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144c05ac005cfe98e9d68941aa70c802b9b6f2c31341cda1ab6b76bcca1c9a92
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2026-01-21T00:00:00-05:00
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Attached Submanifolds Beyond Symmetric Spaces
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arXiv:2601.13461v1 Announce Type: new Abstract: We study submanifold geometry in the presence of symmetry, focusing on submanifolds of solvmanifolds with an unusual property relative to Ricci curvature. We generalize work of H. Tamaru \cite{tamaru-11} in which he explores the geometry of submanifolds of symmetric spaces of noncompact type constructed from parabolic subgroups of the isometry group. He calls these attached submanifolds. The Ricci curvatures of attached submanifolds coincide with the restrictions of the Ricci curvatures of ambient symmetric spaces. We broaden Tamaru's construction by weakening the hypotheses on the ambient space, allowing a pseudo-Riemannian scalar product, and defining attached submanifolds in terms of root spaces. We demonstrate that in this setting, the Ricci curvature restriction property for attached submanifolds holds if and only if the submanifold satisfies an algebraic criterion that we call the Jacobi Star Condition. Like attached submanifolds of symmetric spaces, our attached submanifolds are minimal, and are only totally geodesic under hypotheses analogous to hypotheses in the symmetric space case. Finally, we give an example of a solvmanifold that has an attached submanifold and is not a symmetric space, demonstrating that attached submanifolds are not unique to symmetric spaces.
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https://arxiv.org/abs/2601.13461
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Academic Papers
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8f3a9d08615df4c9a8d7336ce68b2e53bfead53ef09fc2d0adaf57e2a529b591
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2026-01-21T00:00:00-05:00
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Fiber-preserving and orientation-reversing involutions of Seifert fibered 3-manifolds
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arXiv:2601.13469v1 Announce Type: new Abstract: We consider fiber-preserving, orientation-reversing involutions on orientable Seifert fibered 3-manifolds and the conditions on a manifold for admissibility of such involutions. We construct a class $\Psi$ of fiber-preserving, orientation-reversing involutions that act trivially on the base. Each element of $\Psi$ is obtained by extending a product involution across Seifert pieces of type $V(2,2;-1)$ - a solid torus with three fibers filled according to Seifert invariants $(2,1)$, $(2,1)$, and $(1,-1)$. We show that $\Psi$ forms a single conjugacy class under fiber-preserving diffeomorphisms. Our main result establishes that any fiber-preserving, orientation-reversing involution factors as $\psi\circ g$, where $g$ is fiber-preserving and orientation-preserving and $\psi\in\Psi$, thus reducing the problem to the previously known orientation-preserving case. Through the orientable base-space double covering, we further extend the classification to manifolds with non-orientable base orbifold.
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https://arxiv.org/abs/2601.13469
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6ccd871523e241999633664e8a4d84d46bbe2cbe9365c01b7004ed30c466aad9
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2026-01-21T00:00:00-05:00
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Directional Ballistic Transport in Quantum Waveguides
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arXiv:2601.13471v1 Announce Type: new Abstract: We study the transport properties of Schr\"odinger operators on $\mathbb{R}^d$ with potentials that are periodic in some directions and compactly supported in the others. Such systems are known to produce surface states that are weakly confined near the support of the potential. We show that a natural set of surface states exhibits directional ballistic transport, characterized by ballistic transport in the periodic directions and its absence in the others. To prove this, we develop a Floquet theory that captures the analytic variation of surface states. The main idea consists of reformulating the eigenvalue problem for surface states as a Fredholm problem via the Dirichlet-to-Neumann map.
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https://arxiv.org/abs/2601.13471
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288962e4b3918329b930eb5af1538cf1e7cfebc0cb12e3652f97c17d66617654
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2026-01-21T00:00:00-05:00
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iCanonical basis arising from quasi-split rank one iquantum group
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arXiv:2601.13482v1 Announce Type: new Abstract: We compute icanonical basis of the quasi-split rank one modified iquantum group, by obtaining explicit transition matrices among the icanonical basis, monomial basis, and standardized canonical basis; all these bases can be naturally categorified. These transition matrices follow from their counterparts computed in this paper among the icanonical basis, monomial basis, and canonical basis on simple finite-dimensional modules of quantum $\mathfrak{sl}_3$.
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https://arxiv.org/abs/2601.13482
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b8c29e6d14ecaede0b4ad15f5b4ea62efc66e78d8e78540f09604f50bb106445
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2026-01-21T00:00:00-05:00
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Gorenstein Special Fiber Rings of Ladder Determinantal Modules
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arXiv:2601.13483v1 Announce Type: new Abstract: A ladder determinantal module is an arbitrary direct sum of ideals of maximal minors of a generic ladder matrix. In this article, we give necessary and sufficient conditions for the special fiber ring of such modules to be Gorenstein. These conditions are expressed in terms of data obtained from the underlying matrix.
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https://arxiv.org/abs/2601.13483
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5a66efc1f0f03cb403e8d06b1f8fd91da89d84d35a98be900b7d5a04059ec5ed
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2026-01-21T00:00:00-05:00
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Self-Supervised Learning of Parametric Approximation for Security-Constrained DC-OPF
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arXiv:2601.13486v1 Announce Type: new Abstract: This paper introduces a self-supervised learning framework for approximating the Security-Constrained DC Optimal Power Flow (SC-DCOPF) problem using a parametric linear model. The approach preserves the physical structure of the DC-OPF while incorporating demand-dependent tunable parameters that scale transmission line limits. These parameters are predicted via a Graph Neural Network and optimized through differentiable layers, enabling direct training from contingency costs without requiring labeled data. The framework integrates pre- and post-contingency optimization layers into an implicit loss function. Numerical experiments on benchmark systems demonstrate that the proposed method achieves high dispatch accuracy, low cost approximation error, and strong data efficiency, outperforming semi-supervised and end-to-end baselines. This scalable and interpretable approach offers a promising solution for real-time secure power system operations.
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https://arxiv.org/abs/2601.13486
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62e9b7e561434580ef5298e21962b9c932d4e8b5e9b1d027ec0bc6776fe92ae3
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2026-01-21T00:00:00-05:00
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A note on "Higher order linear differential equations for unitary matrix integrals: applications and generalisations"
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arXiv:2601.13488v1 Announce Type: new Abstract: In this note, we briefly introduce the background and motivation of the collaborative work [arXiv:2508.20797], and provide an outline of the main results. The latter relates to matrix and higher order scalar differential equations satisfied by certain Hankel and Toeplitz determinants involving I-Bessel functions, or equivalently certain unitary matrix integrals, and moreover puts this property in a broader context. We also investigate large gaps between zeros of the derivatives of the Hardy $\mathsf{Z}$-function, assuming the validity of a certain joint moments conjecture in random matrix theory.
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https://arxiv.org/abs/2601.13488
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5406dc45713a329b619711e64889a2420711c2028823bdbf4b690fa8372cff03
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2026-01-21T00:00:00-05:00
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Noncommutative Minkowski integral inequality and a unitary categorification criterion for fusion rings
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arXiv:2601.13490v1 Announce Type: new Abstract: We prove a noncommutative analogue of Minkowski's integral inequality for commuting squares of tracial von Neumann algebras. The inequality implies a necessary condition for a quadruple of graphs to be realized as inclusion graphs of a commuting square of multi-matrix algebras. As a corollary, we obtain a unitary categorification criterion for based rings, in particular, fusion rings.
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https://arxiv.org/abs/2601.13490
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dc3b1b0e0fd9cb3573116d37c55d3ea98c3fb163345421311aab4ee653b31eca
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2026-01-21T00:00:00-05:00
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LQ Mean Field Games with Common Noise in Hilbert Spaces: Small and Arbitrary Finite Time Horizons
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arXiv:2601.13493v1 Announce Type: new Abstract: We extend the results of (Liu and Firoozi, 2025), which develops the theory of linear-quadratic (LQ) mean field games in Hilbert spaces, by incorporating a common noise. This common noise is an infinite-dimensional Wiener process affecting the dynamics of all agents. In the presence of common noise, the mean-field consistency condition is characterized by a system of coupled forward-backward stochastic evolution equations (FBSEEs) in Hilbert spaces, whereas in its absence, it is represented by forward-backward deterministic evolution equations. We establish the existence and uniqueness of solutions to the coupled linear FBSEEs associated with the LQ MFG setting for small time horizons and prove the $\epsilon$-Nash property of the resulting equilibrium strategy. Furthermore, for the first time in the literature, we develop an analysis that establishes the well-posedness of these coupled linear FBSEEs in Hilbert spaces, for which only mild solutions exist, over arbitrary finite time horizons.
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https://arxiv.org/abs/2601.13493
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fbb5e08e6404191d3058199ceb13b4086807a46ca6255fca67c14c43b09a05f9
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2026-01-21T00:00:00-05:00
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Double Hall-Littlewood symmetric polynomials
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arXiv:2601.13497v1 Announce Type: new Abstract: We establish a ring isomorphism between the derived Hall algebra of the Jordan quiver and the ring of double symmetric functions (i.e., the ring of symmetric polynomials in two sets of countably many variables, invariant under the respective actions of their symmetric groups) with a parameter $t$. This isomorphism maps the derived Hall basis (the natural basis of the derived Hall algebra) to a class of double Hall-Littlewood (HL) symmetric functions, which are formulated via raising and lowering operators. These double HL functions are parameterized by bipartitions; they reduce to the classical HL functions when one of the partitions is empty, and specialize to Schur Laurent symmetric functions at $t = 0$. We also derive the Pieri rules for these double HL functions. Additionally, we obtain several natural generating functions for the derived Hall algebra as well as their transition relations, which can be transferred to the ring of double symmetric functions via the established ring isomorphism.
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https://arxiv.org/abs/2601.13497
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8a300a9b2c201b33c17f75884617760abb04bb96c54dea8bee9112b49a6771a3
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2026-01-21T00:00:00-05:00
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Hidden convexity of quadratic systems and its application to quadratic programming
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arXiv:2601.13511v1 Announce Type: new Abstract: In this paper, we present sufficient conditions ensuring that the sum of the image of quadratic functions and the nonnegative orthant is convex. The hidden convexity of the trust-region problem with linear inequality constraints is established under a newly proposed assumption, which is compared with the previous one in [{\it Math. Program. 147, 171--206, 2014}]. We also provide a complete proof of the hidden convexity of a system of two quadratic functions in [{\it J. Glob. Optim. 56, 1045--1072, 2013}]. Furthermore, necessary and sufficient conditions for the S-lemma concerning systems of quadratic inequalities are investigated. Finally, we derive necessary and sufficient global optimality conditions and strong duality results for quadratic programming.
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https://arxiv.org/abs/2601.13511
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10110d9bb8c4b2e793aadd9a8defb73f8eec3a5bfac2eb36af190010a064aa41
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2026-01-21T00:00:00-05:00
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Categorical Entropies of Hilbert Schemes of Points on Surfaces and Hyperk\"ahler Manifolds
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arXiv:2601.13526v1 Announce Type: new Abstract: This paper studies the categorical entropy of autoequivalences of derived categories of Hilbert schemes of points on surfaces and hyperk\"ahler manifolds. One of the central questions about categorical entropy is whether it satisfies a Gromov-Yomdin type formula $h_{\mathrm{cat}}(\Phi) = \log\rho(\Phi)$. We say that $X$ has the Gromov-Yomdin (GY) property if this formula holds. We prove that if a surface $S$ fails to satisfy the (GY) property (e.g., K3 surfaces), then so does $\mathrm{Hilb}^n(S)$. Moreover, we show that no hyperk\"ahler or Enriques manifold satisfies the (GY) property by constructing an explicit autoequivalence with positive categorical entropy but unipotent action on the cohomology ring.
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https://arxiv.org/abs/2601.13526
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0e790e832352f029a6c01172fdf31a86e6e1f9bda23f465369b96a4566f6716a
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2026-01-21T00:00:00-05:00
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A construction of smooth varieties admitting small contractions
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arXiv:2601.13527v1 Announce Type: new Abstract: We construct smooth varieties admitting small contractions from arbitrary smooth projective varieties. This construction generalizes Kawamata's four-dimensional example. We also give sufficient conditions for divisors on these varieties to be nef. As an application, we obtain weak Fano fourfolds from products of two del Pezzo surfaces.
