perturbation", which measures the gradient's sensitivity to parameter changes, as the factor controlling fractal dimensions of trainability boundaries. We observed a clear transition from nonfractal to fractal trainability boundaries as roughness increases, with the critical roughness causing the perturbed loss function nonconvex. Thus, we conclude that fractal trainability boundaries can arise from very simple nonconvexity. We anticipate that our findings will enhance the understanding of complex behaviors during neural network training, leading to more consistent and predictable training strategies.
 [319] arXiv:2406.13989 (crosslist from stat.ML) [pdf, html, other]

Title: Random pairing MLE for estimation of item parameters in Rasch modelSubjects: Machine Learning (stat.ML); Information Theory (cs.IT); Machine Learning (cs.LG); Statistics Theory (math.ST)
The Rasch model, a classical model in the item response theory, is widely used in psychometrics to model the relationship between individuals' latent traits and their binary responses on assessments or questionnaires. In this paper, we introduce a new likelihoodbased estimator  random pairing maximum likelihood estimator ($\mathsf{RP\text{}MLE}$) and its bootstrapped variant multiple random pairing MLE ($\mathsf{MRP\text{}MLE}$) that faithfully estimate the item parameters in the Rasch model. The new estimators have several appealing features compared to existing ones. First, both work for sparse observations, an increasingly important scenario in the big data era. Second, both estimators are provably minimax optimal in terms of finite sample $\ell_{\infty}$ estimation error. Lastly, $\mathsf{RP\text{}MLE}$ admits precise distributional characterization that allows uncertainty quantification on the item parameters, e.g., construction of confidence intervals of the item parameters. The main idea underlying $\mathsf{RP\text{}MLE}$ and $\mathsf{MRP\text{}MLE}$ is to randomly pair useritem responses to form itemitem comparisons. This is carefully designed to reduce the problem size while retaining statistical independence. We also provide empirical evidence of the efficacy of the two new estimators using both simulated and real data.
 [320] arXiv:2406.14059 (crosslist from cs.GT) [pdf, other]

Title: Tracking solutions of timevarying variational inequalitiesSubjects: Computer Science and Game Theory (cs.GT); Machine Learning (cs.LG); Optimization and Control (math.OC); Machine Learning (stat.ML)
Tracking the solution of timevarying variational inequalities is an important problem with applications in game theory, optimization, and machine learning. Existing work considers timevarying games or timevarying optimization problems. For strongly convex optimization problems or strongly monotone games, these results provide tracking guarantees under the assumption that the variation of the timevarying problem is restrained, that is, problems with a sublinear solution path. In this work we extend existing results in two ways: In our first result, we provide tracking bounds for (1) variational inequalities with a sublinear solution path but not necessarily monotone functions, and (2) for periodic timevarying variational inequalities that do not necessarily have a sublinear solution pathlength. Our second main contribution is an extensive study of the convergence behavior and trajectory of discrete dynamical systems of periodic timevarying VI. We show that these systems can exhibit provably chaotic behavior or can converge to the solution. Finally, we illustrate our theoretical results with experiments.
 [321] arXiv:2406.14110 (crosslist from physics.fludyn) [pdf, html, other]

Title: Multiobjective optimization of the magnetic wiping process in dipcoatingSubjects: Fluid Dynamics (physics.fludyn); Optimization and Control (math.OC)
Electromagnetic wiping systems allow to premeter the coating thickness of the liquid metal on a moving substrate. These systems have the potential to provide a more uniform coating and significantly higher production rates compared to pneumatic wiping, but they require substantially larger amounts of energy. This work presents a multiobjective optimization accounting for (1) maximal wiping efficiency (2) maximal smoothness of the wiping meniscus, and (3) minimal Joule heating. We present the Pareto front, identifying the best wiping conditions given a set of weights for the three competing objectives. The optimization was based on a 1D steadystate integral model, whose prediction scales according to the Hartmann number (Ha). The optimization uses a multigradient approach, with gradients computed with a combination of finite differences and variational methods. The results show that the wiping efficiency depends solely on Ha and not the magnetic field distribution. Moreover, we show that the liquid thickness becomes insensitive to the intensity of the magnetic field above a certain threshold and that the current distribution (hence the Joule heating) is mildly affected by the magnetic field's intensity and shape.
 [322] arXiv:2406.14126 (crosslist from eess.SP) [pdf, other]

Title: Joint Optimization of Switching Point and Power Control in Dynamic TDD CellFree Massive MIMOComments: Presented at the Asilomar Conference on Signals, Systems, and Computers 2023Subjects: Signal Processing (eess.SP); Information Theory (cs.IT)
We consider a cellfree massive multipleinput multipleoutput (CFmMIMO) network operating in dynamic time division duplex (DTDD). The switching point between the uplink (UL) and downlink (DL) data transmission phases can be adapted dynamically to the instantaneous qualityofservice (QoS) requirements in order to improve energy efficiency (EE). To this end, we formulate a problem of optimizing the DTDD switching point jointly with the UL and DL power control coefficients, and the largescale fading decoding (LSFD) weights for EE maximization. Then, we propose an iterative algorithm to solve the formulated challenging problem using successive convex approximation with an approximate stationary solution. Simulation results show that optimizing switching points remarkably improves EE compared with baseline schemes that adjust switching points heuristically.
 [323] arXiv:2406.14153 (crosslist from quantph) [pdf, html, other]

Title: On random classical marginal problems with applications to quantum information theorySubjects: Quantum Physics (quantph); Mathematical Physics (mathph); Probability (math.PR)
In this paper, we study random instances of the classical marginal problem. We encode the problem in a graph, where the vertices have assigned fixed binary probability distributions, and edges have assigned random bivariate distributions having the incident vertex distributions as marginals. We provide estimates on the probability that a joint distribution on the graph exists, having the bivariate edge distributions as marginals. Our study is motivated by Fine's theorem in quantum mechanics. We study in great detail the graphs corresponding to CHSH and BellWigner scenarios providing rations of volumes between the local and nonsignaling polytopes.
 [324] arXiv:2406.14229 (crosslist from physics.fludyn) [pdf, html, other]

Title: Approaches to conservative Smoothed Particle Hydrodynamics with entropySubjects: Fluid Dynamics (physics.fludyn); Mathematical Physics (mathph)
Smoothed particle hydrodynamics (SPH) is typically used for barotropic fluids, where the pressure depends only on the local mass density. Here, we show how to incorporate the entropy into the SPH, so that the pressure can also depend on the temperature, while keeping the growth of the total entropy, conservation of the total energy, and symplecticity of the reversible part of the SPH equations. The SPH system of ordinary differential equations with entropy is derived by means of the Poisson reduction and the LagrangeEuler transformation. We present several approaches towards SPH with entropy, which are then illustrated on systems with discontinuities, on adiabatic and nonadiabatic expansion, and on the RayleighBeenard convection without the Boussinesq approximation. Finally, we show how to model hyperbolic heat conduction within the SPH, extending the SPH variables with not only entropy but also a heatfluxrelated vector field.
 [325] arXiv:2406.14246 (crosslist from qbio.QM) [pdf, html, other]

Title: NonNegative Universal Differential Equations With Applications in Systems BiologyComments: 6 pages, This work has been submitted to IFAC for possible publication. Initial submission was March 18, 2024Subjects: Quantitative Methods (qbio.QM); Machine Learning (cs.LG); Dynamical Systems (math.DS); Machine Learning (stat.ML)
Universal differential equations (UDEs) leverage the respective advantages of mechanistic models and artificial neural networks and combine them into one dynamic model. However, these hybrid models can suffer from unrealistic solutions, such as negative values for biochemical quantities. We present nonnegative UDE (nUDEs), a constrained UDE variant that guarantees nonnegative values. Furthermore, we explore regularisation techniques to improve generalisation and interpretability of UDEs.
 [326] arXiv:2406.14248 (crosslist from condmat.statmech) [pdf, html, other]

Title: Starving Random WalksComments: 21 pages, 5 figures. Contribution to the book "The Mathematics of Movement: an Interdisciplinary Approach to Mutual Challenges in Animal Ecology and Cell Biology" edited by Luca Giuggioli and Philip MainiSubjects: Statistical Mechanics (condmat.statmech); Mathematical Physics (mathph)
In this chapter, we review recent results on the starving random walk (RW) problem, a minimal model for resourcelimited exploration. Initially, each lattice site contains a single food unit, which is consumed upon visitation by the RW. The RW starves whenever it has not found any food unit within the previous $\mathcal{S}$ steps. To address this problem, the key observable corresponds to the intervisit time $\tau_k$ defined as the time elapsed between the finding of the $k^\text{th}$ and the $(k+1)^\text{th}$ food unit. By characterizing the maximum $M_n$ of the intervisit times $\tau_0,\dots,\tau_{n1}$, we will see how to obtain the number $N_\mathcal{S}$ of food units collected at starvation, as well as the lifetime $T_\mathcal{S}$ of the starving RW.
 [327] arXiv:2406.14292 (crosslist from stat.CO) [pdf, html, other]

Title: Proximal Interacting Particle Langevin AlgorithmsComments: 50 pagesSubjects: Computation (stat.CO); Optimization and Control (math.OC); Machine Learning (stat.ML)
We introduce a class of algorithms, termed Proximal Interacting Particle Langevin Algorithms (PIPLA), for inference and learning in latent variable models whose joint probability density is nondifferentiable. Leveraging proximal Markov chain Monte Carlo (MCMC) techniques and the recently introduced interacting particle Langevin algorithm (IPLA), we propose several variants within the novel proximal IPLA family, tailored to the problem of estimating parameters in a nondifferentiable statistical model. We prove nonasymptotic bounds for the parameter estimates produced by multiple algorithms in the strongly logconcave setting and provide comprehensive numerical experiments on various models to demonstrate the effectiveness of the proposed methods. In particular, we demonstrate the utility of the proposed family of algorithms on a toy hierarchical example where our assumptions can be checked, as well as on the problems of sparse Bayesian logistic regression, sparse Bayesian neural network, and sparse matrix completion. Our theory and experiments together show that PIPLA family can be the de facto choice for parameter estimation problems in latent variable models for nondifferentiable models.
 [328] arXiv:2406.14320 (crosslist from hepth) [pdf, html, other]

Title: Anyon condensation in mixedstate topological orderComments: 52 pages, 14 figuresSubjects: High Energy Physics  Theory (hepth); Strongly Correlated Electrons (condmat.strel); Mathematical Physics (mathph); Category Theory (math.CT); Quantum Physics (quantph)
We discuss anyon condensation in mixedstate topological order. The phases were recently conjectured to be classified by premodular fusion categories. Just like anyon condensation in purestate topological order, a bootstrap analysis shows condensable anyons are given by connected étale algebras. We explain how to perform generic anyon condensation including noninvertible anyons and successive condensations. Interestingly, some condensations lead to purestate topological orders. We clarify when this happens. We also compute topological invariants of equivalence classes.
 [329] arXiv:2406.14327 (crosslist from condmat.statmech) [pdf, html, other]

Title: Application of Haldane's statistical correlation theory in classical systemsSubjects: Statistical Mechanics (condmat.statmech); Mathematical Physics (mathph); Data Analysis, Statistics and Probability (physics.dataan); Quantum Physics (quantph)
This letter investigates the application of Haldane's statistical correlation theory in classical systems. A modified statistical correlation theory has been proposed by including nonlinearity into the original theory of Haldane. It is shown that indistinguishability can be introduced as a form of external statistical correlation into distinguishable systems. It is proved that this modified statistical correlation theory can be used to derive classical fractional exclusion statistics (CFES) using maximum entropy methods for a selfcorrelating system. An extended nonlinear correlation model based on power series expansion is also proposed, which can produce various intermediate statistical models.
 [330] arXiv:2406.14330 (crosslist from quantph) [pdf, html, other]

Title: Promise of Graph Sparsification and Decomposition for Noise Reduction in QAOA: Analysis for TrappedIon CompilationsSubjects: Quantum Physics (quantph); Data Structures and Algorithms (cs.DS); Optimization and Control (math.OC)
We develop new approximate compilation schemes that significantly reduce the expense of compiling the Quantum Approximate Optimization Algorithm (QAOA) for solving the MaxCut problem. Our main focus is on compilation with trappedion simulators using Pauli$X$ operations and alltoall Ising Hamiltonian $H_\text{Ising}$ evolution generated by MolmerSorensen or optical dipole force interactions, though some of our results also apply to standard gatebased compilations. Our results are based on principles of graph sparsification and decomposition; the former reduces the number of edges in a graph while maintaining its cut structure, while the latter breaks a weighted graph into a small number of unweighted graphs. Though these techniques have been used as heuristics in various hybrid quantum algorithms, there have been no guarantees on their performance, to the best of our knowledge. This work provides the first provable guarantees using sparsification and decomposition to improve quantum noise resilience and reduce quantum circuit complexity.
For quantum hardware that uses edgebyedge QAOA compilations, sparsification leads to a direct reduction in circuit complexity. For trappedion quantum simulators implementing alltoall $H_\text{Ising}$ pulses, we show that for a $(1\epsilon)$ factor loss in the MaxCut approximation ($\epsilon>0)$, our compilations improve the (worstcase) number of $H_\text{Ising}$ pulses from $O(n^2)$ to $O(n\log(n/\epsilon))$ and the (worstcase) number of Pauli$X$ bit flips from $O(n^2)$ to $O\left(\frac{n\log(n/\epsilon)}{\epsilon^2}\right)$ for $n$node graphs. We demonstrate significant reductions in noise are obtained in our new compilation approaches using theory and numerical calculations for trappedion hardware. We anticipate these approximate compilation techniques will be useful tools in a variety of future quantum computing experiments.  [331] arXiv:2406.14402 (crosslist from cs.LO) [pdf, html, other]

Title: Logicbased analogical proportionsSubjects: Logic in Computer Science (cs.LO); Discrete Mathematics (cs.DM); Logic (math.LO)
The author has recently introduced an abstract algebraic framework of analogical proportions within the general setting of universal algebra. The purpose of this paper is to lift that framework from universal algebra to the strictly more expressive setting of full firstorder logic. We show that the soobtained logicbased framework preserves all desired properties and we prove novel results in that extended setting.
 [332] arXiv:2406.14420 (crosslist from cs.LG) [pdf, html, other]

Title: Communicationefficient Vertical Federated Learning via Compressed Error FeedbackSubjects: Machine Learning (cs.LG); Distributed, Parallel, and Cluster Computing (cs.DC); Data Structures and Algorithms (cs.DS); Optimization and Control (math.OC)
Communication overhead is a known bottleneck in federated learning (FL). To address this, lossy compression is commonly used on the information communicated between the server and clients during training. In horizontal FL, where each client holds a subset of the samples, such communicationcompressed training methods have recently seen significant progress. However, in their vertical FL counterparts, where each client holds a subset of the features, our understanding remains limited. To address this, we propose an error feedback compressed vertical federated learning (EFVFL) method to train split neural networks. In contrast with previous communicationcompressed methods for vertical FL, EFVFL does not require a vanishing compression error for the gradient norm to converge to zero for smooth nonconvex problems. By leveraging error feedback, our method can achieve a $\mathcal{O}(1/T)$ convergence rate in the fullbatch case, improving over the stateoftheart $\mathcal{O}(1/\sqrt{T})$ rate under $\mathcal{O}(1/\sqrt{T})$ compression error, and matching the rate of uncompressed methods. Further, when the objective function satisfies the PolyakŁojasiewicz inequality, our method converges linearly. In addition to improving convergence rates, our method also supports the use of private labels. Numerical experiments show that EFVFL significantly improves over the prior art, confirming our theoretical results.
 [333] arXiv:2406.14458 (crosslist from cs.LG) [pdf, html, other]

Title: Centimeter Positioning Accuracy using AI/ML for 6G ApplicationsComments: 2 Pages, 2 Figures, ICMLCN Conference, Stockholm, SwedenSubjects: Machine Learning (cs.LG); Artificial Intelligence (cs.AI); Information Theory (cs.IT); Signal Processing (eess.SP)
This research looks at using AI/ML to achieve centimeterlevel user positioning in 6G applications such as the Industrial Internet of Things (IIoT). Initial results show that our AI/MLbased method can estimate user positions with an accuracy of 17 cm in an indoor factory environment. In this proposal, we highlight our approaches and future directions.
 [334] arXiv:2406.14495 (crosslist from cs.LG) [pdf, html, other]

