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bb78ac3bdc4c17d8b6656d05bc4a07e1f9da5370215b967f225594b7f0911f75
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2026-01-01T00:00:00-05:00
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On quadratic Lie algebras containing the Heisenberg Lie algebra
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arXiv:2512.24333v1 Announce Type: new Abstract: In this work we study quadratic Lie algebras that contain the Heisenberg Lie algebra $\h_m$ as an ideal. We give a procedure for constructing these kind of quadratic Lie algebras and prove that any quadratic Lie algebra $\g$ that contains the Heisenberg Lie algebra as an ideal is constructed by using this procedure. We state necessary and sufficiency conditions to determine whether an indecomposable quadratic Lie algebra is the Heisenberg Lie algebra extended by a derivation. In addition, we state necessary and sufficiency conditions to determine whether the quotient $\g/\h_m$ admits an invariant metric and we also study the case when the nilradical of the Lie algebra $\g$ is equal to $\h_m$.
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https://arxiv.org/abs/2512.24333
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3daf15ce65a0cb06f76f6cea45e096b384889ce055c2dfeb0c729551aa107e36
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2026-01-01T00:00:00-05:00
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Backpropagation from KL Projections: Differential and Exact I-Projection Correspondences
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arXiv:2512.24335v1 Announce Type: new Abstract: We establish two exact correspondences between reverse-mode automatic differentiation (backpropagation evaluated at a fixed forward pass) and compositions of projection maps in Kullback--Leibler (KL) geometry. In both settings, message passing defined by alternating KL projections enforces agreement and factorization constraints. In the first setting, backpropagation arises as the differential of a KL projection map on a delta-lifted factorization. In the second setting, on complete and decomposable sum--product networks, backpropagation coincides with exact probabilistic inference, and the backward values admit an interpretation as Lagrange multipliers of a KL I-projection problem.
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https://arxiv.org/abs/2512.24335
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61ee031697c3d0161822abd4c6cde5651ecdf517aae3f5f32b99ffd3c04c9afd
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2026-01-01T00:00:00-05:00
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Charge functions for all dimensional partitions
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arXiv:2512.24343v1 Announce Type: new Abstract: The charge functions for n-dimensional partitions are known for n=2,3,4 in the literature. We give the expression for arbitrary odd dimension in a recent work, and now further conjecture a formula for all even dimensional cases. This conjecture is proved rigorously for 6D, and numerically verified for 8D.
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https://arxiv.org/abs/2512.24343
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0eae0b89b094b42364378f1ba3de662286b31121487f48005b794a1e013ae7b0
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2026-01-01T00:00:00-05:00
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The $k$-Plancherel measure and a Finite Markov Chain
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arXiv:2512.24346v1 Announce Type: new Abstract: Let $\mathcal{P}_k(n)$ denote the set of partitions of $n$ whose largest part is bounded by $k,$ which are in well-known bijection with $(k+1)$-cores $\mathcal{C}_k$. We study a growth process on $\mathcal{C}_k$, whose stationary distribution is the $k$-Plancherel measure, which is a natural extension of the Plancherel measure in the context of $k$-Schur functions. When $k\to\infty$ it converges to the Plancherel measure for partitions, a limit studied first by Vershik-Kerov. However, when $k$ is fixed and $n\to \infty$, we conjecture that it converges to a shape close to the limit shape from the uniform growth of partitions, as studied by Rost. We show that the limiting behavior, for fixed $k$, is governed by a finite Markov chain with $k!$ states over a subset of the $k$-bounded partitions or equivalently as a TASEP over cyclic permutations of length $k+1$. This paper initiates the study of these processes, state some theorems and several intriguing conjectures found by computations of the finite Markov chain.
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https://arxiv.org/abs/2512.24346
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6b612c2a7e497c08bd2452796332e461bc7c658860d1e2f9d4efdf1f1cfcec1e
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2026-01-01T00:00:00-05:00
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The period map from commutative to noncommutative deformations
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arXiv:2512.24347v1 Announce Type: new Abstract: We study the period map from infinitesimal deformations of a scheme $X$ over a perfect field $k$ to those of the associated $k$-linear $\infty$-category $\mathrm{QC}(X)$. For quasicompact, smooth, and separated $X$, we identify the corresponding map on tangent fibres with the dual HKR map $\mathrm{R}\Gamma(X, \mathrm{T}_X)[1] \to \mathrm{HH}^{\bullet}(X/k)[2]$, and give conditions for injectivity on homotopy groups. As applications, we prove liftability along square-zero extensions to be a derived invariant (at least when $\mathrm{char}(k) \ne 2$), and exhibit cases where the entire (classical) deformation functor of $X$ is a derived invariant; this partially answers a question of Lieblich.
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https://arxiv.org/abs/2512.24347
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4773eecf3c554a94636dff4ee153aeba974b46ef5b8f31bdceb630c7618fa78c
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2026-01-01T00:00:00-05:00
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An explicit construction of heat kernels and Green's functions in measure spaces
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arXiv:2512.24348v1 Announce Type: new Abstract: We explicitly construct a heat kernel as a Neumann series for certain function spaces, such as $L^{1}$, $L^{2}$, and Hilbert spaces, associated to a locally compact Hausdorff space $\mathfrak{X}$ with Borel $\sigma$-algebra $\mathcal{B}$, and endowed with additional measure-theoretic data. Our approach is an adaptation of classical work due to Minakshishundaram and Pleijel, and it requires as input a parametrix or small time approximation to the heat kernel. The methodology developed in this article applies to yield new instances of heat kernel constructions, including normalized Laplacians on finite and infinite graphs as well as Hilbert spaces with reproducing kernels.
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https://arxiv.org/abs/2512.24348
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f16a1d68224bc225a7d0daccf22fad45cf52053407a5ae974b8f6c7e00daaf61
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2026-01-01T00:00:00-05:00
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The story of geometry told by coins
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arXiv:2512.24349v1 Announce Type: new Abstract: In three articles published in CNJ in 2012 and 2016 , we discussed some links between mathematical sciences, coin minting and numismatics. This article is a continuation of this cycle. It tells the story of selected important developments in the history of geometry using modern commemorative coins as a background and illustration.
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https://arxiv.org/abs/2512.24349
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d62c7e1be3765f19b85e79528a1abc387209651e988b73ce939154fb8510103b
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2026-01-01T00:00:00-05:00
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Heavy-tailed distributions; extreme value theory; large deviations; ruin probabilities; solvency risk
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arXiv:2512.24352v1 Announce Type: new Abstract: We establish sharp large deviation asymptotics for the maximum order statistic of independent and identically distributed heavy-tailed random variables, valid for all Borel subsets of the right tail. This result yields exact decay rates for exceedance probabilities at thresholds that grow faster than the natural extreme-value scaling. As an application, we derive the polynomial rate of decay of ruin probabilities in insurance portfolios where insolvency is driven by a single extreme claim.
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https://arxiv.org/abs/2512.24352
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0543a8030a88742af49fc1ca195b35957ac535094e75b7ea46767f68308d94f6
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2026-01-01T00:00:00-05:00
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Functional models for $\Gamma_n$-contractions
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arXiv:2512.24353v1 Announce Type: new Abstract: This article develops several functional models for a given $\Gamma_n$-contraction. The first model is motivated by the Douglas functional model for a contraction. We then establish factorization results that clarify the relationship between a minimal isometric dilation and an arbitrary isometric dilation of a contraction. Using these factorization results, we obtain a Sz.-Nagy-Foias type functional model for a completely non-unitary $\Gamma_n$-contraction, as well as Sch\"affer type functional model for $\Gamma_n$-contraction.
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https://arxiv.org/abs/2512.24353
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73b37ebf4c31a0776980fe96c419b30a847ddb770888cdf32673685fcb8ba607
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2026-01-01T00:00:00-05:00
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On $R$-equivalence of Automorphism Groups of Associative Algebras
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arXiv:2512.24357v1 Announce Type: new Abstract: Let $A$ be a finite-dimensional associative $k$-algebra with identity. The primary aim of this paper is to study the rationality properties of the group of all $k$-algebra automorphisms of $A$, as an affine algebraic group over an arbitrary field $k$. We investigate mainly the $R$-equivalence property of the identity component of $\mathrm{Aut}_{k}(A)$ over a perfect field $k$.
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https://arxiv.org/abs/2512.24357
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4052daa9d81359129bec9fac01aefc449829d79484ba2490dcbd38ad6e83a27e
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2026-01-01T00:00:00-05:00
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Permutations with only reduced co-BPDs
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arXiv:2512.24361v1 Announce Type: new Abstract: Bumpless pipe dreams (BPDs) are combinatorial objects used in the study of Schubert and Grothendieck polynomials. Weigandt recently introduced a co-BPD object associated to each BPD and used them to give an analogue to the change of bases formulas of Lenart and Lascoux between these polynomials. She posed the problem of characterizing the set of permutations whose BPDs have only reduced co-BPDs. We give a pattern-avoidance characterization for these permutations using a set of seven patterns.
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https://arxiv.org/abs/2512.24361
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343d5a41deb0746367f661175ae6e54176726e0eb514479043f7a4a50a6233b7
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2026-01-01T00:00:00-05:00
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On Solvability of Automorphism Groups of Commutative Algebras
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arXiv:2512.24364v1 Announce Type: new Abstract: Let $A$ be a finite-dimensional commutative associative algebra with unity over an algebraically closed field $\mathbb{K}$. The purpose of the paper is to study the solvability of $G_A$, where $G_A$ is the identity component of $\text{Aut}_\mathbb{K}(A)$. Inspired by Pollack's work, Saor\'in and Asensio have started this study for a commutative associative algebra $A$ when $\text{dim}(R/R^2)=2$, where $R$ is the Jacobson radical of $A$. In this paper, we give new sufficient conditions on $A$ so that $G_A$ is solvable without any restriction on $\text{dim}(R/R^2)$.
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https://arxiv.org/abs/2512.24364
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32b474126cd190110251d43206feea7f8e9cb546a5a746d82476fa106d22c2e3
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2026-01-01T00:00:00-05:00
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Limit theorems for the distance of random points in $l_p^n$-balls
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arXiv:2512.24367v1 Announce Type: new Abstract: In this paper, we prove that the Euclidean distance between two independent random vectors uniformly distributed on $l_p^n$-balls $(1 \leq p \leq \infty)$ or on its boundary satisfies a central limit theorem as $n$ tends to $\infty$. Also, we give a compact proof of the case of the sphere, which was proved by Hammersley. Furthermore, we complement our central limit theorem by providing large deviation principles for the cases $p \geq 2$.
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https://arxiv.org/abs/2512.24367
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f79e37fd59bda755f3d0bec7b1174135e94f3e29b8734795093ac65ed57fe4bf
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2026-01-01T00:00:00-05:00
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On a Bruhat decomposition related to the Shalika subgroup of $GL(2n)$
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arXiv:2512.24368v1 Announce Type: new Abstract: Let $F$ be a non-archimedean local field or a finite field. In this article, we obtain an explicit and complete set of double coset representatives for $S\backslash GL_{2n}(F)/Q$ where $S$ is the Shalika subgroup and $Q$ a maximal parabolic subgroup of the group $GL_{2n}(F)$ of invertible $2n\times 2n$ matrices. We compute the cardinality of $S\backslash GL_{2n}(F)/Q$ and also give an alternate perspective on the double cosets arising intrinsically from certain subgroups which are relevant for applications in representation theory. Finally, if $Q$ is a maximal parabolic subgroup of the type $(r,2n-r),$ we prove that $S\backslash GL_{2n}(F)/Q$ is in one to one correspondence with $\Delta S_n\backslash S_{2n}/S_{r}\times S_{2n-r}$ leading to a Bruhat decomposition. The results and proofs discussed in this article are valid over any arbitrary field $F$ even though our motivation is from representation theory of $p$-adic and finite linear groups.
