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c5e98418c0bfb8fc73a6d7a2084ee2a6e56ffd0a292fa1c9e78798cd1ef176da
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2026-01-07T00:00:00-05:00
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Boundary operators in the Brownian loop soup
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arXiv:2601.02755v1 Announce Type: new Abstract: We obtain infinitely many boundary operators in the Brownian loop soup in the subcritical phase by analyzing the conformal block expansion of the two-point function that computes the probability of having two marked points on the upper half-plane being separated by Brownian loops. The resulting boundary operators are primary operators in a 2D CFT with central charge $c\leq1$ and have conformal dimensions that are non-negative integers. By comparing the above-mentioned conformal block expansion with probabilities in the Brownian loop soup, we provide a physical interpretation of the boundary operators of even dimensions as operators that insert multiple outer boundaries of Brownian loops at points on the real axis.
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https://arxiv.org/abs/2601.02755
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7e4b330f8aeba298b44f83ee0ecc8ee2a916b53d030caeb411de0158cce7aacd
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2026-01-07T00:00:00-05:00
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The automorphism groups of generalized Kausz compactifications and spaces of complete collineations
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arXiv:2601.02768v1 Announce Type: new Abstract: In this paper, we determine the automorphism groups of generalized Kausz compactifications $\mathcal T_{s,p,n}$. By establishing the (semi-)positivity of the anticanonical bundles of $\mathcal T_{s,p,n}$, we also determine the automorphism groups of generalized spaces of complete collineations $\mathcal M_{s,p,n}$. The results in this paper are partially taken from the author's earlier arxiv post (Canonical blow-ups of grassmann manifolds, arxiv:2007.06200).
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https://arxiv.org/abs/2601.02768
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e537fe73f04e0684e1673ff9841483f7ecc39548efc8cf968a3273125d2dd684
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2026-01-07T00:00:00-05:00
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On rates of convergence in central limit theorems of Selberg and Bourgade
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arXiv:2601.02781v1 Announce Type: new Abstract: Based on the recent works of Radziwill-Soundararajan and Roberts, we establish a rate of convergence in Bourgade's central limit theorem for shifted Dirichlet $L$-functions. Our results also indicate that the dependence structure in the components of a random vector could have a dramatic impact on the rate of convergence in such a multivariate central limit theorem.
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https://arxiv.org/abs/2601.02781
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91da5c97877bbdf99cadacfc147058956988cec1069fa4e0730940ee6de0b33d
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2026-01-07T00:00:00-05:00
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Minimal Sets of Involution Generators for Big Mapping Class Groups
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arXiv:2601.02784v1 Announce Type: new Abstract: Let $S(n)$, for $n \in \mathbb{N}$, be the infinite-type surface of infinite genus with $n$ ends, each of which is accumulated by genus. The mapping class groups of these types of surfaces are not countably generated. However, they are Polish groups, so they can be topologically countably generated. This paper focuses on finding minimal topological generating sets of involutions for $\mathrm{Map}(S(n))$. We establish that for $n \geq 16$, $\mathrm{Map}(S(n))$ can be topologically generated by four involutions. Furthermore, we establish that the the mapping class groups of the Loch Ness Monster surface ($n=1$) and the Jacob's Ladder surface ($n=2$) can be topologically generated by three involutions.
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https://arxiv.org/abs/2601.02784
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aeb6893195ce3f269932d1fc60eebabccd9af7ec83d7398327f9d07211bc5699
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2026-01-07T00:00:00-05:00
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Approximate Birkhoff-James orthogonality preserver on Lebesgue-Bochner spaces
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arXiv:2601.02786v1 Announce Type: new Abstract: In this article, we examine an approximate version of Koldobsky-Blanco-Turn\v{s}ek theorem (namely, Property P) in the space of vector-valued integrable functions. More precisely, we prove that the Lebesgue-Bochner spaces $L^p(\mu,X),\;(1\leq p<\infty)$, do not have Property P under certain conditions on $\mu$ and the Banach space $X$.
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https://arxiv.org/abs/2601.02786
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9cfef55e7f19eb5366bc173412f72e5cabca756663d7dcfeb5f651ba290a57d5
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2026-01-07T00:00:00-05:00
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On Liouiville Type Theorem for the 3D Isentropic Navier-Stokes System without D-condition
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arXiv:2601.02791v1 Announce Type: new Abstract: In this paper, we establish Liouville-type theorems for the steady compressible Navier-Stokes system. Assuming a smooth solution \(u \in L^p(\mathbb{R}^3)\), \(3 \le p \le \frac{9}{2}\), with bounded density, one obtains \(u \equiv0\). This generalizes the result of Li-Yu \cite{Li-Yu} by removing the Dirichlet condition \(\int_{\mathbb{R}^3} |\nabla u|^2 \, dx < \infty\). If \(\frac{9}{2} < p < 6\), Liouville-type theorem holds under the additional oscillation condition for momentum \(\rho u \in \dot{B}^{\frac{3}{p} - \frac{3}{2}}_{\infty,\infty}(\mathbb{R}^3)\). For the marginal case \(u \in L^6(\mathbb{R}^3)\), the oscillation condition can be replaced by \(\rho u \in BMO^{-1}(\mathbb{R}^3)\). We also present results in Morrey-type spaces: \(u \in \dot{M}^{s,6}(\mathbb{R}^3)\) and \(\rho u \in \dot{M}_w^{q,3}(\mathbb{R}^3)\) for \(2 \le s \le 6\) and \(\frac{3}{2} < q \le 3\).
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https://arxiv.org/abs/2601.02791
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67804c32563a4ae7d3a95d98232126f1183b8314b00d0beb2a657875a28b08e9
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2026-01-07T00:00:00-05:00
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Adapting Polyhedral Dominance Cones to Ordinal Preference Structures
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arXiv:2601.02796v1 Announce Type: new Abstract: In combinatorial optimization, ordinal costs can be used to model the quality of elements whenever numerical values are not available. When considering, for example, routing problems for cyclists, the safety of a street can be ranked in ordered categories like safe (separate bike lane), medium safe (street with a bike lane) and unsafe (street without a bike lane). However, ordinal optimization may suggest unrealistic solutions with huge detours to avoid unsafe street segments. In this paper, we investigate how partial preference information regarding the relative quality of the ordinal categories can be used to improve the relevance of the computed solutions. By introducing preference weights which describe how much better a category is at least or at most, compared to the subsequent category, we enlarge the ordinal dominance cone. This leads to a smaller set of alternatives, i. e., of ordinally efficient solutions. We show that the corresponding weighted ordinal ordering cone is a polyhedral cone and provide descriptions via its extreme rays and via its facets. The latter implies a linear transformation to an associated multi-objective optimization problem. This paves the way for the application of standard multi-objective solution approaches. We demonstrate the usefulness of the weighted ordinal ordering cone by investigating a safest path problem with different preference weights. Moreover, we investigate the interrelation between the weighted ordering cone to standard dominance concepts of multi-objective optimization, like, e.g., Pareto dominance, lexicographic dominance and weighted sum dominance.
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https://arxiv.org/abs/2601.02796
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05b6f16a0e46d4a90e1fe0a05ed8510e3155abf1c9e848e95aab6738a9d1b741
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2026-01-07T00:00:00-05:00
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Log-Polynomial Optimization
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arXiv:2601.02797v1 Announce Type: new Abstract: We study an optimization problem in which the objective is given as a sum of logarithmic-polynomial functions. This formulation is motivated by statistical estimation principles such as maximum likelihood estimation, and by loss functions including cross-entropy and Kullback-Leibler divergence. We propose a hierarchy of moment relaxations based on the truncated $K$-moment problems to solve log-polynomial optimization. We provide sufficient conditions for the hierarchy to be tight and introduce a numerical method to extract the global optimizers when the tightness is achieved. In addition, we modify relaxations with optimality conditions to better fit log-polynomial optimization with convenient Lagrange multipliers expressions. Various applications and numerical experiments are presented to show the efficiency of our method.
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https://arxiv.org/abs/2601.02797
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2f825ddc4bb808d236cc89e4cc736f5aa1a7dd1f78edaa6bf90cbdfe66f5313f
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2026-01-07T00:00:00-05:00
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An extended symmetric union with multiple tangle regions and its Alexander polynomial
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arXiv:2601.02800v1 Announce Type: new Abstract: The authors recently introduced a new construction of a knot as an extended symmetric union of a knot with a single tangle region. In this paper, we generalize the construction to include multiple tangle regions. The constructed knot $K$ with a partial knot $\hat{K}$ and multiple tangle regions satisfies the following two properties: its Alexander polynomial is the product of the Alexander polynomials of the numerators of these tangles and the square of the Alexander polynomial of the partial knot $\hat{K}$, and there exists a surjective homomorphism from the knot group of $K$ to that of $\hat{K}$ which maps the longitude of $K$ to the trivial element.
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https://arxiv.org/abs/2601.02800
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9e135923d14a2527fbfeb5020c1b309221309cdac1e2c9c61743053f44fed7bb
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2026-01-07T00:00:00-05:00
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Diffusion on homogeneous ultrametric spaces: the contributions of Alessandro Fig\`a-Talamanca
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arXiv:2601.02809v1 Announce Type: new Abstract: Alessandro Fig\`a-Talamanca (1938-2023) was an influential Italian mathematician, scientific leader of the Italian group of harmonic analysis for many years. Since the late 1970ies, his interest focussed on harmonic analysis on free groups and trees. In the later years of his scientific work he became also interested in diffusion processes on homogeneous ultrametric spaces such as local fields and totally disconnected Abelian groups. This is related with the close connection of those spaces with trees and their boundaries and concerns, in particular, the construction of such processes via discrete-time walks on trees. The present notes provide rather detailed comments on this part of his work and the related, quite abundant literature. This is intended to become part of a volume of selected papers by Fig\`a-Talamanca, accompanied by comments such as the present text.
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https://arxiv.org/abs/2601.02809
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e7a3c1c9ecf04fa2e387687f2e13f4430daa94f5af05c0dedc36b5bb79e39f7e
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2026-01-07T00:00:00-05:00
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Decision-Theoretic Robustness for Network Models
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arXiv:2601.02811v1 Announce Type: new Abstract: Bayesian network models (Erdos Renyi, stochastic block models, random dot product graphs, graphons) are widely used in neuroscience, epidemiology, and the social sciences, yet real networks are sparse, heterogeneous, and exhibit higher-order dependence. How stable are network-based decisions, model selection, and policy recommendations to small model misspecification? We study local decision-theoretic robustness by allowing the posterior to vary within a small Kullback-Leibler neighborhood and choosing actions that minimize worst-case posterior expected loss. Exploiting low-dimensional functionals available under exchangeability, we (i) adapt decision-theoretic robustness to exchangeable graphs via graphon limits and derive sharp small-radius expansions of robust posterior risk; under squared loss the leading inflation is controlled by the posterior variance of the loss, and for robustness indices that diverge at percolation/fragmentation thresholds we obtain a universal critical exponent describing the explosion of decision uncertainty near criticality. (ii) Develop a nonparametric minimax theory for robust model selection between sparse Erdos-Renyi and block models, showing-via robustness error exponents-that no Bayesian or frequentist method can uniformly improve upon the decision-theoretic limits over configuration models and sparse graphon classes for percolation-type functionals. (iii) Propose a practical algorithm based on entropic tilting of posterior or variational samples, and demonstrate it on functional brain connectivity and Karnataka village social networks.
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https://arxiv.org/abs/2601.02811
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24481205a3555ecb378e2a14b4f8ebd4f9c983a5c3332e3b6f310ecb2bbf3e14
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2026-01-07T00:00:00-05:00
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An introduction of Berezin sectorial operators and its application to Berezin number inequalities
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arXiv:2601.02817v1 Announce Type: new Abstract: We introduce a new class of operators, called Berezin sectorial operators, which generalizes classical sectorial operators. We provide examples on the Hardy-Hilbert space showing that there exist operators that are Berezin sectorial but not sectorial and that the Berezin sectorial index can be strictly smaller than the classical one. We derive Berezin number inequalities for this class, including a weak version of the power inequality, and study geometric properties of the Berezin range for finite-rank and weighted shift operators on the Dirichlet space. We also raise the question of whether similar constructions are possible for composition-differentiation operators on the Dirichlet space.