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https://arxiv.org/abs/2601.13527
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a14d99b8b4d6d7428dce1cb9c14921eef6cc9b2793b07f3aee5207d1739cac62
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2026-01-21T00:00:00-05:00
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Dean's conjecture and cycles modulo k
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arXiv:2601.13552v1 Announce Type: new Abstract: Dean conjectured three decades ago that every graph with minimum degree at least $k\ge 3$ contains a cycle whose length is divisible by $k$. While the conjecture has been verified for $k\in \{3,4\}$, it remains open for $k\ge 5$. A weaker version, also proposed by Dean, asserting that every $k$-connected graph contains a cycle of length divisible by $k$, was resolved by Gao, Huo, Liu, and Ma using the notion of admissible cycles. In this paper, we resolve Dean's conjecture for all $k\ge 6$. In fact, we prove a stronger result by showing that every graph with minimum degree at least $k$ contains cycles of length $r \pmod k$ for every even integer $r$, unless every end-block belongs to a specific family of exceptional graphs, which fail only to contain cycles of length $2 \pmod k$. We also establish a strengthened result on the existence of admissible cycles. Our proof introduces two sparse graph families, called trigonal graphs and tetragonal graphs, which provide a flexible framework for studying path and cycle lengths and may be of independent interest.
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https://arxiv.org/abs/2601.13552
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2d5d447d9a0fcd5017f2da0147b073664e43b62c106823137b6414305df4457e
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2026-01-21T00:00:00-05:00
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Universality of the Basilica
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arXiv:2601.13553v1 Announce Type: new Abstract: We establish universality of the fat Basilica Julia set $J(z^2-\frac34)$ in conformal dynamics in the following sense: $J(z^2-\frac34)$ is quasiconformally equivalent to the fat Basilica Julia set of any polynomial as well as to the limit set of any geometrically finite closed surface Bers boundary group. We thus obtain the first example of a connected rational Julia set, not homeomorphic to the circle or the sphere, that is quasiconformally equivalent to a Kleinian limit set. It follows that any geometrically finite Bers boundary limit set is conformally removable. Other consequences of this universality result include quasi-symmetric uniformization of polynomial fat Basilicas by round Basilicas, and the existence of infinitely many non-commensurable uniformly quasi-symmetric surface subgroups of the Basilica quasi-symmetry group. We apply our techniques to cuspidal Basilica Julia sets arising from Schwarz reflections and cubic polynomials, yielding further universality classes. We also show that the standard Basilica Julia set $J(z^2-1)$ is the archbasilica in the David hierarchy.
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https://arxiv.org/abs/2601.13553
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8dcd7695800e0acfbb8334944ab8dd924f920c48c934a75b58d78e2cd368a233
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2026-01-21T00:00:00-05:00
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Mirror construction of Hecke correspondence between Nakajima quiver varieties
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arXiv:2601.13555v1 Announce Type: new Abstract: Nakajima constructed geometric representations of a deformed Kac-Moody Lie algebra using Hecke correspondences between quiver varieties. In this paper, we show that Hecke correspondences, which are holomorphic Lagrangians in products of Nakajima quiver varieties, can be obtained by applying the localized mirror construction to the morphism spaces between families of framed Lagrangian branes supported on the core of a plumbing of two-spheres. Moreover, for a non-ADE quiver, we show that the localized mirror functor is fully-faithful.
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https://arxiv.org/abs/2601.13555
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b330bddd70b5f26fd2229311c1cc764f6776204c98ad2842f7f314785c2978dd
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2026-01-21T00:00:00-05:00
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On the radius of analyticity and Gevrey regularity for the Boltzmann equation
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arXiv:2601.13560v1 Announce Type: new Abstract: This paper investigates the non-cutoff Boltzmann equation for hard potentials in a perturbative setting. We first establish a sharp short-time estimate on the radius of analyticity and Gevrey regularity of mild solutions. Furthermore, we obtain a global-in-time radius estimate in Gevrey space. The proof combines hypoelliptic estimates with the macro-micro decomposition.
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https://arxiv.org/abs/2601.13560
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bc157dbd9e48f723633f4b6db0b1a0a8a85b1d9b87acaf3b939f5c429d70e164
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2026-01-21T00:00:00-05:00
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Logarithmic geometry and Infinitesimal Hodge Theory
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arXiv:2601.13568v1 Announce Type: new Abstract: This paper develops a systematic approach to infinitesimal variations of Hodge structure for singular and equisingular families by means of logarithmic geometry and residue theory. The central idea is that logarithmic vector fields encode precisely those deformation directions that preserve singularities and act trivially on Hodge structures, while the effective variation is entirely governed by residue calculus. This viewpoint provides a conceptual reinterpretation of classical results of Griffiths, Green, and Voisin, and extends them to settings involving singular varieties and equisingular deformations. The resulting framework yields a geometric explanation for the appearance of Jacobian rings in infinitesimal Hodge theory and clarifies the structure of deformation spaces underlying Severi varieties and related moduli problems.
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https://arxiv.org/abs/2601.13568
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4b0282c027e36a148e10f2b1621248e15ae948c77b0865353f37fe9c5654e5d6
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2026-01-21T00:00:00-05:00
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Control policies for a two-stage queueing system with parallel and single server options
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arXiv:2601.13576v1 Announce Type: new Abstract: We study a two-stage tandem service queue attended by two servers. Each job-server pair must complete both service phases together, with the server unable to begin a new job until the current one is fully processed after two stages. Immediately after the first phase of service, the server decides whether to send the job/customer to a downstream station that allows parallel processing or to a single-service facility that offers faster or higher-quality service but handles only one job at a time. This choice determines whether the second phase commences immediately or (potentially) after waiting in a queue for the single-service facility to become available. The decision-making scenario is modeled via a Markov decision process formulation, of a clearing system with holding costs at each station. We fully characterize the structural properties of an optimal control policy based on the relationship between the service rates at the downstream stations. A numerical study highlights the significance of optimal control by comparing its performance against several natural heuristic policies.
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https://arxiv.org/abs/2601.13576
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87ced18ade73f29ed7c75e4be0b1eccd94561ce774ce146c597c4f3776b1d5d6
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2026-01-21T00:00:00-05:00
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Balancing Independent and Collaborative Service
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arXiv:2601.13586v1 Announce Type: new Abstract: We study a two-type server queueing system where flexible Type-I servers, upon their initial interaction with jobs, decide in real time whether to process them independently or in collaboration with dedicated Type-II servers. Independent processing begins immediately, as does collaborative service if a Type-II server is available. Otherwise, the job and its paired Type-I server wait in queue for collaboration. Type-I servers are non-preemptive and cannot engage with new jobs until their current job is completed. We provide a complete characterization of the structural properties of the optimal policy for the clearing system. In particular, an optimal control is shown to follow a threshold structure based on the number of jobs in the queue before a Type-I first interaction and on the number of jobs in either independent or collaborative service. We propose simple threshold heuristics, based on linear approximations, for real-time decision-making. In much of the parameter and state spaces, we establish theoretical bounds that compare the thresholds proposed by our heuristics to those of optimal policies and identify parameter configurations where these bounds are attained. Outside of these regions, the optimal thresholds are infinite. Numerical experiments further demonstrate the accuracy and robustness of our heuristics, particularly when the initial queue length is high. Our proposed heuristics achieve costs within 0.5% of the optimal policy on average and significantly outperform benchmark policies that exhibit extreme sensitivity to system parameters, sometimes incurring costs exceeding 100% of the optimal.
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https://arxiv.org/abs/2601.13586
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b4b4747b8379f3d2951861da98302529b1a4aea6621a3ac2a3f4b83ef6ebc4e7
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2026-01-21T00:00:00-05:00
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Quasi-periodic Dynamics for Multi-dimensional Quasi-linear Schr\"{o}dinger Equations via Resonant Mode Control
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arXiv:2601.13611v1 Announce Type: new Abstract: This paper focuses on the problem of quasi-periodic solutions for multi-dimensional quasi-linear Schr\"odinger equation. To address the challenge of unbounded perturbations caused by quasi-linear terms in the equation, we define the resonant mode set $\mathcal{K}$ to control nonlinear resonant effects. Combining KAM (Kolmogorov-Arnold-Moser) ( or Nash-Moser ) theory and Fourier analysis methods, we prove that there are plenty of quasi-periodic solutions of the equation. We also present the Fourier expansion form of the solutions and the estimation of frequency shifts.
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https://arxiv.org/abs/2601.13611
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a8ff3db1f05cf7c39566932bbbdb8acc3cfe56739ca88825301b7d53b65bc49e
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2026-01-21T00:00:00-05:00
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Locally analytic vectors in the completed cohomology of quaternionic Shimura curves
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arXiv:2601.13625v1 Announce Type: new Abstract: We use the methods introduced by Lue Pan to study the locally analytic vectors of the completed cohomology of Shimura curves associated to an indefinite quaternion algebra $D$ which is ramified at a prime number $p$. Let $D_p^{\times}$ be the group of units of $D$ at $p$. Using $p$-adic uniformization of the quaternionic Shimura curves, we compute the Hecke eigenspace of the completed cohomology with the Hecke eigenvalues associated to a classical automorphic form on another quaternion algebra $\bar D$ (switching invariants of $D$ at $p,\infty$). We present this locally analytic $D_p^\times$-representation using the de Rham complex of the Lubin-Tate tower of dimension $1$. This is analogous to the Breuil-Strauch conjecture for the group $\mathrm{GL}_2(\mathbb{Q}_p)$. We show that the locally analytic $D_p^{\times}$-representation does not detect the Hodge filtration of the local de Rham Galois representation at $p$ in the crystalline case, and also give applications for the locally analytic Jacquet--Langlands correspondence for $\mathrm{GL}_2(\mathbb{Q}_p)$ and $D_p^\times$.
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https://arxiv.org/abs/2601.13625
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7787c403138adf7ffcdc5b4f15137418a7f1f3303ca8c2adb01a87db129e191b
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2026-01-21T00:00:00-05:00
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Symmetric multiple Eisenstein series
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arXiv:2601.13626v1 Announce Type: new Abstract: In this paper, we introduce the symmetric multiple Eisenstein series, a variant of the multiple Eisenstein series. As a fundamental result, we show that they satisfy the linear shuffle relation. As a case study, we investigate the vector space spanned by symmetric double Eisenstein series of weight $k$. When $k$ is even, it coincides with the space spanned by modular forms of weight $k$ and the derivative of the Eisenstein series of weight $k-2$. For $k$ odd, we prove that its dimension equals $\lfloor k/3\rfloor$. We further provide an explicit correspondence between the linear shuffle relation and the Fay-shuffle relation satisfied by elliptic double zeta values, which may be of independent interest. In connection with modular forms, we prove that every modular form can be expressed as a linear combination of symmetric triple Eisenstein series. This will serve as a first step toward understanding modular phenomena for symmeric multiple zeta values observed by Kaneko and Zagier.
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https://arxiv.org/abs/2601.13626
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d7883c20d1a19a922ff006ba4ba8f05dba02b7b4514ce25c353c925107ebba4c
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2026-01-21T00:00:00-05:00
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Exact solution of the (2+1)-dimensional damping forcing coupled Burgers equation by using Darboux transformation
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arXiv:2601.13634v1 Announce Type: new Abstract: In this article, we investigate the (2+1)-dimensional damping forcing coupled Burgers equation, which is obtain by adding damping and forcing terms from couple Burgers equation. The Lax pair of the (2+1)-dimensional damping forcing coupled Burgers equation is established. With the help of Lax pair, we derive the $N$-fold Darboux transformation of (2+1)-dimensional damping forcing coupled Burgers equation. Using one fold and two fold Darboux transformation, we demonstrated some wave solutions including solitary wave solution and periodic wave solution. The impact of damping and forcing terms in solitary wave solution and periodic solution is graphically demonstrated.
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https://arxiv.org/abs/2601.13634
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cf9a1c6a2fb62a78e851e8c4c71f5b6f6891af036d39c2b970ae3b2c64611836
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2026-01-21T00:00:00-05:00
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Borcherds products approximating Gersten complex
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arXiv:2601.13643v1 Announce Type: new Abstract: For an orthogonal modular variety, we construct a complex which is defined in terms of lattices and elliptic modular forms, which resembles the Gersten complex in Milnor K-theory, and which has a morphism to the Gersten complex of the modular variety by the Borcherds lifting. This provides a formalism for approaching the higher Chow groups of the modular variety by special cycles and Borcherds products. The construction is an incorporation of the theory of Borcherds products and ideas from Milnor K-theory.
