diff --git "a/raw_rss_feeds/https___arxiv_org_rss_math.xml" "b/raw_rss_feeds/https___arxiv_org_rss_math.xml" --- "a/raw_rss_feeds/https___arxiv_org_rss_math.xml" +++ "b/raw_rss_feeds/https___arxiv_org_rss_math.xml" @@ -7,4704 +7,10051 @@ http://www.rssboard.org/rss-specification en-us - Mon, 22 Dec 2025 05:00:03 +0000 + Tue, 23 Dec 2025 05:00:25 +0000 rss-help@arxiv.org - Mon, 22 Dec 2025 00:00:00 -0500 + Tue, 23 Dec 2025 00:00:00 -0500 - Saturday Sunday + Saturday - The existence of even factors based on the $A_\alpha$-spectral radius of graphs - https://arxiv.org/abs/2512.16932 - arXiv:2512.16932v1 Announce Type: new -Abstract: An even factor of $G$ is a spanning subgraph $F$ such that every vertex in $F$ has a nonzero even degree. Note that $\delta(G)\geq2$ is a trivial necessary condition for a graph to have an even factor, where $\delta(G)$ is the minimum degree of $G$. In this paper, for a connected graph $G$ with minimum degree $\delta$, we establish a lower bound on the $A_\alpha$-spectral radius of $G$ such that $G$ contains an even factor. - oai:arXiv.org:2512.16932v1 - math.CO - Mon, 22 Dec 2025 00:00:00 -0500 + A proof-of-principle experiment on the spontaneous symmetry breaking machine and numerical estimation of its performance on the $K_{2000}$ benchmark problem + https://arxiv.org/abs/2512.17922 + arXiv:2512.17922v1 Announce Type: new +Abstract: In a previous paper, we proposed a unique physically implemented type simulator for combinatorial optimization problems, called the spontaneous symmetry breaking machine (SSBM). In this paper, we first report the results of experimental verification of SSBM using a small-scale benchmark system, and then describe numerical simulations using the benchmark problems (K2000) conducted to confirm its usefulness for large-scale problems. From 1000 samples with different initial fluctuations, it became clear that SSBM can explore a single extremely stable state. This is based on the principle of a phenomenon used in SSBM, and could be a notable advantage over other simulators. + oai:arXiv.org:2512.17922v1 + math.OC + nlin.AO + physics.optics + quant-ph + Tue, 23 Dec 2025 00:00:00 -0500 new - http://creativecommons.org/publicdomain/zero/1.0/ - Caili Jia, Yong Lu + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Toshiya Sato, Takashi Goh - Finite fields whose members are the sum of a potent and a 5-potent - https://arxiv.org/abs/2512.16942 - arXiv:2512.16942v1 Announce Type: new -Abstract: We show that there are only finitely many finite fields whose members are the sum of an $n$-potent element and a $5$-potent element. Combining this with the algorithmic results provided by S.D. Cohen et al., we confirm the conjecture in \cite{Cohen} concerning all finite fields satisfying this condition. Furthermore, we obtain several elementary results for General problem, proving that the number of finite fields satisfying general condition is also finite. - oai:arXiv.org:2512.16942v1 - math.NT + Classification of almost symmetric numerical semigroups with maximal reduced type + https://arxiv.org/abs/2512.17957 + arXiv:2512.17957v1 Announce Type: new +Abstract: This paper determines almost symmetric numerical semigroups with maximal reduced type completely. In addition, this paper classifies MED-semigroups with maximal reduced type. + oai:arXiv.org:2512.17957v1 math.AC - math.RA - Mon, 22 Dec 2025 00:00:00 -0500 + Tue, 23 Dec 2025 00:00:00 -0500 new http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Juncheng Zhou, Hongfeng Wu - - - Spectral properties of Toeplitz operators with harmonic function symbols on the Bergman space - https://arxiv.org/abs/2512.16952 - arXiv:2512.16952v1 Announce Type: new -Abstract: This paper investigates the spectral properties of Toeplitz operators on the Bergman space of unit disk. We present an integral representation of $T^*_{z^m}$, which establishes a connection between the Bergman functions and certain partial differential equations. Moreover, by leveraging the Poincar\'{e} theorem in difference equations, this paper describes the kernels of certain Toeplitz operators with harmonic polynomial symbols and further gives sufficient conditions for the connectedness of the spectra of these operators. The spectral properties of $T_\varphi$ with $\varphi (z) =\overline{z}^{m} + \alpha z^m + \beta$ are characterized, showing that $\sigma(T_\varphi)= \overline{\varphi (\mathbb {D})}$, the Fredholm index of $T_\varphi$ can only be one of $m, -m$ and $0$, and $T_\varphi$ satisfies Coburn's theorem. These findings offer an illuminating example for studying the essential projective spectra of non-commuting operators. - oai:arXiv.org:2512.16952v1 - math.FA - Mon, 22 Dec 2025 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Puyu Cui, Yufeng Lu, Rongwei Yang, Chao Zu + Akihiro Sugawara - Some examples of use of transfinite induction in analysis - https://arxiv.org/abs/2512.16973 - arXiv:2512.16973v1 Announce Type: new -Abstract: It is not uncommon in analysis that existence of extremal objects is obtained via an iterative procedure: we start from a given admissible object, then modify it, then modify again etc... If being extremal means maximimizing a real valued quantity and we are sure to approach the supremum fast enough, after a countable number of steps and a limiting procedure we are done. - In this short note we want to advertise a slightly different line of thought, where rather than trying to approach the supremum fast enough, we: try to increase, if possible, the function to be maximized and, at the same time, index our recursive procedure over ordinals. Since there are no increasing functions from $\omega_1$ to $\R$, the procedure must stop at some countable ordinal and existence is proved anyway. - The advantage of this line of reasoning is that it can be helpful even in situations where it is not so evident how to measure `being maximal' via a real valued function. This is the case, for instance, for existence of a Maximal Globally Hyperbolic Development of an initial data set in General Relativity. - Speaking of this particular example, we also show that such `real-valued quantification' of the size of a development is actually possible, thus existence of a maximal one can be obtained in a countable number of steps using the original argument in [2] together with the standard procedure depicted above. This provides a way alternative to the one given in [5] to `dezornify' the proof in [2]. - oai:arXiv.org:2512.16973v1 - math.DG - gr-qc - math.CA - Mon, 22 Dec 2025 00:00:00 -0500 + Invariance of the Hausdorff Dimension of McMullen-Bedford Carpets under Coordinate Reflections + https://arxiv.org/abs/2512.17960 + arXiv:2512.17960v1 Announce Type: new +Abstract: We analyze a generalization of the self-affine carpets of Bedford and McMullen where the defining iterated function system includes coordinate reflections. We show that the Hausdorff dimension is invariant under such reflections. The upper bound follows from the standard covering argument using approximate squares, while the lower bound is established by constructing a dimension-maximizing Bernoulli measure and applying the Ledrappier-Young formula. The key to the proof is the observation that the fiber entropies determining the dimension are invariant under the action of the reflection group. + oai:arXiv.org:2512.17960v1 + math.DS + math.PR + Tue, 23 Dec 2025 00:00:00 -0500 new http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Nicola gigli + Vyacheslav Koval - Double Yangian, Factorization, and qKZ-equation for Cotangent Lie Algebras - https://arxiv.org/abs/2512.16996 - arXiv:2512.16996v1 Announce Type: new -Abstract: In this paper, we construct the dual $Y^*_\hbar(\mathfrak d)$ and double $DY_\hbar (\mathfrak d)$ of the Yangian $Y_\hbar (\mathfrak d)$ associated with a cotangent Lie algebra $\mathfrak d=T^*\mathfrak g$. We define a coherent factorization algebra version of the dual Yangian $Y_\hbar^*(\mathfrak d)^{\mathrm{co-op}}$ with opposite coproduct. Furthermore, we define a quantum vertex algebra structure on the quantum vacuum module $V_{\hbar,k}(\mathfrak d)$ of central extensions $\widehat{DY}_{\hbar,\ell} (\mathfrak d)$ of this double Yangian and show that its conformal blocks satisfy quantum KZ equations. We discuss examples of $\mathfrak d$ that arise from 3d $N=4$ gauge theories via the work of Costello-Gaiotto. These examples include Takiff Lie algebras $T^*\mathfrak g$, whose affine VOA is a large subalgebra of the chiral differential operator algebra of $G$, as well as the smallest type-A Lie superalgebra $\mathfrak{gl} (1|1)$. - oai:arXiv.org:2512.16996v1 - math.QA - math-ph - math.MP - math.RA - Mon, 22 Dec 2025 00:00:00 -0500 + Voronoi integration of the rendering equation + https://arxiv.org/abs/2512.17974 + arXiv:2512.17974v1 Announce Type: new +Abstract: In photorealistic image rendering, Monte Carlo methods form the foundation for the integration of the rendering equation in modern approaches. However, despite their effectiveness, traditional Monte Carlo methods often face challenges in controlling variance, resulting in noisy visual artifacts in regions that are difficult to render. In this work, we propose a new approach to the integration of the rendering equation by introducing a Voronoi tessellation reweighting scheme combined with a Poisson point process sampling strategy to address some of the limitations of standard Monte Carlo methods. From a theoretical point of view, we show that the variance induced by a Poisson-Voronoi tessellation is smaller than that of the Monte Carlo method when the intensity of the underlying process is arbitrarily large and when the function to be integrated satisfies a Holder continuity condition. + oai:arXiv.org:2512.17974v1 + math.NA + cs.NA + math.PR + Tue, 23 Dec 2025 00:00:00 -0500 new http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Raschid Abedin, Wenjun Niu + Nicolas Chenavier, Samuel Delepoulle, Christophe Renaud, Franck Vandewi\`ele - A geometric realization of liftings of Cartan type - https://arxiv.org/abs/2512.17000 - arXiv:2512.17000v1 Announce Type: new -Abstract: We introduce a novel approach to compute liftings of bosonizations of Nichols algebras of diagonal braided vector spaces of Cartan type which replaces heavy computations with structural maps related to quantum groups. This provides an answer to a question posed by Andruskiewitsch and Schneider, who classified finite-dimensional complex pointed Hopf algebras over finite abelian groups whose order is coprime with 210. As application and in order to give not-too-technical examples, we recover with our method the liftings of type $A_{n}$ computed by Andruskiewitsch and Schneider, of type $B_2$ computed by Beattie, Dascalescu and Raianu, and of type $B_3$ computed by the authors, for Drinfeld-Jimbo type braidings. Moreover, we present all liftings of type $B_{\theta}$ and $D_{\theta}$, for $\theta \geq 2$, giving in this way new explicit infinite families of liftings for Drinfeld-Jimbo type braidings. - oai:arXiv.org:2512.17000v1 - math.QA - Mon, 22 Dec 2025 00:00:00 -0500 + Cohomological Equation on the Discrete Heisenberg Group + https://arxiv.org/abs/2512.17982 + arXiv:2512.17982v1 Announce Type: new +Abstract: Let $\h_1$ be the one-dimensional Heisenberg group. In this paper, we consider some aspects of discrete dynamical systems on $\h_1$ and give a condition for the solution of a cohomological equations on the group. + oai:arXiv.org:2512.17982v1 + math.DS + math.FA + Tue, 23 Dec 2025 00:00:00 -0500 new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - D. Bagio, G. A. Garc\'ia, O. M\'arquez + http://creativecommons.org/licenses/by/4.0/ + M. E. Egwe - An Explicit Sixth Order Runge-Kutta Method for Simple Lawson Integration - https://arxiv.org/abs/2512.17006 - arXiv:2512.17006v1 Announce Type: new -Abstract: Explicit Runge-Kutta schemes become impractical when a stiff linear operator is present in the dynamics. This failure mode is quite common in numerical simulations of fluids and plasmas. Lawson proposed Generalized Runge-Kutta Processes for stiff problems in 1967, in which the stiff linear operator is treated fully implicitly via matrix exponentiation. Any Runge-Kutta scheme induces valid Lawson integration, but a scheme is exceptionally simple to implement if the abscissa $c_i$ are ordered and equally spaced. Classical RK4 satisfies this requirement, but it is difficult to derive efficient higher order schemes with this constraint. Here I present an explicit sixth order method identified with Newton-Raphson iteration that provides simple Lawson integration. - oai:arXiv.org:2512.17006v1 - math.NA - cs.NA - math-ph - math.MP - nlin.CD - Mon, 22 Dec 2025 00:00:00 -0500 + The real Brown-Peterson homology of $\Omega^\rho S^{\rho + 1}$ + https://arxiv.org/abs/2512.18019 + arXiv:2512.18019v1 Announce Type: new +Abstract: We compute the $RO(C_2)$-graded real Brown--Peterson homology of the representation-loop space $\Omega^\rho S^{\rho + 1}$, where $\rho$ is the regular representation of the cyclic group of order two. This calculation gives a $C_2$-equivariant analogue of the classical computation of Brown--Peterson homology of the double loop space $\Omega^2 S^3$ due to Ravenel. + oai:arXiv.org:2512.18019v1 + math.AT + Tue, 23 Dec 2025 00:00:00 -0500 new http://creativecommons.org/licenses/by/4.0/ - Matthew Golden + Christian Carrick, Bertrand Guillou, Sarah Petersen - The Vicsek-Kuramoto model in collective dynamics: macroscopic equations and pattern formation - https://arxiv.org/abs/2512.17035 - arXiv:2512.17035v1 Announce Type: new -Abstract: In this work, we investigate an individual-based model (IBM) for self-propelled agents interacting locally on a plane. Agents are characterized by their position, the angle determining their direction of motion, and their angular velocity. The dynamics combine features of the well-known Vicsek and Kuramoto models, which describe collective dynamics and synchronization, respectively. The evolution of the directions of motion follows a Vicsek model, where agents align their orientations with the mean orientation of their neighbors, subject to some noise. Similarly, the angular velocities relax towards the average angular velocity of the neighboring agents, also subject to noise. From the IBM we derive the corresponding kinetic equation in the limit of a large number of agents and formally obtain the macroscopic equations through a macroscopic (hydrodynamic) limit. Numerical simulations of the IBM reveal a variety of patterns, including rotating clusters, traveling orientation waves, and globally synchronized rotational motion. A qualitative comparison with simulations of the macroscopic system show the ability of the macroscopic model to reproduce some emergent behavior of the IBM. - oai:arXiv.org:2512.17035v1 - math-ph - math.AP - math.MP - nlin.AO - Mon, 22 Dec 2025 00:00:00 -0500 + Derived functors associated to the ideal of compact operators in Banach spaces + https://arxiv.org/abs/2512.18023 + arXiv:2512.18023v1 Announce Type: new +Abstract: We compute the derived functors of (the functors associated to) the ideal of compact operators in Banach spaces and obtain new results about the extension and lifting of compact operators. + oai:arXiv.org:2512.18023v1 + math.FA + Tue, 23 Dec 2025 00:00:00 -0500 new - http://creativecommons.org/licenses/by-nc-sa/4.0/ - Sara Merino-Aceituno, Carmela Moschella + http://creativecommons.org/licenses/by-nc-nd/4.0/ + F\'elix Cabello S\'anchez, Jes\'us M. F. Castillo, Alberto Salguero-Alarc\'on, Nazaret Trejo-Arroyo - EBIF: Exact Bilinearization Iterative Form for Control-Affine Nonlinear Systems - https://arxiv.org/abs/2512.17036 - arXiv:2512.17036v1 Announce Type: new -Abstract: In this paper, we develop a novel framework, Exact Bilinearization Iterative Form (EBIF), for transforming a nonlinear control-affine system into an exact finite-dimensional bilinear representation. In contrast to most existing approaches which generally lead to an infinite-dimensional representation, the proposed EBIF approach yields an iterative procedure for constructing a finite set of smooth coordinate functions that define an embedding, enabling an exact bilinear representation of the original nonlinear dynamics. Leveraging tools from algebra and differential geometry, we establish both necessary and sufficient conditions for a nonlinear system to be exactly bilinearizable. We further illustrate how the EBIF-induced bilinear systems facilitate reachability analysis and control design. Through theoretical analysis and numerical simulations, we demonstrate the effectiveness of the EBIF framework and highlight its potential in simplifying control synthesis for nonlinear systems. - oai:arXiv.org:2512.17036v1 - math.OC - math.DS - Mon, 22 Dec 2025 00:00:00 -0500 + Scalable Multiterminal Key Agreement via Error-Correcting Codes + https://arxiv.org/abs/2512.18025 + arXiv:2512.18025v1 Announce Type: new +Abstract: We explore connections between secret sharing and secret key agreement, which yield a simple and scalable multiterminal key agreement protocol. In our construction, we use error-correcting codes, specifically Reed-Solomon codes with threshold reconstruction, to ensure no information is leaked to an eavesdropper. We then derive novel bounds for both full-rank maximum distance separable codes and our scheme's secret key capacity, using key capacity's duality with multivariate mutual information. + oai:arXiv.org:2512.18025v1 + cs.IT + cs.CR + math.IT + Tue, 23 Dec 2025 00:00:00 -0500 new http://creativecommons.org/licenses/by/4.0/ - Yuan-Hung Kuan, Jr-Shin Li + Benjamin D. Kim, Daniel Alabi, Lav R. Varshney - Fej\'er and Fej\'er* Monotonicity: New Results and Limiting Examples - https://arxiv.org/abs/2512.17039 - arXiv:2512.17039v1 Announce Type: new -Abstract: Many algorithms in convex optimization and variational analysis can be analyzed using Fej\'er monotone sequences. In 2024, Behling, Bello-Cruz, Iusem, Alves Ribeiro, and Santos introduced a new, more general, notion: Fej\'er* monotonicity. They obtained basic results and discussed applications in optimization. - In this work, we complement Behling et al.'s work by presenting a thorough study of Fej\'er* monotonicity. We reveal striking similarities and differences between these notions, including descriptions of the maximal Fej\'er* set. Moreover, we also touch upon Opial sequences and quasi-Fej\'er monotonicity. Throughout this paper, we provide numerous limiting examples and counterexamples. - oai:arXiv.org:2512.17039v1 - math.OC - Mon, 22 Dec 2025 00:00:00 -0500 + Assembly Addition Chains + https://arxiv.org/abs/2512.18030 + arXiv:2512.18030v1 Announce Type: new +Abstract: In this paper we extend the notion of Addition Chains over Z+ to a general set S. We explain how the algebraic structure of Assembly Multi-Magma over the pairs (S,BB proper subset of S) allows to define the concept of Addition Chain over S, called Assembly Addition Chains of S with Building Blocks BB. Analogously to the Z+ case, we introduce the concept of Optimal Assembly Addition Chains over S and prove lower and upper bounds for their lengths, similar to the bounds found by Schonhage for the Z+ case. In the general case the unit 1 is in set Z+ is replaced by the subset BB and the mentioned bounds for the length of an Optimal Assembly Addition Chain of O is in set S are defined in terms of the size of O (i.e. the number of Building Blocks required to construct O). The main examples of S that we consider through this papers are (i) j-Strings (Strings with an alphabeth of j letters), (ii) Colored Connected Graphs and (iii) Colored Polyominoes. + oai:arXiv.org:2512.18030v1 + math.CO + cs.CC + Tue, 23 Dec 2025 00:00:00 -0500 new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Aleksandr Arakcheev, Heinz H. Bauschke + http://creativecommons.org/licenses/by/4.0/ + Leroy Cronin, Juan Carlos Morales Parra, Keith Y. Patarroyo - Geometry for Kleinian Groups Generated by a Parabolic Pair - https://arxiv.org/abs/2512.17044 - arXiv:2512.17044v1 Announce Type: new -Abstract: This paper develops a computational framework for studying the hyperbolic geometry of 2-bridge link complements and Kleinian groups generated by two parabolic elements. The framework is built on Sakuma-Weeks triangulations and introduces a family of Farey recursive polynomials. - For a rational number which determines a hyperbolic 2-bridge link, this paper provides a simple recursive algorithm to determine a Farey recursive polynomial which has a root which determines the geometry of the link complement. This root is the shape parameter for a pair of tetrahedra in the Sakuma-Weeks triangulation. Using the same polynomials, the paper defines rational functions, one function for each edge in the Farey graph. Evaluating these rational functions at the root yields the complete collection of shape parameters for the triangulation. - The Riley slice and its exterior lie in the complex plane and are traditionally viewed as parameter spaces for two-parabolic generated subgroups of PSL(2,C). This paper's triangulation-based approach provides a more geometric perspective to these parameter spaces. The Farey recursive methods apply throughout the Riley slice and its exterior, enabling computations for quotient orbifolds of Kleinian groups generated by parabolic pairs, incomplete hyperbolic spaces with non-discrete parameter groups, and others. Explicit calculations include the cusp groups in the boundary of the Riley slice and singly augmented 2-bridge link complements. - Applications include: explicit computation of fundamental domains and holonomies for 2-bridge link complements; a precise correspondence between group elements and crossing circles in tangle diagrams; determination of cusp fields for 2-bridge links; geometric analysis of algebraic and geometric limits arising from Dehn surgery on singly augmented 2-bridge links; and explicit triangulations of Heckoid orbifolds. - oai:arXiv.org:2512.17044v1 - math.GT - Mon, 22 Dec 2025 00:00:00 -0500 + A local Fortin projection for the Scott-Vogelius elements on general meshes + https://arxiv.org/abs/2512.18033 + arXiv:2512.18033v1 Announce Type: new +Abstract: We construct a local Fortin projection for the Scott-Vogelius finite element pair for polynomial degree $k \ge 4$ on general shape-regular triangulations in two dimensions. In particular, the triangulation may contain singular vertices. In addition to preserving the divergence in the dual of the pressure space, the projection preserves discrete boundary data and satisfies local stability estimates. + oai:arXiv.org:2512.18033v1 + math.NA + cs.NA + Tue, 23 Dec 2025 00:00:00 -0500 new http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Eric Chesebro + Franziska Eickmann, Johnny Guzm\'an, Michael Neilan, L. Ridgway Scott, Tabea Tscherpel - Un caract\`ere relatif pond\'er\'e - https://arxiv.org/abs/2512.17056 - arXiv:2512.17056v1 Announce Type: new -Abstract: Let $p\geq 1$. The symmetric space $S=GL(2p+1)/GL(p+1)\times GL(p)$ (over a number field) is not cuspidal in the sense that its automorphic spectrum does not contain any cuspidal representation of $GL(2p+1)$. In this article, we compute the spectral decomposition of its relatively cuspidal part: this is, by definition, the part of the spectrum that is induced from the cuspidal part of the symmetric space $(GL(1)\times GL(2p)) / (GL(1)\times GL(p)\times GL(p))$. As an application, we obtain the expression of the contribution of this relatively cuspidal part to the Guo-Jacquet trace formula (established by H. Li and the author) in terms of a weighted relative character. - oai:arXiv.org:2512.17056v1 + Problems on the conductor of finite group characters + https://arxiv.org/abs/2512.18042 + arXiv:2512.18042v1 Announce Type: new +Abstract: This paper reviews recent results and open problems on the conductor of finite group characters, highlighting their connections to one another and to broader topics in the representation theory of finite groups. + oai:arXiv.org:2512.18042v1 math.RT - math.NT - Mon, 22 Dec 2025 00:00:00 -0500 + math.GR + Tue, 23 Dec 2025 00:00:00 -0500 new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Pierre-Henri Chaudouard + http://creativecommons.org/licenses/by/4.0/ + Nguyen N. Hung - Absence of twisting for non-trivial discrete torsion - https://arxiv.org/abs/2512.17068 - arXiv:2512.17068v1 Announce Type: new -Abstract: We study discrete torsion for the $n$--torus with finite symmetry group $G$ from the Dijkgraaf--Witten viewpoint. A class in $H^n(G,U(1))$ assigns a phase to each flat $G$--bundle, equivalently to each commuting $n$--tuple in $G$ up to conjugation. We introduce the subgroup $\Br^n(G)\subseteq H^n(G,U(1))$ of \emph{untwisted} classes, those whose Dijkgraaf--Witten phases are trivial on all commuting tuples, and derive a universal coefficient exact sequence involving this invariant. In degree $2$ this recovers the Bogomolov multiplier / unramified Brauer group. We implement algorithms for computing $\Br^n(G)$ and corresponding torus partition functions, and report on computations for families of finite subgroups of $\SU(4)$. - oai:arXiv.org:2512.17068v1 - math.GR - math-ph - math.MP - Mon, 22 Dec 2025 00:00:00 -0500 + Shortest distance between observed orbits in distinct Dynamical Systems + https://arxiv.org/abs/2512.18050 + arXiv:2512.18050v1 Announce Type: new +Abstract: In this paper, we investigate the asymptotic behavior of the shortest distance between observed orbits in two distinct dynamical systems. Given two measure-preserving transformations $(X, T, \mu)$ and $(X, S, \eta)$ and a Lipschitz observation function $f$, we define \[ \widehat{m}_n^f(x,y) = \min_{i=0,\ldots,n-1} d\big(f(T^i x), f(S^i y)\big). \] %Under suitable mixing assumptions, we show that the asymptotic rate of decay of $\widehat{m}_n^f(x,y)$ is governed by the correlation dimensions of the pushforward measures $f_*\mu$ and $f_*\eta$. Under suitable mixing assumptions, we show that the asymptotic rate of decay of $\widehat{m}_n^f(x,y)$ is governed by the symmetric R\'enyi divergence of the pushforward measures $f_*\mu$ and $f_*\eta$. Our results generalize previous work that consider either a single system or the unobserved case. In addition, we discuss the extension of these results to random dynamical systems and illustrate the applicability of the approach with an example. + oai:arXiv.org:2512.18050v1 + math.DS + Tue, 23 Dec 2025 00:00:00 -0500 new - http://creativecommons.org/publicdomain/zero/1.0/ - Primoz Moravec + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Vanessa Barros, Adriana Coutinho - Graphings with few circulations - https://arxiv.org/abs/2512.17071 - arXiv:2512.17071v1 Announce Type: new -Abstract: In 2021, motivated by graph limit theory Lov\'asz extended most of the theory of flows to a measure theoretic setting. Using this framework, the first author constructed $d$-regular treeings that are measurably bipartite, and have no nonzero measurable circulations, that is, flows without sources or sinks. In particular, these treeings do not admit a measurable perfect matching. - In this paper, we develop tools to build $d$-regular treeings where the space of circulations is exactly $k$-dimensional for any positive integer $k$. As applications, we construct 1) a treeing with a single balanced orientation, but no Schreier decoration; 2) a treeing with a single Schreier decoration; 3) and a treeing with a proper edge $d$-coloring, but no further perfect matchings. - The first answers a question raised by Lov\'asz, as this particular balanced orientation does not decompose as a linear combination of finite cycles and infinite paths. - oai:arXiv.org:2512.17071v1 - math.CO - math.DS - Mon, 22 Dec 2025 00:00:00 -0500 + Cheeger's Constant for the Gabor Transform and Ripples + https://arxiv.org/abs/2512.18058 + arXiv:2512.18058v1 Announce Type: new +Abstract: We discover a new instability mechanism for short-time Fourier transform phase retrieval which yields that for any reasonable window function $\phi$ in any dimension $d$, the local stability constant $c(f)$ defined via + \begin{equation*} + \inf_{|\lambda|=1}\|f- \lambda g\|_{M^p(\mathbb{R}^{d})}\leq c(f)\| |V_\phi f|-|V_{\phi} g|\|_\mathcal{D}, \hspace{5mm} \forall g\in M^p(\mathbb{R}^d), + \end{equation*} + is infinite on a dense set of vectors for all weighted fractional Sobolev norms $\mathcal{D}$, up to the sharp maximal regularity level ensuring that the problem is well-defined. This, in particular, answers an open problem of Rathmair, who asked whether exponential concentration of the Gabor transform on $\mathbb{R}^2$ guaranteed a finite local stability constant. For the specific case of Gabor phase retrieval, we further show that there is a complementary dense set where the local stability constant on $\mathbb{R}^{2d}$ is finite. + Our results extend and complement a series of fundamental stability theorems for Gabor phase retrieval which have been proven over the last ten years. Of particular note is the work of Grohs and Rathmair, who showed that for sufficiently strong weighted Sobolev norms $\mathcal{D}$ on $\mathbb{R}^{2d}$, the local stability constant for Gabor phase retrieval is bounded by the inverse of the Cheeger constant of the flat metric conformally multiplied by $|V_\phi f|$. As a consequence of our analysis, we determine two dense families of functions, one of which has associated Cheeger constant zero and the other strictly positive. We also revisit the stability problem for STFT phase retrieval on bounded subsets of the time-frequency plane, for more general windows, and for restricted signal classes, extending and simplifying many influential results in the literature. + oai:arXiv.org:2512.18058v1 + math.CA + Tue, 23 Dec 2025 00:00:00 -0500 new - http://creativecommons.org/licenses/by/4.0/ - G\'abor Kun, L\'aszl\'o M\'arton T\'oth + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Rima Alaifari, Ben Pineau, Mitchell A. Taylor, Matthias Wellershoff - Universal Cancellations in Uniform Random Waves - https://arxiv.org/abs/2512.17076 - arXiv:2512.17076v1 Announce Type: new -Abstract: A vast literature over the past fifteen years has been devoted to the study of the geometric properties of Gaussian random waves. In this work, we investigate the geometric behavior of \emph{uniform random waves}, a much less studied non-Gaussian model in which the $L^2$ norm is constrained to be exactly equal to one in every realization (a normalization that is natural from the standpoint of quantum mechanics). We show that this norm-constrained formulation has deep consequences for the universality of the so-called \emph{Berry's cancellation phenomenon}, as well as for novel high-frequency asymptotic variance estimates. These effects manifest themselves in both local geometric functionals, such as the Lipschitz--Killing curvatures, and global ones, such as the number of connected components above a fixed threshold. A key byproduct of our analysis is a new explicit relation between Hermite expansions and spherical harmonic decompositions for $0$-homogeneous functionals of Gaussian vectors, which enables a systematic chaos-based analysis of non-Gaussian random waves. - oai:arXiv.org:2512.17076v1 - math.PR - Mon, 22 Dec 2025 00:00:00 -0500 + Approximation and learning with compositional tensor trains + https://arxiv.org/abs/2512.18059 + arXiv:2512.18059v1 Announce Type: new +Abstract: We introduce compositional tensor trains (CTTs) for the approximation of multivariate functions, a class of models obtained by composing low-rank functions in the tensor-train format. This format can encode standard approximation tools, such as (sparse) polynomials, deep neural networks (DNNs) with fixed width, or tensor networks with arbitrary permutation of the inputs, or more general affine coordinate transformations, with similar complexities. This format can be viewed as a DNN with width exponential in the input dimension and structured weights matrices. Compared to DNNs, this format enables controlled compression at the layer level using efficient tensor algebra. On the optimization side, we derive a layerwise algorithm inspired by natural gradient descent, allowing to exploit efficient low-rank tensor algebra. This relies on low-rank estimations of Gram matrices, and tensor structured random sketching. Viewing the format as a discrete dynamical system, we also derive an optimization algorithm inspired by numerical methods in optimal control. Numerical experiments on regression tasks demonstrate the expressivity of the new format and the relevance of the proposed optimization algorithms. Overall, CTTs combine the expressivity of compositional models with the algorithmic efficiency of tensor algebra, offering a scalable alternative to standard deep neural networks. + oai:arXiv.org:2512.18059v1 + math.NA + cs.LG + cs.NA + Tue, 23 Dec 2025 00:00:00 -0500 new http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Louis Gass, Domenico Marinucci, Giovanni Peccati, Francesca Pistolato, Michele Stecconi + Martin Eigel, Charles Miranda, Anthony Nouy, David Sommer - On the existence of solutions to the multi-species Landau equation - https://arxiv.org/abs/2512.17082 - arXiv:2512.17082v1 Announce Type: new -Abstract: We consider the spatially homogeneous Landau equation for multiple species with different masses. As in the single-species case, the singularity of the collision operator is determined by a parameter $\gamma \in [-3,1]$, where $\gamma = -3$ corresponds to Coulomb interactions. We prove that if $\gamma\geq -\sqrt{8}$ in the cross-interaction operators, then there exists a natural multi-species generalization of the Fisher information which is a Lyapunov functional for the multi-species Landau system. On the other hand, we give a counterexample showing that the Fisher information is in general no longer a Lyapunov functional below the threshold $(\gamma < - \sqrt{8})$ for the two-species system if one species has infinite mass. However, we are able to provide a new method to show global well-posedness, by constructing a different Lyapunov functional based on the spherical Fisher information. - oai:arXiv.org:2512.17082v1 + Incompressible limits at large Mach number for a reduced compressible MHD system + https://arxiv.org/abs/2512.18078 + arXiv:2512.18078v1 Announce Type: new +Abstract: This paper studies a singular limit problem for a reduced model for compressible non-resistive MHD which was first introduced in \cite{Li-Sun_JDE, Li-Sun} in a two-dimensional setting. This system can also be related to a certain class of two-fluid models. By a suitable rescaling of the magnetic pressure in terms of some parameter $\varepsilon>0$, by letting $\varepsilon\to 0$ we perform the incompressible limit while keeping the Mach number of order $O(1)$. + The study is conducted in the framework of global in time finite energy weak solutions and for ill-prepared initial data. We also consider a similar problem in presence of a strong Coriolis term. The key ingredient of the proof, based on a compensated compactness argument, is the use of the transport equation (well-known in the context of two-fluid models) underlying the dynamics. Thanks to it, and differently from previous studies about the incompressible limit, we are able to identify the asymptotics of the terms of order $O(\varepsilon)$ and to characterise their dynamics; such an information is in fact crucial to obtain a closed system in the limit. + oai:arXiv.org:2512.18078v1 math.AP - math-ph - math.MP - Mon, 22 Dec 2025 00:00:00 -0500 + Tue, 23 Dec 2025 00:00:00 -0500 new http://creativecommons.org/licenses/by/4.0/ - Jonathan Junn\'e, Raphael Winter, Havva Yolda\c{s} + Francesco Fanelli, Young-Sam Kwon, Aneta Wr\'oblewska-Kami\'nska - Hyperbolic Simplices of Maximal Inradius - https://arxiv.org/abs/2512.17096 - arXiv:2512.17096v1 Announce Type: new -Abstract: For $n\in \mathbb{N}$, consider a hyperbolic $n$-dimensional simplex $\Delta$, defined by $1+n$ points in the compactified hyperbolic space $\mathbf{H}^n \sqcup \partial \mathbf{H}^n$. For each integer $m\le n$, denote $\delta^n_m(\Delta)\in [0,+\infty]$ the Hausdorff distance between its skeleta of dimensions $n$ and $m$. In particular, $\delta^n_{n-1}(\Delta)$ is its inradius. The maximum of $\delta^n_m(\Delta)$ over $\Delta\in (\mathbf{H}^n \sqcup \partial \mathbf{H}^n)^{1+n}$ is denoted $\mu^n_m\in [0,+\infty]$. We first show that $\Delta$ has maximal inradius $\delta^n_{n-1}(\Delta)=\mu^n_m$ if and only if its is (total) ideal and regular; for which the inradius is given by $\tanh \mu^n_{n-1} = 1/n$. We deduce that $\Delta$ has maximal $\delta^n_{n-1}(\Delta)=\mu^n_m$ if and only if it is (total) ideal and regular. We compute that the maximal distance to the $1$-skeleton $\mu^n_1$ is given by $\left(\tanh \mu^n_1\right)^2 = (n-1)/(2n)$ and deduce that those are uniformly bounded by $\lim_{n} \mu^n_1 = \log(1+\sqrt{2})$. - oai:arXiv.org:2512.17096v1 - math.MG - Mon, 22 Dec 2025 00:00:00 -0500 + Packing Independent Cliques in $K_4$-minor-free Graphs + https://arxiv.org/abs/2512.18090 + arXiv:2512.18090v1 Announce Type: new +Abstract: Let $G$ be a graph and $S$ be a set of cliques of $G$. The set $S$ is an indeque set if every component of $G[S]$, the subgraph induced by vertices of $S$, is a clique. In this paper, we prove that the indeque ratio of $K_4$-minor-free graphs is $\frac 1 2$, which settle two conjectures of Biro, Collado and Zamora. We also show that the indeque ratio of subcubic graphs is $\frac 1 2$. + oai:arXiv.org:2512.18090v1 + math.CO + Tue, 23 Dec 2025 00:00:00 -0500 new http://creativecommons.org/licenses/by/4.0/ - Bruno Duchesne, Christopher-Lloyd Simon + Benjamin Xiao, Dong Ye - Constant potentials do not minimise the fundamental gap on convex domains in hyperbolic space - https://arxiv.org/abs/2512.17103 - arXiv:2512.17103v1 Announce Type: new -Abstract: We show that for every $n \geq 2$ and $D > 0$ there exist a convex domain $\Omega \subseteq \mathbb H^n$ with diameter $D$ and a convex potential $V$ on $\Omega$ such that the fundamental gap of the operator $-\Delta+V$ is strictly smaller than the fundamental gap of $-\Delta$. In comparison to previous work, this result requires more refined control of the eigenfunctions. - oai:arXiv.org:2512.17103v1 - math.AP + Relative analytic reciprocity laws + https://arxiv.org/abs/2512.18106 + arXiv:2512.18106v1 Announce Type: new +Abstract: We study reciprocity laws involving complex line bundles on fibrations in oriented circles. In particularly, we prove the following reciprocity law. Let $B$ be a complex manifold and $\pi_i : M_i \to B$ be a fibration in oriented circles, where $i$ runs through a finite set. Let $L_i$ and $N_i$ be complex line bundles on every $M_i$. The reciprocity law states that the sum of all $(\pi_i)_* \left(c_1(L_i) \cup c_1(N_i) \right)$, where $(\pi_i)_*$ is the Gysin map and $c_1$ is the first Chern class, equals zero in $H^3(B, {\mathbb Z})$ when the disjoint union of all $M_i$ is embedded into a holomorphic family of compact Riemann surfaces over the base $B$ such that in every fiber of this family the disjoint union of the embedded circles is the boundary of an embedded compact Riemann surface with boundary, and all $L_i$ and all $N_i$ are restrictions of holomorphic line bundles on this family. + oai:arXiv.org:2512.18106v1 + math.CV + math.AG math.DG - Mon, 22 Dec 2025 00:00:00 -0500 + Tue, 23 Dec 2025 00:00:00 -0500 new - http://creativecommons.org/licenses/by-nc-nd/4.0/ - Julie Clutterbuck, Frieder J\"ackel, Xuan Hien Nguyen + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Denis V. Osipov - On the convergence of the Born series for Coulomb potentials - https://arxiv.org/abs/2512.17104 - arXiv:2512.17104v1 Announce Type: new -Abstract: We provide a short proof of the convergence of the Born series on asymptotically conic manifolds, at sufficiently high energy. The potential is allowed to have multiple Coulomb singularities. This is handled using powerful semiclassical estimates recently proven by Hintz for the case of a single dipole (or better) singularity. The potential is also allowed to be long-range, like the actual Coulomb potential $1/r$; long-range potentials are handled using anisotropic semiclassical (sc-) Sobolev spaces. As a consequence of the above estimates, we show the existence of a resonance-free region for Hamiltonians with multiple screened Coulomb singularities. - oai:arXiv.org:2512.17104v1 - math-ph - math.MP - Mon, 22 Dec 2025 00:00:00 -0500 + Unordered resolutions and homological stability for linear groups + https://arxiv.org/abs/2512.18110 + arXiv:2512.18110v1 Announce Type: new +Abstract: In this paper, we develop a modified proof strategy for homological stability of linear groups, with the general linear groups serving as a primary example. Our arguments are more direct than those in the classical works of Quillen and Suslin--Nesterenko, although they apply only with localized coefficients. The localization at (n-1)! that arises in our approach appears to be closely related to several conjectures of Mirzaii as well as to Suslin's injectivity conjecture. + oai:arXiv.org:2512.18110v1 + math.KT + Tue, 23 Dec 2025 00:00:00 -0500 new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Ethan Sussman, Jared Wunsch + http://creativecommons.org/licenses/by-nc-sa/4.0/ + Ivan Vasilev, Serge Yagunov - On Matrix Product Factorization of Cayley graphs - https://arxiv.org/abs/2512.17110 - arXiv:2512.17110v1 Announce Type: new -Abstract: We study when the adjacency matrix of a Cayley graph factors as the product of two adjacency matrices of Cayley graphs. Let $G$ be a finite group and let $U\subseteq G\setminus \{e\}$ be symmetric. Writing $A(G;U)$ for the adjacency matrix of the Cayley graph of $G$ with respect to $U$, we prove that for symmetric subsets $S,T,U$ of $G\setminus \{e\}$, $A(G;U)=A(G;S)\,A(G;T)$ if and only if $U=ST$ and each $u\in U$ has a unique representation $u=st$, equivalently $\bigl(\sum_{s\in S}s\bigr)\bigl(\sum_{t\in T}t\bigr)=\sum_{u\in U}u$ in the group algebra. When $S,T,U$ are unions of conjugacy classes, this is characterized character-theoretically by $\chi(U)=\chi(S)\chi(T)/\chi(1)$ for all $\chi\in\mathrm{Irr}(G)$. In addition, for abelian groups, we identify $A(G;S)A(G;T)$ with the $0\!-\!1$ convolution $\mathbf{1}_S*\mathbf{1}_T$, so factorability is equivalent to $(S,T)$ being a Sidon pair, i.e., $(S-S)\cap(T-T)=\{0\}$. For cyclic groups, we reformulate factorability via mask polynomials and reduce to prime-power components using the Chinese Remainder Theorem. We also analyze dihedral groups $D_{2n}$, presenting infinite families of factorable generating sets, and give explicit constructions of subsets whose Cayley graphs do and do not admit such factorizations. - oai:arXiv.org:2512.17110v1 - math.CO - math.GR - Mon, 22 Dec 2025 00:00:00 -0500 + Esakia's theorem for the amended monadic intuitionistic calculus + https://arxiv.org/abs/2512.18111 + arXiv:2512.18111v1 Announce Type: new +Abstract: We show that the amended monadic Grzegorczyk logic $\mathsf{M^+Grz}$ is the largest modal companion of the amended monadic intuitionistic logic $\mathsf{M^+IPC}$. Thus, unlike the monadic intuitionistic logic $\mathsf{MIPC}$, Esakia's theorem does extend to $\mathsf{M^+IPC}$. + oai:arXiv.org:2512.18111v1 + math.LO + Tue, 23 Dec 2025 00:00:00 -0500 new - http://creativecommons.org/licenses/by/4.0/ - Allen W. Herman, Bobby Miraftab + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Guram Bezhanishvili, Luca Carai - Verifying Hadwiger's Conjecture for Examples of Graphs with $\alpha(G) = 2$ - https://arxiv.org/abs/2512.17114 - arXiv:2512.17114v1 Announce Type: new -Abstract: Hadwiger's Conjecture states that every graph with chromatic number $k$ contains a complete graph on $k$ vertices as a minor. This conjecture is a tremendous strengthening of the Four-Colour Theorem and is regarded as one of the most important open problems in graph theory. The case of Hadwiger's Conjecture for graphs with $\alpha(G) = 2$ has garnered much attention. Seymour writes: ``My own belief is, if Hadwiger's Conjecture is true for graphs with stability number two then it is probably true in general, so it would be very nice to decide this case.'' - This paper presents several tools useful for proving that a graph $G$ with $\alpha(G) = 2$ satisfies Hadwiger's Conjecture. In doing so, we survey and generalise several classical results on the $\alpha(G) = 2$ case of Hadwiger's Conjecture. Further, we apply these tools to prove variants of Hadwiger's Conjecture for several noteworthy classes of graphs with $\alpha(G) = 2$. In particular, we prove Hadwiger's Conjecture for inflations of the complements of the following graphs: graphs with girth at least $5$, triangle-free Kneser graphs, and the Clebsch, Mesner, and Gewirtz graphs. This paper also highlights classes of graphs with $\alpha(G) = 2$ where it is unknown if Hadwiger's Conjecture holds. - oai:arXiv.org:2512.17114v1 - math.CO - Mon, 22 Dec 2025 00:00:00 -0500 + Classical Ap\'ery-Like Series and Their Cyclotomic Parametric Analogues via Contour Integration + https://arxiv.org/abs/2512.18121 + arXiv:2512.18121v1 Announce Type: new +Abstract: In this paper, we present a method based on contour integration to investigate a class of cyclotomic parametric Ap\'ery-like series. The general term of such series involves a parametric central binomial coefficient, which is defined via the Gamma function. Using this approach, we express a family of cyclotomic Ap\'ery-like series in terms of multiple polylogarithms, cyclotomic Hurwitz zeta values, Riemann zeta values and $\log(2)$. In particular, we provide several illustrative examples and corollaries, which enable us to recover a number of known results on Ap\'ery-like series. At the same time, we have also left open two questions regarding Ap\'ery-like series. Moreover, by considering integrals of the generating function for Fuss-Catalan numbers, we derive an alternative expression for a classical Ap\'ery-like series. Combining this with known results allows us to establish several identities for multiple polylogarithm functions. + oai:arXiv.org:2512.18121v1 + math.NT + Tue, 23 Dec 2025 00:00:00 -0500 new http://creativecommons.org/licenses/by/4.0/ - Jofre Costa, Eric Liu, David R. Wood, Jung Hon Yip + Ce Xu - Bloch-Suslin Complex and Strong $\mathbb{A}^{1}-$invariance - https://arxiv.org/abs/2512.17118 - arXiv:2512.17118v1 Announce Type: new -Abstract: We prove that there is no extension of the abelian groups appearing in the Bloch-Suslin complex to strongly $\mathbb{A}^{1}-$invariant sheaves on $Sm_{k}$ (char($k$)=0) that also extend the canonical symbol maps from the respective $\mathbb{G}_{m}^{\wedge n}$ (i.e., from $\mathbb{G}_{m}$ and $\mathbb{G}_{m}^{\wedge2}$). - oai:arXiv.org:2512.17118v1 - math.AG - math.KT - Mon, 22 Dec 2025 00:00:00 -0500 + Structure and Symmetry of Sally Type Semigroup Rings + https://arxiv.org/abs/2512.18136 + arXiv:2512.18136v1 Announce Type: new +Abstract: Consider a numerical semigroup minimally generated by a subset of the interval $[e,2e-1]$ with multiplicity $e$ and width $e-1$. Such numerical semigroups are called Sally type semigroups. We show that the defining ideals of these semigroup rings, when the embedding dimension is $e-2$, generically have the structure of the sum of two determinantal ideals. More generally, Sally type numerical semigroups with multiplicity $e$ and embedding dimension $d=e-k$ are obtained by introducing $k$ gaps in the interval $[e,2e-1]$. It is known that for $k =2$, there is precisely one such semigroup that is Gorenstein, and it happens when one deletes consecutive integers. Let $S^e_k(j)$ denote the Sally type numerical semigroup of multiplcity $e$, embedding dimension $e-k$ obtained by deleting the $k$ consecutive integers $j, j+1, \ldots, j+k-1$.We prove that for any $1\le k < e/2$, the semigroup $S^e_k(j)$ is Gorenstein if and only if $j=k$. We construct an explicit minimal free resolution of the semigroup ring of $S^e_k(k)$ and compute the Betti numbers. In general, we characterize when $S^e_k(j)$ are symmetric and construct minimal resolutions for these Gorenstein semigroups rings. + oai:arXiv.org:2512.18136v1 + math.AC + math.RA + Tue, 23 Dec 2025 00:00:00 -0500 new - http://creativecommons.org/licenses/by/4.0/ - Saheb Mohapatra + http://creativecommons.org/publicdomain/zero/1.0/ + Srishti Singh, Hema Srinivasan - Beyond Low Rank: Fast Low-Rank + Diagonal Decomposition with a Spectral Approach - https://arxiv.org/abs/2512.17120 - arXiv:2512.17120v1 Announce Type: new -Abstract: Low-rank plus diagonal (LRPD) decompositions provide a powerful structural model for large covariance matrices, simultaneously capturing global shared factors and localized corrections that arise in covariance estimation, factor analysis, and large-scale kernel learning. We introduce an alternating low-rank then diagonal (Alt) algorithm that provably reduces approximation error and significantly outperforms gradient descent while remaining cheaper than majorization-minimization methods~\cite{sun2016majorization}. To scale to large matrices, we develop a randomized LRPD variant that combines fixed-rank Nystrom sketching~\cite{tropp2017fixed} for the low-rank component with Diag++ stochastic diagonal estimation~\cite{baston2022stochastic}. This hybrid algorithm achieves machine precision decomposition error using a number of matrix-vector products far smaller than the ambient dimension, and comes with rigorous non-asymptotic error bounds. On synthetic data, it exactly recovers LRPD structured matrices with high efficiency, and on real-world S&P 500 stock return covariances, where the spectrum decays slowly and strong sector structure exists, it achieves substantially lower error than pure low-rank approximations. - oai:arXiv.org:2512.17120v1 - math.NA - cs.NA - Mon, 22 Dec 2025 00:00:00 -0500 + An Aubin continuity path for asymptotically conical toric shrinking gradient K\"ahler-Ricci solitons: openness and a solution for $t=0$ + https://arxiv.org/abs/2512.18137 + arXiv:2512.18137v1 Announce Type: new +Abstract: We show that any toric asymptotically conical shrinking gradient K\"ahler-Ricci soliton on an anti-canonically polarised resolution of a K\"ahler cone satisfies a complex Monge-Amp\`ere equation. We then set up an Aubin continuity path to solve the resulting equation and show that it has a solution at the initial value of the path parameter in the toric case. This we do by implementing another continuity method. Finally, we prove openness of the initial value of the path parameter independent of the toricity. + oai:arXiv.org:2512.18137v1 + math.DG + Tue, 23 Dec 2025 00:00:00 -0500 new - http://creativecommons.org/licenses/by/4.0/ - Kingsley Yeon, Mihai Anitescu + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Ivin Babu, Ronan J. Conlon, Alix Deruelle - Hermitian Hulls of Rational Algebraic Geometry Codes and Applications in Quantum Codes - https://arxiv.org/abs/2512.17128 - arXiv:2512.17128v1 Announce Type: new -Abstract: Interest in the hulls of linear codes has been growing rapidly. More is known when the inner product is Euclidean than Hermitian. A shift to the latter is gaining traction. The focus is on a code whose Hermitian hull dimension and dual distance can be systematically determined. Such a code can serve as an ingredient in designing the parameters of entanglement-assisted quantum error-correcting codes (EAQECCs). - We use tools from algebraic function fields of one variable to efficiently determine a good lower bound on the Hermitian hull dimensions of generalized rational algebraic geometry (AG) codes. We identify families of AG codes whose hull dimensions can be well estimated by a lower bound. Given such a code, the idea is to select a set of evaluation points for which the residues of the Weil differential associated with the Hermitian dual code has an easily verifiable property. - The approach allows us to construct codes with designed Hermitian hull dimensions based on known results on Reed-Solomon codes and their generalization. Using the Hermitian method on these maximum distance separable (MDS) codes with designed hull dimensions yields two families of MDS EAQECCs. We confirm that the excellent parameters of the quantum codes from these families are new. - oai:arXiv.org:2512.17128v1 - cs.IT - math.IT - Mon, 22 Dec 2025 00:00:00 -0500 + Prime degree irreducible representations of simple algebraic groups and finite simple groups of Lie type + https://arxiv.org/abs/2512.18145 + arXiv:2512.18145v1 Announce Type: new +Abstract: We show that finite quasisimple groups of Lie type in characteristic $p$ with an irreducible representation of prime degree $r$ over a finite field of characteristic $p$ have orders bounded above by a function of $r$, independent of $p$. We also bound the number of such groups in terms of $r$. Apart from being notable in their own right, these results have a significant application in a computational version of the strong approximation theorem for finitely generated Zariski-dense subgroups of $SL_r(\mathbb{P})$, where $\mathbb{P}$ is a number field. + oai:arXiv.org:2512.18145v1 + math.GR + Tue, 23 Dec 2025 00:00:00 -0500 new - http://creativecommons.org/publicdomain/zero/1.0/ - Lin Sok, Martianus Frederic Ezerman, Ling San + http://creativecommons.org/licenses/by/4.0/ + D. L. Flannery, A. E. Zalesski - An Asymptotic Approach for Modeling Multiscale Complex Fluids at the Fast Relaxation Limit - https://arxiv.org/abs/2512.17134 - arXiv:2512.17134v1 Announce Type: new -Abstract: We present a new asymptotic strategy for general micro-macro models which analyze complex viscoelastic fluids governed by coupled multiscale dynamics. In such models, the elastic stress appearing in the macroscopic continuum equation is derived from the microscopic kinetic theory, which makes direct numerical simulations computationally expensive. To address this challenge, we introduce a formal asymptotic scheme that expands the density function around an equilibrium distribution, thereby reducing the high computational cost associated with the fully coupled microscopic processes while still maintaining the dynamic microscopic feedback in explicit expressions. The proposed asymptotic expansion is based on a detailed physical scaling law which characterizes the multiscale balance at the fast relaxation limit of the microscopic state. An asymptotic closure model for the macroscopic fluid equation is then derived according to the explicit asymptotic density expansion. Furthermore, the resulting closure model preserves the energy-dissipation law inherited from the original fully coupled multiscale system. Numerical experiments are performed to validate the asymptotic density formula and the corresponding flow velocity equations in several micro-macro models. This new asymptotic strategy offers a promising approach for efficient computations of a wide range of multiscale complex fluids. - oai:arXiv.org:2512.17134v1 - math-ph - math.MP - physics.comp-ph - physics.flu-dyn - Mon, 22 Dec 2025 00:00:00 -0500 + Fixed points of extended tensor products + https://arxiv.org/abs/2512.18150 + arXiv:2512.18150v1 Announce Type: new +Abstract: For a $p$-permutation equivalence between two block algebras of finite groups, we introduce new square diagrams that link the $p$-permutation equivalence via the Brauer construction to local equivalences between stabilizers of corresponding Brauer pairs. These diagrams can be viewed as lifts of the square diagrams in the definition of isotypies. The proof of the commutativity requires new technical tools, namely a formula for how taking fixed points commutes with extended tensor products of finite sets with group actions and how the Brauer construction commutes with taking extended tensor products of $p$-permutation modules. These fundamental formulas, generalizing earlier results by Boltje-Danz and by Boltje-Perepelitsky, should be of independent interest. + oai:arXiv.org:2512.18150v1 + math.RT + Tue, 23 Dec 2025 00:00:00 -0500 new - http://creativecommons.org/licenses/by-nc-nd/4.0/ - Xuenan Li, Chun Liu, Di Qi + http://creativecommons.org/licenses/by-nc-sa/4.0/ + Robert Boltje, John Revere McHugh - Conditional Expectation Backward Stochastic Differential Equations and Related Backward Stochastic Differential Equations with Conditional Reflection - https://arxiv.org/abs/2512.17135 - arXiv:2512.17135v1 Announce Type: new -Abstract: In this paper, we introduce a new type of backward stochastic differential equations (BSDEs), called conditional expectation BSDEs, whose drivers depend not only on the value of the solutions but also on their conditional expectations with respect to a certain sub-{\sigma}-algebra. The collection of these sub-{\sigma}-algebra forms a subfiltration, which stands for partial information that is common for decision making applications. The classical BSDEs and the mean-field BSDEs can be regarded as two special and extreme cases of conditional expectation BSDEs. We establish the well-posedness for conditional expectation BSDEs under mild conditions and discuss the comparison results. Then, we provide an alternative construction for the solutions to conditional reflected BSDEs without the continuity assumption for the subfiltration, which can be seen as the limit of a sequence of penalized conditional expectation BSDEs. - oai:arXiv.org:2512.17135v1 - math.PR - Mon, 22 Dec 2025 00:00:00 -0500 + The Orbit-Summable Fixed Point Criterion and its Relation to Caristi's Theorem + https://arxiv.org/abs/2512.18153 + arXiv:2512.18153v1 Announce Type: new +Abstract: The relationship between geometric and variational principles remains central to Nonlinear Analysis. This paper introduces the \textbf{Orbit-Summability Fixed Point Criterion}, a novel, purely dynamical condition, and establishes its profound connection to \textbf{Caristi`s Fixed Point Theorem} in complete metric spaces. + Our criterion, which requires only that the total displacement along a single orbit be finite (the orbit-summability property), provides a practical and concrete tool for checking the existence of fixed points without relying on the construction of an abstract potential function. + We demonstrate that, under a minimal regularity assumption involving the function $f$ and the metric $d$, the Orbit-Summability Criterion is \textbf{precisely equivalent} to Caristi`s Fixed Point Theorem. This equivalence is conceptually significant as it creates a direct bridge between the geometric principle of \textbf{dynamical gap summability} (akin to the core idea in Banach`s Contraction Principle) and the variational principle of Caristi. + As a direct consequence of this equivalence, the classical \textbf{Banach Contraction Principle is recovered as a straightforward corollary}. The methodology is designed to merge key elements from the Banach proof (orbit convergence via summability) with the structural requirements of Caristi (majorization by a well-behaved functional), effectively providing a unifying framework for these fundamental results. Thus, the Orbit-Summability Criterion offers an attractive and accessible perspective on these established cornerstones of Fixed Point Theory. + oai:arXiv.org:2512.18153v1 + math.FA + Tue, 23 Dec 2025 00:00:00 -0500 new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Hanwu Li + http://creativecommons.org/licenses/by/4.0/ + Robl\^edo Mak's Miranda Sette - Higher Weight Generalized Dedekind Sums - https://arxiv.org/abs/2512.17139 - arXiv:2512.17139v1 Announce Type: new -Abstract: Building upon the work of Stucker, Vennos, and Young we derive generalized Dedekind sums arising from period integrals applied to holomorphic Eisenstein series attached to pairs of primitive non-trivial Dirichlet characters. Furthermore, we explore a variety of properties of these generalized Dedekind sums: we develop a finite sum formula, demonstrate their behavior as quantum modular forms, provide a Fricke reciprocity law, and characterize analytic and arithmetic aspects of their image. Particularly, for the arithmetic aspect of the image, we generalize an existing conjecture to the higher weight case and provide significant computational evidence to support this generalized conjecture. - oai:arXiv.org:2512.17139v1 + Alternating Power Difference and Matrix Symmetry: Closed-Form Formulas for the First Appearance Degree $m_1$ + https://arxiv.org/abs/2512.18169 + arXiv:2512.18169v1 Announce Type: new +Abstract: This paper focuses on an integer-valued function $f_A(\sigma) := \operatorname{tr}(A P_\sigma)$ defined uniformly from a specific square matrix $A$ of order $n$ and a permutation $\sigma$ on the symmetric group $S_n$. The main objective of this study is to investigate in detail the algebraic behavior of the Alternating Power Difference (APD), denoted as $APD_m(f_A)$, and its first appearance degree $m_1(f_A)$ for this function $f_A$ across various matrix classes. Specifically, we address special matrices such as shifted $r$-th power lattices, Vandermonde matrices, and circulant matrices, analyzing the phenomenon where the value of $APD_m(A)$ remains zero as $m$ increases until a specific degree (the first appearance phenomenon). In particular, we explore closed-form formulas for the first appearance degree $m_1(A)$ and the first appearance value $APD_{m_1}(A)$, presenting Conjectures that hold across multiple matrix classes. These results suggest a deep relationship between the structure of matrices and the analytical properties of functions on the symmetric group, providing new perspectives in matrix theory and combinatorics. + oai:arXiv.org:2512.18169v1 + math.CO math.NT - Mon, 22 Dec 2025 00:00:00 -0500 + Tue, 23 Dec 2025 00:00:00 -0500 new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Preston Tranbarger + http://creativecommons.org/licenses/by/4.0/ + Kenichi Takemura - Harmonic band theory: rigidity of non-zero degree harmonic maps from 2-torus to complex projective space - https://arxiv.org/abs/2512.17150 - arXiv:2512.17150v1 Announce Type: new -Abstract: We prove the rigidity of isotropic harmonic maps from a 2-torus to a complex projective space, when they are constructed from holomorphic embeddings associated to complete linear systems. We also prove that this rigidity holds for any holomorphic embeddings without special hyperosculation points, with an extra assumption on the pullbacks of Fubini--Study symplectic forms. These results ensure the rigidity of towers of harmonic bands in condensed matter physics. - oai:arXiv.org:2512.17150v1 - math-ph - math.AG - math.DG - math.MP - Mon, 22 Dec 2025 00:00:00 -0500 + On well-posedness of the s-Schr\"odinger maps in the subcritical regime + https://arxiv.org/abs/2512.18170 + arXiv:2512.18170v1 Announce Type: new +Abstract: We study well-posedness of the $s$-Schr\"odinger map equation in dimension $n \geq 3$ in the subcritical regime, more precisely we establish a local well-posedness result when the initial data is $u_{0} \in B^{\sigma}_{2,1}$ with $ \sigma \geq \frac{n+1}{2}$ and $ \Vert u_{0} \Vert_{B^{\sigma}_{2,1}} \ll 1.$ + oai:arXiv.org:2512.18170v1 + math.AP + Tue, 23 Dec 2025 00:00:00 -0500 new http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Yoshinori Hashimoto, Bruno Mera, Tomoki Ozawa + Ahmed Dughayshim - High-Voltage Ionized Gas with Spherical Cathode Emission - https://arxiv.org/abs/2512.17159 - arXiv:2512.17159v1 Announce Type: new -Abstract: We consider a plasma that is created by a high voltage difference, which is known as a Townsend gas discharge. The plasma is confined to the region between two concentric spheres, one of which is a cathode and the other an anode. Ion-electron pairs are created by collisions inside the plasma. Additional electrons enter the plasma by collisions of ions with the cathode. We prove under certain conditions that there are many steady states exhibiting gas discharge, beginning with a `sparking' voltage. In fact, there is an analytic one-parameter family of them that connects the non-ionized gas to a plasma with arbitrarily high ionization or arbitrarily high potential, or else the family ends at an `anti-sparking' voltage. - oai:arXiv.org:2512.17159v1 + On scattering behavior of corner domains with anisotropic inhomogeneities: part II + https://arxiv.org/abs/2512.18175 + arXiv:2512.18175v1 Announce Type: new +Abstract: We study the scattering behavior of an anisotropic inhomogeneous Lipschitz medium at a fixed wave number, continuing our previous work [SIAM J. Math. Anal., 56(4):4834-4853, 2024] and using free boundary techniques from [arXiv:2506.22328]. Our main results can be categorized into two distinct cases. In the first case, we show that in two dimensions, piecewise $C^{1}$ or convex penetrable obstacles with corners, and in higher dimensions, obstacles with edge points, always induce nontrivial scattering for any incoming wave. In the second case, we prove that piecewise $C^{1}$ obstacles with corners in two dimensions (and with edge points in higher dimensions) with angles $\notin\pi\mathbb{Q}$ always produce nontrivial scattering for any incoming wave. + oai:arXiv.org:2512.18175v1 math.AP - math-ph - math.MP - Mon, 22 Dec 2025 00:00:00 -0500 + Tue, 23 Dec 2025 00:00:00 -0500 new http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Walter A. Strauss, Masahiro Suzuki + Pu-Zhao Kow, Mikko Salo, Henrik Shahgholian - Capacities Characterizing Removable Sets for Various Function Spaces in Carnot Groups - https://arxiv.org/abs/2512.17167 - arXiv:2512.17167v1 Announce Type: new -Abstract: We study removable sets for the Campanato, H\"{o}lder continuous, $L^p_{\text{loc}}$, and Lipschitz functions in Carnot groups. In the former three cases, we characterize removability through the use of capacities with respect to any left-invariant linear differential operator $\mathcal{L}$ for which $\mathcal{L}$ and $\mathcal{L}^t$ are hypoelliptic and satisfy a homogeneity condition, while in the latter case we characterize Lipschitz functions with respect to the sub-Laplacian. - oai:arXiv.org:2512.17167v1 - math.CA - Mon, 22 Dec 2025 00:00:00 -0500 + A Domain Decomposition Deep Neural Network Method with Multi-Activation Functions for Solving Elliptic and Parabolic Interface Problems + https://arxiv.org/abs/2512.18178 + arXiv:2512.18178v1 Announce Type: new +Abstract: We present a domain decomposition-based deep learning method for solving elliptic and parabolic interface problems with discontinuous coefficients in two to ten dimensions. Our Multi-Activation Function (MAF) approach employs two independent neural networks, one for each subdomain, coupled through interface conditions in the loss function. The key innovation is a multi-activation mechanism within each subdomain network that adaptively blends multiple activation functions (e.g., $\tanh$ and Gaussian-type) with interface-aware weighting, enhancing learning efficiency near interfaces where coupling constraints are most demanding. We prove conditional error bounds relating solution accuracy to trained loss values and quadrature errors. Numerical experiments on elliptic and parabolic interface problems with various interface geometries (2D--10D) validate the effectiveness and accuracy of the proposed method. + oai:arXiv.org:2512.18178v1 + math.NA + cs.NA + Tue, 23 Dec 2025 00:00:00 -0500 new http://creativecommons.org/licenses/by/4.0/ - Zack Boone + Qijia Zhai - Ethic Duality: A Homological Framework for Primal-Dual Problems - https://arxiv.org/abs/2512.17170 - arXiv:2512.17170v1 Announce Type: new -Abstract: We develop a homological duality framework based on a contravariant functor $D=\operatorname{Hom}_E(-,R)$ with dualizing object $R$. A morphism is called ethic when it satisfies the canonical double-dual compatibility $D^2(f)\eta=\eta f$. In the derived setting, the functor $\mathrm{RHom}_E(-,R)$ produces a graded family of Ext-groups that measure all failures of this compatibility. The first layer $\operatorname{Ext}^1$ identifies primal-dual gaps, while higher $\operatorname{Ext}^k$ provide a systematic hierarchy of derived obstructions to exactness. - This formulation specializes uniformly across several classical domains. In linear and conic optimization, Farkas- and Slater-type exactness criteria correspond to the vanishing of $\operatorname{Ext}^1$, and integer duality gaps coincide with torsion Ext-classes. In graph theory, Kirchhoff- and Baker-Norine-type dualities arise as instances of ethic exactness. In dynamical systems, the higher derived layers encode nonvanishing persistence phenomena. Additional examples include social-choice configurations, categorical factorization in scattering formalisms, coding-theoretic duality, and Bellman-type recurrences, all appearing as concrete instances of Ext-controlled exactness. - All resulting invariants are stable under derived Morita equivalence and depend only on the dualizing pair $(E,R)$. The framework therefore provides a substrate-independent criterion for primal-dual exactness and a uniform homological description of its obstructions. - oai:arXiv.org:2512.17170v1 - math.CT - math.OC - Mon, 22 Dec 2025 00:00:00 -0500 + Boundary Stabilization of a Degenerate Euler-Bernoulli Beam under Axial Force and Time Delay + https://arxiv.org/abs/2512.18179 + arXiv:2512.18179v1 Announce Type: new +Abstract: This paper provides a qualitative analysis of a non-uniform Euler-Bernoulli beam with degenerate flexural rigidity, subjected to axial force and boundary control with time delay $\tau > 0$. By reformulating the system as an abstract evolution problem in an augmented Hilbert space incorporating weighted Sobolev spaces, we employ semigroup theory to ensure well-posedness. Using the energy multiplier method and a non-standard Lyapunov functional featuring weighted integral terms, we establish uniform exponential energy decay and provide a precise decay rate estimate. This work extends the results of Salhi et al. \cite{salhi2025} and Siriki et al. \cite{siriki2025} by incorporating axial force and generalized control laws, including rotational velocity control. The proposed framework offers a robust approach for analyzing complex distributed systems. + oai:arXiv.org:2512.18179v1 + math.AP + Tue, 23 Dec 2025 00:00:00 -0500 new http://creativecommons.org/licenses/by/4.0/ - Dmitry Pasechnyuk-Vilensky, Martin Tak\'a\v{c} + Ben Bakary Junior Siriki, Adama Coulibaly - Intersecting well approximable and missing digit sets - https://arxiv.org/abs/2512.17173 - arXiv:2512.17173v1 Announce Type: new -Abstract: Let $b\geq3$ be an integer and $C(b,D)$ be the set of real numbers in $[0,1]$ whose $b$-ary expansion consists of digits restricted to a given set $D\subseteq\{0,\ldots,b-1\}$. Given an integer $t\geq2$ and a real, positive function $\psi$, let $W_{t}(\psi)$ denote the set of $x$ in $[0,1]$ for which $|x-p/t^{n}|<\psi(n)$ for infinitely many $(p,n)\in\mathbb{Z}\times\mathbb{N}$. We prove a general Hausdorff dimension result concerning the intersection of $W_{t}(\psi)$ with an arbitrary self similar set which implies that $\dim_{\rm H}(W_{t}(\psi)\cap C(b,D))\le\dim_{\rm H}W_{t}(\psi)\times \dim_{\rm H}C(b,D)$. When $b$ and $t$ have the same prime divisors, under certain restrictions on the digit set $D$, we give a sufficient condition for the Hausdorff measure of $W_{t}(\psi)\cap C(b,D)$ to be zero. This closes a gap in a result of Li, Li and Wu \cite{LLW2025} and shows that the dimension of the intersection can be strictly less than the product of the dimensions. The latter disproves the product conjecture of Li, Li and Wu. - oai:arXiv.org:2512.17173v1 - math.NT - math.DS - Mon, 22 Dec 2025 00:00:00 -0500 + Decay estimates for one Aharonov-Bohm solenoid in a uniform magnetic field III: Product cones + https://arxiv.org/abs/2512.18183 + arXiv:2512.18183v1 Announce Type: new +Abstract: The goal of a recently launched project is to extend the Euclidean models in \cite{Wang24,WZZ25-AHP,WZZ25-JDE} to a more general setting of conically singular spaces. In this paper, the main results include a weighted dispersive inequality for the Schr\"odinger equation and a dispersive estimate for the wave equation both with one Aharonov-Bohm solenoid in a uniform magnetic field on the product cone $X=\mathcal{C}(\mathbb{S}_\sigma^1)=(0,+\infty)_r\times\mathbb{S}_\sigma^1$ endowed with the flat metric $g=dr^2+r^2d\theta^2$, where $\mathbb{S}_\sigma^1\simeq\mathbb{R}/2\pi\sigma\mathbb{Z}$ denotes the circle of radius $\sigma\geq1$ in the Euclidean plane $\mathbb{R}^2$. As a byproduct, we also give the corresponding Strichartz estimates for these equations via the abstract argument of Keel-Tao. + oai:arXiv.org:2512.18183v1 + math.AP + Tue, 23 Dec 2025 00:00:00 -0500 new - http://creativecommons.org/licenses/by/4.0/ - Bing Li, Sanju Velani, Bo Wang + http://creativecommons.org/licenses/by-nc-nd/4.0/ + Haoran Wang - Distributed Rotary Coverage Control of Multi-Agent Systems in Uncertain Environments - https://arxiv.org/abs/2512.17174 - arXiv:2512.17174v1 Announce Type: new -Abstract: It is always a challenging task for multi-agent systems to achieve efficient and robust coverage in uncertain environments. The absence of global positioning information on the uncertain environment introduces significant complexity to the spatially distributed design of coverage control algorithms. To address this issue, this paper proposes a coverage control formulation based on beacon-free rotary pointer partition mechanism. A partition dynamics is designed to enable the asymptotical consensus of multi-agent reference points, as well as the workload-balanced subdivision of coverage region. On this basis, a distributed coverage control algorithm is developed to drive each agent toward the optimal deployment of their respective subregions, thereby minimizing the coverage cost. Simulation results demonstrate that the proposed coverage control method can significantly improve overall coverage efficiency with workload balance among agents, and exhibit strong adaptability and robustness in uncertain environments. - oai:arXiv.org:2512.17174v1 - math.OC - Mon, 22 Dec 2025 00:00:00 -0500 + Pseudo-Legendrian and Legendrian Simplicity of Links in 3-Manifolds + https://arxiv.org/abs/2512.18185 + arXiv:2512.18185v1 Announce Type: new +Abstract: We construct infinite families of non-simple isotopy classes of links in overtwisted contact structures on $S^1$-bundles over surfaces. These examples include: (1) a pair of Legendrian links that are not Legendrian isotopic, but which are isotopic as framed links, homotopic as Legendrian immersed multi-curves, and have Legendrian-isotopic components and (2) a pair of Legendrian links that are not Legendrian isotopic, but are isotopic as framed links, homotopic as Legendrian immersed multi-curves, and which are link-homotopic as Legendrian links. Moreover, we construct examples showing that both of these non-simplicity phenomena can occur in the same smooth isotopy class. To construct these examples, we develop the theory of links transverse to a nowhere-zero vector field in a 3-manifold, and construct analogous examples in the category of links transverse to a vector field. + oai:arXiv.org:2512.18185v1 + math.GT + math.SG + Tue, 23 Dec 2025 00:00:00 -0500 new http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Chao Zhai, Yanlin Li + Patricia Cahn, Rima Chatterjee, Vladimir Chernov - A remark about Calder\'on-Hardy spaces with variable exponents - https://arxiv.org/abs/2512.17175 - arXiv:2512.17175v1 Announce Type: new -Abstract: In this note we improve the parameter $q$ that appears in Theorem 1 obtained by the author in [Math. Ineq. \& appl., Vol 19 (3) (2016), 1013-1030]. - oai:arXiv.org:2512.17175v1 + On suprema of convolutions on discrete cubes + https://arxiv.org/abs/2512.18188 + arXiv:2512.18188v1 Announce Type: new +Abstract: We find the optimal constant $C$ such that \begin{equation*} \|f_1*f_2*\dots*f_{k}\|_{\infty}\geq C\prod_{i=1}^{k}\|f_i\|_1 \end{equation*} for functions $f_i:\{0,1\}^d\to\mathbb{R}$. As applications, we derive bounds for Sidon sets on hypercubes, and, we also obtain bounds for the continuous analogue problem. + oai:arXiv.org:2512.18188v1 math.CA - Mon, 22 Dec 2025 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Pablo Rocha - - - Generalized diagram categories and monoids, and their representations - https://arxiv.org/abs/2512.17177 - arXiv:2512.17177v1 Announce Type: new -Abstract: Classical diagram categories and monoids, including the Temperley--Lieb, Brauer, and partition cases, arise as special instances of the category of two dimensional cobordisms and admit additional twists that produce a large new family of diagram categories and monoids. In this paper we introduce this family and develop a unified approach to their representation theory. - oai:arXiv.org:2512.17177v1 - math.RT math.CO - math.GR - Mon, 22 Dec 2025 00:00:00 -0500 + Tue, 23 Dec 2025 00:00:00 -0500 new http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Matthias Fresacher, Willow Stewart, Daniel Tubbenhauer + Jos\'e Gaitan, Jos\'e Madrid - Long-time stability and convergence analysis of an IMEX BDF3 scheme for 2-D incompressible Navier-Stokes equation - https://arxiv.org/abs/2512.17182 - arXiv:2512.17182v1 Announce Type: new -Abstract: High-order time-stepping schemes are crucial for simulating incompressible fluid flows due to their ability to capture complex turbulent behavior and unsteady motion. In this work, we propose a third-order accurate numerical scheme for the two-dimensional incompressible Navier-Stokes equation. Spatial and temporal discretization is achieved using Fourier pseudo-spectral approximation and the BDF3 stencil, combined with the Adams-Bashforth extrapolation for the nonlinear convection term, resulting in a semi-implicit, fully discrete formulation. This approach requires solving only a single Poisson-like equation per time step while maintaining the desired temporal accuracy. Classical numerical experiments demonstrate the advantage of our scheme in terms of permissible time step sizes. Moreover, we establish uniform-in-time bounds for the vorticity in both $L^2$ and higher-order $H^m$ norms ($m \geq 1$), provided the time step is sufficiently small. These bounds, in turn, facilitate the derivation of optimal convergence rates. - oai:arXiv.org:2512.17182v1 + A Singularity Guided Nystr\"om Method for Elastostatics on Two Dimensional Domains with Corners + https://arxiv.org/abs/2512.18208 + arXiv:2512.18208v1 Announce Type: new +Abstract: We develop a comprehensive analytical and numerical framework for boundary integral equations (BIEs) of the 2D Lam\'e system on cornered domains. By applying local Mellin analysis on a wedge, we obtain a factorizable characteristic equation for the singular exponents of the boundary densities, and clarify their dependence on boundary conditions. The Fredholm well-posedness of the BIEs on cornered domains is proved in weighted Sobolev spaces. We further construct an explicit density-to-Taylor mapping for the BIE and show its invertibility for all but a countable set of angles. Based on these analytical results, we propose a singularity guided Nystr\"om (SGN) scheme for the numerical solution of BIEs on cornered domains. The SGN uses the computed corner exponents and a Legendre-tail indicator to drive panel refinement. An error analysis that combines this refinement strategy with an exponentially accurate far-field quadrature rule is provided. Numerical experiments across various cornered geometries demonstrate that SGN obtains higher order accuracy than uniform Nystr\"om method and reveal a crowding-limited regime for domains with re-entrant angles. + oai:arXiv.org:2512.18208v1 math.NA cs.NA - Mon, 22 Dec 2025 00:00:00 -0500 + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Baoling Xie, Jun Lai + + + Stability of inverse boundary value problem for the fourth-order Schr\"{o}dinger equation + https://arxiv.org/abs/2512.18212 + arXiv:2512.18212v1 Announce Type: new +Abstract: This paper is concerned with the stability of the inverse boundary value problem for the perturbed fourth-order Schr\"{o}dinger equation in a bounded domain with Cauchy data. We establish stability results for the perturbed potential relying on boundary measurements. The estimates depend on various a priori information regarding the regularity and the support of the inhomogeneity. The proof primarily utilizes the complex geometric optics solution method and Fourier analysis. + oai:arXiv.org:2512.18212v1 + math.AP + Tue, 23 Dec 2025 00:00:00 -0500 new http://creativecommons.org/licenses/by/4.0/ - Kelong Cheng, Jingwei Sun, Hong Zhang + Yang Liu, Yixian Gao - Estimation of First Returning Speed in Null Recurrent Continuous-Time Markov Chains - https://arxiv.org/abs/2512.17190 - arXiv:2512.17190v1 Announce Type: new -Abstract: This paper establishes a novel connection between null-recurrent CTMCs and electric networks, offering a systematic classification of null-recurrent behavior based on the first returning speed. By leveraging techniques from electric network theory, we present a general method for estimating the first returning speed of null recurrent birth-death processes and provide some important examples. - oai:arXiv.org:2512.17190v1 - math.PR - Mon, 22 Dec 2025 00:00:00 -0500 + Fibonacci and Lucas numbers arising from two-component spanning forests of wheel graphs + https://arxiv.org/abs/2512.18214 + arXiv:2512.18214v1 Announce Type: new +Abstract: In this paper, we present a constructive bijection between a conditioned spanning forest of the wheel graph $W_{n+1}$ and a spanning tree of the fan graph $F_n$. In addition, by applying the effective resistance formula obtained by Bapat and Gupta \cite{bapat-gupta}, we derive an explicit formula for the number of two-component spanning forests of $W_{n+1}$ in which two specified vertices $u$ and $v$ lie in distinct components. Based on this result, we obtain explicit formulas for the following three conditioned two-component spanning forests $F_{W_{n+1}}(v_1\mid v_2)$, $F_{W_{n+1}}(v_1\mid v_3)$, and $F_{W_{n+1}}(v_1\mid v_c)$. These formulas are $F_{W_{n+1}}(v_1\mid v_2)=2(f_{2n-1}-1)$, $F_{W_{n+1}}(v_1\mid v_3)=2(\ell_{2n-2}-3)$, $F_{W_{n+1}}(v_1\mid v_c)=f_{2n}$, where $f_i$ and $\ell_j$ denote the $i$-th Fibonacci number and $j$-th Lucas number, respectively. As these identities show, the enumerations naturally lead to formulas involving Fibonacci numbers and Lucas numbers. Taken together, these two approaches show a unified perspective. One is the constructive combinatorial bijection, and the other is the analytic method based on effective resistance. Together they provide a new integrated framework for studying the structure of spanning forests on $W_{n+1}$. + oai:arXiv.org:2512.18214v1 + math.CO + Tue, 23 Dec 2025 00:00:00 -0500 new http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Huixi Wang, Minzhi Zhao + Tsuyoshi Miezaki, Shunya Tamura - Proof of a conjecture of Baruah and Sarma on sign patterns of certain infinite products - https://arxiv.org/abs/2512.17195 - arXiv:2512.17195v1 Announce Type: new -Abstract: Let \[ \sum_{n=0}^{\infty}A(n)q^{n} := \frac{(q^{2};q^{5})_{\infty}^{5}(q^{3};q^{5})_{\infty}^{5}}{(q;q^{5})_{\infty}^{5}(q^{4};q^{5})_{\infty}^{5}}, \] \[ \sum_{n=0}^{\infty} B(n)q^{n} := \frac{(q;q^{5})_{\infty}^{5} (q^{4};q^{5})_{\infty}^{5}} {(q^{2};q^{5})_{\infty}^{5}(q^{3}; q^{5})_{\infty}^{5}}, \] and \[ \sum_{n=0}^{\infty} D(n)q^{n} := \frac{(q^{5};q^{25})_{\infty}(q^{20}; q^{25})_{\infty}} {(q^{10};q^{25})_{\infty}(q^{15}; q^{25})_{\infty}} \frac{(q^{2}; q^{5})_{\infty}^{5}(q^{3};q^{5})_{\infty}^{5}} {(q;q^{5})_{\infty}^{5} (q^{4};q^{5})_{\infty}^{5}} \] where $(a;q)_{\infty} := \prod_{k=0}^{\infty}(1-aq^{k})$ and $|q|<1.$ These sequences are closely related to the celebrated Rogers-Ramanujan continued fraction. - In this paper, we study the sign behavior o of the coefficients $A(n),B(n)$ and $D(n).$ We prove that for all integers $n\geq0,$ \begin{align*} A(5n)<0\quad(n\neq0),\qquad B(5n) < 0\quad(n\neq0),\qquad D(5n+1)>0. \end{align*} This confirms a recent conjecture of Baruah and Sarma. Our proof is different from the previous method of Baruah and Sarma, and combines asymptotic coefficient analysis with symbolic computation for finite case verification. - oai:arXiv.org:2512.17195v1 - math.NT - math.CO - Mon, 22 Dec 2025 00:00:00 -0500 + Stochastic control for Backward Stochastic Differential Equations with semi-Markov chain noises + https://arxiv.org/abs/2512.18218 + arXiv:2512.18218v1 Announce Type: new +Abstract: In this paper, we extend the results of Elliott and Yang \cite{elliott3} and discuss the control of a stochastic process for which the driving noise is provided by a martingale associated with a semi-Markov Chain. An existence and a comparison theorem are obtained. In our discrete time setting, adjoint processes are provided by backward stochastic difference equations. Technical results from partial differential equation theory to establish a verification theorem are not required. + oai:arXiv.org:2512.18218v1 + math.PR + Tue, 23 Dec 2025 00:00:00 -0500 new http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Bing He, Xiongze Zhang + Robert J. Elliott, Zhe Yang - A General Characterization on the Uniqueness Problem of L-Functions and General Meromorphic Functions - https://arxiv.org/abs/2512.17205 - arXiv:2512.17205v1 Announce Type: new -Abstract: In the paper, concerning a question of Yi [23], we study general criterion for the uniqueness of an L-function and a general meromorphic function. Our results improve and extend all the existing results in this direction [23, 18, 17, 4] to the most general setting. Moreover, we have exhibited a handsome number of examples to justify our claims as well as to confirm the wide-ranging applications of our results. - oai:arXiv.org:2512.17205v1 - math.CV - Mon, 22 Dec 2025 00:00:00 -0500 + On potentials for sub-Laplacians and geometric applications + https://arxiv.org/abs/2512.18221 + arXiv:2512.18221v1 Announce Type: new +Abstract: In this paper we extend the research on potential theory and its geometric applications from Euclidean spaces to homogeneous Carnot groups. We introduce a new approach to use the geometric completeness to estimate the Hausdorff dimension of polar sets of potentials of nonnegative Radon measures for sub-Laplacians in homogeneous Carnot groups. Our approach relies on inequalities that are analogous to the classic integral inequalities about Riesz potentials in Euclidean spaces. Our approach also uses extensions of some of geometric measure theory to homogeneous Carnot groups and the polar coordinates with horizontal radial curves constructed by Balogh and Tyson for polarizable Carnot groups. As consequences, we develop applications of potentials for sub-Laplacians in CR geometry, quaternionic CR geometry, and octonionic CR geometry. + oai:arXiv.org:2512.18221v1 + math.DG + math.AP + Tue, 23 Dec 2025 00:00:00 -0500 new http://creativecommons.org/licenses/by/4.0/ - Sanjay Mallick, Ripan Saha + Shiguang Ma, Jie Qing - Approximating geodesics of hyperbolic type metrics on planar domains - https://arxiv.org/abs/2512.17211 - arXiv:2512.17211v1 Announce Type: new -Abstract: We study planar domains $G$ equipped with a hyperbolic type metric and approximate geodesics that join two points $x,y \in G$ and their lengths. We present an algorithm that enables one to approximate the shortest distance in polygonal domains taken with respect to the quasihyperbolic metric. The method is based on Dijkstra's algorithm, and we give several examples demonstrating how the algorithm works and analyze its accuracy. We experimentally demonstrate several previously theoretically observed features of geodesics, such as the relationship between hyperbolic and quasihyperbolic distance in the unit disk. We also investigate bifurcation of geodesics and the connection of this phenomenon to the medial axis of the domain. - oai:arXiv.org:2512.17211v1 - math.MG - cs.NA - math.CV - math.NA - Mon, 22 Dec 2025 00:00:00 -0500 + How often are $ \lfloor {n^{\alpha}} \rfloor $ and $ \lfloor {n^{\beta}} \rfloor $ simultaneously primes? + https://arxiv.org/abs/2512.18233 + arXiv:2512.18233v1 Announce Type: new +Abstract: Let $ \lfloor {x} \rfloor $ denote the greatest integer less than or equal to a real number $x$. Given real numbers $0<\alpha_1 < \alpha_2 < \cdots< \alpha_k < 1$ satisfying a certain condition, we show that there are infinitely many positive integers $n$ for which all of $ \lfloor{n^{\alpha_1}}\rfloor, \lfloor{n^{\alpha_2}}\rfloor,\ldots, \lfloor{n^{\alpha_k}}\rfloor $ are prime numbers. Our approach relies on establishing a simultaneous equidistribution theorem for $ \lfloor{n^{\alpha_i}}\rfloor $ across $k$-many arithmetic progressions. + oai:arXiv.org:2512.18233v1 + math.NT + Tue, 23 Dec 2025 00:00:00 -0500 new http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Shuliang Gao, Anni Hakanen, Antti Rasila, Matti Vuorinen + Anup B. Dixit, Nikhil S Kumar - Enumeration of multipartite series-reduced trees - https://arxiv.org/abs/2512.17216 - arXiv:2512.17216v1 Announce Type: new -Abstract: We obtain a generating function for the degree sequences and colors of rooted multipartite labeled series-reduced trees. As an application of this result, we determine the number of symbolic ultrametrics (introduced by B\"ocker and Dress) and increasingly labeled processes. We also find that the number of multipartite labeled series-reduced trees and the colored chain-increasing binary trees are the same. We obtain the number of rooted multipartite unlabeled series-reduced trees. We also find a refinement of the result of Riordan and Shannon. - oai:arXiv.org:2512.17216v1 - math.CO - Mon, 22 Dec 2025 00:00:00 -0500 + Counting $h^0(D)$ on primary Burniat surfaces + https://arxiv.org/abs/2512.18240 + arXiv:2512.18240v1 Announce Type: new +Abstract: We study the cohomology of divisors on a Burniat surface $X$ with $K_X^2=6$. We provide an algorithm for computing the cohomology groups of arbitrary divisors on $X$. As an application, we prove that there are no Ulrich line bundles\,(with respect to an arbitrary polarization), and that there exists an Ulrich vector bundle of rank 2 with respect to $3K_X$. The existence of Ulrich vector bundle of rank 2 was previously established by Casnati, but our construction yields one that cannot be obtained by his method. + oai:arXiv.org:2512.18240v1 + math.AG + Tue, 23 Dec 2025 00:00:00 -0500 new http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Medet Jumadildayev + Yonghwa Cho - Monotonicity of harmonic functions on $3$-manifolds with an asymptotically flat end - https://arxiv.org/abs/2512.17222 - arXiv:2512.17222v1 Announce Type: new -Abstract: We derive monotone properties of positive harmonic functions on three dimensional manifolds with nonnegative scalar curvature, with an asymptotically flat end. Rigidity characterization of spatial Schwarzschild manifolds with two ends is also given. - oai:arXiv.org:2512.17222v1 - math.DG - Mon, 22 Dec 2025 00:00:00 -0500 + On the Nash Problem over 3-Fold Terminal Singularities of Type cAx/2 + https://arxiv.org/abs/2512.18243 + arXiv:2512.18243v1 Announce Type: new +Abstract: We study Nash valuations on 3-fold terminal singularities, especially in type cAx/2. We find that, in type cAx/2, exceptional prime divisors computing the minimal discrepancy (which is 1/2 in this case) induce Nash valuations. We conjecture this in general for all 3-fold terminal singularities, and provide some evidence in the Gorenstein case. + oai:arXiv.org:2512.18243v1 + math.AG + Tue, 23 Dec 2025 00:00:00 -0500 new http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Pengzi Miao + Keng-Hung Steven Lin - An Induced $A$-Path Theorem - https://arxiv.org/abs/2512.17232 - arXiv:2512.17232v1 Announce Type: new -Abstract: Given a graph $G$ and $\mathcal{A}\subseteq V(G)$, a classical theorem of Gallai (1964) states that for every positive integer $k$, the graph $G$ contains $k$ pairwise vertex-disjoint $\mathcal{A}$-paths, or a set $Z\subseteq V(G)$ of size at most $2(k-1)$ such that $G-Z$ contains no $\mathcal{A}$-paths. We generalise Gallai's theorem to the induced setting: We prove that $G$ contains $k$ pairwise anti-complete $\mathcal{A}$-paths, or a set $Z$ of size at most $78(k-1)$ such that, after removing the closed neighbourhood of $Z$, the resulting graph has no $\mathcal{A}$-path. Here, two paths are anti-complete if they are vertex disjoint and there is no edge in $G$ having one endpoint in each of them. - We further show that the bound $78(k-1)$ on the size of $Z$ can be reduced to $4(k-1)$ if one removes the balls of radius $4$ around the vertices of $Z$ (instead of radius $1$), which is within a factor $2$ of optimal. We also establish analogous results for long induced $\mathcal{A}$-paths. - oai:arXiv.org:2512.17232v1 - math.CO - Mon, 22 Dec 2025 00:00:00 -0500 + Central Limit Theorem for ergodic averages of Markov chains \& the comparison of sampling algorithms for heavy-tailed distributions + https://arxiv.org/abs/2512.18255 + arXiv:2512.18255v1 Announce Type: new +Abstract: Establishing central limit theorems (CLTs) for ergodic averages of Markov chains is a fundamental problem in probability and its applications. Since the seminal work~\cite{MR834478}, a vast literature has emerged on the sufficient conditions for such CLTs. To counterbalance this, the present paper provides verifiable necessary conditions for CLTs of ergodic averages of Markov chains on general state spaces. Our theory is based on drift conditions, which also yield lower bounds on the rates of convergence to stationarity in various metrics. + The validity of the ergodic CLT is of particular importance for sampling algorithms, where it underpins the error analysis of estimators in Bayesian statistics and machine learning. Although heavy-tailed sampling is of central importance in applications, the characterisation of the CLT and the convergence rates are theoretically poorly understood for almost all practically-used Markov chain Monte Carlo (MCMC) algorithms. In this setting our results provide sharp conditions on the validity of the ergodic CLT and establish convergence rates for large families of MCMC sampling algorithms for heavy-tailed targets. Our study includes a rather complete analyses for random walk Metropolis samplers (with finite- and infinite-variance proposals), Metropolis-adjusted and unadjusted Langevin algorithms and the stereographic projection sampler (as well as the independence sampler). By providing these sharp results via our practical drift conditions, our theory offers significant insights into the problems of algorithm selection and comparison for sampling heavy-tailed distributions (see short YouTube presentations~\cite{YouTube_talk} describing our \href{https://youtu.be/m2y7U4cEqy4}{\underline{theory}} and \href{https://youtu.be/w8I_oOweuko}{\underline{applications}}). + oai:arXiv.org:2512.18255v1 + math.PR + stat.CO + stat.ML + Tue, 23 Dec 2025 00:00:00 -0500 new http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Robert Hickingbotham, Gwena\"el Joret + Miha Bre\v{s}ar, Aleksandar Mijatovi\'c, Gareth Roberts - Affine isoperimetric inequalities for the first eigenvalue of the $m$-th order Affine $p$-Laplace Operator - https://arxiv.org/abs/2512.17237 - arXiv:2512.17237v1 Announce Type: new -Abstract: Recently, Haddad, Jim\'enez, and Montenegro introduced the affine $p$-Laplace operator, $p>1$, and studied associated affine versions of the isoperimetric inequalities for the first eigenvalue of the affine $p$-Laplace operator, including the affine Faber-Krahn inequality and affine Talenti inequality. In this work, we introduce the $m$th-order $p$-Laplace operator $\Delta_{Q,p}^\mathcal{A} f$, which recovers the affine $p$-Laplace operator when $m=1$ and $Q$ is a symmetric interval. - Given $n,m \in \mathbb{N}$, a sufficiently smooth convex body $Q \subset \mathbb{R}^m$, a bounded, open set $\Omega \subset \mathbb{R}^n$ and $p >1$, we investigate the eigenvalue problem \[\begin{cases} \Delta_{Q,p}^\mathcal{A} f = \lambda_{1,p}^\mathcal{A}(Q,\Omega) |f|^{p-2} f &\text{ in } \Omega; \\ f=0 & \text{ on } \partial \Omega, \end{cases} \] for $f \in W^{1,p}_0(\Omega)$. Finally, we establish $m$th-order extensions of the affine Talenti inequality and affine Faber-Krahn inequality, which, upon choosing $m=1$, yield new, asymmetric versions of those aforementioned inequalities. - oai:arXiv.org:2512.17237v1 - math.FA + Well-posedness of the Euler system of gas dynamics + https://arxiv.org/abs/2512.18267 + arXiv:2512.18267v1 Announce Type: new +Abstract: We propose a new two-step selection criterion applicable to the dissipative measure--valued solutions of the Euler system of gas dynamics. The process consists of a successive maximisation of the entropy production rate and the total energy defect, i.e. maximisation of the turbulent energy. If the selected solution is a weak solution of the Euler system, then it is identified in the first step. Solutions selected in the second step are truly measure--valued maximising the energy defect. Accordingly, they are called turbulent solutions. The energy defect of turbulent solutions vanishes with growing time. The selected solutions depend in a Borel--measurable way on the initial data. In particular, they are almost continuously dependent on the initial data. + oai:arXiv.org:2512.18267v1 math.AP - math.MG - Mon, 22 Dec 2025 00:00:00 -0500 + Tue, 23 Dec 2025 00:00:00 -0500 new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Dylan Langharst, Michael Roysdon + http://creativecommons.org/licenses/by/4.0/ + Eduard Feireisl, Maria Lukacova-Medvidova - Quasiprimitive and bi-quasiprimitive highly-arc-transitive digraphs and finite simple groups - https://arxiv.org/abs/2512.17244 - arXiv:2512.17244v1 Announce Type: new -Abstract: We extend the notion of an $H$-normal quotient digraph of an $H$-vertex-transitive digraph to that of an $H$-subnormal quotient digraph. Using these concepts, together with bipartite halves of bipartite digraphs, we show that, for each finite connected $H$-vertex-transitive, $(H,s)$-arc-transitive digraph with $s\geqslant6$, either some $H$-normal quotient is a directed cycle of length at least $3$, or there is an $(L,t)$-arc-transitive digraph with $t\geqslant (s-3)/2$, and $L$ a vertex-quasiprimitive almost simple group with socle a composition factor of $H$. This connection demonstrates that, to understand finite $s$-arc-transitive digraphs with large $s$, those admitting a vertex-quasiprimitive almost simple $s$-arc-transitive subgroup of automorphisms play a central role. We show that for each $s$ and each odd valency $k$, there are infinitely many $(H,s)$-arc-transitive digraphs of valency $k$ with $H$ a finite alternating group. - In addition we discovered a novel construction which takes as input a connected non-bipartite $H$-vertex-transitive, $(H,s)$-arc-transitive digraph, and outputs a connected bipartite $G$-vertex-transitive, $(G,2s)$-arc-transitive digraph with $G=(H\times H).2$. This leads to construction of vertex-bi-quasiprimitive $s$-arc-transitive digraphs, for arbitrarily large $s$. Our investigations yield several new open problems. - oai:arXiv.org:2512.17244v1 - math.CO - Mon, 22 Dec 2025 00:00:00 -0500 + Hybrid multiscale method for polymer melts: analysis and simulations + https://arxiv.org/abs/2512.18272 + arXiv:2512.18272v1 Announce Type: new +Abstract: We model the flow behaviour of dense melts of flexible and semiflexible ring polymers in the presence of walls using a hybrid multiscale approach. Specifically, we perform molecular dynamics simulations and apply the Irving-Kirkwood formula to determine an averaged stress tensor for a macroscopic model. For the latter, we choose a Cahn-Hilliard-Navier-Stokes system with dynamic and no-slip boundary conditions. We present numerical simulations of the macroscopic flow that are based on a finite element method. In particular, we present detailed proofs of the solvability and the energy stability of our numerical scheme. Phase segregation under flow between flexible and semiflexible rings, as observed in the microscopic simulations, can be replicated in the macroscopic model by introducing effective attractive forces. + oai:arXiv.org:2512.18272v1 + math.NA + cond-mat.soft + cs.NA + Tue, 23 Dec 2025 00:00:00 -0500 new - http://creativecommons.org/licenses/by/4.0/ - Lei Chen, Cheryl Praeger + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Ranajay Datta, M\'aria Luk\'a\v{c}ov\'a-Medvi\v{d}ov\'a, Andreas Sch\"omer, Peter Virnau - Quasi-recursive MDS Matrices over Galois Rings - https://arxiv.org/abs/2512.17256 - arXiv:2512.17256v1 Announce Type: new -Abstract: Let $p$ be a prime and $s,m,n$ be positive integers. This paper studies quasi-recursive MDS matrices over Galois rings $GR(p^{s}, p^{sm})$ and proposes various direct construction methods for such matrices. The construction is based on skew polynomial rings $GR(p^{s}, p^{sm})[X;\sigma]$, whose rich factorization properties and enlarged class of polynomials are used to define companion matrices generating quasi-recursive MDS matrices. First, two criteria are established for characterizing polynomials that yield recursive MDS matrices, generalizing existing results, and then an additional criterion is derived in terms of the right roots of the associated Wedderburn polynomial. Using these criteria, methods are developed to construct skew polynomials that give rise to quasi-recursive MDS matrices over Galois rings. This framework extends known constructions to the non-commutative setting and significantly enlarges the family of available matrices, with potential applications to efficient diffusion layers in cryptographic primitives. The results are particularly relevant for practical implementations when $s = 1$ and $p = 2$, i.e., over the finite field $\mathbb{F}_{2^m}$, which is of central interest in real-world cryptographic applications. - oai:arXiv.org:2512.17256v1 - cs.IT - math.IT - Mon, 22 Dec 2025 00:00:00 -0500 + Cobordism of nested manifolds + https://arxiv.org/abs/2512.18277 + arXiv:2512.18277v1 Announce Type: new +Abstract: We study cobordisms of nested manifolds, which are manifolds together with embedded submanifolds, which can themselves have embedded submanifolds, etc. We identify a nested analog of the Pontryagin-Thom construction. Moreover, when the highest-dimensional manifold has a normal bundle with a framed direction, we find spaces homotopy equivalent to the nested Pontryagin-Thom spaces that relate nested manifolds up to cobordism with links up to cobordism. This gives rise to nested cobordism invariants coming from previously studied cobordism invariants of links. In addition, we provide an alternative proof of a result by Wall about the splitting of the stable nested cobordism groups. + oai:arXiv.org:2512.18277v1 + math.AT + math.GT + Tue, 23 Dec 2025 00:00:00 -0500 new http://creativecommons.org/licenses/by/4.0/ - Shakir Ali, Atif Ahmad Khan, Abhishek Kesarwani, Susanta Samanta + Alba Send\'on Blanco - Quadratic Embedding Constants of Corona Graphs - https://arxiv.org/abs/2512.17258 - arXiv:2512.17258v1 Announce Type: new -Abstract: The quadratic embedding constant (QEC) of a connected graph is defined to be the maximum of the quadratic function associated with its distance matrix on a certain unit sphere of codimension two. In this paper we derive a formula for the QEC of a corona graph $G\odot H$. It is shown that $\mathrm{QEC}(G\odot H)=\psi_{H*}^{-1}(\mathrm{QEC}(G))$ holds under some spectral assumptions on $H$, where $\psi_{H*}^{-1}$ is the inverse function of the most right branch of the analytic function $\psi_H$ defined by means of the main eigenvalues of the adjacency matrix of $H$. Moreover, if $H$ is a regular graph of which the adjacency matrix has the smallest eigenvalue $-2$, then the formula is written down explicitly. - oai:arXiv.org:2512.17258v1 - math.CO - Mon, 22 Dec 2025 00:00:00 -0500 + Synchronization by degenerate noise + https://arxiv.org/abs/2512.18278 + arXiv:2512.18278v1 Announce Type: new +Abstract: In this paper, we derive several criteria for (weak) synchronization by noise without the global swift transitivity property. Our sufficient conditions for (weak) synchronization are necessary and can be applied to scenarios involving degenerate or non-Gaussian noise. These results partially answer the open question posed by Flandoli et al. (Probab Theory Relat Fields 168:511-556, 2017). As an application, we prove that the weak attractor for stochastic Lorenz 63 systems driven by degenerate noise consists of a single random point provided the noise intensity is small, and there is no weak synchronization if the noise intensity is large. This indicates that a bifurcation occurs in relation to the intensity of the noise. + oai:arXiv.org:2512.18278v1 + math.DS + math.PR + Tue, 23 Dec 2025 00:00:00 -0500 new - http://creativecommons.org/licenses/by/4.0/ - Ferdi, Edy Tri Baskoro, Nobuaki Obata, Aditya Purwa Santika + http://creativecommons.org/publicdomain/zero/1.0/ + Xianming Liu, Xu Sun - Centrally pure C*-algebras - https://arxiv.org/abs/2512.17261 - arXiv:2512.17261v1 Announce Type: new -Abstract: We show that a separable C*-algebra $A$ is $\mathcal{Z}$-stable if and only if its uncorrected central sequence algebra $A' \cap A_{\mathcal{U}}$ is pure, if and only if Kirchberg's central sequence algebra $F(A)$ is pure. - More generally, we show that a C*-algebra $A$ is separably $\mathcal{Z}$-stable if and only if the relative central sequence algebra $B' \cap A_{\mathcal{U}}$ is pure for every separable subalgebra $B \subseteq A_{\mathcal{U}}$. - oai:arXiv.org:2512.17261v1 - math.OA - Mon, 22 Dec 2025 00:00:00 -0500 + Generalized Harmonic Numbers: Identities and Properties + https://arxiv.org/abs/2512.18282 + arXiv:2512.18282v1 Announce Type: new +Abstract: This paper builds on the research initiated by Boyadzhiev, but introduces generalized harmonic numbers, \[ H_n(\alpha)= \sum_{k=1}^n \frac{\alpha^{k}}{k}, \] which enable the derivation of new identities as well as the reformulation of existing ones. We also generalize Gould's identity, allowing classical harmonic numbers to be replaced by their generalized counterparts. Our results contribute to a deeper understanding of the structural properties of these numbers and highlight the effectiveness of elementary techniques in uncovering new mathematical phenomena. In particular, we recover several known identities for generalized harmonic numbers and establish new ones, including identities involving generalized harmonic numbers together with Fibonacci numbers, Laguerre polynomials, and related sequences. + oai:arXiv.org:2512.18282v1 + math.GM + Tue, 23 Dec 2025 00:00:00 -0500 new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Francesc Perera, Hannes Thiel, Eduard Vilalta + http://creativecommons.org/licenses/by/4.0/ + Roberto Sanchez Peregrino - Estimates, asymptotics and trace formulas for periodic vector NLS equations, II - https://arxiv.org/abs/2512.17272 - arXiv:2512.17272v1 Announce Type: new -Abstract: We consider a first order operator with a smooth periodic 3x3 matrix potential on the real line. It is the Lax operator for the periodic vector NLS equation. Its spectrum covers the real line and it is union of the spectral bands of multiplicity 3, separated by intervals (gaps) of multiplicity 1. We prove and describe the following: - \\ $\cdot$ The geometry of the Riemann surface and its branch points. \\ $\cdot$ The asymptotics of branch points are determined and they are real at high energy. \\ $\cdot$ Trace formulas for integral of motions, including the Hamiltonian of the NLS equation. \\ $\cdot$ Estimates of the Hamiltonian in terms of gap lengths. - The proof is based on the analysis of averaged quasi-momentum as a conformal mapping of the upper half plane on the domain on the upper half plane and on the asymptotics of the monodromy matrix and multipliers at high energy. - oai:arXiv.org:2512.17272v1 - math-ph - math.MP - Mon, 22 Dec 2025 00:00:00 -0500 + Sharp large deviation estimates for Gaussian extrema + https://arxiv.org/abs/2512.18297 + arXiv:2512.18297v1 Announce Type: new +Abstract: We establish sharp large-deviation asymptotic estimates for the maximum order statistic of i.i.d.\ standard normal random variables on all Borel subsets of the positive real line. This result yields more accurate tail approximations than the classical Gumbel limit. + oai:arXiv.org:2512.18297v1 + math.PR + Tue, 23 Dec 2025 00:00:00 -0500 new http://creativecommons.org/licenses/by/4.0/ - Evgeny Korotyaev + Jos\'e M. Zapata - Achievement sets -- current results and open problems - https://arxiv.org/abs/2512.17285 - arXiv:2512.17285v1 Announce Type: new -Abstract: We survey recent developments in the theory of achievement sets and present a substantial collection of open problems. - oai:arXiv.org:2512.17285v1 - math.CA - Mon, 22 Dec 2025 00:00:00 -0500 + Sampling elements of a finite group: efficiency of the product replacement algorithm with an accumulator + https://arxiv.org/abs/2512.18302 + arXiv:2512.18302v1 Announce Type: new +Abstract: Let $G$ be a finite group generated by $k$ elements. The well-known product replacement algorithm provides an effective method for sampling generating sets of $G$. We study a refinement of this algorithm that is designed to output individual elements of $G$. We show that after $O(k^2\log|G|)$ steps, the distribution of the output is close to uniform on $G$, which improves upon the best results known to date. The proof proceeds via spectral gap estimates and uses computer assisted calculations. + oai:arXiv.org:2512.18302v1 + math.GR + math.OA + math.PR + Tue, 23 Dec 2025 00:00:00 -0500 new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Szymon G{\l}\k{a}b, Franciszek Prus-Wi\'sniowski + http://creativecommons.org/licenses/by/4.0/ + Micha{\l} Marcinkowski, Piotr Mizerka - A representation for the integral kernel of the composition of multivariate Bernstein-Durrmeyer operators - https://arxiv.org/abs/2512.17297 - arXiv:2512.17297v1 Announce Type: new -Abstract: This paper presents a representation for the kernel of the composition of multivariate Bernstein-Durrmeyer operators for functions defined on the standard simplex in $\mathbb{R}^d$. - oai:arXiv.org:2512.17297v1 - math.CA - Mon, 22 Dec 2025 00:00:00 -0500 + The chiral gyrating H'-T surface family: construction from the dual qtz--qzd nets and existence proof using a toroidal Weierstrass method + https://arxiv.org/abs/2512.18308 + arXiv:2512.18308v1 Announce Type: new +Abstract: This paper provides a construction and existence proof for a 1-parameter family of chiral unbalanced triply-periodic minimal surfaces of genus 4. We name these {\textit{gyrating H'-T} surfaces, because they are related to Schoen's H'-T surfaces in a similar way as the Gyroid is to the Primitive surface. Their chirality is manifest in a screw symmetry of order six. The two labyrinthine domains on either side of the surface are not congruent, rather one representing the quartz net (\texttt{qtz}) and the other one the dual of the quartz net (\texttt{qzd}). The family tends to the Scherk saddle tower in one limit and to the doubly periodic Scherk surface in the other. The motivation for the construction was to construct a chiral tunable unbalanced surface family, originally as a template for photonic materials. The numeric construction is based on reverse-engineering of the tubular surface of two suitably chosen dual nets, using the \textit{Surface Evolver}} to minimize area or curvature variations. The existence is proved using Weierstrass parametrizations defined on the branched torus. + oai:arXiv.org:2512.18308v1 + math.DG + cond-mat.mtrl-sci + Tue, 23 Dec 2025 00:00:00 -0500 new http://creativecommons.org/licenses/by/4.0/ - Ulrich Abel, Ana Maria Acu, Margareta Heilmann, Ioan Rasa + Hao Chen, Shashank G. Markande, Matthias Saba, Gerd E. Schr\"oder-Turk, Elisabetta A. Matsumoto - Euler-Maruyama method for distribution dependent stochastic differential equation driven by multiplicative fractional Brownian motion - https://arxiv.org/abs/2512.17300 - arXiv:2512.17300v1 Announce Type: new -Abstract: In this paper, we establish the propagation of chaos and Euler-Maruyama method of DDSDE driven by multiplicative fractional Brownian motion with Hurst parameter $H\in (\frac{\sqrt{5}-1}{2},1)$. We have not only obtained an upper bound for the error of the Euler-Maruyama method but also verified the correctness of this result via systematic numerical simulation experiments. - oai:arXiv.org:2512.17300v1 + From entropic constraints to reinforced processes: a probabilistic origin of multiscale measures + https://arxiv.org/abs/2512.18313 + arXiv:2512.18313v1 Announce Type: new +Abstract: We investigate multiscale Gibbs measures from a variational and probabilistic viewpoint, focusing on the structural asymmetry among conditional entropies that characterizes their construction. We show how this asymmetry emerges both from variational principles with entropic constraints and from stochastic processes with reinforcement. We thus introduce the reinforced multinomial process and prove a large-deviation principle for its empirical histogram. The associated rate function reproduces precisely the entropy imbalance defining multiscale measures, thereby providing a genuine probabilistic mechanism for their emergence. The reinforced multinomial process thus offers a simple and rigorous stochastic foundation for multiscale Gibbs structures. + oai:arXiv.org:2512.18313v1 + math-ph + cond-mat.stat-mech + math.MP math.PR - Mon, 22 Dec 2025 00:00:00 -0500 + Tue, 23 Dec 2025 00:00:00 -0500 new - http://creativecommons.org/licenses/by/4.0/ - Guangjun Shen, Jiangpeng Wang, Xuekang Zhang + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Francesco Camilli, Pierluigi Contucci, Emanuele Mingione - Circular orders: Topology and continuous actions - https://arxiv.org/abs/2512.17314 - arXiv:2512.17314v1 Announce Type: new -Abstract: We study the topology of abstract circularly ordered sets. While the algebraic notion is classical, the general topological theory has received comparatively little attention. - This work provides a self-contained topological exposition and presents several new directions and results. Specifically, we: - Initiate a systematic study of Generalized Circularly Ordered Topological Spaces (GCOTS). - Analyze in detail Nov\'ak's regular completion and prove that it is the canonical minimal circularly ordered compactification. Provide a convex uniform structure description of circularly ordered compactifications. This implies several new results in the theory of compactifications for topological group actions. Reexamine functions of Bounded Variation on abstract circularly ordered sets and prove generalizations of Helly's selection theorem (for circular and linear orders). - These developments and a systematic analysis of circular order topologies are motivated also by recent applications in topological dynamics, particularly in joint works with E. Glasner, which demonstrate that circularly ordered dynamical systems provide a natural class of ``tame" dynamics. - oai:arXiv.org:2512.17314v1 - math.GN - math.DS + A topological characterization of indecomposable sets of finite perimeter + https://arxiv.org/abs/2512.18319 + arXiv:2512.18319v1 Announce Type: new +Abstract: We prove that a set of finite perimeter is indecomposable if and only if it is, up to a choice of suitable representative, connected in the 1-fine topology. This gives a topological characterization of indecomposability which is new even in Euclidean spaces. Our approach relies crucially on the metric space theory of functions of bounded variation, and we are able to prove our main result in a complete, doubling metric measure space supporting a $1$-Poincar\'{e} inequality and having the two-sidedness property (this class includes all Riemannian manifolds, Carnot groups, and ${\sf RCD}(K,N)$ spaces with $K\in\mathbb R$ and $N<\infty$). As an immediate corollary, we obtain an alternative proof of the decomposition theorem for sets of finite perimeter into maximal indecomposable components. + oai:arXiv.org:2512.18319v1 + math.MG math.FA - Mon, 22 Dec 2025 00:00:00 -0500 + Tue, 23 Dec 2025 00:00:00 -0500 new http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Michael Megrelishvili + Paolo Bonicatto, Panu Lahti, Enrico Pasqualetto - Nowhere-zero flow reconfiguration - https://arxiv.org/abs/2512.17342 - arXiv:2512.17342v1 Announce Type: new -Abstract: We initiate the study of nowhere-zero flow reconfiguration. The natural question is whether any two nowhere-zero $k$-flows of a given graph $G$ are connected by a sequence of nowhere-zero $k$-flows of $G$, such that any two consecutive flows in the sequence differ only on a cycle of $G$. We conjecture that any two nowhere-zero 5-flows in any 2-edge-connected graph are connected in this way. This can be seen as a reconfiguration variant of Tutte's 5-flow conjecture. - We study this problem in the setting of integer flows and group flows, and show that the structure of groups affects the answer, contrary to the existence of nowhere-zero flows. We also highlight a duality with recoloring in planar graphs and deduce that any two nowhere-zero 7-flows in a planar graph are connected, among other results. Finally we show that for any graph $G$, there is an abelian group $A$ such that all nowhere-zero $A$-flows in $G$ are connected, which is a weak form of our original conjecture. We conclude with several problems and conjectures. - oai:arXiv.org:2512.17342v1 - math.CO - cs.DM - Mon, 22 Dec 2025 00:00:00 -0500 + Energy Bounds for Kantorovich Transport Distances with Convex Cost Functions + https://arxiv.org/abs/2512.18324 + arXiv:2512.18324v1 Announce Type: new +Abstract: Energy bounds for Kantorovich transport distances are developed for convex cost functions. The main results extend estimates due to M. Ledoux for the Kantorovich distances $W_p$. + oai:arXiv.org:2512.18324v1 + math.FA + math.PR + Tue, 23 Dec 2025 00:00:00 -0500 new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Louis Esperet, Aur\'elie Lagoutte, Margaux Marseloo + http://creativecommons.org/licenses/by/4.0/ + Sergey Bobkov, Friedrich G\"otze - Endomorphisms of singular del Pezzo surfaces - https://arxiv.org/abs/2512.17345 - arXiv:2512.17345v1 Announce Type: new -Abstract: A natural problem of algebraic dynamics is to classify the complex projective varieties that admit an endomorphism of degree greater than 1. Joshi solved the problem for all canonical del Pezzo surfaces with Picard number 1 except one, a surface with a du Val singularity of type $E_8$. The method of Bott vanishing does not resolve this case. - We show here that the $E_8$ surface has no endomorphism of degree greater than 1. For the proof, we extend the method of Amerik-Rovinsky-Van de Ven, involving Chern number inequalities, from varieties to Deligne-Mumford stacks. This approach should be useful for other hard cases in the classification of varieties with endomorphisms. - oai:arXiv.org:2512.17345v1 - math.AG - Mon, 22 Dec 2025 00:00:00 -0500 + Model Theory of Generic Vector Space Endomorphisms II + https://arxiv.org/abs/2512.18327 + arXiv:2512.18327v1 Announce Type: new +Abstract: This paper further studies the model companion of an endomorphism acting on a vector space, possibly with extra structure. Given a theory $T$ that $\varnothing$-defines an infinite $K$-vector space $\mathbb{V}$ in every model, we set $T_\theta := T \cup \{\text{``$\theta$ defines a $K$-endomorphism of $\mathbb{V}$"}\}$. We previously defined a family $\{T^C_\theta : C \in \mathcal{C}\}$ of extensions of $T_\theta$ which parameterizes all consistent extensions of the form $$ + T_\theta \cup \left\{\sum\nolimits_{k}\bigcap\nolimits_{l}\operatorname{Ker}(\rho_{j, k, l}[\theta]) = \sum\nolimits_{k}\bigcap\nolimits_{l} \operatorname{Ker}(\eta_{j, k, l}[\theta]) : j \in \mathcal{J}\right\}, $$ where all sums and intersections are finite, and all the $\rho[\theta]$'s and $\eta[\theta]$'s are polynomials over $K$ with $\theta$ plugged in. Notice that properties such as $\theta^2 - 2\operatorname{Id} = 0$ or ``$\rho[\theta]$ is injective for every $\rho \in K[X] \setminus \{0\}$" can be expressed in such a manner. We also presented a sufficient condition which implies that every $T^C_\theta$ has a model companion $T\theta^C$. Under this condition, we characterize all definable sets in $T\theta^C$ and use this to study the completions of $T\theta^C$, as well as the algebraic closure. If $T$ is o-minimal and extends $\operatorname{Th}(\mathbb{R}, <)$, we prove that $T\theta^C$ has o-minimal open core. + oai:arXiv.org:2512.18327v1 + math.LO + Tue, 23 Dec 2025 00:00:00 -0500 new http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Burt Totaro + Leon Chini - Unknown input observer design for a class of coupled wave PDE systems - https://arxiv.org/abs/2512.17353 - arXiv:2512.17353v1 Announce Type: new -Abstract: This paper deals with the problem of designing unknown input observers for a class of coupled semilinear wave partial differential equations (PDE) systems. A state observer is designed to estimate the uncertain coupled wave PDE systems. Then, the analysis of the asymptotic stability and $H_{\infty}$ performance for the observer design of coupled wave PDE systems is investigated. Some sufficient conditions of asymptotic stability for the observer error system with disturbance attenuation level are derived via matrix inequalities based on the Lyapunov stability theory. Finally, a numerical simulation is presented to demonstrate the effectiveness of the obtained result. - oai:arXiv.org:2512.17353v1 + Learning Generalized Nash Equilibria in Non-Monotone Games with Quadratic Costs + https://arxiv.org/abs/2512.18330 + arXiv:2512.18330v1 Announce Type: new +Abstract: We study generalized Nash equilibrium (GNE) problems in games with quadratic costs and individual linear equality constraints. Departing from approaches that require strong monotonicity and/or shared constraints, we reformulate the KKT conditions of the (generally non-monotone) games into a tractable convex program whose objective satisfies the Polyak-Lojasiewicz (PL) condition. This PL geometry enables a distributed gradient method over a fixed communication graph with global geometric (linear) convergence to a GNE. When gradient information is unavailable or costly, we further develop a zero-order fully distributed scheme in which each player uses only local cost evaluations and their own constraint residuals. With an appropriate step size policy, the proposed zero-order method converges to a GNE, provided one exists, at rate O(1/t). + oai:arXiv.org:2512.18330v1 math.OC - Mon, 22 Dec 2025 00:00:00 -0500 + Tue, 23 Dec 2025 00:00:00 -0500 new - http://creativecommons.org/licenses/by-nc-sa/4.0/ - Najmeh Ghaderi, Birgit Jacob + http://creativecommons.org/licenses/by/4.0/ + Tatiana Tatarenko, Lucas Wey Hacker - False detection rate control in time series coincidence detection - https://arxiv.org/abs/2512.17372 - arXiv:2512.17372v1 Announce Type: new -Abstract: We study the problem of coincidence detection in time series data, where we aim to determine whether the appearance of simultaneous or near-simultaneous events in two time series is indicative of some shared underlying signal or synchronicity, or might simply be due to random chance. This problem arises across many applications, such as astrophysics (e.g., detecting astrophysical events such as gravitational waves, with two or more detectors) and neuroscience (e.g., detecting synchronous firing patterns between two or more neurons). In this work, we consider methods based on time-shifting, where the timeline of one data stream is randomly shifted relative to another, to mimic the types of coincidences that could occur by random chance. Our theoretical results establish rigorous finite-sample guarantees controlling the probability of false positives, under weak assumptions that allow for dependence within the time series data, providing reassurance that time-shifting methods are a reliable tool for inference in this setting. Empirical results with simulated and real data validate the strong performance of time-shifting methods in dependent-data settings. - oai:arXiv.org:2512.17372v1 - math.ST - stat.TH - Mon, 22 Dec 2025 00:00:00 -0500 + Implementing Transport Coding in OMNeT++ for Message Delay Reduction + https://arxiv.org/abs/2512.18332 + arXiv:2512.18332v1 Announce Type: new +Abstract: Transport coding reduces message delay in packet-switched networks by introducing controlled redundancy at the transport layer: $k$ original packets are encoded into $n\ge k$ coded packets, and the message is reconstructed after the first $k$ successful deliveries, effectively shifting latency from the maximum packet delay to the $k$-th order statistic. We present a concise, reproducible discrete-event implementation of transport coding in OMNeT++, including a multi-hop Kleinrock-type network, FIFO queues, exponential service and link delays, and explicit receiver-side reconstruction that records message delay and deadline violations. Using paired uncoded ($n{=}k$) and coded ($n{>}k$) configurations at the same message generation rate, we compare delay, reliability, and saturation effects across code rates and input loads. Simulation results show consistent reductions of average delay and late-delivery probability for moderate redundancy, while keeping the saturation throughput close to the uncoded baseline. The proposed model provides a transparent bridge between analytical transport-coding formulas and executable simulation for tuning redundancy in low-latency services. + oai:arXiv.org:2512.18332v1 + cs.IT + cs.NI + math.IT + Tue, 23 Dec 2025 00:00:00 -0500 new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Ruiting Liang, Samuel Dyson, Rina Foygel Barber, Daniel E. Holz + http://creativecommons.org/licenses/by-nc-nd/4.0/ + Ilya Petrovanov, Anton Sergeev - Sobolev Algorithm for Local Smoothness Analysis (SALSA) via Sharp Direct and Inverse Statements - https://arxiv.org/abs/2512.17377 - arXiv:2512.17377v1 Announce Type: new -Abstract: We extend sharp direct and inverse approximation statements for kernel-based methods for finitely smooth kernels, i.e. those whose native spaces are norm-equivalent to Sobolev spaces. In particular, our inverse results are now formulated for a broad class of approximation schemes beyond interpolation, extending existing theory. Building on these results, we propose a novel Sobolev Algorithm for Local Smoothness Analysis (SALSA) for detecting local smoothness properties of target data, including their degree of smoothness and non-smoothness. The method is rigorously grounded based on the sharp direct and inverse statements. Numerical experiments in various settings highlight the effectiveness of the proposed algorithm. - oai:arXiv.org:2512.17377v1 - math.NA - cs.NA - Mon, 22 Dec 2025 00:00:00 -0500 + Lorentz Invariance of the Multidimensional Dirac-Hestenes Equation + https://arxiv.org/abs/2512.18347 + arXiv:2512.18347v1 Announce Type: new +Abstract: This paper investigates the Lorentz invariance of the multidimensional Dirac-Hestenes equation, that is, whether the equation remains form-invariant under pseudo-orthogonal transformations of the coordinates. We examine two distinct approaches: the tensor formulation and the spinor formulation. We first present a detailed examination of the four-dimensional Dirac-Hestenes equation, comparing both transformation approaches. These results are subsequently generalized to the multidimensional case with (1,n) signature. The tensor approach requires explicit invariants, while the spinor formulation naturally maintains Lorentz covariance through spin group action. + oai:arXiv.org:2512.18347v1 + math-ph + math.MP + Tue, 23 Dec 2025 00:00:00 -0500 new http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Sara Avesani, Leevan Ling, Francesco Marchetti, Tizian Wenzel + 10.1007/s00006-025-01418-2 + Advances in Applied Clifford Algebras, 36 (2026), 5, 20 pp + S. V. Rumyantseva, D. S. Shirokov - On the membership of two-variable Rational Inner Functions in spaces of Dirichlet-type - https://arxiv.org/abs/2512.17388 - arXiv:2512.17388v1 Announce Type: new -Abstract: We study membership of rational inner functions on the bidisk $\mathbb{D}^2$ in a scale of Dirichlet spaces considered by Bera, Chavan, and Ghara, and in higher-order variants of these spaces. We give a characterization for membership in terms of the geometric concept of contact order of a rational inner function at its singular points, and we further record some consequences and variants of our main result. - oai:arXiv.org:2512.17388v1 - math.CV - Mon, 22 Dec 2025 00:00:00 -0500 + Modified Quasi-Newton Method for Nonconvex Multiobjective Optimization Problems with Barzilai-Borwein diagonal matrix + https://arxiv.org/abs/2512.18348 + arXiv:2512.18348v1 Announce Type: new +Abstract: This paper addresses the challenge of developing efficient algorithms for large-scale nonconvex multiobjective optimization problems (MOPs). While quasi-Newton methods are effective, their traditional application to MOPs is computationally expensive as they require maintaining and inverting separate Hessian approximations for each objective function. To overcome this limitation, we propose a novel Barzilai-Borwein diagonal-type Quasi-Newton method (BB-DQN). Our key innovation is the use of a single, shared, and modified BB-type matrix, updated iteratively using function and gradient information, to approximate the Hessians of all objectives simultaneously. We theoretically demonstrate that this approximation matrix remains positive definite throughout the iterative process. Furthermore, we establish the global convergence of the BB-DQN method without convexity assumptions and prove its R-linear convergence under mild conditions. Numerical experiments on a diverse set of test problems confirm that BB-DQN outperforms existing methods like M-BFGSMO, achieving superior performance in terms of computational time, iteration count, and reliability, especially for large-scale instances. + oai:arXiv.org:2512.18348v1 + math.OC + Tue, 23 Dec 2025 00:00:00 -0500 new - http://creativecommons.org/licenses/by/4.0/ - Athanasios Beslikas, Alan Sola + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Hua Liu - Minimizing numerical radius of weighted cyclic matrices under permutation of the weights - https://arxiv.org/abs/2512.17399 - arXiv:2512.17399v1 Announce Type: new -Abstract: In this article we answer a question asked by Chien et al. in arXiv:2304.06050 in which they study the numerical range of weighted cyclic matrices under permutation of their entries. Namely, we are interested in how $w(A_\sigma)$ fluctuates for various permutations $\sigma\in S_n$ and fixed $0\leq a_1<\cdots<a_n$ with $A_\sigma=\begin{pmatrix} 0&a_{\sigma(1)}&{}&{}&{}\cr {}&0&a_{\sigma(2)}&{}&{}\cr {}&{}&\ddots&\ddots&{}\cr {}&{}&{}&\ddots&a_{\sigma(n-1)}\cr a_{\sigma(n)}&{}&{}&{}&0 \end{pmatrix}$. Previous results of Gau \cite{gau2024proof} and Chang and Wang \cite{chang2012maximizing} made clear the case when $w(A_\sigma)$ is maximal among all the $w(A_\mu)$ with $\mu\in S_n$. Chien et al. in arXiv:2304.06050 ask what the permutation which makes $w(A_\sigma)$ minimal for $n\geq 6$ could be. Answering this question is the aim of this note. - oai:arXiv.org:2512.17399v1 - math.FA - Mon, 22 Dec 2025 00:00:00 -0500 + Know Your Rank! + https://arxiv.org/abs/2512.18349 + arXiv:2512.18349v1 Announce Type: new +Abstract: We study definable ranks of ordered fields, ordered abelian groups, and linear orders. For an arbitrary linear order $\Gamma$, we construct an ordered abelian group $G$ with archimedian spine $\Gamma$ and an ordered field $K$ with natural value group $G$ such that the definable ranks of $K$, $G$ and $\Gamma$ are all isomorphic. This answers a question of Krapp, Kuhlmann, and the second author. + oai:arXiv.org:2512.18349v1 + math.LO + Tue, 23 Dec 2025 00:00:00 -0500 new - http://creativecommons.org/licenses/by/4.0/ - Simon Marionnet + http://creativecommons.org/licenses/by-nc-nd/4.0/ + Blaise Boissonneau, Lasse Vogel - First Eigenvalue and Torsional Rigidity: Isoperimetric Inequalities for the Fractional Laplacian - https://arxiv.org/abs/2512.17400 - arXiv:2512.17400v1 Announce Type: new -Abstract: We present a fractional counterpart of a generalized Kohler-Jobin inequality, showing that, among all bounded, open sets $\Omega\subset \mathbb{R}^N$ with Lipschitz boundary, having the same fractional torsional rigidity, the first Dirichlet eigenvalue $\lambda_1(\Omega)$ of the fractional Laplacian attains its minimum on balls. With the same arguments we also establish a reverse H\"older inequality for an eigenfunction corresponding to $\lambda_1(\Omega)$. - oai:arXiv.org:2512.17400v1 + On the sharp multi-bubble stability for fractional Hardy-Sobolev equations -- A quantitative approach in low dimensions + https://arxiv.org/abs/2512.18350 + arXiv:2512.18350v1 Announce Type: new +Abstract: We establish sharp quantitative multi-bubble stability for non-sign-changing critical points of the fractional Hardy-Sobolev inequality in the low-dimensional regime $2s<N<6s-2t$. For functions whose energy is close to that of a finite superposition of bubbles, we prove that the Euler-Lagrange deficit controls linearly the distance, in the homogeneous fractional Sobolev norm, to the multi-bubble manifold, and we recover the precise bubble configuration. This yields quantitative rigidity under arbitrary finite weak interactions. The proof combines a localization scheme adapted to the Hardy weight, weighted fractional Kato-Ponce commutator estimates, a bubble-wise spectral gap inequality, and a sharp interaction analysis. We also show that the linear rate is optimal by constructing a matching counterexample. + oai:arXiv.org:2512.18350v1 math.AP - Mon, 22 Dec 2025 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Barbara Brandolini, Ida de Bonis, Vincenzo Ferone, Gianpaolo Piscitelli, Bruno Volzone - - - Common positive stabilisation of open book decompositions - https://arxiv.org/abs/2512.17402 - arXiv:2512.17402v1 Announce Type: new -Abstract: The Giroux Correspondence states that two open book decompositions supporting the same contact structure are related by a sequence of positive open book stabilisations and destabilisations. In this note we show that any two open book decompositions supporting isotopic contact structures admit a common positive stabilisation. - oai:arXiv.org:2512.17402v1 - math.GT - Mon, 22 Dec 2025 00:00:00 -0500 + Tue, 23 Dec 2025 00:00:00 -0500 new http://creativecommons.org/licenses/by/4.0/ - Joan Licata, Vera V\'ertesi - - - Graph Isomorphism: Mixed-Integer Convex Optimization from First-Order Methods - https://arxiv.org/abs/2512.17417 - arXiv:2512.17417v1 Announce Type: new -Abstract: The graph isomorphism (GI) problem, which asks whether two graphs are structurally identical, occupies a unique position in computational complexity -- it is neither known to be solvable in polynomial time, nor proven to be NP-complete. We propose a convex mixed-integer formulation of the problem and leverage first-order convex optimization to tackle it, following a stream of recent work on optimization-driven graph isomorphism detection. We strengthen our formulation with variable fixing techniques that prove highly effective while preserving the polyhedral structure. We perform extensive computations evaluating the performance of different families of methods including a mixed-integer convex formulation, mixed-integer linear optimization, local search and spectral heuristics over a collection of challenging GI instances. We find that a high level of symmetry is beneficial for optimization-based methods. On the other hand, presolving techniques that detect local substructures to fix variables are crucial for asymmetric instances. The proposed method outperforms the second best approach, the integer feasibility approach, on 6 of the 12 graphs families and is on par with it on symmetric families. - oai:arXiv.org:2512.17417v1 - math.OC - Mon, 22 Dec 2025 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Wenjie Xiao, Mathieu Besan\c{c}on, Patrick Gel{\ss}, Deborah Hendrych, Stefan Klus, Sebastian Pokutta + Souptik Chakraborty, Utsab Sarkar - Large deviation principle for the absorption time of the Beta-coalescent via integral functionals - https://arxiv.org/abs/2512.17418 - arXiv:2512.17418v1 Announce Type: new -Abstract: We study some aspects of the absorption time of the Beta$(a,b)$-Coalescent starting with $n$ blocks. More precisely, when $a>1$, the absorption time is known to converge to infinity as $n$ goes to infinity, and we prove that it satisfies a large deviation principle. When $a \in (0,1)$, it is known that the coalescent comes down from infinity, and we derive bounds for the convergence in the Kolmogorov distance of the distribution of the absorption time as $n$ goes to infinity. To prove our results we introduce a method, inspired from statistical mechanics, that allows to infer the asymptotic behavior of the Laplace transforms of some integral functionals of the Beta-coalescent as the initial number of blocks $n$ goes to infinity. As a by-product of our proofs we also obtain estimates for the record probabilities of the Beta-coalescent. - oai:arXiv.org:2512.17418v1 + A note on the planar Skorokhod embedding problem + https://arxiv.org/abs/2512.18353 + arXiv:2512.18353v1 Announce Type: new +Abstract: The planar Skorokhod embedding problem was first proposed and solved by R. Gross in 2019 [#gross2019]. Gross worked with probability distributions having finite second moment. In [#boudabra2019remarks, #Boudabra2020], the solutions extended to all distributions with a finite $p^{th}$ moment for $p>1$. The case $p=1$ remained uncovered since then. In this note we show that the planar Skorokhod embedding problem is solvable for $p=1$ when the Hilbert transform of its quantile function is integrable, effectively closing this line of investigation. + oai:arXiv.org:2512.18353v1 math.PR - Mon, 22 Dec 2025 00:00:00 -0500 + Tue, 23 Dec 2025 00:00:00 -0500 new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Gr\'egoire V\'echambre + http://creativecommons.org/licenses/by/4.0/ + Maher Boudabra - Variational Dissipative Mechanics on Lie Algebroids - https://arxiv.org/abs/2512.17424 - arXiv:2512.17424v1 Announce Type: new -Abstract: We formulate a Herglotz-type variational principle on a Lie algebroid and derive the corresponding Euler--Lagrange--Herglotz equations for a Lagrangian depending on an additional scalar variable $z$. This provides a geometric framework for dissipative systems on Lie algebroids and recovers, as special cases, the classical Euler--Lagrange--Herglotz equations on tangent bundles, the Euler--Poincar\'e--Herglotz equations on a Lie algebra, and the Lagrange--Poincar\'e--Herglotz equations on Atiyah algebroids of principal bundles. Starting from the local formulation, we then use Lie algebroid connections to obtain a coordinate-free Euler--Lagrange--Poincar\'e--Herglotz and Hamilton--Pontryagin--Herglotz theory. Finally, we establish energy balance laws and Noether--Herglotz-type results, in which classical conserved quantities are replaced by dissipated invariants. - oai:arXiv.org:2512.17424v1 - math-ph - math.DG - math.DS - math.MP - Mon, 22 Dec 2025 00:00:00 -0500 + Ground states and phase transitions for an aggregation model with fast diffusion on sphere + https://arxiv.org/abs/2512.18358 + arXiv:2512.18358v1 Announce Type: new +Abstract: We consider a free energy on the sphere that contains an entropy associated to nonlinear fast diffusion, and a nonlocal interaction energy. The two components of the free energy compete with each other, as one favours spreading and the other promotes concentration, respectively. The model is a generalization of the Onsager free energy with dipolar potential, used to study polymer orientation. We study the global energy minimizers of the energy functional, and in particular the various phase transitions that occur with respect to the strength of the nonlocal attractive interactions. In the considered regime, diffusion reduces as the density increases, for which reason the global energy minimizers can contain Dirac mass concentrations. We identify various ranges of the fast diffusion exponent and of the interaction strength, which give qualitatively different equilibria and ground states. The theoretical results are supported by numerical illustrations. + oai:arXiv.org:2512.18358v1 + math.AP + Tue, 23 Dec 2025 00:00:00 -0500 new http://creativecommons.org/licenses/by/4.0/ - Alexandre Anahory Simoes, Leonardo Colombo + Razvan C. Fetecau, Hansol Park - On the subcritical Lane-Emden equation on Riemannian models with polynomial volume growth - https://arxiv.org/abs/2512.17428 - arXiv:2512.17428v1 Announce Type: new -Abstract: We focus on the problems of existence and non-existence of positive solutions for the Sobolev-subcritical Lane-Emden equation on certain Riemannian manifolds (mainly models) with asymptotically negative curvature, which, from the viewpoint of the volume growth of geodesic balls, can be regarded as intermediate settings between the Euclidean and the hyperbolic spaces. - A number of interesting phenomena arise: the subcritical regime naturally divides into three further ranges, characterized by existence phenomena (slightly subcritical), non-existence phenomena (strongly subcritical), and by a mixed behavior where existence and non-existence strongly depend on additional assumptions on the manifold (intermediate). In the intermediate regime, we further show that the radial homogeneous Dirichlet problem in geodesics balls may admit multiple positive solutions, thereby revealing substantial differences with respect to both the Euclidean and the hyperbolic settings. - oai:arXiv.org:2512.17428v1 - math.AP - math.DG - Mon, 22 Dec 2025 00:00:00 -0500 + Downlink Power Allocation for STAR-RIS-Assisted Cell-Free Massive MIMO with Multi-antenna Users + https://arxiv.org/abs/2512.18359 + arXiv:2512.18359v1 Announce Type: new +Abstract: This paper investigates the downlink power allocation of the simultaneous transmitting and reflecting reconfigurable intelligent surface (STAR-RIS)-assisted cell-free massive multiple-input multiple-output (MIMO) system with multi-antenna users. We introduce downlink spectral efficiency (SE) and derive novel closed-form SE expressions using linear minimum mean squared error (MMSE) detectors. We also address the downlink power allocation via a sum SE maximization problem framed within an alternating direction method of multipliers (ADMM)-based fractional programming (FP) algorithm. Numerical results demonstrate that systems utilizing multi-antenna users significantly enhance SE, achieving at least a 20% SE increase as the number of antennas increases from one to six. Additionally, our proposed ADMM-based FP algorithm outperforms existing fractional power control approaches, yielding a more than 20% SE increase. These results highlight the necessity for adopting multi-antenna users and efficient power allocation algorithms in STAR-RIS-assisted cell-free massive MIMO systems. + oai:arXiv.org:2512.18359v1 + cs.IT + math.IT + Tue, 23 Dec 2025 00:00:00 -0500 new http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Alessandra De Luca, Matteo Muratori, Nicola Soave + Jun Qian, Ross Murch, Khaled B. Letaief - Qi's problems on classifications of third- and fourth-order symmetric tensors by eigenvalues - https://arxiv.org/abs/2512.17431 - arXiv:2512.17431v1 Announce Type: new -Abstract: This paper addresses two fundamental problems posed by Qi regarding the sufficiency of eigenvalues for the classification of symmetric tensors in the two-dimensional setting. For $2\times2\times2$ and $2\times2\times2\times2$ complex symmetric tensors, we establish their complete set of equivalence classes via a one-to-one correspondence with the canonical forms of their associated binary cubics and quartics. We then prove that these equivalence classes are uniquely determined by spectral invariants, specifically, the number of eigenpair classes and the multiplicities of zero eigenvalues, over the complex domain. We demonstrate that this classification does not hold in the real domain, where distinct equivalence classes can share identical spectral invariants. Finally, we extend this approach to derive canonical forms and complete classification for complex third- and fourth-order linear partial differential equations in two variables using their bijective relationship to binary forms. - oai:arXiv.org:2512.17431v1 - math.RA - Mon, 22 Dec 2025 00:00:00 -0500 + Convexification Numerical Method for Imaging of Moving Targets + https://arxiv.org/abs/2512.18361 + arXiv:2512.18361v1 Announce Type: new +Abstract: The problem of imaging of a moving target is formulated as a Coefficient Inverse Problem for a hyperbolic equation with its coefficient depending on all three spatial variables and time. As the initial condition, the point source running along a straight line is used. Lateral Cauchy data are known for each position of the point source. A truncated Fourier series with respect to a special orthonormal basis is used. First, Lipschitz stability estimate is obtained. Next, a globally convergent numerical method, the so-called convexification method, is developed and its convergence analysis is carried out. The convexification method is based on a Carleman estimate. Results of numerical experiments are presented. + oai:arXiv.org:2512.18361v1 + math.NA + cs.NA + Tue, 23 Dec 2025 00:00:00 -0500 new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Lishan Fang, Hua-Lin Huang + http://creativecommons.org/licenses/by/4.0/ + Michael V. Klibanov, Jingzhi Li, Vladimir G. Romanov, Zhipeng Yang - FAMED by computer: proving the Andersen-Kashaev volume conjecture for 42,000 knots - https://arxiv.org/abs/2512.17437 - arXiv:2512.17437v1 Announce Type: new -Abstract: The FAMED condition is a combinatorial property for ideal triangulations of $3$-manifolds, which was introduced in 2024 by the first and last authors in order to study the Andersen--Kashaev volume conjecture. They notably proved that this conjecture is true for all FAMED geometric triangulations of one-cusped hyperbolic $3$-manifolds with trivial second homology. - In this paper, using a straightforward computer implementation in Regina and Snappy, we find FAMED geometric triangulations for more than 42.000 complements of knots in $S^3$, including all knots with $12$ crossings or fewer and all knots whose complement can be triangulated with $23$ tetrahedra or fewer. As a consequence, the Andersen-Kashaev conjecture is now proven to be true for as many new examples. - Along the way, we find several new insights about the FAMED property, which have great value in the quest of a general proof of the Andersen-Kashaev volume conjecture for every knot complement. - oai:arXiv.org:2512.17437v1 - math.GT - Mon, 22 Dec 2025 00:00:00 -0500 + From Moore-Penrose to Markov via Gauss + https://arxiv.org/abs/2512.18364 + arXiv:2512.18364v1 Announce Type: new +Abstract: Markov categories are the central framework for categorical probability theory. Many important concepts from probability theory can be formalized in terms of Markov categories. In particular, conditional probability distributions and Bayes' theorem are captured via the notion of conditionals in a Markov category. Gaussian probability theory gives an example of a Markov category with conditionals, where the conditionals can be computed using the Moore-Penrose inverse. In this paper, we introduce the Gauss construction on a Moore-Penrose dagger additive category, producing a Markov category with conditionals. Applying the Gauss construction to the category of real matrices recaptures the Gaussian probability theory example, while applying it to the category of complex (resp. quaternionic) matrices gives us new Markov categories of proper complex (resp. quaternionic) Gaussian conditional distributions. Moreover, we also characterize all possible conditionals in the Gauss construction. + oai:arXiv.org:2512.18364v1 + math.CT + Tue, 23 Dec 2025 00:00:00 -0500 new - http://creativecommons.org/licenses/by-nc-sa/4.0/ - Fathi Ben Aribi, Antonin Guilloux, Ka Ho Wong + http://creativecommons.org/licenses/by/4.0/ + Cole Comfort, Jean-Simon Pacaud Lemay - Splitting infinity: a de Finetti game with state-dependent profit rates and singular control for diffusions - https://arxiv.org/abs/2512.17438 - arXiv:2512.17438v1 Announce Type: new -Abstract: We study a game of resource extraction of a common good under one-dimensional diffusive dynamics with player actions corresponding to singular stochastic control up to absorption at $0$, implying a trade-off between profitable resource extraction and sustainability. Unsurprisingly, immediate extraction of all available resources is an equilibrium. A main result is that we characterize and prove the existence of non-trivial equilibria that do not result in immediate absorption, but instead are attained with both players extracting resources according to a state-dependent rate of threshold type, corresponding to the presence of control only when the state process is in an interval $(b,\infty)$. The underlying assumption is, roughly, that the drift coefficient of the uncontrolled state process grows sufficiently fast in relation to the discount rate, implying that the value for the corresponding one-player problem is infinite. We also study a generalization of the game that allows a state-dependent profit rate integrated against the control processes. In this game we again characterize and prove the existence of non-trivial equilibria of threshold type. In particular, a main novelty is that we find equilibria where the state process is controlled with its own local time such that we have reflection points with associated initial jumps, as well as other points in the state space where the control processes increase in a singular manner (skew points). - oai:arXiv.org:2512.17438v1 - math.PR - math.OC - Mon, 22 Dec 2025 00:00:00 -0500 + Generators in the field of hyperelliptic functions + https://arxiv.org/abs/2512.18366 + arXiv:2512.18366v1 Announce Type: new +Abstract: We consider the field of hyperelliptic functions defined for a family of hyperelliptic curves as rational functions in some special functions from Kleinian functions theory. We compare our definition with the classical one. We provide details and references for the result that the field of hyperelliptic functions for a family of hyperelliptic curves of genus $g$ is isomorphic to the field of rational functions with $3g$ generators. The main result of the present work is that there are no algebraic relations between these generators. + oai:arXiv.org:2512.18366v1 + math.CV + math.AT + Tue, 23 Dec 2025 00:00:00 -0500 new http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Piotr Chlebicki, Kristoffer Lindensj\"o + E. Yu. Bunkova - Four special Poncelet triangle families about the incircle - https://arxiv.org/abs/2512.17440 - arXiv:2512.17440v1 Announce Type: new -Abstract: We describe four special families of ellipse-inscribed Poncelet triangles about the incircle which maintain certain triangle centers stationary and which also display interesting conservations. - oai:arXiv.org:2512.17440v1 - math.MG - cs.GR - Mon, 22 Dec 2025 00:00:00 -0500 + The far-flung Gorenstein numerical semigroup rings of type 4 + https://arxiv.org/abs/2512.18370 + arXiv:2512.18370v1 Announce Type: new +Abstract: We classify the far-flung Gorenstein numerical semigroup rings of type 4. + oai:arXiv.org:2512.18370v1 + math.AC + Tue, 23 Dec 2025 00:00:00 -0500 new http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Ronaldo A. Garcia, Mark Helman, Dan Reznik + Teodor I. Grigorescu - Existence and Configuration of Invariant Sets in $C^\infty([a,b])$ on which the Differential Operator Exhibits Devaney's Chaos - https://arxiv.org/abs/2512.17448 - arXiv:2512.17448v1 Announce Type: new -Abstract: In this paper, we investigate the chaotic behavior of the differential operator $\frac{d}{dx}$ on the space of smooth functions $C^\infty([a,b])$ equipped with the $L^p$-norm ($1\le p\le\infty$). We explicitly construct a homeomorphism between a subset of $C^\infty([a,b])$ and the shift space. Moreover, inspired by symbolic dynamics, we demonstrate that invariant sets, on which the differential operator behaves analogously to the shift, are densely configured in $C^\infty([a,b])$. We also prove that the differential operator is chaotic on the entire space $C^\infty([a,b])$ using a similar approach. - oai:arXiv.org:2512.17448v1 - math.DS - math.FA - Mon, 22 Dec 2025 00:00:00 -0500 + Explicit harmonic and wave maps into variable-curvature surfaces + https://arxiv.org/abs/2512.18376 + arXiv:2512.18376v1 Announce Type: new +Abstract: We construct explicit harmonic and wave maps between pseudo-Riemannian surfaces of variable curvature. For a broad class of target metrics, including nonconstant curvature surfaces such as ellipsoids, the harmonic and wave map equations admit a reduction to integrable ordinary differential equations under a natural ansatz. This yields explicit solutions beyond the classical constant-curvature and symmetric-space settings. The method applies uniformly in both elliptic and hyperbolic regimes. + oai:arXiv.org:2512.18376v1 + math.DG + math.AP + Tue, 23 Dec 2025 00:00:00 -0500 new - http://creativecommons.org/licenses/by-nc-nd/4.0/ - Kazutoyo Iketake + http://creativecommons.org/licenses/by/4.0/ + Anestis Fotiadis, Giannis Polychrou - Integrable $\mathbb{Z}_2^2$-graded super-Liouville Equation and Induced $\mathbb{Z}_2^2$-graded super-Virasoro Algebra - https://arxiv.org/abs/2512.17449 - arXiv:2512.17449v1 Announce Type: new -Abstract: We present a framework for enlarging the construction of $\mathbb{Z}_2^2$-graded classical Toda theory from the class of $\mathbb{Z}_2^2$-graded Lie algebras to the class of $\mathbb{Z}_2^2$-graded Lie superalgebras. This scheme is applied to derive a $\mathbb{Z}_2^2$-graded extension of the super-Liouville equation based on a $\mathbb{Z}_2^2$-graded extension of $\mathfrak{osp}(1|2).$ The mathematical tools employed in this work are a $\mathbb{Z}_2^2$-graded version of the zero-curvature formalism and of the Polyakov's soldering procedure. It is demonstrated that both methods yield the same $\mathbb{Z}_2^2$-graded super-Liouville equation. An algebraic construction of solutions to the resulting equations is also presented, together with their B\"acklund transformations. Furthermore, three distinct new $\mathbb{Z}_2^2$-graded extensions of the super-Virasoro algebra are obtained via Hamiltonian reduction of the WZNW currents defined for $\mathbb{Z}_2^2$-$\mathfrak{osp}(1|2).