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https://arxiv.org/abs/2512.24368
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389dd6a85def7aede05f208c427d5d9cce15526c88729670555b395ce13e2f44
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2026-01-01T00:00:00-05:00
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Incremental Certificate Learning for Hybrid Neural Network Verification . A Solver Architecture for Piecewise-Linear Safety Queries
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arXiv:2512.24379v1 Announce Type: new Abstract: Formal verification of deep neural networks is increasingly required in safety-critical domains, yet exact reasoning over piecewise-linear (PWL) activations such as ReLU suffers from a combinatorial explosion of activation patterns. This paper develops a solver-grade methodology centered on \emph{incremental certificate learning}: we maximize the work performed in a sound linear relaxation (LP propagation, convex-hull constraints, stabilization), and invoke exact PWL reasoning only through a selective \emph{exactness gate} when relaxations become inconclusive. Our architecture maintains a node-based search state together with a reusable global lemma store and a proof log. Learning occurs in two layers: (i) \emph{linear lemmas} (cuts) whose validity is justified by checkable certificates, and (ii) \emph{Boolean conflict clauses} extracted from infeasible guarded cores, enabling DPLL(T)-style pruning across nodes. We present an end-to-end algorithm (ICL-Verifier) and a companion hybrid pipeline (HSRV) combining relaxation pruning, exact checks, and branch-and-bound splitting. We prove soundness, and we state a conditional completeness result under exhaustive splitting for compact domains and PWL operators. Finally, we outline an experimental protocol against standardized benchmarks (VNN-LIB / VNN-COMP) to evaluate pruning effectiveness, learned-lemma reuse, and exact-gate efficiency.
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https://arxiv.org/abs/2512.24379
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00ee9a96efeca33c49c7b3dea108319ce68afae7a0ca9b23d9926986cf9364a8
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2026-01-01T00:00:00-05:00
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Equivariant Partially Wrapped Fukaya Categories on Liouville Sectors
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arXiv:2512.24382v1 Announce Type: new Abstract: We develop an equivariant Lagrangian Floer theory for Liouville sectors that have symmetry of a Lie group $G$. Moreover, for Liouville manifolds with $G$-symmetry, we develop a correspondence theory to relate the equivariant Lagrangian Floer cohomology upstairs and Lagrangian Floer cohomology of its quotient. Furthermore, we study the symplectic quotient in the presence of nodal type singularities and prove that the equivariant correspondence gives an isomorphism on cohomologies which was conjectured by Lekili-Segal.
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https://arxiv.org/abs/2512.24382
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134e25cb4ad4f712867a9df8d1df2daca532da1a3cd3458196d996df2c5c7a09
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2026-01-01T00:00:00-05:00
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Mean-Field Limits of Deterministic and Stochastic Flocking Models with Nonlinear Velocity Alignment
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arXiv:2512.24383v1 Announce Type: new Abstract: We study the mean-field limit for a class of agent-based models describing flocking with nonlinear velocity alignment. Each agent interacts through a communication protocol $\phi$ and a non-linear coupling of velocities given by the power law $A(\bv) = |\bv|^{p-2}\bv$, $p > 2$. The mean-field limit is proved in two settings -- deterministic and stochastic. We then provide quantitative estimates on propagation of chaos for deterministic case in the case of the classical fat-tailed kernels, showing an improved convergence rate of the $k$-particle marginals to a solution of the corresponding Vlasov equation. The stochastic version is addressed with multiplicative noise depending on the local interaction intensity, which leads to the associated Fokker-Planck-Alignment equation. Our results extend the classical Cucker-Smale theory to the nonlinear framework which has received considerable attention in the literature recently.
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https://arxiv.org/abs/2512.24383
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469c4ed3f94f8ad33829a74b8cef83d3d47253921bc5658a7a5cf9a3af54820c
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2026-01-01T00:00:00-05:00
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Stability of the reconstruction of the heat reflection coefficient in the phonon transport equation
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arXiv:2512.24394v1 Announce Type: new Abstract: The reflection coefficient is an important thermal property of materials, especially at the nanoscale, and determining this property requires solving an inverse problem based on macroscopic temperature measurements. In this manuscript, we investigate the stability of this inverse problem to infer the reflection coefficient in the phonon transport equation. We show that the problem becomes ill-posed as the system transitions from the ballistic to the diffusive regime, characterized by the Knudsen number converging to zero. Such a stability estimate clarifies the discrepancy observed in previous studies on the well-posedness of this inverse problem. Furthermore, we quantify the rate at which the stability deteriorates with respect to the Knudsen number and confirm the theoretical result with numerical evidence.
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https://arxiv.org/abs/2512.24394
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a2e7c710e43201c12a0b5214a124fd9e212c47551d6762e5d2a08f3c0feb4c84
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2026-01-01T00:00:00-05:00
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Variation of Physical Measures in Nontrivial Mixed Partially Hyperbolic Systems
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arXiv:2512.24409v1 Announce Type: new Abstract: We construct a smooth nontrivial mixed partially hyperbolic system and explicitly identify its skeleton. This example shares characteristics with the classical examples. Moreover, the support of each physical measure contains three fixed points with mutually distinct unstable indices. By appropriately perturbing the skeleton, we provide an example where the number of physical measures varies upper semi-continuously. The general framework of mixed partially hyperbolic systems has been studied in theorem.
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https://arxiv.org/abs/2512.24409
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3eefe1e301431f3adf1a90e871a80f898dd6c327be2fbeac019513380027b105
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2026-01-01T00:00:00-05:00
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A causal Markov category with Kolmogorov products
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arXiv:2512.24417v1 Announce Type: new Abstract: In Fritz & Rischel, Infinite products and zero-one laws in categorical probability, the problem was posed of finding an interesting Markov category which is causal and has all (small) Kolmogorov products (there Problem 6.7). Here we give an example where the deterministic subcategory is the category of Stone spaces (i.e. the dual of the category of Boolean algebras) and the kernels correspond to a restricted class of Kleisli arrows for the Radon monad. We look at this from two perspectives. First via pro-completions and Stone spaces directly. Second via duality with Boolean and algebras and effect algebras.
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https://arxiv.org/abs/2512.24417
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0afce61c8b8a55188c3bed415bf4c90c076528e1edf38ef4ed5ac918d7396026
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2026-01-01T00:00:00-05:00
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The non-backtracking transition probability matrix and its usage for node clustering
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arXiv:2512.24434v1 Announce Type: new Abstract: Relation between the real eigenvalues of the non-backtracking matrix and those of the non-backtracking Laplacian is considered with respect to node clustering. For this purpose we use the real eigenvalues of the transition probability matrix (when the random walk goes through the oriented edges with the rule of ``not going back in the next step'') which have a linear relation to those of the non-backtracking Laplacian of Jost,Mulas. ``Inflation--deflation'' techniques are also developed for clustering the nodes of the non-backtracking graph. With further processing, it leads to the clustering of the nodes of the original graph, which usually comes from a sparse stochastic block model of Bordenave,Decelle.
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https://arxiv.org/abs/2512.24434
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1bdd737d11d4849f0282cc41e121a91fb66400a0f26545071278590db7a7fb4f
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2026-01-01T00:00:00-05:00
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Uniform Continuity in Distribution for Borel Transformations of Random Fields
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arXiv:2512.24441v1 Announce Type: new Abstract: Simple sufficient conditions are given that ensure the uniform continuity in distribution for Borel transformations of random fields.
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https://arxiv.org/abs/2512.24441
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9de396f6054e2c728a87c909926a794cfd619e42318d91b182c7087925472fb6
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2026-01-01T00:00:00-05:00
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On the work of Zhiren Wang on rigidity in dynamics
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arXiv:2512.24444v1 Announce Type: new Abstract: In honor of Zhiren Wang on the occasion of being awarded the Brin Prize, we report on his exciting and deep work on rigidity of higher rank abelian groups and lattices in higher rank semisimple groups.
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https://arxiv.org/abs/2512.24444
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8b73e4fd0ad64457b6bff5b55d5ded33ea7cf483a61b34a840c38b74edec2d44
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2026-01-01T00:00:00-05:00
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Skein relations on punctured surfaces
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arXiv:2512.24447v1 Announce Type: new Abstract: This thesis studies skein relations in cluster algebras arising from punctured surfaces. We introduce skein-type identities expressing cluster variables associated with incompatible curves on a surface in terms of cluster variables corresponding to compatible arcs. Incompatibility arises from phenomena such as intersections, self-intersections, and opposite taggings at punctures. To establish these identities, we develop a combinatorial-algebraic framework that relates loop graphs to certain representations. These skein relations can then be applied to investigate structural properties of cluster algebras from punctured surfaces. In particular, they can be used to prove the existence of bases satisfying natural positivity and compatibility conditions. This extends existing work on surface cluster algebras by incorporating punctures in the interior of the surface, thereby enlarging the class of cluster algebras for which such skein relations and bases can be constructed.
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https://arxiv.org/abs/2512.24447
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cb690673b3400da83368cc97e22ec87576e94a7912c9b092353f89b44a5fe0d2
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2026-01-01T00:00:00-05:00
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Topology, Hyperbolicity, and the Shafarevich Conjecture for Complex Algebraic Varieties
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arXiv:2512.24458v1 Announce Type: new Abstract: This survey presents recent developments concerning the Shafarevich conjecture, non-abelian Hodge theories, hyperbolicity, and the topology of complex algebraic varieties, as well as the interplay among these areas. More precisely, we present the main ideas and techniques involved in the linear versions of the following conjectures: the Shafarevich conjecture, the Chern-Hopf-Thurston conjecture, Koll\'ar's conjecture on the holomorphic Euler characteristic, the de Oliveira-Katzarkov-Ramachandran conjecture, and Campana's nilpotency conjecture. In addition, we discuss characterizations of the hyperbolicity of complex quasi-projective varieties via representations of their fundamental groups, together with the generalized Green-Griffiths-Lang conjecture in the presence of a big local system.
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https://arxiv.org/abs/2512.24458
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35d25cd9ecfc70ba2fed9d0f1e08b24203ca0913078072528bbbb5409067c54a
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2026-01-01T00:00:00-05:00
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Solvability conditions for some non-Fredholm operators with shifted arguments
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arXiv:2512.24476v1 Announce Type: new Abstract: In the first part of the article we establish the existence in the sense of sequences of solutions in $H^{2}(R)$ for some nonhomogeneous linear differential equation in which one of the terms has the argument translated by a constant. It is shown that under the reasonable technical conditions the convergence in $L^{2}(R)$ of the source terms implies the existence and the convergence in $H^{2}(R)$ of the solutions. The second part of the work deals with the solvability in the sense of sequences in $H^{2}(R)$ of the integro-differential equation in which one of the terms has the argument shifted by a constant. It is demonstrated that under the appropriate auxiliary assumptions the convergence in $L^{1}(R)$ of the integral kernels yields the existence and the convergence in $H^{2}(R)$ of the solutions. Both equations considered involve the second order differential operator with or without the Fredholm property depending on the value of the constant by which the argument gets translated.
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https://arxiv.org/abs/2512.24476
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6cd513b2edbe66ce10810bceb836ad174be198abb9e757880ef250b464f8da0b
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2026-01-01T00:00:00-05:00
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Decentralized Optimization over Time-Varying Row-Stochastic Digraphs
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arXiv:2512.24483v1 Announce Type: new Abstract: Decentralized optimization over directed graphs is essential for applications such as robotic swarms, sensor networks, and distributed learning. In many practical scenarios, the underlying network is a Time-Varying Broadcast Network (TVBN), where only row-stochastic mixing matrices can be constructed due to inaccessible out-degree information. Achieving exact convergence over TVBNs has remained a long-standing open question, as the limiting distribution of time-varying row-stochastic mixing matrices depends on unpredictable future graph realizations, rendering standard bias-correction techniques infeasible. This paper resolves this open question by developing the first algorithm that achieves exact convergence using only time-varying row-stochastic matrices. We propose PULM (Pull-with-Memory), a gossip protocol that attains average consensus with exponential convergence by alternating between row-stochastic mixing and local adjustment. Building on PULM, we develop PULM-DGD, which converges to a stationary solution at $\mathcal{O}(\ln(T)/T)$ for smooth nonconvex objectives. Our results significantly extend decentralized optimization to highly dynamic communication environments.
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https://arxiv.org/abs/2512.24483
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028a3d22c3b0dd1b946a31c33add041f1bd86a37767986ca557d5b72d75c36b2
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2026-01-01T00:00:00-05:00
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Minimal Solutions to the Skorokhod Reflection Problem Driven by Jump Processes and an Application to Reinsurance
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arXiv:2512.24491v1 Announce Type: new Abstract: We consider a reflected process in the positive orthant driven by an exogenous jump process. For a given input process, we show that there exists a unique minimal strong solution to the given particle system up until a certain maximal stopping time, which is stated explicitly in terms of the dual formulation of a linear programming problem associated with the state of the system. We apply this model to study the ruin time of interconnected insurance firms, where the stopping time can be interpreted as the failure time of a reinsurance agreement between the firms. Our work extends the analysis of the particle system in Baker, Hambly, and Jettkant (2025) to the case of jump driving processes, and the existence result of Reiman (1984) beyond the case of sub-stochastic reflection matrices.