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https://arxiv.org/abs/2601.13643
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5fe7f8029faf38d6b90ce257d1d1f018cd57fd8d13704458ed35aaa2ee405f52
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2026-01-21T00:00:00-05:00
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Gromov-Hausdorff stability of global attractors of damped wave equations under perturbations of the domain
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arXiv:2601.13650v1 Announce Type: new Abstract: In this paper, we will make use of the Gromov-Hausdorff distance between compact metric spaces to establish the continuous dependence and the Gromov-Hausdorff stability of global attractors for damped wave equations under perturbations of the domain.
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https://arxiv.org/abs/2601.13650
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6f7c84d969c10a8721bc649ee653376dc5d9e2a2bf8dc04e1099e0ab8b2c1b1d
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2026-01-21T00:00:00-05:00
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Kenmotsu Contact Geometry Through the Lens of $\ast-\boldsymbol{\kappa}$-Ricci-Bourguignon Almost Solitons
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arXiv:2601.13661v1 Announce Type: new Abstract: This paper focuses on the study of the newly introduced $\ast-\boldsymbol{\kappa}$-Ricci-Bourguignon almost soliton pertaining to Kenmotsu structure manifolds. Our analysis concerns the characteristics of this soliton and derive the scalar curvature for a Kenmotsu manifold admitting such a structure. Further, we formulate the corresponding vector fields under the assumption that the manifold supports a $\ast-\boldsymbol{\kappa}-$Ricci-Bourguignon soliton. Additionally, we explore applications involving torse-forming vector fields within the framework of the $\ast-\boldsymbol{\kappa}-$Ricci-Bourguignon almost soliton on Kenmotsu structure manifolds. To support the theoretical findings, we provide a concrete illustration belonging to a $\ast-\boldsymbol{\kappa}-$Ricci-Bourguignon almost soliton in a 5D Kenmotsu structure manifold.
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https://arxiv.org/abs/2601.13661
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bd3a4deccfea49bf00a3728dc47bd935fa8259de2fd4da678fd3d4fc9137fc7b
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2026-01-21T00:00:00-05:00
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The norm of the Hilbert matrix operator on Bergman spaces
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arXiv:2601.13672v1 Announce Type: new Abstract: Karapetrovi\'c conjectured that the norm of the Hilbert matrix operator on the Bergman space $A^p_\alpha$ is equal to $\pi/\sin((2+\alpha)\pi/p)$ when $-1<\alpha\frac{1}{47}$ and $\alpha \not=1$.
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https://arxiv.org/abs/2601.13672
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Academic Papers
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2bc19e17aa919d68c614eb9d76c4bd42a4eb5042da931328aeb2fab6a6dffa9f
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2026-01-21T00:00:00-05:00
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The spectral measures of random Jacobi matrices related to beta ensembles at high temperature and Dirichlet processes
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arXiv:2601.13674v1 Announce Type: new Abstract: In a high temperature regime where $\beta N \to 2c$, the empirical distribution of the eigenvalues of Gaussian beta ensembles, beta Laguerre ensembles and beta Jacobi ensembles converges to a limiting measure which is related to associated Hermite polynomials, associated Laguerre polynomials and associated Jacobi polynomials, respectively. Here $\beta$ is the inverse temperature parameter, $N$ is the system size and $c>0$ is a given constant. This paper studies the spectral measure of the random tridiagonal matrix model of the three classical beta ensembles. We show that in the high temperature regime, the spectral measure converges in distribution to a Dirichlet process with base distribution being the limiting distribution, and scaling parameter $c$. Consequently, the spectral measure of a related semi-infinite Jacobi matrix coincides with that Dirichlet process, which provides examples of random Jacobi matrices with explicit spectral measures.
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https://arxiv.org/abs/2601.13674
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79c22b8a15dcfc4cac53705e02d1981824c6e9f5e1859d01cd1cb578912e8204
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2026-01-21T00:00:00-05:00
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Distributed Coverage Control on Poriferous Surface via Poly-Annulus Conformal Mapping
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arXiv:2601.13688v1 Announce Type: new Abstract: The inherent non-convexity of poriferous surfaces typically entraps agents in local minima and complicates workload distribution. To resolve this, we propose a distributed diffeomorphic coverage control framework for the multi-agent system (MAS) in such surfaces. First, we establish a distributed poly-annulus conformal mapping that transforms arbitrary poriferous surfaces into a multi-hole disk. Leveraging this topological equivalence, a collision-free sectorial partition mechanism is designed in the multi-hole disk, which rigorously induces strictly connected subregions and workload balance on the poriferous surfaces. This mechanism utilizes a buffer-based sequence mechanism to ensure strict topological safety when bypassing obstacles. Furthermore, a pull-back Riemannian metric is constructed to define the length metric that encodes safety constraints. Based on this metric, a distributed gradient-based control law is synthesized to drive agents toward optimal configurations, ensuring simultaneous obstacle avoidance and coverage optimization. Theoretical analyses guarantee the Input-to-State Stability (ISS) of the partition dynamics and the asymptotic convergence of the closed-loop system. Numerical simulations confirm the reachability and robustness of the proposed coverage algorithm, offering a scalable solution for distributed coverage in poriferous surfaces.
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https://arxiv.org/abs/2601.13688
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ac51bb746068a32595f511faa39ce5b8e9a5e1c7eb9db2aedfb0b5c5d6f30d58
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2026-01-21T00:00:00-05:00
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On the Birkhoff Spectrum for Hyperbolic Dynamics
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arXiv:2601.13720v1 Announce Type: new Abstract: In this paper, we study the structure of Birkhoff spectra for hyperbolic dynamical systems. For a H\"older observable $f$ on a basic set $\Lambda$, we prove that if the Birkhoff sums take both positive and negative values, then the Birkhoff spectrum $\mathcal{B}(f,\varphi,\Lambda)$ is dense in $\mathbb{R}$. This extends the density result of Gan, Shi, and Xia \cite{Shaobo} from transitive Anosov diffeomorphisms in infranilmanifolds to general basic sets, and yields new density theorems for Axiom~A systems, including Anosov diffeos.\\ Conversely, when the spectrum is not dense, we characterize \emph{concentrated} observables, whose spectrum is confined to one side of zero. For these, we establish two rigidity results: (i) boundedness of the spectrum is equivalent to the function being cohomologous to a zero, which constitutes an extension of the Liv\v sic theorem; (ii) if the spectrum exhibits an arithmetic structure, the function is cohomologous to a constant. Finally, we extend the results to continuous time. For Anosov flows$-$ including geodesic flows on Anosov manifolds$-$we obtain analogous density results for Birkhoff integrals along closed orbits. In particular, we generalize a theorem of Dairbekov--Sharafutdinov (\cite{Dairbekov}) by showing that a bounded or ``arithmetically sparse'' spectrum forces a smooth function to vanish or be constant, respectively.
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https://arxiv.org/abs/2601.13720
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b108d3bd667645af2dc24bd2ffd4036806b770e954e4ea5aa974453ec2cd6802
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2026-01-21T00:00:00-05:00
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Central Limit Theorems in Multiplicative Diophantine Approximation
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arXiv:2601.13726v1 Announce Type: new Abstract: We investigate the number of integer solutions to a multiplicative Diophantine approximation problem and show that the associated counting function converges in distribution to a normal law. Our approach relies on the analysis of correlations of measures on homogeneous spaces, together with estimates for Siegel transforms restricted to subspaces.
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https://arxiv.org/abs/2601.13726
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e5f4e4eae808f30e17ed04d57c0423649eee671fc9d86d15ca56c33ed76029ff
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2026-01-21T00:00:00-05:00
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Some Consequences of the Grunewald-O'Halloran Conjecture for Pseudoquonic Operators
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arXiv:2601.13736v1 Announce Type: new Abstract: Investigating a recent positive solution of a conjecture of Grunewald and O'Halloran for complex finite dimensional nilpotent Lie algebras, we are in the position to find results of existence and uniqueness for the construction of complex nilpotent Lie algebras of arbitrary dimension via pseudobosonic operators. We involve the so-called theory of the deformation of Lie algebras of Gerstenhaber, in order to prove our main results. There isn't a generalized version of the Grunewald-O'Halloran Conjecture when we consider pseudoquonic operators, which specialize to pseudobosonic operators in many cirumstances. Therefore we prove a result of existence (and a direct construction) of pseudobosonic $O^*$-algebras of operators, but leave open the problem of the uniqueness of the construction.
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https://arxiv.org/abs/2601.13736
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c0253bede5377bb51c8ea2e7fdcdc0002f7645fd6703cba46e01bb8bf5bdec22
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2026-01-21T00:00:00-05:00
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Second Order Asymptotics for the Hard Wall Probability of the 2D Harmonic Crystal
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arXiv:2601.13738v1 Announce Type: new Abstract: We estimate the probability that the discrete Gaussian free field on a planar domain with Dirichlet boundary conditions stays positive in the bulk. Improving upon the result by Bolthausen, Deuschel and Giacomin from 2001, we derive the order of the subleading term of this probability when a sequence of discretized scale-ups of given domain and compactly included smooth bulk are considered. A main ingredient in the proof is the double exponential decay of the right tail of the centered minimum of the field in the bulk, conditioned on a certain weighted average of its values to be zero.
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https://arxiv.org/abs/2601.13738
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bf8cab202021974c71ca3d16bcacc8d1d4283e86a5ac6a7a2496cb3b9fa758c0
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2026-01-21T00:00:00-05:00
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Abstract maximal hypoellipticity and applications
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arXiv:2601.13741v1 Announce Type: new Abstract: We prove an abstract theorem of maximal hypoellipticy showing that in an abstract calculus under some natural assumptions, an operator is maximally hypoelliptic if and only if its principal symbol is left invertible. We then show that our theorem implies various known results in the literature like regularity theorem for elliptic operators, Helffer and Nourrigat's resolution of the Rockland conjecture, Rodino's theorem on regularity of operators on products of manifolds, and our resolution of the Helffer-Nourrigat conjecture. Other examples like our resolution of the microlocal Helffer-Nourrigat conjecture will be given in a sequel to this paper. Our arguments are based on the theory of $C^*$-algebras of Type I.
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https://arxiv.org/abs/2601.13741
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caca58c05a4f2c32d3e27609928486b53dc9618e1f116446ffd95eb5a11a1622
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2026-01-21T00:00:00-05:00
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A Note on k-NN Gating in RAG
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arXiv:2601.13744v1 Announce Type: new Abstract: We develop a statistical proxy framework for retrieval-augmented generation (RAG), designed to formalize how a language model (LM) should balance its own predictions with retrieved evidence. For each query x, the system combines a frozen base model q0 ($\times$ x) with a k-nearest neighbor retriever r (k ) ($\times$ x) through a measurable gate k(x). A retrieval-trust weight wfact (x) quantifies the geometric reliability of the retrieved neighborhood and penalizes retrieval in low-trust regions. We derive the Bayes-optimal per-query gate and analyze its effect on a discordance-based hallucination criterion that captures disagreements between LM predictions and retrieved evidence. We further show that this discordance admits a deterministic asymptotic limit governed solely by the structural agreement (or disagreement) between the Bayes rule and the LM. To account for distribution mismatch between queries and memory, we introduce a hybrid geometric-semantic model combining covariate deformation and label corruption. Overall, this note provides a principled statistical foundation for factuality-oriented RAG systems.
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https://arxiv.org/abs/2601.13744
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6aac82597939fcbfe3b4f6c4c46b087f1524dadbdd0a931b5213605362f24fa8
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2026-01-21T00:00:00-05:00
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closed $\mathrm{G}_2$-structures with $\mathbb{T}^3$-symmetry and hypersymplectic structures
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arXiv:2601.13747v1 Announce Type: new Abstract: Closed $\mathrm{G}_2$-structures $\varphi$ with an effective $\mathbb{T}^3$-symmetry on connected manifolds are roughly classified into three types according to the evaluation of $\varphi$ on the principal orbits. Type 1: if there is neither associative nor isotropic orbit, then the action is free and $\varphi$ reduces to a hypersymplectic structure on the quotient manifold admitting three linearly independent closed 1-forms; in particular, it is diffeomorphic to $\mathbb{T}^4$ if the manifold is compact. Type 2: if some orbit is associative, then the action is almost-free and $\varphi$ reduces to a good hypersymplectic orbifold with cyclic isotropic groups. Type 3: if some orbit is isotropic, then the action is locally multi-Hamiltonian for $\varphi$. Moreover, the open and dense subset of principal orbits is foliated by $\mathbb{T}^3$-invariant hypersymplectic manifolds. If $\varphi$ is torsion-free and complete, then the hypersymplectic manifold is flat and $\varphi$ is flat for Type 1; the good hypersymplectic orbifold is good hyperk\"ahler orbifold for Type 2; $\varphi$ is locally toric for Type 3. As shown, hypersymplectic structures have intimate link with closed $\mathrm{G}_2$-structure with effective $\mathbb{T}^3$-symmetry.