$ - oai:arXiv.org:2512.17449v1 - math-ph - hep-th - math.MP - nlin.SI - Mon, 22 Dec 2025 00:00:00 -0500 + Sixth-order explicit one-step methods for stiff ODEs via hybrid deferred correction involving RK2 and RK4: Application to reaction-diffusion equations + https://arxiv.org/abs/2512.18377 + arXiv:2512.18377v1 Announce Type: new +Abstract: In this paper, the fourth-order explicit Runge-Kutta method (RK4) is used to make a Deferred Correction (DC) on the explicit midpoint rule, resulting in an explicit one-step method of order six of accuracy, denoted DC6RK2/4. Convergence and order of accuracy of DC6RK2/4 are proven through a deferred correction condition satisfied by the RK4. The region of absolute stability of this method contains that of a RK6 and is tangent to the region [-5.626,0[x[-4.730,4.730] of the complex plane, containing a significant part of the imaginary axis. Numerical experiments with standard test problems for stiff systems of ODEs show that DC6RK2/4 performs well on problems regarding strong non-linearity and long-term integration, and this method does not require extremely small time steps for accurate numerical solutions of stiff problems. Moreover, this method is better than standard implicit methods like the Backward Differentiation Formulae and the DC methods for the implicit midpoint rule on stiff problems for which Jacobian matrices along the solution curve have complex eigenvalues where imaginary parts have larger magnitudes than real parts. An application of DC6RK2/4 to a class of test problems for reaction-diffusion equations in one dimensional is also carried out. + oai:arXiv.org:2512.18377v1 + math.NA + cs.NA + Tue, 23 Dec 2025 00:00:00 -0500 new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Naruhiko Aizawa, Ichi Fujii, Ren Ito, Toshiya Tanaka, Francesco Toppan + http://creativecommons.org/licenses/by-nc-sa/4.0/ + Saint Cyr E. R. Koyaguerebo-Im\'e - Equality of the critical inverse temperatures for the one- and two-sided Dyson models - https://arxiv.org/abs/2512.17451 - arXiv:2512.17451v1 Announce Type: new -Abstract: We prove that the critical inverse temperatures $\beta_c^{\mathbb N}(\alpha)$ and $\beta_c^{\mathbb Z}(\alpha)$ for the one- and two-sided Dyson models are the same when the power of the interaction strength $\alpha$ satisfies $1<\alpha<2$. We conjecture that this is true also in the remaining case of $\alpha=2$. - oai:arXiv.org:2512.17451v1 - math.PR - math-ph - math.MP - Mon, 22 Dec 2025 00:00:00 -0500 + Strong Central 2-Trees with Tail Degrees {2, 3}: Structural Characterization and Uniqueness Criteria + https://arxiv.org/abs/2512.18378 + arXiv:2512.18378v1 Announce Type: new +Abstract: We study strong $r$-central $2$-trees whose non-central vertices have degrees in $\{2,3\}$, focusing on the cases $r=1,2,3$. For each $r$, we derive exact degree constraints relating the maximum degree $\Delta$ to the numbers of degree-$3$ and degree-$2$ tail vertices. In the unicentral case ($r=1$), we prove that the fan graph is the unique realization for all $n\ge 3$. For bicentral $2$-trees ($r=2$), we show that the number of degree-$3$ vertices is always even, establish sharp uniqueness results for $x\in\{0,2\}$, prove existence for all feasible values of $\Delta$, and obtain linear lower bounds on the number of non-isomorphic realizations. For tricentral $2$-trees ($r=3$), we characterize extremal configurations, establish a divisibility constraint on the tail parameters, and prove a quadratic lower bound on the number of non-isomorphic graphs for infinitely many values of $n$. These results provide a unified structural framework for central $2$-trees with bounded tail degrees and highlight sharp transitions between rigidity and combinatorial growth. + oai:arXiv.org:2512.18378v1 + math.CO + Tue, 23 Dec 2025 00:00:00 -0500 new http://creativecommons.org/licenses/by/4.0/ - Noam Berger, Anders Johansson, Anders \"Oberg + Julian Allagan, Shawn Langley, Weizheng Gao, Mohamed Elbakary - Structure and Topological Generation of Big Mapping Class Groups - https://arxiv.org/abs/2512.17457 - arXiv:2512.17457v1 Announce Type: new -Abstract: Big mapping class groups are the mapping class groups of infinite-type surfaces, that is, surfaces whose fundamental groups are not finitely generated. While mapping class groups of finite-type surfaces have been extensively studied, the theory of big mapping class groups is a recent and rapidly developing area of research. This thesis provides a systematic introduction to the structure and topological generation of big mapping class groups, emphasizing the key differences from the classical finite-type case. It presents a clear exposition of foundational results concerning their topological and algebraic structure, including known results on finite topological generation for a certain family of infinite-type surfaces. - oai:arXiv.org:2512.17457v1 - math.GT - Mon, 22 Dec 2025 00:00:00 -0500 + Kuznecov formulae for fractal measures + https://arxiv.org/abs/2512.18379 + arXiv:2512.18379v1 Announce Type: new +Abstract: Let $(M,g)$ be a compact, connected Riemannian manifold of dimension $n\ge 2$, and let $\{e_j\}_{j=0}^\infty$ be an orthonormal basis of Laplace eigenfunctions $-\Delta_g e_j=\lambda_j^2 e_j$. Given a finite Borel measure $\mu$ on $M$, consider the Kuznecov sum \[ + N_\mu(\lambda):=\sum_{\lambda_j\le \lambda}\Bigl|\int_M e_j\,d\mu\Bigr|^2. \] Assume that $\mu$ is $s$-Ahlfors regular for some $s\in(0,n)$ and admits an averaged $s$-density constant $A_\mu$. We prove that \[ + N_\mu(\lambda) + = (2\pi)^{-(n-s)}\,{\rm vol}\,(B^{\,n-s})\,A_\mu\,\lambda^{n-s} + + o(\lambda^{n-s}) + \qquad (\lambda\to\infty). \] The hypotheses of $s$-Ahlfors regularity and the averaged $s$-density condition are essentially optimal for such a one-term asymptotic, and in general the remainder $o(\lambda^{n-s})$ cannot be improved uniformly to a power-saving error term. This extends the classical Kuznecov formulae of Zelditch for smooth submanifold measures to a broad class of singular and fractal measures. + oai:arXiv.org:2512.18379v1 + math.AP + math.CA + Tue, 23 Dec 2025 00:00:00 -0500 new http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Celal Can Bellek + Yakun Xi - The center of the BMW algebras and an Okounkov-Vershik like approach - https://arxiv.org/abs/2512.17458 - arXiv:2512.17458v1 Announce Type: new -Abstract: We use the Jucys-Murphy elements of the BMW algebra to show that its center over the complex numbers for almost all parameters making it semisimple is given by Wheel Laurent polynomials, a subalgebra of the symmetric Laurent polynomials in the JM elements. As an application, we give an Okounkov-Vershik like approach to its finite dimensional representations. In the non semisimple case related to the type B Lie algebras, the central subalgebra of Wheel Laurent polynomials is large enough to separate blocks of the BMW algebras. - oai:arXiv.org:2512.17458v1 - math.RT - math.QA - Mon, 22 Dec 2025 00:00:00 -0500 + Finite group actions on quasi-Hamiltonian spaces + https://arxiv.org/abs/2512.18380 + arXiv:2512.18380v1 Announce Type: new +Abstract: We organize fundamental properties of quasi-Hamiltonian spaces on which a finite group acts, and we apply them to the theory of moduli spaces of flat connections on an oriented compact surface with boundary. + oai:arXiv.org:2512.18380v1 + math.SG + Tue, 23 Dec 2025 00:00:00 -0500 new http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Christoforos Milionis + Keito Takegoshi - Solutions of the thin film equation obtained in the limit of vanishing slip - https://arxiv.org/abs/2512.17463 - arXiv:2512.17463v1 Announce Type: new -Abstract: We analyze the evolution of thin liquid droplets in the lubrication approximation with different slip conditions at the liquid-solid interface. Motivated by the classical no-slip paradox which states that the Navier-Stokes equations with a no-slip boundary condition require unphysical infinite dissipation during droplet spreading, we focus on the limit of vanishing slip. We show that in the no-slip limit three fundamentally different classes of limiting solutions are approached, each of them corresponding to a different scaling of the microscopic contact angle as the regularization parameter vanishes. These findings suggest that the thin-film equation with no slip supports a rich family of physically admissible solutions, provided one interprets the no-slip thin film equation as the asymptotic limit of models which regularized slip conditions. Even though the large apparent contact angles in some of these solutions seem incompatible with the lubrication approximation, a refined analysis shows that the underlying physical variables remain consistent with the assumptions for the lubrication approximation. - oai:arXiv.org:2512.17463v1 + Studies on the Rao-Nakra Sandwich Beam: Well-Posedness, Dynamics, and Controllability + https://arxiv.org/abs/2512.18381 + arXiv:2512.18381v1 Announce Type: new +Abstract: In this work, we investigate the well-posedness, stabilization, and boundary controllability of a linear Rao-Nakra type sandwich beam. The system consists of three coupled equations that represent the longitudinal displacements of the outer layers and the transverse displacement of the composite beam, all of which are coupled with dynamical boundary conditions. In the first problem, time-dependent weights and delays are considered. Then, we establish the existence and uniqueness of solutions for the Cauchy problem associated with the damped system using semigroup theory and a classical result by Kato. Furthermore, employing a Lyapunov-based approach, we prove that the system's energy decays exponentially, despite the presence of time-varying weights and delays. In the second problem, we consider a boundary linear control system and prove its well-posedness. By deriving an observability inequality for the adjoint system and applying the Hilbert Uniqueness Method (HUM), we show that the system is null controllable. A key contribution of this work lies in handling the full three-equation coupled system, which involves significant difficulty due to the dynamic boundary conditions, resolved via appropriately constructed Lyapunov functionals and intermediate observability inequalities. + oai:arXiv.org:2512.18381v1 + math.OC math.AP - Mon, 22 Dec 2025 00:00:00 -0500 + Tue, 23 Dec 2025 00:00:00 -0500 new - http://creativecommons.org/licenses/by/4.0/ - Hans Knuepfer, Juan Velazquez + http://creativecommons.org/publicdomain/zero/1.0/ + George J. Bautista, Roberto de A. Capistrano-Filho, Boumediene Chentouf, Oscar Sierra Fonseca, Juan L\'imaco - Solution concepts for a model of visco-elasto-plasticity with slight compressibility - https://arxiv.org/abs/2512.17464 - arXiv:2512.17464v1 Announce Type: new -Abstract: We study a model for the deformation of a visco-elasto-plastic material that is nearly incompressible. It originates from geophysics, is given in the Eulerian description and combines a Kelvin-Voigt rheology in the spherical part with a Jeffreys-type rheology in the deviatoric part. Despite a constant density, the model allows for non-isochoric deformation and the propagation of pressure waves. An additive decomposition of the strain rate into elastic and inelastic parts leads to an evolution equation for the small elastic strain, which is coupled with an adapted momentum equation. As plasticity is modeled through a non-smooth dissipation potential, we introduce a weak formulation in terms of a variational inequality. Since the well-posedness in such a weak setting is out of reach, we study two possible modifications: the regularization in terms of stress diffusion, and the relaxation of the solvability concept by transition to energy-variational solutions. In both cases, solutions are constructed by the same time-discrete scheme, consisting of solving a saddle-point problem in each time step. - oai:arXiv.org:2512.17464v1 + A note on stochastic semilinear dissipative evolution equations + https://arxiv.org/abs/2512.18398 + arXiv:2512.18398v1 Announce Type: new +Abstract: Existence and uniqueness of mild solutions to a class of semilinear stochastic evolution equations with additive noise is proved. The linear part of the drift term is the generator of a compact semigroup of contractions, while the nonlinear part is only assumed to be the superposition operator associated to a decreasing function. + oai:arXiv.org:2512.18398v1 + math.PR math.AP - Mon, 22 Dec 2025 00:00:00 -0500 + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Carlo Marinelli + + + Kuznetsov Categories for Gauged Linear Sigma Models + https://arxiv.org/abs/2512.18402 + arXiv:2512.18402v1 Announce Type: new +Abstract: We define Kuznetsov and anti-Kuznetsov categories for gauged linear sigma models. We show that for complete intersections of ample divisors in smooth projective toric varieties, the Kuznetsov category is left orthogonal to an exceptional collection. We prove that any complete intersection of $r \ge 2$ ample divisors in a Fano GIT quotient is a Fano visitor and the derived category of its Fano host is equivalent to an anti-Kuznetsov category of a gauged linear sigma model. + oai:arXiv.org:2512.18402v1 + math.AG + Tue, 23 Dec 2025 00:00:00 -0500 new http://creativecommons.org/licenses/by/4.0/ - Thomas Eiter + David Favero, Daniel Kaplan, Tyler L. Kelly - Minimal Sets of Generators for Big Mapping Class Groups - https://arxiv.org/abs/2512.17465 - arXiv:2512.17465v1 Announce Type: new -Abstract: Let $S(n)$ be the infinite-type surface with infinite genus and $n \in \mathbb{N}$ ends, all of which are accumulated by genus. The mapping class group of this surface, $\mod(S(n))$, is a Polish group that is not countably generated, but it is countably topologically generated. This paper focuses on finding minimal sets of generators for $\mod(S(n))$. We show that for $n \ge 8$, $\mod(S(n))$ is topologically generated by three elements, and for $n \ge 3$, $\mod(S(n))$ is topologically generated by four elements. We also establish a generating set of two elements for the Loch Ness Monster surface ($n=1$) and a generating set of three elements for the Jacob's Ladder surface ($n=2$). - oai:arXiv.org:2512.17465v1 - math.GT - Mon, 22 Dec 2025 00:00:00 -0500 + Explicit sharp bounds for all nodes of Sturm-Liouville operators with potentials in $L^1$ balls + https://arxiv.org/abs/2512.18404 + arXiv:2512.18404v1 Announce Type: new +Abstract: For the classical Sturm-Liouville operators, we prove the sharp bounds for all nodes of eigenfunctions by regarding these nodes as nonlinear functionals of potential $q\in L^1[0,1]$. By studying the optimization problems to minimize or to maximize the nodes $\{ T_{i,m}\}$ subject to the constraint $\|q\|_{1}=r$ with $r>0$ and using the strong continuity of the nodes in potentials, we obtain the explicit expressions for the sharp bounds, which are given as elementary functions. + oai:arXiv.org:2512.18404v1 + math.SP + Tue, 23 Dec 2025 00:00:00 -0500 new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - T\"ulin Altun\"oz, Celal Can Bellek, Emir G\"ul, Mehmetcik Pamuk, O\u{g}uz Y{\i}ld{\i}z + http://creativecommons.org/licenses/by/4.0/ + Jifeng Chu, Shuyuan Guo, Gang Meng, Meirong Zhang - An inverse theorem for all finite abelian groups via nilmanifolds - https://arxiv.org/abs/2512.17468 - arXiv:2512.17468v1 Announce Type: new -Abstract: We prove a first inverse theorem for Gowers norms on all finite abelian groups that uses only nilmanifolds (rather than possibly more general nilspaces). This makes progress toward confirming the Jamneshan--Tao conjecture. The correlating function in our theorem is a projected nilsequence, obtained as the fiber-wise average of a nilsequence defined on a boundedly-larger abelian group extending the original abelian group. This result is tight in the following sense: we prove also that $k$-step projected nilsequences of bounded complexity are genuine obstructions to having small Gowers $U^{k+1}$-norm. This inverse theorem relies on a new result concerning compact finite-rank (CFR) nilspaces, which is the main contribution in this paper: every $k$-step CFR nilspace is a factor of a $k$-step nilmanifold. This new connection between the classical theory of nilmanifolds and the more recent theory of nilspaces has applications beyond arithmetic combinatorics. We illustrate this with an application in topological dynamics, by proving the following result making progress on a question of Jamneshan, Shalom and Tao: every minimal $\mathbb{Z}^\omega$-system of order $k$ is a factor of an inverse limit of $\mathbb{Z}^\omega$-polynomial orbit systems of order $k$, these being natural generalizations of nilsystems alternative to translational systems. - oai:arXiv.org:2512.17468v1 - math.DS - math.CO - Mon, 22 Dec 2025 00:00:00 -0500 + A Phase Space Representation of the Metaplectic Group + https://arxiv.org/abs/2512.18415 + arXiv:2512.18415v1 Announce Type: new +Abstract: The symplectic group Sp(n) acts on phase space while the unitary representation of its double cover, Mp(n), the metaplectic group, acts on functions defined on configuration space. We will construct an extension Mp(n) of Mp(n) acting on square integrable functions on phase space. This is performed using previous results of ours involving explicit expressions of the twisted Weyl symbols of metaplectic operators and Bopp pseudodifferential operators, which are phase space extensions of the usual Weyl operators. + oai:arXiv.org:2512.18415v1 + math-ph + math.AP + math.MP + math.SG + quant-ph + Tue, 23 Dec 2025 00:00:00 -0500 new http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Pablo Candela, Diego Gonz\'alez-S\'anchez, Bal\'azs Szegedy + Maurice de Gosson - Extending Chevalley's Theorem: A Topological Characterization of Constructibility and its Generalization Beyond Noetherian Spaces - https://arxiv.org/abs/2512.17481 - arXiv:2512.17481v1 Announce Type: new -Abstract: We introduce the notion of a \textit{good map} between topological spaces: a continuous map $f:X\to Y$ is good if for every non-empty irreducible locally closed subset $U\subseteq X$, there exists a non-empty open subset $W\subseteq Y$ such that $W\cap f(U)=W\cap\overline{f(U)}\neq\varnothing$. In Noetherian spaces, this condition is equivalent to preserving constructible subsets (Theorem 2.3), giving a topological characterization of Chevalley's theorem. Without the Noetherian assumption, the good property continues to make sense and serves as a reasonable generalization. We establish basic properties of good maps and introduce a weaker variant, \textit{weak good maps}. In algebraic geometry, affine space projections $\mathbb{A}_{S}^{n}\to S$ are good for any base scheme $S$ (Theorem 4.1). From this we obtain: - $\bullet$ A generalization of Chevalley's theorem for morphisms of finite type with Noetherian underlying topological spaces (Theorem 4.2); - $\bullet$ The fact that every morphism locally of finite type is weak good (Proposition 4.4), yielding an elementary proof of Jacobson ascent (Corollary 4.5). - The good property is stable under formal completion, suggesting extensions to other non-Hausdorff geometries. - oai:arXiv.org:2512.17481v1 - math.AG - Mon, 22 Dec 2025 00:00:00 -0500 + Intrinsic homological algebra for triangulated categories + https://arxiv.org/abs/2512.18417 + arXiv:2512.18417v1 Announce Type: new +Abstract: We propose a new framework for the study of homological properties for (compactly generated) triangulated categories such as regularity, finiteness of global or finitistic dimension, gorensteinness or injective generation and the relation between them. Our approach focuses on distinguished, intrinsically defined, subcategories and our main tool is the new notion of far-away orthogonality. We observe that these homological properties generalise previously studied properties on derived categories of modules over rings, and we use the generality of our theory to also examine those same attributes for the homotopy category of injectives and the big singularity category (in the sense of Krause) of an Artin algebra, as well as the derived category of a non-positive differential graded algebra. Finally, using our theory we recover and generalise various results in the theory of recollements of triangulated categories. + oai:arXiv.org:2512.18417v1 + math.RT + math.CT + math.KT + Tue, 23 Dec 2025 00:00:00 -0500 new http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Jiawei Sheng + Panagiotis Kostas, Chrysostomos Psaroudakis, Jorge Vit\'oria - Koenigs functions in the subcritical and critical Markov branching processes with Poisson probability reproduction of particles - https://arxiv.org/abs/2512.17485 - arXiv:2512.17485v1 Announce Type: new -Abstract: Special functions have always played a central role in physics and in mathematics, arising as solutions of nonlinear differential equations, as well as in the theory of branching processes, which extensively uses probability generating functions. The theory of iteration of real functions leads to limit theorems for the discrete-time and real-time Markov branching processes. The Poisson reproduction of particles in real time is analysed through the integration of the Kolmogorov equation. These results are further extended by employing graphical representations of Koenigs functions under subcritical and critical branching mechanisms. The limit conditional law in the subcritical case and the invariant measure for the critical case are discussed, as well. The obtained explicit solutions contain the exponential Bell polynomials and the modified exponential-integral function $\rm{Ein} (z)$. - oai:arXiv.org:2512.17485v1 - math.PR - stat.CO - Mon, 22 Dec 2025 00:00:00 -0500 + Development of Testing Methodology in Mathematics Education in the Context of Digitalization + https://arxiv.org/abs/2512.18418 + arXiv:2512.18418v1 Announce Type: new +Abstract: Access to quality education remains a global challenge, particularly in crisis-affected regions. This study examines the decline in students' mathematical proficiency and proposes an innovative Moodle-based testing system that incorporates step-by-step solution verification and interactive exercises. Unlike traditional assessments, this approach ensures a more structured evaluation process, reducing student errors and improving feedback quality. Additionally, the study integrates Classical Test Theory to analyze test reliability, offering a novel perspective on the effectiveness of digital assessments. The approach enhances assessment accuracy and conceptual understanding while addressing the limitations of traditional testing. Using Classical Test Theory, the study evaluates the reliability of the proposed assessment methods. + oai:arXiv.org:2512.18418v1 + math.HO + Tue, 23 Dec 2025 00:00:00 -0500 new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Penka Mayster, Assen Tchorbadjieff + http://creativecommons.org/licenses/by/4.0/ + I. V. Orlovskyi, O. A. Tymoshenko - Centers and Orderability of Certain Quotients of quasi-isometry groups of Euclidean spaces - https://arxiv.org/abs/2512.17487 - arXiv:2512.17487v1 Announce Type: new -Abstract: In this article, we study the algebraic and dynamical structure of certain normal subgroups of the quasi-isometry group of Euclidean space $QI(\mathbb{R}^n)$. For \[ H = \Big\{ [f] \in QI(\mathbb{R}^n) : \lim_{\|x\|\to\infty} \frac{\|f(x)-x\|}{\|x\|} = 0 \Big\}, \] and for $0<\alpha<1$, \[ H_\alpha = \Big\{ [f] \in QI(\mathbb{R}^n) : \|f(x)-x\| \le K\|x\|^\alpha \text{ for large } \|x\| \Big\}, \] we show that each $H_\alpha$ is a nontrivial normal subgroup of $QI(\mathbb{R}^n)$, satisfying \[ H_\alpha \subset H_\beta \subset H \qquad \text{for } 0<\alpha<\beta<1. \] We prove that the centers of $QI(\mathbb{R}^n)/H$ and $QI(\mathbb{R}^n)/H_\alpha$ are trivial, while these quotients admit nontrivial torsion elements. Consequently, they are neither left-orderable nor locally indicable. Finally, we introduce an asymptotic topology on $QI(\mathbb{R}^n)$ and show that the family $\{H_\alpha\}_{0<\alpha<1}$ is dense in $H$. - oai:arXiv.org:2512.17487v1 - math.GT - Mon, 22 Dec 2025 00:00:00 -0500 + A least-squares meshfree method for the incompressible Navier-Stokes equations: A satisfactory solenoidal velocity field via a staggered-variable arrangement + https://arxiv.org/abs/2512.18422 + arXiv:2512.18422v1 Announce Type: new +Abstract: Incompressible flow solvers based on strong-form meshfree methods represent arbitrary geometries without the need for a global mesh system. However, their local evaluations make it difficult to satisfy incompressibility at the discrete level. Moreover, the collocated arrangement of velocity and pressure variables tends to induce a zero-energy mode, leading to decoupling between the two variables. In projection-based approaches, a spatial discretization scheme based on a conventional node-based moving least-squares method for the pressure causes inconsistency between the discrete operators on both sides of the Poisson equation. Thus, a solenoidal velocity field cannot be ensured numerically. In this study, a numerical method for the incompressible Navier-Stokes equations is developed by introducing a local primal-dual grid into the mesh-constrained discrete point method, enabling consistent discrete operators. The \textit{virtual} dual cell constructed is based on the local connectivity among nodes, and therefore our method remains truly meshfree. To achieve a consistent coupling between velocity and pressure variables under the primal-dual arrangement, time evolution converting is applied to evolve the velocity on cell interfaces. For numerical validation, a linear acoustic equation is solved to confirm the effectiveness of the staggered-variable arrangement based on the local primal-dual grid. Then, incompressible Navier-Stokes equations are solved, and the proposed method is demonstrated to satisfy the condition of a solenoidal velocity field at the discrete level, achieve the expected spatial convergence order, and accurately reproduce flow features over a wide range of Reynolds numbers. + oai:arXiv.org:2512.18422v1 + math.NA + cs.NA + physics.flu-dyn + Tue, 23 Dec 2025 00:00:00 -0500 new http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Swarup Bhowmik, Deblina Das + Takeharu Matsuda, Satoshi Ii - On Coorbit Fr\'echet Spaces - https://arxiv.org/abs/2512.17502 - arXiv:2512.17502v1 Announce Type: new -Abstract: This paper is concerned with a new approach to coorbit space theory. Usually, coorbit spaces are defined by collecting all distributions for which the voice transform associated with a square-integrable group representation possesses a certain decay, usually measured in a Banach space norm such as weighted $L_p$-norms. Unfortunately, in cases where the representation does not satisfy certain integrability conditions, one is faced with a bottleneck, namely that the discretization of the coorbit spaces is surprisingly difficult. It turns out that in these cases the construction of coorbit spaces as Fr\'echet spaces is much more convenient since then atomic decompositions can be established in a very natural way. - oai:arXiv.org:2512.17502v1 - math.FA - Mon, 22 Dec 2025 00:00:00 -0500 + A Flanking Pattern in a Sum-of-Divisors Congruence + https://arxiv.org/abs/2512.18424 + arXiv:2512.18424v1 Announce Type: new +Abstract: We consider composite $n$ satisfying the congruence $$n \cdot \sigma_k(n) \equiv 2 \pmod{\phi(n)},$$ and show a "flanking" structure: $14$ appears in both $S_{k-1}$ and $S_{k+1}$ whenever certain values of $n$ appear in $S_k$; and, moreover, $14$ is the only (nontrivial) case of this property. Along the way, we derive a new characterization of the $n$ that appear in the sets $S_{k}$. + oai:arXiv.org:2512.18424v1 + math.NT + Tue, 23 Dec 2025 00:00:00 -0500 new http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - S. Dahlke, F. De Mari, E. De Vito, M. Hansen, G. Steidl, G. Teschke + Scott Duke Kominers - Absorbing Markov Decision Processes: Geometric Properties and Sufficiency of Finite Mixtures of Deterministic Policies - https://arxiv.org/abs/2512.17511 - arXiv:2512.17511v1 Announce Type: new -Abstract: In this paper we investigate several geometric properties of the set of occupancy measures. In particular, we analyse the structure of the faces generated by a given occupancy measure, together with their relative algebraic interior. We also determine the affine hulls of these faces and describe the associated parallel linear subspaces. It is shown that these structures can be fully characterised in terms of the parameters that define the underlying Markov decision process (MDP). Moreover, we establish that the class of finite mixtures of deterministic stationary policies constitutes a sufficient class of policies for uniformly absorbing MDPs with a measurable state space and multiple criteria. We also provide a characterisation of the minimal order required for a finite mixture of deterministic stationary policies to represent the performance vector of an arbitrary policy. - oai:arXiv.org:2512.17511v1 - math.OC - Mon, 22 Dec 2025 00:00:00 -0500 + Imaging nonlinearity coefficient and sound speed with the JMGT equation in frequency domain + https://arxiv.org/abs/2512.18431 + arXiv:2512.18431v1 Announce Type: new +Abstract: In this paper we prove uniqueness and stability of reconstruction of two coefficients (sound speed and nonlinearity parameter) in the Jordan-Moore-Gibson-Thompson JMGT equation of nonlinear acoustics, relying on observations resulting from only two sources. A key tool for this purpose is a multiharmonic expansion of the PDE solution, which reflects the physical phenomenon of higher harmonics appearing due to nonlinearity and allows us to work in frequency domain. Based on this result, we derive a regularization property of reconstruction with JMGT as the relexation time tends to zero (in the spirit of a quasi reversibility method) for reconstruction from the classical Westervelt equation. + oai:arXiv.org:2512.18431v1 + math.AP + Tue, 23 Dec 2025 00:00:00 -0500 new http://creativecommons.org/licenses/by/4.0/ - Francois Dufour, Tomas Prieto-Rumeau + Barbara Kaltenbacher - On surface polariton resonance and its curvature concentration effects from 3D elastic nanorods - https://arxiv.org/abs/2512.17513 - arXiv:2512.17513v1 Announce Type: new -Abstract: This paper investigates surface polariton resonance (SPR) in three-dimensional elastic metamaterials with nanorod geometry. The primary motivation is to surpass the physical limitations imposed by the quasi-static approximation for SPRs through anisotropic geometric design. The analysis boils down to analyzing the spectral properties of the matrix-valued elastic Neumann-Poincar\'e (NP) operator defined on the nanorod boundary. We develop novel analytical techniques and conduct a rigorous asymptotic analysis of elastic layer potential operators, specifically adapted for highly anisotropic structures. Within this framework, we derive precise asymptotic formulas for the scattered field in the quasi-static regime. A thorough examination of these expressions yields explicit resonance conditions that intricately link three fundamental parameters: elastic material parameters, wave frequency, and nanorod geometry. Furthermore, we characterize the intrinsic relationship between these parameters and the associated energy blow-up rate of the resonant field. This analysis explicitly establishes a sharp curvature concentration effect at the nanorod extremities, where field enhancement is locally maximized. Our work provides a rigorous theoretical foundation for harnessing elastic SPRs through anisotropic geometric engineering, with implications for sensing, wave focusing, and metamaterial applications. - oai:arXiv.org:2512.17513v1 - math-ph - math.MP - Mon, 22 Dec 2025 00:00:00 -0500 + Indistinguishability for recurrent clusters + https://arxiv.org/abs/2512.18435 + arXiv:2512.18435v1 Announce Type: new +Abstract: We introduce a general framework to show the indistinguishability of infinite clusters (ergodicity of the cluster subrelation) in group-invariant percolation processes with a weaker version of the finite energy property: the possibility of moving infinite branches from one infinite cluster to another. Crucially, this removes the necessity for the infinite clusters to be transient, present in most previous works. Our method also applies to more general random graphs, whenever a stationary sequence of vertices is definable. + We use this to show the indistinguishability of infinite clusters (or permutation cycles) in the interchange process (a.k.a.~random stirring process), the loop $O(n)$ model on amenable Cayley graphs, biased corner percolation on $\mathbb{Z}^2$, and the Poisson Zoo process. + Finally, we show that infinite clusters in any invariant process on a Cayley graph are indistinguishable for any ``not essentially tail'' property, i.e., properties that depend only on the local structure of the cluster. + oai:arXiv.org:2512.18435v1 + math.PR + math.DS + Tue, 23 Dec 2025 00:00:00 -0500 new http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Youjun Deng, Hongyu Liu, Wanjing Tang, Guang-Hui Zheng + Damis El Alami, G\'abor Pete, \'Ad\'am Tim\'ar - A cut finite element method for the Biot system of poroelasticity - https://arxiv.org/abs/2512.17521 - arXiv:2512.17521v1 Announce Type: new -Abstract: We propose a novel cut finite element method for the numerical solution of the Biot system of poroelasticity. The Biot system couples elastic deformation of a porous solid with viscous fluid flow and commonly arises on domains with complex geometries that make high-quality volumetric meshing challenging. To address this issue, we employ the cut finite element framework, where the domain boundary is represented independently of the background mesh, which significantly simplifies the meshing process. Our approach builds upon a parameter robust total pressure formulation of the Biot system, which we combine with the cut finite element method to develop a geometrically robust solution scheme, while preserving the parameter robustness. A key ingredient in the theoretical analysis is a modified inf-sup condition which also holds for mixed boundary conditions, leading to stability and optimal error estimates for the proposed formulation. Finally, we provide numerical evidence demonstrating the theoretical properties of the method and showcasing its capabilities by solving the Biot system on a realistic brain geometry. - oai:arXiv.org:2512.17521v1 - math.NA - cs.NA - Mon, 22 Dec 2025 00:00:00 -0500 + On the Lavrentiev gap for manifold-valued maps + https://arxiv.org/abs/2512.18447 + arXiv:2512.18447v1 Announce Type: new +Abstract: We investigate the validity and the failure of modular density of smooth maps on compact manifolds. + oai:arXiv.org:2512.18447v1 + math.AP + math.DS + Tue, 23 Dec 2025 00:00:00 -0500 new http://creativecommons.org/licenses/by/4.0/ - Nanna Berre, Kent-Andre Mardal, Andr\'e Massing, Ivan Yotov + Carlo Alberto Antonini, Filomena De Filippis, Cintia Pacchiano Camacho - Comparison of two statistical image reconstruction algorithms for quantitative assessment of pathological lesions using gamma emission tomography - https://arxiv.org/abs/2512.17523 - arXiv:2512.17523v1 Announce Type: new -Abstract: This study compares two statistical approaches to image reconstruction in single-photon emission computed tomography (SPECT). We evaluated the widely used Ordered Subset Expectation Maximization (OSEM) algorithm and the newer Maximum a Posteriori approach with Entropy prior (MAP-Ent) approach in the context of quantifying radiopharmaceutical uptake in pathological lesions. Numerical experiments were performed using a digital twin of the standardized NEMA IEC phantom, which contains six spheres of varying diameters to simulate lesions. Quantitative accuracy was assessed using the maximum recovery coefficient (RCmax), defined as the ratio of the reconstructed maximum activity to the true value. The study shows that OSEM exhibits unstable convergence during iterations, leading to noise and edge artifacts in lesion images. Post-filtering stabilizes the reconstruction and ensures convergence, producing RCmax-size curves that could be used as correction factors in clinical evaluations. However, this approach significantly underestimates uptake in small lesions and may even lead to the complete loss of small lesions on reconstructed images. In contrast, MAP-Ent demonstrates fundamentally different behavior: it achieves stable convergence and preserves quantitative accuracy without post-filtering, while maintaining the contrast of even the smallest lesions. However, the iteration number at which accurate reconstruction is achieved depends strongly on the choice of a single global regularization parameter, which limits optimal performance across lesions of different sizes. These results demonstrate the need for locally adaptive regularization in MAP-Ent to improve quantitative accuracy in lesion reconstruction. - oai:arXiv.org:2512.17523v1 - math.NA - cs.NA - Mon, 22 Dec 2025 00:00:00 -0500 + Age of Information with Age-Dependent Server Selection + https://arxiv.org/abs/2512.18457 + arXiv:2512.18457v1 Announce Type: new +Abstract: In this paper, we consider a single-source multi-server generate-at-will discrete-time non-preemptive status update system where update packets are transmitted using {\em only one} of the available servers, according to a server selection policy. In particular, when a transmission is complete, the update system makes a threshold-based decision on whether to wait or transmit, and if latter, which server to use for transmissions, on the basis of the instantaneous value of the age of information (AoI) process. In our setting, servers have general heterogeneous discrete phase-type (DPH) distributed service times, and also heterogeneous transmission costs. The goal is to find an age-dependent multi-threshold policy that minimizes the AoI cost with a constraint on transmission costs, the former cost defined in terms of the time average of an arbitrary function of AoI. For this purpose, we propose a novel tool called \emph{multi-regime absorbing Markov chain} (MR-AMC) in discrete time. Using the MR-AMC framework, we exactly obtain the distribution of AoI, and subsequently the costs associated with AoI and transmissions. With the exact analysis in hand, optimum thresholds can be obtained in the case of a few servers, by exhaustive search. We validate the proposed analytical model, and also demonstrate the benefits of age-dependent server selection, with numerical examples. + oai:arXiv.org:2512.18457v1 + cs.IT + cs.PF + math.IT + Tue, 23 Dec 2025 00:00:00 -0500 new - http://creativecommons.org/licenses/by/4.0/ - A. V. Nesterova, N. V. Denisova + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Nail Akar, Ismail Cosandal, Sennur Ulukus - Tightness of Stationary Nodal Measures - https://arxiv.org/abs/2512.17524 - arXiv:2512.17524v1 Announce Type: new -Abstract: We study the rescaled nodal volume field $\xi_R$ associated with a smooth, stationary Gaussian field on $[0,R]^d$, whose covariance satisfies adequate integrability conditions. Our main theorem shows that, as $R \to \infty$, the process $\xi_R$ converges in distribution, in an appropriate space of c\`adl\`ag mappings, to a standard Brownian sheet. The proof relies on a recent finite-dimensional CLT by Ancona, Gass, Letendre, and Stecconi (2025), as well as on a multidimensional Kolmogorov--Chentsov criterion for tightness due to Bickel and Wichura (1971). The application of the latter requires new moment estimates that are of independent interest. Our results stand in sharp contrast with Berry's random wave model, where the required integrability conditions fail and the question of tightness remains open. - oai:arXiv.org:2512.17524v1 - math.PR - Mon, 22 Dec 2025 00:00:00 -0500 + Prioritized Constraints in Optimization-Based Control + https://arxiv.org/abs/2512.18458 + arXiv:2512.18458v1 Announce Type: new +Abstract: We provide theoretical foundations and computational tools for the systematic design of optimization-based control laws with constraints that have different priorities. By introducing the concept of prioritized intersections, we extend and unify previous work on the topic. Moreover, to enable the use of prioritized intersection in real-time applications, we propose an efficient solver for forming such intersections for polyhedral constraints. The solver in question is a tailored implementation of a dual active-set quadratic programming solver that leverages the particular problem structure of the optimization problems arising for prioritized intersections. The method is validated in a real-time MPC application for autonomous driving, where it successfully resolves six different levels of conflicting constraints, confirming its efficiency and practicality for control. Furthermore, we show that the proposed solver outperforms existing solvers for hierarchical quadratic programming, making it relevant beyond control applications. + oai:arXiv.org:2512.18458v1 + math.OC + cs.SY + eess.SY + Tue, 23 Dec 2025 00:00:00 -0500 new - http://creativecommons.org/licenses/by/4.0/ - Louis Gass, Giovanni Peccati + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Daniel Arnstr\"om, Gianluca Garofalo - Stochastic Maximum Principle for Optimal Control of Anticipated Backward Stochastic Systems with Delays - https://arxiv.org/abs/2512.17529 - arXiv:2512.17529v1 Announce Type: new -Abstract: This paper investigates optimal control problems for delayed systems governed by Infinitely Anticipated Backward Stochastic Differential Equations (IABSDEs). Unlike existing frameworks limited to bounded delays, we introduce a generalized formulation utilizing $\sigma$-finite measures that accommodates both long-term memory effects and forward-looking anticipation. - Employing a new type of infinitely delayed stochastic differential equations as adjoint equations, we derive the necessary conditions of the maximum principle for optimal control. Under appropriate assumptions, the sufficiency of the maximum principle is also established. As illustrative examples, a climate policy model, a consumption optimization problem and a linear quadratic control problem are discussed, and all optimal controls are derived explicitly. - oai:arXiv.org:2512.17529v1 - math.OC - Mon, 22 Dec 2025 00:00:00 -0500 + Quantitative polynomial cohomology and applications to $\textrm L^p$-measure equivalence + https://arxiv.org/abs/2512.18463 + arXiv:2512.18463v1 Announce Type: new +Abstract: We introduce a quantitative version of polynomial cohomology for discrete groups and show that it coincides with usual group cohomology when combinatorial filling functions are polynomially bounded. As an application, we show that Betti numbers of nilpotent groups are invariant by mutually cobounded $\textrm L^p$-measure equivalence. We also use this to obtain new vanishing results for non-cocompact lattices in rank 1 simple Lie groups. + oai:arXiv.org:2512.18463v1 + math.GR + math.DS + math.MG + Tue, 23 Dec 2025 00:00:00 -0500 new http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Guanwei Cheng + Antonio L\'opez Neumann, Juan Paucar - The stable trees revisited - https://arxiv.org/abs/2512.17533 - arXiv:2512.17533v1 Announce Type: new -Abstract: We introduce a new, relatively simple, line-breaking construction of the $\alpha$-stable tree which realises its random finite-dimensional distributions. This is a direct analogue of Aldous' line-breaking construction of the Brownian continuum random tree, which is based on an inhomogeneous Poisson process. Here, we replace the deterministic rate function from the Brownian setting by a random rate process, given by a certain measure-changed $(\alpha-1)$-stable subordinator. Rather than attaching uniformly, the line-segments now connect to locations chosen with probability proportional to the sizes of the jumps of the rate process. - We also give a new proof of an invariance principle originally due to Duquesne, which states that the family tree of a Bienaym\'e branching process with critical offspring distribution in the domain of attraction of an $\alpha$-stable law (for $\alpha \in (1,2))$, conditioned to have $n$ vertices, converges on rescaling distances appropriately to the $\alpha$-stable tree. Our proof makes use of a discrete line-breaking construction of the branching process tree, which we show converges to our continuous line-breaking construction. - oai:arXiv.org:2512.17533v1 - math.PR - Mon, 22 Dec 2025 00:00:00 -0500 + A cop-robber game on metric graphs + https://arxiv.org/abs/2512.18468 + arXiv:2512.18468v1 Announce Type: new +Abstract: We study a variant of the classical cop-robber game played on compact metric graphs, where each edge is assigned a positive length and identified with a real interval of corresponding length. In this setting, both the cop and the robber move continuously along the edges, subject to upper bounds on their speeds. The cop has no knowledge of the robber's location and must choose a continuous path through the graph that is guaranteed to intersect the robber's trajectory at some point in time. We show that for every compact metric graph, there exists a constant s > 0 such that if the cop's speed exceeds s times the robber's speed, then the cop can guarantee capture. + oai:arXiv.org:2512.18468v1 + math.CO + Tue, 23 Dec 2025 00:00:00 -0500 new http://creativecommons.org/licenses/by/4.0/ - Christina Goldschmidt, Liam Hill + Daniel Berend, Michael D. Boshernitzan - A distance-free approach to generalized weights - https://arxiv.org/abs/2512.17542 - arXiv:2512.17542v1 Announce Type: new -Abstract: We propose a unified theory of generalized weights for linear codes endowed with an arbitrary distance. Instead of relying on supports or anticodes, the weights of a code are defined via the intersections of the code with a chosen family of spaces, which we call a test family. The choice of test family determines the properties of the corresponding generalized weights and the characteristics of the code that they capture. In this general framework, we prove that generalized weights are weakly increasing and that certain subsequences are strictly increasing. We also prove a duality result reminiscent of Wei's Duality Theorem. The corresponding properties of generalized Hamming and rank-metric weights follow from our general results by selecting optimal anticodes as a test family. For sum-rank metric codes, we propose a test family that results in generalized weights that are closely connected to -- but not always the same as -- the usual generalized weights. This choice allows us to extend the known duality results for generalized sum-rank weights to some sum-rank-metric codes with a nonzero Hamming component. Finally, we explore a family of generalized weights obtained by intersecting the underlying code with MDS or MRD codes. - oai:arXiv.org:2512.17542v1 - cs.IT - cs.DM - math.IT - Mon, 22 Dec 2025 00:00:00 -0500 + Stochastic homogenization of coarse-grained elliptic equations + https://arxiv.org/abs/2512.18469 + arXiv:2512.18469v1 Announce Type: new +Abstract: We prove quenched stochastic homogenization for divergence-form elliptic equations, under the assumption that the coefficients are stationary, ergodic, integrable, and satisfy a coarse-grained ellipticity assumption. The ellipticity assumption requires that the coefficients remain bounded in a negative regularity sense on large scales. As a corollary, we recover a sufficient joint integrability condition on the symmetric and skew-symmetric parts of the coefficient field. + oai:arXiv.org:2512.18469v1 + math.AP + Tue, 23 Dec 2025 00:00:00 -0500 new http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Andrea Di Giusto, Elisa Gorla, Alberto Ravagnani + Aidan Lau - A quantitative Hopf-Oleinik lemma for degenerate fully nonlinear operators and applications to free boundary problems - https://arxiv.org/abs/2512.17543 - arXiv:2512.17543v1 Announce Type: new -Abstract: We prove a quantitative inhomogeneous Hopf-Oleinik lemma for viscosity solutions of $$|\nabla u|^{\alpha}F(D^{2}u)=f $$ and, more generally, for viscosity supersolutions of $|\nabla u|^{\alpha}\,{M}^-_{\lambda,\Lambda}(D^{2}u)\le f$. The result yields linear boundary growth with universal constants depending only on the structural data. We also exhibit a counterexample showing that the Hopf lemma fails for equations that act only in the large-gradient regime (in the sense of Imbert and Silvestre), thereby delineating the scope of our theorem. As applications, we obtain Lipschitz regularity for viscosity solutions of one-phase Bernoulli free boundary problems driven by these degenerate fully nonlinear operators and derive $\varepsilon$-uniform Lipschitz bounds for a one-phase flame propagation model. - oai:arXiv.org:2512.17543v1 - math.AP - Mon, 22 Dec 2025 00:00:00 -0500 + Canonical tree-decompositions of chordal graphs + https://arxiv.org/abs/2512.18480 + arXiv:2512.18480v1 Announce Type: new +Abstract: Halin characterised the chordal locally finite graphs as those that admit a tree-decomposition into cliques. We show that these tree-decompositions can be chosen to be canonical, that is, so that they are invariant under all the graph's automorphisms. As an application, we show that a locally finite, connected graph $G$ is $r$-locally chordal (that is, its $r/2$-balls are chordal) if and only if the unique canonical graph-decomposition $\mathcal{H}_r(G)$ of $G$ which displays its $r$-global structure is into cliques. Our results also serve as tools for further characterisations of $r$-locally chordal graphs. + oai:arXiv.org:2512.18480v1 + math.CO + Tue, 23 Dec 2025 00:00:00 -0500 new - http://creativecommons.org/licenses/by/4.0/ - Davide Giovagnoli, Enzo Maria Merlino, Diego Moreira + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Raphael W. Jacobs, Paul Knappe - Forbidding just one intersection for short integer sequences - https://arxiv.org/abs/2512.17544 - arXiv:2512.17544v1 Announce Type: new -Abstract: In this paper, we study the famous Erd\H{o}s--S\'os forbidden intersection problem for words over alphabet of size $m$: what is the maximal size of a subfamily $\mathcal{F}$ of $[m]^n$ that does not contain two vectors $x, y$ that coincide on exactly $t - 1$ coordinates? We answer this question provided $m \ge poly(t)$, $n \ge poly(t)$ for some polynomial function $poly(\cdot)$ of $t$, greatly extending the recent result of Keevash, Lifshitz, Long and Minzer. Our proof combines some of the recently developed methods in extremal combinatorics, including the spread approximation technique of Kupavskii and Zakharov, and the hypercontractivity approach developed in a series of works of Keevash, Keller, Lifshitz, Long, Marcus and Minzer. - oai:arXiv.org:2512.17544v1 - math.CO - Mon, 22 Dec 2025 00:00:00 -0500 + Uniqueness Theorem: With Normal Components Specified on External Spherical Surface + https://arxiv.org/abs/2512.18486 + arXiv:2512.18486v1 Announce Type: new +Abstract: A uniqueness theorem for time-harmonic electromagnetic fields which requires the normal components of electromagnetic fields specified on a spherical surface is proposed and proved. The statement of the theorem is : "For a spherical volume $V$ that contains only perfect conductors and homogeneous lossless materials and for which the impressed currents $\mathbf{J}$ are specified, a time-harmonic solution to the Maxwell's equations within the volume, having outgoing waves alone, is uniquely specified by the values of the radial components of both $\mathbf{E}$ and $\mathbf{B}$ over the exterior spherical surface $V$ and the tangential components of either $\mathbf{E}$ or $\mathbf{B}$ on the interior surfaces." The proof of this theorem relies on the uniqueness of multipole expansion of electromagnetic fields outside the enclosing sphere. The conventional uniqueness theorem for the volume $V$ having loss-less materials is considered to be the case of lossy materials in the limit the dissipation approaching zero. + oai:arXiv.org:2512.18486v1 + math-ph + math.MP + Tue, 23 Dec 2025 00:00:00 -0500 new http://creativecommons.org/licenses/by/4.0/ - Elizaveta Iarovikova, Fedor Noskov, Georgy Sokolov, Nikolai Terekhov + Rajavardhan Talashila - Clique Probing for Mixed-Integer Programs - https://arxiv.org/abs/2512.17551 - arXiv:2512.17551v1 Announce Type: new -Abstract: Probing is an important presolving technique in mixed-integer programming solvers. It selects binary variables, tentatively fixes them to 0 and 1, and performs propagation to deduce additional variable fixings, bound tightenings, substitutions, and implications. In this work, we propose clique probing instead of probing on individual variables, we select cliques, a set of binary variables of which at most one can be set to one, and systematically probe on all variables of a clique. Experiments with our implementation in the open-source presolve library PaPILO demonstrate that exploiting clique information in this form significantly increases the number of reductions. When integrated into the MIP solver SCIP, we observe a 3% performance improvement on MIPLIB instances containing cliques. - oai:arXiv.org:2512.17551v1 - math.OC - Mon, 22 Dec 2025 00:00:00 -0500 + Limits in categories of \'etale groupoids and pseudogroups + https://arxiv.org/abs/2512.18487 + arXiv:2512.18487v1 Announce Type: new +Abstract: We show that the category of sober \'etale groupoids and actors admits all small limits. This is achieved by computing the limits in the equivalent category of pseudogroups with pseudogroup morphisms, which we show admits a forgetful functor to the category of sets which creates limits. We give an alternative proof of the adjunction of Cockett and Garner in the specific setting of \'etale groupoids and pseudogroups which is a central tool for computing limits of sober \'etale groupoids. + oai:arXiv.org:2512.18487v1 + math.CT + math.GN + math.OA + Tue, 23 Dec 2025 00:00:00 -0500 new http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Jacob von Holly-Ponientzietz, Alexander Hoen, Mark Turner, Ambros Gleixner + Jonathan Taylor - Journey Through the World of Dynamical Systems on Networks - https://arxiv.org/abs/2512.17571 - arXiv:2512.17571v1 Announce Type: new -Abstract: We present a subjective selection of methods for complex systems analysis ranging from statistical tools through numerical methods based on AI to both linear and non-linear ODEs and PDEs. All the notions apply the network structure and are presented in the context of applied problems to visualise the strengths and drawbacks of the approach. The major aim of capturing such a broad overview is to understand the interrelations between network theories that seem to be distant from the mathematical perspective. - oai:arXiv.org:2512.17571v1 + Limit theorems for inhomogeneous random walks on $GL(d,\mathbb R)$ + https://arxiv.org/abs/2512.18494 + arXiv:2512.18494v1 Announce Type: new +Abstract: We prove Berry-Esseen theorems, almost sure invariance principle rates and large deviations for products of independent but not identically distributed invertible matrices with some average (logarithmic) projective contraction and uniform boundedness assumptions. We also characterize the divergence of the variance of the logarithm of the norm of the product. Our approach is based on verifying the conditions of \cite{NewBE} after reversing time. + oai:arXiv.org:2512.18494v1 + math.PR math.DS - Mon, 22 Dec 2025 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - A. Puchalska, M. N. Cartier van Dissel, P. Gora, M. Iskrzy\'nski, M. Kramar Fijav\v{z}, D. Manea, A. Mauroy, I. Naki\'c, S. Nicaise, M. B. Paradowski, G. Rotundo, E. Sikolya - - - A Separation Principle for Conditional Mean-Field Type Linear Quadratic Optimal Control Problem - https://arxiv.org/abs/2512.17572 - arXiv:2512.17572v1 Announce Type: new -Abstract: This paper investigates a conditional mean-field type linear quadratic (LQ) optimal control problem with partial observation and regime switching, where the conditional expectations of the state and control given the history of Markov chain enter into the dynamics and cost. The exact regime of Markov chain is accessible, whereas the system state can only be partially observed. A separation principle is established, showing that the estimate and control procedures can be separated and implemented independently. It extends the classical separation principle to conditional mean-field system. Utilizing two sets of Riccati equations and a set of first-order ordinary differential equations, we derive the feedback representation of the optimal control. To illustrate the effectiveness of the theoretical results, two applications with numerical simulations are provided, including a one-dimensional LQ example and a coupled electrical machines control problem. - oai:arXiv.org:2512.17572v1 - math.OC - Mon, 22 Dec 2025 00:00:00 -0500 + Tue, 23 Dec 2025 00:00:00 -0500 new http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Zhongbin Guo, Guangchen Wang + Yeor Hafouta - Extending Fibrations of the $3$-Torus and Applications to Torus Surgery in $4$-Manifolds - https://arxiv.org/abs/2512.17595 - arXiv:2512.17595v1 Announce Type: new -Abstract: Suppose that $W$ and $W'$ are smooth, compact, and oriented $4$-manifolds that are either diffeomorphic to $S^1$ times the exterior $E_Y(K)$ of a fibered knot $K$ in a closed, connected, orientable $3$-manifold $Y$, or are diffeomorphic to $\Sigma_{g,1}$ bundles over the $2$-torus with monodromy fixing the boundary of the fiber pointwise. If $f: \partial W' \to \partial W$ is an orientation-preserving diffeomorphism of the $3$-torus boundaries, we have that $X = W \cup_f W'$ is a closed, oriented $4$-manifold that fibers over $S^1$. In particular, if $W' = T^2 \times D^2$ and $W = S^1 \times E_Y(K)$, then our result shows that the result of doing torus surgery in $S^1\times Y$ along $S^1 \times K$ is a $4$-manifold that fibers over $S^1$. Furthermore, we extend work of Zentner by showing that the result of torus surgery along $S^1$ times the unknot $\mathcal{U}$ in $S^1 \times S^3$ is diffeomorphic to $S^1$ times a lens space. - oai:arXiv.org:2512.17595v1 - math.GT - Mon, 22 Dec 2025 00:00:00 -0500 + Derivation of stochastic Burgers on the line with a Dirichlet boundary condition at the origin + https://arxiv.org/abs/2512.18497 + arXiv:2512.18497v1 Announce Type: new +Abstract: We analyze the \emph{equilibrium fluctuations} of a Hamiltonian chain of oscillators on \(\mathbb{Z}\) with an exponential potential, perturbed by a conservative, symmetric noise. Under the canonical \emph{diffusive scaling} \(t \mapsto t n^2\) and an interaction strength tuned by \(n^{-1/2}\), the fluctuation field is known to converge to the \emph{energy solution} of the stochastic Burgers equation (SBE) on the torus~\cite{ABGS22}. We introduce a \emph{coupled moving heat bath} of strength \(n^{-\delta}\) acting on the particle system. We prove that for \(\delta \leq 1\) (the \emph{strong-coupling regime}), the equilibrium fluctuation field converges to the \emph{energy solution of the SBE with a Dirichlet boundary condition at zero}. We provide two distinct analytical characterizations of these boundary solutions, corresponding to different spaces of test functions. Conversely, for \(\delta > 1\) (the \emph{weak-coupling regime}), the heat bath becomes irrelevant in the scaling limit: the fluctuations converge to the standard SBE on the full line without any boundary condition, reproducing the full-line result of~\cite{GJ14}. Our analysis thus reveals a sharp \emph{critical scaling} in the coupling strength \(\delta\), which dictates the emergence -- or absence -- of a macroscopic boundary condition from the microscopic perturbation. + oai:arXiv.org:2512.18497v1 + math.PR + math-ph + math.MP + Tue, 23 Dec 2025 00:00:00 -0500 new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Nicholas Meyer + http://creativecommons.org/licenses/by/4.0/ + C\'edric Bernardin, Ana Djurdjevac, Patricia Gon\c{c}alves, Leander Schnee - Les Houches Lectures on Exact WKB Analysis and Painlev\'e Equations - https://arxiv.org/abs/2512.17599 - arXiv:2512.17599v1 Announce Type: new -Abstract: The first part of these lecture notes is devoted to an introduction to the theory of exact WKB analysis for second-order Schr\"odinger-type ordinary differential equations. It reviews the construction of the WKB solution, Borel summability, connection formulas, and their application to direct monodromy problems. - In the second part, we discuss recent developments in applying exact WKB analysis to the study of Painlev\'e equations. By combining exact WKB analysis with topological recursion, it becomes possible to explicitly compute the monodromy of linear differential equations associated with Painlev\'e equations, assuming Borel summability and other conditions. Furthermore, by using isomonodromy deformations (integrability of the Painlev\'e equations), the resurgent structure of the $\tau$-function and partition function is analyzed. - These lecture notes accompanied a series of lectures at the Les Houches school, ``Quantum Geometry (Mathematical Methods for Gravity, Gauge Theories and Non-Perturbative Physics)'' in Summer 2024. - oai:arXiv.org:2512.17599v1 + Electromagnetic Modes in Spherical Cavities: Complete Theory of Angular Spectra, Dispersion Relations, and Self-Adjoint Extensions + https://arxiv.org/abs/2512.18498 + arXiv:2512.18498v1 Announce Type: new +Abstract: We present a complete theory of electromagnetic modes in spherical cavities, resolving fundamental questions about the nature of angular quantization. The standard result that angular indices $(\ell,m)$ must be integers is shown to be a consequence of domain constraints -- regularity at both poles and single-valuedness in the azimuthal coordinate -- rather than a requirement imposed by Maxwell's equations themselves. We prove that, for the sectoral case $\nu=m$, the function $\sin^{m}\theta$ exactly solves the angular eigenvalue equation for any real $m>0$, giving rise to a continuous dispersion curve. We demonstrate why non-sectoral modes (tesseral and zonal) appear only at isolated integer points on the full sphere, and show how boundary modifications such as cones and wedges convert these isolated points into continuous families of modes. Complete field solutions, wave impedances, and energy integrability conditions are derived. At the limiting point $(\nu, m) = (0, 0)$, the electromagnetic field vanishes identically while the underlying Debye potential remains non-trivial -- a distinction with implications for mode counting that connects to longstanding questions in gauge theory and cavity quantization. Full-wave simulations validate the theoretical predictions with sub-percent accuracy. These results raise the possibility of structural analogues in wave equations on curved spacetimes, where conical deficits or horizon excisions similarly modify the angular domain. + oai:arXiv.org:2512.18498v1 math-ph - hep-th - math.CA math.MP - nlin.SI - Mon, 22 Dec 2025 00:00:00 -0500 + Tue, 23 Dec 2025 00:00:00 -0500 new http://creativecommons.org/licenses/by/4.0/ - Kohei Iwaki + Mustafa Bakr, Tongyu Zhang, Smain Amari - Locally-APN Binomials with Low Boomerang Uniformity in Odd Characteristic - https://arxiv.org/abs/2512.17603 - arXiv:2512.17603v1 Announce Type: new -Abstract: Recently, several studies have shown that when $q\equiv3\pmod{4}$, the function $F_r(x)=x^r+x^{r+\frac{q-1}{2}}$ defined over $\mathbb{F}_q$ is locally-APN and has boomerang uniformity at most~$2$. In this paper, we extend these results by showing that if there is at most one $x\in \mathbb{F}_q$ with $\chi(x)=\chi(x+1)=1$ satisfying $(x+1)^r - x^r = b$ for all $b\in \mathbb{F}_q^*$ and $\gcd(r,q-1)\mid 2$, then $F_r$ is locally-APN with boomerang uniformity at most $2$. Moreover, we study the differential spectra of $F_3$ and $F_{\frac{2q-1}{3}}$, and the boomerang spectrum of $F_2$ when $p=3$. - oai:arXiv.org:2512.17603v1 - cs.IT - math.IT + Explicit Lower Bounds for Dirichlet Series of Higher Power Representation Functions + https://arxiv.org/abs/2512.18502 + arXiv:2512.18502v1 Announce Type: new +Abstract: We investigate Dirichlet-type series generated by representation functions that count the number of ways an integer can be expressed as a sum of 'k' signed higher even powers. By combining generalized theta generating functions with a family of generalized cotangent series introduced in previous work, we derive two distinct explicit lower bounds for these series. The first estimate arises from a geometric restriction of the lattice to its diagonal, while the second utilizes Holder's inequality on the integral representation of the series. The methods presented here avoid modular techniques and offer a flexible analytic framework for higher-power representation problems. + oai:arXiv.org:2512.18502v1 math.NT - Mon, 22 Dec 2025 00:00:00 -0500 + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://creativecommons.org/licenses/by-nc-nd/4.0/ + Mahipal Gurram + + + Detecting and Quantifying Isolated Singularities over Discrete Valuation Rings + https://arxiv.org/abs/2512.18506 + arXiv:2512.18506v1 Announce Type: new +Abstract: This paper develops a theory of isolated hypersurface singularities in mixed characteristic $(0,p)$, focusing on quotient rings over a Discrete Valuation Ring (DVR). We introduce and study analogues of the classical Tjurina and Milnor numbers for this setting, prove a generalized analogue of the determinacy theorem and the Mather-Yau Theorem for complete Noetherian local rings, and define numerical invariants that provide distinct criteria for detecting isolated singularities in the unramified and ramified cases. + oai:arXiv.org:2512.18506v1 + math.AC + math.AG + Tue, 23 Dec 2025 00:00:00 -0500 new http://creativecommons.org/licenses/by/4.0/ - Namhun Koo, Soonhak Kwon, Minwoo Ko, Byunguk Kim + Yotam Svoray - A characterization of the local structure of two-dimensional sets with positive reach - https://arxiv.org/abs/2512.17606 - arXiv:2512.17606v1 Announce Type: new -Abstract: The main result of the article is a complete characterization of the local structure of two-dimensional sets with positive reach in $R^d$. We also present a more elementary proof of a recent result of A. Lytchak which describes for $k\leq d$ the local structure of $k$-dimensional sets with positive reach $A$ in $R^d$ at points where the tangent cone of $A$ is $k$-dimensional. As an easy corollary of our and Lytchak's results we obtain a characterization of compact two-dimensional sets with positive reach in $R^d$. Our method also shows that, for any set $A\subset R^d$ with positive reach, the set of points at which the tangent cone of $A$ is $k$-dimensional is locally contained in a $k$-dimensional $C^{1,1}$ surface. As a consequence we obtain that if $1\leq k<d$, and $A$ is $k$-dimensional, it can be covered by countably many $k$-dimensional $C^{1,1}$ surfaces. - oai:arXiv.org:2512.17606v1 - math.MG - Mon, 22 Dec 2025 00:00:00 -0500 + Preference-based optimization from noisy pairwise comparisons + https://arxiv.org/abs/2512.18511 + arXiv:2512.18511v1 Announce Type: new +Abstract: In interactive systems, feedback is often provided in the form of preference between queried options rather than precise scores, which motivates optimization methods to learn from such comparisons. In this work, we propose a preference-based optimization algorithm that relies on noisy two-point comparisons. At each iteration, the algorithm employs a uniform-sphere perturbation to generate a perturbed action and queries the resulting loss comparison to estimate a descent direction. We demonstrate that, under standard smoothness and bounded variance assumptions, the algorithm converges to a stationary point when the smoothing and step size parameters are properly chosen. Numerical experiments on an LQG system demonstrate the effectiveness of the preference-based optimization algorithm with comparison feedback. + oai:arXiv.org:2512.18511v1 + math.OC + Tue, 23 Dec 2025 00:00:00 -0500 new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Jan Rataj, Ludek Zajicek + http://creativecommons.org/publicdomain/zero/1.0/ + Siyi Wang, Zifan Wang, Karl Henrik Johanssson - Symmetry Breaking in Biharmonic Equations with Weighted Exponential Nonlinearities - https://arxiv.org/abs/2512.17611 - arXiv:2512.17611v1 Announce Type: new -Abstract: nonlinearities and spatial weights of H\'enon type. Motivated by the symmetry-breaking phenomena observed in semilinear second-order problems -- such as those governed by the H\'enon equation -- we consider weighted functionals of the form \begin{equation*} F_m(u) = \int_B |x|^\alpha \left( e^{\sigma |u|^2} - \sum_{k=0}^m \frac{\sigma^k}{k!} |u|^{2k} \right) dx, \end{equation*} defined on the unit ball \( B \subset \mathbb{R}^4 \), where $m\in \mathbb N_0$ \( \alpha > 0 \), \( \sigma>0\) are suitable parameters. We first establish an Adams-type inequality with weight, characterizing the sharp threshold for the boundedness of \( F \) on the unit sphere of the biharmonic Sobolev space. Then, we prove that for large values of the weight exponent \( \alpha \), radial symmetry of maximizers is broken. %, i.e., the supremum of the functional is strictly larger when taken over the full space compared to the radial subspace. These results extend classical findings in the second-order setting (e.g., Trudinger--Moser-type functionals and the weighted H\'enon equation) - to the biharmonic context and offer new insights into the interplay between weights, nonlinearity, and symmetry in higher-order PDEs. - oai:arXiv.org:2512.17611v1 - math.AP - Mon, 22 Dec 2025 00:00:00 -0500 + The Narrow Corridor of Stable Solutions in an Extended Osipov--Lanchester Model with Constant Total Population + https://arxiv.org/abs/2512.18515 + arXiv:2512.18515v1 Announce Type: new +Abstract: This paper considers a modification of the classical Osipov--Lanchester model in which the total population of the two forces $N=R+B$ is preserved over time. It is shown that the dynamics of the ratio $y=R/B$ reduce to the Riccati equation $\dot y=\alpha y^2-\beta$, which admits a complete analytical study. The main result is that asymptotically stable invariant sets in the positive quadrant $R,B\ge 0$ exist exactly in three sign cases of $(\alpha,\beta)$: (i) $\alpha<0,\beta<0$ (stable interior equilibrium), (ii) $\alpha=0,\beta<0$ (the face $B=0$ is stable), (iii) $\alpha<0,\beta=0$ (the face $R=0$ is stable). For $\alpha>0$ or $\beta>0$ the solutions reach the boundaries of applicability of the model in finite time. Moreover, $\alpha<0,\beta<0$ corresponds to exponential growth of solutions in the original system. Passing to a model perturbed in $\alpha(t),\beta(t)$ requires buffer dynamics repelling from the axes to preserve stability of the solution. + oai:arXiv.org:2512.18515v1 + math.DS + econ.GN + math.OC + q-fin.EC + Tue, 23 Dec 2025 00:00:00 -0500 new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Calanchi M., Tarsi C + http://creativecommons.org/licenses/by-sa/4.0/ + Sergey Salishev - Isomorphic Loday functors of non-homeomorphic spaces - https://arxiv.org/abs/2512.17618 - arXiv:2512.17618v1 Announce Type: new -Abstract: Each commutative algebra $A$ gives rise to a representation $\mathcal{L}_A$, which we call the Loday functor of $A$, of the category $\Omega$ of finite sets and surjective maps. In this paper we present two (infinite-dimensional) non-isomorphic algebras over $\mathbb{C}$ with isomorphic Loday functors -- the algebras of continuous functions on the M\"obius strip and on the cylinder. - oai:arXiv.org:2512.17618v1 - math.AC - math.AT - math.RT - Mon, 22 Dec 2025 00:00:00 -0500 + Localization of the 1D Non-Stationary Anderson Model + https://arxiv.org/abs/2512.18520 + arXiv:2512.18520v1 Announce Type: new +Abstract: This paper considers the family of Schr\"odinger operators on $\ell^2(\mathbb{Z})$ given by independent but not necessarily identically distributed and possibly unbounded potentials. We assume a finite exponential moment and allow the choice of distributions to come from any compact set away from deterministic distributions. With these assumptions we prove spectral localization with exponentially decaying eigenfunctions as well as dynamical localization. One of the main tools is a Furstenberg-type theorem for non-stationary matrix products. + oai:arXiv.org:2512.18520v1 + math-ph + math.DS + math.MP + math.SP + Tue, 23 Dec 2025 00:00:00 -0500 new - http://creativecommons.org/licenses/by/4.0/ - Igor Baskov + http://creativecommons.org/licenses/by-nc-sa/4.0/ + Karl Zieber - Persistent commutative algebra on graphs and hypergraphs - https://arxiv.org/abs/2512.17619 - arXiv:2512.17619v1 Announce Type: new -Abstract: We introduce a persistent commutative algebra for studying the algebraic and combinatorial evolution of edge ideals of graphs and hypergraphs under filtration. Building on the Persistent Stanley--Reisner Theory (PSRT), we develop the notion of persistent edge ideals and analyze their graded Betti numbers across the filtration of graphs or hypergraphs. To enable this analysis, we establish a persistent extension of Hochster's formula, providing a functorial correspondence between algebraic and topological persistence. We further examine the behavior of Betti splittings in the persistent setting, proving a general inequality that extends the classical splitting result to the filtration of monomial ideals. Motivated by graph-theoretic interpretations, we introduce persistent minimal vertex covers, which encode the temporal structure of combinatorial dependencies within evolving graphs or hypergraphs. Applications to alignment-free genomic classification and molecular isomer discrimination demonstrate the interpretability and representatbility of persistent edge ideals as algebraic invariants, bridging combinatorial commutative algebra and data science. - oai:arXiv.org:2512.17619v1 - math.AC - math.CO - Mon, 22 Dec 2025 00:00:00 -0500 + The Euclidean distance degree of curves: from rational to line multiview varieties + https://arxiv.org/abs/2512.18521 + arXiv:2512.18521v1 Announce Type: new +Abstract: The Euclidean distance (ED) degree is an invariant that measures the algebraic complexity of optimizing the distance function of a point to a model. It has been studied in algebraic statistics, machine learning, and computer vision. In this article, we prove a formula for the ED degree of curves parameterized by rational functions with mild genericity assumptions. We apply our results to resolve conjectures on one-dimensional line multiview varieties from computer vision proposed by Duff and Rydell. + oai:arXiv.org:2512.18521v1 + math.AG + Tue, 23 Dec 2025 00:00:00 -0500 new http://creativecommons.org/licenses/by/4.0/ - Faisal Suwayyid, Guo-Wei Wei + Bella Finkel, Jose Israel Rodriguez - Classification of atmospheric traveling waves at cloud level - https://arxiv.org/abs/2512.17627 - arXiv:2512.17627v1 Announce Type: new -Abstract: We classify within the quasi-geostrophic framework all types of traveling waves in zonal bands of the planetary atmosphere at cloud level according to their wave speeds. This classification pertains to waves of all amplitudes, going beyond the small-amplitude perturbative regime. It provides a structurally robust criterion for determining which traveling-wave profiles are dynamically possible and we show that each wave classification type was observed on Jupiter or Saturn. Building on this classification, we also investigate the related rigidity issue for large-amplitude traveling waves and waves propagating near shear flows. Our study offers a unified quantitative characterization of the intrinsic constraints for traveling waves in the quasi-geostrophic regime of planetary atmospheric flow. - oai:arXiv.org:2512.17627v1 - math.AP + Protecting Human Activity Signatures in Compressed IEEE 802.11 CSI Feedback + https://arxiv.org/abs/2512.18529 + arXiv:2512.18529v1 Announce Type: new +Abstract: Explicit channel state information (CSI) feedback in IEEE~802.11 conveys \emph{transmit beamforming directions} by reporting quantized Givens rotation and phase angles that parametrize the right-singular subspace of the channel matrix. Because these angles encode fine-grained spatial signatures of the propagation environment, recent work have shown that plaintext CSI feedback can inadvertently reveal user activity, identity, and location to passive eavesdroppers. In this work, we introduce a standards-compatible \emph{differentially private (DP) quantization mechanism} that replaces deterministic angular quantization with an $\varepsilon$-DP stochastic quantizer applied directly to the Givens parameters of the transmit beamforming matrix. The mechanism preserves the 802.11 feedback structure, admits closed-form sensitivity bounds for the angular representation, and enables principled privacy calibration. Numerical simulations demonstrate strong privacy guarantees with minimal degradation in beamforming performance. + oai:arXiv.org:2512.18529v1 + cs.IT + cs.CR + cs.NI + math.IT + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Mohamed Seif, Atsutse Kludze, Yasaman Ghasempour, H. Vincent Poor, Doru Calin, Andrea J. Goldsmith + + + Group Contractions via Infinite-Dimensional Lie Theory + https://arxiv.org/abs/2512.18530 + arXiv:2512.18530v1 Announce Type: new +Abstract: Contractions are a procedure to construct a new Lie algebra out of a given one via a singular limit. Specifically, the \.In\"on\"u--Wigner construction starts with a Lie algebra $\mathfrak{g}$ with Lie subalgebra $\mathfrak{h} \subseteq \mathfrak{g}$ and complement $\mathfrak{n}$. Then, the vectors in $\mathfrak{h}$ are rescaled by a formal parameter $\varepsilon \in \mathbb{R}_+$, which effectively turns the Lie bracket $[ \, \cdot \, , \cdot \, ]$ into a formal power series. Notably, the limit $\varepsilon \to 0$ trivialises certain relations, such that the complement $\mathfrak{n}$ becomes an abelian ideal. In the present article, we are not only interested in the limiting Lie algebras and groups, but also in the corresponding series expansions in $\varepsilon$ to understand the limiting behaviour. Particularly, we are interested in how to integrate the `power-series-expanded' Lie algebras to the Lie group level. To this end, we reformulate the above procedure using infinite-dimensional Lie algebras of analytic germs. Then, we apply their integration theory to obtain an extensive analysis of this expansion procedure. In particular, we obtain an explicit construction of the resulting Lie algebras and groups. + oai:arXiv.org:2512.18530v1 math-ph + math.DG + math.GR math.MP - Mon, 22 Dec 2025 00:00:00 -0500 + Tue, 23 Dec 2025 00:00:00 -0500 new http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Adrian Constantin, Zhiwu Lin, Hao Zhu + David Prinz, Alexander Schmeding, Philip K. Schwartz - Iterative Gaussian Approximation for Random Spreading Unsourced Random Access - https://arxiv.org/abs/2512.17628 - arXiv:2512.17628v1 Announce Type: new -Abstract: Massive machine-type communications (mMTC) demand robust solutions to support extensive connectivity efficiently. Unsourced random access (URA) has emerged as a promising approach, delivering high spectral and energy efficiency. Among URA code structures, the random spreading (RS) category is a key enabler, providing strong anti-interference capabilities through spectrum spreading gain. Notably, RS-URA approaches theoretical performance limits over the Gaussian multiple access channel in scenarios with few active users. In this paper, we propose an iterative Gaussian approximation decoder designed universally for RS-URA categories. The proposed receiver iterates extrinsic and intrinsic soft information to enhance decoding performance, requiring only a few iterations to converge. Numerical results validate the decoder's effectiveness in terms of performance and robustness. - oai:arXiv.org:2512.17628v1 + Integrated Control and Communication in LQG Systems + https://arxiv.org/abs/2512.18535 + arXiv:2512.18535v1 Announce Type: new +Abstract: In this paper, we study the Integrated Communication and Control (ICAC) problem. Specifically, we investigate how messages can be transmitted from the controller/encoder to the observer/decoder through the control signal in Multiple-Input Multiple-Output (MIMO) vector-state Linear Quadratic Gaussian (LQG) systems under control constraints. We provide a computable capacity expression using semidefinite programming. We further show that it is possible to transmit data at a nonzero rate over an LQG system while maintaining the same optimal control cost as in the case where no information message are transmitted. Finally, we discuss how this framework generalizes communication over MIMO Gaussian channels with feedback, both with and without InterSymbol Interference (ISI). + oai:arXiv.org:2512.18535v1 cs.IT math.IT - Mon, 22 Dec 2025 00:00:00 -0500 + Tue, 23 Dec 2025 00:00:00 -0500 new http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Liandong Hu, Jian Dang, Zaichen Zhang + Sepehr Jahangiri, H. Ali Talebi - M\"obius function is strongly orthogonal to polynomial phases over $\mathbb{F}_p[t]$ - https://arxiv.org/abs/2512.17633 - arXiv:2512.17633v1 Announce Type: new -Abstract: In this paper, we prove power-saving bounds for the corelation of the M\"obius function with polynomial phases of degree $k$ in function fields $\mathbb{F}_p[t]$, when $p > k$. The proof relies on a new approximation result for phases of biased multilinear forms and the recently established strong bounds for the problem of finding bounded codimension varieties inside the dense ones. Along the way, we also obtain polynomial bounds in the inverse theorem for Gowers uniformity norms in the special case of polynomial phases in finite vector spaces. - oai:arXiv.org:2512.17633v1 + Quasipolynomial behavior via constructability in multigraded algebra + https://arxiv.org/abs/2512.18536 + arXiv:2512.18536v1 Announce Type: new +Abstract: Piecewise quasipolynomial growth of Presburger counting functions combines with tame persistent homology module theory to conclude piecewise quasipolynomial behavior of constructible families of finely graded modules over constructible commutative semigroup rings. Functorial preservation of constructibility for families under local cohomology, $\operatorname{Tor}$, and $\operatorname{Ext}$ yield piecewise quasipolynomial, quasilinear, or quasiconstant growth statements for length of local cohomology, $a$-invariants, regularity, depth; length of $\operatorname{Tor}$ and Betti numbers; length of $\operatorname{Ext}$ and Bass numbers; associated primes via $v$-invariants; and extended degrees, including the usual degree, Hilbert--Samuel multiplicity, arithmetic degree, and homological~degree. + oai:arXiv.org:2512.18536v1 + math.AC math.CO - math.NT - Mon, 22 Dec 2025 00:00:00 -0500 + math.LO + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Hailong Dao, Ezra Miller, Jonathan Monta\~no, Christopher O'Neill, Kevin Woods + + + Global Regular Solutions of the Multidimensional Degenerate Compressible Navier-Stokes Equations with Large Initial Data of Spherical Symmetry + https://arxiv.org/abs/2512.18545 + arXiv:2512.18545v1 Announce Type: new +Abstract: A fundamental open problem in the theory of the compressible Navier-Stokes equations is whether regular spherically symmetric flows can develop singularities -- such as cavitation or implosion -- in finite time. A formidable challenge lies in how the well-known coordinate singularity at the origin can be overcome to control the lower or upper bound of the density. For the barotropic Navier-Stokes system with constant viscosity coefficients, recent striking results have shown that such implosions do indeed occur. In this paper, we show that the situation is fundamentally different when the viscosity coefficients are degenerately density-dependent (as in the shallow water equations). We prove that, for general large spherically symmetric initial data with bounded positive density, solutions remain globally regular and cannot undergo cavitation or implosion in two and three spatial dimensions. Our results hold for all adiabatic exponents $\gamma\in (1,\infty)$ in two dimensions, and for physical adiabatic exponents $\gamma\in (1, 3)$ in three dimensions, without any restriction on the size of the initial data. To achieve these results, we make carefully designed weighted radial estimates via a region segmentation method, which is the key for obtaining uniform control over both the density and the effective velocity. The methodology developed here should also be useful for solving other related nonlinear partial differential equations involving similar difficulties. + oai:arXiv.org:2512.18545v1 + math.AP + math-ph + math.MP + physics.flu-dyn + Tue, 23 Dec 2025 00:00:00 -0500 new http://creativecommons.org/licenses/by/4.0/ - Luka Mili\'cevi\'c, \v{Z}arko Ran{\dj}elovi\'c + Gui-Qiang G. Chen, Jiawen Zhang, Shengguo Zhu - A Conjugate Gradient Method for Nonlinear Programming Problems using Caputo Fractional Gradients - https://arxiv.org/abs/2512.17634 - arXiv:2512.17634v1 Announce Type: new -Abstract: The article proposes a Caputo fractional conjugate gradient (CFCG) method for unconstrained optimization problems which is applicable to smooth as well as non-smooth problmes. The proposed method uses a non-adaptive version of the Caputo fractional derivative that provides integer-order derivatives information. A descent direction is obtained using the Caputo fractional gradients of two consecutive iterative points with a parameter ($\beta$). An inexact line search technique based on Armijo-Wolfe line conditions is used to find a suitable step length. Finally, a descent sequence is generated. The convergence results are derived under mild assumptions that ensuring of convergence is at least linear. Moreover, the convergence of the proposed method for quadratic functions is established through a Tikhonov-regularized formulation that can be interpreted as an extension of the least-squares approach. Finally, some numerical experiments, including neural network applications, are performed to justify that the proposed method achieves faster and more stable performance. - oai:arXiv.org:2512.17634v1 + An adaptive adjoint-oriented neural network for solving parametric optimal control problems with singularities + https://arxiv.org/abs/2512.18548 + arXiv:2512.18548v1 Announce Type: new +Abstract: In this work, we present an adaptive adjoint-oriented neural network (adaptive AONN) for solving parametric optimal control problems governed by partial differential equations. The proposed method integrates deep adaptive sampling techniques with the adjoint-oriented neural network (AONN) framework. It alleviates the limitations of AONN in handling low-regularity solutions and enhances the generalizability of deep adaptive sampling for surrogate modeling without labeled data ($\text{DAS}^2$). The effectiveness of the adaptive AONN is demonstrated through numerical examples involving singularities. + oai:arXiv.org:2512.18548v1 math.OC - Mon, 22 Dec 2025 00:00:00 -0500 + Tue, 23 Dec 2025 00:00:00 -0500 new - http://creativecommons.org/licenses/by/4.0/ - Barsha Shawa, Md Abu Talhamainuddin Ansary + http://creativecommons.org/licenses/by-nc-nd/4.0/ + Zikang Yuan, Guanjie Wang, Qifeng Liao - Perturbative Chern-Simons invariants from non-acyclic flat connections - https://arxiv.org/abs/2512.17638 - arXiv:2512.17638v1 Announce Type: new -Abstract: We give a short review of our construction of a higher-loop perturbative invariant of framed 3-manifolds, generalizing the perturbative Chern-Simons invariant of Witten-Axelrod-Singer, associated to an acyclic flat connection, to an invariant given by the integral of a certain "Chern-Simons volume form" over a smooth closed component of the moduli space of flat connections. - oai:arXiv.org:2512.17638v1 - math-ph - hep-th - math.GT - math.MP - Mon, 22 Dec 2025 00:00:00 -0500 + Infinitely many solutions for a class of resonant problems + https://arxiv.org/abs/2512.18562 + arXiv:2512.18562v1 Announce Type: new +Abstract: We consider radially symmetric solutions for a class of resonant problems on a unit ball $B \subset R^n$ around the origin \[ \Delta u+\la _1 u +g(u)=f(r) \s \mbox{for $x \in B$}, \s u=0 \s \mbox{on $\partial B$} \,. \] Here the function $g(u)$ is periodic of mean zero, $x \in R^n$, $r=|x|$, $\la _1$ is the principal eigenvalue of $\Delta$ on $B$. The problem has either infinitely many or finitely many solutions depending on the space dimension $n$. The situation turns out to be different for each of the following cases: $1 \leq n \leq 3$, $n=4$, $n=5$, $n=6$, and $n \geq 7$. + oai:arXiv.org:2512.18562v1 + math.AP + math.DS + Tue, 23 Dec 2025 00:00:00 -0500 new http://creativecommons.org/licenses/by/4.0/ - Pavel Mnev, Konstantin Wernli + NoDEA Nonlinear Differential Equations Appl. 30 (2023), no. 4, Paper No. 48 + Philip Korman - Stability analysis for active Brownian particle models - https://arxiv.org/abs/2512.17649 - arXiv:2512.17649v1 Announce Type: new -Abstract: We carry out a comprehensive linear stability analysis of active Brownian particle systems around a constant homogeneous state. These scalar models, being important prototypes for the continuous description of active matter, are Fokker-Planck type equations in position-orientation and are known to exhibit motility-induced phase separation. We fully characterize the linear stability and instability regimes, with an explicit threshold depending on the effective speed of the particles. In this way, we rigorously confirm a conjecture on phase separation originating in the physics and applied literature. Our sharp and quantitative (in)stability results are valid both in the non-diffusive case and in the case of small angular diffusion. In the stable non-diffusive regime, we uncover a mixing mechanism reminiscent of Landau damping for the Vlasov equation, albeit with significantly weaker decay. This decay is non-integrable in time and gives rise to substantial mathematical difficulties; in particular, it prevents the use of classical perturbative arguments to treat the case of small angular diffusion. - oai:arXiv.org:2512.17649v1 + Sharp criteria for a degenerate diffusion-aggregation system with the intermediate exponent + https://arxiv.org/abs/2512.18576 + arXiv:2512.18576v1 Announce Type: new +Abstract: In this paper, we investigate a multi-dimensional nonlocal degenerate diffusion-aggregation equation with a diffusion exponent $m$ in the intermediate range $\frac{2d}{2d-\gamma}<m<\frac{d+\gamma}{d}$, where the nonlocal aggregation term is given by singular potential $|x|^{-\gamma}$, $0<\gamma\leq d-2$. Under two different assumptions on the initial data, we establish two sharp criteria (i.e., the critical thresholds in Theorem 1.1 and Theorem 1.2) governing the global existence and finite-time blow-up of solutions. Once the initial free energy is less than a constant that depends on the total mass (or depends on the extremum function of the Hardy-Littlewood-Sobolev inequality), the first criterion depends on the relationship between the $L^{\frac{2d}{2d-\gamma}}$-norm of initial data and total mass, while the second relies on the relationship between the $L^m$-norm of initial data and extremal function. + In the discussion of the second criterion, we do not require $L^\infty(\mathbb{R}^d)$ boundedness of the initial data, which is necessary in reference \cite{B}. Furthermore, with the help of moment estimate, we manage to prove the compactness argument on the whole space by using the Lions-Aubin Lemma. + Importantly, we demonstrate that the two initial free energy conditions on which two criteria are based are equivalent. Building on this, we further prove that the two sharp criteria themselves are also equivalent, thereby unifying the classification results obtained from two different approaches. + oai:arXiv.org:2512.18576v1 math.AP - Mon, 22 Dec 2025 00:00:00 -0500 + Tue, 23 Dec 2025 00:00:00 -0500 new http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Michele Coti Zelati, Lucas Ertzbischoff, David Gerard-Varet + Tiantian Zhou, Li Chen, Yutian Lei - Horizons in noncompact fill-ins of nonnegative scalar curvature - https://arxiv.org/abs/2512.17662 - arXiv:2512.17662v1 Announce Type: new -Abstract: Given a complete Riemannian metric of nonnegative scalar curvature on $\Sigma \times (-\infty, 0 ] $, where $\Sigma$ denotes a $2$-sphere, we exhibit conditions that imply the existence of a closed minimal surface homologous to the boundary. - oai:arXiv.org:2512.17662v1 + Mass of $C^0$-asymptotically hyperbolic spaces via the normalized Ricci-DeTurck flow + https://arxiv.org/abs/2512.18578 + arXiv:2512.18578v1 Announce Type: new +Abstract: We define a mass function on asymptotically hyperbolic manifolds with continuous metrics via the normalized Ricci DeTurck flow. + This definition coincides with the classical mass for smooth metrics. We also introduce the scalar curvature lower bound for continuous metrics a key component in establishing the well-definedness of the continuous mass. + oai:arXiv.org:2512.18578v1 math.DG - gr-qc - Mon, 22 Dec 2025 00:00:00 -0500 + Tue, 23 Dec 2025 00:00:00 -0500 new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Pengzi Miao, Sehong Park + http://creativecommons.org/licenses/by-sa/4.0/ + Yuqiao Li - Around Segal conjecture in p-adic geometry - https://arxiv.org/abs/2512.17665 - arXiv:2512.17665v1 Announce Type: new -Abstract: This article records multiple results coming from interplay between de-completed topological periodic cyclic homology, Segal conjecture, and F-smoothness. - We establish completeness of motivic filtration on de-completed topological periodic cyclic homology of commutative rings with weakly finitely generated absolute cotangent complex. When the ring in question is in addition F-smooth, we show that Segal conjecture holds for its topological Hochschild homology. We also identify our de-completed topological periodic cyclic homology with Manam's Frobenius untwisted topological periodic cyclic homology for quasiregular semiperfectoid rings. - We find a crystalline degeneration of Segal conjecture which corresponds to such a statement for F-smoothness. On the other hand, inspired by constructions for topological Hochschild homology, the theory of cyclotomic synthetic spectra allows us to produce a relative conjugate filtration on Hodge--Tate cohomology and its variants, and in the same time, a relative conjugate filtration on topological Hochschild homology and its variants. As a consequence, we deduce transitivity of weak and strong F-smoothness. - oai:arXiv.org:2512.17665v1 - math.KT - math.AG - math.AT - Mon, 22 Dec 2025 00:00:00 -0500 + Geometry-dependent Ekman layer approximations on curved domains: L^{\infty} convergence + https://arxiv.org/abs/2512.18579 + arXiv:2512.18579v1 Announce Type: new +Abstract: The Ekman boundary layer is a fundamental concept in fluid dynamics that describes fluid motion near boundaries affected by Earth's rotation. Most theoretical studies have simplified their analysis by assuming a planar boundary surface, resulting in limited exploration of structures with general smooth boundary conditions. Investigating the impact of boundary geometry in the Ekman boundary layer is essential, as initially suggested by J.L. Lions and further examined in Masmoudi's study [Comm. Pure Appl. Math. 53 (2000), 432-483] under small amplitude periodic boundary conditions. This paper clarifies how boundary geometry influences flow fields and characterizes its effects on near-boundary layer flow. We construct a class of multi-scale approximate solutions based on the boundary's geometric features and establish their convergence in the L^{\infty} framework. Our findings do not require a small-amplitude assumption, only an upper bound on the Gaussian curvature of the boundary surface. Notably, when the boundary is planar, our approach aligns with existing studies. Additionally, in the vanishing-viscosity limit, we derive a limiting-state system dependent on boundary geometric parameters. These contributions extend the theoretical understanding of boundary-layer interactions to general curved geometries and have possible applications in atmospheric, oceanic, and other geophysical flow contexts. + oai:arXiv.org:2512.18579v1 + math-ph + math.DS + math.MP + physics.flu-dyn + Tue, 23 Dec 2025 00:00:00 -0500 new http://creativecommons.org/licenses/by/4.0/ - Zhouhang Mao + Yifei Jia, Yi Du, Lihui Guo - Local h-, p-, and k-Refinement Strategies for the Isogeometric Shifted Boundary Method Using THB-Splines - https://arxiv.org/abs/2512.17666 - arXiv:2512.17666v1 Announce Type: new -Abstract: The concept of trimming, embedding, or immersing geometries into a computational background mesh has gained considerable attention in recent years, particularly in isogeometric analysis (IGA). In this approach, the physical domain is represented independently from the computational mesh, allowing the latter to be generated more easily compared with body-fitted meshes. While this facilitates the treatment of complex geometries, it also introduces challenges, such as ill-conditioning of the stiffness matrix caused by small cut elements and difficulties in accurately enforcing boundary conditions. A recently proposed technique to address these issues is the Shifted Boundary Method (SBM), which represents the computational domain solely through uncut elements and enforces boundary conditions via a Taylor expansion from a surrogate boundary to the true boundary. Previous studies have shown that, for Neumann boundary conditions, the flux evaluation requires additional derivatives in the Taylor expansion, effectively reducing the order of convergence by one. In this work, we investigate for the first time the performance of SBM combined with Truncated Hierarchical B-splines (THB-splines) under various local refinement strategies. In particular, we propose local p- and k-refinement schemes for THB-splines and compare them with local h-refinement and the unmodified SBM. Furthermore, we propose an enhanced shift operator that incorporates mixed partial derivatives, in contrast to the standard operator. The study assesses accuracy, stability, and computational efficiency for benchmark problems on trimmed domains. The results highlight how different refinement strategies affect convergence behavior in trimmed IGA formulations using SBM and demonstrate that targeted degree elevation can mitigate the Neumann boundary limitations of the standard method. - oai:arXiv.org:2512.17666v1 - math.NA - cs.NA - Mon, 22 Dec 2025 00:00:00 -0500 + Trigonometric Determinants via special values of Dirichlet $L$-Functions + https://arxiv.org/abs/2512.18581 + arXiv:2512.18581v1 Announce Type: new +Abstract: In this paper, we investigate the determinants involving some trigonometric functions. We establish a connection between these determinants and the special values of Dirichlet L-functions, thereby extending Guo's results to arbitrary positive integers n. In addition, we also prove a conjecture raised by Zhi-Wei Sun. Our main tool is the spectral decomposition of some linear operators. By the same method we obtain an explicit formula for the determinants of sine matrices. This formula is expressed as a product of Gauss sums attached to Dirichlet characters. + oai:arXiv.org:2512.18581v1 + math.NT + Tue, 23 Dec 2025 00:00:00 -0500 new http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Christoph Hollweck, Andrea Gorgi, Nicolo Antonelli, Marcus Wagner, Roland W\"uchner + Liwen Gao, Xuejun Guo - Prescribing the mean curvature of an achronal hypersurface as a measure: the case of 3D spacetimes - https://arxiv.org/abs/2512.17670 - arXiv:2512.17670v1 Announce Type: new -Abstract: We study the existence problem for achronal hypersurfaces $M \hookrightarrow \overline{M}$ in a globally hyperbolic spacetime, whose mean curvature is a prescribed -- possibly singular -- source, and whose boundary is a given smooth spacelike submanifold. Since $M$ is allowed to go null somewhere, the mean curvature prescription is to be understood in the distributional sense. We prove a general existence and regularity theorem for surfaces in ambient dimension $3$. Although most of our estimates hold in any dimension, recent counterexamples show that some of our conclusions fail in ambient dimension at least $5$. The case of $4$D-spacetimes is an open problem. Our theorems have application to Born-Infeld electrostatics in general static spacetimes. - oai:arXiv.org:2512.17670v1 - math.DG - math-ph - math.MP - Mon, 22 Dec 2025 00:00:00 -0500 + Overcoming Spectral Bias via Cross-Attention + https://arxiv.org/abs/2512.18586 + arXiv:2512.18586v1 Announce Type: new +Abstract: Spectral bias implies an imbalance in training dynamics, whereby high-frequency components may converge substantially more slowly than low-frequency ones. To alleviate this issue, we propose a cross-attention-based architecture that adaptively reweights a scaled multiscale random Fourier feature bank with learnable scaling factors. The learnable scaling adjusts the amplitudes of the multiscale random Fourier features, while the cross-attention residual structure provides an input-dependent mechanism to emphasize the most informative scales. As a result, the proposed design accelerates high-frequency convergence relative to comparable baselines built on the same multiscale bank. Moreover, the attention module supports incremental spectral enrichment: dominant Fourier modes extracted from intermediate approximations via discrete Fourier analysis can be appended to the feature bank and used in subsequent training, without modifying the backbone architecture. + We further extend this framework to PDE learning by introducing a linear combination of two sub-networks: one specialized in capturing high-frequency components of the PDE solution and the other in capturing low-frequency components, with a learnable (or optimally chosen) mixing factor to balance the two contributions and improve training efficiency in oscillatory regimes. Numerical experiments on high-frequency and discontinuous regression problems, image reconstruction tasks, as well as representative PDE examples, demonstrate the effectiveness and robustness of the proposed method. + oai:arXiv.org:2512.18586v1 + math.NA + cs.NA + Tue, 23 Dec 2025 00:00:00 -0500 new http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Lorenzo Maniscalco, Luciano Mari + Xiaodong Feng, Tao Tang, Xiaoliang Wan, Tao Zhou - On the supremum and its location of the standardized uniform empirical process - https://arxiv.org/abs/2512.17674 - arXiv:2512.17674v1 Announce Type: new -Abstract: We show that the maximizing point and the supremum of the standardized uniform empirical process converge in distribution. Here, the limit variable (Z, Y ) has independent components. Moreover, Z attains the values zero and one with equal probability one half and Y follows the Gumbel-distribution. - oai:arXiv.org:2512.17674v1 - math.PR + Graphon-Level Bayesian Predictive Synthesis for Random Network + https://arxiv.org/abs/2512.18587 + arXiv:2512.18587v1 Announce Type: new +Abstract: Bayesian predictive synthesis provides a coherent Bayesian framework for combining multiple predictive distributions, or agents, into a single updated prediction, extending Bayesian model averaging to allow general pooling of full predictive densities. This paper develops a static, graphon level version of Bayesian predictive synthesis for random networks. At the graphon level we show that Bayesian predictive synthesis corresponds to the integrated squared error projection of the true graphon onto the linear span of the agent graphons. We derive nonasymptotic oracle inequalities and prove that least-squares graphon-BPS, based on a finite number of edge observations, achieves the minimax L^2 rate over this agent span. Moreover, we show that any estimator that selects a single agent graphon is uniformly inconsistent on a nontrivial subset of the convex hull of the agents, whereas graphon-level Bayesian predictive synthesis remains minimax-rate optimal-formalizing a combination beats components phenomenon. Structural properties of the underlying random graphs are controlled through explicit Lipschitz bounds that transfer graphon error into error for edge density, degree distributions, subgraph densities, clustering coefficients, and giant component phase transitions. Finally, we develop a heavy tail theory for Bayesian predictive synthesis, showing how mixtures and entropic tilts preserve regularly varying degree distributions and how exponential random graph model agents remain within their family under log linear tilting with Kullback-Leibler optimal moment calibration. + oai:arXiv.org:2512.18587v1 math.ST + stat.ME stat.TH - Mon, 22 Dec 2025 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Dietmar Ferger - - - Upper Bounds for Sequence Saturation - https://arxiv.org/abs/2512.17683 - arXiv:2512.17683v1 Announce Type: new -Abstract: In this paper, we study the saturation function $\mathrm{Sat}(n,u)$ for sequences. Saturation for sequences was introduced by Anand, Geneson, Kaustav, and Tsai (2021), who proved that $\mathrm{Sat}(n,u)=O(n)$ for two-letter sequences $u$ and conjectured that this bound holds for all sequences. We present an algorithm that constructs a $u$-saturated sequence on $n$ letters and apply it to show $\mathrm{Sat}(n,u)=O(n)$ for several families of sequences $u$, including all repetitions of the form $abcabc\dots$. We further establish $\mathrm{Sat}(n,u)=O(n)$ for a broad class of sequences of the form $aa\dots bb$. In addition, we prove that for most sequences $u$, there exists an infinite $u$-saturated sequence. For three-letter sequences of the form $abc\dots xyz$, where $a,b,c$ are distinct and $xyz$ is a permutation of $abc$, we show -- under certain structural assumptions on $u$ -- that $\mathrm{Sat}(n,u)=O(n)$. Finally, we describe a linear program that computes the exact value of $\mathrm{Sat}(n,u)$ for arbitrary $n$ and $u$. - oai:arXiv.org:2512.17683v1 - math.CO - Mon, 22 Dec 2025 00:00:00 -0500 + Tue, 23 Dec 2025 00:00:00 -0500 new http://creativecommons.org/licenses/by/4.0/ - Shihan Kanungo + Marios Papamichalis, Regina Ruane - Cartan subproduct systems - https://arxiv.org/abs/2512.17690 - arXiv:2512.17690v1 Announce Type: new -Abstract: Given a semisimple compact Lie group $G$ and a nonzero dominant integral weight $\lambda$, the highest weight $G_q$-modules $V_{n\lambda}$ form a subproduct system of finite dimensional Hilbert spaces. Using a conjectural asymptotic behavior of Clebsch-Gordan coefficients we identify the corresponding Cuntz-Pimsner algebras with algebras of quantized functions on homogeneous spaces of $G$. We also show that the gauge-invariant part of the Toeplitz algebra provides a model for convergence of full matrix algebras to quantum flag manifolds, complementing and generalizing results of Landsman and Rieffel for $q=1$ and results of Vaes-Vergnioux in the rank one case for $q\ne1$. - We verify our conjecture on Clebsch-Gordan coefficients for $G=SU(n)$ and all weights that are either regular or multiples of the fundamental weight $\omega_1$. For $\lambda=\omega_1$, we also provide a detailed description of the Toeplitz and Cuntz-Pimsner algebras, generalizing results of Arveson on symmetric subproduct systems. - oai:arXiv.org:2512.17690v1 - math.OA - math.QA - math.RT - Mon, 22 Dec 2025 00:00:00 -0500 + On the subgaussian comparison theorem + https://arxiv.org/abs/2512.18588 + arXiv:2512.18588v1 Announce Type: new +Abstract: The aim of this expository note is to prove that any $1$-subgaussian random vector is dominated in the convex ordering by a universal constant times a standard Gaussian vector. This strengthens Talagrand's celebrated subgaussian comparison theorem. The proof combines a tensorization argument due to J. Liu with ideas that date back to the work of Fernique. + oai:arXiv.org:2512.18588v1 + math.PR + Tue, 23 Dec 2025 00:00:00 -0500 new http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Suvrajit Bhattacharjee, Olof Giselsson, Sergey Neshveyev + Ramon van Handel - Antimagicness of graphs with a dominating clique - https://arxiv.org/abs/2512.17693 - arXiv:2512.17693v1 Announce Type: new -Abstract: A graph $G = (V, E)$ is called antimagic if there exists a bijective labelling $f : E \rightarrow \{1, 2, \ldots, |E|\}$ such that the vertex-sums of labels over edges incident to a given vertex are all distinct. In this paper, we extend the antimagicness results over graphs with a dominating clique. We also introduce an alternative to the usual definition of antimagic graphs, called C-antimagic, allowing for the labelling to be injective in $\{1, 2, . . . , |E| + C\}$ instead of bijective, and show that almost all graphs with a dominating clique are 3-antimagic. - oai:arXiv.org:2512.17693v1 - math.CO - Mon, 22 Dec 2025 00:00:00 -0500 + On the mapping class groups of $\mathbb{P}^1$-bundles over $\mathbb{P}^2$ + https://arxiv.org/abs/2512.18590 + arXiv:2512.18590v1 Announce Type: new +Abstract: In this article we compute the mapping class group of the total space $S(\xi)$ of the sphere bundle of a 3-dimensional real vector bundle $\xi$ over the complex projective plane $\mathbb{P}^2$ with $\langle p_1(\xi), [\mathbb{P}^2] \rangle =8n+5$. Examples of these manifolds include the Milnor hypersurface $M_1$ and its generalizations $M_k=\{(x,y)\in \mathbb{P}^2\times\mathbb{P}^2 \ | \ \sum x^k_iy_i=0\}$ with $k$ odd. + oai:arXiv.org:2512.18590v1 + math.GT + math.AT + Tue, 23 Dec 2025 00:00:00 -0500 new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Gr\'egoire Beaudoire, C\'edric Bentz, Christophe Picouleau + http://creativecommons.org/licenses/by/4.0/ + TengLin Hu - A bound on the equivariant unknotting number - https://arxiv.org/abs/2512.17700 - arXiv:2512.17700v1 Announce Type: new -Abstract: We study how the equivariant signature of strongly invertible knots changes under each type of equivariant unknotting move. We prove that the absolute value of the equivariant signature is bounded above by three times the type A equivariant unknotting number, by twice the type C equivariant unknotting number for some $(1,2)$-knots, and, for admissible diagrams, by twice the type B equivariant unknotting number. These results provide analogues in the equivariant setting of the classical bound relating the knot signature to the unknotting number. - oai:arXiv.org:2512.17700v1 - math.GT - Mon, 22 Dec 2025 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Sarah Zampa - - - Characterizations of Almost Ricci Bourguignon Solitons - https://arxiv.org/abs/2512.17704 - arXiv:2512.17704v1 Announce Type: new -Abstract: In this paper, we revisit the study of almost Ricci-Bourguignon solitons by clarifying their position in the broader context of Einstein-type metrics. Motivated by known rigidity results for compact almost Ricci solitons, we aim to identify conditions under which a compact almost RB-soliton is trivial or exhibits special geometric properties. We compare our results with classical theorems of Barros and Ribeiro, and explain explicitly how our work extends or complements these earlier findings. - oai:arXiv.org:2512.17704v1 - math.DG - Mon, 22 Dec 2025 00:00:00 -0500 + Wavelet Latent Position Exponential Random Graphs + https://arxiv.org/abs/2512.18592 + arXiv:2512.18592v1 Announce Type: new +Abstract: Many network datasets exhibit connectivity with variance by resolution and large-scale organization that coexists with localized departures. When vertices have observed ordering or embedding, such as geography in spatial and village networks, or anatomical coordinates in connectomes, learning where and at what resolution connectivity departs from a baseline is crucial. Standard models typically emphasize a single representation, i.e. stochastic block models prioritize coarse partitions, latent space models prioritize global geometry, small-world generators capture local clustering with random shortcuts, and graphon formulations are fully general and do not solely supply a canonical multiresolution parameterization for interpretation and regularization. We introduce wavelet latent position exponential random graphs (WL-ERGs), an exchangeable logistic-graphon framework in which the log-odds connectivity kernel is represented in compactly supported orthonormal wavelet coordinates and mapped to edge probabilities through a logistic link. Wavelet coefficients are indexed by resolution and location, which allows multiscale structure to become sparse and directly interpretable. Although edges remain independent given latent coordinates, any finite truncation yields a conditional exponential family whose sufficient statistics are multiscale wavelet interaction counts and conditional laws admit a maximum-entropy characterization. These characteristics enable likelihood-based regularization and testing directly in coefficient space. The theory is naturally scale-resolved and includes universality for broad classes of logistic graphons, near-minimax estimation under multiscale sparsity, scale-indexed recovery and detection thresholds, and a band-limited regime in which canonical coefficient-space tilts are non-degenerate and satisfy a finite-dimensional large deviation principle. + oai:arXiv.org:2512.18592v1 + math.ST + stat.ME + stat.TH + Tue, 23 Dec 2025 00:00:00 -0500 new http://creativecommons.org/licenses/by/4.0/ - Mohammad Aqib, Hemangi Madhusudan Shah, Dhriti Sundar Patra + Marios Papamichalis, Regina Ruane - On the Complexity of Bipartite Degree Realizability - https://arxiv.org/abs/2512.17709 - arXiv:2512.17709v1 Announce Type: new -Abstract: We study the \emph{Bipartite Degree Realization} (BDR) problem: given a graphic degree sequence $D$, decide whether it admits a realization as a bipartite graph. While bipartite realizability for a fixed vertex partition can be decided in polynomial time via the Gale--Ryser theorem, the computational complexity of BDR without a prescribed partition remains unresolved. We address this question through a parameterized analysis. - For constants $0 \le c_1 \le c_2 \le 1$, we define $\mathrm{BDR}_{c_1,c_2}$ as the restriction of BDR to degree sequences of length $n$ whose degrees lie in the interval $[c_1 n, c_2 n]$. Our main result shows that $\mathrm{BDR}_{c_1,c_2}$ is solvable in polynomial time whenever $0 \le c_1 \le c_2 \le \frac{\sqrt{c_1(c_1+4)}-c_1}{2}$, as well as for all $c_1 > \tfrac12$. The proof relies on a reduction to extremal \emph{least balanced degree sequences} and a detailed verification of the critical Gale--Ryser inequalities, combined with a bounded subset-sum formulation. - We further show that, assuming the NP-completeness of unrestricted BDR, the problem $\mathrm{BDR}_{c_1,c_2}$ remains NP-complete for all $0 < c_2 < \tfrac12$ and $c_1 < 1 - c_2 - \sqrt{1-2c_2}$. Our results clarify the algorithmic landscape of bipartite degree realization and contribute to the broader study of potentially bipartite graphic degree sequences. - oai:arXiv.org:2512.17709v1 - math.CO - cs.CC - Mon, 22 Dec 2025 00:00:00 -0500 + On a Theorem of Grzegorek and Labuda + https://arxiv.org/abs/2512.18594 + arXiv:2512.18594v1 Announce Type: new +Abstract: This paper presents a generalized version of a theorem of Grzegorek and Labuda in category bases and also endeavours to establish a variant formulation of the same in Marczewski structures. + oai:arXiv.org:2512.18594v1 + math.GN + Tue, 23 Dec 2025 00:00:00 -0500 new - http://creativecommons.org/licenses/by-nc-sa/4.0/ - Istv\'an Mikl\'os + http://creativecommons.org/licenses/by-sa/4.0/ + Sanjib Basu, Navdeep Tamang - Study of a TPFA scheme for the stochastic Allen-Cahn equation with constraint though numerical experiments - https://arxiv.org/abs/2512.17712 - arXiv:2512.17712v1 Announce Type: new -Abstract: This contribution provides numerical experiments for a finite volume scheme for an approximation of the stochastic Allen-Cahn equation with homogeneous Neumann boundary conditions. The approximation is done by a Yosida approximation of the subdifferential operator. The problem is set on a polygonal bounded domain in two or three dimensions. The non-linear character of the projection term induces challenges to implement the scheme. To this end, we provide a splitting method for the finite volume scheme. We show that the splitting method is accurate. The computational error estimates induce that the squared $L^2$-error w.r.t. time is of order $1$ as long as the noise term is small enough. For larger noise terms the order of convergence w.r.t. time might become worse. - oai:arXiv.org:2512.17712v1 - math.NA - cs.NA - math.AP + Long-time reverse transportation inequalities for non-globally-dissipative Langevin dynamics + https://arxiv.org/abs/2512.18598 + arXiv:2512.18598v1 Announce Type: new +Abstract: We establish a dimension-free, uniform-in-time reverse transportation inequality for Langevin dynamics with non-convex potentials. This inequality controls the R\'enyi divergence of arbitrary order between the process distributions starting from distinct initial points and serves as the dual version of the Harnack inequality. Notably, we prove that this inequality retains exponential decay in the long-time regime, thereby extending existing results for log-concave sampling to the non-convex setting. + oai:arXiv.org:2512.18598v1 math.PR - Mon, 22 Dec 2025 00:00:00 -0500 + Tue, 23 Dec 2025 00:00:00 -0500 new http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Niklas Sapountzoglou, Aleksandra Zimmermann + Jianfeng Lu, Yuliang Wang - Preconditioning for the high-order sampling of the invariant distribution of parabolic semilinear SPDEs - https://arxiv.org/abs/2512.17714 - arXiv:2512.17714v1 Announce Type: new -Abstract: For a class of ergodic parabolic semilinear stochastic partial differential equations (SPDEs) with gradient structure, we introduce a preconditioning technique and design high-order integrators for the approximation of the invariant distribution. The preconditioning yields improved temporal regularity of the dynamics while preserving the invariant distribution and allows the application of postprocessed integrators. For the semilinear heat equation driven by space-time white noise in dimension $1$, we obtain new temporal integrators with orders $1$ and $2$ for sampling the invariant distribution with a minor overcost compared to the standard semilinear implicit Euler method of order $1/2$. Numerical experiments confirm the theoretical findings and illustrate the efficiency of the approach. - oai:arXiv.org:2512.17714v1 - math.NA - cs.NA - math.PR - Mon, 22 Dec 2025 00:00:00 -0500 + Embracing Beam-Squint Effects for Wideband LEO Satellite Communications: A 3D Rainbow Beamforming Approach + https://arxiv.org/abs/2512.18600 + arXiv:2512.18600v1 Announce Type: new +Abstract: Low Earth Orbit (LEO) satellite communications (SATCOM) offers high-throughput, low-latency global connectivity to a very large number of users. To accommodate this demand with limited hardware resources, beam hopping (BH) has emerged as a prominent approach in LEO SATCOM. However, its time-domain switching mechanism confines coverage to a small fraction of the service area during each time slot, exacerbating uplink throughput bottlenecks and latency issues as the user density increases. Meanwhile, wideband systems experience the beam-squint effect, where analog beamforming (BF) directions vary with subcarrier frequencies, potentially causing misalignment at certain frequencies, thereby hindering the performance of wideband SATCOM. In this paper, we aim to shift the paradigm in wideband LEO SATCOM from beam-squint as an impairment to beam-squint as an asset. Specifically, we put forth 3D rainbow BF employing a joint phase-time array (JPTA) antenna with true time delay (TTD) to intentionally widen the beam-squint angle, steering frequency-dependent beams toward distributed directions. This novel approach enables the satellite to serve its entire coverage area in a single time slot. By doing so, the satellite simultaneously receives uplink signals from a massive number of users, significantly boosting throughput and reducing latency. To realize 3D rainbow BF, we formulate a JPTA beamformer optimization problem and address the non-convex nature of the optimization problem through a novel joint alternating and decomposition-based optimization framework. Through numerical evaluations incorporating realistic 3D LEO SATCOM geometry, our numerical results demonstrate that the proposed rainbow BF-empowered LEO SATCOM achieves up to 2.8-fold increase in uplink throughput compared to conventional BH systems. These results mark a significant breakthrough for 6G wideband LEO SATCOM. + oai:arXiv.org:2512.18600v1 + cs.IT + math.IT + Tue, 23 Dec 2025 00:00:00 -0500 new - http://creativecommons.org/licenses/by/4.0/ - Charles-Edouard Br\'ehier, Adrien Busnot Laurent, Arnaud Debussche, Gilles Vilmart + http://creativecommons.org/publicdomain/zero/1.0/ + Juha Park, Seokho Kim, Wonjae Shin, H. Vincent Poor - Gaussian random graphs and Ramsey numbers - https://arxiv.org/abs/2512.17718 - arXiv:2512.17718v1 Announce Type: new -Abstract: We give a simple proof of the recent remarkable exponential improvement for Ramsey lower bounds, obtained by Ma, Shen and Xie. Our key ingredient is an alternative construction based on Gaussian random graphs, which allows us to simplify their analysis significantly. As a consequence of this simpler analysis, we also obtain better quantitative bounds. - oai:arXiv.org:2512.17718v1 - math.CO - Mon, 22 Dec 2025 00:00:00 -0500 + Adiabatic Limit and Analytic Torsion of Vector Bundles + https://arxiv.org/abs/2512.18602 + arXiv:2512.18602v1 Announce Type: new +Abstract: For a vector bundle $E^{n+k}$ over a closed manifold $M^n$ with $k$ even and $n$ odd, we equip the metric with an adiabatic parameter, and prove that the index of $E$ is the same as the index of $M$. We also introduce an analog of analytic torsion on $E$ using the Witten Laplacian. Moreover, we prove that the Quillen metric associated with this analytic torsion coincides with that of $M$. + oai:arXiv.org:2512.18602v1 + math.DG + Tue, 23 Dec 2025 00:00:00 -0500 new - http://creativecommons.org/licenses/by/4.0/ - Zach Hunter, Aleksa Milojevi\'c, Benny Sudakov + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Xianzhe Dai, Debin Liu - Inner factors of Dirichlet space functions - https://arxiv.org/abs/2512.17723 - arXiv:2512.17723v1 Announce Type: new -Abstract: Every function in the Dirichlet space on the unit disc has an inner/outer factorization. We study which inner functions occur in this way. For Blaschke products, this is the well known question of which subsets of the disc are zero sets for the Dirichlet space. We also consider singular inner factors, and in particular prove an analogue of the Shapiro-Shields theorem in this setting. Our results on singular inner factors also yield new sufficient conditions for zero sets. - oai:arXiv.org:2512.17723v1 - math.CV - math.FA - Mon, 22 Dec 2025 00:00:00 -0500 + Boundary regularity of a fourth order Alt-Caffarelli problem and applications to the minimization of the critical buckling load + https://arxiv.org/abs/2512.18626 + arXiv:2512.18626v1 Announce Type: new +Abstract: We study a higher order analogue to the Alt-Caffarelli functional that arises in several shape optimization problems, among which the minimization of the critical buckling load of a clamped plate of fixed area. We obtain several regularity results up to the boundary in two dimensions, in particular we prove the full regularity of the boundary (analytic outside angles of opening $\approx 1.43\pi$) near any point of density less than 1 of the optimal shape. These results are based on the monotonicity formula discovered by Dipierro, Karakhanyan, and Valdinoci, which we improve with a new epiperimetric inequality. + oai:arXiv.org:2512.18626v1 + math.AP + math.OC + Tue, 23 Dec 2025 00:00:00 -0500 new http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Michael Hartz, Stefan Richter + Jimmy Lamboley, Micka\"el Nahon - The Fractional Stefan Problem: Global Regularity of the Bounded Selfsimilar Solution" - https://arxiv.org/abs/2512.17725 - arXiv:2512.17725v1 Announce Type: new -Abstract: We study the regularity of the bounded self-similar solution to the one-phase Stefan problem with fractional diffusion posed on the whole line. In terms of the enthalpy $h(x,t)$, the evolution problem reads \[ \begin{cases} \partial_t h + (-\Delta)^s \Phi(h) = 0 & \text{in } \mathbb{R}^n \times (0,T),\\[2mm] h(\cdot,0) = h_0 & \text{in } \mathbb{R}^n , \end{cases} \] where $u = \Phi(h) := (h-L)_+ = \max\{h-L,0\}$ denotes the temperature, $L>0$ is the latent heat, and $s \in (0,1)$. We prove that the regularity of the self-similar solution depends on $s$, with a critical threshold at $s = 1/2$. More precisely, in the subcritical case $0 < s < 1/2$, the self-similar solution exhibits at least $C^{1,\alpha}$ regularity, with H\"older exponent $\alpha >0$. In contrast, we show that the enthalpy of the self-similar solution is not Lipschitz continuous at the free boundary in the critical case $s=1/2$, as well as in the supercritical case $1/2 < s < 1$. Additional results are also established concerning the lateral regularity at the free boundary and the asymptotic behavior of the solution profile as $x \to \pm\infty$. - oai:arXiv.org:2512.17725v1 - math.AP - Mon, 22 Dec 2025 00:00:00 -0500 + Nested Affine Buildings and Their Group Decompositions + https://arxiv.org/abs/2512.18628 + arXiv:2512.18628v1 Announce Type: new +Abstract: In this paper, we construct a higher dimensional generalization of affine buildings and introduce a new structure, which we call Babel buildings. These buildings are non-connected, non-convex metric spaces of non-positive curvature. Despite their non-standard properties, Babel buildings provide an effective framework for studying the structure of groups acting on them. We analyze the metric and nesting structures of Babel buildings and derive key results regarding the group actions consistent with this new framework. + oai:arXiv.org:2512.18628v1 + math.GR + math.NT + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://creativecommons.org/licenses/by-sa/4.0/ + Masaoki Mori + + + On bicanonical maps of threefolds of general type with large volumes + https://arxiv.org/abs/2512.18638 + arXiv:2512.18638v1 Announce Type: new +Abstract: We prove that for any smooth projective $3$-fold of general type with canonical volume greater than $12^6$, the image of its bicanonical map has dimension at least $2$. We also study pluricanonical maps of $3$-folds of general type with large canonical volume and fibered by $(1,2)$-surfaces or $(2,3)$-surfaces. + oai:arXiv.org:2512.18638v1 + math.AG + Tue, 23 Dec 2025 00:00:00 -0500 new http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Marcos Llorca, Juan Luis V\'azquez + Chen Jiang, Ziqi Liu - Stochastic transport equation with L\'evy noise - https://arxiv.org/abs/2512.17727 - arXiv:2512.17727v1 Announce Type: new -Abstract: We study the stochastic transport equation with globally $\beta$-H\"older continuous and bounded vector field driven by a non-degenerate pure-jump L\'evy noise of $\alpha$-stable type. Whereas the deterministic transport equation may lack uniqueness, we prove the existence and pathwise uniqueness of a weak solution in the presence of a multiplicative pure jump noise, assuming $\frac{\alpha}{2}+\beta>1$. Notably, our results cover the entire range $\alpha \in (0,2)$, including the supercritical regime $\alpha\in(0,1)$ where the driving noise exhibits notoriously weak regularization. A key step of our strategy is the development of a \emph{sharp} $C^{1+\delta}$-diffeomorphism - and new regularity results for the Jacobian determinant of the stochastic flow associated to its stochastic characteristic equation. These novel probabilistic results are of independent interest and constitute a substantial component of our work. Our results are the first full generalization of the celebrated paper by Flandoli, Gubinelli, and Priola [Invent. Math. 2010] from the Brownian motion to the pure jump L\'evy noise. To the best of our knowledge, this appears to be the first example of a partial differential equation of fluid dynamics where well-posedness is restored by the influence of a non-degenerate pure-jump noise. - oai:arXiv.org:2512.17727v1 - math.PR - Mon, 22 Dec 2025 00:00:00 -0500 + The Ultra-Radical: Analytic Continuation, Branching, and Stability of the Principal Branch + https://arxiv.org/abs/2512.18643 + arXiv:2512.18643v1 Announce Type: new +Abstract: We study the ultra-radical $\sqrt[{n;a;b}]{x}$, the multi-valued solution to $y^{a}=1+a x y^{b}$. Inside the convergence radius $|x|<R$, every branch is given by a Master-J power series; for $|x|\ge R$, analytic continuation requires switching to one of two conjugate series. We introduce a deterministic geometric criterion that selects, for each branch index $n$, the correct conjugate series, thereby eliminating heuristic search and guaranteeing branch continuity across $|x|=R$. Key finding: Only the principal branch ($n=0$) remains continuous when the parameters $a$, $b$, and $x$ vary smoothly. This includes the critical limits $a\to0$ (transition to an exponential equation) and $b\to0$ (transition to a binomial root), where the principal branch converges to the corresponding classical solution. In contrast, branches with $n\neq0$ exhibit oscillatory divergence as $a\to0$ and lose their identity in these limits. This structural continuity singles out the principal branch for applications where parameters may vary with the system's state, such as in nonlinear media with field-dependent exponents or adaptive dynamical systems. + oai:arXiv.org:2512.18643v1 + math.CA + Tue, 23 Dec 2025 00:00:00 -0500 new http://creativecommons.org/licenses/by/4.0/ - Zdzis{\l}aw Brze\'zniak, Enrico Priola, Jianliang Zhai, Jiahui Zhu + Sergey Viktorovich Berezin - Convergence rates for a finite volume scheme of a stochastic non-linear parabolic equation - https://arxiv.org/abs/2512.17728 - arXiv:2512.17728v1 Announce Type: new -Abstract: In this contribution, we provide convergence rates for a finite volume scheme of a stochastic non-linear parabolic equation with multiplicative Lipschitz noise and homogeneous Neumann boundary conditions. More precisely, we give an error estimate for the $L^2$-norm of the space-time discretization by a semi-implicit Euler scheme with respect to time and a two-point flux approximation finite volume scheme with respect to space and the variational solution. The only regularity assumptions additionally needed is spatial regularity of the initial datum and smoothness of the diffusive term. - oai:arXiv.org:2512.17728v1 - math.NA - cs.NA - math.AP - math.PR - Mon, 22 Dec 2025 00:00:00 -0500 + On the Problem of Mixed-Tateness of the Motives of G-Varieties + https://arxiv.org/abs/2512.18645 + arXiv:2512.18645v1 Announce Type: new +Abstract: Building on earlier work concerning the motives of $G$-bundles, we study the structure of motives associated with certain classes of $G$-varieties. In particular, we show that the corresponding motives lie within the category of mixed-Tate motives, under certain condition on the stabilizers. We further discuss some applications and provide some examples to illustrate the limitations. + oai:arXiv.org:2512.18645v1 + math.AG + Tue, 23 Dec 2025 00:00:00 -0500 new http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Kavin Rajasekaran, Niklas Sapountzoglou + Somayeh Habibi - The Newtonian kernel at the intersection of two discs - https://arxiv.org/abs/2512.17734 - arXiv:2512.17734v1 Announce Type: new -Abstract: We present an exact, closed-form expression for the Newtonian potential of the characteristic function associated with two overlapping discs in the plane. This setting naturally arises when discretising nonlocal interaction terms present in models of phase separation, aggregation dynamics, and quantum systems. We characterise the convolution integral as a piecewise function on the distance between the disc centres, with transitions dictated by the geometry of the overlapping region. Additionally, we derive detailed asymptotic expansions for the small-overlap regime, which allows us to provide stable double-precision codes. - oai:arXiv.org:2512.17734v1 + Inverse problems with integral conditions for the generalized Korteweg-de Vries equation + https://arxiv.org/abs/2512.18649 + arXiv:2512.18649v1 Announce Type: new +Abstract: Results on well-posedness of three inverse problems with integral conditions on a bounded interval for the generalized Korteweg-de Vries equation without any restrictions on the growth rate of nonlinearity are established. Either the right-hand side of equation or the boundary data, or both are chosen as controls. The considered solutions are regular with respect to the time variable. Assumptions on smallness of the input data or smallness of a time interval are required. + oai:arXiv.org:2512.18649v1 math.AP - Mon, 22 Dec 2025 00:00:00 -0500 + Tue, 23 Dec 2025 00:00:00 -0500 new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Andr\'es Miniguano-Trujillo + http://creativecommons.org/publicdomain/zero/1.0/ + Oleg S. Balashov, Andrei V. Faminskii - Pathwise uniqueness by noise for singular stochastic PDEs - https://arxiv.org/abs/2512.17736 - arXiv:2512.17736v1 Announce Type: new -Abstract: Pathwise uniqueness for stochastic PDEs with drift in differential form is a main open problem in the recent literature on regularisation by noise. This paper establishes a self-contained theory in the framework of stochastic evolution equations on separable Hilbert spaces and provides a first result to address such an issue. The singularity of the drift allows to achieve novel uniqueness results for several classes of examples, ranging from fluid-dynamics to phase-separation models. - oai:arXiv.org:2512.17736v1 - math.PR - math.AP - Mon, 22 Dec 2025 00:00:00 -0500 + Hierarchical filtrations of torsion-free sheaves and birational geometry + https://arxiv.org/abs/2512.18654 + arXiv:2512.18654v1 Announce Type: new +Abstract: We introduce the notion of \emph{hierarchical filtrations} of torsion-free sheaves on normal projective varieties and define the associated numerical invariant called \emph{hierarchical depth}. This invariant measures the maximal length of filtrations by saturated subsheaves of equal rank whose successive quotients are torsion sheaves supported in codimension one. + We establish general bounds for hierarchical depth in terms of the divisor class of the determinant and give exact formulas in several basic geometric situations, including the case of smooth projective curves and varieties of Picard rank one. A key technical ingredient is the study of elementary transforms along effective divisors and their commutativity properties. + In dimension two, we analyze the behavior of hierarchical depth under birational morphisms and show that it admits a precise description along the minimal model program. In particular, we prove that hierarchical depth transforms additively with respect to exceptional divisors and is explicitly computable on minimal models. + As an application, we relate hierarchical depth to degeneracies in algebraic--geometric codes and show that birational simplification via the minimal model program leads to effective improvements of code parameters. These results demonstrate that hierarchical depth provides a new bridge between the birational geometry of vector bundles and arithmetic applications. + oai:arXiv.org:2512.18654v1 + math.AG + math.AC + Tue, 23 Dec 2025 00:00:00 -0500 new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Davide Addona, Davide Bignamini, Carlo Orrieri, Luca Scarpa + http://creativecommons.org/licenses/by/4.0/ + Rahim Rahmati-asghar - On orientably-regular maps of Euler characteristic $-2p^2$ - https://arxiv.org/abs/2512.17743 - arXiv:2512.17743v1 Announce Type: new -Abstract: In this article, we study orientably-regular maps of Euler characteristic $-2p^2$ and classify those that admit a group of orientation-preserving automorphisms of order $10p^2$, where $p$ is a prime number. Along the way, we classify all compact Riemann surfaces (or complex algebraic curves) of genus $1+p^2$ endowed with a group of conformal automorphisms of order $5p^2$. - oai:arXiv.org:2512.17743v1 + Cyclic sieving phenomena for trees and tree-rooted maps + https://arxiv.org/abs/2512.18656 + arXiv:2512.18656v1 Announce Type: new +Abstract: We prove cyclic sieving phenomena satisfied by corner-rooted plane trees (alias ordered trees). The sets of rooted plane trees that we consider are: (1) all trees with $n$ nodes; (2) all trees with $n$ nodes and $k$ leaves; (3) all trees with a given degree distribution of the nodes. Moreover, we consider four different cyclic group actions: (1) the root is moved to the next corner along a tour of the tree; (2) only trees in which the root is at a leaf are considered, and the action moves the root to the next leaf; (3) only trees in which the root is at a non-leaf are considered, and the action moves the root to the next non-leaf corner; (4) only trees in which the root is at a node of degree $\delta$ are considered, for a fixed $\delta$, and the action moves the root to the next corner of this type. We prove a cyclic sieving phenomenon for each meaningful combination of these sets and actions. As a bonus, we also establish corresponding cyclic sieving phenomena for tree-rooted planar maps. + oai:arXiv.org:2512.18656v1 math.CO - math.AG - Mon, 22 Dec 2025 00:00:00 -0500 + Tue, 23 Dec 2025 00:00:00 -0500 new http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Tom\'as Foncea E., Sebasti\'an Reyes-Carocca} + Mireille Bousquet-M\'elou, Christian Krattenthaler - On exponentially height-penalized random trees - https://arxiv.org/abs/2512.17747 - arXiv:2512.17747v1 Announce Type: new -Abstract: Given $n \in \mathbb{N}$ and $\mu \in \mathbb{R}$, a $\textit{$\mu$-height-biased tree of size $n$}$ is a random plane tree $\mathbf{\mathbf{T}}_n$ with $n$ vertices with law given by $\mathbb{P}(\mathbf{T}=t) \propto e^{-\mu h(t)}$, where $t$ ranges over fixed plane trees with $n$ vertices, and $h(t)$ is the height of $t$. Fix a sequence $(\mu_n)_{n \ge 1}$ of real numbers, and for $n \ge 1$ let $\mathbf{T}_n$ be a $\mu$-height-biased tree of size $n$. Durhuus and \"Unel (2023) described the asymptotic behaviour of $h(\mathbf{T}_n)$ when $\mu_n \equiv \mu \in \mathbb{R}$ is fixed. In this work, we extend their results to arbitrary sequences of positive parameters depending on $n$. Most notably, we show that such a tree behaves like a height-biased Continuum Random Tree (CRT) when $\mu_n$ is of order $1/\sqrt{n}$; that its height is asymptotically $(2\pi^2n/\mu_n)^{1/3}$ when $\mu_n$ is of larger order than $1/\sqrt{n}$ and of smaller order than $n$; and that its height converges to a fixed constant when $\mu_n$ is of order at least $n$, with some random jumps under specific conditions on $\mu_n$. We additionally prove various results on second order behaviours, and large deviation principles for the height, for different regimes of $\mu_n$. Finally, we describe new statistics of these trees, covering their widths, their root degrees, and the local structure around their roots. - oai:arXiv.org:2512.17747v1 - math.PR - math.CO - Mon, 22 Dec 2025 00:00:00 -0500 + Qualitative analysis of multi-peak solutions for Nonlinear Schr\"{o}dinger equations with nearly critical Sobolev exponents + https://arxiv.org/abs/2512.18663 + arXiv:2512.18663v1 Announce Type: new +Abstract: In this paper, we are concerned with qualitative properties of multi-peak solutions of the following nonlinear Schr\"{o}dinger equations \begin{equation*} -\Delta u+V(x)u= u^{p-\varepsilon},\,\,\,u>0,\,\,\,\text{in}\,\,\,\mathbb{R}^N, \end{equation*} where $V(x)$ is a nonnegative continuous function, $\varepsilon>0$, $p=\frac{N+2}{N-2}$, $N\geq6$. The existence of multi-peak solutions has been obtained by Cao et al. (Calc. Var. Partial Differential Equations, 64: 139, 2025). The main objective in this paper is to establish the local uniqueness and Morse index of the multi-peak solutions in \cite{CLl1} provided that $V(x)$ possesses $k$ non-degenerate critical points by using the blow-up analysis based on Pohozaev identities. + oai:arXiv.org:2512.18663v1 + math.AP + math.CT + Tue, 23 Dec 2025 00:00:00 -0500 new - http://creativecommons.org/licenses/by/4.0/ - Louigi Addario-Berry, Beno\^it Corsini, Neeladri Maitra, Meltem \"Unel + http://creativecommons.org/publicdomain/zero/1.0/ + Zhongyuan Liu, Shuying Tian, Huafei Xie, Pingping Yang - A matrix approach to the enumeration of naturally labeled posets - https://arxiv.org/abs/2512.17749 - arXiv:2512.17749v1 Announce Type: new -Abstract: We propose a matrix approach for enumerating naturally labeled posets by representing each poset $P$ on $[n]$ as a Boolean poset matrix $A$. This algebraic representation enables a systematic handling of partial orderings through $v$-extensions of the form $A^v=\bigl[\begin{smallmatrix}A&0\\ v&1\end{smallmatrix}\bigr]$. We show that $A^v$ defines a valid poset matrix if and only if the Boolean vector $v$ represents an order ideal of the poset $P$ associated to $A$, equivalently satisfying the fixed-point equation $vA=v$. Furthermore, we explore the twin-class decomposition of $A$, which partitions the elements of $P$ according to identical down- and up-sets. Finally, we present an algorithmic generation scheme for the posets based on the topological growth of their distribution lattices, offering a new approach to constructive enumeration of poset families. - oai:arXiv.org:2512.17749v1 + Pieri rule for classical groups, a new perspective + https://arxiv.org/abs/2512.18668 + arXiv:2512.18668v1 Announce Type: new +Abstract: We study a new perspective on a certain Pieri rules for classical groups. Furthermore, we extend a fundamental theorem of Kostant concerning tensor products for classical groups. We show that a certain form of the Pieri rule is equivalent to the converse of this extended version of Kostant's theorem. In addition, we show an equivalence between the Pieri rule and the branching rule for general linear groups. + oai:arXiv.org:2512.18668v1 + math.RT math.CO - Mon, 22 Dec 2025 00:00:00 -0500 + math.GR + Tue, 23 Dec 2025 00:00:00 -0500 new http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Gi-Sang Cheon, Samuele Giraudo + Dibyendu Biswas - Sharp Favard length of random Cantor sets - https://arxiv.org/abs/2512.17753 - arXiv:2512.17753v1 Announce Type: new -Abstract: We show that for a large class of planar $1$-dimensional random fractals $S$, the Favard length $\Fav(S(r))$ of the neighborhood $S(r)$ is comparable to $\log^{-1}(1/r)$, matching a universal lower bound; up to now, this was only known in expectation for a few concrete models. In particular, we show that there exist $1$-Ahlfors regular sets with the fastest possible Favard length decay. For a wide class of planar one-dimensional ``grid random fractals'', including fractal percolation and its Ahlfors-regular variants, we further show that $\Fav(S(r))/\log(1/r)$ converges almost surely, and we identify the limit explicitly. Furthermore, we prove that for some $1$-dimensional Ahlfors-regular random fractals $S$, the Favard length of $S(r)$ decays instead like $\log\log(1/r)/\log(1/r)$, showing that the $1/\log(1/r)$ decay is not universal among random fractals, as might be expected from previous results. - oai:arXiv.org:2512.17753v1 - math.CA - math.MG - math.PR - Mon, 22 Dec 2025 00:00:00 -0500 + Rough Weighted Ideal Convergence and Korovkin-Type Approximation via weighted equi-ideal convergence + https://arxiv.org/abs/2512.18676 + arXiv:2512.18676v1 Announce Type: new +Abstract: If $\omega_t > \beta$ for every $t \in \mathbb{N}$ and for some $\beta > 0$, then the sequence $\{\omega_t\}_{t \in \mathbb{N}}$ represents a weighted sequence of real numbers. In this article, we primarily introduce the concepts of rough weighted ideal limit set and rough weighted ideal cluster points set associated with sequences in normed spaces. Building on these concepts, we derive several important results, including a characterization of maximal ideals, a representation of closed sets in normed spaces, and an analysis of the minimal convergent degree required for the rough weighted ideal limit set to be non-empty. Furthermore, we demonstrate that for an analytic $P$-ideal, the rough weighted ideal limit set forms an $F_{\sigma\delta}$ subset of the normed space. Finally, we introduce the concept of weighted equi-ideal convergence for sequences of functions with respect to analytic $P$-ideals, extending the notion of equi-statistical convergence [Balcerzak et al., J. Math. Anal. Appl. {328} (1) (2007)]. As an application of this notion, we establish a Korovkin-type approximation theorem that serves both as a generalization of [Theorem 2.4, Karaku{\c{s}} et al., J. Math. Anal. Appl. {339} (2) (2008)] and a correction to [Theorem 2.2, Akda\u{g}, Results Math. {72} (3) (2017)]. + oai:arXiv.org:2512.18676v1 + math.FA + Tue, 23 Dec 2025 00:00:00 -0500 new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Alan Chang, Pablo Shmerkin, Ville Suomala + http://creativecommons.org/licenses/by/4.0/ + Tamim Aziz, Sanjoy Ghosal - On a new condition implying that an achievement set is a Cantorval and its applications - https://arxiv.org/abs/2512.17761 - arXiv:2512.17761v1 Announce Type: new -Abstract: Given a nonincreasing sequence of positive numbers $(a_n)$ such that the series $\sum a_n$ is convergent, by $E(a_n)$ we denote the set of all subsums of the series $\sum a_n$ and call it the achievement set of $(a_n)$. It is well known that such a set can be a finite union of closed intervals, a Cantor set or a Cantorval. We give a new condition implying that the last possibility occurs. We also show how we can use this condition to produce new achievable Cantorvals. In particular, we prove that Kakeya conditions cannot tell us more about the form of the achievement set than it was proved by Kakeya. - oai:arXiv.org:2512.17761v1 + The basis functions of Fourier interpolation + https://arxiv.org/abs/2512.18677 + arXiv:2512.18677v1 Announce Type: new +Abstract: The basis functions of the Fourier interpolation formula of Radchenko and Viazovska, constructed by means of weakly holomorphic modular forms for the Hecke theta group, are entire functions of order $2$ having interesting time-frequency properties. We give precise size estimates and study the distribution of zeros of these functions. We give in particular asymptotic estimates for the location and the number of extraneous zeros on or close to the real line. This result reveals the surprising existence of Fourier nonuniqueness pairs whose apparent ``excess'' compared to the Fourier uniqueness pair of Radchenko and Viazovska may be made arbitrarily large. Our estimates also show that the basis functions fail to yield a Riesz basis in the Hilbert space used by Kulikov, Nazarov, and Sodin in their recent study of Fourier uniqueness pairs. Some numerical data are presented, suggesting additional fine scale properties. + oai:arXiv.org:2512.18677v1 + math.NT math.CA - Mon, 22 Dec 2025 00:00:00 -0500 + Tue, 23 Dec 2025 00:00:00 -0500 new http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Piotr Nowakowski - - - Trapped modes in electromagnetic waveguides - https://arxiv.org/abs/2512.17763 - arXiv:2512.17763v1 Announce Type: new -Abstract: We consider the Maxwell's equations with perfect electric conductor boundary conditions in three-dimensional unbounded domains which are the union of a bounded resonator and one or several semi-infinite waveguides. We are interested in the existence of electromagnetic trapped modes, i.e. $L^2$ solutions of the problem without source term. These trapped modes are associated to eigenvalues of the Maxwell's operator, that can be either below the essential spectrum or embedded in it. First for homogeneous waveguides, we present different families of geometries for which we can prove the existence of eigenvalues. Then we exhibit certain non homogeneous waveguides with local perturbations of the dielectric constants that support trapped modes. Let us mention that some of the mechanisms we propose are very specific to Maxwell's equations and have no equivalent for the scalar Dirichlet or Neumann Laplacians. - oai:arXiv.org:2512.17763v1 - math.AP - math.SP - Mon, 22 Dec 2025 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Anne-Sophie Bonnet-Ben Dhia, Lucas Chesnel, Sonia Fliss + David Berghaus, Andriy Bondarenko, Danylo Radchenko, Kristian Seip, Qihang Sun - A Short Report on Importance Sampling for Rare Event Simulation in Diffusions - https://arxiv.org/abs/2512.17766 - arXiv:2512.17766v1 Announce Type: new -Abstract: In this manuscript, we investigate importance sampling methods for rare-event simulation in diffusion processes. We show, from a large-deviation perspective, that the resulting importance sampling estimator is log-efficient. This connection is established via a stochastic optimal control formulation, and the associated Hamilton--Jacobi--Bellman (HJB) equation is derived using dynamic programming. To approximate the optimal control, we adopt a spectral parameterization and employ the cross-entropy method to estimate the parameters by solving a least-squares problem. Finally, we present a numerical example to validate the effectiveness of the cross-entropy approach and the efficiency of the resulting importance sampling estimator. - oai:arXiv.org:2512.17766v1 - math.NA - cs.NA - math.PR - Mon, 22 Dec 2025 00:00:00 -0500 + Standard modules and intertwining operators for reductive p-adic groups + https://arxiv.org/abs/2512.18685 + arXiv:2512.18685v1 Announce Type: new +Abstract: Consider a reductive group G over a non-archimedean local field. The Galois group Gal(C/Q) acts naturally on the category of smooth complex G-representations. We prove that this action stabilizes the class of standard modules. This generalizes and relies on an analogous result about essentially square-integrable representations. + Other important objects in the proof of our main result are intertwining operators between parabolically induced G-representations, and the associated Harish-Chandra \mu-functions. We determine an explicit formula for the \mu-function of any irreducible representation of any Levi subgroup of G. + oai:arXiv.org:2512.18685v1 + math.RT + Tue, 23 Dec 2025 00:00:00 -0500 new - http://creativecommons.org/publicdomain/zero/1.0/ - Zhiwei Gao + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Maarten Solleveld - Existence, uniqueness, and time-asymptotics of regular solutions in multidimensional thermoelasticity on domains with boundary - https://arxiv.org/abs/2512.17767 - arXiv:2512.17767v1 Announce Type: new -Abstract: In the paper, we investigate the nonlinear thermoelasticity model in two- and three-dimensional convex and bounded domains. We propose new boundary conditions for the displacement. These conditions are not usual in thermoelasticity. Whereas, we posit the Neumann boundary condition for the temperature. We prove the existence of global, unique solutions for small initial data. The temperature positivity is also shown. Next, we investigate the long-time behavior of solutions. We show that the divergence-free part of the displacement oscillates. On the other hand, we prove that the potential part and the temperature are strongly coupled. The non-rotation part is strongly affected by heat propagation. It turns out that it tends to $0$ as $x$ approaches infinity. Additionally, the temperature converges to a constant function. - Our techniques are firmly based on the functional $\F$ adopted from {\sc Bies, P. M., Cie\'slak, T., Fuest, M., Lankeit, J., Muha, B., and Trifunovi\'c, S.}, \emph{Existence, uniqueness, and long-time asymptotic behavior of regular solutions in multidimensional thermoelasticity}, arXiv: 2507.20794, 2025. The functional is based on the Fisher information and higher-order derivatives of the displacement. It responds well to the new boundary conditions. It allows us to close a priori estimates. We also need $L^{\infty}$-estimates for the temperature here. The Moser iterative procedure ensures it. - The Helmholtz decomposition is applied to the displacement. The boundary conditions are crucial here. The boundary integrals that appear in the calculations at this point disappear thanks to these conditions. This allows us to split the problem into two separate ones. Each of them is associated with one part of the Helmholtz decomposition. - oai:arXiv.org:2512.17767v1 + Best constants for Hardy inequalities in Triebel--Lizorkin spaces + https://arxiv.org/abs/2512.18688 + arXiv:2512.18688v1 Announce Type: new +Abstract: We find sharp constants in fractional Hardy inequalities for weighted Triebel--Lizorkin seminorms on the whole space and half-spaces. Our results generalize recently obtained weighted fractional Hardy inequalities for Gagliardo seminorms, but are new even for the unweighted case. + oai:arXiv.org:2512.18688v1 math.AP - Mon, 22 Dec 2025 00:00:00 -0500 + math.FA + Tue, 23 Dec 2025 00:00:00 -0500 new - http://creativecommons.org/licenses/by/4.0/ - Piotr Micha{\l} Bies + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Micha{\l} Kijaczko - A Li-Yau and Aronson-B\'enilan approach for the Keller-Segel system with critical exponent - https://arxiv.org/abs/2512.17772 - arXiv:2512.17772v1 Announce Type: new -Abstract: We prove Li-Yau and Aronson-B\'enilan type estimates for the parabolic-elliptic Keller-Segel system with critical exponent $m=2-\frac 2d$, i.e. lower bounds on the Laplacian of a suitable notion of pressure in any dimension. We show that these estimates entail $L^{\infty}$ bounds on the density, depending on its initial mass, up to the critical mass case for $d \in \{ 2, 3 \}$. We deduce from these results the global existence of smooth solutions in two cases: first, when the initial data is merely a measure but has sufficiently small mass; and second, when the initial free energy is bounded, and the mass is subcritical or critical. Our argument requires a careful study of the subsolutions of the Liouville and Lane-Emden equations arising in the model. - oai:arXiv.org:2512.17772v1 + Multiscale homogenization of non-local energies of convolution-type + https://arxiv.org/abs/2512.18697 + arXiv:2512.18697v1 Announce Type: new +Abstract: We analyze a family of non-local integral functionals of convolution-type depending on two small positive parameters $\varepsilon,\delta$: the first rules the length-scale of the non-local interactions and produces a `localization' effect as it tends to $0$, the second is the scale of oscillation of a finely inhomogeneous periodic structure in the domain. We prove that a separation of the two scales occurs and that the interplay between the localization and homogenization effects in the asymptotic analysis is determined by the parameter $\lambda$ defined as the limit of the ratio $\varepsilon/\delta$. We compute the $\Gamma$-limit of the functionals with respect to the strong $L^p$-topology for each possible value of $\lambda$ and detect three different regimes, the critical scale being obtained when $\lambda\in(0,+\infty)$. + oai:arXiv.org:2512.18697v1 math.AP - Mon, 22 Dec 2025 00:00:00 -0500 + math.OC + Tue, 23 Dec 2025 00:00:00 -0500 new http://creativecommons.org/licenses/by/4.0/ - Charles Elbar, Alejandro Fern\'andez-Jim\'enez, Filippo Santambrogio + Giuseppe Cosma Brusca - A linear upper bound for zero-sum Ramsey numbers of bounded degree graphs - https://arxiv.org/abs/2512.17790 - arXiv:2512.17790v1 Announce Type: new -Abstract: Let $G$ be a graph and $\Gamma$ a finite abelian group. The zero-sum Ramsey number of $G$ over $\Gamma$, denoted by $R(G, \Gamma)$, is the smallest positive integer $t$ (if it exists) such that any edge-colouring $c:E(K_t)\to\Gamma$ contains a copy of $G$ with $\sum_{e\in E(G)}c(e)=0_\Gamma$. - We prove a linear upper bound $R(G, \Gamma)\leq Cn$ that holds for every $n$-vertex graph $G$ with bounded maximum degree and every finite abelian group $\Gamma$ with $|\Gamma|$ dividing $e(G)$. - oai:arXiv.org:2512.17790v1 - math.CO - Mon, 22 Dec 2025 00:00:00 -0500 + Real-Time Remote Monitoring of Correlated Markovian Sources + https://arxiv.org/abs/2512.18698 + arXiv:2512.18698v1 Announce Type: new +Abstract: We investigate real-time tracking of two correlated stochastic processes over a shared wireless channel. The joint evolution of the processes is modeled as a two-dimensional discrete-time Markov chain. Each process is observed by a dedicated sampler and independently reconstructed at a remote monitor according to a task-specific objective. Although both processes originate from a common underlying phenomenon (e.g., distinct features of the same source), each monitor is interested only in its corresponding feature. A reconstruction error is incurred when the true and reconstructed states mismatch at one or both monitors. To address this problem, we propose an error-aware joint sampling and transmission policy, under which each sampler probabilistically generates samples only when the current process state differs from the most recently reconstructed state at its corresponding monitor. We adopt the time-averaged reconstruction error as the primary performance metric and benchmark the proposed policy against state-of-the-art joint sampling and transmission schemes. For each policy, we derive closed-form expressions for the resulting time-averaged reconstruction error. We further formulate and solve an optimization problem that minimizes the time-averaged reconstruction error subject to an average sampling cost constraint. Analytical and numerical results demonstrate that the proposed error-aware policy achieves the minimum time-averaged reconstruction error among the considered schemes while efficiently utilizing the sampling budget. The performance gains are particularly pronounced in regimes with strong inter-process correlation and stringent tracking requirements, where frequent sampling by both samplers is necessary. + oai:arXiv.org:2512.18698v1 + cs.IT + cs.NI + cs.SY + eess.SY + math.IT + Tue, 23 Dec 2025 00:00:00 -0500 new - http://creativecommons.org/licenses/by/4.0/ - Jasmin Katz, Xiaopan Lian, Alexandru Malekshahian, Andrey Shapiro + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Mehrdad Salimnejad, Marios Kountouris, Nikolaos Pappas - Convergence of Empirical Measures for i.i.d. samples in $W^{-{\alpha}, p}$ - https://arxiv.org/abs/2512.17794 - arXiv:2512.17794v1 Announce Type: new -Abstract: Given $N$ i.i.d. samples from a probability measure $\mu$ on $\mathbf{R}^d$, we study the rate of convergence of the empirical measure $\mu_N \to \mu$ in the negative Sobolev space $W^{-\alpha, p}$. When $W^{-\alpha, p}$ contains point measures (i.e. when $\alpha p > (p-1)d$), we show $\mathbf{E} \| \mu_N - \mu \|_{W^{-\alpha, p}}^p \leq C_d / N^{p/2}$ for an explicit dimensional constant $C_d$, and obtain a Gaussian tail bound. When $0 < \alpha p \leq d(p-1)$, we prove a similar result for Gaussian regularizations. - oai:arXiv.org:2512.17794v1 - math.PR - Mon, 22 Dec 2025 00:00:00 -0500 + Rigidity for homogeneous solutions to the two-dimensional Euler equations in sector-type domains + https://arxiv.org/abs/2512.18700 + arXiv:2512.18700v1 Announce Type: new +Abstract: We study the rigidity problem for $(-\alpha)$-homogeneous solutions to the two-dimensional incompressible stationary Euler equations in sector-type domains $\Omega_{a, b, \theta_0}:= \{(r,\theta): a<r<b, \ 0<\theta<\theta_0\}$, where $\alpha\in\mathbb{R}$, $0\leqslant a < b \leqslant +\infty$ and $0< \theta_0 \leqslant 2\pi$. For each type of domains, depending on whether $a = 0$ or $a > 0$, and $b = +\infty$ or $b < +\infty$, we show that if a solution satisfies some homogeneity assumptions on the boundary of $\Omega_{a, b, \theta_0}$ and if the radial or angular component of the velocity does not vanish in $\overline{\Omega_{a, b, \theta_0}}\setminus\{\bm{0}\}$, then it must be homogeneous throughout $\overline{\Omega_{a, b, \theta_0}}\setminus\{\bm{0}\}$. + oai:arXiv.org:2512.18700v1 + math.AP + Tue, 23 Dec 2025 00:00:00 -0500 new http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Gautam Iyer, Raghavendra Venkatraman + Li Li, Xukai Yan, Zhibo Yang - A measure-$L^\infty$ div-curl lemma - https://arxiv.org/abs/2512.17798 - arXiv:2512.17798v1 Announce Type: new -Abstract: In this note we give a very short proof of the div-curl lemma in the limit conjugate case $\mathcal M-L^\infty$, where $\mathcal{M}$ is the set of Radon measures on $\mathbb{R}^d$. The proof follows the classical approach by defining here the product in the sense of distributions via a non unique microlocal Hodge's decomposition. The result is valid for many other spaces than $\mathcal M-L^\infty$, including the classical div-curl lemma spaces $L^p-L^{p'}$ for $1<p<\infty$, and spaces of non conjugated regularity. - oai:arXiv.org:2512.17798v1 + Nonlocal conservation laws with p-norm, the singular limit problem and applications to traffic flow + https://arxiv.org/abs/2512.18701 + arXiv:2512.18701v1 Announce Type: new +Abstract: In this contribution, we study scalar nonlocal conservation laws with the $p$-norm. Here, 'nonlocal' means that the velocity of the conservation law depends on an integral term in space. Typically, the nonlocal term consists of integrating the solution in $L^{1}$, whereas here we will study the case when the solution is integrated in the $L^{p}$-norm. We consider even the case of the $L^{p}$ metric when $p\in (0,1)$ and establish, for an initial datum which is uniformly bounded away from zero, the existence and uniqueness of weak solutions. We then demonstrate that there are also solutions to the initial datum being zero under more restrictive assumptions. Furthermore, we investigate the singular limit, i.e., what happens when the nonlocal kernel converges to a Dirac distribution. Indeed, for the one-sided exponential kernel, we recover the (entropy) solution of the corresponding local conservation law for all $p\in(0,\infty)$ with further restrictions for $p\in(0,1)$. This generalizes the celebrated singular limit result for nonlocal conservation laws for $p=1$ significantly and showcases the robustness of the approximation of local conservation laws by nonlocal ones. We investigate also the monotonicity of the solution when assuming that the initial datum is monotone. Finally, we prove the convergence of solutions for $p\rightarrow 0$ on a small time horizon, resulting in a different kind of nonlocal conservation law. Numerical studies showcasing the effect of $p$ on the singular limit convergence and more conclude the contribution. + oai:arXiv.org:2512.18701v1 math.AP - Mon, 22 Dec 2025 00:00:00 -0500 + Tue, 23 Dec 2025 00:00:00 -0500 new - http://creativecommons.org/publicdomain/zero/1.0/ - Valeria Banica, Nicolas Burq + http://creativecommons.org/licenses/by/4.0/ + Felisia Angela Chiarello, Alexander Keimer, Lukas Pflug - Deconstructible classes of modules and stability - https://arxiv.org/abs/2512.17799 - arXiv:2512.17799v1 Announce Type: new -Abstract: We show that every deconstructible class of modules with all embeddings, all pure embedding and all RD-embeddings is stable. The argument is presented in the context of abstract classes of modules without amalgamation and the key idea is to construct a stable-like independence relation. - In particular, the following classes of modules with all embeddings, all pure embedding and all RD-embeddings are shown to be stable: all free and torsion-free modules over any ring, and for each $n \geq 0$, the classes of all modules of projective and flat dimension $\leq n$ over any ring, and the class of all modules of injective dimension $\leq n$ over any right noetherian ring. - oai:arXiv.org:2512.17799v1 - math.LO - math.RA - Mon, 22 Dec 2025 00:00:00 -0500 + A note on the projective representations of $p$-solvable and $\pi$-separable group + https://arxiv.org/abs/2512.18704 + arXiv:2512.18704v1 Announce Type: new +Abstract: Following Gluck and Wolf we complete the It\^o--Michler's Theorem for the projective representations of a $p$-solvable or $\pi$-separable group, and then we relate the projective irreducible modules of such a group with those of its Sylow $p$-subroups and Hall $\pi$-subgroups. + oai:arXiv.org:2512.18704v1 + math.RT + math.GR + Tue, 23 Dec 2025 00:00:00 -0500 new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Marcos Mazari-Armida, Jan Trlifaj + http://creativecommons.org/licenses/by/4.0/ + Mariagrazia Bianchi, Nicola Sambonet - Towards Sharp Minimax Risk Bounds for Operator Learning - https://arxiv.org/abs/2512.17805 - arXiv:2512.17805v1 Announce Type: new -Abstract: We develop a minimax theory for operator learning, where the goal is to estimate an unknown operator between separable Hilbert spaces from finitely many noisy input-output samples. For uniformly bounded Lipschitz operators, we prove information-theoretic lower bounds together with matching or near-matching upper bounds, covering both fixed and random designs under Hilbert-valued Gaussian noise and Gaussian white noise errors. The rates are controlled by the spectrum of the covariance operator of the measure that defines the error metric. Our setup is very general and allows for measures with unbounded support. A key implication is a curse of sample complexity which shows that the minimax risk for generic Lipschitz operators cannot decay at any algebraic rate in the sample size. We obtain essentially sharp characterizations when the covariance spectrum decays exponentially and provide general upper and lower bounds in slower-decay regimes. - oai:arXiv.org:2512.17805v1 + A pivotal transform for the high-dimensional location-scale model + https://arxiv.org/abs/2512.18705 + arXiv:2512.18705v1 Announce Type: new +Abstract: We study the high-dimensional linear model with noise distribution known up to a scale parameter. With an $\ell_1$-penalty on the regression coefficients, we show that a transformation of the log-likelihood allows for a choice of the tuning parameter not depending on the scale parameter. This transformation is a generalization of the square root Lasso for quadratic loss. The tuning parameter can asymptotically be taken at the detection edge. We establish an oracle inequality, variable selection and asymptotic efficiency of the estimator of the scale parameter and the intercept. The examples include Subbotin distributions and the Gumbel distribution. + oai:arXiv.org:2512.18705v1 math.ST - cs.NA - math.NA - stat.ML stat.TH - Mon, 22 Dec 2025 00:00:00 -0500 + Tue, 23 Dec 2025 00:00:00 -0500 new - http://creativecommons.org/licenses/by/4.0/ - Ben Adcock, Gregor Maier, Rahul Parhi + http://creativecommons.org/publicdomain/zero/1.0/ + Sara van de Geer, Sylvain Sardy, Maxim\k{e} van Cutsem - Funnel control with input filter for nonlinear systems of relative degree two - https://arxiv.org/abs/2512.17806 - arXiv:2512.17806v1 Announce Type: new -Abstract: We address the problem of output reference tracking for unknown nonlinear multi-input, multi-output systems with relative degree two and bounded-input bounded-state (BIBS) stable internal dynamics. We propose a novel model-free adaptive controller that ensures the evolution of the tracking error within prescribed performance funnel boundaries. By applying an output filter, the control objective is achieved without utilizing derivative information of system's output. The controller is illustrated by a numerical example. - oai:arXiv.org:2512.17806v1 + Tight Lower Bounds and Optimal Algorithms for Stochastic Nonconvex Optimization with Heavy-Tailed Noise + https://arxiv.org/abs/2512.18713 + arXiv:2512.18713v1 Announce Type: new +Abstract: We study stochastic nonconvex optimization under heavy-tailed noise. In this setting, the stochastic gradients only have bounded $p$--th central moment ($p$--BCM) for some $p \in (1,2]$. Building on the foundational work of Arjevani et al. (2022) in stochastic optimization, we establish tight sample complexity lower bounds for all first-order methods under \emph{relaxed} mean-squared smoothness ($q$-WAS) and $\delta$-similarity ($(q, \delta)$-S) assumptions, allowing any exponent $q \in [1,2]$ instead of the standard $q = 2$. These results substantially broaden the scope of existing lower bounds. To complement them, we show that Normalized Stochastic Gradient Descent with Momentum Variance Reduction (NSGD-MVR), a known algorithm, matches these bounds in expectation. Beyond expectation guarantees, we introduce a new algorithm, Double-Clipped NSGD-MVR, which allows the derivation of high-probability convergence rates under weaker assumptions than previous works. Finally, for second-order methods with stochastic Hessians satisfying bounded $q$-th central moment assumptions for some exponent $q \in [1, 2]$ (allowing $q \neq p$), we establish sharper lower bounds than previous works while improving over Sadiev et al. (2025) (where only $p = q$ is considered) and yielding stronger convergence exponents. Together, these results provide a nearly complete complexity characterization of stochastic nonconvex optimization in heavy-tailed regimes. + oai:arXiv.org:2512.18713v1 math.OC - Mon, 22 Dec 2025 00:00:00 -0500 + Tue, 23 Dec 2025 00:00:00 -0500 new - http://creativecommons.org/licenses/by-sa/4.0/ - D. Dennst\"adt (Institut f\"ur Mathematik, Universit\"at Paderborn), J. Schaa (Institut f\"ur Mathematik, Universit\"at Paderborn), T. Berger (Institut f\"ur Mathematik, Universit\"at Paderborn) + http://creativecommons.org/licenses/by/4.0/ + Adrien Fradin, Abdurakhmon Sadiev, Laurent Condat, Peter Richt\'arik - Zeros of polynomial powers under the heat flow - https://arxiv.org/abs/2512.17808 - arXiv:2512.17808v1 Announce Type: new -Abstract: We study the evolution of zeros of high polynomial powers under the heat flow. For any fixed polynomial $P(z)$, we prove that the empirical zero distribution of its heat-evolved $n$-th power converges to a distribution on the complex plane as $n$ tends to infinity. We describe this limit distribution $\mu_t$ as a function of the time parameter $t$ of the heat evolution: For small time, zeros start to spread out in approximately semicircular distributions, then intricate curves start to form and merge, until for large time, the zero distribution approaches a widespread semicircle law through the initial center of mass. The Stieltjes transform of the limit distribution $\mu_t$ satisfies a self-consistent equation and a Burgers' equation. The present paper deals with general complex-rooted polynomials for which, in contrast to the real-rooted case, no free-probabilistic representation for $\mu_t$ is available. - oai:arXiv.org:2512.17808v1 - math.PR - math.CA + Modulus estimates and cavitation in higher dimensions + https://arxiv.org/abs/2512.18731 + arXiv:2512.18731v1 Announce Type: new +Abstract: We explore the phenomenon of cavitation in higher-dimensional elasticity, defining it as the mapping of a punctured ball onto a non-degenerate ring domain. Crucially, for the class of locally quasiconformal mappings (or more general mappings) defined on the punctured ball $0<|x|<1$ in $\mathbb R^n$ that we examine, cavitation is equivalent to a failure of continuous extension to the origin. While existing modulus estimates prove insufficient for reliably detecting cavitation in this setting, our study establishes refined modulus bounds. This is achieved by introducing a novel directional dilatation which, in conjunction with the known angular dilatation, overcomes the limitations of previous methods. We illustrate our theoretical findings with several examples that demonstrate both cavitation occurrence and its absence. + oai:arXiv.org:2512.18731v1 math.CV - Mon, 22 Dec 2025 00:00:00 -0500 + Tue, 23 Dec 2025 00:00:00 -0500 new http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Antonia H\"ofert, Jonas Jalowy, Zakhar Kabluchko + Anatoly Golberg, Vladimir Gutlyanski\u{i}, Vladimir Ryazanov, Toshiyuki Sugawa - On the classification of capillary graphs in Euclidean and non-Euclidean spaces - https://arxiv.org/abs/2512.17813 - arXiv:2512.17813v1 Announce Type: new -Abstract: We prove some rigidity and classification results for graphs with prescribed mean curvature and locally constant Dirichlet and Neumann data, for instance as they appear in capillarity problems. We consider domains in Riemannian manifolds, with emphasis on $\mathbb{R}^2$ and $\mathbb{R}^3$. We classify both the underlying domain and the resulting solution, providing general splitting theorems in this setting. - oai:arXiv.org:2512.17813v1 - math.DG + Non-homogeneous conormal derivative problem for quasilinear elliptic equations with Morrey data + https://arxiv.org/abs/2512.18742 + arXiv:2512.18742v1 Announce Type: new +Abstract: A non-homogeneous conormal derivative problem is considered for quasilinear divergence form elliptic equations modeled on the $m$-Laplacian operator. The nonlinear terms are given by Carath\'eodory functions and satisfy controlled growth structure conditions with respect to the solution and its gradient, while their $x$-behaviour is controlled in terms of suitable Morrey spaces. + Global essential boundedness is proved for the weak solutions, generalizing thus the classical $L^p$-result of Ladyzhenskaya and Ural'tseva to the framework of the Morrey scales. + oai:arXiv.org:2512.18742v1 math.AP - Mon, 22 Dec 2025 00:00:00 -0500 + Tue, 23 Dec 2025 00:00:00 -0500 new http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Giulio Colombo, Alberto Farina, Marco Magliaro, Luciano Mari, Marco Rigoli + Dian K. Palagachev, Lubomira G. Softova - Cubes from products of terms in progression with one term missing - https://arxiv.org/abs/2512.17821 - arXiv:2512.17821v1 Announce Type: new -Abstract: Let $5 \leq k \leq 11$ and $0\leq i \leq k-1$ be integers. We determine all solutions to the equation \begin{align*} n(n+d)(n+2d)\cdots(n+(i-1)d)(n+(i+1)d) \cdots (n+(k-1)d) = y^3 \end{align*} in integers $n,d,y$ with $ny \neq 0$, $d\geq 1$, and $\text{gcd}(n,d) = 1$. Our method relies on the theory of elliptic curves, including elliptic curve Chabauty over a number field. As an application, we answer a question of Das, Laishram, Saradha, and Sharma concerning rational points on a certain superelliptic curve. - oai:arXiv.org:2512.17821v1 - math.NT - Mon, 22 Dec 2025 00:00:00 -0500 + Quantum Hamiltonian reductions for W-algebras + https://arxiv.org/abs/2512.18743 + arXiv:2512.18743v1 Announce Type: new +Abstract: In this paper, we establish a general criterion for good pairs, namely pairs consisting of a nilpotent orbit and an even good grading in a simple Lie algebra, which guarantees the existence of a quantum Hamiltonian reduction between associated affine W-algebras. In particular, we show that for type A, any two affine W-algebras associated with two adjacent nilpotent orbits are related by quantum Hamiltonian reductions in full generality. + oai:arXiv.org:2512.18743v1 + math.RT + Tue, 23 Dec 2025 00:00:00 -0500 new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Kyle Pratt + http://creativecommons.org/licenses/by/4.0/ + Justine Fasquel, Shigenori Nakatsuka - van den Berg-Kesten--type correlation inequalities for disjoint polymers in the KPZ universality class - https://arxiv.org/abs/2512.17823 - arXiv:2512.17823v1 Announce Type: new -Abstract: In classical percolation theory, the van den Berg-Kesten (BK) inequality is a fundamental tool that shows that disjoint events induce negative conditionings on each other. The inequality also holds in the context of last passage percolation (LPP), which is the zero temperature limit of polymer models and an important subclass in the Kardar-Parisi-Zhang (KPZ) universality class. Recently, an analog of the BK inequality was discovered in the context of zero temperature line ensembles and the scaling limit of LPP, where it was used to study upper tail probabilities of the weight and the scaling limit of geodesics under such upper tail conditionings. However, while it has become apparent that such an inequality in the positive temperature setting would have a number of applications, it seems likely that a direct generalization of the zero temperature inequality would not hold. In this work we prove a version of the BK inequality for the KPZ line ensemble and the continuum directed random polymer. We do so by working with the log gamma polymer, making use of its integrability and the geometric RSK correspondence. Our inequality serves as a key input in analyzing the KPZ line ensemble and proving sharp upper tail estimates of the KPZ equation in arXiv:2208.08922, and proving convergence of the continuum directed random polymer to Brownian bridge under the upper tail event in arXiv:2311.12009. The crucial role of integrability in the validity of such an inequality is highlighted via a counter-example for a non-integrable model. - oai:arXiv.org:2512.17823v1 - math.PR + Higher-Rank Mathieu Opers, Toda Chain, and Analytic Langlands Correspondence + https://arxiv.org/abs/2512.18744 + arXiv:2512.18744v1 Announce Type: new +Abstract: We study the Riemann-Hilbert problem associated to flat sections of oper connections of arbitrary rank on the twice-punctured Riemann sphere with irregular singularities of the mildest type. We construct the solutions in terms of