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Cluster tori over $\mathbb{F}_2$, hexagonal moves on triangulations, and minimal coverings of cluster manifolds
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arxiv.org
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2 days ago
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eng
arXiv:2509.04614v2 Announce Type: replace Abstract: We study cluster algebras over $\mathbb{F}_2$. By the Laurent phenomenon there is a map from the set of seeds of the cluster algebra to the corresponding cluster variety. We show that in type $A$, fibers of this map can be described in terms of certain edges of the universal polytope of triangulations of a polygon. Moreover, we show that there is a section of this map giving seeds whose c..
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On Hardy's $Z$-function and its derivatives associated with the extended Selberg class
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arxiv.org
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2 days ago
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eng
arXiv:2509.06248v2 Announce Type: replace Abstract: Hardy's $Z$-function $Z(t)$ is a real-valued function of the real variable $t$, and whose zeros correspond exactly to the zeros of the Riemann zeta-function on the critical line. In 2012, K. Matsuoka showed that for every non-negative integer $k$, there exists a $T=T(k)>0$ such that $Z^{(k+1)}(t)$ has exactly one zero between consecutive zeros of $Z^{(k)}(t)$ for $t\ge T$ under the Rieman..
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arXiv:2509.06527v3 Announce Type: replace Abstract: The aim of this article is to study basic structures and interrelations of $\delta$-rings, perfectoid towers, and lim Cohen--Macaulay sequences over Noetherian rings in positive or mixed characteristic. We also discuss the deformation of perfectoid purity via perfectoid towers. In the latter part of this paper, we discuss some methods for constructing perfectoid towers, dealing with $p$-t..
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A Statistical Test for Comparing the Linkage and Admixture Model Based on Central Limit Theorems
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arxiv.org
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2 days ago
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eng
arXiv:2509.12734v4 Announce Type: replace Abstract: In the Admixture Model, the probability that an individual carries a certain allele at a specific marker depends on the allele frequencies in $K$ ancestral populations and the proportion of the individual's genome originating from these populations. The markers are assumed to be independent. The Linkage Model is a Hidden Markov Model (HMM) that extends the Admixture Model by incorporating....
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Constructive Proof of Two Characterizations of Arithmetic Equivalence
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arxiv.org
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2 days ago
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eng
arXiv:2509.14083v2 Announce Type: replace Abstract: We prove a formula governing the combinatorics of cyclic group actions and extend a lemma of Hasse to account for ramification. Consequently, we obtain the first constructive proofs of two theorems characterizing arithmetic equivalence. We are also able to give new proofs of strong multiplicity one type theorems for Kronecker equivalence and weak Kronecker equivalence.
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Variable-preconditioned transformed primal-dual method for generalized Wasserstein Gradient Flows
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arxiv.org
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2 days ago
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eng
arXiv:2509.15385v3 Announce Type: replace Abstract: We propose a Variable-Preconditioned Transformed Primal-Dual (VPTPD) method for solving generalized Wasserstein gradient flows based on the structure-preserving JKO scheme. This is a nontrivial extension of the TPD method [Chen et al. (2025) SIAM J. Sci. Comput.] incorporating proximal splitting techniques to address the challenges arising from the nonsmoothness of the objective function.....
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arXiv:2509.18025v2 Announce Type: replace Abstract: One can see deep-learning models as compositions of functions within the so-called tame geometry. In this expository note, we give an overview of some topics at the interface of tame geometry (also known as o-minimality), optimization theory, and deep learning theory and practice. To do so, we gradually introduce the concepts and tools used to build convergence guarantees for stochastic g..
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A hybrid scheme for fixed points of a countable family of generalized non-expansive-type maps and generalized mixed equilibrium problems
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arxiv.org
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2 days ago
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eng
arXiv:2509.23076v2 Announce Type: replace Abstract: Let $\Omega$ be a nonempty closed and convex subset of a uniformly smooth and uniformly convex real Banach space $\mathcal{X}$ with dual space $\mathcal{X}^*$. This article presents a hybrid algorithm for finding a common element of the set of solutions to a generalized mixed equilibrium problem and the set of common fixed points of a family of a general class of nonlinear nonexpansive ma..