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https://arxiv.org/abs/2601.13747
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85502ba1f16fe73fe094a6d0ef68bafd8cf3020ab54c249eb73350845d2da3cf
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2026-01-21T00:00:00-05:00
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Sharp Quantitative Forms of the Hardy Inequality on Cartan-Hadamard Manifolds via Sobolev-Lorentz Embeddings
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arXiv:2601.13750v1 Announce Type: new Abstract: In this article, we investigate the quantitative form of the classical Hardy inequality. In our first result, we prove the following quantitative bound under the assumption that the $\mathbb{M}^N$ is a Riemannian model satisfying the centered isoperimetric inequality: We prove that $$ \|\nabla_g u\|^2_{L^{2}(\mathbb{M}^N)} - \frac{(N-2)^2}{4}\left\|\frac{u}{r(x)}\right\|^2_{L^2(\mathbb{M}^N)} \geq C [\mbox{dist}(u, Z)]^{\frac{4N}{N-2}}\left\|\frac{u}{r(x)}\right\|^2_{L^2(\mathbb{M}^N)},$$ for every real-valued weakly differentiable function $u$ on $\mathbb{M}^N$ such that $|\nabla_g u| \in L^2(\mathbb{M}^N)$ and $u$ decays to zero at infinity. Here $r(x) = d_g(x,x_0)$ denotes the geodesic distance from a fixed pole $x_0,$ the set $Z$ represents the family of virtual extremals, and the distance is understood in an appropriate generalized Lorentz-type space. Our approach is built on the symmetrization technique on manifolds, combined with a novel Jacobian-type transformation that provides a precise way for comparing volume growth, level sets, and gradient terms across the two geometries of Euclidean and manifold settings. When coupled with symmetrization, this framework yields sharp control over the relevant functionals and reveals how the underlying curvature influences extremal behavior. Our result generalizes the seminal result of Cianchi-Ferone [Ann. Inst. H. Poincar\'e C Anal. Non Lin\'eaire 25 (2008)] to the curved spaces. Moreover, building upon this transformation, we succeed in extending Sobolev-Lorentz embedding-classically formulated in the Euclidean setting to the broader framework of Cartan-Hadamard models and we establish an optimal Sobolev-Lorentz embedding in this geometric setting. Finally, we establish a quantitative correspondence between the Hardy deficit on the manifold and an appropriate weighted Hardy deficit in Euclidean space, showing that each controls the other.
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https://arxiv.org/abs/2601.13750
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f9e59d07d28eacba2813024ddabf8e244ad2f91dd5ee48cd9adf7a9ca6fc52ee
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2026-01-21T00:00:00-05:00
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A turnpike property in an eigenvalue optimization problem
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arXiv:2601.13756v1 Announce Type: new Abstract: We consider a constrained eigenvalue optimization problem that arises in an important nonlinear dynamical model for mRNA translation in the cell. We prove that the ordered list of optimal parameters admits a turnpike property, namely, it includes three parts with the first and third part relatively short, and the values in the middle part are all approximately equal. Turnpike properties have attracted considerable attention in econometrics and optimal control theory, but to the best of our knowledge this is the first rigorous proof of such a structure in an eigenvalue optimization problem.
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https://arxiv.org/abs/2601.13756
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0c6255810ed9e7675de25e7cf9847162634da6e7d5d4c7b8fd92979afafe2216
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2026-01-21T00:00:00-05:00
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Quantum Entanglement Geometry on Severi-Brauer Schemes: Subsystem Reductions of Azumaya Algebras
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arXiv:2601.13764v1 Announce Type: new Abstract: We formulate pure-state entanglement in families as a geometric obstruction. In standard quantum information, entanglement is defined relative to a chosen tensor-product factorization of a fixed Hilbert space. In contrast, for a twisted family of pure-state spaces, which can be described by Azumaya algebras $A$ of degree $n$ on $X$ and their Severi-Brauer schemes \[ SB(A)=P\times^{PGL_n}\mathbb{P}^{n-1}\to X, \] such a subsystem choice may fail to globalize. We formalize this algebro-geometrically: fixing a factorization type $\mathbf d=(d_1,\dots,d_s)$ with $n=\prod_i d_i$, the existence of a global product-state locus of type $\mathbf d$ is equivalent to a reduction of the underlying $PGL_n$-torsor $P\to X$ to the stabilizer $G_{\mathbf d}\subset PGL_n$. Thus, entanglement is the obstruction to the existence of a relative Segre subscheme inside $SB(A)$. Writing $\Sigma_{\mathbf d}\subset \mathbb{P}^{n-1}$ for the Segre variety, we call a reduction to $G_{\mathbf d}$ a $\mathbf d$-subsystem structure. Our first main result identifies the moduli of $\mathbf d$-subsystem structures with the quotient $P/G_{\mathbf d}$. Moreover, we realize naturally $P/G_{\mathbf d}$ as a locally closed subscheme of the relative Hilbert scheme, \[ \text{Hilb}^{\Sigma_{\mathbf d}}\!\bigl(SB(A)/X\bigr)\ \subset\ \text{Hilb}\bigl(SB(A)/X\bigr), \] parametrizing relative closed subschemes fppf-locally isomorphic to $\Sigma_{\mathbf d}\times X$.
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https://arxiv.org/abs/2601.13764
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ca475a7abbaf86cea9b8f3ea3a189d41a6c4d04ba41a2ad63c86dea8d312d563
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2026-01-21T00:00:00-05:00
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The Harnack inequality without convexity for curve shortening flow
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arXiv:2601.13767v1 Announce Type: new Abstract: In 1995, Hamilton introduced a Harnack inequality for convex solutions of the mean curvature flow. In this paper we prove an alternative Harnack inequality for curve shortening flow, i.e. one-dimensional mean curvature flow, that does not require any assumption of convexity. For an initial proper curve in the plane whose ends are radial lines but which is otherwise arbitrarily wild, we use the Harnack inequality to give an explicit time by which the curve shortening flow evolution must become graphical. This gives a new instance of delayed parabolic regularity. The Harnack inequality also gives estimates describing how a polar graphical flow with radial ends settles down to an expanding solution. Finally, we relate our Harnack inequality to Hamilton's by identifying a pointwise curvature estimate implied by both Harnack inequalities in the special case of convex flows.
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https://arxiv.org/abs/2601.13767
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eac5c4225ac8393e2275f59e743ffddfff7c1f286a27c4335f1b1909ef72a092
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2026-01-21T00:00:00-05:00
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Existence and regularity of minimizers for a variational problem of species population density
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arXiv:2601.13771v1 Announce Type: new Abstract: We study a variational problem motivated by models of species population density in a nonhomogeneous environment. We first analyze local minimizers and the structure of the saturated region (where the population attains its maximal density) from a free boundary perspective. By comparing the original problem with a radially symmetric minimization problem and studying its properties, we then establish the existence and structure of a global solution. Analytic examples of radially symmetric solutions and numerical simulations illustrate the theoretical results and provide insight into spatial saturation patterns in population models. We further highlight an unresolved question regarding the quasiconcavity of minimizers.
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https://arxiv.org/abs/2601.13771
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0592054e95a42e9fcd9b821b53f57e58428d0f0e2ead012157d488f564ff866f
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2026-01-21T00:00:00-05:00
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Bialgebraic structures on boolean functions
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arXiv:2601.13773v1 Announce Type: new Abstract: We study several bialgebraic structures on boolean functions, that is to say maps defined on the set of subsets of a finite set $X$, taking the value $0$ on $\emptyset$. Examples of boolean functions are given by the indicator function of the hyperedges of a given hypergraph, or the rank function of a matroid. We give the species of boolean functions a two-parameters family of products and a coproduct, and this defines a two-parameters family of twisted bialgebras. We then try to define a second coproduct on boolean functions, based on contractions, in order to obtain a double bialgebra. We show that this is not possible on the whole species of boolean functions, but that there exists a maximal subspecies where this is possible. This subspecies being rather mysterious, we introduce rigid boolean functions and show that this subspecies has indeed a second coproduct, as wished, and that it contains rank functions of matroids and indicator functions associated to hypergraphs. As a consequence, we obtain a unique polynomial invariant on rigid boolean functions, which is a generalization of the chromatic polynomial of graphs.
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https://arxiv.org/abs/2601.13773
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91fec16573a15a2720549f7a319fd6964386c5332559d148f133ce7798f58918
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2026-01-21T00:00:00-05:00
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Extension of the Fundamental Theorem of Algebra to Polynomial Matrix Equations over $Q$-Circulant Matrices
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arXiv:2601.13775v1 Announce Type: new Abstract: In this paper, we establish an analogue of the Fundamental Theorem of Algebra for polynomial matrix equations, where both the coefficient matrices and the unknown matrix are $Q$-circulant matrices. This result generalizes Abramov's result for circulant matrices.
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https://arxiv.org/abs/2601.13775
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7e52940d054e0d046d72403068697542819d0ec0d54e668c0ff01e35c00a4f7c
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2026-01-21T00:00:00-05:00
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Maximum spanning trees in normed planes
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arXiv:2601.13779v1 Announce Type: new Abstract: Extending some properties from the Euclidean plane to any normed plane, we show the validity of the Monma-Paterson-Suri-Yao algorithm for finding the maximum-weighted spanning tree of a set of $n$ points, where the weight of an edge is the distance between the end points measured by the norm and there are not repeated distances. For strictly convex normed planes, we expose an strategy for moving slightly the points of the set in order to obtain distinct distances.
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https://arxiv.org/abs/2601.13779
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9f02c00f52b389808bf122703f0f4478bc2b4fc2af2dcf3bc587bcbf8565fb0e
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2026-01-21T00:00:00-05:00
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Moving Least Squares without Quasi-Uniformity: A Stochastic Approach
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arXiv:2601.13782v1 Announce Type: new Abstract: Local Polynomial Regression (LPR) and Moving Least Squares (MLS) are closely related nonparametric estimation methods, developed independently in statistics and approximation theory. While statistical LPR analysis focuses on overcoming sampling noise under probabilistic assumptions, the deterministic MLS theory studies smoothness properties and convergence rates with respect to the \textit{fill-distance} (a resolution parameter). Despite this similarity, the deterministic assumptions underlying MLS fail to hold under random sampling. We begin by quantifying the probabilistic behavior of the fill-distance $h_n$ and \textit{separation} $\delta_n$ of an i.i.d. random sample. That is, for a distribution satisfying a mild regularity condition, $h_n\propto n^{-1/d}\log^{1/d} (n)$ and $\delta_n \propto n^{-1/d}$. We then prove that, for MLS of degree $k\!-\!1$, the approximation error associated with a differential operator $Q$ of order $|m|\le k-1$ decays as $h_n^{\,k-|m|}$ up to logarithmic factors, establishing stochastic analogues of the classical MLS estimates. Additionally, We show that the MLS approximant is smooth with high probability. Finally, we apply the stochastic MLS theory to manifold estimation. Assuming that the sampled Manifold is $k$-times smooth, we show that the Hausdorff distance between the true manifold and its MLS reconstruction decays as $h_n^k$, extending the deterministic Manifold-MLS guarantees to random samples. This work provides the first unified stochastic analysis of MLS, demonstrating that -- despite the failure of deterministic sampling assumptions -- the classical convergence and smoothness properties persist under natural probabilistic models
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https://arxiv.org/abs/2601.13782
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6d3ef3aa221336b0cc6d33f9b743c7970ab3d31f21d2974e10eb073170f11617
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2026-01-21T00:00:00-05:00
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The Genus-Decreasing Property of Mean Curvature Flow, I
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arXiv:2601.13787v1 Announce Type: new Abstract: This paper proves that, in mean curvature flow of a compact surface in a complete $3$-manifold with Ricci curvature bounded below, the genus of the regular set is a decreasing function of time as long as the only singularities are given by shrinking sphere and shrinking cylinder tangent flows. The paper also proves some local versions of that fact.