Title: rKAN: Rational KolmogorovArnold NetworksComments: The implementations are available at this https URLSubjects: Machine Learning (cs.LG); Neural and Evolutionary Computing (cs.NE); Numerical Analysis (math.NA)
The development of KolmogorovArnold networks (KANs) marks a significant shift from traditional multilayer perceptrons in deep learning. Initially, KANs employed Bspline curves as their primary basis function, but their inherent complexity posed implementation challenges. Consequently, researchers have explored alternative basis functions such as Wavelets, Polynomials, and Fractional functions. In this research, we explore the use of rational functions as a novel basis function for KANs. We propose two different approaches based on Pade approximation and rational Jacobi functions as trainable basis functions, establishing the rational KAN (rKAN). We then evaluate rKAN's performance in various deep learning and physicsinformed tasks to demonstrate its practicality and effectiveness in function approximation.
 [335] arXiv:2406.14506 (crosslist from cs.DS) [pdf, html, other]

Title: Online Matching and Contention Resolution for Edge Arrivals with Vanishing ProbabilitiesJournalref: In EC 2024Subjects: Data Structures and Algorithms (cs.DS); Discrete Mathematics (cs.DM); Combinatorics (math.CO)
We study the performance of sequential contention resolution and matching algorithms on random graphs with vanishing edge probabilities. When the edges of the graph are processed in an adversariallychosen order, we derive a new OCRS that is $0.382$selectable, attaining the "independence benchmark" from the literature under the vanishing edge probabilities assumption. Complementary to this positive result, we show that no OCRS can be more than $0.390$selectable, significantly improving upon the upper bound of $0.428$ from the literature. We also derive negative results that are specialized to bipartite graphs or subfamilies of OCRS's. Meanwhile, when the edges of the graph are processed in a uniformly random order, we show that the simple greedy contention resolution scheme which accepts all active and feasible edges is $1/2$selectable. This result is tight due to a known upper bound. Finally, when the algorithm can choose the processing order, we show that a slight tweak to the random order  give each vertex a random priority and process edges in lexicographic order  results in a strictly better contention resolution scheme that is $1\ln(21/e)\approx0.510$selectable. Our positive results also apply to online matching on $1$uniform random graphs with vanishing (nonidentical) edge probabilities, extending and unifying some results from the random graphs literature.
 [336] arXiv:2406.14535 (crosslist from stat.ME) [pdf, html, other]

Title: On estimation and order selection for multivariate extremes via clusteringComments: 31 pages, 12 figuresSubjects: Methodology (stat.ME); Statistics Theory (math.ST)
We investigate the estimation of multivariate extreme models with a discrete spectral measure using spherical clustering techniques. The primary contribution involves devising a method for selecting the order, that is, the number of clusters. The method consistently identifies the true order, i.e., the number of spectral atoms, and enjoys intuitive implementation in practice. Specifically, we introduce an extra penalty term to the wellknown simplified average silhouette width, which penalizes small cluster sizes and small dissimilarities between cluster centers. Consequently, we provide a consistent method for determining the order of a maxlinear factor model, where a typical informationbased approach is not viable. Our second contribution is a largedeviationtype analysis for estimating the discrete spectral measure through clustering methods, which serves as an assessment of the convergence quality of clusteringbased estimation for multivariate extremes. Additionally, as a third contribution, we discuss how estimating the discrete measure can lead to parameter estimations of heavytailed factor models. We also present simulations and realdata studies that demonstrate order selection and factor model estimation.
Cross submissions for Friday, 21 June 2024 (showing 55 of 55 entries )
 [337] arXiv:1501.04519 (replaced) [pdf, html, other]

Title: Transcendental Brauer groups of products of CM elliptic curvesComments: Corrigendum added as an appendixJournalref: Journal of the London Mathematical Society, 93: 397419 (2016)Subjects: Number Theory (math.NT)
Let $L$ be a number field and let $E/L$ be an elliptic curve with complex multiplication by the ring of integers $\mathcal{O}_K$ of an imaginary quadratic field $K$. We use class field theory and results of Skorobogatov and Zarhin to compute the transcendental part of the Brauer group of the abelian surface $E\times E$. The results for the odd order torsion also apply to the Brauer group of the K3 surface $\textrm{Kum}(E\times E)$. We describe explicitly the elliptic curves $E/\mathbb{Q}$ with complex multiplication by $\mathcal{O}_K$ such that the Brauer group of $E\times E$ contains a transcendental element of odd order. We show that such an element gives rise to a BrauerManin obstruction to weak approximation on $\textrm{Kum}(E\times E)$, while there is no obstruction coming from the algebraic part of the Brauer group.
 [338] arXiv:1610.04505 (replaced) [pdf, html, other]

Title: Schertz style class invariants for higher degree CM fieldsSubjects: Number Theory (math.NT)
Special values of Siegel modular functions for $\operatorname{Sp} (\mathbb{Z})$ generate class fields of CM fields. They also yield abelian varieties with a known endomorphism ring. Smaller alternative values of modular functions that lie in the same class fields (class invariants) thus help to speed up the computation of those mathematical objects.
We show that modular functions for the subgroup $\Gamma^0 (N)\subseteq \operatorname{Sp}(\mathbb{Z})$ yield class invariants under some splitting conditions on $N$, generalising results due to Schertz from classical modular functions to Siegel modular functions. We show how to obtain all Galois conjugates of a class invariant by evaluating the same modular function in CM period matrices derived from an \emph{$N$system}. Such a system consists of quadratic polynomials with coefficients in the realquadratic subfield satisfying certain congruence conditions modulo $N$. We also examine conditions under which the minimal polynomial of a class invariant is real.
Examples show that we may obtain class invariants that are much smaller than in previous constructions.  [339] arXiv:1707.00199 (replaced) [pdf, html, other]

Title: Utility maximization in constrained and unbounded financial markets: Applications to indifference valuation, regime switching, consumption and EpsteinZin recursive utilityComments: 90 pagesSubjects: Probability (math.PR); Mathematical Finance (qfin.MF)
This memoir presents a systematic study of utility maximization problems for an investor in constrained and unbounded financial markets. Building upon the foundational work of Hu et al. (2005) [Ann. Appl. Probab., 15, 16911712] in a bounded framework, we extend our analysis to more challenging unbounded cases. Our methodology combines quadratic backward stochastic differential equations with unbounded solutions and convex duality methods. Central to our approach is the verification of the finite entropy condition, which plays a pivotal role in solving the underlying utility maximization problems and establishing the martingale property and convex duality representation of the value processes. Through four distinct applications, we first study utility indifference valuation of financial derivatives with unbounded payoffs, uncovering novel asymptotic behavior as the risk aversion parameter approaches zero or infinity. Furthermore, we study the regime switching market model with unbounded random endowments and consumptioninvestment problems with unbounded random endowments, both constrained to portfolios chosen from a convex and closed set. Finally, we investigate investmentconsumption problems involving an investor with EpsteinZin recursive utility in an unbounded financial market.
 [340] arXiv:1708.07300 (replaced) [pdf, html, other]

Title: Stability results of octahedrality in tensor product spacesComments: 6 pagesSubjects: Functional Analysis (math.FA)
We prove that there exists a finitedimensional Banach space $X$ such that $L_1^\mathbb C([0,1])\widehat{\otimes}_\varepsilon X$ fails the strong diameter two property and $L_\infty^\mathbb C([0,1])\widehat{\otimes}_\pi X^*$ fails to have octahedral norm. This proves that the octahedrality of the norm (respectively the strong diameter two property) is not automatically inherited from one factor by taking projective tensor product (respectively injective tensor product), which answers [16,Question 4.4].
 [341] arXiv:1906.05757 (replaced) [pdf, html, other]

Title: The rank of sparse random matricesComments: This article supersedes arXiv:1810.07390Subjects: Combinatorics (math.CO); Discrete Mathematics (cs.DM); Probability (math.PR)
We determine the rank of a random matrix over an arbitrary field with prescribed numbers of nonzero entries in each row and column. As an application we obtain a formula for the rate of lowdensity parity check codes. This formula vindicates a conjecture of Lelarge (2013). The proofs are based on coupling arguments and a novel random perturbation, applicable to any matrix, that diminishes the number of short linear relations.
 [342] arXiv:2002.01904 (replaced) [pdf, html, other]

Title: An upper bound conjecture for the Yokota invariantComments: 39 pages, 26 figures. Updated with new title and a stronger focus on the upper bound conjecture; accepted for publication at Algebraic and Geometric TopologySubjects: Geometric Topology (math.GT)
We conjecture an upper bound on the growth of the Yokota invariant of polyhedral graphs, extending a previous result on the growth of the $6j$symbol. Using Barrett's Fourier transform we are able to prove this conjecture in a large family of examples. As a consequence of this result, we prove the TuraevViro Volume Conjecture for a new infinite family of hyperbolic manifolds.
 [343] arXiv:2007.07130 (replaced) [pdf, html, other]

Title: Moduli of hybrid curves I: Variations of canonical measuresComments: 65 pages, 4 figures, change of title. Final version, to appear in Annales scientifiques de l'ENSSubjects: Algebraic Geometry (math.AG); Combinatorics (math.CO); Complex Variables (math.CV); Differential Geometry (math.DG); Number Theory (math.NT)
The present paper is the first in a series devoted to the study of asymptotic geometry of Riemann surfaces and their moduli spaces.
We introduce the moduli space of hybrid curves as a new compactification of the moduli space of curves, refining the one obtained by Deligne and Mumford. This is the moduli space for multiscale geometric objects which mix complex and higher rank tropical and nonArchimedean geometries, reflecting both discrete and continuous features.
We define canonical measures on hybrid curves which combine and generalize ArakelovBergman measures on Riemann surfaces and Zhang measures on metric graphs.
We then show that the universal family of canonically measured hybrid curves over this moduli space varies continuously. This provides a precise link between the nonArchimedean Zhang measure and variations of ArakelovBergman measures in families of Riemann surfaces, answering a question which has been open since the pioneering work of Zhang on admissible pairing in the nineties.  [344] arXiv:2009.10125 (replaced) [pdf, html, other]

Title: Spectral geometry on manifolds with fibred boundary metrics I: Low energy resolventComments: minor corrections compared to the journal version, in particular in Theorem 8.4Journalref: Journal de l'Ecole polytechnique  Mathematiques 9 (2022), p. 9591019Subjects: Differential Geometry (math.DG); Analysis of PDEs (math.AP); Spectral Theory (math.SP)
We study the low energy resolvent of the Hodge Laplacian on a manifold equipped with a fibred boundary metric. We determine the precise asymptotic behavior of the resolvent as a fibred boundary (aka $\phi$) pseudodifferential operator when the resolvent parameter tends to zero. This generalizes previous work by Guillarmou and Sher who considered asymptotically conic metrics, which correspond to the special case when the fibres are points. The new feature in the case of nontrivial fibres is that the resolvent has different asymptotic behavior on the subspace of forms that are fibrewise harmonic and on its orthogonal complement. To deal with this, we introduce an appropriate 'split' pseudodifferential calculus, building on and extending work by Grieser and Hunsicker. Our work sets the basis for the discussion of spectral invariants on $\phi$manifolds.
 [345] arXiv:2104.02993 (replaced) [pdf, html, other]

Title: The homomorphism defect of an extended LevineTristram signature via twisted homologyAlice Merz (Università di Pisa)Comments: 39 pages, 12 figures; Revised version published in Journal of Knot Theory and its RamificationsSubjects: Geometric Topology (math.GT)
Taking the LevineTristram signature of the closure of a braid defines a map from the braid group to the integers. A formula of Gambaudo and Ghys provides an evaluation of the homomorphism defect of this map in terms of the Burau representation and the Meyer cocycle. In 2017 Cimasoni and Conway generalized this formula to the multivariable signature of the closure of coloured tangles. In the present paper, we extend even further their result by using a different 4dimensional interpretation of the signature. We obtain an evaluation of the additivity defect in terms of the Maslov index and the isotropic functor $\mathscr{F}_\omega$. We also show that in the case of coloured braids this defect can be rewritten in terms of the Meyer cocycle and the coloured Gassner representation, making it a direct generalization of the formula of Gambaudo and Ghys.
 [346] arXiv:2107.01874 (replaced) [pdf, html, other]

Title: Schubert Eisenstein series and Poisson summation for Schubert varietiesComments: Minor revision due to referee comments. Short appendix addedSubjects: Number Theory (math.NT); Representation Theory (math.RT)
The first author and Bump defined Schubert Eisenstein series by restricting the summation in a degenerate Eisenstein series to a particular Schubert variety. In the case of $\mathrm{GL}_3$ over $\mathbb{Q}$ they proved that these Schubert Eisenstein series have meromorphic continuations in all parameters and conjectured the same is true in general. We revisit their conjecture and relate it to the program of Braverman, Kazhdan, Lafforgue, Ngô, and Sakellaridis aimed at establishing generalizations of the Poisson summation formula. We prove the Poisson summation formula for certain schemes closely related to Schubert varieties and use it to refine and establish the conjecture of the first author and Bump in many cases.
 [347] arXiv:2110.06721 (replaced) [pdf, html, other]

Title: Asymptotically rigid mapping class groups II: strand diagrams and nonpositive curvatureComments: 47 pages, 23 figures. Minor revisions. Accepted for publication in Transactions of the AMSSubjects: Group Theory (math.GR); Geometric Topology (math.GT)
In this article, we introduce a new family of groups, called Chambord groups and constructed from braided strand diagrams associated to specific semigroup presentations. It includes the asymptotically rigid mapping class groups previously studied by the authors such as the braided HigmanThompson groups and the braided Houghton groups. Our main result shows that polycyclic subgroups in Chambord groups are virtually abelian and undistorted.
 [348] arXiv:2112.08205 (replaced) [pdf, html, other]

Title: Odd moments for the trace of Frobenius and the SatoTate conjecture in arithmetic progressionsSubjects: Number Theory (math.NT)
In this paper, we consider the moments of the trace of Frobenius of elliptic curves if the trace is restricted to a fixed arithmetic progression. We determine the asymptotic behavior for the ratio of the $(2k+1)$th moment to the zeroeth moment as the size of the finite field $\mathbb{F}_{p^r}$ goes to infinity. These results follow from similar asymptotic formulas relating sums and moments of Hurwitz class numbers where the sums are restricted to certain arithmetic progressions. As an application, we prove that the distribution of the trace of Frobenius in arithmetic progressions is equidistributed with respect to the SatoTate measure.
 [349] arXiv:2112.13275 (replaced) [pdf, html, other]

Title: A note on the induced Ramsey theorem for spacesComments: Comments are welcomeSubjects: Combinatorics (math.CO)
The aim of this note is to give a simplified proof of the induced version of the Ramsey theorem for vector spaces first proved by H. J. Prömel.
 [350] arXiv:2201.02717 (replaced) [pdf, html, other]

Title: FrobeniusPoincar\'e function and HilbertKunz multiplicityComments: v3: In Rmk 5.5, an error is fixed; a reference is added. Other minor changesSubjects: Commutative Algebra (math.AC); Algebraic Geometry (math.AG); Complex Variables (math.CV)
We generalize the notion of HilbertKunz multiplicity of a graded triple $(M,R,I)$ in characteristic $p>0$ by proving that for any complex number $y$, the limit
$$\underset{n \to \infty}{\lim}(\frac{1}{p^n})^{\text{dim}(M)}\sum \limits_{j= \infty}^{\infty}\lambda \left( (\frac{M}{I^{[p^n]}M})_j\right)e^{iyj/p^n}$$ exists. We prove that the limiting function in the complex variable $y$ is entire and name this function the \textit{FrobeniusPoincaré function}. We establish various properties of FrobeniusPoincaré functions including its relation with the tight closure of the defining ideal $I$; and relate the study FrobeniusPoincaré functions to the behaviour of graded Betti numbers of $\frac{R}{I^{[p^n]}} $ as $n$ varies. Our description of FrobeniusPoincaré functions in dimension one and two and other examples raises questions on the structure of FrobeniusPoincaré functions in general.  [351] arXiv:2201.06375 (replaced) [pdf, html, other]