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https://arxiv.org/abs/2601.02817
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33730baaf3a5ca0519cb3252acf1dcd5c382bb5ff143a95b84372300edee925e
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2026-01-07T00:00:00-05:00
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Effective Disjunction and Effective Interpolation in Suffciently Strong Proof Systems
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arXiv:2601.02821v1 Announce Type: new Abstract: In this article, we deal with the uniform effective disjunction property and the uniform effective interpolation property, which are weaker versions of the classical effective disjunction property and the effective interpolation property.\\ The main result of the paper is as follows: Suppose the proof system $EF$ (Extended Frege) has the uniform effective disjunction property, then every sufficiently strong proof system $S$ that corresponds to a theory $T$, which is a theory in the same language as the theory $V_{1}^{1}$, also has the uniform effective disjunction property. Furthermore, if we assume that $EF$ has the uniform effective interpolation property, then the proof system $S$ also has the uniform effective interpolation property.\\ From this, it easily follows that if $EF$ has the uniform effective interpolation property, then for every disjoint $NE$-pair, there exists a set in $E$ that separates this pair. Thus, if $EF$ has the uniform effective interpolation property, it specifically holds that $NE \cap coNE = E$. Additionally, at the end of the article, the following is proven: Suppose the proof system $EF$ has the uniform effective interpolation property, and let $A_{1}$ and $A_{2}$ be a (not necessarily disjoint) NE-pair such that $A_{1} \cup A_{2} = \mathbb{N}$; then there exists an exponential time algorithm which for every input $n$ (of length $O(\log n)$) finds $i\in\{1,2\}$ such that $n\in A_{i}$.
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https://arxiv.org/abs/2601.02821
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2b47d1bfea3ad99d84646ab315509f0dd0db52a3513183fadd6d19d2b51eaf04
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2026-01-07T00:00:00-05:00
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Collapsed Structured Block Models for Community Detection in Complex Networks
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arXiv:2601.02828v1 Announce Type: new Abstract: Community detection seeks to recover mesoscopic structure from network data that may be binary, count-valued, signed, directed, weighted, or multilayer. The stochastic block model (SBM) explains such structure by positing a latent partition of nodes and block-specific edge distributions. In Bayesian SBMs, standard MCMC alternates between updating the partition and sampling block parameters, which can hinder mixing and complicate principled comparison across different partitions and numbers of communities. We develop a collapsed Bayesian SBM framework in which block-specific nuisance parameters are analytically integrated out under conjugate priors, so the marginal likelihood p(Y|z) depends only on the partition z and blockwise sufficient statistics. This yields fast local Gibbs/Metropolis updates based on ratios of closed-form integrated likelihoods and provides evidence-based complexity control that discourages gratuitous over-partitioning. We derive exact collapsed marginals for the most common SBM edge types-Beta-Bernoulli (binary), Gamma-Poisson (counts), and Normal-Inverse-Gamma (Gaussian weights)-and we extend collapsing to gap-constrained SBMs via truncated conjugate priors that enforce explicit upper bounds on between-community connectivity. We further show that the same collapsed strategy supports directed SBMs that model reciprocity through dyad states, signed SBMs via categorical block models, and multiplex SBMs where multiple layers contribute additive evidence for a shared partition. Across synthetic benchmarks and real networks (including email communication, hospital contact counts, and citation graphs), collapsed inference produces accurate partitions and interpretable posterior block summaries of within- and between-community interaction strengths while remaining computationally simple and modular.
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https://arxiv.org/abs/2601.02828
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5131fd500d9811e8c866bdea65ad3a28b94d98654f41ab2506a626dbacefcdbd
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2026-01-07T00:00:00-05:00
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Varadhan Functions, Variances, and Means on Compact Riemannian Manifolds
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arXiv:2601.02832v1 Announce Type: new Abstract: Motivated by Varadhan's theorem, we introduce Varadhan functions, variances, and means on compact Riemannian manifolds as smooth approximations to their Fr\'echet counterparts. Given independent and identically distributed samples, we prove uniform laws of large numbers for their empirical versions. Furthermore, we prove central limit theorems for Varadhan functions and variances for each fixed $t\ge0$, and for Varadhan means for each fixed $t>0$. By studying small time asymptotics of gradients and Hessians of Varadhan functions, we build a strong connection to the central limit theorem for Fr\'echet means, without assumptions on the geometry of the cut locus.
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https://arxiv.org/abs/2601.02832
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3c1a3685ec87841ad004061a0df7febe1ad3ffa20debb8d8dce08e5337d67db4
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2026-01-07T00:00:00-05:00
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Une br\`eve histoire des perturbations non-hermitiennes de rang un
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arXiv:2601.02834v1 Announce Type: new Abstract: Les perturbations de faible rang de matrices al\'eatoires ont \'et\'e au c{\oe}ur de nombreux travaux ces vingt derni\`eres ann\'ees. En particulier, les cas non-hermitiens, moins repr\'esent\'es dans la litt\'erature en r\`egle g\'en\'erale, font ici l'objet d'une attention sp\'eciale en raison de leurs applications \`a la physique et \`a l'\'etude des r\'eseaux de neurones. Petit tour d'horizon. -- A brief history of non-Hermitian perturbations of rank one: Low-rank perturbations of random matrices have been the focus of active research over the past twenty years. We give an overview of different non-Hermitian models, which are generally less represented in the literature, as well as some of their applications in physics and the study of neural networks.
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https://arxiv.org/abs/2601.02834
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656bdd2f485fb400ee8075bc5566a2c7955c9429fe4059770f093cf9b6165e13
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2026-01-07T00:00:00-05:00
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Quantum isometry groups of log-Laplacians on Cuntz--Krieger algebras
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arXiv:2601.02835v1 Announce Type: new Abstract: We compute the quantum isometry groups of Cuntz--Krieger algebras endowed with the spectral triples coming from the Ahlfors regular structure of the underlying topological Markov chain. This allows us to exhibit a new family of compact quantum groups, mixing features from quantum automorphism groups of graphs and easy quantum groups. Contrary to the classical isometry groups, whose actions on the Cuntz--Krieger algebras are never ergodic, the quantum isometry group acts ergodically in the case of the Cuntz algebra. This also leads to the construction of a (genuinely quantum) ergodic and faithful action of a compact matrix quantum group on the Cantor space.
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https://arxiv.org/abs/2601.02835
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6fcb7bbd38e34556c24afe01046d07491cbcac14d16f6bfacec519d4c9a4b349
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2026-01-07T00:00:00-05:00
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Data-Driven Modeling of Global Bifurcations and Chaos in a Mechanical System under Delayed and Quantized Control
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arXiv:2601.02838v1 Announce Type: new Abstract: We illustrate how the recent theory of Spectral Submanifolds (SSM) can capture global bifurcations and complex dynamics in mechanical systems even under delay and spatial discretization. Specifically, we build a parameter-dependent SSM-reduced model that predicts global heteroclinic and local bifurcations in a Furuta pendulum under control with delay, and verify these predictions numerically. Under additional spatial discretization of the digital controller, we also obtain an SSM-reduced model that correctly reproduces a numerically and experimentally observed microchaotic attractor in the system.
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https://arxiv.org/abs/2601.02838
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ff585903d326b09418fac1c1cb085944040585cc15cf8fdbb5fd025520cad6c6
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2026-01-07T00:00:00-05:00
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Large-scale geometry of graphs interpolating between curve graphs and pants graphs
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arXiv:2601.02839v1 Announce Type: new Abstract: We study two types of graphs interpolating between the curve graph and the pants graph from the viewpoint of large-scale geometry. One was introduced by Erlandsson and Fanoni, and the other by Mahan Mj. These graphs were developed independently in different contexts. In this paper, we provide explicit formulae for computing their quasi-flat ranks. These formulae depend on the genus and the number of boundary components of the underlying surface, as well as the interpolation parameter. We also classify geometries of the interpolating graphs into the hyperbolic, relatively hyperbolic, and thick cases. Our approach relies on the theory of twist-free graphs of multicurves, which is developed by Vokes and Russel.
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https://arxiv.org/abs/2601.02839
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e916abf198ea8c4e456c7ea4177609d0ba8bc806e196b027cf7478a5b74f50d9
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2026-01-07T00:00:00-05:00
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Constructing $\lambda$-Angenent curve by flow method
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arXiv:2601.02853v1 Announce Type: new Abstract: Using a modified curve shortening flow, we construct $\lambda$-Angenent curve, which was first constructed by the shooting method.
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https://arxiv.org/abs/2601.02853
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6e096ff023f368db7421b7420c536c596a1237b2ae2938e27506719055554a8d
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2026-01-07T00:00:00-05:00
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Higher order H{\"o}lder approximation by solutions of second order elliptic equations
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arXiv:2601.02859v1 Announce Type: new Abstract: For a given second order elliptic operation $\mathcal{L}$ in a domain $\Omega\subset{\mathbb{R}}^\mathbf{N}$, $\mathbf{N}\ $, and a compact set $\mathbf{K}\subset\Omega$, order $\mathbf{N}$-$2$-Ahlfors-David regular, we define the space $\mathcal{H}^{\mathbf{r}+\omega}_{\mathcal{L}}(\mathbf{K})$ of continuous functions $f(x),\, x\in\mathbf{K}$, admitting, for any $\delta>0$, a local approximation in the $\delta $-neighborhood of any point $x\in\mathbf{K}$, with $\delta^{\mathbf{r}}\omega(\delta)$-error estimate, by solutions of the equation $\mathcal{L} u=0$. For such functions, we prove the existence of a global approximation $v_\delta$ on $\mathbf{K}$ with the same order of error estimate, by a solution of the same equation in a $\delta$-neighborhood of $\mathbf{K}$. A number of properties of these functions $v_\delta$ and their derivatives are established.
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https://arxiv.org/abs/2601.02859
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cbc8e8c6e054338a49e0cac2897a2590bdb639c251246a6205190c3bf3a5d785
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2026-01-07T00:00:00-05:00
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Morse index of min-max stationary integral varifolds
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arXiv:2601.02860v1 Announce Type: new Abstract: We prove an upper bound for the Morse index of min-max stationary integral varifolds realizing the $d$-dimensional $p$-width of a closed Riemannian manifold.
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https://arxiv.org/abs/2601.02860
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16f1d0d4e0b48643cdc92eb3a8803c037fa914b83dd8f391a6e31020d76c2897
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2026-01-07T00:00:00-05:00
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Global H\"{o}lder Solvability of parabolic equations on domains with capacity density conditions
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arXiv:2601.02863v1 Announce Type: new Abstract: We investigate the Cauchy-Dirichlet problem for linear parabolic equations in divergence form. Under mild assumptions on the source term and the domain, we prove the existence of globally H\"{o}lder continuous solutions. Notably, our results accommodate data exhibiting singularities nearly as critical as the inverse square of the distance from the boundary.
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https://arxiv.org/abs/2601.02863
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5f5742dd1c76adc3baf8eb37da3fd4cc383255fc503a78c5306cf58dbc36ee79
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2026-01-07T00:00:00-05:00
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New biorthogonal sequences generated by index integrals of the weight functions
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arXiv:2601.02866v1 Announce Type: new Abstract: We exhibit new biorthogonal sequences generated by index integrals of the squares of the modified Bessel functions and gamma functions. The composition orthogonality, involving differential operators is employed. Generalized Wilson polynomials are introduced. Some properties are investigated.
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https://arxiv.org/abs/2601.02866
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e44ae5161f58a3c4665e4778de6e7c5a0bee4211ef5d95e2a417b2b7c4d698a9
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2026-01-07T00:00:00-05:00
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The W-Operator: A Volterra Fractional Time Operator with Non-Bernstein Symbol
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arXiv:2601.02876v1 Announce Type: new Abstract: We introduce a new two-parameter fractional time operator with Volterra structure, denoted by the W-operator, defined through a generalized Laplace symbol. The operator preserves the Caputo-type high-frequency behavior while allowing a controlled modification of the low-frequency regime through an additional parameter, leading to regularized memory effects. We develop a complete symbolic and Volterra theory, including explicit Prabhakar-type kernels, a left-inverse Volterra integral, and a fractional fundamental theorem of calculus. We show that the natural factorization of the Laplace symbol does not fit the classical Bernstein product mechanism and that the symbol is not a Bernstein function in general. Despite this non-Bernstein character, we establish well-posedness of abstract fractional Cauchy problems with sectorial generators by resolvent estimates and Laplace inversion, yielding a W-resolvent family with temporal regularity and smoothing properties. As an illustration, we apply the theory to a W-fractional diffusion model and discuss the influence of the modulation parameter on the relaxation of spectral modes.
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https://arxiv.org/abs/2601.02876
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f07df46536d7ed269622d34a1c8f2c11b2895c3843742fa1e9fa9376cecfe8ce
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2026-01-07T00:00:00-05:00
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Constructing Cospectral Vertices Through Orbits of Subgraphs
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arXiv:2601.02892v1 Announce Type: new Abstract: A constructive method is given for obtaining cospectral vertices in undirected graphs, along with an operation that preserves this construction. We prove that the construction yields cospectral vertices, as well as strongly cospectral vertices under additional conditions. Furthermore, we generalize cospectral vertices to the case of the graph Laplacian and provide an analogous construction.