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Spectral Tur\'an-type problems for the $\alpha$-spectral radius of hypergraphs with degree stability
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arxiv.org
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2 days ago
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eng
arXiv:2509.24354v2 Announce Type: replace Abstract: An $r$-pattern $P$ is an ordered pair $P=([l],E)$, where $l$ is a positive integer and $E$ is a set of $r$-multisets with elements from $[l]$. An $r$-graph $H$ is said to be $P$-colorable if there is a homomorphism $\phi$: $V(H)\rightarrow [l]$ such that $\{\phi(v_{1}),\ldots,\phi(v_{r})\}\in E$ for every edge $\{v_{1},\ldots,v_{r}\}\in E(H)$. Let $\mathrm{Col}(P)$ denote the family of al....
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Good Ramanujan Expansions: A suitably enhanced decay of coefficients has important consequences
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arxiv.org
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2 days ago
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eng
arXiv:2509.24456v2 Announce Type: replace Abstract: In this self-contained short note, we introduce the new definition of Good Ramanujan Expansion, say G.R.E., for a fixed arithmetic function $F$, building upon a good decay of its coefficients $G$; this, gains $\log-$powers w.r.t. the trivial bound for $G$ and precisely $\log^{1+\eta}$, where the present parameter $\eta>0$ is real. This property alone has important consequences for all the....
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arXiv:2510.02501v2 Announce Type: replace Abstract: It was shown by Barron--Shafiee that an analogue of Gromov's non-squeezing theorem holds for affine maps which preserve a power $\omega^k$ of the symplectic form $\omega$ on $\mathbb{R}^{2n}$. In the present paper, we state and prove in two ways an improved version of their result which is closer to the classical affine non-squeezing theorem. One proof closely follows their argument, and ....
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Spectral extremal problems for the $(p,Q)$-spectral radius of hypergraphs
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arxiv.org
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2 days ago
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eng
arXiv:2510.02776v3 Announce Type: replace Abstract: Let $Q$ be an $s$-vertex $r$-uniform hypergraph, and let $H$ be an $n$-vertex $r$-uniform hypergraph. Denote by $\mathcal{N}(Q,H)$ the number of isomorphic copies of $Q$ in $H$. For a hereditary family $\mathcal{P}$ of $r$-uniform hypergraphs, define $$\pi(Q,\mathcal{P}):=\lim\limits_{n\to \infty}\binom{n}{s}^{-1}\max\{\mathcal{N}(Q,H): H\in \mathcal{P}~~\mbox{and}~~|V(H)|=n\}.$$ For $p\g....
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Optimal Network Pricing for Oblivious Users under Projected Decision-Dependent Distributions
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arxiv.org
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2 days ago
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eng
arXiv:2510.07157v5 Announce Type: replace Abstract: Efficient large-scale network allocation requires data-driven pricing mechanisms that internalize stochastic, nonlinear user behavior. We move beyond the classic fully strategic agents to study oblivious users (agents with bounded rationality and imperfect information). Rather than assuming an infinite horizon, our regime acknowledges that real-world flows are too transient to equilibrate....
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A Morphology-Adaptive Random Feature Method for Inverse Source Problem of the Helmholtz Equation
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arxiv.org
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2 days ago
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eng
arXiv:2510.09213v2 Announce Type: replace Abstract: The inverse source problem for the Helmholtz equation poses significant challenges, particularly when sources exhibit complex or discontinuous geometries. Traditional numerical methods suffer from prohibitive computational costs, while machine learning-based approaches such as Physics-Informed Neural Networks (PINNs) and the Random Feature Method (RFM) -- though computationally efficient ....
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Explicitly Computing with Fukaya Categories of Surfaces with Boundary
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arxiv.org
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2 days ago
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eng
arXiv:2510.10867v2 Announce Type: replace Abstract: Fukaya categories are deep and rich invariants of symplectic manifolds which are notoriously difficult to compute explicitly. In the case of surfaces, however, the situation is simple, combinatorial,and is very well understood (at least by experts). In this expository paper we will give an introduction with many examples to welcome newcomers to the area and hopefully equip them with the t..
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arXiv:2510.12035v3 Announce Type: replace Abstract: Webs are planar directed graphs that encode invariant vectors for tensor products of fundamental $U_q(\mathfrak{sl}_n)$-representations. For $\mathfrak{sl}_2$ and $\mathfrak{sl}_3$, web calculus is governed by effective reduction rules and well-understood reduced-web bases, but these features break down in higher rank. In this paper we develop a global combinatorial framework for untagged....