Title: Eigenvalue estimates on weighted manifoldsJournalref: Results Math. 79 (2024), no. 5, Paper No. 187Subjects: Differential Geometry (math.DG)
We derive various eigenvalue estimates for the Hodge Laplacian acting on differential forms on weighted Riemannian manifolds. Our estimates unify and extend various results from the literature and we provide a number of geometric applications. In particular, we derive an inequality which relates the eigenvalues of the Jacobi operator for (f)minimal hypersurfaces and the spectrum of the Hodge Laplacian.
 [352] arXiv:2203.04362 (replaced) [pdf, html, other]

Title: The Sobolev Wavefront Set of the Causal Propagator in Finite RegularitySubjects: Analysis of PDEs (math.AP); Mathematical Physics (mathph)
Given a globally hyperbolic spacetime $M=\mathbb{R}\times \Sigma$ of dimension four and regularity $C^\tau$, we estimate the Sobolev wavefront set of the causal propagator $K_G$ of the KleinGordon operator. In the smooth case, the propagator satisfies $WF'(K_G)=C$, where $C\subset T^*(M\times M)$ consists of those points $(\tilde{x},\tilde{\xi},\tilde{y},\tilde{\eta})$ such that $\tilde{\xi},\tilde{\eta}$ are cotangent to a null geodesic $\gamma$ at $\tilde{x}$ resp. $\tilde{y}$ and parallel transports of each other along $\gamma$.
We show that for $\tau>2$, $WF'^{2+\tau{\epsilon}}(K_G)\subset C$ for every ${\epsilon}>0$. Furthermore, in regularity $C^{\tau+2}$ with $\tau>2$, $C\subset WF'^{\frac{1}{2}}(K_G)\subset WF'^{\tau\epsilon}(K_G)\subset C$ holds for $0<\epsilon<\tau+\frac{1}{2}$.
In the ultrastatic case with $\Sigma$ compact, we show $WF'^{\frac{3}{2}+\tau\epsilon}(K_G)\subset C$ for $\epsilon >0$ and $\tau>2$ and $WF'^{\frac{3}{2}+\tau\epsilon}(K_G)= C$ for $\tau>3$ and $\epsilon<\tau3$. Moreover, we show that the global regularity of the propagator $K_G$ is $H^{\frac{1}{2}\epsilon}_{loc}(M\times M)$ as in the smooth case.  [353] arXiv:2203.16441 (replaced) [pdf, html, other]

Title: Mixed state representability of entropydensity pairsSubjects: Mathematical Physics (mathph); Quantum Physics (quantph)
We show the representability of densityentropy pairs with canonical and grandcanonical states, and we provide bounds on the kinetic energy of the representing states.
 [354] arXiv:2206.14183 (replaced) [pdf, html, other]

Title: Nonergodicity on SU(2) and SU(3) character varieties of the oncepunctured torusComments: 36 pages, 4 figures, accepted for publication in the journal Annales Henri LebesgueSubjects: Dynamical Systems (math.DS); Algebraic Geometry (math.AG); Symplectic Geometry (math.SG)
Utilizing KAM theory, we show that there are certain levels in relative SU(2) and SU(3) character varieties of the oncepunctured torus where the action of a single hyperbolic element is not ergodic.
 [355] arXiv:2207.02030 (replaced) [pdf, html, other]

Title: Uniform convergence of the FlemingViot process in a hard killing metastable caseSubjects: Probability (math.PR)
We study the longtime convergence of a FlemingViot process, in the case where the underlying process is a metastable diffusion killed when it reaches some level set. Through a coupling argument, we establish the longtime convergence of the FlemingViot process toward some stationary measure at an exponential rate independent of $N$, the size of the system, as well as uniform in time propagation of chaos estimates.
 [356] arXiv:2208.03439 (replaced) [pdf, html, other]

Title: Remark on a special class of Finsler $p$Laplacian equationComments: to appear in DCDSSSubjects: Analysis of PDEs (math.AP)
We investigate the anisotropic elliptic equation $\Delta_p^H u = g(u)$. Recently, Esposito, Riey, Sciunzi, and Vuono introduced an anisotropic Kelvin transform in their work \cite{ERSV2022} under the $(H_M)$ condition, where $H(\xi)=\sqrt{\langle M\xi,\xi\rangle}$ with a positive definite symmetric matrix $M$. Here, we emphasize that under the $(H_M)$ assumption, the Finsler $p$Laplacian and the classical $p$Laplacian operator are equivalent following a linear transformation. This equivalence offers us a more direct route to derive the pivotal findings presented in \cite{ERSV2022}. While this equivalence is crucial and noteworthy, to our knowledge, it has not been explicitly stated in the current literature.
 [357] arXiv:2208.06929 (replaced) [pdf, html, other]

Title: Discrete sets definable in strong expansions of ordered Abelian groupsComments: 45 pages. This newly revised version corrects some errors from the original version (pointed out by the anonymous referee) and some arguments have been significantly revised for claritySubjects: Logic (math.LO)
We study the structure of infinite discrete sets D definable in expansions of ordered Abelian groups whose theories are strong and definably complete, with particular emphasis on the set D' comprised of differences between successive elements. In particular, if the burden of the structure is at most n, then the result of applying the operation taking D to D' n times must be a finite set (Theorem 1.1). In the case when the structure is densely ordered and has burden 2, we show that any definable unary discrete set must be definable in some elementary extension of the structure (R; <, +, Z) (Theorem 1.3).
 [358] arXiv:2208.07540 (replaced) [pdf, html, other]

Title: LargeScale Minimization of the Pseudospectral AbscissaComments: 31 pages, 5 figuresSubjects: Numerical Analysis (math.NA); Optimization and Control (math.OC)
This work concerns the minimization of the pseudospectral abscissa of a matrixvalued function dependent on parameters analytically. The problem is motivated by robust stability and transient behavior considerations for a linear control system that has optimization parameters. We describe a subspace procedure to cope with the setting when the matrixvalued function is of large size. The proposed subspace procedure solves a sequence of reduced problems obtained by restricting the matrixvalued function to small subspaces, whose dimensions increase gradually. It possesses desirable features such as a superlinear convergence exhibited by the decay in the errors of the minimizers of the reduced problems. In mathematical terms, the problem we consider is a largescale nonconvex minimax eigenvalue optimization problem such that the eigenvalue function appears in the constraint of the inner maximization problem. Devising and analyzing a subspace framework for the minimax eigenvalue optimization problem at hand with the eigenvalue function in the constraint require special treatment that makes use of a Lagrangian and dual variables. There are notable advantages in minimizing the pseudospectral abscissa over maximizing the distance to instability or minimizing the $\mathcal{H}_\infty$ norm; the optimized pseudospectral abscissa provides quantitative information about the worstcase transient growth, and the initial guesses for the parameter values to optimize the pseudospectral abscissa can be arbitrary, unlike the case to optimize the distance to instability and $\mathcal{H}_\infty$ norm that would normally require initial guesses yielding asymptotically stable systems.
 [359] arXiv:2208.10572 (replaced) [pdf, html, other]

Title: Balanced supersaturation and Turan numbers in random graphsComments: 26 pagesJournalref: Advances in Combinatorics, 2024:3, 26ppSubjects: Combinatorics (math.CO)
In a groundbreaking paper solving a conjecture of Erdős on the number of $n$vertex graphs not containing a given even cycle, Morris and Saxton \cite{MS} made a broad conjecture on socalled balanced supersaturation property of a bipartite graph $H$. Ferber, McKinley, and Samotij \cite{FMS} established a weaker version of this conjecture and applied it to derive farreaching results on the enumeration problem of $H$free graphs.
In this paper, we show that Morris and Saxton's conjecture holds under a very mild assumption about $H$, which is widely believed to hold whenever $H$ contains a cycle. We then use our theorem to obtain enumeration results and general upper bounds on the Turán number of a bipartite $H$ in the random graph $G(n,p)$, the latter being first of its kind.  [360] arXiv:2208.13296 (replaced) [pdf, html, other]

Title: Polynomial time guarantees for sampling based posterior inference in highdimensional generalised linear modelsComments: Revised and updated versionSubjects: Statistics Theory (math.ST); Analysis of PDEs (math.AP); Numerical Analysis (math.NA); Probability (math.PR); Computation (stat.CO)
The problem of computing posterior functionals in general highdimensional statistical models with possibly nonlogconcave likelihood functions is considered. Based on the proof strategy of [49], but using only local likelihood conditions and without relying on Mestimation theory, nonasymptotic statistical and computational guarantees are provided for a gradient based MCMC algorithm. Given a suitable initialiser, these guarantees scale polynomially in key algorithmic quantities. The abstract results are applied to several concrete statistical models, including density estimation, nonparametric regression with generalised linear models and a canonical statistical nonlinear inverse problem from PDEs.
 [361] arXiv:2209.00373 (replaced) [pdf, html, other]

Title: Characterizations and models for the $C_{1,r}$ class and quantum annulusComments: 14 pages, Thoroughly revised, Title changed, New results addedSubjects: Functional Analysis (math.FA); Complex Variables (math.CV)
For fixed $0<r<1$, let $A_r=\{z \in \mathbb{C} : r<z<1\}$ be the annulus with boundary $\partial \overline{A}_r=\mathbb{T} \cup r\mathbb{T}$, where $\mathbb T$ is the unit circle in the complex plane $\mathbb C$. An operator having $\ov{A}_r$ as a spectral set is called an $A_r$\textit{contraction}. Also, a normal operator with its spectrum lying in the boundary $\partial \overline{A}_r$ is called an \textit{$A_r$unitary}. The \textit{$C_{1,r}$ class} was introduced by Bello and Yakubovich in the following way: \[ C_{1, r}=\{T: T \ \mbox{is invertible and} \ \T\, \rT^{1}\ \leq 1\}. \] McCullough and Pascoe defined the \textit{quantum annulus} $\mathbb Q \mathbb A_r$ by \[ \mathbb Q\mathbb A_r = \{T \,:\, T \text{ is invertible and } \, \rT\, \rT^{1}\ \leq 1 \}. \] If $\mathcal A_r$ denotes the set of all $A_r$contractions, then $\mathcal A_r \subsetneq C_{1,r} \subsetneq \mathbb Q \mathbb A_r$. We first find a model for an operator in $C_{1,r}$ and also characterize the operators in $C_{1,r}$ in several different ways. We prove that the classes $C_{1,r}$ and $\mathbb Q\mathbb A_r$ are equivalent. Then, via this equivalence, we obtain analogous model and characterizations for an operator in $\mathbb Q \mathbb A_r$.
 [362] arXiv:2209.02027 (replaced) [pdf, html, other]

Title: Emergence of quantum dynamics from chaos: The case of prequantum cat mapsComments: v2: We clarify some statements and simplify the arguments by restricting to the case of cat maps in checkerboard form. 24 pagesSubjects: Dynamical Systems (math.DS); Mathematical Physics (mathph); Spectral Theory (math.SP)
Faure and Tsujii have recently proposed a novel quantization procedure, named natural quantization, for smooth symplectic Anosov diffeomorphisms. Their method starts with prequantization, which is also the first step of geometric quantization as proposed by KostantSouriauKirillov, and then relies on the RuellePollicott spectrum of the prequantum transfer operator, which they show to have a particular band structure. The appeal of this new quantization scheme resides in its naturalness: the quantum behavior appears dynamically in the classical correlation functions of the prequantum transfer operator. In this paper, we explicitly work out the case of cat maps on the $2n$dimensional torus, showing in particular that the outcome is equivalent to that of the usual Weyl quantization. We also provide a concrete construction of all the prequantum cat maps.
 [363] arXiv:2209.09107 (replaced) [pdf, html, other]

Title: Listavoiding orientationsSubjects: Combinatorics (math.CO)
Given a graph $G$ with a set $F(v)$ of forbidden values at each $v \in V(G)$, an $F$avoiding orientation of $G$ is an orientation in which $deg^+(v) \not \in F(v)$ for each vertex $v$. Akbari, Dalirrooyfard, Ehsani, Ozeki, and Sherkati conjectured that if $F(v) < \frac{1}{2} deg(v)$ for each $v \in V(G)$, then $G$ has an $F$avoiding orientation, and they showed that this statement is true when $\frac{1}{2}$ is replaced by $\frac{1}{4}$. In this paper, we take a step toward this conjecture by proving that if $F(v) < \lfloor \frac{1}{3} deg(v) \rfloor$ for each vertex $v$, then $G$ has an $F$avoiding orientation. Furthermore, we show that if the maximum degree of $G$ is subexponential in terms of the minimum degree, then this coefficient of $\frac{1}{3}$ can be increased to $\sqrt{2}  1  o(1) \approx 0.414$. Our main tool is a new sufficient condition for the existence of an $F$avoiding orientation based on the Combinatorial Nullstellensatz of Alon and Tarsi.
 [364] arXiv:2209.10364 (replaced) [pdf, html, other]

Title: Some strong limit theorems in averagingSubjects: Probability (math.PR)
The paper deals with the fastslow motions setups in the discrete time $X^\epsilon((n+1)\epsilon)=X^\epsilon(n\epsilon)+\epsilon B(X^\epsilon(n\epsilon),\xi(n))$, $n=0,1,...,[T/\epsilon]$ and the continuous time $\frac {dX^\epsilon(t)}{dt}=B(X^\epsilon(t),\xi(t/\epsilon)).\, t\in [0,T]$ where $B$ is a smooth vector function and $\xi$ is a sufficiently fast mixing stationary stochastic process. It is known since 1966 (Khasminskii) that if $\bar X$ is the averaged motion then $G^\epsilon=\epsilon^{1/2}(X^\epsilon\bar X)$ weakly converges to a Gaussian process $G$. We will show that for each $\epsilon$ the processes $\xi$ and $G$ can be redefined on a sufficiently rich probability space without changing their distributions so that $E\sup_{0\leq t\leq T}G^\epsilon(t)G(t)^{2M} =O(\epsilon^{\delta})$, $\delta>0$ which gives also $O(\epsilon^{\delta/3})$ Prokhorov distance estimate between the distributions of $G^\epsilon$ and $G$. In the product case $B(x,\xi)=\Sigma(x)\xi$ we obtain almost sure convergence estimates of the form $\sup_{0\leq t\leq T}G^\epsilon(t)G(t)=O(\epsilon^\delta)$ a.s., as well as the functional form of the law of iterated logarithm for $G^\epsilon$.
We note that our mixing assumptions are adapted to fast motions generated by important classes of dynamical systems.  [365] arXiv:2209.11920 (replaced) [pdf, html, other]

Title: Tradeoffs between convergence rate and noise amplification for momentumbased accelerated optimization algorithmsComments: 23 pages; 7 figuresSubjects: Optimization and Control (math.OC); Machine Learning (cs.LG); Systems and Control (eess.SY); Dynamical Systems (math.DS)
We study momentumbased firstorder optimization algorithms in which the iterations utilize information from the two previous steps and are subject to an additive white noise. This setup uses noise to account for uncertainty in either gradient evaluation or iteration updates, and it includes Polyak's heavyball and Nesterov's accelerated methods as special cases. For strongly convex quadratic problems, we use the steadystate variance of the error in the optimization variable to quantify noise amplification and identify fundamental stochastic performance tradeoffs. Our approach utilizes the Jury stability criterion to provide a novel geometric characterization of conditions for linear convergence, and it reveals the relation between the noise amplification and convergence rate as well as their dependence on the condition number and the constant algorithmic parameters. This geometric insight leads to simple alternative proofs of standard convergence results and allows us to establish ``uncertainty principle'' of strongly convex optimization: for the twostep momentum method with linear convergence rate, the lower bound on the product between the settling time and noise amplification scales quadratically with the condition number. Our analysis also identifies a key difference between the gradient and iterate noise models: while the amplification of gradient noise can be made arbitrarily small by sufficiently decelerating the algorithm, the best achievable variance for the iterate noise model increases linearly with the settling time in the decelerating regime. Finally, we introduce two parameterized families of algorithms that strike a balance between noise amplification and settling time while preserving orderwise Pareto optimality for both noise models.
 [366] arXiv:2210.01321 (replaced) [pdf, html, other]