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https://arxiv.org/abs/2601.02892
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2d0be67941feec74d9299abf69cea590f995e6be1e89737ff64177ecc1b426d0
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2026-01-07T00:00:00-05:00
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Rational-Kernel Fractional Evolution Equations with Almost Sectorial Operators: A Resolvent Framework Unifying ABC and W Dynamics
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arXiv:2601.02894v1 Announce Type: new Abstract: We study fractional evolution equations driven by rational-kernel time operators with non-singular memory, including the Atangana-Baleanu-Caputo operator and a generalized W-operator. These operators are characterized by Laplace symbols that do not necessarily belong to the classical Bernstein class. The analysis is carried out in the framework of almost sectorial operators, which allows resolvent estimates beyond standard analytic semigroup theory. Existence, uniqueness, and temporal regularity of mild solutions are established by Laplace transform techniques and contour integration, leading to the construction of associated resolvent families. A unified resolvent framework is developed, enabling a precise comparison between ABC and W dynamics and clarifying the influence of rational memory kernels on decay and smoothing properties. Several examples, including fractional diffusion-type equations, illustrate the abstract theory and highlight the impact of non-singular memory on long-time behavior.
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https://arxiv.org/abs/2601.02894
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9442a3b9983da52a5eb07b89c6a25855b426aa4ae98ce86145878be3e29335d0
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2026-01-07T00:00:00-05:00
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Homotopical algebra of Lie-Rinehart pairs
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arXiv:2601.02895v1 Announce Type: new Abstract: Dwyer-Kan localization at pairs of quasi-isomorphisms of the category of dg Lie-Rinehart pairs $(A,M)$, where $A$ is a semi-free cdga over a field $k$ of characteristic zero and $M$ a cell complex in $A$-modules, is shown to be equivalent to that of strong homotopy Lie-Rinehart (SH LR) pairs satisfying the same cofibrancy condition. Latter is a category of fibrant objects. We introduce cofibrations of SH LR pairs, construct factorizations, and prove lifting properties. Applying them, we show uniqueness up to homotopy of certain BV-type resolutions. Restricting to dg LR pairs whose underlying cdga is of finite type, and using a different (co)fibrancy condition, we show that the functor $(A,M)\mapsto A$ is a Cartesian fibration with presentable fibers. The two (co)fibrancy conditions yield equivalent $\infty$-categories under Dwyer-Kan localization.
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https://arxiv.org/abs/2601.02895
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c402b3fb593ad2b8f473c4b15d55f102c9240fcf9264770861d34d4041b45ada
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2026-01-07T00:00:00-05:00
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Relating Checkpoint Update Probabilities to Momentum Parameters in Single-Loop Variance Reduction Methods
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arXiv:2601.02899v1 Announce Type: new Abstract: Variance reduction (VR) is a crucial tool for solving finite-sum optimization problems, including the composite general convex setting, which is the focus of this work. On the one hand, denoting the number of component functions as $n$ and the target accuracy as $\epsilon$, some VR methods achieve the near-optimal complexity $\widetilde{\mathcal{O}}\left(n+\sqrt{n}/\sqrt{\epsilon}\right)$, but they all have nested structure and fail to provide convergence guarantee for the iterate sequence itself. On the other hand, single-loop VR methods, being free from the aforementioned disadvantages, have complexity no better than $\mathcal{O}\left(n+n/\sqrt{\epsilon}\right)$ which is the complexity of the deterministic method FISTA, thus leaving a critical gap unaddressed. In this work, we propose the \textit{Harmonia} technique which relates checkpoint update probabilities to momentum parameters in single-loop VR methods. Based on this technique, we further propose to vary the growth rate of the momentum parameter, creating a novel continuous trade-off between acceleration and variance reduction, controlled by the key parameter $\alpha\in[0,1]$. The proposed techniques lead to following favourable consequences. First, several known complexity of quite different algorithms are re-discovered under the proposed unifying algorithmic framework Katyusha-H. Second, under an extra mild condition, Katyusha-H achieves the near-optimal complexity for $\alpha$ belonging to a certain interval, highlighting the effectiveness of the acceleration-variance reduction trade-off. Last, without extra conditions, Katyusha-H achieves the complexity $\widetilde{\mathcal{O}}(n+\sqrt{n}/\sqrt{\epsilon})$ with $\alpha=1$ and proper mini-batch sizes. The proposed idea and techniques may be of general interest beyond the considered problem in this work.
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https://arxiv.org/abs/2601.02899
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0f416996c44894a8ee597d2847d25a044fdd2bdf5fc269322e3746444ce115df
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2026-01-07T00:00:00-05:00
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The 2-systole on compact K\"ahler surfaces with positive scalar curvature
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arXiv:2601.02901v1 Announce Type: new Abstract: We study the 2-systole on compact K\"ahler surfaces of positive scalar curvature. For any such surface $(X,\omega)$, we prove the sharp estimate \(\min_X S(\omega)\cdot\syst_2(\omega)\le12\pi\), with equality if and only if $X=\PP^2$ and $\omega$ is the Fubini--Study metric. Using the classification of positive scalar curvature K\"ahler surfaces by their minimal models, we also determine the optimal constant in each case and describe the corresponding rigid models: $12\pi$ when the minimal model is $\PP^2$, $8\pi$ for Hirzebruch surfaces, and $4\pi$ for non-rational ruled surfaces. In the non-rational ruled case, we also give an independent analytic proof, adapting Stern's level set method to the holomorphic fibration in K\"ahler setting.
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https://arxiv.org/abs/2601.02901
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ad84e38d78e63bc1efc195609d93b531d733a5dd5038320341ad3805cd96e044
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2026-01-07T00:00:00-05:00
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Inhomogeneous nonlinear Schr\"odinger equations with competing singular nonlinearities
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arXiv:2601.02909v1 Announce Type: new Abstract: We study nonlinear elliptic equations arising as stationary states of inhomogeneous nonlinear Schr\"odinger equations with competing singular nonlinearities. Working in a weighted Sobolev space that combines the homogeneous Sobolev space with a weighted Lebesgue term, we establish continuous and compact embeddings of Caffarelli--Kohn--Nirenberg type. These embeddings, together with a model that displays a natural scaling, allow us to apply the abstract critical point framework of Mercuri and Perera (2025), yielding a sequence of nonlinear eigenvalues for the associated problem. This scaling property leads to a classification of weighted power-type nonlinearities into subscaled, scaled, and superscaled regimes. Within this variational setting, we obtain broad existence and multiplicity results for equations driven by sums of weighted power nonlinearities, covering superscaled, scaled, and subscaled interactions, both in the subcritical and critical cases. We also provide a nonexistence result as a consequence of a Pohozaev-type identity. Finally, in the radial setting we employ improved radial CKN inequalities to enlarge the admissible embedding ranges. This yields strengthened radial versions of all our main results, including two-dimensional configurations with more singular weights, where no compact embeddings are available in the nonradial case.
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https://arxiv.org/abs/2601.02909
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330c68525498fcf04a58a9ebb7e2f4b386cffff94d6d865ad11baaefacbf4267
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2026-01-07T00:00:00-05:00
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Diamond Open Access: The AMR Experiment
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arXiv:2601.02910v1 Announce Type: new Abstract: Diamond open access journals charge neither readers nor authors. Despite long-standing support for this ideal within mathematics, relatively few such journals exist. This article documents the Association for Mathematical Research's experience building and operating diamond open access journals, focusing on the infrastructure, cost, and editorial practices that make the model viable. It aims to clarify why earlier reform efforts have been difficult to replicate and how a lightweight institutional framework can lower the barrier to adoption.
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https://arxiv.org/abs/2601.02910
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21902fa44663817ed24c4fe7cdc5343ef7fe6d31e4751f6e7dc0ba38765a582b
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2026-01-07T00:00:00-05:00
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Characteristic quasi-polynomials of truncated arrangements
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arXiv:2601.02912v1 Announce Type: new Abstract: Given an (affine) integral arrangement $\mathcal{A}$ in $\mathbb{R}^n$, the reduction of $\mathcal{A}$ modulo an arbitrary positive integer $q$ naturally yields an arrangement $\mathcal{A}_q$ in $\mathbb{Z}_q^n$. Our primary objective is to study the combinatorial aspects of the restriction $\mathcal{A}^{(B,\bm b)}$ to the solution space of $B\bm x=\bm b$, and its reduction $\mathcal{A}_q^{(B,\bm b)}$ modulo $q$. This work generalizes the earlier results of Kamiya, Takemura and Terao, as well as Chen and Wang. The purpose of this paper is threefold as follows. Firstly, we derive an explicit counting formula for the cardinality of the complement $M\big(\mathcal{A}_q^{(B,\bm b)}\big)$ of $\mathcal{A}_q^{(B,\bm b)}$; and prove that for all positive integers $q>q_0$, this cardinality coincides with a quasi-polynomial $\chi^{\text{quasi}}\big(\mathcal{A}^{(B,\bm b)},q\big)$ in $q$ with a period $\rho_C$. Secondly, we weaken Chen and Wang's original hypothesis $a \mid b$ to a strictly more general condition $\gcd(a,\rho_C)\mid \gcd(b,\rho_C)$, and introduce the concept of combinatorial equivalence for positive integers. Within this framework, we establish three unified comparison relations: between the unsigned coefficients of $\chi^{\text{quasi}}\big(\mathcal{A}^{(B,\bm b)},a\big)$ and $\chi^{\text{quasi}}\big(\mathcal{A}^{(B,\bm b)},b\big)$; between the unsigned coefficients of distinct constituents of $\chi^{\text{quasi}}\big(\mathcal{A}^{(B,\bm b)},q\big)$; and between the cardinalities of $M\big(\mathcal{A}_q^{(B,\bm b)}\big)$ and $M\big(\mathcal{A}_{pq}^{(B,\bm b)}\big)$. Thirdly, using our method, we revisit the enumerative aspects of group colorings and nowhere-zero nonhomogeneous form flows from the early work of Forge, Zaslavsky and Kochol.
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https://arxiv.org/abs/2601.02912
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7c301da9c6ed2aa91290a7239aa9b257121b0f55a67ea989a64ca93b85c95085
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2026-01-07T00:00:00-05:00
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The compositional inverses of some permutation polynomials of the form $x+\gamma\operatorname{Tr}_q^{q^2}(h(x))$
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arXiv:2601.02919v1 Announce Type: new Abstract: Recently, Jiang et al. \cite{JIANG2025102522} obtained several classes of Permutation Polynomial of the form $x+\gamma\operatorname{Tr}_q^{q^2}(h(x))$ over finite fields $\mathbb{F}_{q^2},q=2^n$. In this paper, we find the compositional inverse of six classes of permutation polynomials of this form.
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https://arxiv.org/abs/2601.02919
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03acd3d8c0d3d1e0517c939642a2d5dbdaf4a02388ad50d264a718a0c71104df
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2026-01-07T00:00:00-05:00
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Ramaswami Type translation formulae for the polylogarithm functions
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arXiv:2601.02921v1 Announce Type: new Abstract: In 1934, Ramaswami proved a number of curious translation formulae satisfied by the Riemann zeta function. Such translation formulae, in turn give the meromorphic extension of the Riemann zeta function. In 1954, Apostol extended those identities to establish a family of such similar translation formulae. In this article, we establish many such Ramaswami and Apostol type translation formulae for the Dirichlet series defining the polylogarithm functions. This extended set up has many interesting applications, for example, it allows us to also find some (seemingly new) recurrence relations between the Bernoulli numbers, and use them to deduce some congruence properties of the tangent numbers.
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https://arxiv.org/abs/2601.02921
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6937b1efcc32dbb8cd058857f9575552637a882b5c5273e50f4a69063898c4df
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2026-01-07T00:00:00-05:00
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Consistency of square bracket partition relation
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arXiv:2601.02923v1 Announce Type: new Abstract: Characteristic earlier results were of the form CON$(2^{\aleph_0} \to [\lambda]^2_{n, 2})$, with $2^{\aleph_0} $ an ex-large cardinal, in the best case the first weakly Mahlo cardinal. Characteristic new results are CON$((2^{\aleph_0} = \aleph_m) + \aleph_l \to [\aleph_k]^2_{n, 2})$, for suitable $k < l < m$. So we improve in three respects: the continuum may be small (e.g. not a weakly Mahlo), we use no large cardinal, and the cardinals $\lambda$ involved are $ < 2^{\aleph_0}$ after the forcing.
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https://arxiv.org/abs/2601.02923
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ae1098812694085265652e8a4be4b009b66a0a52a786c7f7b7289a554effee73
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2026-01-07T00:00:00-05:00
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With what probability does an inscribed triangle contain a given point?
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arXiv:2601.02929v1 Announce Type: new Abstract: Three points uniformly selected on the unit circle form a triangle containing a point $X$ at distance $r \in [0; 1]$ from its center with probability $P(r) = \frac{1}{4} - \frac{3}{2 \pi^2}\textrm{Li}_2(r^2)$, where $\textrm{Li}_2$ is the dilogarithm function (Jeremy Tan Jie Rui, 2018). In this paper we present an alternative proof of this fact. We also discuss a couple of other geometric probability problems where the dilogarithm function arises.