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https://arxiv.org/abs/2512.24491
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d6fc627bf0f1904403d62aa28b6b9d05c8a3476c0456d4dab15fafa6f3bcdae9
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2026-01-01T00:00:00-05:00
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From Yang-Mills to Yang-Baxter: In Memory of Rodney Baxter and Chen--Ning Yang
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arXiv:2512.24494v1 Announce Type: new Abstract: The year 2025 marked the passing of two towering figures of twentieth-century mathematical physics, Rodney Baxter and Chen-Ning Yang. Yang reshaped modern physics through the introduction of non-abelian gauge theory and, independently, through the consistency conditions underlying what is now called the Yang-Baxter equation. Baxter transformed those conditions into a systematic theory of exact solvability in statistical mechanics and quantum integrable systems. This article is written in memory of Baxter and Yang, whose work revealed how local consistency principles generate global mathematical structure. We review the Yang-Mills formulation of gauge theory, its mass obstruction and resolution via symmetry breaking, and the geometric framework it engendered, including instantons, Donaldson-Floer theory, magnetic monopoles, and Hitchin systems. In parallel, we trace the emergence of the Yang-Baxter equation from factorised scattering to solvable lattice models, quantum groups, and Chern-Simons theory. Rather than separate narratives, gauge theory and integrability are presented as complementary manifestations of a shared coherence principle, an ongoing journey from gauge symmetry toward mathematical unity.
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https://arxiv.org/abs/2512.24494
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8327bec1fb54591fc2fbcf8610bbf39c99dd802db00ebf4dc7c50f7f4ead4fa5
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2026-01-01T00:00:00-05:00
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Steady Self-Propelled Motion of a Rigid Body in a Viscous Fluid with Navier-Slip Boundary Conditions
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arXiv:2512.24510v1 Announce Type: new Abstract: We investigate the steady self-propelled motion of a rigid body immersed in a three-dimensional incompressible viscous fluid governed by the Navier-Stokes equations. The analysis is performed in a body-fixed reference frame, so that the fluid occupies an exterior domain and the propulsion mechanism is modeled through nonhomogeneous Navier-slip boundary conditions at the fluid-body interface. Such conditions provide a realistic description of propulsion in microfluidic and rough-surface regimes, where partial slip effects are significant. Under suitable smallness assumptions on the boundary flux and on the normal component of the prescribed surface velocity, we establish the existence of weak steady solutions to the coupled fluid-structure system. A key analytical ingredient is the derivation of a Korn-type inequality adapted to exterior domains with rigid-body motion and Navier-slip interfaces, which yields uniform control of both the fluid velocity and the translational and rotational velocities of the body. Beyond existence, we provide a necessary and sufficient condition under which a prescribed slip velocity on the body surface induces nontrivial translational or rotational motion of the rigid body. This is achieved through the introduction of a finite-dimensional thrust space, defined via auxiliary exterior Stokes problems with Navier boundary conditions, which captures the effective contribution of boundary-driven flows to the rigid-body motion. Our results clarify how boundary effects generate propulsion and extend the classical Dirichlet-based theory to the Navier-slip setting.
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https://arxiv.org/abs/2512.24510
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43ed450785f7b945abcc3b4c3e8bd4c45637e42b43c9c6e266ea8e98a6d527c0
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2026-01-01T00:00:00-05:00
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Classification of ancient cylindrical mean curvature flows and the Mean Convex Neighborhood Conjecture
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arXiv:2512.24524v1 Announce Type: new Abstract: We resolve the Mean Convex Neighborhood Conjecture for mean curvature flows in all dimensions and for all types of cylindrical singularities. Specifically, we show that if the tangent flow at a singular point is a multiplicity-one cylinder, then in a neighborhood of that point the flow is mean-convex, its time-slices arise as level sets of a continuous function, and all nearby tangent flows are cylindrical. Moreover, we establish a canonical neighborhood theorem near such points, which characterizes the flow via local models. We also obtain a more uniform version of the Mean Convex Neighborhood Conjecture, which only requires closeness to a cylinder at some initial time and yields a quantitative version of this structural description. Our proof relies on a complete classification of ancient, asymptotically cylindrical flows. We prove that any such flow is non-collapsed, convex, rotationally symmetric, and belongs to one of three canonical families: ancient ovals, the bowl soliton, or the flying wing translating solitons. Central to our method is a refined asymptotic analysis and a novel \emph{leading mode condition,} together with a new ``induction over thresholds'' argument. In addition, our approach provides a full parameterization of the space of asymptotically cylindrical flows and gives a new proof of the existence of flying wing solitons. Our method is independent of prior work and, together with our prequel paper, this work is largely self-contained.
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https://arxiv.org/abs/2512.24524
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53cfa445441b7cd095e4c019b0ddff5c2cdcc5519145fb961c11c263924de079
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2026-01-01T00:00:00-05:00
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Dimension-free estimators of gradients of functions with(out) non-independent variables
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arXiv:2512.24527v1 Announce Type: new Abstract: This study proposes a unified stochastic framework for approximating and computing the gradient of every smooth function evaluated at non-independent variables, using $\ell_p$-spherical distributions on $\R^d$ with $d, p\geq 1$. The upper-bounds of the bias of the gradient surrogates do not suffer from the curse of dimensionality for any $p\geq 1$. Also, the mean squared errors (MSEs) of the gradient estimators are bounded by $K_0 N^{-1} d$ for any $p \in [1, 2]$, and by $K_1 N^{-1} d^{2/p}$ when $2 \leq p \ll d$ with $N$ the sample size and $K_0, K_1$ some constants. Taking $\max\left\{2, \log(d) \right\} < p \ll d$ allows for achieving dimension-free upper-bounds of MSEs. In the case where $d\ll p< +\infty$, the upper-bound $K_2 N^{-1} d^{2-2/p}/ (d+2)^2$ is reached with $K_2$ a constant. Such results lead to dimension-free MSEs of the proposed estimators, which boil down to estimators of the traditional gradient when the variables are independent. Numerical comparisons show the efficiency of the proposed approach.
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https://arxiv.org/abs/2512.24527
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3b9c094f8419d71238bd6df4d5e8d28796ce0e8fb78376b9fd996d3b0cd2a2ca
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2026-01-01T00:00:00-05:00
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On semisimplicity criteria and non-semisimple representation theory for the Kadar-Yu algebras
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arXiv:2512.24535v1 Announce Type: new Abstract: The Kadar--Yu algebras are a physically motivated sequence of towers of algebras interpolating between the Brauer algebras and Temperley--Lieb algebras. The complex representation theory of the Brauer and Temperley--Lieb algebras is now fairly well understood, with each connecting in a different way to Kazhdan--Lusztig theory. The semisimple representation theory of the KY algebras is also understood, and thus interpolates, for example, between the double-factorial and Catalan combinatorial realms. However the non-semisimple representation theory has remained largely open, being harder overall than the (already challenging) Brauer case. In this paper we determine generalised Chebyshev-like forms for the determinants of gram matrices of contravariant forms for standard modules. This generalises the root-of-unity paradigm for Temperley--Lieb algebras (and many related algebras); interpolating in various ways between this and the `integral paradigm' for Brauer algebras. The standard module gram determinants give a huge amount of information about morphisms between standard modules, making thorough use of the powerful homological machinery of towers of recollement (ToR), with appropriate gram determinants providing the ToR `bootstrap'. As for the Brauer and TL cases the representation theory has a strongly alcove-geometric flavour, but the KY cases guide an intriguing generalisation of the overall geometric framework.
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https://arxiv.org/abs/2512.24535
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d878508f6d10036ec79869318b02bf0d212169433da82fd4453d8e1b85a5230d
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2026-01-01T00:00:00-05:00
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The square of a subcubic planar graph without a 5-cycle is 7-choosable
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arXiv:2512.24536v1 Announce Type: new Abstract: The square of a graph $G$, denoted $G^2$, has the same vertex set as $G$ and has an edge between two vertices if the distance between them in $G$ is at most $2$. Thomassen [12] showed that $\chi(G^2) \leq 7$ if $G$ is a subcubic planar graph. A natural question is whether $\chi_{\ell}(G^2) \leq 7$ or not if $G$ is a subcubic planar graph. Recently Kim and Lian [11] showed that $\chi_{\ell}(G^2) \leq 7$ if $G$ is a subcubic planar graph of girth at least 6. And Jin, Kang, and Kim [10] showed that $\chi_{\ell}(G^2) \leq 7$ if $G$ is a subcubic planar graph without 4-cycles and 5-cycles. In this paper, we show that the square of a subcubic planar graph without 5-cycles is 7-choosable, which improves the results of [10] and [11].
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https://arxiv.org/abs/2512.24536
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a8564afe99c4a0633f7bf56c00b061e929fcd508d2f90bf0bcc7ceb071f17dca
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2026-01-01T00:00:00-05:00
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The Diagrammatic Spherical Category
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arXiv:2512.24541v1 Announce Type: new Abstract: We construct a diagrammatic categorification of the spherical module over the Hecke algebra. We establish a basis for the morphism spaces of this category, and prove that it is equivalent to an existing algebraic spherical category.
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https://arxiv.org/abs/2512.24541
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e0042e3fd855f71ec6f2f3827ca58fe8b61e2989af31afe1306d1847504be457
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2026-01-01T00:00:00-05:00
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Non-isomorphic metacyclic $p$-groups of split type with the same group zeta function
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arXiv:2512.24546v1 Announce Type: new Abstract: For a finite group $G$, let $a_n(G)$ be the number of subgroups of order $n$ and define $\zeta_G(s)=\sum_{n\ge 1} a_n(G)n^{-s}$. Examples are known of non-isomorphic finite groups with the same group zeta function. However, no general criterion is known for when two finite groups have the same group zeta function. Fix integers $m,n\ge 1$ and a prime $p$, and consider the metacyclic $p$-groups of split type $G(p,m,n,k)$ defined by $ G(p,m,n,k)=\langle a,b \mid a^{p^{m}}=b^{p^{n}}=\mathrm{id}, b^{-1}ab=a^{k}\rangle$. For fixed $m$ and $n$, we characterize the pairs of parameters $k_1,k_2$ for which $\zeta_{G(p,m,n,k_1)}(s)=\zeta_{G(p,m,n,k_2)}(s)$.
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https://arxiv.org/abs/2512.24546
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a8d8cf0dda6054c4b97d3d1914761e2e2c316408b2d3baae840b37b0064be9fa
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2026-01-01T00:00:00-05:00
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Poincar\'e duality for singular tropical hypersurfaces
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arXiv:2512.24548v1 Announce Type: new Abstract: We find a partial extension of Poincar\'e duality theorem of Jell-Rau-Shaw to hypersurfaces obtained by non-primitive Viro's combinatorial patchworking. We define a classification of the triangulations of a lattice polytope by a level of primitivity and we find a partial Poincar\'e duality for patchworkings depending on the level of primitivity of the triangulation. Our notion of primitivity is defined modulo a certain integral domain, it is weaker than the classical definition of primitivity. We obtain also a generalization of the complete Poincar\'e duality over a certain integral domain to hypersurfaces obtained by patchworkings which are primitive modulo this integral domain. In particular, our corollary is that any tropical hypersurface obtained by patchworking from a triangulation of a simple lattice polytope satisfies complete Poincar\'e duality over the field of rationals, which is a converse of a theorem of Aksnes.
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https://arxiv.org/abs/2512.24548
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f703c3ce2d29dfbcdc14d2c009ac6abfc1a3583d748bce30dcf7d112e0630a87
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2026-01-01T00:00:00-05:00
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Anomalous Dissipation at Onsager-Critical Regularity
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arXiv:2512.24568v1 Announce Type: new Abstract: We construct solutions to the three-dimensional Euler equations exhibiting anomalous dissipation in finite time through a vanishing viscosity limit. Inspired by \cite{BDL23} and \cite{cheskidov2023dissipation}, we extend the \(2\frac{1}{2}\)-dimensional constructions and establish an Onsager-critical energy criterion adapted to such flows, showing its sharpness. Moreover, we provide a fully three-dimensional dissipative Euler example, sharp in Onsager's sense, driven by a slightly rough external force, following the framework of \cite{CL21}.