Title: On Decomposition of the Last Passage Time of DiffusionsSubjects: Probability (math.PR)
For a regular transient diffusion, we provide a decomposition of its last passage time to a certain state $\alpha$. This is accomplished by transforming the original diffusion into two diffusions using the occupation time of the area above and below $\alpha$. Based on these two processes, both having a reflecting boundary at $\alpha$, we derive the decomposition formula of the Laplace transform of the last passage time explicitly in a simple form in terms of Green functions. This equation also leads to the Green function's decomposition formula. We demonstrate an application of these formulas to a diffusion with twovalued parameters.
 [367] arXiv:2210.03309 (replaced) [pdf, html, other]

Title: Localization for general HelmholtzJournalref: Journal of Differential Equations 393 (2024) 139154Subjects: Analysis of PDEs (math.AP)
In \cite{gmw2022}, Guan, Murugan and Wei established the equivalence of the classical Helmholtz equation with a ``fractional Helmholtz" equation in which the Laplacian operator is replaced by the nonlocal fractional Laplacian operator. More general equivalence results are obtained for symbols which are complete Bernstein and satisfy additional regularity conditions. In this work we introduce a novel and general setup for this Helmholtz equivalence problem. We show that under very mild and easytocheck conditions on the Fourier multiplier, the general Helmholtz equation can be effectively reduced to a localization statement on the support of the symbol.
 [368] arXiv:2210.04003 (replaced) [pdf, html, other]

Title: Tropical functions on a skeletonComments: 47 pagesSubjects: Algebraic Geometry (math.AG); Logic (math.LO)
We prove a general finiteness statement for the ordered abelian group of tropical functions on skeleta in Berkovich analytifications of algebraic varieties. Our approach consists in working in the framework of stable completions of algebraic varieties, a modeltheoretic version of Berkovich analytifications, for which we prove a similar result, of which the former one is a consequence.
 [369] arXiv:2210.12569 (replaced) [pdf, html, other]

Title: Generic curves and noncoprime CatalansComments: 28 pages, revised versionJournalref: Algebraic Geometry, 2024Subjects: Algebraic Geometry (math.AG); Combinatorics (math.CO)
We compute the Poincaré polynomials of the compactified Jacobians for plane curve singularities with Puiseaux exponents $(nd,md,md+1)$, and relate them to the combinatorics of $q,t$Catalan numbers in the noncoprime case. We also confirm a conjecture of Cherednik and Danilenko for such curves.
 [370] arXiv:2210.13056 (replaced) [pdf, html, other]

Title: DonohoLogan large sieve principles for the wavelet transformSubjects: Functional Analysis (math.FA)
In this paper we formulate Donoho and Logan's large sieve principle for the wavelet transform on the Hardy space, adapting the concept of maximum Nyquist density to the hyperbolic geometry of the underlying space. The results provide deterministic guarantees for $L_{1}$minimization methods and hold for a class of mother wavelets which constitute an orthonormal basis of the Hardy space and can be associated with higher hyperbolic Landau levels. Explicit calculations of the basis functions reveal a connection with the Zernike polynomials. We prove a novel local reproducing formula for the spaces in consideration and use it to derive concentration estimates of the large sieve type for the corresponding wavelet transforms. We conclude with a discussion of optimality of localization in the analytic case by building on recent results of Kulikov, Ramos and Tilli based on the groundbreaking methods of Nicola and Tilli. This leads to a sharp uncertainty principle and a local Lieb inequality for the wavelet transform.
 [371] arXiv:2210.15047 (replaced) [pdf, other]

Title: A stochastic analysis of subcritical Euclidean fermionic field theoriesComments: The errors in some proofs have been corrected and some typos have been fixed. Accepted for publication in Annals of ProbabilitySubjects: Probability (math.PR); Mathematical Physics (mathph)
Building on previous work on the stochastic analysis for Grassmann random variables, we introduce a forwardbackward stochastic differential equation (FBSDE) which provides a stochastic quantisation of Grassmann measures. Our method is inspired by the socalled continuous renormalisation group, but avoids the technical difficulties encountered in the direct study of the flow equation for the effective potentials. As an application, we construct a family of weakly coupled subcritical Euclidean fermionic field theories and prove exponential decay of correlations.
 [372] arXiv:2211.04776 (replaced) [pdf, html, other]

Title: Regularized R\'enyi divergence minimization through Bregman proximal gradient algorithmsSubjects: Statistics Theory (math.ST)
We study the variational inference problem of minimizing a regularized Rényi divergence over an exponential family, and propose a relaxed momentmatching algorithm, which includes a proximallike step. Using the informationgeometric link between Bregman divergences and the KullbackLeibler divergence, this algorithm is shown to be equivalent to a Bregman proximal gradient algorithm. This novel perspective allows us to exploit the geometry of our approximate model while using stochastic blackbox updates. We use this point of view to prove strong convergence guarantees including monotonic decrease of the objective, convergence to a stationary point or to the minimizer, and geometric convergence rates. These new theoretical insights lead to a versatile, robust, and competitive method, as illustrated by numerical experiments.
 [373] arXiv:2212.05804 (replaced) [pdf, html, other]

Title: Arithmetic degrees and Zariski dense orbits of cohomologically hyperbolic mapsComments: v2. Minor changes. Final version, to appear in Transactions of the AMSSubjects: Algebraic Geometry (math.AG); Dynamical Systems (math.DS); Number Theory (math.NT)
A dominant rational selfmap on a projective variety is called $p$cohomologically hyperbolic if the $p$th dynamical degree is strictly larger than other dynamical degrees. For such a map defined over $\overline{\mathbb{Q}}$, we study lower bounds of the arithmetic degrees, existence of points with Zariski dense orbit, and finiteness of preperiodic points. In particular, we prove that, if $f$ is an $1$cohomologically hyperbolic map on a smooth projective variety, then (1) the arithmetic degree of a $\overline{\mathbb{Q}}$point with generic $f$orbit is equal to the first dynamical degree of $f$; and (2) there are $\overline{\mathbb{Q}}$points with generic $f$orbit. Applying our theorem to the recently constructed rational map with transcendental dynamical degree, we confirm that the arithmetic degree can be transcendental.
 [374] arXiv:2212.07255 (replaced) [pdf, html, other]

Title: A mechanism of threedimensional quadratic termination for the gradient method with applicationsComments: 23 pagesSubjects: Optimization and Control (math.OC)
Recent studies show that the twodimensional quadratic termination property has great potential in improving performance of the gradient method. However, it is not clear whether higherdimensional quadratic termination leads further benefits. In this paper, we provide an affirmative answer by introducing a mechanism of threedimensional quadratic termination for the gradient method. A novel stepsize is derived from the mechanism such that a family of delayed gradient methods equipping with the novel stepsize have the threedimensional quadratic termination property. When applied to the BarzilaiBorwein (BB) method, the novel stepsize does not require the use of any exact line search or the Hessian, and can be computed by stepsizes and gradient norms in previous iterations. Using long BB steps and some short steps associated with the novel stepsize in an adaptive manner, we develop an efficient gradient method for quadratic optimization and further extend it to general unconstrained optimization. Numerical experiments show that the threedimensional quadratic termination property can significantly improve performance of the BB method, and the proposed method outperforms gradient methods that use stepsizes with the twodimensional quadratic termination property.
 [375] arXiv:2212.13149 (replaced) [pdf, html, other]

Title: Solutions to a system of YangBaxter matrix equationsSubjects: Rings and Algebras (math.RA); Representation Theory (math.RT)
In this article, a system of YangBaxtertype matrix equations is studied, $XAX=BXB$, $XBX=AXA$, which "generalizes" the matrix YangBaxter equation and exhibits a broken symmetry. We investigate the solutions of this system from various geometric and topological points of view. We analyze the existence of doubly stochastic solutions and intertwining solutions to the system and describe the conditions for their existence. Furthermore, we characterize the case when $A$ and $B$ are idempotent orthogonal complements. i.e., $A^2 =A, B^2= B, AB = BA =0$. We also completely characterize the set of solutions for $n=2$ using commutative algebraic techniques.
 [376] arXiv:2212.13854 (replaced) [pdf, other]

Title: A DRL Approach for RISAssisted FullDuplex UL and DL Transmission: Beamforming, Phase Shift and Power OptimizationSubjects: Information Theory (cs.IT); Signal Processing (eess.SP)
We propose a deep reinforcement learning (DRL) approach for a fullduplex (FD) transmission that predicts the phase shifts of the reconfigurable intelligent surface (RIS), base station (BS) active beamformers, and the transmit powers to maximize the weighted sum rate of uplink and downlink users. Existing methods require channel state information (CSI) and residual selfinterference (SI) knowledge to calculate exact active beamformers or the DRL rewards, which typically fail without CSI or residual SI. Especially for timevarying channels, estimating and signaling CSI to the DRL agent is required at each time step and is costly. We propose a twostage DRL framework with minimal signaling overhead to address this. The first stage uses the least squares method to initiate learning by partially canceling the residual SI. The second stage uses DRL to achieve performance comparable to existing CSIbased methods without requiring the CSI or the exact residual SI. Further, the proposed DRL framework for quantized RIS phase shifts reduces the signaling from BS to the RISs using $32$ times fewer bits than the continuous version. The quantized methods reduce action space, resulting in faster convergence and $7.1\%$ and $22.28\%$ better UL and DL rates, respectively than the continuous method.
 [377] arXiv:2301.02634 (replaced) [pdf, html, other]

Title: On disjoint stationary sequencesComments: Substantial changes to the proof of Theorem 3, which is now only stated for $\aleph_2$. Amendments to the proof of Theorem 2. Other small changes throughoutSubjects: Logic (math.LO)
We answer a question of Krueger by obtaining disjoint stationary sequences on successive cardinals. The main idea is an alternative presentation of a mixed support iteration, using it even more explicitly as a variant of Mitchell forcing. We also use a Mahlo cardinal to obtain a model in which $\aleph_2 \notin I[\aleph_2]$ and there is no disjoint stationary sequence on $\aleph_2$, answering a question of Gilton.
 [378] arXiv:2301.10762 (replaced) [pdf, html, other]

Title: Optimisation of seismic imaging via bilevel learningSubjects: Numerical Analysis (math.NA)
Full Waveform Inversion (FWI) is a standard algorithm in seismic imaging. Its implementation requires the a priori choice of a number of "design parameters", such as the positions of sensors for the actual measurements and one (or more) regularisation weights. In this paper we describe a novel algorithm for determining these design parameters automatically from a set of training images, using a (supervised) bilevel learning approach. In our algorithm, the upper level objective function measures the quality of the reconstructions of the training images, where the reconstructions are obtained by solving the lower level optimisation problem  in this case FWI. Our algorithm employs (variants of) the BFGS quasiNewton method to perform the optimisation at each level, and thus requires the repeated solution of the forward problem  here taken to be the Helmholtz equation.
This paper focuses on the implementation of the algorithm. The novel contributions are: (i) an adjointstate method for the efficient computation of the upperlevel gradient; (ii) a complexity analysis for the bilevel algorithm, which counts the number of Helmholtz solves needed and shows this number is independent of the number of design parameters optimised; (iii) an effective preconditioning strategy for iteratively solving the linear systems required at each step of the bilevel algorithm; (iv) a smoothed extraction process for point values of the discretised wavefield, necessary for ensuring a smooth upper level objective function. The algorithm also uses an extension to the bilevel setting of classical frequencycontinuation strategies, helping avoid convergence to spurious stationary points. The advantage of our algorithm is demonstrated on a problem derived from the standard Marmousi test problem.  [379] arXiv:2301.12659 (replaced) [pdf, html, other]

Title: GPU Accelerated Newton for Taylor Series Solutions of Polynomial Homotopies in Multiple Double PrecisionComments: Accepted by CASC 2024, the 27th International Workshop on Computer Algebra in Scientific ComputingSubjects: Numerical Analysis (math.NA); Distributed, Parallel, and Cluster Computing (cs.DC); Mathematical Software (cs.MS); Algebraic Geometry (math.AG)
A polynomial homotopy is a family of polynomial systems, typically in one parameter $t$. Our problem is to compute power series expansions of the coordinates of the solutions in the parameter $t$, accurately, using multiple double arithmetic. One application of this problem is the location of the nearest singular solution in a polynomial homotopy, via the theorem of Fabry. Power series serve as input to construct Padé approximations.
Exploiting the massive parallelism of Graphics Processing Units capable of performing several trillions floatingpoint operations per second, the objective is to compensate for the cost overhead caused by arithmetic with power series in multiple double precision. The application of Newton's method for this problem requires the evaluation and differentiation of polynomials, followed by solving a blocked lower triangular linear system. Experimental results are obtained on NVIDIA GPUs, in particular the RTX 2080, RTX 4080, P100, V100, and A100.
Code generated by the CAMPARY software is used to obtain results in double double, quad double, and octo double precision. The programs in this study are self contained, available in a public github repository under the GPLv3.0 License.  [380] arXiv:2302.07150 (replaced) [pdf, html, other]

Title: A Lipschitz metric for $\alpha$dissipative solutions to the HunterSaxton equationComments: 43 pagesSubjects: Analysis of PDEs (math.AP)
We explore the Lipschitz stability of solutions to the HunterSaxton equation with respect to the initial data. In particular, we study the stability of $ \alpha $dissipative solutions constructed using a generalised method of characteristics approach, where $\alpha$ is a function determining the energy loss at each position in space.
 [381] arXiv:2302.08176 (replaced) [pdf, html, other]

Title: The logic behind desirable sets of things, and its filter representationComments: 38 pages, 1 figure, 2 tablesSubjects: Logic (math.LO); Artificial Intelligence (cs.AI)
We identify the (filter representation of the) logic behind the recent theory of coherent sets of desirable (sets of) things, which generalise coherent sets of desirable (sets of) gambles as well as coherent choice functions, and show that this identification allows us to establish various representation results for such coherent models in terms of simpler ones.
 [382] arXiv:2302.13068 (replaced) [pdf, html, other]

Title: Classifying solutions of ${\rm SU}(n+1)$ Toda system around a singular sourceComments: In this new version, we have added some references, indicated how our results align with those of Bryant, and addressed additional queries raised by the reviewersJournalref: Proc. Amer. Math. Soc. 152 (2024), 25852600Subjects: Analysis of PDEs (math.AP); Complex Variables (math.CV); Differential Geometry (math.DG)
Consider a positive integer $n$ and $\gamma_1>1,\cdots,\gamma_n>1$. Let $D=\{z\in {\Bbb C}:z<1\}$, and let $(a_{ij})_{n\times n}$ denote the Cartan matrix of $\frak{su}(n+1)$. Utilizing the ordinary differential equation of $(n+1)$th order around a singular source of ${\rm SU}(n+1)$ Toda system, as discovered by LinWeiYe ({\it Invent Math}, {\bf 190}(1):169207, 2012), we precisely characterize a solution $(u_1,\cdots, u_n)$ to the ${\rm SU}(n+1)$ Toda system \begin{equation*}
\begin{cases} \frac{\partial^2 u_i}{\partial z\partial \bar z}+\sum_{j=1}^n a_{ij} e^{u_j}&=\pi \gamma _i\delta _0\,\,{\rm on}\,\, D\\
\frac{\sqrt{1}}{2}\,\int_{D\backslash \{0\}} e^{u_{i} }{\rm d}z\wedge {\rm d}\bar z &< \infty
\end{cases} \quad \text{for all}\quad i=1,\cdots, n \end{equation*} using $(n+1)$ holomorphic functions that satisfy the normalized condition. Additionally, we demonstrate that for each $1\leq i\leq n$, $0$ represents the cone singularity with angle $2\pi(1+\gamma_i)$ for the metric $e^{u_i}{\rm d}z^2$ on $D\backslash\{0\}$, which can be locally characterized by $(n1)$ nonvanishing holomorphic functions at $0$.  [383] arXiv:2302.13098 (replaced) [pdf, html, other]