the solutions to a single non-linear integral equation. It follows from this construction that the generating function of the submanifold of opers coincides with the Yang-Yang function of the quantum Toda chain, proving a conjecture by Nekrasov, Rosly and Shatashvili. In this way we may furthermore reformulate the quantization conditions of the Toda chain in terms of the connection problem, for which we also provide a solution. We finally interpret our results as a variant of the Analytic Langlands Correspondence for the real version of the Hitchin system corresponding to the Toda chain. + oai:arXiv.org:2512.18744v1 math-ph + hep-th + math.CA math.MP - Mon, 22 Dec 2025 00:00:00 -0500 + math.SP + nlin.SI + Tue, 23 Dec 2025 00:00:00 -0500 new http://creativecommons.org/licenses/by/4.0/ - Shirshendu Ganguly, Milind Hegde, Lingfu Zhang + Jonah Baerman, Giovanni Ravazzini, Joerg Teschner - Sharp pressure estimates for the Navier-Stokes system in thin porous media - https://arxiv.org/abs/2512.17826 - arXiv:2512.17826v1 Announce Type: new -Abstract: A relevant problem for applications is to model the behavior of Newtonian fluids through thin porous media, which is a domain with small thickness $\epsilon$ and perforated by periodically distributed cylinders of size and period $\epsilon^\delta$, with $\delta>0$. Depending on the relation between thickness and the size of the cylinders, it was introduced in (Fabricius et al., Transp. Porous Media, 115, 473-493, 2016), (Anguiano and Su\'arez-Grau, Z. Angew. Math. Phys., 68:45, 2017) and (Anguiano and Su\'arez-Grau, Mediterr. J. Math., 15:45, 2018) that there exist three regimes depending on the value of $\delta$: $\delta\in (0,1)$, $\delta=1$ and $\delta>1$. In each regime, the asymptotic behavior of the fluid is governed by a lower-dimensional Darcy's law. - In previous studies, the Reynolds number is considered to be of order one and so, the question that arises is for what range of values of the Reynolds number the lower-dimensional Darcy laws are still valid in each regime, which represents the main the goal of this paper. In this sense, considering a fluid governed by the Navier-Stokes system and assuming the Reynolds number written in terms of the thickness $\epsilon$, we prove that, for each regime, there exists a critical Reynolds number $Re_c$ such that for every Reynolds number $Re$ with order smaller or equal than $Re_c$, the lower-dimensional Darcy law is still valid. On the contrary, for Reynolds numbers $Re$ greater than $Re_c$, the inertial term of the Navier-Stokes system has to be taken into account in the asymptotic behavior and so, the Darcy law is not valid. - oai:arXiv.org:2512.17826v1 - math.AP - Mon, 22 Dec 2025 00:00:00 -0500 + Critical metrics for the quadratic curvature functional on complete four-dimensional manifolds + https://arxiv.org/abs/2512.18758 + arXiv:2512.18758v1 Announce Type: new +Abstract: We study critical metrics of the curvature functional $\A(g)=\int_M |R|^2\, \vol$, on complete four-dimensional Riemannian manifolds $(M,g)$ with finite energy, that is, $\A(g)<\infty$. Under the natural inequality condition on the curvature operator of the second kind associated with the trace-free Ricci tensor, we prove that $(M,g)$ is either Einstein or locally isometric to a Riemannian product of two-dimensional manifolds of constant Gaussian curvatures $c$ and $-c$ $(c\ne 0)$. This extends the compact classification of four-dimensional $\mathcal{A}$-critical metrics obtained in earlier work to the complete setting. + oai:arXiv.org:2512.18758v1 + math.DG + Tue, 23 Dec 2025 00:00:00 -0500 new - http://creativecommons.org/licenses/by-nc-nd/4.0/ - 10.1007/s40840-023-01514-1 - Bulletin of the Malaysian Mathematical Sciences Society, Volume 46, 2023, article number 117 - Mar\'ia Anguiano, Francisco J. Su\'arez-Grau + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Yunhee Euh, JeongHyeong Park - Roughness-induced effects on the thermomicropolar fluid flow through a thin domain - https://arxiv.org/abs/2512.17829 - arXiv:2512.17829v1 Announce Type: new -Abstract: In this paper, we study the asymptotic behavior of the thermomicropolar fluid flow through a thin channel with rough boundary. The flow is governed by the prescribed pressure drop between the channel's ends and the heat exchange through the rough wall is allowed. Depending on the limit of the ratio between channel's thickness and the wavelength of the roughness, we rigorously derive different asymptotic models clearly showing the roughness-induced effects on the average velocity and microrotation. To accomplish that, we employ the adaptation of the unfolding method to a thin-domain setting. - oai:arXiv.org:2512.17829v1 - math.AP - Mon, 22 Dec 2025 00:00:00 -0500 + Cohomology of varieties over the maximal Kummer extension of a number field + https://arxiv.org/abs/2512.18759 + arXiv:2512.18759v1 Announce Type: new +Abstract: Let $X$ be a smooth projective geometrically connected variety defined over a number field $K$. We prove that the geometric \'etale cohomology of $X$ with $\mathbb{Q}/\mathbb{Z}$-coefficients has finitely many classes invariant under the Galois group of the maximal Kummer extension of $K$ in odd degrees. In particular, every abelian variety has finite torsion over the maximal Kummer extension. This improves results by R\"ossler and the second author as well as Murotani and Ozeki. We also show that finiteness of torsion of a given abelian variety over non-abelian solvable extensions of $K$ is not controlled by the Galois group of the extension. + oai:arXiv.org:2512.18759v1 + math.AG + math.NT + Tue, 23 Dec 2025 00:00:00 -0500 new http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - 10.1111/sapm.12611 - Stud. Appl. Math.151 (2023) 716-751 - Igor Pa\v{z}anin, Francisco J. Su\'arez-Grau + Davide Lombardo, Tam\'as Szamuely - Paravortices: loop braid representations with both generators involutive - https://arxiv.org/abs/2512.17830 - arXiv:2512.17830v1 Announce Type: new -Abstract: We first motivate the study of a certain quotient of the loop braid category, both for the mathematics underpinning recent approaches to topological quantum computation; and as a key example in non-semisimple higher representation theory. For reasons that will become clear, we call this quotient the mixed doubles category, $MD$. Then our main result is a theorem classifying all mixed doubles representations in rank-2. - Each representation yields a mixed doubles group representation for every loop braid group $LB_n$, and we are able to analyse the unified linear representation theory of many of these sequences of representations, using a mixture of very classical, classical, and new techniques. In particular this is a motivating example for the `glue' generalisation of charge-conserving representation theory (a form of rigid higher non-semisimplicity) introduced recently. - oai:arXiv.org:2512.17830v1 - math.QA - Mon, 22 Dec 2025 00:00:00 -0500 + Sharp Fractional Sobolev Embeddings on Closed Manifolds + https://arxiv.org/abs/2512.18770 + arXiv:2512.18770v1 Announce Type: new +Abstract: We develop an intrinsic, heat-kernel based fractional Sobolev framework on closed Riemannian manifolds and study the critical fractional Sobolev embedding. We determine the optimal coefficient of the lower-order $L^{p}$ term and prove that the fully sharp $p$-power inequality cannot hold globally in the superquadratic range. We further establish an almost sharp inequality whose leading constant is arbitrarily close to the Euclidean best constant, and we derive improved inequalities under finitely many orthogonality constraints with respect to sign-changing test families. + oai:arXiv.org:2512.18770v1 + math.AP + math.DG + Tue, 23 Dec 2025 00:00:00 -0500 new http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Paul P. Martin, Fiona Torzewska, Eric C. Rowell + Hao Tan, Zetian Yan, Zhipeng Yang - A generalized Reynolds equation for micropolar flows past a ribbed surface with nonzero boundary conditions - https://arxiv.org/abs/2512.17837 - arXiv:2512.17837v1 Announce Type: new -Abstract: Inspired by the lubrication framework, in this paper we consider a micropolar fluid flow through a rough thin domain, whose thickness is considered as the small parameter $\varepsilon$ while the roughness at the bottom is defined by a periodical function with period of order $\varepsilon^{\ell}$ and amplitude $\varepsilon^{\delta}$, with $\delta>\ell>1$. Assuming nonzero boundary conditions on the rough bottom and by means of a version of the unfolding method, we identify a critical case $\delta={3\over 2}\ell-{1\over 2}$ and obtain three macroscopic models coupling the effects of the rough bottom and the nonzero boundary conditions. In every case we provide the corresponding micropolar Reynolds equation. We apply these results to carry out a numerical study of a model of squeeze-film bearing lubricated with a micropolar fluid. Our simulations reveal the impact of the roughness coupled with the nonzero boundary conditions on the performance of the bearing, and suggest that the introduction of a rough geometry may contribute to enhancing the mechanical properties of the device. - oai:arXiv.org:2512.17837v1 + On the effects of surface roughness in non-isothermal porous medium flow + https://arxiv.org/abs/2512.18787 + arXiv:2512.18787v1 Announce Type: new +Abstract: We analyze a non-isothermal Darcy-Brinkman thin-film flow with a periodically oscillating boundary and viscous dissipation acting as a heat source. Using asymptotic analysis and the periodic unfolding method, we establish the convergence of velocity, pressure, and temperature fields as the small parameter (related to the film thickness and the period of the roughness) tends to zero. The limit problems depend on the relative scaling of the roughness wavelength and consist of coupled elliptic systems combining Reynolds-type equations with Darcy-Brinkman cell problems and reduced energy equation. In the critical roughness regime, the effective model exhibits a strong coupling induced by the oscillatory geometry, which does not occur in a smooth-boundary case. + oai:arXiv.org:2512.18787v1 math.AP - Mon, 22 Dec 2025 00:00:00 -0500 + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://creativecommons.org/licenses/by-nc-nd/4.0/ + Mar\'ia Anguiano, Igor Pa\v{z}anin, Francisco J. Su\'arez-Grau + + + Irrational pencils, and characterization of Varieties isogenous to a product, via the Profinite completion of the Fundamental group + https://arxiv.org/abs/2512.18798 + arXiv:2512.18798v1 Announce Type: new +Abstract: We give a very short proof of two Theorems, whose content is outlined in the title, and where $\Pi_g$ is the fundamental group of a compact complex curve of genus $g$: + (1) Theorem 2.1 of the irrational pencil in the profinite version, saying that for a compact K\"ahler manifold an irrational pencil, that is, a fibration onto a curve of genus $g \geq 2$, corresponds to a surjection of the profinite completion $\widehat{\pi}_1(X) \twoheadrightarrow \widehat{\Pi_g}$, which satisfies a maximality property; + (2) Theorem 1.4 on the characterization of varieties isogenous to a product, profinite version, giving in particular a criterion for $X$ a compact K\"ahler manifold to be isomorphic to a product of curves of genera at least 2: if and only if $\widehat{\pi}_1(X) \cong \prod_1^n \widehat{\Pi_{g_i}}$, and some volume or cohomological condition is satisfied. + Theorem 1.4 yields a stronger result than the Main Theorem A of a recent article by 5 authors. + oai:arXiv.org:2512.18798v1 + math.AG + math.AT + math.CV + Tue, 23 Dec 2025 00:00:00 -0500 new http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - 10.1051/m2an/2022039 - ESAIM: Math. Model. Numer. Anal. 56 (2022) 1255-1305 - Matthieu Bonnivard, Igor Pa\v{z}anin, Francisco J. Su\'arez-Grau + Fabrizio Catanese (Bayreuth University), appendix by Pavel Zalesskii (University of Brasilia) - Time Optimal Control Problem for the Landau-Lifshitz-Bloch equation - https://arxiv.org/abs/2512.17839 - arXiv:2512.17839v1 Announce Type: new -Abstract: This paper investigates the time-optimal control problem for the Landau-Lifshitz-Bloch (LLB) equation, a macroscopic model that characterizes magnetization dynamics in ferromagnetic materials across a wide temperature range, including near and above the Curie temperature. We analyze the LLB system on bounded domains in one, two, and three dimensions, establishing the existence of optimal controls that drive the magnetization to a desired target state within a minimal time frame. Utilizing a Lagrange multiplier approach and an adjoint-based framework, we derive first-order necessary optimality conditions. Furthermore, we establish second-order sufficient conditions for local optimality, addressing the mathematical challenges posed by the system's inherent nonlinearities and the nonlinear appearance of the control in the effective magnetic field. These results provide a rigorous theoretical basis for the rapid manipulation of magnetic states, offering insights into the fundamental limits of control for nonlinear diffusion-relaxation processes in magnetism. Such findings are essential for advancing high-speed magnetic memory technologies and optimizing thermal magnetic switching in next-generation storage technologies. - oai:arXiv.org:2512.17839v1 - math.OC + Positivity and long-term behaviour of a diffusion model with measure-valued nonlocal reaction term + https://arxiv.org/abs/2512.18799 + arXiv:2512.18799v1 Announce Type: new +Abstract: The behaviour is investigated of solutions to a diffusion equation on the real line with nonlocal and singular reaction term, i.e., given by a Dirac source or sink at the origin. It gives a simplified representation of for example a control system that senses concentration at a distance, but "intervenes" at the origin. Positivity of solutions (for positive initial conditions) cannot be guaranteed for all parameter settings in the model. We determine a parameter regime and conditions on the positive initial condition in terms of monotonicity and symmetry, that do allow us to conclude the positivity of the solution for all time. In addition, we provide conditions that ensure convergence of the system to a constant steady state (pointwise), outside the region of observation. Technically, we extensively use Laplace transform arguments to achieve these results. + oai:arXiv.org:2512.18799v1 math.AP - Mon, 22 Dec 2025 00:00:00 -0500 + Tue, 23 Dec 2025 00:00:00 -0500 new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Sidhartha Patnaik, Kumarasamy Sakthivel + http://creativecommons.org/licenses/by/4.0/ + Xiao Yang, Qiyao Peng, Sander C. Hille - On the generalized Fermat equation of signature $(5,p,3)$ - https://arxiv.org/abs/2512.17845 - arXiv:2512.17845v1 Announce Type: new -Abstract: In this article we study solutions to the generalized Fermat equation $x^q+y^p+z^r=0 $ using hypergeometric motives within the framework of the modular method. In doing so, we give an explicit description of the ramification behavior at primes dividing $2qr$ and analyze the contribution of trivial solutions. We identify a general obstruction to the modular method that accounts for its failure in many instances. As an application, assuming a standard large image conjecture, we prove that the previous equation admits no nontrivial primitive solutions $(a,b,c)$ with $3 \nmid c$, when $q=5,$ $r=3$ and $p$ is a prime sufficiently large. - oai:arXiv.org:2512.17845v1 - math.NT - Mon, 22 Dec 2025 00:00:00 -0500 + Legendrian barriers in prequantization bundles + https://arxiv.org/abs/2512.18808 + arXiv:2512.18808v1 Announce Type: new +Abstract: We show that prequantization bundles have explicit Legendrian barriers, whose removal obstruct the embedding of long cylinders over Legendrian submanifolds. + oai:arXiv.org:2512.18808v1 + math.SG + Tue, 23 Dec 2025 00:00:00 -0500 new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Ariel Pacetti, Lucas Villagra Torcomian + http://creativecommons.org/licenses/by/4.0/ + Emmanuel Opshtein - The Semi-Classical Limit from the Dirac Equation with Time-Dependent External Electromagnetic Field to Relativistic Vlasov Equations - https://arxiv.org/abs/2512.17849 - arXiv:2512.17849v1 Announce Type: new -Abstract: We prove the mathematically rigorous (semi-)classical limit $\hbar \to 0$ of the Dirac equation with time-dependent external electromagnetic field to relativistic Vlasov equations with Lorentz force for electrons and positrons. In this limit antimatter and spin remain as intrinsically relativistic effects on a classical level. Our global-in-time results use Wigner transforms and a Lagrange multiplier viewpoint of the matrix-valued Wigner equation. In particular, we pass to the limit in the ''full" Wigner matrix equation without projecting on the eigenspaces of the matrix-valued symbol of the Dirac operator. In the limit, the Lagrange multiplier maintains the constraint that the Wigner measure and the symbol of the Dirac operator commute and vanishes when projected on the electron or positron eigenspace. This is a different approach to the problem as discussed in [P. G\'erard, P. Markowich, N.J. Mauser, F. Poupaud: Comm. Pure Appl. Math. 50(4):323--379, 1997], where the limit is taken in the projected Wigner equation. By explicit calculation of the remainder term in the expansion of the Moyal product we are able to generalize to time-dependent potentials with much less regularity. We use uniform $L^2$ bounds for the Wigner transform, which are only possible for a special class of mixed states as initial data. - oai:arXiv.org:2512.17849v1 - math.AP - math-ph - math.MP - Mon, 22 Dec 2025 00:00:00 -0500 + $SL_2$-tilings with translational symmetry + https://arxiv.org/abs/2512.18810 + arXiv:2512.18810v1 Announce Type: new +Abstract: An $SL_2$-tiling is a bi-infinite matrix in which all adjacent $2 \times 2$ minors are equal to $1$. Positive integral $SL_2$-tilings were introduced by Assem, Reutenauer and Smith as generalisations of classical Conway--Coxeter frieze patterns. We show that positive integral $SL_2$-tilings with translational symmetry are in bijection with triangulations of annuli. We use this correspondence to study the properties of periodic positive integral $SL_2$-tilings. + oai:arXiv.org:2512.18810v1 + math.CO + math.RA + Tue, 23 Dec 2025 00:00:00 -0500 new http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Fran\c{c}ois Golse, Nikolai Leopold, Norbert J. Mauser, Jakob M\"oller, Chiara Saffirio + Veronique Bazier-Matte, Marie-Anne Bourgie, Anna Felikson, Pavel Tumarkin - Conformal invariants for the zero mode equation - https://arxiv.org/abs/2512.17854 - arXiv:2512.17854v1 Announce Type: new -Abstract: For non-trivial solutions to the zero mode equation on a closed spin manifold \[D \varphi=iA\cdot \varphi,\] we first provide a simple proof for the sharp inequality \eq{ \norm{A}_{L^n}^2 \ge \frac {n}{4(n-1)} Y(M,[g]), } where $Y(M,[g])$ is the Yamabe constant of $(M,g)$, which was obtained by Frank-Loss and Reuss. Then we classify completely the equality case by proving that equality holds if and only if $\varphi$ is a Killing spinor, and if and only if $(M,g)$ is a Sasaki-Einstein manifold with $A$ (up to scaling) as its Reeb field and $\varphi$ a vacuum up to a conformal transformation. More generalizations have been also studied. - oai:arXiv.org:2512.17854v1 - math.DG - math-ph - math.AP - math.MP - Mon, 22 Dec 2025 00:00:00 -0500 + Enriques surfaces with non-generic non-degeneracy + https://arxiv.org/abs/2512.18812 + arXiv:2512.18812v1 Announce Type: new +Abstract: We study the non-degeneracy invariant $\mathrm{nd}(Y)$ of complex Enriques surfaces in families. Our first main result shows that $\mathrm{nd}(Y)$ cannot increase under specialization. The second main result is the conclusion of the computation of the non-degeneracy invariant for the $155$ families of $(\tau,\overline{\tau})$-generic surfaces introduced by Brandhorst and Shimada. Of the previously known $144$ cases, only $3$ satisfy $\mathrm{nd}(Y)\neq10$, which is the non-degeneracy invariant of a general Enriques surface. The remaining $11$ families studied in this article also have non-generic non-degeneracy. To compute this, we produce upper bounds on $\mathrm{nd}(Y)$ by refining this invariant into two others: the Fano and Mukai non-degeneracy invariants, which are related to two different classes of projective realizations of Enriques surfaces. As a result, we find the first known examples of Enriques surfaces with $\mathrm{nd}(Y)=9$. + oai:arXiv.org:2512.18812v1 + math.AG + Tue, 23 Dec 2025 00:00:00 -0500 new http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Guofang Wang, Mingwei Zhang + Riccardo Moschetti, Franco Rota, Luca Schaffler - On General Linearly Implicit Quantized State System Methods - https://arxiv.org/abs/2512.17855 - arXiv:2512.17855v1 Announce Type: new -Abstract: This work proposes a methodology to develop new numerical integration algorithms for ordinary differential equations based on state quantization, generalizing the notions of Linearly Implicit Quantized State Systems (LIQSS) methods. Using this idea, two novel sub-families of algorithms are designed that improve the performance of current LIQSS methods while preserving their properties regarding stability, global error bound and efficient event handling capabilities. The features of the new algorithms are studied in two application examples where the advantages over classic numerical integration algorithms is also analyzed. - oai:arXiv.org:2512.17855v1 - math.NA - cs.NA - cs.PF - Mon, 22 Dec 2025 00:00:00 -0500 + On finite quotients of surface braid groups having order at most $127$ + https://arxiv.org/abs/2512.18817 + arXiv:2512.18817v1 Announce Type: new +Abstract: Let $\Sigma_b$ be a compact Riemann surface of genus $b \geq 2$ and let $\mathsf{P}_2(\Sigma_b)=\pi_1(\Sigma_b \times \Sigma_b - \Delta)$ be the corresponding pure braid group on two strands. A finite quotient $\varphi \colon \mathsf{P}_2(\Sigma_b) \to G$ is called "admissible" if $\varphi$ does not factor through $\pi_1(\Sigma_b \times \Sigma_b)$. In this work we classify all admissible quotients of $\mathsf{P}_2(\Sigma_b)$ such that $|G| \leq 127$. + oai:arXiv.org:2512.18817v1 + math.GR + math.AG + math.GT + Tue, 23 Dec 2025 00:00:00 -0500 new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Mariana Bergonzi, Joaqu\'in Fern\'andez, Ernesto Kofman + http://creativecommons.org/licenses/by/4.0/ + Francesco Polizzi, Pietro Sabatino - Optimal Control Problems with Nonlocal Conservation Laws: Existence of Optimizers and Singular Limits in Approximations of Local Conservation Laws - https://arxiv.org/abs/2512.17870 - arXiv:2512.17870v1 Announce Type: new -Abstract: This contribution considers optimal control problems subject to nonlocal conservation laws -- those in which the velocity depends nonlocally (i.e., via a convolution) on the solution -- and the so-called singular limit. First, the existence of minimizers is demonstrated for a broad class of optimal control problems, involving optimization over the initial datum, velocity, and nonlocal kernel for classical tracking-type $L^2$ cost functionals. Then, it is proven that the obtained minimizers converge to minimizers of the corresponding local optimal control problem when the kernel function of the convolution is of exponential type and approaches a Dirac distribution. Finally, some numerical results are presented. - oai:arXiv.org:2512.17870v1 + Locational Marginal Emissions for Carbon-Aware Data Center Operations in Large-Scale Power Grids + https://arxiv.org/abs/2512.18819 + arXiv:2512.18819v1 Announce Type: new +Abstract: Carbon accounting methods for electricity consumption face challenges regarding physical deliverability, double counting, additionality, and impact magnitude. Locational Marginal Emissions (LMEs) show potential to address many of these key issues. However, their use in a large-scale power grids remains understudied. We analyze the properties of LMEs from a data center's perspective in a 1493-bus Western Interconnection over one year of hourly operation. We find that LME characteristics create three distinct regions: the hydropower-dominated Pacific Northwest, with low and stable LMEs; the coal-heavy Intermountain West, containing often high LMEs; and the Sunbelt, where mixed generation leads to variable LMEs correlated with solar output. This characterization provides analytical guidance for data center emission reduction. In particular, LME-guided emission reduction interventions through data center temporal-spatial load shifting, siting, and renewable procurement display over 85% accuracy with respect to actual emission reduction. Moreover, large-scale, nodal grid simulation is shown to be critical to accurate evaluation. + oai:arXiv.org:2512.18819v1 math.OC - Mon, 22 Dec 2025 00:00:00 -0500 + Tue, 23 Dec 2025 00:00:00 -0500 new http://creativecommons.org/licenses/by/4.0/ - Alexander Keimer, Lukas Pflug, Jakob Rodestock + Luc Cote, Andy Sun - Cellular free resolutions for normalizations of toric ideals - https://arxiv.org/abs/2512.17871 - arXiv:2512.17871v1 Announce Type: new -Abstract: For any toric ideal $I$ in a polynomial ring $S$, we provide a combinatorial description of a free resolution of the integral closure of the $S$-module $S/I$. These new complexes arise from an extension of Bayer--Sturmfels' theory of cellular free resolutions. As applications, we unify several constructions for a resolution of the diagonal embedding of a toric variety, and compare the locally free resolutions for toric subvarieties introduced by Hanlon--Hicks--Lazarev and Brown--Erman. - oai:arXiv.org:2512.17871v1 - math.AC - math.AG - Mon, 22 Dec 2025 00:00:00 -0500 + The Minkowski dimension of the image of an arboreal Galois representation + https://arxiv.org/abs/2512.18825 + arXiv:2512.18825v1 Announce Type: new +Abstract: Let $f:\mathbb{P}^1\to\mathbb{P}^1$ be a rational map of degree $d\geq2$ defined over a number field $K$ and let $\alpha\in\mathbb{P}^1(K)$. We consider the lower and upper Minkowski dimensions of the arboreal Galois group $G_{f,\alpha}$ associated to the pair $(f,\alpha)$, which is naturally a subgroup of the automorphism group of the infinite $d$-ary rooted tree whose vertices are indexed by the backward orbit $f^{-\infty}(\alpha)$. We state conjectures on the existence of Minkowski dimension, as well as proposed characterizations of cases in which it takes its minimal and maximal values. We establish basic cases in which the upper Minkowski dimension of $G_{f,\alpha}$ is not maximal, and we establish basic cases in which it is minimal. We show that abelian automorphism groups always have vanishing Minkowski dimension, and as a consequence, that one of our conjectures implies a conjecture of Andrews-Petsche on pairs $(f,\alpha)$ with abelian arboreal Galois group. + oai:arXiv.org:2512.18825v1 + math.NT + Tue, 23 Dec 2025 00:00:00 -0500 new http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Christine Berkesch, Lauren Cranton Heller, Gregory G. Smith, Jay Yang + Chifan Leung, Clayton Petsche - A note on Poincar\'e-Sobolev type inequalities on compact manifolds - https://arxiv.org/abs/2512.17872 - arXiv:2512.17872v1 Announce Type: new -Abstract: We prove a Poincar\'e-Sobolev type inequality on compact Riemannian manifolds where the deviation of a function from a biased average, defined using a density, is controlled by the unweighted Lebesgue norm of its gradient. Unlike classical weighted Poincar\'e inequalities, the density does not enter the measure or the Sobolev norms, but only the reference average. We show that the associated Poincar\'e constant depends quantitatively on the Lebesgue norm of the density. This framework naturally arises in the analysis of coupled elliptic systems and seems not to have been addressed in the existing literature. - oai:arXiv.org:2512.17872v1 - math.AP - math.DG - Mon, 22 Dec 2025 00:00:00 -0500 + Pairwise Attraction-Repulsion on Multilayer Social Networks + https://arxiv.org/abs/2512.18833 + arXiv:2512.18833v1 Announce Type: new +Abstract: We introduce a probabilistic pairwise \emph{attraction--repulsion} model for opinion dynamics on multilayer social networks, in which agents hold layer-specific states and interact through random matchings that couple multiple, time-varying layers. At each time step, interacting pairs update their layer-specific states using layer-dependent, time-varying interaction rates and a random sign (attractive or repulsive), and the resulting updates are averaged across layers. This framework generalizes classical gossip and Deffuant-type models while capturing heterogeneous cross-layer influences and antagonistic interactions. + Under mild graph-theoretic and moment assumptions, we establish almost sure global consensus. Specifically, when the expected net effect of interactions is strictly attractive and random matchings ensure sufficient cross-layer connectivity, all agents' layer states converge almost surely to the global average. We further identify a purely attractive regime in which consensus holds even under intermittent connectivity and without any moment assumptions on the initial states. Numerical experiments illustrate the dynamical regimes predicted by the theory, including consensus, metastability, and polarization. Together, these results provide a rigorous foundation for understanding how multilayer structure, stochastic interactions, and mixed-sign influence shape collective outcomes in social and engineered networked systems. + oai:arXiv.org:2512.18833v1 + math.DS + Tue, 23 Dec 2025 00:00:00 -0500 new http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Romain Gicquaud + Hsin-Lun Li - Plane Strong Connectivity Augmentation - https://arxiv.org/abs/2512.17904 - arXiv:2512.17904v1 Announce Type: new -Abstract: We investigate the problem of strong connectivity augmentation within plane oriented graphs. - We show that deciding whether a plane oriented graph $D$ can be augmented with (any number of) arcs $X$ such that $D+X$ is strongly connected, but still plane and oriented, is NP-hard. - This question becomes trivial within plane digraphs, like most connectivity augmentation problems without a budget constraint. - The budgeted version, Plane Strong Connectivity Augmentation (PSCA) considers a plane oriented graph $D$ along with some integer $k$, and asks for an $X$ of size at most $k$ ensuring that $D+X$ is strongly connected, while remaining plane and oriented. - Our main result is a fixed-parameter tractable algorithm for PSCA, running in time $2^{O(k)} n^{O(1)}$. - The cornerstone of our procedure is a structural result showing that, for any fixed $k$, each face admits a bounded number of partial solutions "dominating" all others. - Then, our algorithm for PSCA combines face-wise branching with a Monte-Carlo reduction to the polynomial Minimum Dijoin problem, which we derandomize. - To the best of our knowledge, this is the first FPT algorithm for a (hard) connectivity augmentation problem constrained by planarity. - oai:arXiv.org:2512.17904v1 + Induced minors and subpolynomial treewidth + https://arxiv.org/abs/2512.18835 + arXiv:2512.18835v1 Announce Type: new +Abstract: Given a family $\mathcal{H}$ of graphs, we say that a graph $G$ is $\mathcal{H}$-induced-minor-free if no induced minor of $G$ is isomorphic to a member of $\mathcal{H}$, We denote by $W_{t\times t}$ the $t$-by-$t$ hexagonal grid, and by $K_{t,t}$ the complete bipartite graph with both sides of the bipartition of size $t$. We show that the class of $\{K_{t,t},W_{t\times t}\}$-induced minor-free graphs with bounded clique number has subpolynomial treewidth. Specifically, we prove that for every integer $t$ there exist $\epsilon \in (0,1]$ and $c \in \nat$ such that every $n$-vertex $\{K_{t,t},W_{t\times t}\}$-induced minor-free graph with no clique of size $t$ has treewidth at most $2^{c\log^{1-\epsilon}n}$. + oai:arXiv.org:2512.18835v1 math.CO - cs.CG - cs.DM - Mon, 22 Dec 2025 00:00:00 -0500 + Tue, 23 Dec 2025 00:00:00 -0500 new - http://creativecommons.org/licenses/by/4.0/ - St\'ephane Bessy, Daniel Gon\c{c}alves, Amadeus Reinald, Dimitrios M. Thilikos + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Maria Chudnovsky, Julien Codsi, David Fischer, Daniel Lokshtanov - Fisher information for the multi-species Landau system - https://arxiv.org/abs/2512.17905 - arXiv:2512.17905v1 Announce Type: new -Abstract: We consider the Fisher information for spatially homogeneous multi-species Landau system. We show that the mass-weighted Fisher information is monotone decreasing in time along the solutions of the Landau system with a general class of interaction potentials. - oai:arXiv.org:2512.17905v1 - math.AP - Mon, 22 Dec 2025 00:00:00 -0500 + Convergence of the adapted empirical measure for mixing observations + https://arxiv.org/abs/2512.18838 + arXiv:2512.18838v1 Announce Type: new +Abstract: The adapted Wasserstein distance $\mathcal{AW}$ is a modification of the classical Wasserstein metric, that provides robust and dynamically consistent comparisons of laws of stochastic processes, and has proved particularly useful in the analysis of stochastic control problems, model uncertainty, and mathematical finance. In applications, the law of a stochastic process $\mu$ is not directly observed, and has to be inferred from a finite number of samples. As the empirical measure is not $\mathcal{AW}$-consistent, Backhoff, Bartl, Beiglb\"ock and Wiesel introduced the adapted empirical measure $\widehat{\mu}^N$, a suitable modification, and proved its $\mathcal{AW}$-consistency when observations are i.i.d. + In this paper we study $\mathcal{AW}$-convergence of the adapted empirical measure $\widehat{\mu}^N$ to the population distribution $\mu$, for observations satisfying a generalization of the $\eta$-mixing condition introduced by Kontorovich and Ramanan. We establish moment bounds and sub-exponential concentration inequalities for $\mathcal{AW}(\mu,\widehat{\mu}^N)$, and prove consistency of $\widehat{\mu}^N$. In addition, we extend the Bounded Differences inequality of Kontorovich and Ramanan for $\eta$-mixing observations to uncountable spaces, a result that may be of independent interest. Numerical simulations illustrating our theory are also provided. + oai:arXiv.org:2512.18838v1 + math.PR + math.ST + stat.TH + Tue, 23 Dec 2025 00:00:00 -0500 new http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Yuzhe Zhu + Ruslan Mirmominov, Johannes Wiesel - Axial Symmetric Navier Stokes Equations and the Beltrami /anti Beltrami spectrum in view of Physics Informed Neural Networks - https://arxiv.org/abs/2512.08846 - arXiv:2512.08846v2 Announce Type: cross -Abstract: In this paper, I further continue an investigation on Beltrami Flows began in 2015 with A. Sorin and amply revised and developed in 2022 with M. Trigiante. Instead of a compact $3$-torus $T^3=\mathbb{R}^3/\Lambda$ where $\Lambda$ is a crystallographic lattice, as done in previous work, here I considered flows confined in a cylinder with identified opposite bases. In this topology I considered axial symmetric flows and found a complete basis of axial symmetric harmonic $1$-forms that, for each energy level, decomposes into six components: two Beltrami, two anti-Beltrami and two closed forms. These objects, that are written in terms of trigonometric and Bessel functions, constitute a function basis for an $L^2$ space of axial symmetric flows. I have presented a general scheme for the search of axial symmetric solutions of Navier Stokes equation by reducing the latter to an hierachy of quadratic relations on the development coefficients of the flow in the above described functional basis. It is proposed that the coefficients can be determined by means of a Physics Informed like Neural Network optimization recursive algorithm. Indeed the present paper provides the theoretical foundations for such a algorithmic construction that is planned for a future publication. - oai:arXiv.org:2512.08846v2 - physics.flu-dyn - cs.IT - math-ph - math.GR - math.IT - math.MP - math.OC - Mon, 22 Dec 2025 00:00:00 -0500 - cross + Continuous in time bubbling and Soliton Resolution for Non-negative Solutions of the Energy-Critical Heat Flow + https://arxiv.org/abs/2512.18840 + arXiv:2512.18840v1 Announce Type: new +Abstract: We show that any finite energy solution of the energy-critical nonlinear heat flow in dimensions $d\geq 3$ asymptotically resolves into a sum of possibly time-dependent solitons, a radiation term, and an error term that vanishes in the energy space. As a consequence, when the initial data has finite energy and is non-negative, we settle the Soliton Resolution Conjecture for all dimensions $d\geq 3.$ + oai:arXiv.org:2512.18840v1 + math.AP + Tue, 23 Dec 2025 00:00:00 -0500 + new http://creativecommons.org/licenses/by/4.0/ - Pietro Fr\'e + Shrey Aryan - Legendrian Non-simple Whitehead Doubles Of The Trefoil - https://arxiv.org/abs/2512.15914 - arXiv:2512.15914v1 Announce Type: cross -Abstract: Ozsv\'ath and Stipsicz showed that some Eliashberg-Chekanov twist knots, which are Whitehead doubles of the unknot, are not Legendrian simple. We extend their result by considering some Whitehead doubles of the trefoil: Using properties of knot Floer homology and the distinguished surgery triangle, we show that this family of knots is Legendrian non-simple in the standard contact 3-sphere. - oai:arXiv.org:2512.15914v1 - math.GT - math.SG - Mon, 22 Dec 2025 00:00:00 -0500 - cross + Generalized Chebyshev acceleration on the unit disc + https://arxiv.org/abs/2512.18848 + arXiv:2512.18848v1 Announce Type: new +Abstract: Generalized Chebyshev acceleration is a semi-iterative technique applicable to a basic iterative method only when the eigenvalues of the iteration matrix satisfy a highly restrictive inclusion condition. In this work, we relax this requirement by introducing an alternative iterative scheme that converges to the same solution. The effectiveness of the proposed approach is examined through its application to a large-scale sparse normal matrix. + oai:arXiv.org:2512.18848v1 + math.NA + cs.NA + Tue, 23 Dec 2025 00:00:00 -0500 + new http://creativecommons.org/licenses/by/4.0/ - Saliha K{\i}van\c{c} + Nurg\"ul G\"okg\"oz - Thermodynamics a la Souriau on K\"ahler Non Compact Symmetric Spaces for Cartan Neural Networks - https://arxiv.org/abs/2512.16772 - arXiv:2512.16772v1 Announce Type: cross -Abstract: In this paper, we clarify several issues concerning the abstract geometrical formulation of thermodynamics on non compact symmetric spaces $\mathrm{U/H}$ that are the mathematical model of hidden layers in the new paradigm of Cartan Neural Networks. We introduce a distinction between the generalized thermodynamics associated with Dynamical Systems and the challenging proposal of Gibbs probability distributions on $\mathrm{U/H}$ provided by generalized thermodynamics {\`a} la Souriau. Main result is the proof that $\mathrm{U/H}$.s supporting Gibbs distributions are only the K\"ahler ones. For the latter, we solve the problem of determining the space of temperatures, namely of Lie algebra elements for which the partition function converges. The space of generalized temperatures is the orbit under the adjoint action of $\mathrm{U}$ of a positivity domain in the Cartan subalgebra $C_c\subset\mathbb{H}$ of the maximal compact subalgebra $\mathbb{H}\subset\mathbb{U}$. We illustrate how our explicit constructions for the Poincar\'e and Siegel planes might be extended to the whole class of Calabi-Vesentini manifolds utilizing Paint Group symmetry. Furthermore we claim that Rao's, Chentsov's, Amari's Information Geometry and the thermodynamical geometry of Ruppeiner and Lychagin are the very same thing. The most important property of the Gibbs probability distributions provided by the here introduced setup is their covariance with respect to the action of the full group of symmetries $\mathrm{U}$. The partition function is invariant against $\mathrm{U}$ transformations and the set of its arguments, namely the generalized temperatures, can be always reduced to a minimal set whose cardinality is equal to the rank of the compact denominator group $\mathrm{H}\subset \mathrm{U}$. - oai:arXiv.org:2512.16772v1 - cs.IT - math-ph - math.DG + Merge on workspaces as Hopf algebra Markov chain + https://arxiv.org/abs/2512.18861 + arXiv:2512.18861v1 Announce Type: new +Abstract: We study the dynamical properties of a Hopf algebra Markov chain with state space the binary rooted forests with labelled leaves. This Markovian dynamical system describes the core computational process of structure formation and transformation in syntax via the Merge operation, according to Chomsky's Minimalism model of generative linguistics. The dynamics decomposes into an ergodic dynamical system with uniform stationary distribution, given by the action of Internal Merge, while the contributions of External Merge and (a minimal form of) Sideward Merge reduce to a simpler Markov chain with state space the set of partitions and with combinatorial weights. The Sideward Merge part of the dynamics prevents convergence to fully formed connected structures (trees), unless the different forms of Merge are weighted by a cost function, as predicted by linguistic theory. Results on the asymptotic behavior of the Perron-Frobenius eigenvalue and eigenvector in this weighted case, obtained in terms of an associated Perron-Frobenius problem in the tropical semiring, show that the usual cost functions (Minimal Search and Resource Restrictions) proposed in the linguistic literature do not suffice to obtain convergence to the tree structures, while an additional optimization property based on the Shannon entropy achieves the expected result for the dynamics. We also comment on the introduction of continuous parameters related to semantic embedding and other computational models, and also on some filtering of the dynamics by coloring rules that model the linguistic filtering by theta roles and phase structure, and on parametric variation and the process of parameter setting in Externalization. + oai:arXiv.org:2512.18861v1 math.DS - math.IT - math.MP - math.SG - Mon, 22 Dec 2025 00:00:00 -0500 - cross - http://creativecommons.org/licenses/by/4.0/ - Pietro G. Fr\'e, Alexander S. Sorin, Mario Trigiante + cs.CL + math.QA + math.RA + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Matilde Marcolli, David Skigin - New Theoretical Insights and Algorithmic Solutions for Reconstructing Score Sequences from Tournament Score Sets - https://arxiv.org/abs/2512.16961 - arXiv:2512.16961v1 Announce Type: cross -Abstract: The score set of a tournament is defined as the set of its distinct out-degrees. In 1978, Reid proposed the conjecture that for any set of nonnegative integers $D$, there exists a tournament $T$ with a degree set $D$. In 1989, Yao presented an arithmetical proof of the conjecture, but a general polynomial-time construction algorithm is not known. This paper proposes a necessary and sufficient condition and a separate necessary condition, based on the existing Landau's theorem for the problem of reconstructing score sequences from score sets of tournament graphs. The necessary condition introduces a structured set that enables the use of group-theoretic techniques, offering not only a framework for solving the reconstruction problem but also a new perspective for approaching similar problems. In particular, the same theoretical approach can be extended to reconstruct valid score sets given constraints on the frequency of distinct scores in tournaments. Based on these conditions, we have developed three algorithms that demonstrate the practical utility of our framework: a polynomial-time algorithm and a scalable algorithm for reconstructing score sequences, and a polynomial-time network-building method that finds all possible score sequences for a given score set. Moreover, the polynomial-time algorithm for reconstructing the score sequence of a tournament for a given score set can be used to verify Reid's conjecture. These algorithms have practical applications in sports analysis, ranking prediction, and machine learning tasks such as learning-to-rank models and data imputation, where the reconstruction of partial rankings or sequences is essential for recommendation systems and anomaly detection. - oai:arXiv.org:2512.16961v1 - cs.DS - math.CO - Mon, 22 Dec 2025 00:00:00 -0500 - cross + An Algebraic-Symmetric Analysis of Counterpoint and Modulation in the Music of Claudio Monteverdi + https://arxiv.org/abs/2512.18862 + arXiv:2512.18862v1 Announce Type: new +Abstract: We address certain structural innovations in the music of Claudio Monteverdi, which defined the pivotal transition from the Renaissance prima pratica to the Baroque seconda pratica. To formalize this analysis, we employ Mazzola's symmetry-based framework for counterpoint and quantum/duality models for tonal modulation. Our findings demonstrate that these mathematical structures provide a rigorous validation of Monteverdi's compositional choices, revealing an underlying logic to harmonic and contrapuntal treatments that were heavily criticized by contemporaries such as Giovanni Artusi. By quantifying concepts like compositional parsimony and modulations, we aim to provide a precise and analytical lens to the interdisciplinary field of mathematical musicology, bridging the gap between historical interpretation and formal theory. + oai:arXiv.org:2512.18862v1 + math.HO + Tue, 23 Dec 2025 00:00:00 -0500 + new http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Bowen Liu + Octavio A. Agust\'in-Aquino, Brandon J. Curiel-L\'opez - Generating temporal networks with the Ascona model - https://arxiv.org/abs/2512.16972 - arXiv:2512.16972v1 Announce Type: cross -Abstract: We introduce a new sampling method for continuous-time temporal networks based on queueing processes. In particular, we focus on a Markovian version of the model where the links between nodes are Poisson distributed in time and have exponential duration. We highlight the stochastic properties of these temporal structures and leverage them to design synthetic temporal networks with a controllable level of smoothness, which follow patterns relevant for the validation and interpretation of methods for community, scale, change-point, and periodicity detection. Additionally, we show that imposing assortativity constraints on the samples leads to a continuous-time generalization of stochastic block models. Finally, we describe how variations of the model can be used to sample alternative types of structure and temporal networks, especially discrete-time ones. - oai:arXiv.org:2512.16972v1 - physics.soc-ph + Diffusion Approximations to Schr\"{o}dinger Bridges on Manifolds + https://arxiv.org/abs/2512.18867 + arXiv:2512.18867v1 Announce Type: new +Abstract: We present a collection of explicit diffusion approximations to small temperature Schr\"{o}dinger bridges on manifolds. Our most precise results are when both marginals are the same and the Schr\"{o}dinger bridge is on a manifold with a reference process given by a reversible diffusion. In the special case that the reference process is the manifold Brownian motion, we use the small time heat kernel asymptotics to show that the gradient of the corresponding Schr\"{o}dinger potential converges in $L^2$, as the temperature vanishes, to a manifold analogue of the score function of the marginal. As an application of the previous result we show that the Euclidean Schr\"{o}dinger bridge, computed for the quadratic cost, between two different marginal distributions can be approximated by a transformation of a two point distribution of a stationary Mirror Langevin diffusion. + oai:arXiv.org:2512.18867v1 math.PR - physics.data-an - Mon, 22 Dec 2025 00:00:00 -0500 - cross - http://creativecommons.org/licenses/by-sa/4.0/ - Samuel Koovely - - - Bundling of bipartite entanglement - https://arxiv.org/abs/2512.16979 - arXiv:2512.16979v1 Announce Type: cross -Abstract: We investigate bipartite entanglement and prove that in constrained energy subspaces, the entanglement spectra of multiple bipartitions are the same across the whole subspace. We show that in quantum many-body systems the bipartite entanglement entropy is affected in such a way that it forms "bundles" under unitary time evolution. Leveraging the structure of the subspace, we present methods to verify whether the entanglement spectrum of two bipartitions is identical throughout the entire subspace. For the subspace defined by the parity embedding, we further provide an algorithm that can determine this in polynomial time. - oai:arXiv.org:2512.16979v1 - quant-ph - math-ph - math.MP - Mon, 22 Dec 2025 00:00:00 -0500 - cross + Tue, 23 Dec 2025 00:00:00 -0500 + new http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Maike Drieb-Schoen, Florian Dreier, Wolfgang Lechner + Garrett Mulcahy, Soumik Pal - Continuum canonical purifications - https://arxiv.org/abs/2512.17014 - arXiv:2512.17014v1 Announce Type: cross -Abstract: We construct and characterize canonical purifications for general algebraic states, extending prior constructions by Woronowicz and by Dutta/Faulkner to general quantum field theories. Given a quantum state on a *-algebra, the canonical purification is a state on a "doubled" algebra that admits an interpretation in terms of CRT reflection. We study the conditions under which these enlarged states are "pure" in the technical sense, compute their modular conjugations, and relate them to GNS and natural-cone purifications in certain settings. In a forthcoming paper with Caminiti and Capeccia, we provide an application of this general theory to the problem of excitability in quantum field theory. - oai:arXiv.org:2512.17014v1 - hep-th - math-ph - math.MP - math.OA - quant-ph - Mon, 22 Dec 2025 00:00:00 -0500 - cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Jonathan Sorce + On the generalized Bass--Quillen conjecture in dimension 2 + https://arxiv.org/abs/2512.18868 + arXiv:2512.18868v1 Announce Type: new +Abstract: Let $A$ be a regular ring of dimension $\le 2$. Let $G$ be a reductive group over $A$ such that its derived group is a split, i.e. a Chevalley--Demazure, semisimple group. We prove that every Zariski-locally trivial principal $G$-bundle over $A[x_1,\ldots,x_n]$ is extended from $A$, for any $n\ge 1$. This result generalizes to split reductive groups the dimension $2$ case of the Bass--Quillen conjecture on finitely generated projective modules, settled in positive by M. P. Murthy. + oai:arXiv.org:2512.18868v1 + math.AG + math.KT + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://creativecommons.org/licenses/by/4.0/ + Anastasia Stavrova - Subsystems (in)dependence in GIE proposals - https://arxiv.org/abs/2512.17024 - arXiv:2512.17024v1 Announce Type: cross -Abstract: Recent proposals suggest that detecting entanglement between two spatially superposed masses would establish the quantum nature of gravity. However, these gravitationally induced entanglement (GIE) experiments rely on assumptions about subsystem independence. We sharpen the theoretical underpinnings of such proposals by examining them through the lens of algebraic quantum field theory (AQFT), distinguishing distinct operational and algebraic notions of independence. We argue that state and measurement independence of subsystems, essential to the experimental logic, is nontrivial in the presence of gauge constraints and gravitational dressing. Using gravitationally dressed fields, we recall that commutation relations between spacelike separated observables are nontrivial, undermining strict Hilbert space factorization. We further explore the implications for entanglement witnesses, investigating the Tsirelson bound when subsystem algebras fail to commute, and showing that the Tsirelson bound persists for a suitably symmetrized CHSH observable even though the operational status of such "joint" observables becomes delicate when commensurability fails. Our analysis highlights how even within linearized covariant quantum gravity, violations of microcausality may affect both the interpretation, modelling, and design of proposed laboratory tests of quantum gravity, despite remaining negligible for current experimental regimes. Although we consider GIE-style protocols as a concrete case study, the subsystem-independence issues we highlight are generic to low-energy (perturbative) quantum gravity. Finally, we derive estimates for dressing-induced microcausality violations, which suggest a complementary avenue to current proposals: in principle, bounding dressing-induced microcausality violations themselves as a probe of the quantum nature of gravity. - oai:arXiv.org:2512.17024v1 - quant-ph - hep-th - math-ph - math.MP - physics.hist-ph - Mon, 22 Dec 2025 00:00:00 -0500 - cross + The Gr\"unbaum--Rigby configuration as a special K\'arteszi configuration + https://arxiv.org/abs/2512.18872 + arXiv:2512.18872v1 Announce Type: new +Abstract: In 1990, Branko Gr\"unbaum and John Rigby presented a 4-configuration, known today as the \emph{Gr\"unbaum--Rigby configuration}; it is denoted by $\mathrm{GR}(21_4)$. Independently and earlier, in 1986, Ferenc K\'arteszi published a paper in which he proved a theorem in real geometry that gives rise to a series of 4-configurations $\mathrm{K}(n;\ell,m)$. In an even earlier paper from 1964, he presented a figure which is essentially the same as that given by Gr\"unbaum and Rigby. In this paper, we explore some properties of the \emph{K\'arteszi configurations} and in particular show that $\mathrm{GR}(21_4)$ is isomorphic to $\mathrm{K}(7;2,3)$. We present a theorem that gives necessary and sufficient conditions on parameters $n,\ell,m$ such that the corresponding configuration $\mathrm{K}(n;\ell,m)$ is realisable as a geometric polycyclic configuration with $n$-fold rotational symmetry and no extra incidences. + oai:arXiv.org:2512.18872v1 + math.CO + math.MG + Tue, 23 Dec 2025 00:00:00 -0500 + new http://creativecommons.org/licenses/by/4.0/ - Nicolas Boulle, Guilherme Franzmann + G\'abor G\'evay, Gy\"orgy Kiss, Toma\v{z} Pisanski - Flux-Preserving Adaptive Finite State Projection for Multiscale Stochastic Reaction Networks - https://arxiv.org/abs/2512.17064 - arXiv:2512.17064v1 Announce Type: cross -Abstract: The Finite State Projection (FSP) method approximates the Chemical Master Equation (CME) by restricting the dynamics to a finite subset of the (typically infinite) state space, enabling direct numerical solution with computable error bounds. Adaptive variants update this subset in time, but multiscale systems with widely separated reaction rates remain challenging, as low-probability bottleneck states can carry essential probability flux and the dynamics alternate between fast transients and slowly evolving stiff regimes. We propose a flux-based adaptive FSP method that uses probability flux to drive both state-space pruning and time-step selection. The pruning rule protects low-probability states with large outgoing flux, preserving connectivity in bottleneck systems, while the time-step rule adapts to the instantaneous total flux to handle rate constants spanning several orders of magnitude. Numerical experiments on stiff, oscillatory, and bottleneck reaction networks show that the method maintains accuracy while using substantially smaller state spaces. - oai:arXiv.org:2512.17064v1 - cs.CE - math.ST - stat.CO - stat.TH - Mon, 22 Dec 2025 00:00:00 -0500 - cross + The Most General Brownian Motion on the Line and on Two Closed Half-Lines + https://arxiv.org/abs/2512.18874 + arXiv:2512.18874v1 Announce Type: new +Abstract: In the 1950s, W. Feller characterized the most general Brownian motion on the closed half-line. He showed that any such process is a mixture of reflected, sticky, and killed Brownian motions. By most general Brownian motion, we mean a strong Markov process whose excursions away from zero coincide with those of standard Brownian motion, and which may be sent to the cemetery state upon hitting zero. In this work, we fully characterize the most general Brownian motion on the whole real line and on the union of two closed half-lines. Our results are twofold. First, we show that the most general Brownian motion on the line is the process known in the literature as the Skew Sticky Brownian Motion Killed at Zero (see Borodin and Salminen's book). Second, we prove that the most general Brownian motion on two closed half-lines is a process, which we call the Skew Sticky Killed at Zero Snapping Out Brownian Motion. This process extends the Snapping Out Brownian Motion introduced by A. Lejay in 2016. + oai:arXiv.org:2512.18874v1 + math.PR + Tue, 23 Dec 2025 00:00:00 -0500 + new http://creativecommons.org/licenses/by/4.0/ - Aditya Dendukuri, Shivkumar Chandrasekaran, Linda Petzold + Dirk Erhard, Tertuliano Franco, Wanessa Muricy - fractional-time deformation of quantum coherence in open systems: a non-markovian framework beyond lindblad dynamics - https://arxiv.org/abs/2512.17144 - arXiv:2512.17144v1 Announce Type: cross -Abstract: In this paper, we propose a fractional time extension of the Quan tum Master Equation. We introduce a Caputo-type fractional derivative in time as an extension of the exponential decay of the Lindblad framework through the incorporation of fractional derivatives into the Lindblad framework. We show that the analytical and numerical results of our analytical and numerical models, demonstrate that fractional dynamics produces long-memory coherence decay naturally and provides an interpretable and flexible model of non-Markovianity. - oai:arXiv.org:2512.17144v1 - quant-ph + Genus~0 Gromov-Witten theory of even dimensional complete intersections of two quadrics: the final step + https://arxiv.org/abs/2512.18875 + arXiv:2512.18875v1 Announce Type: new +Abstract: Even dimensional complete intersections $X$ of two quadrics in projective space are exceptional from the point of view of the Gromov-Witten theory: they are (together with qubic surfaces) the only complete intersections whose Gromov-Witten theory is not invariant under the full orthogonal or symplectic group acting on the primitive cohomology. The genus~0 Gromov-Witten theory of $X$ was studied by Xiaowen Hu. He used geometric arguments and the WDVV equation to compute all genus~0 correlators except one, which cannot be determined by his methods. In this paper we compute the remaining Gromov-Witten invariant of $X$ using Jun Li's degeneration formula. + oai:arXiv.org:2512.18875v1 + math.AG + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://creativecommons.org/licenses/by/4.0/ + Danil Gubarevich + + + Schurian-finiteness of blocks of type B Hecke algebras + https://arxiv.org/abs/2512.18877 + arXiv:2512.18877v1 Announce Type: new +Abstract: Schurian-finiteness, also known as $\tau$-tilting finiteness, is equivalent to the finiteness of various representation theoretic objects such as wide subcategories. The first three authors classified Schurian-finite blocks of type A Hecke algebras in [ALS23]. Here we study the Schurian-finiteness of blocks of type B Hecke algebras, and determine the Schurian-finiteness of all blocks if the Hecke algebra is `non-integral', and for almost all blocks in the integral case. The only remaining cases are a small number of blocks in defect $3$ when $e=3$, and a family of blocks in defects $3$ and $4$ for $e\geqslant4$. The classification is mostly achieved by methods using decomposition numbers, with many degenerate cases requiring direct study using standard methods from the representation theory of quivers. + oai:arXiv.org:2512.18877v1 + math.RT + math.RA + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Susumu Ariki, Sin\'ead Lyle, Liron Speyer, Qi Wang + + + Elementary $\infty$-Toposes from Type Theory + https://arxiv.org/abs/2512.18891 + arXiv:2512.18891v1 Announce Type: new +Abstract: We prove that every categorical model of dependent type theory with dependent sums and products, intensional identity types and univalent universes presents via its $\infty$-localisation an elementary $\infty$-topos, that is, a finitely complete, locally cartesian closed $\infty$-category with enough univalent universal morphisms. We also show that elementary $\infty$-toposes have small subobject classifiers. To achieve this, we extend Joyal's theory of tribes by introducing the notion of a univalent tribe and a univalent fibration in a tribe. + oai:arXiv.org:2512.18891v1 + math.CT + math.AT + math.LO + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Maximilian Petrowitsch + + + Universal categories + https://arxiv.org/abs/2512.18896 + arXiv:2512.18896v1 Announce Type: new +Abstract: The category of models of any theory $T$ in any first-order language $L$ has the surprising property that any small category that is elementarily equivalent with it, already embeds in it. The proof uses an abstract argument via ultrapowers, leaving one wonder which concrete categorical axioms, depending on $T$ and $L$, are responsible for this embedding result. + We also propose a first-order logic for which equivalent categories are always elementarily equivalent. + oai:arXiv.org:2512.18896v1 + math.LO + math.CT + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://creativecommons.org/licenses/by/4.0/ + Hans Schoutens + + + Model-Agnostic Bounds for Augmented Inverse Probability Weighted Estimators' Wald-Confidence Interval Coverage in Randomized Controlled Trials + https://arxiv.org/abs/2512.18898 + arXiv:2512.18898v1 Announce Type: new +Abstract: Nonparametric estimators, such as the augmented inverse probability weighted (AIPW) estimator, have become increasingly popular in causal inference. Numerous nonparametric estimators have been proposed, but they are all asymptotically normal with the same asymptotic variance under similar conditions, leaving little guidance for practitioners to choose an estimator. In this paper, I focus on another important perspective of their asymptotic behaviors beyond asymptotic normality, the convergence of the Wald-confidence interval (CI) coverage to the nominal coverage. Such results have been established for simpler estimators (e.g., the Berry-Esseen Theorem), but are lacking for nonparametric estimators. I consider a simple but practical setting where the AIPW estimator based on a black-box nuisance estimator, with or without cross-fitting, is used to estimate the average treatment effect in randomized controlled trials. I derive non-asymptotic Berry-Esseen-type bounds on the difference between Wald-CI coverage and the nominal coverage. I also analyze the bias of variance estimators, showing that the cross-fit variance estimator might overestimate while the non-cross-fit variance estimator might underestimate, which might explain why cross-fitting has been empirically observed to improve Wald-CI coverage. + oai:arXiv.org:2512.18898v1 + math.ST + stat.ME + stat.TH + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Hongxiang Qiu + + + Weak Galerkin finite element methods for elliptic interface problems on nonconvex polygonal partitions + https://arxiv.org/abs/2512.18905 + arXiv:2512.18905v1 Announce Type: new +Abstract: This paper proposes a weak Galerkin (WG) finite element method for elliptic interface problems defined on nonconvex polygonal partitions. The method features a built-in stabilizer and retains a simple, symmetric, and positive definite formulation. An optimal-order error estimate is rigorously derived in the discrete $H^1$ norm. Furthermore, a series of numerical experiments are provided to verify the theoretical results and to demonstrate the robustness and effectiveness of the proposed WG method for elliptic interface problems. + oai:arXiv.org:2512.18905v1 + math.NA + cs.NA + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://creativecommons.org/publicdomain/zero/1.0/ + Chunmei Wang, Shangyou Zhang + + + The lifespan of strong solutions to the compressible MHD equations with entropy transport in the presence of vacuum + https://arxiv.org/abs/2512.18911 + arXiv:2512.18911v1 Announce Type: new +Abstract: In this paper, we investigate the finite time blow-up of strong solutions to the compressible magnetohydrodynamic (MHD) system (without magnetic diffusion) coupled with entropy transport, and derive an upper bound for the lifespan of such solutions. We first establish the local well-posedness of strong solutions for bounded domains and study the mechanism of finite-time singularity formation in the 2D radially symmetric case and 3D cylindrically symmetric case. We prove that if the initial density vanishes in an interior region containing the origin and the magnetic field is non-trivial within this vacuum region, the strong solution must blow up in finite time. These results generalize and improve the previous results of Huang-Xin-Yan [Math. Ann. 392 (2025) 2365-2394] for the compressible isentropic MHD equations. Significantly, we extend this blow-up result to the free boundary problem. Our analysis of the boundary's expansion allows us to explicitly estimate the maximum lifespan of the solution. + oai:arXiv.org:2512.18911v1 + math.AP + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://creativecommons.org/publicdomain/zero/1.0/ + Yongteng Gu, Xiangdi Huang + + + On the Ban-Linial Conjecture + https://arxiv.org/abs/2512.18913 + arXiv:2512.18913v1 Announce Type: new +Abstract: Let $G$ be a graph and let $\{X_0,X_1\}$ be a partition of $V(G)$. This partition is called external or unfriendly if every $x \in X_i$ has at least as many neighbours in $X_{1-i}$ as in $X_i$. Every maximum edge-cut gives rise to an external partition, so these partitions are always guaranteed to exist. However, it remains a challenge to find such partitions with additional restrictions. Ban and Linial have conjectured that in the case when $G$ is cubic, there always exists an external partition $\{X_0,X_1\}$ for which $-2 \le |X_0| - |X_1| \le 2$. We prove this in two special cases: whenever $G$ can be decomposed into a cycle and a tree, and whenever $G$ has a cubic tree $T$ for which $G - E(T)$ is bipartite. + oai:arXiv.org:2512.18913v1 + math.CO + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://creativecommons.org/licenses/by/4.0/ + Matt DeVos, Kathryn Nurse + + + Singularities of base loci on abelian varieties + https://arxiv.org/abs/2512.18919 + arXiv:2512.18919v1 Announce Type: new +Abstract: We prove that the log canonical threshold of the base ideal of a complete linear system on an abelian variety is $\ge 1$, and equality holds if and only if the base locus has divisorial components. + oai:arXiv.org:2512.18919v1 + math.AG + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Giuseppe Pareschi + + + Nowhere-zero 8-flows in 3-edge-connected signed graphs + https://arxiv.org/abs/2512.18923 + arXiv:2512.18923v1 Announce Type: new +Abstract: In 1983, A. Bouchet extended W.T. Tutte's notion of nowhere-zero flows to signed graphs, and conjectured that every flow-admissible signed graph has a nowhere-zero 6-flow. In this paper we prove that every flow-admissible signed graph that is 3-edge-connected has a nowhere-zero 8-flow. This is a continuation of a previous paper where we proved the same conclusion under stronger assumptions. + oai:arXiv.org:2512.18923v1 + math.CO + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://creativecommons.org/licenses/by/4.0/ + Matt DeVos, Kathryn Nurse, Robert \v{S}\'amal + + + Stochastic quantization of the weighted exponential QFT + https://arxiv.org/abs/2512.18927 + arXiv:2512.18927v1 Announce Type: new +Abstract: We consider the stochastic quantization equation associated with the weighted exponential quantum field model (or the H{\o}egh-Krohn model) on the two dimensional torus. Unlike in the case of the usual (unweighted) exponential model, the drift term of the stochastic quantization equation can be both positive and negative, and that makes the equation more difficult to treat. We prove the unique existence of the time-global solution under a certain initial condition by a pathwise PDE argument in the so-called $L^2$-regime. We also see that this solution is properly associated with a Dirichlet form canonically constructed from the weighted exponential quantum field measure. + oai:arXiv.org:2512.18927v1 + math.PR + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Seiichiro Kusuoka, Hirotatsu Nagoji + + + Fake Mobius-type Functions on the Unit Circle + https://arxiv.org/abs/2512.18936 + arXiv:2512.18936v1 Announce Type: new +Abstract: We study \emph{fake Mobius-type functions on the unit circle}: multiplicative functions $\mathfrak f$ determined by prime-power data $\mathfrak f(p^k)=\varepsilon_k\in\mathbb S^1\cup\{0\}$, independent of $p$. Their Dirichlet series admits the Euler product \[ F_{\mathfrak f}(s)=\sum_{n\ge1}\frac{\mathfrak f(n)}{n^s} =\prod_p g(p^{-s}),\qquad g(u)=\sum_{k\ge0}\varepsilon_k u^k, \] and a canonical zeta-factorization \[ F_{\mathfrak f}(s)=\zeta(s)^{\,z}\,\zeta(2s)^{\,w}\,G_{\mathfrak f}(s), \qquad z=\varepsilon_1,\ \ w=\varepsilon_2-\frac{\varepsilon_1(\varepsilon_1+1)}{2}, \] where $G_{\mathfrak f}(s)$ is a holomorphic Euler product on $\Re s>1/3$. + Assuming the Riemann hypothesis and simplicity of zeros, we derive a general explicit and asymptotic formula for the summatory functions $A_{\mathfrak f}^{\exp}(x)$ of the form \[ A_{\mathfrak f}^{\exp}(x) -\Delta_1(x;z,w) = \Delta_{1/2}(x;z,w)\;+\;\sum_{\rho}\Delta_\rho(x;z,w,\mathfrak f)\;+\;\mathcal E(x), \] where $\Delta_1(x;z,w)$ is the main term coming from the $s=1$ singularity of $\zeta(s)^z$, $\Delta_{1/2}(x;z,w)$ is the secondary term coming from the $s=1/2$ singularity of $\zeta(2s)^w$, the terms $\Delta_\rho(x;z,w,\mathfrak f)$ arise from the singularities of $\zeta(s)^z$ at nontrivial zeros $\rho$ of $\zeta(s)$, and $\mathcal E(x)$ is an error term. Furthermore, we introduce a notion of \emph{bias} at the natural scale $x^{1/2}(\Log x)^{w-1}$ and obtain an explicit criteria distinguishing persistent, apparent, and unbiased behavior in this regime. + The present paper treats the major parameter region \[ -1\le\Re(z)\le1,\qquad -2\le\Re(w)<1, \] and we choose to defer the complementary ranges of $(z,w)$ to a future paper as a different analytic approach is required there. + oai:arXiv.org:2512.18936v1 + math.NT + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Ali Saraeb + + + The critical percolation window in growing random graphs + https://arxiv.org/abs/2512.18937 + arXiv:2512.18937v1 Announce Type: new +Abstract: We describe the critical window for percolation in the universality class of sparse growing random graphs. In our models, vertices arrive sequentially and connect independently to each earlier vertex $v$ with probability proportional to a nonpositive power of the arrival time of $v$, continuing until the graph has $n$ vertices. This class includes uniformly grown random graphs and inhomogeneous random graphs of preferential-attachment type. Whenever the critical percolation threshold is positive, we show that the critical window has width of order $(\log n)^{-2}$ and a secondary phase transition at its finite upper boundary. Inside this window the largest component has size of order $\sqrt{n}/\log n$, and the susceptibility remains finite and independent of the position in the window. The proofs couple component explorations to branching random walks killed outside an interval of length $\log n$, allowing sharp control of the barely subcritical and critical regimes. + oai:arXiv.org:2512.18937v1 + math.PR + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://creativecommons.org/licenses/by/4.0/ + Joost Jorritsma, Pascal Maillard, Peter M\"orters + + + An asymptotically compatible unfitted finite element methods for nonlocal elliptic Interfaces: local limits and sharp error estimates + https://arxiv.org/abs/2512.18939 + arXiv:2512.18939v1 Announce Type: new +Abstract: This paper presents the development and analysis of an asymptotically compatible (AC) unfitted finite element method for one-dimensional nonlocal elliptic interface problems. The proposed method achieves optimal error estimates through three principal contributions: (i) an extended maximum principle, coupled with an asymptotic consistency analysis of the flux operator, which establishes second-order convergence of nonlocal solutions to their local counterparts in the maximum norm; (ii) a Nitsche-type formulation that directly incorporates nonlocal jump conditions into the weak form, enabling high accuracy without body-fitted meshes; and (iii) a rigorous proof of optimal convergence rates in both the energy and L2 norms via the nonlocal maximum principle, flux consistency, and a newly derived nonlocal Poincare inequality. Numerical experiments confirm the theoretical findings and demonstrate the robustness and efficiency of the proposed approach, thereby providing a foundation for extensions to higher dimensions. + oai:arXiv.org:2512.18939v1 + math.NA + cs.NA + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Haixia Dong, Ziqing Xie, Jiwei Zhang + + + Finitely presented simple groups with no piecewise projective actions + https://arxiv.org/abs/2512.18943 + arXiv:2512.18943v1 Announce Type: new +Abstract: We construct an explicit infinite family of pairwise non-isomorphic infinite simple groups of type $\mathrm{F}_\infty$ (in particular, they are finitely presented) that act faithfully on the circle by orientation-preserving homeomorphisms, but that admit no non-trivial piecewise affine nor piecewise projective actions on the projective line. Our examples are certain forest-skein groups which, informally, are a mixture of Richard Thompson's groups with Vaughan Jones' planar algebras. + oai:arXiv.org:2512.18943v1 + math.GR + math.DS + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Arnaud Brothier, Ryan Seelig + + + A solution to the problem of the existence of a dense plastic subset $X\subseteq\mathbb{R}$ of cardinality $|X|<\frak{c}$ + https://arxiv.org/abs/2512.18948 + arXiv:2512.18948v1 Announce Type: new +Abstract: Under the assumption $\frak{c}\geqslant \omega_2$, we give an example of a dense plastic subset $X\subseteq\mathbb{R}$ of cardinality $|X|<\frak{c}$. This answers Problem 1 of arXiv:2510.10537. + oai:arXiv.org:2512.18948v1 + math.GN + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://creativecommons.org/licenses/by/4.0/ + Wojciech Bielas + + + A TraceFEM $C^0$ Interior Penalty Method for the Surface Biharmonic Equation + https://arxiv.org/abs/2512.18949 + arXiv:2512.18949v1 Announce Type: new +Abstract: We construct and analyze a TraceFEM discretization for the surface biharmonic problem. The method utilizes standard quadratic Lagrange finite element spaces defined on a three-dimensional background mesh and a symmetric $C^0$ interior penalty formulation posed on a second-order polyhedral approximation of the surface. Stability is achieved through a combination of surface edge penalties and bulk-facet penalization of gradient and Hessian jumps. We prove optimal first-order convergence in a discrete $H^2$ norm and quadratic convergence in the $L^2$ norm. + oai:arXiv.org:2512.18949v1 + math.NA + cs.NA + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://creativecommons.org/licenses/by/4.0/ + Michael Neilan, Hongzhi Wan + + + A Spectral Low-Mode Reduced Method for Elliptic Problems + https://arxiv.org/abs/2512.18955 + arXiv:2512.18955v1 Announce Type: new +Abstract: We develop a spectral low-mode reduced solver for second-order elliptic boundary value problems with spatially varying diffusion coefficients. The approach projects standard finite difference or finite element discretization onto a global coarse space spanned by the lowest Dirichlet Laplacian eigenmodes, yielding an analytic reduced model that requires no training data and preserves coefficient heterogeneity through an exact Galerkin projection. The reduced solution is energy-optimal in the selected subspace and, for $H^2$-regular solutions, the truncation error associated with discarded modes satisfies a $\sqrt{\log M}/M$ decay in the $H_0^1$ norm. For uniformly stable reduced bases, the projected operator is well conditioned with respect to mesh refinement, and numerical experiments corroborate the predicted accuracy and demonstrate meaningful speedups over sparse direct solvers, with favorable performance relative to multigrid and deflation-based Krylov methods for heterogeneous coefficients in the tested setups. + oai:arXiv.org:2512.18955v1 + math.NA + cs.NA + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Prosper Torsu + + + On $\ell_1$ embeddings of finite metric spaces, and sphere-of-influence graphs + https://arxiv.org/abs/2512.18975 + arXiv:2512.18975v1 Announce Type: new +Abstract: We introduce the {\em pair-cut cone $PCUT_n$} of metrics on sets with $n\ge 3$ elements, that correspond to linear combinations with non-negative coefficients of the cut-metrics resulting from cuts that are pairs. Given a metric, we fully characterize membership in the pair-cut cone in terms of quantities computed from the metric directly. We also prove a new result by which a metric $d$ that satisfies a system of inequalities, lies in the (full) cut cone of metrics, making it $\ell_1$-embeddable into Euclidean space. + We give applications of our results to the $\ell_1$-embeddability of simple graphs into Euclidean space as {\em sphere-of-influence graphs}. We exhibit an example of a simple graph that admits no such $\ell_1$-metric in the pair-cut cone. + oai:arXiv.org:2512.18975v1 + math.MG + math.CO + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://creativecommons.org/licenses/by/4.0/ + Stanislav Jabuka, Ehsan Mirbagheri + + + Global strong solutions and asymptotic behavior for arbitrarily large initial data of the 2D compressible Navier-Stokes equations with transport entropy + https://arxiv.org/abs/2512.18976 + arXiv:2512.18976v1 Announce Type: new +Abstract: In 1995, Kazhikhov and Vaigant introduced a particular class of isentropic compressible Navier-Stokes equations with variable viscosity coefficients and, for the first time, established the existence of global smooth solutions for arbitrarily large initial data in bounded two-dimensional domains. This result was subsequently extended and refined to accommodate more general constraints on the viscosity coefficients. However, because the proofs in this line of work [17,14,15,8] rely heavily on the structure of the isentropic equations, they could not be generalized to the broader setting of multidimensional compressible heat-conductive Navier-Stokes-Fourier systems. In this paper, we consider a special class of non-isentropic compressible fluids governed by the two-dimensional compressible Navier-Stokes equations with variable entropy. In this system, the pressure depends nonlinearly on both density and entropy, and the entropy evolves solely through a transport equation-a feature that distinguishes it from the standard Navier-Stokes-Fourier model. We establish, for the first time, the global existence of strong solutions for arbitrarily large initial data on both two-dimensional periodic domains and bounded domains endowed with Navier-slip boundary conditions. For the bounded-domain case, a key step in our analysis is the derivation of new commutator estimates compatible with the slip condition. Our results hold even when the initial density may contain vacuum and require no smallness assumption on the initial data, provided the shear viscosity is constant and the bulk viscosity follows a power-law form $\lambda(\rho)=\rho^\beta$ with $\beta > 4/3$. Moreover, we demonstrate that the density remains uniformly bounded for all time. Consequently, the solution converges to an equilibrium state as time tends to infinity. + oai:arXiv.org:2512.18976v1 + math.AP + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://creativecommons.org/publicdomain/zero/1.0/ + Jie Fan, Xiangdi Huang + + + Bi-Level Optimal Control Framework For Missed-Thrust-Design With First-Order Bounds On Maximum Missed-Thrust-Duration + https://arxiv.org/abs/2512.18984 + arXiv:2512.18984v1 Announce Type: new +Abstract: In this paper, we present a bi-level optimal control framework for designing low-thrust spacecraft trajectories with robustness against missed-thrust-events. The upper-level (UL) problem generates a nominal trajectory assuming full control authority, while each lower-level (LL) problem computes the optimal recovery maneuver following a missed-thrust-event along the nominal solution. Under suitable regularity conditions ensuring uniqueness and smoothness of the LL response, the hierarchy admits a single-level reformulation by embedding the LL first-order optimality conditions within the UL constraints. We further establish a robustness certificate, which provides an upper bound on the maximum admissible missed-thrust-duration for which the structural assumptions remain valid for the LL problem. The bound depends explicitly on precomputable dynamical quantities along the nominal solution, enabling rapid evaluation over large ensembles without iterative solves. Numerical experiments show that while the certificate identifies when modeling assumptions are valid, it does not fully characterize recoverability after missed-thrust-events. A finite-horizon controllability-energy analysis is therefore used to interpret recovery beyond the theoretical bounds. Collectively, these results provide a deterministic, certifiable approach for incorporating robustness directly into trajectory design, replacing post-hoc margin allocation techniques with formal guarantees. + oai:arXiv.org:2512.18984v1 + math.OC + math.DS + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://creativecommons.org/licenses/by/4.0/ + Amlan Sinha, Ryne Beeson + + + On-Orbit Servicing-Integrated Maintenance Strategy for Satellite Constellation + https://arxiv.org/abs/2512.18985 + arXiv:2512.18985v1 Announce Type: new +Abstract: This paper proposes a maintenance strategy for a satellite constellation that utilizes on-orbit servicing (OOS). Under this strategy, the constellation operator addresses satellite failures in two ways: by deploying new satellites and by recovering failed satellites through OOS. We develop an inventory management model with a parametric replenishment policy for the maintenance process, which can evaluate the performance of the satellite constellation system. Based on this model, we formulate two single-objective optimization problems representing the decision-making contexts of two main stakeholders -- the constellation operator and the OOS provider -- and a bi-objective optimization problem that can reflect the tension between the two. A case study of the OOS-supported maintenance for a real-world-scale constellation provides valuable insights that help explain the behaviors of the stakeholders. + oai:arXiv.org:2512.18985v1 + math.OC + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://creativecommons.org/licenses/by-nc-nd/4.0/ + Jaewoo Kim, Taehyun Sung, Woonam Hwang, Jaemyung Ahn + + + Hybrid Stochastic Functional Differential Equations with Infinite Delay: Approximations and Numerics + https://arxiv.org/abs/2512.18990 + arXiv:2512.18990v1 Announce Type: new +Abstract: This paper is to investigate if the solution of a hybrid stochastic functional differential equation (SFDE) with infinite delay can be approximated by the solution of the corresponding hybrid SFDE with finite delay. A positive result is established for a large class of highly nonlinear hybrid SFDEs with infinite delay. Our new theory makes it possible to numerically approximate the solution of the hybrid SFDE with infinite delay, via the numerical solution of the corresponding hybrid SFDE with finite delay. + oai:arXiv.org:2512.18990v1 + math.PR + cs.NA + math.NA + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Guozhen Li, Xiaoyue Li, Xuerong Mao, Guoting Song + + + Global well-posedness for the generalized intermediate NLS with a nonvanishing condition at infinity + https://arxiv.org/abs/2512.18998 + arXiv:2512.18998v1 Announce Type: new +Abstract: The Intermediate Nonlinear Schr\"odinger equation models quasi-harmonic internal waves in two-fluid layer system, and admits dark solitons, that is, solutions with nonvanishing boundary conditions at spatial infinity. These solutions fall outside existing well-posedness theories. We establish local and global well-posedness in a Zhidkov-type space naturally suited to such non-trivial boundary conditions, and extend these results to a generalized defocusing equation. This appears to be the first well-posedness result for the equation in a functional setting adapted to its dark soliton structure. + oai:arXiv.org:2512.18998v1 + math.AP + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://creativecommons.org/licenses/by/4.0/ + Takafumi Akahori, Rana Badreddine, Slim Ibrahim, Nobu Kishimoto + + + A Quantitative Entropy Power Inequality for Dependent Random Vectors + https://arxiv.org/abs/2512.19002 + arXiv:2512.19002v1 Announce Type: new +Abstract: The entropy power inequality for independent random vectors is a foundational result of information theory, with deep connections to probability and geometric functional analysis. Several extensions of the entropy power inequality have been developed for settings with dependence, including by Takano, Johnson, and Rioul. We extend these works by developing a quantitative version of the entropy power inequality for dependent random vectors. A notable consequence is that an entropy power inequality stated using conditional entropies holds for random vectors whose joint density is log-supermodular. + oai:arXiv.org:2512.19002v1 + cs.IT + math.IT + math.PR + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Mokshay Madiman, James Melbourne, Cyril Roberto + + + The stability of log-supermodularity under convolution + https://arxiv.org/abs/2512.19003 + arXiv:2512.19003v1 Announce Type: new +Abstract: We study the behavior of log-supermodular functions under convolution. In particular we show that log-concave product densities preserve log-supermodularity, confirming in the special case of the standard Gaussian density, a conjecture of Zartash and Robeva. Additionally, this stability gives a ``conditional'' entropy power inequality for log-supermodular random variables. We also compare the Ahlswede-Daykin four function theorem and a recent four function version of the Prekopa-Leindler inequality due to Cordero-Erausquin and Maurey and giving transport proofs for the two theorems. In the Prekopa-Leindler case, the proof gives a generalization that seems to be new, which interpolates the classical three and the recent four function versions. + oai:arXiv.org:2512.19003v1 + math.PR + cs.IT + math.FA + math.IT + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Mokshay Madiman, James Melbourne, Cyril Roberto + + + On finding formal power-logarithmic expansions of solutions to $q$-difference equations + https://arxiv.org/abs/2512.19006 + arXiv:2512.19006v1 Announce Type: new +Abstract: An algebraic $q$-difference equation is considered. A sufficient condition for the existence of a formal power-logarithmic expansion of a solution to such an equation in the neighborhood of zero is proposed. An example of applying this sufficient condition for constructing a formal expansion of a solution to a certain $q$-difference analogue of the fifth Painlev\'{e} equation for specific values of the equation parameters is given; two different values of the number $q$ are considered, leading to qualitatively different formal asymptotic expansions of the solutions of the fifth Painlev\'{e} equation. + oai:arXiv.org:2512.19006v1 + math.CA + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Nikita Gaianov, Anastasia Parusnikova + + + Relative Bruhat decomposition of wonderful compactification + https://arxiv.org/abs/2512.19008 + arXiv:2512.19008v1 Announce Type: new +Abstract: In the seminal paper of Borel and Tits about reductive groups, they show some fundamental results about Bruhat cells with respect to a minimal parabolic subgroup, e.g., relative Bruhat decomposition and its geometrization, relative Bruhat order and the relation of Zariski closure and topological closure. In this paper, we show analogous results for Bruhat cells of wonderful group compactification in the sense of De Concini and Procesi. Our results can be viewed as the version at infinity of those of Borel and Tits. Our main focus is general base field. When the base field is algebraically closed, most of our results are proved by Brion and Springer. + oai:arXiv.org:2512.19008v1 + math.AG + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Fei Chen, Shang Li + + + Randomized time stepping of nonlinearly parametrized solutions of evolution problems + https://arxiv.org/abs/2512.19009 + arXiv:2512.19009v1 Announce Type: new +Abstract: The Dirac-Frenkel variational principle is a widely used building block for using nonlinear parametrizations in the context of model reduction and numerically solving partial differential equations; however, it typically leads to time-dependent least-squares problems that are poorly conditioned. This work introduces a randomized time stepping scheme that solves at each time step a low-dimensional, random projection of the parameter vector via sketching. The sketching has a regularization effect that leads to better conditioned least-squares problems and at the same time reduces the number of unknowns that need to be solved for at each time step. Numerical experiments with benchmark examples demonstrate that randomized time stepping via sketching achieves competitive accuracy and outperforms standard regularization in terms of runtime efficiency. + oai:arXiv.org:2512.19009v1 + math.NA + cs.NA + math.DS + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Yijun Dong, Paul Schwerdtner, Benjamin Peherstorfer + + + On the basin of attraction for the free boundary free elastic flow + https://arxiv.org/abs/2512.19015 + arXiv:2512.19015v1 Announce Type: new +Abstract: The free boundary free elastic flow is the steepest descent gradient flow for the elastic energy of curves meeting parallel lines perpendicularly. In this article we prove that the straight line has, measured in Euler's scale-invariant bending energy, a basin of attraction at least to the level $1.9615\, \pi$. We show that our method of proof cannot be pushed to the previously conjectured level $2\pi$, and in addition present numerical evidence that this conjecture may in fact be false. + oai:arXiv.org:2512.19015v1 + math.AP + math.DG + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Klaus Deckelnick, Hans-Christoph Grunau, Robert N\"urnberg, Glen Wheeler, Valentina-Mira Wheeler + + + Operator Tail Densities of Multivariate Copulas + https://arxiv.org/abs/2512.19023 + arXiv:2512.19023v1 Announce Type: new +Abstract: Operator regular variation of a multivariate distribution can be decomposed into the operator tail dependence of the underlying copula and the regular variation of the univariate marginals. In this paper, we introduce operator tail densities for copulas and show that an operator-regularly-varying density can be characterized through the operator tail density of its copula together with the marginal regular variation. As an example, we demonstrate that although a Liouville copula is not available in closed form, it nevertheless admits an explicit operator tail-dependence function. + oai:arXiv.org:2512.19023v1 + math.ST + math.PR + stat.TH + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Haijun Li + + + Brion atoms for classical types + https://arxiv.org/abs/2512.19034 + arXiv:2512.19034v1 Announce Type: new +Abstract: Let $G$ be a classical group defined over the complex numbers with a Borel subgroup $B$. Choose a holomorphic involution of $G$ and let $K$ be its set of fixed points. The group $K$ acts on the flag variety $G/B$ with finitely many orbits and Brion has derived a general formula for the cohomology classes of the corresponding orbit closures as linear combinations of Schubert classes. This article provide a uniform description of the sets of Weyl group elements (which we refer to as Brion atoms) indexing the terms in this formula. This builds on prior work addressing types A, B, and C. The main novelty of our results is a thorough treatment of type D. As one application, we introduce a notion of involution Schubert polynomials for all classical types and present several conjectures related to these objects. + oai:arXiv.org:2512.19034v1 + math.RT + math.CO + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Eric Marberg + + + Locally interval graphs are circular-arc graphs + https://arxiv.org/abs/2512.19040 + arXiv:2512.19040v1 Announce Type: new +Abstract: Circular-arc graphs are graphs that can be represented as intersection graphs of subpaths of a cycle. Interval graphs are graphs that can be represented as intersection graphs of subpaths of a path. Since cycles are locally paths, every circular-arc graph is locally interval. In this paper, we prove that the converse holds as well: every locally interval graph is a circular-arc graph. This result and its proofs are connected to a recent broader study of structural local-global theory and build on previous work on locally chordal graphs. + oai:arXiv.org:2512.19040v1 + math.CO + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Tara Abrishami, Sandra Albrechtsen, Nathan Bowler, Paul Knappe, Jana Katharina Nickel + + + The global structure of locally chordal graphs + https://arxiv.org/abs/2512.19044 + arXiv:2512.19044v1 Announce Type: new +Abstract: A graph is locally chordal if each of its small-radius balls is chordal. In an earlier work [AKK25], the authors and Kobler proved that locally chordal graphs can be characterized by having chordal local covers, by forbidding short cycles and wheels as induced subgraphs, and by the property that each of their minimal local separators is a clique. In this paper, we address the global structure of locally chordal graphs. The global structure of chordal graphs is given by the following characterizations: a graph is chordal if and only if it is the intersection graph of subtrees of a tree, if and only if it admits a tree-decomposition into cliques. We prove a local analog of this characterization, which essentially says that a graph is locally chordal if and only if it is the intersection graph of special subtrees of a high-girth graph, if and only if it admits a special graph-decomposition over a high-girth graph into cliques. We also prove that these global representations of locally chordal graphs can be efficiently computed. + This paper has two major contributions. The first is to exhibit for locally chordal graphs an ideal "local to global" analysis: given a graph class defined by restricted local structure, we fully describe the global structure of graphs in the class. The second is to develop the theory of graph-decompositions. Much of the work in this paper is devoted to properties of graph-decompositions that represent the global structure of graphs. This theory will be useful to find global decompositions for graph classes beyond locally chordal graphs. + oai:arXiv.org:2512.19044v1 + math.CO + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Tara Abrishami, Paul Knappe + + + Classical double Grothendieck transitions + https://arxiv.org/abs/2512.19045 + arXiv:2512.19045v1 Announce Type: new +Abstract: Kirillov and Naruse have constructed double Grothendieck polynomials to represent the equivariant K-theory classes of Schubert varieties in the complete flag manifolds of types B, C, and D. We derive a recursive formula for these polynomials, extending certain K-theoretic transition equations known in type A to all classical types. As an application, we obtain an identity that expands the K-Stanley symmetric functions in types B, C, and D into positive linear combinations of K-theoretic Schur P- and Q-functions. We also resolve several positivity conjectures related to the skew generalizations of the latter functions. + oai:arXiv.org:2512.19045v1 + math.RT + math.CO + math.KT + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Eric Marberg + + + The cyclicity of period annulus of cubic isochronous Hamiltonian systems + https://arxiv.org/abs/2512.19046 + arXiv:2512.19046v1 Announce Type: new +Abstract: Cima, Ma\~{n}osas and Villadelprat (J. Differ. Equations, 157, 373--413, 1999) proved that a cubic Hamiltonian system possesses an isochronous center at the origin if and only if its Hamiltonian function can be expressed as \begin{eqnarray*}H_1(x,y)=k_1^2x^2+(k_2y+k_3x+k_4x^2)^2, \end{eqnarray*} where $k_1,k_2,k_3,k_4\in\mathbb{R}$, $k_1k_2\neq0$. This paper is devoted to investigating the weak Hilbert's 16th problem for the dynamical system associated with the above Hamiltonian function. We show that the maximum number of limit cycles is $n-1$. Furthermore, this number is reached. That is, we solve the weak Hilbert's 16th problem restricted to cubic Hamiltonian systems with an isochronous center at the origin. + oai:arXiv.org:2512.19046v1 + math.DS + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://creativecommons.org/licenses/by/4.0/ + Jihua Yang + + + On the curvature operator in dimensions $4n$ + https://arxiv.org/abs/2512.19050 + arXiv:2512.19050v1 Announce Type: new +Abstract: We study oriented Riemannian $4n$-manifolds whose Thorpe $2n^{\text{th}}$ curvature operator $\hat{R}_{2n}\colon\Lambda^{2n} \longrightarrow \Lambda^{2n}$, or its Weyl analogue $\hat{W}_{2n}$, commutes with the Hodge star. For pure curvature operators this commuting condition becomes a finite system of hafnian identities in the eigenvalues of the curvature operator, which we analyze in two subclasses, including the locally conformally flat case. We further observe that $*\hat{W}_{2n} = \hat{W}_{2n}*$ is a new conformal invariant in dimensions $4n$, providing higher-dimensional analogues of self-duality. Finally, we give sufficient conditions ensuring nonnegativity of the Euler characteristic and relate these conditions to normal forms. + oai:arXiv.org:2512.19050v1 + math.DG + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://creativecommons.org/licenses/by/4.0/ + Amir Babak Aazami + + + Sharp Decoupling Inequalities for the Variances and Second Moments of Sums of Dependent Random Variables + https://arxiv.org/abs/2512.19063 + arXiv:2512.19063v1 Announce Type: new +Abstract: Both complete decoupling and tangent decoupling are classical tools aiming to compare two random processes where one has a weaker dependence structure. We give a new proof for the complete decoupling inequality, which provides a lower bound for the sum of dependent square-integrable nonnegative random variables $\sum\limits^n_{i=1} d_i$ \[ \frac{1}{2} \mathbb E \left( \sum\limits^n_{i=1} z_i \right)^2 \leq \mathbb E \left( \sum\limits^n_{i=1} d_i \right)^2, \] where $z_i \stackrel{\mathcal{L}}{=} d_i$ for all $i\leq n$ and $z_i$'s are mutually independent. We will then provide the following sharp tangent decoupling inequalities \[\mathbb Var \left( \sum\limits^n_{i=1} d_i\right) \leq 2 \mathbb Var \left( \sum\limits^n_{i=1} e_i\right),\] and \[\mathbb E \left( \sum\limits^n_{i=1} d_i\right)^2 \leq 2 \mathbb E \left( \sum\limits^n_{i=1} e_i\right)^2 - \left[ \mathbb E \left( \sum\limits^n_{i=1} e_i\right) \right]^2,\] where $\{e_i\}$ is the decoupled sequences of $\{d_i\}$ and $d_i$'s are not forced to be nonnegative. Applications to construct Chebyshev-type inequality and Paley-Zygmund-type inequality, and to bound the second moments of randomly stopped sums will be provided. + oai:arXiv.org:2512.19063v1 + math.PR + math.ST + stat.TH + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://creativecommons.org/licenses/by/4.0/ + Victor H. de la Pena, Heyuan Yao, Demissie Alemayehu + + + On Cost-Aware Sequential Hypothesis Testing with Random Costs and Action Cancellation + https://arxiv.org/abs/2512.19067 + arXiv:2512.19067v1 Announce Type: new +Abstract: We study a variant of cost-aware sequential hypothesis testing in which a single active Decision Maker (DM) selects actions with positive, random costs to identify the true hypothesis under an average error constraint, while minimizing the expected total cost. The DM may abort an in-progress action, yielding no sample, by truncating its realized cost at a smaller, tunable deterministic limit, which we term a per-action deadline. We analyze how this cancellation option can be exploited under two cost-revelation models: ex-post, where the cost is revealed only after the sample is obtained, and ex-ante, where the cost accrues before sample acquisition. + In the ex-post model, per-action deadlines do not affect the expected total cost, and the cost-error tradeoffs coincide with the baseline obtained by replacing deterministic costs with cost means. In the ex-ante model, we show how per-action deadlines inflate the expected number of times actions are applied, and that the resulting expected total cost can be reduced to the constant-cost setting by introducing an effective per-action cost. We characterize when deadlines are beneficial and study several families in detail. + oai:arXiv.org:2512.19067v1 + cs.IT + cs.LG + math.IT + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + George Vershinin, Asaf Cohen, Omer Gurewitz + + + Cyclotomic points on varieties and all rational $a^3b$-monotiles + https://arxiv.org/abs/2512.19071 + arXiv:2512.19071v1 Announce Type: new +Abstract: By computing all cyclotomic points on some algebraic varieties, we get an independent and efficient way to find all rational $a^3b$-monotiles for the sphere, thereby completing the classification of edge-to-edge monohedral quadrilateral tilings. Both of the previous classifications \cite{lw2} and \cite{cl} depended on many old works of different authors while quite a few typos and gaps were found. + oai:arXiv.org:2512.19071v1 + math.CO + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://creativecommons.org/licenses/by-nc-sa/4.0/ + Jinjin Liang, Yixi Liao, Erxiao Wang + + + On Factoring and Power Divisor Problems via Rank-3 Lattices and the Second Vector + https://arxiv.org/abs/2512.19076 + arXiv:2512.19076v1 Announce Type: new +Abstract: We propose a deterministic algorithm based on Coppersmith's method that employs a rank-3 lattice to address factoring-related problems. An interesting aspect of our approach is that we utilize the second vector in the LLL-reduced basis to avoid trivial collisions in the Baby-step Giant-step method, rather than the shortest vector as is commonly used in the literature. Our results are as follows: + 1. Compared to the result by Harvey and Hittmeir (Math. Comp. 91 (2022), 1367 - 1379), who achieved a complexity of O( N^(1/5) log^(16/5) N / (log log N)^(3/5)) for factoring a semiprime N = pq, we demonstrate that in the balanced p and q case, the complexity can be improved to O( N^(1/5) log^(13/5) N / (log log N)^(3/5) ). + 2. For factoring sums and differences of powers, that is, numbers of the form N = a^n plus or minus b^n, we improve Hittmeir's result (Math. Comp. 86 (2017), 2947 - 2954) from O( N^(1/4) log^(3/2) N ) to O( N^(1/5) log^(13/5) N ). + 3. For the problem of finding r-power divisors, that is, finding all integers p such that p^r divides N, Harvey and Hittmeir (Proceedings of ANTS XV, Research in Number Theory 8 (2022), no. 4, Paper No. 94) recently directly applied Coppersmith's method and achieved a complexity of O( N^(1/(4r)) log^(10+epsilon) N / r^3 ). By using faster LLL-type algorithms and sieving on small primes, we improve their result to O( N^(1/(4r)) log^(7+3 epsilon) N / ((log log N minus log(4r)) r^(2+epsilon)) ). The worst-case running time for their algorithm occurs when N = p^r q with q on the order of N^(1/2). By focusing on this case and employing our rank-3 lattice approach, we achieve a complexity of O( r^(1/4) N^(1/(4r)) log^(5/2) N ). + In conclusion, we offer a new perspective on these problems, which we hope will provide further insights. + oai:arXiv.org:2512.19076v1 + math.NT + cs.DS + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://creativecommons.org/licenses/by/4.0/ + Yiming Gao, Yansong Feng, Honggang Hu, Yanbin Pan + + + Upper-semicontinuity of uniform attractors for the non-autonomous viscoelastic Kirchhoff plate equation with memory + https://arxiv.org/abs/2512.19079 + arXiv:2512.19079v1 Announce Type: new +Abstract: This paper delves into the long-time dynamics of a non-autonomous viscoelastic Kirchhoff plate equation with memory effects, described by + $$ + u_{t t}-\Delta u_{t t}+a_\epsilon(t) u_t+\alpha \Delta^2 u-\int_0^{\infty} \mu(s) \Delta^2 u(t-s) \mathrm{d} s-\Delta u_t+f(u)=g(x,t), + $$ + in bounded domain $\Omega \subset \mathbb{R}^N$ with smooth boundary and nonlinear terms. Initially, the global existence of a weak solution that induces a continuous process is established. Subsequently, the existence of a uniform attractor is demonstrated in both subcritical and critical growth scenarios, utilizing operator techniques and an innovative analytical approach. Finally, the upper semicontinuity of the family of uniform attractors as the pert parameterurbation $\epsilon \to 0^+$ is proven through delicate energy estimates and a contradiction argument. Our results not only extend classical attractor theory to more general non-autonomous viscoelastic systems but also resolve open questions regarding the limiting behavior of attractors in the presence of both memory and critical nonlinearity. + oai:arXiv.org:2512.19079v1 + math.AP + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://creativecommons.org/licenses/by/4.0/ + Yuming Qin, Hongli Wang + + + Chromatic numbers for contact graphs of congruent cuboids + https://arxiv.org/abs/2512.19080 + arXiv:2512.19080v1 Announce Type: new +Abstract: We initiate the study of chromatic numbers for contact graphs of configurations of integer-sized cuboids in three dimensions, all of which are mutually congruent. Disallowing rotations, we show a global upper bound of 8 for the chromatic numbers, which implies that there is a global upper bound of 48 when the cuboids may be rotated freely. Specializing further to cuboids that are required to have a side length of one we obtain more precise upper bounds. + Such upper bounds are compared to examples of configurations having relatively large chromatic numbers, leading to a complete determination of some of these chromatic numbers, but in general, the gaps between our upper and lower bounds are rather wide. In particular, we know of no such configuration of any size leading to a chromatic number above 6. + oai:arXiv.org:2512.19080v1 + math.CO + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://creativecommons.org/licenses/by/4.0/ + S{\o}ren Eilers, Rune Johansen, Rasmus Veber Rasmussen, Carsten Thomassen + + + Low-Latency and Low-Complexity MLSE for Short-Reach Optical Interconnects + https://arxiv.org/abs/2512.19094 + arXiv:2512.19094v1 Announce Type: new +Abstract: To meet the high-speed, low-latency, and low-complexity demand for optical interconnects, simplified layered 2-step maximum likelihood sequence estimation (L2S-MLSE) is proposed in this paper. Simplified L2S-MLSE combines computational simplification and reduced state in L2S-MLSE. L2S-MLSE with a parallel sliding block architecture reduces latency from linear order to logarithmic order. Computational simplification reduces the number of multipliers from exponential order to linear order. Incorporating the reduced state with computational simplification further decreases the number of adders and comparators. The simplified L2S-MLSE is evaluated in a 112-Gbit/s PAM4 transmission over 2-km standard single-mode fiber. Experimental results show that the simplified L2S-MLSE significantly outperforms the FFE-only case in bit error ratio (BER) performance. Compared with simplified 1-step MLSE, the latency of simplified L2S-MLSE is reduced from 34 delay units in linear order to 7 delay units in logarithmic order. The simplified scheme in L2S-MLSE reduces the number of variable multipliers from 512 in exponential order to 33 in linear order without BER performance deterioration, while reducing the number of adders and comparators to 37.2% and 8.4%, respectively, with nearly identical BER performance. + oai:arXiv.org:2512.19094v1 + cs.IT + math.IT + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Mengqi Guo, Ji Zhou, Haide Wang, Changyuan Yu, Xiangjun Xin, Liangchuan Li + + + On large queue lengths in generalised Jackson networks + https://arxiv.org/abs/2512.19098 + arXiv:2512.19098v1 Announce Type: new +Abstract: This paper proves a large deviation principle (LDP) for the stationary distribution of queue lengths in a subcritical generalised Jackson network assuming a Cramer condition on the interarrival and service times. The deviation function is given by the quasipotential. + oai:arXiv.org:2512.19098v1 + math.PR + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://creativecommons.org/licenses/by/4.0/ + Anatolii A. Puhalskii + + + Explicit and Non-asymptotic Query Complexities of Rank-Based Zeroth-order Algorithm on Stochastic Smooth Functions + https://arxiv.org/abs/2512.19104 + arXiv:2512.19104v1 Announce Type: new +Abstract: Zeroth-order (ZO) optimization with ordinal feedback has emerged as a fundamental problem in modern machine learning systems, particularly in human-in-the-loop settings such as reinforcement learning from human feedback, preference learning, and evolutionary strategies. While rank-based ZO algorithms enjoy strong empirical success and robustness properties, their theoretical understanding, especially under stochastic objectives and standard smoothness assumptions, remains limited. In this paper, we study rank-based zeroth-order optimization for stochastic functions where only ordinal feedback of the stochastic function is available. We propose a simple and computationally efficient rank-based ZO algorithm. Under standard assumptions including smoothness, strong convexity, and bounded second moments of stochastic gradients, we establish explicit non-asymptotic query complexity bounds for both convex and nonconvex objectives. Notably, our results match the best-known query complexities of value-based ZO algorithms, demonstrating that ordinal information alone is sufficient for optimal query efficiency in stochastic settings. Our analysis departs from existing drift-based and information-geometric techniques, offering new tools for the study of rank-based optimization under noise. These findings narrow the gap between theory and practice and provide a principled foundation for optimization driven by human preferences. + oai:arXiv.org:2512.19104v1 + math.OC + cs.LG + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://creativecommons.org/licenses/by/4.0/ + Haishan Ye + + + Polyhedra of Constant Gaussian Curvature + https://arxiv.org/abs/2512.19106 + arXiv:2512.19106v1 Announce Type: new +Abstract: Topology and geometry are deeply intertwined in the study of surfaces, though their interaction manifests differently in smooth and discrete settings. In the smooth category, a classical result asserts that any closed smooth surface embedded in $\mathbb{R}^3$ with constant Gaussian curvature must be a sphere, reflecting the strong rigidity of differential geometry. In contrast, the discrete setting, where curvature is represented as an angular defect concentrated at vertices, admits far greater flexibility. For instance, a flat torus can be realised as a polyhedral surface in $\mathbb{R}^3$ with zero curvature at every vertex. We establish a general result: any closed surface, whether orientable or non-orientable and of arbitrary genus, can be realised in $\mathbb{R}^3$ as a (possibly self-intersecting) polyhedral surface in which every vertex has the same angular defect. This highlights a fundamental distinction between discrete and smooth settings, showing that curvature constraints in the discrete realm impose fewer restrictions. Our proof is constructive and, once recognised, entirely elementary. Yet this fundamental fact appears to have gone unnoticed in the existing literature. + oai:arXiv.org:2512.19106v1 + math.DG + math.GT + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://creativecommons.org/licenses/by/4.0/ + Soto Hisakawa, Shizuo Kaji, Ryo Kawai + + + The geometric Merkurjev-Panin Conjecture for the Cox category + https://arxiv.org/abs/2512.19112 + arXiv:2512.19112v1 Announce Type: new +Abstract: We show that a strong version of the geometric Merkurjev-Panin conjecture holds for the Cox category of a projective toric variety. That is, we prove that the full strong exceptional collection of Bondal-Thomsen line bundles is invariant under the group of lattice automorphisms that permute the rays of the toric variety's fan. Our result is meant to further illustrate that the Cox category is a natural repository for homological algebra on toric varieties. + oai:arXiv.org:2512.19112v1 + math.AG + math.AC + math.CO + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Daniel Erman, Andrew Hanlon, Gaku Liu, Hailun Zheng + + + An analogue of Rognes' connectivity conjecture for free groups + https://arxiv.org/abs/2512.19128 + arXiv:2512.19128v1 Announce Type: new +Abstract: We show that the common basis complex of a free group of rank $n$ has the homotopy type of a wedge of spheres of dimension $2n-3$. This establishes an $\mathrm{Aut}(F_n)$-analogue of the connectivity conjecture that Rognes originally stated for $\mathrm{GL}_n(R)$. To prove this, we provide several homotopy-equivalent models of the common basis complex, both in terms of free factors in free groups and in terms of sphere systems in 3-manifolds. + oai:arXiv.org:2512.19128v1 + math.AT + math.CO + math.GR + math.GT + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Benjamin Br\"uck, Jeremy Miller, Kevin Ivan Piterman + + + The Kontsevich invariant and the action of the Grothendieck--Teichm\"{u}ller group on $2$-component string links + https://arxiv.org/abs/2512.19132 + arXiv:2512.19132v1 Announce Type: new +Abstract: The Kontsevich invariant of links is independent of the choice of associator, whereas for tangles this is not the case in general. In this paper, we focus on $2$-component string links and investigate to what extent the Kontsevich invariant depends on the choice of associator. As an application, we show that the action of the unipotent part of the Grothendieck--Teichm\"{u}ller group on the algebra of proalgebraic $2$-component string links is non-trivial, which provides a partial answer to a problem posed by Furusho. + oai:arXiv.org:2512.19132v1 + math.QA + math.GT + math.NT + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Hisatoshi Kodani, Yuta Nozaki + + + The Snapshot Problem for Wave Equations on Homogeneous Trees + https://arxiv.org/abs/2512.19136 + arXiv:2512.19136v1 Announce Type: new +Abstract: By definition, a wave on a homogeneous tree $\mathfrak X$ is a solution to the discrete wave equation on $\mathfrak{X}$; that is, a family $\{f_k\}_{k\in\mathbb Z}$ of complex-valued functions on $\mathfrak X$ satisfying the partial difference equation $\mu_1 f_k=(f_{k+1}+f_{k-1})/2$ for all $k$, where $\mu_1$ is the mean value operator on $\mathfrak X$ of radius $1$. The function $f_k$ is called the snapshot of the wave at time $k$. For $k\geq 2$, we will show that there exist infinitely many waves having given snapshots at times $0$ and $k$, but that all such waves have the same snapshots at times which are multiples of $k$. For integers $0<k<\ell$, we then consider necessary and sufficient conditions for the existence and uniqueness of a wave with given snapshots at times $0,\,k,\,\ell$. + oai:arXiv.org:2512.19136v1 + math.CO + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Fulton Gonzalez, Adelaide Nebeker, Katie Hallet, Andew Sailstad + + + Minimizing movements for quasilinear Keller--Segel systems with nonlinear mobility in weighted Wasserstein metrics + https://arxiv.org/abs/2512.19137 + arXiv:2512.19137v1 Announce Type: new +Abstract: We prove the global existence of weak solutions to quasilinear Keller--Segel systems with nonlinear mobility by minimizing movements (JKO scheme) in the product space of the weighted Wasserstein space and $L^2$ space. While minimizing movements for Keller--Segel systems with linear mobility are adapted in the product space of the Wasserstein space and $L^2$ space, due to the nonlinearity of mobility, we need to use the weighted Wasserstein space instead of the Wasserstein space. Moreover, since the mobility function is not Lipschitz, we first find solutions to the Keller--Segel systems whose mobility is apporoximated by a Lipschitz function, and then we establish additional uniform estimates and convergences to derive solutions to the Keller--Segel systems. + oai:arXiv.org:2512.19137v1 + math.AP + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Kyogo Murai + + + Quiver braid group action for a 3-fold crepant resolution + https://arxiv.org/abs/2512.19140 + arXiv:2512.19140v1 Announce Type: new +Abstract: The 3-fold cyclic quotient singularity denoted $\tfrac{1}{7}(1,2,4)$ admits a crepant resolution X with three exceptional Hirzebruch surfaces intersecting pairwise along curves. We show that the derived category D(X) carries a faithful action of a quiver braid group, where the relevant quiver is a 3-cycle encoding the intersection data. + oai:arXiv.org:2512.19140v1 + math.AG + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Will Donovan, Luyu Zheng + + + Solving Stengle's Example in Rational Arithmetic: Exact Values of the Moment-SOS Relaxations + https://arxiv.org/abs/2512.19141 + arXiv:2512.19141v1 Announce Type: new +Abstract: We revisit Stengle's classical univariate polynomial optimization example $min 1 - x^2 s.t. (1 - x^2)^3 \geq 0$ whose constraint description is degenerate at the minimizers. We prove that the moment-SOS hierarchy of relaxation order $r \geq 3$ has the exact value $-1/r(r - 2)$. For this we construct in rational arithmetic a dual polynomial sum-of-squares (SOS) certificate and a primal moment sequence representing a finitely atomic measure. The key ingredients are elementary trigonometric properties of Chebyshev and Gegenbauer polynomial, and a Christoffel-Darboux kernel argument. + oai:arXiv.org:2512.19141v1 + math.OC + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Didier Henrion (LAAS-POP) + + + Classical billiards can compute + https://arxiv.org/abs/2512.19156 + arXiv:2512.19156v1 Announce Type: new +Abstract: We show that two-dimensional billiard systems are Turing complete by encoding their dynamics within the framework of Topological Kleene Field Theory. Billiards serve as idealized models of particle motion with elastic reflections and arise naturally as limits of smooth Hamiltonian systems under steep confining potentials. Our results establish the existence of undecidable trajectories in physically natural billiard-type models, including billiard-type models arising in hard-sphere gases and in collision-chain limits of celestial mechanics. + oai:arXiv.org:2512.19156v1 + math.DS + cs.CC + math-ph + math.MP + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Eva Miranda, Isaac Ramos + + + A Characterization of Law-Invariant and Coherent Risk Measures through Optimal Transport + https://arxiv.org/abs/2512.19157 + arXiv:2512.19157v1 Announce Type: new +Abstract: In this article, we propose a novel characterization of law-invariant and coherent risk measures, based on a generalized optimal transport problem in which the second marginal of the admissible plans is not fixed, but required to lie within a target set of probability measures. One of the main contributions of this work is a general representation formula for such risk measures, which is closely related to Kusuoka's theorem. When the aforementioned target set is convex, our representation result allows for the systematic derivation of general duality formulas. To illustrate our findings, we explicitly compute the target sets associated with several classical law-invariant coherent risk measures, including the prototypical conditional value at risk and higher moment measures. + oai:arXiv.org:2512.19157v1 + math.OC + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Riccardo Bonalli (L2S, CNRS), Beno\^it Bonnet-Weill (CNRS, L2S), Laurent Pfeiffer (DISCO, L2S) + + + Kirwan polytopes in the real setting + https://arxiv.org/abs/2512.19158 + arXiv:2512.19158v1 Announce Type: new +Abstract: This is a monograph devoted to the study of Kirwan's real polytopes, i.e., in the context where there are involutions on the group and on the symplectic manifold. This work brings together and completes the two preprints I had already written on this subject (arXiv:2012.08837, arXiv:2111.13399). + oai:arXiv.org:2512.19158v1 + math.DG + math.SG + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Paul-Emile Paradan (IMAG) + + + Centralisers of semi-simple elements are semidirect products + https://arxiv.org/abs/2512.19164 + arXiv:2512.19164v1 Announce Type: new +Abstract: Let G be a reductive algebraic group over an algebraically closed field, and let s $\in$ G be a semisimple element. We show that the centraliser of s is the semi-direct product of its identity component by its group of components. We then look at the case where G is defined over an algebraic closure of a finite field Fq, and F is a Frobenius endomorphism attached to an Fqstructure on G. Under the additional assumption that s lies in an F -stable maximal torus such that F acts trivially on the Weyl group, we show that if the centraliser of s is F -stable we can make the above semi-direct product decomposition F -stable. + oai:arXiv.org:2512.19164v1 + math.GR + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Fran\c{c}ois Digne (LAMFA), Jean Michel (IMJ-PRG) + + + Optimal stabilization rate for the wave equation with hyperbolic boundary condition + https://arxiv.org/abs/2512.19167 + arXiv:2512.19167v1 Announce Type: new +Abstract: We show that the energy of classical solutions to the wave equation with hyperbolic boundary condition (i.e., dynamic Wentzell boundary condition) and damping on the boundary decays like 1/t. In fact we allow mixed boundary conditions: a possibly empty, disjoint part of the boundary may be kept at rest provided that the dynamic part satisfies the geometric control condition. We also prove that this decay rate is sharp. Our results follow from resolvent estimates, which we establish by studying high-frequency quasimodes. + oai:arXiv.org:2512.19167v1 + math.AP + math.OC + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Hugo Parada (IECL), Nicolas Vanspranghe (L2S) + + + Finite-sample guarantees for data-driven forward-backward operator methods + https://arxiv.org/abs/2512.19172 + arXiv:2512.19172v1 Announce Type: new +Abstract: We establish finite sample certificates on the quality of solutions produced by data-based forward-backward (FB) operator splitting schemes. As frequently happens in stochastic regimes, we consider the problem of finding a zero of the sum of two operators, where one is either unavailable in closed form or computationally expensive to evaluate, and shall therefore be approximated using a finite number of noisy oracle samples. Under the lens of algorithmic stability, we then derive probabilistic bounds on the distance between a true zero and the FB output without making specific assumptions about the underlying data distribution. We show that under weaker conditions ensuring the convergence of FB schemes, stability bounds grow proportionally to the number of iterations. Conversely, stronger assumptions yield stability guarantees that are independent of the iteration count. We then specialize our results to a popular FB stochastic Nash equilibrium seeking algorithm and validate our theoretical bounds on a control problem for smart grids, where the energy price uncertainty is approximated by means of historical data. + oai:arXiv.org:2512.19172v1 + math.OC + cs.LG + cs.SY + eess.SY + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Filippo Fabiani, Barbara Franci + + + On some local rings + https://arxiv.org/abs/2512.19197 + arXiv:2512.19197v1 Announce Type: new +Abstract: Given two seprable irreducible polynomials $P_1$ and $P_2$ over a filed $\mathbb{K}$. We show that the rings $\mathbb{K}[X]/(P_1^n)$ and $\mathbb{K}[X]/(P_2^n)$ are isomorphic if and only if their residue fields $\mathbb{K}[X]/(P_1)$ and $\mathbb{K}[X]/(P_2)$ are isomorphic. Partial results in this direction are obtained for the case where the polynomials are not seprable. We note that, given a seprable irreducible polynomial $P$, we prove that we have an isomorphism between $\mathbb{K}[X]/(P^n)$ and $(\mathbb{K}[X](P))[Y]/(Y^n)$. + oai:arXiv.org:2512.19197v1 + math.AC + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://creativecommons.org/licenses/by/4.0/ + Mohamad Maassarani + + + Parallelization of Welded Links + https://arxiv.org/abs/2512.19198 + arXiv:2512.19198v1 Announce Type: new +Abstract: The notion of a welded link was introduced by Fenn, Rim\'anyi, and Rourke as an analogue of welded braids. A welded link is defined as an equivalence class of link diagrams that may contain virtual crossings, where the equivalence is generated by the classical and virtual Reidemeister moves together with the welded moves. In this paper, we introduce a parallelization construction for welded link diagrams and show that it is well defined: if two diagrams represent equivalent welded links, then the corresponding parallel diagrams obtained by our construction are also equivalent. When the two parallel strands are given parallel orientations, the resulting diagram admits a checkerboard coloring, whereas if they are assigned opposite orientations, the diagram is almost classical. Our construction further yields a decomposition in which one component is a copy of the original diagram and the other is a diagram representing a trivial welded link. We also investigate quandle colorings and the fundamental quandle of the parallel diagram, deriving a presentation from that of the original diagram. Finally, we examine conditions under which the parallel diagram is non-split. + oai:arXiv.org:2512.19198v1 + math.GT + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Naoko Kamada, Seiichi Kamada + + + Mean-field optimal control with stochastic leaders + https://arxiv.org/abs/2512.19201 + arXiv:2512.19201v1 Announce Type: new +Abstract: We consider interacting agent systems with a large number of stochastic agents (or particles) influenced by a fixed number of external stochastic lead agents. Such examples arise, for example in models of opinion dynamics, where a small number of leaders (influencers) can steer the behaviour of a large population of followers. In this context, we study a partial mean-field limit where the number of followers tends to infinity, while the number of leaders stays constant. The partial mean-field limit dynamics is then given by a McKean-Vlasov stochastic differential equation (SDE) for the followers, coupled to a controlled It\^o-SDE governing the dynamics of the lead agents. For a given cost functional that the lead agents seek to minimise, we show that the unique optimal control of the finite agent system convergences to the optimal control of the limiting system. This establishes that the low-dimensional control of the partial (mean-field) system provides an effective approximation for controlling the high-dimensional finite agent system. In addition, we propose a stochastic gradient descent algorithm that can efficiently approximate the mean-field control. Our theoretical results are illustrated on opinion dynamics model with lead agents, where the control objective is to drive the followers to reach consensus in finite time. + oai:arXiv.org:2512.19201v1 + math.OC + math.PR + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://creativecommons.org/licenses/by/4.0/ + Sebastian Zimper, Ana Djurdjevac, Carsten Hartmann, Christof Sch\"utte, Nata\v{s}a Djurdjevac Conrad + + + Nonlinear Lebesgue spaces: Dense subspaces, completeness and separability + https://arxiv.org/abs/2512.19208 + arXiv:2512.19208v1 Announce Type: new +Abstract: L^p spaces of mappings taking values in arbitrary metric spaces, which we call nonlinear Lebesgue spaces, play an important role in several fields of mathematics. For instance, membership in these spaces is typically required for transport maps in optimal transport theory and for stochastic processes in probability theory. Nonlinear Lebesgue spaces also arise naturally in applications such as medical imaging, where the physical signals at play often exhibit little regularity and take their values in nonlinear spaces. Yet, these spaces remain little studied in the literature, likely due to their lack of differential structure outside the case where mappings are valued in a linear space. This paper is the first in a series by the authors devoted to the study of geometric and analytic properties of nonlinear Lebesgue spaces. The present article exposes a systematic treatment of their measure-theoretic properties, unifying and refining scattered results from the literature while also extending classical results from the linear setting to this broader nonlinear framework -- including the characterizations of their completeness and their separability as well as the density of some of their subspaces: the spaces of simple, continuous and smooth mappings. + oai:arXiv.org:2512.19208v1 + math.FA + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Guillaume S\'erieys (MAP5), Alain Trouv\'e (CB) + + + Existence of positive solutions for a class of almost critical problems on an annulus + https://arxiv.org/abs/2512.19209 + arXiv:2512.19209v1 Announce Type: new +Abstract: In this paper we will consider multi-peaks positive solutions for a class of slightly subcritical or slightly supercritical elliptic problems on an annulus with Dirichlet boundary conditions. By using the explicit form of the Green function and of the Robin function on the annulus, we prove that the annulus becomes thinner and thinner when the number of bumps increases for the slightly subcritical case, while the hole of the annulus is very small for the slightly supercritical case. + oai:arXiv.org:2512.19209v1 + math.AP + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Gabriele Mancini, Giuseppe Mario Rago, Giusi Vaira + + + On Dirichlet Spaces of Homogeneous Type Via Heat Kernel + https://arxiv.org/abs/2512.19216 + arXiv:2512.19216v1 Announce Type: new +Abstract: This paper considers the properties of Dirichlet Spaces of Homogeneous type which consist of band limited functions that are nearly exponential localizations on $\mathbb{R}^k.$ This is a powerful tool in harmonic analysis and it makes various spaces of functions and distributions more approachable, utilizable and providing non-zero representation of natural function spaces, such as Besov space, on $\mathbb{R}^k$. Spheres and homogeneous spaces can also admit such frames on the intervals and balls. Here, we present mainly the band limited frames that are well-localized in the general setting of Dirichlet spaces of Homogeneous type which have doubling measure and a local scale-invariant Poincare inequality which generates heat kernels through the Gaussian bounds and H$\ddot{o}$lder's continuity. As an application of this build-up, band limited frames are generated in the context of Lie groups which are homogeneous in nature with polynomial volume growth, complete Riemannian manifolds with Ricci curvature bounded from below and admits the volume doubling property, together with other settings. In this general setting, decomposition of Besov spaces was done with the new frames. + oai:arXiv.org:2512.19216v1 + math.FA + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://creativecommons.org/licenses/by/4.0/ + J. I. Opadara, M. E. Egwe + + + Make an optimization problem multidisciplinary + https://arxiv.org/abs/2512.19217 + arXiv:2512.19217v1 Announce Type: new +Abstract: Despite the abundance of benchmark problems for optimization algorithms, there is a notable scarcity of such problems in multidisciplinary design optimization (MDO). To address this gap, we introduce a novel methodology that enables the transformation of any optimization problem with a known solution into an equivalent MDO problem. This equivalence holds for a large class of coupling functions, including non-linear ones. The proposed methodology exploits a ''link function'' that effectively eliminates the coupling variables from the MDO problem, without influencing the solution. This approach allows for the creation of benchmark problems with reference solutions, facilitating the comparison and evaluation of various MDO algorithms. Moreover, it is adaptable to scalable optimization problems, where the dimensions of the search and constraint spaces can be configured. We also present a variant tailored to linear coupling functions with constant coefficients sampled independently at random, for which we derive a closed-form solution to the coupling equations. For the sake of illustration, we put our approach into action on a multidimensional Rosenbrock problem, varying the number of disciplines and design variable sizes. This example showcases the versatility and applicability of our methodology in generating benchmark problems for MDO. + oai:arXiv.org:2512.19217v1 + math.OC + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Matthias De Lozzo (INSA, EPE UT, IMT), Olivier Roustant (INSA, EPE UT, IMT), Amine Aziz-Alaoui + + + Graded embeddings, root generated subalgebras and $\pi$-systems for quasisimple Kac-Moody superalgebras + https://arxiv.org/abs/2512.19222 + arXiv:2512.19222v1 Announce Type: new +Abstract: Motivated by a construction of Gorelik and Shaviv, we show that the real roots of a root generated subalgebra associated with a $\pi$-system contained in the positive roots are obtained by successive applications of even and odd reflections to the $\pi$-system, and that they form a real closed subroot system. Using this result, we establish an analogue of Dynkins bijection in the setting of symmetrizable quasisimple Kac-Moody superalgebras. In addition, we obtain several results on root strings in the super setting, analogous to those of Billig and Pianzola, and show that graded embeddings arise as root generated subalgebras associated with linearly independent $\pi$-systems. + oai:arXiv.org:2512.19222v1 + math.RA + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://creativecommons.org/licenses/by/4.0/ + Irfan Habib, Deniz Kus, Chaithra Pilakkat + + + Global boundedness of weak solutions with finite energy to a general class of Dirichlet problems + https://arxiv.org/abs/2512.19224 + arXiv:2512.19224v1 Announce Type: new +Abstract: As explained in detail in the prologue to this manuscript, boundedness of weak solutions for general classes of elliptic equations in divergence form is a classic tool for achieving higher regularity. We propose here some global boundedness results under general assumptions that can be applied to several cases studied in the recent and extensive literature on partial differential equations \textit{under general growth}. In particular, we propose the class of \textit{weak solutions with finite energy} in which to search for solutions and in which regularity can be studied and achieved. We emphasize that we are not limited to minimizers of certain integral functionals, as often considered recently in this context of general growth, but to the broader class of weak solutions to Dirichlet problems for general nonlinear elliptic equations in divergence form. + oai:arXiv.org:2512.19224v1 + math.AP + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Giovanni Cupini, Paolo Marcellini + + + The space-time-Grassmann measure of the Brakke flow + https://arxiv.org/abs/2512.19227 + arXiv:2512.19227v1 Announce Type: new +Abstract: For a $k$-dimensional Brakke flow on an open subset $U \subset \mathbf{R}^{n}$, over an open time interval $J$, we prove the existence of a canonical space-time-Grassmann measure $\lambda$, over $J \times \mathbf{G}_{k} (U)$, and give a characterisation of the flow with respect to the space-time weight of this measure. This results in a new definition of the Brakke flow, as that of a space-time measure which satisfies the Brakke inequality in a distributional sense. Each such space-time measure corresponds to a class of equivalent (classical) Brakke flows, thus yielding an equivalence between the classical definitions of the Brakke flow, and this new definition. Moreover, we prove that the mean curvature vector, density, and tangent map along the flow, are all measurable with respect to this space-time weight measure. + oai:arXiv.org:2512.19227v1 + math.DG + math.AP + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://creativecommons.org/licenses/by/4.0/ + Yu Tong Liu, Myles Workman + + + 3-path-connectivity of Cayley graphs generated by wheel graphs + https://arxiv.org/abs/2512.19233 + arXiv:2512.19233v1 Announce Type: new +Abstract: Let $G = (V(G), E(G))$ be a simple connected graph and $\Omega$ a subset of $ V(G)$ with $|\Omega|\geq2$. An $\Omega$-path in $G$ is a path that connects all vertices of $\Omega$. Two $\Omega$-paths $P_i$ and $P_j$ are said to be internally disjoint if $V(P_i)\cap V(P_j)=\Omega$ and $E(P_i)\cap E(P_j)=\emptyset$. Denote $\pi_G(\Omega)$ by the maximum number of internally disjoint $\Omega$-paths in $G$. For an integer $k\geq2$, the $k$-path-connectivity $\pi_k(G)$ of $G$ is defined as $\min\{\pi_G(\Omega)\mid\Omega\subseteq V(G)$ and $|\Omega|=k\}$. Let $CW_n$ denote the Cayley graph generated by the $n$-vertex wheel graph. In this paper, we investigate the $3$-path-connectivity of $CW_n$ and prove that $\pi_3(CW_n)=\lfloor\frac{6n-9}4\rfloor$ for all $n\geq4$. + oai:arXiv.org:2512.19233v1 + math.CO + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Yi-Lu Luo, Yun-Ping Deng, Yuan Sun + + + Lagrangian fibrations on Nikulin-type orbifolds + https://arxiv.org/abs/2512.19244 + arXiv:2512.19244v1 Announce Type: new +Abstract: We classify lagrangian fibrations on Nikulin orbifolds, a well studied class of singular irreducible holomorphic symplectic varieties, and prove they verify the SYZ conjecture. + oai:arXiv.org:2512.19244v1 + math.AG + math.SG + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Giacomo Nanni + + + Small eigenvalues of pseudo-Laplacians + https://arxiv.org/abs/2512.19248 + arXiv:2512.19248v1 Announce Type: new +Abstract: We extend the Otal-Rosas bound on the number of small eigenvalues of the Laplacian on a hyperbolic surface to the small eigenvalues of pseudo-Laplacians. In the process, we extend the work of Colin de Verdi\`ere on the spectral theory of pseudo-Laplacians to hyperbolic surfaces with more than one cusp. + oai:arXiv.org:2512.19248v1 + math.DG + math.SP + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Werner Ballmann, Sugata Mondal, Panagiotis Polymerakis + + + Obstacle problems for the fractional $p$-Laplacian on fractal domains: well-posedness and asymptotics + https://arxiv.org/abs/2512.19252 + arXiv:2512.19252v1 Announce Type: new +Abstract: We study obstacle problems for the regional fractional $p$-Laplacian in a domain $\Omega\subset\mathbb{R}^2$ having as fractal boundary the Koch snowflake. We prove well-posedness results for the solution of the obstacle problem, as well as two equivalent formulations. Moreover, we study corresponding approximating obstacle problems in a sequence of domains $\Omega_n\subset\mathbb{R}^2$ having as boundary the $n$-th pre-fractal approximation of the Koch snowflake, for $n\in\mathbb{N}$. After proving the well-posedness of the approximating obstacle problems, we perform the asymptotic analysis for both $n\to+\infty$ and $p\to+\infty$. + oai:arXiv.org:2512.19252v1 + math.AP + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Simone Creo, Salvatore Fragapane + + + The number of rooted spanning forests of bicirculant graphs + https://arxiv.org/abs/2512.19256 + arXiv:2512.19256v1 Announce Type: new +Abstract: A bi-Cayley graph over the cyclic group $(\mathbb{Z}_n, +)$ is called a bicirculant graph. Let $\Gamma=BC(\mathbb{Z}_n; R,T,S)$ be a bicirculant graph with $R=-R\subseteq \mathbb{Z}_n\setminus \{0\}$ and $T={-}T\subseteq \mathbb{Z}_n\setminus \{0\}$ and $S\subseteq \mathbb{Z}_n$. In this paper, using Chebyshev polynomials, we obtain a closed formula for the number of rooted spanning forests of $\Gamma$. Moreover, we investigate some arithmetic properties of the number of rooted spanning forests of $\Gamma$, and find its asymptotic behaviour as $n$ tends infinity. + oai:arXiv.org:2512.19256v1 + math.CO + math.AG + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Jing Yang, Lihua Feng, Rongrong Lu, Tingzeng Wu + + + A classification of four-tuples of spinors of a ten dimensional space + https://arxiv.org/abs/2512.19257 + arXiv:2512.19257v1 Announce Type: new +Abstract: We use the theory of theta-groups developed by Vinberg, along with computations in the computer algebra system GAP4, to classify the orbits of Spin(10,C)x SL(4,C) acting on the tensor product of the half spin module of Spin(10,C) and the natural module of SL(4,C). + oai:arXiv.org:2512.19257v1 + math.RT + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://creativecommons.org/licenses/by/4.0/ + Willem de Graaf, Alexander Elashvili, Mamuka Jibladze + + + On Lie-holomorphs of Leibniz algebras + https://arxiv.org/abs/2512.19276 + arXiv:2512.19276v1 Announce Type: new +Abstract: We study the notion of the Lie-holomorph of a Leibniz algebra, recently introduced by N. P. Souris as a generalisation of the classical holomorph construction for Lie algebras. We establish a connection between the Lie-holomorph construction and the Leibniz algebra of biderivations defined by J.-L. Loday, and we prove that a linear endomorphism is a Lie-derivation if and only if it is simultaneously a derivation and an anti-derivation. As an application, we classify the Lie-holomorph algebras of all low-dimensional non-Lie Leibniz algebras over a field of characteristic different from $2$. + oai:arXiv.org:2512.19276v1 + math.RA + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://creativecommons.org/licenses/by-sa/4.0/ + Gianmarco La Rosa, Manuel Mancini + + + Open XOR-magic odd graphs and closed XOR-magic even graphs + https://arxiv.org/abs/2512.19278 + arXiv:2512.19278v1 Announce Type: new +Abstract: XOR-magic graph labelings form a special subclass of group distance magic labelings. A simple connected graph of order $2^n$ is called an open (respectively, closed) XOR-magic graph of power $n$ if its vertices can be labeled bijectively with vectors from $(\mathbb{Z}_2)^n$ such that the sum (over $(\mathbb{Z}_2)^n$) of labels in each open (respectively, closed) neighborhood of every vertex is equal to the zero vector. In one paper, Batal asked whether there exists any odd-regular open XOR-magic graph or any even-regular closed XOR-magic graph. In this paper, with partial help of MILP solver, we answer this question in the affirmative. More precisely, we prove that for every integer $n>3$, there exists an odd-regular open XOR-magic graph of power $n$ and an even-regular closed XOR-magic graph of power $n$. We also show some applications of the spectra of graphs for an open XOR-magic labeling. + oai:arXiv.org:2512.19278v1 + math.CO + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Sylwia Cichacz, Hubert Grochowski, Rita Zuazua + + + On the large time behavior of the 2D inhomogeneous incompressible viscous flows + https://arxiv.org/abs/2512.19281 + arXiv:2512.19281v1 Announce Type: new +Abstract: This paper studies the two-dimensional inhomogeneous Navier--Stokes equations governing stratified flows in a bounded domain under a gravitational potential \(f\). Our main results are as follows. First, we provide a rigorous characterization of steady states, proving that under the Dirichlet condition \(\mathbf{u}|_{\partial \Omega} = \mathbf{0}\), all admissible equilibria are hydrostatic and satisfy \(\nabla p_s = -\rho_s \nabla f\). Second, through a perturbative analysis around arbitrary hydrostatic profiles, we show that despite possible transient growth induced by the Rayleigh--Taylor mechanism, the system always relaxes to a hydrostatic equilibrium. Third, we identify a necessary and sufficient condition on the initial density perturbation for convergence to a linear hydrostatic density profile of the form \(\rho_s = -\gamma f + \beta\), with \(\gamma > 0\) and \(\beta > 0\). Finally, we establish improved regularity estimates for strong solutions corresponding to initial data in the Sobolev space \(H^3(\Omega)\). + oai:arXiv.org:2512.19281v1 + math.AP + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Song Jiang, Quan Wang + + + Local Topological Constraints on Berry Curvature in Spin--Orbit Coupled BECs + https://arxiv.org/abs/2512.19282 + arXiv:2512.19282v1 Announce Type: new +Abstract: We establish a local topological obstruction to flattening Berry curvature in spin-orbit-coupled Bose-Einstein condensates (SOC BECs), valid even when the global Chern number vanishes. For a generic two-component SOC BEC, the extended parameter space $M=T^{2}_{\mathrm{BZ}}\times S^{1}_{\phi_{+}}\times S^{1}_{\phi_{-}}$ carries a natural metric connection $\nabla^{C}$ whose torsion 3-form encodes the synthetic gauge fields. Its harmonic part defines a mixed cohomology class $ +[\omega]\in\bigl(H^{2}(T^{2}_{\mathrm{BZ}})\otimes H^{1}(S^{1}_{\phi_{+}})\bigr)\oplus\bigl(H^{2}(T^{2}_{\mathrm{BZ}})\otimes H^{1}(S^{1}_{\phi_{-}})\bigr), $ +whose mixed tensor rank equals one. Using a general geometric bound for metric connections with totally skew torsion on product manifolds, we show that the obstruction kernel $\mathcal{K}$ vanishes, yielding the sharp inequality $\dim\mathfrak{hol}^{\mathrm{off}}(\nabla^{C})\geq 1$. This forces at least one off-diagonal curvature operator, preventing complete gauging-away of Berry phases even when the total Chern number is zero. This provides the first cohomological lower bound certifying locally irremovable curvature in SOC BECs beyond the Chern-number paradigm. + oai:arXiv.org:2512.19282v1 + math.DG + math.AT + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://creativecommons.org/licenses/by/4.0/ + Alexander Pigazzini, Magdalena Toda + + + Holomorphic Deformations of Hyperbolicity Notions on Compact Complex Manifolds + https://arxiv.org/abs/2512.19284 + arXiv:2512.19284v1 Announce Type: new +Abstract: We investigate deformation properties of balanced hyperbolicity, with particular emphasis on degenerate balanced manifolds and their behavior under modifications. + In this context, we introduce two new notions of hyperbolicity for compact non-K\"ahler manifolds $X$ of complex dimension $\dim_{\mathbb{C}}X=n$ in degree $1 \leq p \leq n-1$, inspired by the work of F. Haggui and S. Marouani on $p$-K\"ahler hyperbolicity. The first notion, called \emph{p-SKT hyperbolicity}, generalizes the notions of SKT hyperbolicity and Gauduchon hyperbolicity introduced by S. Marouani. The second notion, called \emph{p-HS hyperbolicity}, extends the notion of sG hyperbolicity defined by Y. Ma. + We investigate the relationship between these notions of analytic nature and their geometric counterparts, namely Kobayashi hyperbolicity and \emph{p-cyclic hyperbolicity} for $2 \leq p \leq n-1$, and we examine the openness under holomorphic deformations of both $p$-HS hyperbolicity and $p$-K\"ahler hyperbolicity. + oai:arXiv.org:2512.19284v1 + math.CV + math.AG + math.DG + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Abdelouahab Khelifati + + + Locally constrained inverse curvature flow and Alexandrov-Fenchel type inequalities in de Sitter space + https://arxiv.org/abs/2512.19285 + arXiv:2512.19285v1 Announce Type: new +Abstract: In this paper, we study the behavior of some locally constrained inverse curvature flow in de Sitter space, with initial value any closed spacelike $k$-convex hypersurface satisfying some pinching condition. Assume further the Heintze-Karcher inequality for any closed spacelike mean convex hypersurface in de Sitter space, we derive a class of Alexandrov-Fenchel inequalities. + oai:arXiv.org:2512.19285v1 + math.DG + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://creativecommons.org/licenses/by-nc-nd/4.0/ + Kuicheng Ma + + + On a finite quasi birth-death process with catastrophes and its diffusion approximation + https://arxiv.org/abs/2512.19293 + arXiv:2512.19293v1 Announce Type: new +Abstract: We study a multi-type Ehrenfest process modeled as a finite quasi-birth-death (QBD) process. We assume that the transitions are allowed only to the two adjacent levels of the same phase and are characterized by linear rates. The crucial element lies in the phase switching mechanism at the origin, which is governed by an irreducible stochastic matrix. The process evolution is interrupted by catastrophic events, whose occurrences are controlled by a Poisson process. Each catastrophe resets the system state to zero, initiating a new cycle of evolution until the next resetting event. We conduct a comprehensive analysis, addressing both the transient and long-term behavior of this process. Furthermore, we derive a diffusive approximation, by proving its convergence to a reflected Ornstein-Uhlenbeck jump diffusion process. + oai:arXiv.org:2512.19293v1 + math.PR + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Giulia Di Nunno, Barbara Martinucci, Serena Spina + + + The Profinite Rigidity of Torsion-Free Lamplighter Groups + https://arxiv.org/abs/2512.19295 + arXiv:2512.19295v1 Announce Type: new +Abstract: We prove that the torsion-free lamplighter group $\Gamma = \mathbb{Z}^n \wr \mathbb{Z}$ of any rank $n \in \mathbb{N}$ is profinitely rigid in the absolute sense: the finite quotients of $\Gamma$ determine its isomorphism type uniquely among all finitely generated residually finite groups. The proof combines the theory of profinite rigidity for modules over Noetherian domains with an analysis of the algebraic properties of the lower central series of groups with the same profinite completion as $\Gamma$. + oai:arXiv.org:2512.19295v1 + math.GR + math.AC + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://creativecommons.org/licenses/by/4.0/ + Nikolay Nikolov, Julian Wykowski + + + Almost split sequences in abelian categories + https://arxiv.org/abs/2512.19296 + arXiv:2512.19296v1 Announce Type: new +Abstract: Using the Nakayama duality induced by a Nakayama functor, we provide a novel and concise account of the existence of Auslander-Reiten dualities and almost split sequences in abelian categories with enough projective objects or enough injective objects. As an example, we establish the existence of almost split sequences ending with finitely presented modules and those starting with finitely copresented modules in the category of all modules over a small endo-local Hom-reflexive category. Specializing to algebras given by (not necessarily finite) quivers with relations, we further investigate when the categories of finitely presented modules, finitely copresented modules and finite dimensional modules have almost split sequences on either or both sides. + oai:arXiv.org:2512.19296v1 + math.RT + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://creativecommons.org/licenses/by/4.0/ + Zetao Lin, Shiping Liu + + + Simple Cubic Variance Functions on $\R^n$, Part one + https://arxiv.org/abs/2512.19303 + arXiv:2512.19303v1 Announce Type: new +Abstract: The classification of natural exponential families started with the paper \cite {Morri} where Carl Morris unifies six very familiar families by the fact that their variance functions are polynomials of degree less or equal to two. Extension of this classification to $\R^n$ and to degree three is the subject of this paper. + Keywords: Actions of the group $GL(n+1,\R)$, classification of natural exponential families, multivariate Lagrange formula. variance functions. + oai:arXiv.org:2512.19303v1 + math.ST + stat.TH + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://creativecommons.org/licenses/by/4.0/ + Abdelhanid Hassairi, G\'erard Letac + + + On the construction of Cauchy MDS matrices over Galois rings via nilpotent elements and Frobenius maps + https://arxiv.org/abs/2512.19306 + arXiv:2512.19306v1 Announce Type: new +Abstract: Let $s,m$ be the positive integers and $p$ be any prime number. Next, let $GR(p^s,p^{sm})$ be a Galois ring of characteristic $p^s$ and cardinality $p^{sm}$. In the present paper, we explore the construction of Cauchy MDS matrices over Galois rings. Moreover, we introduce a new approach that considers nilpotent elements and Teichm\"uller set of Galois ring $GR(p^s,p^{sm})$ to reduce the number of entries in these matrices. Furthermore, we construct $p^{(s-1)m}(p^m-1)$ distinct functions with the help of Frobenius automorphisms. These functions preserve MDS property of matrices. Finally, we prove some results using automorphisms and isomorphisms of the Galois rings that can be used to generate new Cauchy MDS matrices. + oai:arXiv.org:2512.19306v1 + cs.IT + math.IT + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://creativecommons.org/licenses/by/4.0/ + Shakir Ali, Atif Ahmad Khan, Abhishek Kesarwani + + + Spinorial Heat Flow and Geometric Singularities on 3-Manifolds + https://arxiv.org/abs/2512.19308 + arXiv:2512.19308v1 Announce Type: new +Abstract: We present a spinor-based geometric heat flow formulated within the Clifford algebra framework, exploring a continuous framework alternative to the surgery-dependent Ricci flow program for 3-manifolds. By taking the spinor field $\psi$ as the fundamental dynamical variable, we derive a linear parabolic evolution equation that remains well-defined even at metric degenerations. We demonstrate that neck-pinch singularities correspond to zero-amplitude nodes in the spinor field, suggesting a mechanism for analytic continuation without curvature blow-up. The flow minimizes a spinorial Dirichlet energy monotonically and distinguishes between spherical and toroidal topologies via the existence of parallel versus Killing spinors. This work provides an analytically regularized approach to geometric flows that aims to preserve topological information while navigating through metric singularities. + oai:arXiv.org:2512.19308v1 + math.DG + math.GT + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://creativecommons.org/licenses/by/4.0/ + Ferhat Ta\c{s} + + + On induced subgraphs with degree parity conditions in Paley graphs and Paley tournaments + https://arxiv.org/abs/2512.19312 + arXiv:2512.19312v1 Announce Type: new +Abstract: In this paper, we investigate the number of induced subgraphs and subdigraphs of Paley graphs and Paley tournaments where the (out-)degree of each vertex has the same parity. For Paley graphs, we establish a lower bound for the number of large even induced subgraphs, particularly those containing a constant proportion of vertices. We determine the number of even-even partitions of Paley graphs, showing it is exponential if $q\equiv 1\Mod{8}$ and is trivial if $q\equiv 5\Mod{8}$, while proving the non-existence of even-even partition for Paley tournaments. Furthermore, we derive asymptotic formulas for the numbers of even induced sub(di)graphs of order $r=o(q^{1/4})$ in Paley graphs and Paley tournaments, demonstrating their concentration around the expected values in the corresponding random (di)graph models. + In the context of coding theory, we establish a correspondence between even/odd induced sub(di)graphs of Paley graphs (tournaments) and maximum distance separable (MDS) self-dual codes that can be constructed via (extended) generalized Reed-Solomon codes from subsets of finite fields. As a consequence, our contribution on induced subgraphs leads to new existence and counting results about MDS self-dual codes. + oai:arXiv.org:2512.19312v1 + math.CO + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Qilong Li, Yue Zhou + + + A New Approach to Defining Cochain Complexes for dual Leibniz algebra + https://arxiv.org/abs/2512.19319 + arXiv:2512.19319v1 Announce Type: new +Abstract: We construct a cochain map embedding the cohomology complex of any dual Leibniz algebra $B$ into the Lie algebra cochain complex of $\mathfrak{g} \otimes B$, where $\mathfrak{g}$ is a Leibniz algebra. This reduces the study of dual Leibniz cohomology to classical Lie algebra cohomology, yielding computational simplifications and new structural insights. + oai:arXiv.org:2512.19319v1 + math.RA + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://creativecommons.org/licenses/by/4.0/ + Hassan Alhussein + + + A New Approach to Defining Cochain Complexes for Tri-dendriform algebra + https://arxiv.org/abs/2512.19322 + arXiv:2512.19322v1 Announce Type: new +Abstract: Our constructions provide a systematic way to study cohomology tri-dendriform algebra via classical cohomology, simplifying computations and enabling the use of established techniques. + oai:arXiv.org:2512.19322v1 + math.RA + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://creativecommons.org/licenses/by/4.0/ + Hassan Alhussein + + + A new family of maximum linear symmetric rank-distance codes + https://arxiv.org/abs/2512.19324 + arXiv:2512.19324v1 Announce Type: new +Abstract: Let $\mathscr{S}_n(q)$ denote the set of symmetric bilinear forms over an $n$-dimensional $\mathbb{F}_q$-vector space. A subset $\mathcal{C}$ of $\mathscr{S}_n(q)$ is called a $d$-code if the rank of $A-B$ is larger than or equal to $d$ for any distinct $A$ and $B$ in $\mathcal{C}$. If $\mathcal{C}$ is further closed under matrix addition, then $|\mathcal{C}|$ is sharply upper bounded by $q^{n(n-d+2)/2}$ if $n-d$ is even and $q^{(n+1)(n-d+1)/2}$ if $n-d$ is odd. Additive codes meeting these upper bounds are called maximum. There are very few known constructions of them. In this paper, we obtain a new family of maximum $\mathbb{F}_q$-linear $(n-2)$-codes in $\mathscr{S}_n(q)$ for $n=6,8$ and $10$ which are not equivalent to any known constructions. Furthermore, we completely determine the equivalence between distinct members in this new family. + oai:arXiv.org:2512.19324v1 + math.CO + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Wei Tang, Yue Zhou + + + An overdetermined problem related to the p-Laplacian on Riemannian manifolds + https://arxiv.org/abs/2512.19329 + arXiv:2512.19329v1 Announce Type: new +Abstract: In this paper, we study the overdetermined problem for the p-Laplacian equation on a compact Riemannian manifold with positive Ricci curvature. By introducing a new P-function which is related to the first nonzero eigenvalue for p-Laplacian, we obtain some integral identities. As their applications, the Heintze-Karcher type inequality and the Soap Bubble Theorem have been achieved. + oai:arXiv.org:2512.19329v1 + math.AP + math.DG + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Guangyue Huang, Chunlei Luo, Hongru Song + + + Orthogonal Approximate Message Passing with Optimal Spectral Initializations for Rectangular Spiked Matrix Models + https://arxiv.org/abs/2512.19334 + arXiv:2512.19334v1 Announce Type: new +Abstract: We propose an orthogonal approximate message passing (OAMP) algorithm for signal estimation in the rectangular spiked matrix model with general rotationally invariant (RI) noise. We establish a rigorous state evolution that precisely characterizes the algorithm's high-dimensional dynamics and enables the construction of iteration-wise optimal denoisers. Within this framework, we accommodate spectral initializations under minimal assumptions on the empirical noise spectrum. In the rectangular setting, where a single rank-one component typically generates multiple informative outliers, we further propose a procedure for combining these outliers under mild non-Gaussian signal assumptions. For general RI noise models, the predicted performance of the proposed optimal OAMP algorithm agrees with replica-symmetric predictions for the associated Bayes-optimal estimator, and we conjecture that it is statistically optimal within a broad class of iterative estimation methods. + oai:arXiv.org:2512.19334v1 + cs.IT + cs.LG + math.IT + math.ST + stat.TH + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Haohua Chen, Songbin Liu, Junjie Ma + + + An Almost Flat Spin$^c$ Manifold Bounds + https://arxiv.org/abs/2512.19335 + arXiv:2512.19335v1 Announce Type: new +Abstract: We prove that every almost flat spin^$c$ manifold bounds a compact orientable manifold, thereby settling, in the spin^$c$ case, a long-standing conjecture of Farrell--Zdravkovska and S. T. Yau. + oai:arXiv.org:2512.19335v1 + math.AT + math.DG + math.GT + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Fei Han, Ruizhi Huang, Weiping Zhang + + + A hybrid-Hill estimator enabled by heavy-tailed block maxima + https://arxiv.org/abs/2512.19338 + arXiv:2512.19338v1 Announce Type: new +Abstract: When analysing extreme values, two alternative statistical approaches have historically been held in contention: the seminal block maxima method (or annual maxima method, spurred by hydrological applications) and the peaks-over-threshold. Clamoured amongst statisticians as wasteful of potentially informative data, the block maxima method gradually fell into disfavour whilst peaks-over-threshold-based methodologies were ushered to the centre stage of extreme value statistics. This paper proposes a hybrid method which reconciles these two hitherto disconnected approaches. Appealing in its simplicity, our main result introduces a new universal limiting characterisation of extremes that eschews the customary requirement of a sufficiently large block size for the plausible block maxima-fit to an extreme value distribution. We advocate that inference should be drawn solely on larger block maxima, from which practice the mainstream peaks-over-threshold methodology coalesces. The asymptotic properties of the promised hybrid-Hill estimator herald more than its efficiency, but rather that a fully-fledged unified semi-parametric stream of statistics for extreme values is viable. A finite sample simulation study demonstrates that a reduced-bias off-shoot of the hybrid-Hill estimator fares exceptionally well against the incumbent maximum likelihood estimation that relies on a numerical fit to the entire sample of block maxima. + oai:arXiv.org:2512.19338v1 + math.ST + stat.AP + stat.ME + stat.TH + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://creativecommons.org/licenses/by/4.0/ + Claudia Neves, Chang Xu + + + Enhancing PLS of Indoor IRS-VLC Systems for Colluding and Non-Colluding Eavesdroppers + https://arxiv.org/abs/2512.19339 + arXiv:2512.19339v1 Announce Type: new +Abstract: Most intelligent reflecting surface (IRS)-aided indoor visible light communication (VLC) studies ignore the time delays introduced by reflected paths, even though these delays are inherent in practical wideband systems. In this work, we adopt a realistic assumption of IRS-induced time delay for physical layer security (PLS) enhancement. We consider an indoor VLC system where an IRS is used to shape the channel so that the reflected signals add constructively at the legitimate user and create intersymbol interference at eavesdroppers located inside the coverage area. The resulting secrecy capacity maximisation over the IRS element allocation is formulated as a complex combinatorial optimisation problem and is solved using deep reinforcement learning with proximal policy optimisation (PPO). The approach is evaluated for both colluding eavesdroppers, which combine their received signals, and non-colluding eavesdroppers, which act independently. Simulation results are shown for various simulation setups, which demonstrate significant secrecy capacity gains. In a worst-case scenario, where the eavesdroppers have stronger channels than the legitimate user, the proposed PPO-based IRS allocation improves secrecy capacity by 107\% and 235\% in the colluding and non-colluding cases, respectively, compared with allocating all IRS elements to the legitimate user. These results demonstrate that time-delay-based IRS control can provide a strong secrecy advantage in practical indoor VLC scenarios. + oai:arXiv.org:2512.19339v1 + cs.IT + math.IT + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Rashid Iqbal, Ahmed Zoha, Salama Ikki, Muhammad Ali Imran, Hanaa Abumarshoud + + + Parameters and Theta lifts + https://arxiv.org/abs/2512.19344 + arXiv:2512.19344v1 Announce Type: new +Abstract: In this note, we make explicit the correspondence between Harish-Chandra parameters and Langlands-Vogan parameters for symplectic groups and orthogonal groups of equal rank over reals. As an application, we reformulate Moeglin's results and Paul's work on the Howe correspondence for symplectic-orthogonal dual pairs using Langlands-Vogan parameters. + oai:arXiv.org:2512.19344v1 + math.RT + math.NT + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Zhe Li, Shanwen Wang, Zhiqi Zhu + + + p-Groups in which kernels of the non-linear irreducible characters are of equal order + https://arxiv.org/abs/2512.19345 + arXiv:2512.19345v1 Announce Type: new +Abstract: For an irreducible character $\chi$ of a finite group $G$, its kernel is defined as $\text{ker }\chi=\{g\in G: \chi(g)=\chi(1)\}$. In this paper we characterize the finite groups of prime power order(for odd prime) in which kernels of all of the non-linear irreducible characters are of the same order. + oai:arXiv.org:2512.19345v1 + math.GR + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Nabajit Talukdar + + + De Rham affineness of the Nygaard filtered prismatization in positive characteristic + https://arxiv.org/abs/2512.19348 + arXiv:2512.19348v1 Announce Type: new +Abstract: Let $k$ be a perfect ring of characteristic $p>0$, and let $R$ be an animated $k$-algebra. This note aims to show that the Nygaard filtered prismatization $R^{\mathrm{Nyg}}$ of $R$ is naturally isomorphic, as a stack over $k^{\mathrm{Nyg}}$, to the relative spectrum over $k^{\mathrm{Nyg}}$ of the Rees algebra of the Nygaard filtered prismatic cohomology of $R$ relative to $k$. In doing so, we axiomatise the functorial affineness property displayed by the relative Nygaard filtered prismatization, and dub it de Rham affineness after the fundamental example of the functor sending an animated ring to its relative de Rham stack. While we treat this concept as an organising tool for the author's forthcoming work on the syntomification of Frobenius liftable schemes, we are able to frame some questions based on a structural theorem of independent interest: a functor to stacks which is de Rham affine often arises via ring stacks through transmutation. + oai:arXiv.org:2512.19348v1 + math.AG + math.AT + math.NT + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://creativecommons.org/licenses/by/4.0/ + Shubhankar Sahai + + + Decoupling for Markov Chains + https://arxiv.org/abs/2512.19351 + arXiv:2512.19351v1 Announce Type: new +Abstract: Consider a Markov chain $(X_i)_{i\ge0}$ with invariant measure $\mu$ that admits the representation $X_{i+1}=\Phi(X_i,U_i)$, where $(U_i)_{i\ge0}$ are i.i.d. random variables and $\Phi$ is a measurable map. We introduce a tangent-decoupled process $(\widetilde X_i)_{i\ge0}$ obtained by replacing $(U_i)$ with an independent copy. Conditional on the realized backbone $(X_i)$, the sequence $(f(\widetilde X_i))$ is independent. Although $(\widetilde X_i)$ is not Markovian, under the same ergodicity assumptions that ensure a law of large numbers for $(X_i)$, the empirical averages $n^{-1}\sum_{i=1}^n f(\widetilde X_i)$ converge almost surely to $\mu(f)$. In addition, for every $f\in L^2(\mu)$ and every $N\ge1$, $$ \operatorname{Var}\!