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https://arxiv.org/abs/2512.24568
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05dbc0f92b5ec26e11cb66942dbaccab1bd44ea39623e46897282c126b59ea62
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2026-01-01T00:00:00-05:00
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Isomorphisms between Covering-Induced Lattices and Classical Geometric Lattices
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arXiv:2512.24569v1 Announce Type: new Abstract: Lattices induced by coverings arise naturally in matroid theory and combinatorial optimization, providing a structured framework for analyzing relationships between independent sets and closures. In this paper, we explore the structural properties of such lattices, with a particular focus on their rank structure, covering relations, and enumeration of elements per level. Leveraging these structural insights, we investigate necessary and sufficient conditions under which the lattice induced by a covering is isomorphic to classical geometric lattices, including the lattice of partitions, the lattice of subspaces of a vector space over a finite field, and the Dowling lattice. Our results provide a unified framework for comparing these combinatorial structures and contribute to the broader study of lattice theory, matroids, and their applications in combinatorics.
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https://arxiv.org/abs/2512.24569
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3314d48e82177fa399323b4f2bc78bca60307a95052017027490157e15a37479
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2026-01-01T00:00:00-05:00
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Functional Calculi, Positivity, and Convolution of Matrices
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arXiv:2512.24575v1 Announce Type: new Abstract: Convolution admits a natural formulation as a functional operation on matrices. Motivated by the functional and entrywise calculi, this leads to a framework in which convolution defines a matrix transform that preserves positivity. Within this setting, we establish results parallel to the classical theories of P\'olya--Szeg\H{o}, Schoenberg, Rudin, Loewner, and Horn in the context of entrywise calculus. The structure of our transform is governed by a Cayley--Hamilton-type theory valid in commutative rings of characteristic zero, together with a novel polynomial-matrix identity specific to convolution. Beyond these analytic aspects, we uncover an intrinsic connection between convolution and the Bruhat order on the symmetric group, illuminating the combinatorial aspect of this functional operation. This work extends the classical theory of entrywise positivity preservers and operator monotone functions into the convolutional setting.
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https://arxiv.org/abs/2512.24575
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e148805bb4db70d43c32c16645a3fa7645e45ff23932431d63dff0f389b19d49
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2026-01-01T00:00:00-05:00
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The Dual Majorizing Measure Theorem for Canonical Processes
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arXiv:2512.24576v1 Announce Type: new Abstract: We completely characterize the boundedness of the log-concave-tailed canonical processes. The corresponding new majorizing measure theorem for log-concave-tailed canonical processes is proved using the new tree structure. Moreover, we introduce the new growth condition. Combining this condition with the dual majorizing measure theorem proven in the paper, we have developed a polynomial-time algorithm for computing expected supremum of the log-concave canonical processes.
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https://arxiv.org/abs/2512.24576
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94b6d6e6a4e5db2ccff4c4b7279984db644a08f7bb4b6aff4ce0eea4e8ac05c0
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2026-01-01T00:00:00-05:00
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Propagation of space-time singularities for perturbed harmonic oscillators
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arXiv:2512.24582v1 Announce Type: new Abstract: We discuss propagation of space-time singularities for the quantum harmonic oscillator with time-dependent metric and potential perturbations. Reformulating the quasi-homogeneous wave front set according to Lascar (1977) in a semiclassical manner, we obtain a characterization of its appearance in comparison with the unperturbed system. The idea of our proof is based on the argument of Nakamura (2009), which was originally devised for the analysis of spatial singularities of the Schr\"odinger equation, however, the application is non-trivial since the time is no more a parameter, but takes a part in the base variables.
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https://arxiv.org/abs/2512.24582
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f6084e52566e4e842d59bd923082536408a9ee7d17a61adf244e4fe0ce130591
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2026-01-01T00:00:00-05:00
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Two-Distance Sets over Finite Fields
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arXiv:2512.24590v1 Announce Type: new Abstract: We show that Blokhuis' quadratic upper bound for two-distance sets is sharp over finite fields in almost all dimensions. Our construction complements Lison\v{e}k's higher-dimensional maximal constructions that were carried out in Lorentz spaces.
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https://arxiv.org/abs/2512.24590
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85d19445574c56a41c92c7d438c5365a0d3c0dab1777c5deb6f41986e74fa0b5
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2026-01-01T00:00:00-05:00
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On the number of pairwise touching cylinders in $\mathbb{R}^d$
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arXiv:2512.24595v1 Announce Type: new Abstract: John E. Littlewood posted the question {\em ``Is it possible in 3-space for seven infinite circular cylinders of unit radius each to touch all the others? Seven is the number suggested by counting constants.''} Boz\'oki, Lee, and R\'onyai constructed a configuration of 7 mutually touching unit cylinders. The best-known upper bounds show that at most 10 unit cylinders in $\mathbb{R}^3$ can mutually touch. We consider this problem in higher dimensions, and obtain exponential (in $d$) upper bounds on the number of mutually touching cylinders in $\mathbb{R}^d$. Our method is fairly flexible, and it makes use of the fact that cylinder touching can be expressed as a combination of polynomial equalities and non-equalities.
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https://arxiv.org/abs/2512.24595
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85c148498e3893962e50d40c26cb286bc2496bdb25c22f178766f2bd4dbfcea9
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2026-01-01T00:00:00-05:00
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A categorical proof of the nonexistence of (120, 35, 10)-difference sets
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arXiv:2512.24597v1 Announce Type: new Abstract: A difference set with parameters $(v, k, \lambda)$ is a subset $D$ of cardinality $k$ in a finite group $G$ of order $v$, such that the number $\lambda$ of occurrences of $g \in G$ as the ratio $d^{-1}d'$ in distinct pairs $(d, d')\in D\times D$ is independent of $g$. We prove the nonexistence of $(120, 35, 10)$-difference sets, which has been an open problem for 70 years since Bruck introduced the notion of nonabelian difference sets. Our main tools are 1. a generalization of the category of finite groups to that of association schemes (actually, to that of relation partitions), 2. a generalization of difference sets to equi-distributed functions and its preservation by pushouts along quotients, 3. reduction to a linear programming in the nonnegative integer lattice with quadratic constraints.
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https://arxiv.org/abs/2512.24597
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467668b2d3cc85a31111971932bc4d4f92b7a3dbe3d999395bb57e7b999a54d0
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2026-01-01T00:00:00-05:00
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Phase transition thresholds and chiral magnetic fields of general degree
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arXiv:2512.24598v1 Announce Type: new Abstract: We study a variational problem for the Landau--Lifshitz energy with Dzyaloshinskii--Moriya interactions arising in 2D micromagnetics, focusing on the Bogomol'nyi regime. We first determine the minimal energy for arbitrary topological degree, thereby revealing two types of phase transitions consistent with physical observations. In addition, we prove the uniqueness of the energy minimizer in degrees $0$ and $-1$, and nonexistence of minimizers for all other degrees. Finally, we show that the homogeneous state remains stable even beyond the threshold at which the skyrmion loses stability, and we uncover a new stability transition driven by the Zeeman energy.
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https://arxiv.org/abs/2512.24598
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b4df7ee0e9b947327d0f57777a12d68d1889d1f50bc56d2669b2b7d0b22ecf20
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2026-01-01T00:00:00-05:00
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Polynomial mixing for the stochastic Schr\"odinger equation with large damping in the whole space
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arXiv:2512.24599v1 Announce Type: new Abstract: We study the long-time mixing behavior of the stochastic nonlinear Schr\"odinger equation in $\mathbb{R}^d$, $d\le 3$. It is well known that, under a sufficiently strong damping force, the system admits unique ergodicity, although the rate of convergence toward equilibrium has remained unknown. In this work, we address the mixing property in the regime of large damping and establish that solutions are attracted toward the unique invariant probability measure at polynomial rates of arbitrary order. Our approach is based on a coupling strategy with pathwise Strichartz estimates.
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https://arxiv.org/abs/2512.24599
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8f1489b93a88a9ad177944309c71cdc2def52fd920ff65bb22b12715364ba12f
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2026-01-01T00:00:00-05:00
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Intermediate topological entropies for subsets of nonautonomous dynamical systems
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arXiv:2512.24606v1 Announce Type: new Abstract: Motivated by the notion of intermediate dimensions introduced by Falconer et al., we introduce a continuum of topological entropies that are intermediate between the (Bowen) topological entropy and the lower and upper capacity topological entropies. This is achieved by restricting the families of allowable covers in the definition of topological entropy by requiring that the lengths of all strings used in a particular cover satisfy \( N \le n < N/\theta + 1 \), where \( \theta \in [0,1] \) is a parameter. When \( \theta = 1 \), only covers using strings of the same length are allowed, and we recover the lower and upper capacity topological entropies; when \( \theta = 0 \), there are no restrictions, and the definition coincides with the topological entropy. We first establish a quantitative inequality for the upper and lower intermediate topological entropies, which mirrors the corresponding result for intermediate dimensions. As a consequence, the intermediate topological entropies are continuous on $(0,1]$, though discontinuity may arise at $0$; an illustrative example is provided to demonstrate this phenomenon. We then investigate several fundamental properties of the intermediate topological entropies for nonautonomous dynamical systems, including the power rule, monotonicity and product formulas. Finally, we derive an inequality relating intermediate entropies with respect to factor maps.
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https://arxiv.org/abs/2512.24606
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98dc924376115e12e55be8c3f4a0c9d19ba1945d43debd894020305632b2c137
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2026-01-01T00:00:00-05:00
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Regulators on some abelian coverings of $\mathbb{P}^1$ minus $n+2$ points
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arXiv:2512.24607v1 Announce Type: new Abstract: In this paper, we construct certain rational or integral elements in the motivic cohomology of superelliptic curves which are quotient curves of abelian coverings of $\mathbb{P}^1$ minus $n+2$ points, and prove that these elements are non-trivial by expressing their regulators in terms of Appell-Lauricella hypergeometric functions. We also check that such elements are integral under a mild assumption. We also give various numerical examples for the Beilinson conjecture on special values of $L$-functions of the superelliptic curves by using hypergeometric expressions.
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https://arxiv.org/abs/2512.24607
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f1113c3264e499a91b50ed4a6731d3e183dc743d9b58974ff97971bb4ade74eb
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2026-01-01T00:00:00-05:00
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Injective hom-complexity between groups
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arXiv:2512.24608v1 Announce Type: new Abstract: We present the notion of injective hom-complexity, leading to a connection between the covering number of a group and the sectional number of a group homomorphism, and provide estimates for computing this invariant.
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https://arxiv.org/abs/2512.24608
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e3dd4079d76b36f3533d58b9cf0e4ae5b6a929d4e64dbe4741c6a79aa30f5182
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2026-01-01T00:00:00-05:00
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Half-space minimizing solutions of a two dimensional Allen-Cahn system
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arXiv:2512.24610v1 Announce Type: new Abstract: This paper studies minimizing solutions to a two dimensional Allen-Cahn system on the upper half plane, subject to Dirichlet boundary conditions, \begin{equation*} \Delta u-\nabla_u W(u)=0, \quad u: \mathbb{R}_+^2\to \mathbb{R}^2,\ u=u_0 \text{ on } \partial \mathbb{R}_+^2, \end{equation*} where $W: \mathbb{R}^2\to [0,\infty)$ is a multi-well potential. We give a complete classification of such half-space minimizing solutions in terms of their blow-down limits at infinity. In addition, we characterize the asymptotic behavior of solutions near the associated sharp interfaces.
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https://arxiv.org/abs/2512.24610
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6c8742140e511936f445b3e23e6e6212a943aa8ac23cae25a3a6062f69a2ffea
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2026-01-01T00:00:00-05:00
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Arithmetic spectral transition for the unitary almost Mathieu operator
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arXiv:2512.24616v1 Announce Type: new Abstract: We study the unitary almost Mathieu operator (UAMO), a one-dimensional quasi-periodic unitary operator arising from a two-dimensional discrete-time quantum walk on $\mathbb Z^2$ in a homogeneous magnetic field. In the positive Lyapunov exponent regime $0\le \lambda_1<\lambda_2\le 1$, we establish an arithmetic localization statement governed by the frequency exponent $\beta(\omega)$. More precisely, for every irrational $\omega$ with $\beta(\omega)0$ denotes the Lyapunov exponent, and every non-resonant phase $\theta$, we prove Anderson localization, i.e. pure point spectrum with exponentially decaying eigenfunctions. This extends our previous arithmetic localization result for Diophantine frequencies (for which $\beta(\omega)=0$) to a sharp threshold in frequency.