Title: Determining skew left braces of size npSubjects: Group Theory (math.GR); Number Theory (math.NT)
We define the twofold semidirect product of two skew left braces, in which both the additive and multiplicative groups are semidirect products of the corresponding groups of the given skew left braces. We consider an odd prime $p$ and an integer $n$ satisfying $p\nmid n$, $p\nmid\mathrm{Aut}(E)$ for every group $E$ of order $n$ and such that each group of order $np$ has a unique $p$Sylow subgroup. Under these conditions, we prove that any skew left brace of size $np$ is either a twofold semidirect product of the trivial brace of size $p$ and a skew brace of size $n$ or a companion skew brace of that one. We develop an algorithm to obtain all skew braces of size $np$ from the braces of size $n$ and provide a formula to count them. We use this result to describe all braces of size $12p$ for $p\geq 7$, which proves a conjecture of V.G. Bardakov, M.V. Neshchadim and M.K. Yadav.
 [384] arXiv:2302.13186 (replaced) [pdf, html, other]

Title: Construction numbers: How to build a graph?Comments: 19 pages; Problem 12401  06 in the June 2023 issue of the American Math Monthly was to find the construction number for $K_n$. Solution is given hereSubjects: Combinatorics (math.CO); Artificial Intelligence (cs.AI)
Counting the number of linear extensions of a partial order was considered by Stanley about 50 years ago. For the partial order on the vertices and edges of a graph given by inclusion, we call a linear extension a {\it construction sequence} for the graph as each edge follows the vertices where it is attached. The number of these csequences is counted for various graph families. We also consider the set of all length$n$ csequences produced by the graphs with $n$ elements, simplified to their structural skeleton: vertex or edge, and further allow the generating graph to be structurally constrained. Efficiency is analyzed.
 [385] arXiv:2302.13772 (replaced) [pdf, html, other]

Title: Solutions to semilinear wave equations of very low regularityComments: 15 pages, to appear in Journal of Differential EquationsSubjects: Analysis of PDEs (math.AP)
This paper finds solutions to semilinear wave equations with strongly anomalous propagation of singularities. For very low Sobolev regularity we obtain solutions whose singular support propagates along any ray inside or outside the light cone. In one dimension these solutions exist for any Sobolev exponent $s<\frac{1}{2}$ in space, while classical results show that the singular support of solutions with higher regularity is contained in the light cone. The spatial Fourier transform of these anomalous solutions is supported in a halfline. We obtain wellposedness results in such function spaces when the problem is illposed for Sobolev data without the support condition and, in some cases, obtain wellposedness below $L^2(\mathbb{R})$. The results are based on new multiplier theorems for Sobolev spaces satisfying the support condition. Extensions to higher space dimensions are given.
 [386] arXiv:2303.09488 (replaced) [pdf, html, other]

Title: Regularity of laws via Dirichlet forms  Application to quadratic forms in independent and identically distributed random variablesComments: 34 pages, comments are welcome, v2 simplifies the technical setting and proofs without changing the main resultsSubjects: Probability (math.PR); Functional Analysis (math.FA)
We study the regularity of the law of a quadratic form $Q(X,X)$, evaluated in a sequence $X = (X_{i})$ of independent and identically distributed random variables, when $X_{1}$ can be expressed as a sufficiently smooth function of a Gaussian field. This setting encompasses a large class of important and frequently used distributions, such as, among others, Gaussian, Beta, for instance uniform, Gamma distributions, or else any polynomial transform of them.
Let us present an emblematic application. Take $X = (X_{i})$ a sequence of independent and identically distributed centered random variables, with unit variance, following such distribution. Consider also $(Q_{n})$ a sequence of quadratic forms, with associated symmetric HilbertSchmidt operators $(\mathsf{A}^{(n)})$. Assume that $\operatorname{Tr}[ (\mathsf{A}^{(n)})^{2} ] = 1/2$, $\mathsf{A}^{(n)}_{ii} =0$, and the spectral radius of $\mathsf{A}^{(n)}$ tends to $0$. Then, $(Q_{n}(X))$ converges in a strong sense to the standard Gaussian distribution. Namely, all derivatives of the densities, which are welldefined for $n$ sufficiently large, converge uniformly on $\mathbb{R}$ to the corresponding derivatives of the standard Gaussian density.
While classical methods, from Malliavin calculus or $\Gamma$calculus, generally consist in bounding negative moments of the socalled \emph{carré du champ} operator $\Gamma(Q(X),Q(X))$, we provide a new paradigm through a secondorder criterion involving the eigenvalues of a Hessiantype matrix related to $Q(X)$. This Hessian is built by iterating twice a tailormade gradient, the \emph{sharp operator} $\sharp$, obtained via a Gaussian representation of the carré du champ. We believe that this method, recently developed by the authors in the current paper and in their companion paper [AoP 52 n°3 (2024)] , is of independent interest and could prove useful in other settings.  [387] arXiv:2303.10082 (replaced) [pdf, other]

Title: Scaling limits and universality: Critical percolation on weighted graphs converging to an $L^3$ graphonComments: 67 pages, 1 figure, to appear in Transactions of the American Mathematical Society. The universality principle (Theorem 3.4) from arXiv:1411.3417 has now been included in this paperSubjects: Probability (math.PR); Combinatorics (math.CO)
We develop a general universality technique for establishing metric scaling limits of critical random discrete structures exhibiting meanfield behavior that requires four ingredients: (i) from the barely subcritical regime to the critical window, components merge approximately like the multiplicative coalescent, (ii) asymptotics of the susceptibility functions are the same as that of the ErdosRenyi random graph, (iii) asymptotic negligibility of the maximal component size and the diameter in the barely subcritical regime, and (iv) macroscopic averaging of distances between vertices in the barely subcritical regime.
As an application of the general universality theorem, we establish, under some regularity conditions, the critical percolation scaling limit of graphs that converge, in a suitable topology, to an $L^3$ graphon. In particular, we define a notion of the critical window in this setting. The $L^3$ assumption ensures that the model is in the ErdosRenyi universality class and that the scaling limit is Brownian. Our results do not assume any specific functional form for the graphon. As a consequence of our results on graphons, we obtain the metric scaling limit for AldousPittel's RGIV model [9] inside the critical window.
Our universality principle has applications in a number of other problems including in the study of noise sensitivity of critical random graphs [52]. In [10], we use our universality theorem to establish the metric scaling limit of critical bounded size rules. Our method should yield the critical metric scaling limit of Rucinski and Wormald's random graph process with degree restrictions [56] provided an additional technical condition about the barely subcritical behavior of this model can be proved.  [388] arXiv:2303.11706 (replaced) [pdf, html, other]

Title: Lower bounds for the tradeoff between bias and mean absolute deviationComments: This is an extended version of Section 7 of arXiv:2006.00278v3. The material has been removed from later versions of arXiv:2006.00278Journalref: Statistics and Probability Letters, Volume 213, 110182, 2024Subjects: Statistics Theory (math.ST)
In nonparametric statistics, rateoptimal estimators typically balance bias and stochastic error. The recent work on overparametrization raises the question whether rateoptimal estimators exist that do not obey this tradeoff. In this work we consider pointwise estimation in the Gaussian white noise model with regression function $f$ in a class of $\beta$Hölder smooth functions. Let 'worstcase' refer to the supremum over all functions $f$ in the Hölder class. It is shown that any estimator with worstcase bias $\lesssim n^{\beta/(2\beta+1)}=: \psi_n$ must necessarily also have a worstcase mean absolute deviation that is lower bounded by $\gtrsim \psi_n.$ To derive the result, we establish abstract inequalities relating the change of expectation for two probability measures to the mean absolute deviation.
 [389] arXiv:2303.14144 (replaced) [pdf, html, other]

Title: On 3generated 6transposition groupsComments: 12 pagesSubjects: Group Theory (math.GR)
We study $6$transposition groups, i.e. groups generated by a normal set of involutions $D$, such that the order of the product of any two elements from $D$ does not exceed $6$. We classify most of the groups generated by $3$ elements from $D$, two of which commute, and prove they are finite.
 [390] arXiv:2303.17843 (replaced) [pdf, html, other]

Title: TambaraYamagami Categories over the Reals: The NonSplit CaseComments: 44 pages; Corrected Theorem 7.1, added Example 7.5Subjects: Quantum Algebra (math.QA); Category Theory (math.CT)
Tambara and Yamagami investigated a simple set of fusion rules with only one noninvertible object, and proved under which circumstances those rules could be given a coherent associator. We consider a generalization of such fusion rules to the setting where simple objects are no longer required to be split simple. Over the real numbers, this means that objects are either real, complex, or quaternionic. In this context, we prove a similar categorification result to the one of Tambara and Yamagami.
 [391] arXiv:2304.00773 (replaced) [pdf, html, other]

Title: Narayana's cows numbers which are concatenations of three repdigits in base $\rho$Subjects: Number Theory (math.NT)
Narayana's sequence is a ternary recurrent sequence defined by the recurrence relation $\mathcal{N}_n=\mathcal{N}_{n1}+\mathcal{N}_{n3}$ with initial terms $\mathcal{N}_0=0$ and $\mathcal{N}_1=\mathcal{N}_2=\mathcal{N}_3=1$. Let $\rho\geqslant2$ be a positive integer. In this study, it is proved that the $n$th Narayana's number $ \mathcal{N}_n$ which is concatenations of three repdigits in base $\rho$ satisfies $n<5.6\cdot 10^{48}\cdot \log^7\rho$. Moreover, it is shown that the largest Narayana's number which is concatenations of three repdigits in base $\rho$ with $1 \leqslant \rho \leqslant 10$ is $58425=\mathcal{N}_{31}=\overline{3332200}_5=\overline{332223}_7.$
 [392] arXiv:2304.03886 (replaced) [pdf, html, other]

Title: Convergence Rate Bounds for the Mirror Descent Method: IQCs, Popov Criterion and Bregman DivergenceComments: Resubmitted. arXiv admin note: substantial text overlap with arXiv:2204.00502Subjects: Optimization and Control (math.OC)
This paper presents a comprehensive convergence analysis for the mirror descent (MD) method, a widely used algorithm in convex optimization. The key feature of this algorithm is that it provides a generalization of classical gradientbased methods via the use of generalized distancelike functions, which are formulated using the Bregman divergence. Establishing convergence rate bounds for this algorithm is in general a nontrivial problem due to the lack of monotonicity properties in the composite nonlinearities involved. In this paper, we show that the Bregman divergence from the optimal solution, which is commonly used as a Lyapunov function for this algorithm, is a special case of Lyapunov functions that follow when the Popov criterion is applied to an appropriate reformulation of the MD dynamics. This is then used as a basis to construct an integral quadratic constraint (IQC) framework through which convergence rate bounds with reduced conservatism can be deduced. We also illustrate via examples that the convergence rate bounds derived can be tight.
 [393] arXiv:2304.04451 (replaced) [pdf, html, other]

Title: Quantitative contraction rates for Sinkhorn algorithm: beyond bounded costs and compact marginalsComments: 34 pages, simplified presentation of main results, added explicit expression for the exponential convergence rates and added stronger results in the logconcave settingSubjects: Probability (math.PR); Optimization and Control (math.OC)
We show nonasymptotic exponential convergence of Sinkhorn iterates to the Schrödinger potentials, solutions of the quadratic Entropic Optimal Transport problem on $\mathbb{R}^d$. Our results hold under mild assumptions on the marginal inputs: in particular, we only assume that they admit an asymptotically positive logconcavity profile, covering as special cases logconcave distributions and bounded smooth perturbations of quadratic potentials. Up to the authors' knowledge, these are the first results which establish exponential convergence of Sinkhorn's algorithm in a general setting without assuming bounded cost functions or compactly supported marginals.
 [394] arXiv:2304.08099 (replaced) [pdf, html, other]

Title: A characterization of halffactorial orders in algebraic number fieldsSubjects: Commutative Algebra (math.AC); Number Theory (math.NT)
We give an algebraic characterization of halffactorial orders in algebraic number fields. This generalizes prior results for seminormal orders and for orders in quadratic number fields.
 [395] arXiv:2304.09452 (replaced) [pdf, html, other]

Title: Support and distribution inference from noisy dataSubjects: Statistics Theory (math.ST)
We consider noisy observations of a distribution with unknown support. In the deconvolution model, it has been proved recently [19] that, under very mild assumptions, it is possible to solve the deconvolution problem without knowing the noise distribution and with no sample of the noise. We first give general settings where the theory applies and provide classes of supports that can be recovered in this context. We then exhibit classes of distributions over which we prove adaptive minimax rates (up to a log log factor) for the estimation of the support in Hausdorff distance. Moreover, for the class of distributions with compact support, we provide estimators of the unknown (in general singular) distribution and prove maximum rates in Wasserstein distance. We also prove an almost matching lower bound on the associated minimax risk.
 [396] arXiv:2304.10615 (replaced) [pdf, other]

Title: Calder\'onZygmund type estimate for the parabolic doublephase systemComments: References are correctedSubjects: Analysis of PDEs (math.AP)
This paper provides a local and global CalderónZygmund type estimate of a weak solution to the parabolic doublephase system. The proof of local estimate is based on comparison estimates and the scaling invariant property of the parabolic doublephase system in the intrinsic cylinders of the stopping time argument setting. For the proof of the global estimate, we have applied the reflection and approximation techniques.
 [397] arXiv:2304.13213 (replaced) [pdf, html, other]

Title: Exact values and improved bounds on the clique number of cyclotomic graphsComments: 10 pages, typos correctedSubjects: Combinatorics (math.CO); Number Theory (math.NT)
Let $q$ be an odd power of a prime $p$, and $S \subset \mathbb{F}_q^*$ such that $S=S$ and $S/S \neq \mathbb{F}_q^*$. We show that the clique number of the Cayley graph $\operatorname{Cay}(\mathbb{F}_q^+,S)$ is at most $\sqrt{S/S}+\sqrt{q/p}$, improving the bestknown $\sqrt{q}$ upper bound for many families of such graphs substantially. Such a new bound is strongest for cyclotomic graphs and in particular, it implies the first nontrivial upper bound on the clique number of all generalized Paley graphs of nonsquare order, extending the work of Hanson and Pertidis. Moreover, our new bound is asymptotically sharp for an infinite family of generalized Paley graphs, and we further discover the first nontrivial family among them for which the clique number can be exactly determined. We also obtain a new lower bound on the number of directions determined by a large Cartesian product in the affine Galois plane $AG(2,q)$, which is sharp for infinite families.
 [398] arXiv:2305.01320 (replaced) [pdf, other]

Title: HigherOrder Generalized Finite Differences for Variable Coefficient Diffusion OperatorsSubjects: Numerical Analysis (math.NA)
We present a novel approach of discretizing variable coefficient diffusion operators in the context of meshfree generalized finite difference methods. Our ansatz uses properties of derived operators and combines the discrete Laplace operator with reconstruction functions approximating the diffusion coefficient. Provided that the reconstructions are of a sufficiently high order, we prove that the order of accuracy of the discrete Laplace operator transfers to the derived diffusion operator. We show that the new discrete diffusion operator inherits the diagonal dominance property of the discrete Laplace operator. Finally, we present the possibility of discretizing anisotropic diffusion operators with the help of derived operators. Our numerical results for Poisson's equation and the heat equation show that even loworder reconstructions preserve the order of the underlying discrete Laplace operator for sufficiently smooth diffusion coefficients. In experiments, we demonstrate the applicability of the new discrete diffusion operator to interface problems with point clouds not aligning to the interface and numerically show firstorder convergence.
 [399] arXiv:2305.06237 (replaced) [pdf, html, other]

Title: Weyl laws for interacting particlesComments: 37 pagesSubjects: Mathematical Physics (mathph); Spectral Theory (math.SP)
We study grandcanonical interacting fermionic systems in the meanfield regime, in a trapping potential. We provide the first order term of integrated and pointwise Weyl laws, but in the case with interaction. More precisely, we prove the convergence of the densities of the grandcanonical HartreeFock ground state to the ThomasFermi ground state in the semiclassical limit $\hbar\to 0$. For the proof, we write the grandcanonical version of the results of Fournais, Lewin and Solovej (Calc. Var. Partial Differ. Equ., 2018) and of Conlon (Commun. Math. Phys., 1983).
 [400] arXiv:2305.08928 (replaced) [pdf, html, other]