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https://arxiv.org/abs/2601.02929
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c5a62380cce1cb2deb9aa69f75bbde40fc3c4e8b3e7ed537f0fc03cfe847489d
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2026-01-07T00:00:00-05:00
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Dimension-decaying diffusion processes as the scaling limit of condensing zero-range processes
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arXiv:2601.02935v1 Announce Type: new Abstract: In this article, we prove that, on the diffusive time scale, condensing zero-range processes converge to a dimension-decaying diffusion process on the simplex \[ \Sigma = \{(x_1,\dots,x_S) : x_i \ge 0,\; \sum_{i\in S} x_i = 1\}, \] where $S$ is a finite set. This limiting diffusion has the distinctive feature of being absorbed at the boundary of the simplex. More precisely, once the process reaches a face \[ \Sigma_A = \{(x_1,\dots,x_S) : x_i \ge 0,\; \sum_{i\in A} x_i = 1\}, \qquad A \subset S, \] it remains confined to this set and evolves in the corresponding lower-dimensional simplex according to a new diffusion whose parameters depend on the subset $A$. This mechanism repeats itself, leading to successive reductions of the dimension, until one of the vertices of the simplex is reached in finite time. At that point, the process becomes permanently trapped. The proof relies on a method to extend the domain of the associated martingale problem, which may be of independent interest and useful in other contexts.
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https://arxiv.org/abs/2601.02935
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8fb76e3a177e0f60992db2e321ae34c251d48fef84b93213852c6ee19022a81a
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2026-01-07T00:00:00-05:00
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On a theorem Dan Rudolph: Part II: Amenable groups
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arXiv:2601.02939v1 Announce Type: new Abstract: We prove an analog of Rudolph's theorem for actions of countable amenable groups, which asserts that among invariant measures with entropy at least c on the $G$-shift $(\Lambda^G,\sigma)$, a typical measure has entropy $c$ and is Bernoulli. We also address a relative version of this theorem.
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https://arxiv.org/abs/2601.02939
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58888382a270b8bc2c332aaede791c5e3878a3685b64799b09f3f65f90c5bdc9
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2026-01-07T00:00:00-05:00
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Rational stable homotopy type of equivariant projective spaces and Grassmannians
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arXiv:2601.02940v1 Announce Type: new Abstract: We prove explicit rational stable splittings of equivariant complex projective spaces $\mathbb{C}P(V)$ and Grassmannians $Gr_n(V)$, for complex representations $V$. When $V$ is a sum of one-dimensional representations, both $\mathbb{C}P(V)$ and $Gr_n(V)$ are rationally a wedge of representation spheres. For general finite groups $G$ and $V$ a sum of irreducible representations which are not necessarily one-dimensional, we show that $\mathbb{C}P(V)$ splits rationally as a wedge of Thom spaces over irreducible factors in $V$. For $Gr_n(V)$, the factors in the corresponding rational splitting are a smash product of Thom spaces over lower Grassmannians on irreducible factors in $V$.
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https://arxiv.org/abs/2601.02940
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a58fadc362ceb58c6f81d156b545b71a13d8789b04957b6a1c239617cc381222
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2026-01-07T00:00:00-05:00
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Hopfield neural networks as port-Hamiltonian and gradient systems
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arXiv:2601.02951v1 Announce Type: new Abstract: The structure of continuous Hopfield networks is revisited from a system-theoretic point of view. After adopting a novel electrical network interpretation involving nonlinear capacitors, it is shown that Hopfield networks admit a port-Hamiltonian formulation provided an extra passivity condition is satisfied. Subsequently it is shown that any Hopfield network can be represented as a gradient system, with Riemannian metric given by the inverse of the Hessian matrix of the total energy stored in the nonlinear capacitors. On the other hand, the well-known 'energy' function employed by Hopfield turns out to be the dissipation potential of the gradient system, and this potential is shown to satisfy a dissipation inequality that can be used for analysis and interconnection.
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https://arxiv.org/abs/2601.02951
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65994d3ec30248924b1c91c19bdf746aede554648440494f0f6c6de90d36fe18
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2026-01-07T00:00:00-05:00
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The left-to-right minima basis of the group algebra of the symmetric group (updated version)
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arXiv:2601.02952v1 Announce Type: new Abstract: We introduce a new basis of the group algebra of the symmetric group, built using the left-to-right minima sets of permutations. We show that on this basis, the descent algebra acts by triangular operators, thus making it an analogue of a cellular basis. The proof involves Dynkin elements (nested commutators) of the free algebra and their interactions with the $\mathbf B$-basis.
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https://arxiv.org/abs/2601.02952
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640ed5bec7460c46a50a4d20bcaeadf730b7e633fad93d5d4e9d1b61d7783dfb
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2026-01-07T00:00:00-05:00
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K-stability of Fano weighted hypersurfaces via plt flags and convex geometry
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arXiv:2601.02974v1 Announce Type: new Abstract: We develop a framework to study the K-stability of weighted Fano hypersurfaces based on a combination of birational and convex-geometric techniques. As an application, we prove that all quasi-smooth weighted Fano hypersurfaces of index 1 with at most two weights greater than 1 are K-stable. We also construct several examples of K-unstable quasi-smooth weighted Fano hypersurfaces of low indices. To prove these results, we establish lower bounds for stability thresholds using the method of Abban-Zhuang, which reduces the problem to lower-dimensional cases. A key feature of our approach is the use of plt flags that are not necessarily admissible.
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https://arxiv.org/abs/2601.02974
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5ecc6ff8440f714bad4dff7e6c5d77d29828590e9d6f4b98e8cf346e93373d74
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2026-01-07T00:00:00-05:00
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Uniform distribution of saddle connection lengths in all $\mathsf{SL}(2,\mathbb{R})$ orbits
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arXiv:2601.02979v1 Announce Type: new Abstract: For every flat surface, almost every flat surface in its $\mathsf{SL}(2,\mathbb{R})$ orbit has the following property: the sequence of its saddle connection lengths in non-decreasing order is uniformly distributed in the unit interval.
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https://arxiv.org/abs/2601.02979
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4122b22ed6eaf5b15cdbfc40181127213e36d6c95a0469f6e6574dd9532cbc49
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2026-01-07T00:00:00-05:00
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Coupling Brownian loop soups and random walk loop soups at all polynomial scales
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arXiv:2601.02992v1 Announce Type: new Abstract: Lawler and Trujillo Ferreras constructed a well-known coupling between the Brownian loop soups in $\mathbb{R}^2$ and the random walk loop soups on $\mathbb{Z}^2$ (one rescales the random walk loops by $1/N$, their time parametrizations by $1/(2N^2)$, and let $N\to \infty$), which led to numerous applications. It nevertheless only holds for loops with time length at least $N^{\theta-2}$ for $\theta \in(2/3,2)$. In particular, there is no control on mesoscopic loops with time length less than $N^{-4/3}$ (i.e.\ roughly diameter less than $N^{-2/3}$). In this paper, we find a simple way to remove the restriction $\theta>2/3$, so that such a coupling works for all $\theta\in (0,2)$, i.e. for loops at all polynomial scales. We also establish this coupling in any dimension $d\ge 1$ (i.e. for random walk loop soups on $\mathbb{Z}^d$ and Brownian loop soups on $\mathbb{R}^d$).
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https://arxiv.org/abs/2601.02992
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68c07c0cb651c716982aa234866a4217a126074d6cc79233491e3a17698b55de
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2026-01-07T00:00:00-05:00
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Remarks on $d$-independent topological groups
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arXiv:2601.03000v1 Announce Type: new Abstract: A non-trivial topological group is called \emph{$d$-independent} if for every subgroup of cardinality less than the continuum there exists a countable dense subgroup intersecting it trivially. This notion was introduced by M\'arquez and Tkachenko and has been intensively studied in the metrizable setting. In particular, they proved that a second-countable locally compact abelian group is $d$-independent if and only if it is algebraically an $M$-group, and asked whether the same conclusion holds for all separable locally compact groups. In this paper we give an affirmative answer to this question. We show that every separable locally compact abelian $M$-group is $d$-independent, thereby removing the metrizability assumption from the result of M\'arquez and Tkachenko. In addition, we investigate several further aspects of $d$-independence. We study its behaviour under taking powers of topological groups and extend the notion of $d$-independence to the non-abelian setting. Moreover, we prove that every separable connected compact group is $d$-independent, thereby answering another question posed by M\'arquez and Tkachenko.
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https://arxiv.org/abs/2601.03000
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144041ca3e66e24bb52050f67bded3458174b67053000623e33cf75d70420035
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2026-01-07T00:00:00-05:00
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G-BSDEs with time-varying monotonicity condition
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arXiv:2601.03006v1 Announce Type: new Abstract: In this paper, we study backward stochastic differential equations driven by G-Brownian motion where the generator has time-varying monotonicity with respect to y and Lipsitz property with respect to z. Through the Yosida approximation, we have proved the existence and uniqueness of the solutions to these equations.
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https://arxiv.org/abs/2601.03006
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828bc2566f5eedd04dda87ef992b8f581f48eba8b698177cd286a60f7f8bf6c2
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2026-01-07T00:00:00-05:00
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A Relaxation Method for Nonsmooth Nonlinear Optimization with Binary Constraints
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arXiv:2601.03008v1 Announce Type: new Abstract: We study binary optimization problems of the form \( \min_{x\in\{-1,1\}^n} f(Ax-b) \) with possibly nonsmooth loss \(f\). Following the lifted rank-one semidefinite programming (SDP) approach\cite{qian2023matrix}, we develop a majorization-minimization algorithm by using the difference-of-convexity (DC) reformuation for the rank-one constraint and the Moreau envelop for the nonsmooth loss. We provide global complexity guarantees for the proposed \textbf{D}ifference of \textbf{C}onvex \textbf{R}elaxation \textbf{A}lgorithm (DCRA) and show that it produces an approximately feasible binary solution with an explicit bound on the optimality gap. Numerical experiments on synthetic and real datasets confirm that our method achieves superior accuracy and scalability compared with existing approaches.
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https://arxiv.org/abs/2601.03008
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c334bd4d2739da9e51bf78b5475dd616561a6f0280e94c108cfbf63437c95d62
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2026-01-07T00:00:00-05:00
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Adaptive Control of Unknown Linear Switched Systems via Policy Gradient Methods
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arXiv:2601.03016v1 Announce Type: new Abstract: We consider the policy gradient adaptive control (PGAC) framework, which adaptively updates a control policy in real time, by performing data-based gradient descent steps on the linear quadratic regulator cost. This method has empirically shown to react to changing circumstances, such as model parameters, efficiently. To formalize this observation, we design a PGAC method which stabilizes linear switched systems, where both model parameters and switching time are unknown. We use sliding window data for the policy gradient estimate and show that under a dwell time condition and small dynamics variation, the policy can track the switching dynamics and ensure closed-loop stability. We perform simulations to validate our theoretical results.
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https://arxiv.org/abs/2601.03016
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6333d8f15a311bdf6d274666a7f5640dc47e6b2f439a7acfa2a36c39f8a297b9
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2026-01-07T00:00:00-05:00
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Periodicity of traces of Hecke operators modulo prime powers
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arXiv:2601.03029v1 Announce Type: new Abstract: We study traces of Hecke operators on spaces of elliptic cusp forms and Drinfeld cusp forms and show that, modulo any prime power, these traces are periodic in the weight.
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https://arxiv.org/abs/2601.03029
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57bfe1b6bb6bb63f5a92bbd1c29346959ea634d0dfc4bd4c30be73ba84e6be69
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2026-01-07T00:00:00-05:00
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On the Hilbert-Chow crepant resolution conjecture
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arXiv:2601.03036v1 Announce Type: new Abstract: We prove the Hilbert-Chow crepant resolution conjecture in the exceptional curve classes for all projective surfaces and all genera. In particular, this confirms Ruan's cohomological Hilbert-Chow crepant resolution conjecture. The proof exploits Fulton-MacPherson compactifications, reducing the conjecture to the case of the affine plane. As an application, using previous results of the author, we also deduce the families DT/GW correspondence for threefolds $S \times C$ in classes that are zero on the first factor, yielding a wall-crossing proof of the correspondence in this case. Finally, we speculate on the relationship between Hilbert schemes and Fulton-MacPherson compactifications beyond the topics considered in this work.
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https://arxiv.org/abs/2601.03036
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ae4c91d860b7513da78cf5aac5eb1861869933f813d9af7cc55afffee6567f51
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2026-01-07T00:00:00-05:00
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Generalized Toeplitz determinants for Starlike Mappings in Several Complex Variables
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arXiv:2601.03039v1 Announce Type: new Abstract: This paper establishes sharp bounds for the second and third-order Toeplitz determinants associated with starlike functions $f$ in the unit disk such that $f(z)-z$ has a zero of order $k+1$ at $z=0$. These bounds are further extended to starlike mappings defined on the unit ball in a complex Banach space and on bounded starlike circular domains in $\mathbb{C}^n$. The derived results generalize several known bounds as special cases.