\Bigl(\sum_{i=1}^N f(X_i)\Bigr) \;\le\; 2\,\operatorname{Var}\!\Bigl(\sum_{i=1}^N f(\widetilde X_i)\Bigr), $$ and therefore $\sigma_f^2 \le 2\,\widetilde\sigma_f^{\,2}$ for the corresponding time-average variance constants. The inequality requires neither reversibility nor mixing assumptions. Its proof identifies the two sequences as tangent in the sense of decoupling theory and applies the sharp $L^2$ tangent decoupling inequality of de la Pe\~na, Yao, and Alemayehu (2025). + oai:arXiv.org:2512.19351v1 + math.PR + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Nawaf Bou-Rabee, Victor H. de la Pe\~na + + + Left invariant complex Finsler metrics on a complex Lie group + https://arxiv.org/abs/2512.19353 + arXiv:2512.19353v1 Announce Type: new +Abstract: In this paper, we consider a left invariant complex Finsler metric $F$ on a complex Lie group. Using the technique of invariant frames, we prove the following properties for $(G,F)$. First, the metric $F$ must be a complex Berwald metric. Second, its complex spray $\chi=w^i\delta_{z^i}$ on $T^{1,0}G\backslash0$ can be extended to a holomorphic tangent field on $T^{1,0}G$. If we view $\chi$ as a real tangent field on $TG$, it coincides with the canonical bi-invariant spray structure on $G$. Third, we prove that the strongly K\"{a}hler, K\"{a}hler, and weakly K\"{a}hler properties for $F$ are equivalent. More over, $F$ is K\"{a}hler if and only if $G$ has an Abelian Lie algebra. Finally, we prove that the holomorphic sectional curvature vanishes. + oai:arXiv.org:2512.19353v1 + math.DG + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://creativecommons.org/licenses/by/4.0/ + Xiyun Xu, Ming Xu + + + On the number of maximal independent sets and maximal induced bipartite subgraphs in $K_4$-free graphs + https://arxiv.org/abs/2512.19356 + arXiv:2512.19356v1 Announce Type: new +Abstract: Let $G$ be a $K_4$-free graph of order $n$ and let $k$ be an integer with $0\leq k\leq n$. We show the existence of positive constants $\eta$ and $\nu$ such that $G$ has at most $(4-\eta)^{(5-\eta)k-n}(5-\eta)^{n-(4-\eta)k}$ maximal independent sets of order $k$ and at most $O\left((12-\nu)^{\frac{n}{4}}\right)$ maximal induced bipartite subgraphs. + oai:arXiv.org:2512.19356v1 + math.CO + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Thilo Hartel, Lucas Picasarri-Arrieta, Dieter Rautenbach + + + Newton's method in adaptive iteratively linearized FEM + https://arxiv.org/abs/2512.19357 + arXiv:2512.19357v1 Announce Type: new +Abstract: This paper concerns the inclusion of Newton's method into an adaptive finite element method (FEM) for the solution of nonlinear partial differential equations (PDEs). It features an adaptive choice of the damping parameter in the Newton iteration for the discretized nonlinear problems on each level ensuring both global linear and local quadratic convergence. In contrast to energy-based arguments in the literature, a novel approach in the analysis considers the discrete dual norm of the residual as a computable measure for the linearization error. As a consequence, this paper provides the first convergence analysis with optimal rates of an adaptive iteratively linearized FEM beyond energy-minimization problems. The presented theory applies to strongly monotone operators with locally Lipschitz continuous Fr\'echet derivative. We present a class of semilinear PDEs fitting into this framework and provide numerical experiments to underline the theoretical results. + oai:arXiv.org:2512.19357v1 + math.NA + cs.NA + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Philipp Bringmann, Maximilian Brunner, Dirk Praetorius + + + The Semiclassical Limit of the 2D Dirac--Hartree Equation with Periodic Potentials + https://arxiv.org/abs/2512.19362 + arXiv:2512.19362v1 Announce Type: new +Abstract: We study the semiclassical limit of the two-dimensional Dirac--Hartree equation in the presence of a periodic external potential. The spinor dynamics are formulated using the matrix-valued Wigner transform together with spectral projectors onto the positive and negative energy bands. Under suitable assumptions on the initial data and the potentials, we rigorously derive Vlasov-type transport equations describing the evolution of the band-resolved phase-space densities in both the massive and massless regimes. In the massless case, the limiting dynamics propagate ballistically with constant speed, while in the massive case the velocity is relativistic. Our analysis justifies the emergence of relativistic Vlasov equations from Dirac--Hartree dynamics in the semiclassical regime. As a corollary, we recover the relativistic Vlasov--Poisson equation from the Dirac equation with a regularized Coulomb interaction when the regularization vanishes together with the semiclassical parameter. + oai:arXiv.org:2512.19362v1 + math-ph + math.MP + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Jinyeop Lee, Kunlun Qi + + + Regular Cyclic $(q+1)$-Arcs in $\PG(3,2^m)$: Spectral Rigidity, Descent, and an MDS Criterion + https://arxiv.org/abs/2512.19371 + arXiv:2512.19371v1 Announce Type: new +Abstract: Let $q=2^m$ with $m\ge 3$ and set $n:=q+1$. We investigate $(q+1)$-arcs $\mathcal A\subset \mathrm{PG}(3,q)$ that admit a regular cyclic subgroup $C\le \mathrm{PGL}(4,q)$ of order $n$. Over $K=\mathbb{F}_{q^2}$, such an action can be conjugated to a diagonal one, producing explicit cyclic monomial models \[ \mathcal M_a = \{[1:t:t^a:t^{a+1}]:t\in U_n\}\subset \mathrm{PG}(3,K), \qquad U_n=\{u\in K^\times:u^n=1\}, \] with $a\in(\mathbb{Z}/n\mathbb{Z})^\times$. We develop + a spectral rigidity principle to + obtain a precise descent criterion: $\mathcal M_a$ is $K$-projectively equivalent to a $(q+1)$-arc defined over $\mathbb{F}_q$ if and only if $a\equiv \pm 2^e \pmod n$ for some integer $e$ with $\gcd(e,m)=1$. Consequently, regular cyclic pairs $(\mathcal A,C)$ fall into exactly $\varphi(m)/2$ $K$-projective equivalence classes. As an immediate coding-theoretic application, we resolve the remaining AMDS/MDS dichotomy for the BCH family $\mathcal C_{(q,q+1,3,h)}$ studied by Xu et al.: $\mathcal C_{(q,q+1,3,h)}$ is MDS + if and only if $2h+1\equiv \pm 2^e \pmod n$ for some $e$ with $\gcd(e,m)=1$. The underlying spectral rigidity step is formulated in a general setting for diagonal regular cyclic pairs in $\mathrm{PG}(r,K)$, providing a portable reduction of projective equivalence questions to explicit congruences on exponent data. + oai:arXiv.org:2512.19371v1 + math.CO + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Bocong Chen, Jing Huang, Hao Wu + + + Asymptotic and monodromy problems for higher-order Painlev\'e III equations + https://arxiv.org/abs/2512.19381 + arXiv:2512.19381v1 Announce Type: new +Abstract: In this paper, we study the isomonodromy deformation equations for the $n\times n$ system of first order meromorphic linear ordinary differential equations with two second order poles. We analyze the asymptotic behaviour of the solutions at a boundary point of the isomonodromic deformation space, and derive a parameterization of the solutions via asymptotic parameters. We then derive the explicit formula for the Stokes matrices and connection matrix of the associated linear system in terms of the asymptotic parameters. In the end, we apply the results to the study of the $tt^{*}$ equations. + oai:arXiv.org:2512.19381v1 + math.CA + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Zikang Wang, Xiaomeng Xu + + + Nevanlinna-Pick norms and local theory of commutative Banach algebras + https://arxiv.org/abs/2512.19385 + arXiv:2512.19385v1 Announce Type: new +Abstract: We initiate the study of Nevanlinna-Pick norms for commutative Banach algebras. + oai:arXiv.org:2512.19385v1 + math.FA + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Przemys{\l}aw Ohrysko, Micha{\l} Wojciechowski + + + A survey of edge-spectral-Tur\'an type problems in spectral graph theory: Results, conjectures and open problems + https://arxiv.org/abs/2512.19395 + arXiv:2512.19395v1 Announce Type: new +Abstract: The edge-spectral-Tur\'an type problem is also called the Brualdi-Hoffman-Tur\'an type problem, which is a central topic in spectral graph theory, seeking to determine the maximum spectral radius $\lambda(G)$ of an $F$-free graph $G$ with $m$ edges. This problem has attracted significant attention in recent years. In this paper, we will sort out several closely related results in this type of problem and then propose some conjectures for further research. + oai:arXiv.org:2512.19395v1 + math.CO + math.SP + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Yuantian Yu, Huihui Zhang, Minjie Zhang + + + The Neumann Green function of the annulus + https://arxiv.org/abs/2512.19397 + arXiv:2512.19397v1 Announce Type: new +Abstract: Using Gegenbauer polynomials and the zonal harmonic functions we build an explicit representation formula for the Green function with Neumann boundary conditions in the annulus. + oai:arXiv.org:2512.19397v1 + math.AP + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Giuseppe Mario Rago + + + Undecidability of theories of semirings with fixed points + https://arxiv.org/abs/2512.19401 + arXiv:2512.19401v1 Announce Type: new +Abstract: In this work we prove the undecidability (and $\Sigma^0_1$-completeness) of several theories of semirings with fixed points. The generality of our results stems from recursion theoretic methods, namely the technique of effective inseperability. Our result applies to many theories proposed in the literature, including Conway $\mu$-semirings, Park $\mu$-semirings, and Chomsky algebras. + oai:arXiv.org:2512.19401v1 + math.LO + cs.LO + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://creativecommons.org/licenses/by/4.0/ + Anupam Das, Abhishek De, Stepan L. Kuznetsov + + + On Spectral Properties of Lanzhou Matrix of Graphs + https://arxiv.org/abs/2512.19404 + arXiv:2512.19404v1 Announce Type: new +Abstract: Let $\Gamma$ be a simple graph on $n$ vertices. Lanzhou index is defined as $Lz(\Gamma)=\sum\limits_{u \in V(\Gamma)}d_\Gamma(u)^2d_{\overline{\Gamma}}(u).$ In this manuscript, the Lanzhou matrix, denoted by $A_{Lz}(\Gamma)$, has been defined, and its spectral properties are studied. The $uv^{th}$ entry in $A_{Lz}(\Gamma)$ is $d_\Gamma(u)d_{\overline{\Gamma}}(u)+d_\Gamma(v)d_{\overline{\Gamma}}(v)$ if $u$ and $v$ are adjacent. Otherwise, the entry is zero. Some bounds on Lanzhou energy and spread on the Lanzhou matrix are obtained. Also, Lanzhou eigenvalues and inertia for some standard graphs have been obtained. Additionally, characterizations for the symmetricity of Lanzhou eigenvalues about the origin are obtained. + oai:arXiv.org:2512.19404v1 + math.CO + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://creativecommons.org/licenses/by/4.0/ + Madhumitha K V, Harshitha A, Swati Nayak, Sabitha D'Souza + + + A Cartesian Cut-Cell Two-Fluid Method for Two-Phase Diffusion Problems + https://arxiv.org/abs/2512.19407 + arXiv:2512.19407v1 Announce Type: new +Abstract: We present a Cartesian cut-cell finite-volume method for sharp-interface two-phase diffusion problems in static geometries. The formulation follows a two-fluid approach: independent diffusion equations are discretized in each phase on a fixed staggered Cartesian grid, while the phases are coupled through embedded interface conditions enforcing continuity of normal flux and a general jump law. Cut cells are treated by integrating the governing equations over phase-restricted control volumes and faces, yielding discrete divergence and gradient operators that are locally conservative within each phase. Interface coupling is achieved by introducing a small set of interfacial unknowns per cut cell on the embedded boundary; the resulting algebraic system involves only bulk and interfacial averages. A key feature of the method is the use of a reduced set of geometric information based solely on low-order moments (trimmed volumes, apertures and interface measures/centroids), allowing robust implementation without constructing explicitly cut-cell polytopes. The method supports steady (Poisson) and unsteady (diffusion) regimes and incorporates Dirichlet, Neumann, Robin boundary conditions and general jumps. We validate the scheme on one-, two- and three-dimensional mono- and diphasic benchmarks, including curved embedded boundaries, Robin conditions and strong property/jump contrasts. The results demonstrate the expected convergence behavior, sharp enforcement of interfacial laws and excellent conservation properties. Extensions to moving interfaces and Stefan-type free-boundary problems are natural perspectives of this framework. + oai:arXiv.org:2512.19407v1 + math.NA + cs.NA + physics.comp-ph + physics.flu-dyn + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Louis Libat, Can Sel\c{c}uk, Eric Ch\'enier, Vincent Le Chenadec + + + Mixed formulation and structure-preserving discretization of Cosserat rod dynamics in a port-Hamiltonian framework + https://arxiv.org/abs/2512.19408 + arXiv:2512.19408v1 Announce Type: new +Abstract: An energy-based modeling framework for the nonlinear dynamics of spatial Cosserat rods undergoing large displacements and rotations is proposed. The mixed formulation features independent displacement, velocity and stress variables and is further objective and locking-free. Finite rotations are represented using a director formulation that avoids singularities and yields a constant mass matrix. This results in an infinite-dimensional nonlinear port-Hamiltonian (PH) system governed by partial differential-algebraic equations with a quadratic energy functional. Using a time-differentiated compliance form of the stress-strain relations allows for the imposition of kinematic constraints, such as inextensibility or shear-rigidity. A structure-preserving finite element discretization leads to a finite-dimensional system with PH structure, thus facilitating the design of an energy-momentum consistent integration scheme. Dissipative material behavior (via the generalized-Maxwell model) and non-standard actuation approaches (via pneumatic chambers or tendons) integrate naturally into the framework. As illustrated by selected numerical examples, the present framework establishes a new approach to energy-momentum consistent formulations in computational mechanics involving finite rotations. + oai:arXiv.org:2512.19408v1 + math.NA + cs.CE + cs.NA + cs.RO + cs.SY + eess.SY + math.DS + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Philipp L. Kinon, Simon R. Eugster, Peter Betsch + + + Arithmetic Bohr radius and Local Banach space theory + https://arxiv.org/abs/2512.19411 + arXiv:2512.19411v1 Announce Type: new +Abstract: This article introduces the notion of arithmetic Bohr radius for operator valued pluriharmonic functions on complete Reinhardt domains in $\mathbb{C}^n$. Using tools from local Banach space theory, we determine its asymptotic behavior in both finite and infinite dimensions. The framework developed in this article extends the classical Minkowski-space setting to a much broader class of sequence spaces, such as mixed Minkowski, Lorentz, and Orlicz spaces. This generality allows for a unified and systematic investigation of Bohr's theorem for both holomorphic and pluriharmonic functions. + oai:arXiv.org:2512.19411v1 + math.CV + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://creativecommons.org/licenses/by/4.0/ + Himadri Halder + + + H\"older regularity of doubly nonlinear nonlocal quasilinear parabolic equations in some mixed singular-degenerate regime + https://arxiv.org/abs/2512.19421 + arXiv:2512.19421v1 Announce Type: new +Abstract: We study local H\"older regularity of bounded, weak solutions for the nonlocal quasilinear equations of the form \[ (|u|^{q-2}u)_t + \text{P.V.} \int_{\mathbb{R}^n} \frac{|u(x,t) - u(y,t)|^{p-2}(u(x,t)-u(y,t))}{|x-y|^{n+sp}} dy = 0, \] with $p\in (1,\infty)$, $q\in (1,\infty)$ and $s \in (0,1)$. Analogous H\"older continuity result in the local case is known in the purely singular case $\{1<p<2, p<q\}$, purely degenerate case $\{2<p, q<p\}$, scale invariant case $\{p=q\}$ and translation invariant case $\{q=2,1<p<\infty\}$. In the nonlocal setting, H\"older regularity is known when the equation is either translation invariant $\{q=2, 1<p<\infty\}$ or scale invariant $\{q=p, 1<p<\infty\}$ or purely degenerate case $\{2<p, q<p\}$. Similar strategy can be used to obtain H\"older regularity in the purely singular case $\{1<p<2, p<q\}$. + In this paper, we adapt several ideas developed over the past few years and combine it with a new intrinsic scaling to prove H\"older regularity in the mixed singular-degenerate range $\max\{p,q,2\} < \min\left\{q + \tfrac{p-1}{1+\frac{n}{sp}}, 2 + \tfrac{p-1}{1+\frac{n}{sp}}\right\}$. The proof explicitly makes use of the nonlocal nature of the problem and as a consequence, our estimates are not stable at $s \rightarrow 0$. We note that the analogous regularity in the local problem remains open. + oai:arXiv.org:2512.19421v1 + math.AP + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://creativecommons.org/licenses/by/4.0/ + Karthik Adimurthi, Mitesh Modasiya + + + On the small Schr\"{o}der semigroup $\mathcal{SS}^{\prime}_{n}$ + https://arxiv.org/abs/2512.19422 + arXiv:2512.19422v1 Announce Type: new +Abstract: Let $[n]$ be a finite $n-$chain $\{1, 2, \dots, n\}$, and let $\mathcal{LS}_{n}$ be the Schr\"{o}der monoid, consisting of all isotone and order-decreasing partial transformations on $[n]$. Furthermore, let $\mathcal{SS}^{\prime}_{n} = \{\alpha \in \mathcal{LS}_{n} : \, 1\not\in \text{ Dom } \alpha\}$ be the subsemigroup of $\mathcal{LS}_{n}$, consisting of all transformations in $\mathcal{LS}_{n}$, each of whose domain does not contain $1$. For $1 \leq p \leq n$, let $K(n,p) = \{\alpha \in \mathcal{SS}^{\prime}_{n} : \, |Im \, \alpha| \leq p\}$ + be the two-sided ideal of $\mathcal{SS}^{\prime}_{n}$. Moreover, let ${RSS}^{\prime}_{n}(p)$ denote the Rees quotient of $K(n,p)$. It is shown in this article that for any $S$ in $\{\mathcal{SS}^{\prime}_{n}, K(n,p), {RSS}^{\prime}_{n}(p)\}$, $S$ is right abundant for all values of $n$, but not left abundant for all $n \geq 2$. In addition, the rank of the Rees quotient ${RSS}^{\prime}_{n}(p)$ is shown to be equal to the rank of the two-sided ideal $K(n,p)$, which is equal to $\binom{n-1}{p-1}+\sum\limits_{k=p}^{n-1}\binom{n-1}{k} \binom{k-1}{p-1}$. Finally, the rank of $\mathcal{SS}^{\prime}_{n}$ is determined to be $3n-4$. + oai:arXiv.org:2512.19422v1 + math.GR + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://creativecommons.org/licenses/by/4.0/ + Muhammad Mansur Zubairu, Abdullahi Umar, Fatma Salim Al-Kharousi + + + Critical Poisson hyperplane percolation in hyperbolic space has no unbounded cells + https://arxiv.org/abs/2512.19425 + arXiv:2512.19425v1 Announce Type: new +Abstract: We show that tessellations of hyperbolic space by isometry-invariant Poisson processes of $(d-1)$-dimensional hyperplanes do not have an unbounded cell at the critical intensity. This extends a result by Porret-Blanc for the hyperbolic plane (C. R. Acad. Sci. Paris, Ser. I, Vol. 344 (2007)) to dimensions $d\ge3$. We also show that for intensities strictly below the critical intensity, infinitely many unbounded cells exist, while for intensities larger than or equal to the critical intensity, no unbounded cell exists. This completely describes the basic phase transition of this continuum percolation model. Our proof uses a method from discrete percolation theory which we adapt to the continuum and combine with specific computations for Poisson hyperplane processes. + oai:arXiv.org:2512.19425v1 + math.PR + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://creativecommons.org/licenses/by/4.0/ + Tillmann B\"uhler, Anna Gusakova, Konstantin Recke + + + A survey on the growth rate inequality for sphere endomorphisms + https://arxiv.org/abs/2512.19430 + arXiv:2512.19430v1 Announce Type: new +Abstract: We survey recent results and current challenges concerning the growth rate inequality for sphere endomorphisms, and present a number of open problems and + conjectures arising in this context. + oai:arXiv.org:2512.19430v1 + math.DS + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://creativecommons.org/licenses/by-sa/4.0/ + Juliana Xavier + + + Fourier dimension of imaginary Gaussian multiplicative chaos + https://arxiv.org/abs/2512.19441 + arXiv:2512.19441v1 Announce Type: new +Abstract: We study the Fourier coefficients of imaginary Gaussian multiplicative chaos (GMC) on the unit circle. Under the subcritical phase $\beta\in(0,1)$, we show that the Fourier dimension is $1-\beta^2$ and prove a central limit theorem for the rescaled coefficients. + oai:arXiv.org:2512.19441v1 + math.PR + math-ph + math.FA + math.MP + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://creativecommons.org/licenses/by/4.0/ + Benjamin Bonnefont, Hermanni Rajam\"aki, Vincent Vargas + + + An alternative approach to well-posedness of McKean-Vlasov equations arising in Consensus-Based Optimization + https://arxiv.org/abs/2512.19446 + arXiv:2512.19446v1 Announce Type: new +Abstract: In this work we study the mean-field description of Consensus-Based Optimization (CBO), a derivative-free particle optimization method. Such a description is provided by a non-local SDE of McKean-Vlasov type, whose fields lack of global Lipschitz continuity. We propose a novel approach to prove the well-posedness of the mean-field CBO equation based on a truncation argument. The latter is performed through the introduction of a cut-off function, defined on the space of probability measures, acting on the fields. This procedure allows us to study the well-posedness problem in the classical framework of Sznitman. Through this argument, we recover the established result on the existence of strong solutions, and we extend the class of solutions for which pathwise uniqueness holds. + oai:arXiv.org:2512.19446v1 + math.OC + math.AP + math.PR + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Alessandro Baldi + + + A critical threshold for the cosmological Euler-Poisson system + https://arxiv.org/abs/2512.19454 + arXiv:2512.19454v1 Announce Type: new +Abstract: We consider the gravitational Euler-Poisson system with a linear equation of state on an expanding cosmological model of the Universe. The expansion of the spatial sections introduces an additional dissipating effect in the Euler equation. We prescribe the expansion rate of space by a scale factor $a(t)=t^\alpha$ with $\alpha\in(0,1)$, which describes the growth of length scales over time. This model is regularly applied in cosmology to study classical fluids in an expanding Universe. We study the behaviour of solutions to this system arising from small, near-homogeneous initial data and discover a \emph{critical} change of behaviour near the expansion rate $\alpha=2/3$, which corresponds to the matter-dominated regime in cosmology. In particular, we prove that for $\alpha>2/3$ the fluid variables are global in time and remain small provided they are sufficiently small in a suitable norm initially. In the complementary regime $\alpha\leq2/3$, we present numerical evidence for shock formation of solutions to the Euler equation for arbitrarily small initial data. In combination, this establishes the existence of a critical stability threshold for barotropic fluids in expanding domains. In contrast to our previous work on the corresponding relativistic system, the threshold in the classical system considered here is independent of the speed of sound of the fluid. This establishes that fluids in cosmology behave fundamentally different in the non-relativistic regime than in the relativistic one. + oai:arXiv.org:2512.19454v1 + math.AP + gr-qc + math-ph + math.MP + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://creativecommons.org/licenses/by/4.0/ + David Fajman, Maciej Maliborski, Maximilian Ofner, Todd Oliynyk, Zoe Wyatt + + + Spectral Shinkage of Gaussian Entropic Optimal Transport + https://arxiv.org/abs/2512.19457 + arXiv:2512.19457v1 Announce Type: new +Abstract: We present a functional calculus treatment of Entropic Optimal Transport (EOT) between Gaussian measures on separable Hilbert spaces, providing a unified framework that handles infinite-dimensional degeneracy. By leveraging the notion of proper alignment and the Schur complement, we reveal that the Gaussian EOT solution operates as a precise \textit{spectral shrinkage}: the optimal coupling is uniquely determined by contracting the spectrum of the correlation operator via a universal scalar function. This geometric insight facilitates an algorithmic shift from iterative fixed-point schemes (e.g., Sinkhorn) to direct algebraic computation, enabling efficient multi-scale analysis, where a single spectral decomposition allows for the exact evaluation of entropic costs across arbitrary regularization parameters $\varepsilon > 0$ at negligible additional cost. Furthermore, we investigate the asymptotic behavior as $\varepsilon \downarrow 0$ in settings where the unregularized Optimal Transport problem admits non-unique solutions. We establish a selection principle that the regularized limit converges to the most diffusive optimal coupling --characterized as the centroid of the convex set of optimal Kantorovich plans. This demonstrates that in degenerate regimes, the entropic limit systematically rejects deterministic Monge solutions (extremal points) in favor of the optimal solution with minimal Hilbert-Schmidt correlation, effectively filtering out spurious correlations in the null space. Finally, we derive stability bounds and convergence rates, recovering established parametric rates ($\varepsilon \log(1/\varepsilon)$) in finite dimensions while identifying distinct non-parametric rates dependent on spectral decay in infinite-dimensional settings. + oai:arXiv.org:2512.19457v1 + math.OC + math.FA + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://creativecommons.org/licenses/by/4.0/ + Ho Yun + + + Secondary cohomology operations and sectional category + https://arxiv.org/abs/2512.19461 + arXiv:2512.19461v1 Announce Type: new +Abstract: We show how secondary cohomology operations in the total space of the fibred join can be used to give lower bounds for the sectional category of a fibration. This suggests a refinement of the module weight of Iwase--Kono, which we call the secondary module weight. Examples are given for which the secondary module weight at the prime $2$ detects sectional category while the module weight does not. + oai:arXiv.org:2512.19461v1 + math.AT + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://creativecommons.org/licenses/by/4.0/ + Mark Grant + + + Pattern Avoiding Permutations as Walks + https://arxiv.org/abs/2512.19462 + arXiv:2512.19462v1 Announce Type: new +Abstract: The Stanley-Wilf limit of the pattern 1324 is known to lie between 10.271 and 13.5. We obtain lower bounds on this limit by encoding permutations as walks in directed graphs: building a permutation by successive insertion of maxima corresponds to traversing edges, and the growth rate of walks equals the spectral radius of the adjacency matrix. For 1324, this graph is too large for direct computation, so we pass to a quotient graph with weighted edges. Conditional on a natural conjecture, this yields a lower bound of 10.418. + oai:arXiv.org:2512.19462v1 + math.CO + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Atli Fannar Frankl\'in + + + Fully Asynchronous Unsourced Random Access over Fading Channels + https://arxiv.org/abs/2512.19468 + arXiv:2512.19468v1 Announce Type: new +Abstract: We examine unsourced random access in a fully asynchronous setup, where active users transmit their data without restriction on the start time over a fading channel. In the proposed scheme, the transmitted signal consists of a pilot sequence and a polar codeword, with the polar codeword distributed across the data part of the packet in an on-off pattern. The receiver uses a double sliding-window decoder, where the inner window employs iterative decoding with joint timing and pilot detection, channel estimation, single-user decoding, and successive interference cancellation to recover the message bits, while the outer window enhances interference cancellation. The numerical results indicate that the proposed scheme exhibits only a slight performance loss compared to the synchronous benchmark while being more applicable in practice. + oai:arXiv.org:2512.19468v1 + cs.IT + eess.SP + math.IT + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://creativecommons.org/licenses/by/4.0/ + Mert Ozates, Mohammad Kazemi, Gianluigi Liva, Deniz G\"und\"uz + + + Open Quantum Systems as Regular Holonomic $\mathcal{D}$-Modules: The Mixed Hodge Structure of Spectral Singularities + https://arxiv.org/abs/2512.19487 + arXiv:2512.19487v1 Announce Type: new +Abstract: The geometric description of open quantum systems via the Quantum Geometric Tensor (QGT) traditionally relies on the assumption that the physical states form a differentiable vector bundle over the parameter manifold. This framework becomes ill-posed at spectral singularities, such as Exceptional Points, where the eigen-bundle admits no local trivialization due to dimension reduction. In this work, we resolve this obstruction by demonstrating that the family of Liouvillian superoperators $\mathcal{L}(k)$ over a complex parameter manifold $X$ canonically defines a \textbf{regular holonomic $\mathcal{D}_X$-module} $\mathcal{M}$. By identifying the physical coherence order with the Hodge filtration and the decay rate hierarchy with the \textbf{Kashiwara filtration}, we show that the open quantum system underlies a \textbf{Mixed Hodge Module (MHM)} structure in the sense of Saito. + This identification allows us to apply the \textbf{Grothendieck six-functor formalism} rigorously to dissipative dynamics. We prove that the divergence corresponds to a non-trivial cohomology class in $\text{Ext}^1_{\mathcal{D}_X}$, thereby regularizing the Quantum Geometric Tensor without ad-hoc cutoffs. Specifically, the ``singular component'' of the Complete QGT arises as the residue of the connection on the \textbf{Brieskorn lattice} associated with the vanishing cycles functor. + oai:arXiv.org:2512.19487v1 + math-ph + math.AG + math.MP + quant-ph + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Prasoon Saurabh + + + High-dimensional normal approximations for sums of Langevin Markov chains + https://arxiv.org/abs/2512.19496 + arXiv:2512.19496v1 Announce Type: new +Abstract: Consider the well-known Langevin diffusion on $\mathbb{R}^d$ $$\mathrm{d} X_t = -\nabla U(X_t)\,\mathrm{d} t + \sqrt{2}\mathrm{d} B_t, $$ and its Euler-Maruyama discretization given by $$X_{k+1}=X_k-\eta \nabla U(X_k)+\sqrt{2\eta }\xi_{k+1},$$ where $\eta$ is the step size. Under mild conditions, the Langevin diffusion admits $\pi(\mathrm{d} x)\propto \exp(-U(x))\mathrm{d} x$ as its unique stationary distribution. + In this paper, we mainly study the normal approximation of the normalized partial sum $$ W_n = \eta^{1/2} n^{-1/2} \left( \sum_{i=0}^{n-1} X_i- \int_{\mathbb{R}^d} x\,\pi(\mathrm{d} x) \right).$$ To the best of our knowledge, this work provides the first dimension-explicit convergence rates in high-dimensional settings. Our main tool is a novel upper bound for the 1-Wasserstein distance $W_1(W,\gamma)$ via the exchange pair approach, where $W$ is any random vector of interest and $\gamma$ is a $d$-dimensional standard normal random vector. + oai:arXiv.org:2512.19496v1 + math.PR + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Tian Shen, Zhonggen Su, Xiaolin Wang + + + Computing multiple solutions from knowledge of the critical set + https://arxiv.org/abs/2512.19499 + arXiv:2512.19499v1 Announce Type: new +Abstract: {We explore a simple {\it geometric model} for functions between spaces of the same dimension (in infinite dimensions, we require that Jacobians be Fredholm operators of index zero). The model combines standard results in analysis and topology associated with familiar global and local aspects. Functions are supposed to be proper on bounded sets. The model is valid for a large class of semilinear elliptic differential operators. + It also provides a fruitful context for numerical analysis. + For a function $F: X \to Y$ between real Banach spaces, continuation methods to solve $F(x) = y$ may improve from considerations about the global geometry of $F$. + We consider three classes of examples. First we handle functions from the Euclidean plane to itself, for which the reasoning behind the techniques is visualizable. The second, between spaces of dimension 15, is obtained by discretizing a nonlinear Sturm-Liouville problem for which special right hand sides admit abundant solutions. Finally, we compute the six solutions of a semilinear elliptic equation $-\Delta u - f(u) = g$ studied by Solimini.} + oai:arXiv.org:2512.19499v1 + math.AP + math.FA + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://creativecommons.org/licenses/by-nc-sa/4.0/ + Otavio Kaminski, Diego S. Monteiro, Carlos Tomei + + + Transitive sets of derangements in primitive actions of PSL_2(q) + https://arxiv.org/abs/2512.19500 + arXiv:2512.19500v1 Announce Type: new +Abstract: Problem 8.75 of the Kourovka Notebook [10], attributed to John G. Thompson, asks the following: Suppose $G$ is a finite primitive permutation group on $\Omega$, and $\alpha$, $\beta$ are distinct points of $\Omega$. Does there exist an element $g\in G$ such that $\alpha^g=\beta$ and $g$ fixes no point of $\Omega$? A recent negative example is given in [12], where $G$ is the Steinberg triality group ${}^{3}D_{4}(2)$ acting primitively on 4,064,256 points. At present this is the only negative example known. In this note we show that almost simple primitive permutation groups with socle isomorphic to PSL_2(q) do not give negative examples. + oai:arXiv.org:2512.19500v1 + math.GR + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://creativecommons.org/licenses/by/4.0/ + Peter M\"uller + + + Large deviations for stochastic evolution equations beyond the coercive case + https://arxiv.org/abs/2512.19501 + arXiv:2512.19501v1 Announce Type: new +Abstract: We prove the small-noise large deviation principle (LDP) for stochastic evolution equations in an $L^2$-setting. As the coefficients are allowed to be non-coercive, our framework encompasses a much broader scope than variational settings. To replace coercivity, we require only well-posedness of the stochastic evolution equation and two concrete, verifiable a priori estimates. Furthermore, we accommodate drift nonlinearities satisfying a modified criticality condition, and we allow for vanishing drift perturbations. The latter permits the inclusion of It\^o--Stratonovich correction terms, enabling the treatment of both noise interpretations. In another paper, our results have been applied to the 3D primitive equations with full transport noise. In the current paper, we give an application to a reaction-diffusion system which lacks coercivity, further demonstrating the versatility of the framework. Finally, we show that even in the coercive case, we obtain new LDP results for equations with critical nonlinearities that rely on our modified criticality condition, including the stochastic 2D Allen--Cahn equation in the weak setting. + oai:arXiv.org:2512.19501v1 + math.PR + math.AP + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://creativecommons.org/licenses/by/4.0/ + Esm\'ee Theewis + + + Finite groups and complex projective surfaces + https://arxiv.org/abs/2512.19505 + arXiv:2512.19505v1 Announce Type: new +Abstract: In response to a question raised by Belolipetsky and the first author, we prove that for every finite group $G$ there are infinitely many isomorphism classes of compact complex hyperbolic $2$-manifolds with automorphism group isomorphic to $G$. + oai:arXiv.org:2512.19505v1 + math.GT + math.AG + math.GR + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://creativecommons.org/licenses/by/4.0/ + Alexander Lubotzky, Matthew Stover + + + New universal vertex algebras as glueings of the basic ones + https://arxiv.org/abs/2512.19508 + arXiv:2512.19508v1 Announce Type: new +Abstract: There are three universal $2$-parameter vertex algebras $\mathcal{W}_{\infty}$, $\mathcal{W}^{\text{ev}}_{\infty}$, and $\mathcal{W}^{\mathfrak{sp}}_{\infty}$ which are freely generated of types $\mathcal{W}(2,3,4,\dots)$, $\mathcal{W}(2,4,6,\dots)$, and $\mathcal{W}(1^3, 2, 3^3, 4,\dots)$, respectively. They serve as classifying objects for vertex algebras with these generating types satisfying mild hypotheses. Their $1$-parameter quotients are expected to be the building blocks of all $\mathcal{W}$-algebras of classical Lie types. Furthermore, such $\mathcal{W}$-algebras are expected to be organized into families that are governed by new universal $2$-parameter vertex algebras, which are themselves glueings of copies of $\mathcal{W}_{\infty}$ in type $A$ (together with a Heisenberg algebra), and copies of $\mathcal{W}^{\text{ev}}_{\infty}$ and $\mathcal{W}^{\mathfrak{sp}}_{\infty}$ in types $B$, $C$, and $D$. We denote these universal objects by $\mathcal{W}^{X,S,M}_{\infty}$, where $X$ denotes the Lie type (either $A$, $C$, or $BD$ since types $B$ and $D$ can be treated uniformly), and $S$, $M$ are sets of positive integers that determine certain families of partitions. More precisely, for a partition $P = (n_0^{m_0}, n_1^{m_1},\dots, n_{t}^{m_t})$ of $N = \sum_{i=0}^t n_i m_i$ consisting of $m_i$ parts of size $n_i$, where $n_0> n_1 > \cdots > n_t \geq 2$, $M = \{m_0,\dots, m_t\}$ is the set of multiplicities, and $S = \{d_1,\dots, d_t\}$ is the set of height differences $d_{i+1} = n_i - n_{i+1}$. After introducing this general conjectural picture, we will construct the first nontrivial example $\mathcal{W}^{\mathfrak{so}_2}_{\infty}:=\mathcal{W}^{BD, \emptyset, \{2\}}_{\infty}$, which is a glueing of two copies of $\mathcal{W}^{\text{ev}}_{\infty}$. + oai:arXiv.org:2512.19508v1 + math.QA + math-ph + math.MP + math.RT + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Thomas Creutzig, Vladimir Kovalchuk, Andrew R. Linshaw + + + Spiking and Resetting + https://arxiv.org/abs/2512.19517 + arXiv:2512.19517v1 Announce Type: new +Abstract: We consider a one-dimensional piecewise deterministic Markov process (PDMP) on $[0,1]$ with resetting at $0$ and depending on a small parameter $\varepsilon>0$. In the singular vanishing limit $\varepsilon \to 0$ we prove that the `` resetting '' simple point process associated to the PDMP converges to a point process described by a jump Markov process decorated by ``spikes'' distributed as a time-space Poisson point process with intensity proportional to $dt \otimes x^{-2} dx$. This proves rigorously results appeared previously in \cite{SBDKC25} and also justifies partially a conjecture formulated there. + oai:arXiv.org:2512.19517v1 + math.PR + math.DS + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://creativecommons.org/licenses/by/4.0/ + C\'edric Bernardin, Vsevolod Vladimirovich Tarsamaev + + + On the radii of Voronoi cells of rings of integers + https://arxiv.org/abs/2512.19518 + arXiv:2512.19518v1 Announce Type: new +Abstract: Since the time of Minkowski a basic problem in number theory has been to find lower bounds for the absolute value $\Delta(K)$ of the discriminant of a number field $K$ in terms of the degree $n(K)$ of $K$. In this paper we study another measure of the size of $K$ given by the covering radius $\mu(K)$ of the ring of integers $O_K$ of $K$. Here $\mu(K)$ is the $L^2$ radius $||V_2(K)||_2$ of the $L^2$ Voronoi cell $V_2(K)$ of $O_K$, where $V_2(K)$ is the set of points in $\mathbb{R} \otimes_{\mathbb{Q}} K$ that are at least as close to the origin as they are to any non-zero element of $O_K$. To put a limit on what lower bounds one can prove for $\mu(K)$ in terms of $n(K)$, we study infinite families of $K$ of increasing degree for which $\mu(K)$ can be bounded above by an explicit power of $n(K)$. We also study analogous questions when the $L^2$ norm is replaced by the $L^\infty$ norm. + oai:arXiv.org:2512.19518v1 + math.NT + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Frauke M. Bleher, Ted Chinburg, Xuxi Ding, Nadia Heninger, Daniele Micciancio + + + Evolution of finite temperature Bose-Einstein Condensates: Some rigorous studies on condensate growth + https://arxiv.org/abs/2512.19525 + arXiv:2512.19525v1 Announce Type: new +Abstract: In trapped Bose-Einstein condensates (BECs), \emph{condensate growth} refers to the process in which an increasing number of quasi-particles are immediately transferred from the non-condensate state (the thermal cloud) into the condensate state following the initial formation of the BEC. Despite its physical significance, this phenomenon has not yet been studied rigorously from a mathematical standpoint. + In this work, we investigate a kinetic equation whose collision operator includes three types of wave interactions: one corresponding to a 3-wave process, and two classified as 4-wave processes. This wave kinetic equation models the evolution of the density function of the thermal cloud. We establish the immediate formation of condensation in solutions to this equation, thus providing a rigorous demonstration of the condensate growth phenomenon. + oai:arXiv.org:2512.19525v1 + math-ph + math.MP + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Gigliola Staffilani, Minh-Binh Tran + + + Finite time energy cascade for mixed $3-$ and $4-$wave kinetic equations + https://arxiv.org/abs/2512.19531 + arXiv:2512.19531v1 Announce Type: new +Abstract: In this work we study a kinetic equation whose collision operator comprises three distinct wave interaction mechanisms: one representing a 3-wave process, and two corresponding to 4-wave processes. This wave kinetic equation describes the temporal evolution of the density function of the thermal cloud of a finite temperature trapped Bose gas. We establish that, for a broad class of initial data, solutions exhibit an immediate cascade of energy towards arbitrarily large frequencies. Furthermore, for other classes of initial conditions, we demonstrate that the energy is transferred to infinity in finite time. + oai:arXiv.org:2512.19531v1 + math-ph + math.AP + math.MP + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Gigliola Staffilani, Minh-Binh Tran + + + A perturbed preconditioned gradient descent method for the unconstrained minimization of composite objectives + https://arxiv.org/abs/2512.19532 + arXiv:2512.19532v1 Announce Type: new +Abstract: We introduce a perturbed preconditioned gradient descent (PPGD) method for the unconstrained minimization of a strongly convex objective $G$ with a locally Lipschitz continuous gradient. We assume that $G(v)=E(v)+F(v)$ and that the gradient of $F$ is only known approximately. Our analysis is conducted in infinite dimensions with a preconditioner built into the framework. We prove a linear rate of convergence, up to an error term dependent on the gradient approximation. We apply the PPGD to the stationary Cahn-Hilliard equations with variable mobility under periodic boundary conditions. Numerical experiments are presented to validate the theoretical convergence rates and explore how the mobility affects the computation. + oai:arXiv.org:2512.19532v1 + math.OC + cs.NA + math.NA + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://creativecommons.org/licenses/by/4.0/ + Jea-Hyun Park, Abner J. Salgado, Steven M. Wise + + + A massively parallel non-overlapping Schwarz preconditioner for PolyDG methods in brain electrophysiology + https://arxiv.org/abs/2512.19536 + arXiv:2512.19536v1 Announce Type: new +Abstract: We investigate non-overlapping Schwarz preconditioners for the algebraic systems stemming from high-order discretizations of the coupled monodomain and Barreto-Cressman models, with applications to brain electrophysiology. The spatial discretization is based on a high-order Polytopal Discontinuous Galerkin (PolyDG) method, coupled with the Crank-Nicolson time discretization scheme with explicit extrapolation of the ion term. To improve solver efficiency, we consider additive Schwarz preconditioners within the PolyDG framework, which combines (massively parallel) local subdomain solvers with a coarse-grid correction. Numerical experiments demonstrate robustness with respect to the discretization parameters, as well as a significant reduction in iteration counts compared to the unpreconditioned solver. These features make the proposed approach well-suited for parallel large-scale simulations in brain electrophysiology. + oai:arXiv.org:2512.19536v1 + math.NA + cs.NA + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Caterina B. Leimer Saglio, Stefano Pagani, Paola F. Antonietti + + + On the basic sequence structure of variable exponent Lebesgue spaces + https://arxiv.org/abs/2512.19538 + arXiv:2512.19538v1 Announce Type: new +Abstract: We study the subsymmetric basic sequence structure of variable exponent Lebesgue spaces $L_{P}$ built from index functions $P\colon\Omega\to(0,\infty]$ on $\sigma$-finite measure spaces $(\Omega,\Sigma,\mu)$. Specifically, we prove that if $P$ is bounded away from infinity, then any complemented subsymmetric basic sequence of $L_{P}$ is equivalent to the canonical basis of $\ell_r$ for some $r\ge 1$ in the essential range of $P$. + oai:arXiv.org:2512.19538v1 + math.FA + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Jos\'e L. Ansorena, Glenier Bello + + + The large deviation principle for the stochastic 3D primitive equations with transport noise + https://arxiv.org/abs/2512.19541 + arXiv:2512.19541v1 Announce Type: new +Abstract: We prove the small-noise large deviation principle for the three-dimensional primitive equations with transport noise and turbulent pressure. Transport noise is important for geophysical fluid dynamics applications, as it takes into account the effect of small scales on the large scale dynamics. The main mathematical challenge is that we allow for the transport noise to act on the full horizontal velocity, therefore leading to a non-trivial turbulent pressure, which requires an involved analysis to obtain the necessary energy bounds. Both Stratonovich and It\^o noise are treated. + oai:arXiv.org:2512.19541v1 + math.PR + math-ph + math.AP + math.MP + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Antonio Agresti, Esm\'ee Theewis + + + Ulrich complexity and Picard rank of bidouble Planes + https://arxiv.org/abs/2512.19544 + arXiv:2512.19544v1 Announce Type: new +Abstract: We determine the Picard number and the Ulrich complexity of general bidouble covers of the projective plane, providing the first systematic study of Ulrich bundles on non-cyclic abelian covers. For a bidouble plane branched along three smooth curves of degrees $n_1,n_2,n_3$, we show that $\rho(S)=1$ unless $(n_1,n_2,n_3)$ belongs to an explicit list, thereby extending Buium's classical results on double planes to the non-cyclic case. As an application, we determine the range of branch degrees for which Ulrich line bundles could exist, and we show that every even bidouble plane carries a special rank-two Ulrich bundle. Our method combines the invariant-theoretic decomposition of $H^2(S,\mathbb{Q})$ under the Galois group with cohomological criteria for Ulrich bundles. + oai:arXiv.org:2512.19544v1 + math.AG + math.NT + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Jerson Caro, Juan Cruz-Penagos, Sergio Troncoso + + + Yang-Mills energy quantization over non-collapsed degenerating Einstein manifolds and applications + https://arxiv.org/abs/2512.19552 + arXiv:2512.19552v1 Announce Type: new +Abstract: We investigate a sequence of Yang-Mills connections $A_j$ lying in vector bundles $E_j$ over non-collapsed degenerating closed Einstein 4-manifolds $(M_j, g_ j)$ with uniformly bounded Einstein constants and bounded diameters. We establish a compactness theory modular three types of bubbles. As applications, we get some quantization results for several important topological number associated with the vector bundles, for instance, the first Pontrjagin numbers $p_1(E)$ of vector bundles over Einstein 4-manifolds and the Euler numbers $\chi(M;E)$ of holomorphic vector bundles over K\"{a}hler-Einstein surfaces. Furthermore, we get some quantization results about the volume $v(L_j)$ and certain cohomological numbers (e.g. $dim H^0(M_j;L_j)$) of holomorphic line bundles $L_j$ over non-collapsed degenerating K\"{a}hler-Einstein surfaces $(M_j,J_j,g_j)$ with the aid of the classical vanishing theorems, the classical Hirzebruch-Riemann-Roch type theorems, and the profound convergence theory of K\"{a}hler-Einstein manifolds. In particular, we obtain some interesting identities involving non-collapsed degenerating compact K\"{a}hler-Einstein surfaces with non-zero scalar curvature, which indicate that we can know the Euler number of $M_j$ for large $j$ provided some topological information of the limit orbifold $M_\infty$. For K\"{a}hler-Einstein Del Pezzo surfaces, an interesting implication is that we can provide some preliminary estimates for the number of singularities of various types in $M_\infty$ in an effective way. As an unexpected surprise, we find an identity which connects Milnor numbers for singularities in $M_\infty$ and the correction terms in the Hirzebruch-Riemann-Roch theorem for orbifolds. Some quantization results can be extended to the case of higher dimensional $n$-manifolds. + oai:arXiv.org:2512.19552v1 + math.DG + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Youmin Chen, Miaomiao Zhu + + + Well-posedness and long time dynamics for a quasi-geostrophic ocean-atmosphere model with radiation balance + https://arxiv.org/abs/2512.19556 + arXiv:2512.19556v1 Announce Type: new +Abstract: We investigate a coupled atmosphere-ocean model including the mechanical and thermodynamical interaction between the two fluids for the mid-latitudes. The formulation combines a multilayer quasi-geostrophic dynamical framework with temperature equations incorporating long- and short-wave radiative forcing, as in energy balance models. Within a suitable functional framework, we establish the existence and uniqueness of solutions, and their continuous dependence on the radiation parameters. We also prove that the long-time dynamics are described by a finite-dimensional global attractor and, moreover, that the system possesses a finite set of determining modes that governs its asymptotic behaviour. In particular, we show that the long-term evolution of the ocean's temperature can be reconstructed solely from observations of the velocity fields across the model's layers. + oai:arXiv.org:2512.19556v1 + math.AP + math-ph + math.MP + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Federico Fornasaro, Tobias Kuna, Giulia Carigi + + + Monoidal Ringel duality and monoidal highest weight envelopes + https://arxiv.org/abs/2512.19558 + arXiv:2512.19558v1 Announce Type: new +Abstract: We show that a large class of non-abelian monoidal categories can be realized as subcategories of tilting objects in abelian monoidal categories with a highest weight structure. The construction relies on a monoidal enhancement of Brundan-Stroppel's semi-infinite Ringel duality and applies to many of Sam-Snowden's triangular categories and Knop's tensor envelopes of regular categories. We also explain how monoidal Ringel duality gives rise to monoidal structures on categories of representations of affine Lie algebras at positive levels. + oai:arXiv.org:2512.19558v1 + math.RT + math.CT + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Johannes Flake, Jonathan Gruber + + + Schr{\"o}dinger maps to a K{\"a}hler manifold in two dimensions + https://arxiv.org/abs/2512.19559 + arXiv:2512.19559v1 Announce Type: new +Abstract: We prove a global well--posedness and scattering result for Schr{\"o}dinger maps to a general K{\"a}hler manifold with small initial data in a Besov space. + oai:arXiv.org:2512.19559v1 + math.AP + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Benjamin Dodson, Jeremy L. Marzuola + + + Ground state solutions to the nonlinear Born-Infeld problem + https://arxiv.org/abs/2512.19566 + arXiv:2512.19566v1 Announce Type: new +Abstract: In the paper we show the existence of ground state solutions to the nonlinear Born-Infeld problem \[ \mathrm{div}\, \left( \frac{\nabla u}{\sqrt{1-|\nabla u|^2}} \right) + f(u) = 0, \quad x \in \mathbb{R}^N \] in the zero and positive mass cases. Moreover, we find a new proof of the Sobolev-type inequality \[ \int_{\mathbb{R}^N} \left(1 - \sqrt{1-|\nabla u|^2}\right) \, dx \geq C_{N,p} \left( \int_{\mathbb{R}^N} |u|^p \, dx \right)^{\frac{N}{N+p}}, \] for $p > 2^*$ as well as the characterization of the optimal constant $C_{N,p}$ in terms of the ground state energy level. Previous approaches relied on approximation schemes and/or symmetry assumptions, which typically yield to compact embeddings and may lead to solutions that are not at the ground state energy level. In contrast, neither approximation arguments nor symmetry assumptions are employed in the paper to obtain a ground state solution. Instead, we develop a new direct variational approach based on minimization over a Poho\v{z}aev manifold combined with profile decomposition techniques. Finally, we show that nonradial solutions exist whenever $N \geq 4$; in particular, this settles a previously open problem in the case $N=5$. + oai:arXiv.org:2512.19566v1 + math.AP + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://creativecommons.org/licenses/by/4.0/ + Bartosz Bieganowski, Norihisa Ikoma, Jaros{\l}aw Mederski + + + The Schwarz-Pick Lemma as a Consequence of the Ahlfors-Schwarz-Pick Lemma + https://arxiv.org/abs/2512.19571 + arXiv:2512.19571v1 Announce Type: new +Abstract: The aim of this article is to give an elementary proof of the fact that the Schwarz-Pick Lemma follows from the Ahlfors-Schwarz-Pick Lemma. + oai:arXiv.org:2512.19571v1 + math.CV + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://creativecommons.org/licenses/by-sa/4.0/ + Rafael Benjumea Cejas, Juan Carlos Garc\'ia V\'azquez + + + Convergence analysis for non-iterative sequential schemes for Biot's model + https://arxiv.org/abs/2512.19579 + arXiv:2512.19579v1 Announce Type: new +Abstract: An alternative to the fully implicit or monolithic methods used for the solution of the coupling of fluid flow and deformation in porous media is a sequential approach in which the fully coupled system is broken into subproblems (flow and mechanics problems) that are solved one after the other. This fully explicit coupling approach is a very simple scheme which allows much flexibility in the implementation and has a lower computational cost, making it quite attractive in practice since smaller linear systems need to be solved in order to obtain the solution for the whole coupled poroelastic system. Due to the appealing advantages of these methods, intensive research is currently being carried out in this direction, as in the present work. Although the application of this type of method is very common in practice, there exist only a few works devoted to their theoretical analysis. In this work, we consider the so-called explicit fixed-stress split scheme, which consists of solving the flow problem first with time-lagging the displacement term, followed by the solution of the mechanics problem. To the best of our knowledge, we provide the first convergence analysis of the explicit fixed-stress split scheme for Biot's equations. In particular, we prove that this algorithm is optimally convergent if the considered finite element discretization satisfies an inf-sup condition. In addition, with the aim of designing the simplest scheme for solving Biot's model, we also propose a similar decoupled algorithm for piecewise linear finite elements for both variables which arises from the novel stabilization recently proposed in \cite{Pe2025}, and is demonstrated to be optimally convergent. + oai:arXiv.org:2512.19579v1 + math.NA + cs.NA + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Xiaozhe Hu, Francisco J. Gaspar, Carmen Rodrigo + + + A modified Brinkman penalization fictitious domain method for the unsteady Navier-Stokes equations + https://arxiv.org/abs/2512.19580 + arXiv:2512.19580v1 Announce Type: new +Abstract: This paper investigates a modification of the fictitious domain method with continuation in the lower-order coefficients for the unsteady Navier-Stokes equations governing the motion of an incompressible homogeneous fluid in a bounded 2D or 3D domain. The modification enables {a solution-dependent} choice of the critical parameter. Global-in-time existence and convergence of a weak solution to the auxiliary problem are proved, and local-in-time existence and convergence of a unique strong solution are established. For the strong solution, a new higher-order convergence rate estimate in the penalization parameter is obtained. The introduced framework allows us to apply a pointwise divergence free finite element method as a discretization technique, leading to strongly mass conservative discrete fictitious domain method. A numerical example illustrates the performance of the method. + oai:arXiv.org:2512.19580v1 + math.NA + cs.NA + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://creativecommons.org/licenses/by/4.0/ + Zhanybek Baitulenov, Maxim Olshanskii, Almas Temirbekov, Nurlan Temirbekov, Syrym Kasenov + + + Geometric Progressions meet Zeckendorf Representations + https://arxiv.org/abs/2512.19586 + arXiv:2512.19586v1 Announce Type: new +Abstract: Motivated by Erd\H{o}s' ternary conjecture and by recent work of Cui--Ma--Jiang [``Geometric progressions meet Cantor sets'', \textit{Chaos Solitons Fractals} \textbf{163} (2022), 112567.] on intersections between geometric progressions and Cantor-like sets in standard bases, we study the corresponding problem in the Zeckendorf numeration system. We prove that, for any fixed finite set of forbidden binary patterns, any integers $u\ge 1$, $q\ge 2$, and any window size $M$, the set of exponents $n$ for which the Zeckendorf expansion of $u q^n$ avoids the forbidden patterns within its $M$ least significant digits is either finite or ultimately periodic. + oai:arXiv.org:2512.19586v1 + math.NT + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://creativecommons.org/licenses/by/4.0/ + Diego Marques, Pavel Trojovsky + + + A Complete Characterization of Pythagorean Hodograph Preserving Mappings + https://arxiv.org/abs/2512.19587 + arXiv:2512.19587v1 Announce Type: new +Abstract: We fully characterize the mappings $\Phi$ that send every Pythagorean-hodograph (PH) curve to a PH curve. We prove that in any dimension, such mappings are precisely the conformal functions whose dilation is the square of a real rational function. In the planar case, this implies (up to conjugation) that $\partial\Phi/\partial z = \Psi^{2}$, where $\Psi$ is meromorphic and satisfies $\operatorname{Res}(\Psi^{2}) = 0$ at every pole. In higher dimensions, PH preservation forces $\Phi$ to be a conformal map; for $n \ge 3$, Liouville's theorem then implies that any local diffeomorphism with this property is (anti-)M\"obius. These results subsume the previously known ``(scaled) PH-preserving'' constructions of mappings $\mathbb{R}^2 \to \mathbb{R}^3$ and align with Ueda's conformal viewpoint on isothermal and spherical geometries. At the level of examples, we demonstrate how PH-preserving mappings relate to the construction of rational PH curves and minimal surfaces. + oai:arXiv.org:2512.19587v1 + math.DG + math.CV + math.MG + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Amedeo Altavilla, Hans-Peter Schr\"ocker, byn\v{e}k \v{S}\'ir, Jan Vr\v{s}ek + + + The monodromy of cyclic Pryms + https://arxiv.org/abs/2512.19597 + arXiv:2512.19597v1 Announce Type: new +Abstract: The {\em Prym} of a cyclic covering of smooth projective curves is the ``new'' part of the Jacobian: the quotient of the Jacobian of the covering curve by the Jacobians of the intermediate covers. Given a family of such coverings, the fundamental group of the base of the family acts on the Tate modules of the Pryms, and the image of this representation is a key ingredient in answering arithmetic statistics questions about the distribution of the group structure of the $L$-torsion of a random Prym in the family. (Over ${\mathbb{F}}_q$, the action of Frobenius is roughly uniformly distributed over the {\em arithmetic} monodromy, a coset of the image of the fundamental group of the base change to $\bar{\mathbb{F}}_q$ (the {\em geometric} monodromy).) In the present note, we show for a number of natural families that (with limited exceptions) the geometric monodromy is sandwiched between a certain unitary group and its derived subgroup. In particular, this holds for the one-parameter families obtained by starting with any fixed cover and varying one (tame) ramification point. As an application, we deduce analogous largeness results for the monodromy of the Selmer groups of elliptic surfaces with $j=0$ or $j=1728$, by relating them to cyclic covers of degree 6 or 4 respectively, implying that their Selmer groups do not satisfy the standard heuristics. For instance, for eliptic surfaces with $j=0$ of sufficiently large height over ${\mathbb{P}}^1_{{\mathbb{F}}_q}$, the average size of the $l$-Selmer group is $l+3+o_q(1)$ when $l$ (fixed) and $q$ (large) are both 1 mod 3, compared to $l+1+o_q(1)$ for general elliptic surfaces. + oai:arXiv.org:2512.19597v1 + math.NT + math.AG + math.CA + math.GR + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Eric M. Rains + + + Chromatic Polynomial Evaluation Spectra + https://arxiv.org/abs/2512.19600 + arXiv:2512.19600v1 Announce Type: new +Abstract: Around 10 years ago, Agol and Krushkal showed that the number of chromatic polynomials $P_{G}$ arising from graphs $G$ on $n$ vertices grows exponentially with $n$, by establishing that the (dual) flow polynomial $F_{G}\left(\frac{3+\sqrt{5}}{2}\right)$ already takes on exponentially many values, if one varies $G$ over all planar cubic graphs $G$ on $n$ vertices. We show, more generally, that the size of the set $\{P_G(q): |V(G)|=n\}$ is exponential in $n$, for every fixed real number $q \neq 0,1,2$. In fact, our approach can also be pushed to show that $P_{G}(q)$ already takes on exponentially many values, if we only vary $G$ over all planar graphs on $n$ vertices. The case $q=3$ confirms a conjecture of Agol, which was initially motivated by the $\mathsf{NP}$-completeness of planar $3$-colorability. + oai:arXiv.org:2512.19600v1 + math.CO + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://creativecommons.org/licenses/by/4.0/ + Rafael Miyazaki, Cosmin Pohoata, Michael Zheng + + + The Lorentzian Calder\'{o}n problem on vector bundles + https://arxiv.org/abs/2512.19601 + arXiv:2512.19601v1 Announce Type: new +Abstract: In this paper we study a Lorentzian version of the Calder\'{o}n problem, which is concerned with the determination of a connection and potential on a Hermitian vector bundle over a Lorentzian manifold from the Dirichlet-to-Neumann map of the associated connection wave operator. For a class of Lorentzian manifolds satisfying a curvature bound, including perturbations of Minkowski space over strictly convex domains, the connection and potential is shown to be uniquely determined up to the natural gauge transformations of the problem. The proof is based on ideas from the earlier works arXiv:2008.07508, arXiv:2112.01663 of the second author in the scalar setting. + oai:arXiv.org:2512.19601v1 + math.AP + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Sean Gomes, Lauri Oksanen + + + Static size-effects meet the dynamic scattering properties of finite-sized mechanical metamaterials: a relaxed micromorphic study with parameter identification via two-stage static-dynamic optimization + https://arxiv.org/abs/2512.19604 + arXiv:2512.19604v1 Announce Type: new +Abstract: Mechanical metamaterials exhibit size-effects when a few unit-cells are subjected to static loading because no clear micro-macro scale separation holds and the characteristic length of the deformation becomes comparable to the unit-cell size. These size-effects typically manifest themselves as a strengthening of the response in a form summarized as "smaller is stiffer". Moreover, the dynamical behavior of mechanical metamaterials is very remarkable, featuring unique phenomena such as dispersive behavior and band-gaps where elastic waves cannot propagate over specific frequency ranges. In these frequency ranges, the wavelength becomes gradually comparable to the unit-cell size, giving rise to microstructure related phenomena which become particularly visible in the reflection/transmission patterns where an incident wave hits the metamaterial's interfaces. This raises the question of whether the static size-effects and dynamic reflection/transmission patterns are correlated. + In this work, we investigate the interaction of the static size-effects and the dynamic scattering response of mechanical metamaterials by employing the relaxed micromorphic model. We introduce a two-stage optimization procedure to identify the material parameters. In the first stage, the static material parameters are identified by exploiting the static size-effects through a least squares fitting procedure based on the total energy. The dynamic parameters are determined in the second stage by fitting the dispersion curves of the relaxed micromorphic model to those of the fully discretized microstructure. At this second stage, we assess the results obtained by fitting the dispersion curves in one and in two propagation directions, for both the relaxed micromorphic model (RMM) with curvature and its reduced counterpart (RRMM) without curvature. + The full abstract is presented in the paper. + oai:arXiv.org:2512.19604v1 + math.NA + cs.NA + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://creativecommons.org/licenses/by-nc-nd/4.0/ + Mohammad Sarhil, Leonardo Andres Perez Ramirez, Max Jendrik Voss, Angela Madeo + + + Tensor products of Lie nilpotent associative algebras and applications to codimension sequences + https://arxiv.org/abs/2512.19610 + arXiv:2512.19610v1 Announce Type: new +Abstract: Let $G$ and $H$ be unital associative algebras over a field $K$, such that $G$ satisfies the identity $[x_1, \dots, x_p] = 0$ for some integer $p \geq 3$ and $H$ satisfies the identities $[x_1, x_2, x_3] = 0$ and $[x_1, x_2] \cdots [x_{2k-1}, x_{2k}]=0$ for some $k \geq 2$. In this paper, extending results of Deryabina and Krasilnikov, we show that the tensor product $G \otimes H$ is again a Lie nilpotent associative algebra, i.e., it satisfies $[x_1, \dots, x_{q}] = 0$ for some $q \geq p$. We also determine an explicit value of $q$ in the case $k = 2$, i.e., when $H$ satisfies the identity $[x_1, x_2][x_3, x_4] = 0$. As a corollary, we reprove a result of Drensky saying that any product of Grassmann algebras of the form $E\otimes E_{i_1}\otimes \cdots \otimes E_{i_s}$ or $E_{j_1} \otimes E_{j_2} \otimes \cdots \otimes E_{j_t}$, where $E$ denotes the Grassmann algebra over a countable dimensional vector space and $E_r$ denotes the Grasmann algebra over an $r$-dimensional vector space, satisfies an identity of the form $[x_1, \dots, x_q] = 0$ for some integer $q \geq 3$. In addition, we show that for products of the form $E\otimes E_{i_1}\otimes \cdots \otimes E_{i_s}$ the minimal value of $q$ is always and odd integer. We also provide several particular cases in which a value of $q$ can be explicitly computed. As an application, we consider a field of characteristic zero, the variety $\mathfrak{N}_p$ of Lie nilpotent associative algebras of index at most $p$ and the corresponding relatively free algebras of finite rank, $F_n(\mathfrak{N}_p)$. We exhibit many explicit irreducible $S_n$-modules in the $S_n$-module decomposition of the space of proper multilinear polynomials in $F_n(\mathfrak{N}_p)$ for any $p$. This gives a lower bound for the dimensions of the spaces of multilinear and proper multilinear polynomials in $F_n(\mathfrak{N}_p)$. + oai:arXiv.org:2512.19610v1 + math.RA + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://creativecommons.org/licenses/by/4.0/ + Elitza Hristova + + + Scenario Reduction for the Two-Stage Stochastic Unit Commitment Problem + https://arxiv.org/abs/2512.19614 + arXiv:2512.19614v1 Announce Type: new +Abstract: The two-stage stochastic unit commitment problem has become an important tool to support decision-making under uncertainty in power systems. Representing the uncertainty by a large number of scenarios guarantees accurate results but challenges the solution process. One way to overcome this is by using scenario reduction methods, which aim at finding a distribution supported on fewer scenarios, but leading to similar optimal first-stage decisions. In this paper, we recap the classical scenario reduction theory based on the distance of probability distributions and the optimal mass transportation problem. We then review and compare various formulations of the underlying cost function of the latter used in the literature. Using the Forward Selection Algorithm, we show that a specific formulation of the cost function can be proven to select the best possible scenario from a given sample on the first draw with respect to the Relative Approximation Error. We demonstrate this result and compare the quality of the approximation as well as the computational performance of the different cost functions using a modified version of the IEEE RTS 24-Bus System. In many cases, we find that the optimal solution of the two-stage stochastic unit commitment problem with 200 scenarios can be approximated with around 2% scenarios when using this cost function. + oai:arXiv.org:2512.19614v1 + math.OC + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://creativecommons.org/licenses/by/4.0/ + Yannick Werner, Juan Miguel Morales, Salvador Pineda, Line Roald, Sonja Wogrin + + + Global bifurcation of hollow vortex streets + https://arxiv.org/abs/2512.19619 + arXiv:2512.19619v1 Announce Type: new +Abstract: Vortex streets are periodic configurations of vortices propagating through an irrotational flow. In this paper, we study streets of hollow vortices, which are solutions to the free boundary $2$-d irrotational incompressible Euler equations. Each vortex core is a region of constant pressure in the complement of the fluid domain with a nonzero circulation around it. We prove that any non-degenerate singly-periodic point vortex configuration can be ``desingularized'' to create a global curve of solutions to the steady hollow vortex street problem, and we further characterize the types of singular behavior that can develop as one transverses the curve to its extreme. As specific examples, we study von K\'arm\'an vortex streets, translating vortex arrays, and a two-pair (2P) configuration. Our method is based on analytic global bifurcation theory and adapts the desingularization technique of Chen, Walsh, and Wheeler to the periodic setting. + oai:arXiv.org:2512.19619v1 + math.AP + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Vasileios N. Oikonomou, Samuel Walsh + + + Quantization of Random Homogeneous Self-Similar Measures + https://arxiv.org/abs/2512.19628 + arXiv:2512.19628v1 Announce Type: new +Abstract: In this article, we study a class of invariant measures generated by a random homogeneous self-similar iterated function system. Unlike the deterministic setting, the random quantization problem requires controlling distortion errors across non-uniform scales. For $r>0$, under a suitable separation condition, we precisely determine the almost sure quantization dimension $\kappa_r$ of this class, by utilizing the ergodic theory of the shift map on the symbolic space. By imposing an additional separation condition, we establish almost sure positivity of the $\kappa_r$-dimensional lower quantization coefficient. Furthermore, without assuming any separation condition, we provide a sufficient condition that guarantees almost sure finiteness of the $\kappa_r$-dimensional upper quantization coefficient. We also include some illustrative examples. + oai:arXiv.org:2512.19628v1 + math.DS + math.PR + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Akash Banerjee, Alamgir Hossain, Md. Nasim Akhtar + + + Rank-metric separation in irreducible representations of finite groups + https://arxiv.org/abs/2512.19638 + arXiv:2512.19638v1 Announce Type: new +Abstract: We give a general lower bound on the rank of matrices of the form $\rho(h) - I$ with $\rho : G \rightarrow GL({\mathbb F}^n)$ an irreducible representation of a finite group $G$. The main tool in the proof is a (strengthening) of a reduction due to Efremenko from low rank matrices spanned by a few images of $\rho$ to Locally Decodable Codes (LDCs), which are a special kind of error correcting codes. We then apply the known results on 2-query LDCs to derive our rank bound. + oai:arXiv.org:2512.19638v1 + math.GR + math.CO + math.RT + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Zeev Dvir + + + Subgroups of Cyclically Amalgamated Free Products + https://arxiv.org/abs/2512.19645 + arXiv:2512.19645v1 Announce Type: new +Abstract: Given a group $G = H_1 \ast_A H_2$ which is the free product of two finitely generated groups $H_1$ and $H_2$ with amalgamation over a cyclic subgroup $A$ which is malnormal in $G$, we study relations between the structure of its subgroups and the structure of the group $G$ itself. Firstly, we show that if $H_1$ and $H_2$ are 3-free products of cyclics of rank $\ge 3$ then $G$ is also a 3-free product of cyclics. Secondly, we prove that if $H_1$ and $H_2$ are 4-free products of cyclics of rank $\ge 4$ then every 4-generated subgroup of $G$ is a free product of $\le 4$ cyclics or a 1-relator quotient of a free product of four cyclic groups. Here a group is called an $n$-free product of cyclics if every $n$-generated subgroup is a free product of $\le n$ cyclic groups. These results are based on ubiquitous applications of the Nielsen method for amalgamated free products which we recall carefully. + Lastly, given an infinite, finitely presented group which is not free, but all of its infinite index subgroups are free, a well-known conjecture says that it is isomorphic to a surface group. We revisit and elaborate on predominantly group theoretic proofs of this conjecture for cyclically amalgamated products as above, as well as for certain HNN extensions. + oai:arXiv.org:2512.19645v1 + math.GR + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Martin Kreuzer, Anja Moldenhauer, Gerhard Rosenberger + + + Milstein-type Schemes for Hyperbolic SPDEs + https://arxiv.org/abs/2512.19647 + arXiv:2512.19647v1 Announce Type: new +Abstract: This article studies the temporal approximation of hyperbolic semilinear stochastic evolution equations with multiplicative Gaussian noise by Milstein-type schemes. We take the term hyperbolic to mean that the leading operator generates a contractive, not necessarily analytic $C_0$-semigroup. Optimal convergence rates are derived for the pathwise uniform strong error \[ + E_h^\infty := \Big(\mathbb{E}\max_{1\le j \le M}\|U_{t_j}-u_j\|_X^p\Big)^{1/p} \] on a Hilbert space $X$ for $p\in [2,\infty)$. Here, $U$ is the mild solution and $u_j$ its Milstein approximation at time $t_j=jh$ with step size $h>0$ and final time $T=Mh>0$. For sufficiently regular nonlinearity and noise, we establish strong convergence of order one, with the error satisfying $E_h^\infty\lesssim h\sqrt{\log(T/h)}$ for rational Milstein schemes and $E_h^\infty \lesssim h$ for exponential Milstein schemes. This extends previous results from parabolic to hyperbolic SPDEs and from exponential to rational Milstein schemes. Root-mean-square error estimates are strengthened to pathwise uniform estimates. Numerical experiments validate the convergence rates for the stochastic Schr\"odinger equation. Further applications to Maxwell's and transport equations are included. + oai:arXiv.org:2512.19647v1 + math.NA + cs.NA + math.AP + math.FA + math.PR + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://creativecommons.org/licenses/by/4.0/ + Felix Kastner, Katharina Klioba + + + The Cone Conjecture for Primitive Symplectic Varieties over a Field of Characteristic Zero and an Application + https://arxiv.org/abs/2512.19656 + arXiv:2512.19656v1 Announce Type: new +Abstract: We prove the Kawamata-Morrison cone conjecture for Q-factorial terminal projective primitive symplectic varieties with second Betti number greater than five defined over a field of characteristic zero. As an application, we prove that the relative movable and the relative nef cone conjectures hold for fibrations whose very general fibre is a projective primitive symplectic varieties under certain assumptions. + oai:arXiv.org:2512.19656v1 + math.AG + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Aur\'elien Faucher + + + A genus-zero surface with bounded curvature enclosing less volume than the unit sphere + https://arxiv.org/abs/2512.19659 + arXiv:2512.19659v1 Announce Type: new +Abstract: We produce a family of bodies in $\mathbb R^3$ parameterized by $\varepsilon > 0$, each bounded by a smooth topological sphere with principal curvatures in $[-1, 1]$, and having volume arbitrarily close to $ 16 - 4\sqrt 3 + \left(10 \sqrt 3 - 14\right) \pi - \left(\frac{10}{3} - \sqrt 3\right) \pi^2 \approx 3.70.$ Thus, in contrast to the two-dimensional case, the unit sphere (which bounds a ball of volume $\frac{4 }{ 3} \pi \approx 4.19$) does not enclose the minimal volume among all smooth spheres in $\mathbb R^3$ with principal curvatures in $[-1,1]$. This answers a folklore question of Dmitri Burago and Anton Petrunin. + oai:arXiv.org:2512.19659v1 + math.DG + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://creativecommons.org/licenses/by/4.0/ + Matthew Bolan + + + Birational geometry of del Pezzo surfaces of degree 4 + https://arxiv.org/abs/2512.19660 + arXiv:2512.19660v1 Announce Type: new +Abstract: It is known that any Mori fiber space birational to a minimal smooth del Pezzo surface $S$ of degree $4$ is either a del Pezzo surface of degree $4$ itself, or a smooth cubic surface with a structure of a relatively minimal conic bundle. We show that any del Pezzo surface of degree $4$ birational to $S$ is actually isomorphic to $S$. Also, we sketch an equivariant version of this fact. On the way, we review the biregular classification of del Pezzo surfaces of degree $4$ obtained by A. N. Skorobogatov. + oai:arXiv.org:2512.19660v1 + math.AG + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Constantin Shramov, Andrey Trepalin + + + Quantum upper triangular matrix algebras + https://arxiv.org/abs/2512.19664 + arXiv:2512.19664v1 Announce Type: new +Abstract: Following the ideas in~\cite{yM88} and some inspiration from~\cite{KO24}, we construct a bialgebra $T_q(n)$ and a pointed Hopf algebra $UT_q(n)$ which quantize the coordinate rings of the algebra of upper triangular matrices and of the group of invertible upper triangular matrices of size $n\geq 2$, respectively, where $q$ is a nonzero parameter. The resulting structure on $UT_q(n)$ is neither commutative nor cocommutative, so we obtain a quantum group. The motivation comes from the idea of quantizing the incidence algebra of a finite poset, as the latter can be embedded as a subalgebra of the algebra of upper triangular matrices. After defining the bialgebra $T_q(n)$ and the Hopf algebra $UT_q(n)$, we study and compare their Lie algebras of derivations, their automorphism groups and their low degree Hochschild cohomology, in case $n=2$. + oai:arXiv.org:2512.19664v1 + math.QA + math.RA + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + \'Erica Z. Fornaroli, Mykola Khrypchenko, Samuel A. Lopes, Ednei A. Santulo Jr + + + Pivotal Module Categories, Factorization Homology and Modular Invariant Modified Traces + https://arxiv.org/abs/2512.19669 + arXiv:2512.19669v1 Announce Type: new +Abstract: The algebraic notion of a pivotal module category was developed by Schaumann and Shimizu and is central to the description of boundary conditions in conformal field theory according to a proposal by Fuchs and Schweigert. In this paper, we present a large class of examples of pivotal module categories of topological origin: For a unimodular finite ribbon category $\mathcal{A}$, we prove that the factorization homology $\int_\Sigma \mathcal{A}$ of a compact oriented surface $\Sigma$ with $n$ marked boundary intervals, at least one per connected component, comes with the structure of a pivotal module category over $\mathcal{A}^{\boxtimes n}$. This endows the internal skein algebras of Ben-Zvi-Brochier-Jordan, in particular the elliptic double, with a symmetric Frobenius structure. As application, we obtain, for each choice of $\mathcal{A}$, a family of full open conformal field theories, each of which comes with correlation functions for all surfaces with marked boundary intervals that are explicitly computable using factorization homology. As a further application, we explain how modified traces can be 'integrated' over surfaces: We show that the modified trace for $\mathcal{A}$ extends in a canonical way to the factorization homology of $\Sigma$. The resulting traces have the remarkable property of being modular invariant, i.e. fixed by the mapping class group action. + oai:arXiv.org:2512.19669v1 + math.QA + math-ph + math.CT + math.MP + math.RT + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Jorge Becerra, Lukas Woike + + + Critical percolation on the discrete torus in high dimensions + https://arxiv.org/abs/2512.19672 + arXiv:2512.19672v1 Announce Type: new +Abstract: We consider percolation on the discrete torus $\mathbb{Z}_n^d$ at $p_c(\mathbb{Z}^d)$, the critical value for percolation on the corresponding infinite lattice $\mathbb{Z}^d$, and within the scaling window around it. We assume that $d$ is a large enough constant for the nearest neighbor model, or any fixed $d>6$ for spread-out models. We prove that there exist constants $\mathbf{C},\mathbf{C}'$ depending only on the dimension and the spread-out parameter such that for any $\lambda \in \mathbb{R}$ if the edge probability is $p_c(\mathbb{Z}^d)+\mathbf{C} \lambda n^{-d/3} + o(n^{-d/3})$, then the joint distribution of the largest clusters normalized by $\mathbf{C}' n^{-2d/3}$ converges as $n\to \infty$ to the ordered lengths of excursions above past minimum of an inhomogeneous Brownian motion started at $0$ with drift $\lambda-t$ at time $t\in[0,\infty)$. This canonical limit was identified by Aldous in the context of critical Erd\H{o}s--R\'enyi graphs. + oai:arXiv.org:2512.19672v1 + math.PR + Tue, 23 Dec 2025 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Arthur Blanc-Renaudie, Asaf Nachmias + + + Topological cluster synchronization via Dirac spectral programming on directed hypergraphs + https://arxiv.org/abs/2512.14729 + arXiv:2512.14729v1 Announce Type: cross +Abstract: Collective synchronization in complex systems arises from the interplay between topology and dynamics, yet how to design and control such patterns in higher-order networks remains unclear. Here we show that a Dirac spectral programming framework enables programmable topological cluster synchronization on directed hypergraphs. By encoding tail-head hyperedges into a topological Dirac operator and introducing a tunable mass term, we obtain a spectrum whose isolated eigenvalues correspond to distinct synchronization clusters defined jointly on nodes and hyperedges. Selecting a target eigenvalue allows the system to self-organize toward the associated cluster state without modifying the underlying hypergraph structure. Simulations on directed-hypergraph block models and empirical systems--including higher-order contact networks and the ABIDE functional brain network--confirm that spectral selection alone determines the accessible synchronization patterns. Our results establish a general and interpretable route for controlling collective dynamics in directed higher-order systems. + oai:arXiv.org:2512.14729v1 + physics.soc-ph + cond-mat.stat-mech + math-ph + math.MP + nlin.AO + physics.data-an + Tue, 23 Dec 2025 00:00:00 -0500 + cross + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Yupeng Guo, Ahmed A. A. Zaid, Xueming Liu, Ginestra Bianconi + + + Characterising the sets of quantum states with non-negative Wigner function + https://arxiv.org/abs/2512.14820 + arXiv:2512.14820v2 Announce Type: cross +Abstract: For Hilbert spaces $\mathcal H\subseteq L^2(\mathbb R)$ we consider the convex sets $\mathcal D_+(\mathcal H)$ of Wigner-positive states (WPS), i.e.~density matrices over $\mathcal H$ with non-negative Wigner function. We investigate the topological structure of these sets, namely concerning closure, compactness, interior and boundary (in a relative topology induced by the trace norm). We also study their geometric structure and construct minimal sets of states that generate $\mathcal D_+(\mathcal H)$ through convex combinations. If $\mathcal H$ is finite-dimensional, the existence of such sets follows from a central result in convex analysis, namely the Krein-Milman theorem. In the infinite-dimensional case $\mathcal H=L^2(\mathbb R)$ this is not so, due to lack of compactness of the set $\mathcal D_+(\mathcal H)$. Nevertheless, we prove that a Krein-Milman theorem holds in this case, which allows us to extend most of the results concerning the sets of generators to the infinite-dimensional setting. Finally, we study the relation between the finite and infinite-dimensional sets of WPS, and prove that the former provide a hierarchy of closed subsets, which are also proper faces of the latter. These results provide a basis for an operational characterisation of the extreme points of the sets of WPS, which we undertake in a companion paper. Our work offers a unified perspective on the topological and geometric properties of the sets of WPS in finite and infinite dimensions, along with explicit constructions of minimal sets of generators. + oai:arXiv.org:2512.14820v2 + math-ph + math.FA + math.MP + quant-ph + Tue, 23 Dec 2025 00:00:00 -0500 + cross + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Nicolas J. Cerf, Ulysse Chabaud, Jack Davis, Nuno C. Dias, Jo\~ao N. Prata, Zacharie Van Herstraeten + + + On Quivers Satisfying Quantum Yang-Baxter Equation, and Hecke Condition + https://arxiv.org/abs/2512.16825 + arXiv:2512.16825v1 Announce Type: cross +Abstract: In this paper we initiate the study of quivers satisfying quantum Yang-Baxter equation. By considering quivers with quantum group like relations such as quantum Yang-Baxter equation, Hecke condition, we follow RTT construction to produce examples of Leavitt path algebras that contain quantum matrix algebra as subalgebra. Furthermore, we show that the quantum Yang-Baxter equation, and Hecke condition for our RTT construction can be generated intrinsically from the adjacency matrix of certain quivers. + oai:arXiv.org:2512.16825v1 + math.QA + math.RA + math.RT + Tue, 23 Dec 2025 00:00:00 -0500 + cross + http://creativecommons.org/licenses/by/4.0/ + Cody Gilbert, Ashish K. Srivastava + + + Efficient Beamforming Optimization for STAR-RIS-Assisted Communications: A Gradient-Based Meta Learning Approach + https://arxiv.org/abs/2512.17928 + arXiv:2512.17928v1 Announce Type: cross +Abstract: Simultaneously transmitting and reflecting reconfigurable intelligent surface (STAR-RIS) has emerged as a promising technology to realize full-space coverage and boost spectral efficiency in next-generation wireless networks. Yet, the joint design of the base station precoding matrix as well as the STAR-RIS transmission and reflection coefficient matrices leads to a high-dimensional, strongly nonconvex, and NP-hard optimization problem. Conventional alternating optimization (AO) schemes typically involve repeated large-scale matrix inversion operations, resulting in high computational complexity and poor scalability, while existing deep learning approaches often rely on expensive pre-training and large network models. In this paper, we develop a gradient-based meta learning (GML) framework that directly feeds optimization gradients into lightweight neural networks, thereby removing the need for pre-training and enabling fast adaptation. Specifically, we design dedicated GML-based schemes for both independent-phase and coupled-phase STAR-RIS models, effectively handling their respective amplitude and phase constraints while achieving weighted sum-rate performance very close to that of AO-based benchmarks. Extensive simulations demonstrate that, for both phase models, the proposed methods substantially reduce computational overhead, with complexity growing nearly linearly when the number of BS antennas and STAR-RIS elements grows, and yielding up to 10 times runtime speedup over AO, which confirms the scalability and practicality of the proposed GML method for large-scale STAR-RIS-assisted communications. + oai:arXiv.org:2512.17928v1 + eess.SP + cs.AI + cs.IT + cs.LG + math.IT + Tue, 23 Dec 2025 00:00:00 -0500 + cross + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Dongdong Yang, Bin Li, Jiguang He, Yicheng Yan, Xiaoyu Zhang, Chongwen Huang + + + Risk-Aware Financial Forecasting Enhanced by Machine Learning and Intuitionistic Fuzzy Multi-Criteria Decision-Making + https://arxiv.org/abs/2512.17936 + arXiv:2512.17936v1 Announce Type: cross +Abstract: In the face of increasing financial uncertainty and market complexity, this study presents a novel risk-aware financial forecasting framework that integrates advanced machine learning techniques with intuitionistic fuzzy multi-criteria decision-making (MCDM). Tailored to the BIST 100 index and validated through a case study of a major defense company in T\"urkiye, the framework fuses structured financial data, unstructured text data, and macroeconomic indicators to enhance predictive accuracy and robustness. It incorporates a hybrid suite of models, including extreme gradient boosting (XGBoost), long short-term memory (LSTM) network, graph neural network (GNN), to deliver probabilistic forecasts with quantified uncertainty. The empirical results demonstrate high forecasting accuracy, with a net profit mean absolute percentage error (MAPE) of 3.03% and narrow 95% confidence intervals for key financial indicators. The risk-aware analysis indicates a favorable risk-return profile, with a Sharpe ratio of 1.25 and a higher Sortino ratio of 1.80, suggesting relatively low downside volatility and robust performance under market fluctuations. Sensitivity analysis shows that the key financial indicator predictions are highly sensitive to variations of inflation, interest rates, sentiment, and exchange rates. Additionally, using an intuitionistic fuzzy MCDM approach, combining entropy weighting, evaluation based on distance from the average solution (EDAS), and the measurement of alternatives and ranking according to compromise solution (MARCOS) methods, the tabular data learning network (TabNet) outperforms the other models and is identified as the most suitable candidate for deployment. Overall, the findings of this work highlight the importance of integrating advanced machine learning, risk quantification, and fuzzy MCDM methodologies in financial forecasting, particularly in emerging markets. + oai:arXiv.org:2512.17936v1 + q-fin.ST + cs.LG + math.OC + Tue, 23 Dec 2025 00:00:00 -0500 + cross + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Safiye Turgay, Serkan Erdo\u{g}an, \v{Z}eljko Stevi\'c, Orhan Emre Elma, Tevfik Eren, Zhiyuan Wang, Mahmut Bayda\c{s} + + + Investigating Hamiltonian Dynamics by the Method of Covariant Lyapunov Vectors + https://arxiv.org/abs/2512.17962 + arXiv:2512.17962v1 Announce Type: cross +Abstract: In this thesis, we review the theory of Lyapunov exponents and covariant Lyapunov vectors (CLVs) and use these objects to numerically investigate the dynamics of several autonomous Hamiltonian systems. The algorithm which we use for computing CLVs is the one developed by Ginelli and collaborators (G&C), which is quite efficient and has been used previously in many numerical investigations. Using two low-dimensional Hamiltonian systems as toy models, we develop a method for measuring the convergence rates of vectors and subspaces computed via the G&C algorithm, and we use the time it takes for this convergence to occur to determine the appropriate transient time lengths needed when applying this algorithm to compute CLVs. The tangent dynamics of the centre subspace of the H\'enon-Heiles system is investigated numerically through the use of CLVs, and we propose a method that improves the accuracy of the centre subspace computed with the G&C algorithm. As another application of the method of CLVs to the H\'enon-Heiles system, we find that the splitting subspaces (which form a splitting of the tangent space and define the CLVs) become almost tangent during sticky regimes of motion, an observation which is related to the hyperbolicity of the system. Additionally, we investigate the dynamics of bubbles (i.e. thermal openings between base pairs) in homogeneous DNA sequences using the Peyrard-Bishop-Dauxois lattice model of DNA. For the purpose of studying short-lived bubbles in DNA, the notions of instantaneous Lyapunov vectors (ILVs) are introduced in the context of Hamiltonian dynamics. While we find that the size of the opening between base pairs has no clear relationship with the spatial distribution of the first CLV at that site, we do observe a distinct relationship with various ILV distributions. + oai:arXiv.org:2512.17962v1 + nlin.CD + math.DS + physics.comp-ph + Tue, 23 Dec 2025 00:00:00 -0500 + cross + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Jean-Jacq du Plessis + + + Convolutional-neural-operator-based transfer learning for solving PDEs + https://arxiv.org/abs/2512.17969 + arXiv:2512.17969v1 Announce Type: cross +Abstract: Convolutional neural operator is a CNN-based architecture recently proposed to enforce structure-preserving continuous-discrete equivalence and enable the genuine, alias-free learning of solution operators of PDEs. This neural operator was demonstrated to outperform for certain cases some baseline models such as DeepONet, Fourier neural operator, and Galerkin transformer in terms of surrogate accuracy. The convolutional neural operator, however, seems not to be validated for few-shot learning. We extend the model to few-shot learning scenarios by first pre-training a convolutional neural operator using a source dataset and then adjusting the parameters of the trained neural operator using only a small target dataset. We investigate three strategies for adjusting the parameters of a trained neural operator, including fine-tuning, low-rank adaption, and neuron linear transformation, and find that the neuron linear transformation strategy enjoys the highest surrogate accuracy in solving PDEs such as Kuramoto-Sivashinsky equation, Brusselator diffusion-reaction system, and Navier-Stokes equations. + oai:arXiv.org:2512.17969v1 + cs.LG + cs.AI + math-ph + math.MP + Tue, 23 Dec 2025 00:00:00 -0500 + cross + http://creativecommons.org/licenses/by-nc-nd/4.0/ + Peng Fan, Guofei Pang + + + Sampling from multimodal distributions with warm starts: Non-asymptotic bounds for the Reweighted Annealed Leap-Point Sampler + https://arxiv.org/abs/2512.17977 + arXiv:2512.17977v1 Announce Type: cross +Abstract: Sampling from multimodal distributions is a central challenge in Bayesian inference and machine learning. In light of hardness results for sampling -- classical MCMC methods, even with tempering, can suffer from exponential mixing times -- a natural question is how to leverage additional information, such as a warm start point for each mode, to enable faster mixing across modes. To address this, we introduce Reweighted ALPS (Re-ALPS), a modified version of the Annealed Leap-Point Sampler (ALPS) that dispenses with the Gaussian approximation assumption. We prove the first polynomial-time bound that works in a general setting, under a natural assumption that each component contains significant mass relative to the others when tilted towards the corresponding warm start point. Similarly to ALPS, we define distributions tilted towards a mixture centered at the warm start points, and at the coldest level, use teleportation between warm start points to enable efficient mixing across modes. In contrast to ALPS, our method does not require Hessian information at the modes, but instead estimates component partition functions via Monte Carlo. This additional estimation step is crucial in allowing the algorithm to handle target distributions with more complex geometries besides approximate Gaussian. For the proof, we show convergence results for Markov processes when only part of the stationary distribution is well-mixing and estimation for partition functions for individual components of a mixture. We numerically evaluate our algorithm's mixing performance compared to ALPS on a mixture of heavy-tailed distributions. + oai:arXiv.org:2512.17977v1 + stat.ML + cs.LG + math.PR + math.ST + stat.CO + stat.TH + Tue, 23 Dec 2025 00:00:00 -0500 + cross + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Holden Lee, Matheau Santana-Gijzen + + + An interface crack in 1d piezoelectric quasicrystal under antiplane mechanical loading and electric field + https://arxiv.org/abs/2512.17981 + arXiv:2512.17981v1 Announce Type: cross +Abstract: The present study provides the consideration of a mode III interface crack in one-dimentional (1D) piezoelectric quasicrystal under antiplane phonon and phason loading and inplane electric field. Due to complex function approach all required electromechanical parameters are presented through vector-functions analytic in the whole complex plane except the crack region. The cases of electrically impermeable (insulated) and electrically limited permeable conditions on the crack faces are considered. In the first case a vector Hilbert problem in the complex plane is formulated and solved exactly and in the second one the quadratic equation with respect to the electric flux through the crack region is obtained additionally. Its solution permits to find phonon and phason stresses, displacement jumps (sliding) and also electric characteristics along the material interface. Analytical formulas are also obtained for the corresponding stress intensity factors related to each field. The numerical computations for three selected variants of the loading conditions was conducted and the resulting field distributions are visualised on the crack continuation beyond the crack and also inside of the crack region. + oai:arXiv.org:2512.17981v1 + cond-mat.mtrl-sci + cs.NA + math.AP + math.CV + math.NA + Tue, 23 Dec 2025 00:00:00 -0500 + cross + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + 10.33108/visnyk_tntu2025.03.012 + Scientific Journal of TNTU (Tern.), vol. 119, no. 3, 2025, pp. 12-25 + Mohammed Altoumaimi, V. V. Loboda + + + Equivalent bounded confidence processes + https://arxiv.org/abs/2512.18016 + arXiv:2512.18016v1 Announce Type: cross +Abstract: In the bounded confidence model the opinions of a set of agents evolve over discrete time steps. In each round an agent averages the opinion of all agents whose opinions are at most a certain threshold apart. Here we assume that the opinions of the agents are elements of the real line. The details of the dynamics are determined by the initial opinions of the agents, i.e. a starting configuration, and the mentioned threshold -- both allowing uncountable infinite possibilities. Recently it was observed that for each starting configuration the set of thresholds can be partitioned into a finite number of intervals such that the evolution of opinions does not depend on the precise value of the threshold within one of the intervals. So, we may say that, given a starting configuration of initial opinions, there is only a finite number of equivalence classes of bounded confidence processes (and an algorithm to compute them). Here we systematically study different notions of equivalence. In our widest notion we can also get rid of the initial starting configuration and end up with a finite number of equivalent bounded confidence processes for each given (finite) number of agents. This allows to precisely study the occurring phenomena for small numbers of agents without the jeopardy of missing interesting cases by performing numerical experiments. We exemplarily study the freezing time, i.e. number of time steps needed until the process stabilizes, and the degree of fragmentation, i.e. the number of different opinions that survive once the process has reached its final state. + oai:arXiv.org:2512.18016v1 + physics.soc-ph + math.CO + Tue, 23 Dec 2025 00:00:00 -0500 + cross + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Sascha Kurz + + + Robustness of Delayed Higher Order Sliding Mode Control + https://arxiv.org/abs/2512.18018 + arXiv:2512.18018v1 Announce Type: cross +Abstract: In this paper, the feasibility of recently developed higher order delayed sliding mode controllers is addressed. With this aim the robustness against the measurement noise and mismatched perturbations for the systems governed by such controllers is established using ISS implicit Lyapunov-Razumikhin function approach. To illustrate proposed results, a simulation example validating the efficiency of the method is provided. + oai:arXiv.org:2512.18018v1 + eess.SY + cs.SY + math.OC + Tue, 23 Dec 2025 00:00:00 -0500 + cross + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Moussa Labbadi, Denis Efimov, Leonid Fridman + + + Fast Rational Search via Stern-Brocot Tree + https://arxiv.org/abs/2512.18036 + arXiv:2512.18036v1 Announce Type: cross +Abstract: We revisit the problem of rational search: given an unknown rational number $\alpha = \frac{a}{b} \in (0,1)$ with $b \leq n$, the goal is to identify $\alpha$ using comparison queries of the form ``$\beta \leq \alpha$?''. The problem has been studied several decades ago and optimal query algorithms are known. We present a new algorithm for rational search based on a compressed traversal of the Stern--Brocot tree, which appeared to have been overlooked in the literature. This approach also naturally extends to two related problems that, to the best of our knowledge, have not been previously addressed: (i) unbounded rational search, where the bound $n$ is unknown, and (ii) computing the best (in a precise sense) rational approximation of an unknown real number using only comparison queries. + oai:arXiv.org:2512.18036v1 + cs.DS + math.CO + Tue, 23 Dec 2025 00:00:00 -0500 + cross + http://creativecommons.org/licenses/by/4.0/ + Connor Weyers, N. V. Vinodchandran + + + Real 3-qubit gate decompositions via triality + https://arxiv.org/abs/2512.18049 + arXiv:2512.18049v1 Announce Type: cross +Abstract: We show that any unimodular real 3-qubit gate can be expressed as the product of at most 14 CNOT gates plus single-qubit gates, improving on the bound of 16 CNOTs due to Wei and Di. Our method uses the exotic triality symmetry of $\operatorname{PSO}(8)$, and we explore some of the useful properties of this map in relation to the study of real 3-qubit gates. + oai:arXiv.org:2512.18049v1 + quant-ph + math.GR + Tue, 23 Dec 2025 00:00:00 -0500 + cross + http://creativecommons.org/licenses/by/4.0/ + Brendan Pawlowski + + + Correcting quantum errors one gradient step at a time + https://arxiv.org/abs/2512.18061 + arXiv:2512.18061v1 Announce Type: cross +Abstract: In this work, we introduce a general, gradient-based method that optimises codewords for a given noise channel and fixed recovery. We do so by differentiating fidelity and descending on the complex coefficients using finite-difference Wirtinger gradients with soft penalties to promote orthonormalisation. We validate the gradients on symmetry checks (XXX/ZZZ repetition codes) and the $[[5, 1, 3]]$ code, then demonstrate substantial gains under isotropic Pauli noise with Petz recovery: fidelity improves from 0.783 to 0.915 in 100 steps for an isotropic Pauli noise of strength 0.05. The procedure is deterministic, highly parallelisable, and highly scalable. + oai:arXiv.org:2512.18061v1 + quant-ph + cs.NA + math.NA + Tue, 23 Dec 2025 00:00:00 -0500 + cross + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Manav Seksaria, Anil Prabhakar + + + Optimization of Si/SiGe Heterostructures for Large and Robust Valley Splitting in Silicon Qubits + https://arxiv.org/abs/2512.18064 + arXiv:2512.18064v1 Announce Type: cross +Abstract: The notoriously low and fluctuating valley splitting is one of the key challenges for electron spin qubits in silicon (Si), limiting the scalability of Si-based quantum processors. In silicon-germanium (SiGe) heterostructures, the problem can be addressed by the design of the epitaxial layer stack. Several heuristic strategies have been proposed to enhance the energy gap between the two nearly degenerate valley states in strained Si/SiGe quantum wells (QWs), e.g., sharp Si/SiGe interfaces, Ge spikes or oscillating Ge concentrations within the QW. In this work, we develop a systematic variational optimization approach to compute optimal Ge concentration profiles that boost selected properties of the intervalley coupling matrix element. Our free-shape optimization approach is augmented by a number of technological constraints to ensure feasibility of the resulting epitaxial profiles. The method is based on an effective-mass-type envelope-function theory accounting for the effects of strain and compositional alloy disorder. Various previously proposed heterostructure designs are recovered as special cases of the constrained optimization problem. Our main result is a novel heterostructure design we refer to as the "modulated wiggle well," which provides a large deterministic enhancement of the valley splitting along with a reliable suppression of the disorder-induced volatility. In addition, our new design offers a wide-range tunability of the valley splitting ranging from about 200 $\mu$eV to above 1 meV controlled by the vertical electric field, which offers new perspectives to engineer switchable qubits with on-demand adjustable valley splitting. + oai:arXiv.org:2512.18064v1 + cond-mat.mes-hall + math.OC + physics.app-ph + quant-ph + Tue, 23 Dec 2025 00:00:00 -0500 + cross + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Abel Thayil, Lasse Ermoneit, Lars R. Schreiber, Thomas Koprucki, Markus Kantner + + + Inference in partially identified moment models via regularized optimal transport + https://arxiv.org/abs/2512.18084 + arXiv:2512.18084v1 Announce Type: cross +Abstract: Partial identification often arises when the joint distribution of the data is known only up to its marginals. We consider the corresponding partially identified GMM model and develop a methodology for identification, estimation, and inference in this model. We characterize the sharp identified set for the parameter of interest via a support-function/optimal-transport (OT) representation. For estimation, we employ entropic regularization, which provides a smooth approximation to classical OT and can be computed efficiently by the Sinkhorn algorithm. We also propose a statistic for testing hypotheses and constructing confidence regions for the identified set. To derive the asymptotic distribution of this statistic, we establish a novel central limit theorem for the entropic OT value under general smooth costs. We then obtain valid critical values using the bootstrap for directionally differentiable functionals of Fang and Santos (2019). The resulting testing procedure controls size locally uniformly, including at parameter values on the boundary of the identified set. We illustrate its performance in a Monte Carlo simulation. Our methodology is applicable to a wide range of empirical settings, such as panels with attrition and refreshment samples, nonlinear treatment effects, nonparametric instrumental variables without large-support conditions, and Euler equations with repeated cross-sections. + oai:arXiv.org:2512.18084v1 + econ.EM + math.ST + stat.TH + Tue, 23 Dec 2025 00:00:00 -0500 + cross + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Grigory Franguridi, Laura Liu + + + Copula Entropy: Theory and Applications + https://arxiv.org/abs/2512.18168 + arXiv:2512.18168v1 Announce Type: cross +Abstract: This is the monograph on the theory and applications of copula entropy (CE). This book first introduces the theory of CE, including its background, definition, theorems, properties, and estimation methods. The theoretical applications of CE to structure learning, association discovery, variable selection, causal discovery, system identification, time lag estimation, domain adaptation, multivariate normality test, copula hypothesis test, two-sample test, change point detection, and symmetry test are reviewed. The relationships between the theoretical applications and their connections to correlation and causality are discussed. The framework based on CE for measuring statistical independence and conditional independence is compared to the other similar ones. The advantages of CE based methodologies over the other comparable ones are evaluated with simulations. The mathematical generalizations of CE are reviewed. The real applications of CE to every branch of science and engineering are briefly introduced. + oai:arXiv.org:2512.18168v1 + stat.ME + cs.IT + math.IT + math.PR + math.ST + stat.TH + Tue, 23 Dec 2025 00:00:00 -0500 + cross + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Jian Ma + + + Alternating Minimization for Time-Shifted Synergy Extraction in Human Hand Coordination + https://arxiv.org/abs/2512.18206 + arXiv:2512.18206v1 Announce Type: cross +Abstract: Identifying motor synergies -- coordinated hand joint patterns activated at task-dependent time shifts -- from kinematic data is central to motor control and robotics. Existing two-stage methods first extract candidate waveforms (via SVD) and then select shifted templates using sparse optimization, requiring at least two datasets and complicating data collection. We introduce an optimization-based framework that jointly learns a small set of synergies and their sparse activation coefficients. The formulation enforces group sparsity for synergy selection and element-wise sparsity for activation timing. We develop an alternating minimization method in which coefficient updates decouple across tasks and synergy updates reduce to regularized least-squares problems. Our approach requires only a single data set, and simulations show accurate velocity reconstruction with compact, interpretable synergies. + oai:arXiv.org:2512.18206v1 + cs.RO + math.OC + Tue, 23 Dec 2025 00:00:00 -0500 + cross + http://creativecommons.org/licenses/by/4.0/ + Trevor Stepp, Parthan Olikkal, Ramana Vinjamuri, Rajasekhar Anguluri + + + FedSUM Family: Efficient Federated Learning Methods under Arbitrary Client Participation + https://arxiv.org/abs/2512.18275 + arXiv:2512.18275v1 Announce Type: cross +Abstract: Federated Learning (FL) methods are often designed for specific client participation patterns, limiting their applicability in practical deployments. We introduce the FedSUM family of algorithms, which supports arbitrary client participation without additional assumptions on data heterogeneity. Our framework models participation variability with two delay metrics, the maximum delay $\tau_{\max}$ and the average delay $\tau_{\text{avg}}$. The FedSUM family comprises three variants: FedSUM-B (basic version), FedSUM (standard version), and FedSUM-CR (communication-reduced version). We provide unified convergence guarantees demonstrating the effectiveness of our approach across diverse participation patterns, thereby broadening the applicability of FL in real-world scenarios. + oai:arXiv.org:2512.18275v1 + cs.LG + math.OC + Tue, 23 Dec 2025 00:00:00 -0500 + cross + http://creativecommons.org/licenses/by-nc-sa/4.0/ + Runze You, Shi Pu + + + Stability of nonlinear Dirac solitons under the action of external potential + https://arxiv.org/abs/2512.18284 + arXiv:2512.18284v1 Announce Type: cross +Abstract: The instabilities observed in direct numerical simulations of the Gross-Neveu equation under linear and harmonic potentials are studied. The Lakoba algorithm, based on the method of characteristics, is performed to numerically obtain the two spinor components. We identify non-conservation of energy and charge in simulations with instabilities and we find that all studied solitons are numerically stable, except the low-frequency solitons oscillating in the harmonic potential over long periods of time. These instabilities, as in the case of Gross-Neveu equation without potential, can be removed by imposing absorbing boundary conditions. The dynamics of the soliton is in perfect agreement with the prediction obtained using an ansatz with only two collective coordinates, namely the position and momentum of the center of mass. We use the temporal variation of both field energy and momentum to determine the evolution equations satisfied by the collective coordinates. By applying the same methodology, we also demonstrate the spurious character of the reported instabilities in the Alexeeva-Barashenkov-Saxena model under external potentials. + oai:arXiv.org:2512.18284v1 + nlin.PS + math-ph + math.MP + Tue, 23 Dec 2025 00:00:00 -0500 + cross + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + 10.1063/5.0177392 + Chaos 34, 013140 (2024) + David Mellado-Alcedo, Niurka R. Quintero + + + Geometry of autonomous discrete Painlev\'e equations related to the Weyl group $W(E_8^{(1)})$ + https://arxiv.org/abs/2512.18288 + arXiv:2512.18288v1 Announce Type: cross +Abstract: Discrete Painlev\'e equations are integrable two-dimensional birational maps associated to a family of generalized Halphen surfaces. The latter can be seen either as $\mathbb P^2$ blown up at nine points or as $\mathbb P^1\times\mathbb P^1$ blown up at eight points. These maps become autonomous if the blow-up points are in a special position (support a pencil of cubic curves in $\mathbb P^2$, respectively a pencil of biquadratic curves in $\mathbb P^1\times\mathbb P^1$), so that the generalized Halphen surfaces become rational elliptic surfaces. In the generic case, the symmetry of a discrete Painlev\'e equation is the Weyl group $W(E_8^{(1)})$. One has a system of commuting maps which correspond to translational elements of $W(E_8^{(1)})$ associated to the roots of the lattice $E_8^{(1)}$. In the present note, we give a geometric construction of these commuting maps. For this, we use some novel birational involutions based on the above mentioned pencils of curves. + oai:arXiv.org:2512.18288v1 + nlin.SI + math-ph + math.MP + Tue, 23 Dec 2025 00:00:00 -0500 + cross + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Jaume Alonso, Yuri B. Suris + + + Robust H-infinity control under stochastic requirements: minimizing conditional value-at-risk instead of worst-case performance + https://arxiv.org/abs/2512.18356 + arXiv:2512.18356v1 Announce Type: cross +Abstract: Conventional robust $\mathcal H_2/\mathcal H_\infty$ control minimizes the worst-case performance, often leading to a conservative design driven by very rare and somewhat arbitrary parametric configurations. To reduce this conservatism while taking advantage of the stochastic properties of Monte-Carlo sampling and its compatibility with parallel computing, we introduce an alternative paradigm that optimizes the controller with respect to a stochastic criterion, namely the conditional value at risk. We illustrate the potential of this approach on a realistic satellite benchmark, showing that it can significantly improve overall performance by tolerating some degradation in very rare worst-case scenarios. + oai:arXiv.org:2512.18356v1 + eess.SY + cs.SY + math.OC + Tue, 23 Dec 2025 00:00:00 -0500 + cross + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Ervan Kassarian, Francesco Sanfedino, Daniel Alazard, Andrea Marrazza + + + Towards Guided Descent: Optimization Algorithms for Training Neural Networks At Scale + https://arxiv.org/abs/2512.18373 + arXiv:2512.18373v1 Announce Type: cross +Abstract: Neural network optimization remains one of the most consequential yet poorly understood challenges in modern AI research, where improvements in training algorithms can lead to enhanced feature learning in foundation models, order-of-magnitude reductions in training time, and improved interpretability into how networks learn. While stochastic gradient descent (SGD) and its variants have become the de facto standard for training deep networks, their success in these over-parameterized regimes often appears more empirical than principled. This thesis investigates this apparent paradox by tracing the evolution of optimization algorithms from classical first-order methods to modern higher-order techniques, revealing how principled algorithmic design can demystify the training process. Starting from first principles with SGD and adaptive gradient methods, the analysis progressively uncovers the limitations of these conventional approaches when confronted with anisotropy that is representative of real-world data. These breakdowns motivate the exploration of sophisticated alternatives rooted in curvature information: second-order approximation techniques, layer-wise preconditioning, adaptive learning rates, and more. Next, the interplay between these optimization algorithms and the broader neural network training toolkit, which includes prior and recent developments such as maximal update parametrization, learning rate schedules, and exponential moving averages, emerges as equally essential to empirical success. To bridge the gap between theoretical understanding and practical deployment, this paper offers practical prescriptions and implementation strategies for integrating these methods into modern deep learning workflows. + oai:arXiv.org:2512.18373v1 + cs.LG + math.OC + stat.ML + Tue, 23 Dec 2025 00:00:00 -0500 + cross + http://creativecommons.org/licenses/by/4.0/ + Ansh Nagwekar + + + Topological edge states in two-dimensional $\mathbb{Z}_4$ Potts paramagnet protected by the $\mathbb{Z}_4^{\times 3}$ symmetry + https://arxiv.org/abs/2512.18460 + arXiv:2512.18460v1 Announce Type: cross +Abstract: We construct a two-dimensional bosonic symmetry-protected topological (SPT) paramagnet protected by an on-site $G=\mathbb{Z}_4^{\times 3}$ symmetry, starting from a three-component $\mathbb{Z}_4$ Potts paramagnet on a triangular lattice. Within the group-cohomology framework, $H^{3}(G,U(1))\cong \mathbb{Z}_4^{\times 7}$, we focus on a "colorless" cocycle representative obtained by antisymmetrizing the basic $\mathbb{Z}_4$ three-cocycle, and generate the corresponding SPT Hamiltonian via a cocycle-induced nonlocal unitary transformation followed by symmetry averaging. For open geometry, we derive the boundary theory explicitly: one color sector decouples, while the nontrivial edge reduces to an interacting $\mathbb{Z}_4$ chain with next-to-nearest-neighbor constraints that admits a compact dressed-Potts form. Using DMRG we show that the boundary model is gapless, with the lowest gap scaling as $1/L$ and an entanglement-entropy scaling consistent with a conformal field theory of central charge $c=2.191(4)\simeq 11/5$. The rational value $c=11/5$ matches the coset $SU(3)_3/SU(2)_3$, making it a candidate for the continuum description of the $\mathbb{Z}_4^{\times 3}$ edge; we outline spectral and symmetry-resolved diagnostics needed to test this identification at the level of conformal towers beyond the central charge. + oai:arXiv.org:2512.18460v1 + cond-mat.str-el + hep-th + math-ph + math.MP + Tue, 23 Dec 2025 00:00:00 -0500 + cross + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Hrant Topchyan, Tigran Hakobyan, Mkhitar Mirumyan, Tigran A. Sedrakyan, Ara Sedrakyan + + + Generalized Birkhoff theorems and 2+2 direct pruduct spacetimes in Weyl conformal gravity + https://arxiv.org/abs/2512.18482 + arXiv:2512.18482v1 Announce Type: cross +Abstract: In this paper, we study 2+2 direct product spacetimes sourced by separated electromagnetic and Yang--Mills fields within Weyl conformal gravity. We prove that all such configurations admit at least 2 independent, commuting non-null Killing vectors, which we use to find general solutions. As a special case, we obtain a generalization of the Birkhoff--Riegert theorem to all spacetimes containing a two-dimensional subspace of constant Gaussian curvature, and we also revisit the original formulation of the theorem. We further analyze the resulting solutions in terms of Weyl equivalence classes. Their connections to known solutions in both Weyl conformal gravity and Einstein gravity are established through conformal relations. We also examine the fundamental physical and geometric properties of the newly obtained configurations and their equivalence classes. + oai:arXiv.org:2512.18482v1 + gr-qc + hep-th + math-ph + math.MP + Tue, 23 Dec 2025 00:00:00 -0500 + cross + http://creativecommons.org/licenses/by/4.0/ + Petr Jizba, Tereza Lehe\v{c}kov\'a + + + Sink Proximity: A Novel Approach for Online Vehicle Dispatch in Ride-hailing + https://arxiv.org/abs/2512.18501 + arXiv:2512.18501v1 Announce Type: cross +Abstract: Ride-hailing platforms have a profound impact on urban transportation systems, and their performance largely depends on how intelligently they dispatch vehicles in real time. In this work, we develop a new approach to online vehicle dispatch that strengthens a platform's ability to serve more requests under demand uncertainty. We introduce a novel measure called sink proximity, a network-science-inspired measure that captures how demand and vehicle flows are likely to evolve across the city. By integrating this measure into a shareability-network framework, we design an online dispatch algorithm that naturally considers future network states, without depending on fragile spatiotemporal forecasts. Numerical studies demonstrate that our proposed solution significantly improves the request service rate under peak hours within a receding horizon framework with limited future information available. + oai:arXiv.org:2512.18501v1 + eess.SY + cs.SY + math.OC + Tue, 23 Dec 2025 00:00:00 -0500 + cross + http://creativecommons.org/licenses/by-nc-nd/4.0/ + Ruiting Wang, Jiaman Wu, Fabio Paparella, Scott J. Moura, Marta C. Gonzalez + + + Scaling up Stability: Reinforcement Learning for Distributed Control of Networked Systems in the Space of Stabilizing Policies + https://arxiv.org/abs/2512.18540 + arXiv:2512.18540v1 Announce Type: cross +Abstract: We study distributed control of networked systems through reinforcement learning, where neural policies must be simultaneously scalable, expressive and stabilizing. We introduce a policy parameterization that embeds Graph Neural Networks (GNNs) into a Youla-like magnitude-direction parameterization, yielding distributed stochastic controllers that guarantee network-level closed-loop stability by design. The magnitude is implemented as a stable operator consisting of a GNN acting on disturbance feedback, while the direction is a GNN acting on local observations. We prove robustness of the closed loop to perturbations in both the graph topology and model parameters, and show how to integrate our parameterization with Proximal Policy Optimization. Experiments on a multi-agent navigation task show that policies trained on small networks transfer directly to larger ones and unseen network topologies, achieve higher returns and lower variance than a state-of-the-art MARL baseline while preserving stability. + oai:arXiv.org:2512.18540v1 + eess.SY + cs.LG + cs.SY + math.OC + Tue, 23 Dec 2025 00:00:00 -0500 + cross + http://creativecommons.org/licenses/by/4.0/ + John Cao, Luca Furieri + + + Localized wave solutions of three-component defocusing Kundu-Eckhaus equation with 4x4 matrix spectral problem + https://arxiv.org/abs/2512.18570 + arXiv:2512.18570v1 Announce Type: cross +Abstract: This work focuses on three-component defocusing Kundu-Eckhaus equation, which serves as a significant coupled model for describing complex wave propagation in nonlinear optical fibers. By employing binary Darboux transformation based on 4x4 matrix spectral problem, we derive vector dark soliton solutions, and meanwhile, the exact expressions of asymptotic dark soliton components are obtained through an asymptotic analysis method. Furthermore, breather and Y-shaped breather solutions, absent from single-component defocusing kundu-Eckhaus systems, are obtained due to the mutual coupling effects between different components. The results significantly advance our understanding nonlinear wave phenomenon induced by coupling effects and provide a theoretical reference for subsequent studies on defocusing multi-component systems. + oai:arXiv.org:2512.18570v1 + nlin.PS + math-ph + math.MP + Tue, 23 Dec 2025 00:00:00 -0500 + cross + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Yanan Wang, Min Xue + + + State-Space Modeling of Time-Varying Spillovers on Networks + https://arxiv.org/abs/2512.18584 + arXiv:2512.18584v1 Announce Type: cross +Abstract: Many modern time series arise on networks, where each component is attached to a node and interactions follow observed edges. Classical time-varying parameter VARs (TVP-VARs) treat all series symmetrically and ignore this structure, while network autoregressive models exploit a given graph but usually impose constant parameters and stationarity. We develop network state-space models in which a low-dimensional latent state controls time-varying network spillovers, own-lag persistence and nodal covariate effects. A key special case is a network time-varying parameter VAR (NTVP-VAR) that constrains each lag matrix to be a linear combination of known network operators, such as a row-normalised adjacency and the identity, and lets the associated coefficients evolve stochastically in time. The framework nests Gaussian and Poisson network autoregressions, network ARIMA models with graph differencing, and dynamic edge models driven by multivariate logistic regression. We give conditions ensuring that NTVP-VARs are well-defined in second moments despite nonstationary states, describe network versions of stability and local stationarity, and discuss shrinkage, thresholding and low-rank tensor structures for high-dimensional graphs. Conceptually, network state-space models separate where interactions may occur (the graph) from how strong they are at each time (the latent state), providing an interpretable alternative to both unstructured TVP-VARs and existing network time-series models. + oai:arXiv.org:2512.18584v1 + stat.ME + math.ST + stat.TH + Tue, 23 Dec 2025 00:00:00 -0500 + cross + http://creativecommons.org/licenses/by/4.0/ + Marios Papamichalis, Regina Ruane, Theofanis Papamichalis + + + Partition function and magnetization of two-dimensional Ising models in non-zero magnetic field: A semi-empirical approach + https://arxiv.org/abs/2512.18611 + arXiv:2512.18611v1 Announce Type: cross +Abstract: The partition functions of ferromagnetic Ising models of square lattices in a finite magnetic field is deduced using topological considerations within a heuristic graph-theoretical approach. These equations are derived separately for low and high temperature regimes while the exact solution of Onsager is obtained therefrom when the magnetic field is zero. The derived partition function equations here are almost similar to those given by Onsager, thus indicating a straight-forward protocol, even when the magnetic field is present. The spontaneous magnetization derived here using the Helmholtz free energy is identical with that arising from the exact solution. The partition functions lead to the known series expansions of the magnetization and zero-field susceptibility. + oai:arXiv.org:2512.18611v1 + cond-mat.stat-mech + math-ph + math.MP + Tue, 23 Dec 2025 00:00:00 -0500 + cross + http://creativecommons.org/licenses/by-nc-nd/4.0/ + M V Vismaya, M V Sangaranarayanan + + + Smart nudging for efficient routing through networks + https://arxiv.org/abs/2512.18630 + arXiv:2512.18630v1 Announce Type: cross +Abstract: In this paper, we formulate the design of efficient digitalised deposit return schemes as a control problem. We focus on the recycling of paper cups, though the proposed methodology applies more broadly to reverse logistics systems arising in circular economy R-strategies. Each item is assumed to carry a digital wallet through which monetary rewards are allocated to actors transferring the item across successive stages, incentivising completion of the recycling process. System efficiency is ensured by: (i) decentralised algorithms that avoid congestion at individual nodes; (ii) a decentralised AIMD-based algorithm that optimally splits the deposit across layers; and (iii) a feedback control loop that dynamically adjusts the deposit to achieve a desired throughput. The effectiveness of the framework is demonstrated through extensive simulations using realistic paper cup recycling data. + oai:arXiv.org:2512.18630v1 + eess.SY + cs.SY + econ.GN + math.OC + q-fin.EC + Tue, 23 Dec 2025 00:00:00 -0500 + cross + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Pouria M. Oqaz, Emanuele Crisostomi, Elena Dieckmann, Robert Shorten + + + A Hidden Quantum Markov model framework for Entanglement and Topological Order in the AKLT Chain + https://arxiv.org/abs/2512.18642 + arXiv:2512.18642v1 Announce Type: cross +Abstract: This paper introduces a hidden quantum Markov models (HQMMs) framework to the Affleck-Kennedy-Lieb-Tasaki (AKLT) state-a cornerstone example of a symmetry-protected topological (SPT) phase. The model's observation system is the physical spin-1 chain, which emerges from a hidden spin-1/2 layer through well-defined quantum emission operation. We show that the underlying Markov dynamics caputure maximal entanglement through the use of significant channels relevant to the AKLT state. We also show that SPT order induces a covariance on the observation decoding channels. This establishes an additional bridge between the quantum Machine learning and many-body physics, with promising implication in topological order and quantum information. + oai:arXiv.org:2512.18642v1 + quant-ph + math-ph + math.MP + Tue, 23 Dec 2025 00:00:00 -0500 + cross + http://creativecommons.org/licenses/by/4.0/ + Abdessatar Souissi, Amenallah Andolsi + + + RIS-Enabled Smart Wireless Environments: Fundamentals and Distributed Optimization + https://arxiv.org/abs/2512.18788 + arXiv:2512.18788v1 Announce Type: cross +Abstract: This chapter overviews the concept of Smart Wireless Environments (SWEs) motivated by the emerging technology of Reconfigurable Intelligent Surfaces (RISs). The operating principles and state-of-the-art hardware architectures of programmable metasurfaces are first introduced. Subsequently, key performance objectives and use cases of RIS-enabled SWEs, including spectral and energy efficiency, physical-layer security, integrated sensing and communications, as well as the emerging paradigm of over-the-air computing, are discussed. Focusing on the recent trend of Beyond-Diagonal (BD) RISs, two distributed designs of respective SWEs are presented. The first deals with a multi-user Multiple-Input Single-Output (MISO) system operating within the area of influence of a SWE comprising multiple BD-RISs. A hybrid distributed and fusion machine learning framework based on multi-branch attention-based convolutional Neural Networks (NNs), NN parameter sharing, and neuroevolutionary training is presented, which enables online mapping of channel realizations to the BD-RIS configurations as well as the multi-user transmit precoder. Performance evaluation results showcase that the distributedly optimized RIS-enabled SWE achieves near-optimal sum-rate performance with low online computational complexity. The second design focuses on the wideband interference MISO broadcast channel, where each base station exclusively controls one BD-RIS to serve its assigned group of users. A cooperative optimization framework that jointly designs the base station transmit precoders as well as the tunable capacitances and switch matrices of all metasurfaces is presented. Numerical results demonstrating the superior sum-rate performance of the designed RIS-enabled SWE for multi-cell MISO networks over benchmark schemes, considering non-cooperative configuration and conventional diagonal metasurfaces, are presented. + oai:arXiv.org:2512.18788v1 + eess.SP + cs.IT + cs.LG + math.IT + Tue, 23 Dec 2025 00:00:00 -0500 + cross + http://creativecommons.org/licenses/by-nc-nd/4.0/ + George C. Alexandropoulos, Kostantinos D. Katsanos, George Stamatelis, Ioannis Gavras + + + Families of $k$-positive maps and Schmidt number witnesses from generalized equiangular measurements + https://arxiv.org/abs/2512.18807 + arXiv:2512.18807v1 Announce Type: cross +Abstract: Quantum entanglement is an important resource in many modern technologies, like quantum computation or quantum communication and information processing. Therefore, most interest is given to detect and quantify entangled states. Entanglement degree of bipartite mixed quantum states can be quantified using the Schmidt number. Witnesses of the Schmidt numbers are closely related to $k$-positive linear maps, for which there is no general construction. Here, we use the generalized equiangular measurements to define a family of $k$-positive maps and the corresponding Schmidt number witnesses. + oai:arXiv.org:2512.18807v1 + quant-ph + math-ph + math.MP + Tue, 23 Dec 2025 00:00:00 -0500 + cross + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Katarzyna Siudzi\'nska + + + Painlev\'e Integrability And Shifted Nonlocal Reductions Of A Variable Coefficient Coupled HI Mkdv System + https://arxiv.org/abs/2512.18820 + arXiv:2512.18820v1 Announce Type: cross +Abstract: We analyze a variable coefficient coupled HI mKdV system that has shifted nonlocal reductions. The Weiss Tabor Carnevale test gives us coefficient restrictions to perform a time reparametrization to achieve an autonomous integrable model. We also show a Hirota bilinear form along with a simplified example to demonstrate how the shifted symmetries create new symmetry centers, but do not affect the shape of the soliton. + oai:arXiv.org:2512.18820v1 + nlin.SI + math-ph + math.MP + Tue, 23 Dec 2025 00:00:00 -0500 + cross + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Taylan Demir + + + A logic for default deontic reasoning + https://arxiv.org/abs/2512.18824 + arXiv:2512.18824v1 Announce Type: cross +Abstract: In many real-life settings, agents must navigate dynamic environments while reasoning under incomplete information and acting on a corpus of unstable, context-dependent, and often conflicting norms. We introduce a general, non-modal, proof-theoretic framework for deontic reasoning grounded in default logic. Its central feature is the notion of controlled sequent - a sequent annotated with sets of formulas (control sets) that prescribe what should or should not be entailed by the formulas in the antecedent. When combined with distinct extra-logical rules representing defaults and norms, these control sets record the conditions and constraints governing their applicability, thereby enabling local soundness checks for derived sequents. We prove that controlled sequent calculi satisfies admissibility of contraction and non-analytic cuts, and we establish their strong completeness with respect to credulous consequence in default theories and normative systems. Finally, we illustrate in depth how controlled sequent calculi provide a flexible and expressive basis for resolving deontic conflicts and capturing dynamic deontic notions via appropriate extra-logical rules. + oai:arXiv.org:2512.18824v1 + cs.LO + math.LO + Tue, 23 Dec 2025 00:00:00 -0500 + cross + http://creativecommons.org/licenses/by/4.0/ + Mario Piazza, Andrea Sabatini + + + Vector systems of Painlev\'e type + https://arxiv.org/abs/2512.18828 + arXiv:2512.18828v1 Announce Type: cross +Abstract: The group reduction procedure is applied to vector generalizations of the NLS, mKdV, and KdV equations. The resulting ODE systems admit isomonodromic Lax representations and are multicomponent generalizations of the Painlev\'e equations P$_1$, P$_2$, P$_{34}$, and P$_4$. Some of them can be interpreted as nonautonomous deformations of well-known systems integrable in the Liouville sense, in particular, the Garnier and H\'enon--Heiles systems. In one case, an unexpected connection with the equations of quasiperiodic dressing chain for the Schr\"odinger operator is established. + oai:arXiv.org:2512.18828v1 + nlin.SI + math-ph + math.MP + Tue, 23 Dec 2025 00:00:00 -0500 + cross + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + V. E. Adler, V. V. Sokolov + + + Generative Modeling through Spectral Analysis of Koopman Operator + https://arxiv.org/abs/2512.18837 + arXiv:2512.18837v1 Announce Type: cross +Abstract: We propose Koopman Spectral Wasserstein Gradient Descent (KSWGD), a generative modeling framework that combines operator-theoretic spectral analysis with optimal transport. The novel insight is that the spectral structure required for accelerated Wasserstein gradient descent can be directly estimated from trajectory data via Koopman operator approximation which can eliminate the need for explicit knowledge of the target potential or neural network training. We provide rigorous convergence analysis and establish connection to Feynman-Kac theory that clarifies the method's probabilistic foundation. Experiments across diverse settings, including compact manifold sampling, metastable multi-well systems, image generation, and high dimensional stochastic partial differential equation, demonstrate that KSWGD consistently achieves faster convergence than other existing methods while maintaining high sample quality. + oai:arXiv.org:2512.18837v1 + cs.LG + math.DS + Tue, 23 Dec 2025 00:00:00 -0500 + cross + http://creativecommons.org/licenses/by/4.0/ + Yuanchao Xu, Fengyi Li, Masahiro Fujisawa, Youssef Marzouk, Isao Ishikawa + + + Structure-Preserving Optimal Control of Open Quantum Systems via a Discrete Contact PMP + https://arxiv.org/abs/2512.18879 + arXiv:2512.18879v1 Announce Type: cross +Abstract: We develop a discrete Pontryagin Maximum Principle (PMP) for controlled open quantum systems governed by Lindblad dynamics, and introduce a second--order \emph{contact Lie--group variational integrator} (contact LGVI) that preserves both the CPTP (completely positive and trace--preserving) structure of the Lindblad flow and the contact geometry underlying the discrete PMP. A type--II discrete contact generating function produces a strict discrete contactomorphism under which the state, costate, and cost propagate in exact agreement with the variational structure of the discrete contact PMP. + We apply this framework to the optimal control of a dissipative qubit and compare it with a non--geometric explicit RK2 discretization of the Lindblad equation. Although both schemes have the same formal order, the RK2 method accumulates geometric drift (loss of trace, positivity violations, and breakdown of the discrete contact form) that destabilizes PMP shooting iterations, especially under strong dissipation or long horizons. In contrast, the contact LGVI maintains exact CPTP structure and discrete contact geometry step by step, yielding stable, physically consistent, and geometrically faithful optimal control trajectories. + oai:arXiv.org:2512.18879v1 + quant-ph + cs.NA + math-ph + math.MP + math.NA + math.OC + Tue, 23 Dec 2025 00:00:00 -0500 + cross + http://creativecommons.org/licenses/by/4.0/ + Leonardo Colombo + + + Characterizing Kadison--Schwarz maps on $M_3$ + https://arxiv.org/abs/2512.18900 + arXiv:2512.18900v1 Announce Type: cross +Abstract: Kadison--Schwarz (KS) maps form a natural class of positive linear maps lying between positivity and complete positivity. Despite their relevance in quantum dynamics and operator algebras, a detailed characterization of KS maps beyond low dimensions remains largely open. In this work we analyze unital linear maps on $M_3$ using the Bloch--Gell--Mann representation. Exploiting unitary equivalence and structural properties of the $\mathfrak{su}(3)$ algebra, we derive analytic conditions ensuring the Kadison--Schwarz property. Our approach clarifies the relation between KS maps and completely positive maps on $M_3$. + oai:arXiv.org:2512.18900v1 + quant-ph + math.OA + Tue, 23 Dec 2025 00:00:00 -0500 + cross + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Adam Rutkowski + + + Optical perspective on the time-dependent Dirac oscillator + https://arxiv.org/abs/2512.18904 + arXiv:2512.18904v1 Announce Type: cross +Abstract: The Dirac oscillator is a relativistic quantum system, characterized by its linearity in both position and momentum. Moreover, considering $(1{+}1)$ and $(2{+}1)$ dimensions, the system can be mapped onto the Jaynes-Cummings and anti-Jaynes-Cummings models, as illustrated in an exact manner by Bermudez \emph{et al.} [\href{ https://doi.org/10.1103/PhysRevA.76.041801}{Phys. Rev. A 76, 041801(R)}]. Using the optical counterparts of the Dirac oscillator, we analyze an extension of the model that incorporates a time-dependent frequency. We focus on the consequences of these time modulations on the angular momentum observables and spin-orbit entanglement. Noticeable changes in the \emph{Zitterbewegung} are found. We show that a specific choice of time dependence yields aperiodic evolution of the observables, whereas an alternative choice allows analytical solutions. + oai:arXiv.org:2512.18904v1 + quant-ph + math-ph + math.MP + Tue, 23 Dec 2025 00:00:00 -0500 + cross + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Thiago T. Tsutsui, Alison A. Silva, Antonio S. M. de Castro, Fabiano M. Andrade + + + Testing for latent structure via the Wilcoxon--Wigner random matrix of normalized rank statistics + https://arxiv.org/abs/2512.18924 + arXiv:2512.18924v1 Announce Type: cross +Abstract: This paper considers the problem of testing for latent structure in large symmetric data matrices. The goal here is to develop statistically principled methodology that is flexible in its applicability, computationally efficient, and insensitive to extreme data variation, thereby overcoming limitations facing existing approaches. To do so, we introduce and systematically study certain symmetric matrices, called Wilcoxon--Wigner random matrices, whose entries are normalized rank statistics derived from an underlying independent and identically distributed sample of absolutely continuous random variables. These matrices naturally arise as the matricization of one-sample problems in statistics and conceptually lie at the interface of nonparametrics, multivariate analysis, and data reduction. Among our results, we establish that the leading eigenvalue and corresponding eigenvector of Wilcoxon--Wigner random matrices admit asymptotically Gaussian fluctuations with explicit centering and scaling terms. These asymptotic results enable rigorous parameter-free and distribution-free spectral methodology for addressing two hypothesis testing problems, namely community detection and principal submatrix detection. Numerical examples illustrate the performance of the proposed approach. Throughout, our findings are juxtaposed with existing results based on the spectral properties of independent entry symmetric random matrices in signal-plus-noise data settings. + oai:arXiv.org:2512.18924v1 + stat.ME + math.PR + math.ST + stat.ML + stat.TH + Tue, 23 Dec 2025 00:00:00 -0500 + cross + http://creativecommons.org/licenses/by/4.0/ + Jonquil Z. Liao, Joshua Cape + + + Classical and Quantum Algorithms for Topological Invariants of Torus Bundles + https://arxiv.org/abs/2512.19028 + arXiv:2512.19028v1 Announce Type: cross +Abstract: Computing topological invariants of 3-manifolds is generally intractable, yet specialized algebraic structures can enable efficient algorithms. For Witten-Reshetikhin-Turaev (WRT) invariants of torus bundles, we exploit the non-commutative torus structure to embed the skein algebra of the closed torus into its symmetric subalgebra at roots of unity. This yields a fixed $N^2$-dimensional representation that supports polynomial-time classical computation with $O(N^2)$ space, and a quantum algorithm using only $O(\log N)$ qubits -- an exponential space advantage. We further prove that extracting individual expansion coefficients is #P-complete, yet there is a quantum algorithm that can efficiently approximate these coefficients for a non-negligible fraction of configurations. + oai:arXiv.org:2512.19028v1 + quant-ph + math.GT + math.QA + Tue, 23 Dec 2025 00:00:00 -0500 + cross + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Nelson Abdiel Col\'on Vargas, Carlos Ortiz Marrero + + + Efficient Personalization of Generative Models via Optimal Experimental Design + https://arxiv.org/abs/2512.19057 + arXiv:2512.19057v1 Announce Type: cross +Abstract: Preference learning from human feedback has the ability to align generative models with the needs of end-users. Human feedback is costly and time-consuming to obtain, which creates demand for data-efficient query selection methods. This work presents a novel approach that leverages optimal experimental design to ask humans the most informative preference queries, from which we can elucidate the latent reward function modeling user preferences efficiently. We formulate the problem of preference query selection as the one that maximizes the information about the underlying latent preference model. We show that this problem has a convex optimization formulation, and introduce a statistically and computationally efficient algorithm ED-PBRL that is supported by theoretical guarantees and can efficiently construct structured queries such as images or text. We empirically present the proposed framework by personalizing a text-to-image generative model to user-specific styles, showing that it requires less preference queries compared to random query selection. + oai:arXiv.org:2512.19057v1 + cs.LG + cs.IT + math.IT + Tue, 23 Dec 2025 00:00:00 -0500 + cross + http://creativecommons.org/licenses/by/4.0/ + Guy Schacht, Ziyad Sheebaelhamd, Riccardo De Santi, Mojm\'ir Mutn\'y, Andreas Krause + + + Conditioning Accept-Desirability models in the context of AGM-like belief change + https://arxiv.org/abs/2512.19096 + arXiv:2512.19096v1 Announce Type: cross +Abstract: We discuss conditionalisation for Accept-Desirability models in an abstract decision-making framework, where uncertain rewards live in a general linear space, and events are special projection operators on that linear space. This abstract setting allows us to unify classical and quantum probabilities, and extend them to an imprecise probabilities context. We introduce a new conditioning rule for our Accept-Desirability models, based on the idea that observing an event introduces new indifferences between options. We associate a belief revision operator with our conditioning rule, and investigate which of the AGM axioms for belief revision still hold in our more general framework. We investigate two interesting special cases where all of these axioms are shown to still hold: classical propositional logic and full conditional probabilities. + oai:arXiv.org:2512.19096v1 + cs.AI + math.LO + math.PR + Tue, 23 Dec 2025 00:00:00 -0500 + cross + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Kathelijne Coussement, Gert de Cooman, Keano De Vos + + + Variation of entropy in the Duffing system with the amplitude of the external force + https://arxiv.org/abs/2512.19119 + arXiv:2512.19119v1 Announce Type: cross +Abstract: In this paper, we revisit the well-known perturbed Duffing system and investigate its chaotic dynamics by means of numerical Runge--Kutta method based on topological horseshoe theory. Precisely, we investigate chaos through the topological horseshoes associated with the first, second, and third return maps, obtained by varying the amplitude of an external force term while keeping all other parameters fixed. Our new finding demonstrates that, when the force amplitude exceeds a certain value, the topological (Smale) horseshoe degenerates into a pseudo-horseshoe, while chaotic invariant set persists. This phenomenon indicates that the lower bound of the topological entropy decreases as the force amplitude increases, thereby enriching the dynamics in the perturbed Duffing system. + Furthermore, we identify a critical value of the force amplitude governing the attractivity of the chaotic invariant set. For amplitudes slightly below this value, the basin of attraction of the chaotic invariant set progressively shrinks as the amplitude increases. In contrast, for larger amplitudes, both Lyapunov exponents become negative while the topological horseshoe persists, suggesting that the chaotic invariant set loses attractivity as the amplitude grows. + oai:arXiv.org:2512.19119v1 + nlin.CD + math.DS + Tue, 23 Dec 2025 00:00:00 -0500 + cross + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Junfeng Cheng, Xiao-Song Yang + + + A Convex Loss Function for Set Prediction with Optimal Trade-offs Between Size and Conditional Coverage + https://arxiv.org/abs/2512.19142 + arXiv:2512.19142v1 Announce Type: cross +Abstract: We consider supervised learning problems in which set predictions provide explicit uncertainty estimates. Using Choquet integrals (a.k.a. Lov{\'a}sz extensions), we propose a convex loss function for nondecreasing subset-valued functions obtained as level sets of a real-valued function. This loss function allows optimal trade-offs between conditional probabilistic coverage and the ''size'' of the set, measured by a non-decreasing submodular function. We also propose several extensions that mimic loss functions and criteria for binary classification with asymmetric losses, and show how to naturally obtain sets with optimized conditional coverage. We derive efficient optimization algorithms, either based on stochastic gradient descent or reweighted least-squares formulations, and illustrate our findings with a series of experiments on synthetic datasets for classification and regression tasks, showing improvements over approaches that aim for marginal coverage. + oai:arXiv.org:2512.19142v1 + cs.LG + math.OC + stat.ML + Tue, 23 Dec 2025 00:00:00 -0500 + cross + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Francis Bach (SIERRA) + + + Rapid stabilization of the heat equation with localized disturbance + https://arxiv.org/abs/2512.19160 + arXiv:2512.19160v1 Announce Type: cross +Abstract: This paper studies the rapid stabilization of a multidimensional heat equation in the presence of an unknown spatially localized disturbance. A novel multivalued feedback control strategy is proposed, which synthesizes the frequency Lyapunov method (introduced by Xiang [41]) with the sign multivalued operator. This methodology connects Lyapunov-based stability analysis with spectral inequalities, while the inclusion of the sign operator ensures robustness against the disturbance. The closed-loop system is governed by a differential inclusion, for which well-posedness is proved via the theory of maximal monotone operators. This approach not only guarantees exponential stabilization but also circumvents the need for explicit disturbance modeling or estimation. + oai:arXiv.org:2512.19160v1 + eess.SY + cs.SY + math.AP + math.OC + Tue, 23 Dec 2025 00:00:00 -0500 + cross + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Patricio Guzm\'an (SPHINX, IECL), Hugo Parada (SPHINX, IECL), Christian Calle-C\'ardenas + + + Self-Consistent Probability Flow for High-Dimensional Fokker-Planck Equations + https://arxiv.org/abs/2512.19196 + arXiv:2512.19196v1 Announce Type: cross +Abstract: Solving high-dimensional Fokker-Planck (FP) equations is a challenge in computational physics and stochastic dynamics, due to the curse of dimensionality (CoD) and the bottleneck of evaluating second-order diffusion terms. Existing deep learning approaches, such as Physics-Informed Neural Networks (PINNs), face computational challenges as dimensionality increases, driven by the $O(D^2)$ complexity of automatic differentiation for second-order derivatives. While recent probability flow approaches bypass this by learning score functions or matching velocity fields, they often involve serial computational operations or depend on sampling efficiency in complex distributions. To address these issues, we propose the Self-Consistent Probability Flow (SCPF) method. We reformulate the second-order FP equation into an equivalent first-order deterministic Probability Flow ODE (PF-ODE) constraint. Unlike score matching or velocity matching, SCPF solves this problem by minimizing the residual of the PF-ODE continuity equation, which avoids explicit Hessian computation. We leverage Continuous Normalizing Flows (CNF) combined with the Hutchinson Trace Estimator (HTE) to reduce the training complexity to linear scale $O(D)$, achieving an effective $O(1)$ wall-clock time on GPUs. To address data sparsity in high dimensions, we apply a generative adaptive sampling strategy and theoretically prove that dynamically aligning collocation points with the evolving probability mass is a necessary condition to bound the approximation error. Experiments on diverse benchmarks -- ranging from anisotropic Ornstein-Uhlenbeck (OU) processes and high-dimensional Brownian motions with time-varying diffusion terms, to Geometric OU processes featuring non-Gaussian solutions -- demonstrate that SCPF effectively mitigates the CoD, maintaining high accuracy and constant computational cost for problems up to 100 dimensions. + oai:arXiv.org:2512.19196v1 + physics.comp-ph + cs.LG + cs.NA + math.NA + Tue, 23 Dec 2025 00:00:00 -0500 + cross + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Xiaolong Wu, Qifeng Liao + + + Sprecher Networks: A Parameter-Efficient Kolmogorov-Arnold Architecture + https://arxiv.org/abs/2512.19367 + arXiv:2512.19367v1 Announce Type: cross +Abstract: We present Sprecher Networks (SNs), a family of trainable neural architectures inspired by the classical Kolmogorov-Arnold-Sprecher (KAS) construction for approximating multivariate continuous functions. Distinct from Multi-Layer Perceptrons (MLPs) with fixed node activations and Kolmogorov-Arnold Networks (KANs) featuring learnable edge activations, SNs utilize shared, learnable splines (monotonic and general) within structured blocks incorporating explicit shift parameters and mixing weights. Our approach directly realizes Sprecher's specific 1965 sum of shifted splines formula in its single-layer variant and extends it to deeper, multi-layer compositions. We further enhance the architecture with optional lateral mixing connections that enable intra-block communication between output dimensions, providing a parameter-efficient alternative to full attention mechanisms. Beyond parameter efficiency with $O(LN + LG)$ scaling (where $G$ is the knot count of the shared splines) versus MLPs' $O(LN^2)$, SNs admit a sequential evaluation strategy that reduces peak forward-intermediate memory from $O(N^2)$ to $O(N)$ (treating batch size as constant), making much wider architectures feasible under memory constraints. We demonstrate empirically that composing these blocks into deep networks leads to highly parameter and memory-efficient models, discuss theoretical motivations, and compare SNs with related architectures (MLPs, KANs, and networks with learnable node activations). + oai:arXiv.org:2512.19367v1 + cs.LG + cs.AI + cs.NA + math.NA + Tue, 23 Dec 2025 00:00:00 -0500 + cross + http://creativecommons.org/licenses/by/4.0/ + Christian H\"agg, Kathl\'en Kohn, Giovanni Luca Marchetti, Boris Shapiro + + + Attention Is Not What You Need + https://arxiv.org/abs/2512.19428 + arXiv:2512.19428v1 Announce Type: cross +Abstract: We revisit a basic question in sequence modeling: is explicit self-attention actually necessary for strong performance and reasoning? We argue that standard multi-head attention is best seen as a form of tensor lifting: hidden vectors are mapped into a high-dimensional space of pairwise interactions, and learning proceeds by constraining this lifted tensor through gradient descent. This mechanism is extremely expressive but mathematically opaque, because after many layers it becomes very hard to describe the model with a small family of explicit invariants. + To explore an alternative, we propose an attention-free architecture based on Grassmann flows. Instead of forming an L by L attention matrix, our Causal Grassmann layer (i) linearly reduces token states, (ii) encodes local token pairs as two-dimensional subspaces on a Grassmann manifold via Plucker coordinates, and (iii) fuses these geometric features back into the hidden states through gated mixing. Information therefore propagates by controlled deformations of low-rank subspaces over multi-scale local windows, so the core computation lives on a finite-dimensional manifold rather than in an unstructured tensor space. + On the Wikitext-2 language modeling benchmark, purely Grassmann-based models with 13 to 18 million parameters achieve validation perplexities within about 10 to 15 percent of size-matched Transformers. On the SNLI natural language inference task, a Grassmann-Plucker head on top of DistilBERT slightly outperforms a Transformer head, with best validation and test accuracies of 0.8550 and 0.8538 compared to 0.8545 and 0.8511. We analyze the complexity of Grassmann mixing, show linear scaling in sequence length for fixed rank, and argue that such manifold-based designs offer a more structured route toward geometric and invariant-based interpretations of neural reasoning. + oai:arXiv.org:2512.19428v1 + cs.LG + cs.AI + math.AG + Tue, 23 Dec 2025 00:00:00 -0500 + cross + http://creativecommons.org/licenses/by/4.0/ + Zhang Chong + + + Binary Kernel Logistic Regression: a sparsity-inducing formulation and a convergent decomposition training algorithm + https://arxiv.org/abs/2512.19440 + arXiv:2512.19440v1 Announce Type: cross +Abstract: Kernel logistic regression (KLR) is a widely used supervised learning method for binary and multi-class classification, which provides estimates of the conditional probabilities of class membership for the data points. Unlike other kernel methods such as Support Vector Machines (SVMs), KLRs are generally not sparse. Previous attempts to deal with sparsity in KLR include a heuristic method referred to as the Import Vector Machine (IVM) and ad hoc regularizations such as the $\ell_{1/2}$-based one. Achieving a good trade-off between prediction accuracy and sparsity is still a challenging issue with a potential significant impact from the application point of view. In this work, we revisit binary KLR and propose an extension of the training formulation proposed by Keerthi et al., which is able to induce sparsity in the trained model, while maintaining good testing accuracy. To efficiently solve the dual of this formulation, we devise a decomposition algorithm of Sequential Minimal Optimization type which exploits second-order information, and for which we establish global convergence. Numerical experiments conducted on 12 datasets from the literature show that the proposed binary KLR approach achieves a competitive trade-off between accuracy and sparsity with respect to IVM, $\ell_{1/2}$-based regularization for KLR, and SVM while retaining the advantages of providing informative estimates of the class membership probabilities. + oai:arXiv.org:2512.19440v1 + cs.LG + math.OC + Tue, 23 Dec 2025 00:00:00 -0500 + cross + http://creativecommons.org/licenses/by-nc-nd/4.0/ + Antonio Consolo, Andrea Manno, Edoardo Amaldi + + + A Mathematical Framework for Misinformation Propagation in Complex Networks: Topology-Dependent Distortion and Control + https://arxiv.org/abs/2512.19465 + arXiv:2512.19465v1 Announce Type: cross +Abstract: Misinformation is pervasive in natural, biological, social, and engineered systems, yet its quantitative characterization remains challenging.We develop a general mathematical framework for quantifying information distortion in distributed systems by modeling how local transmission errors accumulate along network geodesics and reshape each agent's perceived global state. Through a drift-fluctuation decomposition of pathwise binomial noise, we derive closed-form expressions for node-level perception distributions and show that directional bias induces only a uniform shift in the mean, preserving the fluctuation structure. Applying the framework to canonical graph ensembles, we uncover strong topological signatures of misinformation: Erd\H{o}s--R\'enyi random graphs exhibit a double-peaked distortion profile driven by connectivity transitions and geodesic-length fluctuations, scale-free networks suppress misinformation through hub-mediated integration, and optimally rewired small-world networks achieve comparable suppression by balancing clustering with short paths. A direct comparison across regular lattices, Erd\H{o}s--R\'enyi random graphs, Watts--Strogatz small-world networks, and Barab\'asi--Albert scale-free networks reveals a connectivity-dependent crossover. In the extremely sparse regime, scale-free and Erd\H{o}s--R\'enyi networks behave similarly. At intermediate sparsity, Watts--Strogatz small-world networks exhibit the lowest misinformation. In contrast, Barab\'asi--Albert scale-free networks maintain low misinformation in sparse and dense regimes, while regular lattices produce the highest distortion across connectivities. We additionally show how sparsity constraints, structural organization, and connection costs delineate regimes of minimal misinformation. + oai:arXiv.org:2512.19465v1 + physics.soc-ph + math.PR + Tue, 23 Dec 2025 00:00:00 -0500 + cross + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Saikat Sur, Rohitashwa Chattopadhyay, Jens Christian Claussen, Archan Mukhopadhyay + + + Fare Zone Assignment + https://arxiv.org/abs/2512.19493 + arXiv:2512.19493v1 Announce Type: cross +Abstract: Tariff setting in public transportation networks is an important challenge. A popular approach is to partition the network into fare zones ("zoning") and fix journey prices depending on the number of traversed zones ("pricing"). In this paper, we focus on finding revenue-optimal solutions to the zoning problem for a given concave pricing function. We consider tree networks with $n$ vertices, since trees already pose non-trivial algorithmic challenges. Our main results are efficient algorithms that yield a simple $\mathcal{O}(\log n)$-approximation as well as a more involved $\mathcal{O}(\log n/\log \log n)$-approximation. We show how to solve the problem exactly on rooted instances, in which all demand arises at the same source. For paths, we prove strong NP-hardness and outline a PTAS. Moreover, we show that computing an optimal solution is in FPT or XP for several natural problem parameters. + oai:arXiv.org:2512.19493v1 + cs.DS + cs.GT + math.OC + Tue, 23 Dec 2025 00:00:00 -0500 + cross + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Martin Hoefer, Lennart Kauther, Philipp Pabst, Britta Peis, Khai Van Tran + + + Initialization of a Polyharmonic Cascade, Launch and Testing + https://arxiv.org/abs/2512.19524 + arXiv:2512.19524v1 Announce Type: cross +Abstract: This paper concludes a series of studies on the polyharmonic cascade, a deep machine learning architecture theoretically derived from indifference principles and the theory of random functions. A universal initialization procedure is proposed, based on symmetric constellations in the form of hyperoctahedra with a central point. This initialization not only ensures stable training of cascades with tens and hundreds of layers (up to 500 layers without skip connections), but also radically simplifies the computations. Scalability and robustness are demonstrated on MNIST (98.3% without convolutions or augmentations), HIGGS (AUC approximately 0.885 on 11M examples), and Epsilon (AUC approximately 0.963 with 2000 features). All linear algebra is reduced to 2D operations and is efficiently executed on GPUs. A public repository and an archived snapshot are provided for full reproducibility. + oai:arXiv.org:2512.19524v1 + cs.LG + cs.NA + math.NA + Tue, 23 Dec 2025 00:00:00 -0500 + cross + http://creativecommons.org/licenses/by/4.0/ + Yuriy N. Bakhvalov + + + Universal BPS Structure of Scalar Kinks in Static Geometries + https://arxiv.org/abs/2512.19574 + arXiv:2512.19574v1 Announce Type: cross +Abstract: We present a geometric extension of the Bogomolny-Prasad-Sommerfield (BPS) construction for scalar kinks in (1+1) dimensions embedded in static curved spacetimes. By introducing a nonminimal coupling between the scalar prepotential and the extrinsic curvature of the static foliation, the flat-space first-order Bogomolny equation remains exactly valid for arbitrary static backgrounds. As a consequence, the kink profile is unchanged, while the effective potential and vacuum structure acquire a controlled geometric dependence. We show that these curved-space BPS kinks are always linearly stable. However, the existence of the translational zero mode is not guaranteed: its normalizability depends on the competition between the intrinsic length scale of the kink and the asymptotic curvature scale of the geometry. When the geometric scale dominates, the zero mode is removed and the soliton becomes geometrically pinned, despite remaining an exact BPS solution. Explicit realizations in AdS2 demonstrate how different static slicings of the same spacetime lead to qualitatively distinct physical outcomes, ranging from preserved translational invariance to its complete removal by horizons. These results establish geometry as a precise mechanism for controlling solitonic moduli without compromising linear stability. + oai:arXiv.org:2512.19574v1 + hep-th + gr-qc + math-ph + math.MP + Tue, 23 Dec 2025 00:00:00 -0500 + cross + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + G. Luchini, G. B. Sant'Anna, U. Camara da Silva + + + Towards a universal phase diagram of planar chiral magnets + https://arxiv.org/abs/2512.19590 + arXiv:2512.19590v1 Announce Type: cross +Abstract: In planar chiral magnets, the competition of the positive definite Heisenberg exchange and Zeeman energies with the indefinite Dzyaloshinskii-Moriya interaction (DMI) energy allows for the possibility of negative energy ground states, and leads to an intricate dependence of the ground states on the parameters of the theory. In this paper, we consider arbitrary spiralization tensors for the DMI interaction and arbitrary directions for the external magnetic field, and study the nature of the ground states in this parameter space, using a combination of analytical and numerical methods. Classifying ground states by their symmetry into ferromagnetic (invariant under under arbitrary translations in the plane), spiral (invariant under arbitrary translations in one direction) and skyrmion lattice ground states (invariant under a two dimensional lattice group), we give a complete description of the phase diagram of this class of theories. + oai:arXiv.org:2512.19590v1 + cond-mat.str-el + hep-th + math-ph + math.MP + Tue, 23 Dec 2025 00:00:00 -0500 + cross + http://creativecommons.org/licenses/by/4.0/ + Bernd Schroers, Martin Speight, Thomas Winyard + + + Topological Flux on a Context Manifold Generates Nonreciprocal Collective Dynamics + https://arxiv.org/abs/2512.19598 + arXiv:2512.19598v1 Announce Type: cross +Abstract: Non-reciprocal interactions, where the influence of agent $i$ on $j$ differs from that of $j$ on $i$, are fundamental in active and living matter. Yet, most models implement such asymmetry phenomenologically. Here we show that non-reciprocity can emerge from internal topology alone. Agents evolve on an internal ``context manifold'' coupled to a Chern-Simons gauge field. Because the gauge field is first order in time, it relaxes rapidly; eliminating it yields an effective transverse, antisymmetric interaction kernel that generically produces chiral waves, persistent vorticity, and irreversible state transitions. Numerical simulations reveal clear signatures of broken reciprocity: long-lived vortex cores, finite circulation, asymmetric information flow, and a nonzero reciprocity residual. The dynamics further exhibit pronounced hysteresis under parameter sweeps, demonstrating memory effects that cannot occur in reciprocal or potential-driven systems. These results identify Chern-Simons gauge fields as a minimal and universal source of directional influence and robust non-reciprocal collective behavior. + oai:arXiv.org:2512.19598v1 + nlin.AO + math-ph + math.MP + physics.bio-ph + physics.soc-ph + Tue, 23 Dec 2025 00:00:00 -0500 + cross + http://creativecommons.org/licenses/by/4.0/ + Jyotiranjan Beuria, Venkatesh H. Chembrolu + + + Higher lattice gauge theory from representations of 2-groups and 3+1D topological phases + https://arxiv.org/abs/2512.19608 + arXiv:2512.19608v1 Announce Type: cross +Abstract: We construct a higher lattice gauge theory based on the representation of 2-groups described by a category of crossed modules on a lattice model described by path 2-groupoids. Using these lattice gauge representations, an exactly solvable Hamiltonian for topological phases in 3+1 dimensions is constructed. We show that the ground states of this model are topological observables. + oai:arXiv.org:2512.19608v1 + hep-lat + cond-mat.other + hep-th + math-ph + math.MP + quant-ph + Tue, 23 Dec 2025 00:00:00 -0500 + cross + http://creativecommons.org/licenses/by-nc-nd/4.0/ + Lat\'evi M. Lawson, Prince K. Osei + + + Non-invertible defects from the Conway SCFT to K3 sigma models II: duality and Fibonacci defects + https://arxiv.org/abs/2512.19640 + arXiv:2512.19640v1 Announce Type: cross +Abstract: We continue the study, initiated in [hep-th:2504.18619], of topological defect lines (TDLs) in the Conway module $V^{f \natural}$ and K3 non-linear sigma models (NLSMs). In the case of $V^{f \natural}$, we fully classify the potential $N=1$ (and $N=4$)--preserving duality defects for cyclic Tambara--Yamagami categories TY$(\mathbb{Z}_N)$, noting a curious relation to genus zero groups of monstrous moonshine. We use the correspondence with Leech lattice endomorphisms, discovered in [hep-th:2504.18619], to construct a number of non-trivial examples of TDLs in $V^{f \natural}$, including examples of irrational quantum dimension. In particular, we fully classify and construct defects for the TY$(\mathbb{Z}_2)$ and TY$(\mathbb{Z}_3)$ cases, and provide examples of duality defects for TY$(\mathbb{Z}_2\times \mathbb{Z}_2)$ and Fibonacci fusion categories as well. In the case of K3 NLSMs, we describe a duality defect of irrational quantum dimension $\sqrt{2}$ for the category TY$(\mathbb{Z}_2, -1)$ in a particular torus orbifold, which exists on a 16-dimensional slice of the moduli space. We also provide a detailed analysis of spectral flow--preserving TDLs in Gepner models of K3, of independent interest, and use this to construct non-invertible defects for Fibonacci and $Rep(S_3)$ categories in particular examples. Finally we provide evidence for our conjecture in [hep-th:2504.18619] that special subcategories of such TDLs in $V^{f \natural}$ correspond to $N=(4,4)$ and spectral flow--preserving defect lines in a corresponding K3 NLSM. In particular, we compute defect--twined elliptic genera for all non-invertible defects constructed in this article, demonstrating that for each defect found in a K3 NLSM, there is a corresponding defect in $V^{f \natural}$ with coincident twining genus, and making a prediction for a number of TDLs in K3 NLSMs yet to be found. + oai:arXiv.org:2512.19640v1 + hep-th + math.QA + Tue, 23 Dec 2025 00:00:00 -0500 + cross + http://creativecommons.org/licenses/by/4.0/ + Roberta Angius, Stefano Giaccari, Sarah M. Harrison, Roberto Volpato + + + Escape from heterogeneous diffusion + https://arxiv.org/abs/2512.19646 + arXiv:2512.19646v1 Announce Type: cross +Abstract: Many physical processes depend on the time it takes a diffusing particle to find a target. Though this classical quantity is now well-understood in various scenarios, little is known if the diffusivity depends on the location of the particle. For such heterogeneous diffusion, an ambiguity arises in interpreting the stochastic process, which reflects the well-known It\^{o} versus Stratonovich controversy. Here we analytically determine the mean escape time and splitting probabilities for an arbitrary heterogeneous diffusion in an arbitrary three-dimensional domain with small targets that can be perfectly or imperfectly absorbing. Our analysis reveals general principles for how search depends on heterogeneous diffusion and its interpretation (e.g. It\^{o}, Stratonovich, or kinetic). An intricate picture emerges in which, for instance, increasing the diffusivity can decrease, not affect, or even increase the escape time. Our results could be used to determine the appropriate interpretation for specific physical systems. + oai:arXiv.org:2512.19646v1 + cond-mat.stat-mech + math.AP + math.PR + Tue, 23 Dec 2025 00:00:00 -0500 + cross + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Hwai-Ray Tung, Sean D Lawley + + + Deep Legendre Transform + https://arxiv.org/abs/2512.19649 + arXiv:2512.19649v1 Announce Type: cross +Abstract: We introduce a novel deep learning algorithm for computing convex conjugates of differentiable convex functions, a fundamental operation in convex analysis with various applications in different fields such as optimization, control theory, physics and economics. While traditional numerical methods suffer from the curse of dimensionality and become computationally intractable in high dimensions, more recent neural network-based approaches scale better, but have mostly been studied with the aim of solving optimal transport problems and require the solution of complicated optimization or max-min problems. Using an implicit Fenchel formulation of convex conjugation, our approach facilitates an efficient gradient-based framework for the minimization of approximation errors and, as a byproduct, also provides a posteriori error estimates for the approximation quality. Numerical experiments demonstrate our method's ability to deliver accurate results across different high-dimensional examples. Moreover, by employing symbolic regression with Kolmogorov--Arnold networks, it is able to obtain the exact convex conjugates of specific convex functions. + oai:arXiv.org:2512.19649v1 + cs.LG + math.OC + Tue, 23 Dec 2025 00:00:00 -0500 + cross + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Aleksey Minabutdinov, Patrick Cheridito + + + Finite-gap potentials as a semiclassical limit of the thermodynamic Bethe Ansatz + https://arxiv.org/abs/2512.19655 + arXiv:2512.19655v1 Announce Type: cross +Abstract: We show that the semiclassical limit of thermodynamic Bethe Ansatz equations naturally reconstructs the algebro-geometric spectra of finite-gap periodic potentials. This correspondence is illustrated using the traveling-wave (snoidal) solution of the defocusing modified Korteweg--de Vries equation. In this framework, the Bethe-root distribution of the associated quantum field theory yields an Abelian differential of the second kind on the elliptic Riemann surface specified by the spectral endpoints, a structure central to the algebro-geometric theory of solitons. The semiclassical parameter is identified with the large-rank limit of the internal symmetry group ($O(2N)$) of the underlying quantum field theory (the Gross-Neveu model with a chemical potential). Our analysis indicates that the analytic structure of the spectrum is dictated solely by the Dynkin diagram ($D_N$) and its large-rank limit ($D_\infty$), independently of the particular integrable model used to realize it. + oai:arXiv.org:2512.19655v1 + hep-th + cond-mat.str-el + math-ph + math.MP + Tue, 23 Dec 2025 00:00:00 -0500 + cross + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Valdemar Melin, Paul Wiegmann, Konstantin Zarembo + + + Extremizing Measures of Magic on Pure States by Clifford-stabilizer States + https://arxiv.org/abs/2512.19657 + arXiv:2512.19657v1 Announce Type: cross +Abstract: Magic states are essential resources enabling universal, fault-tolerant quantum computation within the stabilizer framework. Their non-stabilizerness provides the additional resource required to overcome the constraints of stabilizer codes, as formalized by the Eastin-Knill theorem, while still permitting fault-tolerant distillation. Although numerous measures of magic have been introduced, not every state with nonzero magic has been shown to be distillable by a stabilizer code, and all currently known distillable states arise as special cases of Clifford-stabilizer states, defined as pure states uniquely stabilized by finite subgroups of the Clifford group. In this work, we develop a general framework for group-covariant functionals on the real manifold of Hermitian operators. We formalize the notions of $G$-stabilizer spaces, states, and codes for arbitrary finite subgroups $G \subset \mathrm{U}(\mathcal{H})$, and introduce analytic families of $G$-covariant functionals. Our main theorem shows that any $G$-invariant pure state is an extremal point of a broad class of derived functionals, including symmetric, max-type, and R\'enyi-type functionals, provided the underlying family is $G$-covariant. This extremality holds for variations restricted to directions orthogonal to the stabilized subspace while preserving purity. Specializing to the Pauli and Clifford groups, our framework unifies the extremality structure of several canonical magic measures, including mana, stabilizer R\'enyi entropies, and stabilizer fidelity. In particular, Clifford-stabilizer states extremize these measures. We classify such states for qubits, qutrits, ququints, and two-qubit systems, identifying new candidates for magic distillation protocols. We further propose an inefficient distillation protocol for a two-qubit magic state with stabilizer fidelity exceeding that of standard benchmark states. + oai:arXiv.org:2512.19657v1 + quant-ph + math-ph + math.MP + Tue, 23 Dec 2025 00:00:00 -0500 + cross + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Muhammad Erew, Moshe Goldstein + + + The modal logic of arithmetic potentialism and the universal algorithm + https://arxiv.org/abs/1801.04599 + arXiv:1801.04599v5 Announce Type: replace +Abstract: I investigate the modal commitments of various conceptions of the philosophy of arithmetic potentialism. Specifically, I shall consider the potentialist conceptions arising from a model-theoretic view of the models of arithmetic as possible arithmetic realms of feasibility, considering them under their natural extension concepts, such as end-extensions, arbitrary extensions, conservative extensions and more, which in effect express distinct potentialist ideas. In these potentialist systems, I show, the propositional modal assertions that are valid with respect to all arithmetic assertions with parameters are exactly the assertions of S4. With respect to sentences, however, the validities of a model lie between S4 and S5, and these bounds are sharp in that there are models realizing both endpoints. For a model of arithmetic to validate S5 is precisely to fulfill the arithmetic maximality principle, which asserts that every possibly necessary statement is already true, and these models are equivalently characterized as those satisfying a maximal $\Sigma_1$ theory. The main S4 analysis makes fundamental use of the universal algorithm, of which this article provides a simplified, self-contained account. The main philosophical point is that fundamentally different potentialist conceptions -- linear inevitability, convergent potentialism and radical branching possibility -- are revealed by the precise modal validities of the corresponding potentialist systems in which those attitudes are expressed, and so it is important to discover them. + oai:arXiv.org:1801.04599v5 + math.LO + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Joel David Hamkins + + + Non-convex Mather's theory and the Conley conjecture on the cotangent bundle of the torus + https://arxiv.org/abs/1807.09461 + arXiv:1807.09461v3 Announce Type: replace +Abstract: The aim of this paper is to use the methods and results of symplectic homogenization (see [V4]) to prove existence of periodic orbits and invariant measures with rotation number depending on the differential of the Homogenized Hamiltonian. We also prove the Conley conjecture on the cotangent bundle of the torus. Both proofs rely on Symplectic Homogenization and a refinement of it. + oai:arXiv.org:1807.09461v3 + math.DS + math.SG + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Claude Viterbo (Laboratoire de Math\'ematiques d'Orsay, Universit\'e de Paris-Saclay, Orsay, FRANCE) + + + Equivariant discretizations of diffusions, random walks, and harmonic functions + https://arxiv.org/abs/1906.11716 + arXiv:1906.11716v3 Announce Type: replace +Abstract: For covering spaces and properly discontinuous actions with compatible diffusion processes, we discuss Lyons-Sullivan discretizations of the processes and the associated function theory. + oai:arXiv.org:1906.11716v3 + math.DG + math.PR + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Werner Ballmann, Panagiotis Polymerakis + + + A cubical model for $(\infty, n)$-categories + https://arxiv.org/abs/2005.07603 + arXiv:2005.07603v3 Announce Type: replace +Abstract: We propose a new model for the theory of $(\infty,n)$-categories (including the case $n=\infty$) in the category of marked cubical sets with connections, similar in flavor to complicial sets of Verity. The model structure characterizing our model is shown to be monoidal with respect to suitably defined (lax and pseudo) Gray tensor products; in particular, these tensor products are both associative and biclosed. Furthermore, we show that the triangulation functor to pre-complicial sets is a left Quillen functor and is strong monoidal with respect to both Gray tensor products. + oai:arXiv.org:2005.07603v3 + math.AT + math.CT + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + 10.2140/gt.2025.29.1115 + Geom. Topol. 