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https://arxiv.org/abs/2512.24616
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393b8d7a7019801358b9b35c9366789a5607c94e39f62456732a298b990b61d9
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2026-01-01T00:00:00-05:00
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A Study on the Algorithm and Implementation of SDPT3
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arXiv:2512.24623v1 Announce Type: new Abstract: This technical report presents a comprehensive study of SDPT3, a widely used open-source MATLAB solver for semidefinite-quadratic-linear programming, which is based on the interior-point method. It includes a self-contained and consistent description of the algorithm, with mathematical notation carefully aligned with the implementation. The aim is to offer a clear and structured reference for researchers and developers seeking to understand or build upon the implementation of SDPT3.
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https://arxiv.org/abs/2512.24623
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3e5956faca79c9b2e3c7da23b656c6801bd221aa8d4d03ed6a0848beb1116813
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2026-01-01T00:00:00-05:00
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Geometric Quantization by Paths Part II: The General Case
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arXiv:2512.24627v1 Announce Type: new Abstract: In Part I, we established the construction of the Prequantum Groupoid for simply connected spaces. This second part extends the theory to arbitrary connected parasymplectic diffeological spaces $(\mathrm{X}, \omega)$. We identify the obstruction to the existence of the Prequantum Groupoid as the non-additivity of the integration of the prequantum form $\mathbf{K}\omega$ on the space of loops. By defining a Total Group of Periods $\mathrm{P}_\omega$ directly on the space of paths, which absorbs the periods arising from the algebraic relations of the fundamental group, we construct a Prequantum Groupoid $\mathbf{T}_\omega$ with connected isotropy isomorphic to the torus of periods $\mathrm{T}_\omega = \mathbf{R}/\mathrm{P}_\omega$. Furthermore, we propose that this groupoid $\mathbf{T}_\omega$ constitutes the Quantum System itself. The classical space $\mathrm{X}$ is embedded as the Skeleton of units, surrounded by a Quantum Fog of non-identity morphisms. We prove that the group of automorphisms of the Quantum System is isomorphic to the group of symmetries of the Dynamical System, $\mathrm{Diff}(\mathrm{X}, \omega)$.
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https://arxiv.org/abs/2512.24627
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ba78bbc516178040c332f73255690eeaa66d45140c36a9cf752decf2c4667199
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2026-01-01T00:00:00-05:00
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Average first-passage times for character sums
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arXiv:2512.24631v1 Announce Type: new Abstract: Let $\varepsilon>0$ and, for an odd prime $p$, set $$ S_\ell(p):=\sum_{n\le \ell}\left(\frac{n}{p}\right). $$ Define the first-passage time $$ f_\varepsilon(p):=\min\{\ell\ge 1:\ S_\ell(p)0$ such that, as $x\to\infty$, $$ \sum_{p\le x} f_\varepsilon(p)\sim c_\varepsilon \frac{x}{\log x}. $$
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https://arxiv.org/abs/2512.24631
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47e7d029ea7a0e5508c0a34dd04f17dce7a69905e9f94123eb13cd7318547dc2
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2026-01-01T00:00:00-05:00
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A Differential Game with Symmetric Incomplete Information on Probabilistic Initial Condition and with Signal Revelation
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arXiv:2512.24640v1 Announce Type: new Abstract: In this paper, we investigate the existence and characterization of the value for a two-player zero-sum differential game with symmetric incomplete information on a continuum of initial positions and with signal revelation. Before the game starts, the initial position is chosen randomly according to a probability measure with compact support, and neither player is informed of the chosen initial position. However, they observe a public signal revealing the current state as soon as the trajectory of the dynamics hits a target set. We prove that, under a suitable notion of signal-dependent strategies, the value of the game exists, and the extended value function of the game is the unique viscosity solution of an associated Hamilton-Jacobi-Isaacs equation that satisfies a boundary condition.
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https://arxiv.org/abs/2512.24640
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41e5f499660d9c0cde31d96fd4d43e58115d5c3759db710bd56ede0b76bbb53b
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2026-01-01T00:00:00-05:00
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Periodic Beurling-Ahlfors Extensions and Quasisymmetric Rigidity of Carpets
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arXiv:2512.24649v1 Announce Type: new Abstract: We establish periodic quasiconformal extension theorems for periodic orientation-preserving quasisymmetric self homeomorphisms of quasicircles or quasi-round carpets. As applications, we prove that, if $f$ is a periodic orientation-preserving quasisymmetric self homeomorphism of a quasi-round carpet $S$ of measure zero in $\mathbb{C}$, which has a fixed point in the outer peripheral circle of $S$, then $f$ is the identity on $S$. Moreover, we prove that, if $f$ is a quasisymmetric self homeomorphism of a square carpet $S$ of measure zero in a rectangle ring, which fixes each of the four vertices of the outer peripheral circle of $S$, then $f$ is the identity on $S$. An analogous rigidity problem for the $\mathbb{C}^*$-square carpets is discussed.
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https://arxiv.org/abs/2512.24649
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1286117ff61a7fda03e9e6c5ee77730f9be43e8c245039e363d41140a74154af
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2026-01-01T00:00:00-05:00
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Rational Angle Bisection Problem in Higher Dimensional Spaces and Incenters of Simplices over Fields
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arXiv:2512.24660v1 Announce Type: new Abstract: In this article, we generalize the following problem, which is called the rational angle bisection problem, to the $n$-dimensional space $k^n$ over a subfield $k$ of $\mathbb R$: on the coordinate plane, for which rational numbers $a$ and $b$ are the slopes of the angle bisectors between two lines with slopes $a$ and $b$ rational? First, we give a few characterizations of when the angle bisectors between two lines with direction vectors in $k^n$ have direction vectors in $k^n.$ To find solutions to the problem in the case when $k = \mathbb Q,$ we also give a formula for the integral solutions of $x_1{}^2+\dots +x_n{}^2 = dx_{n+1}{}^2,$ which is a generalization of the negative Pell's equation $x^2-dy^2 = -1,$ where $d$ is a square-free positive integer. Second, by applying the above characterizations, we give a necessary and sufficient condition for the incenter of a given $n$-simplex with $k$-rational vertices to be $k$-rational. On the coordinate plane, we prove that every triangle with $k$-rational vertices and incenter can be obtained by scaling a triangle with $k$-rational side lengths and area, which is a generalization of a Heronian triangle. We also state certain fundamental properties of a few centers of a given triangle with $k$-rational vertices.
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https://arxiv.org/abs/2512.24660
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39e3596698a5db26b5807e06f560f8d6826128e9c08abf7c3f49daba357727eb
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2026-01-01T00:00:00-05:00
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Collective behaviors of an electron gas in the mean-field regime
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arXiv:2512.24666v1 Announce Type: new Abstract: In this paper, we study the momentum distribution of an electron gas in a $3$-dimensional torus. The goal is to compute the occupation number of Fourier modes for some trial state obtained through random phase approximation. We obtain the mean-field analogue of momentum distribution formulas for electron gas in [Daniel and Voskov, Phys. Rev. 120, (1960)] in high density limit and [Lam, Phys. Rev. \textbf{3}, (1971)] at metallic density. Our findings are related to recent results obtained independently by Benedikter, Lill and Naidu, and our analysis applies to a general class of singular potentials rather than just the Coulomb case.
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https://arxiv.org/abs/2512.24666
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910bc9d1294e1d978510525f64c2880ecce17139a7660a7c7fb168ceb5f05493
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2026-01-01T00:00:00-05:00
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Nonparametric Bandits with Single-Index Rewards: Optimality and Adaptivity
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arXiv:2512.24669v1 Announce Type: new Abstract: Contextual bandits are a central framework for sequential decision-making, with applications ranging from recommendation systems to clinical trials. While nonparametric methods can flexibly model complex reward structures, they suffer from the curse of dimensionality. We address this challenge using a single-index model, which projects high-dimensional covariates onto a one-dimensional subspace while preserving nonparametric flexibility. We first develop a nonasymptotic theory for offline single-index regression for each arm, combining maximum rank correlation for index estimation with local polynomial regression. Building on this foundation, we propose a single-index bandit algorithm and establish its convergence rate. We further derive a matching lower bound, showing that the algorithm achieves minimax-optimal regret independent of the ambient dimension $d$, thereby overcoming the curse of dimensionality. We also establish an impossibility result for adaptation: without additional assumptions, no policy can adapt to unknown smoothness levels. Under a standard self-similarity condition, however, we construct a policy that remains minimax-optimal while automatically adapting to the unknown smoothness. Finally, as the dimension $d$ increases, our algorithm continues to achieve minimax-optimal regret, revealing a phase transition that characterizes the fundamental limits of single-index bandit learning.
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https://arxiv.org/abs/2512.24669
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4721086b87ae97c481edfca60b87024d871127c1483c884b3de59e3f0cbc9a4a
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2026-01-01T00:00:00-05:00
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Some inequalities related to Heinz mean constant with Birkhoff orthogonality
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arXiv:2512.24675v1 Announce Type: new Abstract: Motivated by the work of Baronti et al. [J. Math. Anal. Appl. 252(2000) 124-146], where they defined the supremum of an arithmetic mean of the side lengths of a triangle, summing antipodal points on the unit sphere, we introduce a new geometric constant for Banach spaces, utilizing the Heinz means that interpolate between the geometric and arithmetic means associated with Birkhoff orthogonality. We discuss the bounds in Banach spaces and find the values of constant in Hilbert spaces. We obtain the characterization of uniformly non-square spaces. We investigate the correlation between our notion of the Heinz mean constant and other well-known terms, viz., the modulus of convexity, modulus of smoothness, and rectangular constant. Furthermore, we also give a characterization of the Radon plane with an affine regular hexagonal unit sphere.
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https://arxiv.org/abs/2512.24675
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a957cbf1cd459382c37b22bef19b29a591992403b72ac93bb964faf57ca7def5
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2026-01-01T00:00:00-05:00
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Small 3-fold blocking sets in $\mathrm{PG}(2,p^n)$
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arXiv:2512.24689v1 Announce Type: new Abstract: A $t$-fold blocking set of the finite Desarguesian plane $\mathrm{PG}(2,p^n)$, $p$ prime, is a set of points meeting each line of the plane in at least $t$ points. The minimum size of such sets is of interest for numerous reasons; however, even the minimum size of nontrivial blocking sets (i.e. $1$-fold blocking sets not containing a line) in \(\mathrm{PG}(2,p^n)\) is an open question when $n\geq 5$ is odd. For $n>1$ the conjectured lower bound for this size is $(p^n+p^{n(s-1)/s}+1)$, where $p^{n/s}$ is the size of the largest proper subfield of $\mathbb{F}_{p^n}$. Since the union of $t$ pairwise disjoint nontrivial blocking sets is a $t$-fold blocking set, it is conjectured that when $p^{n/s}$ is large enough w.r.t. $t$, then the minimum size of a $t$-fold blocking set in $\mathrm{PG}(2,p^n)$ is $t(p^n+p^{n(s-1)/s}+1)$. If $n$ is even, then the decomposition of the plane into disjoint Baer subplanes gives a $t$-fold blocking set of this size. However, for odd $n$, the existence of such sets is an unsolved problem in most cases. In this paper, we construct $3$-fold blocking sets of conjectured size. These blocking sets are obtained as the disjoint union of three linear blocking sets of R\'edei type, and they lie on the same orbit of the projectivity $(x:y:z)\mapsto (z:x:y)$.
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https://arxiv.org/abs/2512.24689
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1e55fb5731d8963985b3cf4bc31832be86a1c0fb22f340204489123d7f141a5b
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2026-01-01T00:00:00-05:00
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Counting Lattices with Local Hecke Series
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arXiv:2512.24690v1 Announce Type: new Abstract: We count the maximal lattices over $p$-adic fields and the rational number field. For this, we use the theory of Hecke series for a reductive group over nonarchimedean local fields, which was developed by Andrianov and Hina-Sugano. By treating the Euler factors of the counting Dirichlet series for lattices, we obtain zeta functions of classical groups, which were earlier studied with $p$-adic cone integrals. When our counting series equals the existing zeta functions of groups, we recover the known results in a simple way. Further we obtain some new zeta functions for non-split even orthogonal and odd orthogonal groups.
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https://arxiv.org/abs/2512.24690
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e3c9893fcc9c3452ab2bf429fa971a56f941a1eb987d8630c29827bb76dcd602
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2026-01-01T00:00:00-05:00
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Strict germs on normal surface singularities
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arXiv:2512.24699v1 Announce Type: new Abstract: We show that any holomorphic germ $f \colon (X,x_0) \to (Y,y_0)$ of topological degree $1$ between normal surface singularities can be written as $f=\pi \circ \sigma$, where $\pi \colon Y' \to (Y,y_0)$ is a modification and $\sigma \colon (X,x_0) \to (Y',y_1)$ is a local isomorphism sending $x_0$ to a point $y_1 \in \pi^{-1}(y_0)$. A result by Fantini, Favre and myself guarantees that when $f$ is a selfmap, then $(X,x_0)$ is a sandwiched singularity. We give here an alternative proof based on the construction of the associated Kato surfaces, and valuative dynamics.