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arXiv:2510.12673v2 Announce Type: replace Abstract: In this work, we establish a local smoothing result on metrics with small curvature concentration with respect to Sobolev constants and volume growth. In contrast with all previous works, we remove the Ricci curvature condition and completely localize the smoothing. As an application, we prove the compactness of the space of compact manifolds with bounded curvature concentration under Ahl..
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On leveraging constrained smooth additive regression models for global optimization
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arxiv.org
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2 days ago
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eng
arXiv:2510.14122v2 Announce Type: replace Abstract: Many real-world decision-making processes rely on solving mixed-integer nonlinear programs (MINLPs). However, finding high-quality solutions to MINLPs is often computationally demanding, motivating the development of specialized algorithms to improve their tractability. In this work, we propose Mixed-Integer Smoothing Surrogate Optimization with Constraints (MISSOC), a novel optimization ....
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Optimality-Based Control Space Reduction for Infinite-Dimensional Control Spaces
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arxiv.org
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2 days ago
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eng
arXiv:2510.14479v2 Announce Type: replace Abstract: We consider linear model reduction in both the control and state variables for unconstrained linear-quadratic optimal control problems subject to time-varying parabolic PDEs. The first-order optimality condition for a state-space reduced model naturally leads to a reduced structure of the optimal control. Thus, we consider a control- and state-reduced problem that admits the same minimize..
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The high-dimension limit of characters of compact reductive Lie groups and restrictions on the production of quantum randomness
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arxiv.org
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2 days ago
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eng
arXiv:2510.24553v2 Announce Type: replace Abstract: For any element $g$ of compact reductive group $G$ we investigate the asymptotic behavior of its normalized irreducible character in the high-dimension limit, $\frac{\chi_\lambda(g)}{d_\lambda}$. We show that when $G$ is simple the limit vanishes besides identity element. For semisimple groups one gets the same results under the additional assumption that dimensions of irreducible represe..
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$L^{\alpha-1}$ distance between two one-dimensional stochastic differential equations with drift terms driven by a symmetric $\alpha$-stable process
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arxiv.org
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2 days ago
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eng
arXiv:2510.25151v3 Announce Type: replace Abstract: This paper establishes a quantitative stability theory for one-dimensional stochastic differential equations (SDEs) with non-zero drift, driven by a symmetric $\alpha$-stable process for $\alpha\in(1,2)$. Our work generalizes the classical pathwise comparison method, pioneered by Komatsu for uniqueness problems, to address the stability of SDEs featuring both non-zero drift and, crucially....
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Weight distributions of two classes of linear codes with few weights derived from Weil sums
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arxiv.org
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2 days ago
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eng
arXiv:2510.25578v2 Announce Type: replace Abstract: Linear codes with few weights have been a subject of study for many years, as they have applications in secret sharing, authentication codes, association schemes, and strongly regular graphs. In this article, two distinct classes of $p$-ary linear codes are constructed through the selection of two specific defining sets. Their weight distributions are completely determined for each case b..
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Gradient bounds for a widely degenerate orthotropic parabolic equation
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arxiv.org
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2 days ago
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eng
arXiv:2511.01480v2 Announce Type: replace Abstract: In this paper, we consider the following nonlinear parabolic equation \[ \partial_{t}u\,=\,\sum_{i=1}^{n}\partial_{x_{i}}\left[(\vert u_{x_{i}}\vert-\delta_{i})_{+}^{p-1}\frac{u_{x_{i}}}{\vert u_{x_{i}}\vert}\right]\,\,\,\,\,\,\,\,\,\,\mathrm{in}\,\,\,\Omega\times I, \] where $\Omega$ is a bounded open subset of $\mathbb{R}^{n}$ for $n\geq2$, $I\subset\mathbb{R}$ is a bounded open interva....
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Higher representation infinite algebras and toric Fano stacks of Picard number one or two
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arxiv.org
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2 days ago
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eng
arXiv:2511.02641v4 Announce Type: replace Abstract: Tilting bundles translate geometry into non-commutative algebra via derived equivalences. We prove the existence of, and classify, $d$-tilting bundles consisting of line bundles on $d$-dimensional smooth toric Fano stacks of Picard number one or two. Their endomorphism algebras give natural examples of $d$-representation infinite algebras and are closely related to the derived McKay corre....