Title: Handle numbers of guts of sutured manifolds and nearly fibered knotsComments: 17 pages, 4 figuresSubjects: Geometric Topology (math.GT)
Extending Haken's Theorem to product annuli and disks for Heegaard splittings of sutured manifolds, we show that the handle number of an irreducible sutured manifold equals the handle number of its guts. We further show that reduced sutured manifolds with torus boundary contained in $S^3$ fall in to three types that generalize the three models of guts of knots that are nearly fibered in the instanton or Heegaard Floer sense. In conjunction with these results and another concerning uniqueness of incompressible Seifert surfaces, we show that while many nearly fibered knots have handle number $2$ and a unique incompressible Seifert surface, some have handle number $4$ and others have extra incompressible Seifert surfaces. Examples of nearly fibered knots with nonisotopic incompressible Seifert surfaces are exhibited.
 [401] arXiv:2305.14762 (replaced) [pdf, html, other]

Title: The finite frame property of some extensions of the pure logic of necessitationComments: 24 pagesSubjects: Logic (math.LO)
We study the finite frame property of some extensions of Fitting, Marek, and Truszczyński's pure logic of necessitation $\mathbf{N}$. For any natural numbers $m, n$, we introduce the logic $\mathbf{N}^+\mathbf{A}_{m, n}$ by adding the single axiom scheme $\Box^n \varphi \to \Box^m \varphi$ and the rule $\dfrac{\neg \Box \varphi}{\neg \Box \Box \varphi}$ (Ros$^\Box$) into $\mathbf{N}$. We prove the finite frame property of $\mathbf{N}^+\mathbf{A}_{m, n}$ with respect to Fitting, Marek, and Truszczyński's relational semantics. We also prove that for $n \ge 2$, the logic obtained by removing the rule Ros$^\Box$ from $\mathbf{N}^+\mathbf{A}_{0, n}$ is incomplete with respect to that semantics.
 [402] arXiv:2305.19095 (replaced) [pdf, html, other]

Title: Matroidal Mixed Eulerian NumbersComments: revised version. 32 pagesSubjects: Combinatorics (math.CO); Algebraic Geometry (math.AG)
We make a systematic study of matroidal mixed Eulerian numbers which are certain intersection numbers in the matroid Chow ring generalizing the mixed Eulerian numbers introduced by Postnikov. These numbers are shown to be valuative and obey a logconcavity relation. We establish recursion formulas and use them to relate matroidal mixed Eulerian numbers to the characteristic and Tutte polynomials, reproving results of HuhKatz and BergetSpinkTseng. Generalizing Postnikov, we show that these numbers are equal to certain weighted counts of binary trees. Lastly, we study these numbers for perfect matroid designs, proving that they generalize the remixed Eulerian numbers of NadeauTewari.
 [403] arXiv:2306.04230 (replaced) [pdf, html, other]

Title: Distributed accelerated proximal conjugate gradient methods for multiagent constrained optimization problemsComments: page 30Subjects: Optimization and Control (math.OC)
The purpose of this paper is to introduce two new classes of accelerated distributed proximal conjugate gradient algorithms for multiagent constrained optimization problems; given as minimization of a function decomposed as a sum of M number of smooth and M number of nonsmooth functions over the common fixed points of M number of nonlinear mappings. Exploiting the special properties of the cost component function of the objective function and the nonlinear mapping of the constraint problem of each agent, a new inertial accelerated incremental and parallel computing distributed algorithms will be presented based on the combinations of computations of proximal, conjugate gradient and Halpern methods. Some numerical experiments and comparisons are given to illustrate our results.
 [404] arXiv:2306.05486 (replaced) [pdf, html, other]

Title: Multilevel domain decompositionbased architectures for physicsinformed neural networksComments: 42 pages, 12 figuresSubjects: Numerical Analysis (math.NA)
Physicsinformed neural networks (PINNs) are a powerful approach for solving problems involving differential equations, yet they often struggle to solve problems with high frequency and/or multiscale solutions. Finite basis physicsinformed neural networks (FBPINNs) improve the performance of PINNs in this regime by combining them with an overlapping domain decomposition approach. In this work, FBPINNs are extended by adding multiple levels of domain decompositions to their solution ansatz, inspired by classical multilevel Schwarz domain decomposition methods (DDMs). Analogous to typical tests for classical DDMs, we assess how the accuracy of PINNs, FBPINNs and multilevel FBPINNs scale with respect to computational effort and solution complexity by carrying out strong and weak scaling tests. Our numerical results show that the proposed multilevel FBPINNs consistently and significantly outperform PINNs across a range of problems with high frequency and multiscale solutions. Furthermore, as expected in classical DDMs, we show that multilevel FBPINNs improve the accuracy of FBPINNs when using large numbers of subdomains by aiding global communication between subdomains.
 [405] arXiv:2306.07110 (replaced) [pdf, html, other]

Title: Invariant measures on padic Lie groups: the padic quaternion algebra and the Haar integral on the padic rotation groupsComments: 49 pages, minor changesJournalref: Lett. Math. Phys. 114, 78 (2024)Subjects: Mathematical Physics (mathph); Functional Analysis (math.FA); Number Theory (math.NT)
We provide a general expression of the Haar measure $$ that is, the essentially unique translationinvariant measure $$ on a $p$adic Lie group. We then argue that this measure can be regarded as the measure naturally induced by the invariant volume form on the group, as it happens for a standard Lie group over the reals. As an important application, we next consider the problem of determining the Haar measure on the $p$adic special orthogonal groups in dimension two, three and four (for every prime number $p$). In particular, the Haar measure on $\mathrm{SO}(2,\mathbb{Q}_p)$ is obtained by a direct application of our general formula. As for $\mathrm{SO}(3,\mathbb{Q}_p)$ and $\mathrm{SO}(4,\mathbb{Q}_p)$, instead, we show that Haar integrals on these two groups can conveniently be lifted to Haar integrals on certain $p$adic Lie groups from which the special orthogonal groups are obtained as quotients. This construction involves a suitable quaternion algebra over the field $\mathbb{Q}_p$ and is reminiscent of the quaternionic realization of the real rotation groups. Our results should pave the way to the development of harmonic analysis on the $p$adic special orthogonal groups, with potential applications in $p$adic quantum mechanics and in the recently proposed $p$adic quantum information theory.
 [406] arXiv:2306.09213 (replaced) [pdf, html, other]

Title: Stationarity and Fredholm Theory in Subextremal Kerrde Sitter SpacetimesComments: This paper is dedicated to Christian Bär's 60th birthday. The paper is a continuation and generalization of arXiv:2112.01355. Correspondingly, some assumptions and theorems are formulated the same wayJournalref: SIGMA 20 (2024), 052, 11 pagesSubjects: Analysis of PDEs (math.AP); General Relativity and Quantum Cosmology (grqc); Differential Geometry (math.DG)
In a recent paper, we proved that solutions to linear wave equations in a subextremal Kerrde Sitter spacetime have asymptotic expansions in quasinormal modes up to a decay order given by the normally hyperbolic trapping, extending the results of Vasy (2013). One central ingredient in the argument was a new definition of quasinormal modes, where a nonstandard choice of stationary Killing vector field had to be used in order for the Fredholm theory to be applicable. In this paper, we show that there is in fact a variety of allowed choices of stationary Killing vector fields. In particular, the horizon Killing vector fields work for the analysis, in which case one of the corresponding ergoregions is completely removed.
 [407] arXiv:2306.09748 (replaced) [pdf, html, other]

Title: Liouville comparison theory for blowup of EulerArnold equationsComments: 32 pagesSubjects: Analysis of PDEs (math.AP)
In this article we introduce a new blowup criterion for (generalized) EulerArnold equations on $\mathbb R^n$. Our method is based on treating the equation in Lagrangian coordinates, where it is an ODE on the diffeomorphism group, and comparison with the Liouville equation; in contrast to the usual comparison approach at a single point, we apply comparison in an infinite dimensional function space. We thereby show that the Jacobian of the Lagrangian flow map of the solution reaches zero in finite time, which corresponds to $C^1$blowup of the velocity field solution. We demonstrate the applicability of our result by proving blowup of smooth solutions to some higherorder versions of the EPDiff equation in all dimensions $n\geq 3$. Previous results on blowup of higher dimensional EPDiff equations were only for versions where the geometric description corresponds to a Sobolev metric of order zero or one. In these situations the behavior does not depend on the dimension and thus already solutions to the onedimensional version were exhibiting blowup. In the present paper blowup is proved even in situations where the onedimensional equation has global solutions, such as the EPDiff equation corresponding to a Sobolev metric of order two.
 [408] arXiv:2306.10800 (replaced) [pdf, other]

Title: Multilevel Surrogatebased Control VariatesMohamed Reda El Amri (IFPEN), Paul Mycek (CERFACS, CONCACE), Sophie Ricci (CERFACS), Matthias De LozzoSubjects: Statistics Theory (math.ST)
Monte Carlo (MC) sampling is a popular method for estimating the statistics (e.g. expectation and variance) of a random variable. Its slow convergence has led to the emergence of advanced techniques to reduce the variance of the MC estimator for the outputs of computationally expensive solvers. The control variates (CV) method corrects the MC estimator with a term derived from auxiliary random variables that are highly correlated with the original random variable. These auxiliary variables may come from surrogate models. Such a surrogatebased CV strategy is extended here to the multilevel Monte Carlo (MLMC) framework, which relies on a sequence of levels corresponding to numerical simulators with increasing accuracy and computational cost. MLMC combines output samples obtained across levels, into a telescopic sum of differences between MC estimators for successive fidelities. In this paper, we introduce three multilevel variance reduction strategies that rely on surrogatebased CV and MLMC. MLCV is presented as an extension of CV where the correction terms devised from surrogate models for simulators of different levels add up. MLMCCV improves the MLMC estimator by using a CV based on a surrogate of the correction term at each level. Further variance reduction is achieved by using the surrogatebased CVs of all the levels in the MLMCMLCV strategy. Alternative solutions that reduce the subset of surrogates used for the multilevel estimation are also introduced. The proposed methods are tested on a test case from the literature consisting of a spectral discretization of an uncertain 1D heat equation, where the statistic of interest is the expected value of the integrated temperature along the domain at a given time. The results are assessed in terms of the accuracy and computational cost of the multilevel estimators, depending on whether the construction of the surrogates, and the associated computational cost, precede the evaluation of the estimator. It was shown that when the lower fidelity outputs are strongly correlated with the highfidelity outputs, a significant variance reduction is obtained when using surrogate models for the coarser levels only. It was also shown that taking advantage of preexisting surrogate models proves to be an even more efficient strategy.
 [409] arXiv:2306.13342 (replaced) [pdf, other]

Title: On the (super)additivity of simplicial volumeComments: 55 pages; completely rewritten; Lemma 3.6 in previous versions is falseSubjects: Geometric Topology (math.GT)
We show that the simplicial volume is superadditive with respect to gluings along certain submanifolds of the boundary. Our criterion applies to boundary connected sums and 1handle attachments. Moreover, we generalize a wellknown additivity result in the case of aspherical manifolds. Our arguments are based on new results about relative bounded cohomology and pairs of multicomplexes, which are of independent interest.
 [410] arXiv:2307.01334 (replaced) [pdf, html, other]

Title: Finitely generated subgroups of algebraic elements of plane Cremona groups are boundedComments: Minor revisionsSubjects: Group Theory (math.GR); Algebraic Geometry (math.AG); Dynamical Systems (math.DS)
We prove that any finitely generated subgroup of the plane Cremona group consisting only of algebraic elements is of bounded degree. This follows from a more general result on `decent' actions on infinite direct sums. We apply our results to describe the degree growth of finitely generated subgroups of the plane Cremona group.
 [411] arXiv:2307.04323 (replaced) [pdf, html, other]

Title: Optimal $(2,\delta)$ Locally Repairable Codes via Punctured Simplex CodesSubjects: Information Theory (cs.IT)
Locally repairable codes (LRCs) have attracted a lot of attention due to their applications in distributed storage systems. In this paper, we provide new constructions of optimal $(2, \delta)$LRCs over $\mathbb{F}_q$ with flexible parameters. Firstly, employing techniques from finite geometry, we introduce a simple yet useful condition to ensure that a punctured simplex code becomes a $(2, \delta)$LRC. It is worth noting that this condition only imposes a requirement on the size of the puncturing set. Secondly, utilizing character sums over finite fields and Krawtchouk polynomials, we determine the parameters of more punctured simplex codes with puncturing sets of new structures. Several infinite families of LRCs with new parameters are derived. All of our new LRCs are optimal with respect to the generalized CadambeMazumdar bound and some of them are also Griesmer codes or distanceoptimal codes.
 [412] arXiv:2307.05125 (replaced) [pdf, html, other]

Title: Linearizing Binary Optimization Problems Using Variable Posets for Ising MachinesComments: 22 pages. This article has been accepted for publication in IEEE Transactions on Emerging Topics in ComputingSubjects: Optimization and Control (math.OC); Statistical Mechanics (condmat.statmech); Emerging Technologies (cs.ET); Applied Physics (physics.appph)
Ising machines are nextgeneration computers expected to efficiently sample nearoptimal solutions of combinatorial optimization problems. Combinatorial optimization problems are modeled as quadratic unconstrained binary optimization (QUBO) problems to apply an Ising machine. However, current stateoftheart Ising machines still often fail to output nearoptimal solutions due to the complicated energy landscape of QUBO problems. Furthermore, the physical implementation of Ising machines severely restricts the size of QUBO problems to be input as a result of limited hardware graph structures. In this study, we take a new approach to these challenges by injecting auxiliary penalties preserving the optimum, which reduces quadratic terms in QUBO objective functions. The process simultaneously simplifies the energy landscape of QUBO problems, allowing the search for nearoptimal solutions, and makes QUBO problems sparser, facilitating encoding into Ising machines with restriction on the hardware graph structure. We propose linearization of QUBO problems using variable posets as an outcome of the approach. By applying the proposed method to synthetic QUBO instances and to multidimensional knapsack problems, we empirically validate the effects on enhancing minorembedding of QUBO problems and the performance of Ising machines.
 [413] arXiv:2307.06379 (replaced) [pdf, html, other]

Title: Induced subgraph density. III. Cycles and subdivisionsComments: 20 pagesSubjects: Combinatorics (math.CO)
We show that for every two cycles $C,D$, there exists $c>0$ such that if $G$ is both $C$free and $\overline{D}$free then $G$ has a clique or stable set of size at least $G^c$. ("$H$free" means with no induced subgraph isomorphic to $H$, and $\overline{D}$ denotes the complement graph of $D$.) Since the fivevertex cycle $C_5$ is isomorphic to its complement, this extends the earlier result that $C_5$ satisfies the ErdősHajnal conjecture. It also unifies and strengthens several other results.
The results for cycles are special cases of results for subdivisions, as follows. Let $H,J$ be obtained from smaller graphs by subdividing every edge exactly twice. We will prove that there exists $c>0$ such that if $G$ is both $H$free and $\overline{J}$free then $G$ has a clique or stable set of size at least $G^c$. And the same holds if $H$ and/or $J$ is obtained from a graph bychoosing a forest $F$ and subdividing every edge not in $F$ at least five times. Our proof uses the framework of iterative sparsification developed in other papers of this series.
Along the way, we will also give a short and simple proof of a celebrated result of Fox and Sudakov, that says that for all $H$, every $H$free graph contains either a large stable set or a large complete bipartite subgraph.  [414] arXiv:2307.07357 (replaced) [pdf, html, other]

Title: Inverse Optimization for Routing ProblemsSubjects: Optimization and Control (math.OC); Machine Learning (cs.LG)
We propose a method for learning decisionmakers' behavior in routing problems using Inverse Optimization (IO). The IO framework falls into the supervised learning category and builds on the premise that the target behavior is an optimizer of an unknown cost function. This cost function is to be learned through historical data, and in the context of routing problems, can be interpreted as the routing preferences of the decisionmakers. In this view, the main contributions of this study are to propose an IO methodology with a hypothesis function, loss function, and stochastic firstorder algorithm tailored to routing problems. We further test our IO approach in the Amazon Last Mile Routing Research Challenge, where the goal is to learn models that replicate the routing preferences of human drivers, using thousands of realworld routing examples. Our final IOlearned routing model achieves a score that ranks 2nd compared with the 48 models that qualified for the final round of the challenge. Our examples and results showcase the flexibility and realworld potential of the proposed IO methodology to learn from decisionmakers' decisions in routing problems.
 [415] arXiv:2307.08878 (replaced) [pdf, html, other]