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https://arxiv.org/abs/2601.03039
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8c20d1513e559641d35b3337e9e0fd42d63b83639a0ec34d535fe78f984379b8
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2026-01-07T00:00:00-05:00
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Egorov-Type Semiclassical Limits for Open Quantum Systems with a Bi-Lindblad Structure
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arXiv:2601.03041v1 Announce Type: new Abstract: This paper develops a bridge between bi-Hamiltonian structures of Poisson-Lie type, contact Hamiltonian dynamics, and the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) formalism for quantum open systems. On the classical side, we consider bi-Hamiltonian systems defined by a Poisson pencil with non-trivial invariants. Using an exact symplectic realization, these invariants are lifted and projected onto a contact manifold, yielding a completely integrable contact Hamiltonian system in terms of dissipated quantities and a Jacobi-commutative algebra of observables. On the quantum side, we introduce a class of contact-compatible Lindblad generators: GKSL evolutions whose dissipative part preserves a commutative $C^\ast$-subalgebra generated by the quantizations of the classical dissipated quantities, and whose Hamiltonian part admits an Egorov-type semiclassical limit to the contact dynamics. This construction provides a mathematical mechanism compatible with the semiclassical limit for pure dephasing, compatible with integrability and contact dissipation. An explicit Euler-top-type Poisson-Lie pencil, inspired by deformed Euler top models, is developed as a fully worked-out example illustrating the resulting bi-Lindblad structure and its semiclassical behavior.
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https://arxiv.org/abs/2601.03041
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44c451154e690d15111f01ec68ffcf629a34f637c75b9c8fe3e72fac4593eaa4
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2026-01-07T00:00:00-05:00
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Classification of reductive homogeneous spaces satisfying strict inequality for Benoist-Kobayashi's $\rho$ functions
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arXiv:2601.03049v1 Announce Type: new Abstract: Let $G$ be a real reductive Lie group and $H$ a reductive subgroup of $G$. Benoist-Kobayashi studied when $L^2(G/H)$ is a tempered representation of $G$. They introduced the functions $\rho$ on Lie algebras and gave a necessary and sufficient condition for the temperedness of $L^2(G/H)$ in terms of an inequality on $\rho$. In a joint work with Y. Oshima, we considered when $L^2(G/H)$ is equivalent to a unitary subrepresentation of $L^2(G)$ and gave a sufficient condition for this in terms of a strict inequality of $\rho$. In this paper, we will classify the pairs $(\mathfrak{g}, \mathfrak{h})$ with $\mathfrak{g}$ complex reductive and $\mathfrak{h}$ complex semisimple which satisfy that strict inequality of $\rho$.
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https://arxiv.org/abs/2601.03049
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d518d1c30be3aac20ac5aa773e0f7fd9744e1ee6c150e9d1fe9c34be2aa50ba4
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2026-01-07T00:00:00-05:00
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Existence and concentration of ground state solutions for an exponentially critical Choquard equation involving mixed local-nonlocal operators
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arXiv:2601.03060v1 Announce Type: new Abstract: We study the Choquard equation involving mixed local and nonlocal operators \[-\varepsilon^{2}\Delta u+\varepsilon^{2s}(-\Delta)^{s}u+V(x)u=\varepsilon^{\mu-2}\left(\frac{1}{|x|^{\mu}}*F(u)\right)f(u)\quad \text{in }\R^{2},\] where \(\varepsilon>0\), \(s\in(0,1)\), \(0<\mu<2\), \(f\) has Trudinger--Moser critical exponential growth, and \(F(t)=\int_{0}^{t}f(\tau)\,d\tau\). By variational methods, combined with the Trudinger--Moser inequality and compactness arguments adapted to the critical growth and the nonlocal interaction term, we prove the existence of ground state solutions and describe their concentration behavior as \(\varepsilon\to0^{+}\).
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https://arxiv.org/abs/2601.03060
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901bebe7ee3c81235be75f39ff45d211bc5b5b7e7a3ce9cef34b23ad1876ba82
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2026-01-07T00:00:00-05:00
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Hamiltonian reductions as affine closures of cotangent bundles
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arXiv:2601.03068v1 Announce Type: new Abstract: For an irreducible non-singular affine $G$-variety $Y$ whose action is $2$-large, we prove that the Hamiltonian reduction $T^*Y/\!\!/\!\!/G$ is a symplectic variety with terminal singularities, isomorphic to the affine closure of $T^*Z_{\text{reg}}$ for $Z:=Y/\!/G$. As applications, we provide a short proof of G. Schwarz's theorem on the graded surjectivity of the push-forward map $\mathcal{D}(Y)^G\rightarrow \mathcal{D}(Z)$, and we establish the surjectivity of the symbol map on $Z$.
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https://arxiv.org/abs/2601.03068
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abc0518b046ff40045475ec64f3747495cc0ccb9a927a24e6775b5bb079e08ca
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2026-01-07T00:00:00-05:00
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Average gradient localisation for degenerate elliptic equations in the plane
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arXiv:2601.03078v1 Announce Type: new Abstract: We consider Lipschitz solutions to the possibly highly degenerate elliptic equation $ \dv G(\nabla u)=0 $ in $B_1\subset\R^2 $, for any continuous strictly monotone vector field $ G\colon\R^2\to\R^2$. We show that $u$ is either $C^1$ at $0$, or any blowup limit $v(x)=\lim \frac{u(\delta x)-u(0)}{\delta} $ along a sequence $\delta\to 0$ satisfies $ \nabla v\in \mathcal{D}\cap \mathcal{S} \text{ a.e} $. Here, $ \mathcal{D}$ and $\mathcal{S}$ can be roughly interpreted as the sets where ellipticity degenerates from below and above, that is, the symmetric parts of $ \nabla G$ and $(\nabla G)^{-1}$ have a zero eigenvalue. This is a strong indication in favor of the expected continuity of $H(\nabla u)$ for any continuous $H$ vanishing on $\mathcal{D}\cap \mathcal{S}$. In contrast with previous results in the same spirit, we do not make any assumption on the structure of $G$ besides its continuity and strict monotony.
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https://arxiv.org/abs/2601.03078
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bc0c8783484824c1cf62a30a4d2826afcada04390c8cbaddde398af87b2c484e
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2026-01-07T00:00:00-05:00
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A proof of Xin-Zhang's tridiagonal determinant conjecture
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arXiv:2601.03082v1 Announce Type: new Abstract: We confirm a recent conjecture by Xin and Zhang, which establishes a simple product formula for the characteristic polynomial of an $(n-1) \times (n-1)$ tridiagonal matrix $C$. This characteristic polynomial arises from a recurrence relation that enumerates $n \times n$ nonnegative integer matrices with all row and column sums equal to $t$, also called the Ehrhart polynomial of the $n$th Birkhoff polytope. The proof relies on an unexpected observation: shifting $C$ by a scalar multiple of the identity matrix yields a matrix similar to a lower triangular matrix. In triangular form, the characteristic polynomial reduces to the product of the diagonal entries, leading to the desired closed-form expression. Moreover, we extend this method to broader families of tridiagonal matrices. This provides a new approach for deriving exact enumeration formulas.
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https://arxiv.org/abs/2601.03082
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8edb5d51a9b88d0d6237f9fb4d210b59f6bad94e41dbcccbd6a120877fbed569
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2026-01-07T00:00:00-05:00
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Pseudo-differential operators associated with the fractional Hankel-Bessel transform
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arXiv:2601.03091v1 Announce Type: new Abstract: We introduce and study a new class of pseudo-differential operators associated with a fractional Hankel--Bessel transform. Motivated by the classical Hankel transform and the pseudo-differential operators associated with Bessel operators studied by Pathak and Pandey \cite{PathakPandey1995}, we define a fractional variant by inserting a fractional Fourier-type phase into the Hankel kernel. We then introduce global Shubin-type symbol classes adapted to this transform, derive kernel estimates and integral representations, and establish boundedness results on weighted L^{p}-spaces and on fractional Hankel--Sobolev spaces. This provides a new framework parallel to the classical Hankel pseudo-differential calculus, but in a fractional and global setting.
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https://arxiv.org/abs/2601.03091
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7d3cf3fb14c230f4e5073b872e1b5ddeb0af4f0c83b06df1028bfd8cf39914ce
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2026-01-07T00:00:00-05:00
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Stability of Hyperk\"ahler Flow
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arXiv:2601.03092v1 Announce Type: new Abstract: In this work, we discuss the stability of Donaldson's flow of surfaces in a hyperk\"ahler 4-manifold. In \cite{WT2}, Wang and Tsai proved a uniqueness theorem and $C^1$ dynamic stability theorem of the mean curvature flow for minimal surface. We extend their results and obtain a similar dynamic stability theorem of the hyperk\"ahler flow.
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https://arxiv.org/abs/2601.03092
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721295af6650ab11d0a01da58a0ede9aed908c5bb1fc7ee89fef0cc3a28e0f41
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2026-01-07T00:00:00-05:00
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Submanifolds of almost quaternionic skew-Hermitian manifolds
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arXiv:2601.03094v1 Announce Type: new Abstract: We investigate several classes of submanifolds of almost quaternionic skew-Hermitian manifolds $(M^{4n}, Q, \omega)$, including almost symplectic, almost complex, almost pseudo-Hermitian and almost quaternionic submanifolds. In the torsion-free case, we realize each type of submanifold considered in the theoretical part by constructing explicit examples of submanifolds of semisimple quaternionic skew-Hermitian symmetric spaces.
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https://arxiv.org/abs/2601.03094
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740d6ea0a1a59ff6369cb1e5efc5477886d8274c952a026fb4caeea8ccb33df4
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2026-01-07T00:00:00-05:00
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A Kirchhoff equation with infinite conservation laws
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arXiv:2601.03095v1 Announce Type: new Abstract: We show here that the quasilinear Kirchhoff-Pokhozhaev equation $$u_{tt}-\big(a\int_{\mathbb{R}^n} |\nabla u |^2 dx + b \big)^{-2} \Delta u = 0,$$ with $a\neq0$, admits conservation laws of all orders.
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https://arxiv.org/abs/2601.03095
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64f180e2fb860b224d6ba47915f775467de499eb39e9e5daf7121a337f6172d1
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2026-01-07T00:00:00-05:00
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Spherical Ricci tori with rotational symmetry
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arXiv:2601.03096v1 Announce Type: new Abstract: In this article, we study $c$-spherical Ricci metrics, that is, Riemannian metrics whose Gaussian curvature $K$ satisfies \begin{equation*} (K - c)\Delta K - |\nabla K|^2 - 4K(K - c)^2 = 0, \end{equation*} for some $c>0$. We explicitly construct a two-parameter family of such metrics with rotational symmetry and show that infinitely many non-isometric examples can be realized on the same torus. Moreover, we investigate their realization as induced metrics on compact rotational surfaces in $\mathbb{S}^3$, establishing the existence of embedded compact spherical Ricci surfaces by controlling a period function associated with the isometric immersion.
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https://arxiv.org/abs/2601.03096
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8e93f5d90478eea26dbf23d492c7cac7a30e5c4aeecc49ae69e8107fb936174d
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2026-01-07T00:00:00-05:00
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Point-set models for homotopy coherent coalgebras
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arXiv:2601.03101v1 Announce Type: new Abstract: We show a first rectification result for homotopy chain coalgebras over a field. On the one hand, we consider the $\infty$-category obtained by localizing differential graded coalgebras over an operad with respect to quasi-isomorphisms; on the other, we give a general definition of an $\infty$-category of coalgebras over an enriched $\infty$-operad. We show by induction over cell attachments that these two $\infty$-categories are in fact equivalent when the operad is cofibrant. This yields explicit point-set models for $E_n$-coalgebras and $E_\infty$-coalgebras in the derived $\infty$-category of chain complexes over a field, and an explicit point-set model for the cellular chains functor with its $E_\infty$-coalgebra structure. After Bachmann--Burklund, this gives a point-set algebraic model for nilpotent $p$-adic homotopy types.
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https://arxiv.org/abs/2601.03101
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a5eef2adeefb91f185f47047128a982b25dd48110fda3f1e20c214e5655ab433
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2026-01-07T00:00:00-05:00
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On the monotonicity of the entropy production in the Landau-Maxwell equation
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arXiv:2601.03107v1 Announce Type: new Abstract: We study the homogeneous Landau equation with Maxwell molecules and prove that the entropy production is non-increasing provided the directional temperatures are well-distributed and the solution admits a moment of order $\ell$, for some $\ell$ arbitrarily close to $2$. It implies that for an initial condition with finite moment of order $\ell$, the entropy production is guaranteed to be non-increasing after a certain time, that we explicitly compute. This is the first partial answer to a conjecture made by Henry P. McKean in 1966 on the sign of the time-derivatives of the entropy. We also obtain algebraic decay estimates for the entropy production for large time; as well as a short-time estimate without moment assumptions.