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https://arxiv.org/abs/2601.13787
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e7d651d394dc395320f5a62388baf4e3770fb66e962ba7ab34cbe7cee9cedfac
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2026-01-21T00:00:00-05:00
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Characterizations of a class of Musielak--Orlicz BMO spaces via commutators of Riesz potential operators
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arXiv:2601.13788v1 Announce Type: new Abstract: The fractional integral operators $I_\alpha$ can be used to characterize the Musielak--Orlicz Hardy spaces. This paper shows that for $b\in \rm BMO(\mathbb R^n)$, the commutators $[b,I_\alpha]$ generated by fractional integral operators $I_\alpha$ with $b$ are bounded from the Musielak--Orlicz Hardy spaces $H^{\varphi_1}(\mathbb R^n)$ to the Musielak--Orlicz spaces $L^{\varphi_2}(\mathbb R^n)$ (where $1<\infty$ and $\varphi_1$, $\varphi_2$ are growth functions) if and only if $b\in \mathcal {BMO}_{\varphi_1,u}(\mathbb R^n)$, which are a class of non-trivial subspaces of $\rm BMO(\mathbb R^n)$. Additionally, we obtain the boundedness of the commutator $[b,I_\alpha]$ from $H^{\varphi_1}(\mathbb R^n)$ to $H^{\varphi_2}(\mathbb R^n)$. The corresponding results are also provided for commutators of fractional integrals associated with general homogeneous kernels.
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https://arxiv.org/abs/2601.13788
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056092ee9a6f136e4d14dcb05ad5a276f9fec9cbd4d6b652ef18e714b4973742
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2026-01-21T00:00:00-05:00
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Nijenhuis BiHom-Lie bialgebras and differential Lie bialgebras
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arXiv:2601.13791v1 Announce Type: new Abstract: In this paper, we first introduce the concept of Nijenhuis BiHom-Lie algebras. We then establish the equivalence relations between the Manin triples of Nijenhuis BiHom-Lie algebras, Nijenhuis BiHom-Lie bialgebras, and matched pairs of Nijenhuis BiHom-Lie algebras. Furthermore, we show that such an equivalence also holds for differential Lie bialgebras, together with their associated Manin triples and corresponding matched pairs.
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https://arxiv.org/abs/2601.13791
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098e8425681848e7bc59efe5c0fe841353809af26a70b862370bf3b63233033e
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2026-01-21T00:00:00-05:00
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Asymptotic Properties of Filtrations of Ideals
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arXiv:2601.13794v1 Announce Type: new Abstract: We introduce a unified framework for studying persistence phenomena in commutative algebra via filtrations of ideals. For a filtration $\mathcal{F} = \{I_i\}_{i \in \mathbb{N}}$, we define $\mathcal{F}$-persistence and $\mathcal{F}$-strong persistence, extending the classical notions for ordinary and symbolic powers of ideals. We show that if $\mathcal{F}$ is strongly persistent, then $\mathcal{F}_{\mathrm{sym}}$ is strongly persistent, where $\mathcal{F}_{\mathrm{sym}}$ denotes the symbolic filtration associated with the filtration $\mathcal{F}$. In addition, we prove that if $\mathcal{F}$ is strongly persistent, then $\mathcal{F}$ is persistent.
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https://arxiv.org/abs/2601.13794
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c86066234152ab79b80f651585c92a712dd87d9d70e06bf69db7e77db113d017
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2026-01-21T00:00:00-05:00
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On rough ideal convergence
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arXiv:2601.13805v1 Announce Type: new Abstract: We continue the study of ideal convergence for sequences $(x_n)$ with values in a topological space $X$ with respect to a family $\{F_\eta:\eta\in X\}$ of subsets of $X$ with $\eta\in F_\eta$, where each $F_\eta$ measures the allowed ``roughness'' of convergence toward $\eta$. More precisely, after introducing the corresponding notions of cluster and limit points, we prove several inclusion and invariance properties, discuss their structural properties, and give examples showing that the rough notions are genuinely different from the classical ideal ones.
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https://arxiv.org/abs/2601.13805
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5d6c5ea248f140f9ee788e7dc42a4b42bca612b817b37a5f761f0d35afa0f286
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2026-01-21T00:00:00-05:00
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Non-finitely generated $(\mathbb{Z}_2)^k$-equivariant bordism ring
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arXiv:2601.13807v1 Announce Type: new Abstract: In 1998, Mukherjee and Sankaran posed two problems concerning the algebraic structure of the equivariant bordism ring of smooth closed $(\mathbb{Z}_2)^k$-manifolds with only isolated fixed points. One is the property of being finitely generated as a $\mathbb{Z}_2$-algebra, and the other is the existence of indecomposable elements. This paper definitively resolves both problems for the fully effective case. Specifically, let $\mathcal{Z}_*((\mathbb{Z}_2)^k)$ denote the equivariant bordism ring of smooth closed manifolds equipped with fully effective smooth $(\mathbb{Z}_2)^k$-actions having only isolated fixed points. We prove that $\mathcal{Z}_*((\mathbb{Z}_2)^k)$ is not finitely generated as a $\mathbb{Z}_2$-algebra for all $k\geqslant 3$. Moreover, the proof explicitly constructs an infinite family of indecomposable elements with unbounded degrees, thereby settling the second problem simultaneously.
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https://arxiv.org/abs/2601.13807
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259a00db0570ec2d1aa7899d301a941007a2b21be7aab2ee00f3dc035b8fa9eb
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2026-01-21T00:00:00-05:00
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Dimensional Constraints from SU(2) Representation Theory in Graph-Based Quantum Systems
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arXiv:2601.13828v1 Announce Type: new Abstract: We investigate dimensional constraints arising from representation theory when abstract graph edges possess internal degrees of freedom but lack geometric properties. We prove that such internal degrees of freedom can only encode directional information, necessitating quantum states in $\mathbb{C}^2$ (qubits) as the minimal representation. Any geometrically consistent projection of these states maps necessarily to $\mathbb{R}^3$ via the Bloch sphere. This dimensional constraint $d=3$ emerges through self-consistency: edges without intrinsic geometry force directional encoding ($\mathbb{C}^2$), whose natural symmetry group $SU(2)$ has three-dimensional Lie algebra, yielding emergent geometry that validates the hypothesis via Bloch sphere correspondence ($S^2 \subset \mathbb{R}^3$). We establish uniqueness (SU($N>2$) yields $d>3$) and robustness (dimensional saturation under graph topology changes). The Euclidean metric emerges canonically from the Killing form on $\mathfrak{su}(2)$. A global gauge consistency axiom is justified via principal bundle trivialization for finite graphs. Numerical simulations verify theoretical predictions. This result demonstrates how dimensional structure can be derived from information-theoretic constraints, with potential relevance to quantum information theory, discrete geometry, and quantum foundations.
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https://arxiv.org/abs/2601.13828
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b90ae6a2401249d20a54b7c37d2b3d5705b6fff86a01231c9679a354fbf8ba11
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2026-01-21T00:00:00-05:00
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Analytic description of the moving moisture front in soils
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arXiv:2601.13833v1 Announce Type: new Abstract: The fact that moisture propagates in soils at a finite speed is confirmed by natural everyday experience as well as by controlled laboratory tests. In this text, we rigorously derive analytical upper bounds for the speed of moisture front propagation under gravity for the solution to the Richards equation with compactly supported initial data. The main result is an explicit criterion describing a competition between gravity and capillarity, where the dominant effect is determined by the characteristics of the soil. If capillarity prevails, the initially wet regions remain wet for all times, while if gravity is dominant, moisture travels downward at a speed that is asymptotically bounded from below and above. As a by-product, we prove the existence and uniqueness of a solution to an initial value problem for the degenerate Richards equation on the whole space. Numerical simulations based on the proposed model confirm the theoretical predictions, with results that closely match experimental observations.
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https://arxiv.org/abs/2601.13833
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7d9878d0c1769a062967ecb01b168af4edeebbac7653b70d7962ca2ccec0391e
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2026-01-21T00:00:00-05:00
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Derivative free data-driven stabilization of continuous-time linear systems from input-output data
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arXiv:2601.13848v1 Announce Type: new Abstract: This letter presents a data-driven framework for the design of stabilizing controllers from input-output data in the continuous-time, linear, and time-invariant domain. Rather than relying on measurements or reliable estimates of input and output time derivatives, the proposed approach uses filters to derive a parameterization of the system dynamics. This parameterization is amenable to the application of linear matrix inequalities enabling the design of stabilizing output feedback controllers from input-output data and the knowledge of the order of the system.
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https://arxiv.org/abs/2601.13848
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200cced42f3dbb758e07fce592575ae3d51ed29a77c7db565a83bc6f4bf1a561
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2026-01-21T00:00:00-05:00
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Basic Albanese maps of regular Riemannian foliations
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arXiv:2601.13853v1 Announce Type: new Abstract: In the paper we introduce the notion of basic Albanese map which we define for foliated Riemannian manifolds using basic 1-forms. We relate this mapping to the classical Albanese map for the ambient manifold. The study of general properties is supplemented with the description of several important examples.
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https://arxiv.org/abs/2601.13853
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7fc133646aa143d8a42dab61cc7716e0f3cb74d9da5d3827b3a07b8698daafe5
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2026-01-21T00:00:00-05:00
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Jacob's ladders, point of contact of the remained in the prime-number law with the Fermat-Wiles theorem and multiplicative puzzles on some sets of integrals
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arXiv:2601.13855v1 Announce Type: new Abstract: In this paper we prove, on the Riemann hypothesis, the existence of such increments of the Ingham integral (1932) that generate new functionals together with corresponding new $P\zeta$-equivalents of the Fermat-Wiles theorem. We obtain also new results in this direction.
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https://arxiv.org/abs/2601.13855
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997e43939bf16b5bb953f3fd9385c99f97b7e230e40c0d2e557ca7b077fbec24
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2026-01-21T00:00:00-05:00
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Patterns and Tracks
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arXiv:2601.13861v1 Announce Type: new Abstract: Patterns in triangulated $2$-spheres and $3$-spheres are investigated. A new proof of a lemma in Abigail Thompson's proof of the Recognition Algorithm for $3$-spheres is obtained.
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https://arxiv.org/abs/2601.13861
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81fdf58e5c97633af8f8f4c47369ea7ed37a18888286675f09d041ffaad3d2aa
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2026-01-21T00:00:00-05:00
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Optimizing the Geometry of an L-Shaped Building to Enhance Energy Efficiency and Sustainability
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arXiv:2601.13884v1 Announce Type: new Abstract: The geometric form of a building strongly influences its material use, heat losses, and energy efficiency. This paper presents an analytical optimization of L-shaped residential buildings aimed at minimizing the external surface area for a prescribed volume. Both symmetric and asymmetric configurations are examined under realistic design constraints, including fixed or bounded wing aspect ratios and fixed building height. Using explicit optimization methods and Karush-Kuhn-Tucker conditions, closed-form expressions for the optimal geometric parameters and minimal envelope area are derived. The results show that unconstrained optimization leads to degenerate cuboid shapes, highlighting the importance of geometric constraints to preserve the L-shaped form. The obtained results provide practical design guidelines for architects and engineers, supporting informed early stage decisions that balance functional requirements, regulatory constraints, architectural intent, and energy performance. Case studies of existing houses demonstrate that the proposed approach can reduce external surface area or confirm near-optimality of practical designs, supporting energy-efficient early-stage architectural decisions.
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https://arxiv.org/abs/2601.13884
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604f79a1b1596b6b99ca2726fa21056fc3f403ed4ec0537af143d1d3a357cab4
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2026-01-21T00:00:00-05:00
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From geometry to sustainability: Optimal shapes of hip roof houses
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arXiv:2601.13896v1 Announce Type: new Abstract: In this paper, we develop a rigorous mathematical framework for the optimization of hip roof house geometry, with the primary goal of minimizing the external surface of the building envelope for a given set of design constraints. Five optimization scenarios are systematically analyzed: fixed volume, fixed footprint ratio, fixed slenderness ratio, fixed floor area, and constrained height. For each case, explicit formulas for the optimal dimensions are derived, offering architects and engineers practical guidelines for improving material efficiency, reducing construction costs, and enhancing energy performance. To illustrate the practical relevance of the theoretical results, case studies of real-world hip roof houses are presented, revealing both inefficiencies in common practice and near-optimal examples. Furthermore, a freely available software application has been developed to support designers in applying the optimization methods directly to architectural projects. The findings confirm that square-based footprints combined with balanced slenderness ratios yield the most efficient forms, while deviations toward elongated or flattened proportions significantly increase energy and material demands. This work demonstrates how mathematical modeling and architectural design can be integrated to support sustainable architecture, providing both theoretical insight and practical tools for shaping energy-efficient, cost-effective, and aesthetically coherent residential buildings.