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https://arxiv.org/abs/2512.24699
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7956a406235a486da5f22ea4df0565690c89a142c252335e566433db75937d93
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2026-01-01T00:00:00-05:00
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Reformulating Confidence as Extended Likelihood
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arXiv:2512.24701v1 Announce Type: new Abstract: Fisher's fiducial probability has recently received renewed attention under the name confidence. In this paper, we reformulate it within an extended-likelihood framework, a representation that helps to resolve many long-standing controversies. The proposed formulation accommodates multi-dimensional parameters and shows how higher-order approximations can be used to refine standard asymptotic confidence statements.
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https://arxiv.org/abs/2512.24701
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3d8659ec37920152809589ad8a6e3356f77f34038eabbaf623cc399020d7f55a
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2026-01-01T00:00:00-05:00
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$L_p$-estimates for nonlocal equations with general L\'evy measures
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arXiv:2512.24704v1 Announce Type: new Abstract: We consider nonlocal operators of the form \begin{equation*} L_t u(x) = \int_{\mathbb{R}^d} \left( u(x+y)-u(x)-\nabla u(x)\cdot y^{(\sigma)} \right) \nu_t(dy), \end{equation*} where $\nu_t$ is a general L\'evy measure of order $\sigma \in(0,2)$. We allow this class of L\'evy measures to be very singular and impose no regularity assumptions in the time variable. Continuity of the operators and the unique strong solvability of the corresponding nonlocal parabolic equations in $L_p$ spaces are established. We also demonstrate that, depending on the ranges of $\sigma$ and $d$, the operator can or cannot be treated in weighted mixed-norm spaces.
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https://arxiv.org/abs/2512.24704
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d988d508e76540ec32cd527c981797f8e0cb3f056829dbe347672b656585089c
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2026-01-01T00:00:00-05:00
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On $\mathscr{M}$-arrangements of conics and lines with ordinary singularities
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arXiv:2512.24707v1 Announce Type: new Abstract: In this paper, we study combinatorial aspects of reduced plane curves known as $\mathscr{M}$-curves. This notation is a natural generalization of maximizing plane curves which are well-known in the theory of algebraic surfaces. We focus here on $\mathscr{M}$-arrangements of conics and lines with ordinary singularities of multiplicity less than five and we provide various numerical constraints on their existence, particularly in terms of their weak combinatorics. Moreover, we study in detail the scenario when our $\mathscr{M}$-arrangements consist of lines and just one conic.
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https://arxiv.org/abs/2512.24707
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c56564b336786abde941c0dc0ff3b67caea4ddb69a5c35fa08ac2af466b011fe
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2026-01-01T00:00:00-05:00
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Absolutely Summing Toeplitz operators on Bergman spaces in the unit ball of $\mathbb{C}^n$
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arXiv:2512.24710v1 Announce Type: new Abstract: In this paper, for $p> 1 $ and $r \ge 1$ we provide a complete characterization of the positive Borel measures $\mu$ on the unit ball $\B_n$ of $\mathbb {C}^n$ for which the induced Toeplitz operator $T_\mu$ is $r$-summing on the Bergman space $A^{p}$. We prove that the $r$-summing norm of $T_\mu: A^p\to A^p$ is equivalent to $\|\widetilde{\mu}\|_{L^{\kappa}(d\lambda)}$, where $\kappa$ is a positive number determined by $p$ and $r$. As some preliminary, we describe when a Carleson embedding $J_\mu: A^p \to L^q(\mu) (1\le p, q\le 2)$ is $r$-summing, which extends the main result in [B. He, et al, Absolutely summing Carleson embeddings on Bergman spaces, Adv. Math., 439, 109495 (2024)].
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https://arxiv.org/abs/2512.24710
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478b657acb2c79cb90df2dbc64f3615eacf022c3a117f1ddd17e61b5a5c2c020
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2026-01-01T00:00:00-05:00
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A proximal subgradient algorithm for constrained multiobjective DC-type optimization
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arXiv:2512.24717v1 Announce Type: new Abstract: In this paper, we consider a class of constrained multiobjective optimization problems, where each objective function can be expressed by adding a possibly nonsmooth nonconvex function and a differentiable function with Lipschitz continuous gradient, then subtracting a weakly convex function. This encompasses multiobjective optimization problems involving difference-of-convex (DC) functions, which are prevalent in various applications due to their ability to model nonconvex problems. We first establish necessary and sufficient optimality conditions for these problems, providing a theoretical foundation for algorithm development. Building on these conditions, we propose a proximal subgradient algorithm tailored to the structure of the objectives. Under mild assumptions, the sequence generated by the proposed algorithm is bounded and each of its cluster points is a stationary solution.
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https://arxiv.org/abs/2512.24717
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e0f3da7d80485d25d65e7c11d080ef080c5608e0bd9bc9eae798023b6e83f940
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2026-01-01T00:00:00-05:00
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Products of random Hermitian matrices and brickwork Hurwitz numbers. Products of normal matrices
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arXiv:2512.24720v1 Announce Type: new Abstract: We consider products of $n$ random Hermitian matrices which generalize the one-matrix model and show its relation to Hurwitz numbers which count ramified coverings of certain type. Namely, these Hurwitz numbers count $2k$-fold ramified coverings of the Riemann sphere with arbitrary ramification type over $0$ and $\infty$ and ramifications related to the partition $(2^k)$ (``brickworks'' - involution without fixed points) elsewhere. Products of normal random matrices are also considered.
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https://arxiv.org/abs/2512.24720
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e59fbf081cd8c1552604c4d7c6bb4d23b5c40f1ad20a66a0e71464a39c3deb48
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2026-01-01T00:00:00-05:00
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Isocapacitary constants for the $p$-Laplacian on compact manifolds
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arXiv:2512.24725v1 Announce Type: new Abstract: In this paper, we introduce Steklov and Neumann isocapacitary constants for the $p$-Laplacian on compact manifolds. These constants yield two-sided bounds for the $(p,\alpha)$-Sobolev constants, which degenerate to upper and lower bounds for the first nontrivial Steklov and Neumann eigenvalues of the $p$-Laplacian when $\alpha= 1$.
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https://arxiv.org/abs/2512.24725
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a8dde0be27af04202e00d28d9d187604e055c27703225efad9b51cc29b2e8bdd
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2026-01-01T00:00:00-05:00
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Isomorphism between Hopf algebras for multiple zeta values
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arXiv:2512.24732v1 Announce Type: new Abstract: The classical quasi-shuffle algebra for multiple zeta values have a well-known Hopf algebra structure. Recently, the shuffle algebra for multiple zeta values are also equipped with a Hopf algebra structure. This paper shows that these two Hopf algebras are isomorphic.
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https://arxiv.org/abs/2512.24732
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6d3e0e6e2e854ef24edb411b10540fbc3386193a87211fa371da1a64ff6d360f
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2026-01-01T00:00:00-05:00
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From boundary random walks to Feller's Brownian Motions
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arXiv:2512.24734v1 Announce Type: new Abstract: We establish an invariance principle connecting boundary random walks on $\mathbb N$ with Feller's Brownian motions on $[0,\infty)$. A Feller's Brownian motion is a Feller process on $[0,\infty)$ whose excursions away from the boundary $0$ coincide with those of a killed Brownian motion, while its behavior at the boundary is characterized by a quadruple $(p_1,p_2,p_3,p_4)$. This class encompasses many classical models, including absorbed, reflected, elastic, and sticky Brownian motions, and further allows boundary jumps from $0$ governed by the measure $p_4$. For any Feller's Brownian motion that is not purely driven by jumps at the boundary, we construct a sequence of boundary random walks whose appropriately rescaled processes converge weakly to the given Feller's Brownian motion.
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https://arxiv.org/abs/2512.24734
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cc0d4a33baf7d7b0890d53d3b3bd2adc2e5dd96e8791ae325a12e722bcc04b9e
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2026-01-01T00:00:00-05:00
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Some Studies on Stochastic Optimization based Quantitative Risk Management
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arXiv:2512.24736v1 Announce Type: new Abstract: Risk management often plays an important role in decision making under uncertainty. In quantitative risk management, assessing and optimizing risk metrics requires efficient computing techniques and reliable theoretical guarantees. In this paper, we introduce several topics on quantitative risk management and review some of the recent studies and advancements on the topics. We consider several risk metrics and study decision models that involve the metrics, with a main focus on the related computing techniques and theoretical properties. We show that stochastic optimization, as a powerful tool, can be leveraged to effectively address these problems.
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https://arxiv.org/abs/2512.24736
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2f1ec5b51ca43dc6dde8d1c0a1eabe0cf70902a8fc47aa5ce61364b832f7f5eb
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2026-01-01T00:00:00-05:00
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Structure of twisted Jacquet modules of principal series representations of $GL_{2n}(F)$
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arXiv:2512.24737v1 Announce Type: new Abstract: Let $F$ be a non-archimedean local field or a finite field. Let $\pi$ be a principal series representation of $GL_{2n}(F)$ induced from any of its maximal standard parabolic subgroups. Let $N$ be the unipotent radical of the maximal parabolic subgroup $P$ of $GL_{2n}(F)$ corresponding to the partition $(n,n).$ In this article, we describe the structure of the twisted Jacquet module $\pi_{N,\psi}$ of $\pi$ with respect to $N$ and a non-degenerate character $\psi$ of $N.$ We also provide a necessary and sufficient condition for $\pi_{N,\psi}$ to be non-zero and show that the twisted Jacquet module is non-zero under certain assumptions on the inducing data. As an application of our results, we obtain the structure of twisted Jacquet modules of certain non-generic irreducible representations of $GL_{2n}(F)$ and establish the existence of their Shalika models. We conclude our article with a conjecture by Dipendra Prasad classifying the smooth irreducible representations of $GL_{2n}(F)$ with a non-zero twisted Jacquet module.
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https://arxiv.org/abs/2512.24737
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2ba9ede121287927986301f6a06df61d045d96e49bd5dee5300f4f4be8d22ff4
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2026-01-01T00:00:00-05:00
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The Radon--Nikodym topography of acyclic measured graphs
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arXiv:2512.24741v1 Announce Type: new Abstract: We study locally countable acyclic measure-class-preserving (mcp) Borel graphs by analyzing their "topography" -- the interaction between the geometry and the associated Radon--Nikodym cocycle. We identify three notions of topographic significance for ends in such graphs and show that the number of nonvanishing ends governs both amenability and smoothness. More precisely, we extend the Adams dichotomy from the pmp to the mcp setting, replacing the number of ends with the number of nonvanishing ends: an acyclic mcp graph is amenable if and only if a.e. component has at most two nonvanishing ends, while it is nowhere amenable exactly when a.e. component has a nonempty perfect (closed) set of nonvanishing ends. We also characterize smoothness: an acyclic mcp graph is essentially smooth if and only if a.e. component has no nonvanishing ends. Furthermore, we show that the notion of nonvanishing ends depends only on the measure class and not on the specific measure. At the heart of our analysis lies the study of acyclic countable-to-one Borel functions. Our critical result is that, outside of the essentially two-ended setting, all back ends in a.e. orbit are vanishing and admit cocycle-finite geodesics. We also show that the number of barytropic ends controls the essential number of ends for such functions. This leads to a surprising topographic characterization of when such functions are essentially one-ended. Our proofs utilize mass transport, end selection, and the notion of the Radon--Nikodym core for acyclic mcp graphs, a new concept that serves as a guiding framework for our topographic analysis.
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https://arxiv.org/abs/2512.24741
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7d295ed4b40366d91e3ee584a7205ca4cec2a7086a30612e366d8e9e628050cb
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2026-01-01T00:00:00-05:00
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The function-operator convolution algebra over the Bergman space of the ball and its Gelfand theory
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arXiv:2512.24746v1 Announce Type: new Abstract: We investigate the structure of the commutative Banach algebra formed as the direct sum of integrable radial functions on the disc and the radial operators on the Bergman space, endowed with the convolution from quantum harmonic analysis as the product. In particular, we study the Gelfand theory of this algebra and discuss certain properties of the appropriate Fourier transform of operators which naturally arises from the Gelfand transform.