Title: The Poisson boundary of lampshuffler groupsComments: 25 pages, no figuresSubjects: Group Theory (math.GR); Probability (math.PR)
We study random walks on the lampshuffler group $\mathrm{FSym}(H)\rtimes H$, where $H$ is a finitely generated group and $\mathrm{FSym}(H)$ is the group of finitary permutations of $H$. We show that for any step distribution $\mu$ with a finite first moment that induces a transient random walk on $H$, the permutation coordinate of the random walk almost surely stabilizes pointwise. Our main result states that for $H=\mathbb{Z}$, the above convergence completely describes the Poisson boundary of the random walk $(\mathrm{FSym}(\mathbb{Z})\rtimes \mathbb{Z},\mu)$.
 [416] arXiv:2307.12044 (replaced) [pdf, html, other]

Title: Kinetic description of swarming dynamics with topological interaction and transient leadersSubjects: Numerical Analysis (math.NA); Populations and Evolution (qbio.PE)
In this paper, we present a model describing the collective motion of birds. The model introduces spontaneous changes in direction which are initialized by few agents, here referred as leaders, whose influence act on their nearest neighbors, in the following referred as followers. Starting at the microscopic level, we develop a kinetic model that characterizes the behaviour of large flocks with transient leadership. One significant challenge lies in managing topological interactions, as identifying nearest neighbors in extensive systems can be computationally expensive. To address this, we propose a novel stochastic particle method to simulate the mesoscopic dynamics and reduce the computational cost of identifying closer agents from quadratic to logarithmic complexity using a $k$nearest neighbours search algorithm with a binary tree. Lastly, we conduct various numerical experiments for different scenarios to validate the algorithm's effectiveness and investigate collective dynamics in both two and three dimensions.
 [417] arXiv:2307.13540 (replaced) [pdf, html, other]

Title: Scattering theory of topologically protected edge transportComments: 30 pagesSubjects: Spectral Theory (math.SP)
This paper develops a scattering theory for the asymmetric transport observed at interfaces separating twodimensional topological insulators. Starting from the spectral decomposition of an unperturbed interface Hamiltonian, we present a limiting absorption principle and construct a generalized eigenfunction expansion for perturbed systems. We then relate a physical observable quantifying the transport asymmetry to the scattering matrix associated to the generalized eigenfunctions. In particular, we show that the observable is concretely expressed as a difference of transmission coefficients and is stable against perturbations. We apply the theory to systems of perturbed Dirac equations with asymptotically linear domain wall.
 [418] arXiv:2307.14181 (replaced) [pdf, html, other]

Title: Convex semiinfinite programming algorithms with inexact separation oraclesSubjects: Optimization and Control (math.OC)
Solving convex SemiInfinite Programming (SIP) problems is challenging when the separation problem, i.e., the problem of finding the most violated constraint, is computationally hard. We propose to tackle this difficulty by solving the separation problem approximately, i.e., by using an inexact oracle. Our focus lies in two algorithms for SIP, namely the CuttingPlanes (CP) and the InnerOuter Approximation (IOA) algorithms. We prove the CP convergence rate to be in O(1/k), where k is the number of calls to the limitedaccuracy oracle, if the objective function is strongly convex. Compared to the CP algorithm, the advantage of the IOA algorithm is the feasibility of its iterates. In the case of a semiinfinite program with Quadratically Constrained Quadratic Programming separation problem, we prove the convergence of the IOA algorithm toward an optimal solution of the SIP problem despite the oracle's inexactness.
 [419] arXiv:2308.01642 (replaced) [pdf, html, other]

Title: Weak uniqueness by noise for singular stochastic PDEsComments: 43 pagesSubjects: Probability (math.PR); Analysis of PDEs (math.AP)
We prove weak uniqueness of mild solutions for general classes of SPDEs on a Hilbert space. The main novelty is that the drift is only defined on a Sobolevtype subspace and no Höldercontinuity assumptions are required. This framework turns out to be effective to achieve novel uniqueness results for several specific examples. Such wide range of applications is obtained by exploiting either coloured or rougherthancylindrical noises.
 [420] arXiv:2308.06158 (replaced) [pdf, html, other]

Title: Infinitesimal Modular Group: $q$Deformed $\mathfrak{sl}_2$ and Witt AlgebraJournalref: SIGMA 20 (2024), 053, 16 pagesSubjects: Quantum Algebra (math.QA); Mathematical Physics (mathph)
We describe new $q$deformations of the 3dimensional Heisenberg algebra, the simple Lie algebra $\mathfrak{sl}_2$ and the Witt algebra. They are constructed through a realization as differential operators. These operators are related to the modular group and $q$deformed rational numbers defined by MorierGenoud and Ovsienko and lead to $q$deformed Möbius transformations acting on the hyperbolic plane.
 [421] arXiv:2309.00931 (replaced) [pdf, other]

Title: Parametric Finite Element Discretization of the Surface Stokes EquationsComments: 40 pages, 14 figuresSubjects: Numerical Analysis (math.NA)
We study a higherorder surface finite element (SFEM) penaltybased discretization of the tangential surface Stokes problem. Several discrete formulations are investigated which are equivalent in the continuous setting. The impact of the choice of discretization of the diffusion term and of the divergence term on numerical accuracy and convergence, as well as on implementation advantages, is discussed. We analyze the infsup stability of the discrete scheme in a generic approach by lifting stable finite element pairs known from the literature. A discretization error analysis in tangential norms then shows optimal order convergence of an isogeometric setting that requires only geometric knowledge of the discrete surface.
 [422] arXiv:2309.12875 (replaced) [pdf, html, other]

Title: A secondorder in time, BGNbased parametric finite element method for geometric flows of curvesComments: 37 pages, 8 figuresSubjects: Numerical Analysis (math.NA)
Over the last two decades, the field of geometric curve evolutions has attracted significant attention from scientific computing. One of the most popular numerical methods for solving geometric flows is the socalled BGN scheme, which was proposed by Barrett, Garcke, and Nürnberg (J. Comput. Phys., 222 (2007), pp.~441467), due to its favorable properties (e.g., its computational efficiency and the good mesh property). However, the BGN scheme is limited to firstorder accuracy in time, and how to develop a higherorder numerical scheme is challenging. In this paper, we propose a fully discrete, temporal secondorder parametric finite element method, which integrates with two different mesh regularization techniques, for solving geometric flows of curves. The scheme is constructed based on the BGN formulation and a semiimplicit CrankNicolson leapfrog time stepping discretization as well as a linear finite element approximation in space. More importantly, we point out that the shape metrics, such as manifold distance and Hausdorff distance, instead of function norms, should be employed to measure numerical errors. Extensive numerical experiments demonstrate that the proposed BGNbased scheme is secondorder accurate in time in terms of shape metrics. Moreover, by employing the classical BGN scheme as mesh regularization techniques, our proposed secondorder schemes exhibit good properties with respect to the mesh distribution. In addition, an unconditional interlaced energy stability property is obtained for one of the mesh regularization techniques.
 [423] arXiv:2309.14169 (replaced) [pdf, html, other]

Title: Extrapolated regularization of nearly singular integrals on surfacesSubjects: Numerical Analysis (math.NA)
We present a method for computing nearly singular integrals that occur when single or double layer surface integrals, for harmonic potentials or Stokes flow, are evaluated at points nearby. Such values could be needed in solving an integral equation when one surface is close to another or to obtain values at grid points. We replace the singular kernel with a regularized version having a length parameter $\delta$ in order to control discretization error. Analysis near the singularity leads to an expression for the error due to regularization which has terms with unknown coefficients multiplying known quantities. By computing the integral with three choices of $\delta$ we can solve for an extrapolated value that has regularization error reduced to $O(\delta^5)$, uniformly for target points on or near the surface. In examples with $\delta/h$ constant and moderate resolution we observe total error about $O(h^5)$ close to the surface. For convergence as $h \to 0$ we can choose $\delta$ proportional to $h^q$ with $q < 1$ to ensure the discretization error is dominated by the regularization error. With $q = 4/5$ we find errors about $O(h^4)$. For harmonic potentials we extend the approach to a version with $O(\delta^7)$ regularization; it typically has smaller errors but the order of accuracy is less predictable.
 [424] arXiv:2309.15001 (replaced) [pdf, html, other]

Title: Convergence guarantees for forward gradient descent in the linear regression modelComments: 17 pagesJournalref: Journal of Statistical Planning and Inference, Volume 233, 106174, 2024Subjects: Statistics Theory (math.ST); Neural and Evolutionary Computing (cs.NE)
Renewed interest in the relationship between artificial and biological neural networks motivates the study of gradientfree methods. Considering the linear regression model with random design, we theoretically analyze in this work the biologically motivated (weightperturbed) forward gradient scheme that is based on random linear combination of the gradient. If d denotes the number of parameters and k the number of samples, we prove that the mean squared error of this method converges for $k\gtrsim d^2\log(d)$ with rate $d^2\log(d)/k.$ Compared to the dimension dependence d for stochastic gradient descent, an additional factor $d\log(d)$ occurs.
 [425] arXiv:2309.15967 (replaced) [pdf, html, other]

Title: Classification of irreducible representations of affine group superschemes and the division superalgebras of their endomorphismsComments: The continuity arguments of cocycles and Galois actions are added in Sections 35. The definition of (super) quasirationality is changed to let Theorem 3.16 (2) hold true for imperfect fields. Theorem 4.14 is revised. The notion of basic supergroups is introduced instead of absolutely basic supergroups. Other typos are correctedSubjects: Representation Theory (math.RT)
In this paper, we classify irreducible representations of affine group superschemes over fields $F$ of characteristic not two in terms of those over a separable closure $F^{\mathrm{sep}}$ and their Galois twists. We also compute the division superalgebras of their endomorphisms mainly when they are central. Finally, we give numerical conclusions for quasireductive algebraic supergroups under certain conditions, based on Shibata's BorelWeil theory for split quasireductive algebraic supergroups.
 [426] arXiv:2309.16763 (replaced) [pdf, other]

Title: Higher multiplier idealsComments: v5, some typos are fixedSubjects: Algebraic Geometry (math.AG); Commutative Algebra (math.AC); Complex Variables (math.CV)
We associate a family of ideal sheaves to any Qeffective divisor on a complex manifold, called the higher multiplier ideals, using the theory of mixed Hodge modules and Vfiltrations. This family is indexed by two parameters, an integer indicating the Hodge level and a rational number, and these ideals admit a weight filtration. When the Hodge level is zero, they recover the usual multiplier ideals. We study the local and global properties of higher multiplier ideals systematically. In particular, we prove vanishing theorems and restriction theorems, and provide criteria for the nontriviality of the new ideals. The main idea is to exploit the global structure of the Vfiltration along an effective divisor using the notion of twisted Hodge modules. In the local theory, we introduce the notion of the center of minimal exponent, which generalizes the notion of minimal log canonical center. As applications, we prove some cases of conjectures by Debarre, CasalainaMartin and Grushevsky on singularities of theta divisors on principally polarized abelian varieties and the geometric RiemannSchottky problem.
 [427] arXiv:2310.05583 (replaced) [pdf, html, other]

Title: New Volume Comparison Results and Volume Growth Rigidity of Gradient Ricci Almost SolitonsSubjects: Differential Geometry (math.DG)
In this paper, we establish a new volume comparison theorem for a complete manifold with a function $\rho(x)$ as the lower bound of the BakryEmery Ricci curvature. As applications, we obtain a new volume rigidity result of the gradient Ricci almost solitons. Furthermore, we extend the results of Cao and Zhou \cite{CZ} to shrinking gradient Ricci almost solitons and get the rigidity result with respect to the maximal volume growth.
 [428] arXiv:2310.05781 (replaced) [pdf, html, other]

Title: On variational inference and maximum likelihood estimation with the {\lambda}exponential familySubjects: Statistics Theory (math.ST)
The {\lambda}exponential family has recently been proposed to generalize the exponential family. While the exponential family is wellunderstood and widely used, this it not the case of the {\lambda}exponential family. However, many applications require models that are more general than the exponential family. In this work, we propose a theoretical and algorithmic framework to solve variational inference and maximum likelihood estimation problems over the {\lambda}exponential family. We give new sufficient optimality conditions for variational inference problems. Our conditions take the form of generalized momentmatching conditions and generalize existing similar results for the exponential family. We exhibit novel characterizations of the solutions of maximum likelihood estimation problems, that recover optimality conditions in the case of the exponential family. For the resolution of both problems, we propose novel proximallike algorithms that exploit the geometry underlying the {\lambda}exponential family. These new theoretical and methodological insights are tested on numerical examples, showcasing their usefulness and interest, especially on heavytailed target distributions.
 [429] arXiv:2310.08472 (replaced) [pdf, other]

Title: The prismatic realization functor for Shimura varieties of abelian typeComments: 90 pages. Significant changes. The results are upgraded to prismatic Fgauges. This has several interesting consequences, including the SerreTate theory for Shimura varieties of abelian type. Additionally, some portion of the original article has been detached and is now included in arXiv:2406.08259Subjects: Number Theory (math.NT); Algebraic Geometry (math.AG)
For the integral canonical model $\mathscr{S}_{\mathsf{K}^p}$ of a Shimura variety $\mathrm{Sh}_{\mathsf{K}_0\mathsf{K}^p}(\mathbf{G},\mathbf{X})$ of abelian type at hyperspecial level $K_0=\mathcal{G}(\mathbb{Z}_p)$, we construct a prismatic $F$gauge model for the `universal' $\mathcal{G}(\mathbb{Z}_p)$local system on $\mathrm{Sh}_{\mathsf{K}_0\mathsf{K}^p}(\mathbf{G},\mathbf{X})$. We use this to obtain new $p$adic information about these Shimura varieties, notably an abeliantype analogue of the SerreTate theorem (realizing an expectation of Drinfeld in the abeliantype case) and a prismatic characterization of these models. To do this, we make several advances in integral $p$adic Hodge theory, notably the development of an integral analogue of the functor $D_{\mathrm{crys}}$.
 [430] arXiv:2311.00381 (replaced) [pdf, html, other]

Title: Timeinconsistent meanfield stopping problems: A regularized equilibrium approachSubjects: Optimization and Control (math.OC); Probability (math.PR)
This paper studies the meanfield Markov decision process (MDP) with the centralized stopping under the nonexponential discount. The problem differs fundamentally from most existing studies on meanfield optimal control/stopping due to its time inconsistency by nature. We look for the subgame perfect relaxed equilibria, namely the randomized stopping policies that satisfy the timeconsistent planning with future selves from the perspective of the social planner. On the other hand, unlike many previous studies on timeinconsistent stopping where the decreasing impatience plays a key role, we are interested in the general discount function without imposing any conditions. As a result, the study on the relaxed equilibrium becomes necessary as the purestrategy equilibrium may not exist in general. We formulate relaxed equilibria as fixed points of a complicated operator, whose existence is challenging by a direct method. To overcome the obstacles, we first introduce the auxiliary problem under the entropy regularization on the randomized policy and the discount function, and establish the existence of the regularized equilibria as fixed points to an auxiliary operator via Schauder fixed point theorem. Next, we show that the regularized equilibrium converges as the regularization parameter $\lambda$ tends to $0$ and the limit corresponds to a fixed point to the original operator, and hence is a relaxed equilibrium. We also establish some connections between the meanfield MDP and the Nagent MDP when $N$ is sufficiently large in our timeinconsistent setting.
 [431] arXiv:2311.01190 (replaced) [pdf, html, other]

Title: Noncanonical maximum cliques without a design structure in the block graphs of 2designsSubjects: Combinatorics (math.CO)
In this note we answer positively a question of Chris Godsil and Karen Meagher on the existence of a 2design whose block graph has a noncanonical maximum clique without a design structure.
 [432] arXiv:2311.01264 (replaced) [pdf, html, other]