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https://arxiv.org/abs/2601.13896
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d61d5c5008829c78eae13c9967754a6efdd104c7eb5f995a8289512f7621a8e7
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2026-01-21T00:00:00-05:00
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Homogeneous substructures in random ordered uniform matchings
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arXiv:2601.13906v1 Announce Type: new Abstract: An ordered $r$-uniform matching of size $n$ is a collection of $n$ pairwise disjoint $r$-subsets of a linearly ordered set of $rn$ vertices. For $n=2$, such a matching is called an $r$-pattern, as it represents one of $\tfrac12\binom{2r}r$ ways two disjoint edges may intertwine. Given a set $\mathcal{P}$ of $r$-patterns, a $\mathcal{P}$-clique is a matching with all pairs of edges belonging to $\mathcal{P}$. In this paper we determine the order of magnitude of the size of a largest $\mathcal{P}$-clique in a random ordered $r$-uniform matching for several sets $\mathcal{P}$, including all sets of size $|\mathcal{P}|\le2$ and the set $\mathcal{R}^{(r)}$ of all $2^{r-1}$ $r$-partite $r$-patterns.
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https://arxiv.org/abs/2601.13906
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c482884c03f7f9371011a4d074f8113831dbeb1bb8def93989ec3500e74546cc
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2026-01-21T00:00:00-05:00
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Designing sustainable barn-type houses: Optimal shapes for minimal envelope and energy use
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arXiv:2601.13911v1 Announce Type: new Abstract: Barn-type houses have become one of the most popular single-family housing typologies in Poland and across Europe due to their simplicity, functionality, and potential for energy efficiency. Despite their widespread use, systematic methods for optimizing their geometry in terms of envelope area and energy performance remain limited. This paper develops a rigorous mathematical framework for determining the optimal proportions of barn-type houses with respect to minimizing the external surface area while satisfying constraints of either fixed volume or fixed floor area. Closed-form solutions for the optimal width, length, and height are derived as explicit functions of the roof slope, together with formulas for the minimal achievable surface. A recently introduced dimensionless compactness measure is also calculated, allowing quantitative assessment of how far a given design deviates from the theoretical optimum. The methodology is applied to case studies of three existing houses, showing that while some designs deviate significantly from optimal compactness, others already closely approximate it. The results confirm that theoretical optimization can lead to meaningful reductions in construction costs and energy demand. To support practical implementation, two original freely available software tools were developed, enabling architects and engineers to perform optimization analyses.
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https://arxiv.org/abs/2601.13911
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72c7c6998f90637255bf233cda423ad0d4d2c61221bb2717e631dc134b020d3f
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2026-01-21T00:00:00-05:00
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Geometry-Driven Conditioning of Multivariate Vandermonde Matrices in High-Degree Regimes
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arXiv:2601.13915v1 Announce Type: new Abstract: We study multivariate monomial Vandermonde matrices $V_N(Z)$ with arbitrary distinct nodes $Z=\{z_1,\dots,z_s\}\subset B_2^n$ in the high-degree regime $N\ge s-1$. Introducing a projection-based geometric statistic -- the \emph{max-min projection separation} $\rho(Z,j)$ and its minimum $\kappa(Z)=\min_j\rho(Z,j)$ -- we construct Lagrange polynomials $Q_j\in\mathcal P_N^n$ with explicit coefficient bounds $$ \|Q_j\|_\infty \lesssim s\Bigl(\frac{4n}{\rho(Z,j)}\Bigr)^{s-1}. $$ These polynomials yield quantitative distance-to-span estimates for the rows of $V_N(Z)$ and, as consequences, $$ \sigma_{\min}(V_N(Z)) \gtrsim \frac{\kappa(Z)^{s-1}}{(4n)^{s-1} s\sqrt{s \nu(n,N)}}, \quad \nu(n,N)={N+n\choose N}, $$ and an explicit right inverse $V_N(Z)^+$ with operator-norm control $$ \|V_N(Z)^+\| \lesssim s^{3/2}\sqrt{\nu(n,N)}\Bigl(\frac{4n}{\kappa(Z)}\Bigr)^{s-1}. $$ Our estimates are dimension-explicit and expressed directly in terms of the local geometry parameter $\kappa(Z)$; they apply to \emph{every} distinct node set $Z\subset B_2^n$ without any \emph{a priori} separation assumptions. In particular, $V_N(Z)$ has full row rank whenever $N\ge s-1$. The results complement the Fourier-type theory (on the complex unit circle/torus), where lower bounds for $\sigma_{\min}$ hinge on uniform separation or cluster structure; here stability is quantified instead via high polynomial degree and the projection geometry of $Z$.
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https://arxiv.org/abs/2601.13915
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119d2e55a85941ae6faf0a45f4d9b3fe20e57fe50c19c0c59172f3dc422f3d9b
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2026-01-21T00:00:00-05:00
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Wiener Algebras Methods for Liouville Theorems on the Stationary Navier-Stokes System
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arXiv:2601.13916v1 Announce Type: new Abstract: We prove some Liouville theorems for the stationary Navier-Stokes system for incompressible fluids. We provide some sufficient conditions on the low frequency part of the solution, using some properties of classical singular integrals with respect to Wiener algebras.
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https://arxiv.org/abs/2601.13916
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739a5d03b59357e081d5b1021a2d4d069ce2e9a41a318c24243e2b103a359596
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2026-01-21T00:00:00-05:00
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A finiteness result on representations of Nori's fundamental group scheme
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arXiv:2601.13917v1 Announce Type: new Abstract: Let $(X,x)$ be a pointed geometrically connected smooth projective variety over a sub-$p$-adic field $K$. For any given rank $n$, we prove that there are only finitely many isomorphism classes of representations $\pi_{1}^{EF}(X,x)\rightarrow \mathrm{GL}_{n}$, where $\pi_{1}^{EF}(X,x)$ is Nori's fundamental group of essentially finite bundles. Equivalently, there are only finitely many isomorphism classes of essentially finite bundles of rank $n$. This answers a question from C.Gasbarri.
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https://arxiv.org/abs/2601.13917
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715f1498a684bacb6ac54b2f7856aebdc76b2a4764d87304ed201a8092fca387
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2026-01-21T00:00:00-05:00
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The properad of quadratic Poisson structures is Koszul
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arXiv:2601.13921v1 Announce Type: new Abstract: In this paper, we suggest a sufficient condition on the properadic envelope of a quadratic dioperad to be Koszul in terms of twisted associative algebras. As a particular new example, we show that the properad of quadratic Poisson structures is Koszul.
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https://arxiv.org/abs/2601.13921
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dacdf84f19fa5c9a2a03fff5ac3be04d8918bd96a5336ed0bcd01e14d9ca856c
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2026-01-21T00:00:00-05:00
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On spectral clustering under non-isotropic Gaussian mixture models
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arXiv:2601.13930v1 Announce Type: new Abstract: We evaluate the misclustering probability of a spectral clustering algorithm under a Gaussian mixture model with a general covariance structure. The algorithm partitions the data into two groups based on the sign of the first principal component score. As a corollary of the main result, the clustering procedure is shown to be consistent in a high-dimensional regime.
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https://arxiv.org/abs/2601.13930
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66e39507c09e6979b2694663b545c71fd3009dacf941f6f91e44c6105fa54cd8
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2026-01-21T00:00:00-05:00
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Packing minima of convex bodies
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arXiv:2601.13937v1 Announce Type: new Abstract: In 2021, Henk, Schymura and Xue introduced packing minima, associated with a convex body and a lattice, as packing counterparts to the covering minima of Kannan and Lov\'asz. Motivated by conjectures on the volume inequalities for the successive minima, we generalized the definition of the packing minima to the class of all convex bodies that contain the origin in their interior. For these packing minima, we presented several novel volume inequalities and calculated the specific values of the packing minima for several special convex bodies.
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https://arxiv.org/abs/2601.13937
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c6b4649ff1e4fa43886f67e6a53a12a5d36e9255a58720868362be7d5db95247
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2026-01-21T00:00:00-05:00
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Gallai-Ramsey Numbers for $\ell$-Connected Graphs
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arXiv:2601.13944v1 Announce Type: new Abstract: Given a nonempty graph $G$, a collection of nonempty graphs $\cal{H}$, and a positive integer $k$, the Gallai-Ramsey number $\mathrm{gr}_k(G:\mathcal{H})$ is defined to be the minimum positive integer $n$ such that every exact $k$-edge-coloring of a complete graph $K_n$ contains either a rainbow copy of $G$ or a monochromatic copy of some element in $\mathcal{H}$. In this paper, we obtain some exact values and general lower and upper bounds for $\mathrm{gr}_k(G:\mathcal{F}^\ell)$, where $\mathcal{F}^\ell$ is the set of $\ell$-connected graphs and $G\in\{P_5, K_{1,3}\}$.
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https://arxiv.org/abs/2601.13944
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a6cfe1646968e0918f6644c3108a64b0c6f2ae1228ae2a18cb5b9f8c126e6cb8
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2026-01-21T00:00:00-05:00
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Topological Criteria for Hypothesis Testing with Finite-Precision Measurements
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arXiv:2601.13946v1 Announce Type: new Abstract: We establish topological necessary and sufficient conditions under which a pair of statistical hypotheses can be consistently distinguished when i.i.d. observations are recorded only to finite precision. Requiring the test's decision regions to be open in the sample-space topology to accommodate finite-precision data, we show that a pair of null- and alternative hypotheses $H_0$ and $H_1$ admits a consistent test if and only if they are $F_\sigma$ in the weak topology on the space of probability measures $W := H_0\cup H_1$. Additionally, the hypotheses admit uniform error control under $H_0$ and/or $H_1$ if and only if $H_0$ and/or $H_1$ are closed in $W$. Under compactness assumptions, uniform consistency is characterised by $H_0$ and $H_1$ having disjoint closures in the ambient space of probability measures. These criteria imply that - without regularity assumptions - conditional independence is not consistently testable. We introduce a Lipschitz-continuity assumption on the family of conditional distributions under which we recover testability of conditional independence with uniform error control under the null, with testable smoothness constraints.
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https://arxiv.org/abs/2601.13946
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13fee2ec0c94b5847ff2bcd2648e111d1e1eb3d923b0f5c9cecb657056b6c841
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2026-01-21T00:00:00-05:00
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Wold-type decomposition for doubly twisted left-invertible covariant representations
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arXiv:2601.13950v1 Announce Type: new Abstract: We will introduce the notion of a near-isometric covariant representation of a $C^*$-correspondence and prove its Wold-type decomposition. Wold-type decomposition for doubly twisted left-invertible covariant representations of a product system is also obtained.
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https://arxiv.org/abs/2601.13950
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c2d7a0cb128ee47b955a6c0232903914c2b66edc76b34d242853e526d6509e98
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2026-01-21T00:00:00-05:00
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Hypercube subgroups of (outer) reduced Weyl groups of the Cuntz algebras
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arXiv:2601.13952v1 Announce Type: new Abstract: We develop some tools, of an algebraic and combinatorial nature, which enable us to obtain a detailed description of certain quadratic subgroups of the (outer) reduced Weyl group of the Cuntz algebra ${\mathcal O}_n$. In particular, for $n=4$ our findings give a self-contained theoretical interpretation of the groups tabulated in [AJS18], which were obtained with the help of a computer. For each of these groups we provide a set of generators. A prominent role in our analysis is played by a certain family of subgroups of the symmetric group of a discrete square which we call bicompatible.
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https://arxiv.org/abs/2601.13952
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926b18183099055b0400b4dd1a3bb51783e83430de116edd31a625a0d402aa81
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2026-01-21T00:00:00-05:00
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Uniform Consistency of Generalized Cross-Validation for Ridge Regression in High-Dimensional Misspecified Linear Models
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arXiv:2601.13955v1 Announce Type: new Abstract: This study examines generalized cross-validation for the tuning parameter selection for ridge regression in high-dimensional misspecified linear models. The set of candidates for the tuning parameter includes not only positive values but also zero and negative values. We demonstrate that if the second moment of the specification error converges to zero, generalized cross-validation is still a uniformly consistent estimator of the out-of-sample prediction risk. This implies that generalized cross-validation selects the tuning parameter for which ridge regression asymptotically achieves the smallest prediction risk among the candidates if the degree of misspecification for the regression function is small. Our simulation studies show that ridge regression tuned by generalized cross-validation exhibits a prediction performance similar to that of optimally tuned ridge regression and outperforms the Lasso under correct and incorrect model specifications.