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https://arxiv.org/abs/2512.24746
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6d055944b01979afbf70c01b95a112c7309eacd48339a81349a5a0e4a4b77cb0
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2026-01-01T00:00:00-05:00
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Non-Commutative Maximal Inequalities for State-Preserving Actions of amenable groups
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arXiv:2512.24751v1 Announce Type: new Abstract: In this article, we establish maximal inequalities and deduce ergodic theorems for state-preserving actions of amenable, locally compact, second-countable groups on tracial non-commutative $L^1$-spaces. As a further consequence, in combination with the Neveu decomposition, we obtain a stochastic ergodic theorem for amenable group actions.
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https://arxiv.org/abs/2512.24751
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b5761944e9291418a771231482fe3bed4ee403f2922b3444c7739874a027bfdb
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2026-01-01T00:00:00-05:00
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Phase Reduction of Limit Cycle Oscillators: A Tutorial Review with New Perspectives on Isochrons and an Outlook to Higher-Order Reductions
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arXiv:2512.24775v1 Announce Type: new Abstract: The phase reduction technique is essential for studying rhythmic phenomena across various scientific fields. It allows the complex dynamics of high-dimensional oscillatory systems to be expressed by a single phase variable. This paper provides a detailed review and synthesis of phase reduction with two main goals. First, we develop a solid geometric framework for the theory by creating isochrons, which are the level sets of the asymptotic phase, using the Graph Transform theorem. We show that isochrons form an invariant, continuous structure of the basin of attraction of a stable limit cycle, helping to clarify the concept of the asymptotic phase. Second, we systematically explain how to derive the first-order phase reduction for weakly perturbed and coupled systems. In the end, we discuss the limitations of the first-order approach, particularly its restriction to very small perturbations and the issue of vanishing coupling terms in certain networks. We finish by outlining the framework and importance of higher-order phase reductions. This establishes a clear link from classical theory to modern developments and sets the stage for a more in-depth discussion in a future publication.
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https://arxiv.org/abs/2512.24775
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b31b92f1b2031f3e38b953c104289e8ed92b427989341b1ac3b61b13db797c16
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2026-01-01T00:00:00-05:00
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The tournament ratchet's clicktime process, and metastability in a Moran model
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arXiv:2512.24779v1 Announce Type: new Abstract: Muller's ratchet, in its prototype version, models a haploid, asexual population whose size~$N$ is constant over the generations. Slightly deleterious mutations are acquired along the lineages at a constant rate, and individuals carrying less mutations have a selective advantage. In the classical variant, an individual's selective advantage is proportional to the difference between the population average and the individual's mutation load, whereas in the ratchet with {\em tournament selection} only the signs of the differences of the individual mutation loads matter. In a parameter regime which leads to slow clicking (i.e. to a loss of the currently fittest class at a rate $\ll 1/N$) we prove that the rescaled process of click times of the tournament ratchet converges as $N\to \infty$ to a Poisson process. Central ingredients in the proof are a thorough analysis of the metastable behaviour of a two-type Moran model with selection and deleterious mutation (which describes the size of the fittest class up to its extinction time) and a lower estimate on the size of the new fittest class at a clicktime.
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https://arxiv.org/abs/2512.24779
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20671ab39d87f08757f1ca88a91c5c6bc2ec26fbc33325ab89eecd67de95a804
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2026-01-01T00:00:00-05:00
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On rational orbits in some prehomogeneous vector spaces
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arXiv:2512.24783v1 Announce Type: new Abstract: Let $k$ be a field with characteristic different from $2$. In this paper, we describe the $k$-rational orbit spaces in some irreducible prehomogeneous vector spaces $(G,V)$ over $k$, where $G$ is a connected reductive algebraic group defined over $k$ and $V$ is an irreducible rational representation of $G$ with a Zariski dense open orbit. We parametrize all composition algebras over the field $k$ in terms of the orbits in some of these representations. This leads to a parametric description of the reduced Freudenthal algebras of dimensions $6$ and $9$ over $k$ (if $\text{char}(k)\neq 2,3$). We also get a parametrization for the involutions of the second kind defined on a central division $K$-algebra $B$ with center $K$, a quadratic extension of the underlying field $k$.
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https://arxiv.org/abs/2512.24783
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Academic Papers
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6cb7c08aa7c267d3ed3d1c60e6b73b452b8e54e5dabea02ddcbc37c32c580e30
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2026-01-01T00:00:00-05:00
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Coarse geometry of extended admissible groups
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arXiv:2512.24784v1 Announce Type: new Abstract: Extended admissible groups belong to a particular class of graphs of groups that admit a decomposition generalizing those of non-geometric 3-manifold groups and Croke-Kleiner admissible groups. In this paper, we study several coarse-geometric aspects of extended admissible groups. We show that changing the gluing edge isomorphisms does not affect the quasi-isometry type of these groups. We also prove that, under mild conditions on the vertex groups, extended admissible groups exhibit large-scale nonpositive curvature, thereby answering a question posed by Nguyen-Yang. As an application, our results enlarge the class of extended admissible groups known to admit well-defined quasi-redirecting boundaries, a notion recently introduced by Qing-Rafi. In addition, we compute the divergence of extended admissible groups, generalizing a result of Gersten from non-geometric 3-manifold groups to this broader setting. Finally, we study several aspects of subgroup structure in extended admissible groups.
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https://arxiv.org/abs/2512.24784
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e1f9d8aa2195b4781fabd47a49257a78a743d908687711c59b6ff590493f8dd2
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2026-01-01T00:00:00-05:00
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A first approximation algorithm for the Bin Packing Problem with Setups
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arXiv:2512.24785v1 Announce Type: new Abstract: We study constant-factor approximation algorithms for the Bin Packing Problem with Setups (BPPS). First, we show that adaptations of classical BPP heuristics can have arbitrarily poor worst-case performance on BPPS instances. Then, we propose a two-phase heuristic for the BPPS that applies an {\alpha}-approximation algorithm for the BPP to the items of each class and then performs a merging phase on the open bins. We prove that this heuristic is a 2 {\alpha}-approximation algorithm for the BPPS.
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https://arxiv.org/abs/2512.24785
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Academic Papers
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23c39bc3eb5a8dade2d51e8ed2dbf5dbe6d954e8d7592a3fc5181990035e7206
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2026-01-01T00:00:00-05:00
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Rational orbits in some prehomogeneous vector spaces associated to $Sp_{6}$ revisited
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arXiv:2512.24789v1 Announce Type: new Abstract: Let $k$ be a field with $\text{char}(k)\neq 2$. We prove that all maximal flags of composition algebras over $k$, appear as the $k$-rational $Sp_{6}$-orbits in a Zariski-dense $Sp_{6}$-invariant subset $V^{ss}\subset V=\wedge^{3}V_{6}$, where $V_{6}$ is the standard $6$-dimensional irreducible representation of $Sp_{6}$. This gives an arithmetic interpretation for the orbit spaces of the semi-stable sets in the prehomogeneous vector spaces $(Sp_{6}\times GL_{1}^{2},V)$ and $(GSp_{6}\times GL_{1}^{2},V)$. We also get all reduced Freudenthal algebras of dimensions $6$ and $9$, represented by the same orbit spaces.
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https://arxiv.org/abs/2512.24789
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074e5641dd065e4057248a57c6688b479f638b5266c5410f73ed6849e49f0230
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2026-01-01T00:00:00-05:00
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Curvature of left-invariant complex Finsler metric on Lie groups
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arXiv:2512.24791v1 Announce Type: new Abstract: Let $ G $ be a connected Lie group with real Lie algebra $ \mathfrak{g}$. Suppose $G$ is also a complex manifold. We obtain explicit holomorphic sectional and bisectional curvature formulas of left-invariant strongly pseudoconvex complex Finsler metrics $F$ on $G$ in terms of the complex Lie algebra $\mathfrak{g}^{1,0}$; we also obtain a necessary and sufficient condition for $F$ to be a K\"ahler-Finsler metric and a weakly K\"ahler-Finsler metric, respectively. As an application, we obtain the rigidity result: if $F$ is a left-invariant strongly pseudoconvex complex Finsler metric on a complex Lie group $G$, then $F$ must be a complex Berwald metric with vanishing holomorphic bisectional curvature; moreover, $F$ is a K\"ahler-Berwald metric iff $G$ is an Abelian complex Lie group.
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https://arxiv.org/abs/2512.24791
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9b86b47ea07ca9e9f60864809e243091c70a29c09fd6160b33501e3c3c1ebd4f
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2026-01-01T00:00:00-05:00
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Geometric approaches to Lie bialgebras, their classification, and applications
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arXiv:2512.24795v1 Announce Type: new Abstract: This PhD Thesis consists of two parts. The first part focuses on novel algebraic and geometric approaches to the classification problem of coboundary Lie bialgebras up to Lie algebra automorphisms. More specifically, Grassmann, graded algebra and algebraic invariant techniques are discussed. Using these algebraic methods, equivalence classes of r-matrices for three-dimensional coboundary Lie bialgebras are studied. Moreover, particular higher-dimensional cases, e.g. $\mathfrak{so}(2,2)$ and $\mathfrak{so}(3,2)$, are partially analysed. From the geometric perspective, the main role is played by the newly introduced notion: the Darboux family. This powerful tool allows an efficient and thorough study of equivalence classes of r-matrices for four-dimensional indecomposable coboundary Lie bialgebras. In order to showcase its ability to tackle decomposable examples, $\mathfrak{gl}_2$ is additionally studied. The second part of the Thesis sketches interesting directions for applications of r-matrices. Firstly, it is illustrated how r-matrices might be useful to describe foliated Lie-Hamilton systems. Secondly, the role of r-matrices in deformations of certain cases of Lie systems is discussed. In particular, based on the general procedure for deformations of Lie-Hamilton systems, its extension to Jacobi-Lie systems is suggested and supported by the detailed computation of the deformed Schwarz equation.
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https://arxiv.org/abs/2512.24795
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Academic Papers
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b335e710a7c1009391ac5b9bdbbf2ff80d0aa76e82107407571b56e0b86584e5
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2026-01-01T00:00:00-05:00
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Global spherically symmetric classical solutions for arbitrary large initial data of the multi-dimensional non-isentropic compressible Navier-Stokes equations
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arXiv:2512.24799v1 Announce Type: new Abstract: In 1871, Saint-Venant introduced the shallow water equations. Since then, the global classical solutions for arbitrary large initial data of the multi-dimensional viscous Saint-Venant system have remained a well-known open problem. It was only recently that [Huang-Meng-Zhang, http:arXiv:2512.15029, 2025], under the assumption of radial symmetry, first proved the existence of global classical solutions for arbitrary large initial data to the initial-boundary value problem of the two-dimensional viscous shallow water equations. At the same time, [Chen-Zhang-Zhu, http:arXiv:2512.18545, 2025] also independently proved the existence of global large solutions to the Cauchy problem of this system. Notably, in the work of Huang-Meng-Zhang, they also established the existence of global classical solutions for arbitrary large initial data to the isentropic compressible Navier-Stokes equations satisfying the BD entropy equality in both two and three dimensions, and the viscous shallow water equations are precisely a specific class of isentropic compressible fluids subject to the BD entropy equality. In this paper, we prove a new BD entropy inequality for a class of non-isentropic compressible fluids, which can be regarded as a generalization of the shallow water equations with transported entropy. Employing new estimates on the lower bound of density different from that of Huang-Meng-Zhang's work, we show the "viscous shallow water system with transport entropy" will admit global classical solutions for arbitrary large initial data to the spherically symmetric initial-boundary value problem in both two and three dimensions. Our results also relax the restrictions on the dimension and adiabatic index imposed in Huang-Meng-Zhang's work on the shallow water equations, extending the range from $N=2,\ \gamma \ge \frac{3}{2}$ to $N=2,\ \gamma > 1$ and $N=3,\ 1<\gamma<3$.
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https://arxiv.org/abs/2512.24799
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8cf5f9bf46572dc895e9acce6f4bcca47a977dbfec3e31b1a3d9997c6de79926
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2026-01-01T00:00:00-05:00
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A Study of S-primary Ideals in Commutative Semirings
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arXiv:2512.24800v1 Announce Type: new Abstract: In this article, we define the concept of an $S$-$k$-irreducible ideal and $S$-$k$-maximal ideal in a commutative semiring. We also establish several results concerning $S$-$k$-primary ideals and prove the existence theorem and the $S$-version of the uniqueness theorem using localization, for $S$-$k$-primary decompositions. Also we show that the $S$-radical of every $S$-primary ideal is a prime ideal of $R$. Moreover, we investigate the structure of $S$-primary ideals in principal ideal semidomain and prove that each such ideal can be expressed of the form, $I = (vp^n)$, $n\in \mathbf{N}$ and for some $p \in \mathbf P -\mathbf P_S$ and $v\in R$ such that $(v)\cap S\neq \varnothing $, where $\mathbf P$ is the set of all irreducible (prime) elements of R and for a multiplicative subset $S\subsetneq R$, the set $\mathbf P_S$ defined by $\mathbf P_S=\{p\in \mathbf P : (p) \cap S \neq \varnothing \}$.