29 (2025) 1115-1170 + Tim Campion, Chris Kapulkin, Yuki Maehara + + + The Uniform Boundedness and Dynamical Lang Conjectures for polynomials + https://arxiv.org/abs/2105.05240 + arXiv:2105.05240v3 Announce Type: replace +Abstract: We give a conditional proof of the Uniform Boundedness Conjecture of Morton and Silverman in the case of polynomials over number fields, assuming a standard conjecture in arithmetic geometry. Our technique simultaneously yields a dynamical analogue of Lang's conjecture on minimal canonical heights for these maps. We obtain similar results for non-isotrivial polynomials over a function field of characteristic zero. When the latter are unicritical of degree at least 5, the results hold unconditionally. + oai:arXiv.org:2105.05240v3 + math.NT + math.DS + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Nicole R. Looper + + + Equivalence of cubical and simplicial approaches to $(\infty,n)$-categories + https://arxiv.org/abs/2106.09428 + arXiv:2106.09428v4 Announce Type: replace +Abstract: We prove that the marked triangulation functor from the category of marked cubical sets equipped with a model structure for ($n$-trivial, saturated) comical sets to the category of marked simplicial set equipped with a model structure for ($n$-trivial, saturated) complicial sets is a Quillen equivalence. Our proof is based on the theory of cones, previously developed by the first two authors together with Lindsey and Sattler. + oai:arXiv.org:2106.09428v4 + math.AT + math.CT + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Adv. Math. 416 (2023), Paper No. 108902, 81 pp + Brandon Doherty, Chris Kapulkin, Yuki Maehara + + + The multiple points of maps from sphere to Euclidean space + https://arxiv.org/abs/2109.11575 + arXiv:2109.11575v5 Announce Type: replace +Abstract: In this paper, we obtain some sufficient conditions to guarantee the existence of multiple points of maps from $S^m$ to $\mathbb{R}^d$. Our main tool is the ideal-valued index of $G$-space defined by E. Fadell and S. Husseini. We obtain more detailed relative positional relationship of multiple points. It is proved that for a continuous real value function $f: S^m\rightarrow \mathbb{R}$ such that $f(-p)=-f(p)$, if $m+1$ is a power of $2$, then there are $m+1$ points $p_1, \ldots, p_{m+1}$ in $S^m$ such that $f(p_1)=\cdots=f(p_{m+1})$, where $p_1, \ldots, p_{m+1}$ are linearly dependent and any $m$ points of $p_1, \ldots, p_{m+1}$ are linearly independent. As a generalization of Hopf's theorem, we also prove that for any continuous map $f: S^m\rightarrow \mathbb{R}^d$, if $m> d$, then there exists a pair of mutually orthogonal points having the same image in addition to the antipodal points. + oai:arXiv.org:2109.11575v5 + math.AT + math.GT + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by-nc-nd/4.0/ + Jun Wang, Xuezhi Zhao + + + Limit of trees with fixed degree sequence + https://arxiv.org/abs/2110.03378 + arXiv:2110.03378v2 Announce Type: replace +Abstract: We show, under natural conditions, that uniform rooted trees with fixed degree sequence converge after renormalization toward inhomogeneous continuum random trees (ICRT). We also provide a sharp upper-bound for the tail of their heights. We also extend our results to P-trees, ICRT, and trees with random degree sequence. In passing we confirm a conjecture of Aldous, Miermont, and Pitman stating that L\'evy trees are ICRT with random parameters. + oai:arXiv.org:2110.03378v2 + math.PR + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Arthur Blanc-Renaudie + + + Pointwise rotation for homeomorphisms with integrable distortion and controlled compression + https://arxiv.org/abs/2110.12809 + arXiv:2110.12809v2 Announce Type: replace +Abstract: We obtain sharp rotation bounds for homeomorphisms $f:\mathbb{C}\to\mathbb{C}$ whose distortion is in $L^p_{loc}$, $p\geq1$, and whose inverse have controlled modulus of continuity. The motivation to study this class of maps comes from so-called Yudovich solutions to planar Euler equations. Furthermore, we present examples proving sharpness in a strong sense, thereby settling the borderline case $p=1$ in \cite[Theorem 3]{CHS}. + oai:arXiv.org:2110.12809v2 + math.DS + math.AP + math.CV + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by/4.0/ + Lauri Hitruhin, Banhirup Sengupta + + + List chromatic numbers and singular compactness + https://arxiv.org/abs/2111.11826 + arXiv:2111.11826v4 Announce Type: replace +Abstract: We prove that the list chromatic number of graphs satisfies singular compactness at strong limit singular cardinals. + oai:arXiv.org:2111.11826v4 + math.CO + math.LO + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by/4.0/ + Shimon Garti + + + Divisible Codes + https://arxiv.org/abs/2112.11763 + arXiv:2112.11763v4 Announce Type: replace +Abstract: A linear code over $\mathbb{F}_q$ with the Hamming metric is called $\Delta$-divisible if the weights of all codewords are divisible by $\Delta$. They have been introduced by Harold Ward a few decades ago. Applications include subspace codes, partial spreads, vector space partitions, and distance optimal codes. The determination of the possible lengths of projective divisible codes is an interesting and comprehensive challenge. + oai:arXiv.org:2112.11763v4 + cs.IT + math.CO + math.IT + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Sascha Kurz + + + Constructions and bounds for subspace codes + https://arxiv.org/abs/2112.11766 + arXiv:2112.11766v3 Announce Type: replace +Abstract: Subspace codes are the $q$-analog of binary block codes in the Hamming metric. Here the codewords are vector spaces over a finite field. They have e.g. applications in random linear network coding, distributed storage, and cryptography. In this chapter we survey known constructions and upper bounds for subspace codes. + oai:arXiv.org:2112.11766v3 + cs.IT + math.CO + math.IT + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Sascha Kurz + + + Logarithmic Akizuki--Nakano vanishing theorems on weakly pseudoconvex K\"{a}hler manifolds + https://arxiv.org/abs/2201.11458 + arXiv:2201.11458v2 Announce Type: replace +Abstract: In this paper, we establish a logarithmic vanishing theorem on weakly pseudoconvex K\"ahler manifolds, where the divisor may have infinitely many irreducible components. This result serves as a generalization of Norimatsu's findings on compact K\"ahler manifolds. We derive vanishing theorems for certain direct image sheaves as a direct corollary. + oai:arXiv.org:2201.11458v2 + math.CV + math.AG + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Yongpan Zou + + + Homotopy groups of cubical sets + https://arxiv.org/abs/2202.03511 + arXiv:2202.03511v3 Announce Type: replace +Abstract: We define and study homotopy groups of cubical sets. To this end, we give four definitions of homotopy groups of a cubical set, prove that they are equivalent, and further that they agree with their topological analogues via the geometric realization functor. We also provide purely combinatorial proofs of several classical theorems, including: product preservation, commutativity of higher homotopy groups, the long exact sequence of a fibration, and Whitehead's theorem. This is a companion paper to our "Cubical setting for discrete homotopy theory, revisited" in which we apply these results to study the homotopy theory of simple graphs. + oai:arXiv.org:2202.03511v3 + math.AT + math.CT + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Expo. Math. 41 (2023), no. 4, Paper No. 125518, 55 pp + Daniel Carranza, Chris Kapulkin + + + Cubical setting for discrete homotopy theory, revisited + https://arxiv.org/abs/2202.03516 + arXiv:2202.03516v3 Announce Type: replace +Abstract: We construct a functor associating a cubical set to a (simple) graph. We show that cubical sets arising in this way are Kan complexes, and that the A-groups of a graph coincide with the homotopy groups of the associated Kan complex. We use this to prove a conjecture of Babson, Barcelo, de Longueville, and Laubenbacher from 2006, and a strong version of the Hurewicz theorem in discrete homotopy theory. + oai:arXiv.org:2202.03516v3 + math.CO + math.AT + math.CT + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Compos. Math. 160 (2024), no. 12, 2856-2903 + Daniel Carranza, Chris Kapulkin + + + On Rota-Baxter and Nijenhuis anti-flexible algebras + https://arxiv.org/abs/2207.11834 + arXiv:2207.11834v2 Announce Type: replace +Abstract: We define and derive basic properties of the notion of Rota-Baxter operator on anti-flexible algebra. Starting from a Rota-Baxter operator on an anti-flexible algebra, we construct pre-anti-flexible algebra structure and associated left(right)-symmetric algebra as well. The notion of O-operator on anti-flexible algebra is recalled and used to build left(right)- symmetric algebra as well as related properties. Furthermore, we introduce Nijenhuis anti-flexible algebra and derive associated properties. Nijenhuis operator on anti-flexible algebra is used to build pre-anti-flexible algebra structure and related left(right)-symmetric algebra. + oai:arXiv.org:2207.11834v2 + math.RA + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by/4.0/ + Mafoya Landry Dassoundo + + + A refinement of bounded cohomology + https://arxiv.org/abs/2208.03168 + arXiv:2208.03168v3 Announce Type: replace +Abstract: We introduce a refinement of bounded cohomology and prove that the suitable comparison homomorphisms vanish for an amenable group. We investigate in this context Thompson's group F and provide further evidence towards its amenability. We provide a source of nontrivial 1-bounded classes and show that the space of 1-bounded classes in degree 2 is huge. + oai:arXiv.org:2208.03168v3 + math.GR + math.MG + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by/4.0/ + \'Swiatos{\l}aw R. Gal, Jarek K\k{e}dra + + + Sharp upper tail behavior of line ensembles via the tangent method + https://arxiv.org/abs/2208.08922 + arXiv:2208.08922v3 Announce Type: replace +Abstract: We develop a new probabilistic and geometric method to obtain several sharp results pertaining to the upper tail behavior of continuum Gibbs measures on infinite ensembles of random continuous curves, also known as line ensembles, satisfying some natural assumptions. The arguments make crucial use of Brownian resampling invariance properties and correlation inequalities admitted by such Gibbs measures. We obtain sharp one-point upper tail estimates showing that the probability of the value at zero being larger than $\theta$ is $\exp(-\frac{4}{3}\theta^{3/2}(1+o(1)))$. A key intermediate step is developing a precise understanding of the profile when conditioned on the value at zero equaling $\theta$. Our method further allows one to obtain multi-point asymptotics which were out of reach of previous approaches. As an example, we prove sharp explicit two-point upper tail estimates. This framework is then used to establish the corresponding results for the KPZ equation, which are all new. Even for the zero-temperature case of the Airy$_2$ process, our arguments yield new proofs for one-point estimates previously known due to its connections to random matrix theory, as well as new two-point asymptotics. To showcase the reach of the method, we obtain the same results in a purely non-integrable setting under only assumptions of stationarity and extremality in the class of Gibbs measures. Our method bears resemblance to the tangent method introduced by Colomo-Sportiello and mathematically realized by Aggarwal in the context of the six-vertex model. + oai:arXiv.org:2208.08922v3 + math.PR + math-ph + math.MP + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Shirshendu Ganguly, Milind Hegde + + + Multibubble blow-up analysis for the Brezis-Nirenberg problem in three dimensions + https://arxiv.org/abs/2208.12337 + arXiv:2208.12337v4 Announce Type: replace +Abstract: For a smooth bounded domain $\Omega \subset \mathbb R^3$ and smooth functions $a$ and $V$, we consider the asymptotic behavior of a sequence of positive solutions $u_\epsilon$ to $-\Delta u_\epsilon + (a+\epsilon V) u_\epsilon = u_\epsilon^5$ on $\Omega$ with zero Dirichlet boundary conditions, which blow up as $\epsilon \to 0$. We derive the sharp blow-up rate and characterize the location of concentration points in the general case of multiple blow-up, thereby obtaining a complete picture of blow-up phenomena in the framework of the Brezis-Peletier conjecture in dimension $N=3$. + oai:arXiv.org:2208.12337v4 + math.AP + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Tobias K\"onig, Paul Laurain + + + An arithmetic Yau-Zaslow formula + https://arxiv.org/abs/2210.15788 + arXiv:2210.15788v4 Announce Type: replace +Abstract: We prove an arithmetic refinement of the Yau-Zaslow formula by replacing the classical Euler characteristic in Beauville's argument by a "motivic Euler characteristic", related to the work of Levine. Our result implies similar formulas for other related invariants, including a generalisation of a formula of Kharlamov and Rasdeaconu on counting real rational curves on real K3 surfaces, and Saito's determinant of cohomology. + oai:arXiv.org:2210.15788v4 + math.AG + math.NT + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Jesse Pajwani, Ambrus P\'al + + + Fine multibubble analysis in the higher-dimensional Brezis-Nirenberg problem + https://arxiv.org/abs/2211.00595 + arXiv:2211.00595v3 Announce Type: replace +Abstract: For a bounded set $\Omega \subset \mathbb R^N$ and a perturbation $V \in C^1(\overline{\Omega})$, we analyze the concentration behavior of a blow-up sequence of positive solutions to \[ -\Delta u_\epsilon + \epsilon V = N(N-2) u_\epsilon^\frac{N+2}{N-2} \] for dimensions $N \geq 4$, which are non-critical in the sense of the Brezis--Nirenberg problem. + For the general case of multiple concentration points, we prove that concentration points are isolated and characterize the vector of these points as a critical point of a suitable function derived from the Green's function of $-\Delta$ on $\Omega$. Moreover, we give the leading order expression of the concentration speed. This paper, with a recent one by the authors (arXiv:2208.12337) in dimension $N = 3$, gives a complete picture of blow-up phenomena in the Brezis-Nirenberg framework. + oai:arXiv.org:2211.00595v3 + math.AP + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + 10.4171/AIHPC/95 + Ann. Inst. H. Poincar\'e, Anal. Non Lin\'eaire 41 (2024), no. 5, 1239-1281 + Tobias K\"onig, Paul Laurain + + + Cofibration category of digraphs for path homology + https://arxiv.org/abs/2212.12568 + arXiv:2212.12568v2 Announce Type: replace +Abstract: We prove that the category of directed graphs and graph maps carries a cofibration category structure in which the weak equivalences are the graph maps inducing isomorphisms on path homology. + oai:arXiv.org:2212.12568v2 + math.CO + math.AT + math.CT + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Algebr. Comb. 7 (2024), no. 2, 475-514 + Daniel Carranza, Brandon Doherty, Chris Kapulkin, Morgan Opie, Maru Sarazola, Liang Ze Wong + + + Multiperiodic Processes: Ergodic Sources with a Sublinear Entropy + https://arxiv.org/abs/2302.09049 + arXiv:2302.09049v3 Announce Type: replace +Abstract: We construct multiperiodic processes -- a simple example of stationary ergodic (but not mixing) processes over natural numbers that enjoy the vanishing entropy rate under a mild condition. Multiperiodic processes are supported on randomly shifted deterministic sequences called multiperiodic sequences, which can be efficiently generated using an algorithm called the Infinite Clock. Under a suitable parameterization, multiperiodic sequences exhibit relative frequencies of particular numbers given by Zipf's law. Exactly in the same setting, the respective multiperiodic processes satisfy an asymptotic power-law growth of block entropy, called Hilberg's law. Hilberg's law is deemed to hold for statistical language models, in particular. + oai:arXiv.org:2302.09049v3 + cs.IT + cs.LG + math.IT + math.ST + stat.TH + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + {\L}ukasz D\k{e}bowski + + + The fundamental group in discrete homotopy theory + https://arxiv.org/abs/2303.06029 + arXiv:2303.06029v2 Announce Type: replace +Abstract: We develop a robust foundation for studying the fundamental group(oid) in discrete homotopy theory, including: equivalent definitions and basic properties, the theory of covering graphs, and the discrete version of the Seifert-van Kampen theorem. + oai:arXiv.org:2303.06029v2 + math.CO + math.AT + math.CT + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Adv. in Appl. Math. 164 (2025), Paper No. 102838, 56 pp + Chris Kapulkin, Udit Mavinkurve + + + Finite sample rates of convergence for the Bigraphical and Tensor graphical Lasso estimators + https://arxiv.org/abs/2304.00441 + arXiv:2304.00441v4 Announce Type: replace +Abstract: Many modern datasets exhibit dependencies among observations as well as variables. A decade ago, Kalaitzis et. al. (2013) proposed the Bigraphical Lasso, an estimator for precision matrices of matrix-normals based on the Cartesian product of graphs; they observed that the associativity of the Kronecker sum yields an approach to the modeling of datasets organized into 3 or higher-order tensors. Subsequently, Greenewald, Zhou and Hero (2019) explored this possibility to a great extent, by introducing the tensor graphical Lasso (TeraLasso) for estimating sparse $L$-way decomposable inverse covariance matrices for all $L \ge 2$, and showing the rates of convergence in the Frobenius and operator norms for estimating this class of inverse covariance matrices for sub-gaussian tensor-valued data. In this paper, we provide sharper rates of convergence for both Bigraphical and TeraLasso estimators for inverse covariance matrices. This improves upon the rates presented in GZH 2019. In particular, (a) we strengthen the bounds for the relative errors in the operator and Frobenius norm by a factor of approximately $\log p$; (b) Crucially, this improvement allows for finite sample estimation errors in both norms to be derived for the two-way Kronecker sum model. This closes the gap between the low single-sample error for the two-way model as observed in GZH 2019 and the lack of theoretical guarantee for this particular case. The two-way regime is important because it is the setting that is the most theoretically challenging, and simultaneously the most common in applications. In the current paper, we elaborate on the Kronecker Sum model, highlight the proof strategy and provide full proofs of all main theorems. + oai:arXiv.org:2304.00441v4 + math.ST + stat.TH + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by-nc-nd/4.0/ + Shuheng Zhou, Kristjan Greenewald + + + Optimal Sobolev inequalities in the hyperbolic space + https://arxiv.org/abs/2305.06797 + arXiv:2305.06797v2 Announce Type: replace +Abstract: We find the optimal function norm on the left-hand side of the $m$th order Sobolev type inequality $\|u\|_{Y(\mathbb{H}^n)} \leq C \|\nabla_g^m u\|_{X(\mathbb{H}^n)}$ in the $n$-dimensional hyperbolic space $\mathbb{H}^n$, $1\leq m < n$. The optimal function norm in the inequality among all rearrangement-invariant function norms is completely characterized. A variety of concrete examples of optimal function norms is provided. The examples include delicate limiting cases, and especially when $m\geq3$, seem to provide new, improved inequalities in these limiting cases. + oai:arXiv.org:2305.06797v2 + math.FA + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Zden\v{e}k Mihula + + + Sampling recovery in $L_2$ and other norms + https://arxiv.org/abs/2305.07539 + arXiv:2305.07539v5 Announce Type: replace +Abstract: We study the recovery of functions in various norms, including $L_p$ with $1\le p\le\infty$, based on function evaluations. We obtain worst case error bounds for general classes of functions in terms of the best $L_2$-approximation from a given nested sequence of subspaces and the Christoffel function of these subspaces. In the case $p=\infty$, our results imply that linear sampling algorithms are optimal up to a constant factor for many reproducing kernel Hilbert spaces. + oai:arXiv.org:2305.07539v5 + math.NA + cs.CC + cs.NA + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by/4.0/ + 10.1090/mcom/4148 + Math. Comp. Published Online: October 6, 2025 + David Krieg, Kateryna Pozharska, Mario Ullrich, Tino Ullrich + + + Diagonal Lemma for Presheaves on Eilenberg-Zilber Categories + https://arxiv.org/abs/2306.02217 + arXiv:2306.02217v2 Announce Type: replace +Abstract: The diagonal lemma asserts that if a map of bisimplicial sets is a levelwise weak equivalence in the Kan-Quillen model structure, then it induces a weak equivalence of the diagonal simplicial sets. In this short note, we observe that the standard proof of this fact works for an arbitrary Eilenberg-Zilber category in place of the simplex category. + oai:arXiv.org:2306.02217v2 + math.AT + math.CT + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Theory Appl. Categ. 44 (2025), Paper No. 11, 326-343 + Daniel Carranza, Chris Kapulkin, Liang Ze Wong + + + On the $L^{p}$-spaces of projective limits of probability measures + https://arxiv.org/abs/2307.12178 + arXiv:2307.12178v2 Announce Type: replace +Abstract: The present article describes the precise structure of the $L^{p}$-spaces of projective limit measures by introducing a category theoretical perspective. This analysis is applied to measures on vector spaces and in particular to Gaussian measures on nuclear topological vector spaces. A simple application to constructive Quantum Field Theory (QFT) is given through the Osterwalder-Schrader axioms. + oai:arXiv.org:2307.12178v2 + math.PR + math.FA + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Juan Carlos Sampedro + + + The well-posedness of three-dimensional Navier-Stokes and magnetohydrodynamic equations with partial fractional dissipation + https://arxiv.org/abs/2308.06489 + arXiv:2308.06489v2 Announce Type: replace +Abstract: It is well-known that if one replaces standard velocity and magnetic dissipation by $(-\Delta)^\alpha u$ and $(-\Delta)^\beta b$ respectively, the magnetohydrodynamic equations are well-posed for $\alpha\ge\frac{5}{4}$ and $\alpha + \beta \ge \frac{5}{2}$. This paper considers the 3D Navier-Stokes and magnetohydrodynamic equations with partial fractional hyper-dissipation. It is proved that when each component of the velocity and magnetic field lacks dissipation along some direction, the existence and conditional uniqueness of the solution still hold. This paper extends the previous results in (Yang, Jiu and Wu J. Differential Equations 266(1): 630-652, 2019) to a more general case. + oai:arXiv.org:2308.06489v2 + math.AP + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Discrete and Continuous Dynamical Systems - Series B, 2025 + Qibo Ma, Li Li + + + A new decision method for Intuitionistic Logic by 3-valued non-deterministic truth-tables (pre-print version) + https://arxiv.org/abs/2308.13664 + arXiv:2308.13664v4 Announce Type: replace +Abstract: Kurt G\"odel proved that it is not possible to characterize Intuitionistic Propositional Logic (IPL) by means of finite and deterministic truth-tables. After extending the same result with respect to non-deterministic matrices, we provide a semantical characterization of IPL by means of a 3-valued non-deterministic matrix with a restricted set of valuations. This structure allows to define an algorithm to delete unsound rows from the non-deterministic truth-tables generated for each formula, which constitutes a new and very simple decision procedure for IPL. This method can be seen as truth-tables in a broader sense, and a way to overcome G\"odel's limiting result. + oai:arXiv.org:2308.13664v4 + math.LO + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by/4.0/ + 10.1017/jsl.2025.10174 + Renato Leme, Marcelo Coniglio, Bruno Lopes + + + Ordering sampling rules for sequential anomaly identification under sampling constraints + https://arxiv.org/abs/2309.14528 + arXiv:2309.14528v2 Announce Type: replace +Abstract: We consider the problem of sequential anomaly identification over multiple independent data streams, under the presence of a sampling constraint. The goal is to quickly identify those that exhibit anomalous statistical behavior, when it is not possible to sample every source at each time instant. Thus, in addition to a stopping rule that determines when to stop sampling, and a decision rule that indicates which sources to identify as anomalous upon stopping, one needs to specify a sampling rule that determines which sources to sample at each time instant. We focus on the family of ordering sampling rules that select the sources to be sampled at each time instant based not only on the currently estimated subset of anomalous sources as the probabilistic sampling rules \cite{Tsopela_2022}, but also on the ordering of the sources' test-statistics. We show that under an appropriate design specified explicitly, an ordering sampling rule leads to the optimal expected time for stopping among all policies that satisfy the same sampling and error constraints to a first-order asymptotic approximation as the false positive and false negative error thresholds go to zero. This is the first asymptotic optimality result for ordering sampling rules, when more than one sources can be sampled per time instant, and it is established under a general setup where the number of anomalous sources is not required to be known. A novel proof technique is introduced that encompasses all different cases of the problem concerning sources' homogeneity, and prior information on the number of anomalies. Simulations show that ordering sampling rules have better performance in finite regime compared to probabilistic sampling rules. + oai:arXiv.org:2309.14528v2 + math.ST + stat.TH + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by/4.0/ + Aristomenis Tsopelakos, Georgios Fellouris + + + Lagrangian formalism and classical statistical ensemble + https://arxiv.org/abs/2309.16288 + arXiv:2309.16288v4 Announce Type: replace +Abstract: We present a formulation of classical statistical mechanics based on a Lagrangian description on the tangent bundle. In this approach, a Wick rotation from real time to imaginary time is employed as a technical device that facilitates the construction of a Hamiltonian structure expressed in velocity variables. The resulting dynamics preserves a natural measure induced by the associated symplectic form on the tangent bundle. This measure-preserving property enables the consistent definition of classical statistical ensembles directly in terms of Lagrangian variables. + oai:arXiv.org:2309.16288v4 + math-ph + math.MP + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by/4.0/ + Sikarin Yoo-Kong + + + Closed symmetric monoidal structures on the category of graphs + https://arxiv.org/abs/2310.00493 + arXiv:2310.00493v2 Announce Type: replace +Abstract: We show that the category of (reflexive) graphs and graph maps carries exactly two closed symmetric monoidal products: the box product and the categorical product. + oai:arXiv.org:2310.00493v2 + math.CT + math.CO + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Theory Appl. Categ. 41 (2024), Paper No. 23, 760-784 + Chris Kapulkin, Nathan Kershaw + + + Extensional concepts in intensional type theory, revisited + https://arxiv.org/abs/2310.05706 + arXiv:2310.05706v2 Announce Type: replace +Abstract: Revisiting a classic result from M. Hofmann's dissertation, we give a direct proof of Morita equivalence, in the sense of V. Isaev, between extensional type theory and intensional type theory extended by the principles of functional extensionality and of uniqueness of identity proofs. + oai:arXiv.org:2310.05706v2 + math.LO + cs.LO + math.CT + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Theoret. Comput. Sci. 1029 (2025), Paper No. 115051, 29 pp + Chris Kapulkin, Yufeng Li + + + Chaotic Dynamics and Zero Distribution: Implications and Applications in Control Theory for Yitang Zhang's Landau Siegel Zero Theorem + https://arxiv.org/abs/2310.14127 + arXiv:2310.14127v2 Announce Type: replace +Abstract: This study delves into the realm of chaotic dynamics derived from Dirichlet L-functions, drawing inspiration from Yitang Zhang's groundbreaking work on Landau-Siegel zeros. The dynamic behavior reveals profound chaos, corroborated by the calculated Lyapunov exponents and entropy, attesting to the system's inherent unpredictability. + Furthermore, we establish a novel connection between Fractal geometry and Quantum chaos, predicting the distributions of zeros for both Yitang dynamics and Riemann dynamics. These findings offer indirect support for Zhang's groundbreaking theorem concerning Landau-Siegel zeros and suggest that these chaotic dynamics could find application in engineering and control systems, demonstrating the potential to harness chaos for beneficial purposes. + The exploration of stability within electrical systems further uncovers the instability of fixed points, highlighting both the challenges and opportunities for harnessing chaotic behavior to achieve specific control objectives. + This study not only contributes to our understanding of chaotic dynamics but also opens new avenues for exploring the potential applications of Yitang dynamics in the field of electrical control systems. It paves the way for innovative approaches to address real-world engineering challenges and may be considered as a new consequence for the generalized Riemann hypothesis. + oai:arXiv.org:2310.14127v2 + math.DS + nlin.CD + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by/4.0/ + 10.1140/epjp/s13360-024-05000-w + Eur. Phys. J. Plus 139, 217 (2024) + Zeraoulia Rafik, Alvaro Humberto Salas + + + Synthetic approach to the Quillen model structure on topological spaces + https://arxiv.org/abs/2310.14235 + arXiv:2310.14235v2 Announce Type: replace +Abstract: We provide an axiomatic treatment of Quillen's construction of the model structure on topological spaces to make it applicable to a wider range of settings, including $\Delta$-generated spaces and pseudotopological spaces. We use this axiomatization to construct a model structure on the category of locales. + oai:arXiv.org:2310.14235v2 + math.AT + math.CT + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + 10.2140/agt.2025.25.1227 + Algebr. Geom. Topol. 25 (2025) 1227-1264 + Sterling Ebel, Chris Kapulkin + + + Level-raising of even representations of tetrahedral type and equidistribution of lines in the projective plane + https://arxiv.org/abs/2310.14352 + arXiv:2310.14352v2 Announce Type: replace +Abstract: The distribution of primes raising the level of even Galois representations of tetrahedral type is studied. Data are presented on primes $v\leq 10^8$ raising the level of $3$-adic even representations of various conductors. Based on the data, a conjecture is formulated concerning the distribution of certain lines in the plane. By an application of Wiles' formula, the conjecture is shown to imply that the density of primes raising the level of a $p$-adic even representation is $$\frac{p-1}{p},$$ in agreement with the density of $2/3$ for $p=3$ observed in the data. + oai:arXiv.org:2310.14352v2 + math.NT + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Peter Vang Uttenthal + + + Local well-posedness for the quasilinear Schr\"odinger equations via the generalized energy method + https://arxiv.org/abs/2311.02556 + arXiv:2311.02556v3 Announce Type: replace +Abstract: We study the Cauchy problem of quasilinear Schr\"odinger equations, for which Kenig et al. (Invent Math, 2004; Adv Math, 2006) obtained large data local well-posedness by pseudo-differential techniques and viscosity methods, while Marzuola et al. (Adv Math, 2012; Kyoto J Math, 2014; Arch Ration Mech Anal, 2021) and Ben et al. (Arch Ration Mech Anal, 2024) improved the results by dispersive arguments. In this paper, we introduce a generalized energy method that combines momentum and energy estimates to close the bounds, thereby obtaining our results through viscosity methods. If the data is small, the proof relies mainly on integration by parts and Sobolev embeddings, much like the classical local existence theory for semilinear Schr\"odinger equations. For large data, the framework remains applicable with the incorporation of certain pseudo-differential tools. In the case of quadratic interactions, we establish low regularity local well-posedness for both small and large data in the same function spaces as in works of Kenig et al. For cubic interactions with small initial data, we recover the low regularity results obtained by Marzuola et al. (Kyoto J Math, 2014). + oai:arXiv.org:2311.02556v3 + math.AP + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by/4.0/ + Jie Shao, Yi Zhou + + + Affine extended weak order is a lattice + https://arxiv.org/abs/2311.05737 + arXiv:2311.05737v3 Announce Type: replace +Abstract: Coxeter groups are equipped with a partial order known as the weak order, such that $u \leq v$ if the inversions of $u$ are a subset of the inversions of $v$. In finite Coxeter groups, weak order is a complete lattice, but in infinite Coxeter groups it is only a meet semi-lattice. Motivated by questions in Kazhdan-Lusztig theory, Matthew Dyer introduced a larger poset, now known as extended weak order, which contains the weak order as an order ideal and coincides with it for finite Coxeter groups. The extended weak order is the containment order on certain sets of positive roots: those which satisfy a geometric condition making them "biclosed". The finite biclosed sets are precisely the inversion sets of Coxeter group elements. Generalizing the result for finite Coxeter groups, Dyer conjectured that the extended weak order is always a complete lattice, even for infinite Coxeter groups. + In this paper, we prove Dyer's conjecture for Coxeter groups of affine type. To do so, we introduce the notion of a clean arrangement, which is a hyperplane arrangement where the regions are in bijection with biclosed sets. We show that root poset order ideals in a finite or rank 3 untwisted affine root system are clean. We set up a general framework for reducing Dyer's conjecture to checking cleanliness of certain subarrangements. We conjecture this framework can be used to prove Dyer's conjecture for all Coxeter groups. + oai:arXiv.org:2311.05737v3 + math.CO + math.GR + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Grant T. Barkley, David E Speyer + + + Thermodynamic formalism for correspondences + https://arxiv.org/abs/2311.09397 + arXiv:2311.09397v3 Announce Type: replace +Abstract: In this article, we investigate the Variational Principle and develop thermodynamic formalism for correspondences. We define the measure-theoretic entropy for transition probability kernels and topological pressure for correspondences. Based on these two notions, we establish the following results: + The Variational Principle holds and equilibrium states exist for continuous potential functions, provided that the correspondence satisfies some expansion property called forward expansiveness. If, in addition, the correspondence satisfies the specification property and the potential function is Bowen summable, then the equilibrium state is unique. On the other hand, for a distance-expanding, open, strongly transitive correspondence and a H\"{o}lder continuous potential function, there exists a unique equilibrium state, and the backward orbits are equidistributed. Furthermore, we investigate the Variational Principle for general correspondences. + In complex dynamics, we establish the Variational Principle for the Lee-Lyubich-Markorov-Mukherjee anti-holomorphic correspondences, which are matings of some anti-ho\-lo\-mor\-phic rational maps with anti-Hecke groups and are not forward expansive. We also show a Ruelle-Perron-Frobenius theorem for a family of hyperbolic holomorphic correspondences of the form $\boldsymbol{f}_c (z)= z^{q/p}+c$. + oai:arXiv.org:2311.09397v3 + math.DS + math.CV + math.GR + math.PR + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Xiaoran Li, Zhiqiang Li, Yiwei Zhang + + + Brownian bridge limit of path measures in the upper tail of KPZ models + https://arxiv.org/abs/2311.12009 + arXiv:2311.12009v2 Announce Type: replace +Abstract: For models in the KPZ universality class, such as the zero temperature model of planar last passage-percolation (LPP) and the positive temperature model of directed polymers, its upper tail behavior has been a topic of recent interest, with particular focus on the associated path measures (i.e., geodesics or polymers). For Exponential LPP, diffusive fluctuation had been established in Basu-Ganguly. In the directed landscape, the continuum limit of LPP, the limiting Gaussianity at one point, as well as of related finite-dimensional distributions of the KPZ fixed point, were established, using exact formulas in Liu and Wang-Liu. It was further conjectured in these works that the limit of the corresponding geodesic should be a Brownian bridge. We prove it in both zero and positive temperatures; for the latter, neither the one-point limit nor the scale of fluctuations was previously known. Instead of relying on formulas (which are still missing in the positive temperature literature), our arguments are geometric and probabilistic, using the results on the shape of the weight and free energy profiles under the upper tail from Ganguly-Hegde as a starting point. Another key ingredient involves novel coalescence estimates, developed using the recently discovered shift-invariance Borodin-Gorin-Wheeler in these models. Finally, our proof also yields insight into the structure of the polymer measure under the upper tail conditioning, establishing a quenched localization exponent around a random backbone. + oai:arXiv.org:2311.12009v2 + math.PR + math-ph + math.MP + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by/4.0/ + Shirshendu Ganguly, Milind Hegde, Lingfu Zhang + + + Using Machine Learning to Design Time Step Size Controllers for Stable Time Integrators + https://arxiv.org/abs/2312.01796 + arXiv:2312.01796v2 Announce Type: replace +Abstract: We present a new method for developing time step controllers based on a technique from the field of machine learning. This method is applicable to stable time integrators that have an embedded scheme, i.e., that have local error estimation similar to Runge-Kutta pairs. To design good time step size controllers using these error estimates, we propose to use Bayesian optimization. In particular, we design a novel objective function that captures important properties such as tolerance convergence and computational stability. We apply our new approach to several modified Patankar--Runge--Kutta (MPRK) schemes and a Rosenbrock-type scheme, equipping them with controllers based on digital signal processing which extend classical PI and PID controllers. We demonstrate that the optimization process yields controllers that are at least as good as the best controllers chosen from a wide range of suggestions available for classical explicit and implicit time integration methods by providing work-precision diagrams on a variety of ordinary and partial differential equations. + oai:arXiv.org:2312.01796v2 + math.NA + cs.NA + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Thomas Izgin, Hendrik Ranocha + + + On the connected (sub)partition polytope + https://arxiv.org/abs/2401.01716 + arXiv:2401.01716v2 Announce Type: replace +Abstract: Let $k$ be a positive integer and let $G$ be a graph with $n$ vertices. A connected $k$-subpartition of $G$ is a collection of $k$ pairwise disjoint sets (a.k.a. classes) of vertices in $G$ such that each set induces a connected subgraph. The connected $k$-subpartition polytope of $G$, denoted by $\poly(G,k)$, is defined as the convex hull of the incidence vectors of all connected $k$-subpartitions of $G$. Many applications arising in off-shore oil-drilling, forest planning, image processing, cluster analysis, political districting, police patrolling, and biology are modeled in terms of finding connected (sub)partitions of a graph. This study focuses on the facial structure of~$\poly(G,k)$ and the computational complexity of the corresponding separation problems. We first propose a set of valid inequalities having non-zero coefficients associated with a single class that extends and generalizes the ones in the literature of related problems, show sufficient conditions for these inequalities to be facet-defining, and design a polynomial-time separation algorithm for them. We also devise two sets of inequalities that consider multiple classes, prove when they define facets, and study the computational complexity of associated separation problems. Finally, we report on computational experiments showing the usefulness of the proposed inequalities. + oai:arXiv.org:2401.01716v2 + math.CO + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by/4.0/ + Phablo F. S. Moura, Hande Yaman, Roel Leus + + + On the strong Feller property of the heat equation on quantum graphs with Kirchoff noise + https://arxiv.org/abs/2401.11866 + arXiv:2401.11866v4 Announce Type: replace +Abstract: We consider a so-called quantum graph with standard continuity and Kirchhoff vertex conditions where the Kirchhoff vertex condition is perturbed by Gaussian noise. We show that the quantum graph setting is very different from the classical one dimensional boundary noise setting, where the transition semigroup is known to be strong Feller, by giving examples and counterexamples to the strong Feller property. In particular, when the graph is a tree, and there is noise present in all of the boundary vertices except one, then the transition semigroup associated with the problem is strong Feller at any time T > 0. This turns out to be also a necessary condition for equilateral star graphs. We also comment on the existence and uniqueness of the invariant measure and the regularity of the solution. + oai:arXiv.org:2401.11866v4 + math.DS + math.PR + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://creativecommons.org/publicdomain/zero/1.0/ + 10.1007/s00233-025-10599-y + Mohamed Fkirine, Mih\'aly Kov\'acs, Eszter Sikolya + + + Ahlfors regularity of Patterson-Sullivan measures of Anosov groups and applications + https://arxiv.org/abs/2401.12398 + arXiv:2401.12398v5 Announce Type: replace +Abstract: For all Zarski dense Anosov subgroups of a semisimple real algebraic group, we prove that their limit sets are Ahlfors regular for intrinsic conformal premetrics. As a consequence, we obtain that a Patterson-Sullivan measure is Ahlfors regular (and hence equal to the Hausdorff measure) if and only if the associated linear form is symmetric. We also discuss several applications, including analyticity of $(p,q)$-Hausdorff dimensions on the Teichm\"uller spaces, new upper bounds on the growth indicator, and $L^2$-spectral properties of associated locally symmetric manifolds. + oai:arXiv.org:2401.12398v5 + math.GR + math.DS + math.GT + math.MG + math.SP + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Subhadip Dey, Dongryul M. Kim, Hee Oh + + + Novel approaches for the reliable and efficient numerical evaluation of Landau-type operators + https://arxiv.org/abs/2402.02247 + arXiv:2402.02247v2 Announce Type: replace +Abstract: When applying Hamiltonian operator splitting methods for the time integration of multi-species Vlasov-Maxwell-Landau systems, the reliable and efficient numerical approximation of the Landau equation represents a fundamental component of the entire algorithm. Substantial computational issues arise from the treatment of the physically most relevant three-dimensional case with Coulomb-type interaction. This work is concerned with the introduction and numerical comparison of novel approaches for the evaluation of the Landau collision operator and related integral operators with more general kernels. In the spirit of collocation, common tools are the identification of fundamental integrals, series expansions of the integral kernel and the density function on the main part of the velocity domain, and interpolation as well as quadrature approximation nearby the singularity of the kernel. Focusing on the favourable choice of the Fourier spectral method, their practical implementation uses the reduction to basic integrals, fast Fourier techniques, and summations along certain directions. Moreover, an important observation is that a significant percentage of the overall computational effort can be transferred to precomputations which are independent of the density function. For the purpose of exposition and numerical validation, the cases of constant, regular, and singular integral kernels are distinguished, and the procedure is adapted accordingly to the increasing complexity of the problem. + oai:arXiv.org:2402.02247v2 + math.NA + cs.NA + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Jose Antonio Carrillo, Mechthild Thalhammer + + + Variance Reduction and Low Sample Complexity in Stochastic Optimization via Proximal Point Method + https://arxiv.org/abs/2402.08992 + arXiv:2402.08992v3 Announce Type: replace +Abstract: High-probability guarantees in stochastic optimization are often obtained only under strong noise assumptions such as sub-Gaussian tails. We show that such guarantees can also be achieved under the weaker assumption of bounded variance by developing a stochastic proximal point method. This method combines a proximal subproblem solver, which inherently reduces variance, with a probability booster that amplifies per-iteration reliability into high-confidence results. The analysis demonstrates convergence with low sample complexity, without restrictive noise assumptions or reliance on mini-batching. + oai:arXiv.org:2402.08992v3 + math.OC + cs.LG + stat.ML + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by/4.0/ + Jiaming Liang + + + Symplectic fillings of unit cotangent bundles of spheres and applications + https://arxiv.org/abs/2402.10363 + arXiv:2402.10363v3 Announce Type: replace +Abstract: We prove the uniqueness, up to diffeomorphism, of symplectically aspherical fillings of the unit cotangent bundle of odd-dimensional spheres. As applications, we first show the non-existence of exact symplectic cobordisms between some 5-dimensional Brieskorn manifolds. We also determine the diffeomorphism types of closed symplectic 6-manifolds with certain codimension 2 symplectic submanifolds. + oai:arXiv.org:2402.10363v3 + math.SG + math.GT + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Myeonggi Kwon, Takahiro Oba + + + Scattering and localized states for defocusing nonlinear Schr\"odinger equations with potential + https://arxiv.org/abs/2402.11366 + arXiv:2402.11366v5 Announce Type: replace +Abstract: We study the large-time behavior of global energy class ($H^1$) solutions of the one-dimensional nonlinear Schr\"odinger equation with a general localized potential term and a defocusing nonlinear term. By using a new type of interaction Morawetz estimate localized to an exterior region, we prove that these solutions decompose into a free wave and a weakly localized part which is asymptotically orthogonal to any fixed free wave. We further show that the $L^2$ norm of this weakly localized part is concentrated in the region $|x| \leq t^{1/2+}$, and that the energy ($\dot{H}^1$) norm is concentrated in $|x| \leq t^{1/3+}$. Our results hold for solutions with arbitrarily large initial data. + oai:arXiv.org:2402.11366v5 + math.AP + math-ph + math.MP + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by/4.0/ + 10.1007/s00023-025-01646-z + Avy Soffer, Gavin Stewart + + + The sandpile model on the complete split graph: $q,t$-Schr\"oder polynomials, sawtooth polyominoes, and a cycle lemma + https://arxiv.org/abs/2402.15372 + arXiv:2402.15372v3 Announce Type: replace +Abstract: This paper studies sorted recurrent configurations of the Abelian sandpile model on the complete split graph. We introduce two natural toppling processes, CTI and ITC toppling, on the recurrent configurations and use these to define two toppling delay statistics, wtopple$_{CTI}$ and wtopple$_{ITC}$. These new toppling delay statistics are time-weighted sums for the number of vertices that topple during each iteration of the toppling processes. We then introduce the bivariate $q,t$-CTI and $q,t$-ITC polynomials that are the generating functions of the bistatistics (level,wtopple$_{ITC}$) and (level,wtopple$_{CTI}$), where level is the well-established sandpile level statistic. + We prove the bistatistic (level,wtopple$_{ITC}$) maps to a bistatistic (area,bounce) on Schr\"oder paths that was introduced by Egge, Haglund, Killpatrick and Kremer (2003). This establishes equality of the $q,t$-ITC polynomial and the $q,t$-Schr\"oder polynomial of those same authors. This connection allows us to relate the $q,t$-ITC polynomial to the theory of symmetric functions and also establishes symmetry of the $q,t$-ITC polynomials. We conjecture equality of the $q,t$-CTI and $q,t$-ITC polynomials. + We also present and prove a characterization of sorted recurrent configurations as a new class of polyominoes that we call sawtooth polyominoes. The CTI and ITC toppling processes on sorted recurrent configurations are proven to correspond to bounce paths within the polyominoes. The main difference between the two bounce paths is the initial direction in which they travel. In addition to this, and building on the results of Aval, D'Adderio, Dukes, and Le Borgne (2016), we present a cycle lemma for a slight extension of stable configurations that allows for an enumeration of sorted recurrent configurations within the framework of the sandpile model. + oai:arXiv.org:2402.15372v3 + math.CO + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Henri Derycke, Mark Dukes, Yvan Le Borgne + + + The critical disordered pinning measure + https://arxiv.org/abs/2402.17642 + arXiv:2402.17642v3 Announce Type: replace +Abstract: In this paper, we study a disordered pinning model induced by a random walk whose increments have a finite $(2+\kappa)$-th moment for some $\kappa>0$. It is known that this model is marginally relevant, and moreover, it undergoes a phase transition in an intermediate disorder regime. We show that, in the critical window, the point-to-point partition functions converge to a unique limiting random measure, which we call the critical disordered pinning measure. We also obtain an analogous result for a continuous counterpart to the pinning model, which is closely related to two other models: one is a critical stochastic Volterra equation that gives rise to a rough volatility model, and the other is a critical stochastic heat equation with multiplicative noise that is white in time and delta in space. + oai:arXiv.org:2402.17642v3 + math.PR + q-fin.MF + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by/4.0/ + 10.1214/25-AAP2207 + Ann. Appl. Probab. 35(6): 3844-3905 (December 2025) + Ran Wei, Jinjiong Yu + + + Sharp one-point estimates and Minkowski content for the scaling limit of three-dimensional loop-erased random walk + https://arxiv.org/abs/2403.07256 + arXiv:2403.07256v2 Announce Type: replace +Abstract: In this work, we consider the scaling limit of loop-erased random walk (LERW) in three dimensions and prove that the limiting occupation measure is equivalent to its $\beta$-dimensional Minkowski content, where $\beta \in (1, 5/3]$ is its Hausdorff dimension. In doing this we also establish the existence of the two-point function and provide some sharp estimates on one-point function and ball-hitting probabilities for 3D LERW in any scale, which is a considerable improvement of previous results. + oai:arXiv.org:2403.07256v2 + math.PR + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Sarai Hernandez-Torres, Xinyi Li, Daisuke Shiraishi + + + A nontrivial uniform algebra Dirichlet on its maximal ideal space + https://arxiv.org/abs/2403.19583 + arXiv:2403.19583v5 Announce Type: replace +Abstract: It is shown that there exists a nontrivial uniform algebra that is Dirichlet on its maximal ideal space and has a dense set of elements that are exponentials. This answers a 65-year-old question of John Wermer and a 17-year-old question of Garth Dales and Joel Feinstein. Our example is P(X) for a certain compact set X in complex Euclidean 2-space ($\mathbb{C}^2$). It is also shown that there exists a logmodular uniform algebra with proper Shilov boundary but with no nontrivial Gleason parts. This answers a modification of another 65-year-old question of Wermer. + oai:arXiv.org:2403.19583v5 + math.CV + math.FA + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Alexander J. Izzo + + + On Classification of compact complex surfaces of class VII + https://arxiv.org/abs/2403.20178 + arXiv:2403.20178v2 Announce Type: replace +Abstract: Let $S$ be a minimal compact complex surface with Betti numbers $b_1(S)=1$ and $b_2(S)\ge 1$ i.e. a compact surface in class VII$_0^+$. We show that if there exists a twisted logarithmic 1-form $\tau\in H^0(S,\Omega^1(\log D)\otimes \mathcal L_\lambda)$, where $D$ is a non zero divisor and $\mathcal L\in H^1(S,\mathbb C^\star)$, then $S$ is a Kato surface. It is known that $\lambda$ is in fact real and we show that $\lambda\ge 1$ and unique if $S$ is not a Inoue-Hirzebruch surface. Moreover $\lambda=1$ if and only if $S$ is a Enoki surface. When $\lambda>1$ these conditions are equivalent to the existence of a negative PSH function $\hat \tau$ on the cyclic covering $p:\hat S\to S$ of $S$ which is PH outside $\hat D:=p^{-1}(D)$ with automorphy constant being the same automorphy constant $\lambda$ for a suitable automorphism of $\hat S$. With previous results obtained with V.Apostolov it suggests a strategy to prove the GSS conjecture. + oai:arXiv.org:2403.20178v2 + math.CV + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://creativecommons.org/publicdomain/zero/1.0/ + 10.1007/978-3-031-92297-8_7 + Georges Dloussky + + + Thin Simplices via Modular Arithmetic + https://arxiv.org/abs/2404.03975 + arXiv:2404.03975v2 Announce Type: replace +Abstract: The local $h^*$-polynomial is a natural invariant of a lattice polytope appearing in Ehrhart theory and Hodge theory. In this work, we study the question posed in [GKZ94] concerning the classification of lattice simplices with vanishing local $h^*$-polynomial. Such simplices are called thin. We relate this question to linear codes and hyperplane arrangements over finite rings. This allows us to obtain a complete classification of the $4$-dimensional thin simplices, extending the previously known results in dimensions up to $3$. + oai:arXiv.org:2404.03975v2 + math.CO + math.AG + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by/4.0/ + Vadym Kurylenko + + + An adaptive finite element multigrid solver using GPU acceleration + https://arxiv.org/abs/2405.05047 + arXiv:2405.05047v3 Announce Type: replace +Abstract: Adaptive finite elements combined with geometric multigrid solvers are one of the most efficient numerical methods for problems such as the instationary Navier-Stokes equations. Yet despite their efficiency, computations remain expensive and the simulation of, for example, complex flow problems can take many hours or days. GPUs provide an interesting avenue to speed up the calculations due to their very large theoretical performance. However, the large degree of parallelism and non-standard API make the use of GPUs in scientific computing challenging. In this work, we develop a GPU acceleration for the adaptive finite element library Gascoigne and study its effectiveness for different systems of partial differential equations. Our goal is thereby to integrate the GPU acceleration into the existing code with minimal changes, even when this requires a penalty in the GPU acceleration. Through the systematic formulation of all computations as linear algebra operations, we can employ GPU-accelerated linear algebra libraries, which simplifies the implementation and ensures the maintainability of the code while achieving very efficient GPU utilizations. Our results for a transport-diffusion equation, linear elasticity, and the instationary Navier-Stokes equations show substantial speedups of up to 20X compared to multi-core CPU implementations. + oai:arXiv.org:2405.05047v3 + math.NA + cs.NA + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by-sa/4.0/ + Manuel Liebchen, Robert Jendersie, Utku Kaya, Christian Lessig, Thomas Richter + + + Results on Dynamics of Bungee set of Composite Entire Functions in the Eremenko-Lyubich Class + https://arxiv.org/abs/2405.11217 + arXiv:2405.11217v4 Announce Type: replace +Abstract: In this paper, we have discussed the dynamics of composite entire functions in terms of relationship between bungee set, escaping set and filled-in Julia set. We have established some relation between the dynamics of composition of entire functions and the functions taken for composition. We have shown that the union of the bungee set of two entire functions contains the bungee set of the composite function. In addition, it is shown that the filled-in Julia set of composite entire functions contains the filled-in Julia set of functions used for the composition. The results have been illustrated with several examples. We have mostly dealt with permutable(commuting) functions. + oai:arXiv.org:2405.11217v4 + math.DS + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by/4.0/ + Dinesh Kumar, Soumyajeet Das + + + Periodic Waves for the Regularized Camassa-Holm Equation: Existence and Spectral Stability + https://arxiv.org/abs/2406.00433 + arXiv:2406.00433v5 Announce Type: replace +Abstract: In this paper, we investigate the existence and spectral stability of periodic traveling wave solutions for the regularized Camassa-Holm equation. To establish the existence of periodic waves, we employ tools from bifurcation theory to construct solutions with the zero-mean property. We also prove that such waves may not exist for the well-known Camassa-Holm equation. Regarding spectral stability, we analyze the difference between the number of negative eigenvalues of the second variation of the Lyapunov functional at the wave, restricted to the space of zero-mean periodic functions, and the number of negative eigenvalues of the matrix formed from the tangent space associated with the low-order conserved quantities of the evolution model. Finally, we address the problem of orbital stability as a consequence of the spectral stability. + oai:arXiv.org:2406.00433v5 + math.AP + math.CA + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by/4.0/ + 10.13140/RG.2.2.29071.39843 + Fabio Natali + + + Polynomial Poisson Algebras and Superintegrable Systems from Cartan centralisers of Types $B_3$, $C_3$ and $D_3$ + https://arxiv.org/abs/2406.01958 + arXiv:2406.01958v2 Announce Type: replace +Abstract: In this work, we construct explicit formulas for the generators of the Cartan centralisers of complex semisimple Lie algebras $B_n,C_n$ and $D_n$, the case $A_n$ being already known \cite{campoamor2023algebraic}. The precise structures for the cases of rank-three simple Lie algebras ($B_3,C_3$ and $D_3$) are provided, and the inclusion relations between the corresponding polynomial Poisson algebras (finitely generated Poisson algebras over $\mathbb{C}[\mathfrak{h}^*]$) are illustrated. We develop the idea of constructing algebraic superintegrable systems and their integrals from the generators of these polynomial Poisson algebras. In particular, we explicitly present the algebraic superintegrable systems corresponding to the Cartan reduction chains $\mathfrak{h} \subset \mathfrak{so}(6,\mathbb{C})$, $\mathfrak{h} \subset \mathfrak{so}(7,\mathbb{C})$, and $\mathfrak{h} \subset \mathfrak{sp}(6,\mathbb{C})$. + oai:arXiv.org:2406.01958v2 + math-ph + math.MP + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by/4.0/ + Rutwig Campoamor-Stursberg, Danilo Latini, Ian Marquette, Junze Zhang, Yao-Zhong Zhang + + + Exotically knotted 2-spheres and the fundamental groups of their complements + https://arxiv.org/abs/2406.07093 + arXiv:2406.07093v2 Announce Type: replace +Abstract: We show that for any finitely presented group $G$, there is a simply connected closed 4-manifold containing an infinite family of topologically isotopic but smoothly inequivalent 2-links whose 2-link group is $G$. We also show that, if $G$ satisfies the necessary topological conditions, these 2-links have nullhomotopic components. + oai:arXiv.org:2406.07093v2 + math.GT + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Younes Benyahia + + + Bifurcation analysis of figure-eight choreography in the three-body problem based on crystallographic point groups + https://arxiv.org/abs/2406.07717 + arXiv:2406.07717v3 Announce Type: replace +Abstract: The bifurcation of figure-eight choreography is analyzed by its symmetry group based on the variational principle of the action. The irreducible representations determine the symmetry and the dimension of the Lyapunov-Schmidt reduced action, which yields four types of bifurcations in the sequence of the bifurcation cascade. Type 1 bifurcation, represented by trivial representation, bifurcates two solutions. Type 2, by non-trivial one-dimensional representation, bifurcates two congruent solutions. Type 3 and 4, by two-dimensional irreducible representations, bifurcate two sets of three and six congruent solutions, respectively. We analyze numerical bifurcation solutions previously published and four new ones: non-symmetric choreographic solution of type 2, non-planar solution of type 2, $y$-axis symmetric solution of type 3, and non-symmetric solution of type 4. + oai:arXiv.org:2406.07717v3 + math-ph + math.MP + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by/4.0/ + 10.1088/1751-8121/ad97fc + J. Phys. A: Math. Theor. 58 (2025) 025203 (21pp) + Hiroshi Fukuda, Hiroshi Ozaki + + + Cycle conjectures and birational invariants over finite fields + https://arxiv.org/abs/2406.14438 + arXiv:2406.14438v2 Announce Type: replace +Abstract: 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 Grothendieck--Serre semi-simplicity conjecture for all smooth projective varieties over finite fields. We further show that the Tate, Beilinson, and 1-semi-simplicity conjecture in half of the degrees implies those conjectures in all degrees. + oai:arXiv.org:2406.14438v2 + math.AG + math.NT + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Samet Balkan, Stefan Schreieder + + + Parabolic vector bundles and Lie algebroid connections + https://arxiv.org/abs/2406.15828 + arXiv:2406.15828v2 Announce Type: replace +Abstract: Given a holomorphic Lie algebroid on an m-pointed Riemann surface, we define parabolic Lie algebroid connections on any parabolic vector bundle equipped with parabolic structure over the marked points. An analogue of the Atiyah exact sequence for parabolic Lie algebroids is constructed. For any Lie algebroid whose underlying holomorphic vector bundle is stable, we give a complete characterization of all the parabolic vector bundles that admit a parabolic Lie algebroid connection. + oai:arXiv.org:2406.15828v2 + math.AG + math.DG + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + David Alfaya, Indranil Biswas, Pradip Kumar, Anoop Singh + + + Small Volume Bodies of Constant Width with Tetrahedral Symmetries + https://arxiv.org/abs/2406.18428 + arXiv:2406.18428v2 Announce Type: replace +Abstract: For every $n\ge 2$, we construct a body $U_n$ of constant width $2$ in $\mathbb{E}^n$ with small volume and symmetries of a regular $n$-simplex. $U_2$ is the Reuleaux triangle. To the best of our knowledge, $U_3$ was not previously constructed, and its volume is smaller than the volume of other three-dimensional bodies of constant width with tetrahedral symmetries. While the volume of $U_3$ is slightly larger than the volume of Meissner's bodies of width $2$, it exceeds the latter by less than $0.137\%$. For all large $n$, the volume of $U_n$ is smaller than the volume of the ball of radius $0.891$. + oai:arXiv.org:2406.18428v2 + math.MG + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by/4.0/ + 10.3842/SIGMA.2025.109 + SIGMA 21 (2025), 109, 8 pages + Andrii Arman, Andriy Bondarenko, Andriy Prymak, Danylo Radchenko + + + Voting Profiles Admitting All Candidates as Knockout Winners + https://arxiv.org/abs/2407.01382 + arXiv:2407.01382v3 Announce Type: replace +Abstract: A set of $2^n$ candidates is presented to a commission. At every round, each member of this commission votes by pairwise comparison, and one-half of the candidates is deleted from the tournament, the remaining ones proceeding to the next round until the $n$-th round (the final one) in which the final winner is declared. The candidates are arranged on a board in a given order, which is maintained among the remaining candidates at all rounds. A study of the size of the commission is carried out in order to obtain the desired result of any candidate being a possible winner. For $2^n$ candidates with $n \geq 3$, we identify a voting profile with $4n -3$ voters such that any candidate could win simply by choosing a proper initial order of the candidates. Moreover, in the setting of a random number of voters, we obtain the same results, with high probability, when the expected number of voters is large. + oai:arXiv.org:2407.01382v3 + math.CO + math.PR + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Bernard De Baets, Emilio De Santis + + + Asymptotic theory for nonparametric testing of $k$-monotonicity in discrete distributions + https://arxiv.org/abs/2407.01751 + arXiv:2407.01751v2 Announce Type: replace +Abstract: In shape-constrained nonparametric inference, it is often necessary to perform preliminary tests to verify whether a probability mass function (p.m.f.) satisfies qualitative constraints such as monotonicity, convexity, or in general $k$-monotonicity. In this paper, we are interested in nonparametric testing of $k$-monotonicity of a finitely supported discrete distribution. We consider a unified testing framework based on a natural statistic which is directly derived from the very definition of $k$-monotonicity. The introduced framework allows us to design a new consistent method to select the unknown knot points that are required to consistently approximate the limit distribution of several test statistics based either on the empirical measure or the shape-constrained estimators of the p.m.f. We show that the resulting tests are asymptotically valid and consistent for any fixed alternative. Additionally, for the test based solely on the empirical measure, we study the asymptotic power under contiguous alternatives and derive a quantitative separation result that provides sufficient conditions to achieve a given power. We employ this test to design an estimator for the largest parameter $k \in \mathbb N_0$ such that the p.m.f. is $j$-monotone for all $j = 0, \ldots, k$, and show that the estimator is different from the true parameter with probability which is asymptotically smaller than the nominal level of the test. A detailed simulation study is performed to assess the finite sample performance of all the proposed tests, and applications to several real datasets are presented to illustrate the theory. + oai:arXiv.org:2407.01751v2 + math.ST + stat.ME + stat.TH + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Fadoua Balabdaoui, Antonio Di Noia + + + Asymptotic expansions for semilinear waves on asymptotically flat spacetimes + https://arxiv.org/abs/2407.08997 + arXiv:2407.08997v3 Announce Type: replace +Abstract: We establish precise asymptotic expansions for solutions to semilinear wave equations with power-type nonlinearities on asymptotically flat spacetimes. Our analysis focuses on two key cases: cubic nonlinearities and higher-order power nonlinearities. For cubic nonlinearities of the form $a(t,x)\phi^3$, we prove asymptotic expansions for the solution globally in the spacetime. In the special case of compact spatial regions, solutions exhibit the asymptotic behavior $\phi(t,x) = ct^{-2} + O(t^{-3+})$. For higher-order nonlinearities $a(t,x)\phi^p$ with $p\geq 4$, we prove the solution satisfies $\phi(t, x)= d t^{-3} + O(t^{-4+})$, thereby extending the classical Price's law (a late-time tail postulated in 1972) to nonlinear settings in a precise fashion. These results sharpen previous decay estimates for nonlinear waves. We develop a radiation field expansion and a low-energy resolvent expansion adapted to conormal asymptotic inputs, extending Hintz's approach for linear waves to the semilinear setting. Our methods connect geometric microlocal analysis (b-calculus) with classical physical-space techniques, providing a convenient tool for analyzing asymptotic behavior of nonlinear waves. + oai:arXiv.org:2407.08997v3 + math.AP + gr-qc + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by/4.0/ + Shi-Zhuo Looi, Haoren Xiong + + + Overlapping substitutions and tilings + https://arxiv.org/abs/2407.18666 + arXiv:2407.18666v3 Announce Type: replace +Abstract: We generalize the notion of (geometric) substitution rule to obtain overlapping substitutions. Our motivating example is the substitution presented in Ziherl, Dotera and Bekku \cite{DBZ}, which features a substitution matrix with non-integer entries. We give the meaning of such a matrix by showing that the right Perron--Frobenius eigenvector encodes the patch frequency of the resulting tiling. The patch frequencies are shown to be uniformly convergent, implying that the corresponding dynamical system is uniquely ergodic. Under mild assumptions, we further prove that the associated expansion constant is always an algebraic integer. In general, overlapping substitutions may yield a patch with illegal (partial) overlaps of tiles, even if it is locally consistent. We provide a sufficient condition for an overlapping substitution to be consistent, ensuring that no such illegal tiles emerge. Finally, we construct many intriguing one-dimensional overlapping substitutions and present higher dimensional examples from Delone multi-sets with inflation symmetry. + oai:arXiv.org:2407.18666v3 + math.CO + math.DS + math.NT + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by/4.0/ + Shigeki Akiyama, Yasushi Nagai, Shu-Qin Zhang + + + Semihomogenous vector bundles, $\mathbb Q$-twisted sheaves, duality, and linear systems on abelian varieties + https://arxiv.org/abs/2407.20646 + arXiv:2407.20646v2 Announce Type: replace +Abstract: In this paper we point out the natural relation between $\mathbb Q$-twisted objects of the derived category of abelian varieties, cohomological rank functions, and semihomogeneous vector bundles. We apply this to two basic classes of objects, corresponding to each other via the Fourier-Mukai-Poincar\'e transform: positive twists of the ideal sheaf of one point and of the evaluation complexes of ample simple semihomogeneous vector bundles. This naturally leads to the introduction of $\mathbb Q^{\ge 0}$- graded section modules associated to line bundles on abelian varieties built by means of semihomogeneous vector bundles (containing the usual section rings). We prove a duality relation between such modules associated to dual polarizations, which is not visible at the level of the usual section rings. Other applications include formulas relating the thresholds of relevant cohomological rank functions appearing in this context. As a consequence we show a lower bound for the base point free threshold of a polarization in function of its type, and some obstructions to surjectivity of multiplication maps of global sections of certain line bundles. + oai:arXiv.org:2407.20646v2 + math.AG + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Nelson Alvarado, Giuseppe Pareschi + + + Enumerating Finite Braid Group Orbits on $SL_2(\C)$-Character Varieties + https://arxiv.org/abs/2407.21180 + arXiv:2407.21180v3 Announce Type: replace +Abstract: We analyze finite orbits of the natural braid group action on the character variety of the $n$ times punctured sphere. Building on recent results relating middle convolution and finite complex reflection groups, our work implements Katz's middle convolution to explicitly classify finite orbits in the $SL_2(\C)$-character variety of the punctured sphere. We provide theoretical results on the existence of finite orbits arising from the imprimitive finite complex reflection groups and formulas for constructing such examples when they exist. In the primitive finite complex reflection groups, we perform an exhaustive search and provide computational results. Our contributions also include Magma computer code for middle convolution and for computing the orbit under this action when it is known to be finite. + oai:arXiv.org:2407.21180v3 + math.AG + math.DS + math.RT + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by-sa/4.0/ + Amal Vayalinkal + + + Homotopy $n$-types of cubical sets and graphs + https://arxiv.org/abs/2408.05289 + arXiv:2408.05289v2 Announce Type: replace +Abstract: We give a new construction of the model structure on the category of simplicial sets for homotopy $n$-types, originally due to Elvira-Donazar and Hernandez-Paricio, using a right transfer along the coskeleton functor. We observe that an analogous model structure can be constructed on the category of cubical sets, and use it to equip the category of (simple) graphs with a fibration category structure whose weak equivalences are discrete $n$-equivalences. + oai:arXiv.org:2408.05289v2 + math.CT + math.AT + math.CO + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Chris Kapulkin, Udit Mavinkurve + + + Products of two sober dcpo's need not be sober + https://arxiv.org/abs/2408.08587 + arXiv:2408.08587v4 Announce Type: replace +Abstract: We construct two dcpo's whose Scott spaces are sober, but the Scott space of their order product is not sober. This answers an open problem on the sobriety of Scott spaces. Meantime, we show that if $M$ and $N$ are special type of sober complete lattices, then the Scott space of their order product $M\times N$ is sober. + oai:arXiv.org:2408.08587v4 + math.GN + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://creativecommons.org/publicdomain/zero/1.0/ + Hualin Miao, Xiaoyong Xi, Xiaodong Jia, Qingguo Li, Dongsheng Zhao + + + Hecke $L$-values, definite Shimura sets and Mod $\ell$ non-vanishing + https://arxiv.org/abs/2408.13932 + arXiv:2408.13932v3 Announce Type: replace +Abstract: Let $\lambda$ be a self-dual Hecke character over an imaginary quadratic field $K$ of infinity type $(1,0)$. Let $\ell$ and $p$ be primes which are coprime to $6N_{K/\mathbb{Q}}({\mathrm cond}(\lambda))$. We determine the $\ell$-adic valuation of Hecke $L$-values $L(1,\lambda\chi)/\Omega_K$ as $\chi$ varies over $p$-power order anticyclotomic characters over $K$. As an application, for $p$ inert in $K$, we prove the vanishing of the $\mu$-invariant of Rubin's $p$-adic $L$-function, leading to the first results on the $\mu$-invariant of imaginary quadratic fields at non-split primes. + Our approach and results complement the work of Hida and Finis. The approach is rooted in the arithmetic of a CM form on a definite Shimura set.The application to Rubin's $p$-adic $L$-function also relies on the proof of his conjecture. Along the way, we present an automorphic view on Rubin's theory. + oai:arXiv.org:2408.13932v3 + math.NT + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by/4.0/ + Ashay A. Burungale, Wei He, Shinichi Kobayashi, Kazuto Ota + + + Fast convergence rates for estimating the stationary density in SDEs driven by a fractional Brownian motion with semi-contractive drift + https://arxiv.org/abs/2408.15904 + arXiv:2408.15904v2 Announce Type: replace +Abstract: We study the estimation of the invariant density of additive fractional stochastic differential equations with Hurst parameter $H \in (0,1)$. We first focus on continuous observations and develop a kernel-based estimator achieving faster convergence rates than previously available. This result stems from a martingale decomposition combined with new bounds on the (conditional) convergence in total variation to equilibrium of fractional SDEs. For $H<1/2$, we further refine the rates based on recent bounds on the marginal density. We then extend the methodology to discrete observations, showing that the same convergence rates can be attained. Moreover, we establish concentration inequalities for the estimator and introduce a data-driven bandwidth selection procedure that adapts to unknown smoothness. Numerical experiments for the fractional Ornstein-Uhlenbeck process illustrate the estimator's practical performance. Finally, our results weaken the usual convexity assumptions on the drift component, allowing us to consider settings where strong convexity only holds outside a compact set. + oai:arXiv.org:2408.15904v2 + math.ST + math.PR + stat.TH + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by/4.0/ + Chiara Amorino, Eulalia Nualart, Fabien Panloup, Julian Sieber + + + Effective estimate and Central Limit Theorem for Diophantine approximation on spheres + https://arxiv.org/abs/2409.02970 + arXiv:2409.02970v2 Announce Type: replace +Abstract: We study the counting function of rational approximations with given bounds on the denominator and satisfying the critical Dirichlet exponent on the sphere $S^d$, $d\geq 3$. We give an effective estimate for this counting function, with an error term of square root order, analogous to the optimal estimate in the Euclidean setting. We also show that the counting function has vanishing third and higher correlations and derive a Central Limit Theorem describing its fluctuations. We prove these results using arguments from homogeneous dynamics on the space of orthogonal lattices, in particular effective multiple equidistribution of all orders, which we establish for spherical averages and which could be useful for other applications. + oai:arXiv.org:2409.02970v2 + math.NT + math.DS + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by-nc-nd/4.0/ + Zouhair Ouaggag + + + Periodic solutions to nonlocal pseudo-differential equations. A bifurcation theoretical perspective + https://arxiv.org/abs/2409.04253 + arXiv:2409.04253v3 Announce Type: replace +Abstract: In this paper we use abstract bifurcation theory for Fredholm operators of index zero to deal with periodic even solutions of the one-dimensional equation $\mathcal{L}u=\lambda u+|u|^{p}$, where $\mathcal{L}$ is a nonlocal pseudodifferential operator defined as a Fourier multiplier and $\lambda$ is the bifurcation parameter. Our general setting includes the fractional Laplacian $\mathcal{L}\equiv(-\Delta)^{s}$ and sharpens the results obtained for this operator to date. As a direct application, we establish the existence of traveling waves for general nonlocal dispersive equations for some velocity ranges. + oai:arXiv.org:2409.04253v3 + math.AP + math.CA + math.FA + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Juan Carlos Sampedro + + + Optimal control for coupled sweeping processes under minimal assumptions + https://arxiv.org/abs/2409.07722 + arXiv:2409.07722v2 Announce Type: replace +Abstract: In this paper, the study of nonsmooth optimal control problems (P) involving a controlled sweeping process with three main characteristics is launched. First, the sweeping sets are nonsmooth, time-dependent, and uniformly prox-regular. Second, the sweeping process is coupled with a controlled differential equation. Third, a joint-state endpoints constraint set S is present. This general model incorporates different important controlled submodels, such as a class of second order sweeping processes, and coupled evolution variational inequalities. A full form of the nonsmooth Pontryagin maximum principle for strong local minimizers in (P) is derived for bounded or unbounded moving sweeping sets satisfying local constraint qualifications (CQ) without any additional restriction. The existence and uniqueness of a Lipschitz solution for the Cauchy problem of our dynamic is established and the existence of an optimal solution for (P) is obtained. Two of the novelties in achieving the first goal are (i) the construction of a problem over truncated sweeping sets and truncated joint endpoints constraint set that has the same strong local minimizer as (P) and its (CQ) automatically holds, and (ii) the complete redesign of the exponential-penalty approximation technique for problems with moving sweeping sets that do not require any special assumption on the sets, their corners, or on the gradients of their generators. The utility of the optimality conditions is illustrated with an example. + oai:arXiv.org:2409.07722v2 + math.OC + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Samara Chamoun, Vera Zeidan + + + Optimal sequencing depth for single-cell RNA-sequencing in Wasserstein space + https://arxiv.org/abs/2409.14326 + arXiv:2409.14326v2 Announce Type: replace +Abstract: How many samples should one collect for an empirical distribution to be as close as possible to the true population? This question is not trivial in the context of single-cell RNA-sequencing. With limited sequencing depth, profiling more cells comes at the cost of fewer reads per cell. Therefore, one must strike a balance between the number of cells sampled and the accuracy of each measured gene expression profile. In this paper, we analyze an empirical distribution of cells and obtain upper and lower bounds on the Wasserstein distance to the true population. Our analysis holds for general, non-parametric distributions of cells, and is validated by simulation experiments on a real single-cell dataset. + oai:arXiv.org:2409.14326v2 + math.ST + stat.ME + stat.TH + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Jakwang Kim, Sharvaj Kubal, Geoffrey Schiebinger + + + On the Kodaira dimension of some algebraic fiber spaces + https://arxiv.org/abs/2409.19981 + arXiv:2409.19981v4 Announce Type: replace +Abstract: In this paper, we study the descent of positivity of the canonical bundle along fiber spaces. As a consequence, we prove a conjecture of Schnell, establishing the equivalence between the Non-vanishing Conjecture and its generalized version proposed by Campana and Peternell. + oai:arXiv.org:2409.19981v4 + math.AG + math.CV + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Yongpan Zou + + + Equivalences in diagrammatic sets + https://arxiv.org/abs/2410.00123 + arXiv:2410.00123v2 Announce Type: replace +Abstract: We show that diagrammatic sets, a topologically sound alternative to polygraphs and strict $\omega$-categories, admit an internal notion of equivalence in the sense of coinductive weak invertibility. We prove that equivalences have the expected properties: they include all degenerate cells, are closed under 2-out-of-3, and satisfy an appropriate version of the "division lemma", which ensures that enwrapping a diagram with equivalences at all sides is an invertible operation up to higher equivalence. On the way to this result, we develop methods, such as an algebraic calculus of natural equivalences, for handling the weak units and unitors which set this framework apart from strict $\omega$-categories. + oai:arXiv.org:2410.00123v2 + math.CT + math.AT + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by-nc-nd/4.0/ + 10.1016/j.jpaa.2025.108165 + Journal of Pure and Applied Algebra, Volume 230, Issue 1, 2026, 108165 + Cl\'emence Chanavat, Amar Hadzihasanovic + + + Accessible Complexity Bounds for Restarted PDHG on Linear Programs with a Unique Optimizer + https://arxiv.org/abs/2410.04043 + arXiv:2410.04043v3 Announce Type: replace +Abstract: The restarted primal-dual hybrid gradient method (rPDHG) has recently emerged as an important tool for solving large-scale linear programs (LPs). For LPs with unique optima, we present an iteration bound of $O\left(\kappa\Phi\cdot\ln\left(\frac{\kappa\Phi\|w^*\|}{\varepsilon}\right)\right)$, where $\varepsilon$ is the target tolerance, $\kappa$ is the standard matrix condition number, $\|w^*\|$ is the norm of the optimal solution, and $\Phi$ is a geometric condition number of the LP sublevel sets. This iteration bound is "accessible" in the sense that computing it is typically no more difficult than computing the optimal solution itself. Indeed, we present a closed-form and tractably computable expression for $\Phi$. This enables an analysis of the "two-stage performance" of rPDHG: we show that the first stage identifies the optimal basis in ${O}\left(\kappa\Phi\cdot\ln(\kappa\Phi)\right)$ iterations, and the second stage computes an $\varepsilon$-optimal solution in $O\left(\|B^{-1}\|\|A\|\cdot\ln\left(\frac{\xi}{\varepsilon}\right)\right)$ additional iterations, where $A$ is the constraint matrix, $B$ is the optimal basis and $\xi$ is the smallest nonzero in the optimal solution. Furthermore, computational tests mostly confirm the tightness of our iteration bounds. We also show a reciprocal relation between the iteration bound and stability under data perturbation, which is also equivalent to (i) proximity to multiple optima, and (ii) the LP sharpness of the instance. Finally, we analyze an "optimized" primal-dual reweighting which offers some intuition concerning the step-size heuristics used in practice. + oai:arXiv.org:2410.04043v3 + math.OC + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Zikai Xiong + + + Log Canonical Minimal Model Program for corank one foliations on Threefolds + https://arxiv.org/abs/2410.05178 + arXiv:2410.05178v3 Announce Type: replace +Abstract: If $(X, \mcF, \D)$ is a projective rank two foliated log canonical triple such that $(X,B)$ is klt for some $0 \leq B \leq \D$, we show that we can run a $(K_\mcF +\Delta)$-MMP and any such MMP terminates with either a minimal model or Mori fiber space. Next, we establish a Bertini type lemma and adjunction for generalized foliated quadruples. Using these, we extend the full log canonical MMP to the setting of rank two NQC generalized foliated quadruples. Finally, we apply the generalized MMP to study the relation between different minimal models, namely, any two minimal models of a given foliated log canonical triple can be connected by a sequence of flops and in the boundary polarized case, the minimal models are good and only finitely many in number. + oai:arXiv.org:2410.05178v3 + math.AG + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Priyankur Chaudhuri, Roktim Mascharak + + + Improving the Accuracy of DC Optimal Power Flow Formulations via Parameter Optimization + https://arxiv.org/abs/2410.11725 + arXiv:2410.11725v3 Announce Type: replace +Abstract: DC Optimal Power Flow (DC-OPF) problems optimize the generators' active power setpoints while satisfying constraints based on the DC power flow linearization. The computational tractability advantages of DC-OPF problems come at the expense of inaccuracies relative to AC Optimal Power Flow (AC-OPF) problems that accurately model the nonlinear steady-state behavior of power grids. This paper proposes an algorithm that significantly improves the accuracy of the generators' active power setpoints from DC-OPF problems with respect to the corresponding AC-OPF problems over a specified range of operating conditions. Using sensitivity information in a machine learning-inspired methodology, this algorithm tunes coefficient and bias parameters in the DC power flow approximation to improve the accuracy of the resulting DC-OPF solutions. Employing the Truncated Newton Conjugate-Gradient (TNC) method, a Quasi-Newton optimization technique, this parameter tuning occurs during an offline training phase, with the resulting parameters then used in online computations. Numerical results underscore the algorithm's efficacy with accuracy improvements in squared two-norm and $\infty$-norm losses of up to $90\%$ and $79\%$, respectively, relative to traditional DC-OPF formulations. + oai:arXiv.org:2410.11725v3 + math.OC + cs.SY + eess.SY + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by-nc-sa/4.0/ + Babak Taheri, Daniel K. Molzahn + + + Graded supermanifolds and homogeneity + https://arxiv.org/abs/2411.00537 + arXiv:2411.00537v3 Announce Type: replace +Abstract: We introduce the concept of a homogeneity supermanifold, which is, roughly speaking, a supermanifold equipped with a privileged atlas whose coordinates carry prescribed (real) homogeneity degrees. This structure defines a sheaf of graded algebras on the supermanifold, regarded as an additional geometric structure. The guiding principle of this approach is that grading is ultimately related to homogeneity. Assigning homogeneity degrees to coordinates in a consistent way is equivalent to fixing a global vector field, the weight vector field. This approach is simple and substantially more general than most existing approaches to graded manifolds. In particular, the homogeneity degrees may be arbitrary real numbers, and the resulting category includes compact supermanifolds. + We systematically study homogeneity submanifolds, homogeneity Lie supergroups, tangent and cotangent lifts of homogeneity structures, homogeneous distributions and codistributions, as well as related notions such as double homogeneity. The main achievements of this framework include proofs of the homogeneous Poincar\'e Lemma, the homogeneous Frobenius Theorem, and the homogeneous symplectic Darboux Theorem, results that are of independent interest even in the purely even case. + oai:arXiv.org:2411.00537v3 + math.DG + math-ph + math.MP + math.SG + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Katarzyna Grabowska, Janusz Grabowski + + + Optimal control under unknown intensity with Bayesian learning + https://arxiv.org/abs/2411.04917 + arXiv:2411.04917v3 Announce Type: replace +Abstract: We investigate an optimal control problem motivated by neuroscience, where the dynamics is driven by a Poisson process with a controlled stochastic intensity and an unknown parameter. Given a prior distribution for the unknown parameter, we describe its evolution using Bayes' rule. We reformulate the optimization problem by applying Girsanov's theorem and establish a dynamic programming principle. Finally, we characterize the value function as the unique viscosity solution to a finite-dimensional Hamilton-Jacobi-Bellman equation, which can be solved numerically. + oai:arXiv.org:2411.04917v3 + math.OC + math.PR + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Nicolas Baradel, Quentin Cormier + + + Mutually non isomorphic mixed-norm Lebesgue spaces + https://arxiv.org/abs/2411.10576 + arXiv:2411.10576v2 Announce Type: replace +Abstract: We prove that for $1\le p,q\le\infty$ the mixed-norm spaces $L_q(L_p)$ are mutually non-isomorphic, with the only exception that $L_q(L_2)$ is isomorphic to $L_q(L_q)$ for all $1<q<\infty$. + oai:arXiv.org:2411.10576v2 + math.FA + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Jos\'e L. Ansorena, Glenier Bello + + + Functional normalizing flow for statistical inverse problems of partial differential equations + https://arxiv.org/abs/2411.13277 + arXiv:2411.13277v2 Announce Type: replace +Abstract: Inverse problems of partial differential equations are ubiquitous across various scientific disciplines and can be formulated as statistical inference problems using Bayes' theorem. To address large-scale problems, it is crucial to develop discretization-invariant algorithms, which can be achieved by formulating methods directly in infinite-dimensional space. We propose a novel normalizing flow based infinite-dimensional variational inference method (NF-iVI) to extract posterior information efficiently. Specifically, by introducing well-defined transformations, the prior in Bayes' formula is transformed into post-transformed measures that approximate the posterior. To circumvent the issue of mutually singular probability measures, we formulate general conditions for the employed transformations. As guiding principles, these conditions yield four concrete transformations. Additionally, to minimize computational demands, we have developed a conditional normalizing flow variant, termed CNF-iVI, which is adapt at processing measurement data of varying dimensions while requiring minimal computational resources. We apply the proposed algorithms to three typical inverse problems governed by the simple smooth equation, the steady-state Darcy flow equation, and the electric impedance tomography. Numerical results confirm our theoretical findings, illustrate the efficiency of our algorithms, and verify the discretization-invariant property. + oai:arXiv.org:2411.13277v2 + math.NA + cs.NA + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Yang Zhao, Haoyu Lu, Junxiong Jia, Tao Zhou + + + Quantum unique ergodicity for magnetic Laplacians on T^2 + https://arxiv.org/abs/2411.18449 + arXiv:2411.18449v2 Announce Type: replace +Abstract: Given a smooth integral two-form and a smooth potential on the flat torus of dimension 2, we study the high energy properties of the corresponding magnetic Schr\"odinger operator. Under a geometric condition on the magnetic field, we show that every sequence of high energy eigenfunctions satisfies the quantum unique ergodicity property even if the Liouville measure is not ergodic for the underlying classical flow (the Euclidean geodesic flow on the 2-torus). + oai:arXiv.org:2411.18449v2 + math.SP + math-ph + math.AP + math.MP + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by/4.0/ + L\'eo Morin, Gabriel Rivi\`ere + + + Global hyperbolicity in higher signatures + https://arxiv.org/abs/2411.19519 + arXiv:2411.19519v2 Announce Type: replace +Abstract: We provide a generalization of global hyperbolicity in pseudo-Riemannian spaces of signature (p, q) for p ___ q ___ 2. We then prove the compactness of causal diamonds in globally hyperbolic spaces and deduce the existence of solutions to a Plateau problem for this class of spaces. Finally, caracterize GH-regular representations in SO(p,q+1) as holonomies of some globally hyperbolic spaces modeled on H^{p,q}. + oai:arXiv.org:2411.19519v2 + math.DG + math.GT + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Rom\'eo Troubat (IRMA) + + + Implementation of neural network operators with applications to remote sensing data + https://arxiv.org/abs/2412.00375 + arXiv:2412.00375v2 Announce Type: replace +Abstract: In this paper, we provide two algorithms based on the theory of multidimensional neural network (NN) operators activated by hyperbolic tangent sigmoidal functions. Theoretical results are recalled to justify the performance of the here implemented algorithms. Specifically, the first algorithm models multidimensional signals (such as digital images), while the second one addresses the problem of rescaling and enhancement of the considered data. We discuss several applications of the NN-based algorithms for modeling and rescaling/enhancement remote sensing data (represented as images), with numerical experiments conducted on a selection of remote sensing (RS) images from the (open access) RETINA dataset. A comparison with classical interpolation methods, such as bilinear and bicubic interpolation, shows that the proposed algorithms outperform the others, particularly in terms of the Structural Similarity Index (SSIM). + oai:arXiv.org:2412.00375v2 + math.NA + cs.CV + cs.NA + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by-nc-nd/4.0/ + Danilo Costarelli, Michele Piconi + + + Large deviations for invariant measures of multivalued stochastic differential equations with jumps + https://arxiv.org/abs/2412.01225 + arXiv:2412.01225v2 Announce Type: replace +Abstract: This work focuses on multivalued stochastic differential equations with jumps. First, by employing the weak convergence approach, we establish the Freidlin-Wentzell uniform large deviation principle and the Dembo-Zeitouni uniform large deviation principle for these equations. Subsequently, based on these results, we derive both upper and lower bounds for the large deviations of invariant measures associated with the equations. + oai:arXiv.org:2412.01225v2 + math.PR + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Huijie Qiao + + + Linear minimum-variance approximants for noisy data + https://arxiv.org/abs/2412.01287 + arXiv:2412.01287v4 Announce Type: replace +Abstract: Inspired by recent developments in subdivision schemes founded on the Weighted Least Squares technique, we construct linear approximants for noisy data in which the weighting strategy minimizes the output variance, thereby establishing a direct correspondence with the Generalized Least Squares and the Minimum-Variance Formulas methodologies. By introducing annihilation-operators for polynomial spaces, we derive usable formulas that are optimal for general correlated non-uniform noise. We show that earlier subdivision rules are optimal for uncorrelated non-uniform noise and, finally, we present numerical evidence to confirm that, in the correlated case, the proposed approximants are better than those currently used in the subdivision literature. + oai:arXiv.org:2412.01287v4 + math.NA + cs.NA + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by/4.0/ + 10.1016/j.apnum.2025.12.002 + Sergio L\'opez Ure\~na, Dionisio F. Y\'a\~nez + + + Hypergeometric Motives from Euler Integral Representations + https://arxiv.org/abs/2412.03257 + arXiv:2412.03257v2 Announce Type: replace +Abstract: We revisit certain one-parameter families of affine covers arising naturally from Euler's integral representation of hypergeometric functions. We introduce a partial compactification of this family. We show that the zeta function of the fibers in the family can be written as an explicit product of $L$-series attached to nondegenerate hypergeometric motives and zeta functions of tori, twisted by Hecke Grossencharacters. This permits a combinatorial algorithm for computing the Hodge numbers of the family. + oai:arXiv.org:2412.03257v2 + math.NT + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by/4.0/ + Tyler L. Kelly, John Voight + + + On growth of Sobolev norms for periodic nonlinear Schr\"{o}dinger and generalised Korteweg-de Vries equations under critical Gibbs dynamics + https://arxiv.org/abs/2412.08630 + arXiv:2412.08630v2 Announce Type: replace +Abstract: We prove logarithmic growth bounds on Sobolev norms of the focusing mass-critical NLS and gKdV equations on the torus, which hold almost surely under the focusing Gibbs measure with optimal mass threshold constructed by Oh, Sosoe, and Tolomeo [Invent. Math. 227 (2022), no. 3, 1323--1429]. More precisely, we will establish almost sure growth bounds for solutions $u(t)$ of the equations of the form \[ \sup_{t \in [-T,T]} \lVert u(t) \rVert_{H^s(\mathbb{T})} \lesssim_{s, u_0} \log(2+T)\] with initial data $u_0 \in H^s(\mathbb{T})$ for $s< \frac{1}{2}$. The proof uses a generalisation of Bourgain's invariant measure argument for measures in a suitable Orlicz space. + oai:arXiv.org:2412.08630v2 + math.AP + math.PR + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + 10.1090/proc/17379 + Proc. Amer. Math. Soc. 153 (2025), 5215-5230 + Fabian H\"ofer, Niko A. Nikov + + + A Riemannian Optimization Perspective of the Gauss-Newton Method for Feedforward Neural Networks + https://arxiv.org/abs/2412.14031 + arXiv:2412.14031v5 Announce Type: replace +Abstract: In this work, we establish non-asymptotic convergence bounds for the Gauss-Newton method in training neural networks with smooth activations. In the underparameterized regime, the Gauss-Newton gradient flow in parameter space induces a Riemannian gradient flow on a low-dimensional embedded submanifold of the function space. Using tools from Riemannian optimization, we establish geodesic Polyak-Lojasiewicz and Lipschitz-smoothness conditions for the loss under appropriately chosen output scaling, yielding geometric convergence to the optimal in-class predictor at an explicit rate independent of the conditioning of the Gram matrix. In the overparameterized regime, we propose adaptive, curvature-aware regularization schedules that ensure fast geometric convergence to a global optimum at a rate independent of the minimum eigenvalue of the neural tangent kernel and, locally, of the modulus of strong convexity of the loss. These results demonstrate that Gauss-Newton achieves accelerated convergence rates in settings where first-order methods exhibit slow convergence due to ill-conditioned kernel matrices and loss landscapes. + oai:arXiv.org:2412.14031v5 + math.OC + cs.AI + cs.LG + cs.SY + eess.SY + stat.ML + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Semih Cayci + + + Introducing irrational enumeration: analytic combinatorics for objects of irrational size + https://arxiv.org/abs/2412.14682 + arXiv:2412.14682v3 Announce Type: replace +Abstract: We extend the scope of analytic combinatorics to classes containing objects that have irrational sizes. The generating function for such a class is a power series that admits irrational exponents (which we call a Ribenboim series). A transformation then yields a generalised Dirichlet series from which the asymptotics of the coefficients can be extracted by singularity analysis using an appropriate Tauberian theorem. In practice, the asymptotics can often be determined directly from the original generating function. We illustrate the technique with a variety of applications, including tilings with tiles of irrational area, ordered integer factorizations, lattice walks enumerated by Euclidean length, and plane trees with vertices of irrational size. We also explore phase transitions in the asymptotics of families of irrational combinatorial classes. + oai:arXiv.org:2412.14682v3 + math.CO + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by/4.0/ + David Bevan, Julien Cond\'e + + + Universal approximation on non-geometric rough paths and applications to financial derivatives pricing + https://arxiv.org/abs/2412.16009 + arXiv:2412.16009v2 Announce Type: replace +Abstract: We present a novel perspective on the universal approximation theorem for rough path functionals, introducing a polynomial-based approximation class. We extend universal approximation to non-geometric rough paths within the tensor algebra. This development addresses critical needs in finance, where no-arbitrage conditions necessitate It\^o integration. Furthermore, our findings motivate a hypothesis for payoff functionals in financial markets, allowing straightforward analysis of signature payoffs proposed in \cite{arribas2018derivativespricingusingsignature}. + oai:arXiv.org:2412.16009v2 + math.FA + math.PR + q-fin.PR + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Fabian A. Harang, Fred Espen Benth, Fride Straum + + + Efficient Implementation of Third-Order Tensor Methods with Adaptive Regularization for Unconstrained Optimization + https://arxiv.org/abs/2501.00404 + arXiv:2501.00404v3 Announce Type: replace +Abstract: High-order tensor methods that employ local Taylor models of degree $p$ within adaptive regularization frameworks (AR$p$) have recently received significant attention, due to their optimal/improved global and local rates of convergence, for both convex and nonconvex optimization problems. In this paper, we showcase the numerical performance of standard second- and third-order variants ($p=2,3$) and propose novel techniques for key algorithmic aspects when $p\geq 3$. In particular, we extend the interpolation-based updating strategy for the regularization parameter introduced in [Gould, Porcelli and Toint, Comput Optim Appl (2012) 53:1--22] for $p=2$, to the case when $p \geq 3$. We identify fundamental differences between the different local minima of the regularised subproblems for $p=2$ and $p \geq 3$ and their effect on algorithm performance. For $p\geq 3$, we introduce a novel pre-rejection technique that rejects poor/unsuccessful subproblem minimizers prior to any function evaluation. Numerical studies showcase the efficiency improvements generated by our proposed modifications of the AR$3$ algorithm. We also assess numerically, the effect of different subproblem termination conditions and the choice of the initial regularization parameter on the overall algorithm performance. Finally, we benchmark our best-performing AR$3$ variants, as well as those in [Birgin et al., Optim Lett (2020) 14:815--838], against second-order ones (AR$2$). Encouraging results on standard test problems are obtained, confirming that AR$3$ variants can be made to outperform second-order variants in terms of objective evaluations, derivative evaluations, and number of subproblem solves. We provide an efficient, extensive and modular software package in MATLAB that includes many AR$2$ and AR$3$ variants, including Hessian- and tensor-free ones, allowing ease of use and experimentation for interested users. + oai:arXiv.org:2501.00404v3 + math.OC + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by/4.0/ + Coralia Cartis, Raphael Hauser, Yang Liu, Karl Welzel, Wenqi Zhu + + + Anytime Validity is Free: Inducing Sequential Tests + https://arxiv.org/abs/2501.03982 + arXiv:2501.03982v5 Announce Type: replace +Abstract: Anytime valid sequential tests permit us to stop testing based on the current data, without invalidating the inference. Given a maximum number of observations $N$, one may believe this must come at the cost of power when compared to a conventional test that waits until all $N$ observations have arrived. Our first contribution is to show that this is false: for any valid test based on $N$ observations, we show how to construct an anytime valid sequential test that matches it after $N$ observations. Our second contribution is that we may continue testing by using the outcome of a $[0, 1]$-valued test as a conditional significance level in subsequent testing, leading to an overall procedure that is valid at the original significance level. This shows that anytime validity and optional continuation are readily available in traditional testing, without requiring explicit use of e-values. We illustrate this by deriving the anytime valid sequentialized $z$-test and $t$-test, which at time $N$ coincide with the traditional $z$-test and $t$-test. Finally, we characterize the SPRT by invariance under test induction, and also show under an i.i.d. assumption that the SPRT is induced by the Neyman-Pearson test for a tiny significance level and huge $N$. + oai:arXiv.org:2501.03982v5 + math.ST + stat.ME + stat.TH + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Nick W. Koning, Sam van Meer + + + Constructing Riemannian metrics with prescribed nodal sets for Laplacian eigenfunctions + https://arxiv.org/abs/2501.06352 + arXiv:2501.06352v2 Announce Type: replace +Abstract: Let $C$ be a configuration of $n$ ovals in $\mathbb{S}^2$. We show that there is a Riemannian metric $g$ over $\mathbb{S}^2$ with a Laplacian eigenfunction whose zero set is $C$, and the corresponding eigenvalue is the $k$-th eigenvalue for $n\leq k \leq \alpha_1 n$. We also have that $\lambda\operatorname{Vol}_g\left(\mathbb{S}^2\right) = \Theta(n)$. + Additionally, assuming $C$ can be drawn as a topological minor of the $m\times m$ grid graph, we show that there is an infinitesimal perturbation of the round metric on $\mathbb{S}^2$ and a corresponding Laplacian eigenfunction $f$ with eigenvalue $\Theta(m^2)$ such that the zero set of $f$ is equivalent to $C$. + oai:arXiv.org:2501.06352v2 + math.SP + math.AP + math.DG + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by/4.0/ + 10.1093/imrn/rnaf362 + International Mathematics Research Notices, Volume 2025, Issue 24, December 2025 + Yoav Krauz + + + Estimates for short character sums evaluated at homogeneous polynomials + https://arxiv.org/abs/2501.12325 + arXiv:2501.12325v2 Announce Type: replace +Abstract: Let $p$ be a prime. We prove bounds on short Dirichlet character sums evaluated at a class of homogeneous polynomials in arbitrary dimensions. In every dimension, this bound is nontrivial for sums over boxes with side lengths as short as $p^{1/4 + \kappa}$ for any $\kappa>0$. Our methods capitalize on the relationship between characters mod $p$ and characters over finite field extensions as well as bounds on the multiplicative energy of sets in products of finite fields. + oai:arXiv.org:2501.12325v2 + math.NT + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + J. Lond. Math. Soc. (2) 112 (2025), no. 5, Paper No. e70351 + Rena Chu + + + Continuous Algebra: Algebraic Semantics for Continuous Propositional Logic + https://arxiv.org/abs/2501.13114 + arXiv:2501.13114v3 Announce Type: replace +Abstract: We present algebraic semantics for Continuous Propositional Logic, CPL, introduced by Itai Ben Yaacov, viewed as {\L}ukasiewicz propositional logic with a reversed truth-falsity orientation and enriched by a unary halving connective. We introduce continuous algebras as MV-algebras together with an unary operator $\kappa$ analogous to the halving operator introduced in CPL and analyze their core structural properties, including ideals, quotient constructions, and subdirect representations. We further establish a correspondence between continuous algebras and the class of 2-divisible $\ell u$-groups, extending Mundici's representation theory to the continuous setting. This correspondence leads to a purely algebraic proof of the weak completeness theorem for CPL. + oai:arXiv.org:2501.13114v3 + math.LO + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://creativecommons.org/publicdomain/zero/1.0/ + Purbita Jana, Prateek + + + Network Oblivious Transfer via Noisy Channels: Limits and Capacities + https://arxiv.org/abs/2501.17021 + arXiv:2501.17021v4 Announce Type: replace +Abstract: In this paper, we study the information-theoretic limits of oblivious transfer via noisy channels. We also investigate oblivious transfer over a noisy multiple-access channel with two non-colluding senders and a single receiver. The channel is modeled through correlations among the parties, who may be honest-but-curious or, in the case of the receiver, potentially malicious. We first revisit the information-theoretic limits of two-party oblivious transfer and then extend these results to the multiple-access setting. For honest-but-curious participants, we introduce a multiparty protocol that reduces a general multiple access channel to a suitable correlation model. In scenarios with a malicious receiver, we characterize an achievable oblivious transfer rate region. + oai:arXiv.org:2501.17021v4 + cs.IT + cs.CR + math.IT + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by/4.0/ + Hadi Aghaee, Bahareh Akhbari, Christian Deppe + + + Locally chordal graphs + https://arxiv.org/abs/2501.17320 + arXiv:2501.17320v2 Announce Type: replace +Abstract: In this paper we study locally chordal graphs, i.e. graphs where every small-radius ball is chordal. We prove four characterizations of locally chordal graphs. Two are counterparts of the classic descriptions of chordal graphs via induced subgraphs and via minimal separators. For the latter, we rely on the local separators introduced in [CJKK25]. Another characterization is via the local covering, which was introduced in [DJKK22] to study local-global characteristics of graphs using coverings from topology. Our final characterization of locally chordal graphs is in terms of their binary cycle spaces. This gives a new characterization of chordal graphs as wheel-free graphs whose binary cycle space is generated by triangles. + Together, these results demonstrate the potential of local-global tools to uncover rich new properties. Our results in this paper also form the basis of our local-global analysis of locally chordal graphs [AKb], where we develop a local-global perspective into structural characterizations. + oai:arXiv.org:2501.17320v2 + math.CO + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Tara Abrishami, Paul Knappe, Jonas Kobler + + + Optimal sensor placement under model uncertainty in the weak-constraint 4D-Var framework + https://arxiv.org/abs/2502.00150 + arXiv:2502.00150v3 Announce Type: replace +Abstract: In data assimilation, the model may be subject to uncertainties and errors. The weak-constraint data assimilation framework enables incorporating model uncertainty in the dynamics of the governing equations. We propose a new framework for near-optimal sensor placement in the weak-constrained setting. This is achieved by first deriving a design criterion based on the expected information gain, which involves the Kullback-Leibler divergence from the forecast prior to the posterior distribution. An explicit formula for this criterion is provided, assuming that the model error and background are independent and Gaussian and the dynamics are linear. We discuss algorithmic approaches to efficiently evaluate this criterion through randomized approximations. To provide further insight and flexibility in computations, we also provide alternative expressions for the criteria. We provide an algorithm to find near-optimal experimental designs using column subset selection, including a randomized algorithm that avoids computing the adjoint of the forward operator. Through numerical experiments in one and two spatial dimensions, we show the effectiveness of our proposed methods. + oai:arXiv.org:2502.00150v3 + math.NA + cs.NA + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by/4.0/ + Alen Alexanderian, Hugo D\'iaz, Vishwas Rao, Arvind K. Saibaba + + + Even Hypergeometric Polynomials and Finite Free Commutators + https://arxiv.org/abs/2502.00254 + arXiv:2502.00254v2 Announce Type: replace +Abstract: We study in detail the class of even polynomials and their behavior with respect to finite free convolutions. To this end, we use some specific hypergeometric polynomials and a variation of the rectangular finite free convolution to understand even real-rooted polynomials in terms of positive-rooted polynomials. Then, we study some classes of even polynomials that are of interest in finite free probability, such as even hypergeometric polynomials, symmetrizations, and finite free commutators. Specifically, we provide many new examples of these objects, involving classical families of special polynomials (such as Laguerre, Hermite, and Jacobi). Finally, we relate the limiting root distributions of sequences of even polynomials with the corresponding symmetric measures that arise in free probability. + oai:arXiv.org:2502.00254v2 + math.CA + math.OA + math.PR + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by/4.0/ + 10.3842/SIGMA.2025.108 + SIGMA 21 (2025), 108, 33 pages + Jacob Campbell, Rafael Morales, Daniel Perales + + + Compactification of homology cells, Fujita's conjectures and the complex projective space + https://arxiv.org/abs/2502.01072 + arXiv:2502.01072v3 Announce Type: replace +Abstract: We show that a compact K\"ahler manifold $M$ containing a smooth connected divisor $D$ such that $M \setminus D$ is a homology cell, e.g., contractible, must be projective space with $D$ a hyperplane, provided $\dim M \not \equiv 3 \pmod 4$. This answers conjectures of Fujita in these dimensions. + oai:arXiv.org:2502.01072v3 + math.AG + math.DG + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Ping Li, Thomas Peternell + + + Strong law of large numbers for a function of the local times of a transient random walk on groups + https://arxiv.org/abs/2502.04792 + arXiv:2502.04792v2 Announce Type: replace +Abstract: This paper presents the strong law of large numbers for a function of the local times of a transient random walk on groups, extending the research of Asymont and Korshunov for random walks on the integer lattice $\mathbb{Z}^d$. Under some weaker conditions, we prove that certain function of the local times converges almost surely and in $L^1$ and $L^2$. The proof is mainly based on the subadditive ergodic theorem. + oai:arXiv.org:2502.04792v2 + math.PR + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Yinshan Chang, Qinwei Chen, Qian Meng, Xue Peng + + + A Bose-Laskar-Hoffman theory for $\mu$-bounded graphs with fixed smallest eigenvalue + https://arxiv.org/abs/2502.05520 + arXiv:2502.05520v2 Announce Type: replace +Abstract: In 2018, by Ramsey and Hoffman theory, Koolen, Yang, and Yang presented a structural result on graphs with smallest eigenvalue at least $-3$ and large minimum degree. In this study, we depart from the conventional use of Ramsey theory and instead employ a novel approach that combines the Bose-Laskar type argument with Hoffman theory to derive structural insights into $\mu$-bounded graphs with fixed smallest eigenvalue. Our method establishes a reasonable bound on the minimum degree. Note that local graphs of distance-regular graphs are $\mu$-bounded. We apply these results to characterize the structure for any local graph of a distance-regular graph with classical parameters $(D,b,\alpha,\beta)$. Consequently, we show that the parameter $\alpha$ is bounded by a cubic polynomial in $b$ if $D \geq 9$ and $b \geq 2$. We also show that $\alpha \leq 2$ if $b =2$ and $D \geq 12$. + oai:arXiv.org:2502.05520v2 + math.CO + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Jack H. Koolen, Hong-Jun Ge, Chenhui Lv, Qianqian Yang + + + Proof of a conjecture of Green and Liebeck on codes in symmetric groups + https://arxiv.org/abs/2502.10744 + arXiv:2502.10744v2 Announce Type: replace +Abstract: Let $A$ and $B$ be subsets of a finite group $G$ and $r$ a positive integer. If for every $g\in G$, there are precisely $r$ pairs $(a,b)\in A\times B$ such that $g=ab$, then $B$ is called a code in $G$ with respect to $A$ and we write $r G=A\boldsymbol{\cdot}B$. If in addition $B$ is a subgroup of $G$, then we say that $B$ is a subgroup code in $G$. In this paper we resolve a conjecture by Green and Liebeck \cite[Conjecture 2.3]{Green20} on certain subgroup codes in the symmetric group $S_n$. Let $n>2k$ and let $j$ be such that $2^j\leqslant k<2^{j+1}$. Suppose that $X$ is a conjugacy class in $S_n$ containing $x$, and $Y_k$ is the subgroup $S_k\times S_{n-k}$ of $S_n$, where the factor $S_k$ permutes the subset $\{1,\ldots,k\}$ and the factor $S_{n-k}$ permutes the subset $\{k+1,\ldots,n\}$. We prove that $r S_n=X\boldsymbol{\cdot}Y_k$ for some positive integer $r$ if and only if the cycle type of $x$ has exactly one cycle of length $2^i$ for $0\leqslant i\leqslant j$ and all other cycles have length at least $k+1$. We also propose several problems concerning the existence of certain subgroup codes in a finite group $G$ with respect to a conjugation-closed subset in $G$. + oai:arXiv.org:2502.10744v2 + math.CO + math.GR + math.RT + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Teng Fang, Jinbao Li + + + A Reynolds-semi-robust method with hybrid velocity and pressure for the unsteady incompressible Navier--Stokes equations + https://arxiv.org/abs/2502.15293 + arXiv:2502.15293v2 Announce Type: replace +Abstract: In this paper we propose and analyze a new Finite Element method for the solution of the two- and three-dimensional incompressible Navier--Stokes equations based on a hybrid discretization of both the velocity and pressure variables. The proposed method is pressure-robust, i.e., irrotational forcing terms do not affect the approximation of the velocity, and Reynolds-quasi-robust, with error estimates that, for smooth enough exact solutions, do not depend on the inverse of the viscosity. We carry out an in-depth convergence analysis highlighting pre-asymptotic convergence rates and validate the theoretical findings with a complete set of numerical experiments. + oai:arXiv.org:2502.15293v2 + math.NA + cs.NA + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by-nc-nd/4.0/ + 10.1137/25M1736104 + SIAM J. Numer. Anal., 2025, Vol. 63, No. 6, pp. 2317--2342 + Louren\c{c}o Beir\~ao da Veiga, Daniele A. Di Pietro, J\'er\^ome Droniou, Kirubell B. Haile, Thomas J. Radley + + + Noncommutative invariants of finite and classical groups + https://arxiv.org/abs/2502.16675 + arXiv:2502.16675v2 Announce Type: replace +Abstract: We investigate the structure of the invariant subring of the tensor algebra $T(W)$ of a $G$-representation $W$, viewed as a twisted commutative algebra (tca). For a faithful representation $W$ of a finite group $G$ over a field $k$, we show that if char$(k) \mid \#G$, then $T(W)^G$ is not finitely generated as a tca. In contrast, for a representation $W$ of a classical group $G_{\mathbb{Z}}$, we prove that the invariant subring $T(W_k)^{G_k}$ is finitely generated as a tca when $k$ is algebraically closed of sufficiently large characteristic, provided that $W$ admits a good filtration over $\mathbb{Z}$. Finally, we introduce a categorical variant of the Gelfand--Kirillov dimension and compute its value to be $\binom{n+1}{2}$ for $T(\mathbb{C}^n)$ as a tca. Our key insight is to use the Schur functor to reduce questions about noncommutative invariants to those concerning vector invariants. + oai:arXiv.org:2502.16675v2 + math.RA + math.AC + math.RT + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Karthik Ganapathy + + + The Golden Ratio Primal-Dual Algorithm with Two New Stepsize Rules for Convex-Concave Saddle Point Problems + https://arxiv.org/abs/2502.17918 + arXiv:2502.17918v3 Announce Type: replace +Abstract: In this paper, we present two stepsize strategies for the extended Golden Ratio primal-dual algorithm (E-GRPDA) designed to address structured convex optimization problems in finite-dimensional real Hilbert spaces. The first rule features a non-increasing primal stepsize that remains bounded below by a positive constant and is updated adaptively at each iteration, eliminating the need to compute the Lipschitz constant of the gradient of the function and the norm of the operator, without using backtracking. The second stepsize rule is adaptive, adjusting based on the local smoothness of the smooth component function and the norm of the operator involved. In other words, we present an adaptive version of the E-GRPDA algorithm. We prove that E-GRPDA achieves an ergodic sublinear convergence rate with both stepsize rules, based on the function-value residual and constraint violation rather than on the so-called primal-dual gap function. Additionally, we establish an R-linear convergence rate for E-GRPDA with the first stepsize rule, under standard assumptions and with appropriately chosen parameters. Through numerical experiments on various convex optimization problems, we demonstrate the effectiveness of our approaches and compare their performance to existing ones. + oai:arXiv.org:2502.17918v3 + math.OC + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Santanu Soe, Matthew K. Tam, V. Vetrivel + + + Weighted composition operators on Hilbert function spaces on the ball + https://arxiv.org/abs/2502.18301 + arXiv:2502.18301v2 Announce Type: replace +Abstract: A weighted composition operator on a reproducing kernel Hilbert space is given by a composition, followed by a multiplication. We study unitary and co-isometric weighted composition operators on unitarily invariant spaces on the Euclidean unit ball $\mathbb B_d$. We establish a dichotomy between the spaces $\mathcal{H}_\gamma$ with reproducing kernel $(1 - \langle z,w \rangle)^{-\gamma}$ for $\gamma > 0$, and all other spaces. Whereas the former admit many unitary weighted composition operators, the latter only admit trivial ones. This extends results of Mart\'in, Mas and Vukoti\'c from the disc to the ball. Some of our results continue to hold when $d = \infty$. + oai:arXiv.org:2502.18301v2 + math.FA + math.CV + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Michael Hartz, Maximilian Tornes + + + The Offended Voter Model + https://arxiv.org/abs/2502.18619 + arXiv:2502.18619v2 Announce Type: replace +Abstract: We study a variant of the voter model on a coevolving network in which interactions of two individuals with differing opinions only lead to an agreement on one of these opinions with a fixed probability $q$. Otherwise, with probability $1-q$, both individuals become offended in the sense that they never interact again, i.e. the corresponding edge is removed from the underlying network. Eventually, these dynamics reach an absorbing state at which there is only one opinion present in each connected component of the network. If globally both opinions are present at absorption we speak of "segregation'', otherwise of "consensus''. We rigorously show that segregation and a weaker form of consensus both occur with positive probability for every $q \in (0,1)$ and that the segregation probability tends to $1$ as $q \to 0$. Furthermore, we establish that, if $q \to 1$ fast enough, with high probability the population reaches consensus while the underlying network is still densely connected. We provide results from simulations to assess the obtained bounds and to discuss further properties of the limiting state. + oai:arXiv.org:2502.18619v2 + math.PR + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Raphael Eichhorn, Felix Hermann, Marco Seiler + + + Representations of Hamiltonian Lie algebras + https://arxiv.org/abs/2503.00749 + arXiv:2503.00749v3 Announce Type: replace +Abstract: We consider the Shen-Larsson functor from the category of modules for the symplectic Lie algebra $\s$ to the category of modules for the Hamiltonian Lie algebra and show that it preserves the irreducibility except in the finite number of cases. The obtained irreducible modules for the Hamiltonian Lie algebra are cuspidal, whose weight multiplicities equal the dimension of the corresponding module of the symplectic Lie algebra. This extends well-known results for other Cartan type Lie algebras to the Hamiltonian case. + oai:arXiv.org:2503.00749v3 + math.RT + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by/4.0/ + Vyacheslav Futorny, Santanu Tantubay + + + The $\ell_{\infty}$ Directed Spanning Forest + https://arxiv.org/abs/2503.02594 + arXiv:2503.02594v2 Announce Type: replace +Abstract: We study the $\ell_{\infty}$\textit{ directed spanning forest}(DSF), which is a directed forest with vertex set given by a homogeneous Poisson point process such that each Poisson point connects to the nearest Poisson point (in $\ell_{\infty}$ distance) with a strictly larger $y$-coordinate. In this paper, we prove that the $\ell_{\infty}$ DSF is connected and we find optimal estimates on the tail distribution of coalescing time of two $\ell_{\infty}$ DSF paths. Similar estimates were earlier obtained in \cite{coupier20212d} for the $\ell_2$ (Euclidean) DSF and showed that when properly scaled, it converges in distribution to the Brownian web. The geometry of $\ell_\infty$ balls compel us to develop new argument. + oai:arXiv.org:2503.02594v2 + math.PR + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Dipranjan Pal, Kumarjit Saha + + + Intersection of Positive Closed Currents + https://arxiv.org/abs/2503.06964 + arXiv:2503.06964v4 Announce Type: replace +Abstract: We investigate the intersection of positive closed currents in a general setting, employing tangent currents alongside King's residue formula. Our main result establishes a natural condition for the intersection--namely, the Dinh-Sibony product--of positive closed currents on domains and derives an integral representation of this intersection. In parallel, we study the existence, $h$-dimension, and shadow of tangent currents, extending our approach to the study of the self-intersection of analytic subsets. We also present a local version of superpotentials and a regularization of positive closed currents, explore the connections with slicing theory, and examine classical examples. Our work extends to general complex manifolds, including compact K\"ahler manifolds. + oai:arXiv.org:2503.06964v4 + math.CV + math.DG + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Taeyong Ahn + + + The linearized Korteweg-de Vries equation on the line with metric graph defects + https://arxiv.org/abs/2503.12639 + arXiv:2503.12639v2 Announce Type: replace +Abstract: We study the small amplitude linearization of the Korteweg de Vries equation on the line with a local defect scattering waves represented by a metric graph domain adjoined at one point. For a representative collection of examples, we derive explicit solution formulae expressed as contour integrals and obtain existence and unicity results for piecewise absolutely continuous data. In so doing, we implement the unified transform method on metric graphs comprising both bonds and leads for a third order differential operator. + oai:arXiv.org:2503.12639v2 + math.AP + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Dave Smith + + + Adjoint-based optimization of the Rayleigh-B\'enard instability with melting boundary + https://arxiv.org/abs/2503.23831 + arXiv:2503.23831v4 Announce Type: replace +Abstract: In this work, we propose an adjoint-based optimization procedure to control the onset of the Rayleigh-B\'enard instability with a melting front. A novel cut cell method is used to solve the Navier-Stokes equations in the Boussinesq approximation and the convection-diffusion equation in the fluid layer, as well as the heat equation in the solid phase. To track the interface we use the level set method where its evolution is simply governed by an advection equation. An incomplete continuous adjoint problem is then derived by treating the velocity field obtained from the forward problem as a known variable in the adjoint convection-diffusion equation, thereby avoiding the need to solve a Navier-Stokes adjoint in the fluid phase. To the best of our knowledge, this provides the first adjoint-based optimization framework for Rayleigh-B\'enard instability with a melting boundary. Two optimization problems, together with a comparison against a derivative-free particle-swarm method, demonstrate that the proposed incomplete adjoint yields gradients accurate enough to control the front shape while reducing the number of expensive function evaluations by about an order of magnitude. + oai:arXiv.org:2503.23831v4 + math-ph + math.MP + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by/4.0/ + Tomas Fullana, Alejandro Quir\'os Rodr\'iguez, Vincent Le Chenadec, Taraneh Sayadi + + + Unifying Different Theories of Conformal Prediction + https://arxiv.org/abs/2504.02292 + arXiv:2504.02292v2 Announce Type: replace +Abstract: This paper presents a unified framework for understanding the methodology and theory behind several different methods in the conformal prediction literature, which includes standard conformal prediction (CP), weighted conformal prediction (WCP), nonexchangeable conformal prediction (NexCP), and randomly-localized conformal prediction (RLCP), among others. At the crux of our framework is the idea that conformal methods are based on revealing partial information about the data at hand, and positing a conditional distribution for the data given the partial information. Different methods arise from different choices of partial information, and of the corresponding (approximate) conditional distribution. In addition to recovering and unifying existing results, our framework leads to both new theoretical guarantees for existing methods, and new extensions of the conformal methodology. + oai:arXiv.org:2504.02292v2 + math.ST + stat.TH + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Rina Foygel Barber, Ryan J. Tibshirani + + + Estimation of Population Linear Spectral Statistics by Marchenko--Pastur Inversion + https://arxiv.org/abs/2504.03390 + arXiv:2504.03390v4 Announce Type: replace +Abstract: A new method of estimating population linear spectral statistics from high-dimensional data is introduced. When the dimension $d$ grows with the sample size $n$ such that $\frac{d}{n} \to c>0$, the proposed method is the first with proven convergence rate of $\mathcal{O}(n^{\varepsilon - 1})$ for any $\varepsilon > 0$ in a general nonparametric setting. For Gaussian data, a CLT for the estimation error with normalization factor $n$ is shown. + oai:arXiv.org:2504.03390v4 + math.ST + stat.TH + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Ben Deitmar + + + Stochastic Optimization with Optimal Importance Sampling + https://arxiv.org/abs/2504.03560 + arXiv:2504.03560v2 Announce Type: replace +Abstract: Importance Sampling (IS) is a widely used variance reduction technique for enhancing the efficiency of Monte Carlo methods, particularly in rare-event simulation and related applications. Despite its effectiveness, the performance of IS is highly sensitive to the choice of the proposal distribution and often requires stochastic calibration. While the design and analysis of IS have been extensively studied in estimation settings, applying IS within stochastic optimization introduces a fundamental challenge: the decision variable and the importance sampling distribution are mutually dependent, creating a circular optimization structure. This interdependence complicates both convergence analysis and variance control. We consider convex stochastic optimization problems with linear constraints and propose a single-loop stochastic approximation algorithm, based on a joint variant of Nesterov's dual averaging, that jointly updates the decision variable and the importance sampling distribution, without time-scale separation or nested optimization. The method is globally convergent and achieves minimal asymptotic variance among stochastic gradient schemes, matching the performance of an oracle sampler adapted to the optimal solution. + oai:arXiv.org:2504.03560v2 + math.OC + cs.LG + math.ST + stat.ML + stat.TH + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by/4.0/ + Liviu Aolaritei, Bart P. G. Van Parys, Henry Lam, Michael I. Jordan + + + Feature Selection for Data-driven Explainable Optimization + https://arxiv.org/abs/2504.12184 + arXiv:2504.12184v2 Announce Type: replace +Abstract: Mathematical optimization, although often leading to NP-hard models, is now capable of solving even large-scale instances within reasonable time. However, the primary focus is often placed solely on optimality. This implies that while obtained solutions are globally optimal, they are frequently not comprehensible to humans, in particular when obtained by black-box routines. In contrast, explainability is a standard requirement for results in Artificial Intelligence, but it is rarely considered in optimization yet. There are only a few studies that aim to find solutions that are both of high quality and explainable. In recent work, explainability for optimization was defined in a data-driven manner: A solution is considered explainable if it closely resembles solutions that have been used in the past under similar circumstances. To this end, it is crucial to identify a preferably small subset of features from a presumably large set that can be used to measure instance similarity. In this work, we formally define the feature selection problem for explainable optimization and prove that its decision version is NP-complete. We introduce mathematical models for optimized feature selection. As their global solution requires significant computation time with modern mixed-integer linear solvers, we employ local heuristics. Our computational study using data that reflect real-world scenarios demonstrates that the problem can be solved practically efficiently for instances of reasonable size. + oai:arXiv.org:2504.12184v2 + math.OC + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by/4.0/ + Kevin-Martin Aigner, Marc Goerigk, Michael Hartisch, Frauke Liers, Arthur Miehlich, Florian R\"osel + + + Mean Curvature Flow for Isoparametric Submanifolds in Hyperbolic Spaces + https://arxiv.org/abs/2504.15602 + arXiv:2504.15602v2 Announce Type: replace +Abstract: Mean curvature flows of isoparametric submanifolds in Euclidean spaces and spheres have been studied by Liu and Terng. In particular, it was proved that such flows always have ancient solutions. This is also true for mean curvature flows of isoparametric hypersurfaces in hyperbolic spaces by a result of Reis and Tenenblat. In this paper, we study mean curvature flows of isoparametric submanifolds in hyperbolic spaces with arbitrary codimension. In particular, we will show that they always have ancient solutions and study their limiting behaviors. + oai:arXiv.org:2504.15602v2 + math.DG + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by/4.0/ + Xiaobo Liu, Wanxu Yang + + + Linear Regression Using Principal Components from General Hilbert-Space-Valued Covariates + https://arxiv.org/abs/2504.16780 + arXiv:2504.16780v3 Announce Type: replace +Abstract: We consider linear regression with covariates that are random elements in a general Hilbert space. We first develop a principal component analysis for Hilbert-space-valued covariates based on finite-dimensional projections of the covariance operator, and establish asymptotic linearity and joint Gaussian limits for the leading eigenvalues and eigenfunctions under mild moment conditions. We then propose a principal component regression framework that combines Euclidean and Hilbert-space-valued covariates, obtain root-n consistent and asymptotically normal estimators of the regression parameters, and establish the validity of nonparametric and wild bootstrap procedures for inference. Simulation studies with two- and three-dimensional imaging predictors demonstrate accurate recovery of eigenstructures, regression coefficients, and bootstrap coverage. The methodology is further illustrated with neuroimaging data, in both a standard regression setting and a precision-medicine formulation. + oai:arXiv.org:2504.16780v3 + math.ST + stat.ME + stat.TH + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Xinyi Li, Margaret Hoch, Michael R. Kosorok + + + On Commutative Analogues of Clifford Algebras and Their Decompositions + https://arxiv.org/abs/2504.19763 + arXiv:2504.19763v2 Announce Type: replace +Abstract: We investigate commutative analogues of Clifford algebras -- algebras whose generators square to $\pm1$ but commute, instead of anti-commuting as they do in Clifford algebras. We observe that commutativity allows for elegant results. We note that these algebras generalise multicomplex spaces -- we show that a commutative analogue of Clifford algebra is either isomorphic to a multicomplex space or to `multi split-complex space' (space defined just like multicomplex numbers but uses split-complex numbers instead of complex numbers). We do a general study of commutative analogues of Clifford algebras and use tools like operations of conjugation and idempotents to give a tensor product decomposition and a direct sum decomposition for them. Tensor product decomposition follows relatively easily from the definition. For the direct sum decomposition, we give explicit basis using new techniques. + oai:arXiv.org:2504.19763v2 + math.RA + math.AC + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + 10.1007/s00006-025-01422-6 + Advances in Applied Clifford Algebras, 36 (2026), 9, 29 pp + Heerak Sharma, Dmitry Shirokov + + + Differentiable Nonlinear Model Predictive Control + https://arxiv.org/abs/2505.01353 + arXiv:2505.01353v2 Announce Type: replace +Abstract: The efficient computation of parametric solution sensitivities is a key challenge in the integration of learning-enhanced methods with nonlinear model predictive control (MPC), as their availability is crucial for many learning algorithms. This paper discusses the computation of solution sensitivities of general nonlinear programs (NLPs) using the implicit function theorem (IFT) and smoothed optimality conditions treated in interior-point methods (IPM). We detail sensitivity computation within a sequential quadratic programming (SQP) method which employs an IPM for the quadratic subproblems. Previous works presented in the machine learning community are limited to convex or unconstrained formulations, or lack an implementation for efficient sensitivity evaluation. The publication is accompanied by an efficient open-source implementation within the acados framework, providing both forward and adjoint sensitivities for general optimal control problems, achieving speedups exceeding 3x over the state-of-the-art solvers mpc.pytorch and cvxpygen. + oai:arXiv.org:2505.01353v2 + math.OC + cs.AI + cs.LG + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Jonathan Frey, Katrin Baumg\"artner, Gianluca Frison, Dirk Reinhardt, Jasper Hoffmann, Leonard Fichtner, Sebastien Gros, Moritz Diehl + + + Sharp stability of the Heisenberg Uncertainty Principle: Second-Order and Curl-Free Field Cases + https://arxiv.org/abs/2505.02758 + arXiv:2505.02758v2 Announce Type: replace +Abstract: Using techniques from harmonic analysis, we derive several sharp stability estimates for the second order Heisenberg Uncertainty Principle. We also present the explicit lower and upper bounds for the sharp stability constants and compute their exact limits when the dimension $N\rightarrow\infty$. Our proofs rely on spherical harmonics decomposition and Fourier analysis, differing significantly from existing approaches in the literature. Our results substantially improve the stability constants of the second order Heisenberg Uncertainty Principle recently obtained in [27]. As direct consequences of our main results, we also establish the sharp stability, with exact asymptotic behavior of the stability constants, of the Heisenberg Uncertainty Principle with curl-free vector fields and a sharp version of the second order Poincar\'{e} type inequality with Gaussian measure. + oai:arXiv.org:2505.02758v2 + math.AP math-ph + math.CA math.MP - Mon, 22 Dec 2025 00:00:00 -0500 - cross + Tue, 23 Dec 2025 00:00:00 -0500 + replace http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Taylan Demir + Anh Xuan Do, Nguyen Lam, Guozhen Lu - Solomonoff-Inspired Hypothesis Ranking with LLMs for Prediction Under Uncertainty - https://arxiv.org/abs/2512.17145 - arXiv:2512.17145v1 Announce Type: cross -Abstract: Reasoning under uncertainty is a key challenge in AI, especially for real-world tasks, where problems with sparse data demands systematic generalisation. Existing approaches struggle to balance accuracy and simplicity when evaluating multiple candidate solutions. We propose a Solomonoff-inspired method that weights LLM-generated hypotheses by simplicity and predictive fit. Applied to benchmark (Mini-ARC) tasks, our method produces Solomonoff-weighted mixtures for per-cell predictions, yielding conservative, uncertainty-aware outputs even when hypotheses are noisy or partially incorrect. Compared to Bayesian Model Averaging (BMA), Solomonoff scoring spreads probability more evenly across competing hypotheses, while BMA concentrates weight on the most likely but potentially flawed candidates. Across tasks, this highlights the value of algorithmic information-theoretic priors for interpretable, reliable multi-hypothesis reasoning under uncertainty. - oai:arXiv.org:2512.17145v1 - cs.AI - cs.IT - math.IT - Mon, 22 Dec 2025 00:00:00 -0500 - cross + Global-local mixing for infinite measure dynamical systems + https://arxiv.org/abs/2505.07492 + arXiv:2505.07492v2 Announce Type: replace +Abstract: We prove global-local mixing for a large class of dynamical systems with infinite invariant measure. In particular, we treat intermittent maps including maps with multiple neutral fixed points, nonMarkovian intermittent maps, and multidimensional nonMarkovian intermittent maps. We also prove global-local mixing for parabolic rational maps of the complex plane. + oai:arXiv.org:2505.07492v2 + math.DS + Tue, 23 Dec 2025 00:00:00 -0500 + replace http://creativecommons.org/licenses/by/4.0/ - Josh Barber (QUT), Rourke Young (QUT), Cameron Coombe (QUT,CSIRO), Will Browne (QUT) + Douglas Coates, Ian Melbourne - Learning solution operator of dynamical systems with diffusion maps kernel ridge regression - https://arxiv.org/abs/2512.17203 - arXiv:2512.17203v1 Announce Type: cross -Abstract: Many scientific and engineering systems exhibit complex nonlinear dynamics that are difficult to predict accurately over long time horizons. Although data-driven models have shown promise, their performance often deteriorates when the geometric structures governing long-term behavior are unknown or poorly represented. We demonstrate that a simple kernel ridge regression (KRR) framework, when combined with a dynamics-aware validation strategy, provides a strong baseline for long-term prediction of complex dynamical systems. By employing a data-driven kernel derived from diffusion maps, the proposed Diffusion Maps Kernel Ridge Regression (DM-KRR) method implicitly adapts to the intrinsic geometry of the system's invariant set, without requiring explicit manifold reconstruction or attractor modeling, procedures that often limit predictive performance. Across a broad range of systems, including smooth manifolds, chaotic attractors, and high-dimensional spatiotemporal flows, DM-KRR consistently outperforms state-of-the-art random feature, neural-network and operator-learning methods in both accuracy and data efficiency. These findings underscore that long-term predictive skill depends not only on model expressiveness, but critically on respecting the geometric constraints encoded in the data through dynamically consistent model selection. Together, simplicity, geometry awareness, and strong empirical performance point to a promising path for reliable and efficient learning of complex dynamical systems. - oai:arXiv.org:2512.17203v1 - cs.LG + Surface stability of a layered magnetoelastic half-space + https://arxiv.org/abs/2505.10660 + arXiv:2505.10660v2 Announce Type: replace +Abstract: We evaluate the conditions for surface stability of a layered magnetoelastic half-space subjected to large deformations and a magnetic field. After reviewing the fundamental measures of deformation and summarizing the magnetostatic equations in Eulerian and Lagrangian forms, we derive the constitutive relations from a total energy function dependent on the deformation gradient and Lagrangian magnetic induction. Energy principles yield the equilibrium equations, magnetic field equations, and boundary conditions. The second variation of the energy functional provides the incremental equations and conditions for stability analysis. Surface instability is studied by linearizing increments of deformation and magnetic induction about a finitely deformed state under a magnetic field normal to the surface. Four illustrative cases are considered: (i) a layered non-magnetizable half-space with varying stiffness contrast; (ii) the critical stretch of a magnetoelastic half-space as a function of magnetic induction; (iii) surface stability of a magneto-sensitive layer atop a non-magnetizable substrate; and (iv) bifurcation conditions in a two-layered magnetoelastic solid with different stiffness ratios. Graphical results are provided throughout. + oai:arXiv.org:2505.10660v2 + math.NA + cond-mat.mtrl-sci cs.NA + math-ph + math.MP + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + 10.1016/j.ijsolstr.2025.113807 + International Journal of Solids and Structures, 327, 113807 + Davood Shahsavari, Luis Dorfmann, Prashant Saxena + + + Mean Curvature Flow of Closed Curves Evolving in Two Dimensional Manifolds + https://arxiv.org/abs/2505.12775 + arXiv:2505.12775v2 Announce Type: replace +Abstract: We investigate the motion of a family of closed curves evolving according to the geometric evolution law on a given two dimensional manifold which is embedded or immersed in the three-dimensional Euclidean space. We derive a system of nonlinear parabolic equations describing the motion of curves belonging to a given two-dimensional manifold. Using the abstract theory of analytic semiflows, we prove the local existence, uniqueness of H\"older smooth solutions to the governing system of nonlinear parabolic equations for the position vector parametrization of evolving curves. We apply the method of flowing finite volumes in combination with the methods of lines for numerical approximation of the governing equations. Qualitative analytical results are illustrated by various numerical experiments. + oai:arXiv.org:2505.12775v2 + math.AP + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by/4.0/ + Miroslav Kolar, Daniel Sevcovic + + + Free monoids and Riguet congruences + https://arxiv.org/abs/2505.15767 + arXiv:2505.15767v2 Announce Type: replace +Abstract: We begin by associating to $\mathbf{A}^{\star}$, the free monoid on a set $A$, a category $\mathsf{C}(\mathbf{A}^{\star})$ -- an instance of the free coproduct completion of a discrete category -- which is in general non-skeletal, and by proving that it is equivalent to $\mathsf{Set}^{A}_{\mathrm{f}}$, the category of finite $A$-sorted sets. Next, as a step toward constructing a skeletal quotient category of $\mathsf{C}(\mathbf{A}^{\star})$ via the notion of a Riguet congruence on a category, we recall this notion, correct and complete