Title: Structure preserving discontinuous Galerkin approximation of a hyperbolicparabolic systemSubjects: Numerical Analysis (math.NA)
We study the numerical approximation of a coupled hyperbolicparabolic system by a family of discontinuous Galerkin spacetime finite element methods. The model is rewritten as a firstorder evolutionary problem that is treated by the unified abstract solution theory of R. Picard. For the discretization in space, generalizations of the distribution gradient and divergence operators on broken polynomial spaces are defined. Since their skewselfadjointness is perturbed by boundary surface integrals, adjustments are introduced such that the skewselfadjointness of the firstorder differential operator in space is recovered. Wellposedness of the fully discrete problem and error estimates for the discontinuous Galerkin approximation in space and time are proved.
 [433] arXiv:2311.02596 (replaced) [pdf, html, other]

Title: Embedding of Markov matrices for $d\leqslant 4$Michael Baake (Bielefeld), Jeremy Sumner (Hobart)Comments: 41 pages, 2 tables; revised and improved versionSubjects: Probability (math.PR); Populations and Evolution (qbio.PE)
The embedding problem of Markov matrices in Markov semigroups is a classic problem that regained a lot of impetus and activities through recent needs in phylogeny and population genetics. Here, we give an account for dimensions $d\leqslant 4$, including a complete and simplified treatment of the case $d=3$, and derive the results in a systematic fashion, with an eye on the potential applications.
Further, we reconsider the setup of the corresponding problem for timeinhomogeneous Markov chains, which is needed for realworld applications because transition rates need not be constant over time. Additional cases of this more general embedding occur for any $d\geqslant 3$. We review the known case of $d=3$ and describe the setting for future work on $d=4$.  [434] arXiv:2311.07142 (replaced) [pdf, html, other]

Title: Numerical integrator for highly oscillatory differential equations based on the Neumann seriesComments: 19 pages, 6 figuresJournalref: Numerical Algorithms, 2024Subjects: Numerical Analysis (math.NA)
We propose a thirdorder numerical integrator based on the Neumann series and the Filon quadrature, designed mainly for highly oscillatory partial differential equations. The method can be applied to equations that exhibit small or moderate oscillations; however, counterintuitively, large oscillations increase the accuracy of the scheme. With the proposed approach, the convergence order of the method can be easily improved. Error analysis of the method is also performed. We consider linear evolution equations involving first and secondtime derivatives that feature elliptic differential operators, such as the heat equation or the wave equation. Numerical experiments consider the case in which the space dimension is greater than one and confirm the theoretical study.
 [435] arXiv:2311.08757 (replaced) [pdf, other]

Title: A Scalable TwoLevel Domain Decomposition Eigensolver for Periodic Schr\"odinger Eigenstates in Anisotropically Expanding DomainsComments: 26 pages, 7 figures, 2 tablesSubjects: Numerical Analysis (math.NA); Computational Engineering, Finance, and Science (cs.CE)
Accelerating iterative eigenvalue algorithms is often achieved by employing a spectral shifting strategy. Unfortunately, improved shifting typically leads to a smaller eigenvalue for the resulting shifted operator, which in turn results in a high condition number of the underlying solution matrix, posing a major challenge for iterative linear solvers. This paper introduces a twolevel domain decomposition preconditioner that addresses this issue for the linear Schrödinger eigenvalue problem, even in the presence of a vanishing eigenvalue gap in nonuniform, expanding domains. Since the quasioptimal shift, which is already available as the solution to a spectral cell problem, is required for the eigenvalue solver, it is logical to also use its associated eigenfunction as a generator to construct a coarse space. We analyze the resulting twolevel additive Schwarz preconditioner and obtain a condition number bound that is independent of the domain's anisotropy, despite the need for only one basis function per subdomain for the coarse solver. Several numerical examples are presented to illustrate its flexibility and efficiency.
 [436] arXiv:2311.10625 (replaced) [pdf, html, other]

Title: Central limit theorems for Soft random simplicial complexesComments: 28 pagesSubjects: Probability (math.PR); Algebraic Topology (math.AT)
A soft random graph $G(n,r,p)$ can be obtained from the random geometric graph $G(n,r)$ by keeping every edge in $G(n,r)$ with probability $p$. The soft random simplicial complexes is a model for random simplicial complexes built over the soft random graph $G(n,r,p)$. This new model depends on a probability vector $\rho$ which allows the simplicial complexes to present randomness in all dimensions. In this article, we use a normal approximation theorem to prove central limit theorems for the number of $k$faces and for the Euler's characteristic for soft random simplicial complexes.
 [437] arXiv:2311.11078 (replaced) [pdf, other]

Title: A Lie group analog for the Monster Lie algebraSubjects: Representation Theory (math.RT); Group Theory (math.GR)
The Monster Lie algebra $\frak m $, which admits an action of the Monster finite simple group $\mathbb{M}$, was introduced by Borcherds as part of his work on the ConwayNorton Monstrous Moonshine conjecture. Here we construct an analog~$G(\frak m)$ of a Lie group or KacMoody group, associated to~$\frak m$. The group~$G(\frak m)$ is given by generators and relations, analogous to a construction of a KacMoody group given by Tits. In the absence of local nilpotence of the adjoint representation of $\frak m$, we introduce the notion of prosummability of an infinite sum of operators. We use this to construct a complete prounipotent group $\Uhp$ of automorphisms of a completion $\widehat{\mathfrak{m}}=\frak n^\ \oplus\ \frak h\ \oplus\ \widehat{\frak n}^+$ of~$\mathfrak{m}$, where $\widehat{\frak n}^+$ is the formal product of the positive root spaces of $\frak m$. The elements of $\widehat{U}^+$ are prosummable infinite series with constant term 1. The group $\widehat{U}^+$ has a subgroup~$\widehat{U}^+_\text{im}$, which is an analog of a complete unipotent group corresponding to the positive imaginary roots of~$\frak m$.We construct analogs $\text{Exp}: \widehat{\mathfrak{n}}^+\to\widehat{U}^+$ and $\text{Ad} :\widehat{U}^+ \to \Aut(\widehat{\frak{n}}^+)$ of the classical exponential map and adjoint representation. We show that the action of $\mathbb{M}$ on $\mathfrak m$ induces an action of~$\mathbb{M}$ on~$\widehat{\frak m}$, and that this in turn induces an action of $\mathbb{M}$ on~$\widehat{U}^+$. We also show that the action of $\mathbb{M}$ on $\widehat{\mathfrak n}^+$ is compatible with the action of $\widehat{U}^+$ on $\widehat{\mathfrak n}^+$.
 [438] arXiv:2311.11703 (replaced) [pdf, html, other]

Title: The delay feedback control for the McKeanVlasov stochastic differential equations with common noiseSubjects: Probability (math.PR)
Since response lags are essential in the feedback loops and are required by most physical systems, it is more appropriate to stabilize McKeanVlasov stochastic differential equations (MVSDEs) with common noise through the implementation of delay feedback control mechanisms. The aim of this paper is to design delay feedback control functions of the system state such that the controlled system to be boundedness in infinite horizon and further exponentially stable in the mean square. The designed controller, which depends only on the system state is easier to implement than that in [27] which was designed to depend on both system state and measure. The existence and uniqueness of the global solution of the controlled system is proved. The Itô formula with respect to both state and measure is derived. The proposed delay feedback control strategies are rendered viable for effective stabilization of MVSDEs with common noise. Furthermore, the moment Lyapunov exponent, which is intricately linked to the time delays, is meticulously estimated.
 [439] arXiv:2311.12635 (replaced) [pdf, html, other]

Title: A new approach to weighted Sobolev spacesComments: Submitted to Monatshefte für MathematikSubjects: Analysis of PDEs (math.AP)
We present in this paper a new way to define weighted Sobolev spaces when the weight functions are arbitrary small. This new approach can replace the old one consisting in modifying the domain by removing the set of points where at least one of the weight functions is very small. The basic idea is to replace the distributional derivative with a new notion of weak derivative. In this way, nonlocally integrable functions can considered in these spaces. Indeed, assumptions under which a degenerate elliptic partial differential equation has a unique nonlocally integrable solution are given. Tools like a Poincaré Inequality and a trace operator are developed, and density results of smooth function are established.
 [440] arXiv:2311.14616 (replaced) [pdf, html, other]

Title: Using MultiPrecisonArrays.jl: Iterative Refinement in JuliaSubjects: Numerical Analysis (math.NA)
MultiPrecisionArrays.jl is a Julia package. This package provides data structures and solvers for several variants of iterative refinement. It will become much more useful when half precision (aka Float16) is fully supported in LAPACK/BLAS. For now, its only generalpurpose application is classical iterative refinement with double precision equations and single precision factorizations.
 [441] arXiv:2311.17573 (replaced) [pdf, html, other]

Title: A stability result for Berge$K_{3,t}$ $r$graphs and its applicationsComments: 18 pagesSubjects: Combinatorics (math.CO)
An $r$uniform hypergraph ($r$graph) is linear if any two edges intersect at most one vertex. For a graph $F$, a hypergraph $H$ is Berge$F$ if there is a bijection $\phi:E(F)\rightarrow E(H)$ such that $e\subseteq \phi(e)$ for all $e$ in $E(F)$. In this paper, a kind of stability result for Berge$K_{3,t}$ linear $r$graphs is established. Based on this stability result, an upper bound for the linear Turán number of Berge$K_{3,t}$ is determined. For an $r$graph $H$, let $\mathcal{A}(H)$ be the adjacency tensor of $H$. The spectral radius of $H$ is the spectral radius of the tensor $\mathcal{A}(H)$. Some bounds for the maximum spectral radius of connected Berge$K_{3,t}$free linear $r$graphs are obtained.
 [442] arXiv:2312.01613 (replaced) [pdf, html, other]

Title: Local Cohomology Modules of Binomial Edge Ideals of Complements of Connected Graphs of Girth at Least 5Comments: 15 pages, v2 includes small changes to several proofs for clarity, and reorganises section 2 into subsections (whilst some numbering has changed, no new results are introduced)Subjects: Commutative Algebra (math.AC)
We calculate the local cohomology modules of the binomial edge ideals of the complements of connected graphs of girth at least 5 using the tools introduced by Àlvarez Montaner in arXiv:1901.08645. We then use this calculation to compute the depth, dimension, and regularity of these binomial edge ideals.
 [443] arXiv:2312.02277 (replaced) [pdf, other]

Title: ALEXR: An Optimal SingleLoop Algorithm for Convex FiniteSum Coupled Compositional Stochastic OptimizationComments: Fixed several typosSubjects: Optimization and Control (math.OC); Machine Learning (cs.LG)
This paper revisits a class of convex FiniteSum Coupled Compositional Stochastic Optimization (cFCCO) problems with many applications, including group distributionally robust optimization (GDRO), learning with imbalanced data, reinforcement learning, and learning to rank. To better solve these problems, we introduce an efficient singleloop primaldual blockcoordinate proximal algorithm, dubbed ALEXR. This algorithm leverages blockcoordinate stochastic mirror ascent updates for the dual variable and stochastic proximal gradient descent updates for the primal variable. We establish the convergence rates of ALEXR in both convex and strongly convex cases under smoothness and nonsmoothness conditions of involved functions, which not only improve the best rates in previous works on smooth cFCCO problems but also expand the realm of cFCCO for solving more challenging nonsmooth problems such as the dual form of GDRO. Finally, we present lower complexity bounds to demonstrate that the convergence rates of ALEXR are optimal among firstorder blockcoordinate stochastic algorithms for the considered class of cFCCO problems.
 [444] arXiv:2312.02461 (replaced) [pdf, html, other]

Title: Conjugate gradient methods without line search for multiobjective optimizationSubjects: Optimization and Control (math.OC)
This paper addresses an unconstrained multiobjective optimization problem where two or more continuously differentiable functions have to be minimized. We delve into the conjugate gradient methods proposed by Lucambio Pérez and Prudente (SIAM J Optim, 28(3): 26902720, 2018) for this type of problem. Instead of the Wolfetype line search procedure used in their work, we employ a fixed stepsize formula (or nolinesearch scheme), which can mitigate the selection pressure caused by multiple inequalities and avoid the computational cost associated with objective function evaluations in specific applications. The nolinesearch scheme is utilized to derive the condition of Zoutendijk's type. Global convergence encompasses the vector extensions of FletcherReeves, conjugate descent, DaiYuan, PolakRibièrePolyak and HestenesStiefel parameters, subject to certain mild assumptions. Additionally, numerical experiments are conducted to demonstrate the practical performance of the proposed stepsize rule, and comparative analyses are made with the multiobjective steepest descent methods using the Armijo line search and the multiobjective conjugate gradient methods using the Wolfetype line search.
 [445] arXiv:2312.03354 (replaced) [pdf, html, other]

Title: Calibrating FiguresSubjects: Metric Geometry (math.MG); Algebraic Geometry (math.AG)
It is known that a camera can be calibrated using three pictures of either squares, spheres, or surfaces of revolution. We give a new method to calibrate a camera with the picture of a single torus.
 [446] arXiv:2312.05843 (replaced) [pdf, html, other]

Title: Nonlinear Inverse Optimal Transport: Identifiability of the Transport Cost from its Marginals and Optimal ValuesComments: 23 pagesSubjects: Optimization and Control (math.OC)
The inverse optimal transport problem is to find the underlying cost function from the knowledge of optimal transport plans. While this amounts to solving a linear inverse problem, in this work we will be concerned with the nonlinear inverse problem to identify the cost function when only a set of marginals and its corresponding optimal values are given. We focus on absolutely continuous probability distributions with respect to the $d$dimensional Lebesgue measure and classes of concave and convex cost functions. Our main result implies that the cost function is uniquely determined from the union of the ranges of the gradients of the optimal potentials. Since, in general, the optimal potentials may not be observed, we derive sufficient conditions for their identifiability  if an open set of marginals is observed, the optimal potentials are then identified via the value of the optimal costs. We conclude with a more indepth study of this problem in the univariate case, where an explicit representation of the transport plan is available. Here, we link the notion of identifiability of the cost function with that of statistical completeness.
 [447] arXiv:2312.07197 (replaced) [pdf, html, other]

Title: Resolutions as directed colimitsComments: LaTeX 2e, 37 pages; v.2: several misprints corrected, references added and updated; v.3: small additions and corrections; v.4: small corrections, references updatedSubjects: Commutative Algebra (math.AC); Category Theory (math.CT); Rings and Algebras (math.RA)
A general principle suggests that "anything flat is a directed colimit of countably presentable flats". In this paper, we consider resolutions and coresolutions of modules over a countably coherent ring $R$ (e.g., any coherent ring or any countably Noetherian ring). We show that any $R$module of flat dimension $n$ is a directed colimit of countably presentable $R$modules of flat dimension at most $n$, and any flatly coresolved $R$module is a directed colimit of countably presentable flatly coresolved $R$modules. If $R$ is a countably coherent ring with a dualizing complex, then any Ftotally acyclic complex of flat $R$modules is a directed colimit of Ftotally acyclic complexes of countably presentable flat $R$modules. The proofs are applications of an even more general categorytheoretic principle going back to an unpublished 1977 preprint of Ulmer. Our proof of the assertion that every Gorensteinflat module over a countably coherent ring is a directed colimit of countably presentable Gorensteinflat modules uses a different technique, based on results of Saroch and Stovicek. We also discuss totally acyclic complexes of injectives and Gorensteininjective modules, obtaining various cardinality estimates for the accessibility rank under various assumptions.
 [448] arXiv:2312.07847 (replaced) [pdf, html, other]

Title: Floertype bipersistence modules and rectangle barcodesComments: 35 pages, 7 figures; (v4) main results (Theorems 4.2, 5.3, 6.5 and 6.10) improved, Examples 5.5 and 5.6 modifiedSubjects: Symplectic Geometry (math.SG); Algebraic Topology (math.AT)
In this paper, we show that the pointwise finitedimensional twoparameter persistence module $\mathbb{HF}_*^{(\bullet,\bullet]}$, defined in terms of inter