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https://arxiv.org/abs/2601.13955
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be8b044008fdf599696134631ab112a106169b5ecfa1c3ee2c9c156d7590c32f
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2026-01-21T00:00:00-05:00
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A Bregman Regularized Proximal Point Method for Solving Equilibrium Problems on Hadamard Manifolds
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arXiv:2601.13959v1 Announce Type: new Abstract: In this paper we develop a Bregman regularized proximal point algorithm for solving monotone equilibrium problems on Hadamard manifolds. It has been shown that the regularization term induced by a Bregman function is, in general, nonconvex on Hadamard manifolds unless the curvature is zero. Nevertheless, we prove that the proposed Bregman regularization scheme does converge to a solution of the equilibrium problem on Hadamard manifolds in the presence of a strong assumption on the convexity of the set formed by the regularization term. Moreover, we employ a coercivity condition on the Bregman function which is weaker than those typically assumed in the existing literature on Bregman regularization. Numerical experiments on illustrative examples demonstrate the practical effectiveness of our proposed method.
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https://arxiv.org/abs/2601.13959
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1b2f522dec9a75e2b9730622eaa0b3aa5887ecc416dbcb96eb15eb0c9a43e657
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2026-01-21T00:00:00-05:00
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Information-Theoretic and Computational Limits of Correlation Detection under Graph Sampling
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arXiv:2601.13966v1 Announce Type: new Abstract: Correlation analysis is a fundamental problem in statistics. In this paper, we consider the correlation detection problem between a pair of Erdos-Renyi graphs. Specifically, the problem is formulated as a hypothesis testing problem: under the null hypothesis, the two graphs are independent; under the alternative hypothesis, the two graphs are edge-correlated through a latent permutation. We focus on the scenario where only two induced subgraphs are sampled, and characterize the sample size threshold for detection. At the information-theoretic level, we establish the sample complexity rates that are optimal up to constant factors over most parameter regimes, and the remaining gap is bounded by a subpolynomial factor. On the algorithmic side, we propose polynomial-time tests based on counting trees and bounded degree motifs, and identify the regimes where they succeed. Moreover, leveraging the low-degree conjecture, we provide evidence of computational hardness that matches our achievable guarantees, showing that the proposed polynomial-time tests are rate-optimal. Together, these results reveal a statistical--computational gap in the sample size required for correlation detection. Finally, we validate the proposed algorithms on synthetic data and a real coauthor network, demonstrating strong empirical performance.
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https://arxiv.org/abs/2601.13966
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71011a37190c1cdf3b2977121cea1e28a5145f727644d953063bb3652b88c7bd
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2026-01-21T00:00:00-05:00
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Dispersive estimate for quasi-periodic Klein-Gordon equation on 1-d lattices
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arXiv:2601.13967v1 Announce Type: new Abstract: The dispersive estimate plays a pivotal role in establishing the long-term behavior of solutions to the nonlinear equation, thereby being crucial for investigating the well-posedness of the equation.In this work we prove that the solutions to Klein-Gordon equation on 1-d lattices follow the dispersive estimate provided that potential is quasi-periodic with Diophantine frequencies and closed to positive constants.
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https://arxiv.org/abs/2601.13967
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5cc08825cf2c7f8c89b2c9e422366d65bae7e9476e49c0f6b049cca19cbd52d9
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2026-01-21T00:00:00-05:00
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Toric Euler-Jacobi vanishing theorem and zeros at infinity
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arXiv:2601.13977v1 Announce Type: new Abstract: Residues appear naturally in various questions in complex and algebraic geometry: interpolation, duality, representation problems, and obstructions. The first global vanishing result in the projective plane, known as the Euler-Jacobi theorem, was established by Jacobi in 1835. In the toric case, the input is a system of n Laurent sparse polynomials with fixed Newton polytopes, and the first version of the Euler-Jacobi toric vanishing theorem for residues in the n-torus is due to Khovanskii in 1978, under restrictive genericity assumptions. In this paper, we provide geometric conditions on the input Newton polytopes to ensure that this global vanishing is equivalent to the existence of zeros at infinity in the associated compact toric variety. We relate these conditions to the dimension at the toric critical degree of the quotient of the Cox ring by the ideal generated by the (multi)homogenizations of the input polynomials. We also relate the existence of zeros at infinity to interpolation questions.
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https://arxiv.org/abs/2601.13977
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81ba600cc9036b358d65b721ddcc5584c2edfa05234c200b8b0a3613e59a2826
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2026-01-21T00:00:00-05:00
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Spectral Gaps on Large Hyperbolic Surfaces
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arXiv:2601.13988v1 Announce Type: new Abstract: In this expository paper, we review the history and the recent breakthroughs in the spectral theory of large volume hyperbolic surfaces. More precisely, we focus mostly on the investigation of the first non-trivial eigenvalue $\lambda_1$ and its possible behaviour in the large volume regime.
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https://arxiv.org/abs/2601.13988
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a384f33cf5daf0e0416998f9266aa174cef661762c914c93f446fb8f8207eb81
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2026-01-21T00:00:00-05:00
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Eigensets of switching dynamical systems
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arXiv:2601.13990v1 Announce Type: new Abstract: Reachability sets of linear switching dynamical systems (systems of ODE with time-dependent matrices that take values from a given compact set) are analysed. An eigenset is a non-trivial compact set M that possesses the following property: the closure of the set of points reachable by trajectories starting in M in time t is equal to exp(at)M. This concept introduced in a recent paper of E.Viscovini is an analogue of an eigenvector for compact sets of matrices. We prove the existence of eigensets, analyse their structure and properties, and find ``eigenvalues'' a for an arbitrary system. The question which compact sets, in particular, which convex sets and polyhedra, can be presented as eigensets of suitable systems, is studied.
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https://arxiv.org/abs/2601.13990
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905658f636792000a85535b398ee77cfe5c37df4b42b431b92dc74f232471213
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2026-01-21T00:00:00-05:00
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Computing Crystalline Cohomology and p-Divisible Groups for Curves over Finite Fields
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arXiv:2601.14006v1 Announce Type: new Abstract: Let $X$ be a smooth projective curve over a finite field of characteristic $p$. We describe and implement a practical algorithm for computing the $p$-divisible group $Jac(X)[p^\infty]$ via computing its Dieudonn\'{e} module, or equivalently computing the Frobenius and Verschiebung operators on the first crystalline cohomology of $X$. We build on Tuitman's $p$-adic point counting algorithm, which computes the rigid cohomology of $X$ and requires a ``nice'' lift of $X$ to be provided.
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https://arxiv.org/abs/2601.14006
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51fc7502da9308b4b10b2a3b94db9525e688bf171bfecc7e379122a23c1f1199
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2026-01-21T00:00:00-05:00
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Robustness for free: asymptotic size and power of max-tests in high dimensions
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arXiv:2601.14013v1 Announce Type: new Abstract: Consider testing a zero restriction on the mean of a $d$-dimensional random vector based on an i.i.d. sample of size $n$. Suppose further that the coordinates are only assumed to possess $m>2$ moments. Then, max-tests based on arithmetic means and critical values derived from Gaussian approximations are not guaranteed to be asymptotically valid unless $d$ is relatively small compared to $n$, because said approximation faces a polynomial growth barrier of $d=o(n^{m/2-1})$. We propose a max-test based on winsorized means, and show that it holds the desired asymptotic size even when $d$ grows at an exponential rate in $n$ and the data are adversarially contaminated. Our characterization of its asymptotic power function shows that these benefits do not come at the cost of reduced asymptotic power: the robustified max-test has identical asymptotic power to that based on arithmetic means whenever the stronger assumptions underlying the latter are satisfied. We also investigate when -- and when not -- data-driven (bootstrap) critical values can strictly increase asymptotic power of the robustified max-test.
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https://arxiv.org/abs/2601.14013
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Academic Papers
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db0a9a5c5851497e2a8b2c0f1b03de22c8d259ef91de68d9573935f713d4adb5
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2026-01-21T00:00:00-05:00
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Non-linear traces of Choquet type on AF algebras
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arXiv:2601.14016v1 Announce Type: new Abstract: We study non-linear traces of Choquet type on AF algebras. Building on the characterization of Choquet traces on matrix algebras due to Nagisa--Watatani, we generalize the construction to arbitrary unital AF algebras. We show that there is a one-to-one correspondence between such traces and increasing functions on the dimension scale, and we obtain explicit Choquet formulas in terms of the spectrum and ranks of spectral projections along a fixed AF filtration.
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https://arxiv.org/abs/2601.14016
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Academic Papers
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3716ef160f4639caddf24e821b5254a2b43d9ee90ff4ea54622de7525750ab6c
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2026-01-21T00:00:00-05:00
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Principal $p-$frequency estimates on non-compact manifolds with negative Ricci curvature
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arXiv:2601.14018v1 Announce Type: new Abstract: We establish a lower bound for the principal $p-$frequency $\lambda_{1,p}(\Omega)$ on a bounded domain $\Omega$ in a non-compact Riemannian manifold of dimension $n.$ Under the assumption that the Ricci curvature satisfies $\operatorname{Ric} \geq (n-1)K$ with $K \bar{\lambda}_{D,K,n}$, where $D$ is the diameter of $\Omega$ and $\bar{\lambda}_{D,K,n}$ is explicitly defined as the first eigenvalue of an associated one-dimensional ordinary differential equation model that incorporates both $D$ and $K.$ Moreover, the estimate is sharp. This work extends previous results for the case $K=0$ to the geometrically more complex setting of negative Ricci curvature, and providing a new quantitative connection between the eigenvalue, the diameter of domains, and the curvature lower bound.
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https://arxiv.org/abs/2601.14018
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Academic Papers
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52d98a4f285181a1d8cb91c0b895300e62d8725cf82158a08fe813d80e2e58ca
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2026-01-21T00:00:00-05:00
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Tensor Abelian geometry of VI-modules
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arXiv:2601.14020v1 Announce Type: new Abstract: In this short note, we study the spectrum of prime Serre ideals of global represen tations for noetherian families. In particular, we prove that the spectrum of prime Serre ideals of finitely generated VI-modules is homeomorphic to N^{*}, the one-point compactification of N, which differs from the Balmer spectrum of derived VI-modules. Our method could also be applied to the category of finitely generated FI-modules and the category of global representations for the family of cyclic p-groups.
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https://arxiv.org/abs/2601.14020
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Academic Papers
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1ead9fb1cae0e88567ae39a7c320564b907cf7fcdfbc7b96b5cdb11736febbe9
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2026-01-21T00:00:00-05:00
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Some Results on Causal Modalities in General Spacetimes
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arXiv:2601.14029v1 Announce Type: new Abstract: Causality is one of the fundamental structures of spacetimes, it determines the possible behaviour and propagation of physical information through different relations. Causal structure can be analysed through the various modal logics it induces. The modal logics for the standard chronological and causal relations of the archetypal Minkowski spacetime have been classified. However only partial results have been achieved for the strict variant of the causal relation, also known as the after relation. The present work continues this analysis towards arbitrary spacetimes. By utilizing the definition of the causal relations through causal paths, we can lift known results about the modal logics of Minkowski spacetime to general spacetimes. In particular, for the after relation, we show that a previously studied formula within the logics of Minkowski spacetime holds in arbitrary spacetimes. We introduce a related modal formula that demonstrates that the logic of two-dimensional spacetimes are more expressive than higher dimensional ones. Lastly, we study the interrelation between the logical properties and physical properties along the causal ladder, a classification of causal structures according to a hierarchy of physically relevant properties.
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https://arxiv.org/abs/2601.14029
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Academic Papers
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3a476c8bcbd72d57b52ab2bb6d1e19578d660cf643217031ffbdf9bca5891001
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2026-01-21T00:00:00-05:00
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Classification of invariant tight contact structures on the 3-space, -ball and -sphere
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arXiv:2601.14040v1 Announce Type: new Abstract: We prove some classification results for tight contact structure in the 3-space, -ball and -sphere that are invariant with respect to some arbitrary involution, that is conjugated to the standard rotation around the x-axis. Unlike the classical scenario, a new integral torsion appears, dictating a splitting between equivalence classes. These tools could be useful fur future classification results regarding strongly invertible Legendrian knots.