it, and examine its relationship with generalized congruences from both lattice-theoretic and category-theoretic perspectives. In particular, after introducing the notion of strong generalized congruence on a category, we prove that, for any category $\mathsf{C}$, there exists an isotone and Scott continuous morphism from $(\mathrm{RCgr}(\mathsf{C}),\subseteq)$, the bounded directed-complete ordered set of Riguet congruences on $\mathsf{C}$ to $(\mathrm{GCgr}(\mathsf{C}),\subseteq)$, the algebraic lattice of generalized congruences on $\mathsf{C}$, that sends a Riguet congruence $\Phi$ on $\mathsf{C}$ to the strong generalized congruence $\Phi^{\natural}$ on $\mathsf{C}$. Finally, for a suitable Riguet congruence on $\mathsf{C}(\mathbf{A}^{\star})$, denoted by $\equiv^{A}$, we construct a skeletal quotient category $\mathsf{Q}(\mathbf{A}^{\star})$ of $\mathsf{C}(\mathbf{A}^{\star})$ and prove that it is equivalent to $\mathsf{Set}^{A}_{\mathrm{f}}$ and also to $\mathsf{C}(\mathbf{A}^{\star})/{\equiv^{A\natural}}$, where $\equiv^{A\natural}$ is the strong generalized congruence on $\mathsf{C}(\mathbf{A}^{\star})$ canonically associated to $\equiv^{A}$. + oai:arXiv.org:2505.15767v2 + math.CT + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by-nc-nd/4.0/ + Juan Climent Vidal, Enric Cosme Ll\'opez, Ra\'ul Ruiz Mora + + + Normal Singular Geodesics of a Conformally Generic Sub-Riemannian Metric + https://arxiv.org/abs/2505.17319 + arXiv:2505.17319v2 Announce Type: replace +Abstract: We prove that a Ma\~n\'e generic real-analytic $D$-Hamiltonian H, subjected to a totally non-holonomic real-analytic distribution $D$, has no non-trivial normal $D$-singular orbits of minimal rank. If $D$ has co-rank 1, this implies that $H + u$, where $u$ is a generic real-analytic potential, does not admit non-trivial normal $D$-singular orbits. + oai:arXiv.org:2505.17319v2 + math.DS + math.DG + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by-nc-nd/4.0/ + 10.1088/1361-6544/ae2057 + 2025 Nonlinearity 38 125010 + Shahriar Aslani + + + A sparse $hp$-finite element method for piecewise-smooth differential equations with periodic boundary conditions + https://arxiv.org/abs/2505.17849 + arXiv:2505.17849v2 Announce Type: replace +Abstract: We develop an efficient $hp$-finite element method for piecewise-smooth differential equations with periodic boundary conditions, using orthogonal polynomials defined on circular arcs. The operators derived from this basis are banded and achieve optimal complexity regardless of $h$ or $p$, both for building the discretisation and solving the resulting linear system in the case where the operator is symmetric positive definite. The basis serves as a useful alternative to other bases such as the Fourier or integrated Legendre bases, especially for problems with discontinuities. We relate the convergence properties of these bases to regions of analyticity in the complex plane, and further use several differential equation examples to demonstrate these properties. The basis spans the low order eigenfunctions of constant coefficient differential operators, thereby achieving better smoothness properties for time-evolution partial differential equations. + oai:arXiv.org:2505.17849v2 math.NA - Mon, 22 Dec 2025 00:00:00 -0500 - cross + cs.NA + Tue, 23 Dec 2025 00:00:00 -0500 + replace http://creativecommons.org/licenses/by/4.0/ - Jiwoo Song, Daning Huang, John Harlim + Daniel VandenHeuvel, Sheehan Olver - MINPO: Memory-Informed Neural Pseudo-Operator to Resolve Nonlocal Spatiotemporal Dynamics - https://arxiv.org/abs/2512.17273 - arXiv:2512.17273v1 Announce Type: cross -Abstract: Many physical systems exhibit nonlocal spatiotemporal behaviors described by integro-differential equations (IDEs). Classical methods for solving IDEs require repeatedly evaluating convolution integrals, whose cost increases quickly with kernel complexity and dimensionality. Existing neural solvers can accelerate selected instances of these computations, yet they do not generalize across diverse nonlocal structures. In this work, we introduce the Memory-Informed Neural Pseudo-Operator (MINPO), a unified framework for modeling nonlocal dynamics arising from long-range spatial interactions and/or long-term temporal memory. MINPO, employing either Kolmogorov-Arnold Networks (KANs) or multilayer perceptron networks (MLPs) as encoders, learns the nonlocal operator and its inverse directly through neural representations, and then explicitly reconstruct the unknown solution fields. The learning is guarded by a lightweight nonlocal consistency loss term to enforce coherence between the learned operator and reconstructed solution. The MINPO formulation allows to naturally capture and efficiently resolve nonlocal spatiotemporal dependencies governed by a wide spectrum of IDEs and their subsets, including fractional PDEs. We evaluate the efficacy of MINPO in comparison with classical techniques and state-of-the-art neural-based strategies based on MLPs, such as A-PINN and fPINN, along with their newly-developed KAN variants, A-PIKAN and fPIKAN, designed to facilitate a fair comparison. Our study offers compelling evidence of the accuracy of MINPO and demonstrates its robustness in handling (i) diverse kernel types, (ii) different kernel dimensionalities, and (iii) the substantial computational demands arising from repeated evaluations of kernel integrals. MINPO, thus, generalizes beyond problem-specific formulations, providing a unified framework for systems governed by nonlocal operators. - oai:arXiv.org:2512.17273v1 + Ordine privo di periodicit\`a: il fascino matematico delle tassellazioni + https://arxiv.org/abs/2505.21379 + arXiv:2505.21379v2 Announce Type: replace +Abstract: This is a review (in Italian) on aperiodic tilings of the plane intended for a general audience. First, we recall some basic results about lattices and periodic tilings. Then, we move on to one-dimensional (domino) tilings and Wang tilings. We present a beautiful proof of the existence of an aperiodic set of Wang prototiles due to J. Kari. Next, we discuss Penrose tilings and their properties. Finally, we briefly present the recent discovery by D. Smith and his collaborators of an aperiodic monotile. + oai:arXiv.org:2505.21379v2 + math.HO + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Francesco D'Andrea + + + Attractor learning for spatiotemporally chaotic dynamical systems using echo state networks with transfer learning + https://arxiv.org/abs/2505.24099 + arXiv:2505.24099v2 Announce Type: replace +Abstract: In this paper, we explore the predictive capabilities of echo state networks (ESNs) for the generalized Kuramoto-Sivashinsky (gKS) equation, an archetypal nonlinear PDE that exhibits spatiotemporal chaos. Our research focuses on predicting changes in long-term statistical patterns of the gKS model that result from varying the dispersion relation or the length of the spatial domain. We use transfer learning to adapt ESNs to different parameter settings and successfully capture changes in the underlying chaotic attractor. Previous work has shown that transfer learning can be used effectively with ESNs for single-orbit prediction. The novelty of our paper lies in our use of this pairing to predict the long-term statistical properties of spatiotemporally chaotic PDEs. We also show that transfer learning nontrivially improves the length of time that predictions of individual gKS trajectories remain accurate. + oai:arXiv.org:2505.24099v2 + math.DS + cs.AI cs.LG - cs.NA - math-ph - math.MP - math.NA - Mon, 22 Dec 2025 00:00:00 -0500 - cross + nlin.CD + stat.ML + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Mohammad Shah Alam, William Ott, Ilya Timofeyev + + + On the geometry of holomorphic curves and complex surface + https://arxiv.org/abs/2505.24719 + arXiv:2505.24719v2 Announce Type: replace +Abstract: We investigate the geometry of holomorphic curves and complex surfaces from the perspective of singularity theory. We show that, with a suitable choice of a complex bilinear symmetric form, the families of functions and mappings that measure the contact between curves or surfaces and model objects become holomorphic. This allows the application of singularity theory, yielding analogues of classical results from the real case. Our approach enables the definition of geometric invariants of curves, which we call the $C$-curvature and $C$-torsion, as well as surface invariants such as the $C$-principal curvature and $C$-Gaussian curvature. It also gives geometric meaning to the complexification of of the families measuring contact of analytic surfaces in $\mathbb R^3$ with lines, planes and spheres. + oai:arXiv.org:2505.24719v2 + math.DG + Tue, 23 Dec 2025 00:00:00 -0500 + replace http://creativecommons.org/licenses/by/4.0/ - Farinaz Mostajeran, Aruzhan Tleubek, Salah A Faroughi + Amanda Dias Falqueto, Farid Tari - A Synthetic Instrumental Variable Method: Using the Dual Tendency Condition for Coplanar Instruments - https://arxiv.org/abs/2512.17301 - arXiv:2512.17301v1 Announce Type: cross -Abstract: Traditional instrumental variable (IV) methods often struggle with weak or invalid instruments and rely heavily on external data. We introduce a Synthetic Instrumental Variable (SIV) approach that constructs valid instruments using only existing data. Our method leverages a data-driven dual tendency (DT) condition to identify valid instruments without requiring external variables. SIV is robust to heteroscedasticity and can determine the true sign of the correlation between endogenous regressors and errors--an assumption typically imposed in empirical work. Through simulations and real-world applications, we show that SIV improves causal inference by mitigating common IV limitations and reducing dependence on scarce instruments. This approach has broad implications for economics, epidemiology, and policy evaluation. - oai:arXiv.org:2512.17301v1 - stat.ME + Axioms of Quantum Mechanics in light of Continuous Model Theory + https://arxiv.org/abs/2506.02029 + arXiv:2506.02029v2 Announce Type: replace +Abstract: The aim of this note is to recast somewhat informal axiom system of quantum mechanics used by physicists (Dirac calculus) in the language of Continuous Logic. + We note an analogy between Tarski's notion of cylindric algebras, as a tool of algebraisation of first order logic, and Hilbert spaces which can serve the same purpose for continuous logic of physics. + oai:arXiv.org:2506.02029v2 + math.LO + quant-ph + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by/4.0/ + Boris Zilber + + + On the orbital stability of periodic snoidal waves for the $\phi^4-$equation + https://arxiv.org/abs/2506.05547 + arXiv:2506.05547v4 Announce Type: replace +Abstract: The main purpose of this paper is to investigate the global well-posedness and orbital stability of odd periodic traveling waves for the $\phi^4$-equation in the Sobolev space of periodic functions with zero mean. We establish new results on the global well-posedness of weak solutions by combining a semigroup approach with energy estimates. As a consequence, we prove the orbital stability of odd periodic waves by applying a Morse index theorem to the constrained linearized operator defined in the Sobolev space with the zero mean property. + oai:arXiv.org:2506.05547v4 + math.AP + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by/4.0/ + B. S. Lonardoni, F. Natali + + + The Dixmier problem for skew PBW extensions and rings + https://arxiv.org/abs/2506.09285 + arXiv:2506.09285v5 Announce Type: replace +Abstract: In this paper we discuss for skew $PBW$ extensions the famous Dixmier problem formulated by Jacques Dixmier in 1968. The skew $PBW$ extensions are noncommutative rings of polynomial type and covers several algebras and rings arising in mathematical physics and noncommutative algebraic geometry. For this purpose, we introduce the Dixmier algebras and we will study the Dixmier problem for algebras over commutative rings, in particular, for $\mathbb{Z}$-algebras, i.e., for arbitrary rings. The results are focused on the investigation of the Dixmier problem for matrix algebras, product of algebras, tensor product of algebras and also on the Dixmier question for the following particular key skew $PBW$ extension: Let $K$ be a field of characteristic zero and let $\mathcal{CSD}_n(K)$ be the $K$-algebra generated by $n\geq 2$ elements $x_1,\dots,x_n$ subject to relations + $$x_jx_i=x_ix_j+d_{ij}, \ for \ all \ 1\leq i<j\leq n, \ with \ d_{ij}\in K-\{0\}$$. + We prove that the algebra $\mathcal{CSD}_n(K)$ is central and simple. In the last section we present a matrix-computational approach to the problem formulated by Jacques Dixmier and also we compute some concrete nontrivial examples of automorphisms of the first Weyl algebra $A_1(K)$ and $\mathcal{CSD}_n(K)$ using the MAPLE library SPBWE developed for the first author. We compute the inverses of these automorphisms, and for $A_1(K)$, its factorization through some elementary automorphisms. For $n$ odd, we found some endomorphisms of $\mathcal{CSD}_n(K)$ that are not automorphisms. We conjecture that $\mathcal{CSD}_n(K)$ is Dixmier when $n$ is even. + oai:arXiv.org:2506.09285v5 + math.RA + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + William Fajardo, Oswaldo Lezama + + + Density estimation via periodic scaled Korobov kernel method with exponential decay condition + https://arxiv.org/abs/2506.15419 + arXiv:2506.15419v2 Announce Type: replace +Abstract: We propose the periodic scaled Korobov kernel (PSKK) method for nonparametric density estimation on $\mathbb{R}^d$. By first wrapping the target density into a periodic version through modulo operation and subsequently applying kernel ridge regression in scaled Korobov spaces, we extend the kernel approach proposed by Kazashi and Nobile (SIAM J. Numer. Anal., 2023) and eliminate its requirement for inherent periodicity of the density function. This key modification enables effective estimation of densities defined on unbounded domains. We establish rigorous mean integrated squared error (MISE) bounds, proving that for densities with smoothness of order $\alpha$ and exponential decay, our PSKK method achieves an $\mathcal{O}(M^{-1/(1+1/(2\alpha)+\epsilon)})$ MISE convergence rate with an arbitrarily small $\epsilon>0$. While matching the convergence rate of the previous kernel approach, our method applies to non-periodic distributions at the cost of stronger differentiability and exponential decay assumptions. Numerical experiments confirm the theoretical results and demonstrate a significant improvement over traditional kernel density estimation in large-sample regimes. + oai:arXiv.org:2506.15419v2 math.ST + cs.NA + math.NA stat.TH - Mon, 22 Dec 2025 00:00:00 -0500 - cross - http://creativecommons.org/licenses/by-sa/4.0/ - Ratbek Dzhumashev, Ainura Tursunalieva + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by/4.0/ + Ziyang Ye, Haoyuan Tan, Xiaoqun Wang, Zhijian He - Spin minimum uncertainty states for refined uncertainty relations - https://arxiv.org/abs/2512.17307 - arXiv:2512.17307v1 Announce Type: cross -Abstract: Minimum uncertainty states of the conventional Heisenberg uncertainty relation have been extensively studied and are often regarded as the most classical quantum states from the perspective of uncertainty, providing valuable insight into the nature of quantumness and its potential applications. In this work, we investigate the minimum uncertainty states associated with an information-theoretic refinement of the Heisenberg uncertainty relation in general spin systems. Using two different approaches, the matrix formulation and the Wick symbol representation, we derive explicit expressions for the states that saturate the uncertainty bound. We show that spin coherent states indeed achieve minimum uncertainty, consistent with their conventional identification as the classical states of spin systems. Moreover, we also identify additional classes of minimum uncertainty states beyond the coherent family. Finally, we compare the spin-system results with the previously studied bosonic case and elucidate the origin of the differences between the two settings. - oai:arXiv.org:2512.17307v1 - quant-ph + A diagrammatic approach to reflection functors + https://arxiv.org/abs/2506.17070 + arXiv:2506.17070v3 Announce Type: replace +Abstract: We construct reflection functors for quiver Hecke algebras associated with arbitrary symmetrizable Kac-Moody algebras, from a higher representation-theoretic viewpoint. These functors provide a categorification of Lusztig's braid group action on the quantum group. Similar functors were recently constructed independently by Kashiwara-Kim-Oh-Park via a different approach. Moreover, we prove that our reflection functors satisfy the braid relations as natural isomorphisms. + oai:arXiv.org:2506.17070v3 + math.RT + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Haruto Murata + + + Normal Typicality and Dynamical Typicality for a Random Block-Band Matrix Model + https://arxiv.org/abs/2506.17177 + arXiv:2506.17177v2 Announce Type: replace +Abstract: We prove normal typicality and dynamical typicality for a (centered) random block-band matrix model with block-dependent variances. A key feature of our model is that we achieve intermediate equilibration times, an aspect that has not been proven rigorously in any model before. Our proof builds on recently established concentration estimates for products of resolvents of Wigner type random matrices [arXiv:2403.10359] and an intricate analysis of the deterministic approximation. + oai:arXiv.org:2506.17177v2 math-ph math.MP - Mon, 22 Dec 2025 00:00:00 -0500 - cross + math.PR + Tue, 23 Dec 2025 00:00:00 -0500 + replace http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Hao Dai, Yue Zhang + L\'aszl\'o Erd\H{o}s, Joscha Henheik, Cornelia Vogel - Active RIS-Aided Anti-Jamming Wireless Communications: A Stackelberg Game Perspective - https://arxiv.org/abs/2512.17335 - arXiv:2512.17335v1 Announce Type: cross -Abstract: The pervasive threat of jamming attacks, particularly from adaptive jammers capable of optimizing their strategies, poses a significant challenge to the security and reliability of wireless communications. This paper addresses this issue by investigating anti-jamming communications empowered by an active reconfigurable intelligent surface. The strategic interaction between the legitimate system and the adaptive jammer is modeled as a Stackelberg game, where the legitimate user, acting as the leader, proactively designs its strategy while anticipating the jammer's optimal response. We prove the existence of the Stackelberg equilibrium and derive it using a backward induction method. Particularly, the jammer's optimal strategy is embedded into the leader's problem, resulting in a bi-level optimization that jointly considers legitimate transmit power, transmit/receive beamformers, and active reflection. We tackle this complex, non-convex problem by using a block coordinate descent framework, wherein subproblems are iteratively solved via convex relaxation and successive convex approximation techniques. Simulation results demonstrate the significant superiority of the proposed active RIS-assisted scheme in enhancing legitimate transmissions and degrading jamming effects compared to baseline schemes across various scenarios. These findings highlight the effectiveness of combining active RIS technology with a strategic game-theoretic framework for anti-jamming communications. - oai:arXiv.org:2512.17335v1 - eess.SP - cs.IT - math.IT - Mon, 22 Dec 2025 00:00:00 -0500 - cross + Tolerants + https://arxiv.org/abs/2506.22897 + arXiv:2506.22897v3 Announce Type: replace +Abstract: We study a generalization of the discriminant of a polynomial, which we call the tolerant. The tolerant differs by multiplication by a square from the duplicant, which was discovered in recent work on $\mathbb{P}^1$-loop spaces in motivic homotopy theory. We show that the tolerant is rational by deriving a formula in terms of discriminants. This allows us to formulate a conjectural unstable Poincar\'e--Hopf formula over an arbitrary locus of points. We also show that the tolerant satisfies many of the same properties as the discriminant. A notable difference between the two is that the discriminant is inversion invariant for all polynomials, whereas the tolerant is only inversion invariant on a proper multiplicative subset of polynomials. + oai:arXiv.org:2506.22897v3 + math.AG + math.AC + Tue, 23 Dec 2025 00:00:00 -0500 + replace http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Xiao Tang, Zhen Ma, Bin Li, Cong Li, Qinghe Du, Dusit Niyato, Zhu Han + Swechchha Adhikari, Brent Hall, Stephen McKean - Sharp Structure-Agnostic Lower Bounds for General Functional Estimation - https://arxiv.org/abs/2512.17341 - arXiv:2512.17341v1 Announce Type: cross -Abstract: The design of efficient nonparametric estimators has long been a central problem in statistics, machine learning, and decision making. Classical optimal procedures often rely on strong structural assumptions, which can be misspecified in practice and complicate deployment. This limitation has sparked growing interest in structure-agnostic approaches -- methods that debias black-box nuisance estimates without imposing structural priors. Understanding the fundamental limits of these methods is therefore crucial. This paper provides a systematic investigation of the optimal error rates achievable by structure-agnostic estimators. We first show that, for estimating the average treatment effect (ATE), a central parameter in causal inference, doubly robust learning attains optimal structure-agnostic error rates. We then extend our analysis to a general class of functionals that depend on unknown nuisance functions and establish the structure-agnostic optimality of debiased/double machine learning (DML). We distinguish two regimes -- one where double robustness is attainable and one where it is not -- leading to different optimal rates for first-order debiasing, and show that DML is optimal in both regimes. Finally, we instantiate our general lower bounds by deriving explicit optimal rates that recover existing results and extend to additional estimands of interest. Our results provide theoretical validation for widely used first-order debiasing methods and guidance for practitioners seeking optimal approaches in the absence of structural assumptions. This paper generalizes and subsumes the ATE lower bound established in \citet{jin2024structure} by the same authors. - oai:arXiv.org:2512.17341v1 - stat.ML - cs.LG - econ.EM - math.ST - stat.ME - stat.TH - Mon, 22 Dec 2025 00:00:00 -0500 - cross - http://creativecommons.org/licenses/by-nc-sa/4.0/ - Jikai Jin, Vasilis Syrgkanis + An Equivalence Between Erd\H{o}s's Square Packing Conjecture and the Convergence of an Infinite Series + https://arxiv.org/abs/2506.23284 + arXiv:2506.23284v2 Announce Type: replace +Abstract: Let $f(n)$ denote the maximum sum of the side lengths of $n$ non-overlapping squares packed inside a unit square. We prove that $f(n^2+1) = n$ for all positive integers $n$ if and only if the sum $\sum_{k\geq 1}(f(k^2+1)-k)$ converges. We also show that if $f(k^2+1) = k$, for infinitely many positive integers then $f(k^2+1) = k$ for all positive integers. + oai:arXiv.org:2506.23284v2 + math.CO + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Anshul Raj Singh - Generative modeling of conditional probability distributions on the level-sets of collective variables - https://arxiv.org/abs/2512.17374 - arXiv:2512.17374v1 Announce Type: cross -Abstract: Given a probability distribution $\mu$ in $\mathbb{R}^d$ represented by data, we study in this paper the generative modeling of its conditional probability distributions on the level-sets of a collective variable $\xi: \mathbb{R}^d \rightarrow \mathbb{R}^k$, where $1 \le k<d$. We propose a general and effcient learning approach that is able to learn generative models on different level-sets of $\xi$ simultaneously. To improve the learning quality on level-sets in low-probability regions, we also propose a strategy for data enrichment by utilizing data from enhanced sampling techniques. We demonstrate the effectiveness of our proposed learning approach through concrete numerical examples. The proposed approach is potentially useful for the generative modeling of molecular systems in biophysics, for instance. - oai:arXiv.org:2512.17374v1 - stat.ML - math.OC - Mon, 22 Dec 2025 00:00:00 -0500 - cross + Degenerate symplectic fixed points and Gromov-Witten invariants + https://arxiv.org/abs/2507.04191 + arXiv:2507.04191v5 Announce Type: replace +Abstract: We establish a connection between Gromov-Witten invariants and the number of fixed points of Hamiltonian diffeomorphisms on a closed rational symplectic manifold via deformed Hamiltonian spectral invariants. We generalize Givental's symplectic fixed point theorem for Fano toric manifolds to closed rational symplectic manifolds which admit nonzero Gromov-Witten invariants with fixed marked points and one point insertion. We prove a new cuplength estimate of symplectic fixed points involved in deformed spectral invariants. We extend Schwarz's quantum cuplength to the notion of deformed quantum cuplength for symplectic periods and employ it to estimate the number of fixed points of Hamiltonian diffeomorphisms on monotone symplectic manifolds with nonzero mixed Gromov-Witten invariants. + oai:arXiv.org:2507.04191v5 + math.SG + math.AG + math.DS + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Wenmin Gong + + + Jacobi-Haantjes manifolds, integrability and dissipative mechanical systems + https://arxiv.org/abs/2507.11715 + arXiv:2507.11715v3 Announce Type: replace +Abstract: The notion of Jacobi-Haantjes manifold, consisting of a Jacobi manifold endowed with an algebra of extended Haantjes operator fields, is proposed as a natural geometric framework which allows us to define the notion of integrability of both conservative and dissipative Hamiltonian systems, in a unified way. As a reduction, contact-Haantjes manifolds are defined. We prove that the integrability of a contact Hamiltonian system is equivalent to the existence of a suitable Abelian extended Haantjes algebra associated with the system. This result allows us to define a large class of new, completely integrable contact Hamiltonian systems from a given extended Haantjes algebra. + Moreover, we propose a theory of separation of variables for dissipative systems. This result is achieved by lifting a dissipative system into a higher-dimensional manifold, obtained as the symplectization of the Jacobi-Haantjes structure associated with the system. This new manifold naturally acquires the structure of a symplectic-Haantjes manifold. We prove that the Darboux-Haantjes coordinates which separate the Hamilton-Jacobi equation of the higher-dimensional symplectic-Haantjes manifold are in fact separation variables for the Hamilton equations associated with the original dissipative system. + oai:arXiv.org:2507.11715v3 + math-ph + math.DG + math.MP + Tue, 23 Dec 2025 00:00:00 -0500 + replace http://creativecommons.org/licenses/by/4.0/ - Fatima-Zahrae Akhyar, Wei Zhang, Gabriel Stoltz, Christof Sch\"utte + Rafael Azuaje, Piergiulio Tempesta - Timely Information Updating for Mobile Devices Without and With ML Advice - https://arxiv.org/abs/2512.17381 - arXiv:2512.17381v1 Announce Type: cross -Abstract: This paper investigates an information update system in which a mobile device monitors a physical process and sends status updates to an access point (AP). A fundamental trade-off arises between the timeliness of the information maintained at the AP and the update cost incurred at the device. To address this trade-off, we propose an online algorithm that determines when to transmit updates using only available observations. The proposed algorithm asymptotically achieves the optimal competitive ratio against an adversary that can simultaneously manipulate multiple sources of uncertainty, including the operation duration, the information staleness, the update cost, and the availability of update opportunities. Furthermore, by incorporating machine learning (ML) advice of unknown reliability into the design, we develop an ML-augmented algorithm that asymptotically attains the optimal consistency-robustness trade-off, even when the adversary can additionally corrupt the ML advice. The optimal competitive ratio scales linearly with the range of update costs, but is unaffected by other uncertainties. Moreover, an optimal competitive online algorithm exhibits a threshold-like response to the ML advice: it either fully trusts or completely ignores the ML advice, as partially trusting the advice cannot improve the consistency without severely degrading the robustness. Extensive simulations in stochastic settings further validate the theoretical findings in the adversarial environment. - oai:arXiv.org:2512.17381v1 - cs.NI - cs.IT - cs.LG - math.IT - Mon, 22 Dec 2025 00:00:00 -0500 - cross + A Block Reduction Method for Random Band Matrices with General Variance Profiles + https://arxiv.org/abs/2507.11945 + arXiv:2507.11945v2 Announce Type: replace +Abstract: We present a novel block reduction method for the study of a general class of random band matrices (RBM) defined on the $d$-dimensional lattice $\mathbb{Z}_{L}^d:=\{1,2,\ldots,L\}^{d}$ for $d\in \{1,2\}$, with band width $W$ and an almost arbitrary variance profile subject to a core condition. We prove the delocalization of bulk eigenvectors for such RBMs under the assumptions $W\ge L^{1/2+\varepsilon}$ in one dimension and $W\geq L^{\varepsilon}$ in two dimensions, where $\varepsilon$ is an arbitrarily small constant. This result extends the findings of arXiv:2501.01718 and arXiv:2503.07606 on block RBMs to models with general variance profiles. Furthermore, we generalize our results to Wegner orbital models with small interaction strength $\lambda\ll 1$. Under the sharp condition $\lambda\gg W^{-d/2}$, we establish optimal lower bounds for the localization lengths of bulk eigenvectors, thereby extending the results of arXiv:2503.11382 to settings with nearly arbitrary potential and hopping terms. Our block reduction method provides a powerful and flexible framework that reduces both the dynamical analysis of the loop hierarchy and the derivation of deterministic estimates for general RBMs to the corresponding analysis of block RBMs, as developed in arXiv:2501.01718, arXiv:2503.07606 and arXiv:2503.11382. + oai:arXiv.org:2507.11945v2 + math.PR + Tue, 23 Dec 2025 00:00:00 -0500 + replace http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Yu-Pin Hsu, Yi-Hsuan Tseng + Jiaqi Fan, Fan Yang, Jun Yin - Alternating Direction Method of Multipliers for Nonlinear Matrix Decompositions - https://arxiv.org/abs/2512.17473 - arXiv:2512.17473v1 Announce Type: cross -Abstract: We present an algorithm based on the alternating direction method of multipliers (ADMM) for solving nonlinear matrix decompositions (NMD). Given an input matrix $X \in \mathbb{R}^{m \times n}$ and a factorization rank $r \ll \min(m, n)$, NMD seeks matrices $W \in \mathbb{R}^{m \times r}$ and $H \in \mathbb{R}^{r \times n}$ such that $X \approx f(WH)$, where $f$ is an element-wise nonlinear function. We evaluate our method on several representative nonlinear models: the rectified linear unit activation $f(x) = \max(0, x)$, suitable for nonnegative sparse data approximation, the component-wise square $f(x) = x^2$, applicable to probabilistic circuit representation, and the MinMax transform $f(x) = \min(b, \max(a, x))$, relevant for recommender systems. The proposed framework flexibly supports diverse loss functions, including least squares, $\ell_1$ norm, and the Kullback-Leibler divergence, and can be readily extended to other nonlinearities and metrics. We illustrate the applicability, efficiency, and adaptability of the approach on real-world datasets, highlighting its potential for a broad range of applications. - oai:arXiv.org:2512.17473v1 - eess.SP - cs.LG - math.OC - stat.ML - Mon, 22 Dec 2025 00:00:00 -0500 - cross + Characterizations of certain matroids by maximizing valuative invariants + https://arxiv.org/abs/2507.14720 + arXiv:2507.14720v3 Announce Type: replace +Abstract: Luis Ferroni and Alex Fink recently introduced a polytope of all unlabeled matroids of rank $r$ on $n$ elements, and they showed that the vertices of this polytope come from matroids that can be characterized by maximizing a sequence of valuative invariants. We prove that a number of the matroids that they conjectured to yield vertices indeed do (these include cycle matroids of complete graphs, projective geometries, and Dowling geometries), and we give additional examples (including truncations of cycle matroids of complete graphs, Bose-Burton geometries, and binary and free spikes with tips). We prove a special case of a conjecture of Ferroni and Fink by showing that direct sums of uniform matroids yield vertices of their polytope, and we prove a similar result for direct sums whose components are in certain restricted classes of extremal matroids. + oai:arXiv.org:2507.14720v3 + math.CO + Tue, 23 Dec 2025 00:00:00 -0500 + replace http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Atharva Awari, Nicolas Gillis, Arnaud Vandaele + Joseph E. Bonin + + + Comparing four definitions of cotilting modules + https://arxiv.org/abs/2507.18860 + arXiv:2507.18860v2 Announce Type: replace +Abstract: In contrast to the theory of tilting modules, the dual theory lacks a unified definition. Nevertheless, several notions of cotilting modules have been proposed. In this paper, we compare four of the main definitions of cotilting modules that have appeared in the literature. We show that, in the setting of finitely generated right modules, three of these definitions coincide over right Artinian Noetherian algebras, and all four coincide over Artin algebras. + oai:arXiv.org:2507.18860v2 + math.RT + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Kamran Divaani-Aazar, Ali Mahin Fallah, Massoud Tousi + + + Local and global well-posedness for the kinetic derivative NLS on $\mathbb{R}$ + https://arxiv.org/abs/2507.20271 + arXiv:2507.20271v2 Announce Type: replace +Abstract: We investigate the local and global well-posedness of the kinetic derivative nonlinear Schr\"odinger equation (KDNLS) on $\mathbb{R}$, described by \[ i\partial_t u + \partial_x^2 u = i\alpha \partial_x (|u|^2 u) + i\beta \partial_x (H(|u|^2) u), \] where $\alpha, \beta \in \mathbb{R}$, and $H$ represents the Hilbert transformation. For KDNLS, the $L^2$ norm of a solution is decreasing (resp. increasing, conserved) when $\beta$ is negative (resp. positive, zero). Focusing on the Sobolev spaces $H^2$ and $H^2 \cap H^{1,1}$, we establish local well-posedness via the energy method combined with gauge transformations to address resonant interactions in both cases of negative and positive $\beta$. For the dissipative case $\beta < 0$, we further demonstrate global well-posedness by deriving an a priori bound in $H^2$. + oai:arXiv.org:2507.20271v2 + math.AP + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by/4.0/ + Nobu Kishimoto, Kiyeon Lee - On the abelian structure of noncompetitive chemical reaction networks - https://arxiv.org/abs/2512.17491 - arXiv:2512.17491v1 Announce Type: cross -Abstract: Chemical reaction networks (CRNs) are foundational models for describing complex biochemical processes. We study noncompetitive CRNs, a class of networks whose static states are rate-independent, and that can implement ReLU neural networks. A central contribution of this work is that noncompetitive CRNs are special instances of Abelian networks (ANs) a well-established framework for self-organized criticality. CRNs of interest in biochemistry and systems biology are embedded in complex networks so that local CRNs have to respond to internal and environmental cues. We describe the network's response to such perturbations using a sandpile Markov chain whose state space is the set of CRN's static states, from where no reaction is possible. The addition of molecules to a static state induces reactions that move the system into a new static state. For noncompetitive CRNs of finite state space, we use AN theory to get that the fraction of static states that are recurrent is in one to one correspondence with the critical group of the AN. - Overall, this work establishes a unified algebraic and probabilistic framework for analyzing the long-term behavior of noncompetitive CRNs. - oai:arXiv.org:2512.17491v1 - q-bio.MN + Growth rates for the H\"older coefficients of the linear stochastic fractional heat equation with rough dependence in space + https://arxiv.org/abs/2507.22379 + arXiv:2507.22379v2 Announce Type: replace +Abstract: We study the linear stochastic fractional heat equation $$ + \frac{\partial}{\partial t}u(t,x)=-(-\Delta)^{\frac{\alpha}2}u (t,x)+\dot{W}(t,x),\ \ t> 0,\ \ x\in\RR, $$ where $-(-\Delta)^{\frac{\alpha}{2}}$ denotes the fractional Laplacian with power $\alpha\in (1, 2)$, and the driving noise $\dot W$ is a centered Gaussian field which is white in time and has the covariance of a fractional Brownian motion with Hurst parameter $H\in\left(\frac {2-\alpha}2,\frac 12\right)$. We establish exact asymptotics for the solution as both time and space variables tend to infinity and derive sharp growth rates for the H\"older coefficients. The proofs are based on Talagrand's majorizing measure theorem and Sudakov's minoration theorem. + oai:arXiv.org:2507.22379v2 math.PR - Mon, 22 Dec 2025 00:00:00 -0500 - cross + Tue, 23 Dec 2025 00:00:00 -0500 + replace http://creativecommons.org/licenses/by/4.0/ - Louis Faul, Xavier Richard, Mary Betrisey, Christian Mazza + Chang Liu, Bin Qian, Ran Wang - Yet another cubical type theory, but via a semantic approach - https://arxiv.org/abs/2512.17548 - arXiv:2512.17548v1 Announce Type: cross -Abstract: We propose a new cubical type theory, termed (self-deprecatingly) the naive cubical type theory, and study its semantics using the universe category framework, which is similar to Uemura's categories with representable morphisms. In particular, we show that this new type theory admits an interpretation in a wide variety of settings, including simplicial sets and cartesian cubical sets. - oai:arXiv.org:2512.17548v1 - cs.LO + Local smoothing and maximal estimates for average over surfaces of codimension 2 in $\mathbb R^4$ + https://arxiv.org/abs/2507.22695 + arXiv:2507.22695v2 Announce Type: replace +Abstract: In this paper, we obtain local smoothing estimates for the averages over nondegenerate surfaces of codimension $2$ in $\mathbb R^4$. We make use of multilinear restriction estimates and decoupling inequalities for a hypersurface in $\mathbb R^5$, a conical extension of a two-dimensional nondegenerate surface along two flat directions. We also establish sharp $L^p$--$L^q$ estimates for maximal averages over nondegenerate surfaces of half the ambient dimension in $\mathbb R^{2n}$ for even $n \ge 2$. + oai:arXiv.org:2507.22695v2 + math.CA + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by/4.0/ + Seheon Ham, Hyerim Ko + + + On plus-one generated arrangements of plane conics + https://arxiv.org/abs/2507.23024 + arXiv:2507.23024v3 Announce Type: replace +Abstract: In this paper, we examine the combinatorial properties of conic arrangements in the complex projective plane that possess certain quasi-homogeneous singularities. First, we introduce a new tool that enables us to characterize the property of being plus-one generated within the class of conic arrangements with some naturally chosen quasi-homogeneous singularities. Next, we present a classification result on plus-one generated conic arrangements admitting only nodes and tacnodes as singularities. Building on results regarding conic arrangements with nodes and tacnodes, we present new examples of strong Ziegler pairs of conic-line arrangements -- that is, arrangements having the same strong combinatorics but distinct derivation modules. + oai:arXiv.org:2507.23024v3 + math.AG + math.CO + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by/4.0/ + Artur Bromboszcz, Bartosz Jaros{\l}awski, Piotr Pokora + + + On multiple null-series in the Walsh system, M- and U- sets + https://arxiv.org/abs/2508.00182 + arXiv:2508.00182v2 Announce Type: replace +Abstract: A family of M-sets and null-series for the d-dimensional Walsh system is constructed if we consider convergence over rectangles, cubes, or iterated convergence. Non-empty portions of the constructed M-sets are also M-sets. The question of the rate of convergence to zero of the coefficients of zero-series that realize the constructed M-sets is studied, and it is shown how to modify the construction of the latter to turn them into U-sets + oai:arXiv.org:2508.00182v2 + math.CA + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by/4.0/ + A. D. Kazakova, M. G. Plotnikov + + + Locally finitely presented Grothendieck categories with a flat generator + https://arxiv.org/abs/2508.00670 + arXiv:2508.00670v2 Announce Type: replace +Abstract: A problem raised by Cuadra and Simson in 2007 asks whether any locally finitely presented Grothendieck category with enough flat objects also has enough projectives. In this paper, we start from a key observation: a locally finitely presented Grothendieck category has enough flat objects if, and only if, it has exact products. This enables several equivalent reformulations of the problem, allowing us to identify a counterexample (thus providing a negative solution to the problem), while also connecting it to a classical ring-theoretical question posed by Miller in 1975, and even to the Telescope Conjecture for compactly generated triangulated categories. Moreover, we describe several classes of Grothendieck categories where the problem can be answered affirmatively. For example, we show that a locally finitely presented Grothendieck category whose category of finitely presented objects is Krull--Schmidt has enough flats if, and only if, it is generated by a family of finitely generated projectives. + oai:arXiv.org:2508.00670v2 math.CT - math.LO - Mon, 22 Dec 2025 00:00:00 -0500 - cross + math.AG + math.RA + math.RT + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by/4.0/ + Lorenzo Martini, Carlos E. Parra, Manuel Saor\'in, Simone Virili + + + A new conjecture on the inertia of graphs + https://arxiv.org/abs/2508.01163 + arXiv:2508.01163v3 Announce Type: replace +Abstract: Let $G$ be a graph with adjacency matrix $A(G)$. We conjecture that \[2n^+(G) \le n^-(G)(n^-(G) + 1),\] where $n^+(G)$ and $n^-(G)$ denote the number of positive and negative eigenvalues of $A(G)$, respectively. This conjecture generalizes to all graphs the well-known absolute bound for strongly regular graphs. The conjecture also relates to a question posed by Torga\v{s}ev. We prove the conjecture for special graph families, including line graphs and planar graphs, and provide examples where the conjecture is exact. We also conjecture that for any connected graph $G$, its line graph $L(G)$ satisfies $n^+(L(G)) \le n^-(L(G)) + 1$, and obtain partial results. + oai:arXiv.org:2508.01163v3 + math.CO + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by/4.0/ + Saieed Akbari, Clive Elphick, Hitesh Kumar, Shivaramakrishna Pragada, Quanyu Tang + + + Lagrangian Fibrations onto Varieties with Isolated Quotient Singularities + https://arxiv.org/abs/2508.04135 + arXiv:2508.04135v2 Announce Type: replace +Abstract: In this note, we show that if $f\colon M\rightarrow X$ is a germ of a projective Lagrangian fibration from a holomorphic symplectic manifold $M$ onto a normal analytic variety $X$ with isolated quotient singularities, then $X$ is smooth. In particular, if $f\colon M\rightarrow X$ is a Lagrangian fibration from a hyper-K\"ahler fourfold $M$ onto a normal surface $X$, then $X\cong \mathbb{P}^2$, which recovers a recent result of Huybrechts--Xu and Ou. + oai:arXiv.org:2508.04135v2 + math.AG + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Niklas M\"uller, Zheng Xu + + + Fast and Simple Multiclass Data Segmentation: An Eigendecomposition and Projection-Free Approach + https://arxiv.org/abs/2508.09738 + arXiv:2508.09738v2 Announce Type: replace +Abstract: Graph-based machine learning has seen an increased interest over the last decade with many connections to other fields of applied mathematics. Learning based on partial differential equations, such as the phase-field Allen-Cahn equation, allows efficient handling of semi-supervised learning approaches on graphs. The numerical solution of the graph Allen-Cahn equation via a convexity splitting or the Merriman-Bence-Osher (MBO) scheme, albeit being a widely used approach, requires the calculation of a graph Laplacian eigendecomposition and repeated projections over the unit simplex to maintain valid partitions. The computational efficiency of those methods is hence limited by those two bottlenecks in practice, especially when dealing with large-scale instances. In order to overcome these limitations, we propose a new framework combining a novel penalty-based reformulation of the segmentation problem, which ensures valid partitions (i.e., binary solutions) for appropriate parameter choices, with an eigendecomposition and projection-free optimization scheme, which has a small per-iteration complexity (by relying primarily on sparse matrix-vector products) and guarantees good convergence properties. Experiments on synthetic and real-world datasets related to data segmentation in networks and images demonstrate that the proposed framework achieves comparable or better accuracy than the CS and MBO methods while being significantly faster, particularly for large-scale problems. + oai:arXiv.org:2508.09738v2 + math.NA + cs.NA + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by/4.0/ + Chiara Faccio, Margherita Porcelli, Francesco Rinaldi, Martin Stoll + + + Unconditional uniqueness for the derivative nonlinear Schr\"{o}dinger equation by normal form approach + https://arxiv.org/abs/2508.09740 + arXiv:2508.09740v2 Announce Type: replace +Abstract: We prove uniqueness of solutions to the Cauchy problem for the derivative nonlinear Schr\"odinger equation in $L^\infty_tH^{1/2}_x$. Our proof is based on the method of normal form reduction (NFR), which has been employed to obtain the uniqueness in $C_tH^s_x$, $s>1/2$. To overcome logarithmic divergences at the $H^{1/2}$ regularity, we exploit the $B^{0+}_{\infty,1}$ control of solutions provided by a refined Strichartz estimate. Our NFR argument consists of two stages: we first use NFR finitely many times to derive an intermediate equation in which the main cubic nonlinearity is restricted to a certain type of frequency interaction; we then apply the infinite NFR scheme to the intermediate equation. Moreover, we modify the usual NFR argument relying on continuity in time of solutions so that the uniqueness in the class $L^\infty_tH^{1/2}_x$ can be obtained directly. + oai:arXiv.org:2508.09740v2 + math.AP + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by/4.0/ + Nobu Kishimoto + + + An alternative solvability criterion for the Dirichlet problem for the minimal surface equation and an application to the mean curvature flow + https://arxiv.org/abs/2508.09806 + arXiv:2508.09806v2 Announce Type: replace +Abstract: We propose an alternative condition for the solvability of the Dirichlet problem for the minimal surface equation that applies to non-mean convex domains. This condition is derived from a second-order ordinary differential equation whose solution produces a barrier that appears to be novel in the context of barrier constructions. It admits an explicit formulation and, in the setting of Hadamard manifolds, reveals a direct and transparent relationship between the geometry of the domain and the behavior of the boundary data required for solvability.The condition also extends naturally to unbounded domains. + In the Euclidean case, it is not only more practical to verify but also less restrictive than the classical Jenkins-Serrin criterion, ensuring the existence of solutions in situations where that approach fails. Furthermore, unlike the Jenkins-Serrin condition, our approach separates the domain's geometric properties from its boundary data, providing a clearer and more manageable framework for solvability analysis. + Beyond the stationary theory, the same ODE-generated barrier also yields natural sub- and supersolutions for the mean curvature flow of graphs, allowing one to initiate and control the short-time evolution from non-mean convex boundaries. As a consequence, our barrier construction extends a classical result of Ecker-Huisken on short-time existence of graphical mean curvature flow to domains with non-mean convex boundary; in the mean convex setting, their original result is recovered. + oai:arXiv.org:2508.09806v2 + math.AP + math.DG + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by/4.0/ + Ari J. Aiolfi, Giovanni da Silva Nunes, Jaime Ripoll, Lisandra Sauer, Rodrigo Soares + + + On Random Fields Associated with Analytic Wavelet Transform + https://arxiv.org/abs/2508.10495 + arXiv:2508.10495v2 Announce Type: replace +Abstract: Despite the broad application of the analytic wavelet transform (AWT), a systematic statistical characterization of its magnitude and phase as inhomogeneous random fields on the time-frequency domain when the input is a random process remains underexplored. In this work, we study the magnitude and phase of the AWT as random fields on the time-frequency domain when the observed signal is a deterministic function plus additive stationary Gaussian noise. We derive their marginal and joint distributions, establish concentration inequalities that depend on the signal-to-noise ratio (SNR), and analyze their covariance structures. Based on these results, we derive an upper bound on the probability of incorrectly identifying the time-scale ridge of the clean signal, explore the regularity of scalogram contours, and study the relationship between AWT magnitude and phase. Our findings lay the groundwork for developing rigorous AWT-based algorithms in noisy environments. + oai:arXiv.org:2508.10495v2 + math.ST + math.PR + stat.TH + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Gi-Ren Liu, Yuan-Chung Sheu, Hau-Tieng Wu + + + On a nonnegativity conjecture of Andrews + https://arxiv.org/abs/2508.10871 + arXiv:2508.10871v2 Announce Type: replace +Abstract: I settle a conjecture of Andrews related to the Alladi-Schur polynomials. In addition, I give further relations and implications to two families of polynomials related to the Alladi-Schur polynomials. + oai:arXiv.org:2508.10871v2 + math.NT + math.CO + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by/4.0/ + Yazan Alamoudi + + + Silting correspondences and Calabi-Yau dg algebras + https://arxiv.org/abs/2508.12836 + arXiv:2508.12836v2 Announce Type: replace +Abstract: This paper is devoted to studying two important classes of objects in triangulated categories; silting objects and $d$-cluster tilting objects, and their correspondences. First, we introduce the notion of $d$-silting objects as a generalization tilting objects whose endomorphism algebras have global dimension at most $d$. For a smooth dg algebra $A$ and its $(d+1)$-Calabi-Yau completion $\Pi$, we show that the induction functor gives an embedding from the poset $\operatorname{silt}^dA$ of $d$-silting objects of $A$ to the poset $\operatorname{silt}\Pi$ of silting objects of $\Pi$. Moreover, when $H^0\Pi$ is finite dimensional, this functor identifies the Hasse quiver of $\operatorname{silt}^dA$ as a full subquiver of the Hasse quiver of $\operatorname{silt}\Pi$. In this case, we also prove that each $d$-silting object $P$ of $A$ gives a $d$-cluster tilting subcategory of $\operatorname{per} A$ as the $\nu[-d]$-orbit of $P$. Secondly, for a connective Calabi-Yau dg algebra $\Pi$, we study the map from $\operatorname{silt}\Pi$ to the set $d\text{-}\operatorname{ctilt}\mathcal{C}(\Pi)$ of $d$-cluster tilting objects in the cluster category $\mathcal{C}(\Pi)$. We call $\Pi$ $\mathcal{F}$-liftable if the induced map $\operatorname{silt}\Pi\cap\mathcal{F}\to d\text{-}\operatorname{ctilt}\mathcal{C}(\Pi)$ is bijective, where $\mathcal{F}$ is the fundamental domain in $\operatorname{per}\Pi$. We prove that $\mathcal{F}$-liftable Calabi-Yau dg algebras $\Pi$ such that $H^0\Pi$ is hereditary are precisely the Calabi-Yau completions of hereditary algebras. As an application, we obtain counter-examples to an open question posed in [IYa1]. We also study Calabi-Yau dg algebras such that the map $\operatorname{silt}\Pi\to d\text{-}\operatorname{ctilt}\mathcal{C}(\Pi)$ is surjective, which we call liftable. We explain our results by polynomial dg algebras and Calabi-Yau completions of type $A_2$. + oai:arXiv.org:2508.12836v2 + math.RT + math.RA + Tue, 23 Dec 2025 00:00:00 -0500 + replace http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Chris Kapulkin, Yufeng Li + Norihiro Hanihara, Osamu Iyama - Group-theoretical analysis of quantum complexity: the oscillator group case - https://arxiv.org/abs/2512.17552 - arXiv:2512.17552v1 Announce Type: cross -Abstract: Motivated by the recent rapid development of complexity theory applied to quantum mechanical processes we present the complete derivation of Nielsen's complexity of unitaries belonging to the representations of oscillator group. Our approach is based on the observation that the whole problem refers to the structure of the underlying group. The questions concerning the complexity of particular unitaries are solved by lifting the abstract structure to the operator level by considering the relevant unitary representation. For the class of right-invariant metrics obeying natural invariance condition we solve the geodesic equations on oscillator group. The solution is given explicitly in terms of elementary functions. Imposing the boundary conditions yield a transcendental equation and the length of the geodesic is given in terms of the solutions to the latter. Since the unitary irreducible representations of oscillator group are classified this allows us to compute, at least in principle, the complexity of any unitary operator belonging to the representation. - oai:arXiv.org:2512.17552v1 - quant-ph - hep-th - math-ph - math.MP - Mon, 22 Dec 2025 00:00:00 -0500 - cross + Fixed points of the Berezin transform on Fock-type spaces + https://arxiv.org/abs/2508.13115 + arXiv:2508.13115v2 Announce Type: replace +Abstract: We study the fixed points of the Berezin transform on the Fock-type spaces $F_m^2$ with the weight $e^{-|z|^m}, m > 0.$ It is known that the Berezin transform is well-defined on the polynomials in $z$ and $\overline{z}$. In this paper we focus on the polynomial fixed points and we show that these polynomials must be harmonic, except possibly for countably many $m \in (0, \infty).$ We also show that, in some particular cases, the fixed point polynomials are harmonic for all $m.$ + oai:arXiv.org:2508.13115v2 + math.CV + math.FA + Tue, 23 Dec 2025 00:00:00 -0500 + replace http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - K. Andrzejewski, K. Bolonek-Laso\'n, P. Kosi\'nski + Ghazaleh Asghari, Zeljko Cuckovic, Sonmez Sahutoglu - Bayesian Optimisation: Which Constraints Matter? - https://arxiv.org/abs/2512.17569 - arXiv:2512.17569v1 Announce Type: cross -Abstract: Bayesian optimisation has proven to be a powerful tool for expensive global black-box optimisation problems. In this paper, we propose new Bayesian optimisation variants of the popular Knowledge Gradient acquisition functions for problems with \emph{decoupled} black-box constraints, in which subsets of the objective and constraint functions may be evaluated independently. In particular, our methods aim to take into account that often only a handful of the constraints may be binding at the optimum, and hence we should evaluate only relevant constraints when trying to optimise a function. We empirically benchmark these methods against existing methods and demonstrate their superiority over the state-of-the-art. - oai:arXiv.org:2512.17569v1 - cs.LG + Strong Lyapunov functions for rough systems + https://arxiv.org/abs/2508.14559 + arXiv:2508.14559v4 Announce Type: replace +Abstract: We extend the Lyapunov function technique, a fundamental tool for investigating asymptotic stability and existence of attractors for ordinary differential equations, by introducing the notion of a {\it strong Lyapunov function} for an autonomous drift under stochastic perturbation driven by general H\"older-continuous multiplicative noise, not necessarily Brownian. + The mathematical setting within which our method proceeds consists of rough path calculus and the framework of random dynamical systems. We conclude that if such a function exists for the drift then the perturbed system admits a global random pullback attractor that is upper semi-continuous w.r.t. the noise intensity coefficient and the dyadic approximation of the noise. Moreover, in case the drift is globally Lipschitz continuous, then there exists a numerical attractor for the discretization which is upper semi-continuous w.r.t. the noise intensity and converges to the continuous attractor as the step size tends to zero. Several applications, including dissipative systems, the pendulum, the FitzHugh-Nagumo neuro-system and the Lorenz system, demonstrate the power of our approach. We also prove that strong Lyapunov functions can be approximated in practice by Lyapunov neural networks. + oai:arXiv.org:2508.14559v4 + math.DS math.OC - Mon, 22 Dec 2025 00:00:00 -0500 - cross - http://creativecommons.org/licenses/by/4.0/ - Xietao Wang Lin, Juan Ungredda, Max Butler, James Town, Alma Rahat, Hemant Singh, Juergen Branke + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Luu Hoang Duc, J\"urgen Jost - A Unified Representation of Neural Networks Architectures - https://arxiv.org/abs/2512.17593 - arXiv:2512.17593v1 Announce Type: cross -Abstract: In this paper we consider the limiting case of neural networks (NNs) architectures when the number of neurons in each hidden layer and the number of hidden layers tend to infinity thus forming a continuum, and we derive approximation errors as a function of the number of neurons and/or hidden layers. Firstly, we consider the case of neural networks with a single hidden layer and we derive an integral infinite width neural representation that generalizes existing continuous neural networks (CNNs) representations. Then we extend this to deep residual CNNs that have a finite number of integral hidden layers and residual connections. Secondly, we revisit the relation between neural ODEs and deep residual NNs and we formalize approximation errors via discretization techniques. Then, we merge these two approaches into a unified homogeneous representation of NNs as a Distributed Parameter neural Network (DiPaNet) and we show that most of the existing finite and infinite-dimensional NNs architectures are related via homogeneization/discretization with the DiPaNet representation. Our approach is purely deterministic and applies to general, uniformly continuous matrix weight functions. Differences and similarities with neural fields are discussed along with further possible generalizations and applications of the DiPaNet framework. - oai:arXiv.org:2512.17593v1 - cs.LG + Lower Bounds on the Haraux Function + https://arxiv.org/abs/2508.15735 + arXiv:2508.15735v3 Announce Type: replace +Abstract: The Haraux function is an important tool in monotone operator theory and its applications. One of its salient properties for maximally monotone operators is to be valued in $[0,+\infty]$ and to vanish only on the graph of the operator. Sharper lower bounds for this function were recently proposed in specific cases. We derive lower bounds in the general context of set-valued operators in reflexive Banach spaces. These bounds are new, even for maximally monotone operators acting on Euclidean spaces, a scenario in which we show that they can be better than existing ones. As a by-product, we obtain lower bounds for the Fenchel--Young function in variational analysis. Several examples are given and applications to composite monotone inclusions are discussed. + oai:arXiv.org:2508.15735v3 math.OC - Mon, 22 Dec 2025 00:00:00 -0500 - cross + Tue, 23 Dec 2025 00:00:00 -0500 + replace http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Christophe Prieur, Mircea Lazar, Bogdan Robu + Patrick L. Combettes, Julien N. Mayrand - A Systems-Theoretic View on the Convergence of Algorithms under Disturbances - https://arxiv.org/abs/2512.17598 - arXiv:2512.17598v1 Announce Type: cross -Abstract: Algorithms increasingly operate within complex physical, social, and engineering systems where they are exposed to disturbances, noise, and interconnections with other dynamical systems. This article extends known convergence guarantees of an algorithm operating in isolation (i.e., without disturbances) and systematically derives stability bounds and convergence rates in the presence of such disturbances. By leveraging converse Lyapunov theorems, we derive key inequalities that quantify the impact of disturbances. We further demonstrate how our result can be utilized to assess the effects of disturbances on algorithmic performance in a wide variety of applications, including communication constraints in distributed learning, sensitivity in machine learning generalization, and intentional noise injection for privacy. This underpins the role of our result as a unifying tool for algorithm analysis in the presence of noise, disturbances, and interconnections with other dynamical systems. - oai:arXiv.org:2512.17598v1 - cs.LG - math.OC - stat.ML - Mon, 22 Dec 2025 00:00:00 -0500 - cross - http://creativecommons.org/licenses/by-nc-sa/4.0/ - Guner Dilsad Er, Sebastian Trimpe, Michael Muehlebach + Spectral representations of interpolation spaces of reproducing kernel Hilbert spaces + https://arxiv.org/abs/2508.16492 + arXiv:2508.16492v2 Announce Type: replace +Abstract: In statistical learning theory, interpolation spaces of the form $[\mathrm{L}^2,H]_{\theta,r}$, where $H$ is a reproducing kernel Hilbert space, are in widespread use. So far, however, they are only well understood for fine index $r=2$. We generalise existing results from $r=2$ to all possible values of $r$. In particular, we present a spectral decomposition of such spaces, analyse their embedding properties, and describe connections to the theory of Banach spaces of functions. We additionally present example applications of our results to regularisation error estimation in statistical learning. + oai:arXiv.org:2508.16492v2 + math.FA + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by/4.0/ + Michael Bitzer, Ingo Steinwart - St\"ackel problem for non-diagonal Killing tensors: Yano-Patterson lifts, algebra of strong symmetries and quadratic in momenta integrals - https://arxiv.org/abs/2512.17609 - arXiv:2512.17609v1 Announce Type: cross -Abstract: We construct integrable Hamiltonian systems such that functionally independent Poisson commuting integrals are quadratic in the momenta. Unlike the classical St\"ackel setting, we allow the associated self-adjoint $(1,1)$-tensors $K_\alpha$ to be non-diagonalisable and have Jordan blocks and points where the Segre characteristic changes. Our construction is covariant and is based on Nijenhuis geometry: starting from a gl-regular Nijenhuis operator $L$ and its symmetry algebra, we obtain a large class of such integrable systems in a coordinate-free and signature-independent way; it is explicit once we have chosen a gl-regular Nijnhuis operator. In the diagonalisable case, our construction reproduces the St\"ackel construction, and in dimension $n=2$ it recovers all known systems of this type; for $n\ge 3$ most of our systems are new. Finally, we establish applications to infinite-dimensional integrable systems of hydrodynamic type: namely, we show that for Killing $(1,1)$-tensors $ K_\alpha$ corresponding to our example the evolutionarly PDE system of hydrodynamic type $u_t = K_\alpha(u)u_x$ is integrable. We describe its symmetries, and use generalised reciprocal transformations to reduce it to a system with constant coefficient matrices. - oai:arXiv.org:2512.17609v1 - nlin.SI - math-ph - math.DG + Dimensions and dimension spectra of Non-autonomous iterated function systems + https://arxiv.org/abs/2508.20632 + arXiv:2508.20632v3 Announce Type: replace +Abstract: Non-autonomous iterated function systems are a generalization of iterated function systems. If the contractions in the system are conformal mappings, it is called a non-autonomous conformal iterated function system, and its attractor is called a non-autonomous conformal set. In this paper, we study intermediate dimension spectra of non-autonomous conformal sets which provide a unifying framework for Hausdorff and box-counting dimensions. First, we obtain the intermediate dimension spectra formula of non-autonomous conformal sets by using upper and lower topological pressures. As a consequence, we obtain simplified forms of their Hausdorff, packing and box dimensions. Finally, we explore the Hausdorff dimensions of the non-autonomous infinite conformal iterated function systems which consists of countably many conformal mappings at each level, and we provide the Hausdorff dimension formula under certain conditions. + oai:arXiv.org:2508.20632v3 math.DS - math.MP - Mon, 22 Dec 2025 00:00:00 -0500 - cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Alexey V. Bolsinov, Andrey Yu. Konyaev, Vladimir S. Matveev + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by/4.0/ + Junjie Miao, Tianrui Wang - The threshold for quantum-classical correspondence is $D \sim \hbar^{\frac43}$ - https://arxiv.org/abs/2512.17623 - arXiv:2512.17623v1 Announce Type: cross -Abstract: In chaotic quantum systems, an initially localized quantum state can deviate strongly from the corresponding classical phase-space distribution after the Ehrenfest time $t_{\mathrm{E}} \sim \log(\hbar^{-1})$, even in the limit $\hbar \to 0$. Decoherence by the environment is often invoked to explain the persistence of the quantum-classical correspondence at longer timescales. Recent rigorous results for Lindblad dynamics with phase-space diffusion strength $D$ show that quantum and classical evolutions remain close for times that are exponentially longer than the Ehrenfest time whenever $D \gg \hbar^{\frac43}$, in units set by the classical Hamiltonian. At the same time, some heuristic arguments have suggested the weaker condition $D \gg \hbar^{2}$ always suffices. Here we construct an explicit Lindbladian that demonstrates that the scaling $D \sim \hbar^{\frac43}$ is indeed the threshold for quantum-classical correspondence beyond the Ehrenfest time. Our example uses a smooth time-dependent Hamiltonian and linear Lindblad operators generating homogeneous isotropic diffusion. It exhibits an $\hbar$-independent quantum-classical discrepancy at the Ehrenfest time whenever $D \ll \hbar^{\frac43}$, even for $\hbar$-independent "macroscopic" smooth observables. - oai:arXiv.org:2512.17623v1 - quant-ph - math-ph - math.AP - math.MP - Mon, 22 Dec 2025 00:00:00 -0500 - cross + Separating subsets from their images + https://arxiv.org/abs/2508.20731 + arXiv:2508.20731v2 Announce Type: replace +Abstract: Let $G$ be a transitive permutation group acting on $\Omega$. In this paper, we introduce and study the parameter ${\bf m}(G)$, which denotes the size of the smallest set of points $A$ such that, for every permutation $g\in G$, $A \cap A^g$ is nonempty. In particular, we focus on deriving general bounds for arbitrary transitive groups, and on the asymptotic behaviour of certain families of primitive groups. We also provide a classification of transitive groups with ${\bf m}(G)$ largest possible, namely with ${\bf m}(G)=\lceil (|\Omega|+1) / 2 \rceil$. + oai:arXiv.org:2508.20731v2 + math.GR + math.CO + Tue, 23 Dec 2025 00:00:00 -0500 + replace http://creativecommons.org/licenses/by/4.0/ - Felipe Hern\'andez, Daniel Ranard, C. Jess Riedel + Marco Barbieri, Maru\v{s}a Lek\v{s}e, Primo\v{z} Poto\v{c}nik, Kamilla Rekv\'enyi - Estimation and model errors in Gaussian-process-based Sensitivity Analysis of functional outputs - https://arxiv.org/abs/2512.17635 - arXiv:2512.17635v1 Announce Type: cross -Abstract: Global sensitivity analysis (GSA) of functional-output models is usually performed by combining statistical techniques, such as basis expansions, metamodeling and sampling based estimation of sensitivity indices. By neglecting truncation error from basis expansion, two main sources of errors propagate to the final sensitivity indices: the metamodeling related error and the sampling-based, or pick-freeze (PF), estimation error. This work provides an efficient algorithm to estimate these errors in the frame of Gaussian processes (GP), based on the approach of Le Gratiet et al. [16]. The proposed algorithm takes advantage of the fact that the number of basis coefficients of expanded model outputs is significantly smaller than output dimensions. Basis coefficients are fitted by GP models and multiple conditional GP trajectories are sampled. Then, vector-valued PF estimation is used to speed-up the estimation of Sobol indices and generalized sensitivity indices (GSI). We illustrate the methodology on an analytical test case and on an application in non-Newtonian hydraulics, modelling an idealized dam-break flow. Numerical tests show an improvement of 15 times in the computational time when compared to the application of Le Gratiet et al. [16] algorithm separately over each output dimension. - oai:arXiv.org:2512.17635v1 - stat.ME - math.ST - stat.AP - stat.TH - Mon, 22 Dec 2025 00:00:00 -0500 - cross + On Zero-sum Game Representation for Replicator Dynamics + https://arxiv.org/abs/2508.21299 + arXiv:2508.21299v2 Announce Type: replace +Abstract: Replicator dynamics have been widely used in evolutionary game theory to model how strategy frequencies evolve over time in large populations. The so-called payoff matrix encodes the pairwise fitness that each strategy obtains when interacting with every other strategy, and it solely determines the replicator dynamics. If the payoff matrix is unknown, we show in this paper that it cannot be inferred from observed strategy frequencies alone -- distinct payoff matrices can induce the same replicator dynamics. We thus look for a canonical representative of the payoff matrix in the equivalence class. The main result of the paper is to show that for every polynomial replicator dynamics (i.e., the vector field is a polynomial), there always exists a skew-symmetric, polynomial payoff matrix that can induce the given dynamics. + oai:arXiv.org:2508.21299v2 + math.DS + cs.SY + eess.SY + Tue, 23 Dec 2025 00:00:00 -0500 + replace http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Yuri Taglieri S\'ao, Olivier Roustant, Geraldo de Freitas Maciel + Haoyu Yin, Xudong Chen, Bruno Sinopoli - Polyharmonic Cascade - https://arxiv.org/abs/2512.17671 - arXiv:2512.17671v1 Announce Type: cross -Abstract: This paper presents a deep machine learning architecture, the "polyharmonic cascade" -- a sequence of packages of polyharmonic splines, where each layer is rigorously derived from the theory of random functions and the principles of indifference. This makes it possible to approximate nonlinear functions of arbitrary complexity while preserving global smoothness and a probabilistic interpretation. For the polyharmonic cascade, a training method alternative to gradient descent is proposed: instead of directly optimizing the coefficients, one solves a single global linear system on each batch with respect to the function values at fixed "constellations" of nodes. This yields synchronized updates of all layers, preserves the probabilistic interpretation of individual layers and theoretical consistency with the original model, and scales well: all computations reduce to 2D matrix operations efficiently executed on a GPU. Fast learning without overfitting on MNIST is demonstrated. - oai:arXiv.org:2512.17671v1 - cs.LG - cs.NA + Development of numerical methods for nonlinear hybrid stochastic functional differential equations with infinite delay + https://arxiv.org/abs/2509.00475 + arXiv:2509.00475v3 Announce Type: replace +Abstract: This paper addresses the challenging numerical simulation of nonlinear hybrid stochastic functional differential equations with infinite delays. We first propose an explicit scheme using space and time truncation, requiring only finite historical storage. Leveraging approximation theory, we prove the boundedness of the numerical solution's $p$th moment and establish its convergence, achieving a rate of $1/2$ order under polynomially growing coefficients. Furthermore, we refine the scheme to better capture the underlying exponential stability of the exact solution, in both moment and almost sure senses. Finally, numerical experiments are presented to validate our theoretical results. + oai:arXiv.org:2509.00475v3 math.NA - Mon, 22 Dec 2025 00:00:00 -0500 - cross - http://creativecommons.org/licenses/by/4.0/ - Yuriy N. Bakhvalov + cs.NA + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Guozhen Li, Xiaoyue Li, Xuerong Mao - Relative arbitrage problem under eigenvalue lower bounds - https://arxiv.org/abs/2512.17702 - arXiv:2512.17702v1 Announce Type: cross -Abstract: We give a new formulation of the relative arbitrage problem from stochastic portfolio theory that asks for a time horizon beyond which arbitrage relative to the market exists in all ``sufficiently volatile'' markets. In our formulation, ``sufficiently volatile'' is interpreted as a lower bound on an ordered eigenvalue of the instantaneous covariation matrix, a quantity that has been studied extensively in the empirical finance literature. Upon framing the problem in the language of stochastic optimal control, we characterize the time horizon in question through the unique upper semicontinuous viscosity solution of a fully nonlinear elliptic partial differential equation (PDE). In a special case, this PDE amounts to the arrival time formulation of the Ambrosio-Soner co-dimension mean curvature flow. Beyond the setting of stochastic portfolio theory, the stochastic optimal control problem is analyzed for arbitrary compact, possibly non-convex, domains, thanks to a boundedness assumption on the instantaneous covariation matrix. - oai:arXiv.org:2512.17702v1 - q-fin.MF - math.AP - math.DG + Trust-region Filter Algorithms utilising Hessian Information for Grey-Box Optimisation + https://arxiv.org/abs/2509.01651 + arXiv:2509.01651v3 Announce Type: replace +Abstract: Optimising industrial processes often involves grey-box models that couple algebraic glass-box equations with black-box components lacking analytic derivatives. Such hybrid systems challenge derivative-based solvers. The classical trust-region filter (TRF) algorithm provides a robust framework but requires extensive parameter tuning and numerous black-box evaluations. This work introduces four Hessian-informed TRF variants (A1-A4) that use projected positive definite Hessians for automatic step scaling and minimal tuning, combined with both low-fidelity (linear, quadratic) and high-fidelity (Taylor series, Gaussian process) surrogates for local black-box approximation. Tested on 25 grey-box benchmarks and five engineering case studies (Himmelblau, liquid-liquid extraction, pressure vessel design, alkylation, and spring design), the new variants achieved up to an order-of-magnitude reduction in iterations and black-box evaluations, with reduced sensitivity to tuning parameters relative to the classical TRF algorithm. High-fidelity surrogates solved 92-100 % problems, compared to 72-84 % for the low-fidelity surrogates. Developed TRF methods also outperformed classical derivative-free optimisation solvers. The results show that new variants offer robust and scalable alternatives for grey-box process systems optimisation. + oai:arXiv.org:2509.01651v3 math.OC - math.PR - Mon, 22 Dec 2025 00:00:00 -0500 - cross + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by/4.0/ + Gul Hameed, Tao Chen, Antonio del Rio Chanona, Lorenz T. Biegler, Michael Short + + + Practical Channel Estimation for Pinching-Antenna Systems: Serial vs. Parallel and Downlink vs. Uplink? + https://arxiv.org/abs/2509.02403 + arXiv:2509.02403v2 Announce Type: replace +Abstract: The practical channel estimation for pinching-antenna networks is investigated, in which an electromagnetic-compliant in-waveguide transmission model is exhibited, incorporating bidirectional power splitting, cumulative power leakage, and waveguide attenuation. Based on this model, the paper investigates two antenna activation protocols for channel estimation: a serial protocol based on one-by-one antenna activation and a parallel protocol utilizing a binary S-Matrix activation. The serial protocol is characterized by its superior numerical stability but a lack of array gain, whereas the parallel protocol theoretically offers array gain but suffers from severe performance degradation due to structural crosstalk from the non-orthogonal S-Matrix and ill-conditioning from cumulative leakage. Furthermore, the paper analyzes the fundamental commonalities and asymmetries between uplink and downlink channel estimation in pinching-antenna systems. Numerical results demonstrate that 1) in an ideal lossless model, the parallel protocol is superior to the serial protocol due to the array gain from simultaneous energy collection in uplink transmission; 2) in a practical model with physical losses, the serial protocol outperforms the parallel protocol, as the performance of the parallel protocol is degraded by the numerical instability from cumulative leakage, which outweighs the benefit of array gain; 3) For downlink channel estimation, the serial protocol is more suitable because its strategy of concentrating the entire power budget on one measurement, while the parallel protocol is more suitable for the uplink as it can make full use of array gain. + oai:arXiv.org:2509.02403v2 + cs.IT + math.IT + Tue, 23 Dec 2025 00:00:00 -0500 + replace http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Jou-Hua Lai, Mykhaylo Shkolnikov, H. Mete Soner + Jian Xiao - Inclusion constants for free spectrahedra with applications to quantum incompatibility - https://arxiv.org/abs/2512.17706 - arXiv:2512.17706v1 Announce Type: cross -Abstract: Building on the matrix cube problem, inclusions of free spectrahedra have been used successfully to obtain relaxations of hard spectrahedral inclusion problems. The quality of such a relaxation is quantified by the inclusion constant associated with each free spectrahedron. While optimal values of inclusion constants were known in certain highly symmetric cases, no general method for computing them was available. In this work, we show that inclusion constants for Cartesian products of free simplices can be computed using methods from non-commutative polynomial optimization, together with a detailed analysis of the extreme points of the associated free spectrahedra. This analysis also yields new closed-form analytic expressions for these constants. As an application to quantum information theory, we prove new bounds on the amount of white noise that incompatible measurements can tolerate before they become compatible. In particular, we study the case of one dichotomic and one $k$-outcome measurement, as well as the case of four dichotomic qubit measurements. - oai:arXiv.org:2512.17706v1 - quant-ph - math-ph - math.FA - math.MP + Safe Navigation in the Presence of Range-Limited Pursuers + https://arxiv.org/abs/2509.04258 + arXiv:2509.04258v3 Announce Type: replace +Abstract: This paper examines the degree to which an evader seeking a safe and efficient path to a target location can benefit from increasing levels of knowledge regarding one or more range-limited pursuers seeking to intercept it. Unlike previous work, this research considers the time of flight of the pursuers actively attempting interception. It is shown that additional knowledge allows the evader to safely steer closer to the threats, shortening paths without accepting additional risk of capture. A control heuristic is presented, suitable for real-time implementation, which capitalizes on all knowledge available to the evader. + oai:arXiv.org:2509.04258v3 math.OC - Mon, 22 Dec 2025 00:00:00 -0500 - cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Andreas Bluhm, Eric Evert, Igor Klep, Victor Magron, Ion Nechita + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by-nc-nd/4.0/ + 10.1109/LCSYS.2025.3645221 + Thomas Chapman, Alexander Von Moll, Isaac E. Weintraub - Certified bounds on optimization problems in quantum theory - https://arxiv.org/abs/2512.17713 - arXiv:2512.17713v1 Announce Type: cross -Abstract: Semidefinite relaxations of polynomial optimization have become a central tool for addressing the non-convex optimization problems over non-commutative operators that are ubiquitous in quantum information theory and, more in general, quantum physics. Yet, as these global relaxation methods rely on floating-point methods, the bounds issued by the semidefinite solver can - and often do - exceed the global optimum, undermining their certifiability. To counter this issue, we introduce a rigorous framework for extracting exact rational bounds on non-commutative optimization problems from numerical data, and apply it to several paradigmatic problems in quantum information theory. An extension to sparsity and symmetry-adapted semidefinite relaxations is also provided and compared to the general dense scheme. Our results establish rational post-processing as a practical route to reliable certification, pushing semidefinite optimization toward a certifiable standard for quantum information science. - oai:arXiv.org:2512.17713v1 - quant-ph - cs.SC + What is the Best Way to Do Something? A Discreet Tour of Discrete Optimization + https://arxiv.org/abs/2509.05932 + arXiv:2509.05932v3 Announce Type: replace +Abstract: In mathematical optimization, we want to find the best possible solution for a decision-making problem. Curiously, these problems are harder to solve if they have discrete decisions. Imagine that you would like to buy chocolate: you can buy no chocolate or one chocolate bar, but typically you cannot buy just half of a bar. Now imagine that you could also buy many other items, and that you need to meet nutritional needs while minimizing the grocery bill. With more options and more demands, finding the best solution becomes trickier. But since many real-world settings benefit from mathematical optimization, such as scheduling trains and flights, planning truck deliveries, and making better investment decisions, these problems are widely studied in a branch of mathematics called Operations Research (OR). Sometimes we can simply write the mathematical model and find an optimal solution with OR software, but for larger problems we may need to develop new mathematical models and even write our own algorithms. We explore both cases with a simple and well-known problem (the traveling salesperson problem), some computer programming (in Python), and software that is free for academic use (Gurobi). All the code and data used is available at: https://github.com/thserra/discreet + oai:arXiv.org:2509.05932v3 math.OC - Mon, 22 Dec 2025 00:00:00 -0500 - cross + Tue, 23 Dec 2025 00:00:00 -0500 + replace http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Younes Naceur, Jie Wang, Victor Magron, Antonio Ac\'in + Thiago Serra - Near-Maturity Asymptotics of Critical Prices of American Put Options under Exponential L\'{e}vy Models - https://arxiv.org/abs/2512.17791 - arXiv:2512.17791v1 Announce Type: cross -Abstract: In the present paper, we study the near-maturity ($t\rightarrow T^{-}$) convergence rate of the optimal early-exercise price $b(t)$ of an American put under an exponential L\'{e}vy model with a {\it nonzero} Brownian component. Two important settings, not previous covered in the literature, are considered. In the case that the optimal exercise price converges to the strike price ($b(T^{-})=K$), we contemplate models with negative jumps of unbounded variation (i.e., processes that exhibit high activity of negative jumps or sudden falls in asset prices). In the second case, when the optimal exercise price tend to a value lower than $K$, we consider infinite activity jumps (though still of bounded variations), extending existing results for models with finite jump activity (finitely many jumps in any finite interval). In both cases, we show that $b(T^{-})-b(t)$ is of order $\sqrt{T-t}$ with explicit constants proportionality. Furthermore, we also derive the second-order near-maturity expansion of the American put price around the critical price along a certain parabolic branch. - oai:arXiv.org:2512.17791v1 - q-fin.MF - math.PR - Mon, 22 Dec 2025 00:00:00 -0500 - cross - http://creativecommons.org/licenses/by/4.0/ - Jos\'e E. Figueroa-L\'opez, Ruoting Gong + Degeneration of Riemann surfaces and small eigenvalues of the Laplacian + https://arxiv.org/abs/2509.06151 + arXiv:2509.06151v2 Announce Type: replace +Abstract: For a one-parameter degeneration of compact Riemann surfaces endowed with the K\"ahler metric induced from the K\"ahler metric on the total space of the family, we determine the exact magnitude of the small eigenvalues of the Laplacian as a function on the parameter space, under the assumption that the singular fiber is reduced. The novelty in our approach is that we compute the asymptotic behavior of certain difference of (logarithm of) analytic torsions in the degeneration in two ways. On the one hand, via heat kernel estimates, it is shown that the leading asymptotic is determined by the product of the small eigenvalues. On the other hand, using Quillen metrics, the leading asymptotic is connected with the period integrals, which we explicitly evaluate. + oai:arXiv.org:2509.06151v2 + math.DG + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Xianzhe Dai, Ken-Ichi Yoshikawa - Quantum Wasserstein distance for Gaussian states - https://arxiv.org/abs/2512.17809 - arXiv:2512.17809v1 Announce Type: cross -Abstract: Optimal transport between classical probability distributions has been proven useful in areas such as machine learning and random combinatorial optimization. Quantum optimal transport, and the quantum Wasserstein distance as the minimal cost associated with transforming one quantum state to another, is expected to have implications in quantum state discrimination and quantum metrology. In this work, following the formalism introduced in [De Palma, G. and Trevisan, D. Ann. Henri Poincar\'e, {\bf 22} (2021), 3199-3234] to compute the optimal transport plan between two quantum states, we give a general formula for the Wasserstein distance of order 2 between any two one-mode Gaussian states. We discuss how the Wasserstein distance between classical Gaussian distributions and the quantum Wasserstein distance by De Palma and Trevisan for thermal states can be recovered from our general formula for Gaussian states. This opens the path to directly compare various known distance measures with the Wasserstein distance through their closed-form solutions. - oai:arXiv.org:2512.17809v1 - quant-ph - math-ph - math.MP - Mon, 22 Dec 2025 00:00:00 -0500 - cross + Isoperimetric-type inequalities for Mather's $\beta$-function of convex billiards + https://arxiv.org/abs/2509.06915 + arXiv:2509.06915v2 Announce Type: replace +Abstract: In this article we discuss pointwise spectral rigidity results for several billiard systems (e.g., Birkhoff billiards, symplectic billiards and $4$-th billiards), showing that a single value of Mather's $\beta$-function can determine whether a strongly convex smooth planar domain is a disk (or an ellipse, in the affine-invariant case of symplectic billiards). Evoking the famous question "Can you hear the shape of a billiard?", one could say that circular billiards can be heard by a single whisper! More specifically, we prove isoperimetric-type inequalities comparing the $\beta$-function associated to the billiard map of domain to that of a disk with the same perimeter or area, and investigate what are the consequences of having an equality. Surprisingly, this rigidity fails for outer billiards, where explicit counterexamples are constructed for rotation numbers $1/3$ and $1/4$. The results are framed within Aubry-Mather theory and provide a modern dynamical reinterpretation and extension of classical geometric inequalities for extremal polygons. + oai:arXiv.org:2509.06915v2 + math.DS + Tue, 23 Dec 2025 00:00:00 -0500 + replace http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Anaelle Hertz, Mohammad Ahmadpoor, Oleksandr Dzhenzherov, Augusto Gerolin, Khabat Heshami + Stefano Baranzini, Misha Bialy, Alfonso Sorrentino - Mathematical Modeling of Biofilm Eradication Using Optimal Control - https://arxiv.org/abs/2512.17825 - arXiv:2512.17825v1 Announce Type: cross -Abstract: We propose and analyze a model for antibiotic resistance transfer in a bacterial biofilm and examine antibiotic dosing strategies that are effective in bacterial elimination. In particular, we consider a 1-D model of a biofilm with susceptible, persistor and resistant bacteria. Resistance can be transferred to the susceptible bacteria via horizontal gene transfer (HGT), specifically via conjugation. We analyze some basic properties of the model, determine the conditions for existence of disinfection and coexistence states, including boundary equilibria and their stability. Numerical simulations are performed to explore different modeling scenarios and support our theoretical findings. Different antibiotic dosing strategies are then studied, starting with a continuous dosing; here we note that high doses of antibiotic are needed for bacterial elimination. We then consider periodic dosing, and again observe that insufficient levels of antibiotic per dose may lead to treatment failure. Finally, using an extended version of Pontryagin's maximum principle we determine efficient antibiotic dosing protocols, which ensure bacterial elimination while keeping the total dosing low; we note that this involves a tapered dosing which reinforces results presented in other clinical and modeling studies. We study the optimal dosing for different parameter values and note that the optimal dosing schedule is qualitatively robust. - oai:arXiv.org:2512.17825v1 - q-bio.PE - math.AP - Mon, 22 Dec 2025 00:00:00 -0500 - cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Rehan Akber, Adnan Khan + SCA-LLM: Spectral-Attentive LLM-Based Wireless World Modeling for Agentic Communications + https://arxiv.org/abs/2509.08139 + arXiv:2509.08139v2 Announce Type: replace +Abstract: Future AI-native wireless networks are moving from reactive optimization to agentic decision-making that can sense, predict, and plan under fast-varying channels. This calls for wireless world models that can predict and roll out channel dynamics, for which multi-step channel state information (CSI) prediction offers a practical short-horizon look-ahead. Recent advances in foundation sequence models further motivate large language models (LLMs) as general-purpose dynamics learners when suitably adapted to non-text time-series signals. However, bridging CSI to LLMs is non-trivial because an effective adapter must expose informative spectral and temporal evolution patterns, while prior designs provide limited inductive bias to capture such channel structures. To this end, we propose SCA-LLM, a spectral-attentive LLM-based wireless world modeling framework that bridges CSI to LLMs via a spectral-channel attention (SCA) adapter. Specifically, the SCA adapter performs multi-spectral representation learning to extract informative channel features and align CSI with the LLM's sequence modeling capability, enabling parameter-efficient adaptation while keeping the LLM backbone largely frozen. Extensive simulations show that SCA-LLM achieves state-of-the-art prediction performance and strong zero-shot generalization, yielding up to -2.4 dB normalized mean squared error (NMSE) advantage over the previous LLM based method. Our ablation studies further confirm the effectiveness of the proposed SCA adapter in mitigating domain mismatch. + oai:arXiv.org:2509.08139v2 + cs.IT + cs.LG + math.IT + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by/4.0/ + Ke He, Le He, Lisheng Fan, Xianfu Lei, Thang X. Vu, George K. Karagiannidis, Symeon Chatzinotas - Delayed Acceptance Slice Sampling - https://arxiv.org/abs/2512.17868 - arXiv:2512.17868v1 Announce Type: cross -Abstract: Slice sampling is a well-established Markov chain Monte Carlo method for (approximate) sampling of target distributions which are only known up to a normalizing constant. The method is based on choosing a new state on a slice, i.e., a superlevel set of the given unnormalized target density (with respect to a reference measure). However, slice sampling algorithms usually require per step multiple evaluations of the target density, and thus can become computationally expensive. This is particularly the case for Bayesian inference with costly likelihoods. In this paper, we exploit deterministic approximations of the target density, which are relatively cheap to evaluate, and propose delayed acceptance versions of hybrid slice samplers. We show ergodicity of the resulting slice sampling methods, discuss the superiority of delayed acceptance (ideal) slice sampling over delayed acceptance Metropolis-Hastings algorithms, and illustrate the benefits of our novel approach in terms improved computational efficiency in several numerical experiments. - oai:arXiv.org:2512.17868v1 - stat.CO - math.ST - stat.TH - Mon, 22 Dec 2025 00:00:00 -0500 - cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Kevin Bitterlich, Daniel Rudolf, Bj\"orn Sprungk + Approximation by Neural Network operators in $L^p$ spaces associated with an arbitrary measure + https://arxiv.org/abs/2509.09377 + arXiv:2509.09377v2 Announce Type: replace +Abstract: In this paper, we investigate the approximation behavior of both one and multidimensional neural network type operators for functions in $L^p(I^d,\rho)$, where $1\leq p<\infty$, associated with a general measure $\rho$ defined over a hypercube. First, we prove the uniform approximation for a continuous function and the $L^p$ approximation theorem by the NN operators in one and multidimensional settings. In addition, we also obtain the $L^p$ error bounds in terms of $\mathcal{K}$-functionals for these neural network operators. Finally, we consider the logistic and tangent hyperbolic activation functions and verify the hypothesis of the theorems. We also show the implementation of continuous and integrable functions by NN operators with respect to the Lebesgue and Jacobi measures defined on $[0,1]\times[0,1]$ with logistic and tangent hyperbolic activation functions. + oai:arXiv.org:2509.09377v2 + math.FA + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by/4.0/ + Nitin Bartwal, A. Sathish Kumar - Regularized Random Fourier Features and Finite Element Reconstruction for Operator Learning in Sobolev Space - https://arxiv.org/abs/2512.17884 - arXiv:2512.17884v1 Announce Type: cross -Abstract: Operator learning is a data-driven approximation of mappings between infinite-dimensional function spaces, such as the solution operators of partial differential equations. Kernel-based operator learning can offer accurate, theoretically justified approximations that require less training than standard methods. However, they can become computationally prohibitive for large training sets and can be sensitive to noise. We propose a regularized random Fourier feature (RRFF) approach, coupled with a finite element reconstruction map (RRFF-FEM), for learning operators from noisy data. The method uses random features drawn from multivariate Student's $t$ distributions, together with frequency-weighted Tikhonov regularization that suppresses high-frequency noise. We establish high-probability bounds on the extreme singular values of the associated random feature matrix and show that when the number of features $N$ scales like $m \log m$ with the number of training samples $m$, the system is well-conditioned, which yields estimation and generalization guarantees. Detailed numerical experiments on benchmark PDE problems, including advection, Burgers', Darcy flow, Helmholtz, Navier-Stokes, and structural mechanics, demonstrate that RRFF and RRFF-FEM are robust to noise and achieve improved performance with reduced training time compared to the unregularized random feature model, while maintaining competitive accuracy relative to kernel and neural operator tests. - oai:arXiv.org:2512.17884v1 - cs.LG - cs.NA - math.NA - stat.ML - Mon, 22 Dec 2025 00:00:00 -0500 - cross + $H^\infty$-calculus for the Dirichlet Laplacian on conical domains + https://arxiv.org/abs/2509.09519 + arXiv:2509.09519v2 Announce Type: replace +Abstract: We establish boundedness of the $H^\infty$-calculus for the Dirichlet Laplacian on conical domains in $\mathbb{R}^d$ and corresponding wedges on $L^p$-spaces with mixed weights. The weights are based on both the distance to the boundary and the distance to the tip/edge of the cone/wedge. Our main motivation comes from the study of stochastic partial differential equations and associated degenerate deterministic parabolic equations on non-smooth domains. As a consequence of our analysis, we also obtain maximal $L^p$-regularity for the Poisson equation on conical domains in appropriate weighted Sobolev spaces. + oai:arXiv.org:2509.09519v2 + math.FA + math.AP + Tue, 23 Dec 2025 00:00:00 -0500 + replace http://creativecommons.org/licenses/by/4.0/ - Xinyue Yu, Hayden Schaeffer + Petru A. Cioica-Licht, Emiel Lorist, P. Tobias Werner - Extra-Dimensional \eta-Invariants and Anomaly Theories - https://arxiv.org/abs/2512.17906 - arXiv:2512.17906v1 Announce Type: cross -Abstract: Anomalies of a quantum field theory (QFT) constitute fundamental non-perturbatively robust data. In this paper we extract anomalies of 5D superconformal field theories (SCFTs) directly from the underlying extra-dimensional geometry. We show that all of this information can be efficiently extracted from extra-dimensional $\eta$-invariants, bypassing previously established approaches based on computationally cumbersome blowup / resolution techniques. We illustrate these considerations for 5D SCFTs engineered in M-theory by non-compact geometries $X=\mathbb{C}^3/\Gamma$ with finite subgroup $\Gamma\subset SU(3)$, where the anomalies are determined by the $\eta$-invariants of the asymptotic boundary $\partial X=S^5/\Gamma$. Our results apply equally to Abelian and non-Abelian $\Gamma$, as well as isolated and non-isolated singularities. In the setting of non-isolated singularities we further analyze the interplay of anomaly structures across different strata of the singular locus. Our considerations extend readily to backgrounds which are not global orbifolds, as well as those which do not preserve supersymmetry. - oai:arXiv.org:2512.17906v1 - hep-th + The Deligne-Simpson Problem + https://arxiv.org/abs/2509.11998 + arXiv:2509.11998v4 Announce Type: replace +Abstract: Given k similarity classes of invertible matrices, the Deligne-Simpson problem asks to determine whether or not one can find matrices in these classes whose product is the identity and with no common invariant subspace. The first author conjectured an answer in terms of an associated root system, and proved one implication in joint work with Shaw. In this paper we prove the other implication, thus confirming the conjecture. + oai:arXiv.org:2509.11998v4 + math.RA + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by/4.0/ + William Crawley-Boevey, Andrew Hubery + + + On distributional topological complexity of groups and manifolds + https://arxiv.org/abs/2509.16759 + arXiv:2509.16759v3 Announce Type: replace +Abstract: We prove the equality $\dTC(\Gamma)=\TC(\Gamma)$ for distributional topological complexity of torsion free hyperbolic and of torsion free nilpotent groups. + For the distributional topological complexity of lens spaces we prove the inequality $\dTC(L^n_p)\le 2p-1$ and for the distributional LS-category the inequality $d\cat(L^n_p)\le p-1$ which turns into equality for prime $p$ and $n>p$. We use these inequalities to bring counter-examples to the product formula for $d\cat$ and $\dTC$. + oai:arXiv.org:2509.16759v3 + math.GT math.AT - math.DG - Mon, 22 Dec 2025 00:00:00 -0500 - cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Mirjam Cveti\v{c}, Ron Donagi, Jonathan J. Heckman, Max H\"ubner + math.GR + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by/4.0/ + Alexander Dranishnikov - Adiabatic Limit and Deformations of Complex Structures - https://arxiv.org/abs/1901.04087 - arXiv:1901.04087v5 Announce Type: replace -Abstract: Based on our recent adaptation of the adiabatic limit construction to the case of complex structures, we prove the fact that the deformation limiting manifold of any holomorphic family of Moishezon manifolds is Moishezon. Two new ingredients, hopefully of independent interest, are introduced. The first one associates with every compact complex manifold $X$, in every degree $k$, a holomorphic vector bundle over $\C$ of rank equal to the $k$-th Betti number of $X$. This vector bundle, previously given an algebraic construction in the literature, shows that the degenerating page of the Fr\"olicher spectral sequence of $X$ is the holomorphic limit, as $h\in\C^\star$ tends to $0$, of the $d_h$-cohomology of $X$, where $d_h=h\partial + \bar\partial$. A relative version of this vector bundle is then associated with every holomorphic family of compact complex manifolds. The second ingredient is a relaxation of the notion of strongly Gauduchon (sG) metric that we introduced in 2009. For a given positive integer $r$, a Gauduchon metric $\gamma$ on an $n$-dimensional compact complex manifold $X$ is said to be $E_r$-sG if $\partial\gamma^{n-1}$ represents the zero cohomology class on the $r$-th page of the Fr\"olicher spectral sequence of $X$. Strongly Gauduchon metrics coincide with $E_1$-sG metrics. - oai:arXiv.org:1901.04087v5 + Stability conditions on Calabi-Yau threefolds via Brill-Noether theory of curves + https://arxiv.org/abs/2509.24990 + arXiv:2509.24990v2 Announce Type: replace +Abstract: Fix a polarised Calabi-Yau threefold $(X,H)$. We reduce a version of the Bayer-Macr\`i-Toda conjecture for $(X,H)$, which ensures the existence of Bridgeland stability conditions on $X$, to verifying a Brill-Noether-type inequality for curves on $X$. We then prove this inequality for a broad class of Calabi-Yau threefolds, including complete intersection Calabi-Yau threefolds in weighted projective spaces. + oai:arXiv.org:2509.24990v2 math.AG + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Soheyla Feyzbakhsh, Naoki Koseki, Zhiyu Liu, Nick Rekuski + + + The shrinking target and recurrence problem for non-autonomous systems + https://arxiv.org/abs/2510.02586 + arXiv:2510.02586v2 Announce Type: replace +Abstract: We investigate the shrinking target and recurrence set associated to non-autonomous measure-preserving systems on compact metric spaces, establishing zero-one criteria in the spirit of classical Borel-Cantelli results. + Our first main theorem gives a quantitative shrinking target result for non-autonomous systems under a uniform mixing condition, providing asymptotics with an optimal error term. This general result is applicable to certain families of inner functions, yielding concrete applications such as patterns of zeros in the multibase expansion. Turning to recurrence, we establish new zero-measure laws for non-autonomous systems. In the autonomous case, we prove a zero-one criterion for recurrence sets of centred, one-component inner functions via Markov partitions and distortion estimates. Together, these results provide a unified framework for shrinking target and recurrence problems in both autonomous and non-autonomous dynamics. + oai:arXiv.org:2510.02586v2 + math.DS math.CV + math.NT + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by/4.0/ + Ayesha Bennett + + + Carrollian Lie Algebroids: Taming Singular Carrollian Geometries + https://arxiv.org/abs/2510.03877 + arXiv:2510.03877v3 Announce Type: replace +Abstract: Developments in Carrollian gravity and holography necessitate the use of singular Carroll vector fields, a feature that cannot be accommodated within standard Carrollian geometry. We introduce Carrollian Lie algebroids as a framework to study such singular Carrollian geometries. In this approach, we define the Carroll distribution as the image of the kernel of the degenerate metric under the anchor map. The Carroll distribution is, in general, a singular Stefan--Sussmann distribution that will fluctuate between rank-1 and rank-0, and so captures the notion of a singular Carroll vector field. As an example, we show that an invariant Carrollian structure on a principal bundle leads to a Carrollian structure on the associated Atiyah algebroid that will, in general, have a singular Carroll distribution. Mixed null-spacelike hypersurfaces, under some simplifying assumptions, also lead to examples of Carrollian Lie algebroids. Furthermore, we establish the existence of compatible connections on Carrollian Lie algebroids, and as a direct consequence, we conclude that Carrollian manifolds can always be equipped with compatible affine connections. + oai:arXiv.org:2510.03877v3 math.DG - Mon, 22 Dec 2025 00:00:00 -0500 + gr-qc + hep-th + math-ph + math.MP + Tue, 23 Dec 2025 00:00:00 -0500 replace http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Dan Popovici + Andrew James Bruce - On the ROF Model in Rectilinear Anisotropy: Piecewise Constant Approximation and Universal Minimality - https://arxiv.org/abs/1910.05186 - arXiv:1910.05186v2 Announce Type: replace -Abstract: We prove that the $L^2$ distance between the minimizer of the $\ell^1$-anisotropic Rudin-Osher-Fatemi (ROF) functional and its minimizer over the space of piecewise constant functions on a rectilinear grid is $\mathcal{O}(h^{\frac12 - \frac{q'}{2q}})$, where $h$ is the grid's mesh size and the datum belongs to $L^q$, $q \ge 2$. These convergence rates are valid in any dimension $d\ge 1$. However, in dimension $d = 1$ they can be further improved to $\mathcal{O}(h^{\frac12 - \frac{1}{2q}})$. To establish the error bounds, $L^q$ estimates of the ROF minimizer in terms of the datum are critical. Such estimates are particular cases of a universal minimality property of the ROF minimizer derived in the second part of the paper. There it is shown, in both the finite-dimensional and infinite-dimensional settings, that the minimizer simultaneously minimizes a broad class of convex functionals over a neighbourhood of the datum arising in the convex dual of the ROF problem. This extends previous results of similar type about taut strings and the ROF problem. - oai:arXiv.org:1910.05186v2 - math.OC - Mon, 22 Dec 2025 00:00:00 -0500 + Inverse scattering for $N$-body time-decaying harmonic oscillators + https://arxiv.org/abs/2510.04702 + arXiv:2510.04702v3 Announce Type: replace +Abstract: In the previous study (Ishida, 2025), the author proved the uniqueness of short-range potential functions using the Enss-Weder time-dependent method (Enss and Weder, 1995) for a two-body quantum system described by time-decaying harmonic oscillators. In this study, we extend the result of Ishida (2025) to the $N$-body case. We use the approaches developed in Enss and Weder (1995), Weder (1996), and Valencia and Weder (2012) to prove that the high-velocity limit of the scattering operator uniquely determines all the pairwise interaction potentials among the $N$ particles, focusing respectively on each fixed pair of particles. + oai:arXiv.org:2510.04702v3 + math-ph + math.AP + math.MP + Tue, 23 Dec 2025 00:00:00 -0500 replace http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Clemens Kirisits, Eric Setterqvist + Atsuhide Ishida - The number of maximal unrefinable partitions - https://arxiv.org/abs/2206.04261 - arXiv:2206.04261v2 Announce Type: replace -Abstract: This paper completes the classification of maximal unrefinable partitions, extending a previous work of Aragona et al. devoted only to the case of triangular numbers. We show that the number of maximal unrefinable partitions of an integer coincides with the number of suitable partitions into distinct parts, depending on the distance from the successive triangular number. - oai:arXiv.org:2206.04261v2 - math.CO + Multiplicative dependence in the denominators of points of elliptic curves + https://arxiv.org/abs/2510.05462 + arXiv:2510.05462v2 Announce Type: replace +Abstract: Let $E_1, \ldots, E_s $ be $s$, not necessary distinct, elliptic curves over $\mathbb{Q}$. We give upper bounds on the frequency of $s$-tuples of points in $E_1(\mathbb{Q})\times \ldots \times E_s(\mathbb{Q})$ whose denominators or $x$-coordinates are multiplicatively dependent. More precisely, we give such bounds in two scenarios: one in which we fix $s$ non-torsion $\mathbb{Q}$-rational points $P_i \in E_i(\mathbb{Q})$ and arbitrary $\mathbb{Q}$-rational points $Q_i \in E_i(\mathbb{Q})$, $i =1, \ldots, s$, and we count $s$-tuples \[ (n_1P_1+Q_1,\ldots, n_sP_s+Q_s) \in E_1(\mathbb{Q}) \times \ldots \times E_s(\mathbb{Q}) \] with $n_1, \ldots, n_s$ in an arbitrary interval of length $N$, and the second in which we count points $(P_1,\ldots,P_s) \in E_1(\mathbb{Q}) \times \ldots \times E_s(\mathbb{Q})$ of bounded canonical height. + oai:arXiv.org:2510.05462v2 math.NT - Mon, 22 Dec 2025 00:00:00 -0500 + Tue, 23 Dec 2025 00:00:00 -0500 replace http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Riccardo Aragona, Lorenzo Campioni, Roberto Civino + Attila B\'erczes, Subham Bhakta, Lajos Hajdu, Alina Ostafe, Igor E. Shparlinski - Hyperbolicity in presence of a large local system - https://arxiv.org/abs/2207.03283 - arXiv:2207.03283v2 Announce Type: replace -Abstract: We prove that the projective complex algebraic varieties admitting a large complex local system satisfy a strong version of the Green-Griffiths-Lang conjecture. - oai:arXiv.org:2207.03283v2 - math.AG - math.CV - Mon, 22 Dec 2025 00:00:00 -0500 + Automorphically Equivalent Elements of Finite Abelian Groups + https://arxiv.org/abs/2510.06013 + arXiv:2510.06013v2 Announce Type: replace +Abstract: Given a finite abelian group $G$ and elements $x, y \in G$, we prove that there exists $\phi \in \text{Aut}(G)$ such that $\phi(x) = y$ if and only if $G/\langle x \rangle \cong G/\langle y \rangle$. This result leads to our development of the two fastest known algorithms to determine if two elements of a finite abelian group are automorphic images of one another. The second algorithm also computes $G/\langle x \rangle$ in a near-linear time algorithm for groups, most feasible when the group has exponent at most $10^{20}$. We conculde with an algorithm that computes the automorphic orbits of finite abelian groups. + oai:arXiv.org:2510.06013v2 + math.GR + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by/4.0/ + Arjun Agarwal, Rachel Chen, Rohan Garg, Jared Kettinger + + + Fully nonlinear prescribed curvature problems on closed manifolds with negative curvature + https://arxiv.org/abs/2510.06577 + arXiv:2510.06577v2 Announce Type: replace +Abstract: In this manuscript, we investigate fully nonlinear prescribed curvature problems for the modified Schouten tensor on closed Riemannian manifolds with negative curvature. We prove that whenever the corresponding concave elliptic operator satisfies a structural Condition $T$, which encompasses all $O(n)$-invariant G\r{a}rding-Dirichlet operator, such prescribed curvature problems are always solvable. + oai:arXiv.org:2510.06577v2 + math.DG + math.AP + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by/4.0/ + Jiaogen Zhang + + + Nested superposition principle for random measures and the geometry of the Wasserstein on Wasserstein space + https://arxiv.org/abs/2510.07523 + arXiv:2510.07523v2 Announce Type: replace +Abstract: We study the geometric structure of the space of random measures $\mathcal{P}_p(\mathcal{P}_p(X))$, endowed with the Wasserstein on Wasserstein metric, where $(X, d)$ is a complete separable metric space. In this setting, we prove a metric superposition principle, in the spirit of the result by S. Lisini, that will allow us to recover important geometric features of the space. When $X$ is $\mathbb{R}^d$, we study the differential structure of $\mathcal{P}_p(\mathcal{P}_p(\mathbb{R}^d))$ in analogy with the simpler Wasserstein space $\mathcal{P}_p(\mathbb{R}^d)$. We show that continuity equations for random measures involving the abstract concept of derivation acting on cylinder functions can be more conveniently described by suitable non-local vector fields $b:[0,T]\times \mathbb{R}^d \times \mathcal{P}(\mathbb{R}^d) \to \mathbb{R}^d$. In this way, we can: 1) characterize the absolutely continuous curves on the Wasserstein on Wasserstein space; 2) define and characterize its tangent bundle; 3) prove a superposition principle for the solutions to the standard non-local continuity equation in terms of solutions of interacting particle systems. + oai:arXiv.org:2510.07523v2 + math.FA + math.MG + math.PR + Tue, 23 Dec 2025 00:00:00 -0500 replace http://creativecommons.org/licenses/by/4.0/ - Yohan Brunebarbe + Alessandro Pinzi, Giuseppe Savar\'e - Emergence of quantum dynamics from chaos: The case of prequantum cat maps - https://arxiv.org/abs/2209.02027 - arXiv:2209.02027v3 Announce Type: replace -Abstract: Faure and Tsujii recently proposed a new quantization theory for symplectic Anosov diffeomorphisms. It combines prequantization with the study of the Pollicott--Ruelle resonances of an associated transfer operator. We apply this framework to the hyperbolic symplectic automorphisms of the $2n$-dimensional torus, the so-called cat maps. Our main result gives an explicit relation between the resonances of the prequantum transfer operator and the eigenvalues of the standard quantum cat maps, generalizing the case $n=1$ previously treated by Faure. - oai:arXiv.org:2209.02027v3 + Limit cycles and invariant algebraic curves + https://arxiv.org/abs/2510.11705 + arXiv:2510.11705v2 Announce Type: replace +Abstract: We give lower bounds in terms of~$n,$ for the number of limit cycles of polynomial vector fields of degree~$n,$ having any prescribed invariant algebraic curve. By applying them when the ovals of this curve are also algebraic limit cycles we obtain a new recurrent property for the Hilbert numbers. Finally, we apply our results to two important families of models: Kolmogorov systems and a general family of systems appearing in Game Theory. + oai:arXiv.org:2510.11705v2 math.DS - math-ph - math.MP - math.SP - Mon, 22 Dec 2025 00:00:00 -0500 + Tue, 23 Dec 2025 00:00:00 -0500 replace http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - 10.3934/dcds.2026016 - Javier Echevarr\'ia Cuesta + Armengol Gasull, Paulo Santana - Cell Modules for Type $A$ Webs - https://arxiv.org/abs/2210.09639 - arXiv:2210.09639v3 Announce Type: replace -Abstract: We examine the cell modules for the category of type An webs and their natural cellular forms. We modify the bases of these modules, as described by Elias, to obtain an orthogonal basis of each cell module. Hence, we calculate the determinant of the Gram matrix with respect to such bases. - These Gram determinants are given in terms of intersection forms, computed from certain traces of clasps - higher order Jones-Wenzl morphisms. Additionally, the modified basis is constructed using these clasps, and each clasp is constructed using traces of smaller clasps. - Elias conjectures a value for these intersection forms and verifies it in types $A_1$, $A_2$ and $A_3$. This paper concludes with a proof of the conjecture in type $A_n$. - oai:arXiv.org:2210.09639v3 - math.RT - Mon, 22 Dec 2025 00:00:00 -0500 + Homogeneous spaces over an abelian variety + https://arxiv.org/abs/2510.15126 + arXiv:2510.15126v2 Announce Type: replace +Abstract: In this paper, we study a question of Colliot-Th\'el\`ene and Iyer concerning the existence of rational sections in families of homogeneous spaces over an abelian variety, after base change by a suitable \'etale isogeny of the abelian variety. Assuming characteristic zero and that the homogeneous spaces arise from connected reductive groups, the problem is reformulated in terms of torsors under reductive groups over an abelian variety $A$. + Building on work of Moonen and Polishchuk, we construct a filtration on the motive of a Jacobian variety to analyze the action of isogenies on unramified cohomology and Witt groups. This approach allows for a positive response to the question for reductive groups whose root data do not contain a factor of type~$E_8$ when $\dim A > 2$ and $\mathrm{cd}(k) \leqslant 1$, and for all reductive groups when $\dim A = 2$ and $k$ is algebraically closed. + oai:arXiv.org:2510.15126v2 + math.AG + Tue, 23 Dec 2025 00:00:00 -0500 replace http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Stuart Martin, Robert A. Spencer + Margot Bruneaux - openCFS-Data: Data Pre-Post-Processing Tool for openCFS - https://arxiv.org/abs/2302.03637 - arXiv:2302.03637v3 Announce Type: replace -Abstract: Many numerical simulation tools have been developed and are on the market, but there is still a strong need for appropriate tools capable to simulate multi-field problems, especially in aeroacoustics. Therefore, openCFS provides an open-source framework for implementing partial differential equations using the finite element method. Since 2000, the software has been developed continuously. The result of is openCFS (before 2020 known as CFS++ Coupled Field Simulations written in C++). In this paper, we present for the first time the CFS-Data, the open-source pre-post-processing part of openCFS with a focus on the aeroacoustic source computation (called filters). - oai:arXiv.org:2302.03637v3 + General transformation neural networks: A class of parametrized functions for high-dimensional function approximation + https://arxiv.org/abs/2510.20142 + arXiv:2510.20142v2 Announce Type: replace +Abstract: We propose a novel class of neural network-like parametrized functions, i.e., general transformation neural networks (GTNNs), for high-dimensional approximation. Conventional deep neural networks sometimes perform less accurately in approximation problems under gradient descent training, especially when the target function is oscillatory. To improve accuracy, we generalize the affine transformation of the abstract neuron to more general functions that serve as complex shape functions and have greater capacity. Specifically, we discuss three types of GTNNs in detail: the cubic, quadratic, and trigonometric transformation neural networks (CTNNs, QTNNs, and TTNNs). We perform an approximation error analysis of GTNNs, presenting their universal approximation properties for continuous functions and error bounds for smooth and Barron-type functions. Several numerical examples of regression problems and partial differential equations are presented, demonstrating that CTNNs/QTNNs/TTNNs achieve higher accuracy and greater robustness than conventional fully connected neural networks. + oai:arXiv.org:2510.20142v2 math.NA cs.NA - Mon, 22 Dec 2025 00:00:00 -0500 + Tue, 23 Dec 2025 00:00:00 -0500 replace http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Stefan Schoder, Klaus Roppert + Xiaoyang Wang, Yiqi Gu - Sparsity of postcritically finite maps of $\mathbb{P}^k$ and beyond: A complex analytic approach - https://arxiv.org/abs/2305.02246 - arXiv:2305.02246v2 Announce Type: replace -Abstract: An endomorphism $f:\mathbb{P}^k\to\mathbb{P}^k$ of degree $d\geq2$ is said to be postcritically finite (or PCF) if its critical set $\mathrm{Crit}(f)$ is preperiodic, i.e. if there are integers $m>n\geq0$ such that $f^m(\mathrm{Crit}(f))\subseteq f^n(\mathrm{Crit}(f))$. When $k\geq2$, it was conjectured by Ingram, Ramadas and Silverman that, in the space $\mathrm{End}_d^k$ of all endomorphisms of degree $d$ of $\mathbb{P}^k$, such endomorphisms are not Zariski dense. We prove this conjecture. Further, in the space $\mathrm{Poly}_d^2$ of all regular polynomial endomorphisms of degree $d\geq2$ of the affine plane $\mathbb{A}^2$, we construct a dense and Zariski open subset where we have a uniform bound on the number of preperiodic points lying in the critical set. - The proofs are a combination of the theory of heights in arithmetic dynamics and methods from real dynamics to produce open subsets with maximal bifurcation. - oai:arXiv.org:2305.02246v2 - math.DS - math.CV - math.NT - Mon, 22 Dec 2025 00:00:00 -0500 + Error Estimates for Sparse Tensor Products of B-spline Approximation Spaces + https://arxiv.org/abs/2510.21517 + arXiv:2510.21517v3 Announce Type: replace +Abstract: This work introduces and analyzes B-spline approximation spaces defined on general geometric domains obtained through a mapping from a parameter domain. These spaces are constructed as sparse-grid tensor products of univariate spaces in the parameter domain and are mapped to the physical domain via a geometric parametrization. Both the univariate approximation spaces and the geometric mapping are built using maximally smooth B-splines. We construct two such spaces, employing either the sparse-grid combination technique or the hierarchical subspace decomposition of sparse-grid tensor products, and we prove their mathematical equivalence. Furthermore, we derive approximation error estimates and inverse inequalities that highlight the advantages of sparse-grid tensor products. Specifically, under suitable regularity assumptions on the solution, these spaces achieve the same approximation order as standard tensor product spaces while using significantly fewer degrees of freedom. Additionally, our estimates indicate that, in the case of non-tensor-product domains, stronger regularity assumptions on the solution -- particularly concerning isotropic (non-mixed) derivatives -- are required to achieve optimal convergence rates compared to sparse-grid methods defined on tensor-product domains. + oai:arXiv.org:2510.21517v3 + math.NA + cs.NA + Tue, 23 Dec 2025 00:00:00 -0500 replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Thomas Gauthier, Johan Taflin, Gabriel Vigny + http://creativecommons.org/licenses/by/4.0/ + Cl\'ement Guillet - Generalized Feynman-Kac Formula and Associated Heat Kernel - https://arxiv.org/abs/2306.04740 - arXiv:2306.04740v4 Announce Type: replace -Abstract: Let M be a smooth closed (compact without boundary) Riemannian manifold of dimension n and P a q-dimensional smooth submanifold of M. U will denote the tubular neighborhood of P in M. Let E be a smooth vector bundle over M. Here we will obtain a vector bundle Generalized Feynman-Kac formula associated to U and the vector bundle differential operator L consisteing of half the generalized Laplacian, a vector field X on M and a potential term V on M. From this formula, we shall deduce the usual Feynman-Kac formula as well as a stochastic representation of the Generalized Elworthy-Truman Heat Kernel formula, and ultimately the heat kernel formula. The Feynman-Kac expression can be expanded and from this expansion we shall deduce both the generalized heat trace and heat content expansions. The Generalized Feynman-Kac Formula is thus at the center of several other previously known results. - oai:arXiv.org:2306.04740v4 - math.GM - Mon, 22 Dec 2025 00:00:00 -0500 + Multiscale analysis of the conductivity in the Lorentz mirrors model + https://arxiv.org/abs/2510.24091 + arXiv:2510.24091v3 Announce Type: replace +Abstract: We consider the mirrors model in $d$ dimensions on an infinite slab and with unit density. This is a deterministic dynamics in a random environment. We argue that the crossing probability of the slab goes like $\kappa/(\kappa+N)$ where $N$ is the width of the slab. We are able to compute $\kappa$ perturbatively by using a multiscale approach. The only small parameter involved in the expansion is the inverse of the size of the system. This approach rests on an inductive process and a closure assumption adapted to the mirrors model. For $d=3$, we propose the recursive relation for the conductivity $\kappa_n$ at scale $n$ : $\kappa_{n+1}=\kappa_n(1+\frac{\kappa_n}{2^{n}}\alpha)$, up to $o(1/2^n)$ terms and with $\alpha\simeq 0.0374$. This sequence has a finite limit. + oai:arXiv.org:2510.24091v3 + math.PR + cond-mat.stat-mech + math-ph + math.MP + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by-nc-nd/4.0/ + Raphael Lefevere + + + A Single-Loop First-Order Algorithm for Linearly Constrained Bilevel Optimization + https://arxiv.org/abs/2510.24710 + arXiv:2510.24710v2 Announce Type: replace +Abstract: We study bilevel optimization problems where the lower-level problems are strongly convex and have coupled linear constraints. To overcome the potential non-smoothness of the hyper-objective and the computational challenges associated with the Hessian matrix, we utilize penalty and augmented Lagrangian methods to reformulate the original problem as a single-level one. Especially, we establish a strong theoretical connection between the reformulated function and the original hyper-objective by characterizing the closeness of their values and derivatives. Based on this reformulation, we propose a single-loop, first-order algorithm for linearly constrained bilevel optimization (SFLCB). We provide rigorous analyses of its non-asymptotic convergence rates, showing an improvement over prior double-loop algorithms -- form $O(\epsilon^{-3}\log(\epsilon^{-1}))$ to $O(\epsilon^{-3})$. The experiments corroborate our theoretical findings and demonstrate the practical efficiency of the proposed SFLCB algorithm. Simulation code is provided at https://github.com/ShenGroup/SFLCB. + oai:arXiv.org:2510.24710v2 + math.OC + cs.IT + cs.LG + math.IT + stat.ML + Tue, 23 Dec 2025 00:00:00 -0500 replace http://creativecommons.org/licenses/by/4.0/ - Martin N. Ndumu + Wei Shen, Jiawei Zhang, Minhui Huang, Cong Shen - Reverse isoperimetric inequalities for Lagrangian intersection Floer theory - https://arxiv.org/abs/2306.04761 - arXiv:2306.04761v2 Announce Type: replace -Abstract: We extend Groman and Solomon's reverse isoperimetric inequality to pseudoholomorphic curves with punctures at the boundary and whose boundary components lie in a collection of Lagrangian submanifolds with intersections locally modelled on $\mathbb{R}^n\cap (\mathbb{R}^{k}\times \sqrt{-1}\mathbb{R}^{n-k})$ inside $\mathbb{C}^n$. Our construction closely follows the methods used by Duval (2016) and Abouzaid (2021) and corrects an error appearing in the latter approach. - oai:arXiv.org:2306.04761v2 - math.CV - math.SG - Mon, 22 Dec 2025 00:00:00 -0500 + Fractional Iterates and Oscillatory Convergence + https://arxiv.org/abs/2510.25606 + arXiv:2510.25606v3 Announce Type: replace +Abstract: The simple continued fractions for the Golden & Silver means are well-known. It is astonishing that, as far as we know, no one has published half-iterates (let alone quarter-iterates) for the corresponding algorithms. We also examine the cosine and logistic maps (with parameter $2 < \lambda < 3$). + oai:arXiv.org:2510.25606v3 + math.NT + cs.DM + Tue, 23 Dec 2025 00:00:00 -0500 replace http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Jean-Philippe Chass\'e, Jeff Hicks, Yoon Jae Nick Nho + Steven Finch - On Watanabe's characterisation and change of intensity \`{a} la Girsanov for Cox processes - https://arxiv.org/abs/2308.05080 - arXiv:2308.05080v3 Announce Type: replace -Abstract: We discuss the equivalence of definitions for conditional Poisson processes, Cox processes, and stochastic intensities of point processes on the real line. We show that Watanabe's characterisation of conditional Poisson processes in terms of local martingales is necessary and sufficient. Additionally, we consider conditions enabling the measure change method a la Girsanov to alter the intensity of Cox processes to a desired new target intensity, e.g. for the probability reference approach in filtering. Such a measure change exists if a corresponding stochastic exponential is a proper martingale. We show that this holds if the new locally integrable target intensity is the product of the original intensity and another non-negative process. - oai:arXiv.org:2308.05080v3 - math.PR - Mon, 22 Dec 2025 00:00:00 -0500 + On the class of exponential statistical structures of type B + https://arxiv.org/abs/2510.26863 + arXiv:2510.26863v2 Announce Type: replace +Abstract: The article is devoted to the study of exponential statistical structures of type B, which constitute a subclass of exponential families of probability distributions. This class is characterized by a number of analytical and probabilistic properties that make it a convenient tool for solving both theoretical and applied problems in statistics. The relevance of this research lies in the need to generalize known classes of distributions and to develop a unified framework for their analysis, which is essential for applications in stochastic modeling, machine learning, financial mathematics. The paper proposes a formal definition of type B. Necessary and sufficient conditions for a statistical structure to belong to class B are established, and it is proved that such structures can be represented through a dominating measure with an explicit Laplace transform. The obtained results make it possible to describe a wide range of well-known one-dimensional and multivariate distributions, including the Binomial, Poisson, Normal, Gamma, Polynomial, Logarithmic distributions, as well as specific cases such as the Borel-Tanner and Random Walk distributions. Particular attention is given to the proof of structural theorems that determine the stability of class B under linear transformations and the addition of independent random vectors. Recursive relations for initial and central moments as well as for semi-invariants are obtained, providing an efficient analytical and computational framework for their evaluation. Furthermore, the tails of type B distributions are investigated using the properties of the Laplace transform. New exponential inequalities for estimating the probabilities of large deviations are derived. The obtained results can be applied in theoretical studies and in practical problems of stochastic modeling. + oai:arXiv.org:2510.26863v2 + math.ST + stat.TH + Tue, 23 Dec 2025 00:00:00 -0500 replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Dirk Becherer, Thomas Bernhardt, Pavel Gapeev + http://creativecommons.org/licenses/by/4.0/ + Oleksandr Volkov, Yurii Volkov - Unifying Distributionally Robust Optimization via Optimal Transport Theory - https://arxiv.org/abs/2308.05414 - arXiv:2308.05414v2 Announce Type: replace -Abstract: In recent years, two prominent paradigms have shaped distributionally robust optimization (DRO), modeling distributional ambiguity through $\phi$-divergences and Wasserstein distances, respectively. While the former focuses on ambiguity in likelihood ratios, the latter emphasizes ambiguity in outcomes and uses a transportation cost function to capture geometric structure in the outcome space. This paper proposes a unified framework that bridges these approaches by leveraging optimal transport (OT) with conditional moment constraints. Our formulation enables adversarial distributions to jointly perturb likelihood ratios and outcomes, yielding a generalized OT coupling between the nominal and perturbed distributions. We further establish key duality results and develop tractable reformulations that highlight the practical power of our unified approach. - oai:arXiv.org:2308.05414v2 + Nonasymptotic Convergence Rates for Plug-and-Play Methods With MMSE Denoisers + https://arxiv.org/abs/2510.27211 + arXiv:2510.27211v4 Announce Type: replace +Abstract: It is known that the minimum-mean-squared-error (MMSE) denoiser under Gaussian noise can be written as a proximal operator, which suffices for asymptotic convergence of plug-and-play (PnP) methods but does not reveal the structure of the induced regularizer or give convergence rates. We show that the MMSE denoiser corresponds to a regularizer that can be written explicitly as an upper Moreau envelope of the negative log-marginal density, which in turn implies that the regularizer is 1-weakly convex. Using this property, we derive (to the best of our knowledge) the first sublinear convergence guarantee for PnP proximal gradient descent with an MMSE denoiser. We validate the theory with a one-dimensional synthetic study that recovers the implicit regularizer. We also validate the theory with imaging experiments (deblurring and computed tomography), which exhibit the predicted sublinear behavior. + oai:arXiv.org:2510.27211v4 math.OC + eess.SP stat.ML - Mon, 22 Dec 2025 00:00:00 -0500 + Tue, 23 Dec 2025 00:00:00 -0500 replace http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Jose Blanchet, Daniel Kuhn, Jiajin Li, Bahar Taskesen + Henry Pritchard, Rahul Parhi - On flat even deformation rings - https://arxiv.org/abs/2309.01871 - arXiv:2309.01871v2 Announce Type: replace -Abstract: In the presence of a nontrivial dual Selmer group, certain global even deformation rings are shown to be finite and flat over $\mathbb{Z}_p$. Previously, flatness was only known in established cases of Langlands reciprocity in the odd parity. By techniques from global class field theory, explicit examples of even representations are computed to which the results apply. For even representations $\overline{\rho}$ in an explicit family, it is observed that if Leopoldt's conjecture is true for a certain number field attached to $\overline{\rho}$, then the global even deformation ring is flat at the minimal level. - oai:arXiv.org:2309.01871v2 - math.NT - Mon, 22 Dec 2025 00:00:00 -0500 + Study of power series distributions with specified covariances + https://arxiv.org/abs/2511.01081 + arXiv:2511.01081v2 Announce Type: replace +Abstract: This paper presents a study of power series distributions (PSD) with prescribed covariance characteristics. Such distributions constitute a fundamental class in probability theory and mathematical statistics, as they generalize a wide range of well-known discrete distributions and enable the description of various stochastic phenomena with a predetermined variance structure. The aim of the research is to develop analytical methods for constructing power series distributions with given covariances and to establish the conditions under which a particular function can serve as the covariance of a certain PSD. The paper derives a first-order differential equation for the generating function of the distribution, which determines the relationship between its parameters and the form of the covariance function. It is shown that the choice of an analytical or polynomial covariance completely specifies the structure of the corresponding generating function. The analysis made it possible to construct new families of PSDs that generalize the classical Bernoulli, Poisson, geometric, and other distributions while preserving a given covariance structure. The proposed approach is based on the analytical relationship between the generating function and the covariance function, providing a framework for constructing stochastic models with predefined dispersion properties. The results obtained expand the theoretical framework for describing discrete distributions and open up opportunities for practical applications in statistical estimation, modeling of complex systems, financial processes, machine learning where it is crucial to control the dependence between the mean and the variation. Further research may focus on constructing continuous analogues of such distributions, studying their limiting properties, and applying them to problems of regression and Bayesian analysis. + oai:arXiv.org:2511.01081v2 + math.ST + stat.TH + Tue, 23 Dec 2025 00:00:00 -0500 replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Peter Vang Uttenthal + http://creativecommons.org/licenses/by/4.0/ + Oleksandr Volkov, Yurii Volkov, Nataliia Voinalovych + + + Dirac - von Neumann axioms in the setting of Continuous Model Theory + https://arxiv.org/abs/2511.01900 + arXiv:2511.01900v2 Announce Type: replace +Abstract: We recast the well-known axiom system of quantum mechanics used by physicists (the Dirac calculus) in the language of Continuous Logic. For the basic version of the axiomatic system we prove that along with the canonical continuous model the axioms have approximate finite models of large sizes, in fact the continuous model is isomorphic to an ultraproduct of finite models. We analyse the continuous logic quantifier corresponding to Dirac integration and show that in finite context it has two versions, local and global, which coincide on Gaussian wave-functions. + oai:arXiv.org:2511.01900v2 + math.LO + quant-ph + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by/4.0/ + Boris Zilber - Langevin dynamics of lattice Yang-Mills-Higgs and applications - https://arxiv.org/abs/2401.13299 - arXiv:2401.13299v3 Announce Type: replace -Abstract: In this paper, we investigate the Langevin dynamics of various lattice formulations of the Yang--Mills--Higgs model, with an inverse Yang--Mills coupling $\beta$ and a Higgs parameter $\kappa$. The Higgs component is either a bounded field taking values in a compact target space, or an unbounded field taking values in a vector space in which case the model also has a Higgs mass parameter $m$. We study the regime where $(\beta,\kappa)$ are small in the first case or $(\beta,\kappa/m)$ are small in the second case. We prove the exponential ergodicity of the dynamics on the whole lattice via functional inequalities. We establish exponential decay of correlations for a broad class of observables, namely, the infinite volume measure exhibits a strictly positive mass gap. Moreover, when the target space of the Higgs field is compact, appropriately rescaled observables exhibit factorized correlations in the large $N$ limit. These extend the earlier results \cite{SZZ22} on pure lattice Yang--Mills to the case with a coupled Higgs field. - Unlike pure lattice Yang--Mills where the field is always bounded, in the case where the coupled Higgs component is unbounded, the control of its behavior is much harder and requires new techniques. Our approach involves a disintegration argument and a delicate analysis of correlations to effectively control the unbounded Higgs component. - oai:arXiv.org:2401.13299v3 + Estimates of transport distance in the central limit theorem + https://arxiv.org/abs/2511.02033 + arXiv:2511.02033v2 Announce Type: replace +Abstract: Let $X_1,\ldots,X_n$ be $d$-dimensional independent random vectors bounded with probability one. For simplicity, we assume that they have zero mean values: \begin{equation} \mathbf{P}\{\|X_{j}\|\le\tau\}=1,\quad\mathbf{E}\,X_{j}=0,\quad j=1,\ldots, n.\nonumber \end{equation} We study the distribution behavior of the sum $S=X_{1}+\cdots+X_{n}$ as a function of the bounding value $\tau$. + From the non-uniform Bikelis estimate in the one-dimensional central limit theorem it follows that $$ W_1(F,\Phi_{\sigma})\le c\tau. $$ with an absolute constant $c$, where $W_1$ is the Kantorovich--Rubinstein--Wasserstein transport distance, $F$ is the distribution of the sum $S$, and $\Phi_{\sigma}$ is the corresponding normal distribution. The main result of the paper is significantly stronger and more precise. It is claimed that $$ \rho(F,\Phi_{\sigma}) =\inf\int\exp(|x-y|/c\tau)\,d\pi(x,y)\le c, $$ where the infimum is taken over all bivariate probability distributions $\pi$ with marginal distributions $F$ and $\Phi_{\sigma}$. The result has also been generalized to distributions with sufficiently slowly growing cumulants from the class $\mathcal{A}_{1}(\tau )$, introduced in the author's 1986 paper. The possibility of generalizing the result to the multivariate case is discussed. + oai:arXiv.org:2511.02033v2 math.PR - math-ph - math.MP - Mon, 22 Dec 2025 00:00:00 -0500 + Tue, 23 Dec 2025 00:00:00 -0500 replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Hao Shen, Rongchan Zhu, Xiangchan Zhu + http://creativecommons.org/licenses/by/4.0/ + Andrei Yu. Zaitsev - Non-semisimple Crane-Yetter theory varying over the character stack - https://arxiv.org/abs/2404.19667 - arXiv:2404.19667v3 Announce Type: replace -Abstract: We construct a relative version of the Crane-Yetter topological quantum field theory in four dimensions, from non-semisimple data. Our theory is defined relative to the classical $G$-gauge theory in five dimensions -- this latter theory assigns to each manifold $M$ the appropriate linearization of the moduli stack of $G$-local systems, called the character stack. Our main result is to establish a relative invertibility property for our construction. This invertibility generalizes the key invertibility property of the original Crane-Yetter theory which allowed it to capture the framing anomaly of the celebrated Witten-Reshetikhin-Turaev theory. In particular our invertibilty statement at the level of surfaces implies a categorical, stacky version of the unicity theorem for skein algebras; at the level of 3-manifolds it equips the character stack with a canonical line bundle. Regarded as a topological symmetry defect of classical gauge theory, our work establishes invertibility of this defect by a gauging procedure. - oai:arXiv.org:2404.19667v3 - math.QA - math.AG - math.GT - math.RT - Mon, 22 Dec 2025 00:00:00 -0500 + Weighted wave envelope estimates for the parabola + https://arxiv.org/abs/2511.04503 + arXiv:2511.04503v2 Announce Type: replace +Abstract: In this paper, we extend the C\'ordoba-Fefferman square function estimate for the parabola to a weighted setting. Our weighted square function estimate is derived from a weighted wave envelope estimate for the parabola. The bounds are formulated in terms of families of multiscale tubes together with weight parameters that quantify the distribution of the weight. As an application, we obtain some weighted L^p-estimates for a class of Fourier multiplier operators and for solutions to free Schr\"odinger equation. + oai:arXiv.org:2511.04503v2 + math.CA + math.AP + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by/4.0/ + Jongchon Kim, Hyerim Ko + + + Uniform pathwise stability of additive singular SDEs driven by fractional Brownian motion + https://arxiv.org/abs/2511.05262 + arXiv:2511.05262v2 Announce Type: replace +Abstract: We study the long-time behaviour of solutions to a class of $d$-dimensional stochastic differential equations driven by fractional Brownian motion with Hurst parameter $H \in (0,1)$. The drift consists of a dissipative Lipschitz term and a singular term of regularity $\gamma >1-1/(2H)$ in Besov-H\"older scales. We establish well-posedness and, through a Markovian enhancement, existence of an invariant measure. If the singular contribution is sufficiently small, we prove exponential contraction of solutions, and thereby, uniqueness of the invariant measure. Our methods rely on uniform pathwise estimates which utilise together the dissipativity of the drift and the regularisation effect of the noise. + oai:arXiv.org:2511.05262v2 + math.PR + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Konstantinos Dareiotis, El Mehdi Haress, Khoa L\^e + + + Partition Principle without Choice via Symmetric Iterations and Sheaf-Toposes + https://arxiv.org/abs/2511.07675 + arXiv:2511.07675v2 Announce Type: replace +Abstract: We study the topos $\mathcal{E}=\mathsf{Sh}(H\ltimes 2^{\mathbb{N}})$ arising from a nontrivial finite group $H$ acting freely on Cantor space. + Using a local embedding property for the relevant epimorphisms together with effective descent for monomorphisms, + we show that the \emph{internal} set universe $V$ obtained from algebraic set theory (AST) inside $\mathcal{E}$ + satisfies the Partition Principle. + On the other hand, the quotient $q:X\to X/H$ is a small epimorphism in $\mathcal{E}$ with no section, + and this yields (via the display interpretation) an internal surjection in $V$ with no internal section; hence $V\models\neg\mathsf{AC}$. + In summary, $\mathcal{E}$ contains an internal model of $\mathsf{IZF}+\mathsf{PP}+\neg\mathsf{AC}$ + (and if $\mathcal{E}$ is Boolean, equivalently after $\neg\neg$-sheafification, this upgrades to $\mathsf{ZF}+\mathsf{PP}+\neg\mathsf{AC}$). + oai:arXiv.org:2511.07675v2 + math.LO + math.CT + Tue, 23 Dec 2025 00:00:00 -0500 replace http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Patrick Kinnear + Frank Gilson - Degree sequence condition for Hamiltonicity in tough graphs - https://arxiv.org/abs/2405.04728 - arXiv:2405.04728v4 Announce Type: replace -Abstract: Generalizing both Dirac's condition and Ore's condition for Hamilton cycles, Chv\'atal in 1972 established a degree sequence condition for the existence of a Hamilton cycle in a graph. Ho\`ang in 1995 generalized Chv\'atal's degree sequence condition for 1-tough graphs and conjectured a $t$-tough analogue for any positive integer $t\ge 1$. Ho\`ang in the same paper verified his conjecture for $t\le 3$ and recently Ho\`ang and Robin verified the conjecture for $t=4$. In this paper, we confirm the conjecture for all $t\ge 4$. The proof depends on two newly established results on cycle structures in tough graphs, which hold independent interest. - oai:arXiv.org:2405.04728v4 + Combinatorial degree version of a generalized $\mathbb{Z}_p$-Tucker's lemma with a combinatorial proof + https://arxiv.org/abs/2511.10319 + arXiv:2511.10319v2 Announce Type: replace +Abstract: Combinatorial analogues of classical Borsuk-Ulam-type theorems (e.g., Tucker's lemma, $\mathbb{Z}_p$-Tucker's lemma, etc.) have numerous important applications in combinatorics. In this paper, we formulate a combinatorial degree version of a generalized $\mathbb{Z}_p$-Tucker's lemma. Our proof is purely combinatorial in the sense that it does not involve homology, cohomology or any other notions from continuous topology. In order to prove the aforementioned degree theorem, as a main technical tool, we prove a Hopf trace-type formula, which is also purely combinatorial and involves no homology. This combinatorial Hopf trace formula is of independent interest. + oai:arXiv.org:2511.10319v2 math.CO - Mon, 22 Dec 2025 00:00:00 -0500 + math.AT + Tue, 23 Dec 2025 00:00:00 -0500 replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Songling Shan, Arthur Tanyel + http://creativecommons.org/licenses/by/4.0/ + Sajal Mukherjee, Pritam Chandra Pramanik - One-level densities in families of Gr\"ossencharakters associated to CM elliptic curves - https://arxiv.org/abs/2405.15597 - arXiv:2405.15597v3 Announce Type: replace -Abstract: We study the low-lying zeros of a family of $L$-functions attached to the CM elliptic curve $E_d \;:\; y^2 = x^3 - dx$, for each odd and square-free integer $d$. Specifically, upon writing the $L$-function of $E_d$ as $L(s-\frac12, \xi_d)$ for the appropriate Gr\"ossencharakter $\xi_d$ of conductor $\mathfrak{f}_d$, we consider the collection $\mathcal{F}_d$ of $L$-functions attached to $\xi_{d,k}$, $k \geq 1$, where for each integer $k$, $\xi_{d, k}$ denotes the primitive character inducing $\xi_d^k$. We observe that $25\%$ of the $L$-functions in $\mathcal{F}_d$ have negative root number. $\mathcal{F}_d$ is thus not one of the essentially homogeneous families of the Universality Conjecture of Sarnak, Shin and Templier, with unitary, symplectic or orthogonal (odd or even) symmetry type. By computing the one-level density in the family of $L$-functions in $\mathcal{F}_{d}$ with conductor at most $K^2 \mathrm N (\mathfrak{f}_d)$, we find that $\mathcal{F}_d$ naturally decomposes into subfamilies: more specifically, a collection of symplectic ($L(s, \xi_{d,k})$ for $k \equiv \alpha \bmod 8$, $\alpha$ even) and orthogonal ($L(s, \xi_{d,k})$ for $k \equiv \alpha \bmod 8$, $\alpha$ odd) subfamilies. For each such subfamily, we moreover compute explicit lower order terms in decreasing powers of $\log (K^2 \mathrm N(\mathfrak{f}_d))$. - oai:arXiv.org:2405.15597v3 - math.NT - Mon, 22 Dec 2025 00:00:00 -0500 + Splitting obstructions and $\mathbb{Z}_2$ invariants in time-reversal symmetric topological insulators + https://arxiv.org/abs/2511.10444 + arXiv:2511.10444v2 Announce Type: replace +Abstract: The Fu-Kane-Mele $\mathbb{Z}_2$ index characterizes two-dimensional time-reversal symmetric topological phases of matter. We shed some light on some features of this index by investigating projection-valued maps endowed with a fermionic time-reversal symmetry. + Our main contributions are threefold. First, we establish a decomposition theorem, proving that any such projection-valued map admits a splitting into two projection-valued maps that are related to each other via time-reversal symmetry. Second, we provide a complete homotopy classification theorem for these maps, thereby clarifying their topological structure. Third, by means of the previous analysis, we connect the Fu-Kane-Mele index to the Chern number of one of the factors in the previously-mentioned decomposition, which in turn allows to exhibit how the $\mathbb{Z}_2$-valued topological obstruction to constructing a periodic and smooth Bloch frame for the projection-valued map, measured by the Fu-Kane-Mele index, can be concentrated in a single pseudo-periodic Kramers pair. + oai:arXiv.org:2511.10444v2 + math-ph + cond-mat.mes-hall + math.MP + Tue, 23 Dec 2025 00:00:00 -0500 replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - 10.1112/mtk.70067 - Mathematika (72)1 (Jan. 2026) - Chantal David, Lucile Devin, Ezra Waxman + http://creativecommons.org/licenses/by-nc-sa/4.0/ + Alessandro Ferreri, Domenico Monaco, Gabriele Peluso - Viscous shock fluctuations in KPZ - https://arxiv.org/abs/2406.06502 - arXiv:2406.06502v3 Announce Type: replace -Abstract: We study ``V-shaped'' solutions to the KPZ equation, those having opposite asymptotic slopes $\theta$ and $-\theta$, with $\theta>0$, at positive and negative infinity, respectively. Answering a question of Janjigian, Rassoul-Agha, and Sepp\"al\"ainen, we show that the spatial increments of V-shaped solutions cannot be statistically stationary in time. This completes the classification of statistically time-stationary spatial increments for the KPZ equation by ruling out the last case left by those authors. - To show that these V-shaped time-stationary measures do not exist, we study the location of the corresponding ``viscous shock,'' which, roughly speaking, is the location of the bottom of the V. We describe the limiting rescaled fluctuations, and in particular show that the fluctuations of the shock location are not tight, for both stationary and flat initial data. We also show that if the KPZ equation is started with V-shaped initial data, then the long-time limits of the time-averaged laws of the spatial increments of the solution are mixtures of the laws of the spatial increments of $x\mapsto B(x)+\theta x$ and $x\mapsto B(x)-\theta x$, where $B$ is a standard two-sided Brownian motion. - oai:arXiv.org:2406.06502v3 + Stochastic Burgers Equation Driven by a Hermite Sheet with Additive Noise: Existence, Uniqueness, and Regularity + https://arxiv.org/abs/2511.10463 + arXiv:2511.10463v2 Announce Type: replace +Abstract: We study the stochastic Burgers equation driven by a Hermite sheet of order \( q \geq 1 \) with \textbf{additive noise}, establishing the well-posedness of mild solutions via a fixed-point argument in suitable Banach spaces. Under appropriate conditions on the Hurst parameters \( \mathbf{H} = (H_0, H_1, \dots, H_d) \in (1/2, 1)^{d+1} \), we prove existence and uniqueness of solutions through a Picard iteration scheme. The solution exhibits spatial and temporal H\"older regularity, with exponents determined by the Hurst parameters of the driving noise. Furthermore, we demonstrate that the solution inherits the self-similarity property from the Hermite sheet, providing explicit scaling exponents. Uniform moment estimates in space and time are derived, forming the foundation for the regularity analysis. The additive noise formulation allows us to use the standard Wiener integral construction for Hermite processes, thereby avoiding the technical complications of Malliavin calculus required for multiplicative noise. This restriction is mathematically justified as it circumvents the need for Malliavin derivative bounds essential for random integrands with Hermite processes of order \(q \geq 2\), a key difficulty highlighted in recent literature. The work develops the stochastic integration theory with respect to Hermite sheets for deterministic integrands and establishes a complete framework for analyzing nonlinear SPDEs with non-Gaussian noise, contributing to the understanding of stochastic systems with long-range dependence and non-Gaussian fluctuations. + oai:arXiv.org:2511.10463v2 math.PR - Mon, 22 Dec 2025 00:00:00 -0500 + Tue, 23 Dec 2025 00:00:00 -0500 replace http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Alexander Dunlap, Evan Sorensen + Atef Lechiheb - Almost sharp local Bernstein estimates for Laplace eigenfunctions on compact Riemannian manifolds - https://arxiv.org/abs/2406.16036 - arXiv:2406.16036v3 Announce Type: replace -Abstract: We study local growth properties of Laplace eigenfunctions on compact Riemannian manifolds. Following the paradigm introduced by Donnelly and Fefferman in the late 1980s, an eigenfunction is expected to behave locally like a polynomial of degree comparable to the square root of the eigenvalue. In this direction we establish almost sharp local $L^{p}$--Bernstein inequalities, $p\in[1,\infty]$, conjectured by Donnelly--Fefferman in 1990. We also derive analogous estimates for $A$-harmonic functions, with the square root of the eigenvalue replaced by the doubling index. - Our argument refines the original Donnelly--Fefferman method based on $L^{2}$--Carleman estimates. At the $L^{2}$--level, we first prove a uniform bound for the doubling index on annuli of width comparable to the wavelength. This implies, with an arbitrarily small polynomial loss, the corresponding property at the $L^{p}$--level for all $p\in[1,\infty]$. The latter step relies on a bootstrap scheme combining elliptic regularity with a patching of local Carleman estimates on small balls. - oai:arXiv.org:2406.16036v3 + Guaranteeing Higher Order Convergence Rates for Accelerated Wasserstein Gradient Flow Schemes + https://arxiv.org/abs/2511.10884 + arXiv:2511.10884v2 Announce Type: replace +Abstract: In this paper, we study higher-order-accurate-in-time minimizing movements schemes for Wasserstein gradient flows. We introduce a novel accelerated second-order scheme, leveraging the differential structure of the Wasserstein space in both Eulerian and Lagrangian coordinates. For sufficiently smooth energy functionals, we show that our scheme provably achieves an optimal quadratic convergence rate. Under the weaker assumptions of Wasserstein differentiability and $\lambda$-displacement convexity (for any $\lambda\in \mathbb{R}$), we show that our scheme still achieves a first-order convergence rate and has strong numerical stability. In particular, we show that the energy is nearly monotone in general, while when the energy is $L$-smooth and $\lambda$-displacement convex (with $\lambda>0$), we prove the energy is non-increasing and the norm of the Wasserstein gradient is exponentially decreasing along the iterates. Taken together, our work provides the first fully rigorous proof of accelerated second-order convergence rates for smooth functionals and shows that the scheme performs no worse than the classical scheme JKO scheme for functionals that are $\lambda$-displacement convex and Wasserstein differentiable. + oai:arXiv.org:2511.10884v2 math.AP - math.CA - math.DG - math.SP - Mon, 22 Dec 2025 00:00:00 -0500 + cs.NA + math.NA + Tue, 23 Dec 2025 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by/4.0/ + Raymond Chu, Matt Jacobs + + + Ordinal Analysis of Well-Ordering Principles, Well Quasi-Orders Closure Properties, and $\Sigma_n$-Collection Schema + https://arxiv.org/abs/2511.11196 + arXiv:2511.11196v2 Announce Type: replace +Abstract: The study of well quasi-orders, wqo, is a cornerstone of combinatorics and within wqo theory Kruskal's theorem plays a crucial role. Extending previous proof-theoretic results, we calculate the $\Pi^1_1$ ordinals of two different versions of labelled Kruskal's theorem: $\forall n \,$ $\mbox{KT}_\ell(n)$ and $\mbox{KT}_{\omega}(n)$; denoting, respectively, all the cases of labelled Kruskal's theorem for trees with an upper bound on the branching degree, and the standard Kruskal's theorem for labelled trees. + In order to reach these computations, a key step is to move from Kruskal's theorem, which regards preservation of wqo's, to an equivalent Well-Ordering Principle (WOP), regarding instead preservation of well-orders. Given an ordinal function $g$, WOP$(g)$ amounts to the following principle $\forall X\, [\mbox{WO}(X) \rightarrow \mbox{WO}(g(X))]$, where $WO(X)$ states that ``$X$ is a well-order''. In our case, the two ordinal functions involved are ${g}_{\forall}(X)=\sup_{n}\vartheta(\Omega^n \cdot X)$ and ${g}_{\omega}(X)=\vartheta(\Omega^{\omega}\! \cdot X)$. + In addition to the ordinal analysis of Kruskal's theorem and its related WOP, a series of Well Quasi-orders Principles (WQP) is considered. Given a set operation $G$ that preserves the property of being a wqo, its Well Quasi-orders closure Property, WQP$(G)$, is given by the principle $\forall Q\, [Q\, \mbox{wqo} \rightarrow G(Q)\, \mbox{wqo}]$. Conducting this study, unexpected connections with different principles arising from Ramsey and Computational theory, such as RT$^2_{<\infty}$, CAC, ADS, RT$^1_{<\infty}$, turn up. + Lastly, extending and combining previous results, we achieve also the ordinal analysis of the collection schema $\mbox{B}\Sigma_n$. + oai:arXiv.org:2511.11196v2 + math.LO + Tue, 23 Dec 2025 00:00:00 -0500 replace - http://creativecommons.org/publicdomain/zero/1.0/ - K\'evin Le Balc'h + http://creativecommons.org/licenses/by-nc-nd/4.0/ + Gabriele Buriola, Andreas Weiermann - Polarization and Gorenstein liaison - https://arxiv.org/abs/2406.19985 - arXiv:2406.19985v2 Announce Type: replace -Abstract: A major open question in the theory of Gorenstein liaison is whether or not every arithmetically Cohen--Macaulay subscheme of $\mathbb{P}^n$ can be G-linked to a complete intersection. Migliore and Nagel showed that, if such a scheme is generically Gorenstein (e.g., reduced), then, after re-embedding so that it is viewed as a subscheme of $\mathbb{P}^{n+1}$, indeed it can be G-linked to a complete intersection. Motivated by this result, we consider techniques for constructing G-links on a scheme from G-links on a closely related reduced scheme. - Polarization is a tool for producing a squarefree monomial ideal from an arbitrary monomial ideal. Basic double G-links on squarefree monomial ideals can be induced from vertex decompositions of their Stanley--Reisner complexes. Given a monomial ideal $I$ and a vertex decomposition of the Stanley--Reisner complex of its polarization $P(I)$, we give conditions that allow for the lifting of an associated basic double G-link of $P(I)$ to a basic double G-link of $I$ itself. We use the relationship we develop in the process to show that the Stanley--Reisner complexes of polarizations of stable Cohen--Macaulay monomial ideals are vertex decomposable. - We then introduce and study polarization of a Gr\"obner basis of an arbitrary homogeneous ideal and give a relationship between geometric vertex decomposition of a polarization and elementary G-biliaison that is analogous to our result on vertex decomposition and basic double G-linkage. - oai:arXiv.org:2406.19985v2 - math.AC - math.AG - math.CO - Mon, 22 Dec 2025 00:00:00 -0500 + On the entropy for indeterminate moment problems + https://arxiv.org/abs/2511.12684 + arXiv:2511.12684v2 Announce Type: replace +Abstract: For an indeterminate Hamburger moment problem we consider an infinite family of analytic densities solving the moment problem and we prove that they all have finite (Shannon) entropy. These densities are either all bounded or all unbounded. The result is illustrated by the Al-Salam--Carlitz moment problem, where all the densities in the family are bounded. + oai:arXiv.org:2511.12684v2 + math.PR + math.CA + math.DS + Tue, 23 Dec 2025 00:00:00 -0500 replace http://creativecommons.org/licenses/by/4.0/ - 10.1112/jlms.70319 - J. Lond. Math. Soc. (2) 112 (2025), no. 6, Paper No. e70319 - Sara Faridi, Patricia Klein, Jenna Rajchgot, Alexandra Seceleanu + Christian Berg - Weyls's law for Compact Rank One Symmetric Spaces - https://arxiv.org/abs/2407.05274 - arXiv:2407.05274v2 Announce Type: replace -Abstract: Weyls law is a fundamental result governing the asymptotic behaviour of the eigenvalues of teh Laplacian. It states that for a compact d dimensional manifold M (without boundary), the eigenvalue counting function has an asymptotic growth, whose leading term is of the order of d and the error term is no worse than order d-1. - A natural question is: when is the error term sharp and when can it be improved? It has been known for a long time that the error term is sharp for the round sphere (since 1968). In contrast, it has only recently been shown (in 2019) by Iosevich and Wyman that for the product of spheres, the error term can be polynomially improved. They conjecture that a polynomial improvement should be true for products in general. In this paper we extend both these results to Compact Rank One Symmetric Spaces (CROSSes). We show that for CROSSes, the error term is sharp. Furthermore, we show that for a product of CROSSes, the error term can be polynomially improved. This gives further evidence to the conjecture made by Iosevich and Wyman. - oai:arXiv.org:2407.05274v2 - math.DG - Mon, 22 Dec 2025 00:00:00 -0500 + Pointwise bounds on Dirichlet Green's functions for a singular drift term + https://arxiv.org/abs/2511.12741 + arXiv:2511.12741v2 Announce Type: replace +Abstract: We introduce a technique to obtain pointwise upper and lower bounds for the Green's function of elliptic operators whose principal part is the Laplacian and that include a drift term diverging near the boundary like a power of the inverse distance with exponent less than 1, in the unit ball B(0,1) \subset \mathbb{R}^n, n \ge 3. The constants in the upper estimates are uniform in B(0,r) for each r < 1, with explicit dependence on r. The drift here belongs to C^{1,\alpha}_{\mathrm{loc}} and may, more generally, be majorized by a function radially integrable up to the boundary. These appear to be the first such estimates for non-coercive drifts and remain new even for smooth drifts, suggesting extensions to singular potentials and other settings where energy methods fail. + oai:arXiv.org:2511.12741v2 + math.AP + Tue, 23 Dec 2025 00:00:00 -0500 replace http://creativecommons.org/licenses/by/4.0/ - Sai Sriharsha Indukuri, Ritwik Mukherjee + Aritro Pathak - The long way of a viscous vortex dipole - https://arxiv.org/abs/2407.13562 - arXiv:2407.13562v2 Announce Type: replace -Abstract: We consider the evolution of a viscous vortex dipole in $R^2$ originating from a pair of point vortices with opposite circulations. At high Reynolds number $Re >> 1$, the dipole can travel a very long way, compared to the distance between the vortex centers, before being slowed down and eventually destroyed by diffusion. In this regime we construct an accurate approximation of the solution in the form of a two-parameter asymptotic expansion involving the aspect ratio of the dipole and the inverse Reynolds number. We then show that the exact solution of the Navier-Stokes equations remains close to the approximation on a time interval of length $O(Re^\sigma)$, where $\sigma < 1$ is arbitrary. This improves upon previous results which were essentially restricted to $\sigma = 0$. As an application, we provide a rigorous justification of an existing formula which gives the leading order correction to the translation speed of the dipole due to finite size effects. - oai:arXiv.org:2407.13562v2 + Exponential Lower Bounds for the Advection-Diffusion Equation with Shear Flows + https://arxiv.org/abs/2511.14512 + arXiv:2511.14512v2 Announce Type: replace +Abstract: In this paper, we prove that the $L^2$ norm of spatial mean-free solutions to the advection--diffusion equation on $\mathbb{T}^2$ with shear drifts satisfies an \emph{exponential lower bound} in time. This lower bound shows that diffusion can fundamentally suppress passive-scalar mixing. + oai:arXiv.org:2511.14512v2 math.AP - physics.flu-dyn - Mon, 22 Dec 2025 00:00:00 -0500 + Tue, 23 Dec 2025 00:00:00 -0500 replace http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Michele Dolce, Thierry Gallay + Yupei Huang, Xiaoqian Xu - Prosoluble subgroups of the profinite completion of the fundamental group of compact 3-manifolds - https://arxiv.org/abs/2408.04152 - arXiv:2408.04152v3 Announce Type: replace -Abstract: We give a description of finitely generated prosoluble subgroups of the profinite completion of $3$-manifold groups and virtually compact special groups. - oai:arXiv.org:2408.04152v3 - math.GT - math.GR - Mon, 22 Dec 2025 00:00:00 -0500 + Cutting a Pancake with an Exotic Knife + https://arxiv.org/abs/2511.15864 + arXiv:2511.15864v2 Announce Type: replace +Abstract: In the first chapter of their classic book "Concrete Mathematics", Graham, Knuth, and Patashnik consider the maximum number of pieces that can be obtained from a pancake by making n cuts with a knife blade that is straight, or bent into a V, or bent twice into a Z. We extend their work by considering knives, or "cookie-cutters", of even more exotic shapes, including a k-armed V, a chain of k connected line segments, a long-legged version of one of the letters A, E, H, L, M, T, W, or X, a convex polygon, a circle, a phi, a figure 8, a pentagram, a hexagram, or a lollipop. In many cases a counting argument combined with Euler's formula produces an explicit expression for the maximum number of pieces. "Constrained" versions of the long-legged letters A and T are also considered, for which we have only conjectural formulas. + oai:arXiv.org:2511.15864v2 + math.CO + Tue, 23 Dec 2025 00:00:00 -0500 replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - 10.1112/jlms.70330 - Journal of the London Mathematical Society, 2025, 112: e70330 - Lucas C. Lopes, Pavel A. Zalesskii + http://creativecommons.org/licenses/by/4.0/ + David O. H. Cutler, Neil J. A. Sloane - Optimal Ratcheting of Dividends with Irreversible Reinsurance - https://arxiv.org/abs/2408.16989 - arXiv:2408.16989v2 Announce Type: replace -Abstract: This paper considers an insurance company that faces two key constraints: a ratcheting dividend constraint and an irreversible reinsurance constraint. The company allocates part of its reserve to pay dividends to its shareholders while strategically purchasing reinsurance for its claims. The ratcheting dividend constraint ensures that dividend cuts are prohibited at any time. The irreversible reinsurance constraint ensures that reinsurance contracts cannot be prematurely terminated or sold to external entities. The dividend rate and reinsurance level are modeled as nondecreasing processes, thereby satisfying the constraints. Claims are modeled using a Brownian risk model. The main objective is to maximize the cumulative expected discounted dividend payouts until the time of ruin. The reinsurance and dividend levels are restricted to a finite set. The optimal value function is shown to be the unique viscosity solution of the corresponding Hamilton-Jacobi-Bellman equation. A threshold strategy is constructed and shown to be optimal. Finally, numerical examples are presented to illustrate the optimality conditions and optimal strategies. - oai:arXiv.org:2408.16989v2 - math.OC - math.PR - q-fin.PR - q-fin.RM - Mon, 22 Dec 2025 00:00:00 -0500 + On the asymptotic dynamics for the $L^2$-supercritical gKDV equation + https://arxiv.org/abs/2511.16847 + arXiv:2511.16847v2 Announce Type: replace +Abstract: We study the $L^2$-supercritical generalized Korteweg-de Vries equation (gKdV) with nonlinearities $p>5$. While local well-posedness in $H^1$ is classical, the long-time dynamics in the supercritical regime remains largely unexplored beyond small data global solutions, the construction of multi-solitons for any power and self-similar blow-up near the critical power $p=5$. We develop a unified description of the non-solitonic region for arbitrary $H^1$ solutions, both global and blowing up. Our analysis shows that the asymptotic $L^2$ and $L^p$ dynamics in this region is completely determined by the growth rate of the $L^2$ norm of the gradient (or, equivalently, the critical $H^{s_p}$ norm). In particular, we prove sharp far-field decay on both half-lines and establish normalized local vanishing along sequences of times, with improved estimates in the case of even-power nonlinearities. A key ingredient is a new virial method that compensates for the possible unboundedness of the $H^1$ norm by exploiting the conservation of mass and a careful localization of the nonlinear flux. This yields quantitative versions of decay phenomena previously known only in subcritical settings, and it applies without any smallness or proximity-to-soliton assumptions. + oai:arXiv.org:2511.16847v2 + math.AP + Tue, 23 Dec 2025 00:00:00 -0500 replace http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Tim J. Boonen, Engel John C. Dela Vega + Ricardo Freire, Claudio Mu\~noz - Blockwise Gluings And Amalgamation Failures in Integral Residuated Lattices - https://arxiv.org/abs/2408.17400 - arXiv:2408.17400v3 Announce Type: replace -Abstract: We introduce the blockwise gluing construction. This describes residuated integral chains which can be decomposed into (possibly) partial algebras, stacked one on top of the other, and such that elements in a certain component multiply in blocks, i.e., in the same way, with respect to lower components. This construction generalizes that of 1-sums (or ordinal sums). As first main results, we provide finite axiomatizations for varieties generated by particular chains that are gluings of their archimedean components. For such varieties we also prove the finite embeddability property, and as a consequence, the decidability of their universal theory. Moreover, we solve in the negative several longstanding open problems in the literature about the amalgamation property (AP). Indeed, we provide denumerably many new examples of varieties lacking the AP, including: semilinear (commutative) integral residuated lattices, semilinear FL_w-algebras, MTL-algebras, involutive and pseudocomplemented MTL-algebras, and all their n-potent subvarieties for n > 1. For the commutative varieties, this also entails that the associated substructural logics do not have the deductive interpolation property. - oai:arXiv.org:2408.17400v3 - math.LO - Mon, 22 Dec 2025 00:00:00 -0500 + A Robust GPU-Accelerated Kernel Compensation Solver with Novel Discretization for Photonic Crystals in Pseudochiral Media + https://arxiv.org/abs/2511.17107 + arXiv:2511.17107v2 Announce Type: replace +Abstract: This paper develops a robust solver for the Maxwell eigenproblem in 3D photonic crystals (PCs) with pseudochiral media. The solver employs the Kernel Compensation technique under the framework of Yee's scheme to eliminate null space and enable matrix-free, GPU-accelerated operations via 3D discrete Fourier transform (DFT). Furthermore, we propose a novel discretization for permittivity tensor containing off-diagonal entries and rigorously prove that the resulting matrix is Hermitian positive definite (HPD), which ensures the correctness of the kernel compensation technique. Numerical experiments on several benchmark examples are demonstrated to validate the robustness and accuracy of our scheme. + oai:arXiv.org:2511.17107v2 + math.NA + cs.NA + Tue, 23 Dec 2025 00:00:00 -0500 replace http://creativecommons.org/licenses/by/4.0/ - Valeria Giustarini, Sara Ugolini + Chenhao Jin, Hehu Xie - Tykhyy's Conjecture on finite mapping class group orbits - https://arxiv.org/abs/2409.04379 - arXiv:2409.04379v3 Announce Type: replace -Abstract: We classify the finite orbits of the mapping class group action on the character variety of Deroin--Tholozan representations of punctured spheres. In particular, we prove that the action has no finite orbits if the underlying sphere has 7 punctures or more. When the sphere has six punctures, we show that there is a unique 1-parameter family of finite orbits. Our methods also recover Tykhyy's classification of finite orbits for 5-punctured spheres. The proof is inductive and uses Lisovyy--Tykhyy's classification of finite mapping class group orbits for 4-punctured spheres as the base case for the induction. - Our results on Deroin--Tholozan representations cover the last missing cases to complete the proof of Tykhyy's Conjecture on finite mapping class group orbits for $\mathrm{SL}_2\mathbb{C}$ representations of punctured spheres, after the recent work by Lam--Landesman--Litt. - oai:arXiv.org:2409.04379v3 + Asymptotics of motion planning complexity for control-affine systems + https://arxiv.org/abs/2511.17130 + arXiv:2511.17130v2 Announce Type: replace +Abstract: In this paper, we study the complexity of the approximation of nonadmissible curves for nonlinear control-affine systems satisfying the strong H{\"o}rmander condition. Focusing on tubular approximation complexities, we provide asymptotic equivalences, with explicit constants, for all generic situations where the distribution, i.e., the linear part of the control system, is of co-rank one. Namely, we consider curves in step 2 distributions and any dimension. In the 3 dimensional case, we also consider the case of distributions with Martinet-type singularities that are crossed by the curve at isolated points. + oai:arXiv.org:2511.17130v2 math.DS - math.AG - math.GT - Mon, 22 Dec 2025 00:00:00 -0500 + math.OC + Tue, 23 Dec 2025 00:00:00 -0500 replace - http://creativecommons.org/licenses/by/4.0/ - Samuel Bronstein, Arnaud Maret + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Michele Motta (SISSA / ISAS), Dario Prandi (L2S) - Expression Rates of Neural Operators for Linear Elliptic PDEs in Polytopes - https://arxiv.org/abs/2409.17552 - arXiv:2409.17552v3 Announce Type: replace -Abstract: We study the approximation rates of a class of deep neural network approximations of operators which arise as data-to-solution maps $\mathcal{S}$ of linear elliptic partial differential equations (PDEs), and act between pairs $X,Y$ of suitable infinite-dimensional spaces. We prove expression rate bounds for approximate neural operators $\mathcal{G}$ with the structure $\mathcal{G} = \mathcal{R} \circ \mathcal{A} \circ \mathcal{E}$, with linear encoders $\mathcal{E}$ and decoders $\mathcal{R}$. We focus in particular on deepONets emulating the coefficient-to-solution maps for elliptic PDEs set in polygons and in some polyhedra. Exploiting the regularity of the solution sets of elliptic PDEs in polytopes, we show algebraic rates of convergence for problems with data with finite regularity, and exponential rates for analytic data. - oai:arXiv.org:2409.17552v3 - math.NA - cs.NA - Mon, 22 Dec 2025 00:00:00 -0500 + Quadratic Mean-Field BSDEs and Exponential Utility Maximization + https://arxiv.org/abs/2511.17214 + arXiv:2511.17214v2 Announce Type: replace +Abstract: In this paper, we study a class of real-valued mean-field backward stochastic differential equations (BSDEs) with generator of quadratic growth in the control variable and the mean-field term. Under this assumption, together with a bounded terminal condition, we establish existence and uniqueness of solutions. Our approach departs from classical fixed-point arguments and instead combines Malliavin calculus with refined BMO and stability estimates. The result bridges the gap between the quadratic BSDE results of Cheridito and Nam (2017) and Hao et al. (2025). Moreover, motivated by the structure of the mean-field exponential utility maximization problem introduced in our paper, we extend our framework to generators satisfying a weaker quadratic condition on the generator. This relaxation is designed to accommodate the additional mean-field terms that arise in our utility maximization setting and that fall outside the scope of previous quadratic assumptions. Within this more general regime, we establish existence and uniqueness of solutions under a smallness condition on the terminal random variable. We then apply this extended theory to solve a mean-field exponential utility maximization problem, thereby generalizing the classical framework of Hu et al. (2005)to a fully coupled quadratic mean-field setting. + oai:arXiv.org:2511.17214v2 + math.OC + math.PR + Tue, 23 Dec 2025 00:00:00 -0500 replace http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Carlo Marcati, Christoph Schwab + Yining Ding, Kihun Nam, Jiaqiang Wen - Fluctuation exponents of the half-space KPZ at stationarity - https://arxiv.org/abs/2410.01653 - arXiv:2410.01653v3 Announce Type: replace -Abstract: We study the half-space KPZ equation with a Neumann boundary condition, starting from stationary Brownian initial data. We derive a variance identity that links the fluctuations of the height function to the transversal fluctuations of a half-space polymer model. Utilizing this identity, we obtain estimates for the polymer endpoints, leading to optimal fluctuation exponents for the height function in both the subcritical and critical regimes, as well as an optimal upper bound for the fluctuation exponents in the extended critical regime. We also compute the average growth rate as a function of the boundary parameter. - oai:arXiv.org:2410.01653v3 - math.PR - math-ph - math.MP - Mon, 22 Dec 2025 00:00:00 -0500 + Invariant and Coinvariant Morse Homologies for Orbifolds + https://arxiv.org/abs/2511.17811 + arXiv:2511.17811v2 Announce Type: replace +Abstract: In this note, we construct invariant and coinvariant Morse chain complexes with integer coefficients for any compact effective orbifold. We conjecture that the homology of the coinvariant chain complex computes the homology of the underlying topological space over $\mathbb{Z}$, improving a construction of \cite{cho2014orbifold}, which is isomorphic to the homology of the underlying topological space over $\mathbb{Q}$. In contrast, the homology of the invariant Morse chain complex is sensitive to the orbifold structure itself. + oai:arXiv.org:2511.17811v2 + math.GT + math.AT + Tue, 23 Dec 2025 00:00:00 -0500 replace http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Yu Gu, Ran Tao + Erkao Bao, Lina Liu - A convergence not metrizable - https://arxiv.org/abs/2410.12792 - arXiv:2410.12792v2 Announce Type: replace -Abstract: Certain notions of convergence of sequences functions such as pointwise convergence and (uniform) convergence on compact or bounded sets come from suitable topological function spaces; see [1]. Under certain conditions these topologies involved are metrizable, which in an advantage since there is an extensive theory on convergence in metric spaces. However, the case of pointwise convergence is delicate, since it is shown that under certain hypotheses this form of convergence of sequences of functions is not equivalent to convergence in metric. - oai:arXiv.org:2410.12792v2 - math.GM - Mon, 22 Dec 2025 00:00:00 -0500 + Hilbert properties of varieties + https://arxiv.org/abs/2511.18431 + arXiv:2511.18431v2 Announce Type: replace +Abstract: This is a survey of results on the Hilbert property of algebraic varieties, and variants of it. + oai:arXiv.org:2511.18431v2 + math.AG + math.AC + math.NT + Tue, 23 Dec 2025 00:00:00 -0500 replace http://creativecommons.org/licenses/by/4.0/ - Luis David Rivera + Arno Fehm, Ariyan Javanpeykar - Bounding the parameter $\beta$ of a distance-regular graph with classical parameters - https://arxiv.org/abs/2410.22994 - arXiv:2410.22994v2 Announce Type: replace -Abstract: Let $\Gamma$ be a distance-regular graph with classical parameters $(D, b, \alpha, \beta)$ satisfying $b\geq 2$ and $D\geq 3$. Let $r=1+b+b^2+\cdots+b^{D-1}$. In 1999, K. Metsch showed that there exists a positive constant $C(\alpha,b)$ only depending on $\alpha$ and $b$, such that if $\beta \geq C(\alpha, b)r^2$, then either $\Gamma$ is a Grassmann graph or a bilinear forms graph. - In this work, we show that for $b\geq 2$ and $D\geq 3$, then there exists a constant $C_1(\alpha, b)$ only depending on $\alpha$ and $b$, such that if $\beta \geq C_1(\alpha, b)r$, then either $\Gamma$ is a Grassmann graph, or a bilinear forms graph. - oai:arXiv.org:2410.22994v2 - math.CO - Mon, 22 Dec 2025 00:00:00 -0500 + On Some Generalisations of Gauss Sequences + https://arxiv.org/abs/2511.19503 + arXiv:2511.19503v2 Announce Type: replace +Abstract: In this paper, we introduce integer sequences satisfying new congruence properties inspired by the Euler and Gauss congruences, which we call Euler-Gauss sequences. Noting that every Gauss sequence is an Euler-Gauss sequence, we compare them with certain generalisations of Gauss sequences and provide several counterexamples. In particular, the Smallest Prime Factor (SPF) and Greatest Prime Factor (GPF) sequences (suitably defined at 1) studied extensively by Erdos and Alladi arise as natural examples of Euler-Gauss sequences that are not Gauss sequences. We further extend these congruence-based integer sequences to a q-analog setting and establish characteristic properties that reveal their structure and fill gaps in the literature on q-Gauss sequences. In recent works, q-Gauss sequences have been shown to admit interesting combinatorial interpretations and to exhibit the Cyclic Sieving Phenomenon (CSP). Not only do our q-Euler-Gauss sequences satisfy the standard CSP with some restriction, but we also derive a new CSP condition for the SPF and GPF sequences, not hitherto known in the literature. + oai:arXiv.org:2511.19503v2 + math.NT + Tue, 23 Dec 2025 00:00:00 -0500 replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Chenhui Lv, Jack H. Koolen + http://creativecommons.org/licenses/by/4.0/ + Sathyanarayan Narayan, N. Uday Kiran - Local Well-posedenss of the Bartnik Static Extension Problem near Schwarzschild spheres - https://arxiv.org/abs/2411.02801 - arXiv:2411.02801v2 Announce Type: replace -Abstract: We establish the local well-posedness of the Bartnik static metric extension problem for arbitrary Bartnik data that perturb that of any sphere in a Schwarzschild $\{t=0\}$ slice. Our result in particular includes spheres with arbitrary small mean curvature. We introduce a new framework to this extension problem by formulating the governing equations in a geodesic gauge, which reduce to a coupled system of elliptic and transport equations. Since standard function spaces for elliptic PDEs are unsuitable for transport equations, we use certain spaces of Bochner-measurable functions traditionally used to study evolution equations. In the process, we establish existence and uniqueness results for elliptic boundary value problems in such spaces in which the elliptic equations are treated as evolutionary equations, and solvability is demonstrated using rigorous energy estimates. The precise nature of the expected difficulty of solving the Bartnik extension problem when the mean curvature is very small is identified and suitably treated in our analysis. - oai:arXiv.org:2411.02801v2 + Upper bound estimation for the ratio of the first two eigenvalues of Robin Laplacian + https://arxiv.org/abs/2511.20988 + arXiv:2511.20988v3 Announce Type: replace +Abstract: The celebrated conjecture by Payne, P\'{o}lya and Weinberger (1956) states that for the fixed membrane problem, the ratio of the first two eigenvalues, $\lambda_2/\lambda_1$, is maximized by a disk. A more general dimensional version of this conjecture was later resolved by Ashbaugh and Benguria in the 1990s. For the Robin Laplacian, Payne and Schaefer (2001) formulated an analogous conjecture, positing that the ratio $\mu_2/\mu_1$ is also maximized by a disk for a range of the boundary parameter $\sigma$. This was later restated by Henrot in 2003. In this work, under some suitable conditions, we affirm this conjecture for all dimensions $N\geq2$ and for all $\sigma>0$. Furthermore, we prove that the maximum value of $\mu_2/\mu_1$ is strictly decreasing in $\sigma$ over the entire interval $(0,+\infty)$. Our result provides a positive answer to a variant of Yau's Problem 77: by measuring the ratio of the first two eigenfrequencies, one can determine whether an elastically supported drum is circular. + oai:arXiv.org:2511.20988v3 math.AP - gr-qc - math.DG - Mon, 22 Dec 2025 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - 10.1007/s00526-025-03175-3 - Calc. Var. 65, 7 (2026) - Ahmed Ellithy - - - Matrix representation of Picard--Lefschetz--Pham theory near the real plane in $\mathbb{C}^2$ - https://arxiv.org/abs/2412.02481 - arXiv:2412.02481v3 Announce Type: replace -Abstract: A matrix formalism is proposed for computations based on Picard--Lefschetz theory in a 2D case. The formalism is essentially equivalent to the computation of the intersection indices necessary for the Picard--Lefschetz formula and enables one to prove non-trivial topological identities for integrals depending on parameters. - We introduce the universal Riemann domain $\tilde U$, i.e. a sort of ``compactification'' of the universal covering space $\tilde U_2$ over a small tubular neighborhood $N\mathbb{R}^2$ of $\mathbb{R}^2\backslash\sigma$ in $\mathbb{B}\subset\mathbb{C}^2$, where $\mathbb{B}\subset\mathbb{C}^2$ is a big ball, and $\sigma$ is a one-dimensional complex analytic set (the set of singularities). We compute the Picard-Lefschetz monodromy of the relative homology group of the space $\tilde U$ modulo the singularities and the boundary for the standard local degenerations of type $P_1 ,P_2,P_3$ in Pham's [1] notations and for more complicated configurations in $\mathbb{C}^2$. We consider this homology group as a module over the group ring of the $\pi_1((N\mathbb{R}^2 \cap \mathbb{B})\backslash\sigma)$ over $\mathbb{Z}$. The results of the computations are presented in the form of a matrix of the monodromy operator calculated in a certain natural basis. We prove an ``inflation'' theorem, which states that the integration surfaces of interest (i.e.