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Proof of a conjectured spectral upper bound on the chromatic number of a graph
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arxiv.org
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2 days ago
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eng
arXiv:2511.07712v4 Announce Type: replace Abstract: Let $G$ be a simple graph on $n$ vertices and $m$ edges with chromatic number $\chi$, and let $\lambda_n$ denote the least adjacency eigenvalue. Solving a conjecture of Fan, Yu and Wang~[Electron. J. Combin., 2012], we prove that when $3\le \chi\le n-1$, the chromatic number satisfies the following upper bound: $$ \chi \le \left(\frac{n}{2}+1+\lambda_n\right) + \sqrt{\left(\frac{n}{2}+1+\....
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arXiv:2511.11151v2 Announce Type: replace Abstract: Fix a bounded $3$-polygon $(\Omega; x_1, x_2, x_3)$ with three marked boundary points $x_1, x_2, x_3\in\partial\Omega$ and suppose $(\Omega^{\delta}; x_1^{\delta}, x_2^{\delta}, x_3^{\delta})$ is an approximation of $(\Omega; x_1, x_2, x_3)$ on $\delta$-scaled hexagonal lattice. We consider uniform spanning tree (UST) in $\Omega^{\delta}$ with wired boundary conditions. Conditional on the....
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A monoidal category of dependently sorted algebraic theories I: syntax
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arxiv.org
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2 days ago
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eng
arXiv:2511.13547v2 Announce Type: replace Abstract: This is the first of a pair of papers where we construct and investigate a closed monoidal structure on the category of generalized algebraic theories (in the sense of Cartmell). In the present text, as a starting point, we define the tensor product, $A \otimes B$, between two generalized algebraic theories $A$ and $B$. This is done syntactically via an algorithm that uses the axioms of ....
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arXiv:2511.15212v2 Announce Type: replace Abstract: If a complex $X$ is a subcomplex of a diagrammatically reducible 2-complex $Y$ that has locally indicable fundamental group, then $X$ has locally indicable fundamental group. This is a consequence of the Corson-Trace characterization of diagrammatic reducibility. In this paper we use a Corson-Trace like characterization of diagrammatic reducibility away from a subcomplex to obtain a consi..
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Complexity guarantees and polling strategies for Riemannian direct-search methods
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arxiv.org
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2 days ago
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eng
arXiv:2511.15360v2 Announce Type: replace Abstract: Direct-search algorithms are derivative-free optimization techniques that operate by polling the variable space along specific directions forming positive spanning sets (PSSs). When the problem variables are constrained to lie on a Riemannian manifold, polling must be performed along tangent directions. Although Riemannian variants of direct search have already been proposed and endowed w....
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arXiv:2511.15940v2 Announce Type: replace Abstract: Modeling tumor growth accurately is essential for understanding cancer progression and informing treatment strategies. To estimate the parameters in the tumor growth model described by a nonlinear PDE, we adopt Physics-Informed Neural Networks (PINNs) and DeepONet, which show advantages especially when the observation data is scarce and contains noise. With the help of real-life lab data,..
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arXiv:2511.16889v2 Announce Type: replace Abstract: In this paper, we study the quantization of classical observables (i.e., functions) on a K\"ahler manifold $X$ as differential operators acting on holomorphic sections of tensor powers $L^{\otimes k}$ of the pre-quantum line bundle $L$. We prove two global results as follows. (1). For a general smooth function $f \in C^\infty(X)$, we construct higher order generalizations of Kostant-Sour....
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arXiv:2511.20173v2 Announce Type: replace Abstract: Generalizing work of Marin [12], we construct in a unified way all the "braids and ties'' algebras available in literature and new ones.
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Safeguarded Stochastic Polyak Step Sizes for Non-smooth Optimization: Robust Performance Without Small (Sub)Gradients
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arxiv.org
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2 days ago
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eng
arXiv:2512.02342v3 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 inte....
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arXiv:2512.04428v2 Announce Type: replace Abstract: We investigate the lifespan of solutions to the higher-order semilinear parabolic equation $$u_t+(-\Delta)^m u=|u|^p, \quad x \in \mathbb{R}^n, t>0 $$ with initial data. We focus on the precise asymptotic behavior of the lifespan of nontrivial solutions. By combining the test function method and semigroup estimates, we derive both upper and lower bounds for the lifespan of solutions $$T_{..
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Nonabelian multiplicative integration and curvature obstructions for surface holonomy
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arxiv.org
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2 days ago
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eng
arXiv:2512.05155v2 Announce Type: replace Abstract: Surface holonomy plays a central role in higher gauge theory, bundle gerbes and the geometric formulation of Wess--Zumino terms in string theory. In this work, we consider the relation between surface holonomy and nonabelian multiplicative integration on surfaces. In this framework, we interpret the local Stokes law as a curvature obstruction law for higher holonomy and investigate its co....