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https://arxiv.org/abs/2601.14040
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Academic Papers
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7fe276ce07037e11aaba891a865e45a1f9f06a5f7a1a54f38579f173d65297b1
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2026-01-21T00:00:00-05:00
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On the Diophantine Equation Involving Elementary Symmetric Polynomials and the Decomposition of Unity
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arXiv:2601.14057v1 Announce Type: new Abstract: We consider the equality of the values of the $n$th and $k$th elementary symmetric polynomials of $n$ not necessarily distinct positive integers. For $k < n$, we prove that this equation always has a solution, but only finitely many solutions. Furthermore, we consider the equality of the values of the $n$th and $(n-2)$th elementary symmetric polynomials of $n$ not necessarily distinct positive integers. In particular, we show that the number of solutions of this equation tends to infinity if $n$ tends to infinity.
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https://arxiv.org/abs/2601.14057
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Academic Papers
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497a902c12ad7aea907add63ba4a62c9eec13c4bbffcabd37db36d0b9d92da67
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2026-01-21T00:00:00-05:00
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A Splitting Theorem for non-positively curved Lorentzian spaces
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arXiv:2601.14058v1 Announce Type: new Abstract: We prove a splitting theorem for Lorentzian pre-length spaces with global non-positive timelike curvature. Additionally, we extend the first variation formula to spaces with any timelike curvature bound, either from above or below, and different from 0.
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https://arxiv.org/abs/2601.14058
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Academic Papers
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e0e091ccf67a7a7a0e826535fbb75bd1b0858de22033add94ab33d6b48076f9c
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2026-01-21T00:00:00-05:00
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Frostman dimension of Furstenberg measure for $\mathrm{SL}(2,\mathbb{R})$ random matrix products
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arXiv:2601.14061v1 Announce Type: new Abstract: For compactly supported $\mu \in \mathcal{P}(\mathrm{SL}(2,\mathbb{R}))$ satisfying strong irreducibility and proximality, we obtain a formula for the Frostman dimension of the associated Furstenberg measure. We also describe the left neighbourhood of 0 for which the classical transfer operators defined by Le Page have a spectral gap on H\"older spaces in this setting.
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https://arxiv.org/abs/2601.14061
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Academic Papers
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45e50313f7a56ae90b5f99a0f2e8a5f5a2218b420a5a1082ee4c8511f58d324f
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2026-01-21T00:00:00-05:00
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Time-dependent metrics and connections
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arXiv:2601.14064v1 Announce Type: new Abstract: Time-dependent structures often appear in differential geometry, particularly in the study of non-autonomous differential equations on manifolds. One may study the geodesics associated with a time-dependent Riemannian metric by extremizing the corresponding energy functional, but also through the introduction of a more general concept of time-dependent covariant derivative operator. This relies on the examination of connections on the product manifold $\mathbb{R}\times M$. For these time-dependent covariant derivatives we explore the notions of parallel transport, geodesics and torsion. We also define the derivative of a one-parameter family of connections.
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https://arxiv.org/abs/2601.14064
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Academic Papers
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138b5b0e1b471e9456b99f4a0ab45a21eb8e0306d312100450b959b4d6c9fb24
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2026-01-21T00:00:00-05:00
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LU-type factorizations for birth--death processes and their Darboux transformations
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arXiv:2601.14074v1 Announce Type: new Abstract: We study LU-type factorizations of the infinitesimal generator of a birth--death process on $\mathbb{N}_0$. Our goal is to characterize those factorizations whose Darboux transformations (that is, inverting the order of the factors) yield new infinitesimal generators of birth--death processes. Two types are considered: lower--upper (LU), which is unique and upper--lower (UL), which involves a free parameter. For both cases, we determine the conditions under which such factorizations can occur, derive explicit formulas for their coefficients, and provide a probabilistic interpretation of the factors. The spectral properties and associated orthogonal polynomials of the Darboux transformations are also analyzed. Finally, the general results are applied to classical examples such as the $M/M/1$ and $M/M/\infty$ queues and to different cases of linear birth--death processes.
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https://arxiv.org/abs/2601.14074
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Academic Papers
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bb98b90ce92bd38390dff20f3fe865896cd07856d38f5419c793215f1a17a002
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2026-01-21T00:00:00-05:00
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On large periodic traveling wave solutions to the free boundary Stokes and Navier-Stokes equations
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arXiv:2601.14085v1 Announce Type: new Abstract: We study the free boundary problem for a finite-depth layer of viscous incompressible fluid in arbitrary dimension, modeled by the Stokes or Navier-Stokes equations. In addition to the gravitational field acting in the bulk, the free boundary is acted upon by surface tension and an external stress tensor posited to be in traveling wave form. We prove that for any isotropic stress tensor with periodic profile, there exists a locally unique periodic traveling wave solution, which can have large amplitude. Moreover, we prove that the constructed traveling wave solutions are asymptotically stable for the dynamic free boundary Stokes equations. Our proofs rest on the analysis of the nonlocal normal-stress to normal-Dirichlet operators for the Stokes and Navier-Stokes equations in domains of Sobolev regularity.
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https://arxiv.org/abs/2601.14085
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Academic Papers
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7b718827d19058778523dfef6a67fd2c819e164222c1ded25a4a4282302248c1
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2026-01-21T00:00:00-05:00
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Period collapse of Markov triangles
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arXiv:2601.14090v1 Announce Type: new Abstract: Cristofaro-Gardiner and Kleinman showed the complete period collapse of the Ehrhart quasipolynomial of Fibonacci triangles and their irrational limits, by studying the Fourier-Dedekind sums involved in the Ehrhart function of right-angled rational triangles. We generalize this result using integral affine geometrical methods to all Markov triangles, as defined by Vianna. In particular, we show new occurrences of strong period collapse, namely by constructing for each Markov number $p$ a two-sided sequence of rational triangles and two irrational limits with quasipolynomial Ehrhart function of period $p$.
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https://arxiv.org/abs/2601.14090
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Academic Papers
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59473e952ac2b351f4b647112f291625e77950e8cd58fdd4bf9ba2026ed802c2
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2026-01-21T00:00:00-05:00
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Basis Number and Pathwidth
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arXiv:2601.14095v1 Announce Type: new Abstract: We prove two results relating the basis number of a graph $G$ to path decompositions of $G$. Our first result shows that the basis number of a graph is at most four times its pathwidth. Our second result shows that, if a graph $G$ has a path decomposition with adhesions of size at most $k$ in which the graph induced by each bag has basis number at most $b$, then $G$ has basis number at most $b+O(k\log^2 k)$. The first result, combined with recent work of Geniet and Giocanti shows that the basis number of a graph is bounded by a polynomial function of its treewidth. The second result (also combined with the work of Geniet and Giocanti) shows that every $K_t$-minor-free graph has a basis number bounded by a polynomial function of $t$.
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https://arxiv.org/abs/2601.14095
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Academic Papers
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3b4242adbade3b6252072d18bb9fc01502cc1ec0a8e5bf71d20ff028746f4de2
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2026-01-21T00:00:00-05:00
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Simple subquotients of crossed products by abelian groups and twisted group algebras
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arXiv:2601.14097v1 Announce Type: new Abstract: Motivated by work of Poguntke we study the question under what conditions simple subquotients of crossed products $A\rtimes_{\alpha}G$ by (twisted) actions of abelian groups $G$ are isomorphic to simple twisted group algebras of abelian groups. As a consequence, we recover a theorem of Poguntke's saying that the simple subquotients of group $C^*$-algebras of connected groups are either stably isomorphic to $\mathbb C$ or they are stably isomorphic to simple non-commutative tori.
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https://arxiv.org/abs/2601.14097
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Academic Papers
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2ac6ddfbdab26dd08ca3360634a241fbefd7d481437f02d5ef1bec8bcb521536
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2026-01-21T00:00:00-05:00
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Angular pair-of-pants decompositions of complex varieties
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arXiv:2601.14116v1 Announce Type: new Abstract: We define the notion of torically hyperbolic varieties and we construct pair-of-pants decompositions for these in terms of angle sets of essential projective hyperplane complements. This construction generalizes the classical pair-of-pants decomposition for hyperbolic Riemann surfaces. In our first main theorem, we prove that the natural angle map associated to an essential projective hyperplane complement is a homotopy equivalence, extending earlier work of Salvetti and Bj\"orner-Ziegler. By a topological argument, we further show that the angle map for a finite Kummer covering of an essential projective hyperplane complement is likewise a homotopy equivalence. We then explain how these local building blocks can be glued along the dual intersection complex of a semistable degeneration. Using the theory of Kato-Nakayama spaces, we prove that the resulting space is homotopy equivalent to the original algebraic variety. We make this explicit for complete intersections in projective space using techniques from tropical geometry.
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https://arxiv.org/abs/2601.14116
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Academic Papers
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35ec346b69968191a5c458b4eec5d93ba08d12ab8f75120d5a56f4a1e96ebc17
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2026-01-21T00:00:00-05:00
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Scalar-rigid submersions are Riemannian products
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arXiv:2601.14117v1 Announce Type: new Abstract: Scalar-rigid maps are Riemannian submersions by works of Llarull, Goette--Semmelmann, and the second named author. In this article we show that they are essentially Riemannian products of the base manifold with a Ricci-flat fiber. As an application we obtain a Llarull-type theorem for non-zero degree maps onto products of manifolds of non-negative curvature operator and positive Ricci curvature with some enlargeable manifold. The proof is based on spin geometry for Dirac operators and an analysis connecting Clifford multiplication with the representation theory of the curvature operator.
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https://arxiv.org/abs/2601.14117
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Academic Papers
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94050f06d1dbebf5ab531c656b06d8ad6d72318d23e6a6b48dc7344814a8839a
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2026-01-21T00:00:00-05:00
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Universal Chord Theorem and a Topological Analysis
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arXiv:2601.14120v1 Announce Type: new Abstract: We study the set of chords of a real-valued continuous function on [0,1] with f(0)=f(1)=0. We describe which chords may appear as isolated points and provide examples illustrating our characterization. Maximal Hopf sets are introduced and analyzed.
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https://arxiv.org/abs/2601.14120
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Academic Papers
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88ecffcd857bc51470b97912c1a3a8a85dda0c8e734e40c2e54a742c38a8881b
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2026-01-21T00:00:00-05:00
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$Q_p$-weighted zero-sum constants
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arXiv:2601.14122v1 Announce Type: new Abstract: A sequence $S=(x_1,\ldots, x_k)$ in $\mathbb Z_p$ is called a $(Q_p,\mathbf 1)$-weighted zero-sum sequence if there exist $a_1,\ldots,a_k\in Q_p$ such that $a_1x_1+\cdots+a_kx_k=0$ and $a_1+\cdots+a_k=0$. The constant $E_{Q_p,\mathbf 1}$ is defined to be the smallest positive integer $k$ such that every sequence of length $k$ in $\mathbb Z_p$ has a $(Q_p,\mathbf 1)$-weighted zero-sum subsequence of length $p$. We determine the constant $E_{Q_p,\mathbf 1}$ and the related constants $C_{Q_p,\mathbf 1}$ and $D_{Q_p,\mathbf 1}$. We also study some $(Q_p,B)$-weighted zero-sum constants where $B$ is a subset of $Q_p$.
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https://arxiv.org/abs/2601.14122
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Academic Papers
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ba9ccbeae136bc79de532fe8059c8f48e2466b2cdbf6abfaf2097e3aef0d7452
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2026-01-21T00:00:00-05:00
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Flexible curves and Hausdorff dimension
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arXiv:2601.14125v1 Announce Type: new Abstract: We show that given a log-singular circle homeomorphism $h$ and given any $s\in[1,2]$, there is a flexible curve of Hausdorff dimension $s$ with welding $h$. We also see that there is another curve with welding $h$ and positive area. In particular, this implies that given a flexible curve $\Gamma$, there is a homeomorphism of the plane $\phi\colon\mathbb{C}\to\mathbb{C}$, conformal off $\Gamma$, so that $\phi(\Gamma)$ has positive area. This answers a particular case of the corresponding conjecture for general non-conformally removable sets, for a class of curves that is residual in the space of all Jordan curves.
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https://arxiv.org/abs/2601.14125
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Academic Papers
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