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https://arxiv.org/abs/2601.03107
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87172bd9801233ce48af36fc44bdc447f8f568aa25ebd4b970ddbc5d8e8f8821
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2026-01-07T00:00:00-05:00
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First passage times for decoupled random walks
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arXiv:2601.03109v1 Announce Type: new Abstract: Motivated by a connection to the infinite Ginibre point process, decoupled random walks were introduced in a recent article Alsmeyer, Iksanov and Kabluchko (2025). The decoupled random walk is a sequence of independent random variables, in which the $n$th variable has the same distribution as the position at time $n$ of a standard random walk with nonnegative increments. We prove distributional convergence in the Skorokhod space equipped with the $J_1$-topology of the running maxima and the first passage times of decoupled random walks. We show that there exist five different regimes, in which distinct limit theorems arise. Rather different functional limit theorems for the number of visits of decoupled standard random walk to the interval $[0,t]$ as $t\to\infty$ were earlier obtained in the aforementioned paper Alsmeyer, Iksanov and Kabluchko (2025). While the limit processes for the first passage times are inverse extremal-like processes, the limit processes for the number of visits are stationary Gaussian.
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https://arxiv.org/abs/2601.03109
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1c8f7ea48ce83b189e523a361abe225db508ec58155ab991d34372e38cc5461e
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2026-01-07T00:00:00-05:00
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On derived categories of module categories over multiring categories
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arXiv:2601.03128v1 Announce Type: new Abstract: Let $\mathcal{A}$ and $\mathcal{B}$ be subcategories of tensor categories $\mathcal{C}$ and $\mathcal{D}$, respectively, both of which are abelian categories with finitely many isomorphism classes of simple objects. We prove that if their derived categories $\mathbf{D}^b(\mathcal{A})$ and $\mathbf{D}^b(\mathcal{B})$ are left triangulated tensor ideals and are equivalent as triangulated $\mathbf{D}^b(\mathcal{C})$-module categories via an equivalence induced by a monoidal triangulated functor $F:\mathbf{D}^b(\mathcal{C})\rightarrow \mathbf{D}^b(\mathcal{D})$, then the original module categories $\mathcal{A}$ and $\mathcal{B}$ are themselves equivalent. We then apply this result to smash product algebras. Furthermore, the localization theory of module categories and triangulated module categories is investigated.
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https://arxiv.org/abs/2601.03128
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94ea655b5c73385ab0fc79c7d2515dc4fd20a4063494b35bd8cf04c34e96bd03
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2026-01-07T00:00:00-05:00
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Lipschitz extension and Lipschitz-free spaces over nets in normed spaces
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arXiv:2601.03131v1 Announce Type: new Abstract: We consider subsets $S$ of a metric space $M$ such that Lipschitz mappings defined on $S$ can be extended to Lipschitz mappings on $M$, and we show that the union of such subsets has the same property under appropriate geometric conditions. We then derive several consequences to the isomorphic structure and classification of Lipschitz and Lipschitz-free spaces. Our main result is that the Lipschitz-free space $\mathcal{F}(M)$ is isomorphic to its countable $\ell_1$-sum when $M$ is either a net $N_X$ in any Banach space $X$ or the integer grid $\mathbb{Z}_{\ell_1}$ in $\ell_1$. We also prove that the Lipschitz space $\mathrm{Lip}_0(\mathbb{Z}_{\ell_1})$ is isomorphic to $\mathrm{Lip}_0(\ell_1)$ and that $\mathrm{Lip}_0(N_X)$ contains a complemented copy of $\mathrm{Lip}_0(X)$, among other results. This answers questions raised by Albiac, Ansorena, C\'uth and Doucha and Candido, C\'uth and Doucha, respectively, and extends previous results by the same authors as well as H\'ajek and Novotn\'y.
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https://arxiv.org/abs/2601.03131
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edc4b01e789fee1f13114d60e8bc503262741e96974266552e8f78c206941122
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2026-01-07T00:00:00-05:00
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Freely floating cylinder on a 3D fluid governed by the Boussinesq equations in the axisymmetric without swirl case
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arXiv:2601.03133v1 Announce Type: new Abstract: This paper deals with the interactions of waves governed by a non-linear dispersive Boussinesq type system with the vertical displacement of a cylindrical floating structure in an axisymmetric without swirl situation. The Boussinesq regime is a good approximation of free surface Euler's equations when the non-linear parameter and the shallowness parameter are small. The vertical motion of the floating body is governed by the Newton equation. The full coupled wave-structure interaction problem under consideration is reduced to a boundary problem. The boundary condition satisfied by the discharge is given in terms of the vertical displacement of the floating cylinder. The latter is calculated using an ODE, which requires knowledge of the trace of the surface elevation and its second-time derivative. We use the dispersion in order to exhibit a hidden second order ODE on the trace of the surface elevation. This finally allows us to rewrite the waves-structure interaction problem as a system of non-local conservative PDEs plus bounded radial terms with a dispersive boundary layer, combined with an ODE at the boundary. This is what we call the Augmented formulation. Afterwards we showed that this formulation is well-posed with two different methods. Finally, we study the return to equilibrium situation in the linear regime. In particular, we improved previous results on the explicit time decay. We showed that the center mass of the floating body cannot converge to its equilibrium faster than $\mathcal{O}(t^{-1/2})$ in 2D without viscosity and faster than $\mathcal{O}(t^{-3/2})$ with viscosity.
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https://arxiv.org/abs/2601.03133
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982bcb0336c3c41331b8a1d8af7319ef8b41e967e6d279523555443b0e1232e0
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2026-01-07T00:00:00-05:00
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Classifying the Fine Polyhedral Spectrum
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arXiv:2601.03145v1 Announce Type: new Abstract: In this paper, we examine an analogue of the recently solved spectrum conjecture by Fujita in the setting of Fine polyhedral adjunction theory. We present computational results for lower-dimensional polytopes, which lead to a complete classification of the highest numbers of this Fine spectrum in any dimension. Moreover, we present a classification of the Fine spectrum in dimensions one, two and (almost) three, while providing a framework for general classification results in any dimension.
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https://arxiv.org/abs/2601.03145
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9e04caf8d701fe1ddb44c772bc2d3ded50a00b84bed39c5680187ab791f79df7
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2026-01-07T00:00:00-05:00
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Normalization flow and Poincar\'e-Dulac theory
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arXiv:2601.03147v1 Announce Type: new Abstract: In this article, we develop a new approach to the Poincar\'e--Dulac normal form theory for a system of differential equations near a singular point. Using the continuous averaging method, we construct a normalization flow that moves a vector field to its normal form. We prove that, in the algebra of formal vector fields (given by power series), the normalization procedure achieves full normalization. When convergence is taken into account, we show that the radius of convergence admits a lower bound of order $1/(1+A\delta)$, with $A>0$, as $\delta \to +\infty$. Based on the methods of this work and on the approaches of \cite{Tres2}, we provide a new proof of the Siegel--Brjuno theorem on the convergence of the normalizing transformation.
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https://arxiv.org/abs/2601.03147
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93274f67a3eae7ae85fa06cf9db4c0447940238509b7c6dd07eedf78f7152f0c
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2026-01-07T00:00:00-05:00
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Vaught's Conjecture and Theories of Partial Order Admitting a Finite Lexicographic Decomposition
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arXiv:2601.03155v1 Announce Type: new Abstract: A complete theory ${\mathcal T}$ of partial order is an FLD$_1$-theory iff some (equivalently, any) of its models ${\mathbb X}$ admits a finite lexicographic decomposition ${\mathbb X} =\sum _{{\mathbb I}}{\mathbb X} _i$, where ${\mathbb I}$ is a finite partial order and ${\mathbb X} _i$-s are partial orders with a largest element. Then we write $\sum _{{\mathbb I}}{\mathbb X}_i\in {\mathcal D} ({\mathcal T})$ and call $\sum _{{\mathbb I}}{\mathbb X}_i$ a VC-decomposition (resp. a VC$^\sharp$-decomposition} iff ${\mathbb X} _i$ satisfies Vaught's conjecture (VC) (resp. VC$^\sharp$: $I({\mathbb X} _i)\in \{ 1,{\mathfrak{c}}\}$), for each $i\in I$. ${\mathcal T}$ is called actually Vaught's iff for some $\sum _{{\mathbb I}}{\mathbb X}_i\in {\mathcal D} ({\mathcal T})$ there are sentences $\tau _i\in \mathop{\rm Th}\nolimits ({\mathbb X} _i)$, $i\in I$, providing VC. We prove that: (1) VC is true for ${\mathcal T}$ iff ${\mathcal T}$ is large or its atomic model has a VC decomposition; (2) VC is true for each actually Vaught's FLD$_1$ theory; (3) VC$^\sharp$ is true for ${\mathcal T}$, if there is a VC$^\sharp$-decomposition of a model of ${\mathcal T}$. VC is true for the partial orders from the closure $\langle {\mathcal C} ^{\rm reticle}_0\cup {\mathcal C} ^{\rm ba}\rangle _{\Sigma}$, where $\langle {\mathcal C}\rangle _{\Sigma}$ denotes the closure of a class ${\mathcal C}$ under finite lexicographic sums. VC$^{\sharp}$ is true for a large class of partial orders of the form $\sum _{{\mathbb I}}(\dot{\bigcup}_{j<m_i^j}{\mathbb X} _i^{j,k})_r$, where ${\mathbb X} _i^{j,k}$-s can be linear orders, or Boolean algebras, or belong to a wide class of trees.
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https://arxiv.org/abs/2601.03155
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723c42934027505c329dc9e78aebdf2d7427232ffbb77187338cb724026439a8
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2026-01-07T00:00:00-05:00
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Valuations on polyhedra and topological arrangements
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arXiv:2601.03176v1 Announce Type: new Abstract: We revisit a classical theme of (general or translation invariant) valuations on convex polyhedra. Our setting generalizes the classical one, in a ``dual'' direction to previously considered generalizations: while previous research was mostly concerned with variations of ground fields/rings, over which the vertices of polytopes are defined, we consider more general collections of defining hyperplanes. No algebraic structures are imposed on these collections. This setting allows us to uncover a close relationship between scissors congruence problems on the one hand and finite hyperplane arrangements on the other hand: there are many parallel results in these fields, for which the parallelism seems to have gone unnoticed. In particular, certain properties of the Varchenko--Gelfand algebras for arrangements translate to properties of polytope rings or valuations. Studying these properties is possible in a general topological setting, that is, in the context of the so-called topological arrangements, where hyperplanes do not have to be straight and may even have nontrivial topology.
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https://arxiv.org/abs/2601.03176
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32ece18920e50f938ba4344b0de5eced1588f2fe26b619b437710560fa2a0a48
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2026-01-07T00:00:00-05:00
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Deformations of the connected sum of Gorenstein algebras
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arXiv:2601.03179v1 Announce Type: new Abstract: We prove that the Gorenstein locus of the Hilbert scheme of points on $\mathbb A^n$ is non-reduced for $n>9$; we construct examples of non-reduced points that come from apolar algebras of the sum of general cubics. As a corollary, we get a non-reducedness result for the cactus scheme. We generalise the Bia{\l}ynicki-Birula decomposition to abstract deformation functors, providing a new method of studying deformation theory. Our construction gives us fractal structures on the nested Hilbert scheme.
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https://arxiv.org/abs/2601.03179
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e3fef4f4dec769b6a6dd2a44d2179f898071f9e9bf82fb70a03b620ec9df94f4
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2026-01-07T00:00:00-05:00
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Strongly finitary metric monads are too strong
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arXiv:2601.03180v1 Announce Type: new Abstract: Varieties of quantitative algebras are fully described by their free-algebra monads on the category Met of metric spaces. For a longer time it has been an open problem whether the resulting enriched monads are precisely the strongly finitary ones (determined by their values on finite discrete spaces). We present a counter-example: the variety of algebras on two close binary operations yields a monad which is not strongly finitary. A full characterization of free-algebra monads of varieties is: they are the semi-strongly finitary monads, i.e., weighted colimits of strongly finitary monads (in the category of finitary monads).