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arXiv:2512.06425v2 Announce Type: replace Abstract: We establish complete characterizations of various notions of expansivity for weighted composition operators on a very general class of locally convex spaces of continuous functions. This class includes several classical classes of continuous function spaces, such as the Banach spaces $C_0(X)$ of continuous scalar-valued functions vanishing at infinity on a Hausdorff locally compact space....
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Compact Runge-Kutta flux reconstruction methods for non-conservative hyperbolic equations
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arxiv.org
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2 days ago
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eng
arXiv:2512.08611v2 Announce Type: replace Abstract: Compact Runge-Kutta (cRK) Flux Reconstruction (FR) methods are a variant of RKFR methods for hyperbolic conservation laws with a compact stencil including only immediate neighboring finite elements. We extend cRKFR methods to handle hyperbolic equations with stiff source terms and non-conservative products. To handle stiff source terms, we use IMplicit EXplicit (IMEX) time integration sch....
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arXiv:2512.09167v2 Announce Type: replace Abstract: We show that a bounded planar simply connected domain $\Omega$ is a $W^{1,\,1}$-extension domain if and only if for every pair $x,y$ of points in $\Omega^c$ there exists a curve $\gamma \subset \Omega^c$ connecting $x$ and $y$ with $$ \int_\gamma \frac{1}{\chi_{\mathbb R^2\setminus \partial\Omega}(z)}\,ds(z) \le C|x-y|.$$ Consequently, a planar Jordan domain $\Omega$ is a $W^{1,\,1}$-exte..
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Algebraic approach to the inverse spectral problem for rational matrices
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arxiv.org
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2 days ago
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eng
arXiv:2512.10468v2 Announce Type: replace Abstract: We consider the problem of reconstruction of an $n\times n$ matrix with coefficients depending rationally on $x\in \mathbb P^1$ from the data of: (a) its characteristic polynomial and (b) a line bundle of degree $g+n-1$, with $g$ the geometric genus of the spectral curve, represented by a choice of $g+n+1$ points forming a (non-positive) divisor of the given degree. We thus provide a rec..
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arXiv:2512.11598v3 Announce Type: replace Abstract: We present a numerical method for simulating rarefied gases that interact with moving boundaries and rigid bodies. The gas is described by the BGK equation in Lagrangian form and solved using an Arbitrary Lagrangian-Eulerian method, in which grid points move with the local mean velocity of the gas. The main advantage of the moving grid is that the algorithm can deal well with cases where ....
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Low-degree mod 2 cohomology of classifying spaces of $G_2$-gauge groups
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arxiv.org
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2 days ago
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eng
arXiv:2512.12195v5 Announce Type: replace Abstract: Let $\mathcal{G}_k$ denote the gauge group of the principal $G_2$--bundle over $S^4$ classified by $k\in \pi_4(BG_2)\cong \mathbb Z$. Motivated by the $p$--local homotopy classification of these gauge groups, due to Kishimoto--Theriault--Tsutaya and Kameko, we study the low-degree mod~$2$ cohomology of the classifying spaces $B\mathcal{G}_k$ as unstable modules over the Steenrod algebra. ....
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Mixed-Integer Linear Programming Approximations for the Stochastic Knapsack
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arxiv.org
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2 days ago
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eng
arXiv:2512.14912v2 Announce Type: replace Abstract: We develop mathematical programming approximations to tackle the stochastic knapsack problem. In this problem, the decision maker considers items for which either weights or values, or both, are random. The aim is to select a subset of these items to be included into their knapsack. We study both static and dynamic variants of this problem: in the static setting, the decision about which ....
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arXiv:2512.18891v3 Announce Type: replace 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 s..
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Cylindricity of weighted singular del Pezzo surfaces over fields of characteristic zero
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arxiv.org
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2 days ago
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eng
arXiv:2512.20134v3 Announce Type: replace Abstract: In this paper, we study the cylindricity of $\Bbbk$-forms of singular del Pezzo surfaces obtained by blowing up weighted projective planes $\mathbb{P}(1,1,m)$ over an arbitrary field $\Bbbk$ of characteristic zero. As an application, we obtain vertical cylinders on higher-dimensional fibrations whose generic fibers are such $\Bbbk$-forms.
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