\ the elements of the homology group $H_2(\tilde U_2,\tilde{\partial \mathbb{B}})$) (the surfaces in the branched space possibly passing through singularities) are injectively mapped to the group $H_2(\tilde U,\tilde U'\cup\tilde{\partial \mathbb{B}})$ (the surfaces avoiding the singularities). The matrix formalism obtained describes the behaviour of integrals depending on parameters and can be applied to the study of Wiener-Hopf method in two complex variables. - oai:arXiv.org:2412.02481v3 - math-ph - math.MP - Mon, 22 Dec 2025 00:00:00 -0500 + Tue, 23 Dec 2025 00:00:00 -0500 replace - http://creativecommons.org/licenses/by-nc-sa/4.0/ - A. V. Shanin, A. I. Korolkov, N. M. Artemov, R. C. Assier + http://creativecommons.org/publicdomain/zero/1.0/ + Guowei Dai, Yingxin Sun - Eigenvalue estimates and applications on weighted manifolds - https://arxiv.org/abs/2412.09396 - arXiv:2412.09396v3 Announce Type: replace -Abstract: We will present an estimate for the first eigenvalue of the Dirichlet and Neumann problems in terms of the Bakry-\'Emery Ricci curvature for a compact weighted manifold. As an application we will establish a stability condition for a h-minimal hypersurface. - oai:arXiv.org:2412.09396v3 - math.DG - Mon, 22 Dec 2025 00:00:00 -0500 + A proof of irrationality of $\pi$ based on nested radicals with roots of $2$ + https://arxiv.org/abs/2511.21776 + arXiv:2511.21776v3 Announce Type: replace +Abstract: In this work, we consider three theorems and a lemma to prove the irrationality of $\pi$. This approach is based on nested radicals with roots of $2$ of kind $c_k = \sqrt{2 + c_{k - 1}}$ and $c_0 = 0$. Sample computations showing how the rational approximation tends to $\pi$ with increasing the integer $k$ are presented. + oai:arXiv.org:2511.21776v3 + math.GM + Tue, 23 Dec 2025 00:00:00 -0500 replace http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - A. C. Bezerra, T. Castro Silva, F. Manfio - - - Additive codes attaining the Griesmer bound - https://arxiv.org/abs/2412.14615 - arXiv:2412.14615v2 Announce Type: replace -Abstract: Additive codes may have better parameters than linear codes. However, still very few cases are known and the explicit construction of such codes is a challenging problem. Here we show that a Griesmer type bound for the length of additive codes can always be attained with equality if the minimum distance is sufficiently large. This solves the problem for the optimal parameters of additive codes when the minimum distance is large and yields many infinite series of additive codes that outperform linear codes. - oai:arXiv.org:2412.14615v2 - cs.IT - math.CO - math.IT - Mon, 22 Dec 2025 00:00:00 -0500 + Sanjar M. Abrarov, Rehan Siddiqui, Rajinder Kumar Jagpal, Brendan M. Quine + + + A new interpolation method for metric spaces based on bi-infinite sequences: The $R$-Method + https://arxiv.org/abs/2511.22126 + arXiv:2511.22126v2 Announce Type: replace +Abstract: We introduce a new interpolation method for metric spaces, termed the $R$-method, based + on bi-infinite linking sequences. Although the construction is inspired by the classical + metric functional $J_M$, the resulting interpolated space is generated by a distinct + object that behaves as a multiscale energy functional. This functional measures the + minimal discrete action required to connect two points through $\mathbb{Z}$-indexed + sequences, leading to a new intrinsic metric on $X_0 \cap X_1$. + The associated interpolated space is obtained as the relative completion of this metric + inside $X_0 \cup X_1$ and is genuinely different from those produced by the $J_M$- and + $K_M$-methods. A fundamental structural property of the $R$-method is that the resulting + space embeds continuously into the corresponding $K_M$-interpolated space, situating the + construction naturally within the existing theory of metric interpolation. + When the method is restricted to a normed setting, the $R$-method induces a genuine + interpolation functor. In this framework, it preserves the Lipschitz property of + operators with closed graphs, even in the absence of linearity, thereby extending the + classical scope of interpolation theory, which is traditionally confined to linear + continuous operators. As a consequence, standard compactness properties are also + preserved under mild assumptions. + The $R$-method thus provides a new interpolation framework whose foundations rely + exclusively on intrinsic metric properties and the summability of discrete orbits, + bridging metric interpolation, nonlinear analysis, and classical interpolation theory. + oai:arXiv.org:2511.22126v2 + math.FA + math.AP + Tue, 23 Dec 2025 00:00:00 -0500 replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Sascha Kurz + http://creativecommons.org/licenses/by/4.0/ + Robl\^edo Mak's Miranda Sette - Optimal Error Analysis of Channel Estimation for IRS-assisted MIMO Systems - https://arxiv.org/abs/2412.16827 - arXiv:2412.16827v2 Announce Type: replace -Abstract: As intelligent reflecting surface (IRS) has emerged as a new and promising technology capable of configuring the wireless environment favorably, channel estimation for IRS-assisted multiple-input multiple-output (MIMO) systems has garnered extensive attention in recent years. Despite the development of numerous algorithms to address this challenge, a comprehensive theoretical characterization of the optimal recovery error is still lacking. This paper aims to address this gap by providing theoretical guarantees in terms of stable recovery of channel matrices for noisy measurements. We begin by establishing the equivalence between IRS-assisted MIMO systems in the uplink scenario and a compact tensor train (TT)-based tensor-on-tensor (ToT) regression. Building on this equivalence, we then investigate the restricted isometry property (RIP) for complex-valued subgaussian measurements. Our analysis reveals that successful recovery hinges on the relationship between the number of user terminals and the number of time slots during which channel matrices remain invariant. Utilizing the RIP condition, we establish a theoretical upper bound on the recovery error for solutions to the constrained least-squares optimization problem, as well as a minimax lower bound for the considered model. Our analysis demonstrates that the recovery error decreases inversely with the number of time slots, and increases proportionally with the total number of unknown entries in the channel matrices, thereby quantifying the fundamental trade-offs in channel estimation accuracy. In addition, we explore a multi-hop IRS scheme and analyze the corresponding recovery errors. Finally, we have performed numerical experiments to support our theoretical findings. - oai:arXiv.org:2412.16827v2 - cs.IT - eess.SP - math.IT - Mon, 22 Dec 2025 00:00:00 -0500 + The Reduced Basis Multigrid scheme for the Virtual Element Method + https://arxiv.org/abs/2511.22219 + arXiv:2511.22219v2 Announce Type: replace +Abstract: We present a non-nested W-cycle multigrid scheme for the lowest order Virtual Element Method on polygonal meshes. To avoid the implicit definition of the Virtual Element space, which poses several issues in the computation of intergrid operators that underpin multigrid methods, the proposed scheme uses a fully-conforming auxiliary space constructed by cheaply computing the virtual basis functions via the reduced basis method. + oai:arXiv.org:2511.22219v2 + math.NA + cs.NA + Tue, 23 Dec 2025 00:00:00 -0500 replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Zhen Qin, Zhihui Zhu + http://creativecommons.org/licenses/by-nc-nd/4.0/ + Paola F. Antonietti, Silvia Bertoluzza, Fabio Credali - Stratification in equivariant Kasparov theory - https://arxiv.org/abs/2412.21109 - arXiv:2412.21109v2 Announce Type: replace -Abstract: We study stratification, that is the classification of localizing tensor ideal subcategories by geometric means, in the context of Kasparov's equivariant KK-theory of C*-algebras. We introduce a straightforward countable analog of the notion of stratification by Balmer-Favi supports and conjecture that it holds for the equivariant bootstrap subcategory of every finite group G. We prove this conjecture for groups whose nontrivial elements all have prime order, and we verify it rationally for arbitrary finite groups. In all these cases we also compute the Balmer spectrum of compact objects. In our proofs we use larger versions of the equivariant Kasparov categories which admit not only countable coproducts but all small ones; they are constructed in an Appendix using infinity-categorical enhancements and adapting ideas of Bunke-Engel-Land. - oai:arXiv.org:2412.21109v2 - math.KT - math.OA - Mon, 22 Dec 2025 00:00:00 -0500 + Ratio asymptotics and zero density for orthogonal polynomials with varying Verblunsky coefficients + https://arxiv.org/abs/2511.22507 + arXiv:2511.22507v2 Announce Type: replace +Abstract: We study asymptotic behavior of orthogonal polynomials on the unit circle with varying Verblunsky coefficients $\alpha_{n,N}$ when the ratio $n/N$ converges as $n,N\to\infty$. + First, we give a streamlined proof of ratio asymptotics for orthogonal and paraorthogonal polynomials in the case of asymptotically constant and asymptotically periodic coefficients $\alpha_{n,N}$. + Second, we determine the asymptotic zero distribution of paraorthogonal polynomials in the locally constant and locally periodic regimes. Analogous results are obtained for orthogonal polynomials under a mild additional condition on the varying coefficients. + oai:arXiv.org:2511.22507v2 + math.CA + math-ph + math.MP + math.SP + Tue, 23 Dec 2025 00:00:00 -0500 replace - http://creativecommons.org/licenses/by/4.0/ - Ivo Dell'Ambrogio, Rub\'en Martos + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Rostyslav Kozhan, Franti\v{s}ek \v{S}tampach - Feedback Arc Sets and Feedback Arc Set Decompositions in Weighted and Unweighted Oriented Graphs - https://arxiv.org/abs/2501.06935 - arXiv:2501.06935v3 Announce Type: replace -Abstract: Let $D=(V(D),A(D))$ be a digraph with at least one directed cycle. A set $F$ of arcs is a feedback arc set (FAS) if $D-F$ has no directed cycle. The FAS decomposition number ${\rm fasd}(D)$ of $D$ is the maximum number of pairwise disjoint FASs whose union is $A(D)$. The directed girth $g(D)$ of $D$ is the minimum length of a directed cycle of $D$. Note that ${\rm fasd}(D)\le g(D).$ The FAS decomposition number appears in the well-known and far-from-solved conjecture of Woodall (1978) stating that for every planar digraph $D$ with at least one directed cycle, ${\rm fasd}(D)=g(D).$ The degree of a vertex of $D$ is the sum of its in-degree and out-degree. - Let $D$ be an arc-weighted digraph and let ${\rm fas}_w(D)$ denote the minimum weight of its FAS. In this paper, we study bounds on ${\rm fasd}(D)$, ${\rm fas}_w(D)$ and ${\rm fas}(D)$ for arc-weighted oriented graphs $D$ (i.e., digraphs without opposite arcs) with upper-bounded maximum degree $\Delta(D)$ and lower-bounded $g(D)$. Note that these parameters are related: ${\rm fas}_w(D)\le w(D)/{\rm fasd}(D)$, where $w(D)$ is the total weight of $D$, and ${\rm fas}(D)\le |A(D)|/{\rm fasd}(D).$ In particular, we prove the following: (i) If $\Delta(D)\leq~4$ and $g(D)\geq 3$, then ${\rm fasd}(D) \geq 3$ and therefore ${\rm fas}_w(D)\leq \frac{w(D)}{3}$ which generalizes a known tight bound for an unweighted oriented graph with maximum degree at most 4; (ii) If $\Delta(D)\leq 3$ and $g(D)\in \{3,4,5\}$, then ${\rm fasd}(D)=g(D)$; (iii) If $\Delta(D)\leq 3$ and $g(D)\ge 8$ then ${\rm fasd}(D)<g(D).$ We also give some bounds for the cases when $\Delta$ or $g$ are large and state several open problems and a conjecture. - oai:arXiv.org:2501.06935v3 + Algebraic Obstructions and the Collapse of Elementary Structure in the Kronecker Problem + https://arxiv.org/abs/2511.22856 + arXiv:2511.22856v3 Announce Type: replace +Abstract: While Kronecker coefficients $g(\lambda,\mu,\nu)$ with bounded rows are polynomial-time computable via lattice-point methods, no explicit closed-form formulas have been obtained for genuinely three-row cases in the 87 years since Murnaghan's foundational work. This paper provides such formulas for the first time and identifies a universal structural boundary at parameter value 5 where elementary combinatorial patterns collapse. + We analyze two independent families of genuinely three-row coefficients and establish that for $k \leq 4$, the formulas exhibit elementary structure: oscillation bounds follow the triangular-Hogben pattern, and polynomial expressions factor completely over $\mathbb{Z}$. At the critical threshold $k=5$, this structure collapses: the triangular pattern fails, and algebraic obstructions -- irreducible quadratic factors with negative discriminant -- emerge. + We develop integer forcing, a proof technique exploiting the tension between continuous asymptotics and discrete integrality. As concrete results, we prove that $g((n,n,1)^3) = 2 - (n \mod 2)$ for all $n \geq 3$ -- the first explicit formula for a genuinely three-row Kronecker coefficient -- derive five explicit polynomial formulas for staircase-hook coefficients, and verify Saxl's conjecture for 132 three-row partitions. + oai:arXiv.org:2511.22856v3 math.CO - cs.DM - Mon, 22 Dec 2025 00:00:00 -0500 + cs.CC + math.RT + quant-ph + Tue, 23 Dec 2025 00:00:00 -0500 replace http://creativecommons.org/licenses/by/4.0/ - Gregory Gutin, Mads Anker Nielsen, Anders Yeo, Yacong Zhou + Soong Kyum Lee - The Return Times Theorem, Auto-Correlation and Sequences with an Empty Fourier-Bohr Spectrum - https://arxiv.org/abs/2501.07453 - arXiv:2501.07453v4 Announce Type: replace -Abstract: This paper explores the proof by J. Bourgain, H. Furstenberg, Y. Katznelson and D.S. Ornstein of their return times theorem [2] and lights a corner in it regarding the role of auto-correlation. As for pointwise convergence, this was already observed in [5], and here we exploit the opportunity to write down the proof. This yields a more intrinsic characterization of the sequences satisfying the pointwise theorem. Then we proceed and obtain a characterization linked to auto-correlation also to sequences satisfying the mean theorem - by that theorem those were already known to be exactly the sequences with an empty Fourier-Bohr spectrum. Some further investigation is done and examples are provided regarding generic sequences satisfying the pointwise theorem for which the measure on the circle that the auto-correlation function represents (by Fourier transform) is not atomless, and also regarding the existence of sequences that satisfy the mean theorem but not the pointwise one. - oai:arXiv.org:2501.07453v4 - math.DS - math.PR - Mon, 22 Dec 2025 00:00:00 -0500 + ORBGRAND Is Exactly Capacity-achieving via Rank Companding + https://arxiv.org/abs/2512.00347 + arXiv:2512.00347v3 Announce Type: replace +Abstract: Among guessing random additive noise decoding (GRAND) algorithms, ordered reliability bits GRAND (ORBGRAND) has attracted considerable attention due to its efficient use of soft information and suitability for hardware implementation. It has also been shown that ORBGRAND achieves a rate very close to the capacity of an additive white Gaussian noise channel under antipodal signaling. In this work, it is further established that, for general binary-input memoryless channels under symmetric input distribution, via suitably companding the ranks in ORBGRAND according to the inverse cumulative distribution function (CDF) of channel reliability, the resulting CDF-ORBGRAND algorithm exactly achieves the mutual information, i.e., the symmetric capacity. This result is then applied to bit-interleaved coded modulation (BICM) systems to handle high-order input constellations. Via considering the effects of mismatched decoding due to both BICM and ORBGRAND, it is shown that CDF-ORBGRAND is capable of achieving the BICM capacity, which was initially derived in the literature by treating BICM as a set of independent parallel channels. + oai:arXiv.org:2512.00347v3 + cs.IT + math.IT + Tue, 23 Dec 2025 00:00:00 -0500 replace http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Matan Tal + Zhuang Li, Wenyi Zhang - Scattering, Polyhomogeneity and Asymptotics for Quasilinear Wave Equations From Past to Future Null Infinity - https://arxiv.org/abs/2501.09814 - arXiv:2501.09814v2 Announce Type: replace -Abstract: We present a general construction of semiglobal scattering solutions to quasilinear wave equations in a neighbourhood of spacelike infinity including past and future null infinity, where the scattering data are posed on an ingoing null cone and along past null infinity. More precisely, we prove weighted, optimal-in-decay energy estimates and propagation of polyhomogeneity statements from past to future null infinity for these solutions, we provide an algorithmic procedure how to compute the precise coefficients in the arising polyhomogeneous expansions, and we apply this procedure to various examples. As a corollary, our results directly imply the summability in the spherical harmonic number $\ell$ of the estimates proved for fixed spherical harmonic modes in the papers [Keh22b,KM24] from the series "The Case Against Smooth Null Infinity". - Our (physical space) methods are based on weighted energy estimates near spacelike infinity similar to those of [HV23], commutations with (modified) scaling vector fields to remove leading order terms in the relevant expansions, time inversions, as well as the Minkowskian conservation laws: $$ - \partial_u(r^{-2\ell}\partial_v(r^2\partial_v)^{\ell}(r\phi_{\ell}))=0, $$ which are satisfied if $\Box_\eta\phi=0$. - Our scattering constructions apply to systems of equations as well and go beyond the usual class of finite energy solutions. We use this to also derive a scattering theory and prove propagation of polyhomogeneity for the Einstein vacuum equations in a harmonic gauge. In the process, we also need to introduce a novel ansatz accounting for the stronger-than-Schwarzschildean divergence of the light cones, which, in particular, extends existing exterior stability of Minkowski statements in harmonic gauge to allow for slowly decaying data as considered in [Bie10]. - oai:arXiv.org:2501.09814v2 - math.AP - gr-qc - math-ph - math.MP - Mon, 22 Dec 2025 00:00:00 -0500 + Continuous-time reinforcement learning for optimal switching over multiple regimes + https://arxiv.org/abs/2512.04697 + arXiv:2512.04697v2 Announce Type: replace +Abstract: This paper studies the continuous-time reinforcement learning (RL) for optimal switching problems across multiple regimes. We consider a type of exploratory formulation under entropy regularization where the agent randomizes both the timing of switches and the selection of regimes through the generator matrix of an associated continuous-time finite-state Markov chain. We establish the well-posedness of the associated system of Hamilton-Jacobi-Bellman (HJB) equations and provide a characterization of the optimal policy. The policy improvement and the convergence of the policy iterations are rigorously established by analyzing the system of equations. We also show the convergence of the value function in the exploratory formulation towards the value function in the classical formulation as the temperature parameter vanishes. Finally, a reinforcement learning algorithm is devised and implemented by invoking the policy evaluation based on the martingale characterization. Our numerical examples with the aid of neural networks illustrate the effectiveness of the proposed RL algorithm. + oai:arXiv.org:2512.04697v2 + math.OC + cs.LG + q-fin.CP + Tue, 23 Dec 2025 00:00:00 -0500 replace http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Istvan Kadar, Lionor Kehrberger + Yijie Huang, Mengge Li, Xiang Yu, Zhou Zhou - Zeros and critical points of Gaussian fields: cumulants asymptotics and limit theorems - https://arxiv.org/abs/2501.10226 - arXiv:2501.10226v2 Announce Type: replace -Abstract: Let $f:\mathbb{R}^d \to \mathbb{R}^k$ be a smooth centered stationary Gaussian field and $\mathcal{B} \subset \mathbb{R}^d$ be a bounded Borel set. In this paper, we determine the asymptotics as $R \to \infty$ of all the cumulants of the $(d-k)$-dimensional volume of $f^{-1}(0) \cap R\mathcal{B}$. When $k=1$, we obtain similar asymptotics for the number of critical points of $f$ in $R\mathcal{B}$. Our main hypotheses are some regularity and non-degeneracy of the field, as well as mild integrability conditions on the first derivatives of its covariance kernel. As corollaries of these cumulants estimates, we deduce a strong Law of Large Numbers and a Central Limit Theorem for the nodal volume (resp.~the number of critical points) of a regular and non-degenerate enough field whose covariance decays fast enough at infinity. Our results hold more generally for a one-parameter family $(f_R)$ of Gaussian fields admitting a stationary local scaling limit as $R \to \infty$, for example Kostlan polynomials in the large degree limit. They also hold for the random measures of integration over the vanishing locus of $f_R$ as $R \to +\infty$. - oai:arXiv.org:2501.10226v2 - math.PR - Mon, 22 Dec 2025 00:00:00 -0500 + Interpretation of a Discrete de Rham method as a Finite Element System + https://arxiv.org/abs/2512.05912 + arXiv:2512.05912v2 Announce Type: replace +Abstract: We show that the DDR method can be interpreted as defining a computable consistent discrete $\mathrm{L}^2$ product on a conforming FES defined by PDEs. Without modifying the numerical method itself, this point of view provides an alternative approach to the analysis. The conformity and consistency properties we obtain are stronger than those previously shown, even in low dimensions. We can also recover some of the other results that have been proved about DDR, from those that have already been proved, in principle, in the general context of FES. We also bring VEM, the Virtual Element Method, into the discussion. + oai:arXiv.org:2512.05912v2 + math.NA + cs.NA + Tue, 23 Dec 2025 00:00:00 -0500 replace - http://creativecommons.org/licenses/by-sa/4.0/ - Michele Ancona, Louis Gass, Thomas Letendre, Michele Stecconi + http://creativecommons.org/licenses/by/4.0/ + Snorre H. Christiansen, Francesca Rapetti - Poisson kernels on the half-plane are bell-shaped - https://arxiv.org/abs/2501.16068 - arXiv:2501.16068v2 Announce Type: replace -Abstract: Consider a second-order elliptic operator $L$ in the half-plane $\mathbb R \times (0, \infty)$ with coefficients depending only on the second coordinate. The Poisson kernel for $L$ is used in the representation of positive $L$-harmonic functions, that is, solutions of $L u = 0$. In probabilistic terms, the Poisson kernel is the density function of the distribution of the diffusion in $\mathbb R \times (0, \infty)$ with generator $L$ at the hitting time of the boundary. We prove that the Poisson kernel for $L$ is bell-shaped: its $n$th derivative changes sign $n$ times. In particular, it is unimodal and it has two inflection points (it is concave, then convex, then concave again). - oai:arXiv.org:2501.16068v2 - math.AP - Mon, 22 Dec 2025 00:00:00 -0500 + Mixed exponential statistical structures and their approximation operators + https://arxiv.org/abs/2512.07870 + arXiv:2512.07870v2 Announce Type: replace +Abstract: The paper examines the construction and analysis of a new class of mixed exponential statistical structures that combine the properties of stochastic models and linear positive operators.The relevance of the topic is driven by the growing need to develop a unified theoretical framework capable of describing both continuous and discrete random structures that possess approximation properties. The aim of the study is to introduce and analyze a generalized family of mixed exponential statistical structures and their corresponding linear positive operators, which include known operators as particular cases. We define auxiliary statistical structures B and H through differential relations between their elements, and construct the main Phillips-type structure. Recurrent relations for the central moments are obtained, their properties are established, and the convergence and approximation accuracy of the constructed operators are investigated. The proposed approach allows mixed exponential structures to be viewed as a generalization of known statistical systems, providing a unified analytical and stochastic description. The results demonstrate that mixed exponential statistical structures can be used to develop new classes of positive operators with controllable preservation and approximation properties. The proposed methodology forms a basis for further research in constructing multidimensional statistical structures, analyzing operators in weighted spaces, and studying their asymptotic characteristics. + oai:arXiv.org:2512.07870v2 + math.ST + stat.TH + Tue, 23 Dec 2025 00:00:00 -0500 replace http://creativecommons.org/licenses/by/4.0/ - Mateusz Kwa\'snicki + Yurii Volkov, Oleksandr Volkov, Nataliia Voinalovych - Beyond Eckmann-Hilton: Commutativity in Higher Categories - https://arxiv.org/abs/2501.16465 - arXiv:2501.16465v3 Announce Type: replace -Abstract: We show that in a weak globular $\omega$-category, all composition operations are equivalent and commutative for cells with sufficiently degenerate boundary, which can be considered a higher-dimensional generalisation of the Eckmann-Hilton argument. Our results are formulated constructively in a type-theoretic presentation of $\omega$-categories. The heart of our construction is a family of padding and repadding techniques, which gives an equivalence relation between cells which are not necessarily parallel. Our work has been implemented, allowing us to explicitly compute suitable witnesses, which grow rapidly in complexity as the dimension increases. These witnesses can be exported as inhabitants of identity types in Homotopy Type Theory, and hence are of relevance in synthetic homotopy theory. - oai:arXiv.org:2501.16465v3 - math.CT - cs.LO - Mon, 22 Dec 2025 00:00:00 -0500 + The limit joint distributions of some statistics used in testing the quality of random number generators + https://arxiv.org/abs/2512.08002 + arXiv:2512.08002v2 Announce Type: replace +Abstract: The limit joint distribution of statistics that are generalizations of some statistics from the NIST STS, TestU01, and other packages is found under the following hypotheses $H_0$ and $H_1$. Hypothesis $H_0$ states that the tested sequence is a sequence of independent random vectors with a known distribution, and the simple alternative hypothesis $H_1$ converges in some sense to $H_0$ with increasing sample size. In addition, an analogue of the Berry-Esseen inequality is obtained for the statistics under consideration, and conditions for their asymptotic independence are found. + oai:arXiv.org:2512.08002v2 + math.ST + stat.AP + stat.TH + Tue, 23 Dec 2025 00:00:00 -0500 replace http://creativecommons.org/licenses/by/4.0/ - Thibaut Benjamin, Ioannis Markakis, Wilfred Offord, Chiara Sarti, Jamie Vicary + M. P. Savelov - Varieties of prime tropical ideals and the dimension of the coordinate semiring - https://arxiv.org/abs/2501.18053 - arXiv:2501.18053v2 Announce Type: replace -Abstract: In this note we study the relationship between ideals and congruences of the tropical polynomial and Laurent polynomial semirings. We show that the variety of a non-zero prime ideal of the tropical (Laurent) polynomial semiring consists of at most one point. We also prove a result relating the dimension of an affine tropical variety and the dimension of its coordinate semiring. - oai:arXiv.org:2501.18053v2 - math.AG - math.AC - Mon, 22 Dec 2025 00:00:00 -0500 + Fock Space Tensor Product Categorifications and Multiplicities in Complex Rank Parabolic Category O + https://arxiv.org/abs/2512.08312 + arXiv:2512.08312v2 Announce Type: replace +Abstract: We undertake the study of complex rank analogues of parabolic category O defined using Deligne categories. We regard these categories as a family over an affine space, introduce a stratification on this parameter space, and formulate conjectures on the structural constancy of fibers on each stratum. Using the theory of $\mathfrak{sl}_{\mathbb{Z}}$-categorification, we prove these conjectures for admissible strata. Namely, we axiomatize the notion of multi-Fock tensor product categorifications (MFTPCs), which are interval finite highest weight categories equipped with a compatible action of commuting copies of $\mathfrak{sl}_{\mathbb{Z}}$, categorifying an external tensor product of tensor products of highest and lowest weight Fock space representations. We prove a uniqueness theorem for admissible MFTPCs and show that complex rank parabolic categories O have the structure of MFTPCs. In turn, for suitable choices of parameters, we produce an equivalence of complex rank category O with a stable limit of classical parabolic categories O, resolving our conjecture in the admissible case. These equivalences yield multiplicities of simple objects in Verma modules in terms of stable parabolic Kazhdan--Lusztig polynomials, answering a question posed by Etingof. In particular, for the case of two Levi blocks of non-integral size, we completely describe the structure of the corresponding category O in terms of stable representation theory. As an application, we obtain multiplicities for parabolic analogs of hyperalgebra Verma modules introduced by Haboush in the large rank and large characteristic limit. + oai:arXiv.org:2512.08312v2 + math.RT + math.CT + Tue, 23 Dec 2025 00:00:00 -0500 replace http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - D\'aniel Jo\'o, Kalina Mincheva + Hamilton Wan - Derived representations of quantum character varieties - https://arxiv.org/abs/2502.04267 - arXiv:2502.04267v2 Announce Type: replace -Abstract: Quantum moduli algebras $\mathcal{L}_{g,n}^{\mathrm{inv}}(H)$ were introduced by Alekseev-Grosse-Schomerus and Buffenoir-Roche in the context of quantization of character varieties of surfaces and exist for any quasitriangular Hopf algebra $H$. In this paper we construct representations of $\mathcal{L}_{g,n}^{\mathrm{inv}}(H)$ on cohomology spaces $\mathrm{Ext}_H^m(X,M)$ for all $m \geq 0$, where $X$ is any $H$-module and $M$ is any $\mathcal{L}_{g,n}(H)$-module endowed with a compatible $H$-module structure. As a corollary and under suitable assumptions on $H$, we obtain projective representations of mapping class groups of surfaces on such Ext spaces. This recovers the projective representations constructed by Lentner-Mierach-Schweigert-Sommerh\"auser from Lyubashenko theory, when the category $\mathcal{C} = H\text{-}\mathrm{mod}$ is used in their construction. Other topological applications are matrix-valued invariants of knots in thickened surfaces and representations of skein algebras on Ext spaces. - oai:arXiv.org:2502.04267v2 - math.QA - Mon, 22 Dec 2025 00:00:00 -0500 + The Jordan canonical form of the Fr\'{e}chet derivative of a matrix function + https://arxiv.org/abs/2512.08399 + arXiv:2512.08399v3 Announce Type: replace +Abstract: Let $\mathbb{F}$ be an algebraically closed field of characteristic $0$. Given a square matrix $A \in \mathbb{F}^{n \times n}$ and a polynomial $f \in \mathbb{F}[w]$, we determine the Jordan canonical form of the formal Fr\'{e}chet derivative of $f(A)$, in terms of that of $A$ and of $f$. When $\mathbb{F}\subseteq \mathbb{C}$, via Hermite interpolation, our result provides a solution to [N.J. Higham, \emph{Functions of Matrices: Theory and Computation}, Research Problem 3.11]. A generalization consists of finding the Jordan canonical form of linear combinations of Kronecker products of powers of two square matrices, i.e., $\sum_{i,j} a_{ij} (X^i \otimes Y^j)$. For this generalization, we provide some new partial results, including a partial solution under certain assumptions and general bounds on the number and the sizes of Jordan blocks. + oai:arXiv.org:2512.08399v3 + math.RA + Tue, 23 Dec 2025 00:00:00 -0500 replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Matthieu Faitg + http://creativecommons.org/licenses/by-nc-nd/4.0/ + Vanni Noferini - A measure-valued HJB perspective on Bayesian optimal adaptive control - https://arxiv.org/abs/2502.12957 - arXiv:2502.12957v2 Announce Type: replace -Abstract: We consider a Bayesian adaptive optimal stochastic control problem where a hidden static signal has a non-separable influence on the drift of a noisy observation. Being allowed to control the specific form of this dependence, we aim at optimising a cost functional depending on the posterior distribution of the hidden signal. Our setup is in sharp contrast to existing work: we include costs that depend on the full posterior distribution in a form that admits a large class of non-linear relationships. Expressing the dynamics of this posterior distribution in the observation filtration, we embed our problem into a genuinely infinite-dimensional stochastic control problem using measure-valued martingales. We address this problem by use of viscosity theory and approximation arguments. Specifically, we show equivalence to a corresponding weak formulation, characterise the optimal value of the problem in terms of the unique continuous viscosity solution of an associated HJB equation, and construct a piecewise constant and arbitrarily-close-to-optimal control to our main problem of study. As a byproduct of our analysis, we also provide a novel stability result for a class of measure-valued SDEs which we believe is of independent interest. - oai:arXiv.org:2502.12957v2 - math.OC - math.PR - q-fin.MF - Mon, 22 Dec 2025 00:00:00 -0500 + A Lie-theoretic generalization of some Hilbert schemes + https://arxiv.org/abs/2512.08532 + arXiv:2512.08532v2 Announce Type: replace +Abstract: We define several versions of a class of varieties $X_{\mathfrak{g}}$ attached to a complex reductive Lie algebra $\mathfrak{g}$, generalizing the Hilbert scheme of points on the plane. These include trigonometric and elliptic versions attached to the corresponding groups. We also define the corresponding isospectral varieties $Y_{\mathfrak{g}}$. We prove a Gordon-Stafford localization theorem for $X_{\mathfrak{g}}$ and the corresponding equal-parameter rational Cherednik algebras, relate these varieties to the affine Springer fiber-sheaf correspondence of arXiv:2204.00303, and discuss examples. We conjecture that the torus-fixed points of our varieties are in bijection with two-sided cells in the finite Weyl group and prove this in types $ABC$. We relate these results to known results about Calogero-Moser spaces. + oai:arXiv.org:2512.08532v2 + math.AG + math.RT + Tue, 23 Dec 2025 00:00:00 -0500 replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Alexander M. G. Cox, Sigrid K\"allblad, Chaorui Wang + http://creativecommons.org/licenses/by/4.0/ + Oscar Kivinen - Multispecies inhomogeneous $t$-PushTASEP from antisymmetric fusion - https://arxiv.org/abs/2503.00829 - arXiv:2503.00829v3 Announce Type: replace -Abstract: We investigate the recently introduced inhomogeneous $n$-species $t$-PushTASEP, a long-range stochastic process on a periodic lattice. A Baxter-type formula is established, expressing the Markov matrix as an alternating sum of commuting transfer matrices over all the fundamental representations of $U_t(\widehat{sl}_{n+1})$. This superposition acts as an inclusion-exclusion principle, selectively extracting the sequential particle transitions characteristic of the PushTASEP, while canceling forbidden channels. The homogeneous specialization connects the PushTASEP to ASEP, showing that the two models share eigenstates and a common integrability structure. - oai:arXiv.org:2503.00829v3 - math-ph - math.CO - math.MP - math.PR - math.QA - Mon, 22 Dec 2025 00:00:00 -0500 + Embedding $K$-algebras into Leavitt algebra $L_K(1, 2)$ + https://arxiv.org/abs/2512.09241 + arXiv:2512.09241v2 Announce Type: replace +Abstract: Since the commutative monoid $T = (\{0, 1\}, \vee)$ is a weak terminal object in the category of conical monoids with order units, there is a unital homomorphism from every Bergman $K$-algebra corresponding to a conical finitely generated commutative monoid into the Leavitt algebra $L_K(1,2)$, where $K$ is a field. This fact will be used to give a short proof that Leavitt path algebras associated with finite graphs with condition $(L)$ embed into $L_K(1,2)$, as well as provide criteria for an embedding of $M_s(L_{K}(1, m))$ in $M_s(L_{K}(1, n))$. As our second main result, we show that the Heisenberg equation $xy-yx=1$ cannot be realized in any Steinberg algebra, implying that the first Weyl algebra cannot be embedded into $L_K(1,2)$, giving an affirmative answer to a question of Brownlowe and Sorensen on the embeddability of $K$-algebras with a countable basis inside $L_K(1,2)$. Whereas, $L_K(E)$ cannot be graded-embedded into $L_K(1,2)$ in general, in the final section we show that $L_K(E)$ does admit a graded embedding into $L_K(1,2)\otimes_K L_K(1,2)$. + oai:arXiv.org:2512.09241v2 + math.RA + math.OA + Tue, 23 Dec 2025 00:00:00 -0500 replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - 10.1214/25-EJP1448 - Electron. J. Probab. 30: 1-28 (2025) - Arvind Ayyer, Atsuo Kuniba + http://creativecommons.org/licenses/by/4.0/ + Boris Bilich, Roozbeh Hazrat, Tran Giang Nam - Simultaneous block diagonalization of a set of symmetric matrices via congruence - https://arxiv.org/abs/2503.01166 - arXiv:2503.01166v2 Announce Type: replace -Abstract: This article studies canonical forms derived from the finest simultaneous block diagonalization of a set of symmetric matrices via congruence. Our technique relies on Harrison's center theory, which is extended from a single higher degree form to multiple quadratic forms, hence a set of symmetric matrices. The algebraic structures of centers and the bijective relationship between the simultaneous block diagonalization via congruence and complete sets of orthogonal idempotents of centers are investigated. We provide an algorithm that mainly uses standard linear algebra tasks and several examples to demonstrate its effectiveness. In addition, this technique can be extended verbatim to the simultaneous block diagonalization of a set of Hermitian matrices via $*$-congruence. - oai:arXiv.org:2503.01166v2 + Reductive monoids and cluster algebras + https://arxiv.org/abs/2512.09564 + arXiv:2512.09564v2 Announce Type: replace +Abstract: We show that the coordinate ring of the Vinberg monoid of a simply connected semisimple complex group is an upper cluster algebra. As an application, we construct cluster structures on a large class of flat reductive monoids. After localization, we obtain cluster structures on any connected reductive group whose commutator group is simply connected. + oai:arXiv.org:2512.09564v2 + math.RT math.RA - Mon, 22 Dec 2025 00:00:00 -0500 + Tue, 23 Dec 2025 00:00:00 -0500 replace http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Lishan Fang, Hua-Lin Huang, Jiayan Huang + Jinfeng Song, Jeff York Ye - Heat semigroups on quantum automorphism groups of finite dimensional C*-algebras - https://arxiv.org/abs/2503.03448 - arXiv:2503.03448v4 Announce Type: replace -Abstract: In this paper, we investigate heat semigroups on a quantum automorphism group ${\rm Aut}^+(B)$ of a finite dimensional C*-algebra $B$ and its Plancherel trace. We show ultracontractivity, hypercontractivity, and the spectral gap inequality of the heat semigroups on ${\rm Aut}^+(B)$. Furthermore, we obtain the sharpness of the Sobolev embedding property and the Hausdorff-Young inequality of ${\rm Aut}^+(B)$. - oai:arXiv.org:2503.03448v4 - math.QA - math-ph + Extrapolation for bilinear compact operators in the variable exponent setting + https://arxiv.org/abs/2512.09721 + arXiv:2512.09721v2 Announce Type: replace +Abstract: We establish extrapolation of compactness for bilinear operators in the scale of weighted variable exponent Lebesgue spaces. First, we prove an abstract principle relying on the Cobos-Fern\'{a}ndez-Cabrera-Mart\'{i}nez theorem. Then, as an application we deduce new compactness results for the commutators of bilinear $\omega$-Calder\'{o}n-Zygmund operators, bilinear fractional integrals and bilinear Fourier multipliers acting on weighted variable exponent Lebesgue spaces. Our work extends and unifies among others earlier works of the second named author together with Hyt\"{o}nen as well as Oikari. + oai:arXiv.org:2512.09721v2 + math.CA math.FA - math.MP - math.OA - Mon, 22 Dec 2025 00:00:00 -0500 + Tue, 23 Dec 2025 00:00:00 -0500 replace http://creativecommons.org/licenses/by/4.0/ - Futaba Sato + Spyridon Kakaroumpas, Stefanos Lappas - Quantitative Stability in Fractional Hardy-Sobolev Inequalities: The Role of Euler-Lagrange Equations - https://arxiv.org/abs/2503.06716 - arXiv:2503.06716v3 Announce Type: replace -Abstract: This paper investigates sharp stability estimates for the fractional Hardy-Sobolev inequality: $$\mu_{s,t}\left(\mathbb{R}^N\right) \left(\int_{\mathbb{R}^N} \frac{|u|^{2^*_s(t)}}{|x|^t} \,{\rm d}x \right)^{\frac{2}{2^*_s(t)}} \leq \int_{\mathbb{R}^N} \left|(-\Delta)^{\frac{s}{2}} u \right|^2 \,{\rm d}x, \quad \text{for all } u \in \dot{H}^s\left(\mathbb{R}^N\right),$$ where $N > 2s$, $s \in (0,1)$, $0 < t < 2s < N $, and $2^*_s(t) = \frac{2(N-t)}{N-2s}$. Here, $\mu_{s,t}\left(\mathbb{R}^N\right)$ represents the best constant in the inequality. - The paper focuses on the quantitative stability results of the above inequality and the corresponding Euler-Lagrange equation near a positive ground-state solution. Additionally, a qualitative stability result is established for the Euler-Lagrange equation, offering a thorough characterization of the Palais-Smale sequences for the associated energy functional. These results generalize the sharp quantitative stability results for the classical Sobolev inequality in $\mathbb{R}^N$, originally obtained by Bianchi and Egnell \cite{BE91} as well as the corresponding critical exponent problem in $\mathbb{R}^N$, explored by Ciraolo, Figalli, and Maggi \cite{CFM18} in the framework of fractional calculus. - oai:arXiv.org:2503.06716v3 - math.AP - Mon, 22 Dec 2025 00:00:00 -0500 + Mostow Rigidity Made Easier + https://arxiv.org/abs/2512.09774 + arXiv:2512.09774v3 Announce Type: replace +Abstract: This article gives a self-contained proof of Mostow Rigidity, at least modulo undergrad real analysis. The proof should be accessible to grad students interested in geometry and topology. It has no new research, but I think that this is an unusually clean and analytically light proof of this famous result. I am posting this because I think it will be useful to geometry/topology students. + oai:arXiv.org:2512.09774v3 + math.GT + Tue, 23 Dec 2025 00:00:00 -0500 replace http://creativecommons.org/licenses/by/4.0/ - 10.1017/prm.2025.10096 - Souptik Chakraborty, Utsab Sarkar + Richard Evan Schwartz - Determination of the density in the linear elastic wave equation - https://arxiv.org/abs/2503.12825 - arXiv:2503.12825v3 Announce Type: replace -Abstract: We study the inverse boundary value problem for the linear elastic wave equation in three-dimensional isotropic medium. We show that both the Lam\'e parameters and the density can be uniquely recovered from the boundary measurements under the strictly convex foliation condition. - oai:arXiv.org:2503.12825v3 - math.AP - Mon, 22 Dec 2025 00:00:00 -0500 + On framed rook algebras + https://arxiv.org/abs/2512.11188 + arXiv:2512.11188v2 Announce Type: replace +Abstract: We introduce and study the framed rook algebra, a structure that unifies two significant generalizations of the Iwahori-Hecke algebra. The first one, introduced by Solomon, extends the Hecke algebra to the full matrix monoid, yielding the rook monoid algebra. The second one, developed by Yokonuma, replaces the Borel subgroup with the unipotent subgroup, resulting in the Yokonuma-Hecke algebra. Our concrete algebra is constructed from the double cosets of the unipotent subgroup within the full matrix monoid. We show that this double coset decomposition is indexed by the framed symmetric inverse monoid. We also define the Rook Yokonuma-Hecke algebra as an abstract structure using generators and relations. We then prove the main isomorphism theorem, which establishes that this abstract algebra is isomorphic to the framed rook algebra under a specific parameter specialization. To complete our characterization, we provide a faithful representation on a tensor space and establish a linear basis for the Rook Yokonuma-Hecke algebra. + oai:arXiv.org:2512.11188v2 + math.RT + Tue, 23 Dec 2025 00:00:00 -0500 replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Jian Zhai + http://creativecommons.org/licenses/by/4.0/ + Diego Arcis, Jorge Espinoza, Marcelo Flores - Fractional diffusion in convex domains and reflected isotropic stable processes - https://arxiv.org/abs/2503.13071 - arXiv:2503.13071v2 Announce Type: replace -Abstract: We establish the fractional diffusion limit of the kinetic scattering equation with diffusive boundary condition in a strongly convex bounded domain $\mathcal{D}\subset\mathbb{R}^d$. According to the nature of the boundary condition, two types of fractional heat equations may arise at the limit, corresponding to two types of isotropic stable processes reflected in $\mathcal{D}$. In both cases, when the process tries to jump across the boundary, it is stopped at the unique point where $\partial\mathcal{D}$ intersects the line segment defined by the attempted jump. It then leaves the boundary either continuously (for the first type) or by a power-law distributed jump (for the second type). The construction of these processes is done via an It\^o synthesis: we concatenate their excursions in the domain, which are obtained by translating, rotating and stopping the excursions of some stable processes reflected in the half-space. The key ingredient in this procedure is the construction of the boundary processes, i.e. the processes time-changed by their local time on the boundary, which solve stochastic differential equations driven by some Poisson measures of excursions. The well-posedness of these boundary processes relies on delicate estimates involving some geometric inequalities and the laws of the undershoot and overshoot of the excursion when it leaves the domain. We show that these reflected Markov processes are Markov and Feller, we study their infinitesimal generator and we write down the reflected fractional heat equations satisfied by their time-marginals. - oai:arXiv.org:2503.13071v2 - math.PR - math.AP - Mon, 22 Dec 2025 00:00:00 -0500 + Sphere Decoding Revisited + https://arxiv.org/abs/2512.11195 + arXiv:2512.11195v2 Announce Type: replace +Abstract: In this paper, the paradigm of sphere decoding (SD) for solving the integer least square problem (ILS) is revisited, where extra degrees of freedom are introduced to exploit the decoding potential. Firstly, the equivalent sphere decoding (ESD) is proposed, which is essentially the same with the classic Fincke-Pohst sphere decoding but characterizes the sphere radius $D>0$ with two new parameters named as initial searching size $K>1$ and deviation factor $\sigma>0$. By fixing $\sigma$ properly, we show that given the sphere radius $D\triangleq\sigma\sqrt{2\ln K}$, the complexity of ESD in terms of the number of visited nodes is upper bounded by $|S|<nK$, thus resulting in an explicit and tractable decoding trade-off solely controlled by $K$. To the best of our knowledge, this is the first time that the complexity of sphere decoding is exactly specified, where considerable decoding potential can be explored from it. After that, two enhancement mechanisms named as normalized weighting and candidate protection are proposed to further upgrade the ESD algorithm. On one hand, given the same setups of $K$ and $\sigma$, a larger sphere radius is achieved, indicating a better decoding trade-off. On the other hand, the proposed ESD algorithm is generalized, which bridges suboptimal and optimal decoding performance through the flexible choice of $K$. Finally, further performance optimization and complexity reduction with respect to ESD are also derived, and the introduced tractable and flexible decoding trade-off is verified through large-scale MIMO detection. + oai:arXiv.org:2512.11195v2 + cs.IT + math.IT + Tue, 23 Dec 2025 00:00:00 -0500 replace - http://creativecommons.org/licenses/by/4.0/ - Lo\"ic B\'ethencourt, Nicolas Fournier + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + 10.1109/TCOMM.2023.3278732 + in IEEE Transactions on Communications, vol. 72, no. 1, pp. 85-100, Jan. 2024 + Zheng Wang, Cong Ling, Shi Jin, Yongming Huang, Feifei Gao - Full measure universality for Cantor Sets - https://arxiv.org/abs/2503.21079 - arXiv:2503.21079v2 Announce Type: replace -Abstract: We investigate variants of the Erd\H{o}s similarity problem for Cantor sets. We prove that under a mild Hausdorff or packing logarithmic dimension assumption, Cantor sets are not full measure universal, significantly improving the known fact that sets of positive Hausdorff dimension are not measure universal. We prove a weaker result for all Cantor sets $A$: there is a dense $G_\delta$ set of full measure $X\subset\mathbb{R}^d$, such that for any bi-Lipschitz function $f:\mathbb{R}^d\to \mathbb{R}^d$, the set of translations $t$ such that $f(A)+t\subseteq X$ is of measure zero. Equivalently, there is a null set $B\subset\mathbb{R}^d$ such that $\mathbb{R}^d\setminus (f(A)+B)$ is null for all bi-Lipschitz functions $f$. - oai:arXiv.org:2503.21079v2 - math.CA - Mon, 22 Dec 2025 00:00:00 -0500 + Unit-generated orders of real quadratic fields I. Class number bounds + https://arxiv.org/abs/2512.11311 + arXiv:2512.11311v2 Announce Type: replace +Abstract: Unit-generated orders of a quadratic field are orders of the form $\mathcal{O} = \mathbb{Z}[\varepsilon]$, where $\varepsilon$ is a unit in the quadratic field. If the order $\mathcal{O}$ is a maximal order of a real quadratic field, then the quadratic number field is necessarily of a restricted form, being of narrow Richaud--Degert type. However, every real quadratic field contains infinitely many distinct unit-generated orders. They are parametrized as $\mathcal{O} = \mathcal{O}_{n}^{\pm}$ having quadratic discriminants $\Delta(\mathcal{O}) = \Delta_{n}^{+} = n^2 - 4$ (for $n \geq 3$) and $\Delta(\mathcal{O}) = \Delta_{n}^{-} = n^2 + 4$ (for $n \geq 1$). We show the (wide or narrow) class numbers of unit-generated orders satisfy $\log \left|{\rm Cl}(\mathcal{O})\right| \sim \log \frac{1}{2}\left|\Delta(\mathcal{O})\right|$ as $\left|\Delta(\mathcal{O})\right| \to \infty$, using a result of L.-K. Hua. We deduce that there are finitely many unit-generated quadratic orders of class number one and finitely many unit-generated quadratic orders whose class group is $2$-torsion. We classify all unit-generated real quadratic maximal orders having class number one. We provide numerical lists of quadratic unit-generated orders whose class groups are $2$-torsion for $\Delta \leq 10^{10}$, for both wide and narrow class groups, which are conjecturally complete. + oai:arXiv.org:2512.11311v2 + math.NT + Tue, 23 Dec 2025 00:00:00 -0500 replace - http://creativecommons.org/licenses/by/4.0/ - Pablo Shmerkin, Alexia Yavicoli + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Gene S. Kopp, Jeffrey C. Lagarias - Unified interface flux evaluation in a general discontinuous Galerkin spectral element framework - https://arxiv.org/abs/2504.03573 - arXiv:2504.03573v2 Announce Type: replace -Abstract: High-order discontinuous Galerkin spectral element methods (DGSEM) have received growing attention and development, especially in the regime of computational fluid dynamics in recent years. The inherent flexibility of the discontinuous Galerkin approach in handling non-conforming interfaces, such as those encountered in moving geometries or hp-refinement, presents a significant advantage for real-world simulations. Despite the well-established mathematical framework of DG methods, practical implementation challenges persist to boost performance and capability. Most previous studies only focus on certain choices of element shape or basis type in a structured mesh, although they have demonstrated the capability of DGSEM in complex flow simulations. This work discusses the low-cost and unified interface flux evaluation approaches for general spectral elements in unstructured meshes, alongside their implementations in the open-source spectral element framework, Nektar++. The initial motivation arises from the discretization of Helmholtz equations by the symmetric interior penalty method, in which the system matrix can easily become non-symmetric if the flux is not properly evaluated on non-conforming interfaces. We focus on the polynomial non-conforming case in this work but extending to the geometric non-conforming case is theoretically possible. Comparisons of different approaches, trade-offs, and performance of benchmark of our initial matrix-free implementation are also included, contributing to the broader discourse on high-performance spectral element method implementations. - oai:arXiv.org:2504.03573v2 + Projected Sobolev Natural Gradient Descent for Neural Variational Monte Carlo Solution of the Gross-Pitaevskii Equation + https://arxiv.org/abs/2512.11339 + arXiv:2512.11339v2 Announce Type: replace +Abstract: This paper proposes a neural variational Monte Carlo method based on deep neural networks to solve the Gross-Pitaevskii equation (GPE) via projected Sobolev natural gradient descent (NGD). Adopting an "optimize-then-discretize" strategy, we first apply a constraint-preserving continuous Riemannian gradient flow on an infinite-dimensional Riemannian manifold, which is subsequently mapped to the neural network parameter space via Galerkin projection. This process naturally induces a Sobolev energy metric that incorporates physical information, effectively mitigating stiffness during optimization. To address the explicit dependence on the normalization constant caused by the nonlinear interaction term in the GPE, we design a hybrid sampling strategy combining an integration stream and a MCMC stream to achieve precise estimation of the generalized Gram matrix and energy gradients. Numerical experiments on benchmark cases, including the harmonic oscillator potential in the strong interaction limit and multi-scale optical lattice potentials, demonstrate the high accuracy of the proposed method. Furthermore, it achieves an order-of-magnitude acceleration in convergence compared to standard optimizers like Adam, exhibiting superior robustness in handling strong nonlinearities and complex geometric constraints. + oai:arXiv.org:2512.11339v2 math.NA cs.NA - Mon, 22 Dec 2025 00:00:00 -0500 + Tue, 23 Dec 2025 00:00:00 -0500 replace http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Boyang Xia, David Moxey + Chenglong Bao, Chen Cui, Kai Jiang, Shi Shu - Degenerate complex Hessian equations with arbitrary measure in bounded domains - https://arxiv.org/abs/2504.08592 - arXiv:2504.08592v3 Announce Type: replace -Abstract: Let $\Omega$ be a bounded strictly $m$-pseudoconvex domain of $\mathbb{C}^n$. We solve degenerate complex Hessian equations of the form $(\omega + dd^c \varphi)^m\wedge\beta^{n-m} = \mu$ in the generalized Cegrell classes $\mathcal{K}_m(\Omega,\omega,\phi)$, where $\phi \in \mathcal{E}_m(\Omega)$ is a $m$-maximal function, $\omega$ is a smooth real $(1,1)$-form defined in a neighborhood of $\bar\Omega$ and $\mu$ is a positive Radon measure which is dominated by a Hessian measure of $m$-subharnomic functions in Cegrell class. - oai:arXiv.org:2504.08592v3 - math.CV - Mon, 22 Dec 2025 00:00:00 -0500 + Virtual invariants from the non-associative Hilbert scheme + https://arxiv.org/abs/2512.11538 + arXiv:2512.11538v2 Announce Type: replace +Abstract: We introduce a non-associative model for the Hilbert scheme of points in arbitrary dimension. We define a smooth ambient space, which we call the non-associative Hilbert scheme, containing the classical nested Hilbert scheme $\mathrm{NHilb}^{\underline{d}}(\mathbb{A}^n)$ as the associativity, cut out by an explicit section of an associativity bundle. This construction yields canonical perfect obstruction theories and virtual fundamental classes on $\mathrm{NHilb}^{\underline{d}}(\mathbb{A}^n)$ for all $(n,\underline d)$. Using virtual localization, we obtain closed formulas for these virtual classes as sums over admissible nested partitions. Over the punctual locus, we rewrite these as a single multivariable iterated residue formula governing all virtual integrals. Our construction works for all $n$, produces positive-dimensional virtual classes when $n$ is large compared to the number of points, and we expect that they extend the non-commutative matrix model and virtual class construction on Calabi-Yau threefolds. + oai:arXiv.org:2512.11538v2 + math.AG + Tue, 23 Dec 2025 00:00:00 -0500 replace - http://creativecommons.org/licenses/by-nc-nd/4.0/ - Nguyen Van Phu, Le Mau Hai + http://creativecommons.org/licenses/by/4.0/ + Gergely B\'erczi, Felix Minddal - Hamiltonian cycles in tough $(P_4 \cup P_1)$-free graphs - https://arxiv.org/abs/2504.08936 - arXiv:2504.08936v2 Announce Type: replace -Abstract: In 1973, Chv\'atal conjectured that there exists a constant $t_0$ such that every $t_0$-tough graph on at least three vertices is Hamiltonian. This conjecture has inspired extensive research and has been verified for several special classes of graphs. Notably, Jung in 1978 proved that every 1-tough $P_4$-free graph on at least three vertices is Hamiltonian. However, the problem remains challenging even when restricted to graphs with no induced $P_4\cup P_1$, the disjoint union of a path on four vertices and a one-vertex path. In 2013, Nikoghosyan conjectured that every 1-tough $(P_4\cup P_1)$-free graph on at least three vertices is Hamiltonian. Later in 2015, Broersma remarked that ``this question seems to be very hard to answer, even if we impose a higher toughness." He instead posed the following question: ``Is the general conjecture of Chv\'atal's true for $(P_4\cup P_1)$-free graphs?" We provide a positive answer to Broersma's question by establishing that every $23$-tough $(P_4\cup P_1)$-free graph on at least three vertices is Hamiltonian. - oai:arXiv.org:2504.08936v2 - math.CO - Mon, 22 Dec 2025 00:00:00 -0500 + Lower Bound of Nodal Sets in Elliptic Homogenization and Functions with Strong Maximum Principle + https://arxiv.org/abs/2512.12305 + arXiv:2512.12305v2 Announce Type: replace +Abstract: In this note, we first try to prove a uniform lower bound of nodal volume in elliptic homogenization setting. This lower bound is far from optimal. But, we can prove a constant lower bound in dimension two. Motivated by the proof, we extend this results to more general settings. To be more specific, we prove that the nodal volume has a constant lower bound for all continuous functions with strong maximum principle. Our result works for general functions beyond solutions to elliptic PDEs. + oai:arXiv.org:2512.12305v2 + math.AP + Tue, 23 Dec 2025 00:00:00 -0500 replace http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Songling Shan + Jiahuan Li, Zhichen Ying - Ciliberto-Di Gennaro conjecture for sextic hypersurfaces - https://arxiv.org/abs/2505.06742 - arXiv:2505.06742v2 Announce Type: replace -Abstract: The Ciliberto-Di Gennaro conjecture addresses the factoriality of three-dimensional nodal hypersurfaces, and their geometric properties. We prove this conjecture for hypersurfaces of degree 6 by adapting a recent technique due to R. Kloosterman. - oai:arXiv.org:2505.06742v2 - math.AG - Mon, 22 Dec 2025 00:00:00 -0500 + The maximal length of the Erd\H{o}s--Herzog--Piranian lemniscate in high degree + https://arxiv.org/abs/2512.12455 + arXiv:2512.12455v2 Announce Type: replace +Abstract: Let $n \geq 1$, and let $p : {\bf C} \to {\bf C}$ be a monic polynomial of degree $n$. It was conjectured by Erd\H{o}s, Herzog, and Piranian that the maximal length of lemniscate $\{z \in {\bf C}: |p(z)| = 1\}$ is attained by the polynomial $p(z) = z^n-1$. In this paper, building upon a previous analysis of Fryntov and Nazarov, we establish this conjecture for all sufficiently large $n$. + oai:arXiv.org:2512.12455v2 + math.CV + Tue, 23 Dec 2025 00:00:00 -0500 replace - http://creativecommons.org/licenses/by/4.0/ - Ksenia Kvitko + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Terence Tao - BOLT: Block-Orthonormal Lanczos for Trace estimation of matrix functions - https://arxiv.org/abs/2505.12289 - arXiv:2505.12289v2 Announce Type: replace -Abstract: Efficient matrix trace estimation is essential for scalable computation of log-determinants, matrix norms, and distributional divergences. In many large-scale applications, the matrices involved are too large to store or access in full, making even a single matrix-vector (mat-vec) product infeasible. Instead, one often has access only to small subblocks of the matrix or localized matrix-vector products on restricted index sets. Hutch++ achieves optimal convergence rate but relies on randomized SVD and assumes full mat-vec access, making it difficult to apply in these constrained settings. We propose the Block-Orthonormal Stochastic Lanczos Quadrature (BOLT), which matches Hutch++ accuracy with a simpler implementation based on orthonormal block probes and Lanczos iterations. BOLT builds on the Stochastic Lanczos Quadrature (SLQ) framework, which combines random probing with Krylov subspace methods to efficiently approximate traces of matrix functions, and performs better than Hutch++ in near flat-spectrum regimes. To address memory limitations and partial access constraints, we introduce Subblock SLQ, a variant of BOLT that operates only on small principal submatrices. As a result, this framework yields a proxy KL divergence estimator and an efficient method for computing the Wasserstein-2 distance between Gaussians - both compatible with low-memory and partial-access regimes. We provide theoretical guarantees and demonstrate strong empirical performance across a range of high-dimensional settings. - oai:arXiv.org:2505.12289v2 - math.NA - cs.DS - cs.LG - cs.NA - Mon, 22 Dec 2025 00:00:00 -0500 + Skew 2-Dyck paths via the kernel method + https://arxiv.org/abs/2512.12876 + arXiv:2512.12876v2 Announce Type: replace +Abstract: We continue on a recent concept introduced by Kariuki and Okoth, about skew 2-Dyck paths, introducing an additional down-step $L$, together with the usual steps + $U$ (up) and $D$ down. There is the syntactical condition that $UL$ and $LU$ can never occur. An automaton that checks these conditions is introduced, and the + relevant generating functions are obtained by applying the kernel method to three functional equations. It is briefly discussed how the setting can be extended to $t$-Dyck paths. As a benefit, prefixes of skew $t$-Dyck paths are also enumerated. An approach that scans 2-Dyck paths from right to left is also discussed. + oai:arXiv.org:2512.12876v2 + math.CO + Tue, 23 Dec 2025 00:00:00 -0500 replace http://creativecommons.org/licenses/by/4.0/ - Kingsley Yeon, Promit Ghosal, Mihai Anitescu + Helmut Prodinger - Conditioned stochastic stability of equilibrium states on uniformly hyperbolic sets - https://arxiv.org/abs/2506.15503 - arXiv:2506.15503v2 Announce Type: replace -Abstract: We establish the conditioned stochastic stability of equilibrium states for H\"older potentials on uniformly hyperbolic sets. While standard stochastic stability characterises measures on attractors, we analyse the statistics of transient dynamics on non-attracting sets by conditioning small random perturbations of the dynamics to not escape from our regions of interest. We prove that as the noise intensity vanishes, the quasi-ergodic measure of the $e^\phi$-weighted process generated by $\e$-small random perturbations of the deterministic dynamics converges to the unique equilibrium state associated with the potential $\phi - \log \left|\det \left. D T\right|_{E^u}\right|$. The results are obtained via perturbative spectral analysis of transfer operators acting on anisotropic Banach spaces and topological hyperbolic dynamics arguments. Furthermore, we extend this framework globally to Axiom A diffeomorphisms with multiple basic sets using dynamical filtrations. This work provides a rigorous characterisation of natural measures on uniformly hyperbolic repellers, which are fundamental in the context of transient chaos. - oai:arXiv.org:2506.15503v2 - math.DS - math.PR - Mon, 22 Dec 2025 00:00:00 -0500 + Computing Selmer groups associated to mod p Galois representations + https://arxiv.org/abs/2512.12899 + arXiv:2512.12899v2 Announce Type: replace +Abstract: We present methods to compute Selmer groups associated to mod p Galois representations rho over a number field K, with a particular focus on comparing their ranks with periods coming from cohomology classes associated to rho by Serre's conjecture. This provides evidence for a loose version of a "mod p Bloch-Kato conjecture", where the vanishing of a period is predicted to capture the presence of rank in a Selmer group. Our methods are explicit, and implemented in Magma. + oai:arXiv.org:2512.12899v2 + math.NT + Tue, 23 Dec 2025 00:00:00 -0500 replace - http://creativecommons.org/licenses/by-nc-nd/4.0/ - Bernat Bassols Cornudella, Matheus M. Castro + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Lewis Combes - Optimal solutions employing an algebraic Variational Multiscale approach Part II: Application to Navier-Stokes - https://arxiv.org/abs/2506.21395 - arXiv:2506.21395v3 Announce Type: replace -Abstract: This work presents a non-linear extension of the high-order discretisation framework based on the Variational Multiscale (VMS) method previously introduced for steady linear problems. We build on the concept of an optimal projector defined via the symmetric part of the governing operator. Using this idea, we generalise the formulation to the two-dimensional incompressible Navier-Stokes equations. The approach maintains a clear separation between resolved and unresolved scales, with the fine-scale contribution approximated through the approximate Fine-Scale Greens' operator of the associated symmetric operator. This enables a consistent variational treatment of non-linearity while preserving high-order accuracy. We show that the method yields numerical solutions that closely approximate the optimal projection of the continuous/highly-resolved solution and inherits desirable conservation properties. Particularly, the formulation guarantees discrete conservation of mass, energy, and vorticity, where enstrophy conservation is also achieved when exact or over-integration is employed. Numerical results confirm the methodology's robustness and accuracy, while also demonstrating its computational cost advantage compared to the baseline Galerkin approach for the same accuracy. - oai:arXiv.org:2506.21395v3 - math.NA - cs.NA - Mon, 22 Dec 2025 00:00:00 -0500 + Stopping Rules for Stochastic Gradient Descent via Anytime-Valid Confidence Sequences + https://arxiv.org/abs/2512.13123 + arXiv:2512.13123v3 Announce Type: replace +Abstract: We study stopping rules for stochastic gradient descent (SGD) for convex optimization from the perspective of anytime-valid confidence sequences. Classical analyses of SGD provide convergence guarantees in expectation or at a fixed horizon, but offer no statistically valid way to assess, at an arbitrary time, how close the current iterate is to the optimum. We develop an anytime-valid, data-dependent upper confidence sequence for the weighted average suboptimality of projected SGD, constructed via nonnegative supermartingales and requiring no smoothness or strong convexity. This confidence sequence yields a simple stopping rule that is provably $\varepsilon$-optimal with probability at least $1-\alpha$, with explicit bounds on the stopping time under standard stochastic approximation stepsizes. To the best of our knowledge, these are the first rigorous, time-uniform performance guarantees and finite-time $\varepsilon$-optimality certificates for projected SGD with general convex objectives, based solely on observable trajectory quantities. + oai:arXiv.org:2512.13123v3 + math.OC + cs.LG + math.ST + stat.ML + stat.TH + Tue, 23 Dec 2025 00:00:00 -0500 replace http://creativecommons.org/licenses/by/4.0/ - Suyash Shrestha, Marc Gerritsma, Gonzalo Rubio, Steven Hulshoff, Esteban Ferrer + Liviu Aolaritei, Michael I. Jordan - No arbitrage assumption implies the differentiability of derivative pricing function - https://arxiv.org/abs/2506.22213 - arXiv:2506.22213v2 Announce Type: replace -Abstract: In this article, we show necessary and sufficient conditions for a function to transform a continuous Markov semimartingale to a semimartingale. As a result, the no-arbitrage principle guarantees the differentiability of asset prices with respect to the underlying noise, if the asset prices are continuous and the underlying noise is a continuous Markov semimartingale. - oai:arXiv.org:2506.22213v2 - math.PR - Mon, 22 Dec 2025 00:00:00 -0500 + From Zipf's Law to Neural Scaling through Heaps' Law and Hilberg's Hypothesis + https://arxiv.org/abs/2512.13491 + arXiv:2512.13491v2 Announce Type: replace +Abstract: We inspect the deductive connection between the neural scaling law and Zipf's law -- two statements discussed in machine learning and quantitative linguistics. The neural scaling law describes how the cross entropy rate of a foundation model -- such as a large language model -- changes with respect to the amount of training tokens, parameters, and compute. By contrast, Zipf's law posits that the distribution of tokens exhibits a power law tail. Whereas similar claims have been made in more specific settings, we show that the neural scaling law is a consequence of Zipf's law under certain broad assumptions that we reveal systematically. The derivation steps are as follows: We derive Heaps' law on the vocabulary growth from Zipf's law, Hilberg's hypothesis on the entropy scaling from Heaps' law, and the neural scaling from Hilberg's hypothesis. We illustrate these inference steps by a toy example of the Santa Fe process that satisfies all the four statistical laws. + oai:arXiv.org:2512.13491v2 + cs.IT + cs.LG + math.IT + math.ST + stat.TH + Tue, 23 Dec 2025 00:00:00 -0500 replace http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Kihun Nam, Yunxi Xu + {\L}ukasz D\k{e}bowski - Lieb-Robinson bounds, automorphic equivalence and LPPL for long-range interacting fermions - https://arxiv.org/abs/2507.03319 - arXiv:2507.03319v2 Announce Type: replace -Abstract: We prove a Lieb-Robinson bound for lattice fermion models with polynomially decaying interactions, which can be used to show the locality of the quasi-local inverse Liouvillian. This allows us to prove automorphic equivalence and the local perturbations perturb locally (LPPL) principle for these systems. The proof of the Lieb-Robinson bound is based on the work of Else et al. (2020), and our results also apply to spin systems. We explain why some newer Lieb-Robinson bounds for long-range spin systems cannot be used to prove the locality of the quasi-local inverse Liouvillian, and in some cases may not even hold for fermionic systems. - oai:arXiv.org:2507.03319v2 - math-ph - math.MP - quant-ph - Mon, 22 Dec 2025 00:00:00 -0500 + Conditional means, vector pricings, amenability and fixed points in cones + https://arxiv.org/abs/2512.13829 + arXiv:2512.13829v2 Announce Type: replace +Abstract: We study a generalization of conditional probability for arbitrary ordered vector spaces. A related problem is that of assigning a numerical value to one vector relative to another. + We characterize the groups for which these generalized probabilities can be stationary, respectively invariant. Our results deviate from the setting of classical probability; this leads to a new criterion for amenability and for fixed points in cones. + oai:arXiv.org:2512.13829v2 + math.PR + math.DS + math.FA + math.GR + Tue, 23 Dec 2025 00:00:00 -0500 replace http://creativecommons.org/licenses/by/4.0/ - Stefan Teufel, Tom Wessel + Nicolas Monod - Effective Gaps between singular values of non-stationary matrix products subject to non-degenerate noise - https://arxiv.org/abs/2507.04058 - arXiv:2507.04058v2 Announce Type: replace -Abstract: We study the singular values and Lyapunov exponents of non-stationary random matrix products subject to small, absolutely continuous, additive noise. Consider a fixed sequence of matrices of bounded norm. Independently perturb the matrices by additive noise distributed according to Lebesgue measure on matrices with norm less than $\epsilon$. Then the gaps between the logarithms of the singular values of the random product of $n$ of these matrices are all of order at least $\epsilon^2n$, both in expectation; and almost surely for large $n$. - To prove this, we develop recent work of Gorodetski and Kleptsyn \cite{gorodetski2023nonstationary}. That paper gives a very flexible method, based on relative entropy, for showing that a non-stationary product of matrices in SL(d,R) has a strictly positive Lyapunov exponent. We extend their work in two ways, firstly by making the estimates quantitative in the context of absolutely continuous distributions, giving the universal estimates described above; and secondly by developing a fibered version of their methods, working on flag bundles instead of the projective space to estimate gaps between arbitrary consecutive exponents. Our methods retain much of the flexibility of those of Gorodetski and Kleptsyn, and we hope that they will find application in other related problems. - oai:arXiv.org:2507.04058v2 - math.PR - math.DS - Mon, 22 Dec 2025 00:00:00 -0500 + On the topological Brauer group of generalized Kummer varieties + https://arxiv.org/abs/2512.14262 + arXiv:2512.14262v2 Announce Type: replace +Abstract: We study the topological Brauer group of generalized Kummer varieties. We prove that it vanishes when their dimension is divisible by 4, while for all other dimensions except dimension 10 we prove that it is at most 8-torsion. + oai:arXiv.org:2512.14262v2 + math.AG + Tue, 23 Dec 2025 00:00:00 -0500 replace http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Sam Bednarski, Jonathan DeWitt, Anthony Quas + Moritz Hartlieb, Matteo Verni - Binomial Transforms and the Binomial Convolution of Sequences - https://arxiv.org/abs/2507.04179 - arXiv:2507.04179v5 Announce Type: replace -Abstract: Given any two sequences of complex numbers, we establish simple relations between their binomial convolution and the binomial convolution of their individual binomial transforms. We employ these relations to derive new identities involving Fibonacci numbers, Bernoulli numbers, Catalan numbers, harmonic numbers, odd harmonic numbers, Stirling numbers of the second kind, and binomial coefficients. In addition, we present several results which allow the construction of new binomial-transform pairs from existing ones. Many new relations concerning self-inverse sequences are also derived. - oai:arXiv.org:2507.04179v5 - math.CO - Mon, 22 Dec 2025 00:00:00 -0500 + Sums of four fourth power of primes + https://arxiv.org/abs/2512.14386 + arXiv:2512.14386v3 Announce Type: replace +Abstract: For any sufficiently large positive integer $\ell$, suppose that $\ell$ can be expressed as $ \ell=p_1^4+p_2^4+p_3^4+p_4^4$, where $p_1, p_2,p_3,p_4$ are primes. For such $\ell$, in this paper we will use circle method and sieves to prove that the proportion of $\ell$ in positive integers is at least $\frac{1}{27241.64}$ . + oai:arXiv.org:2512.14386v3 + math.NT + Tue, 23 Dec 2025 00:00:00 -0500 replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Kunle Adegoke + http://creativecommons.org/licenses/by/4.0/ + Yang Qu, Rong Ma - A new upper bound on the specific free energy of dilute Bose gases - https://arxiv.org/abs/2507.20877 - arXiv:2507.20877v2 Announce Type: replace -Abstract: We prove an upper bound for the free energy (per unit volume) of the dilute Bose gas in the thermodynamic limit, showing that the free energy at density $\rho$ and inverse temperature $\beta$ differs from that of the non-interacting system by the correction term $4 \pi \frak{a} (2 \rho^2 - [\rho - \rho_{\textsf{c}}(\beta)]^2_+ )$. Here, $\frak{a}$ denotes the scattering length of the interaction potential, $\rho_{\textsf{c}}(\beta)$ the critical density for Bose-Einstein condensation of the non-interacting gas and $[\cdot]_+=\max\{0,\cdot\}$. This result was previously established by Yin in [37]. Our proof applies to a broader class of interaction potentials, yields a better rate, and we believe it has potential for further extensions. - oai:arXiv.org:2507.20877v2 - math-ph - math.MP - Mon, 22 Dec 2025 00:00:00 -0500 + Hierarchical structure of graded Betti numbers in the quadratic strand + https://arxiv.org/abs/2512.14454 + arXiv:2512.14454v2 Announce Type: replace +Abstract: The classical results, initiated by Castelnuovo and Fano and later refined by Eisenbud and Harris, provide several upper bounds on the number of quadrics defining a nondegenerate projective variety. Recently, it has been revealed that these bounds extend naturally to certain linear syzygies, suggesting the presence of a hierarchical structure governing the quadratic strand of graded Betti numbers. + In this article, we establish such a hierarchy in full generality. We first prove sharp upper bounds for $\beta_{p,1}(X)$ depending on the degree of a projective variety $X$, extending the classical quadratic bounds to all linear syzygies and identifying the extremal varieties in each range. We then introduce geometric conditions that describe how containment of $X$ in low-degree varieties influences syzygies, and we show that these conditions stratify the quadratic strand into a finite sequence of hierarchies. This leads to a complete description of all possible extremal behavior. We also prove a generalized $K_{p,1}$-theorem, demonstrating that the vanishing of $\beta_{p,1}(X)$ detects containment in a variety of minimal degree at each hierarchy. + oai:arXiv.org:2512.14454v2 + math.AG + math.AC + Tue, 23 Dec 2025 00:00:00 -0500 replace http://creativecommons.org/licenses/by-nc-nd/4.0/ - Giulia Basti, Chiara Boccato, Serena Cenatiempo, Andreas Deuchert + Jong In Han, Sijong Kwak, Wanseok Lee - Iwasawa Theory of Elliptic Curves in Quadratic Twist Families - https://arxiv.org/abs/2507.21339 - arXiv:2507.21339v4 Announce Type: replace -Abstract: In this article, we use two different approaches -- one algebraic and the other analytic -- to study the variation of Iwasawa invariants of rational elliptic curves in some quadratic twist families. The analytic approach involves a thorough investigation of half-integral weight modular forms. On the other hand, the algebraic proof requires studying the BDP-Selmer groups and the fine Selmer groups. - oai:arXiv.org:2507.21339v4 - math.NT - Mon, 22 Dec 2025 00:00:00 -0500 + Sharp convergence rates for Spectral methods via the feature space decomposition method + https://arxiv.org/abs/2512.14473 + arXiv:2512.14473v2 Announce Type: replace +Abstract: In this paper, we apply the Feature Space Decomposition (FSD) method developed in [LS24, GLS25, ALSS26] to obtain, under fairly general conditions, matching upper and lower bounds for the population excess risk of spectral methods in linear regression under the squared loss, for every covariance and every signal. This result enables us, for a given linear regression problem, to define a partial order on the set of spectral methods according to their convergence rates, thereby characterizing which spectral algorithm is superior for that specific problem. Furthermore, this allows us to generalize the saturation effect proposed in inverse problems and to provide necessary and sufficient conditions for its occurrence. Our method also shows that, under broad conditions, any spectral algorithm lacks a feature learning property, and therefore cannot overcome the barrier of the information exponent in problems such as single-index learning. + oai:arXiv.org:2512.14473v2 + math.ST + stat.TH + Tue, 23 Dec 2025 00:00:00 -0500 replace - http://creativecommons.org/licenses/by/4.0/ - Debanjana Kundu, Katharina M\"uller + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Guillaume Lecu\'e, Zhifan Li, Zong Shang - Irreducibility of polarized automorphic Galois representations in infinitely many dimensions - https://arxiv.org/abs/2507.22631 - arXiv:2507.22631v2 Announce Type: replace -Abstract: Let $\pi$ be a polarized, regular algebraic, cuspidal automorphic representation of $\operatorname{GL}_n(\mathbb{A}_F)$ where $F$ is totally real or imaginary CM, and let $(\rho_\lambda)_\lambda$ be its associated compatible system of Galois representations. Suppose that $7\nmid n$ and, if $4\mid n$, then $n = 4p$ for some prime number $p$. We prove that there is a Dirichlet density $1$ set of rational primes $\mathcal{L}$ such that whenever $\lambda\mid \ell$ for some $\ell\in \mathcal{L}$, then $\rho_\lambda$ is irreducible. - oai:arXiv.org:2507.22631v2 - math.NT + Staircase Minimality and a Proof of Saxl's Conjecture + https://arxiv.org/abs/2512.15035 + arXiv:2512.15035v2 Announce Type: replace +Abstract: Saxl's conjecture (2012) asserts that for the staircase partition $\rho_k = (k, k-1, \ldots, 1)$, the tensor square of the corresponding irreducible representation of the symmetric group $S_{T_k}$ contains every irreducible representation as a constituent, where $T_k = k(k+1)/2$ is the $k$th triangular number. We prove this conjecture unconditionally. + Our proof introduces the Staircase Minimality Theorem: among all 2-regular partitions of $T_k$, the staircase $\rho_k$ is the unique dominance-minimal element. Combined with Ikenmeyer's theorem on dominance and Kronecker positivity for staircases, this establishes that every 2-regular partition appears in the tensor square. Modular saturation then follows using only the diagonal entries $d_{\mu\mu} = 1$ of the decomposition matrix, and the Bessenrodt--Bowman--Sutton lifting theorem completes the proof. + We further prove that at triangular numbers, staircases are the only Kronecker-universal self-conjugate partitions, providing a complete characterization. + oai:arXiv.org:2512.15035v2 math.RT - Mon, 22 Dec 2025 00:00:00 -0500 + math.CO + Tue, 23 Dec 2025 00:00:00 -0500 replace - http://creativecommons.org/licenses/by/4.0/ - Zachary Feng, Dmitri Whitmore + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Soong Kyum Lee - New conjectures on the inertia of graphs - https://arxiv.org/abs/2508.01163 - arXiv:2508.01163v2 Announce Type: replace -Abstract: Let $G$ be a graph with adjacency matrix $A(G)$. We conjecture that \[2n^+(G) \le n^-(G)(n^-(G) + 1),\] where $n^+(G)$ and $n^-(G)$ denote the number of positive and negative eigenvalues of $A(G)$, respectively. This conjecture generalizes to all graphs the well-known absolute bound for strongly regular graphs. The conjecture also relates to a question posed by Torga\v{s}ev. We prove the conjecture for special graph families, including line graphs and planar graphs, and provide examples where the conjecture is exact. We also conjecture that for any connected graph $G$, its line graph $L(G)$ satisfies $n^+(L(G)) \le n^-(L(G)) + 1$, and obtain partial results. - oai:arXiv.org:2508.01163v2 - math.CO - Mon, 22 Dec 2025 00:00:00 -0500 + No exact on average additive complements of squares + https://arxiv.org/abs/2512.15407 + arXiv:2512.15407v2 Announce Type: replace +Abstract: Let $\mathbb{N}$ be the set of natural numbers and $\mathcal{S}=\big\{1^2, 2^2, 3^2,\cdots\big\}$ the set of squares. Let $\mathcal{W}$ be an additive complement of $\mathcal{S}$ and $$ f(n)=\#\big\{(w,m^2)\in \mathcal{W}\times \mathcal{S}: n=w+m^2\big\}. $$ It is proved that there is a positive constant $c_{\mathcal{W}}$ (depending at most on $\mathcal{W}$) such that $$ \sum_{n\le N}f(n)-N\ge c_{\mathcal{W}}N $$ for infinitely many positive integers $N$, which makes some progress on a 1993 conjecture of Cilleruelo. As consequences of this result, we answer negatively a 2001 problem of Ruzsa as well as a 2017 problem of Ben Green. + oai:arXiv.org:2512.15407v2 + math.NT + Tue, 23 Dec 2025 00:00:00 -0500 replace http://creativecommons.org/licenses/by/4.0/ - Saieed Akbari, Clive Elphick, Hitesh Kumar, Shivaramakrishna Pragada, Quanyu Tang - - - Asymptotic minimality of one-dimensional transition profiles in Aviles-Giga type models: an approach via 1-currents - https://arxiv.org/abs/2508.13753 - arXiv:2508.13753v2 Announce Type: replace -Abstract: For vector fields on a two-dimensional domain, we study the asymptotic behaviour of Modica-Mortola (or Allen-Cahn) type functionals under the assumption that the divergence converges to $0$ at a certain rate, which effectively produces a model of Aviles-Giga type. This problem will typically give rise to transition layers, which degenerate into discontinuities in the limit. We analyse the energy concentration at these discontinuities and the corresponding transition profiles. - We derive an estimate for the energy concentration in terms of a novel geometric variational problem involving the notion of $\mathbb{R}^2$-valued $1$-currents from geometric measure theory. This in turn leads to criteria, under which the energetically favourable transition profiles are essentially one-dimensional. - oai:arXiv.org:2508.13753v2 - math.AP - Mon, 22 Dec 2025 00:00:00 -0500 + Yuchen Ding + + + Matchings avoiding ordered patterns + https://arxiv.org/abs/2512.15461 + arXiv:2512.15461v2 Announce Type: replace +Abstract: A {\it vertex-ordered} graph is a graph equipped with a linear ordering of its vertices. + A pair of independent edges in an ordered graph can exhibit one of the following three patterns: separated, nested or crossing. + We say a pair of independent edges is non-separated if it is either crossing or nested. + Non-nested and non-crossing pairs are defined analogously. + We are interested in the following Tur\'an-type problems: for each of the aforementioned six patterns, determine the maximum number of edges of an $n$-vertex ordered graph that does not contain a $k$-matching such that every pair of edges exhibit the fixed pattern. + Exact answers have already been obtained for four of the six cases. + The main objective of this paper is to investigate the two remaining open cases, namely non-separated and non-nested matchings. + We determine the exact maximum number of edges of an $n$-vertex ordered graph that does not contain a non-separated $k$-matching, which has the form $\frac{3}{2}(k-1)n+\Theta(k^2)$. + For the non-nested case, we show the maximum number of edges lies between $(k-1)n$ and $(k-1)n+\binom{k-1}{2}$. + We also determine the exact maximum number of edges of an $n$-vertex ordered graph that does not contain an alternating path of given length. + We discuss some related problems and raise several + conjectures. + Furthermore, our results and conjectures yield consequences to certain Ramsey-type problems for non-nested matchings and alternating paths. + oai:arXiv.org:2512.15461v2 + math.CO + Tue, 23 Dec 2025 00:00:00 -0500 replace http://creativecommons.org/licenses/by/4.0/ - Radu Ignat, Roger Moser + J\'anos Bar\'at, Andrea Freschi, G\'eza T\'oth - On a relation between Deninger's foliated dynamical systems and Connes-Consani's adelic spaces - https://arxiv.org/abs/2508.15971 - arXiv:2508.15971v4 Announce Type: replace -Abstract: We give a relation between Deninger's foliated dynamical systems associated to abelian number fields and Connes-Consani's adelic spaces. It fits with the analogy between knots and primes in arithmetic topology and lights up a geometric view of class field theory. - oai:arXiv.org:2508.15971v4 - math.NT - Mon, 22 Dec 2025 00:00:00 -0500 + Existence of a Non-Uniquely Ergodic Interval Exchange Transformation with Flips Possessing Three Invariant Measures + https://arxiv.org/abs/2512.15625 + arXiv:2512.15625v2 Announce Type: replace +Abstract: We present the first explicit example of an interval exchange transformation with flips (FIET) possessing three distinct invariant ergodic measures. The proof is based on a generalization of M. Keane's method, using the Rauzy induction adapted for FIETs, which contributes to the study of the ergodic properties of this class of dynamical systems. + oai:arXiv.org:2512.15625v2 + math.DS + Tue, 23 Dec 2025 00:00:00 -0500 replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Masanori Morishita + http://creativecommons.org/licenses/by-nc-nd/4.0/ + Aleksei Kobzev - Ehrhart non-positivity and unimodular triangulations for classes of s-lecture hall simplices - https://arxiv.org/abs/2508.18890 - arXiv:2508.18890v3 Announce Type: replace -Abstract: Counting lattice points and triangulating polytopes is a prominent subject in discrete geometry, yet proving Ehrhart positivity or existence of unimodular triangulations remain of utmost difficulty in general, even for ``easy'' simplices. We study these questions for classes of s-lecture hall simplices. Inspired by a question of Olsen, we present a new natural class of sequences s for which the s-lecture hall simplices are not Ehrhart positive, by explicitly estimating a negative coefficient. Meanwhile, motivated by a conjecture of Hibi, Olsen and Tsuchiya, we extend the previously known classes of sequences s for which the s-lecture hall simplex admits a flag, regular and unimodular triangulation. The triangulations we construct are explicit. - oai:arXiv.org:2508.18890v3 - math.CO - Mon, 22 Dec 2025 00:00:00 -0500 + Deep Reinforcement Learning Optimization for Uncertain Nonlinear Systems via Event-Triggered Robust Adaptive Dynamic Programming + https://arxiv.org/abs/2512.15735 + arXiv:2512.15735v2 Announce Type: replace +Abstract: This work proposes a unified control architecture that couples a Reinforcement Learning (RL)-driven controller with a disturbance-rejection Extended State Observer (ESO), complemented by an Event-Triggered Mechanism (ETM) to limit unnecessary computations. The ESO is utilized to estimate the system states and the lumped disturbance in real time, forming the foundation for effective disturbance compensation. To obtain near-optimal behavior without an accurate system description, a value-iteration-based Adaptive Dynamic Programming (ADP) method is adopted for policy approximation. The inclusion of the ETM ensures that parameter updates of the learning module are executed only when the state deviation surpasses a predefined bound, thereby preventing excessive learning activity and substantially reducing computational load. A Lyapunov-oriented analysis is used to characterize the stability properties of the resulting closed-loop system. Numerical experiments further confirm that the developed approach maintains strong control performance and disturbance tolerance, while achieving a significant reduction in sampling and processing effort compared with standard time-triggered ADP schemes. + oai:arXiv.org:2512.15735v2 + math.OC + cs.AI + cs.SY + eess.SY + Tue, 23 Dec 2025 00:00:00 -0500 replace http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Jhon B. Caicedo, Martina Juhnke, Germain Poullot - - - Mathematical Analysis 1 (Chapters in Univariate Real Analysis) - https://arxiv.org/abs/2508.19405 - arXiv:2508.19405v4 Announce Type: replace -Abstract: Preliminary version of a course in univariate real analysis, with 14 chapters and 1 appendix (Chapters 1-8 complete at present). 1. Infinite sums. Real numbers; 2. Limits of sequences and subsequences; 3. Arithmetic of limits. AK series; 4. Infinite series. Elementary functions; 5. Limits of functions. Asymptotic notation; 6. Continuous functions; 7. Derivatives; 8. Applications of mean value theorems; 9. Taylor polynomials and series. Real analytic functions; 10. Primitives of uniformly continuous functions; 11. Newton integral. Primitives of rational functions; 12. Riemann integral. Transcendence of the number e; 13. Riemann integral. Henstock--Kurzweil integral; 14. More applications of Riemann integral and A. Solutions to exercises. - oai:arXiv.org:2508.19405v4 - math.HO - Mon, 22 Dec 2025 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Martin Klazar + Ningwei Bai, Chi Pui Chan, Qichen Yin, Tengyang Gong, Yunda Yan, Zezhi Tang - Simplicial Approach to Frobenius Algebras in the Category of Relations - https://arxiv.org/abs/2509.06193 - arXiv:2509.06193v2 Announce Type: replace -Abstract: Frobenius algebras in the category of sets and relations ($\mathbf{Rel}$) serve as a unifying framework for various algebraic and combinatorial structures, including groupoids, effect algebras, and abstract circles. Recently, a nerve construction of simplicial sets for Frobenius algebras in $\mathbf{Rel}$ has been introduced. In this work, we investigate the lifting properties of these simplicial sets, linking them to the algebraic properties of Frobenius algebras. We introduce $\epsilon$-simplicial sets -- simplicial sets with marked edges -- that enable the representation of a broader class of structures, such as test spaces from quantum logic. Our main results focus on weakly saturated classes generated by cofibrations, corresponding to specific lifting problems. Furthermore, we provide a characterization of Frobenius algebras in $\mathbf{Rel}$ within the framework of $\epsilon$-simplicial sets. These findings lay the groundwork for the development of a convenient model structure in future research. - oai:arXiv.org:2509.06193v2 - math.CT - Mon, 22 Dec 2025 00:00:00 -0500 + Global weak solutions of 3D compressible magnetohydrodynamic equations subject to large external potential forces with discontinuous initial data and vacuum + https://arxiv.org/abs/2512.16121 + arXiv:2512.16121v2 Announce Type: replace +Abstract: We investigate the compressible magnetohydrodynamic equations subject to large external potential forces with discontinuous initial data in a three-dimensional bounded domain under Navier-slip boundary conditions. We show the global existence of weak solutions for such an initial-boundary value problem provided the initial energy is suitably small. In particular, the initial data may contain vacuum states and possibly exhibit large oscillations. To overcome difficulties brought by boundary and large external forces, some new estimates based on the effective viscous flux play crucial roles. + oai:arXiv.org:2512.16121v2 + math.AP + Tue, 23 Dec 2025 00:00:00 -0500 replace - http://creativecommons.org/publicdomain/zero/1.0/ - Dominik Lachman + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Geyuan Chen, Xin Zhong - The 3D index and Dehn filling - https://arxiv.org/abs/2509.09886 - arXiv:2509.09886v2 Announce Type: replace -Abstract: We provide a rigorous proof of the Gang-Yonekura formula describing the transformation of the 3D index under Dehn filling a cusp in an orientable 3-manifold. The 3D index, originally introduced by Dimofte, Gaiotto and Gukov, is a physically inspired q-series that encodes deep topological and geometric information about cusped 3-manifolds. Building on the interpretation of the 3D index as a generating function over Q-normal surfaces, we introduce a relative version of the index for ideal triangulations with exposed boundary. This notion allows us to formulate a relative Gang-Yonekura formula, which we prove by developing a gluing principle for relative indices and establishing an inductive framework in the case of layered solid tori. Our approach makes use of Garoufalidis-Kashaev's meromorphic extension of the index, along with new identities involving q-hypergeometric functions. As an application, we study the limiting behaviour of the index for large fillings. We also develop code to perform certified computations of the index, guaranteeing correctness up to a specified accuracy. Our extensive computations support the topological invariance of the 3D index and suggest a well-defined extension to closed manifolds. - oai:arXiv.org:2509.09886v2 - math.GT - hep-th + Polynomial densities and Heilbronn's criterion + https://arxiv.org/abs/2512.16220 + arXiv:2512.16220v2 Announce Type: replace +Abstract: Heilbronn gave a sufficient condition for a number field with a totally ramified prime to fail to be norm-Euclidean. We say that Heilbronn's criterion applies to a polynomial $f$ if it applies to the number field $K=\mathbb{Q}[x]/(f)$ generated by $f$. + Suppose $n\geq 3$ is odd and $p\geq 5$ is prime with $\gcd(p-1,n)=1$. Let ${F}_{p,n}$ denote the collection of monic polynomials $f\in\mathbb{Z}[x]$ of degree $n$ that are Eisenstein at the prime $p$. We order our polynomials by the natural height $\mathrm{Ht}(f)$. Define $\delta_{p,n}(X)$ to be the proportion of polynomials $f\in {F}_{p,n}$ with $\mathrm{Ht}(f)\leq X$ for which Heilbronn's criterion applies. One has $$\liminf_{X\to\infty}\delta_{p,n}(X)\geq \max\left\{\frac{2}{27}\,,\;1-\varepsilon(p)\right\}\,,$$ where $\varepsilon(p)\to 0$ and is effectively computable. In particular, the lower density tends to $1$ as $p\to\infty$ uniformly in $n$. We also give a version of this result where we weaken the condition on $\gcd(p-1,n)$. + As a corollary, we show that given an integer $n\geq 2$, a positive proportion of Eisenstein polynomials of degree $n$ fail to generate norm-Euclidean fields. + oai:arXiv.org:2512.16220v2 math.NT - Mon, 22 Dec 2025 00:00:00 -0500 + Tue, 23 Dec 2025 00:00:00 -0500 replace http://creativecommons.org/licenses/by/4.0/ - Daniele Celoria, Craig D. Hodgson, J. Hyam Rubinstein + Alexis Hibbler, Kevin J. McGown, Enrique Trevi\~no - Gromov hyperbolicity III: an improved geometric characterization and its applications - https://arxiv.org/abs/2509.10403 - arXiv:2509.10403v3 Announce Type: replace -Abstract: In the seminal work of Balogh-Buckley [Invent. Math. 2003], the authors asked the following fundamental open problem: for proper subdomains in the Euclidean space $\mathbb{R}^n$, does the ball separation condition alone imply the Gehring-Hayman inequality? - In this paper, via a completely new measure-independent approach, we establish the following geometric characterization of Gromov hyperbolicity in a fairly general setting: The Gromov hyperbolicity of a proper subdomain in a doubling metric space is quantitatively equivalent to the geometric ball separation condition, with explicit dependence on the coefficients. In the special case of Euclidean spaces, it affirmatively solves the above Balogh-Buckely problem. Our result also significantly improves the main result of Koskela-Lammi-Manojlovi\'c [Ann. Sci. \'Ec. Norm. Sup\'er. 2014]. As applications, we obtain the quasiconformal invariance of ball separation condition, a geometric characterization of inner uniformity in terms of ball separation condition, and the Gromov hyperbolicity of quasihyperbolic John length spaces. - oai:arXiv.org:2509.10403v3 + Higher-Order Volterra-Type Integral Operators on Hardy Spaces + https://arxiv.org/abs/2512.16412 + arXiv:2512.16412v2 Announce Type: replace +Abstract: We study the $n$-fold iterated Volterra-type integral operator $T_{g,n}$ acting on Hardy spaces, defined by \[ T_{g,n}[f](z) :=\underbrace{\int_0^z\int_0^{t_1}\cdots\int_0^{t_{n-1}}}_{n\ \text{times}} f(t_n)g'(t_n)\,dt_n\cdots dt_1, \quad z\in\mathbb{D}, \] where $f$ and $g$ are analytic functions in the unit disk $\mathbb{D}$. We show that the boundedness and compactness of $T_{g,n}$ are independent of the order $n$ and depend solely on the symbol $g$. In particular, we obtain sharp norm estimates and characterize boundedness and compactness of $T_{g,n}$ in terms of classical function space conditions on $g$. + oai:arXiv.org:2512.16412v2 math.CV - math.MG - Mon, 22 Dec 2025 00:00:00 -0500 + Tue, 23 Dec 2025 00:00:00 -0500 replace - http://creativecommons.org/licenses/by-nc-sa/4.0/ - Chang-Yu Guo, Manzi Huang, Xiantao Wang + http://creativecommons.org/licenses/by/4.0/ + Rahim Kargar - Nef cones of Hilbert schemes of points on some K3 surfaces - https://arxiv.org/abs/2509.10724 - arXiv:2509.10724v2 Announce Type: replace -Abstract: We illustrate the typical usage of Bayer and Macr\`{i}'s Positivity Lemma to compute the nef cones of the Hilbert schemes $X^{[n]}$ by combining the Bridgeland stability methods (for large $n$) and classical methods (for small $n$). We use Mori dream K3 surfaces $X$ of Picard rank $2$ as our working example. We also compute the nef cones of the nested Hilbert schemes $X^{[n,n+1]}$ for such $X$, for large $n$. - oai:arXiv.org:2509.10724v2 - math.AG - Mon, 22 Dec 2025 00:00:00 -0500 + On the distribution kernels of Toeplitz operators on CR manifolds + https://arxiv.org/abs/2512.16506 + arXiv:2512.16506v2 Announce Type: replace +Abstract: We study the distribution kernel of a Toeplitz operator associated with a classical pseudodifferential operator on a compact, embeddable, strictly pseudoconvex CR manifold. The main result consists of a formula for the values at the diagonal of the second coefficient in the expansion of the symbol of the kernel. We also establish asymptotic expansions for Toeplitz operators on the positive part of a compact not necessary strictly pseudoconvex CR orbifold under certain natural assumptions. + oai:arXiv.org:2512.16506v2 + math.CV + math.AP + math.DG + Tue, 23 Dec 2025 00:00:00 -0500 replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Uttaran Dutta, Sean Edwards, Neelarnab Raha + http://creativecommons.org/licenses/by/4.0/ + Chin-Yu Hsiao, Ood Shabtai - Shanks' bias in function fields - https://arxiv.org/abs/2509.16142 - arXiv:2509.16142v2 Announce Type: replace -Abstract: We study the function field analogue of Shanks bias. For Liouville function $\lambda(f)$, we compare the number of monic polynomials $f$ with $\lambda(f) \chi_m(f) = 1$ and $\lambda(f) \chi_m(f) = -1$ for a nontrivial quadratic character $\chi_m$ modulo a monic square-free polynomial $m$ over a finite field. Under Grand Simplicity Hypothesis (GSH) for $L$-functions, we prove that $\lambda \cdot \chi_m$ is biased towards $+1$. We also give some examples where GSH is violated. - oai:arXiv.org:2509.16142v2 - math.NT - Mon, 22 Dec 2025 00:00:00 -0500 + Regularity for fully nonlinear elliptic equations in generalized Orlicz spaces + https://arxiv.org/abs/2512.16600 + arXiv:2512.16600v2 Announce Type: replace +Abstract: In this paper, we establish an optimal global Calder\'{o}n-Zygmund type estimate for the viscosity solution to the Dirichlet boundary problem of fully nonlinear elliptic equations with possibly nonconvex nonlinearities. We prove that the Hessian of the solution is as integrable as the nonhomogeneous term in the setting of a given generalized Orlicz space even when the nonlinearity is asymptotically convex with respect to the Hessian of the solution. + oai:arXiv.org:2512.16600v2 + math.AP + Tue, 23 Dec 2025 00:00:00 -0500 replace http://creativecommons.org/licenses/by/4.0/ - Seewoo Lee + Sun-Sig Byun, Jeongmin Han, Mikyoung Lee - Ping-pong for basis-conjugating HNN-extension of free group - https://arxiv.org/abs/2509.19635 - arXiv:2509.19635v2 Announce Type: replace -Abstract: We isolate a tractable class of HNN-extensions of a free group, namely, multiple HNN-extensions by basis-conjugating embeddings. For this class, we construct a normal form and establish a practical version of the ping-pong lemma that provides verifiable sufficient conditions for a set of elements to generate a free subgroup. - We then apply these results to the pure braid group $P_{n+1}$, exploiting its well-known decomposition as a semidirect product of free groups. Our approach yields new families of free subgroups within the first two factors $F_n \rtimes F_{n-1}$ of this decomposition. - oai:arXiv.org:2509.19635v2 + A note on the triple product property for finite groups with abelian normal subgroups of prime index + https://arxiv.org/abs/2512.16730 + arXiv:2512.16730v3 Announce Type: replace +Abstract: Three non-empty subsets $S,T,U$ of a group $G$ are said to satisfy the triple product property (TPP) if, for elements $s,s' \in S$, $t,t' \in T$, $u,u' \in U$, the equation $s's^{-1}t't^{-1}u'u^{-1}=1$ holds if and only if $s = s'$, $t = t'$, $u = u'$. In this case $(S,T,U)$ is called a TPP triple of $G$ and $|S||T||U|$ is called the size of the triple. If $G$ is a finite group the triple product ratio of $G$ can be defined as the quantity $\rho(G) := \frac{\beta(G)}{|G|}$, where $\beta(G)$ is the largest size of a TPP triple of $G$, and a special case of this, the subgroup triple product ratio, is the quantity $\rho_0(G) := \frac{\beta_0(G)}{|G|}$, where $\beta_0(G)$ is the largest size of a TPP triple of $G$ composed only of subgroups. There is a conjecture that $\rho(G) \leq \frac{4}{3}$ if $G$ contains a cyclic subgroup of index $2$ \citep[Conjecture 7.6]{HM}. This note proves a more general version of this conjecture for subgroups by showing that $\rho_0(G) \leq \frac{p^2}{2p-1}$ if $G$ is any finite group which contains an abelian normal subgroup of prime index $p$, and discusses its implications for $\rho$ for groups with cyclic normal subgroups of prime index, based on the known data for $\rho$ in such groups of small order. + oai:arXiv.org:2512.16730v3 math.GR - Mon, 22 Dec 2025 00:00:00 -0500 + Tue, 23 Dec 2025 00:00:00 -0500 replace - http://creativecommons.org/publicdomain/zero/1.0/ - Vasily Ionin + http://creativecommons.org/licenses/by/4.0/ + Sandeep R. Murthy - Multi-Order Runge-Kutta Methods or how to numerically solve initial value problems of any order - https://arxiv.org/abs/2509.23513 - arXiv:2509.23513v3 Announce Type: replace -Abstract: When one wishes to numerically solve an initial value problem, it is customary to rewrite it as an equivalent first-order system to which a method, usually from the class of Runge-Kutta methods, is applied. Directly treating higher-order initial value problems without such rewriting, however, allows for significantly greater accuracy. We therefore introduce a new generalization of Runge-Kutta methods, called multi-order Runge-Kutta methods, designed to solve initial value problems of arbitrary order. We establish fundamental properties of these methods, including convergence, order of consistency, and linear stability. We also analyze the structure of the system satisfied by the approximations of a method, which enables us to provide a proper definition of explicit methods and to gain a finer understanding of implicit methods. - oai:arXiv.org:2509.23513v3 - math.NA - cs.NA - Mon, 22 Dec 2025 00:00:00 -0500 + Absence of twisting for non-trivial discrete torsion + https://arxiv.org/abs/2512.17068 + arXiv:2512.17068v2 Announce Type: replace +Abstract: We study discrete torsion for the $n$--torus with finite symmetry group $G$ from the Dijkgraaf--Witten viewpoint. A class in $H^n(G,U(1))$ assigns a phase to each flat $G$--bundle, equivalently to each commuting $n$--tuple in $G$ up to conjugation. We introduce the subgroup $\Br^n(G)\subseteq H^n(G,U(1))$ of \emph{untwisted} classes, those whose Dijkgraaf--Witten phases are trivial on all commuting tuples, and derive a universal coefficient exact sequence involving this invariant. In degree $2$ this recovers the Bogomolov multiplier / unramified Brauer group. We implement algorithms for computing $\Br^n(G)$ and corresponding torus partition functions, and report on computations for families of finite subgroups of $\SU(4)$. + oai:arXiv.org:2512.17068v2 + math.GR + math-ph + math.MP + Tue, 23 Dec 2025 00:00:00 -0500 replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Loris Petronijevic + http://creativecommons.org/publicdomain/zero/1.0/ + Primoz Moravec - Capacitary Muckenhoupt Weights and Weighted Norm Inequalities for Hardy-Littlewood Maximal Operators - https://arxiv.org/abs/2509.23839 - arXiv:2509.23839v2 Announce Type: replace -Abstract: Let $\mathcal H_{\infty}^\delta$ denote the Hausdorff content of dimension $\delta\in(0,n]$ defined on subsets of $\mathbb R^n$. The principal problem, considered in this paper, is to characterize the non-negative function $w$ for which the weighted $L^p$-norm inequality with $p\in(1,\infty)$ and the weighted weak $L^1$-norm inequality on Hardy-Littlewood maximal operators associated with Hausdorff contents hold true. To achieve this, we introduce a class of capacitary Muckenhoupt weights depending on the dimension $\delta$, denoted as $\mathcal A_{p,\delta}$, which enjoys the strict monotonicity on the dimension index $\delta$. Then we show that, for any $p\in(1,\infty)$ and $\delta\in(0,n]$, the weighted $L^p$-norm inequality holds true if and only if $w\in\mathcal A_{p,\delta}$, and the weighted weak $L^1$-norm inequality holds true if and only if $w\in\mathcal A_{1,\delta}$ by a new approach developed in this paper. As the second objective, applying this new approach, the seminal properties of classical Muckenhoupt $A_p$ weights, such as the reverse H\"older inequality [R. R. Coifman and C. Fefferman, Studia Math. 51 (1974), 241-250], the self-improving property [B. Muckenhoupt, Trans. Amer. Math. Soc. 165 (1972), 207-226], and Jones' factorization theorem [P. W. Jones, Ann. of Math. (2) 111 (1980), 511-530], are all established within the framework of capacitary Muckenhoupt weight class $\mathcal A_{p,\delta}$. Finally, we also show that the maximal operator is bounded on the weak weighted Choquet-Lebesgue space $L_w^{p,\infty}(\mathbb R^n,{\mathcal H}_\infty^\delta)$ if and only if $w\in\mathcal A_{p,\delta}$ with $p\in(1,\infty)$ and $\delta\in(0,n]$. - oai:arXiv.org:2509.23839v2 - math.CA - math.FA - Mon, 22 Dec 2025 00:00:00 -0500 + Verifying Hadwiger's Conjecture for Examples of Graphs with $\alpha(G) = 2$ + https://arxiv.org/abs/2512.17114 + arXiv:2512.17114v2 Announce Type: replace +Abstract: Hadwiger's Conjecture states that every graph with chromatic number $k$ contains a complete graph on $k$ vertices as a minor. This conjecture is a tremendous strengthening of the Four-Colour Theorem and is regarded as one of the most important open problems in graph theory. The case of Hadwiger's Conjecture for graphs with $\alpha(G) = 2$ has garnered much attention. Seymour writes: ``My own belief is, if Hadwiger's Conjecture is true for graphs with stability number two then it is probably true in general, so it would be very nice to decide this case.'' + This paper presents several tools useful for proving that a graph $G$ with $\alpha(G) = 2$ satisfies Hadwiger's Conjecture. In doing so, we survey and generalise several classical results on the $\alpha(G) = 2$ case of Hadwiger's Conjecture. Further, we apply these tools to prove variants of Hadwiger's Conjecture for several noteworthy classes of graphs with $\alpha(G) = 2$. In particular, we prove Hadwiger's Conjecture for inflations of the complements of the following graphs: graphs with girth at least $5$, triangle-free Kneser graphs, and the Clebsch, Mesner, and Gewirtz graphs. This paper also highlights classes of graphs with $\alpha(G) = 2$ where it is unknown if Hadwiger's Conjecture holds. + oai:arXiv.org:2512.17114v2 + math.CO + Tue, 23 Dec 2025 00:00:00 -0500 replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Long Huang, Yangzhi Zhang, Ciqiang Zhuo + http://creativecommons.org/licenses/by/4.0/ + Jofre Costa, Eric Luu, David R. Wood, Jung Hon Yip - Dirichlet moment tensors and the correspondence between admixture and mixture of product models - https://arxiv.org/abs/2509.25441 - arXiv:2509.25441v3 Announce Type: replace -Abstract: Understanding posterior contraction behavior in Bayesian hierarchical models is of fundamental importance, but progress in this question is relatively sparse in comparison to the theory of density estimation. In this paper, we study two classes of hierarchical models for grouped data, where observations within groups are exchangeable. Using moment tensor decomposition of the distribution of the latent variables, we establish a precise equivalence between the class of Admixture models (such as Latent Dirichlet Allocation) and the class of Mixture of products of multinomial distributions. This correspondence enables us to leverage the result from the latter class of models, which are more well-understood, so as to arrive at the identifiability and posterior contraction rates in both classes under conditions much weaker than in existing literature. For instance, our results shed light on cases where the topics are not linearly independent or the number of topics is misspecified in the admixture setting. Finally, we analyze individual documents' latent allocation performance via the borrowing of strength properties of hierarchical Bayesian modeling. Many illustrations and simulations are provided to support the theory. - oai:arXiv.org:2509.25441v3 - math.ST - stat.TH - Mon, 22 Dec 2025 00:00:00 -0500 + Four special Poncelet triangle families about the incircle + https://arxiv.org/abs/2512.17440 + arXiv:2512.17440v2 Announce Type: replace +Abstract: We describe four special families of ellipse-inscribed Poncelet triangles about the incircle which maintain certain triangle centers stationary and which also display interesting conservations. + oai:arXiv.org:2512.17440v2 + math.MG + cs.GR + Tue, 23 Dec 2025 00:00:00 -0500 replace http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Dat Do, Sunrit Chakraborty, Jonathan Terhorst, XuanLong Nguyen + Ronaldo A. Garcia, Mark Helman, Dan Reznik - On almost commuting unitary matrices - https://arxiv.org/abs/2510.03674 - arXiv:2510.03674v2 Announce Type: replace -Abstract: A question going back to Halmos asks when two approximately commuting matrices of a certain kind are close to exactly commuting matrices of the same kind. It has long been known that there is a winding number obstruction for approximately commuting unitary matrices to be close, in a dimension-independent way, to genuinely commuting unitary matrices. In this paper, under the vanishing of the said obstruction, we obtain effective bounds for the distance to commuting unitary matrices in terms of the commutator of the original matrices. - oai:arXiv.org:2510.03674v2 - math.OA - math.FA - math.SP - Mon, 22 Dec 2025 00:00:00 -0500 + Extending Chevalley's Theorem: A Topological Characterization of Constructibility and its Generalization Beyond Noetherian Spaces + https://arxiv.org/abs/2512.17481 + arXiv:2512.17481v2 Announce Type: replace +Abstract: We introduce the notion of a good map between topological spaces: a continuous map $f:X \to Y$ is *good* if for every non-empty irreducible locally closed subset $U \subseteq X$, there exists a non-empty open subset $W \subseteq Y$ such that $W \cap f(U) = W \cap \overline{f(U)} \neq \varnothing$. + In Noetherian spaces, this condition is equivalent to preserving constructible subsets (Theorem 2.5), giving a purely topological characterization of Chevalley's theorem. Without the Noetherian assumption, the good property continues to make sense and serves as a reasonable generalization. + We establish basic properties of good maps and introduce a weaker variant, *weak good maps*. In algebraic geometry, we prove that **every morphism locally of finite type is good** (Theorem 4.1). From this we obtain a generalization of Chevalley's theorem for morphisms locally of finite type whose underlying topological spaces are Noetherian (Theorem 4.3), and an elementary proof of Jacobson ascent (Corollary 4.4). + The theory is developed for sober spaces and therefore applies not only to schemes but also to other geometric categories such as **adic spaces** and **perfectoid spaces**. The good property is stable under formal completion, suggesting extensions to formal and non-Archimedean geometry. + oai:arXiv.org:2512.17481v2 + math.AG + Tue, 23 Dec 2025 00:00:00 -0500 replace http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Adam Dor-On, Lucas Hall, Ilya Kachkovskiy + Jiawei Sheng - Refinement-based Christoffel sampling for least squares approximation in non-orthogonal bases - https://arxiv.org/abs/2510.08461 - arXiv:2510.08461v2 Announce Type: replace -Abstract: We introduce a refinement-based Christoffel sampling (RCS) algorithm for least squares approximation in the span of a given, generally non-orthogonal set of functions $\Phi_n = \{\phi_1, \dots, \phi_n\}$. A standard sampling strategy for this problem is Christoffel sampling, which achieves near-best approximations in probability using only $\mathcal{O}(n \log(n))$ samples. However, it requires i.i.d. sampling from a distribution whose density is proportional to the inverse Christoffel function $k_n$, the computation of which requires an orthonormal basis. As a result, existing approaches for non-orthogonal bases $\Phi_n$ typically rely on costly discrete orthogonalization. We propose a new iterative algorithm, inspired by recent advances in approximate leverage score sampling, that avoids this bottleneck. Crucially, while the computational cost of discrete orthogonalization grows proportionally with $\|k_n\|_{L^\infty(X)}$, the cost of our approach increases only logarithmically in $\|k_n\|_{L^\infty(X)}$. In addition, we account for finite-precision effects by considering a numerical variant of the Christoffel function, ensuring that the algorithm relies only on computable quantities. Alongside a convergence proof, we present extensive numerical experiments demonstrating the efficiency and robustness of the proposed method. - oai:arXiv.org:2510.08461v2 - math.NA - cs.NA - Mon, 22 Dec 2025 00:00:00 -0500 - replace + A Consistent ICM-based $\chi^2$ Specification Test + https://arxiv.org/abs/2208.13370 + arXiv:2208.13370v3 Announce Type: replace-cross +Abstract: In spite of the omnibus property of Integrated Conditional Moment (ICM) specification tests, they are not commonly used in empirical practice owing to features such as the non-pivotality of the test and the high computational cost of available bootstrap schemes, especially in large samples. This paper proposes specification and mean independence tests based on ICM metrics. The proposed test exhibits consistency, asymptotic $\chi^2$-distribution under the null hypothesis, and computational efficiency. Moreover, it demonstrates robustness to heteroskedasticity of unknown form and can be adapted to enhance power towards specific alternatives. A power comparison with classical bootstrap-based ICM tests using Bahadur slopes is also provided. Monte Carlo simulations are conducted to showcase the excellent size control and competitive power of the proposed test. + oai:arXiv.org:2208.13370v3 + econ.EM + math.ST + stat.TH + Tue, 23 Dec 2025 00:00:00 -0500 + replace-cross http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Astrid Herremans, Ben Adcock + Feiyu Jiang, Emmanuel Selorm Tsyawo - Quasi-compactness for dominated kernels with application to quasi-stationary distribution theory - https://arxiv.org/abs/2510.19573 - arXiv:2510.19573v2 Announce Type: replace -Abstract: We establish a domination principle for positive operators, which provides an upper bound on the essential spectral radius and yields quasi-compactness criteria on weighted supremum spaces with Lyapunov type functions and local domination. In particular, for kernels acting on such spaces, we obtain $r_{ess}(P)\leq r_{ess}(Q)$ whenever $0\leq P\leq Q$ as kernels, a property that is known to fail in general on $L^p$ spaces, $p<+\infty$. - We then describe the asymptotics of iterates of positive quasi-compact kernels, showing convergence, after suitable renormalization, towards a finite decomposition over eigenelements, and we study the long-time behaviour of quasi-compact continuous-time semigroups. For the latter, we prove that measurability in time and quasi-compactness at a single positive time imply quasi-compactness at all times, exclude periodic behaviour, and entail convergence to eigenelements as time goes to infinity. - Finally, we apply these results to absorbed Markov processes and quasi-stationary distributions. In this setting, the domination and Lyapunov criteria allow one to work in reducible situations and to relax classical regularity assumptions, for instance replacing strong Feller conditions by domination from a regular kernel or by locally uniformly integrable densities on suitable weighted supremum spaces. - oai:arXiv.org:2510.19573v2 - math.PR - math.SP - Mon, 22 Dec 2025 00:00:00 -0500 - replace + Generalized Data Thinning Using Sufficient Statistics + https://arxiv.org/abs/2303.12931 + arXiv:2303.12931v3 Announce Type: replace-cross +Abstract: Our goal is to develop a general strategy to decompose a random variable $X$ into multiple independent random variables, without sacrificing any information about unknown parameters. A recent paper showed that for some well-known natural exponential families, $X$ can be "thinned" into independent random variables $X^{(1)}, \ldots, X^{(K)}$, such that $X = \sum_{k=1}^K X^{(k)}$. These independent random variables can then be used for various model validation and inference tasks, including in contexts where traditional sample splitting fails. In this paper, we generalize their procedure by relaxing this summation requirement and simply asking that some known function of the independent random variables exactly reconstruct $X$. This generalization of the procedure serves two purposes. First, it greatly expands the families of distributions for which thinning can be performed. Second, it unifies sample splitting and data thinning, which on the surface seem to be very different, as applications of the same principle. This shared principle is sufficiency. We use this insight to perform generalized thinning operations for a diverse set of families. + oai:arXiv.org:2303.12931v3 + stat.ME + math.ST + stat.ML + stat.TH + Tue, 23 Dec 2025 00:00:00 -0500 + replace-cross http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Denis Villemonais + 10.1080/01621459.2024.2353948 + Journal of the American Statistical Association, 120(549), 511-523 (2025) + Ameer Dharamshi, Anna Neufeld, Keshav Motwani, Lucy L. Gao, Daniela Witten, Jacob Bien - Finsler geometry in anisotropic superconductivity: a Ginzburg-Landau approach - https://arxiv.org/abs/2510.19720 - arXiv:2510.19720v3 Announce Type: replace -Abstract: We present a rigorous generalization of the classical Ginzburg--Landau model to smooth, compact Finsler manifolds without boundary. This framework provides a natural analytic setting for describing anisotropic superconductivity within Finsler geometry. The model is constructed via the Finsler--Laplacian, defined through the Legendre transform associated with the fundamental function F, and by employing canonical Finsler measures such as the Busemann--Hausdorff and Holmes--Thompson volume forms. We introduce an anisotropic Ginzburg--Landau functional for complex scalar fields coupled to gauge potentials and establish the existence of minimizers in the appropriate Finsler--Sobolev spaces by the direct method in the calculus of variations. Furthermore, we analyze the asymptotic regime as the Ginzburg--Landau parameter epsilon to 0 and prove a precise Gamma--convergence result: the rescaled energies converge to the Finslerian length functional associated with the limiting vortex filaments. In particular, the limiting vortex energy is shown to equal $\pi$ times the Finslerian length of the corresponding current, thereby extending the classical Bethuel--Brezis--He'lein result to anisotropic settings. These findings demonstrate that Finsler geometry unifies metric anisotropy and variational principles in gauge-field models, broadening the geometric scope of the Ginzburg--Landau theory beyond the Riemannian framework. - oai:arXiv.org:2510.19720v3 - math-ph - math.MP - Mon, 22 Dec 2025 00:00:00 -0500 - replace + Persistence diagrams as morphological signatures of cells: A method to measure and compare cells within a population + https://arxiv.org/abs/2310.20644 + arXiv:2310.20644v3 Announce Type: replace-cross +Abstract: Cell biologists study in parallel the morphology of cells with the regulation mechanisms that modify this morphology. Such studies are complicated by the inherent heterogeneity present in the cell population. It remains difficult to define the morphology of a cell with parameters that can quantify this heterogeneity, leaving the cell biologist to rely on manual inspection of cell images. We propose an alternative to this manual inspection that is based on topological data analysis. We characterise the shape of a cell by its contour and nucleus. We build a filtering of the edges defining the contour using a radial distance function initiated from the nucleus. This filtering is then used to construct a persistence diagram that serves as a signature of the cell shape. Two cells can then be compared by computing the Wasserstein distance between their persistence diagrams. Given a cell population, we then compute a distance matrix that includes all pairwise distances between its members. We analyse this distance matrix using hierarchical clustering with different linkage schemes and define a purity score that quantifies consistency between those different schemes, which can then be used to assess homogeneity within the cell population. We illustrate and validate our approach to identify sub-populations in human mesenchymal stem cell populations. + oai:arXiv.org:2310.20644v3 + q-bio.QM + math.AT + stat.AP + Tue, 23 Dec 2025 00:00:00 -0500 + replace-cross http://creativecommons.org/licenses/by/4.0/ - 10.46298/ocnmp.16773 - Open Communications in Nonlinear Mathematical Physics, Volume 5 (November 4, 2025) ocnmp:16773 - Y. Alipour Fakhri + Yossi Bokor Bleile, Pooja Yadav, Patrice Koehl, Florian Rehfeldt - Recurrence, transience and anti-concentration of Rademacher random walks - https://arxiv.org/abs/2510.24568 - arXiv:2510.24568v2 Announce Type: replace -Abstract: The Rademacher random walk associated with a deterministic sequence $(a_n)_{n \geq 1}$ is the walk which starts at zero and, at step $i$, independently steps either up or down by $a_i$ with equal probability. We continue the study begun by Bhattacharya and Volkov in 2023 of the transience or recurrence of one-dimensional Rademacher random walks. In particular, we show that if the sequence of step sizes is bounded, the walk is weakly recurrent, meaning that it returns infinitely often to a random finite interval, while if the step sizes tend to infinity arbitrarily slowly, the walk may be transient. On the other hand, using a construction with integer step sizes, we show that the step sizes may grow arbitrarily fast and still give a weakly recurrent random walk. We also show, using a construction with non-integer step sizes, that the same conclusion holds even if we restrict to strictly increasing step sizes. However, we prove that if $a_n = n^{\alpha + o(1)}$ for some $\alpha > 1/2$, then the walk is transient. We show that the bound on $\alpha$ is tight by giving an example where $a_n = \Theta(n^{1/2})$ and the walk is weakly recurrent. - oai:arXiv.org:2510.24568v2 - math.PR - Mon, 22 Dec 2025 00:00:00 -0500 - replace + Training robust and generalizable quantum models + https://arxiv.org/abs/2311.11871 + arXiv:2311.11871v4 Announce Type: replace-cross +Abstract: Adversarial robustness and generalization are both crucial properties of reliable machine learning models. In this paper, we study these properties in the context of quantum machine learning based on Lipschitz bounds. We derive parameter-dependent Lipschitz bounds for quantum models with trainable encoding, showing that the norm of the data encoding has a crucial impact on the robustness against data perturbations. Further, we derive a bound on the generalization error which explicitly involves the parameters of the data encoding. Our theoretical findings give rise to a practical strategy for training robust and generalizable quantum models by regularizing the Lipschitz bound in the cost. Further, we show that, for fixed and non-trainable encodings, as those frequently employed in quantum machine learning, the Lipschitz bound cannot be influenced by tuning the parameters. Thus, trainable encodings are crucial for systematically adapting robustness and generalization during training. The practical implications of our theoretical findings are illustrated with numerical results. + oai:arXiv.org:2311.11871v4 + quant-ph + cs.LG + math.OC + Tue, 23 Dec 2025 00:00:00 -0500 + replace-cross http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Satyaki Bhattacharya, Edward Crane, Tom Johnston + 10.1103/PhysRevResearch.6.043326 + Physical Review Research 6, 043326, 2024 + Julian Berberich, Daniel Fink, Daniel Pranji\'c, Christian Tutschku, Christian Holm - Probabilities are always axiomatizable - https://arxiv.org/abs/2511.00228 - arXiv:2511.00228v2 Announce Type: replace -Abstract: In this paper we study the interaction between logic and probability. In particular, we show that the convex hull of evaluations of a broad class of logics is always effectively axiomatizable. We define a Birkhoff-style calculus for probability axioms for which compactness, and finite completeness is proved. We give example for a logic for which probabilities are not finitely axiomatizable. - oai:arXiv.org:2511.00228v2 - math.LO - Mon, 22 Dec 2025 00:00:00 -0500 - replace + Majority voting is not good for heaven or hell, with mirrored performance + https://arxiv.org/abs/2401.00592 + arXiv:2401.00592v5 Announce Type: replace-cross +Abstract: Within the ViSE (Voting in Stochastic Environment) model, we study the effectiveness of majority voting in various environments. As shown by the pit-of-losses paradox identified in previous work, majority decisions in apparently hostile environments tend to reduce the capital of society. In such cases, the simple social decision rule of ``rejecting all proposals without voting'' outperforms majority voting. In this paper, we identify another pit of losses appearing in favorable environments; here, the simple social decision rule of ``accepting all proposals without voting'' is superior to majority voting. We prove that, under a version of simple majority called symmetrized majority and under the antisymmetry of the voting body, this second pit of losses is a mirror image of the one arising in hostile environments, and we explain this phenomenon. Technically, we consider a voting society consisting of individualists who support all proposals that increase their personal capital and a group (or groups) whose members vote to increase their group's wealth. According to the key lemma, the expected capital gain of each agent under the social decision rule when the random gain generator is $X$ with mean $\mu>0$ exceeds their expected gain under the reflected generator $-X$ by exactly $\mu$. This extends to location-scale families of generators with distributions symmetric about their mean. This result reveals a mirror symmetry in the performance of the symmetrized majority rule relative to a baseline rule. The baseline rule accepts all proposals in favorable environments and rejects them in unfavorable (hostile) ones. + oai:arXiv.org:2401.00592v5 + physics.soc-ph + cs.GT + math.OC + Tue, 23 Dec 2025 00:00:00 -0500 + replace-cross http://creativecommons.org/licenses/by-nc-nd/4.0/ - Zal\'an Gyenis - - - A Self Propelled Vortex Dipole Model on Surfaces of Variable Negative Curvature - https://arxiv.org/abs/2511.00923 - arXiv:2511.00923v3 Announce Type: replace -Abstract: We investigate vortex dipoles on surfaces of variable negative curvature, focusing on a catenoid of arbitrary throat radius as a concrete example. We construct the effective dynamical system including mutual and geometric self-interaction terms and show that the resulting Hamiltonian dynamics makes dipoles follow catenoid geodesics, in agreement with recent works, Gustafsson (J. Nonlinear Sci. 32, 62, 2022) and by Drivas, Glukhovskiy and Khesin (Int. Math. Res. Not. 2024, 14, 10880-10894). We utilize the symplectic structure to find a conserved momentum map J related to the U(1) symmetry along the azimuthal direction. We verify the conservation of both the Hamiltonian and this momentum for arbitrary throat radius. We then demonstrate direct and exchange scattering of classical vortices on the catenoid, and we contrast this with the collective rotational motion (with azimuthal drift) that arises for chiral pairs. Finally, we build a finite-dipole dynamical system on the catenoid and show that the self-propulsion terms emerge to leading order in the dipole size. This provides a concrete realization, on a curved minimal surface, of the intuitive statement that a finite dipole propels orthogonal to the dipole axis, with a speed modulated by curvature. - oai:arXiv.org:2511.00923v3 - math-ph - cond-mat.quant-gas - hep-th - math.MP - Mon, 22 Dec 2025 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Khushi Banthia, Rickmoy Samanta + Pavel Chebotarev, Vadim Afonkin - A dodecic surface with 320 cusps - https://arxiv.org/abs/2511.01322 - arXiv:2511.01322v3 Announce Type: replace -Abstract: We construct a degree $12$ homogeneous invariant of the complex reflection group $G_{29}$ (in Shephard-Todd's notation) whose associated surface has 320 singularities of type $A_2$, improving previous records for dodecic surfaces. - oai:arXiv.org:2511.01322v3 - math.AG - math.RT - Mon, 22 Dec 2025 00:00:00 -0500 - replace + An overview of systems-theoretic guarantees in data-driven model predictive control + https://arxiv.org/abs/2406.04130 + arXiv:2406.04130v2 Announce Type: replace-cross +Abstract: The development of control methods based on data has seen a surge of interest in recent years. When applying data-driven controllers in real-world applications, providing theoretical guarantees for the closed-loop system is of crucial importance to ensure reliable operation. In this review, we provide an overview of data-driven model predictive control (MPC) methods for controlling unknown systems with guarantees on systems-theoretic properties such as stability, robustness, and constraint satisfaction. The considered approaches rely on the Fundamental Lemma from behavioral theory in order to predict input-output trajectories directly from data. We cover various setups, ranging from linear systems and noise-free data to more realistic formulations with noise and nonlinearities, and we provide an overview of different techniques to ensure guarantees for the closed-loop system. Moreover, we discuss avenues for future research that may further improve the theoretical understanding and practical applicability of data-driven MPC. + oai:arXiv.org:2406.04130v2 + eess.SY + cs.SY + math.OC + Tue, 23 Dec 2025 00:00:00 -0500 + replace-cross http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - C\'edric Bonnaf\'e + 10.1146/annurev-control-030323-024328 + Annual Review of Control, Robotics, and Autonomous Systems 8 (1), pp. 77-100, 2025 + Julian Berberich, Frank Allg\"ower - Arithmetic Geometric Model for the Renormalisation of Bi-critical Irrationally Indifferent Attractors - https://arxiv.org/abs/2511.04656 - arXiv:2511.04656v2 Announce Type: replace -Abstract: In this paper we build a geometric model for the renormalisation of irrationally indifferent fixed points of holomorphic maps with two critical points. The model incorporates arithmetic properties of the rotation number at the fixed point, as well as the ``angle" between the two critical points. Using this model for the renormalisation, we build a topological model for the local dynamics of such maps. We also explain the topology of the maximal invariant set for the model, and the dynamics of the map on the maximal invariant set. - oai:arXiv.org:2511.04656v2 - math.DS - Mon, 22 Dec 2025 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by-sa/4.0/ - Jocelyn Finbar Russell + Variational Markov chain mixtures with automatic component selection + https://arxiv.org/abs/2406.04653 + arXiv:2406.04653v2 Announce Type: replace-cross +Abstract: Markov state modeling has gained popularity in various scientific fields since it reduces complex time-series data sets into transitions between a few states. Yet common Markov state modeling frameworks assume a single Markov chain describes the data, so they suffer from an inability to discern heterogeneities. As an alternative, this paper models time-series data using a mixture of Markov chains, and it automatically determines the number of mixture components using the variational expectation-maximization algorithm.Variational EM simultaneously identifies the number of Markov chains and the dynamics of each chain without expensive model comparisons or posterior sampling. As a theoretical contribution, this paper identifies the natural limits of Markov state mixture modeling by proving a lower bound on the classification error. It then presents numerical experiments where variational EM achieves performance consistent with the theoretically optimal error scaling. The experiments are based on synthetic and observational data sets including Last.fm music listening, ultramarathon running, and gene expression. In each of the three data sets, variational EM leads to the identification of meaningful heterogeneities. + oai:arXiv.org:2406.04653v2 + stat.ME + cs.NA + math.NA + stat.ML + Tue, 23 Dec 2025 00:00:00 -0500 + replace-cross + http://creativecommons.org/licenses/by/4.0/ + Christopher E. Miles, Robert J. Webber - Grothendieck--Teichm\"uller Symmetries of Cyclic Operads and Tangles - https://arxiv.org/abs/2511.05911 - arXiv:2511.05911v2 Announce Type: replace -Abstract: We characterise the profinite Grothendieck-Teichm\"uller group $\widehat{\mathsf{GT}}$ as the group of automorphisms of the profinite completion of a cyclic operad of parenthesised ribbon braids. This operad generates a symmetric monoidal category which is equivalent to the category of framed, oriented tangles, thereby providing an operadic model for profinite tangles and their arithmetic symmetries. As applications, we show that $\widehat{\mathsf{GT}}$ acts naturally on tangles and provide an alternative proof of the formality of the cyclic framed little disks operad. - oai:arXiv.org:2511.05911v2 - math.AT - math.CT - math.QA - Mon, 22 Dec 2025 00:00:00 -0500 - replace + Continuum Attention for Neural Operators + https://arxiv.org/abs/2406.06486 + arXiv:2406.06486v4 Announce Type: replace-cross +Abstract: Transformers, and the attention mechanism in particular, have become ubiquitous in machine learning. Their success in modeling nonlocal, long-range correlations has led to their widespread adoption in natural language processing, computer vision, and time series problems. Neural operators, which map spaces of functions into spaces of functions, are necessarily both nonlinear and nonlocal if they are universal; it is thus natural to ask whether the attention mechanism can be used in the design of neural operators. Motivated by this, we study transformers in the function space setting. We formulate attention as a map between infinite dimensional function spaces and prove that the attention mechanism as implemented in practice is a Monte Carlo or finite difference approximation of this operator. The function space formulation allows for the design of transformer neural operators, a class of architectures designed to learn mappings between function spaces. In this paper, we state and prove the first universal approximation result for transformer neural operators, using only a slight modification of the architecture implemented in practice. The prohibitive cost of applying the attention operator to functions defined on multi-dimensional domains leads to the need for more efficient attention-based architectures. For this reason we also introduce a function space generalization of the patching strategy from computer vision, and introduce a class of associated neural operators. Numerical results, on an array of operator learning problems, demonstrate the promise of our approaches to function space formulations of attention and their use in neural operators. + oai:arXiv.org:2406.06486v4 + cs.LG + cs.NA + math.NA + Tue, 23 Dec 2025 00:00:00 -0500 + replace-cross http://creativecommons.org/licenses/by/4.0/ - Marcy Robertson, Chandan Singh + Edoardo Calvello, Nikola B. Kovachki, Matthew E. Levine, Andrew M. Stuart - Mathematical Analysis and Modeling of Ebola Virus Dynamics via Optimal Control and Neural Network Paradigms - https://arxiv.org/abs/2511.06303 - arXiv:2511.06303v3 Announce Type: replace -Abstract: Ebola virus disease is a severe hemorrhagic fever with rapid transmission through infected fluids and surfaces. We develop a fractional-order model using Caputo derivatives to capture memory effects in disease dynamics. An eight-compartment structure distinguishes symptomatic, asymptomatic, and post-mortem transmission pathways. We prove global well-posedness, derive the basic reproduction number $\mathcal{R}_0$, and establish stability theorems. Sensitivity analysis shows $\mathcal{R}_0$ is most sensitive to transmission rate, incubation period, and deceased infectivity. Treatment-safe burial synergy achieves 86.5\% morbidity-mortality control, with safe burial being most effective. Our disease-informed neural network achieves near-perfect predictive accuracy ($R^2$: 0.991-0.999, 99.1-99.9\% accuracy), closely matching real epidemic behavior. - oai:arXiv.org:2511.06303v3 - math.OC - Mon, 22 Dec 2025 00:00:00 -0500 - replace + A framework for the use of generative modelling in non-equilibrium statistical mechanics + https://arxiv.org/abs/2406.11630 + arXiv:2406.11630v5 Announce Type: replace-cross +Abstract: We discuss an approach to mathematically modelling systems made of objects that are coupled together, using generative models of the dependence relationships between states (or trajectories) of the things comprising such systems. This broad class includes open or non-equilibrium systems and is especially relevant to self-organising systems. The ensuing variational free energy principle (FEP) has certain advantages over using random dynamical systems explicitly, notably, by being more tractable and offering a parsimonious explanation of why the joint system evolves in the way that it does, based on the properties of the coupling between system components. The FEP is a method whose use allows us to build a model of the dynamics of an object as if it were a process of variational inference, because variational free energy (or surprisal) is a Lyapunov function for its dynamics. In short, we argue that using generative models to represent and track relations amongst subsystems leads us to a particular statistical theory of interacting systems. Conversely, this theory enables us to construct nested models that respect the known relations amongst subsystems. We point out that the fact that a physical object conforms to the FEP does not necessarily imply that this object performs inference in the literal sense; rather, it is a useful explanatory fiction which replaces the `explicit' dynamics of the object with an `implicit' flow on free energy gradients -- a fiction that may or may not be entertained by the object itself. + oai:arXiv.org:2406.11630v5 + cond-mat.stat-mech + math-ph + math.MP + nlin.AO + Tue, 23 Dec 2025 00:00:00 -0500 + replace-cross http://creativecommons.org/licenses/by/4.0/ - Noor Muhammad (School of Mathematics, Sichuan University, Chengdu, China), Md. Nur Alam (Department of Mathematics, Pabna University of Science & Technology, Pabna-6600, Bangladesh), Zhang Shiqing (School of Mathematics, Sichuan University, Chengdu, China) + Karl J Friston, Maxwell J D Ramstead, Dalton A R Sakthivadivel - Subset expansions of monoids - https://arxiv.org/abs/2511.13435 - arXiv:2511.13435v2 Announce Type: replace -Abstract: We initiate the study of the expansion $\mathcal{S}(M)$ of a monoid $M$ obtained via the semidirect product of $M$ acting naturally on the left of its power set (regarded as a semilattice under union). We term this the `subset expansion' of $M$. The monoid $\mathcal{S}(M)$ contains the images of several expansions of $M$ of wide interest and use in semigroup theory, in particular the prefix and Szendrei expansions (in the case where $M$ is free, these `smaller' expansions produce free algebras in certain varieties). - We first focus on algebraic properties, specifically those determined by idempotents. Particularly, we show that the expansion $\mathcal{S}$ maps groups to proper inverse monoids, unipotent monoids to proper left restriction monoids, right cancellative monoids to left ample monoids, right abundant monoids to right abundant monoids, and left cancellative monoids to right adequate monoids. - Subsequently, we focus on finitary conditions. We examine the condition of weak left coherence (every finitely generated left ideal has a finite presentation as a left act); the related conditions of property (L), left ideal Howson, finitely left equated, and each of the corresponding left-right dual notions. Each of these conditions is preserved under retract, from which it is immediate that if $\mathcal{S}(M)$ satisfies one of our finitary conditions, then so must $M$, but the converse is not true. For a property to `lift' from $M$ to $\mathcal{S}(M)$ it must undergo a strengthening. Indeed, we show that $\mathcal{S}(M)$ satisfies property (L) (or its left-right dual) if and only if $M$ is finite. We provide exact characterisations of the monoids $M$ such that $\mathcal{S}(M)$ is: left (or right) ideal Howson; finitely left equated; and (consequently) weakly left coherent. We give sufficient conditions for $\mathcal{S}(M)$ to be finitely right equated and hence weakly right coherent. - oai:arXiv.org:2511.13435v2 - math.RA - Mon, 22 Dec 2025 00:00:00 -0500 - replace + Logic-based analogical proportions + https://arxiv.org/abs/2406.14402 + arXiv:2406.14402v2 Announce Type: replace-cross +Abstract: 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 first-order logic. We show that the so-obtained logic-based framework preserves all desired properties and we prove novel results in that extended setting. + oai:arXiv.org:2406.14402v2 + cs.LO + cs.DM + math.LO + Tue, 23 Dec 2025 00:00:00 -0500 + replace-cross http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Victoria Gould, Marianne Johnson + Christian Anti\'c - Chow polynomials of rank-uniform labeled posets - https://arxiv.org/abs/2511.13819 - arXiv:2511.13819v2 Announce Type: replace -Abstract: We introduce and develop the theory of UMEL-shellable posets. These are posets equipped with an edge-lexicographical labeling satisfying certain uniformity and monotonicity properties. This framework encompasses classical families of combinatorial geometries, including uniform matroids, projective and affine geometries, braid matroids of type A and B, and all Dowling geometries. It also comprises all rank-uniform supersolvable lattices, and therefore also all rank-uniform distributive lattices. Our main result establishes real-rootedness phenomena for the Chow polynomials, the augmented Chow polynomials, and the chain polynomials associated with those posets, thus making simultaneous progress towards conjectures by Ferroni--Schr\"oter, Huh--Stevens, and Athanasiadis--Kalampogia-Evangelinou. In the special case of lattices of flats of matroids, the (augmented) Chow polynomials coincide with the Hilbert--Poincar\'e series of the Chow ring associated to the smooth and generally noncompact toric varieties of the (augmented) Bergman fan of the matroid, whereas the chain polynomial encodes the Hilbert--Poincar\'e series of the Stanley--Reisner ring of the Bergman complex of the matroid. Therefore, these real-rootedness results are tightly linked to the study of these algebro-geometric structures in matroid theory. - oai:arXiv.org:2511.13819v2 + Single-cell 3D genome reconstruction in the haploid setting using rigidity theory + https://arxiv.org/abs/2407.10700 + arXiv:2407.10700v2 Announce Type: replace-cross +Abstract: This article considers the problem of 3-dimensional genome reconstruction for single-cell data, and the uniqueness of such reconstructions in the setting of haploid organisms. We consider multiple graph models as representations of this problem, and use techniques from graph rigidity theory to determine identifiability. Biologically, our models come from Hi-C data, microscopy data, and combinations thereof. Mathematically, we use unit ball and sphere packing models, as well as models consisting of distance and inequality constraints. In each setting, we describe and/or derive new results on realisability and uniqueness. We then propose a 3D reconstruction method based on semidefinite programming and apply it to synthetic and real data sets using our models. + oai:arXiv.org:2407.10700v2 + q-bio.GN math.CO - Mon, 22 Dec 2025 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Basile Coron, Luis Ferroni, Shiyue Li - - - Energy functionals on almost K\"ahler manifolds: I - https://arxiv.org/abs/2511.17086 - arXiv:2511.17086v2 Announce Type: replace -Abstract: In this paper, we consider the Donaldson gauge functional and the twisted Aubin functionals on almost K\"ahler manifolds. As in K\"ahler geometry, we generalize the inequality between Aubin functionals. - oai:arXiv.org:2511.17086v2 - math.DG - math.SG - Mon, 22 Dec 2025 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Ken Wang, Zuyi Zhang, Jiuru Zhou - - - Complex invariants of poristic Steiner 4-chains - https://arxiv.org/abs/2511.17613 - arXiv:2511.17613v2 Announce Type: replace -Abstract: We are concerned with the Steiner chains consisting of four circles. More precisely, we deal with the so-called complex moments of Steiner 4-chains introduced in a recent paper by J.Lagarias, C.Mallows and A.Wilks. We compute the invariant complex moments of poristic Steiner 4-chains and establish certain algebraic relations between those invariants. To this end we use the invariance of certain moments of curvatures of poristic Steiner chains established by R.Schwartz and S.Tabachnikov, combined with the computation of these moments for the so-called symmetric Steiner 4-chains. We also present analogous results for poristic Steiner 3-chains and give an application to the feasibility problem for the centers of Steiner 4-chains. - KEYWORDS: Steiner chain, parent circles, Steiner porism, poristic Steiner chains, Descartes circle theorem, invariant bending moments, complex moments of Steiner chains, algebraic relations between invariants - oai:arXiv.org:2511.17613v2 - math.GM - Mon, 22 Dec 2025 00:00:00 -0500 - replace + math.MG + math.OC + Tue, 23 Dec 2025 00:00:00 -0500 + replace-cross http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Ana Diakvnishvili, Giorgi Khimshiashvili + 10.1007/s00285-025-02203-2 + Journal of Mathematical Biology, Volume 90, article number 45, (2025) + Sean Dewar, Georg Grasegger, Kaie Kubjas, Fatemeh Mohammadi, Anthony Nixon - Variations of the Hardy Z-Function and the Montgomery Pair Correlation Conjecture - https://arxiv.org/abs/2511.18275 - arXiv:2511.18275v2 Announce Type: replace -Abstract: In 1973 Montgomery formulated the pair correlation conjecture, predicting that the local spacing statistics of the nontrivial zeros of the Riemann zeta function coincide with those of eigenvalues of large Hermitian matrices from the Gaussian Unitary Ensemble (GUE). The zeta function, however, is a fixed deterministic object, and the mechanism by which its zeros reproduce random matrix statistics has remained unclear. In this paper, assuming the Riemann Hypothesis, we prove Montgomery's pair correlation conjecture for the zeros of Hardy's $Z$-function. Building on earlier works, we use a finite-dimensional variational space of sections $Z_N(t;a)$ that approximate $Z(t)$ on each window $[2N,2N+2]$. Inside this space we define the real hall $\mathcal{RH}_N(\mathbb{R})$, consisting of those sections whose zeros in the corresponding critical rectangle are real, simple, and remain so along any homotopy from the core section. This real hall plays the role of a random matrix ensemble. Equipping it with an admissible probability measure with smooth, positive density on the coefficient space, we construct a Skorokhod-type stochastic differential equation with reflection at the discriminant boundary. We show that the induced dynamics of the unfolded zeros are, in the bulk, equivalent in law to Dyson Brownian motion with $\beta=2$, and by invoking modern universality results for Dyson Brownian motion and log-gases we obtain that, for any such measure, the ensemble-averaged local pair-correlation converges to the GUE sine-kernel law. Finally, using Selberg's probabilistic theory of the argument $S(t)$, we prove that the pair-correlation observables of the canonical approximants $Z_N(t;1)$ over disjoint windows behave asymptotically like decorrelated samples drawn from this GUE-distributed ensemble, and we upgrade the averaged GUE law to a deterministic pair-correlation law for the zeros of $Z(t)$. - oai:arXiv.org:2511.18275v2 - math.NT + Axiomatization of R\'enyi Entropy on Quantum Phase Space + https://arxiv.org/abs/2410.15976 + arXiv:2410.15976v5 Announce Type: replace-cross +Abstract: Phase-space versions of quantum mechanics -- from Wigner's original distribution to modern discrete-qudit constructions -- represent some states with negative quasi-probabilities. Conventional Shannon and R\'enyi entropies become complex-valued in this setting and lose their operational meaning. Building on the axiomatic treatments of R\'enyi (1961) and Dar\'oczy (1963), we develop a conservative extension that applies to signed finite phase spaces and identify a single admissible entropy family, which we call signed R\'enyi $\alpha$-entropy (for a free parameter $\alpha \ge 0$). The obvious signed Shannon candidate is ruled out because it violates extensivity. We prove four results that bolster the usefulness of the new measure. (i) It serves as a witness of the presence of cancellation, detecting the coexistence of positive and negative weight in a signed measure. (ii) For $\alpha > 1$, it is Schur-concave, delivering the intuitive property that mixing increases entropy (iii) The same parametric family obeys a quantum H-theorem, namely, that under de-phasing dynamics entropy cannot decrease. (iv) The $2$-entropy is conserved under discrete Moyal-bracket dynamics, mirroring conservation of von Neumann entropy under unitary evolution on Hilbert space. We also comment on interpreting the R\'enyi order parameter as an inverse temperature. Overall, we believe that our investigation provides good evidence that our axiomatically derived signed R\'enyi entropy may be a useful addition to existing entropy measures employed in quantum information, foundations, and thermodynamics. + oai:arXiv.org:2410.15976v5 + quant-ph math-ph math.MP - math.PR - Mon, 22 Dec 2025 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Yochay Jerby + Tue, 23 Dec 2025 00:00:00 -0500 + replace-cross + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Adam Brandenburger, Pierfrancesco La Mura - An Elementary Proof of a Minimax Theorem - https://arxiv.org/abs/2511.19416 - arXiv:2511.19416v2 Announce Type: replace -Abstract: Here, we give a self-contained and elementary proof of a minimax theorem due to Fan in a simplified setting that can be taught in an advanced undergraduate course. Our proof follows Nikaido's argument with some simplifications. - oai:arXiv.org:2511.19416v2 - math.HO - math.CA - math.OC - Mon, 22 Dec 2025 00:00:00 -0500 - replace + Absence of nontrivial local conserved quantities in the spin-1 bilinear-biquadratic chain and its anisotropic extensions + https://arxiv.org/abs/2411.04945 + arXiv:2411.04945v4 Announce Type: replace-cross +Abstract: We provide a complete classification of the integrability and nonintegrability of the spin-1 bilinear-biquadratic model with a uniaxial anisotropic field, which includes the Heisenberg model and the Affleck-Kennedy-Lieb-Tasaki model. It is rigorously shown that, within this class, the only integrable systems are those that have been solved by the Bethe ansatz method, and that all other systems are nonintegrable, in the sense that they do not have nontrivial local conserved quantities. Here, "nontrivial" excludes quantities like the Hamiltonian or the total magnetization, and "local" refers to sums of operators that act only on sites within a finite distance. This result establishes the nonintegrability of the Affleck-Kennedy-Lieb-Tasaki model and, consequently, demonstrates that the quantum many-body scars observed in this model emerge independently of any conservation laws of local quantities. Furthermore, we extend the proof of nonintegrability to more general spin-1 models that encompass anisotropic extensions of the bilinear-biquadratic Hamiltonian and completely classify the integrability of generic Hamiltonians that possess translational symmetry, $U(1)$ symmetry, time-reversal symmetry, and spin-flip symmetry. Our result accomplishes a breakthrough in nonintegrability proofs by expanding their scope to spin-1 systems. + oai:arXiv.org:2411.04945v4 + cond-mat.stat-mech + math-ph + math.MP + quant-ph + Tue, 23 Dec 2025 00:00:00 -0500 + replace-cross http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Jeff Calder + 10.1103/vjtq-sz21 + Phys. Rev. Research 7, 043297 (2025) + Akihiro Hokkyo, Mizuki Yamaguchi, Yuuya Chiba - Intensity doubling for Brownian loop-soups in high dimensions - https://arxiv.org/abs/2511.21670 - arXiv:2511.21670v2 Announce Type: replace -Abstract: We derive an intensity doubling feature of critical Brownian loop-soups on the cable-graphs of ${\mathbb Z}^d$ for $d \ge 7$ that can be described as follows: In the box $[-N, N]^d$ (and with a probability that goes to $1$ as $N$ goes to infinity), the set of all clusters of Brownian loops that do contain proper self-avoiding cycles of diameter comparable to $N$ can be decomposed into two identically distributed families: (a) The collection of clusters that do contain a large Brownian loop from the loop-soup (and therefore do automatically contain such a large cycle) (b) The collection of clusters that contain no macroscopic loop from the loop-soup (more specifically, no loop of diameter greater than $N^{\beta}$ when $\beta > 4/ (d-2)$ is fixed) but nevertheless contain a large cycle. In particular, due to the fact that these two families are asymptotically identically distributed, large cycles formed in case (b) by chains of small Brownian loops (i.e., all of diameter much smaller than $N$) will look like large Brownian loops themselves, and form a second independent "ghost" critical loop-soup in the scaling limit. Reformulated in terms of the Gaussian free field on such cable-graphs, this shows that large cycles in the collection of its sign clusters will converge in the scaling limit to a Brownian loop-soup with twice the usual critical intensity. This result had been conjectured by the first author in arXiv:2209.07901 [math.PR] ; our proof builds heavily on the second author's switching property for such loop-soups from arXiv:2502.06754 [math.PR] . - oai:arXiv.org:2511.21670v2 - math.PR + Thermodynamic Coupling of Mass and Electromagnetic Fields: Entropic Origin of Parity Asymmetry and the Meissner Effect + https://arxiv.org/abs/2411.16798 + arXiv:2411.16798v2 Announce Type: replace-cross +Abstract: We develop a thermodynamic framework that couples mass dynamics, described by the Newton- Gibbs-van der Waals formalism, with electromagnetic fields beyond the scope of classical Maxwell theory. Classical Newtonian mechanics does not capture density evolution in the momentum balance, while the standard Maxwell equations neglect the contribution of the curl component of the electric field associated with moving charges. Building on an alternative understanding on entropy, we develop a generalized theory for electrodynamics governed by entropy-production constraints. The resulting framework yields a modified Maxwell stress tensor that incorporates the moving-charge contribution, leading to intrinsic parity asymmetry in electromagnetic forces. The theory naturally reproduces key features of superconductivity, including the Meissner effect, and reduces to the conventional Maxwell-Faraday and Maxwell-Ampere equations in an appropriate limit. This entropic formulation provides a unified thermodynamic basis for mass-field coupling and reveals new physical consequences arising from motion-induced electromagnetic effects. + oai:arXiv.org:2411.16798v2 + physics.class-ph math-ph math.MP - Mon, 22 Dec 2025 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Titus Lupu, Wendelin Werner + Tue, 23 Dec 2025 00:00:00 -0500 + replace-cross + http://creativecommons.org/licenses/by/4.0/ + Fei Wang - Maximum Spectral Efficiency With Adaptive MQAM Transmissions Over Terrestrial Coherent FSO Links - https://arxiv.org/abs/2511.22682 - arXiv:2511.22682v2 Announce Type: replace -Abstract: Coherent free-space optical (FSO) communication is recognized as a key enabler for ultra-high-capacity fronthaul and backhaul links in next-generation wireless networks. Spectrally efficient $M$-ary quadrature amplitude modulation (MQAM) formats are well-suited for these links. However, theoretical analyses of adaptive MQAM transmissions over terrestrial FSO channels remain limited. In this letter, we first derive the spectral efficiency limit of adaptive unconstrained MQAM over gamma-gamma turbulence with pointing error. We then show that adaptive transmissions using only six square MQAM constellations performs close to the theoretical limit (within $0.10$-$0.12$ bits/s/Hz) across a wide range of signal-to-noise ratios and channel conditions. - oai:arXiv.org:2511.22682v2 + Generalized free energy and excess/housekeeping decomposition in nonequilibrium systems: from large deviations to thermodynamic speed limits + https://arxiv.org/abs/2412.08432 + arXiv:2412.08432v2 Announce Type: replace-cross +Abstract: In genuine nonequilibrium systems that undergo continuous driving, the thermodynamic forces are nonconservative, meaning they cannot be described by any free energy potential. Nonetheless, we show that the dynamics of such systems are governed by a "generalized free energy" that is derived from a large-deviations variational principle. This variational principle also yields a decomposition of fluxes, forces, and dissipation (entropy production) into a conservative "excess" part and a nonconservative "housekeeping" part. Our decomposition is universally applicable to stochastic master equations, deterministic chemical reaction networks, and open systems. We also show that the excess entropy production obeys a thermodynamic speed limit (TSL), a fundamental thermodynamic constraint on the rate of state evolution and/or external fluxes. We demonstrate our approach on several examples, including real-world metabolic networks, where we derive fundamental dissipation bounds and uncover "futile" metabolic cycles. Our generalized free energy and decomposition are empirically accessible to thermodynamic inference in both stochastic and deterministic systems. We discuss important connections to several theoretical frameworks, including information geometry and Onsager theory, as well as previous excess/housekeeping decompositions. + oai:arXiv.org:2412.08432v2 + cond-mat.stat-mech cs.IT math.IT - Mon, 22 Dec 2025 00:00:00 -0500 - replace + Tue, 23 Dec 2025 00:00:00 -0500 + replace-cross http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Himani Verma, Kamal Singh, Ranjan K. Mallik + Artemy Kolchinsky, Andreas Dechant, Kohei Yoshimura, Sosuke Ito - Action de groupe sur la compactification hybride - https://arxiv.org/abs/2512.00201 - arXiv:2512.00201v2 Announce Type: replace -Abstract: Let $X$ be an algebraic variety over $\mathbb{C}$ and $G$ be an algebraic group acting on $X$ whose action is closed. J. Poineau defined a compactification $X^\urcorner$ of $X(\mathbb{C})$ by using hybrid Berkovich spaces. We will focus on the extension of the action of $G$ on this compactification by characterising the set $\mathcal{U} \subset X^\urcorner$ where the action is well defined. We will also show that the quotient of $\mathcal{U}$ by the action of $G$ is homeomorphic to $(X/G)^\urcorner$, the compactification of $(X/G)(\mathbb{C})$. We then apply these results to $X = \mathrm{Rat}_d$, the space of rational maps and $G = \mathrm{SL}_2$. It gives the results of C. Favre-C. Gong in a more general setting. Furthermore, we get a compactification of $\mathrm{M}_d = \mathrm{Rat}_d/\mathrm{SL}_2$ where the boundary is made of orbits of non-archimedean rational maps. The results still holds if $\mathbb{C}$ is replaced by $k$ a non-trivially valued field and complex analytic spaces by Berkovich spaces over $k$ or if $X$ is the set of stable points of a $k$-variety defined in the sense of GIT. - oai:arXiv.org:2512.00201v2 - math.AG - math.DS - math.NT - Mon, 22 Dec 2025 00:00:00 -0500 - replace + Informativity Conditions for Multiple Signals: Properties, Experimental Design, and Applications (extended version) + https://arxiv.org/abs/2501.10030 + arXiv:2501.10030v4 Announce Type: replace-cross +Abstract: Recent studies highlight the importance of persistently exciting condition in single signal sequence for model identification and data-driven control methodologies. However, maintaining prolonged excitation in control signals introduces significant challenges, as continuous excitation can reduce the lifetime of mechanical devices. In this paper, we introduce three informativity conditions for various types of multi-signal data, each augmented by weight factors. We explore the interrelations between these conditions and their rank properties in linear time-invariant systems. Furthermore, we introduce open-loop experimental design methods tailored to each of the three conditions, which can synthesize the required excitation conditions either offline or online, even in the presence of limited information within each signal segment. We demonstrate the effectiveness of these informativity conditions in least-squares identification. Additionally, all three conditions can extend Willems' fundamental lemma and are utilized to assess the properties of the system. Illustrative examples confirm that these conditions yield satisfactory outcomes in both least-squares identification and the construction of data-driven controllers. + oai:arXiv.org:2501.10030v4 + eess.SY + cs.IT + cs.SY + math.IT + Tue, 23 Dec 2025 00:00:00 -0500 + replace-cross http://creativecommons.org/licenses/by/4.0/ - Alexandre Roy + Ao Cao, Fuyong Wang - A Dual-Mode Framework for Mean-Field Systems: Model-Based $H_2/H_\infty$ Control with Jump Diffusions and Model-Free Reinforcement Learning - https://arxiv.org/abs/2512.01000 - arXiv:2512.01000v4 Announce Type: replace -Abstract: Two approaches for solving the robust control of mean-field systems are investigated in this paper. For the stochastic $H_2/H_\infty$ control problem of continuous-time mean-field stochastic differential equations with Poisson jumps over a finite horizon, the continuous and jump diffusion terms in the system depend not only on the state but also on the control input, external disturbance, and mean-field components. The feasibility of the stochastic $H_2/H_\infty$ control problem is demonstrated to be equivalent to the solvability of four sets of cross-coupled generalized differential Riccati equations. Based on this conclusion, a model-based numerical method is presented. Furthermore, a data-driven, model-free, off-policy reinforcement learning approach is proposed, which can be employed to solve the $H_\infty$ control problem of the linear mean-field (x, u, v)-dependent systems. Two distinct methodologies for designing robust controllers for interacting particle systems are demonstrated in this paper. - oai:arXiv.org:2512.01000v4 - math.OC - Mon, 22 Dec 2025 00:00:00 -0500 - replace + Spearman's rho for zero-inflated count data: formulation and attainable bounds + https://arxiv.org/abs/2503.13148 + arXiv:2503.13148v2 Announce Type: replace-cross +Abstract: We propose an alternative formulation of Spearman's rho for zero-inflated count data. The formulation yields an estimator with explicitly attainable bounds, facilitating interpretation in settings where the standard range [-1,1] is no longer informative. + oai:arXiv.org:2503.13148v2 + stat.ME + math.ST + stat.TH + Tue, 23 Dec 2025 00:00:00 -0500 + replace-cross http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Huimin Han, Shaolin Ji, Weihai Zhang + Jasper Arends, Guanjie Lyu, Mhamed Mesfioui, Elisa Perrone, Julien Trufin - Splash-squeeze singularities and analytic breakdown in ideal incompressible MHD - https://arxiv.org/abs/2512.02144 - arXiv:2512.02144v2 Announce Type: replace -Abstract: We construct splash-squeeze singularities for the free boundary ideal incompressible plasma-vacuum system, in which two arcs of the plasma boundary come together to form a smooth, glancing self-intersection. As the interface self-intersects, Sobolev norms remain bounded, although analyticity is necessarily lost. This contrasts classical splash singularities, in which solutions remain analytic up to the time of self-intersection. - The narrowing gap bounded by these arcs is not occupied by plasma, as squeezing the plasma itself would cause blow-up in Sobolev norms. Instead, the gap represents the region outside the plasma, a vacuum carrying a nontrivial magnetic field. The plasma on either side pinches the field as the gap closes, and, in response, the field flattens to infinite order at the intersection point (and nowhere else), thereby forming an analytic singularity. This gives the first example of analytic breakdown without Sobolev blow-up in a locally well-posed free-boundary incompressible fluid system, and can be viewed as the first rigorous construction of a squeeze-type singularity, in which we study and quantify the precise behavior of an active, incompressible vector field as it is completely pinched off by a free-boundary in finite time. - The proof combines a magnetically-aligned Lagrangian formulation of ideal MHD together with weighted elliptic estimates in the vacuum that remain uniform as the width of the gap tends to zero. Our framework may provide a starting point for the analysis of squeeze-type singularities in other incompressible fluid models. - oai:arXiv.org:2512.02144v2 - math.AP - Mon, 22 Dec 2025 00:00:00 -0500 - replace + Error bounds for composite quantum hypothesis testing and a new characterization of the weighted Kubo-Ando geometric means + https://arxiv.org/abs/2503.13379 + arXiv:2503.13379v4 Announce Type: replace-cross +Abstract: The optimal error exponents of binary composite i.i.d. state discrimination are trivially bounded by the worst-case pairwise exponents of discriminating individual elements of the sets representing the two hypotheses, and in the finite-dimensional classical case, these bounds in fact give exact single-copy expressions for the error exponents. In contrast, in the non-commutative case, the optimal exponents are only known to be expressible in terms of regularized divergences, resulting in formulas that, while conceptually relevant, are practically not very useful. In this paper, we develop further an approach initiated in [Mosonyi, Szil\'agyi, Weiner, IEEE Trans. Inf. Th. 68(2):1032--1067, 2022] to give improved single-copy bounds on the error exponents by comparing not only individual states from the two hypotheses, but also various unnormalized positive semi-definite operators associated to them. Here, we show a number of equivalent characterizations of such operators giving valid bounds, and show that in the commutative case, considering weighted geometric means of the states, and in the case of two states per hypothesis, considering weighted Kubo-Ando geometric means, are optimal for this approach. As a result, we give a new characterization of the weighted Kubo-Ando geometric means as the only $2$-variable operator geometric means that are block additive, tensor multiplicative, and satisfy the arithmetic-geometric mean inequality. We also extend our results to composite quantum channel discrimination, and show an analogous optimality property of the weighted Kubo-Ando geometric means of two quantum channels, a notion that seems to be new. We extend this concept to defining the notion of superoperator perspective function and establish some of its basic properties, which may be of independent interest. + oai:arXiv.org:2503.13379v4 + quant-ph + cs.IT + math-ph + math.FA + math.IT + math.MP + Tue, 23 Dec 2025 00:00:00 -0500 + replace-cross http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Diego C\'ordoba, Alberto Enciso, Matthew Hernandez + P\'eter E. Frenkel, Mil\'an Mosonyi, P\'eter Vrana, Mih\'aly Weiner - Safeguarded Stochastic Polyak Step Sizes for Non-smooth Optimization: Robust Performance Without Small (Sub)Gradients - https://arxiv.org/abs/2512.02342 - arXiv:2512.02342v2 Announce Type: replace -Abstract: The stochastic Polyak step size (SPS) has proven to be a promising choice for stochastic gradient descent (SGD), delivering competitive performance relative to state-of-the-art methods on smooth convex and non-convex optimization problems, including deep neural network training. However, extensions of this approach to non-smooth settings remain in their early stages, often relying on interpolation assumptions or requiring knowledge of the optimal solution. In this work, we propose a novel SPS variant, Safeguarded SPS (SPS$_{safe}$), for the stochastic subgradient method, and provide rigorous convergence guarantees for non-smooth convex optimization with no need for strong assumptions. We further incorporate momentum into the update rule, yielding equally tight theoretical results. On non-smooth convex benchmarks, our experiments are consistent with the theoretical predictions on how the safeguard affects the convergence neighborhood. On deep neural networks the proposed step size achieves competitive performance to existing adaptive baselines and exhibits stable behavior across a wide range of problem settings. Moreover, in these experiments, the gradient norms under our step size do not collapse to (near) zero, indicating robustness to vanishing gradients. - oai:arXiv.org:2512.02342v2 - math.OC + Planning and Learning in Average Risk-aware MDPs + https://arxiv.org/abs/2503.17629 + arXiv:2503.17629v3 Announce Type: replace-cross +Abstract: For continuing tasks, average cost Markov decision processes have well-documented value and can be solved using efficient algorithms. However, it explicitly assumes that the agent is risk-neutral. In this work, we extend risk-neutral algorithms to accommodate the more general class of dynamic risk measures. Specifically, we propose a relative value iteration (RVI) algorithm for planning and design two model-free Q-learning algorithms, namely a generic algorithm based on the multi-level Monte Carlo (MLMC) method, and an off-policy algorithm dedicated to utility-based shortfall risk measures. Both the RVI and MLMC-based Q-learning algorithms are proven to converge to optimality. Numerical experiments validate our analysis, confirm empirically the convergence of the off-policy algorithm, and demonstrate that our approach enables the identification of policies that are finely tuned to the intricate risk-awareness of the agent that they serve. + oai:arXiv.org:2503.17629v3 cs.LG - stat.ML - Mon, 22 Dec 2025 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Dimitris Oikonomou, Nicolas Loizou - - - Indecomposability and beyond via the graph of edge dependencies - https://arxiv.org/abs/2512.05307 - arXiv:2512.05307v2 Announce Type: replace -Abstract: A polytope is called indecomposable if it cannot be expressed (non-trivially) as a Minkowski sum of other polytopes. Since the concept was introduced by Gale in 1954, several increasingly strong criteria have been developed to characterize indecomposability. Our first contribution is a new indecomposability criterion that unifies and generalizes most of the previous techniques. The key new ingredient of our method is the introduction of the graph of (implicit) edge dependencies, which has broader applications in the study of deformation cones of polytopes, beyond indecomposability. One of our main applications is providing new indecomposable deformed permutahedra that are not matroid polytopes. In 1970, Edmonds posed the problem of characterizing the extreme rays of the submodular cone, that is, indecomposable deformed permutahedra. Matroid polytopes from connected matroids give one such family of polytopes. We provide a new infinite disjoint family by taking certain graphical zonotopes and deeply truncating 1 or 2 specific vertices. In this way, we construct $2 \lfloor\frac{n-1}{2}\rfloor$ new indecomposable deformations of the $n$-permutahedron in $\mathbb{R}^n$. We also showcase other applications of our tools. For example, we use them to refute a conjecture by Smilansky (1987) stating that an indecomposable polytope needs to have few vertices with respect to its number of facets. We provide bounds on the dimension of deformation cones and characterize certain of their rays, we introduce parallelogramic Minkowski sums whose deformation cone can be written as a product of deformation cones, and we construct indecomposable polytopes via truncations and stackings. - oai:arXiv.org:2512.05307v2 - math.CO - Mon, 22 Dec 2025 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Arnau Padrol, Germain Poullot + math.OC + Tue, 23 Dec 2025 00:00:00 -0500 + replace-cross + http://creativecommons.org/licenses/by-nc-nd/4.0/ + Weikai Wang, Erick Delage - Affine diagram categories, algebras and monoids - https://arxiv.org/abs/2512.05510 - arXiv:2512.05510v2 Announce Type: replace -Abstract: We introduce and study several affine (=annular in this paper) versions of the classical diagram algebras such as Temperley-Lieb, partition, Brauer, Motzkin, rook Brauer, rook, planar partition, and planar rook algebras. We give generators and relation presentation for them and their associated categories, study their representation theory, and the asymptotic behavior of tensor products of their representations in the monoid case. Under a mild hypothesis, we also prove a previous conjecture concerning the asymptotic growth of the number of indecomposable summands in the tensor powers of representations for finite monoids. - oai:arXiv.org:2512.05510v2 - math.RT - math.CO - math.GR - Mon, 22 Dec 2025 00:00:00 -0500 - replace + Additivity, Haag duality, and non-invertible symmetries + https://arxiv.org/abs/2503.20863 + arXiv:2503.20863v2 Announce Type: replace-cross +Abstract: The algebraic approach to quantum field theory focuses on the properties of local algebras, whereas the study of (possibly non-invertible) global symmetries emphasizes global aspects of the theory and spacetime. We study connections between these two perspectives by examining how either of two core algebraic properties -- "additivity" or "Haag duality" -- is violated in a 1+1D CFT or lattice model restricted to the symmetric sector of a general global symmetry. For the Verlinde symmetry of a bosonic diagonal RCFT, we find that additivity is violated whenever the symmetry algebra contains an invertible element, while Haag duality is violated whenever it contains a non-invertible element. We find similar phenomena for the Kramers-Wannier and Rep(D$_8$) non-invertible symmetries on spin chains. + oai:arXiv.org:2503.20863v2 + hep-th + cond-mat.str-el + math.OA + quant-ph + Tue, 23 Dec 2025 00:00:00 -0500 + replace-cross http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - David He, Daniel Tubbenhauer + 10.1007/JHEP08(2025)009 + J. High Energ. Phys. 2025, 9 (2025) + Shu-Heng Shao, Jonathan Sorce, Manu Srivastava - A recognition criterion for lax-idempotent pseudomonads - https://arxiv.org/abs/2512.05631 - arXiv:2512.05631v2 Announce Type: replace -Abstract: We describe a simple criterion which makes it easy to recognise when a pseudomonad is lax-idempotent. The criterion concerns the behaviour of colax bilimits of arrows - certain comma objects - and is easy to verify in examples. Building on this, we obtain a new characterisation of lax-idempotent pseudomonads on 2-categories with colax bilimits of arrows. - oai:arXiv.org:2512.05631v2 - math.CT - Mon, 22 Dec 2025 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - John Bourke + Stable EEG Source Estimation for Standardized Kalman Filter using Change Rate Tracking + https://arxiv.org/abs/2504.01984 + arXiv:2504.01984v2 Announce Type: replace-cross +Abstract: This article focuses on the measurement and evolution modeling of Standardized Kalman filtering for brain activity estimation using non-invasive electroencephalography data. Here, we propose new parameter tuning and a model that uses the rate of change in the brain activity distribution to improve the stability of otherwise accurate estimates. Namely, we propose a backward-differentiation-based measurement model for the change rate, which notably improves the filtering-parametrization-stability of the tracking. Simulated data and data from a real subject were used in experiments. + oai:arXiv.org:2504.01984v2 + stat.AP + cs.NA + eess.SP + math.NA + Tue, 23 Dec 2025 00:00:00 -0500 + replace-cross + http://creativecommons.org/licenses/by-nc-sa/4.0/ + Joonas Lahtinen - Fast operator learning for mapping correlations - https://arxiv.org/abs/2512.09286 - arXiv:2512.09286v2 Announce Type: replace -Abstract: We propose a fast, optimization-free method for learning the transition operators of high-dimensional Markov processes. The central idea is to perform a Galerkin projection of the transition operator to a suitable set of low-order bases that capture the correlations between the dimensions. Such a discretized operator can be obtained from moments corresponding to our choice of basis without curse of dimensionality. Furthermore, by exploiting its low-rank structure and the spatial decay of correlations, we can obtain a compressed representation with computational complexity of order $\mathcal{O}(dN)$, where $d$ is the dimensionality and $N$ is the sample size. We further theoretically analyze the approximation error of the proposed compressed representation. We numerically demonstrate that the learned operator allows efficient prediction of future events and solving high-dimensional boundary value problems. This gives rise to a simple linear algebraic method for high-dimensional rare-events simulations. - oai:arXiv.org:2512.09286v2 - math.NA - cs.NA - Mon, 22 Dec 2025 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Yuehaw Khoo, Yuguan Wang, Siyao Yang + A stochastic method to estimate a zero-inflated two-part mixed model for human microbiome data + https://arxiv.org/abs/2504.15411 + arXiv:2504.15411v2 Announce Type: replace-cross +Abstract: Human microbiome studies based on genetic sequencing techniques produce compositional longitudinal data of the relative abundances of microbial taxa over time, allowing to understand, through mixed-effects modeling, how microbial communities evolve in response to clinical interventions, environmental changes, or disease progression. In particular, the Zero-Inflated Beta Regression (ZIBR) models jointly and over time the presence and abundance of each microbe taxon, considering the compositional nature of the data, its skewness, and the over-abundance of zeros. However, as for other complex random effects models, maximum likelihood estimation suffers from the intractability of likelihood integrals. Available estimation methods rely on log-likelihood approximation, which is prone to potential limitations such as biased estimates or unstable convergence. In this work we develop an alternative maximum likelihood estimation approach for the ZIBR model, based on the Stochastic Approximation Expectation Maximization (SAEM) algorithm. The proposed methodology allows to model unbalanced data, which is not always possible in existing approaches. We also provide estimations of the standard errors and the log-likelihood of the fitted model. The performance of the algorithm is established through simulation, and its use is demonstrated on two microbiome studies, showing its ability to detect changes in both presence and abundance of bacterial taxa over time and in response to treatment. + oai:arXiv.org:2504.15411v2 + stat.ME + math.ST + stat.TH + Tue, 23 Dec 2025 00:00:00 -0500 + replace-cross + http://creativecommons.org/licenses/by-nc-sa/4.0/ + John Barrera, Cristian Meza, Ana Arribas-Gil - Towards a Mathematical Theory of Adaptive Memory: From Time-Varying to Responsive Fractional Brownian Motion - https://arxiv.org/abs/2512.10057 - arXiv:2512.10057v2 Announce Type: replace -Abstract: This work develops a comprehensive mathematical theory for a class of stochastic processes whose local regularity adapts dynamically in response to their own state. We first introduce and rigorously analyze a time-varying fractional Brownian motion (TV-fBm) with a deterministic, H\"older-continuous Hurst exponent function. Key properties are established, including its exact variance scaling law, precise local increment asymptotics, local non-determinism, large deviation asymptotics for its increments, and a covariance structure that admits a closed-form hypergeometric representation. We then define a novel class of processes termed Responsive Fractional Brownian Motion (RfBm). Here,the Hurst exponent is governed by a Lipschitz-H\"older response function depending on the process state itself, creating an intrinsic feedback mechanism between state and memory. We establish the well-posedness of this definition, prove pathwise H\"older regularity of the induced instantaneous scaling exponent, and analyze associated cumulative memory processes along with their asymptotic convergence. The mathematical structure of RfBm naturally gives rise to a continuous-time, pathwise attention mechanism. We show that its kernel induces a well-defined attention weight distribution, derive fundamental bounds for these weights, and quantify the stability of attentional allocation through residence measures and volatility functionals. This work develops a stochastic-process-theoretic framework for concepts central to adaptive memory and content-sensitive information processing, offering a mathematically grounded perspective that may complement existing empirical approaches. - oai:arXiv.org:2512.10057v2 - math.PR - math-ph - math.MP - Mon, 22 Dec 2025 00:00:00 -0500 - replace + Personalized and Resilient Distributed Learning Through Opinion Dynamics + https://arxiv.org/abs/2505.14081 + arXiv:2505.14081v2 Announce Type: replace-cross +Abstract: In this paper, we address two practical challenges of distributed learning in multi-agent network systems, namely personalization and resilience. Personalization is the need of heterogeneous agents to learn local models tailored to their own data and tasks, while still generalizing well; on the other hand, the learning process must be resilient to cyberattacks or anomalous training data to avoid disruption. Motivated by a conceptual affinity between these two requirements, we devise a distributed learning algorithm that combines distributed gradient descent and the Friedkin-Johnsen model of opinion dynamics to fulfill both of them. We quantify its convergence speed and the neighborhood that contains the final learned models, which can be easily controlled by tuning the algorithm parameters to enforce a more personalized/resilient behavior. We numerically showcase the effectiveness of our algorithm on synthetic and real-world distributed learning tasks, where it achieves high global accuracy both for personalized models and with malicious agents compared to standard strategies. + oai:arXiv.org:2505.14081v2 + cs.MA + cs.LG + eess.SP + math.OC + Tue, 23 Dec 2025 00:00:00 -0500 + replace-cross http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Jiahao Jiang + Luca Ballotta, Nicola Bastianello, Riccardo M. G. Ferrari, Karl H. Johansson - Chern character of Schur bundles - https://arxiv.org/abs/2512.11657 - arXiv:2512.11657v2 Announce Type: replace -Abstract: Given a vector bundle $E$, we give an explicit formula to compute Chern classes of Schur bundles $\operatorname{S}^\alpha E$ in terms of those of $E$. - oai:arXiv.org:2512.11657v2 - math.AG - Mon, 22 Dec 2025 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Alessandro D'Andrea, Enrico Fatighenti, Claudio Onorati + A Tutorial on Beyond-Diagonal Reconfigurable Intelligent Surfaces: Modeling, Architectures, System Design and Optimization, and Applications + https://arxiv.org/abs/2505.16504 + arXiv:2505.16504v2 Announce Type: replace-cross +Abstract: Written by its inventors, this first tutorial on Beyond-Diagonal Reconfigurable Intelligent Surfaces (BD-RISs) provides the readers with the basics and fundamental tools necessary to appreciate, understand, and contribute to this emerging and disruptive technology. Conventional (Diagonal) RISs (D-RISs) are characterized by a diagonal scattering matrix $\mathbf{\Theta}$ such that the wave manipulation flexibility of D-RIS is extremely limited. In contrast, BD-RIS refers to a novel and general framework for RIS where its scattering matrix is not limited to be diagonal (hence, the ``beyond-diagonal'' terminology) and consequently, all entries of $\mathbf{\Theta}$ can potentially help shaping waves for much higher manipulation flexibility. This physically means that BD-RIS can artificially engineer and reconfigure coupling across elements of the surface thanks to inter-element reconfigurable components which allow waves absorbed by one element to flow through other elements. Consequently, BD-RIS opens the door to more general and versatile intelligent surfaces that subsumes existing RIS architectures as special cases. In this tutorial, we share all the secret sauce to model, design, and optimize BD-RIS and make BD-RIS transformative in many different applications. Topics discussed include physics-consistent and multi-port network-aided modeling; transmitting, reflecting, hybrid, and multi-sector mode analysis; reciprocal and non-reciprocal architecture designs and optimal performance-complexity Pareto frontier of BD-RIS; signal processing, optimization, and channel estimation for BD-RIS; hardware impairments (discrete-value impedance and admittance, lossy interconnections and components, wideband effects, mutual coupling) of BD-RIS; benefits and applications of BD-RIS in communications, sensing, power transfer. + oai:arXiv.org:2505.16504v2 + eess.SP + cs.IT + math.IT + Tue, 23 Dec 2025 00:00:00 -0500 + replace-cross + http://creativecommons.org/licenses/by/4.0/ + Hongyu Li, Matteo Nerini, Shanpu Shen, Bruno Clerckx - Spectral Barron spaces of vector-valued functions on compact groups - https://arxiv.org/abs/2512.12382 - arXiv:2512.12382v2 Announce Type: replace -Abstract: In this article, we study spectral Barron spaces whose elements are made up of some vector-valued functions on a compact group whose Fourier transforms admit a certain summability property. We investigate their functional properties and some continuous embeddings of these spaces with respect to other function spaces among which are Sobolev spaces of vector-valued functions and the space of bounded vector-valued functions on compact groups. - oai:arXiv.org:2512.12382v2 - math.FA - Mon, 22 Dec 2025 00:00:00 -0500 - replace + Tight Generalization of Robertson-Type Uncertainty Relations + https://arxiv.org/abs/2505.19861 + arXiv:2505.19861v2 Announce Type: replace-cross +Abstract: We establish the tightest possible Robertson-type preparation uncertainty relation, which explicitly depends on the eigenvalues of the quantum state. The conventional constant $ \tfrac{1}{4} $ is replaced by a state-dependent coefficient $\frac{(\lambda_{\max} + \lambda_{\min})^2}{4(\lambda_{\max} - \lambda_{\min})^2}$, where $ \lambda_{\max} $ and $ \lambda_{\min}$ denote the largest and smallest eigenvalues of the density operator $\rho$, respectively. This coefficient is optimal among all Robertson-type generalizations and does not admit further improvement.Our relation becomes more pronounced as the quantum state becomes more mixed, capturing a trade-off in quantum uncertainty that the conventional Robertson's relation fails to detect. In addition, our result also provides a strict generalization of the Schr\"oedinger's uncertainty relation, showing that the uncertainty trade-off is governed by the sum of the covariance term and a state-dependent improvement over the Robertson bound. As applications, we also refine error-disturbance trade-offs by incorporating spectral information of both the system and the measuring apparatus,thereby generalizing the Arthurs--Goodman and Ozawa inequalities. + oai:arXiv.org:2505.19861v2 + quant-ph + cond-mat.stat-mech + hep-th + math-ph + math.MP + Tue, 23 Dec 2025 00:00:00 -0500 + replace-cross http://creativecommons.org/licenses/by/4.0/ - Yaogan Mensah, Isiaka Aremua + Gen Kimura, Aina Mayumi, Haruki Yamashita - Distribution questions for isogeny graphs over finite fields - https://arxiv.org/abs/2512.14469 - arXiv:2512.14469v2 Announce Type: replace -Abstract: In the first part of the paper, we fix a non-CM elliptic curve $E/\mathbb{Q}$ and an odd prime $\ell$ and investigate the distribution of invariants associated to the $\ell$-volcano containing the reduction $E_p$, as $p$ ranges over primes of good ordinary reduction. Let $H(p)$ be the height of the volcano and let $d'(p)$ denote the relative position of $j(E_p)$ above the floor, and let $r\ge 0$ be an integer. Assuming that the $\ell$-adic Galois representation attached to $E$ is surjective, we derive an explicit formula for the natural density of primes $p$ for which $H(p)=r$ (resp.\ $d'(p)=r$). In the non-surjective case, we show that all sufficiently large heights occur with positive density. In the second part of the paper, we analyze the distribution of $\ell$-volcano heights over a finite field $\mathbb{F}_q$ and consider the limit as $q\to\infty$. Using analytic estimates for sums of Hurwitz class numbers in arithmetic progressions, we compute exact limiting densities for ordinary elliptic curves whose $\ell$-isogeny graph has a prescribed height $r$. - oai:arXiv.org:2512.14469v2 - math.NT - math.CO - Mon, 22 Dec 2025 00:00:00 -0500 - replace + Chimera states on m-directed hypergraphs + https://arxiv.org/abs/2506.12511 + arXiv:2506.12511v4 Announce Type: replace-cross +Abstract: Chimera states are synchronization patterns in which coherent and incoherent regions coexist in systems of identical oscillators. This elusive phenomenon has attracted significant interest and has been widely analyzed, revealing several types of dynamical states. Most studies involve reciprocal pairwise couplings, where each oscillator exerts and receives the same interaction from neighboring ones, thus being modeled via symmetric networks. However, real-world systems often exhibit non-reciprocal, non-pairwise (many-body) interactions. Previous studies have shown that chimera states are more elusive in the presence of non-reciprocal pairwise interactions, while they are easier to observe when the interactions are reciprocal and higher-order (many-body). In this work, we investigate the emergence of chimera states on non-reciprocal higher-order structures, called mdirected hypergraphs, which we compare with their corresponding networks, and we observe that chimera state and specifically amplitude-mediated chimeras can emerge due to directionality, which had not been previously observed in the absence of directionality. We also compare the effect of non-reciprocal interactions between higher-order and pairwise couplings, and we find numerically that chimera states appear over a broader parameter range when considering higher-order interactions than in the corresponding network case, demonstrating the impact of directionality and the effect of higher-order interactions. Finally, the nature of phase chimeras has been further validated through phase reduction theory. + oai:arXiv.org:2506.12511v4 + nlin.PS + math-ph + math.MP + nlin.AO + nlin.CD + physics.comp-ph + Tue, 23 Dec 2025 00:00:00 -0500 + replace-cross http://creativecommons.org/licenses/by/4.0/ - Anwesh Ray + Rommel Tchinda Djeudjo, Timoteo Carletti, Hiroya Nakao, Riccardo Muolo - Optimization of gridding algorithms for FFT by vector optimization - https://arxiv.org/abs/2512.14914 - arXiv:2512.14914v3 Announce Type: replace -Abstract: The Fast Fourier Transform (FFT) is widely used in applications such as MRI, CT, and interferometry; however, because of its dependence on uniformly sampled data, it requires the use of gridding techniques for practical implementation. The performance of these algorithms strongly depends on the choice of the gridding kernel, with the first prolate spheroidal wave function (PSWF) regarded as optimal. This work redefines kernel optimality through the lens of vector optimization (VO), introducing a rigorous framework that characterizes optimal kernels as Pareto-efficient solutions of an error shape operator. We establish the continuity of such operator, study the existence of solutions, and propose a novel methodology to construct kernels tailored to a desired target error function. The approach is implemented numerically via interior-point optimization. Comparative experiments demonstrate that the proposed kernels outperform both the PSWF and the state-of-the-art methods (MIRT-NUFFT) in specific regions of interest, achieving orders-of-magnitude improvements in mean absolute errors. These results confirm the potential of VO-based kernel design to provide customized accuracy profiles aligned with application-specific requirements. Future research will extend this framework to multidimensional cases and relative error minimization, with potential integration of machine learning for adaptive target error selection. - oai:arXiv.org:2512.14914v3 - math.NA - cs.NA - Mon, 22 Dec 2025 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by-sa/4.0/ - Federico Achini, Paola Causin, Sara Vanini, Ke Chen, Simone Scacchi + Lack-of-fit reduction in the path-integral formalism + https://arxiv.org/abs/2506.20242 + arXiv:2506.20242v2 Announce Type: replace-cross +Abstract: We present a new formulation of the lack-of-fit reduction in non-equilibrium thermodynamics using the path-integral formalism. The formulation is based on the Onsager-Machlup variational principle, and it allows us to find reduced dynamical equations by minimizing information discrepancy with respect to the detailed evolution. The reduced evolution consists of a Hamiltonian vector field and a gradient flow. The reduction method is illustrated on the Kac-Zwanzig model, where we show how irreversibility emerges from purely Hamiltonian evolution by ignoring some degrees of freedom. We also show how to generalize the Fisher information matrix and Kullback-Leibler divergence between two probability distributions to the case when the two distributions are related by the principle of maximum entropy, even in the case when the entropy is not of Boltzmann-Gibbs type (for instance Tsallis-Havrda-Charvat entropy). + oai:arXiv.org:2506.20242v2 + cond-mat.stat-mech + math-ph + math.MP + Tue, 23 Dec 2025 00:00:00 -0500 + replace-cross + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Katerina Mlada, Michal Pavelka, Vaclav Klika - Isomorphism between Jacobi forms of index $D_{2n+1}$ and elliptic modular forms of level $2$ - https://arxiv.org/abs/2512.15012 - arXiv:2512.15012v2 Announce Type: replace -Abstract: There are three aims in this paper: (i) We show an isomorphism between Jacobi forms of index $D_{2n+1}$ (lattice index) and elliptic modular forms of level $2$. (ii) We give an explicit formula of Fourier coefficients of Jacobi-Eisenstein series of index $D_{2n+1}$. (iii) We construct a holomorphic modular form of weight $3/2$ of level $8$ from the Zagier Eisenstein series $\mathscr{F}$ of weight $3/2$ of level $4$. Moreover, we show that the four functions $E^*_2$, $\eta^3$, $\theta^3$ and $\mathscr{F}$ have essentially the same Hecke eigenvalue $1+p$ for any odd prime $p$, where $E^*_2$ is the non-holomorphic Eisenstein series of weight $2$, $\eta$ is the Dedekind eta-function and $\theta$ is the usual theta function. This fact follows from a special case of the isomorphism of (i). - As an application, we give a formula for a sum of the numbers $r_3(n)$, where $r_3(n)$ is the number of representations of an integer $n \geq 0$ as a sum of $3$ squares. - oai:arXiv.org:2512.15012v2 - math.NT - Mon, 22 Dec 2025 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Shuichi Hayashida + Classification of four-qubit pure codes and five-qubit absolutely maximally entangled states + https://arxiv.org/abs/2507.02185 + arXiv:2507.02185v3 Announce Type: replace-cross +Abstract: We prove that every 5-qubit absolutely maximally entangled (AME) state is equivalent by a local unitary transformation to a point in the unique ((5,2,3)) quantum error correcting code C. Furthermore, two points in C are equivalent if and only if they are related by a group of order 24 acting on C. There exists a set of 3 invariant polynomials that separates equivalence classes of 5-qubit AME states. We also show that every 4-qubit pure code is equivalent to a subspace of the unique ((4,4,2)) and construct an infinite family of 3-uniform n-qubit states for even $n\geq 6$. The proofs rely heavily on results from Vinberg and classical invariant theory. + oai:arXiv.org:2507.02185v3 + quant-ph + math.RT + Tue, 23 Dec 2025 00:00:00 -0500 + replace-cross + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Ian Tan - A New Fast Finite Difference Scheme for Tempered Time Fractional Advection-Dispersion Equation with a Weak Singularity at Initial Time - https://arxiv.org/abs/2512.15141 - arXiv:2512.15141v3 Announce Type: replace -Abstract: In this paper, we propose a new second-order fast finite difference scheme in time for solving the Tempered Time Fractional Advection-Dispersion Equation. Under the assumption that the solution is nonsmooth at the initial time, we investigate the uniqueness, stability, and convergence of the scheme. Furthermore, we prove that the scheme achieves second-order convergence in both time and space. Finally, corresponding numerical examples are provided. - oai:arXiv.org:2512.15141v3 - math.NA + On the statistical convergence of N-body simulations of the Solar System + https://arxiv.org/abs/2507.04987 + arXiv:2507.04987v2 Announce Type: replace-cross +Abstract: Most direct N-body integrations of planetary systems use a symplectic integrator with a fixed timestep. A large timestep is desirable in order to speed up the numerical simulations. However, simulations yield unphysical results if the timestep is too large. Surprisingly, no systematic convergence study has been performed on long (Gyr) timescales. In this paper we present numerical experiments to determine the minimum timestep one has to use in long-term integrations of the Solar System in order to recover the system's fundamental secular frequencies and instability rate. We find that timesteps of up to 32 days, i.e. a third of Mercury's orbital period, yield physical results in an ensemble of 5 Gyr integrations. We argue that the chaotic diffusion that drives the Solar System's long-term evolution dominates over numerical diffusion and timestep resonances. Our results bolster confidence that the statistical results of most simulations in the literature are indeed physical and provide guidance on how to run time and energy efficient simulations while making sure results can be trusted. + oai:arXiv.org:2507.04987v2 + astro-ph.EP + astro-ph.IM cs.NA - math.AP - Mon, 22 Dec 2025 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Liangcai Huang, Shujuan L\"u + math.NA + Tue, 23 Dec 2025 00:00:00 -0500 + replace-cross + http://creativecommons.org/licenses/by-nc-sa/4.0/ + Hanno Rein, Garett Brown, Mei Kanda - On the Codebook Design for NOMA Schemes from Bent Functions - https://arxiv.org/abs/2512.16560 - arXiv:2512.16560v2 Announce Type: replace -Abstract: Uplink grant-free non-orthogonal multiple access (NOMA) is a promising technology for massive connectivity with low latency and high energy efficiency. In code-domain NOMA schemes, the requirements boil down to the design of codebooks that contain a large number of spreading sequences with low peak-to-average power ratio (PAPR) while maintaining low coherence. When employing binary Golay sequences with guaranteed low PAPR in the design, the fundamental problem is to construct a large set of $n$-variable quadratic bent or near-bent functions in a particular form such that the difference of any two is bent for even $n$ or near-bent for odd $n$ to achieve optimally low coherence. In this work, we propose a theoretical construction of NOMA codebooks by applying a recursive approach to those particular quadratic bent functions in smaller dimensions. The proposed construction yields desired NOMA codebooks that contain $6\cdot N$ Golay sequences of length $N=2^{4m}$ for any positive integer $m$ and have the lowest possible coherence $1/\sqrt{N}$. - oai:arXiv.org:2512.16560v2 - math.CO - cs.DM - cs.IT - math.IT - Mon, 22 Dec 2025 00:00:00 -0500 - replace + A-Type Open ${\rm SL}(2,\mathbb{C})$ Spin Chain + https://arxiv.org/abs/2507.09568 + arXiv:2507.09568v3 Announce Type: replace-cross +Abstract: For the noncompact open ${\rm SL}(2,\mathbb{C})$ spin chain, the eigenfunctions of the special matrix element of monodromy matrix are constructed. The key ingredients of the whole construction are local Yang-Baxter $\mathcal{R}$-operators, $Q$-operator and raising operators obtained by reduction from the $Q$-operator. The calculation of various scalar products and the proof of orthogonality are based on the properties of $Q$-operator and demonstrate its hidden role. The symmetry of eigenfunctions with respect to reflection of the spin variable $s \to 1-s$ is established. The Mellin-Barnes representation for eigenfunctions is derived and equivalence with initial coordinate representation is proved. The transformation from one representation to another is grounded on the application of $A$-type Gustafson integral generalized to the complex field. + oai:arXiv.org:2507.09568v3 + hep-th + math-ph + math.MP + Tue, 23 Dec 2025 00:00:00 -0500 + replace-cross http://creativecommons.org/licenses/by/4.0/ - Chunlei Li, Constanza Riera, Palash Sarkar, Pantelimon Stanica + 10.3842/SIGMA.2025.107 + SIGMA 21 (2025), 107, 48 pages + Pavel V. Antonenko, Sergey \'E. Derkachov, Pavel A. Valinevich - The capillary Christoffel-Minkowski problem - https://arxiv.org/abs/2512.16655 - arXiv:2512.16655v2 Announce Type: replace -Abstract: In this article, we introduce a $k$-th capillary area measure for capillary convex bodies in the Euclidean half-space, which serves as a boundary counterpart to the classical concept of area measure (see, e.g., \cite[Chapter 8]{Sch}). We then propose a Christoffel-Minkowski problem for capillary convex bodies, to find a capillary convex body in the Euclidean half-space with a prescribed $k$-th capillary area measure. This problem is equivalent to solving a Hessian-type equation with a Robin boundary value condition. We then establish the existence and uniqueness of a smooth solution under a natural sufficient condition. - oai:arXiv.org:2512.16655v2 - math.AP - math.DG - Mon, 22 Dec 2025 00:00:00 -0500 - replace + Wi-Fi: Twenty-Five Years and Counting + https://arxiv.org/abs/2507.09613 + arXiv:2507.09613v2 Announce Type: replace-cross +Abstract: Today, Wi-Fi is over 25 years old. Yet, despite sharing the same branding name, today's Wi-Fi boasts entirely new capabilities that were not even on the roadmap 25 years ago. This article aims to provide a holistic and comprehensive technical and historical tutorial on Wi-Fi, beginning with IEEE 802.11b (Wi-Fi 1) and looking forward to IEEE 802.11bn (Wi-Fi 8). This is the first tutorial article to span these eight generations. Rather than a generation-by-generation exposition, we describe the key mechanisms that have advanced Wi-Fi. We begin by discussing spectrum allocation and coexistence, and detailing the IEEE 802.11 standardization cycle. Second, we provide an overview of the physical layer and describe key elements that have enabled data rates to increase by over 1,000x. Third, we describe how Wi-Fi Medium Access Control has been enhanced from the original Distributed Coordination Function to now include capabilities spanning from frame aggregation to wideband spectrum access. Fourth, we describe how Wi-Fi 5 first broke the one-user-at-a-time paradigm and introduced multi-user access. Fifth, given the increasing use of mobile, battery-powered devices, we describe Wi-Fi's energy-saving mechanisms over the generations. Sixth, we discuss how Wi-Fi was enhanced to seamlessly aggregate spectrum across 2.4 GHz, 5 GHz, and 6 GHz bands to improve throughput, reliability, and latency. Finally, we describe how Wi-Fi enables nearby Access Points to coordinate in order to improve performance and efficiency. In the Appendix, we further discuss Wi-Fi developments beyond 802.11bn, including integrated mmWave operations, sensing, security and privacy extensions, and the adoption of AI/ML. + oai:arXiv.org:2507.09613v2 + cs.NI + cs.IT + eess.SP + math.IT + Tue, 23 Dec 2025 00:00:00 -0500 + replace-cross http://creativecommons.org/licenses/by/4.0/ - Xinqun Mei, Guofang Wang, Liangjun Weng + Giovanni Geraci, Francesca Meneghello, Francesc Wilhelmi, David Lopez-Perez, I\~naki Val, Lorenzo Galati Giordano, Carlos Cordeiro, Monisha Ghosh, Edward Knightly, Boris Bellalta - A note on the triple product property for finite groups with abelian normal subgroups of prime index - https://arxiv.org/abs/2512.16730 - arXiv:2512.16730v2 Announce Type: replace -Abstract: Three non-empty subsets $S,T,U$ of a group $G$ are said to satisfy the triple product property (TPP) if, for elements $s,s' \in S$, $t,t' \in T$, $u,u' \in U$, the equation $s's^{-1}t't^{-1}u'u^{-1}=1$ holds if and only if $s = s'$, $t = t'$, $u = u'$. In this case $(S,T,U)$ is called a TPP triple of $G$ and $|S||T||U|$ is called the size of the triple. If $G$ is a finite group the triple product ratio of $G$ can be defined as the quantity $\rho(G) := \frac{\beta(G)}{|G|}$, where $\beta(G)$ is the largest size of a TPP triple of $G$, and a special case of this, the subgroup triple product ratio, is the quantity $\rho_0(G) := \frac{\beta_0(G)}{|G|}$, where $\beta_0(G)$ is the largest size of a TPP triple of $G$ composed only of subgroups. There is a conjecture that $\rho(G) \leq \frac{4}{3}$ if $G$ contains a cyclic subgroup of index $2$ \citep[Conjecture 7.6]{HM}. This note proves a version of this conjecture for subgroups by showing that $\rho_0(G) \leq \frac{p^2}{2p-1}$ if $G$ is any finite group which contains an abelian normal subgroup of prime index $p$. - oai:arXiv.org:2512.16730v2 - math.GR - Mon, 22 Dec 2025 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Sandeep R. Murthy + Optimal Qubit Purification and Unitary Schur Sampling via Random SWAP Tests + https://arxiv.org/abs/2508.05046 + arXiv:2508.05046v2 Announce Type: replace-cross +Abstract: The goal of qubit purification is to combine multiple noisy copies of an unknown pure quantum state to obtain one or more copies that are closer to the pure state. We show that a simple protocol based solely on random SWAP tests achieves the same fidelity as the Schur transform, which is optimal. This protocol relies only on elementary two-qubit SWAP tests, which project a pair of qubits onto the singlet or triplet subspaces, to identify and isolate singlet pairs, and then proceeds with the remaining qubits. For a system of $n$ qubits, we show that after approximately $T \approx n \ln n$ random SWAP tests, a sharp transition occurs: the probability of detecting any new singlet decreases exponentially with $T$. Similarly, the fidelity of each remaining qubit approaches the optimal value given by the Schur transform, up to an error that is exponentially small in $T$. More broadly, this protocol achieves what is known as weak Schur sampling and unitary Schur sampling with error $\epsilon$, after only $2n \ln(n \epsilon^{-1})$ SWAP tests. That is, it provides a lossless method for extracting any information invariant under permutations of qubits, making it a powerful subroutine for tasks such as quantum state tomography and metrology. + oai:arXiv.org:2508.05046v2 + quant-ph + cond-mat.dis-nn + cond-mat.stat-mech + math-ph + math.MP + nucl-th + Tue, 23 Dec 2025 00:00:00 -0500 + replace-cross + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Shrigyan Brahmachari, Austin Hulse, Henry D. Pfister, Iman Marvian - U(1) Gauge Potentials on de Sitter Spacetime - https://arxiv.org/abs/1304.0568 - arXiv:1304.0568v3 Announce Type: replace-cross -Abstract: The smooth 1-form Verma module of $\mathfrak{so}(1,4)$ is acquired, which can be regarded as the U(1) gauge potential on de Sitter spacetime. It is shown that electromagnetic fields could not be source free on de Sitter background. - oai:arXiv.org:1304.0568v3 + Generalized Symmetries and Deformations of Symmetric Product Orbifolds + https://arxiv.org/abs/2509.12180 + arXiv:2509.12180v3 Announce Type: replace-cross +Abstract: We construct generalized symmetries in two-dimensional symmetric product orbifold CFTs $\text{Sym}^N(\mathcal{T}),$ for a generic seed CFT $\mathcal{T}$. These symmetries are more general than the universal and maximally symmetric ones previously constructed. We show that, up to one fine-tuned example when the number of copies $N$ equals four, the only symmetries that can be preserved under twisted sector marginal deformations are invertible and maximally symmetric. The results are obtained in two ways. First, using the mathematical machinery of $G$-equivariantization of fusion categories, and second, via the projector construction of topological defect lines. As an application, we classify all preserved symmetries in symmetric product orbifold CFTs with the seed CFT given by any $A$-series $\mathcal{N}=(2,2)$ minimal model. We comment on the implications of our results for holography. + oai:arXiv.org:2509.12180v3 hep-th - gr-qc + cond-mat.str-el math-ph math.MP - Mon, 22 Dec 2025 00:00:00 -0500 + Tue, 23 Dec 2025 00:00:00 -0500 replace-cross http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Bin Zhou, Shi-Bei Kong, Peng Zhao + Nathan Benjamin, Suzanne Bintanja, Yu-Jui Chen, Michael Gutperle, Conghuan Luo, Dikshant Rathore - Foundations for an Abstract Proof Theory in the Context of Horn Rules - https://arxiv.org/abs/2304.05697 - arXiv:2304.05697v2 Announce Type: replace-cross -Abstract: We introduce a novel, logic-independent framework for the study of sequent-style proof systems, which covers a number of proof-theoretic formalisms and concrete proof systems that appear in the literature. In particular, we introduce a generalized form of sequents, dubbed 'g-sequents,' which are taken to be binary graphs of typical, Gentzen-style sequents. We then define a variety of 'inference rule types' as sets of operations that act over such objects, and define 'abstract (sequent) calculi' as pairs consisting of a set of g-sequents together with a finite set of operations. Our approach permits an analysis of how certain inference rule types interact in a general setting, demonstrating under what conditions rules of a specific type can be permuted with or simulated by others, and being applicable to any sequent-style proof system that fits within our framework. We then leverage our permutation and simulation results to establish generic calculus and proof transformation algorithms, which show that every abstract calculus can be effectively transformed into a lattice of polynomially equivalent abstract calculi. We determine the complexity of computing this lattice and compute the relative sizes of proofs and sequents within distinct calculi of a lattice. We recognize that top and bottom elements in lattices correspond to many known deep-inference nested sequent systems and labeled sequent systems (respectively) for logics characterized by Horn properties. - oai:arXiv.org:2304.05697v2 - cs.LO - cs.DM - cs.DS - math.LO - Mon, 22 Dec 2025 00:00:00 -0500 + GenUQ: Predictive Uncertainty Estimates via Generative Hyper-Networks + https://arxiv.org/abs/2509.21605 + arXiv:2509.21605v2 Announce Type: replace-cross +Abstract: Operator learning is a recently developed generalization of regression to mappings between functions. It promises to drastically reduce expensive numerical integration of PDEs to fast evaluations of mappings between functional states of a system, i.e., surrogate and reduced-order modeling. Operator learning has already found applications in several areas such as modeling sea ice, combustion, and atmospheric physics. Recent approaches towards integrating uncertainty quantification into the operator models have relied on likelihood based methods to infer parameter distributions from noisy data. However, stochastic operators may yield actions from which a likelihood is difficult or impossible to construct. In this paper, we introduce, GenUQ, a measure-theoretic approach to UQ that avoids constructing a likelihood by introducing a generative hyper-network model that produces parameter distributions consistent with observed data. We demonstrate that GenUQ outperforms other UQ methods in three example problems, recovering a manufactured operator, learning the solution operator to a stochastic elliptic PDE, and modeling the failure location of porous steel under tension. + oai:arXiv.org:2509.21605v2 + cs.LG + cs.NA + math.NA + stat.ML + Tue, 23 Dec 2025 00:00:00 -0500 replace-cross http://creativecommons.org/licenses/by/4.0/ - Tim S. Lyon, Piotr Ostropolski-Nalewaja + Tian Yu Yen, Reese E. Jones, Ravi G. Patel - Sparse Anomaly Detection Across Referentials: A Rank-Based Higher Criticism Approach - https://arxiv.org/abs/2312.04924 - arXiv:2312.04924v2 Announce Type: replace-cross -Abstract: Detecting anomalies in large sets of observations is crucial in various applications, such as epidemiological studies, gene expression studies, and systems monitoring. We consider settings where the units of interest result in multiple independent observations from potentially distinct referentials. Scan statistics and related methods are commonly used in such settings, but rely on stringent modeling assumptions for proper calibration. We instead propose a rank-based variant of the higher criticism statistic that only requires independent observations originating from ordered spaces. We show under what conditions the resulting methodology is able to detect the presence of anomalies. These conditions are stated in a general, non-parametric manner, and depend solely on the probabilities of anomalous observations exceeding nominal observations. The analysis requires a refined understanding of the distribution of the ranks under the presence of anomalies, and in particular of the rank-induced dependencies. The methodology is robust against heavy-tailed distributions through the use of ranks. Within the exponential family and a family of convolutional models, we analytically quantify the asymptotic performance of our methodology and the performance of the oracle, and show the difference is small for many common models. Simulations confirm these results. We show the applicability of the methodology through an analysis of quality control data of a pharmaceutical manufacturing process. - oai:arXiv.org:2312.04924v2 - stat.ME - math.ST - stat.TH - Mon, 22 Dec 2025 00:00:00 -0500 + Bridging the Gap Between Scientific Laws Derived by AI Systems and Canonical Knowledge via Abductive Inference with AI-Noether + https://arxiv.org/abs/2509.23004 + arXiv:2509.23004v2 Announce Type: replace-cross +Abstract: Advances in AI have shown great potential in contributing to the acceleration of scientific discovery. Symbolic regression can fit interpretable models to data, but these models are not necessarily derivable from established theory. Recent systems (e.g., AI-Descartes, AI-Hilbert) enforce derivability from prior knowledge. However, when existing theories are incomplete or incorrect, these machine-generated hypotheses may fall outside the theoretical scope. Automatically finding corrections to axiom systems to close this gap remains a central challenge in scientific discovery. We propose a solution: an open-source algebraic geometry-based system that, given an incomplete axiom system expressible as polynomials and a hypothesis that the axioms cannot derive, generates a minimal set of candidate axioms that, when added to the theory, provably derive the (possibly noisy) hypothesis. We illustrate the efficacy of our approach by showing that it can reconstruct key axioms required to derive the carrier-resolved photo-Hall effect, Einstein's relativistic laws, and several other laws. + oai:arXiv.org:2509.23004v2 + cs.AI + cs.SC + math.AG + Tue, 23 Dec 2025 00:00:00 -0500 replace-cross http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - 10.1214/24-AOS2477 - Annals of Statistics 2025, Vol. 53, No. 2, 676--702 - Ivo V. Stoepker, Rui M. Castro, Ery Arias-Castro + Karan Srivastava, Sanjeeb Dash, Ryan Cory-Wright, Barry Trager, Cristina Cornelio, Lior Horesh - SCAFFLSA: Taming Heterogeneity in Federated Linear Stochastic Approximation and TD Learning - https://arxiv.org/abs/2402.04114 - arXiv:2402.04114v3 Announce Type: replace-cross -Abstract: In this paper, we analyze the sample and communication complexity of the federated linear stochastic approximation (FedLSA) algorithm. We explicitly quantify the effects of local training with agent heterogeneity. We show that the communication complexity of FedLSA scales polynomially with the inverse of the desired accuracy $\epsilon$. To overcome this, we propose SCAFFLSA a new variant of FedLSA that uses control variates to correct for client drift, and establish its sample and communication complexities. We show that for statistically heterogeneous agents, its communication complexity scales logarithmically with the desired accuracy, similar to Scaffnew. An important finding is that, compared to the existing results for Scaffnew, the sample complexity scales with the inverse of the number of agents, a property referred to as linear speed-up. Achieving this linear speed-up requires completely new theoretical arguments. We apply the proposed method to federated temporal difference learning with linear function approximation and analyze the corresponding complexity improvements. - oai:arXiv.org:2402.04114v3 - stat.ML + Network-Optimised Spiking Neural Network for Event-Driven Networking + https://arxiv.org/abs/2509.23516 + arXiv:2509.23516v3 Announce Type: replace-cross +Abstract: Time-critical networking requires low-latency decisions from sparse and bursty telemetry, where fixed-step neural inference waste computation. We introduce Network-Optimised Spiking (NOS), a two-state neuron whose variables correspond to normalised queue occupancy and a recovery resource. NOS combines a saturating excitability nonlinearity for finite buffers, service and damping leaks, graph-local inputs with per-link gates and delays, and differentiable resets compatible with surrogate gradients and neuromorphic deployment. We establish existence and uniqueness of subthreshold equilibria, derive Jacobian-based local stability tests, and obtain a scalar network stability threshold that separates topology from node physics through a Perron-mode spectral condition. A stochastic arrival model aligned with telemetry smoothing links NOS responses to classical queueing behaviour while explaining increased variability near stability margins. Across chain, star, and scale-free graphs, NOS improves early-warning F1 and detection latency over MLP, RNN, GRU, and temporal-GNN baselines under a common residual-based protocol, while providing practical calibration and stability rules suited to resource-constrained networking deployments. Code and Demos: https://mbilal84.github.io/nos-snn-networking/ + oai:arXiv.org:2509.23516v3 + cs.NE cs.LG + cs.NI math.OC - Mon, 22 Dec 2025 00:00:00 -0500 + Tue, 23 Dec 2025 00:00:00 -0500 replace-cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Paul Mangold, Sergey Samsonov, Safwan Labbi, Ilya Levin, Reda Alami, Alexey Naumov, Eric Moulines + http://creativecommons.org/licenses/by/4.0/ + Muhammad Bilal - Operational Dosage: Implications of Capacity Constraints for the Design and Interpretation of Experiments - https://arxiv.org/abs/2407.21322 - arXiv:2407.21322v3 Announce Type: replace-cross -Abstract: We study RCTs that evaluate the impact of service interventions, for example, teachers or advisors conducting proactive outreach to at-risk students, medical providers giving medication adherence support by calling or texting, or social workers that conduct home visits. A defining feature of service interventions is that they are delivered by a capacity-constrained resource -- teachers, healthcare providers, or social workers -- whose limited availability creates causal inference complications. Because participants share a finite service capacity, adding more participants can reduce the timeliness or intensity of the service that others receive, introducing interference across participants. This generates hidden variation in the treatment itself, which we term operational dosage. We provide a mathematical model of service interventions using techniques from queueing theory and study the impact of capacity constraints on experimental outcomes. Our main insight is that treatment effects are both capacity- and sample-size-dependent, as well as decreasing in sample size once a critical threshold is exceeded. Interestingly, an implication is that statistical power of service intervention RCTs peaks at intermediate sample sizes -- directly contradicting conventional power calculations that assume monotonically increasing power with sample size. We instantiate our insights using simulations calibrated to a real-world trial evaluating a behavioral health intervention for tuberculosis patients in Kenya. Our simulation results suggest that a trial with high service capacity but limited sample size can obtain the same statistical power as a trial with lower service capacity but large sample size. Taken together, our results highlight the importance of capacity selection in experiment design and provide a mechanism for why experiments may fail to replicate or perform at scale. - oai:arXiv.org:2407.21322v3 - stat.ME - math.OC - Mon, 22 Dec 2025 00:00:00 -0500 + Geometric Learning of Canonical Parameterizations of $2D$-curves + https://arxiv.org/abs/2509.26070 + arXiv:2509.26070v2 Announce Type: replace-cross +Abstract: Most datasets encountered in computer vision and medical applications present symmetries that should be taken into account in classification tasks. A typical example is the symmetry by rotation and/or scaling in object detection. A common way to build neural networks that learn the symmetries is to use data augmentation. In order to avoid data augmentation and build more sustainable algorithms, we present an alternative method to mod out symmetries based on the notion of section of a principal fiber bundle. This framework allows the use of simple metrics on the space of objects in order to measure dissimilarities between orbits of objects under the symmetry group. Moreover, the section used can be optimized to maximize separation of classes. We illustrate this methodology on a dataset of contours of objects for the groups of translations, rotations, scalings and reparameterizations. In particular, we present a $2$-parameter family of canonical parameterizations of curves, containing the constant-speed parameterization as a special case, which we believe is interesting in its own right. We hope that this simple application will serve to convey the geometric concepts underlying this method, which have a wide range of possible applications. The code is available at the following link: $\href{https://github.com/GiLonga/Geometric-Learning}{https://github.com/GiLonga/Geometric-Learning}$. A tutorial notebook showcasing an application of the code to a specific dataset is available at the following link: $\href{https://github.com/ioanaciuclea/geometric-learning-notebook}{https://github.com/ioanaciuclea/geometric-learning-notebook}$ + oai:arXiv.org:2509.26070v2 + cs.CV + math.DG + Tue, 23 Dec 2025 00:00:00 -0500 + replace-cross + http://creativecommons.org/licenses/by/4.0/ + Ioana Ciuclea, Giorgio Longari, Alice Barbara Tumpach + + + One-dimensional long-range Ising model: two (almost) equivalent approximations + https://arxiv.org/abs/2510.02458 + arXiv:2510.02458v2 Announce Type: replace-cross +Abstract: We investigate the critical behavior of the one-dimensional Ising model with long-range interactions using the functional renormalization group in the local potential approximation (LPA), and compare our findings with Dyson's hierarchical model (DHM). While the DHM lacks translational invariance, it admits a field-theoretical description closely resembling the LPA, up to minor but nontrivial differences. After reviewing the real-space renormalization group approach to the DHM, we demonstrate a remarkable agreement in the critical exponent $\nu$ between the two methods across the entire range of power-law decays $1/2 < \sigma < 1$. We further benchmark our results against Monte Carlo simulations and analytical expansions near the upper boundary of the nontrivial regime, $\sigma \lesssim 1$. + oai:arXiv.org:2510.02458v2 + cond-mat.stat-mech + math-ph + math.MP + Tue, 23 Dec 2025 00:00:00 -0500 replace-cross http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Justin Boutilier, Jonas Oddur Jonasson, Hannah Li, Erez Yoeli + 10.1103/2y4k-rvcf + Valerio Pagni, Guido Giachetti, Andrea Trombettoni, Nicol\`o Defenu - A consistent treatment of dynamic contact angles in the sharp-interface framework with the generalized Navier boundary condition - https://arxiv.org/abs/2411.10762 - arXiv:2411.10762v3 Announce Type: replace-cross -Abstract: In this work, we revisit the Generalized Navier Boundary condition (GNBC) introduced by Qian et al.\ in the sharp interface Volume-of-Fluid context. We replace the singular uncompensated Young stress by a smooth function with a characteristic width $\varepsilon > 0$ that is understood as a physical parameter of the model. Therefore, we call the model the ``Contact Region GNBC'' (CR-GNBC). We show that the model is consistent with the fundamental kinematics of the contact angle transport described by Fricke, K\"ohne and Bothe. We implement the model in the geometrical Volume-of-Fluid solver Basilisk using a ``free angle'' approach. This means that the dynamic contact angle is not prescribed but reconstructed from the interface geometry and subsequently applied as an input parameter to compute the uncompensated Young stress. We couple this approach to the two-phase Navier Stokes solver and study the withdrawing tape problem with a receding contact line. It is shown that the model allows for grid-independent solutions and leads to a full regularization of the singularity at the moving contact line, which is in accordance with the thin-film equation subject to this boundary condition. In particular, it is shown that the curvature at the moving contact line is finite and mesh converging. As predicted by the fundamental kinematics, the parallel shear stress component vanishes at the moving contact line for quasi-stationary states (i.e. for $\dot{\theta}_d=0$) and the dynamic contact angle is determined by a balance between the uncompensated Young stress and an effective contact line friction. Furthermore, a non-linear generalization of the model is proposed, which aims at reproducing the Molecular Kinetic Theory of Blake and Haynes for quasi-stationary states. - oai:arXiv.org:2411.10762v3 - physics.flu-dyn + A Flux-Tunable Discrete Angular Filter + https://arxiv.org/abs/2510.06395 + arXiv:2510.06395v2 Announce Type: replace-cross +Abstract: Recent work by Lawrie et al. [PRR 7, 023209 (2025)] introduced a non-diffracting resonant angular filter on a network of thin channels (modelled via quantum graph theory) that exhibits unit transmission of acoustic waves at a discrete, symmetry-paired set of incidence angles determined solely by the graph topology, while transmission at all other angles is strictly forbidden. In the present work, we study the same filtering geometry for waves governed by the magnetic Schr\"odinger equation rather than the classical wave equation. Using a phase shift induced by non-reciprocal wave propagation due to the presence of the magnetic potential and tuning $\delta$-type vertex boundary conditions, we make the previously topology-fixed discrete pass directions continuously tunable: both the transmission angle and the transmission coefficient become control parameters. The resulting flux-tunable angular filtering device replaces topology-constrained passbands with a programmable steering device, broadening the scope for wave-filter and beam-shaping applications. + oai:arXiv.org:2510.06395v2 + physics.app-ph math-ph math.MP - Mon, 22 Dec 2025 00:00:00 -0500 + physics.optics + Tue, 23 Dec 2025 00:00:00 -0500 replace-cross http://creativecommons.org/licenses/by/4.0/ - Tomas Fullana, Yash Kulkarni, Mathis Fricke, St\'ephane Popinet, Shahriar Afkhami, Dieter Bothe, St\'ephane Zaleski + 10.12693/APhysPolA.148.S25 + Acta Physica Polonica A, Vol. 148, No. 5 (2025) + Tristan M. Lawrie, Oliver M. Brown - Asymptotic robustness of entanglement in noisy quantum networks and graph connectivity - https://arxiv.org/abs/2411.12548 - arXiv:2411.12548v2 Announce Type: replace-cross -Abstract: Quantum networks are promising venues for quantum information processing. This motivates the study of the entanglement properties of the particular multipartite quantum states that underpin these structures. In particular, it has been recently shown that when the links are noisy two drastically different behaviors can occur regarding the global entanglement properties of the network. While in certain configurations the network displays genuine multipartite entanglement (GME) for any system size provided the noise level is below a certain threshold, in others GME is washed out if the system size is big enough for any fixed non-zero level of noise. However, this difference has only been established considering the two extreme cases of maximally and minimally connected networks (i.e. complete graphs versus trees, respectively). In this article we investigate this question much more in depth and relate this behavior to the growth of several graph theoretic parameters that measure the connectivity of the graph sequence that codifies the structure of the network as the number of parties increases. The strongest conditions are obtained when considering the degree growth. Our main results are that a sufficiently fast degree growth (i.e. $\Omega(N)$, where $N$ is the size of the network) is sufficient for asymptotic robustness of GME, while if it is sufficiently slow (i.e. $o(\log N)$) then the network becomes asymptotically biseparable. We also present several explicit constructions related to the optimality of these results. - oai:arXiv.org:2411.12548v2 + Entanglement in von Neumann Algebraic Quantum Information Theory + https://arxiv.org/abs/2510.07563 + arXiv:2510.07563v2 Announce Type: replace-cross +Abstract: In quantum systems with infinitely many degrees of freedom, states can be infinitely entangled across a pair of subsystems, but are there different forms of infinite entanglement? To understand entanglement in such systems, we use a framework in which subsystems are described by von Neumann algebras on the full system's Hilbert space. Although this approach has been known for over 50 years, an operational justification has been missing so far. We resolve this by deriving the von Neumann algebraic description of subsystems from operational axioms. This raises the question of how physical properties of the subsystem relate to algebraic properties. Our main result shows a surprisingly strong connection: The type classification of von Neumann algebras (types I, II, III, and their respective subtypes) is in one-to-one correspondence with a family of operational entanglement properties. For instance, Connes' classification of type III factors can be formulated in terms of the smallest achievable error when "embezzling" entanglement from the system. Our findings promote the type classification from algebraic bookkeeping to a classification of infinite quantum systems based on the kind of infinite entanglement that they support. + oai:arXiv.org:2510.07563v2 quant-ph math-ph math.MP - Mon, 22 Dec 2025 00:00:00 -0500 + math.OA + Tue, 23 Dec 2025 00:00:00 -0500 replace-cross http://creativecommons.org/licenses/by/4.0/ - Fernando Lled\'o, Carlos Palazuelos, Julio I. de Vicente + 10.15488/19655 + PhD thesis, Leibniz University Hanover, 2025 + Lauritz van Luijk - Refined Analysis of Federated Averaging and Federated Richardson-Romberg - https://arxiv.org/abs/2412.01389 - arXiv:2412.01389v2 Announce Type: replace-cross -Abstract: In this paper, we present a novel analysis of \FedAvg with constant step size, relying on the Markov property of the underlying process. We demonstrate that the global iterates of the algorithm converge to a stationary distribution and analyze its resulting bias and variance relative to the problem's solution. We provide a first-order bias expansion in both homogeneous and heterogeneous settings. Interestingly, this bias decomposes into two distinct components: one that depends solely on stochastic gradient noise and another on client heterogeneity. Finally, we introduce a new algorithm based on the Richardson-Romberg extrapolation technique to mitigate this bias. - oai:arXiv.org:2412.01389v2 - stat.ML + Network-Optimised Spiking Neural Network (NOS) Scheduling for 6G O-RAN: Spectral Margin and Delay-Tail Control + https://arxiv.org/abs/2510.11291 + arXiv:2510.11291v2 Announce Type: replace-cross +Abstract: This work presents a Network-Optimised Spiking (NOS) delay-aware scheduler for 6G radio access. The scheme couples a bounded two-state kernel to a clique-feasible proportional-fair (PF) grant head: the excitability state acts as a finite-buffer proxy, the recovery state suppresses repeated grants, and neighbour pressure is injected along the interference graph via delayed spikes. A small-signal analysis yields a delay-dependent threshold $k_\star(\Delta)$ and a spectral margin $\delta = k_\star(\Delta) - gH\rho(W)$ that compress topology, controller gain, and delay into a single design parameter. Under light assumptions on arrivals, we prove geometric ergodicity for $\delta>0$ and derive sub-Gaussian backlog and delay tail bounds with exponents proportional to $\delta$. A numerical study, aligned with the analysis and a DU compute budget, compares NOS with PF and delayed backpressure (BP) across interference topologies over a $5$--$20$\,ms delay sweep. With a single gain fixed at the worst spectral radius, NOS sustains higher utilisation and a smaller 99.9th-percentile delay while remaining clique-feasible on integer PRBs. + oai:arXiv.org:2510.11291v2 + cs.NI + cs.IT cs.LG - math.OC - Mon, 22 Dec 2025 00:00:00 -0500 + math.IT + Tue, 23 Dec 2025 00:00:00 -0500 replace-cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Paul Mangold, Alain Durmus, Aymeric Dieuleveut, Sergey Samsonov, Eric Moulines + http://creativecommons.org/licenses/by/4.0/ + Muhammad Bilal, Xiaolong Xu - Saturation-Aware Snapshot Compressive Imaging: Theory and Algorithm - https://arxiv.org/abs/2501.11869 - arXiv:2501.11869v3 Announce Type: replace-cross -Abstract: Snapshot Compressive Imaging (SCI) uses coded masks to compress a 3D data cube into a single 2D snapshot. In practice, multiplexing can push intensities beyond the sensor's dynamic range, producing saturation that violates the linear SCI model and degrades reconstruction. This paper provides the first theoretical characterization of SCI recovery under saturation. We model clipping as an element-wise nonlinearity and derive a finite-sample recovery bound for compression-based SCI that links reconstruction error to mask density and the extent of saturation. The analysis yields a clear design rule: optimal Bernoulli masks use densities below one-half, decreasing further as saturation strengthens. Guided by this principle, we optimize mask patterns and introduce a novel reconstruction framework, Saturation-Aware PnP Net (SAPnet), which explicitly enforces consistency with saturated measurements. Experiments on standard video-SCI benchmarks confirm our theory and demonstrate that SAPnet significantly outperforms existing PnP-based methods. - oai:arXiv.org:2501.11869v3 - eess.IV + Practical hybrid decoding scheme for parity-encoded spin systems + https://arxiv.org/abs/2510.26189 + arXiv:2510.26189v3 Announce Type: replace-cross +Abstract: We propose a practical hybrid decoding scheme for the parity-encoding architecture. This architecture was first introduced by N. Sourlas as a computational technique for tackling hard optimization problems, especially those modeled by spin systems such as the Ising model and spin glasses, and reinvented by W. Lechner, P. Hauke, and P. Zoller to develop quantum annealing devices. We study the specific model, called the SLHZ model, aiming to achieve a near-term quantum annealing device implemented solely through geometrically local spin interactions. Taking account of the close connection between the SLHZ model and a classical low-density-parity-check code, two approaches can be chosen for the decoding: (1) finding the ground state of a spin Hamiltonian derived from the SLHZ model, which can be achieved via stochastic decoders such as a quantum annealer or a classical Monte Carlo sampler; (2) using deterministic decoding techniques for the classical LDPC code, such as belief propagation and bit-flip decoder. The proposed hybrid approach combines the two approaches by applying bit-flip decoding to the readout of the stochastic decoder based on the SLHZ model. We present simulations demonstrating that this approach can reveal the latent potential of the SLHZ model, realizing soft-annealing concept proposed by Sourlas. + oai:arXiv.org:2510.26189v3 + quant-ph cs.IT math.IT - stat.AP - Mon, 22 Dec 2025 00:00:00 -0500 + Tue, 23 Dec 2025 00:00:00 -0500 + replace-cross + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Yoshihiro Nambu + + + Co-algebraic methods for String Field Theory and Quantum Field Theory + https://arxiv.org/abs/2511.02753 + arXiv:2511.02753v2 Announce Type: replace-cross +Abstract: In this work we extend the notion of co-algebra, co-algebraic Wess-Zumino-Witten formulation of Lagrangian Field Theory and the Homotopy transfer theorem to many strings and particle systems. We discuss in detail the construction of higher dimensional co-algebras and the computational methods derived from them with a special interest regarding String Field Theory and Quantum Field Theory. As a result of this work we will be able to effortlessly extend some of the newly developed tools to study the algebraic structure, compute effective actions and compute scattering amplitudes of more complicated QFTs. + oai:arXiv.org:2511.02753v2 + hep-th + math-ph + math.MP + Tue, 23 Dec 2025 00:00:00 -0500 + replace-cross + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Enrico Perron Cabus + + + Mathematical exploration and discovery at scale + https://arxiv.org/abs/2511.02864 + arXiv:2511.02864v3 Announce Type: replace-cross +Abstract: AlphaEvolve (Novikov et al., 2025) is a generic evolutionary coding agent that combines the generative capabilities of LLMs with automated evaluation in an iterative evolutionary framework that proposes, tests, and refines algorithmic solutions to challenging scientific and practical problems. In this paper we showcase AlphaEvolve as a tool for autonomously discovering novel mathematical constructions and advancing our understanding of long-standing open problems. + To demonstrate its breadth, we considered a list of 67 problems spanning mathematical analysis, combinatorics, geometry, and number theory. The system rediscovered the best known solutions in most of the cases and discovered improved solutions in several. In some instances, AlphaEvolve is also able to generalize results for a finite number of input values into a formula valid for all input values. Furthermore, we are able to combine this methodology with Deep Think and AlphaProof in a broader framework where the additional proof-assistants and reasoning systems provide automated proof generation and further mathematical insights. + These results demonstrate that large language model-guided evolutionary search can autonomously discover mathematical constructions that complement human intuition, at times matching or even improving the best known results, highlighting the potential for significant new ways of interaction between mathematicians and AI systems. We present AlphaEvolve as a powerful new tool for mathematical discovery, capable of exploring vast search spaces to solve complex optimization problems at scale, often with significantly reduced requirements on preparation and computation time. + oai:arXiv.org:2511.02864v3 + cs.NE + cs.AI + math.CA + math.CO + math.MG + Tue, 23 Dec 2025 00:00:00 -0500 replace-cross http://creativecommons.org/licenses/by/4.0/ - Mengyu Zhao, Shirin Jalali + Bogdan Georgiev, Javier G\'omez-Serrano, Terence Tao, Adam Zsolt Wagner - Weihrauch problems as containers - https://arxiv.org/abs/2501.17250 - arXiv:2501.17250v3 Announce Type: replace-cross -Abstract: We note that Weihrauch problems can be regarded as containers over the category of projective represented spaces and that Weihrauch reductions correspond exactly to container morphisms. We also show that Bauer's extended Weihrauch degrees and the posetal reflection of containers over partition assemblies are equivalent. Using this characterization, we show how a number of operators over Weihrauch degrees, such as the composition product, also arise naturally from the abstract theory of polynomial functors. - oai:arXiv.org:2501.17250v3 - cs.LO - math.LO - Mon, 22 Dec 2025 00:00:00 -0500 + Enhancing Diffusion Model Guidance through Calibration and Regularization + https://arxiv.org/abs/2511.05844 + arXiv:2511.05844v3 Announce Type: replace-cross +Abstract: Classifier-guided diffusion models have emerged as a powerful approach for conditional image generation, but they suffer from overconfident predictions during early denoising steps, causing the guidance gradient to vanish. This paper introduces two complementary contributions to address this issue. First, we propose a differentiable calibration objective based on the Smooth Expected Calibration Error (Smooth ECE), which improves classifier calibration with minimal fine-tuning and yields measurable improvements in Frechet Inception Distance (FID). Second, we develop enhanced sampling guidance methods that operate on off-the-shelf classifiers without requiring retraining. These include tilted sampling with batch-level reweighting, adaptive entropy-regularized sampling to preserve diversity, and a novel f-divergence-based sampling strategy that strengthens class-consistent guidance while maintaining mode coverage. Experiments on ImageNet 128x128 demonstrate that our divergence-regularized guidance achieves an FID of 2.13 using a ResNet-101 classifier, improving upon existing classifier-guided diffusion methods while requiring no diffusion model retraining. The results show that principled calibration and divergence-aware sampling provide practical and effective improvements for classifier-guided diffusion. + oai:arXiv.org:2511.05844v3 + cs.CV + cs.AI + cs.IT + cs.LG + eess.IV + math.IT + Tue, 23 Dec 2025 00:00:00 -0500 replace-cross - http://creativecommons.org/publicdomain/zero/1.0/ - C\'ecilia Pradic, Ian Price + http://creativecommons.org/licenses/by/4.0/ + Seyed Alireza Javid, Amirhossein Bagheri, Nuria Gonz\'alez-Prelcic - Constraint-based causal discovery with tiered background knowledge and latent variables in single or overlapping datasets - https://arxiv.org/abs/2503.21526 - arXiv:2503.21526v3 Announce Type: replace-cross -Abstract: In this paper we consider the use of tiered background knowledge within constraint based causal discovery. Our focus is on settings relaxing causal sufficiency, i.e. allowing for latent variables which may arise because relevant information could not be measured at all, or not jointly, as in the case of multiple overlapping datasets. We first present novel insights into the properties of the 'tiered FCI' (tFCI) algorithm. Building on this, we introduce a new extension of the IOD (integrating overlapping datasets) algorithm incorporating tiered background knowledge, the 'tiered IOD' (tIOD) algorithm. We show that under full usage of the tiered background knowledge tFCI and tIOD are sound, while simple versions of the tIOD and tFCI are sound and complete. We further show that the tIOD algorithm can often be expected to be considerably more efficient and informative than the IOD algorithm even beyond the obvious restriction of the Markov equivalence classes. We provide a formal result on the conditions for this gain in efficiency and informativeness. Our results are accompanied by a series of examples illustrating the exact role and usefulness of tiered background knowledge. - oai:arXiv.org:2503.21526v3 + Source-Optimal Training is Transfer-Suboptimal + https://arxiv.org/abs/2511.08401 + arXiv:2511.08401v3 Announce Type: replace-cross +Abstract: We prove that training a source model optimally for its own task is generically suboptimal when the objective is downstream transfer. We study the source-side optimization problem in L2-SP ridge regression and show a fundamental mismatch between the source-optimal and transfer-optimal source regularization: outside of a measure-zero set, $\tau_0^* \neq \tau_S^*$. We characterize the transfer-optimal source penalty $\tau_0^*$ as a function of task alignment and identify an alignment-dependent reversal: with imperfect alignment ($0<\rho<1$), transfer benefits from stronger source regularization, while in super-aligned regimes ($\rho>1$), transfer benefits from weaker regularization. In isotropic settings, the decision of whether transfer helps is independent of the target sample size and noise, depending only on task alignment and source characteristics. We verify the linear predictions in a synthetic ridge regression experiment, and we present CIFAR-10 experiments as evidence that the source-optimal versus transfer-optimal mismatch can persist in nonlinear networks. + oai:arXiv.org:2511.08401v3 stat.ML cs.LG math.ST stat.TH - Mon, 22 Dec 2025 00:00:00 -0500 + Tue, 23 Dec 2025 00:00:00 -0500 replace-cross - http://creativecommons.org/licenses/by/4.0/ - Christine W. Bang, Vanessa Didelez + http://creativecommons.org/publicdomain/zero/1.0/ + C. Evans Hedges - Les Houches lectures on non-perturbative Seiberg-Witten geometry - https://arxiv.org/abs/2503.21742 - arXiv:2503.21742v2 Announce Type: replace-cross -Abstract: In these lectures we detail the interplay between the low-energy dynamics of quantum field theories with four supercharges and the exact WKB analysis. This exposition may be the first comprehensive account of this connection and includes new arguments and results. - The lectures start with the introduction of massive two-dimensional $\mathcal{N}=(2,2)$ theories and their spectra of BPS solitons. We place these theories in a two-dimensional cigar background with supersymmetric boundary conditions labelled by a phase $\zeta = e^{i \vartheta}$, while turning on the two-dimensional $\Omega$-background with parameter~$\epsilon$. We show that the resulting partition function $\mathcal{Z}_{\mathrm{2d}}^\vartheta(\epsilon)$ can be characterized as the Borel-summed solution, in the direction $\vartheta$, to an associated Schr\"odinger equation. The partition function $\mathcal{Z}_{\mathrm{2d}}^\vartheta(\epsilon)$ is locally constant in the phase $\vartheta$ and jumps across phases $\vartheta_\textrm{BPS}$ associated with the BPS solitons. Since these jumps are non-perturbative in the parameter~$\epsilon$, we refer to $Z^\vartheta_\mathrm{2d}(\epsilon)$ as the non-perturbative partition function for the original two-dimensional $\mathcal{N}=(2,2)$ theory. We completely determine this partition function $\mathcal{Z}^\vartheta_\mathrm{2d}(\epsilon)$ in two classes of examples, Landau-Ginzburg models and gauged linear sigma models, and show that $\mathcal{Z}^\vartheta_\mathrm{2d}(\epsilon)$ encodes the well-known vortex partition function at a special phase $\vartheta_\textrm{FN}$ associated with the presence of self-solitons. This analysis generalizes to four-dimensional $\mathcal{N}=2$ theories in the $\frac{1}{2} \Omega$-background. - oai:arXiv.org:2503.21742v2 - hep-th - math.CA - math.GT - math.SG - Mon, 22 Dec 2025 00:00:00 -0500 + Exact Stochastic Differential Equations for Quantum Reverse Diffusion + https://arxiv.org/abs/2511.15919 + arXiv:2511.15919v4 Announce Type: replace-cross +Abstract: The ensemble-averaged dynamics of open quantum systems are typically irreversible. We show that this irreversibility need not hold at the level of individually monitored quantum trajectories. Our main results are analytical stochastic differential equations for quantum reverse diffusion, along with corresponding stochastic master equations. These equations describe the exact and approximate stochastic reverse processes for continuously monitored Pauli channels, including time-dependent depolarizing noise. We show that the reverse processes generalize the forward dynamics by combining the noise effects of the forward processes with an additional non-Markovian stochastic drift that dynamically steers a quantum state back to its initial configuration. Consequently, the exact SDEs admit closed-form solutions that can be implemented in real-time without the need for variational techniques. Our findings establish an analytical framework for quantum state recovery, noise-resilient quantum gates, quantum generative modelling, quantum tomography via forward-reverse cycles, and potential paradigms for quantum error correction based on reverse diffusion. + oai:arXiv.org:2511.15919v4 + quant-ph + math-ph + math.MP + Tue, 23 Dec 2025 00:00:00 -0500 replace-cross - http://creativecommons.org/licenses/by-sa/4.0/ - Lo\"ic Bramley, Lotte Hollands, Subrabalan Murugesan + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Einar Gabbassov - Bulk metric reconstruction from entanglement data via minimal surface area variations - https://arxiv.org/abs/2504.07016 - arXiv:2504.07016v2 Announce Type: replace-cross -Abstract: We investigate the reconstruction of asymptotically anti-de Sitter (AdS) bulk geometries from boundary entanglement entropy data for ball-shaped entangling regions. By deriving an explicit inversion formula, we relate variations in entanglement entropy to deviations of the bulk metric about a fixed background. Applying this formula, we recover the Schwarzschild-AdS spacetime in the low-temperature regime to first order. We further extend our analysis to include deformations of the bulk geometry with nontrivial dependence on boundary directions, and propose an iterative reconstruction scheme aimed at recovering the full spacetime starting close to a conformal fixed point. We do this by building on recent advances in the mathematics of inverse problems by introducing the higher-order linearization method as a new tool in the context of holographic bulk reconstruction. - oai:arXiv.org:2504.07016v2 + Generalizing fusion rules by shuffle: Symmetry-based classifications of nonlocal systems constructed from similarity transformations + https://arxiv.org/abs/2512.02139 + arXiv:2512.02139v2 Announce Type: replace-cross +Abstract: We study fusion rings, or symmetry topological field theories (SymTFTs), which lie outside the non-negative integer matrix representation (NIM-rep), by combining knowledge from generalized symmetry and that from pseudo-Hermitian systems. By applying the Galois shuffle operation to the SymTFTs, we reconstruct fusion rings that correspond to nonlocal CFTs constructed from the corresponding local nonunitary CFTs by applying the similarity transformations. The resultant SymTFTs are outside of NIM-rep, whereas they are ring isomorphic to the NIM-rep of the corresponding local nonunitary CFTs. We study the consequences of this correspondence between the nonlocal unitary model and local nonunitary models. We demonstrate the correspondence between their classifications of massive or massless renormalization group flows and the discrepancies between their boundary or domain wall phenomena. Our work reveals a new connection between ring isomorphism and similarity transformations, providing the fundamental implications of ring-theoretic ideas in the context of symmetry in physics. + oai:arXiv.org:2512.02139v2 hep-th + cond-mat.stat-mech + cond-mat.str-el math-ph - math.AP math.MP - Mon, 22 Dec 2025 00:00:00 -0500 + quant-ph + Tue, 23 Dec 2025 00:00:00 -0500 replace-cross http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - JHEP 10 (2025) 079 - Niko Jokela, Tony Liimatainen, Miika Sarkkinen, Leo Tzou + Yoshiki Fukusumi, Taishi Kawamoto - Going deep and going wide: Counting logic and homomorphism indistinguishability over graphs of bounded treedepth and treewidth - https://arxiv.org/abs/2505.01193 - arXiv:2505.01193v2 Announce Type: replace-cross -Abstract: We study the expressive power of first-order logic with counting quantifiers, especially the $k$-variable and quantifier-rank-$q$ fragment $\mathsf{C}^k_q$, using homomorphism indistinguishability. Recently, Dawar, Jakl, and Reggio (2021) proved that two graphs satisfy the same $\mathsf{C}^k_q$-sentences iff they are homomorphism indistinguishable over the class $\mathcal{T}^k_q$ of graphs admitting a $k$-pebble forest cover of depth $q$. After reproving this result using elementary means, we provide a graph-theoretic analysis of $\mathcal{T}^k_q$. This allows us to separate $\mathcal{T}^k_q$ from the intersection $\mathcal{TW}_{k-1} \cap \mathcal{TD}_q$, provided that $q$ is sufficiently larger than $k$. Here $\mathcal{TW}_{k-1}$ is the class of all graphs of treewidth at most $k-1$ and $\mathcal{TD}_q$ is the class of all graphs of treedepth at most $q$. - We are able to lift this separation to a separation of the respective homomorphism indistinguishability relations $\equiv_{\mathcal{T}^k_q}$ and $\equiv_{\mathcal{TW}_{k-1} \cap \mathcal{TD}_q}$. We do this by showing that the classes $\mathcal{TD}_q$ and $\mathcal{T}^k_q$ are homomorphism distinguishing closed, as conjectured by Roberson (2022). - In order to prove Roberson's conjecture for $\mathcal{T}^k_q$, we characterise $\mathcal{T}^k_q$ in terms of a monotone Cops-and-Robber game. The crux is to prove that if Cop has a winning strategy then Cop also has a winning strategy that is monotone. To that end, we transform Cops' winning strategy into a pree-tree-decomposition, which is inspired by decompositions of matroids, and then apply an intricate breadth-first cleaning up procedure along the pree-tree-decomposition (which may temporarily lose the property of representing a strategy). Thereby, we achieve monotonicity while controlling the number of rounds across all branches of the decomposition via a vertex exchange argument. - oai:arXiv.org:2505.01193v2 - cs.LO - cs.DM - math.CO - Mon, 22 Dec 2025 00:00:00 -0500 + Gauge Symmetries, Contact Reduction, and Singular Field Theories + https://arxiv.org/abs/2512.03645 + arXiv:2512.03645v2 Announce Type: replace-cross +Abstract: The reduction of dynamical systems which are invariant under changes of global scale is well-understood, for classical theories of particles, and fields. The excision of the superfluous degree of freedom describing such a scale leads to a dynamically-equivalent theory, which is frictional in nature. In this article, we extend the formalism to physical models, of both particles and fields, described by singular Lagrangians. Our treatment of classical field theory is based on the manifestly covariant Hamilton De-Donder Weyl formalism, in which the Lagrangian density is introduced as a bundle morphism on the pre-multisymplectic velocity phase space $J^1E$. The results obtained are subsequently applied to a number of physically-motivated examples, as well as a discussion presented on the implications of our work for classical General Relativity. + oai:arXiv.org:2512.03645v2 + gr-qc + hep-th + math-ph + math.MP + Tue, 23 Dec 2025 00:00:00 -0500 replace-cross http://creativecommons.org/licenses/by/4.0/ - Isolde Adler, Eva Fluck, Tim Seppelt, Gian Luca Spitzer + Callum Bell, David Sloan - On the performance of multi-fidelity and reduced-dimensional neural emulators for inference of physiological boundary conditions - https://arxiv.org/abs/2506.11683 - arXiv:2506.11683v2 Announce Type: replace-cross -Abstract: Solving inverse problems in cardiovascular modeling is particularly challenging due to the high computational cost of running high-fidelity simulations. In this work, we focus on Bayesian parameter estimation and explore different methods to reduce the computational cost of sampling from the posterior distribution by leveraging low-fidelity approximations. A common approach is to construct a surrogate model for the high-fidelity simulation itself. Another is to build a surrogate for the discrepancy between high- and low-fidelity models. This discrepancy, which is often easier to approximate, is modeled with either a fully connected neural network or a nonlinear dimensionality reduction technique that enables surrogate construction in a lower-dimensional space. A third possible approach is to treat the discrepancy between the high-fidelity and surrogate models as random noise and estimate its distribution using normalizing flows. This allows us to incorporate the approximation error into the Bayesian inverse problem by modifying the likelihood function. We validate five different methods which are variations of the above on analytical test cases by comparing them to posterior distributions derived solely from high-fidelity models, assessing both accuracy and computational cost. Finally, we demonstrate our approaches on two cardiovascular examples of increasing complexity: a lumped-parameter Windkessel model and a patient-specific three-dimensional anatomy. - oai:arXiv.org:2506.11683v2 - stat.ML - cs.CE - cs.LG - math.ST - q-bio.QM - stat.TH - Mon, 22 Dec 2025 00:00:00 -0500 + Configuration-Constrained Tube MPC for Periodic Operation + https://arxiv.org/abs/2512.04239 + arXiv:2512.04239v2 Announce Type: replace-cross +Abstract: Periodic operation often emerges as the economically optimal mode in industrial processes, particularly under varying economic or environmental conditions. This paper proposes a robust model predictive control (MPC) framework for uncertain systems modeled as polytopic linear differential inclusions (LDIs), where the dynamics evolve as convex combinations of finitely many affine control systems with additive disturbances. The robust control problem is reformulated as a convex optimization program by optimizing over configuration-constrained polytopic tubes and tracks a periodic trajectory that is optimal for a given economic criterion. Artificial variables embedded in the formulation ensure recursive feasibility and robust constraint satisfaction when the economic criterion is updated online, while guaranteeing convergence to the corresponding optimal periodic tube when the criterion remains constant. To improve computational efficiency, we introduce a quadratic over-approximation of the periodic cost under a Lipschitz continuity assumption, yielding a Quadratic Program (QP) formulation that preserves the above theoretical guarantees. The effectiveness and scalability of the approach are demonstrated on a benchmark example and a ball-plate system with eight states. + oai:arXiv.org:2512.04239v2 + eess.SY + cs.SY + math.OC + Tue, 23 Dec 2025 00:00:00 -0500 replace-cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - 10.1016/j.compbiomed.2025.111389 - Computers in Biology and Medicine 200, 111389 (2026) - Chloe H. Choi, Andrea Zanoni, Daniele E. Schiavazzi, Alison L. Marsden + http://creativecommons.org/licenses/by/4.0/ + Filippo Badalamenti, Jose A. Borja-Conde, Sampath Kumar Mulagaleti, Boris Houska, Alberto Bemporad, Mario Eduardo Villanueva - Local Well-Posedness for the Bartnik Stationary Extension Problem near Schwarzschild Spheres - https://arxiv.org/abs/2509.03478 - arXiv:2509.03478v2 Announce Type: replace-cross -Abstract: We investigate the Bartnik stationary extension conjecture, which arises from the definition of the spacetime Bartnik mass for a compact region in a general initial data set satisfying the dominant energy condition. This conjecture posits the existence and uniqueness (up to isometry) of an asymptotically flat stationary vacuum spacetime containing an initial data set $(M, \mathfrak{g}, \Pi)$ that realizes prescribed Bartnik boundary data on $\partial M$, consisting of the induced metric, mean curvature, and appropriate components of the spacetime extrinsic curvature $\Pi$. - Building on the analytic framework developed in arXiv:2411.02801 for the static case, we show that, in a double geodesic gauge, the stationary vacuum Einstein equations reduce to a coupled system comprising elliptic and transport-type equations, with the genuinely stationary contributions encoded in an additional boundary value problem for a $1$-form $\theta$. - We establish local well-posedness for the Bartnik stationary metric extension problem for Bartnik data sufficiently close to that of any coordinate sphere in any initial data set (possibly non time-symmetric) in Schwarzschild spacetime. This includes spheres arbitrary close to the apparent horizon in the initial data set. A key feature of our framework is that the linearized equations decouple: the equations for the metric and potential reduce to the previously solved static case, while the boundary value problem for $\theta$ is treated independently. We prove solvability of this boundary value problem in the Bochner-measurable function spaces adapted to the coupled system developed in arXiv:2411.02801, establishing uniform estimates for the vector spherical harmonic decomposition of $\theta$. - oai:arXiv.org:2509.03478v2 - gr-qc - math.AP - math.DG - Mon, 22 Dec 2025 00:00:00 -0500 + Accurate Models of NVIDIA Tensor Cores + https://arxiv.org/abs/2512.07004 + arXiv:2512.07004v2 Announce Type: replace-cross +Abstract: Matrix multiplication is a fundamental operation in for both training of neural networks and inference. To accelerate matrix multiplication, Graphical Processing Units (GPUs) provide it implemented in hardware. Due to the increased throughput over the software-based matrix multiplication, the multipliers are increasingly used outside of AI, to accelerate various applications in scientific computing. However, matrix multipliers targeted at AI are at present not compliant with IEEE 754 floating-point arithmetic behaviour, with different vendors offering different numerical features. This leads to non-reproducible results across different generations of GPU architectures, at the matrix multiply-accumulate instruction level. To study numerical characteristics of matrix multipliers-such as rounding behaviour, accumulator width, normalization points, extra carry bits, and others-test vectors are typically constructed. Yet, these vectors may or may not distinguish between different hardware models, and due to limited hardware availability, their reliability across many different platforms remains largely untested. We present software models for emulating the inner product behavior of low- and mixed-precision matrix multipliers in the V100, A100, H100 and B200 data center GPUs in most supported input formats of interest to mixed-precision algorithm developers: 8-, 16-, and 19-bit floating point. + oai:arXiv.org:2512.07004v2 + cs.MS + cs.AR + cs.NA + math.NA + Tue, 23 Dec 2025 00:00:00 -0500 replace-cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Ahmed Ellithy + http://creativecommons.org/licenses/by/4.0/ + Faizan A. Khattak, Mantas Mikaitis - Combinatorial Control Barrier Functions: Nested Boolean and p-choose-r Compositions of Safety Constraints - https://arxiv.org/abs/2509.10716 - arXiv:2509.10716v2 Announce Type: replace-cross -Abstract: This paper investigates the problem of composing multiple control barrier functions (CBFs) -- and matrix control barrier functions (MCBFs) -- through logical and combinatorial operations. Standard CBF formulations naturally enable conjunctive (AND) combinations, but disjunctive (OR) and more general logical structures introduce nonsmoothness and possibly a combinatorial blow-up in the number of logical combinations. We introduce the framework of combinatorial CBFs that addresses p-choose-r safety specifications and their nested composition. The proposed framework ensures safety for the exact safe set in a scalable way, using the original number of primitive constraints. We establish theoretical guarantees on safety under these compositions, and we demonstrate their use on a patrolling problem in a multi-agent system. - oai:arXiv.org:2509.10716v2 - eess.SY - cs.SY - math.OC - Mon, 22 Dec 2025 00:00:00 -0500 + On the emergence of preferred structures in quantum theory + https://arxiv.org/abs/2512.07468 + arXiv:2512.07468v2 Announce Type: replace-cross +Abstract: We assess the possibilities offered by Hilbert space fundamentalism, an attitude towards quantum physics according to which all physical structures (e.g. subsystems, locality, spacetime, preferred observables) should emerge from minimal quantum ingredients (typically a Hilbert space, Hamiltonian, and state). As a case study, we first mainly focus on the specific question of whether the Hamiltonian can uniquely determine a tensor product structure, a crucial challenge in the growing field of quantum mereology. The present paper reviews, clarifies, and critically examines two apparently conflicting theorems by Cotler et al. and Stoica. We resolve the tension, show how the former has been widely misinterpreted and why the latter is correct only in some weaker version. We then propose a correct mathematical way to address the general problem of preferred structures in quantum theory, relative to the characterization of emergent objects by unitary-invariant properties. Finally, we apply this formalism in the particular case we started with, and show that a Hamiltonian and a state are enough structure to uniquely select a preferred tensor product structure. + oai:arXiv.org:2512.07468v2 + quant-ph + hep-th + math-ph + math.MP + Tue, 23 Dec 2025 00:00:00 -0500 replace-cross - http://creativecommons.org/licenses/by/4.0/ - 10.1109/LCSYS.2025.3640191 - Pio Ong, Haejoon Lee, Tamas G. Molnar, Dimitra Panagou, Aaron D. Ames + http://creativecommons.org/licenses/by-nc-nd/4.0/ + Antoine Soulas, Guilherme Franzmann, Andrea Di Biagio - Cosmological dynamical systems of non-minimally coupled fluids and scalar fields - https://arxiv.org/abs/2509.14440 - arXiv:2509.14440v2 Announce Type: replace-cross -Abstract: We study the cosmological dynamics of non-minimally coupled matter models using the Brown's variational approach to relativistic fluids in General Relativity. After decomposing the Ricci scalar into a bulk and a boundary term, we construct new models by coupling the bulk term to the fluid variables and an external scalar field. Using dynamical systems techniques, we study models of this type and find that they can give rise to both early-time inflationary behaviour and late-time accelerated expansion. Moreover, these models also contain very interesting features that are rarely seen in this context. For example, we find dark energy models which exhibit phantom crossing in the recent past. Other possibilities include models that give a viable past evolution but terminate in a matter-dominated universe. The dynamical systems themselves display an array of mathematically interesting phenomena, including spirals, centres, and non-trivial bifurcations depending on the chosen parameter values. - oai:arXiv.org:2509.14440v2 + The equation of Binet in classical and relativistic orbital mechanics + https://arxiv.org/abs/2512.07485 + arXiv:2512.07485v2 Announce Type: replace-cross +Abstract: Binet's equation provides a direct way to obtain the geometric shape of orbits in a central force field. It is well known that in Newtonian gravitation Binet's equation leads to all the conic curves as solutions for an inverse-square force. In this work, we show how Binet's equation arises from the horizontal and vertical infinitesimal displacements of a body in free fall and in inertial motion. This derivation uses elementary concepts of infinitesimal calculus. Second, we derive the relativistic version of Binet's equation for the Schwarzschild-(anti-)de Sitter metric. This derivation, which is novel, directly relates the coordinates involved in Binet's equation without the need to introduce potentials or the use of Killing vectors. Finally, we tackle some controversies related to the role of the cosmological constant in the trajectory of photons in a Schwarzschild-(anti-)de Sitter or even in Reissner-Nordstr\"om-(anti-)de Sitter spacetimes. + oai:arXiv.org:2512.07485v2 gr-qc math-ph math.MP - Mon, 22 Dec 2025 00:00:00 -0500 + Tue, 23 Dec 2025 00:00:00 -0500 replace-cross http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - 10.1007/s10714-025-03502-1 - General Relativity and Gravitation 57 (2025) 167 - Hala A. Ashi, Christian G. Boehmer, Antonio d'Alfonso del Sordo, Erik Jensko + Jose Luis Alvarez-Perez - Consistent energy-momentum trace couplings of fluids - https://arxiv.org/abs/2509.24843 - arXiv:2509.24843v2 Announce Type: replace-cross -Abstract: Gravitational models with non-minimal couplings involving the trace of the energy-momentum tensor have become increasingly popular. The idea of coupling the trace of the matter tensor to the geometry can be applied to various matter models, including relativistic perfect fluids. However, it is well-known that the variational formulation of perfect fluids involves some technicalities. We carefully derive the field equations including the trace coupling of a perfect fluid using two different approaches, namely, that given by Brown using Lagrange multipliers, and that given by Schutz using velocity potentials. We show that previous results involving such trace couplings do not match the results presented here. We demonstrate that our fluid's equations of motion are consistent with the gravitational field equations. Moreover, we present a simple on-shell argument which further supports the correctness of our results. - Our work implies that a vast amount of the $f(R,T)$ literature using perfect fluids needs to be revisited. - oai:arXiv.org:2509.24843v2 + About possible measures in Quantum Gravity + https://arxiv.org/abs/2512.09191 + arXiv:2512.09191v2 Announce Type: replace-cross +Abstract: Possible measures for Quantum Gravity are considered. Choices that are invariant under diffeomorphisms are analyzed, but the possibility of employing non invariant measures is also taken into account. The last possibility may be accepted if the anomaly in the measure is compensated by counter term redefinitions of the model under analysis. Particular attention is paid to some concrete examples of non covariant looking measures, which may be useful for generalizing the Veltmann identities when quantizing around curved space times. The results are specified for the Stelle gravity model [1]-[2], which is known to be renormalizable in flat space, although not known to be so in curved ones. + oai:arXiv.org:2512.09191v2 gr-qc + hep-th math-ph math.MP - Mon, 22 Dec 2025 00:00:00 -0500 + Tue, 23 Dec 2025 00:00:00 -0500 replace-cross http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Christian G. Boehmer, Eissa Al-Nasrallah + O. P. Santill\'an - A Neural Surrogate-Enhanced Multi-Method Framework for Robust Wing Design Optimization - https://arxiv.org/abs/2510.08582 - arXiv:2510.08582v3 Announce Type: replace-cross -Abstract: This paper introduces a modular and scalable design optimization framework for the wing design process that enables faster early-phase design while ensuring aerodynamic stability. The pipeline starts with the generation of initial wing geometries and then proceeds to optimize the wing using several algorithms. Aerodynamic performance is assessed using a Vortex Lattice Method (VLM) applied to a carefully selected dataset of wing configurations. These results are employed to develop surrogate neural network models, which can predict lift and drag rapidly and accurately. The stability evaluation is implemented by setting the control surfaces and components to fixed positions in order to have realistic flight dynamics. The approach unifies and compares several optimization techniques, including Particle Swarm Optimization (PSO), Genetic Algorithms (GA), gradient-based MultiStart methods, Bayesian optimization, and Lipschitz optimization. Each method ensures constraint management via adaptive strategies and penalty functions, where the targets for lift and design feasibility are enforced. The progression of aerodynamic characteristics and geometries over the optimization iterations will be investigated in order to clarify each algorithm's convergence characteristics and performance efficiency. Our results show improvement in aerodynamic qualities and robust stability properties, offering a mechanism for wing design at speed and precision. In the interest of reproducibility and community development, the complete implementation is publicly available at Github. - oai:arXiv.org:2510.08582v3 - cs.NE + The meaning of "Big Bang" + https://arxiv.org/abs/2512.09950 + arXiv:2512.09950v2 Announce Type: replace-cross +Abstract: What does ``Big Bang'' actually mean? What was the origin of these two words? It has often been said that the expression ``Big Bang'' began as an insult. Even if this were true, it would be just an irrelevant part of the whole issue. There are many more aspects hidden under this name, and which are seldom explained. They will be discussed in this work. In order to frame the analysis, help will be sought from the highly authoritative voices of two exceptional writers: William Shakespeare and Umberto Eco. Both Shakespeare and Eco have explored the tension existing between words and the realities they name. With the conclusion that names are, in general, just labels, simple stickers put to identify things. And this includes those given to great theorems or spectacular discoveries. Stigler's law of eponymy is recalled to further substantiate those statements. These points will be at the heart of the investigation carried out here, concerning the very important concept of ``Big Bang''. Everybody thinks to know what ``the Big Bang'' is, but only very few do know it, in fact. When Fred Hoyle first pronounced these two words together, on a BBC radio program, listeners were actually left with the false image that Hoyle was trying to destroy. That is, the tremendous explosion of Lema\^itre's primeval atom (or cosmic egg), which scattered all its enormous matter and energy content throughout the rest of the Universe. This image is absolutely wrong! As will be concluded, today the label ``Big Bang'' is used in several different contexts: (a) the Big Bang Singularity; (b) as the equivalent of cosmic inflation; (c) speaking of the Big Bang cosmological model; (d) to name a very popular TV program; and more. + oai:arXiv.org:2512.09950v2 + physics.pop-ph + astro-ph.CO + gr-qc + math-ph + math.MP + physics.hist-ph + Tue, 23 Dec 2025 00:00:00 -0500 + replace-cross + http://creativecommons.org/licenses/by/4.0/ + Emilio Elizalde + + + The Moroccan Public Procurement Game + https://arxiv.org/abs/2512.10109 + arXiv:2512.10109v2 Announce Type: replace-cross +Abstract: In this paper, we study the public procurement market through the lens of game theory by modeling it as a strategic game with discontinuous and non-quasiconcave payoffs. We first show that the game admits no Nash equilibrium in pure strategies. We then analyze the two-player case and derive two explicit mixed-strategy equilibria for the symmetric game and for the weighted $(p,1-p)$ formulation. Finally, we study the existence of a symmetric mixed strategies Nash equilibrium in the general $N$-player case by applying the diagonal disjoint payoff matching condition. + oai:arXiv.org:2512.10109v2 + econ.TH math.OC - Mon, 22 Dec 2025 00:00:00 -0500 + Tue, 23 Dec 2025 00:00:00 -0500 + replace-cross + http://creativecommons.org/licenses/by/4.0/ + Nizar Riane + + + Sub-$n^k$ Deterministic algorithm for minimum $k$-way cut in simple graphs + https://arxiv.org/abs/2512.12900 + arXiv:2512.12900v2 Announce Type: replace-cross +Abstract: We present a \emph{deterministic exact algorithm} for the \emph{minimum $k$-cut problem} on simple graphs. + Our approach combines the \emph{principal sequence of partitions (PSP)}, derived canonically from ideal loads, with a single level of \emph{Kawarabayashi--Thorup (KT)} contractions at the critical PSP threshold~$\lambda_j$. + Let $j$ be the smallest index with $\kappa(P_j)\ge k$ and $R := k - \kappa(P_{j-1})$. + We prove a structural decomposition theorem showing that an optimal $k$-cut can be expressed as the level-$(j\!-\!1)$ boundary $A_{\le j-1}$ together with exactly $(R-r)$ \emph{non-trivial} internal cuts of value at most~$\lambda_j$ and $r$ \emph{singleton isolations} (``islands'') inside the parts of~$P_{j-1}$. + At this level, KT contractions yield kernels of total size $\widetilde{O}(n / \lambda_j)$, and from them we build a \emph{canonical border family}~$\mathcal{B}$ of the same order that deterministically covers all optimal refinement choices. + Branching only over~$\mathcal{B}$ (and also including an explicit ``island'' branch) gives total running time + $$ + T(n,m,k) = \widetilde{O}\left(\mathrm{poly}(m)+\Bigl(\tfrac{n}{\lambda_j}+n^{\omega/3}\Bigr)^{R}\right), + $$ + where $\omega < 2.373$ is the matrix multiplication exponent. + In particular, if $\lambda_j \ge n^{\varepsilon}$ for some constant $\varepsilon > 0$, we obtain a \emph{deterministic sub-$n^k$-time algorithm}, running in $n^{(1-\varepsilon)(k-1)+o(k)}$ time. + Finally, combining our PSP$\times$KT framework with a small-$\lambda$ exact subroutine via a simple meta-reduction yields a deterministic $n^{c k+O(1)}$ algorithm for $c = \max\{ t/(t+1), \omega/3 \} < 1$, aligning with the exponent in the randomized bound of He--Li (STOC~2022) under the assumed subroutine. + oai:arXiv.org:2512.12900v2 + cs.DS + math.CO + Tue, 23 Dec 2025 00:00:00 -0500 + replace-cross + http://creativecommons.org/licenses/by-nc-nd/4.0/ + Mohit Daga + + + Universality of high-dimensional scaling limits of stochastic gradient descent + https://arxiv.org/abs/2512.13634 + arXiv:2512.13634v2 Announce Type: replace-cross +Abstract: We consider statistical tasks in high dimensions whose loss depends on the data only through its projection into a fixed-dimensional subspace spanned by the parameter vectors and certain ground truth vectors. This includes classifying mixture distributions with cross-entropy loss with one and two-layer networks, and learning single and multi-index models with one and two-layer networks. When the data is drawn from an isotropic Gaussian mixture distribution, it is known that the evolution of a finite family of summary statistics under stochastic gradient descent converges to an autonomous ordinary differential equation (ODE), as the dimension and sample size go to $\infty$ and the step size goes to $0$ commensurately. Our main result is that these ODE limits are universal in that this limit is the same whenever the data is drawn from mixtures of arbitrary product distributions whose first two moments match the corresponding Gaussian distribution, provided the initialization and ground truth vectors are coordinate-delocalized. We complement this by proving two corresponding non-universality results. We provide a simple example where the ODE limits are non-universal if the initialization is coordinate aligned. We also show that the stochastic differential equation limits arising as fluctuations of the summary statistics around their ODE's fixed points are not universal. + oai:arXiv.org:2512.13634v2 + stat.ML + cs.LG + math.PR + math.ST + stat.TH + Tue, 23 Dec 2025 00:00:00 -0500 replace-cross http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Arash Fath Lipaei, Melika Sabzikari + Reza Gheissari, Aukosh Jagannath - A Homological Separation of $\mathbf{P}$ from $\mathbf{NP}$ via Computational Topology and Category Theory - https://arxiv.org/abs/2510.17829 - arXiv:2510.17829v2 Announce Type: replace-cross -Abstract: This paper establishes the separation of complexity classes $\mathbf{P}$ and $\mathbf{NP}$ through a novel homological algebraic approach grounded in category theory. We construct the computational category $\mathbf{Comp}$, embedding computational problems and reductions into a unified categorical framework. By developing computational homology theory, we associate to each problem $L$ a chain complex $C_{\bullet}(L)$ whose homology groups $H_n(L)$ capture topological invariants of computational processes. Our main result demonstrates that problems in $\mathbf{P}$ exhibit trivial computational homology ($H_n(L) = 0$ for all $n > 0$), while $\mathbf{NP}$-complete problems such as SAT possess non-trivial homology ($H_1(\mathrm{SAT}) \neq 0$). This homological distinction provides the first rigorous proof of $\mathbf{P} \neq \mathbf{NP}$ using topological methods. Our work inaugurates computational topology as a new paradigm for complexity analysis, offering finer distinctions than traditional combinatorial approaches and establishing connections between structural complexity theory and homological invariants. - oai:arXiv.org:2510.17829v2 - cs.CC - math.AC - math.CT - math.RA - Mon, 22 Dec 2025 00:00:00 -0500 + Chirp Delay-Doppler Domain Modulation Based Joint Communication and Radar for Autonomous Vehicles + https://arxiv.org/abs/2512.14432 + arXiv:2512.14432v2 Announce Type: replace-cross +Abstract: This paper introduces a sensing-centric joint communication and millimeter-wave radar paradigm to facilitate collaboration among intelligent vehicles. + We first propose a chirp waveform-based delay-Doppler quadrature amplitude modulation (DD-QAM) that modulates data across delay, Doppler, and amplitude dimensions. + Building upon this modulation scheme, we derive its achievable rate to quantify the communication performance. + We then introduce an extended Kalman filter-based scheme for four-dimensional (4D) parameter estimation in dynamic environments, enabling the active vehicles to accurately estimate orientation and tangential-velocity beyond traditional 4D radar systems. + Furthermore, in terms of communication, we propose a dual-compensation-based demodulation and tracking scheme that allows the passive vehicles to effectively demodulate data without compromising their sensing functions. + Simulation results underscore the feasibility and superior performance of our proposed methods, marking a significant advancement in the field of autonomous vehicles. + Simulation codes are provided to reproduce the results in this paper: \href{https://github.com/LiZhuoRan0/2026-IEEE-TWC-ChirpDelayDopplerModulationISAC}{https://github.com/LiZhuoRan0}. + oai:arXiv.org:2512.14432v2 + eess.SP + cs.IT + math.IT + Tue, 23 Dec 2025 00:00:00 -0500 replace-cross http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Jian-Gang Tang + Zhuoran Li, Zhen Gao, Sheng Chen, Dusit Niyato, Zhaocheng Wang, George K. Karagiannidis - Universal Limits on Quantum Correlations - https://arxiv.org/abs/2510.24950 - arXiv:2510.24950v2 Announce Type: replace-cross -Abstract: Quantum correlations are the singular, defining resource of quantum information science and metrology, forming the basis of every operational advantage that quantum systems hold over classical ones. Yet exact bounds on these correlations-such as the Lieb-Robinson bound on entanglement propagation and the Heisenberg limit on metrological precision-are known only in special cases and have long appeared to arise from unrelated mechanisms. Here we show that these limits share a common geometric origin. We identify a positivity invariant of the block correlation matrix, denoted $\chi$, that quantifies how far a bipartite state lies from the positivity boundary of quantum state space. For any system with a specified observable algebra and parameter-encoding map, every correlation measure determined solely by the positive correlation matrix obeys a $\chi$-dependent inequality. For systems with simple symmetry structures these inequalities take closed analytic form, reproducing the structure of the Heisenberg and Cram\'er-Rao limits and producing new results, including an exact entanglement floor and a universal Fisher-information ceiling even in all-to-all connected quantum networks. We thus demonstrate that positive geometry provides a unified framework for the attainable strength of quantum correlations, linking entanglement, metrological sensitivity, and dynamical causal structure through a single invariant. - oai:arXiv.org:2510.24950v2 - quant-ph - math-ph - math.MP - Mon, 22 Dec 2025 00:00:00 -0500 + Large Model Enabled Embodied Intelligence for 6G Integrated Perception, Communication, and Computation Network + https://arxiv.org/abs/2512.15109 + arXiv:2512.15109v2 Announce Type: replace-cross +Abstract: The advent of sixth-generation (6G) places intelligence at the core of wireless architecture, fusing perception, communication, and computation into a single closed-loop. This paper argues that large artificial intelligence models (LAMs) can endow base stations with perception, reasoning, and acting capabilities, thus transforming them into intelligent base station agents (IBSAs). We first review the historical evolution of BSs from single-functional analog infrastructure to distributed, software-defined, and finally LAM-empowered IBSA, highlighting the accompanying changes in architecture, hardware platforms, and deployment. We then present an IBSA architecture that couples a perception-cognition-execution pipeline with cloud-edge-end collaboration and parameter-efficient adaptation. Subsequently,we study two representative scenarios: (i) cooperative vehicle-road perception for autonomous driving, and (ii) ubiquitous base station support for low-altitude uncrewed aerial vehicle safety monitoring and response against unauthorized drones. On this basis, we analyze key enabling technologies spanning LAM design and training, efficient edge-cloud inference, multi-modal perception and actuation, as well as trustworthy security and governance. We further propose a holistic evaluation framework and benchmark considerations that jointly cover communication performance, perception accuracy, decision-making reliability, safety, and energy efficiency. Finally, we distill open challenges on benchmarks, continual adaptation, trustworthy decision-making, and standardization. Together, this work positions LAM-enabled IBSAs as a practical path toward integrated perception, communication, and computation native, safety-critical 6G systems. + oai:arXiv.org:2512.15109v2 + eess.SP + cs.AI + cs.IT + math.IT + Tue, 23 Dec 2025 00:00:00 -0500 replace-cross http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Samuel Alperin + Zhuoran Li, Zhen Gao, Xinhua Liu, Zheng Wang, Xiaotian Zhou, Lei Liu, Yongpeng Wu, Wei Feng, Yongming Huang - Spectral torsion of the internal noncommutative geometry of the Standard Model - https://arxiv.org/abs/2511.08159 - arXiv:2511.08159v3 Announce Type: replace-cross -Abstract: We compute the nonvanishing spectral torsion functional of the internal part of the noncommutative geometry behind the Standard Model. We show that with a suitable modification of the usual differential graded calculus it matches an analogous functional constructed in terms of the connection. We study also the impact of the torsion on the other spectral fuctionals, which correspond to geometric invariants such as volume integral, metric and Einstein tensors, and scalar curvature. We discuss the impact of the SM Yukawa couplings and the Majorana mass matrix on our results. - oai:arXiv.org:2511.08159v3 - hep-th + A random purification channel for arbitrary symmetries with applications to fermions and bosons + https://arxiv.org/abs/2512.15690 + arXiv:2512.15690v2 Announce Type: replace-cross +Abstract: The random purification channel maps n copies of any mixed quantum state to n copies of a random purification of the state. We generalize this construction to arbitrary symmetries: for any group G of unitaries, we construct a quantum channel that maps states contained in the algebra generated by G to random purifications obtained by twirling over G. In addition to giving a surprisingly concise proof of the original random purification theorem, our result implies the existence of fermionic and bosonic Gaussian purification channels. As applications, we obtain the first tomography protocol for fermionic Gaussian states that scales optimally with the number of modes and the error, as well as an improved property test for this class of states. + oai:arXiv.org:2512.15690v2 + quant-ph math-ph math.MP - math.QA - Mon, 22 Dec 2025 00:00:00 -0500 + Tue, 23 Dec 2025 00:00:00 -0500 replace-cross - http://creativecommons.org/licenses/by/4.0/ - Ludwik D\k{a}browski, Sugato Mukhopadhyay, Filip Po\v{z}ar + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Michael Walter, Freek Witteveen - Generalized infinite dimensional Alpha-Procrustes based geometries - https://arxiv.org/abs/2511.09801 - arXiv:2511.09801v2 Announce Type: replace-cross -Abstract: This work extends the recently introduced Alpha-Procrustes family of Riemannian metrics for symmetric positive definite (SPD) matrices by incorporating generalized versions of the Bures-Wasserstein (GBW), Log-Euclidean, and Wasserstein distances. While the Alpha-Procrustes framework has unified many classical metrics in both finite- and infinite- dimensional settings, it previously lacked the structural components necessary to realize these generalized forms. We introduce a formalism based on unitized Hilbert-Schmidt operators and an extended Mahalanobis norm that allows the construction of robust, infinite-dimensional generalizations of GBW and Log-Hilbert-Schmidt distances. Our approach also incorporates a learnable regularization parameter that enhances geometric stability in high-dimensional comparisons. Preliminary experiments reproducing benchmarks from the literature demonstrate the improved performance of our generalized metrics, particularly in scenarios involving comparisons between datasets of varying dimension and scale. This work lays a theoretical and computational foundation for advancing robust geometric methods in machine learning, statistical inference, and functional data analysis. - oai:arXiv.org:2511.09801v2 - stat.ML + Learning Confidence Ellipsoids and Applications to Robust Subspace Recovery + https://arxiv.org/abs/2512.16875 + arXiv:2512.16875v2 Announce Type: replace-cross +Abstract: We study the problem of finding confidence ellipsoids for an arbitrary distribution in high dimensions. Given samples from a distribution $D$ and a confidence parameter $\alpha$, the goal is to find the smallest volume ellipsoid $E$ which has probability mass $\Pr_{D}[E] \ge 1-\alpha$. Ellipsoids are a highly expressive class of confidence sets as they can capture correlations in the distribution, and can approximate any convex set. This problem has been studied in many different communities. In statistics, this is the classic minimum volume estimator introduced by Rousseeuw as a robust non-parametric estimator of location and scatter. However in high dimensions, it becomes NP-hard to obtain any non-trivial approximation factor in volume when the condition number $\beta$ of the ellipsoid (ratio of the largest to the smallest axis length) goes to $\infty$. This motivates the focus of our paper: can we efficiently find confidence ellipsoids with volume approximation guarantees when compared to ellipsoids of bounded condition number $\beta$? + Our main result is a polynomial time algorithm that finds an ellipsoid $E$ whose volume is within a $O(\beta)^{\gamma d}$ multiplicative factor of the volume of best $\beta$-conditioned ellipsoid while covering at least $1-O(\alpha/\gamma)$ probability mass for any $\gamma < \alpha$. We complement this with a computational hardness result that shows that such a dependence seems necessary up to constants in the exponent. The algorithm and analysis uses the rich primal-dual structure of the minimum volume enclosing ellipsoid and the geometric Brascamp-Lieb inequality. As a consequence, we obtain the first polynomial time algorithm with approximation guarantees on worst-case instances of the robust subspace recovery problem. + oai:arXiv.org:2512.16875v2 + cs.DS cs.LG - math.FA - math.OC - Mon, 22 Dec 2025 00:00:00 -0500 + math.ST + stat.ML + stat.TH + Tue, 23 Dec 2025 00:00:00 -0500 replace-cross http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Salvish Goomanee, Andi Han, Pratik Jawanpuria, Bamdev Mishra + Chao Gao, Liren Shan, Vaidehi Srinivas, Aravindan Vijayaraghavan - Incremental Generation is Necessary and Sufficient for Universality in Flow-Based Modelling - https://arxiv.org/abs/2511.09902 - arXiv:2511.09902v2 Announce Type: replace-cross -Abstract: Incremental flow-based denoising models have reshaped generative modelling, but their empirical advantage still lacks a rigorous approximation-theoretic foundation. We show that incremental generation is necessary and sufficient for universal flow-based generation on the largest natural class of self-maps of $[0,1]^d$ compatible with denoising pipelines, namely the orientation-preserving homeomorphisms of $[0,1]^d$. All our guarantees are uniform on the underlying maps and hence imply approximation both samplewise and in distribution. - Using a new topological-dynamical argument, we first prove an impossibility theorem: the class of all single-step autonomous flows, independently of the architecture, width, depth, or Lipschitz activation of the underlying neural network, is meagre and therefore not universal in the space of orientation-preserving homeomorphisms of $[0,1]^d$. By exploiting algebraic properties of autonomous flows, we conversely show that every orientation-preserving Lipschitz homeomorphism on $[0,1]^d$ can be approximated at rate $O(n^{-1/d})$ by a composition of at most $K_d$ such flows, where $K_d$ depends only on the dimension. Under additional smoothness assumptions, the approximation rate can be made dimension-free, and $K_d$ can be chosen uniformly over the class being approximated. Finally, by linearly lifting the domain into one higher dimension, we obtain structured universal approximation results for continuous functions and for probability measures on $[0,1]^d$, the latter realized as pushforwards of empirical measures with vanishing $1$-Wasserstein error. - oai:arXiv.org:2511.09902v2 - cs.LG - cs.NA - math.CA - math.DS - math.NA - stat.ML - Mon, 22 Dec 2025 00:00:00 -0500 + Subsystems (in)dependence in GIE proposals + https://arxiv.org/abs/2512.17024 + arXiv:2512.17024v2 Announce Type: replace-cross +Abstract: Recent proposals suggest that detecting entanglement between two spatially superposed masses would establish the quantum nature of gravity. However, these gravitationally induced entanglement (GIE) experiments rely on assumptions about subsystem independence. We sharpen the theoretical underpinnings of such proposals by examining them through the lens of algebraic quantum field theory (AQFT), distinguishing distinct operational and algebraic notions of independence. We argue that state and measurement independence of subsystems, essential to the experimental logic, is nontrivial in the presence of gauge constraints and gravitational dressing. Using gravitationally dressed fields, we recall that commutation relations between spacelike separated observables are nontrivial, undermining strict Hilbert space factorization. We further explore the implications for entanglement witnesses, investigating the Tsirelson bound when subsystem algebras fail to commute, and showing that the Tsirelson bound persists for a suitably symmetrized CHSH observable even though the operational status of such "joint" observables becomes delicate when commensurability fails. Our analysis highlights how even within linearized covariant quantum gravity, violations of microcausality may affect both the interpretation, modelling, and design of proposed laboratory tests of quantum gravity, despite remaining negligible for current experimental regimes. Although we consider GIE-style protocols as a concrete case study, the subsystem-independence issues we highlight are generic to low-energy (perturbative) quantum gravity. Finally, we derive estimates for dressing-induced microcausality violations, which suggest a complementary avenue to current proposals: in principle, bounding dressing-induced microcausality violations themselves as a probe of the quantum nature of gravity. + oai:arXiv.org:2512.17024v2 + quant-ph + hep-th + math-ph + math.MP + physics.hist-ph + Tue, 23 Dec 2025 00:00:00 -0500 replace-cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Hossein Rouhvarzi, Anastasis Kratsios + http://creativecommons.org/licenses/by/4.0/ + Nicolas Boulle, Guilherme Franzmann - New Hybrid Heuristics for Pseudo-Boolean Propagation - https://arxiv.org/abs/2511.21417 - arXiv:2511.21417v2 Announce Type: replace-cross -Abstract: In pseudo-boolean solving the currently most successful unit propagation strategy is a hybrid mode combining the watched literal scheme with the counting method. This short paper introduces new heuristics for this hybrid decision, which are able to drastically outperform the current method in the RoundingSAT solver. - oai:arXiv.org:2511.21417v2 + Solomonoff-Inspired Hypothesis Ranking with LLMs for Prediction Under Uncertainty + https://arxiv.org/abs/2512.17145 + arXiv:2512.17145v2 Announce Type: replace-cross +Abstract: Reasoning under uncertainty is a key challenge in AI, especially for real-world tasks, where problems with sparse data demands systematic generalisation. Existing approaches struggle to balance accuracy and simplicity when evaluating multiple candidate solutions. We propose a Solomonoff-inspired method that weights LLM-generated hypotheses by simplicity and predictive fit. Applied to benchmark (Mini-ARC) tasks, our method produces Solomonoff-weighted mixtures for per-cell predictions, yielding conservative, uncertainty-aware outputs even when hypotheses are noisy or partially incorrect. Compared to Bayesian Model Averaging (BMA), Solomonoff scoring spreads probability more evenly across competing hypotheses, while BMA concentrates weight on the most likely but potentially flawed candidates. Across tasks, this highlights the value of algorithmic information-theoretic priors for interpretable, reliable multi-hypothesis reasoning under uncertainty. + oai:arXiv.org:2512.17145v2 cs.AI - math.OC - Mon, 22 Dec 2025 00:00:00 -0500 + cs.IT + math.IT + Tue, 23 Dec 2025 00:00:00 -0500 replace-cross http://creativecommons.org/licenses/by/4.0/ - Mia M\"u{\ss}ig, Jan Johannsen + Josh Barber (QUT), Rourke Young (QUT), Cameron Coombe (QUT,CSIRO), Will Browne (QUT) - Bayesian Optimization of Laser-Wakefield Acceleration via Spectral Pulse Shaping - https://arxiv.org/abs/2512.09125 - arXiv:2512.09125v2 Announce Type: replace-cross -Abstract: In this paper, we investigate the effect of spectral pulse shaping of the laser driver on the performance of channel-guided, laser-plasma accelerators. The study was carried out with the assistance of Bayesian optimization using particle-in-cell simulations. We used a realistic plasma profile based on a novel optical-field-ionized channel technique with ionization injection and low on-axis plasma densities to maximize the energy gain of the electron bunch trailing the laser. Spectral shaping allows us to modify the temporal profile of the laser driver while keeping the laser energy constant, affecting the acceleration and injection processes. Given the complexity and breadth of the parameter space in question, we used numerical optimization to identify high performers. In particular, we found laser profiles with additional spectral content that, when used with optimal plasma channel parameters, result in charge content an order of magnitude higher than the baseline Gaussian case while also increasing the mean energy of the electron bunch. - oai:arXiv.org:2512.09125v2 - physics.plasm-ph + Generative modeling of conditional probability distributions on the level-sets of collective variables + https://arxiv.org/abs/2512.17374 + arXiv:2512.17374v2 Announce Type: replace-cross +Abstract: Given a probability distribution $\mu$ in $\mathbb{R}^d$ represented by data, we study in this paper the generative modeling of its conditional probability distributions on the level-sets of a collective variable $\xi: \mathbb{R}^d \rightarrow \mathbb{R}^k$, where $1 \le k<d$. We propose a general and efficient learning approach that is able to learn generative models on different level-sets of $\xi$ simultaneously. To improve the learning quality on level-sets in low-probability regions, we also propose a strategy for data enrichment by utilizing data from enhanced sampling techniques. We demonstrate the effectiveness of our proposed learning approach through concrete numerical examples. The proposed approach is potentially useful for the generative modeling of molecular systems in biophysics, for instance. + oai:arXiv.org:2512.17374v2 + stat.ML math.OC - physics.acc-ph - Mon, 22 Dec 2025 00:00:00 -0500 + Tue, 23 Dec 2025 00:00:00 -0500 replace-cross http://creativecommons.org/licenses/by/4.0/ - B. Z. Djordjevi\'c, C. Benedetti, A. D. McNaughton, R. Lehe, H. -E. Tsai, S. C. Wilks, B. A. Reagan, G. J. Williams, J. van Tilborg, C. B. Schroeder + Fatima-Zahrae Akhyar, Wei Zhang, Gabriel Stoltz, Christof Sch\"utte - Multiscale modeling of blood circulation with cerebral autoregulation and network pathway analysis for hemodynamic redistribution in the vascular network with anatomical variations and stenosis conditions - https://arxiv.org/abs/2512.15482 - arXiv:2512.15482v2 Announce Type: replace-cross -Abstract: Cerebral hemodynamics is fundamentally regulated by the Circle of Willis (CoW), which redistributes flow through communicating arteries to stabilize perfusion under anatomical variations and vascular stenosis. In this study, we develop a multiscale circulation model by coupling a systemic hemodynamic framework with a cerebral arterial network reconstructed from medical imaging. The model incorporates a cerebral autoregulation mechanism (CAM) and enables quantitative simulation of flow redistribution within the CoW under normal, anatomically varied, and stenotic conditions. Baseline simulations reproduce physiological flow distributions in which communicating arteries remain nearly inactive, showing negligible cross-flow and agreement with clinical measurements. In contrast, anatomical variations reveal distinct collateral activation patterns: the anterior communicating artery (ACoA) emerges as the earliest and most sensitive functional collateral, whereas the posterior communicating arteries (PCoAs) exhibit structure-dependent engagement. Progressive stenosis simulations further demonstrate a transition from a complete CoW to a fetal-type posterior cerebral artery (PCA) configuration, characterized by early ACoA flow reversal followed by ipsilateral PCoA activation, consistent with experimental and transcranial Doppler observations. Finally, a path-based quantitative analysis is introduced to illustrate how the cerebral vascular network dynamically reconfigures collateral pathways in response to structural changes. Overall, the proposed framework provides a physiologically interpretable, image-informed tool for investigating cerebral flow regulation through functional collaterals within the CoW, with potential applications in the diagnosis and treatment planning of cerebrovascular diseases. - oai:arXiv.org:2512.15482v2 - physics.med-ph - cs.NA - math.NA - Mon, 22 Dec 2025 00:00:00 -0500 + Alternating Direction Method of Multipliers for Nonlinear Matrix Decompositions + https://arxiv.org/abs/2512.17473 + arXiv:2512.17473v2 Announce Type: replace-cross +Abstract: We present an algorithm based on the alternating direction method of multipliers (ADMM) for solving nonlinear matrix decompositions (NMD). Given an input matrix $X \in \mathbb{R}^{m \times n}$ and a factorization rank $r \ll \min(m, n)$, NMD seeks matrices $W \in \mathbb{R}^{m \times r}$ and $H \in \mathbb{R}^{r \times n}$ such that $X \approx f(WH)$, where $f$ is an element-wise nonlinear function. We evaluate our method on several representative nonlinear models: the rectified linear unit activation $f(x) = \max(0, x)$, suitable for nonnegative sparse data approximation, the component-wise square $f(x) = x^2$, applicable to probabilistic circuit representation, and the MinMax transform $f(x) = \min(b, \max(a, x))$, relevant for recommender systems. The proposed framework flexibly supports diverse loss functions, including least squares, $\ell_1$ norm, and the Kullback-Leibler divergence, and can be readily extended to other nonlinearities and metrics. We illustrate the applicability, efficiency, and adaptability of the approach on real-world datasets, highlighting its potential for a broad range of applications. + oai:arXiv.org:2512.17473v2 + eess.SP + cs.LG + math.OC + stat.ML + Tue, 23 Dec 2025 00:00:00 -0500 replace-cross - http://creativecommons.org/licenses/by/4.0/ - Jiawei Liu, Atsushi Kanoke, Hidenori Endo, Kuniyasu Niizuma, Hiroshi Suito + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Atharva Awari, Nicolas Gillis, Arnaud Vandaele + + + A Unified Representation of Neural Networks Architectures + https://arxiv.org/abs/2512.17593 + arXiv:2512.17593v2 Announce Type: replace-cross +Abstract: In this paper we consider the limiting case of neural networks (NNs) architectures when the number of neurons in each hidden layer and the number of hidden layers tend to infinity thus forming a continuum, and we derive approximation errors as a function of the number of neurons and/or hidden layers. Firstly, we consider the case of neural networks with a single hidden layer and we derive an integral infinite width neural representation that generalizes existing continuous neural networks (CNNs) representations. Then we extend this to deep residual CNNs that have a finite number of integral hidden layers and residual connections. Secondly, we revisit the relation between neural ODEs and deep residual NNs and we formalize approximation errors via discretization techniques. Then, we merge these two approaches into a unified homogeneous representation of NNs as a Distributed Parameter neural Network (DiPaNet) and we show that most of the existing finite and infinite-dimensional NNs architectures are related via homogenization/discretization with the DiPaNet representation. Our approach is purely deterministic and applies to general, uniformly continuous matrix weight functions. Relations with neural fields and other neural integro-differential equations are discussed along with further possible generalizations and applications of the DiPaNet framework. + oai:arXiv.org:2512.17593v2 + cs.LG + math.OC + Tue, 23 Dec 2025 00:00:00 -0500 + replace-cross + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Christophe Prieur, Mircea Lazar, Bogdan Robu