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https://arxiv.org/abs/2601.03180
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44e0428a79f9e19f95db4b744f71598f815f3a3eea09b25614bb077f6a080763
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2026-01-07T00:00:00-05:00
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Subjective-Objective Median-based Importance Technique (SOMIT) to Aid Multi-Criteria Renewable Energy Evaluation
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arXiv:2601.03182v1 Announce Type: new Abstract: Accelerating the renewable energy transition requires informed decision-making that accounts for the diverse financial, technical, environmental, and social trade-offs across different renewable energy technologies. A critical step in this multi-criteria decision-making (MCDM) process is the determination of appropriate criteria weights. However, deriving these weights often solely involves either subjective assessment from decision-makers or objective weighting methods, each of which has limitations in terms of cognitive burden, potential bias, and insufficient contextual relevance. This study proposes the subjective-objective median-based importance technique (SOMIT), a novel hybrid approach for determining criteria weights in MCDM. By tailoring SOMIT to renewable energy evaluation, the method directly supports applied energy system planning, policy analysis, and technology prioritization under carbon neutrality goals. The practical utility of SOMIT is demonstrated through two MCDM case studies on renewable energy decision-making in India and Saudi Arabia. Using the derived weights from SOMIT, the TOPSIS method ranks the renewable energy alternatives, with solar power achieving the highest performance scores in both cases. The main contributions of this work are five-fold: 1) the proposed SOMIT reduces the number of required subjective comparisons from the conventional quadratic order to a linear order; 2) SOMIT is more robust to outliers in the alternatives-criteria matrix (ACM); 3) SOMIT balances subjective expert knowledge with objective data-driven insights, thereby mitigating bias; 4) SOMIT is inherently modular, allowing both its individual parts and the complete approach to be seamlessly coupled with a wide range of MCDM methods commonly applied in energy systems and policy analysis; 5) a dedicated Python library, pysomit, is developed for SOMIT.
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https://arxiv.org/abs/2601.03182
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fe43da79bdb6c50c1802669ee313b85344860e4d60a7316733831c60e3f3f702
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2026-01-07T00:00:00-05:00
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Flat simplices and kissing polytopes
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arXiv:2601.03183v1 Announce Type: new Abstract: We consider how flat a lattice simplex contained in the hypercube $[0,k]^d$ can be. This question is related to the notion of kissing polytopes: two lattice polytopes contained in the hypercube $[0,k]^d$ are kissing when they are disjoint but their distance is as small as possible. We show that the smallest possible distance of a lattice point $P$ contained in the cube $[0,k]^3$ to a lattice triangle in the same cube that does not contain $P$ is $$ \frac{1}{\sqrt{3k^4-4k^3+4k^2-2k+1}} $$ when $k$ is at least $2$. We also improve the known lower bounds on the distance of kissing polytopes for $d$ at least $4$ and $k$ at least $2$.
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https://arxiv.org/abs/2601.03183
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b81b8879be58cf71e14bcf11154cf69ffc5e19af2855ef1f13c24a0eb046358e
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2026-01-07T00:00:00-05:00
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Subprincipal Controlled Quasimodes and Spectral Instability
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arXiv:2601.03188v1 Announce Type: new Abstract: Here we explore, in a series of articles, semiclassical quasimodes u(h,b), approximative solutions P(h)u(h,b)\sim 0, depending on $0<1$, and on b, the subprincipal symbol. We study a pseudodifferential operator with transversal intersections of bicharacteristics, where the principal symbol has double multiplicity, $p=dp=0$, in a small neigborhood $\Omega$. Because of this fact, we instead study the subprincipal symbol b, and we can conclude that we get transport equations depending on b where sign changes for the imaginary part of b give approximative solutions with small support. These modes are used to estimate spectral instability, or the pseudospectrum. We also investigate the possibility that we can factorize the model operator as $P(h)=h^2P_1P_2,$ in this way actually annihilating the subprincipal symbol, thus there is no condition for the imaginary part of b. In a follow-up article, we examine different cases for more complex operators with tangential intersections of bicharacteristics, thereby generalizing the findings here.
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https://arxiv.org/abs/2601.03188
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a4f4bb386e74d89007361d089e8ebc59f85206822a3d82e82ad1e2ae9b68dbb7
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2026-01-07T00:00:00-05:00
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HOMFLY parabolic restriction, defect skein theory and the Turaev coproduct
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arXiv:2601.03196v1 Announce Type: new Abstract: We define a HOMFLY version of the category $\text{Rep}_q\text{P}$ of quantum representations of a parabolic subgroup $\text{P}\subseteq\text{GL}_{m+n}$ of block triangular matrices. Alongside this category, we construct functors that interpolate the usual restriction functors between $\text{GL}_{m+n}$, $\text{P}$ and the subgroup $\text{L}\subseteq\text{GL}_{m+n}$ of block-diagonal matrices, yielding a universal version of the formalism of parabolic restriction. Based on this formalism, we construct central algebras and centred bimodules which serve as algebraic ingredients for defining a skein theory on $3$-manifolds with surface and line defects. We recover the Turaev coproduct on the HOMFLY skein algebra as a particular instance of this theory. In particular, this coproduct is compatible with the cutting and gluing of surfaces.
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https://arxiv.org/abs/2601.03196
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Academic Papers
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56eb58e4001074db3c74273417d6f81fe3acd2762069c5b1945421ad3cbbb537
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2026-01-07T00:00:00-05:00
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Signature invariants of monomial ideals
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arXiv:2601.03208v1 Announce Type: new Abstract: Let $I$ be a monomial ideal of a polynomial ring $R=K[x_1,\ldots,x_n]$ over a field $K$ and let ${\rm sgn}(I)$ be its signature ideal. If $I$ is not a principal ideal, we show that the depth of $R/I$ is the depth of $R/{\rm sgn}(I)$, and the regularity of $R/{\rm sgn}(I)$ is at most the regularity of $R/I$. For ideals of height at least $2$, we show that the height and the associated primes of $I$ and its signature ${\rm sgn}(I)$ are the same, and we show that $I$ is Cohen--Macaulay (resp. Gorenstein) if and only if ${\rm sgn}(I)$ is Cohen--Macaulay (resp. Gorenstein), and furthermore we show that the v-number of ${\rm sgn}(I)$ is at most the v-number of $I$. We give an algorithm to compute the signature of a monomial ideal using \textit{Macaulay}$2$, and an algorithm to examine given families of monomial ideal by computing their signature ideals and determining which of these are either Cohen--Macaulay or Gorenstein.
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https://arxiv.org/abs/2601.03208
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59397b3e08664f882a87adec9f00e0b9896e208634a248369f5a30157c96e5cf
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2026-01-07T00:00:00-05:00
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Values of ternary quadratic forms at integers and the Berry-Tabor conjecture for 3-tori
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arXiv:2601.03209v1 Announce Type: new Abstract: Berry and Tabor conjectured in 1977 that spectra of generic integrable quantum systems have the same local statistics as a Poisson point process. We verify their conjecture in the case of the two-point spectral density for a quantum particle in a three-dimensional box, subject to a Diophantine condition on the domain's proportions. A permissible choice of width, height and depth is for example $1,2^{1/3},2^{-1/3}$. This extends previous work of Eskin, Margulis and Mozes (Annals of Math., 2005) in dimension two, where the problem reduces to the quantitative Oppenheim conjecture for quadratic forms of signature $(2,2)$. The difficulty in three and higher dimensions is that we need to consider the distribution of indefinite forms in shrinking rather than fixed intervals, which we are able to resolve for special diagonal forms of signature $(3,3)$ in various scalings, including a rate of convergence. A key step of our approach is to represent the relevant counting problem as an average of a theta function on $\mathrm{SL}(2,\mathbb{Z})^3\backslash\mathrm{SL}(2,\mathbb{R})^3$ over an expanding family of one-parameter unipotent orbits. The asymptotic behaviour of these unipotent averages follows from Ratner's measure classification theorem and subtle escape of mass estimates.
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https://arxiv.org/abs/2601.03209
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96f647382b701a33d871691edadc3032b0e4f3d1956c9d62e0f825675e39deb9
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2026-01-07T00:00:00-05:00
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Lattice coverings and homogeneous covering congruences
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arXiv:2601.03212v1 Announce Type: new Abstract: We consider the problem of covering $\mathbb{Z}^2$ with a finite number of sublattices of finite index, satisfying a simple minimality or non-degeneracy condition. We show how this problem may be viewed as a projective (or homogeneous) version of the well-known problem of covering systems of congruences. We give a construction of minimal coverings which produces many, but not all, minimal coverings, and determine all minimal coverings with at most $8$ sublattices.
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https://arxiv.org/abs/2601.03212
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d69fcda07293951bc40421ba8eff64c0c9a910d71b4aacf92a8955ede00c0c6a
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2026-01-07T00:00:00-05:00
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Generalized affine buildings for semisimple algebraic groups over real closed fields
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arXiv:2601.03226v1 Announce Type: new Abstract: We use real algebraic geometry to construct an affine $\Lambda$-building $B$ associated to the $\mathbb{F}$-points of a semisimple algebraic group, where $\mathbb{F}$ is a valued real closed field. We characterize the spherical building at infinity and the local building at a base point. We compute stabilizers of various subsets of $B$ and obtain group decompositions.
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https://arxiv.org/abs/2601.03226
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51f85f1944072d9616feabf0a94d6e6b85b70b02042358e6e12e5017aa4f6fd2
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2026-01-07T00:00:00-05:00
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Sets of Lengths of Integer-Valued Polynomials on Prime Ideals of Principal Ideal Domains
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arXiv:2601.03246v1 Announce Type: new Abstract: Let $D$ be a principal ideal domain with infinite spectrum such that for every nonzero prime ideal $M$ of $D$, the residue field $D/M$ is finite. Let $K$ be the quotient field of $D$. We investigate sets of lengths in the ring of integer-valued polynomials on $M$, $\text{Int}(M, D) = \{f \in K[x] ~ \vert ~ f(M) \subseteq D\}$. For every multiset of integers $1 < z_1 \leq z_2 \leq \cdots \leq z_n$, we explicitly construct an element of $\text{Int}(M, D)$ with exactly $n$ essentially different factorizations into irreducible elements of $\text{Int}(M, D)$ whose lengths are $z_1, z_2, \ldots, z_n$. Furthermore, we show that $\text{Int}(M, D)$ is not a transfer Krull domain. These results spark off the study of sets of lengths in the rings $\text{Int}(S, D) \neq \text{Int}(D)$, where $S$ is an infinite subset of $D$.
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https://arxiv.org/abs/2601.03246
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Academic Papers
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41bc3f2a79812b5ef1f29cbb998039828a28d3d729474740fb5c6f6802d4c061
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2026-01-07T00:00:00-05:00
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Quantum polylogarithms
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arXiv:2601.00472v1 Announce Type: cross Abstract: Multiple polylogarithms are periods of variations of mixed Tate motives. Conjecturally, they deliver all such periods. We introduce deformations of multiple polylogarithms depending on a complex parameter h. We call them quantum polylogarithms. Their asymptotic expansion as h goes to 0 recovers multiple polylogarithms. The quantum dilogarithm was studied by Barnes in the XIX century. Its exponent appears in many areas of Mathematics and Physics. Quantum polylogarithms satisfy a holonomic systems of modular difference equations with coefficients in variations of mixed Hodge-Tate structures of motivic origin. If h is a rational number, the quantum polylogarithms can be expressed via multiple polylogarithms. Otherwise quantum polylogarithms are not periods of variations of mixed motives, i.e. they can not be given by integrals of rational differential forms on algebraic varieties. Instead, quantum polylogarithms are integrals of differential forms built from both rational functions and exponentials of rational functions. We call them rational exponential integrals. We suggest that quantum polylogarithms reflect a very general phenomenon: Periods of variations of mixed motives should have quantum deformations.
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https://arxiv.org/abs/2601.00472
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Academic Papers
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a1949bf69d781b884e207d55464a319e0918e2be5962d15b1f2c6bb4e28ae4db
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2026-01-07T00:00:00-05:00
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Color-kinematics duality from an algebra of superforms
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arXiv:2601.02478v1 Announce Type: cross Abstract: Color-kinematics duality states that the kinematic numerators of the cubic tree-level Yang-Mills scattering amplitudes obey the same symmetry properties that the color factors obey due to the Jacobi identity. We present a novel strategy for deriving this duality, based on the differential forms on a superspace. This space of superforms carries a generalization of a Batalin-Vilkovisky (BV) algebra (BV$^{\square}$ algebra). We show that the homotopy algebra of color-stripped Yang-Mills theory is obtained as a quotient of this space in which a subspace, which is an ideal `up to homotopy', is modded out. This algebra is a subsector of a BV$_{\infty}^{\square}$ algebra. Deriving the latter would provide a first-principle proof of color-kinematics duality from field theory.
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https://arxiv.org/abs/2601.02478
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Academic Papers
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6eb596d256c6051a8936258aa6081599ac8697332608958b5c271f79515ec8bf
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2026-01-07T00:00:00-05:00
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Exact critical-temperature bounds for two-dimensional Ising models
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arXiv:2601.02502v1 Announce Type: cross Abstract: We derive exact critical-temperature bounds for the classical ferromagnetic Ising model on two-dimensional periodic tessellations of the plane. For any such tessellation or lattice, the critical temperature is bounded from a above by a universal number that is solely determined by the largest coordination number on the lattice. Crucially, these bounds are tight in some cases such as the Honeycomb, Square, and Triangular lattices. We prove the bounds using the Feynman--Kac--Ward formalism, confirm their validity for a selection of over two hundred lattices, and construct a two-dimensional lattice with 24-coordinated sites and record high critical temperature.