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https://arxiv.org/abs/2512.24800
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Academic Papers
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95921f27dece6d1b48693e1f25437c5e68b6ea4bfae94f81b20784f5f421bd9b
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2026-01-01T00:00:00-05:00
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Heat kernel estimates for Markov processes with jump kernels blowing-up at the boundary
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arXiv:2512.24807v1 Announce Type: new Abstract: In this paper, we study purely discontinuous symmetric Markov processes on closed subsets of ${\mathbb R}^d$, $d\ge 1$, with jump kernels of the form $J(x,y)=|x-y|^{-d-\alpha}{\mathcal B}(x,y)$, $\alpha\in (0,2)$, where the function ${\mathcal B}(x,y)$ may blow up at the boundary of the state space. This extends the framework developed recently for conservative self-similar Markov processes on the upper half-space to a broader geometric setting. Examples of Markov processes that fall into our general framework include traces of isotropic $\alpha$-stable processes in $C^{1,\rm Dini}$ sets, processes in Lipschitz sets arising in connection with the nonlocal Neumann problem, and a large class of resurrected self-similar processes in the closed upper half-space. We establish sharp two-sided heat kernel estimates for these Markov processes. A fundamental difficulty in accomplishing this task is that, in contrast to the existing literature on heat kernels for jump processes, the tails of the associated jump measures in our setting are not uniformly bounded. Thus, standard techniques in the existing literature used to study heat kernels are not applicable. To overcome this obstacle, we employ recently developed weighted functional inequalities specifically designed for jump kernels blowing up at the boundary.
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https://arxiv.org/abs/2512.24807
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Academic Papers
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67cddd994853e429ec11656708cc3c489a79a9e89555ec82d41bb528ff4601b5
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2026-01-01T00:00:00-05:00
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H\"older continuity of weak solutions to the thin-film equation in $d=2$
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arXiv:2512.24809v1 Announce Type: new Abstract: The thin-film equation $\partial_t u = -\nabla \cdot (u^n \nabla \Delta u)$ describes the evolution of the height $u=u(x,t)\geq 0$ of a viscous thin liquid film spreading on a flat solid surface. We prove H\"older continuity of energy-dissipating weak solutions to the thin-film equation in the physically most relevant case of two spatial dimensions $d=2$. While an extensive existence theory of weak solutions to the thin-film equation was established more than two decades ago, even boundedness of weak solutions in $d=2$ has remained a major unsolved problem in the theory of the thin-film equation. Due the fourth-order structure of the thin-film equation, De Giorgi-Nash-Moser theory is not applicable. Our proof is based on the hole-filling technique, the challenge being posed by the degenerate parabolicity of the fourth-order PDE.
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https://arxiv.org/abs/2512.24809
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22b06ee4649f508a6dc95671c7c5b19d7486889138cb24c2eb121240ce8b7e13
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2026-01-01T00:00:00-05:00
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Four collapsing one-dimensional particles: a dynamical system approach of the spherical billiard reduction
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arXiv:2512.24812v1 Announce Type: new Abstract: We consider a system of four one-dimensional inelastic hard spheres evolving on the real line $\mathbb{R}$, and colliding according to a scattering law characterized by a fixed restitution coefficient $r$. We study the possible orders of collisions when the inelastic collapse occurs, relying on the so-called $\mathfrak{b}$-to-$\mathfrak{b}$ mapping, a two-dimensional dynamical system associated to the original particle system which encodes all the possible collision orders. We prove that the $\mathfrak{b}$-to-$\mathfrak{b}$ mapping is a piecewise projective transformation, which allows one to perform efficient numerical simulations of its orbits. We recover previously known results concerning the one-dimensional four-particle inelastic hard sphere system and we support the conjectures stated in the literature concerning particular periodic orbits. We discover three new families of periodic orbits that coexist depending on the restitution coefficient, we prove rigorously that there exist stable periodic orbits for the $\mathfrak{b}$-to-$\mathfrak{b}$ mapping for restitution coefficients larger than the upper bounds previously known, and we prove the existence of quasi-periodic orbits for this mapping.
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https://arxiv.org/abs/2512.24812
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a29c786dd79fb74cd07a53b1ed25a2ee4c1ca21dd73d311362f1c02f30284ca6
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2026-01-01T00:00:00-05:00
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Number of $K$-rational points with given $j$-invariant on modular curves
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arXiv:2512.24817v1 Announce Type: new Abstract: In this article, we study how to compute the number of $K$-rational points with a given $j$-invariant on an arbitrary modular curve. As an application, for each positive integer $n$, we determine the list of possible numbers of cyclic $n$-isogenies an elliptic curve over some number field can admit. Similarly, for an odd prime power $p^k$, we calculate the possible values for the number of points above some $j$-invariant on Cartan modular curves $X_{\mathrm s}(p^k)$, $X_{\mathrm{ns}}(p^k)$ and their normalizers. Combining known results about images of Galois representations of CM elliptic curves with our work, we also devise a simple algorithm to determine the number of rational CM points on any modular curve.
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https://arxiv.org/abs/2512.24817
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Academic Papers
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2f899b98bfe644b97009086926661acd54cd8deb3eb5b98ceeba4f105d8bc5f4
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2026-01-01T00:00:00-05:00
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A ccc indestructible construction with CH
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arXiv:2512.24821v1 Announce Type: new Abstract: We introduce a variant of the Kurepa family. We then use one such family to construct a ccc indestructible property associated with a complete coherent Suslin tree $S$. Moreover, in every ccc forcing extension that preserves Suslin of $S$, forcing with $S$ induces a strong negative partition relation.
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https://arxiv.org/abs/2512.24821
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99e4cf4e82ed149f690a8d87b5e7aea4048c4536398ccb9d350f0adac2c6d02f
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2026-01-01T00:00:00-05:00
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Bol's type inequality for singular metrics and its application to prescribing $Q$-curvature problems
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arXiv:2512.24828v1 Announce Type: new Abstract: In this article, we study higher-order Bol's inequality for radial normal solutions to a singular Liouville equation. By applying these inequalities along with compactness arguments, we derive necessary and sufficient conditions for the existence of radial normal solutions to a singular $Q$-curvature problem. Moreover, under suitable assumptions on the $Q$-curvature, we obtain uniform bounds on the total $Q$-curvature.
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https://arxiv.org/abs/2512.24828
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Academic Papers
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8b9bd212cb4bda3fc8ee76b912f725e44505465fa9fc27e9b706016a2a6cbcf9
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2026-01-01T00:00:00-05:00
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Penny graphs in the hyperbolic plane
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arXiv:2512.24832v1 Announce Type: new Abstract: We consider the problem of finding the maximum number $e_d(n)$ of pairs of touching circles in a packing of $n$ congruent circles of diameter $d$ in the hyperbolic plane of curvature $-1$. In the Euclidean plane, the maximum comes from a spiral construction of the tiling of the plane with equilateral triangles (Harborth 1974), with a similar result in the hyperbolic plane for the values of $d$ corresponding to the order-$k$ triangular tilings (Bowen 2000). We present various upper and lower bounds for $e_d(n)$ for all values of $d > 0$. In particular, we prove that if $d > 0.66114\dots$ except for $d=0.76217\dots$, then the number of touching pairs is less than the one coming from a spiral construction in the order-$7$ triangular tiling, which we conjecture to be extremal. We also give a lower bound $e_d(n) > (2+\varepsilon_d)n$ where $\varepsilon_d > 1$ for all $d > 0$.
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https://arxiv.org/abs/2512.24832
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4a3ae2dfb9b2233d87c52d380203f913c40c04c7d4e6d5c399d3f9b7cd8b3093
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2026-01-01T00:00:00-05:00
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A Comparison Principle for Bifurcation of Periodic Solutions of Hamiltonian Systems
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arXiv:2512.24835v1 Announce Type: new Abstract: We obtain novel criteria for the existence of local bifurcation for periodic solutions of Hamiltonian systems by a comparison principle of the spectral flow. Our method allows to find the appearance of new solutions by a simple inspection of the coefficients of the system.
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https://arxiv.org/abs/2512.24835
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e7619c8d1e65cc32e72cf603f4aaac6876e0c3f946219948fe39a7dc270f2224
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2026-01-01T00:00:00-05:00
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Boundedness of Fourier Integral Operators with complex phases on Fourier Lebesgue spaces
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arXiv:2512.24854v1 Announce Type: new Abstract: In this paper, we develop boundedness estimates for Fourier integral operators on Fourier Lebesgue spaces when the associated canonical relation is parametrised by a complex phase function. Our result constitutes the complex analogue of those obtained for real canonical relations by Rodino, Nicola, and Cordero. We prove that, under the spatial factorization condition of rank $\varkappa$, the corresponding Fourier integral operator is bounded on the Fourier Lebesgue space $\mathcal{F}L^p,$ provided that the order $m$ of the operator satisfies that $ m \leq -\varkappa\left|\frac{1}{p}-\frac{1}{2}\right|, 1 \leq p \leq \infty. $ This condition on the order $m$ is sharp.
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https://arxiv.org/abs/2512.24854
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918e71ba40eec151c27369c571c528e64d6e02c99d44d276c8e7e69d77e900c6
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2026-01-01T00:00:00-05:00
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On a conjecture of Almgren II: area-minimizing submanifolds with fractal singular sets on almost any manifold
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arXiv:2512.24859v1 Announce Type: new Abstract: This paper is the second in a two-part solution to Almgren's conjecture on the existence of area-minimizing submanifolds with fractal singular sets. In part one, we construct area-minimizing submanifolds with fractal singular sets on certain special manifolds. Here we continue our work and show that area-minimizing submanifolds with fractal singular sets exist on almost any smooth manifold.
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https://arxiv.org/abs/2512.24859
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ac88ba503ec8678068543a608de52ff01aa20497ad4a8f8746a57a1527c8e006
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2026-01-01T00:00:00-05:00
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Configuration Spaces of Finite Representation Type Algebras
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arXiv:2512.24870v1 Announce Type: new Abstract: To every finite-dimensional $\mathbb C$-algebra $\Lambda$ of finite representation type we associate an affine variety. These varieties are a large generalization of the varieties defined by "$u$ variables" satisfying "$u$-equations", first introduced in the context of open string theory and moduli space of ordered points on the real projective line by Koba and Nielsen, rediscovered by Brown as "dihedral co-ordinates", and recently generalized to any finite type hereditary algebras. We show that each such variety is irreducible and admits a rational parametrization. The assignment is functorial: algebra quotients correspond to monomial maps among the varieties. The non-negative real part of each variety has boundary strata that are controlled by Jasso reduction. These non-negative parts naturally define a generalization of open string integrals in physics, exhibiting factorization and splitting properties that do not come from a worldsheet picture. We further establish a family of Rogers dilogarithm identities extending results of Chapoton beyond the Dynkin case.
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https://arxiv.org/abs/2512.24870
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5f0b1e21519d277c5966244651cff718fb57118e31f0f7d2bb22f909237e87fa
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2026-01-01T00:00:00-05:00
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Tensor Based Proximal Alternating Minimization Method for A Kind of Inhomogeneous Quartic Optimization Problem
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arXiv:2512.24872v1 Announce Type: new Abstract: In this paper, we propose an efficient numerical approach for solving a specific type of quartic inhomogeneous polynomial optimization problem inspired by practical applications. The primary contribution of this work lies in establishing an inherent equivalence between the quartic inhomogeneous polynomial optimization problem and a multilinear optimization problem (MOP). This result extends the equivalence between fourth-order homogeneous polynomial optimization and multilinear optimization in the existing literature to the equivalence between fourth-order inhomogeneous polynomial optimization and multilinear optimization. By leveraging the multi-block structure embedded within the MOP, a tensor-based proximal alternating minimization algorithm is proposed to approximate the optimal value of the quartic problem. Under mild assumptions, the convergence of the algorithm is rigorously proven. Finally, the effectiveness of the proposed algorithm is demonstrated through preliminary computational results obtained using synthetic datasets.
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https://arxiv.org/abs/2512.24872
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