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https://arxiv.org/abs/2601.02502
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Academic Papers
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80ea66fe2c53a7b12c3ba818ab817ba5bd2804ad2f3fbefe5abcadc545bd911e
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2026-01-07T00:00:00-05:00
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A novel finite-sample testing procedure for composite null hypotheses via pointwise rejection
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arXiv:2601.02529v1 Announce Type: cross Abstract: We propose a novel finite-sample procedure for testing composite null hypotheses. Traditional likelihood ratio tests based on asymptotic $\chi^2$ approximations often exhibit substantial bias in small samples. Our procedure rejects the composite null hypothesis $H_0: \theta \in \Theta_0$ if the simple null hypothesis $H_0: \theta = \theta_t$ is rejected for every $\theta_t$ in the null region $\Theta_0$, using an inflated significance level. We derive formulas that determine this inflated level so that the overall test approximately maintains the desired significance level even with small samples. Whereas the traditional likelihood ratio test applies when the null region is defined solely by equality constraints--that is, when it forms a manifold without boundary--the proposed approach extends to null hypotheses defined by both equality and inequality constraints. In addition, it accommodates null hypotheses expressed as unions of several component regions and can be applied to models involving nuisance parameters. Through several examples featuring nonstandard composite null hypotheses, we demonstrate numerically that the proposed test achieves accurate inference, exhibiting only a small gap between the actual and nominal significance levels for both small and large samples.
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https://arxiv.org/abs/2601.02529
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19f82e49584f7e32758c560e0724e44a155a461a9ca9c470c919377e763a68a2
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2026-01-07T00:00:00-05:00
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Higher-Dimensional Anyons via Higher Cohomotopy
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arXiv:2601.03150v1 Announce Type: cross Abstract: We highlight that integer Heisenberg groups at level 2 underlie topological quantum phenomena: their group algebras coincide with the algebras of quantum observables of abelian anyons in fractional quantum Hall (FQH) systems on closed surfaces. Decades ago, these groups were shown to arise as the fundamental groups of the space of maps from the surface to the 2-sphere -- which has recently been understood as reflecting an effective FQH flux quantization in 2-Cohomotopy. Here we streamline and generalize this theorem using the homotopy theory of H-groups, showing that for $k \in \{1,2,4\}$, the non-torsion part of $\pi_1 \mathrm{Map}\big({(S^{2k-1})^2, S^{2k}}\big)$ is an integer Heisenberg group of level 2, where we identify this level with 2 divided by the Hopf invariant of the generator of $\pi_{4k-1}(S^{2k})$. This result implies the existence of higher-dimensional analogs of FQH anyons in the cohomotopical completion of 11D supergravity ("Hypothesis H").
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https://arxiv.org/abs/2601.03150
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7a4d095037d9a0cb1518a3baa92c7a8107713bae3a6b5314e0029a60d165e457
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2026-01-07T00:00:00-05:00
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On the Bezrukavnikov-Kaledin quantization of symplectic varieties in characteristic $p$
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arXiv:2011.08259v2 Announce Type: replace Abstract: We prove that after inverting the Planck constant $h$ the Bezrukavnikov-Kaledin quantization $(X, \mathcal{O}_h)$ of symplectic variety $X$ in characteristic $p$ is Morita equivalent to a certain central reduction of the algebra of differential operators on $X$.
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https://arxiv.org/abs/2011.08259
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a05fd6a30a4b6a4ebf0d6f01621177a2f7b01d609cb2a866995c0f03245fde78
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2026-01-07T00:00:00-05:00
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Equivariant Chevalley, Giambelli, and Monk Formulae for the Peterson Variety
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arXiv:2111.15663v2 Announce Type: replace Abstract: We present a formula for the Poincar\'e dual in the flag manifold of the equivariant fundamental class of any regular nilpotent or regular semisimple Hessenberg variety as a polynomial in terms of certain Chern classes. We then develop a type-independent proof of the Giambelli formula for the Peterson variety, and use this formula to compute the intersection multiplicity of a Peterson variety with an opposite Schubert variety corresponding to a Coxeter word. Finally, we develop an equivariant Chevalley formula for the cap product of a divisor class with a fundamental class, and a dual Monk rule, for the Peterson variety.
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https://arxiv.org/abs/2111.15663
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Academic Papers
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604f47d510171ee1b9dc8c009c564544962fa0374c90943d197dc94f90e81ff0
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2026-01-07T00:00:00-05:00
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Stability of three-dimensional stochastic Navier-Stokes equation with Markov switching
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arXiv:2203.15971v2 Announce Type: replace Abstract: A right continuous Markov chain is introduced in the noise terms of the three-dimensional stochastic Navier-Stokes equation, and we call such stochastic system as stochastic Navier-Stokes equation with Markov switching. In the present article, we study the $p$-th moment exponential stability and the almost surely exponential stability of the solution to the equation.
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https://arxiv.org/abs/2203.15971
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3368d0fc4742a4bee5aa1f7a309e98c1e283aab294affe78889b983cc2636527
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2026-01-07T00:00:00-05:00
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A Host--Kra ${\mathbf F}_2^\omega$-system of order $5$ that is not Abramov of order $5$, and non-measurability of the inverse theorem for the $U^6({\mathbf F}_2^n)$ norm
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arXiv:2303.04853v4 Announce Type: replace Abstract: It was conjectured by Bergelson, Tao, and Ziegler \cite{btz} that every Host--Kra $\F_p^\omega$-system of order $k$ is an Abramov system of order $k$. This conjecture has been verified for $k \leq p+1$. In this paper we show that the conjecture fails when $k=5, p=2$. We in fact establish a stronger (combinatorial) statement, in that we produce a bounded function $f: \F_2^n \to \C$ of large Gowers norm $\|f\|_{U^6(\F_2^n)}$ which (as per the inverse theorem for that norm) correlates with a non-classical quintic phase polynomial $e(P)$, but with the property that all such phase polynomials $e(P)$ are ``non-measurable'' in the sense that they cannot be well approximated by functions of a bounded number of random translates of $f$. A simpler version of our construction can also be used to answer a question of Candela, Gonz\'alez-S\'anchez, and Szegedy \cite{CGSS}.
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https://arxiv.org/abs/2303.04853
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Academic Papers
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515e0975fe156c13739723ba07d62d00d22ef44cd19ff34fb5704f7bf3867a79
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2026-01-07T00:00:00-05:00
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Global Koszul duality
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arXiv:2304.08409v4 Announce Type: replace Abstract: We construct a monoidal model structure on the category of all curved coalgebras and show that it is Quillen equivalent, via the extended bar-cobar adjunction, to another model structure we construct on the category of curved algebras. When the coalgebras under consideration are conilpotent and the algebras are dg, i.e. uncurved, this corresponds to the ordinary dg Koszul duality of Positselski and Keller-Lef\`evre. As an application we construct global noncommutative moduli spaces for flat connections on vector bundles, holomorphic structures on almost complex vector bundles, dg modules over a dg algebra, objects in a dg category, and others.
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https://arxiv.org/abs/2304.08409
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Academic Papers
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64ee90b201721c8fcfb9709b8ec2f19c0ebc250b9878e9b55a304c7b04b9ebf1
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2026-01-07T00:00:00-05:00
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An $\mathfrak{sl}_2$ action on link homology of T(2,k) torus links
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arXiv:2307.01910v2 Announce Type: replace Abstract: We determine an $\mathfrak{sl}_2$ module structure on the equivariant Khovanov-Rozansky homology of (2,k)-torus links following the framework defined in arXiv:2306.10729.
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https://arxiv.org/abs/2307.01910
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Academic Papers
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efa698c17a67703cb41eb19ab09885d0f9fc42564a8d2b25af2eb97153ffb830
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2026-01-07T00:00:00-05:00
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The Beauville-Voisin conjecture for double EPW sextics
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arXiv:2307.15240v2 Announce Type: replace Abstract: We prove that the Beauville-Voisin conjecture is true for any double EPW sextic, i.e. the subalgebra of the Chow ring generated by divisors and Chern classes of the tangent bundle injects into cohomology.
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https://arxiv.org/abs/2307.15240
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Academic Papers
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51c42cc72fac9cfb7715b5b604a4ecc1469408293f2467af81e9f6c6917efb22
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2026-01-07T00:00:00-05:00
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Doubly-weighted zero-sum constants
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arXiv:2311.00090v3 Announce Type: replace Abstract: Let $A,B\subseteq\mathbb Z_n$ be given and $S=(x_1,\ldots, x_k)$ be a sequence in $\mathbb Z_n$. We say that $S$ is an $(A,B)$-weighted zero-sum sequence if there exist $a_1,\ldots,a_k\in A$ and $b_1,\ldots,b_k\in B$ such that $a_1x_1+\cdots+a_kx_k=0$ and $b_1a_1+\cdots+b_ka_k=0$. We show that if $S$ has length $2n-1$, then $S$ has an $(A,B)$-weighted zero-sum subsequence of length $n$. The constant $E_{A,B}$ is defined to be the smallest positive integer $k$ such that every sequence of length $k$ in $\mathbb Z_n$ has an $(A,B)$-weighted zero-sum subsequence of length $n$. A sequence in $\mathbb Z_n$ of length $E_{A,B}-1$ which does not have any $(A,B)$-weighted zero-sum subsequence of length $n$ is called an $E$-extremal sequence for $(A,B)$. We determine the constant $E_{A,B}$ and characterize the $E$-extremal sequences for some pairs $(A,B)$. We also study the related constants $C_{A,B}$ and $D_{A,B}$ which are defined in the article.
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https://arxiv.org/abs/2311.00090
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Academic Papers
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e5d76c6a5851d237da9d1a7422582dfb6230e19d398d664e002d295e54070db5
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2026-01-07T00:00:00-05:00
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On the kernels of the pro-$p$ outer Galois representations associated to once-punctured CM elliptic curves
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arXiv:2312.04196v3 Announce Type: replace Abstract: In this paper, we compare a certain field arising from the pro-$p$ outer Galois representation associated to a once-punctured CM elliptic curve over an imaginary quadratic field $K$ with the maximal pro-$p$ Galois extension of the mod-$p$ ray class field $K(p)$ of $K$ unramified outside $p$. We prove that these two fields coincide for every prime $p$ which satisfies certain assumptions, assuming an analogue of the Deligne-Ihara conjecture. This may be regarded as an analogue of a result of Sharifi on the kernel of the pro-$p$ outer Galois representation associated to the projective line minus three points.
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https://arxiv.org/abs/2312.04196
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Academic Papers
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60b2b4df89741fe96571bf186bca58dbd4bc16b655e9581a0a45f5ff6c76687a
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2026-01-07T00:00:00-05:00
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Decision Making under Costly Sequential Information Acquisition: the Paradigm of Reversible and Irreversible Decisions
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arXiv:2401.00569v4 Announce Type: replace Abstract: Decision making in modern stochastic systems, including e-commerce platforms, financial markets and healthcare systems, has evolved into a multifaceted process that combines information acquisition and adaptive information sources. This paper initiates a study on such integrated settings, where these elements are not only fundamental but, also, interact in a complex and stochastically intertwined manner. We introduce a relatively simple model, which, however, captures the involved novel elements. A decision maker (DM) may choose between an established product $A$ of known value and a new product $B$ whose value is unknown. In parallel, the DM observes signals about the unknown value of product $B$ and can, also, opt to exchange it for product $A$ if $B$ is initially chosen. Mathematically, the model gives rise to sequential optimal stopping problems with distinct informational regimes (before and after buying product $B$), differentiated by the initial, coarser signal and the subsequent, more accurate one. We analyze in detail the underlying problems using predominantly viscosity solution techniques, departing from the existing literature on information acquisition which is based on traditional optimal stopping arguments. More broadly, the modeling approach introduced herein offers a novel framework for developing more complex interactions among decisions, information sources and information costs in stochastic environments, through a sequence of nested obstacle problems.
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https://arxiv.org/abs/2401.00569
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Academic Papers
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691eff8d500dc21e8d3aa814e9b1d98976e15c9fa7580e0c7008d65fdc8d6033
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2026-01-07T00:00:00-05:00
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Quasi-ergodic theorems for Feynman-Kac semigroups and large deviation for additive functionals
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arXiv:2401.17997v2 Announce Type: replace Abstract: We study the long-time behavior of an additive functional that takes into account the jumps of a symmetric Markov process. This process is assumed to be observed through a biased observation scheme that includes the survival to events of extinction and the Feynman-Kac weight by another similar additive functional. Under conditioning for the convergence to a quasi-stationary distribution and for two-sided estimates of the Feynmac-Kac semigroup to be obtained, we shall discuss general assumptions on the symmetric Markov process. For the law of additive functionals, we will prove a quasi-ergodic theorem, namely a conditional version of the ergodic theorem and a conditional functional weak law of large numbers. As an application, we also establish a large deviation principle for the mean ratio of additive functionals.
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https://arxiv.org/abs/2401.17997
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